Extracting Principles from Biology for Application to
Running Robots
by
Matthew Daniel Haberland
ARCKNES
B.S., Mechanical Engineering, Cornell University (2007)
M.Eng., Mechanical Engineering, Cornell University (2007)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
I
MASSACHUSETTS INST
OF TECHNOLOGY
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at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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© Massachusetts Institute of Technology 2014. All rights reserved.
Author......................................................................
Department of Mechanical Engineering
December 19, 2013
Certified by.......................................
.....
..........
angbae Kim
Ass tant Professor
Thesis Supervisor
Accepted by ...........................................................
David E. Hardt
Chairman, Department Committee on Graduate Students
E
Extracting Principles from Biology for Application to Running Robots
by
Matthew Daniel Haberland
Submitted to the Department of Mechanical Engineering
on December 19, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
When millions of years of evolution suggest a particular design solution, we may be tempted
to abandon traditional design methods and copy the biological example. However, biological
solutions do not often translate directly into the engineering domain, and even when they
do, copying eliminates the opportunity to improve. A better approach is to extract design
principles relevant to the task of interest and incorporate them in engineering designs when
they outperform conventional solutions. This thesis presents an original, general framework
for extracting engineering design principles from biology.
Case studies involving legged robots introduce important elements of the framework.
An investigation of the effect of swing leg retraction on the energetic efficiency of running
proposes the use of optimal control in the principle extraction process and finds that swing
leg retraction can minimize impact between the foot and the ground, but shows that this
does not necessarily improve overall energetic efficiency. An analysis of the effect of a
bioinspired tail on maneuverability motivates the importance of parameter variation for
developing general conclusions and finds that tails can be more effective than reaction
wheels when the time available for an aerial maneuver is short.
Finally, case studies utilize the complete framework to extract principles regarding the
effects of leg morphology on running robot performance. An examination of the effect of knee
joint type on the energetic efficiency of running finds that telescoping legs tend to be more
efficient than rotary-kneed legs for a class of simple robots and reveals a strong correlation
among step size, speed, and energetic efficiency. A study of the effect of rotary knee joint
direction on the energetic efficiency of running demonstrates that running robots with a
knee joint oriented 'backwards', as bird legs appear, tend to be more efficient than robots
with a knee joint oriented 'forwards', as human legs are. More generally, the case studies
demonstrate the effectiveness of the framework for very complex and widely applicable
principle extraction studies, the results of which can be utilized to save design resources
and improve design performance.
Thesis Supervisor: Sangbae Kim
Title: Assistant Professor
Acknowledgments
Thank you to:
the Defense Advanced Research Projects Agency (DARPA) for providing the funding that
supported this work;
Sangbae Kim, my advisor, for his tremendous intuition and for helping me find my own
path;
Kim Vandiver and Peko Hosoi for their enthusiasm, insightful suggestions, and support as
members of my committee;
the wonderful staff of the mechanical engineering department;
all my teachers along the way;
Dave Lavery for introducing me to engineering;
Andy Ruina for introducing me to legged robots and for suggesting graduate study;
Dani8I Karssen and all my co-authors;
Liz Jordan for suggesting MIT and for all her support as I applied;
Terry Orlando, my mentor at MIT;
my friends at the Thirsty, Ashdown House, the Biomimetic Robotics Lab, and everywhere;
Mike Grenier, my number one email collaborator to date, for his teamwork and friendship;
Hunter McClelland, a close second in email and a most dependable friend; and
Heather, Mom, Dad, and Chris for always being there.
A Poem:
It's a pain that my code takes a long time to run
'cause I only find out that it's bad when it's done.
8
Contents
1
Introduction
25
I
Methodology
29
2
A General Framework for Principle Extraction
31
2.1
Observation of Nature Inspires a Biological Design Question .........
32
2.2
Transfer of the Question from the Biological Domain to the Engineering
Domain Prompts an Engineering Model . . . . . . . . . . . . . . . . . . . .
2.3
2.4
II
3
33
Experiments with the Engineering Model Provide Data Supporting Engineering Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.1
Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.2
Experiment Implementation . . . . . . . . . . . . . . . . . . . . . . .
38
Analysis of the Data Yields Engineering Principles . . . . . . . . . . . . . .
40
Case Studies Incorporating Elements of the Framework
42
The Effect of Swing Leg Retraction on Performance of a Bipedal Running
Robot
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.2
Models and Experimental Platform . . . . . . . . . . . . . . . . . . . . . . .
45
Realistic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Impact Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.2.1
3.3
9
3.4
3.5
3.6
3.3.1
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
3.3.2
Results
3.3.3
Discussion . . . . . . . . . .
. . . . . . . .
52
Overall Energetic Efficiency . . . .
. . . . . . . .
54
3.4.1
Methods . . . . . . . . . . .
. . . . . . . .
54
3.4.2
Results
. . . . . . . . . . .
. . . . . . . .
56
3.4.3
Discussion . . . . . . . . . .
. . . . . . . .
57
Impact Forces and Footing Stability
. . . . . . . .
58
3.5.1
Methods . . . . . . . . . . .
. . . . . . . .
58
3.5.2
Results
. . . . . . . . . . .
. . . . . . . .
59
3.5.3
Discussion . . . . . . . . . .
. . . . . . . .
60
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
. . . . . . . . . . . . .
. . . . . . . . . 61
4 The Effect of Tails on The Maneuverability of Running Robots
63
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Analysis: Comparison of Biological and Artificial Force/Moment Technologies 64
4.3
63
4.2.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.2.2
Actuator Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2.3
Angular Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.4
Orientation Change
. . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.5
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Conclusion
III
5
50
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Case Studies Demonstrating the Complete Framework
72
73
The Effect of a Rotary Knee on the Energetic Efficiency of Running
Robots
75
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.2
M ethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.2.1
Question
10
5.3
5.2.2
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.2.3
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.1
5.4
6
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
The Effect of Leg Morphology on the Efficiency of Running Robots
87
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.2
Method .....
89
...
..
............
.
...
....
.
... ........
6.2.1
Robot Design / Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.2.3
Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
6.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
6.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.4.1
Explanation of Results . . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.4.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
6.5
IV
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Concluding Discussion
112
A Difficulties of Principle Extraction
A.1
115
Domain Transfer Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Hypothesis Testing Challenges and Solutions
115
. . . . . . . . . . . . . . . . .
116
A.2.1
Multiple Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
A.2.2
Constrained Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
A.2.3
Function Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
B Mathematical Preliminaries
B.1 Computational Dynamics
123
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
B.1.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
B.1.2
Impact Equations
125
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
B.1.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . .
127
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
B.2.1
Direct/Indirect Methods . . . . . . . . . . . . . . . . . . . . . . . . .
130
B.2.2
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.2.3
Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . .
134
B.2.4
Derivative Evaluation
. . . . . . . . . . . . . . . . . . . . . . . . . .
135
B.3 Statistics and Design of Experiments . . . . . . . . . . . . . . . . . . . . . .
143
B.3.1
Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
143
B.3.2
Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
B.3.3
Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . .
145
B.3.4
Monte Carlo Integral Estimation . . . . . . . . . . . . . . . . . . . .
146
B.3.5
Other Distributions
. . . . . . . . . . . . . . . . . . . . . . . . . . .
147
B.2 Optimal Control
12
List of Figures
1-1
The biological and engineering domains have different characteristic elements
and constraints. We seek to extract only the principles that are useful in engineering designs. Ideally this would be at the intersection of biology and
engineering (represented as the intersection of all ellipses) but even with
biological inspiration we may discover principles that only apply in the engineering domain (intersection of all 'Engineering' ellipses). Cheetah photo
from [27.
2-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
The principle extraction process filters principles from biology that are applicable to engineering design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3-1
Running robot 'Phides'. This robot consists of a torso and two kneed legs
with small, spherical feet. The robot is attached to a boom with a parallelogram mechanism to achieve planar behavior. The two hip joints are directly
actuated by DC-motors in the torso. The two knee joints are actuated by
DC-motors with a spring in series (torsion bar inside the knee shaft) as well
as a spring in parallel (leaf spring with pulley mechanism).
13
. . . . . . . . .
44
3-2
2-dimensional realistic running model that consists of five rigid bodies. The
left figure shows the robot during the flight phase and labels model parameters, the values of which are given in Table 3.1. The right figure shows the
robot during the stance phase, during which the leg spring is active, and the
6 generalized coordinates used to describe the motion. Note that the torso
orientation is not a degree-of-freedom as the rotation of the torso is fixed
with respect to the world. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
Kneed leg model, a simplification of the realistic model in which the mass of
the opposing leg and the torso are lumped into a point mass at the hip. . .
3-4
48
Prismatic leg model, a simplification of the realistic model with a point mass
49
foot connected to a distributed upper leg mass by a massless spring. ....
3-5
47
The effect of the normalized swing leg retraction rate
energy loss
D on the normalized
Eloss for a prismatic leg model, kneed leg model, and realistic
model. For the simplified models, shaded regions indicate results for a range
of touchdown conditions, including variations in instantaneous velocity and
angle of attack, encountered at touchdown in limit cycles of the realistic
model. Lines indicate the mean result for the range of touchdown conditions
studied.
Lines for simplified models with mass parameters of the Phides
robot (+0% upper leg mass) and ±50% upper leg mass are also shown. Results for simulated limit cycle running of the realistic model with Phides-like
parameters are represented with a thick black stroke. For reference,
Eloss
=
1
would occur if the full mass of the robot translating at 3.37 m/s were to stop
com pletely. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
53
3-6
The effect of the normalized retraction rate
D on the minimal mechanical
cost of transport of limit cycle running at an average normalized speed of
1.36. Each point represents a limit cycle with locally minimal mechanical
cost of transport optimized from a different random initial seed. Though
the initial seeds are entirely unrelated, the locally minimal mechanical cost
of transport of the optimized limit cycles cluster densely about an apparent
boundary believed to be representative of the globally minimal mechanical
cost of transport.........
3-7
..................................
57
The effect of the normalized retraction rate & on the minimal mechanical cost
of transport of limit cycle running for each of several normalized running
speeds 0.84-1.47. Using a wider range on the ordinate axis, the trend of
Figure 3-6 is no longer obvious. In fact, swing leg retraction rate appears to
have very little inherent effect on mechanical cost of transport. . . . . . . .
3-8
58
The effect of the normalized swing leg retraction rate FD on the normalized
touchdown impulse magnitude for the prismatic leg model, kneed leg model,
and realistic model. For the simplified models, shaded regions indicate results
for a range of touchdown conditions, including variations in instantaneous
velocity and angle of attack, encountered at touchdown in limit cycles of the
realistic model. Lines indicate the mean result for the range of touchdown
conditions studied. Lines for simplified models with mass parameters of the
Phides robot (+0% upper leg mass) and ±50%upper leg mass are also shown.
Results for simulated limit cycle running of the realistic model with Phideslike parameters are represented with a thick black stroke. For reference, a
normalized impulse magnitude of 1 would be required if the full mass of the
robot translating at 2.37 m/s were to stop completely. . . . . . . . . . . . .
15
60
3-9
The effect of the normalized swing leg retraction rate 0 on the impulse angle
for a prismatic leg model, kneed leg model, and realistic model. Impulse angle,
or the magnitude of the angle between the impulse and vertical, is defined as
Jatan(I,/Iy)J, where I, and
I, are the horizontal and vertical components of
the impulse vector, respectively. A lower value corresponds with an impulse
closer to vertical, which corresponds with less slipping at touchdown. See
caption of Figure 3-8 for more information about the meaning of the lines
and fills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4-1
Free body diagram of the posterior (body A) and tail/reaction wheel (body
B). Torque T and force R act at the joint between the bodies. Center of mass
points A and B are at positions rA and rB relative to an inertial reference;
point C is at relative position rC/A with respect to point A and rc/B with
respect to point B. Vector rB/A (not illustrated) extends from point A to point
B. Bodies A and B have masses mA and mB, moments of inertia relative to
their respective center of masses IA and IB, and angular velocities WA and
WB. Gravitational acceleration g acts along -j.
4-2
. . . . . . . . . . . . . . . .
66
Results of analysis: maximum achievable angular impulse for given maximum
motor power P* = 200W and body A inertia IA = 4 kg.m 2 as a function
of system effective moment of inertia IE and time of interest tf. Shading
indicates relative angle AO; lines of constant relative angle are shown. For
the MIT Cheetah, the maximum mass for tail or reaction wheel is m = 2kg.
Allowing maximum radius r, =
0.1m for a reaction wheel yields IE,w <
mr2 = 0.02kg-m 2 , and allowing rt = 0.6m for a tail yields IE,t
mr?
0.61kg-m 2 . For the time of interest tf z 0.2s, the optimal tail can produce a
higher maximum angular impulse than the reaction wheel without exceeding
the maximum allowable rotation of AO
16
7r/2. . . . . . . . . . . . . . . . . . . 71
5-1
The model used in this study consists of a point mass m acting under the
influence of gravity g and leg force F. Both telescoping and kneed versions
have the same electric motor which can produce maximum power P*. Consequently, in the telescoping model, the maximum force depends on leg contraction/extension speed but is independent of leg length. For the kneed leg,
however, the maximum force is a function of both leg length and speed. . .
5-2
77
Design space shaded by efficiency of kneed leg relative to telescoping leg.
'Null hypothesis', that there is no efficiency difference between telescoping
and kneed leg, applies where a t-test on the efficiency data for that point in
the design space did not indicate significance at the p = 0.01 level. 'No data'
means that optimization did not converge to any feasible solutions for that
point in the design space.
5-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
The effect of Froude number (Fr)and normalized step length (Sl) on difference in minimal mechanical cost of transport (kneed leg minus telescoping
leg). Where local minima resulting from the multiple random seeds were not
identical (but tightly clustered), the surface represents the difference in the
sample m eans.
5-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
The effect of the ratio of normalized step length to Froude number squared
(Sl/Fr2 ) on the minimal mechanical cost of transport. All data points, that
is, all local minima from the multiple random seeds, are included to show
how tightly the local minima cluster.
6-1
. . . . . . . . . . . . . . . . . . . . .
84
When scaled according to hip height, the knee of the human is at the same
level as the ankle of the ostrich. As a result, if the human leg is considered
to bend 'forwards', then the ostrich leg appears to bend 'backwards'. This
inspires robot designers to ask a question: 'in which direction should a robot's
knee bend to achieve maximum running efficiency?'. Human Skeleton: [118],
Ostrich Skeleton: [28]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
88
6-2
The two leg morphologies considered in the experiment: a 'forwards' knee and
a 'backwards' knee. Rectangles of the same color and number represent rigid
bodies with the same mass, moment of inertia, length, and center of mass
location. Arcs represent actuators with the same torque limit and maximum
torque rate of change. All physical parameters and the average running speed
are randomly selected from their respective populations to define a robot
design. For each of many robot designs and for each knee orientation, initial
conditions and torque rate trajectories are calculated to optimize energetic
efficiency. The differences between forwards and backwards knee energetic
efficiencies for each design reveal the inherent effect of knee direction on
optimal energetic efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-3
90
The physical parameters of a robot design: each segment is defined by total
length
li, mass mi, center of mass location ci, and moment of inertia with
respect to center of mass Ii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6-4 Graphical representation of the design space, including biological and robotic
examples. The position of each example's icon on a number line visually depicts the value of its dimensionless parameter. Solid gray blocks mark regions
excluded from the design space for this study. Black checker patterns mark
hard physical constraints: for example, none of the dimensionless variables
can possibly be less than zero because no mass or length can be negative.
Gray checker patterns mark soft physical constraints: for example, it would
be unusual for c1/li to be greater than unity, as this would suggest that
the center of mass location of the upper leg link is not between the hip and
the knee. Human parameters are from
(71]; ostrich, [58]; Mabel, personal
communication with Hae Won Park; Phides, personal communication with
Dani8l Karssen; Rabbit, [22].
6-5
. . . . . . . . . . . . . . . . . . . . . . . . . .
92
The state of the robot is determined by horizontal and vertical Cartesian
coordinates of the hip, angles of each segment, and their time derivatives.
Control torques act at the hip between the upper leg segments and the torso
and at the knee between the upper and lower leg segments.
18
. . . . . . . . .
95
6-6
Example forwards running seed (random guess).
. . . . . . . . . . . . . . .
9
99
6-7
Example forwards running gait (optimized). . . . . . . . . . . . . . . . . . .
99
6-8
Example backwards running seed (random guess).
. . . . . . . . . . . . . .
99
6-9
Example backwards running gait (optimized). . . . . . . . . . . . . . . . . .
99
6-10 Absolute work cost of transport of running gaits optimized from random
seeds for each robot design. Only results with a low (<
0.7) absolute work
cost of transport are shown. For most robot designs, optimization converged
from random seeds to gaits that cluster above an apparent cost of transport
lower bound. The clusters for backwards knee gaits tend to be lower than
the clusters for the forwards knee gaits, suggesting that backwards knee gaits
tend to be more efficient than forwards knee gaits. . . . . . . . . . . . . . . . 101
6-11 Graphical representation of all robot designs for which optimization converged. Link width represents link mass; links are drawn such that the mass
is equal to the product of link width squared, link length, and a density factor common to all designs. The center of each circle represents the center of
mass of each link, and the circle radius corresponds with the link radius of
gyration. The purpose of this figure is to illustrate the great variety of robot
designs randomly sampled from the design space; this variety allows us to
make inferences about the whole design space . . . . . . . . . . . . . . . . .
102
6-12 Rank of absolute work of transport of running gaits for each robot design.
Robot designs (columns) with many backwards running gaits more efficient
than the most efficient forwards running gait have been moved toward the
left and vice versa. There is a marked tendency for the backwards knee gaits
to have lower numbered ranks, suggesting superior efficiency of backwards
gaits.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-13 Rank of torque squared cost of transport
CT2
103
of running gaits for each robot
design. There is a marked tendency for the backwards knee gaits to have lower
numbered ranks, suggesting superior efficiency of backwards gaits according
to this m etric, too.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
104
6-14 Rank of impact loss cost of transport cil of running gaits for each robot design.
There is a slight tendency for the forwards knee gaits to have lower numbered
ranks, suggesting reduced impact loss of forwards knee gaits, in contrast with
the findings of [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
6-15 Trajectory of hip torque of stance leg plotted against the relative angle between the stance leg and the torso for the example gaits of Figures 6-7 and
6-9. Both peak torque and total angular excursion are much smaller in the
case of the backwards knee. Note also that the backwards knee gait separates periods of high torque from those of significant angular displacement,
whereas the forwards knee gait exhibits considerable hip excursion during
periods of high torque. Because absolute work corresponds with the area
between this curve and the horizontal axis (shaded for forwards gait), it is
apparent that the hip actuator of the stance leg performs far less absolute
work for the backwards knee gait. . . . . . . . . . . . . . . . . . . . . . . . .
106
6-16 Trajectory of hip torque of stance leg plotted against time for the example
gaits of Figures 6-7 and 6-9. Both peak torque and total time are much
smaller in the case of the backwards knee. Note also that periods of high
torque are of much shorter duration for the backwards knee gait, whereas the
forwards knee gait exhibits considerable torque for almost the entire period.
Because the integral of torque squared corresponds with the area between
the square of this curve and the horizontal axis, it is apparent that the hip
actuator of the stance leg contributes far less to the torque squared integral
for the backwards knee gait. . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
6-17 Rank of hip absolute work cost of transport, that is, absolute mechanical
work performed by the hip motors normalized by the product of weight and
distance traveled, of running gaits for each robot design. There is a marked
tendency for the backwards knee gaits to have lower numbered ranks, suggesting superior efficiency at the hip for backwards gaits.
20
. . . . . . . . . .
108
6-18 Rank of knee absolute work cost of transport of running gaits for each robot
design. There is a marked tendency for the forwards knee gaits to have lower
numbered ranks, suggesting superior efficiency at the knee for forwards gaits. 108
6-19 Rank of hip torque squared cost of transport, that is, the time integral of
torque squared performed by the hip motors normalized by the product of
weight and distance traveled, of running gaits for each robot design. There
is a marked tendency for the backwards knee gaits to have lower numbered
ranks, suggesting superior efficiency at the hip for backwards gaits. . . . . .
109
6-20 Rank of knee torque squared cost of transport of running gaits for each robot
design. There is a marked tendency for the forwards knee gaits to have lower
numbered ranks, suggesting superior efficiency at the knee for forwards gaits. 109
6-21 Propulsion advantage of the backwards knee. With the hip joint locked, only
the backwards knee supports propulsion of the running machine; a running
machine with a forwards knee would collapse. . . . . . . . . . . . . . . . . .
110
6-22 Protraction advantage of the backwards knee. With the hip joint locked, only
the backwards knee can lift from the ground and prepare for another step;
a running machine with a forwards knee requires significant hip motion to
complete the running cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
B-1 The 'dartboard function': to every point on the plane, we assign the point
value associated with the corresponding region of the dartboard. Outside
the boundaries of the board, the function is undefined; these points are not
counted in the calculation of the mean . . . . . . . . . . . . . . . . . . . . .
21
148
22
List of Tables
3.1
Parameter values of the realistic model and robot. See Figure 3-2 for the
parameter definitions, in which parameters are given by the symbol (e.g. m
for mass) with a subscript to indicate the segment (e.g. t for torso) . . . . .
49
4.1
Properties of force/moment application technologies: . . . . . . . . . . . . .
64
5.1
Fixed variables, i.e., variables that assume fixed values for the duration of
the experiment. These variables are only used to keep the equations in dimensional, physically-intuitive form; they do not affect the optimal cmt as
a consequence of the Buckingham 1I theorem.
Therefore, the experiment
remains valid for robots with any values of these variables. . . . . . . . . . .
5.2
79
Sampled variables, i.e., variables for which trials are performed at randomly
sampled values to broaden the applicability of the experiment to a wide class
of robots and gaits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
80
Decision variables, i.e., those which are optimized to achieve minimal cmt.
Values in gray are de-emphasized because they do not affect the experiment;
they are sufficiently generous that they are rarely observed to be active.
When a bound happens to be active, the corresponding results should be
discarded to prevent the choice of bound value from affecting results. Values
in black are 'hard' bounds set by physics or by the intent of the experiment, except for the lower bound on duty factor, which is sufficiently large
to prevent numerical difficulties associated with a vanishing stance phase but
sufficiently small to permit approximation of the optimal cmt.... . . . . . . .
23
80
5.4
Summary of results for the full population. As the population now includes
samples with lower Pr, the dominance of the telescoping leg is even more
clear. This might be expected, as the difference between the two configurations only manifests for low Pr; for high Pr the constraints distinguishing
the two are less likely to become active.
6.1
. . . . . . . . . . . . . . . . . . . .
84
The design space is defined by ranges of torso length 13, torso mass M 3 , and
11 dimensionless parameters.
A design is sampled by randomly selecting
values for each parameter. Note that one of the parameters included in the
'design' is the Froude number Fr
[41,
the ratio of running speed squared to
the product of gravitational acceleration and leg length. Sampling from the
design space also determines the average speed of the running gait. . . . . .
6.2
94
Limits on states, controls, and times. Note 10 = 11 + 12 is the leg length, average velocity v, =
VgloFr
respects the Froude number, maximum torque To =
l (m3 + 2m1 + 2m2)9 is that required to raise the entire weight of the robot at
one leg length, and maximum torque rate TRo =
TO
tphasel,m in +tstance,rnin +tphase3,min
limits the torque from sweeping through a range greater than To in the minimum possible step time. In the original code, bounds on AXstride did not
include the factor of lo and bounds on Atflight included Atstep in place of
Atstride. After correcting the code and re-running the experiment with a reduced number of samples, the nature of the results did not change. *Bounds
on 021 and 022 depend on the knee direction; bounds are given by [0, 7r] for
forwards knee and [-7r, 0] for backwards knee. . . . . . . . . . . . . . . . . .
6.3
98
Frequency of Optimization Success. Two hundred robot designs were sampled at random from the design space. For each of the two knee directions,
local optimization was performed for each of 50 random seeds for each design. Optimization converged successfully for a fraction of cases, and some of
these solutions were eliminated for kinematically infeasible trajectories not
precluded by optimization constraints. . . . . . . . . . . . . . . . . . . . . .
100
B.1 Possible outcomes of two coin flips . . . . . . . . . . . . . . . . . . . . . . .
144
24
Chapter 1
Introduction
Nature provides a wealth of design ideas valuable to engineers because it holds the only
examples of design other than our own [126]. Biology has lent the pattern for many original
and successful materials, devices, and manufacturing methods [114, 124], and so 'biomimetics' has become a popular field of engineering research. It promises that by harvesting
biology's secrets, we can apply our understanding of the natural world to improve human
quality of life. But even though humans have looked to nature for technological inspiration for over 3000 years [125], development of biomimetic design methodology is ongoing.
Indeed, [125] states that no general approach has been developed for biomimetics, [123]
admits that while there are many case studies of biologically inspired design, these are only
a first step toward developing a design methodology, and [46] notes that only recently have
researchers begun to develop formal bioinspired design methods and tools. Several of these
approaches are outlined in [124, 85, 52, 24], and all of these address the importance of
extracting principles from biology, often for the purpose of tabulating these principles for
subsequent lookup. However, these methods do not fully address how principles can be
isolated from a biological example, even after one has been identified.
Although experimental biologists extract scientific principles using well-developed experimental practices and careful inductive reasoning, often their objective is to better understand purely biological systems, and so the results are not necessarily suitable for immediate
application to engineering design due to differences between the biological and engineering
domains. Biological systems may rely on unique biological capabilities currently beyond
25
Instincts
Digestion
Muscles
Reproduction
Neurons
Principles
Manufacturing
Transistors
Motors
"
Batteries
Algorithms
Figure 1-1: The biological and engineering domains have different characteristic elements
and constraints. We seek to extract only the principles that are useful in engineering designs. Ideally this would be at the intersection of biology and engineering (represented as
the intersection of all ellipses) but even with biological inspiration we may discover principles that only apply in the engineering domain (intersection of all 'Engineering' ellipses).
Cheetah photo from [27].
the performance limits of their engineering counterparts.
For instance, in perhaps the
most seminal publication regarding the mechanism of gecko adhesion [8], the authors admitted that production of "small, closely packed arrays mimicking setae" was beyond the
manufacturing technology of the time, and years of additional research were necessary to extend the abilities of component technologies before bioinspired prototypes became possible
[45, 86, 67, 106, 73, 72]. Likewise, biological solutions are often limited by constraints not
shared in engineering. A biological example may indeed perform a function of interest to
engineers, but the particular approach taken by biology may not truly be optimized for that
function [65]. Rather, it is the result of evolution, which must consider many other complex
biological objectives and constraints [126], such as the needs to extract energy from food, to
grow, and to reproduce. Consequently, engineers must assess whether a hypothesized bioinspired principle actually applies in the engineering domain and whether the biomimetic
approach has the potential to outperform existing solutions before applying it to design.
The differences between the biological and engineering domains prompt the need to
26
extract principles only as they apply to engineering design, as illustrated in Figure 1-1.
Although I have found no discussion of a general procedure for extracting bio-inspired
principles for application in the engineering domain, one might attempt to generalize the
ad hoc methods used by relevant studies. For instance, researchers have considered the
following questions for legged robots:
" What is the effect of spine flexibility on running speed and efficiency [50, 39, 29]?
" What is the effect of tails on quadruped maneuverability [17, 74]?
" What is the effect of foot shape on energetic efficiency [7]?
" What is the effect of having toes on locomotion stability [2]?
" What is the effect of the upper body on locomotion stability and efficiency [23]?
" What is the effect of leg segmentation on robustness to disturbances [103, 102]?
" What is the effect of leg configuration on energy loss of walking and running [61]?
General answers to these questions could serve as principles to guide robot design. Unfortunately, it would be difficult to generalize the conclusions of any of these studies to a broad
class of realistic robots for one or both of the following reasons:
" the use of a model with fixed parameters limits the applicability of the results to the
particular system studied [7, 2, 23, 50, 39, 29], and/or
" the model used is too simple to provide convincing evidence of direct applicability to
physical robots [103, 102, 17, 61].
Because of these limitations, none of the ad hoc methods used in these studies (or any others
I am aware of) provide a complete basis for a general method for extracting principles from
biology for application to engineering design, prompting the need for this thesis.
Surprisingly, extensions of the rigorous methods used by experimental biologists are
not commonly found in the robot literature.
For example, biologists can carefully test
physiological hypotheses by performing experiments on several specimens of, a given type
of organism. Based on their observation of multiple samples from a population they can
induce a truth about the entire species. I argue that engineers can similarly test design
hypotheses by performing experiments on multiple specimens from a 'species' of robots.
27
Thesis Structure
In the introduction, I have motivated the importance of biomimetic
principle extraction, demonstrated that general biomimetic principle extraction techniques
do not exist, and shown that they cannot be generalized easily from ad hoc methods.
Before continuing, the reader may wish to consider some of the difficulties in developing
a principle extraction framework and potential solutions in Appendix A. Also in Appendix
B, I introduce the mathematical preliminaries required to build a general framework, including computer simulation of rigid body dynamics, optimization and optimal control, and
elementary statistic.
After carefully defining 'design principles' and 'principle extraction', I present the framework itself in Part I.
In Part II, I present case studies that introduce elements of the framework. I begin
in Chapter 3 with a study of the effect of swing leg retraction on the energetic efficiency
of running, introducing the use of optimization/optimal control in the principle extraction
process.
In Chapter 4, I analyze the effect of a bioinspired tail on maneuverability to
introduce the importance of parameter variation for developing a general conclusion.
In Part III, I present case studies that implement the complete framework. Chapter 5
includes a simplified examination of the effect of knee type on the energetic efficiency of
running. Chapter 6 demonstrates that the framework can be scaled to very complex studies
via an investigation of the effect of knee direction on the energetic efficiency of running, and
Part IV concludes.
28
Part I
Methodology
29
30
Chapter 2
A General Framework for Principle
Extraction
and
Consider the design process as a procedure that begins with initial conceptual designs
refines design parameters iteratively until the final solution achieves certain design requirements, including objectives and constraints.
Compare this to the numerical process of
the
solving a system of inequalities: a computer typically begins with an initial guess for
solution, and iteratively changes the independent variables until equality and inequality con-
Biology
Question
Fac
from obseration of
Biologicalsys
p
Modeling
with realistic physics
0ofEnging systm
Experiment
to test hypothiesis
Analysis
to generft d&Sigp
principles from data
Bio-Inspired Engineering Principles
from biology that are applicaFigure 2-1: The principle extraction process filters principles
ble to engineering design.
31
ditions axe satisfied. While computer algorithms perform best with complete, quantitative
relationships or partial derivatives, human designers typically evaluate these only sparsely.
Human designers excel at achieving a final solution based on 'design principles': relationships between input design parameters and output objectives and constraints applicable to
many systems. 'Biologically-inspired design principles' are, in my opinion, design principles
derived in any part from the observation of biological systems1 , and 'principle extraction' is
any process used to turn observation of biological systems into design principles. For example, the MIT Cheetah will employ a tail to assist in aerial maneuvers because the principle
that tails are more effective than reaction wheels has been confirmed under the constraints
of a typical quadrupedal robot
[171.
Figure 2-1 illustrates our approach to the discovery of
such biomimetic design principles, and the remainder of the section details that process.
2.1
Observation of Nature Inspires a Biological Design Question
Nature provides an abundant supply of profound biological examples, but our ability to extract design principles from them relies to some extent on innate human curiosity and talent
at recognizing patterns. Even when a biological example is particularly well adapted for
performing a task similar to one desired of machines, humans must recognize the comparison before the extraction process can begin. Once this link is established, the researcher can
observe and study the biological example further, often in collaboration with experimental
biologists, in an attempt to form a question and a hypothesis about the relationship between
the organism's features and its performance of the task. The researcher may first pose the
question in the biological domain, 'What is the effect of this feature on the performance of
the organism?'.
'This does not require that the design principle actually be observed in biological systems.
32
2.2
Transfer of the Question from the Biological Domain to
the Engineering Domain Prompts an Engineering Model
Although a physiological principle obtained by observation of biology may not be true in
the engineering domain, a question inspired by observation of biology can be transferred to
the engineering domain directly. The question becomes 'What is the effect of an engineering adaptation of this feature on the performance of an engineered system?'. To perform
this transfer, the researcher must define the high-level architecture of the the engineered
system, including the adaptation of the bio-inspired feature. In some cases, such as legged
locomotion, there is already some consensus on the architecture of the engineered system:
the bones of an animal are often replaced with plastic or metal links, the joints with bearings or flexures, and the muscles with electromagnetic or pressurized fluid actuators. In
others systems, this transfer relies more on the researcher's innovation. For instance, when
researchers have considered the effects of geometry and material on the adhesion performance of gecko-inspired adhesives, engineering adaptations of the adhesive gecko hairs have
ranged from millimeter-scale elastomer pillars [107] to carbon nanotubes [110]. As with the
solution of a new differential equation, there may not necessarily be a formulaic approach
to defining the architecture of the engineering system; instead the researcher must make an
educated guess which will be tested. Any guess is admissible, but the utility of the results
will depend on the choice of the engineering system that is studied.
Once the architecture has been established, the researcher has several options for answering the question. Typically, analytical studies are impeded by the complexity of the
system; principles can be deduced only in the most heavily simplified cases. Physical testing
of engineering systems offers the greatest guarantees for the accuracy of conclusions, but
constructing and testing multiple subject machines is usually prohibitively time consuming
and expensive. When systems governed by relatively well-understood physics can be modeled to simulate behavior quickly and inexpensively, computer experiments are often the
method of choice for testing hypotheses. I will focus on in silico testing for this framework
because of its ability to treat complex systems with relatively low resource requirements.
Computer experiments begin with a model of the relationship between the system's
33
inputs and outputs. Simple models with few inputs are general in the sense that by avoiding
representation of any particular system, they represent many different embodiments of a
system simultaneously [115]. This feature would seem to make them particularly suitable
for principle extraction studies. For instance, the SLIP model and variations have been
used to study the effects of certain biologically-inspired features, such swing-leg retraction,
on running performance metrics, such as efficiency [97, 64, 49] and stability [14, 111, 112,
34, 64, 89]. However, Chapter 3 and [63] compare some of these results for simple models,
a detailed model, and a physical robot, and find that the simple SLIP model is unreliable
at predicting such complex behaviors. Therefore, unless simple models have been carefully
verified against more detailed models and physical systems, they should only be used for
explanation of observed phenomena and not be relied on for principle extraction.
The definition of the output of interest is also important. In the study of legged robots,
there are many different ways of quantifying concepts like 'stability' and 'efficiency' [55, 18],
and they may not be interchangeable. For example, the rate of recovery from infinitesimal
disturbances may not at all be indicative of the magnitude of disturbances a robot can
recover from [63]. Likewise, although a portion of a robot's energy expenditure may be due
to impact with the ground, collisional losses and energy use per distance traveled may not
always be strongly correlated [63]. Selection of the method used to quantify the output of
interest cannot be based solely on computational efficiency, but also on its direct relevance
to the performance or ability robot designers desire.
2.3
Experiments with the Engineering Model Provide Data
Supporting Engineering Principles
2.3.1
Experimental Design
In the field of experimental design, it is common to select a value of the input as a 'control'
and refer to other values of the input as different 'treatments'.
The set of all possible
combinations of the other inputs may be referred to as the 'population'. Then it may be
possible to learn enough about the relationship of interest for design purposes by answering
either of the following questions (numbered as in Appendix A.2.1):
34
"
Question 4: What is the effect of the treatment on the percentage of the population
with improved <output>, relative to the control?
" Question 5: What is the effect of the treatment on the average (across the population)
change in <output>, relative to the control?
While answering these questions does not provide detailed information about the coupling
relationships among the inputs, the conclusions can still be applicable to the full class of
systems under study. Furthermore, these questions can be answered using Monte Carlo
methods [75], that is, sampling at random from the population, measuring the effect of the
treatment for these samples, and using the results to make an inference about the entire
population. The computational cost of Monte Carlo methods can be far lower than that
of testing all samples generated by varying parameters on a grid. Critically, we can also
quantify our uncertainty in the conclusion with statistical tests, such as the binomial test
[15] or a resampling method [75], and be sure to collect sufficient data to build confidence
in the result.
Design of experiments [37, 15] is a vast field of applied statistics. Working knowledge of
the fundamentals is imperative for performing principle extraction experiments, and many
advanced topics might be beneficial as well. The purpose of this section is not to summarize
statistics for experimenters or to repeat Section B.3, but to outline steps for answering
questions of the two forms above, chosen because they require only simple techniques and
relatively low computational resources to answer, yet their answers may be very insightful
for design purposes. In Section IV, I will mention other questions and techniques that may
also be useful.
Use of Statistics
An important difference between computer experiments with deterministic simulations and
physical experiments is the absence of random error. Therefore, statistical techniques are
not needed to quantify our uncertainty due to random error [105], but primarily the uncertainty in our conclusions due to the limited number of population members sampled.
We seek to understand a relationship
y =
f
between input of interest x and scalar output
f(x) to inform a design. Specifically, we seek to answer Question 4 and 5 numerically,
35
which are expressed mathematically as:
"
what is the percentage of the population g(xt)
P
H(f(x±,p)-f(xc,p))dP
=
for which
output f(xt, p) > f(x,, p) , where H is the Heaviside step function, and
" what is the average across the population
of the difference
h(xt) = fp f(xtp)-f(xcp)dP
J dP
in outputs f(xt, p) - f(Xe, p),
for treatment xt and control x, as other input parameters p vary across the population P.
We cannot evaluate the integrals exactly, but we can use Monte Carlo methods to develop
a confidence interval in which the the answer is believed to lie.
Answer to Question 4
Using N random samples pi from P, we can approximate the
true percentage g(xt) with the observed percentage
-
Zf=1
H (f(xt, pi)
N
-
f(xc, p
(2.1
a random variable. Then §(xt)N follows the binomial distribution of N experiments and
unknown proportion of success q = g(xt), i.e.
(xt)N
-
B(N, g(xt)). We can use this
information to test a null hypothesis, that is, the likelihood of the observed §(xt) being
produced with an assumed qo, and likewise we can produce a binomial proportion confidence
interval by any one of several methods [1, 60].
Answer to Question 5
Using N random samples pi from P, we can approximate the
true mean h(xt) as the sample mean
(xt) =
-
f(Xt, p,)
N _ f(X,
(2.2)
a random variable which follows an unknown distribution. Nonetheless, we can estimate a
confidence interval for h(xt) using resampling methods, such as the bootstrap [75].
Note that Questions 4 and 5 both call for the practice of blocking: assigning similar,
or in this case identical, test subjects pi to all treatments and the control group [15] in a
given trial. This improves the sensitivity of the experiment because it allows the effect of
the treatments to be measured independently of any confounding factors. By performing
36
trials with many different 'blocks', each with a randomly selected test subject pi, we collect
data to make an inference about the population.
Variable Partitioning
Once the form of the question is understood, the researcher must decide how to treat
each of the input variables, and in doing so begins to determine the scope of the study's
applicability.
Fixed Variables
Variables that assume fixed values for the duration of the experiment
(e.g. the architecture of the model) we call 'fixed variables'. While sometimes necessary,
these typically limit the generality of the study. On the other hand, a certain number of
parameters may be held fixed without loss of generality as a consequence of the Buckingham
II Theorem [16]. Note that we do not list fixed variables as inputs to the function of interest
f because
they do not change; they are instead considered to be part of
Sampled Variables
f.
Those variables p which vary only between trials, e.g. model pa-
rameters, we call 'sampled variables'. Choosing different values for different trials, as by
random sampling from a population, broadens the scope of the study.
Independent Variables
Those variables x which are varied and take on values x, and xt
within a trial, e.g. the inputs of interest, are referred to as 'independent variables'. Multiple
variables independently varied on a grid usually simplify interpretation of results, but can
require extensive data collection. More efficient experiment designs for multiple variables
exist (e.g. Latin hypercube or factorial designs [15]), but we limit our discussion here to a
single, scalar independent variable.
Decision Variables
Variables that vary depending on others within a trial, e.g.
to
optimize an objective and/or satisfy constraints, we call 'decision variables'. We discuss the
use of such variables in 2.3.2. Although these limit the applicability of a study somewhat,
sometimes it is necessary to treat variables as such when handling them as sampled variables
or independent variables is impractical due to the enormous variety of values (or functions)
37
these variables assume. We do not list decision variables as inputs to the function of interest
f
because they are not independent of, but rather determined by, the other inputs p and
X.
Variable Level Selection
After the inputs have been partitioned into different types, the designer of a computer
experiment has the task of defining the levels each will assume, the population from which
they will be sampled, or the space over which they will be optimized. When doing so, the
researcher must bear in mind that the purpose of the experiment is to provide evidence for
or against the applicability of a principle for design purposes. The ranges of values should
be large enough to include all the potential system designs to which the researcher wishes
conclusions to be applicable.
The ranges should be no larger than necessary, however,
because inclusion of irrelevant members in a population can skew the results. Choice of
the populations of sampled variables and bounds on decision variables deserve particular
attention.
2.3.2
Experiment Implementation
The use of computers dramatically simplifies the process of collecting data.
Once the
experiment has been designed, there is code to be written, but computers do most of the
processing.
While the code to simulate the system of interest will be problem specific, and code to
select sampled variables from the population is relatively straightforward to write, the use
of optimization deserves special attention here. Inputs can be treated as decision variables
when treating them as fixed variables limits the generality of the experiment, treating them
as independent variables would require excessive computation, and treating them as sampled
variables is impractical because defining the distribution of input values (or functions) is
difficult. It is also justified when reasonable levels of the input are unknown, but is likely
that the system designer would choose the level of the input to improve the output of
interest. In our experience, inputs that must respect nonlinear constraints and/or inputs
that are functions, rather than scalars or vectors, can often be treated only as either fixed
38
variables or decision variables. Of the two options, treatment as decision variables seems
more satisfying because rather than fixing the inputs at arbitrary values, they are fixed at
optimal values.
Optimization
In the study of legged robots, for example, the robot controller is an input that will affect
the output of interest. The controller is a function that must satisfy certain constraints (e.g.
the robot cannot fall). If there were a standard parametrization for a robot controller with
accepted ranges for the parameters, perhaps those parameters could be treated as sampled
variables, but this is not typical. We would prefer not to choose an arbitrary controller
that satisfies the constraints because this limits the applicability of the study to robots
using that particular controller. It is preferable, then, to treat the controller as a decision
variable; we use the controller which optimizes the output of interest for the given fixed
variables, sampled variables, and independent variables. Fortunately, it is often reasonable
to expect that regardless of the physical robot design, the designer will typically choose a
controller that will nearly optimize the output of interest (e.g. stability, efficiency, etc...),
and so extraction of a principle regarding the optimal value of the output of interest is of
interest to robot designers. This motivates a brief discussion of optimal control.
Analytical optimal control is usually impossible for all but the simplest models, and
thus computational optimal control techniques must be employed [127]. As is common in
numerical algorithms, computational optimal control schemes typically begin with a guess
of the solution, a 'seed'. We believe that it is important for this seed to be generated
automatically, as hand-selecting the seed for many test subjects would be time-consuming,
and more importantly, it is undesirable for human intervention to affect the results (i.e.
which local optimum is found). We prefer that the seed be selected at random from the
space of all possible seeds to minimize the effect of human choice on the output of the
experiment, and we perform optimization from many such seeds ('multistart' [109]) to
reduce the effect of chance occurrences on experimental results.
As explained in Section B.2, computational optimal control is typically divided into direct methods, which discretizes the optimal control problem in order to formulate a discrete
39
nonlinear programming problem, and indirect methods, which use optimal control theory
to generate necessary conditions for optimality in the form of a boundary value problem
that can be discretized and solved numerically [127, 12]. Similar discretization ideas can
be employed in either direct or indirect methods; single shooting, multiple shooting, and
collocation are common discretization methods [127, 12]. We prefer the direct approach,
as it is easier to use, and we discretize using collocation methods because they result in a
numerical problem that is easier to solve without a good initial guess [127]. In particular, we
find pseudospectral methods [10, 99] to be remarkably robust to the initial guess; regularly
converging from a random seed to a solution confirmed by Runge-Kutta integration[113]
to satisfy the dynamics and constraints. Potentially important to their success is use of
accurate partial derivatives; complex step approximation [76], algorithmic differentiation
[47], and analytical differentiation are all more accurate and potentially faster than the typical first-order finite difference approximation. When partial derivatives are not available,
derivative-free optimization [25] techniques and/or metaheuristics [69] might be used.
2.4
Analysis of the Data Yields Engineering Principles
The statistical tests mentioned in 2.3.1 are the primarily analyses as they answer the Questions 4 and 5 the experiment was designed to investigate, and these answers are the primary
conclusions. However the experiment yields a tremendous amount of data which can be used
for exploratory data analysis [121]. For instance, it may be informative to plot histograms
of the output of interest, as a distribution with an unusual structure may be worthy of
deeper investigation. Also, the output of interest and possibly other outputs can by plotted
against each input, or more generally functions of inputs and outputs can be plotted against
one another. Any notable trends may prompt conclusions or suggest ideas for further study.
While the experiment may reveal correlation between the inputs and outputs, they do
not necessarily illuminate the relationship in an intuitive way. Correlation may be perfectly
sufficient information for design purposes, but understanding of the relationship is more
satisfying to human curiosity, is generally considered to be a 'deeper' and more fundamental
type of knowledge, and can have greater predictive power. Studies which reveal correlation
40
may be refined, however, to improve understanding. For instance, if the output of interest
can be subdivided into several parts, and similar experiments are performed with each part
as the output of interest, then more about the original experiment can be explained. Such
subdivisions may also make it easier to relate the results of the experiment to physical
intuition.
41
Part II
Case Studies Incorporating
Elements of the Framework
42
Chapter 3
The Effect of Swing Leg Retraction
on Performance of a Bipedal
Running Robot
Using simple running models, researchers have argued that swing leg retraction can improve
running robot performance. In this chapter, which consists primarily of my contributions
to [63], I investigate whether this holds for more realistic simulation models based on a
physical running robot. In addition, I introduce the use of optimization to study the effect
of swing leg retraction on overall energetic efficiency; the use of optimization removes dependence on a particular controller and enables the extraction of information inherent to
the physical robot design. The study finds that for this robot, swing leg retraction can indeed simultaneously reduce touchdown forces, slipping likelihood, and impact energy losses,
however, the retraction rates at which the effects are maximized are strongly model dependent, suggesting that robot designers cannot always rely on simplified models to accurately
predict such complex behaviors. Also, while swing leg retraction can also improve overall
energetic efficiency, the improvement is very small and does not correlate well with impact
loss, in contrast with the assumptions of previous work.
43
Figure 3-1: Running robot 'Phides'. This robot consists of a torso and two kneed legs with
small, spherical feet. The robot is attached to a boom with a parallelogram mechanism
to achieve planar behavior. The two hip joints are directly actuated by DC-motors in the
torso. The two knee joints are actuated by DC-motors with a spring in series (torsion bar
inside the knee shaft) as well as a spring in parallel (leaf spring with pulley mechanism).
3.1
Introduction
Legged locomotion is an important topic in robotics because legged robots promise improved
mobility in unstructured environments [94, 108, 66]. Intuition regarding the sensitivities
of robot performance to hardware and controller parameters is essential for the design of
effective legged mobility systems. In addition, knowledge of these sensitivities can give
insight into human locomotion, which is useful for the design of better prostheses and
orthoses.
The goal of this chapter is to develop intuition about the inherent effects of
a particular control parameter, swing leg retraction rate, on several running performance
metrics if possible, and otherwise identify trends that defy simple description.
Swing leg retraction (SLR) is a behavior exhibited by humans and animals in which the
airborne front leg rotates rearward prior to touchdown
[83].
It is hypothesized that SLR
enhances performance of biological systems [14], and that SLR might be used to improve
the performance of legged robots, such as the Phides robot shown in Figure 3-1.
Use of
SLR to improve controller performance is attractive, because it is a conceptually simple
extension to any foot placement controller, such as the constant angle of attack controller
[111] and the neutral point controller [98].
The effect of swing leg retraction on limit cycle walking [54] is relatively well studied
44
and has been shown to improve energy efficiency, small disturbance stability, and large
disturbance rejection [6, 131, 56]. These results are illuminating for walking systems, but
fundamental differences between walking and running gaits preclude the direct application
of these results to running systems. Certain aspects of swing leg retraction have been studied
using relatively simple running models; for instance, multiple authors agree that swing leg
retraction can improve the stability of running [14, 111, 112, 34, 64, 89]; conclude that low
retraction rates yield better stability, but high retraction rates minimize peak forces [30, 64];
and suggest that swing leg retraction can improve energetic efficiency [97, 64, 49]. However,
the existing literature leaves an important open question: Do simple models accurately
predict the effects of SLR on a realistic running robot?
The present chapter provides answers to this question through a study of the effects of
SLR on running performance using simple models and a realistic model validated against
physical hardware.
The remainder of this chapter is organized as follows: Section 3.2
introduces the simulation models used in this study. Section 3.3 and Section 3.4 present
the effects of the retraction rate on the impact losses and energetic efficiency of the models,
respectively. Section 3.5 discusses the effect of the retraction rate on the impact impulse
and footing stability, and conclusions are presented in Section 3.6.
3.2
3.2.1
Models and Experimental Platform
Realistic model
Simulation
The realistic simulation model is designed to closely resemble the physical running robot
Phides, shown in Figure 3-1. The model, shown in Figure 3-2, is 2-dimensional (planar)
and consists of five rigid bodies with distributed mass: a torso, two upper legs, and two
lower legs. The sizes and mass distributions of the rigid bodies are given in Table 3.1. The
feet are simply points at the ends of the lower legs. As in the physical robot, the torso
orientation is fixed with respect to the world to eliminate the need to control it. This leaves
the model with 6 degrees-of-freedom: the
x- and y-position of the hip, the rotations of two
hip joints, and the rotations of two knee joints. Torques, limited to 21.4 Nm to represent
45
the actuator limitations of the robot, act at all four joints.
The running motion of the realistic model has two distinct phases: a flight phase during
which the robot is airborne and the center of mass follows a ballistic trajectory, and a stance
phase in which one foot acts as a pin joint fixed to the ground.
When a foot touches the ground, the impact is modeled as an impulsive, perfectlyinelastic collision. The state after impact is calculated according to angular momentum
conservation. Under certain conditions, the impulse of the ground acting on the robot is
negative; that is, the floor pulls on the robot. This is common in such models [79, 49].
While the negative impulse is physically unrealizable,
[80]
and personal experiments found
that this collision model predicts the state after touchdown reasonably well and is far less
computationally expensive compared to a finite friction model, which might represent the
physics more faithfully.
During stance, a spring of constant stiffness 5.06 kN/m produces a force linearly proportional to the distance between the stance foot and the hip, storing and releasing energy
exactly as in the SLIP model. The knee is equipped with an end stop at 0.52 rad that
prevents the leg from extending further than the rest length of the leg spring of 0.58 m.
This knee end stop is modeled as a unidirectional spring-damper, that is, a shock absorber.
As in flight, joint torques control the robot during stance.
Liftoff occurs and flight resumes when the normal force between the foot and the ground
falls to zero. There is no impulsive collision involved, so there is no instantaneous change
in state at liftoff.
The equations of motion and impact equations were derived using the TMT method
[122] and independently verified using Lagrangian mechanics and conservation of angular
momentum. The resulting equations are far too long to be included in this chapter. Integration is performed using MATLAB's ode45 () function with absolute and relative tolerances
of 10-5.
Control of Realistic model
The swing leg retraction rate is defined as the angular velocity of the "virtual leg", that is,
the line connecting the hip and the foot. This swing leg retraction is not to be confused
46
10
11
1
a
toL
0
L*St
Figure 3-2: 2-dimensional realistic running model that consists of five rigid bodies. The
left figure shows the robot during the flight phase and labels model parameters, the values
of which are given in Table 3.1. The right figure shows the robot during the stance phase,
during which the leg spring is active, and the 6 generalized coordinates used to describe the
motion. Note that the torso orientation is not a degree-of-freedom as the rotation of the
torso is fixed with respect to the world.
with swing leg contraction, which is the time derivative of the length of the virtual leg.
Note that in this study, the swing leg contraction rate at touchdown is set to zero for two
reasons:
" Although both swing leg retraction and swing leg contraction is required for perfect
ground speed matching (and zero impact loss), humans tend to exhibit much more
swing leg retraction than swing leg contraction [14].
" The physical Phides robot does not permit swing leg contraction when the leg spring
is engaged shortly before touchdown because the motors cannot backdrive the stiff leg
spring.
During the flight phase, all joints are PD-controlled to follow quadratic spline trajectories. These trajectories are chosen such that they start at the joint state at the moment of
liftoff and go to a manually tuned desired state after a preset time, while minimizing the
maximal acceleration magnitude. During the last part of the flight phase, the swing leg
knee joint is locked and the hip joint rotates with a constant angular rate w, resulting in
overall swing leg retraction rate w.
During the stance phase, the hip and knee joint of the swing leg follow quadratic spline
trajectories as during the flight phase. No torque is applied to the hip of the stance leg
during the stance phase. Also, the stance knee actuator does not apply any torque during
the first part of the stance phase from touchdown to maximal knee compression. However,
47
7nt+mnu+mi
1/h
U)z
Figure 3-3: Kneed leg model, a simplification of the realistic model in which the mass of
the opposing leg and the torso are lumped into a point mass at the hip.
the stance knee pushes off during the second half of the stance phase, from maximal knee
compression to liftoff. This push off is used to control the energy level of the system. The
push-off torque is chosen such that energy supplied by the push off is equal to the difference
between the system energy at maximal knee compression and a desired energy level.
Simplified Impact Equations
In addition to the realistic model, two simplifications of the realistic model are used to study
the effects of swing leg retraction on the impact event.
In the simple kneed leg model shown in Figure 3-3, the touchdown leg is entirely preserved, but the other leg and the torso are lumped into a point mass at the hip. The impact
equations are derived according to conservation of angular momentum, assuming the foot
sticks upon landing. These equations are not used in the dynamic simulation of the realistic model; they are only used in post-processing to illuminate whether the particular state
of the second leg and torso have a significant effect on the impact dynamics. If not, the
simplified equations may be used to simplify study of the impact dynamics.
In the simple prismatic leg model shown in Figure 3-4, much of the mass of the realistic model is moved into the upper segment of the leg, and a massless spring connects a
point mass foot in place of the lower leg. The impact equations are derived according to
conservation of angular momentum assuming a perfectly inelastic collision, that is, the foot
sticks upon landing. These equations are not used in the dynamic simulation of the realistic
model; they are only used in post-processing to illuminate whether this simplified leg model
used in previous studies [64, 49] agrees with the realistic model.
48
'S
Figure 3-4: Prismatic leg model, a simplification of the realistic model with a point mass
foot connected to a distributed upper leg mass by a massless spring.
Table 3.1: Parameter values of the realistic model and robot. See Figure 3-2 for the parameter definitions, in which parameters are given by the symbol (e.g. m for mass) with a
subscript to indicate the segment (e.g. t for torso)
torso
npper leg
lower leg
Mass m [kgj
7.41
2.54
0.51
Moment of inertia I [kgm 2 ]
-
0.036
0.005
Length 1 [m]
-
0.3
0.3
Vertical offset CoM c [m]
-
0.183
0.139
Horizontal offset CoM w [m]
-
0
0
49
To fairly compare the results of the simulation models and the robot, the model parameters are matched to the extent permitted by their structures. All parameters and results
are normalized with the total mass M, leg length LO and gravitational acceleration g to get
dimensionless numbers. The swing leg retraction rate w (in rad/s) is normalized as:
O = W
0
Unless otherwise noted, the models and the robot run at normalized average speed
0.42 (Froude number of
0.18 =
±avg).
(3.1)
,vg of
This is a slow speed for a running gait, but it is near
the maximal speed of the robot.
This finishes the description of the models that are used in this study. Next, I study
the effect of the swing leg retraction rate on the impact losses, energetic efficiency, impact
impulse, and footing stability of these models.
3.3
Impact Losses
Swing leg retraction rate affects the energy usage of running systems. The most obvious
reason is that the speed of the foot, and thus the energy loss as the foot impacts the ground,
is greatly influenced by the rate of swing leg retraction. While there are other sources of
losses in running, I first investigate the effect of swing leg retraction on the impact losses of
the realistic model of Section 3.2.1 and the two simplified impacts models of Sections 3.2.1
and 3.2.1 to test intuition regarding the most apparent link between swing leg retraction
and efficiency. The results from both simplified and realistic models are presented to assess
whether the trends of the simplified models are shared by the realistic model.
3.3.1
Methods
Impact losses are determined by taking the difference between the kinetic energy of the
system immediately before and immediately after the instant of touchdown. The impact
losses of the realistic model are determined while running under the hand-tuned controller
of Section 3.2.1. All elements of the realistic model touchdown states except for retraction
rate are mapped onto the prismatic and kneed leg models. To investigate the sensitivity
of impact loss with respect to typical state variations, impact loss is calculated for the
50
simplified models in all of these touchdown states for a variety of retraction rates. In order
to investigate the sensitivity of the results for the prismatic and kneed leg models with
respect to parameter variations, impact losses are calculated for the models with ±50%
upper leg mass.
3.3.2
Results
Prismatic Leg
In [64], we presented simple analytical relationship between running velocity, leg length, leg
touchdown angle, leg retraction rate, and energy loss of a prismatic leg. The energy loss of
the prismatic leg model of Section 3.2.1 upon impulsive, perfectly-inelastic collision between
the foot and the ground is
Eloss = mlv +
S(in +
(3.2)
m)
u + mu (L2 + c2 - 2cLO)
+
in which the meaning of all symbols is illustrated in Figure 3-4. The symbols va and vt
represent the axial and tangential foot speed, respectively, and are functions of the hip
velocity and the retraction rate:
Va = -Zh
cos ao + Yh
sin
a,
(3.3)
Vt = -h sin aO + N cos ao - Low.
Because there is control over swing leg retraction w, and thus the tangential foot speed
Vt, but no control over swing leg contraction, or axial foot speed Va, the energy loss is
minimized when vt = 0. Note that this is not, in general, the same as zeroing the horizontal
component of the relative velocity between the foot and the ground, v.
Figure 3-5 shows
the effect of the swing leg retraction rate on impact loss of the prismatic leg for a range
of typical touchdown states and varying upper leg mass. The curves agree very closely; all
have a minimum at the retraction rate that zeros the foot tangential speed.
51
Kneed Leg
For a kneed leg, the impact equations are more complicated than for the prismatic leg.
The symbolic expression is omitted because of its length and difficulty of interpretation;
I could not find a simple relationship to describe the location of the minimum as for the
prismatic leg. The graphical representation of Figure 3-5 shows that the location of minimal
impact loss for the kneed leg is very different from the normalized retraction rate for zero
foot tangential speed. Also, the minimum impact loss is achieved at significantly different
normalized retraction rates depending on the mass distribution of the leg.
Robot Simulation
The robot is simulated using the realistic model and controller described in Section 3.2.1 and
the resulting energy losses for limit cycle running is measured for a range of retraction rates.
The results are superposed on Figure 3-5 and show close agreement with the simplified
kneed leg model.
3.3.3
Discussion
Figure 3-5 shows that variations within the characteristic range of touchdown horizontal
speeds, touchdown vertical speeds, and angles-of-attack have little effect on the normalized
energy loss, as indicated by the narrow bands. Also, for prismatic legs, mass distribution
of the leg has little effect on the trend, as the minima of the lines for i50% upper leg mass
share a minimum with the band for normal robot mass parameters.
For kneed legs, however, the minima of the curves lie at a much lower retraction rate
than that of horizontal or tangential speed matching. That is, intuition about lower impact
loss stemming from reduced relative speed between the foot and the ground, or ground speed
matching, does not hold for kneed legs when there is no swing leg contraction during flight.
However, we can adjust our intuition by considering that the impact loss is not only due to
abrupt changes in the translation of the foot or the leg segments, but also due to abrupt
changes in the rotation of the leg segments. Note that immediately before touchdown, the
rate of swing leg contraction is very low (by constraint), but immediately after touchdown,
there must be significant leg contraction as the knee bends. This sudden increase in leg
52
0.1
.
rd 0.08
Kneed leg model
Prismatic leg model
Realistic model
+5ow%
1 0.06
Vt=0 V =0
%upper leg masm
and inertia
,
A%
0.04
0.02
0
-1.2
-
-0.8
-0.4
0
0.4
Normalized retraction rate
0.8
1.2
[]
Figure 3-5: The effect of the normalized swing leg retraction rate M on the normalized energy
loss E,08 for a prismatic leg model, kneed leg model, and realistic model. For the simplified
models, shaded regions indicate results for a range of touchdown conditions, including
variations in instantaneous velocity and angle of attack, encountered at touchdown in limit
cycles of the realistic model. Lines indicate the mean result for the range of touchdown
conditions studied. Lines for simplified models with mass parameters of the Phides robot
(+0% upper leg mass) and ±50% upper leg mass are also shown. Results for simulated
limit cycle running of the realistic model with Phides-like parameters are represented with
a thick black stroke. For reference, E10 , 8 = 1 would occur if the full mass of the robot
translating at 3.37 m/s were to stop completely.
contraction is manifested as a decrease in the angular velocity of the upper leg segment and
an increase in the angular velocity of the lower leg segment. The impact losses due to these
sudden jumps in angular velocity are lower if the more massive upper segment has a less
positive, or even more negative, angular velocity prior to touchdown. Indeed, the minima
of the impact loss curves for the kneed leg models occur at much lower retraction rates than
for prismatic leg models. Furthermore, the more massive the upper leg segment the more
pronounced this effect is: as upper leg mass is increased, the retraction rate at which the
minimum occurs is decreased.
The curves for a kneed and prismatic leg are quite different, so we cannot consider the
prismatic leg a representative simplification of a kneed leg when studying impact losses.
This suggests that while the results of [49] may be valid for prismatic legged robots, they
do not necessarily generalize to kneed legs as originally suspected. On the other hand, the
kneed leg model agrees closely with the realistic model, indicating that the state of the
second leg at impact does not significantly affect the energy loss.
53
3.4
Overall Energetic Efficiency
Impact loss is not the only factor in running energetic efficiency; another consideration is
that the forward and rearward acceleration of the swing leg is accomplished, at least in part,
by actuator work, and thus the swing leg retraction rate is the result of a certain energy
expenditure. Perhaps subtler still is that the state of the leg as it touches down sets the
initial conditions for the stance phase, during which much of the work of running is done,
and the ensuing dynamics are significantly affected by the initial conditions. Since impact
losses are only a portion of the energy expenditure of the robot in running, it is unclear a
priori whether the reduction in impact loss due to a given retraction rate leads to an overall
efficiency improvement. Therefore, we should consider the effect of the retraction rate on
overall
3.4.1
limit cycle energetic efficiency of the Phides robot in simulation.
Methods
I measure the overall energetic efficiency of the realistic model of Section 3.2.1 in limit cycle
motion for a range of swing leg retraction rates. The energy efficiency is quantified using
mechanical cost of transport cmt, which is the energy consumed by the actuators normalized
by the robot's weight and distance traveled. The energy consumed is assumed to be the
integral of the absolute mechanical power of the actuators. Thus,
-
fttep
CMt
dt
d .
M - g xstep
(3.4)
in which tstep is the temporal duration of a step, xtep is the distance traveled in a step,
is the vector of instantaneous joint torques,
r
is a vector of the angular velocities of the
joints, M is the total robot mass, and g is gravitational acceleration.
Initially, I analyzed the effect of the retraction rate on efficiency under the particular,
hand-tuned controller of Section 3.2.1 and found that swing leg retraction rate had only a
small effect on cost of transport, which ranged from 0.64 - 0.7. To determine whether this
limited effect of swing leg retraction is particular to the hand-tuned controller or inherent to
all controllers, we wish to study the effect of the retraction rate on the maximum possible
efficiency of limit cycle running. To do this, we seek the initial conditions of the robot
54
state and control trajectories of the four actuators that yield minimal cmt by means of
optimization.
Optimization
General Pseudospectral Optimal Control Software (GPOPS) [99, 11, 44, 43] is used to find
the torque profiles and initial conditions for energetically-optimal limit cycle running under
the constraints of
" limit cycle periodicity,
" specified touchdown swing leg retraction rate,
" specified average speed of limit cycle running,
" 30*(+00, -2.9*) touchdown knee angle,
* >30*1eg extension at all times,
* foot-ground clearance as appropriate,
* no swing leg contraction (leg segment angular velocities match within 0.1 rad/s at
touchdown),
* bounds on state and control variables to match physical robot where applicable (except
a 50 Nm torque limit is used to enable higher speed running), and
" upper and lower bounds on all other state and control variables for better optimization
performance.
Note that no path constraint was placed on the ground reaction forces. Thus, it is possible,
indeed typical, for the foot to pull against its pivot at the ground for a short time after
touchdown. While physically unrealizable, this is neglected similarly to the pulling impulses
at touchdown, which is explained in Section 3.2.1.
For better numerical performance, the control variables in GPOPS are the time rate
of change of the joint torques; GPOPS optimizes the initial joint torques along with these
control profiles. For good convergence properties, the cost function must be smooth, so the
absolute value function abs(x) of Equation 3.4 is approximated as
VX+
e,
where c is a
small positive real.
GPOPS implements a pseudospectral method to discretize the optimal control problem.
Like other direct collocation methods, pseudospectral methods treat both state and control
55
variables as unknowns. A nonlinear programming routine solves for both simultaneously to
minimize the cost function and enforce constraints, including those imposed by the equations
of motion. Accordingly, GPOPS requires an initial guess, or seed, for both control and state
trajectories in each phase, i.e. stance and flight. A seed includes, for each phase, an array
of time points and arrays of state and control values corresponding with each time point.
Such a seed might be actual control and state trajectories from simulation, in which case
the constraints imposed by the equations of motion are satisfied with the initial guess.
However, we do not wish to constrain the search for energetically optimal running with
any preconceived notions about running.
To avoid this, we generate random state and
control arrays using MATLAB'S rando function and scale them according to the bounds
placed on the state and control variables. The time arrays contain equally spaced points,
but are of random duration within generous bounds. From these random seeds, GPOPS
robustly converges to optimal state trajectories that satisfy the constraints and agree closely
with 'propagated' state trajectories, that is, states integrated in simulation by MATLAB'S
ode450 from the initial conditions and under the open loop controls provided by GPOPS.
However, the problem is quite complicated and is plagued by local minima. To more closely
approximate the globally optimal mechanical cost of transport for each swing leg retraction
rate, the optimization problem is repeated from many random seeds.
3.4.2
Results
Typical optimization results are presented in Figure 3-6.
For each value of normalized
retraction rate between -1 and +1 at 0.025 intervals, optimization is performed 10 times.
That is, each point in Figure 3-6 represents a locally optimal limit cycle produced from a
different random seed. While there are some scattered local optima that do not exhibit any
discernible trend, most points tend to cluster densely around a lower bounding curve. From
hundreds of random seeds, no limit cycles were produced with a cost of transport below this
curve. Thus, we consider this curve to be a close approximation of the globally minimal
mechanical cost of transport as a function of swing leg retraction rate. For limit cycle
running at an average normalized speed of 1.36, the figure shows that optimal energetic
efficiency is achieved with a relatively high, positive retraction rate.
56
l RA
Locally optimal limit cycle
0.35-.
--- Boundary of limit cycles found
0
0.34
0
8
0.33 -*
-*
0.32 -
0.31
-1
0.5
0
-0.5
Normalized retraction rate Q [-
1
Figure 3-6: The effect of the normalized retraction rate D on the minimal mechanical cost
of transport of limit cycle running at an average normalized speed of 1.36. Each point
represents a limit cycle with locally minimal mechanical cost of transport optimized from
a different random initial seed. Though the initial seeds are entirely unrelated, the locally
minimal mechanical cost of transport of the optimized limit cycles cluster densely about an
apparent boundary believed to be representative of the globally minimal mechanical cost of
transport.
However, the absolute effect of the retraction rate on the cost of transport is small, with
a maximal difference of only 3% over the whole range of retraction rates. This becomes
particularly clear when the points for optimally efficient limit cycle running at several
speeds are plotted against the same axes, as in Figure 3-7. Swing leg retraction rate no
longer appears to affect the optimal efficiency in any significant way. Running speed, for
example, has a much more obvious effect on running efficiency than swing leg retraction
rate.
Though the energetic cost of the hand-tuned intuitive controller is much higher than
that of the optimized trajectory, the result is qualitatively the same: there is no strong
correlation between retraction rate and overall energetic efficiency.
3.4.3
Discussion
At first glance, Figure 3-7 shows that speed matters: as speed increases, retraction rate for
minimal impact loss increases. More significantly, it shows that despite the effect of swing
leg retraction rate on impact losses, there is little inherent effect of swing leg retraction rate
on the overall efficiency of limit cycle running. This appears to be because impact losses
57
-
f4A
Normalied
running
Rpeed H
~
L. -. 1.47
0.35
..
baAu
.
.:
1.26
1.05
-'
0.3
-
-
9.
;.
0.
Z0.25
0.2
-1
0.5
0
-0.5
Normalized retraction rate C [H
1
Figure 3-7: The effect of the normalized retraction rate 0 on the minimal mechanical cost
of transport of limit cycle running for each of several normalized running speeds 0.84-1.47.
Using a wider range on the ordinate axis, the trend of Figure 3-6 is no longer obvious. In
fact, swing leg retraction rate appears to have very little inherent effect on mechanical cost
of transport.
are a small portion of the total energy budget. Other factors, such as the energy required
to swing the legs and produce vertical impulse, level the curve.
3.5
Impact Forces and Footing Stability
When designing a robot controller, it may be necessary to minimize the magnitude of impact
forces at touchdown to avoid damaging the robot, and it is often important to limit sliding
between the foot and the ground to avoid slipping and falling. By changing the relative
speed between the foot and the ground at touchdown, SLR can have a significant effect on
the extent to which these risks are mitigated. In this section, I analyze the effect of SLR
on impact forces and slippage.
3.5.1
Methods
The realistic model and the two simplified impact models are not appropriate for predicting
the magnitude of touchdown forces, because touchdown is modeled as an instantaneous
event with impulses rather than finite forces. However, we can assume that the risk of
damage is roughly proportional to the magnitude of the total touchdown impulse. Likewise,
58
the simulation does not predict how much slipping will occur at touchdown because the foot
is assumed to stick to the point of ground contact. However, we can assume that slipping at
touchdown will depend on the angle of the touchdown impulse: if the impulse angle is zero,
the impulse is purely vertical, and slipping is impossible; if the impulse angle is 7r/2, the
impulse is purely horizontal, and slipping is certain. At intermediate angles, slipping will
occur if the impulse angle exceeds the effective friction angle, the arctangent of the effective
friction coefficient.
The impulses are measured by taking the difference between the total linear momentum
of the robot immediately after and immediately before the instant of touchdown.
The
magnitude and angle of the impulse vector are computed during simulation of the realistic
model in limit cycle motion under the action of the hand-tuned controller. They are also
calculated for the simple prismatic and kneed leg models for a range of retraction rates and
for a variety of typical touchdown horizontal speeds, touchdown vertical speeds, angles-ofattack, and swing leg retraction rates typical of limit cycle running.
3.5.2
Results
Figure 3-8 shows the magnitude of the touchdown impulse as a function of the retraction
rate for the prismatic leg model, kneed leg model, and realistic model. For the prismatic leg,
the magnitude of the touchdown impulse is minimal at the retraction rate for which foot
tangential speed is zero, as anticipated by analysis presented in [64]. However, the minimal
touchdown impulses for the kneed leg occur at negative retraction rates very different from
the retraction rate of zero foot tangential speed. Again, the realistic robot model shows
close agreement with the simplified kneed leg model.
Figure 3-9 shows the angle of the touchdown impulse as a function of the retraction rate.
The larger this angle is, the more likely it is that slipping will occur. For the prismatic
leg, the retraction rate has a large influence on the impulse angle and the minimal angle
occurs near the retraction rate for which the foot tangential speed is zero. For the kneed
model, however, the retraction rate has a much smaller effect on the impulse angle. For
all normalized retraction rates between -0.8 and 1.2, the impulse angle is low enough that
an effective friction coefficient of 0.2 would be sufficient to prevent slipping. Also for the
59
9-
He'
,
0.2 -
0 -15
Kneed leg model
Prismatic leg model
Realistic model
H
upper leg mass
and inertia
+5M
0.5
0.050
-1.2
=0
!V.=o
-0.8
-0.4
0
0.4
Normalized retraction rate
0.8
1.2
[]
Figure 3-8: The effect of the normalized swing leg retraction rate 0 on the normalized
touchdown impulse magnitude for the prismatic leg model, kneed leg model, and realistic
model. For the simplified models, shaded regions indicate results for a range of touchdown
conditions, including variations in instantaneous velocity and angle of attack, encountered
at touchdown in limit cycles of the realistic model. Lines indicate the mean result for the
range of touchdown conditions studied. Lines for simplified models with mass parameters of
the Phides robot (+0% upper leg mass) and ±50% upper leg mass are also shown. Results
for simulated limit cycle running of the realistic model with Phides-like parameters are
represented with a thick black stroke. For reference, a normalized impulse magnitude of
1 would be required if the full mass of the robot translating at 2.37 m/s were to stop
completely.
impulse, the realistic robot model shows close agreement with the simplified kneed leg
model.
3.5.3
Discussion
Many of the same conclusions of Section 3.3.3 can be drawn from Figures 3-8 and 3-9. Variations within the characteristic range of touchdown horizontal speeds, touchdown vertical
speeds, and angles-of-attack have little effect on the magnitude and angle of the touchdown
impulse, as indicated by the narrow shaded regions. For prismatic legs, mass distribution
of the leg has little effect on the trend; for kneed legs, the greater the mass of the upper
leg segment, the greater the disagreement with the prismatic leg results. In addition, the
simplified kneed leg model is representative of the realistic robot model.
Also, it is interesting that the impulse angle observed for the prismatic leg rises sharply
from the minimum in either direction, and thus the foot tangential speed must be matched
60
-.-
-
Kneed leg model
Prismatic leg model
Realistic model
Sir
+5
+50
and ineti
"50%upper
-------
g
0L
-1.2
0.8
0.4
0
-0.4
-0.8
Normalized retraction rate C [-]
1.2
Figure 3-9: The effect of the normalized swing leg retraction rate E on the impulse angle
for a prismatic leg model, kneed leg model, and realistic model. Impulse angle, or the
magnitude of the angle between the impulse and vertical, is defined as Jatan(I,/I,)I, where
I. and I, are the horizontal and vertical components of the impulse vector, respectively.
A lower value corresponds with an impulse closer to vertical, which corresponds with less
slipping at touchdown. See caption of Figure 3-8 for more information about the meaning
of the lines and fills.
relatively precise to avoid slipping. For example, with a coefficient of friction between a
rubber robot foot and concrete of 1, slipping will occur at an impulse angle greater than
arctan 1 = r/4. For the prismatic leg model, the normalized retraction rate must be within
±0.3 of that required for vt = 0 to prevent slipping. For the kneed leg, on the other hand,
slipping is unlikely for all but the most negative retraction rates. This may be an inherent
advantage of a rotary knee over a telescoping joint.
3.6
Conclusion
In this chapter, I showed how the benefits of swing leg retraction depend on the retraction
rate for simple and realistic mathematical models validated against a physical robot. Based
on these results, we can conclude that for the kneed leg morphology and parameters used
in this study:
" Swing leg retraction can be used to decrease the impact energy loss, but the overall
effect on efficiency, as measured by mechanical cost of transport, is small.
" Swing leg retraction can decrease touchdown forces and increase footing stability, as
61
estimated by impact impulses.
* The optimal retraction rate depends heavily on which metrics of running performance
are valued and the specifics of the systems; generalizations are difficult to generate.
More generally, we conclude that a prismatic leg is not a satisfactory simplification of a
kneed leg when considering the touchdown impact event; the impact dynamics of the two
models are fundamentally different. Indeed, principles for legged robots may be strongly
dependent on the robot morphology, parameters, and controller; robot designers must use
accurate models of their own machines to predict the effects of inputs unless the principle
extraction study carefully considers these variables. The following chapter gives an example
study where such variability is explicitly considered.
62
Chapter 4
The Effect of Tails on The
Maneuverability of Running
Robots
This chapter, consisting primarily of my contributions to [17], analytically compares the
performance of a tail against a reaction wheel for reorienting a robot in midair. It also
introduces the importance of parameter variation in a study, as it finds that a tail only
outperforms a reaction wheel when the time available for the maneuver is sufficiently long
and space permits a tail of sufficient inertia.
4.1
Introduction
The field of bio-inspired robotics continually mines nature's abundant store of solutions to
engineering challenges. However, one aspect of animal motion that has not been thoroughly
explored is the incredible variety of ways the tail is used in nature, including balance,
swimming, flight control, running, hopping, climbing, defense, warning, courtship, and
thermoregulation. Many of these examples could provide engineers with solutions to difficult
unsolved problems in robot design.
Only a few robots have incorporated tails for more than aesthetic purposes or as simple
fixed inertia. One of the first robots to use an active tail was the Uniroo developed at the
63
Table 4.1: Properties of force/moment application technologies:
Technology
Force
Torque
A CoM
Tail
/
V
V
Reaction Wheel [129]
X
/
x
Control Moment Gyroscope [129]
X
/
X
Reaction Mass
/
X
/
Thruster
/
Gas Jet [129]
xX
/
Propeller [87]/ Rotor
/
/
X
Turbojet [87]/ Turbofan [87]
/
X
X
Capabilities of technologies to apply controlled forces and
moments at their attachment point and adjust system center of mass.
Leg Lab [133]. This robot emulated the motion of a hopping kangaroo and actuated the
tail to cancel the motion of the leg in order to maintain constant body pitch. Simple robots
with tails have been used to reproduce and better understand tail motion in geckos and
other lizards [62][74]. These robots demonstrated how tails could be used for orientation
control during leaping and falling when no ground reaction is available. Other robots have
used swinging appendages to climb stairs [51] and hop [59].
4.2
Analysis: Comparison of Biological and Artificial Force/Moment
Technologies
In this chapter, I study the principle that an appendage for applying forces/moments without requiring ground contact is useful for legged locomotion. Animals use a tail for this
purpose, but in engineering we should not neglect alternative mechanisms on the sole basis
that they are not found in nature.
Indeed, several artificial mechanisms are used to apply forces/moments without reacting
those forces against fixed solids: turbojets propel planes through the air, reaction masses
stabilize skyscrapers, and gyroscopes reorient spacecraft in orbit. Table I summarizes some
of these mechanisms and key properties of each.
Several of them are dismissed as candidates for use on the MIT Cheetah on qualitative
grounds. Thrusters, turbofans, turbojets, and propellers are all very capable, but have
64
relatively high minimum complexity for practical implementation.
Gas jets are simpler,
but less powerful for a given mass, and require a source of compressed gas not otherwise
needed by the robot. The effectiveness of a translating reaction mass would be limited by
the stroke of its linear guide, and a long linear guide is an undesirable use of space. The
analysis of control moment gyroscopes is more complicated than other technologies; they
deserve further study, but they are beyond the scope of this chapter.
I consider in greater detail the relative advantages of a reaction wheel. As noted in
[21], the principal difference between a tail and a reaction wheel is that a reaction wheel is
designed to fit entirely within a volume such that it can rotate continuously, whereas a tail
is allowed to have greater maximum dimensions, and thus considerably greater moment of
inertia, but it cannot rotate continuously without colliding with other parts of the body or
the ground.
Consider the goal of reorienting the roll angle of a cheetah's posterior to change running
direction while in pursuit of evasive prey. It is hypothesized that cheetahs rapidly swing
their tails to provide the moment needed to rotate their posterior, especially when the rear
legs are not in contact with the ground. On the MIT Cheetah robot, either a tail or a
reaction wheel could react such a moment, but we wish to quantify which is capable of
applying a greater average moment over a given amount of time. It is assumed that at the
end of this time, the legs will be in contact with the ground, and any angular momentum
accumulated by the reaction wheel or tail can be 'dumped' to the ground through the legs.
Analysis begins with derivation of the system equations of motion. After finding the
differential equation describing the relative rotation of the reaction wheel or tail and the
robot's posterior, we will consider the torque limitation imposed by an electric actuator
with a given speed/torque relationship. We will then study how the optimal design of the
actuator's transmission depends on parameters such as the motor's peak power P*, the
reaction wheel or tail's effective inertia IE, and the time of interest tf. The result allows
us to compare whether a reaction wheel or tail is more appropriate for rolling the cheetah
posterior under the given constraints.
65
Body A (Body)
Body B (Tail or R.W.)
-R
A
CCB
B
R
r
C /
mBg
rB
Figure 4-1: Free body diagram of the posterior (body A) and tail/reaction wheel (body B).
Torque T and force R act at the joint between the bodies. Center of mass points A and B
are at positions rA and rB relative to an inertial reference; point C is at relative position
rC/A with respect to point A and rC/B with respect to point B. Vector rB/A (not illustrated)
extends from point A to point B. Bodies A and B have masses mA and mB, moments of
inertia relative to their respective center of masses IA and IB, and angular velocities WA
and WB. Gravitational acceleration g acts along -j.
4.2.1
Equations of Motion
A tail or reaction wheel, hereafter referred to as body B, is joined to the posterior of the
cheetah, body A, which is assumed to be decoupled from the rest of the cheetah by the
flexibility of the spine. A rotational actuator produces torque T between between body A
and body B. The system is split at the joint between the two bodies, and the resulting free
body diagrams are shown in Figure 4-1.
Applying Newton's second law to body A yields
Fi = -mAgj - R = mArA.
(4.1)
A
Euler's second law for body A is
5T
dt
= -T +rc/A x -R = +(IA.-WA).
(4.2)
A
Likewise, for body B,
Fi = -mBgj + R = mBPB,
B
66
(4.3)
=T+rCB x R=+
STi
B
Expressing rB =
(IB -Wo) .
(4.4)
fA + PB/A, Equation 4.1 and Equation 4.3 sum and rearrange to give
A
-9
-
m
MA + MB
rB/A.
(4.5)
Substituting Equation 4.5 back into Equation 4.1 yields an expression for the reaction force
R = mEiB/A,
where mE =
mAmB
(4.6)
is the 'effective mass' of the bodies. Surely the dynamics can be
derived without this constraint reaction force entering the equations explicitly, but it is
instructive to see that the reaction force acting on body A is linearly proportional to the
relative acceleration of B with respect to A. In the general case, this potentially complicated
R acts at a distance from the center of mass of body A, creating a moment about the center
of mass. This presents an essential difference between a reaction wheel and a tail, which
is also noted in [77]. For a reaction wheel, rB/A = 0, and the moment applied about body
A center of mass is simply T.
The motion of a tail, however, imparts a force on body
A, which creates a moment about the body A center of mass. Depending on the state at
each instant, this may either augment or diminish the resultant torque produced by the tail,
complicating analysis considerably.
We continue by reducing our equations to the planar case and assuming that point A
projects onto point C in the plane, eliminating the rC/A x -R term of Equation 4.1. This
simplification is justified when considering a real cheetah in the mediolateral plane M (from
behind), in which the projection of the center of mass is likely to be only slightly above
or below the attachment point of the tail. Also, since we are interested in biology-inspired
design rather than copying biology, we are free to choose the location of the tail joint on the
cheetah if we deem analytical tractability to be sufficiently important. Adding Equation
'However, T itself can be complicated in three dimensions, when the accumulated angular momentum of
body B begets gyroscopic torques.
67
4.2 and Equation 4.4 yields
rC/B X
R = IAWAk
(4-7)
+ IBWBf,
where some vector and tensor quantities have been replaced by their corresponding vector
and scalar quantities in the plane, and k is the unit vector normal to the plane M. Defining
b = projM (rc/B) and using Equation 4.6, the term rc/B x R evaluates to -mEb2JBk, and
thus
IAWA
-
(IB + meb2 ) WB-
(4-8)
Note that this can be integrated, yielding a statement of angular momentum conservation,
IAwA + (IB + meb 2 ) WB = H,
(4.9)
where H is the constant of integration. If the initial angular velocities are zero, then integrating once more yields
AA = -
IB &meb2
A
(4.10)
AOB,
which reveals that the change in the absolute angle of body A AOA is linearly related to the
change in angle of body B AOB and opposite in direction.
Solving Equation 4.2 and Equation 4.4 for
WA
and
6B,
respectively, and taking the
difference, we get an equation for the relative angular acceleration
B
where IE =IAIB+ME
4.2.2
Jis
-
A =
(4.11)
=-,
the effective inertia of bodies A and B.
Actuator Limitation
Assuming the actuator, including a fixed gear transmission, has a linear speed-torque curve
while operating at its maximum voltage, the maximum motor power P*is produced at half
its no load speed
and half its stall torque
9, that is P* =
68
O.
Then the torque of the
motor as a function of the speed can be expressed as
T =TO 1
4p,
(4.12)
Substituting this relationship into Equation 4.11 completes the equation of motion,
L +
4
WT 2
0
P*IE
To
IE
(4.13)
=0
the solution of which is
w(t)
4.2.3
=
AP*
w(0) -
e
T4P*E
t
TO
+
4P*
T
TO
(4.14)
Angular Impulse
We wish to design our system to maximize the average torque over time of interest tf,
which is equivalent to maximizing angular impulse, the integral of the torque over the time
of interest. Assuming that the initial relative angular velocity w(0) = 0, Equation 4.14 is
substituted into Equation 4.12 and integrated for the total impulse
J
ftf
(Tdt=
4
P*IE
T
/2
t_
- e
(4.15)
4P*IEJ,
the quantity we seek to maximize.
4.2.4
Orientation Change
It is important to note that the integral of Equation 4.12 can also be expressed
J = To
where AO
= AOB -
t) -
4
(4.16)
) ,
AOA is the relative angle between the two bodies. Equation 4.16 can be
equated to Equation 4.15 and rearranged to give a closed form expression for AO, which, for
a tail, may need to be limited to avoid a collision with the legs or the ground. Combined
with Equation 4.10, we also have closed form time trajectories for
69
AOA
and A0B.
4.2.5
Optimization
In order to properly compare the reaction wheel and tail, we must maximize J(TO) for
each by appropriate design of the transmission, or choice of stall torque To. We look for a
maximum J*
maxTo J(T) by taking the derivative
VJ
dTo
and solving 4
-
4IEP*
(2tf +
T
2
e
-
T2-tf
EP
-
4IEP*
4
(4.17)
0
= 0 for an extreme point.
The MATLAB Symbolic Toolbox [78]
provides the unique positive real solution
TO* =
1+2w
P-
1
)
,
(4.18)
where W is the Lambert W-function [128], which evaluates at the given argument to
W_ 1
(-1) = -1.7564.
This is verified to be a maximum, as
d2 Jd2=
-- 2
-_7e
1+
dT2 TO=T*
t - < 0.
W-1
_
(4.19)
T*
The optimal To = To* of Equation 4.18 can be substituted into Equation 4.16, yielding an
analytical expression for maximum J = P
as a function of P*,
IE, and t 1 . A plot of this
function is shown in Figure 4-2.
4.2.6
Discussion
This analysis provides a means for assessing whether a tail or reaction wheel is more appropriate for the given task. Given a body inertia IA, geometry and mass constraints, and
maximum power P* that determines the motor to be used, the rest of the system can be
optimized roughly as follows:
* Design a reaction wheel of maximal moment of inertia (under the constraints) that is
free to rotate continuously.
" Design a maximal moment of inertia tail. Determine the maximum allowable relative
70
Optimal MIT Cheetah:
A rail 0 Reaction Wheel
Relative Rotation
Cl)
152
AO (rad)
-J
10
a)
U,
8
%
/2
T/32:
.
0
0.8
0.6
Time tf(s) 0.4 0.2
0.6
0. 8
0.4
0.2
2 Effective Moment )f
Inertia IE ft'22)
0.
Figure 4-2: Results of analysis: maximum achievable angular impulse for given maximum
motor power P*= 200W and body A inertia IA = 4 kg.m 2 as a function of system effective
moment of inertia 1E and time of interest tf. Shading indicates relative angle AO; lines of
constant relative angle are shown. For the MIT Cheetah, the maximum mass for tail or
reaction wheel is m = 2kg. Allowing maximum radius r. = 0.1m for a reaction wheel yields
IE,w
mrw2
0.02kg'm 2 , and allowing rt = 0.6m for a tail yields IEt < mrt2 = 0.61kg.m 2.
For the time of interest tf ~ 0.2s, the optimal tail can produce a higher maximum angular
impulse than the reaction wheel without exceeding the maximum allowable rotation of
AO ~: 7r/2.
rotation of the tail.
" Use Equation 4.18 for actuator design, that is, to determine the optimal stall torques
for the tail actuator and the reaction wheel actuator.
" Use Equation 4.15 to determine the maximum angular impulse J* in the time of
interest tf (and within the maximum allowable AO, for the tail) of each system. The
system capable of a higher impulse under the constraints is likely the better option.
Of course, many different constraints can complicate the analysis. There is always a finite
selection of motor and gearbox combinations, it is common for total system mass rather than
power to drive motor selection, and adding length to a tail may simultaneously increase its
moment of inertia but decrease its range of motion. Other complexities, such as a nonlinear
71
motor speed/torque curve, and a large distance between the actuator axis and body center
of mass, can preclude closed-form analysis.
Still, the preceding analysis outlines a path for numerical studies under such complications, and reveals several important trends when the assumptions made herein are nearly
true.
" The maximum impulse that can be delivered over a long time period is proportional
to the square root of the product of motor power, system effective moment of inertia,
and time (Equations 4.15 and 4.18).
" The longer spatial dimension of a tail provides the advantage of a greater moment of
inertia at the cost of a constraint on maximum allowable relative rotation.
* A reaction wheel is appropriate when there are tightly confining geometry constraints
and the time of interest is very long.
" A tail is appropriate when space is available for a high moment of inertia and the time
of interest is very short.
For the MIT Cheetah robot, the time of interest is very short, and there is plenty of
space behind the robot for a high effective moment of inertia tail. As illustrated in Figure
4-2, a tail offers a considerably higher impulse in this application.
4.3
Conclusion
In this chapter, I demonstrated how a tail can be more effective than a reaction wheel in
reorienting the body in cases when space, power, and time are limited. More generally,
whether a tail or reaction wheel was more appropriate depended on the power of the robot,
the time of the maneuver, and the moment of inertia of the robot, highlighting the importance of parameter variation in a principle extraction study. However, this problem was
simplified because the choice of controller was obvious. The next part of thesis includes
case studies for which parameter variation and optimal control-and indeed, all elements
of the principle extraction framework of Chapter 2-are essential.
72
Part III
Case Studies Demonstrating the
Complete Framework
73
74
Chapter 5
The Effect of a Rotary Knee on the
Energetic Efficiency of Running
Robots
5.1
Introduction
Several early, influential legged robots from the MIT Leg Lab [81, 96, 95, 57, 90] and more
recent examples from other labs [92, 66] have used legs with telescoping, or prismatic joints.
However, all animals and many other influential legged robots [133, 94, 48, 5] are equipped
with articulated legs, that is, legs with a revolute joint. While several advantages of revolutejointed legs have been suggested, including simpler mechanical design and construction
[133], improved dynamic stability and robustness [103], and variable mechanical advantage
[13, 70], the relative advantages of these two options are still not entirely understood [103].
To elucidate the performance differences between these two leg types, this chapter uses the
framework to guide an investigation of the effect of leg joint type on the energetic efficiency
of running robots.
75
5.2
5.2.1
Method
Question
All running animals have legs with revolute joints. While nature's unique constraints may
have discouraged the evolution of prismatic joints [13], revolute joints may also have offered
biological systems particular advantages. We ask: what is the effect of revolute joints on
the energetic efficiency of running?
5.2.2
Model
The question can be transferred to the engineering domain as: what is the relative efficiency
of revolute joints compared to telescoping joints for running robots? We begin with the
running robot model used in [115] and illustrated in Figure 5-1; a point body of mass m,
massless point foot, and massless leg actuated to apply a force F between the foot and
body.
For this first case study of the full framework, I have chosen a simple model against
recommendations in section 2.2 only to demonstrate the effectiveness of the approach presented in Section 2 without being distracted by the complexity of a very high-dimensional
problem. Using a model and output similar to [115] has the added advantage of allowing
us to compare the results to those of the reference. (The framework itself scales to higherdimensional systems with modest increase in computational complexity, as demonstrated
in Chapter 6.)
The body accelerates according to gravity g during the 'flight' phase and is influenced
by both gravity and the leg force during 'stance'. During stance, the state of the model is
parametrized by the length of the leg 1, the speed of leg extension
i, the angle of the leg with
respect to vertical 0, and the angular velocity of the leg 0. The controls are the angle Otd
and length ltd of the leg at touchdown, when the foot touches the ground, the flight phase
ends, and the stance phase begins; the length of the leg at takeoff
lao, when the foot leaves
the ground, the stance phase ends, and the flight phase begins; and the force trajectory
F(t) of the leg during stance.
We modify the model to make it more representative of running robots: the source of
76
I
apex
Flight
I
touchdown
Stance
Flight I
takeoff
apex
AX
% P*
10
~P*%%%
Telescoping
Kneed
Figure 5-1: The model used in this study consists of a point mass m acting under the
influence of gravity g and leg force F. Both telescoping and kneed versions have the same
electric motor which can produce maximum power P*. Consequently, in the telescoping
model, the maximum force depends on leg contraction/extension speed but is independent
of leg length. For the kneed leg, however, the maximum force is a function of both leg
length and speed.
actuation is an electric motor whose maximum torque is constrained by an affine function
of its speed, as required by the equilibrium equations of a direct current motor [82]. In the
'prismatic leg' version of the model, the massless leg is endowed with a prismatic joint and
a massless, ideal rack and pinion to convert the motor's torque and rotation to a force and
translation. In the 'kneed leg' model, the motor has a massless, ideal gearbox which acts
on a revolute joint positioned in the center of the leg.
The maximum positive or negative voltage that can be applied to the motor, and thus
the maximum mechanical power P* the motor can produce, is the same for the two models.
However, the effective force limits of the two legs are different, although both can be written
in the same form,
Fo(Al) (1
where 1lq
=
-
4F ,
F < Fo(A) (1 -
4P
,,
(5.1)
/i7 - 2 is the compression of the leg from its maximum length, Fo(A) is the
stall force, and
i is the instantaneous rate of change in leg length. The difference is that
for the prismatic leg Fo(Al) = FO is a constant (independent of displacement) while for the
kneed leg Fo(A) =
T,
where To is the stall torque of its actuator (motor and gearbox).
77
The inefficiency is quantified using the absolute mechanical cost of transport (CoT)
Cmt
jtatep jFi dt
the integral of the absolute mechanical power
(5.2)
IFiI over a the step time tlep normalized by
the product of the robot's weight mg and distance traveled A,.
5.2.3
Experiment
The questions can now be formulated more specifically as:
1. For what percentage of the design space of running robots would a 'kneed leg' robot
have lower optimal mechanical cost of transport than a 'prismatic leg' robot?
2. Across the design space of running robots, what is the average improvement in the
optimal mechanical cost of transport for a 'kneed leg' over a 'prismatic leg'?
The results of the experiment are intended to be applicable to all running robots that are
well-approximated by the model, but there are some limitations:
1. The experiment will measure the optimal cost of transport ct
cycles (periodic motions) of 1(t),
over all possible limit
i(t), 0(t), 9(t) that can be achieved with force tra-
jectories F(t) under the dynamics. Because these variables are treated as decision
variables, the results are most applicable to robots following efficient trajectories.
2. Additionally, rack and pinion stall force FO is optimized for the prismatic leg, and
gearbox stall torque To is optimized (independently) for the kneed leg. Therefore, the
results are applicable to robots with properly tuned transmissions.
3. As explained in [115], the dimensionless Froude number Fr = Vx,"g must be con-
strained or the optimal solution would tend toward zero average horizontal speed
Vxavg, which is not useful. We limit the experiment to Fr in the range [1,5], which
encompasses much of the range of Fr normally observed in nature [4].
4. Likewise, the dimensionless step length Si = Ax must be constrained or the optimal
solution would tend toward zero-length steps, which is not realistic [115]. We limit
78
Symbol
Description
Range
Units
m
Body Mass
80
kg
10
Max Leg Length
1
m
g
Gravity
9.81
m/s
Table 5.1: Fixed variables, i.e., variables that assume fixed values for the duration of the
experiment. These variables are only used to keep the equations in dimensional, physicallyintuitive form; they do not affect the optimal cmt as a consequence of the Buckingham
II theorem. Therefore, the experiment remains valid for robots with any values of these
variables.
the experiment to Si in the range [1,5], which encompasses the range of step lengths
normally observed in nature [4].
5. The dimensionless duty factor D.F. =
a
tstep
must be constrained or the optimal
solution would converge toward impulsive running, that is, infinitely high forces F
oo over a vanishingly short stance tstance
-+
-
0. However, cmt approaches its optimal
value asymptotically, so D.F. is bounded below by a value considered small enough to
be representative of the optimal, impulsive gait.
6. Only four parameters remain: m, g,
l, and P*. According to the Buckingham H
theorem, the relationship among the N variables can be expressed as a relationship
among N - M dimensionless parameters, where M is the number of fundamental
physical units. In this case M = 3 corresponding with the physical units of mass,
length, and time. Consequently, we fix m, g, and 10 without loss of generality and
constrain dimensionless power ratio Pr =
. We limit the experiment to Pr in
the range [0.25,1.25]. At the lower limit, the robot inefficiency is guaranteed to be less
than 5x the average human cost of transport, while the upper limit would provide
sufficient power for the machine to climb against gravity at a rate 25% greater than
the average horizontal speed.
The different types of variables and their ranges are tabulated in Tables 5.1, 5.2, and 5.3.
For each of many sets of sampled variables, we choose the decision variables to minimize
the mechanical cost of transport for a robot with a 'kneed leg' and again, independently, for
a robot with a 'prismatic leg'. For each point in the design space, optimization is performed
79
Symbol
Description
Range
Fr
Froude Number
[1,5]
-
Si
Step Length
[1,51
-
Pr
Power Ratio
[0.25,1.251
-
Units
Table 5.2: Sampled variables, i.e., variables for which trials are performed at randomly
sampled values to broaden the applicability of the experiment to a wide class of robots and
gaits.
Symbol
Description
Range
F0
Stall Force
[0, 100mg]
To
Stall Torque
[0,100mglo]
1(t)
Leg Length
[0.751o,lo]
i(t)
Leg Length Rate
[1i,
0(t)
Leg Angle
[j,
0(t)
Units
N
Nm
m
bJ
m/s
']
rad
Leg Angle Rate
[Os, Oj]
rad/s
F(t)
Leg Force
[0, 100mgl]
N
D.F.
Duty Factor
[0.05,0.49]
-
Table 5.3: Decision variables, i.e., those which are optimized to achieve minimal Cmt. Values
in gray are de-emphasized because they do not affect the experiment; they are sufficiently
generous that they are rarely observed to be active. When a bound happens to be active,
the corresponding results should be discarded to prevent the choice of bound value from
affecting results. Values in black are 'hard' bounds set by physics or by the intent of the
experiment, except for the lower bound on duty factor, which is sufficiently large to prevent
numerical difficulties associated with a vanishing stance phase but sufficiently small to
permit approximation of the optimal cmt.
using General Pseudospectral Optimal Control Software (GPOPS) [99, 11, 44, 43] from
many seeds randomly generated from within the decision variables' bounds to avoid the
potential for single local optima to skew the results.
80
5.3
5.3.1
Analysis
Results and Discussion
As expected from the results of [115], optimizations consistently converged to an impulsive
running gate for Fr > 1.5: a bouncing gait characterized by a single, short burst of high
force from the actuator during stance followed by a long ballistic trajectory. Results reported
in [115] for running at Fr = 1, S1 = 1 also match those of the present study:
1. The optimal gait is a 'pendular run' characterized by bursts of high force at the
beginning and end of a relatively long stance followed by a moderate-length ballistic
flight.
2. The cost of transport reported in [115] of - 0.1 is comparable to that of ; 0.3 found in
the present study after considering that we include not only positive but also negative
work in the CoT definition and that the addition of actuator constraints and a duty
factor lower bound can only increase cost.
Finally, all optimizations from randomly selected seeds resulted in tightly clustered CoT
values (as shown in Figure 5-4), which suggests that the local minima found are actually close
approximations to global optima. Together, these are taken as evidence that the dynamics
are correctly implemented and that the optimal control algorithm is working properly. Also,
optimization performed with randomly selected fixed variables, instead of those reported in
Table 5.1, resulted in nearly identical CoT values so long as the dimensionless parameters
remained constant. This confirms that the results of the study are applicable to robots
without dependence on m,
lo, and g as expected according to the Buckingham II theorem.
Figure 5-2 presents the relative efficiency of the kneed leg with respect to the telescoping
leg for a range of Froude number Fr and normalized step length S1 at fixed power ratio
Pr = 1. It graphically answers Question 4 for the Pr = 1 cross-section of the population by
shading regions according to whether the kneed leg or telescoping leg are more efficient. Data
from the 17 x 17 grid suggests that the kneed leg is more efficient for approximately 11.2%
of 'successful samples', that is, points in the space for which both kneed and telescoping
legs had running solutions.
81
Design Space Cross Section (Pr= 1) Colored by Favored Leg Morphology
5 4.5
4
0 3.5
23.5
1
1
1.5
2
3
2.5
3.5
4
4.5
5
Froude Number (Fr)
EIENE_______
leg. 'Null
Figure 5-2: Design space shaded by efficiency of kneed leg relative to telescoping
leg, applies
hypothesis', that there is no efficiency difference between telescoping and kneed
not indicate
where a t-test on the efficiency data for that point in the design space did
converge to
not
did
optimization
that
means
data'
significance at the p = 0.01 level. 'No
space.
any feasible solutions for that point in the design
well; the
The Monte Carlo approximation with only 100 samples agrees surprisingly
samples.
kneed leg outperformed the telescoping leg in 10 (11.1%) of the N = 90 successful
percentage
Referring to the binomial distribution, the 95% confidence interval for the true
of the space that is more efficient with a telescoping leg is 5.5% to 19.5%.
CoT of
Figure 5-3 shows the difference in minimal CoT of the kneed leg and the minimal
length Si at
the telescoping leg as as a function of Froude number Fr and normalized step
answers
fixed power ratio Pr = 1. The sample mean of the height of the grid points, 0.0055,
the minimal CoT
Question 5: the minimal CoT of the kneed leg is about 4.6% higher than
of the telescoping leg, on average.
the two
The Monte Carlo approximation agrees, with an average difference between
CoTs of 0.0068.
The bootstrap reference distribution is generated by randomly selecting
82
0.09,
0.08-
0.07,
d
0.06,
0.05,
U004
0.03,
0.02-
.
25
3.5
3
2.5
3.
Froude Number (Fr)
2.5
4.5
1.5
5
2
3.5
4
3
Normalized Step Length (Si)
1
Figure 5-3: The effect of Froude number (Fr)and normalized step length (Sl) on difference
in minimal mechanical cost of transport (kneed leg minus telescoping leg). Where local
minima resulting from the multiple random seeds were not identical (but tightly clustered),
the surface represents the difference in the sample means.
N = 90 'resamples' with replacement from the observed N = 90 sample differences, taking
the mean of the resamples, and repeating this process 10000 times. Comparing the sample
mean of the observed samples to this reference distribution permits the generation of an
approximate 95% confidence interval of the true mean difference between 0.0045 and 0.0105.
The close agreement between estimates from data evaluated on the grid and from the
Monte Carlo data demonstrate that Monte Carlo approximation can be a useful tool to
estimate such statistics with acceptable confidence intervals, but with far lower computation. For this problem, evaluating the minimal cost of transport on the coarse grid in two
dimensions as in Figure 5-2 was reasonable, but producing a fine grid in all three dimensions (Pr,Fr,Sl) would be very time-consuming. For realistic models in higher dimensions,
use of grids is entirely infeasible, but we can approximate statistics using the Monte Carlo
method with no additional computation. The number of samples needed depends only on
the desired width or confidence of the interval; it is independent of the dimensionality of
the population space. Table 5.4 presents the answers to Questions 4 and 5 using a Monte
Carlo approximation with 100 sample points from the full Fr-Si-Prpopulation space.
83
% population for which kneed leg is more efficient
Sample %
C.I. Lower limit
C.I. Upper Limit
3%
1%
10%
Mean additional CoT of kneed leg
Sample Mean
C.1. Lower limit
C.I. Upper Limit
0.0060
0.0044
0.0081
Table 5.4: Summary of results for the full population. As the population now includes
samples with lower Pr,the dominance of the telescoping leg is even more clear. This might
be expected, as the difference between the two configurations only manifests for low Pr; for
high Pr the constraints distinguishing the two are less likely to become active.
0.4 0
0.35
0.3
0
'
0V 0.25
*;
0.2
-
*
0.1
a'.
*
0.15
*
a'
S
Telescoping
Kneed
'I
0.05
V
0
0.5
1
2
1.5
Ratio SlIFr2
2.5
3
Figure 5-4: The effect of the ratio of normalized step length to Froude number squared
(Si/Fr2 ) on the minimal mechanical cost of transport. All data points, that is, all local
minima from the multiple random seeds, are included to show how tightly the local minima
cluster.
84
While the Monte Carlo method presented does not yield detailed spatial information,
neither is analysis restricted to the calculation of scalar statistics. Exploratory analysis
of the data can produce additional interesting results: for instance, Figure 5-4 shows the
minimal mechanical cost of transport as a function of ratio s.
The trend appears quite
linear with little scatter, suggesting that the mechanical cost of transport does not depend
strongly on the speed and step length independently, but only on a function of the two
dimensionless parameters. Both the telescoping-leg and kneed-leg runner are efficient when
running fast with small steps, but less efficient when running slowly with large steps. Furthermore, even at the magnification level in the figure it is possible to see that none of the
points where the kneed leg is more efficient occur with a CoT greater than 0.1 or a ratio
Si/Fr2 greater than about 0.7. This tells a great deal about where in the population space
the few points that kneed legs are more efficient lie.
5.4
Conclusion
The results support the following design principle, applicable to robots of all sizes and
weights running according to the model studied: the optimal efficiency of telescoping legs
tends to match or exceed that of kneed legs. Although this is only a strong statistical
tendency and may not hold for all possible robots within the range studied, it serves as a
useful guideline to improve performance when resources do not permit more detailed study
for design of a particular machine.
85
86
Chapter 6
The Effect of Leg Morphology on
the Efficiency of Running Robots
6.1
Introduction
While many animal legs have three joints we might refer to as hip, knee, and ankle, the
relative shapes and masses of the links between them (bones) differ dramatically. Two
extremes are observed in the comparison between the leg of a bird, such as an ostrich, and
the leg of a human. From exterior inspection, one would suspect that the 'knee' of an
ostrich bends in the opposite direction of the human knee [20], when in fact this 'tibiotarsal articulation' of the ostrich is more akin to the human ankle, and the actual knee joint
between the tibia and femur is hidden within the body. The laws of mechanics, however, do
not govern the performance of legs on an anatomical basis, but according to the locations
of the joints and the mass distributions of the segments. To a legged robot designer, this
suggests a question: when designing a legged robot, in which direction should the knee
bend?
Although the question is inspired by biology, biology does not answer this question for
engineers. Ostriches can run faster and more efficiently than humans [101], but many studies have shown that humans run less efficiently backwards than forwards [38, 130, 132, 26].
These results would suggest opposing answers to the question, and it is difficult to say which
provides a more direct answer. In the interspecial comparison, there are many confound87
-Hip
Knee
Ankle
Figure 6-1: When scaled according to hip height, the knee of the human is at the same level
as the ankle of the ostrich. As a result, if the human leg is considered to bend 'forwards',
then the ostrich leg appears to bend 'backwards'. This inspires robot designers to ask
a question: 'in which direction should a robot's knee bend to achieve maximum running
efficiency?'. Human Skeleton: [118], Ostrich Skeleton: [28]
ing factors other than leg morphology that might explain the superiority of ostriches over
humans, such as differences in energy storage and release structures. On the other hand,
there are many differences between a human running backwards and a human running with
a knee that bends backwards, such as the direction of the foot. Besides, the question is
not about biology but about robots, and there are many differences between biology and
engineering [126] which make the results of physiological experiments difficult to transfer
into the engineering domain.
Instead, we wish take this question inspired by biology and answer it for robots specifically. Several studies consider the effects of leg morphology on robot locomotion performance, but to our understanding, none answer the question about the inherent effects of
knee bend direction. For instance, [88] optimized leg parameters and control simultaneously
88
to maximize walking performance, but the knee was not allowed to bend backwards. In [93],
the backwards knee of Spring Flamingo was reversed to exploit passive dynamic protraction of the leg, but this does not suggest which configuration is more efficient overall for a
running machine.
[61] specifically considers the effect of knee direction on energy economy
of running and walking, but only quantifies impact loss, which accounts for only a portion
of total energy expenditure of a locomotion gait.
The goal of this chapter is to provide insights about the effect of the 'knee' joint articulation direction on running efficiency that apply to a wide range of possible robot designs.
The purpose is to provide robot designers with important information with which to make
a well-supported decision about their robot's knee articulation direction. Without having
to perform detailed analysis on their particular robot or pursue multiple design directions
simultaneously, robot designers can save time and other resources while producing a more
efficient design. The remainder of the chapter is organized as follows: Section II outlines
the method, including the rigid body dynamics models used to compare the relative efficiencies of particular robots running according to particular gaits, and the optimization and
statistical techniques used to generalize the utility of results beyond these particular cases.
Section III presents the results, which suggest that backwards running robots tend to be
more efficient for the great majority of possible robot designs. Section IV suggests intuitive
reasons for the results and presents plans for future work, and Section V concludes.
6.2
Method
We wish to measure an inherent effect of the two leg morphologies shown in Figure 6-2
on the energetic efficiency of a biped.
However, robots with either leg morphology can
be designed with a variety of different link lengths, mass distributions, motor parameters,
and control systems, all of which could potentially change the effect of knee direction on
efficiency. It is impossible to test the effect of knee direction on efficiency for the infinitude
of possible combinations of other system inputs, or even on a reasonably fine grid of possible
inputs. Therefore, we ask a question which addresses our interests, can be precisely defined,
and can be answered within reasonable time: "For what fraction of the space of possible
89
Forwards Knee
Backwards Knee
1/14K
Figure 6-2: The two leg morphologies considered in the experiment: a 'forwards' knee and a
'backwards' knee. Rectangles of the same color and number represent rigid bodies with the
same mass, moment of inertia, length, and center of mass location. Arcs represent actuators
with the same torque limit and maximum torque rate of change. All physical parameters
and the average running speed are randomly selected from their respective populations to
define a robot design. For each of many robot designs and for each knee orientation, initial
conditions and torque rate trajectories are calculated to optimize energetic efficiency. The
differences between forwards and backwards knee energetic efficiencies for each design reveal
the inherent effect of knee direction on optimal energetic efficiency.
bipedal robot designs do 'forwards' gaits tend to be more efficient than 'backwards' gaits?"
The experimental procedure is summarized:
For each of n trials,
1. randomly select a robot design from a pre-defined population, or design space.
2. For each of the two knee bending directions, and
(a) for each of m optimization iterations,
i. generate a guess, at random, for a periodic running motion, or gait, and
ii. optimize to determine the (locally) most efficient gait.
90
1
iM3 ,
13
C3
ml, I]
C2
M2, I2
12
Figure 6-3: The physical parameters of a robot design: each segment is defined by total
length li, mass mi, center of mass location ci, and moment of inertia with respect to center
of mass I,.
(b) Compare the efficiency of the best locally optimal 'forwards' gait to the best
locally optimal 'backwards' gait.
3. Count the number of robot designs for which the best 'forwards' gait is more efficient
than the best 'backwards' gait.
The implementation of this procedure is detailed in the following subsections.
6.2.1
Robot Design / Model
The overall architecture of the robot under study is shown in Figure 6-3; a body segment
of distributed mass is joined at a hip to two legs, each with an upper and lower segment
joined at a knee. Each segment is characterized by its length, mass, center of mass location,
and moment of inertia with respect to its center of mass.
The populations from which these parameters are selected must be defined carefully.
91
O
Animals:
Ostrich
Human
*
Robots:
>- Mabel X Phides *
1
>-4MX021
00
+12
11 +12+r3
Rabbit
11+12
frX**E
M1, +M2
2(mI +rM2 )
ma+2(mi+m 2 )
C,
11
12
1a
13
rn,3)131
0-2
0.4
0.6
0.8
I
Figure 6-4: Graphical representation of the design space, including biological and robotic
examples. The position of each example's icon on a number line visually depicts the value of
its dimensionless parameter. Solid gray blocks mark regions excluded from the design space
for this study. Black checker patterns mark hard physical constraints: for example, none
of the dimensionless variables can possibly be less than zero because no mass or length can
be negative. Gray checker patterns mark soft physical constraints: for example, it would
be unusual for cl /li to be greater than unity, as this would suggest that the center of mass
location of the upper leg link is not between the hip and the knee. Human parameters
are from [71]; ostrich, [58]; Mabel, personal communication with Hae Won Park; Phides,
personal communication with Dani81 Karssen; Rabbit, [22].
Consider if all of these physical parameters were to be chosen independently, at random,
from all physically possibly values. Then one could expect many designs not only to appear
strange but also to be incapable of running. Much computing time would be wasted investigated designs that human designers would not be interested in. On the other hand, ranges
that are too narrow are too heavily influenced by the experiment designer's preconceptions
of a running robot and do not allow for all the possibilities that other designers might wish
to consider.
After careful consideration, the space of all robots considered in this study is defined by
assigning upper and lower limits to the values of functions relating the the physical param92
eters, as presented in Table 6.1. The limits were chosen to encompass typical parameters
for humans and ostriches, as well as those of running robots Mabel, Phides, and Rabbit,
as illustrated in Figure 6-4. In most cases, bounds are quite loose relative to the robot
examples, or they are chosen to respect physical limits (i.e. the radius of gyration cannot
be negative), with a few exceptions:
" upper bounds on torso mass and length are near human scale, which is considered to
be a reasonable size limit on the robots under study;
" the fraction of machine mass contained in the legs is limited to 50%, as practical
robots will likely reserve more mass for a payload, such as a head, on the torso; and
" the moments of inertia reported for Rabbit each correspond with a radius of gyration
greater than half the length of the segment. This is physically possible only if some of
the mass is distributed beyond the reported length of the segment, or if high actuator
inertia is included in the reported figure. The upper limits on relative radii of gyration
chosen for this study are high enough to encompass Rabbit, but the bound is relatively
tight.
Now that the space of possible designs has been defined, we can treat each of the parameters that define the design space is an independent and identically distributed random
variable. A robot design is selected from the population by random sampling of each parameter, which uniquely determines all physical parameters of a design.
6.2.2
Dynamics
Once a design is sampled, its equations of motion must be derived. The running motion
has two phases: a flight phase in which the robot is airborne, and a stance phase in which
one foot acts as a pin joint fixed to the ground. Gravity acts throughout both phases.
During flight, there are seven degrees of freedom, as shown in Figure 6-5: one angle for
each segment, and two Cartesian coordinates for the hip. During stance, the foot is pinned
with respect to the ground, eliminating two degrees of freedom; the Cartesian coordinates
x and y of the hip are uniquely determined as functions of the stance leg angle variables.
The motion of the robot is influenced by torques produced by actuators at each joint.
The actuators have first-order dynamics to represent the inductance of electric motors. This
93
Parameter
min
max
units
13
0
1
m
33.3
66.6
%
11+12
11+1 2 +C3
66.6
100
%
m3
5
50
kg
Mlm
30
90
%
2(m1+m2)
10
50
%
25
75
%
25
75
%
0
75
%
0
66.6
%
0
66.6
%
0
66.6
%
1
4
-
-111+12
m3+2(ml+m2)
13
V
m212
13
F-
Table 6.1: The design space is defined by ranges of torso length 13, torso mass m3, and
11 dimensionless parameters. A design is sampled by randomly selecting values for each
parameter. Note that one of the parameters included in the 'design' is the Froude number
Fr [4], the ratio of running speed squared to the product of gravitational acceleration and
leg length. Sampling from the design space also determines the average speed of the running
gait.
physical limitation not only makes the dynamics more realistic but also improves numerical
performance of the computations.
During flight and stance, the state of the robot is parametrized by the coordinates,
their derivatives, and the instantaneous torques. The time rate of change of each torque is
used as a control. For each robot design, the continuous equations of motion are derived
using Lagrangian mechanics. When a foot touches the ground, the state after impact is
calculated according to angular momentum conservation assuming an impulsive, perfectlyinelastic collision.
94
03
12
012
011
22
y
21 021
022!
I
Figure 6-5: The state of the robot is determined by horizontal and vertical Cartesian
coordinates of the hip, angles of each segment, and their time derivatives. Control torques
act at the hip between the upper leg segments and the torso and at the knee between the
upper and lower leg segments.
6.2.3
Optimal Control
Once the equations of motion have been derived, optimization can be performed to determine the minimal mechanical cost of transport, defined similarly to that in Chapter 3 and
Chapter 5, as
Caw
Mgvi
Mgv ,
=10
fo
(6.1)
dt,
where tstep is the time of a running step from apex to apex, indices i and
j
select among
the hip and knee actuators of each leg, Tij and jj are the actuator torques and speeds,
M is the total mass of the robot, g is acceleration due to gravity, and v. is the average
running speed. In words, this is a measure of the absolute work done by the actuators
normalized by the product of weight and distance traveled. Because of the ambiguity of the
word 'mechanical', this metric will be referred to as 'absolute work cost of transport' and
the symbol caw for the remainder of this chapter.
95
General Pseudospectral Optimal Control Software (GPOPS) [99, 11, 44, 43] is used
to find the torque rate profiles and initial conditions that yield minimal caw under the
constraints of
* gait periodicity (limit cycle);
" specified average running speed, within 0.1 m/s;
" bounded knee angles, according to selected knee direction;
" ground clearance of swing foot > 0.021o during stance; and
" bounds on all phase durations, state, and control variables.
These bounds, summarized in Table 6.2, were carefully selected to encompass a wide
variety of possible running motions, but to reject physically impossible or highly unusual
locomotion styles. For instance,
" bounds on O1 and 612 ensure that the legs oscillate rather than rotating continuously,
" the lower bound on the hip height y prevents the hip from passing through the ground,
* bounds on
63 ensure that the torso remains upright,
" bounds on Atstep ensure that step lengths remain within the naturally observed range
[4], and
* bounds on Atstance and Atflight restrict the duty factor of running to values observed
in nature when the step time bounds are active [35, 3].
Remaining bounds are set at levels intended to be generous. For instance, it seems unlikely
" that the instantaneous horizontal velocity of the hip ± should ever be negative or more
than twice the average running speed,
" that the instantaneous vertical velocity of the hip between apex and touchdown
yphasel
should be positive or greater in magnitude than maximum possible speed of the center
of mass at touchdown, or
" that the instantaneous angular velocity of any segment should ever be more than that
required to complete a full rotation in the minimum allowed step time.
Once bounds have been defined, GPOPS requires an initial guess, or seed, for both
control and state trajectories during a step.
A seed includes, for each phase, an array
of time points and arrays of state and control values corresponding with each time point.
To automate the procedure of specifying seeds, and to minimize human influence over the
96
results, guesses for phase durations and state and control variables are randomly sampled
from their respective ranges defined in Table 6.2. For each robot design and knee direction,
optimization is performed from 50 randomly generated seeds.
6.3
Results
Table 6.3 summarizes the extent to which optimization succeeded.
Example seeds are
shown in Figure 6-6 and 6-8 with the corresponding optimized gaits in Figure 6-7 and 6-9.
Optimization converged, that is, the optimization algorithm found a locally optimal gait
satisfying all constraints, including the dynamics, for about 12% of the cases attempted.
Gaits with kinematically infeasible trajectories that were not precluded by the constraints
(e.g. knee passing below the ground) were eliminated. The absolote work cost of transport of
the remaining gaits are represented in Figure 6-10. All robot designs for which optimization
converged to feasible trajectories are illustrated in Figure 6-11.
97
Variables
min
max
units
aLXstride
10
61o
m
DF
0.2
0.4
A tstride
AIstride,min
vx
J4stride,max
~Atstride,min
Atstride,max
Atstep
Atstep2
2
Atstance
DFminAtstride,min
DFmaxAtstride,max
Atnight
(I
(I
Atphasel
"A tfight,min
Atphase3
-
S
DFmax) Atstride,min
DFmin) Atstridemax
-
s
tflight,max
S
'Atflight,min
3Atflight,max
s
Xphasel
0
2
Xphase3
0
2vxAtstepmax
m
0
2vx
m/s
y
0.3l0
210
m
Yphasel
-gAtphasel,max
0
M/s
Zphase3
0
011, 012
0+03
7r
021, 022
0*
7r*
03
37r
57r
9
ii
3'
m
m/s
+ 03
27r
tstrde,min
rad
Atstride,min
-TRO
rad
rad
27r
Ti
T Rij
vxAtphasel,max
rad/s
TO
Nm
T Ro
Nm/s
Table 6.2: Limits on states, controls, and times. Note l = 11 + 12 is the leg length, average
velocity v. = \g,/-gloFr respects the Froude number, maximum torque To = l0(m3 + 2m 1 +
2m2)g is that required to raise the entire weight of the robot at one leg length, and maximum
torque rate TRO = tphase1,min+tstance,min
TO
limits the torque from sweeping through a
+tphase3,min
range greater than To in the minimum possible step time. In the original code, bounds on
AXstride did not include the factor of l and bounds on Atnight included Atstep in place of
Atstride. After correcting the code and re-running the experiment with a reduced number
of samples, the nature of the results did not change. *Bounds on 021 and 022 depend on the
knee direction; bounds are given by [0, i-] for forwards knee and [-7, 0] for backwards knee.
98
L4 1
41
AM
Figure 6-6: Example forwards running seed (random guess).
Figure 6-7: Example forwards running gait (optimized).
Figure 6-8: Example backwards running seed (random guess).
Figure 6-9: Example backwards running gait (optimized).
99
Forwards
Backwards
Optimization Attempts
10000
10000
Optimization Successes
1170
1223
Kinematically Valid Soins
934
1156
Table 6.3: Frequency of Optimization Success. Two hundred robot designs were sampled at
random from the design space. For each of the two knee directions, local optimization was
performed for each of 50 random seeds for each design. Optimization converged successfully for a fraction of cases, and some of these solutions were eliminated for kinematically
infeasible trajectories not precluded by optimization constraints.
100
0
0.4
0.5
0
0.1
0.2
x
-I
H
I
I
-Ax
x
-A x
0
x
0
I I I
Sx
I I
o
8
x
x
x
I
I
I
0
I
x
X
I
~
I
0 X
x
I
I
0
I
0
6
80
I
I
I
I
I
I I
Backward Knee Gait
Forward Knee Gait
8
5
x
X
01
0
0
0
x
0
x
x
I
0
0
0
0
0
0
: x
x
X
i
0
0
I
00
L
0
0
08
I I I I I I I I I I I II I I I I I I I II I I
x
Robot Design (each column)
I
X
x 40
0~
00
Figure 6-10: Absolute work cost of transport of running gaits optimized frona random seeds for each robot design. Only results
with a low (< 0.7) absolute work cost of transport are shown. For most robot designs, optimization converged from random seeds
to gaits that cluster above an apparent cost of transport lower bound. The clusters for backwards knee gaits tend to be lower
than the clusters for the forwards knee gaits, suggesting that backwards knee gaits tend to be more efficient than forwards knee
gaits.
U
0
Q 0.3
0
0
rj2
0
0.6
0.7
I'
'7-
I
/7
cI1
-
Figure 6-11: Graphical representation of all robot designs for which optimization converged. Link width represents link mass;
links are drawn such that the mass is equal to the product of link width squared, link length, and a density factor common to
all designs. The center of each circle represents the center of mass of each link, and the circle radius corresponds with the link
radius of gyration. The purpose of this figure is to illustrate the great variety of robot designs randomly sampled from the design
space; this variety allows us to make inferences about the whole design space.
I
CL/
80
U Forward Knee Gait
Backward Knee Gait
70 60
50
40
0
20
10
Robot Design (each column)
Figure 6-12: Rank of absolute work of transport of running gaits for each robot design.
Robot designs (columns) with many backwards running gaits more efficient than the most
efficient forwards running gait have been moved toward the left and vice versa. There is a
marked tendency for the backwards knee gaits to have lower numbered ranks, suggesting
superior efficiency of backwards gaits.
While Figure 6-10 has the advantage of showing the data in a very raw form, it is not
very compact and trends can be difficult to read. The same data is shown in a different form
in Figure 6-12: instead of depicting the magnitude of the absolute work cost of transport
of each gait, Figure 6-12 shows only the rank of each gait relative to other gaits for the
same robot design. In this representation, it is clear at a glance that backwards running
gaits tend to have a lower cost of transport than forwards running gaits for the same robot
design. For instance, of these 57 designs for which both valid forwards and valid backwards
gaits were found, the most efficient gait for 80% of them was a backwards running gait.
Consider a different metric for the energetic efficiency based on the integral of torque
squared over time rather than absolute mechanical work:
cT2 =C2= tstep
So
EijgMg
T,,(t)2
t(62
~dt,
Mg
for
w
(6.2)
Such a performance metric may be more appropriate for robots with electric actuators, for
103
80
SForward Knee Gait
Backward KneeGait
"t~~
~70
60 60
50
40
0
1
Robot Design (each column)
Figure 6-13: Rank of torque squared cost of transport C7 2 of running gaits for each robot
design. There is a marked tendency for the backwards knee gaits to have lower numbered
ranks, suggesting superior efficiency of backwards gaits according to this metric, too.
which the total energy usage can depend more on the resistive heating losses, which are
proportional to current squared and thus torque squared, than on absolute work. Figure
6-13 shows that for this metric, again backwards running gaits tend to be more efficient
than forwards running gaits for a given robot design.
6.4
6.4.1
Discussion
Explanation of Results
The results clearly show that it is unlikely for a robot design chosen at random from the
design space to have more efficient forwards running gaits than the most efficient backwards
running gaits. Instead, it is much more likely to have more efficient backwards running gaits
than the most efficient forwards running gaits. But why? We seek support for an intuitive
explanation by further analysis of the data.
One notable hypothesis would be that backwards running gaits tend to be more efficient
because the system energy losses at impact tend to be lower when the knee faces backwards,
104
80r
N Forward Knee Gait
70 - N Backward Knee Gait
tT60 -
r:50
40
30
20
0
1
Robot Design (each column)
Figure 6-14: Rank of impact loss cost of transport cji of running gaits for each robot
design. There is a slight tendency for the forwards knee gaits to have lower numbered
ranks, suggesting reduced impact loss of forwards knee gaits, in contrast with the findings
of [61].
as suggested by [61]. More important to running efficiency than the absolute impact energy
loss is the impact loss cost of transport, or
cii =
AEte
MgAxst' p
(6.3)
the energy loss at touchdown AEtd normalized by the weight Mg and distance traveled
Axstep. However, Figure 6-14 does not even suggest that the impact losses tend to be lower
when the knee faces backwards, so this hypothesis is not supported by the data.
An alternative is suggested by comparing Figure 6-7 and 6-9: in backwards running,
the upper leg seems to move relatively little, and running is enabled almost entirely by
the reciprocating motion of the lower leg, whereas in forwards running, running is enabled
by considerable hip motion, while the knee remains nearly fixed. Indeed, Figure 6-15 and
Figure 6-16 confirm that the absolute work and the integral of torque squared of the hip
actuator are considerably lower for backwards running for this gait. While these savings
105
120.
.
.
.
.
.
Forwards Knee
--- Backwards Knee
100:-
60 -
a)
40
0
H
a)
U
--
20
0
-20 -
-40 -L
-60-
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Stance Hip Angle (rad)
Figure 6-15: Trajectory of hip torque of stance leg plotted against the relative angle between
the stance leg and the torso for the example gaits of Figures 6-7 and 6-9. Both peak torque
and total angular excursion are much smaller in the case of the backwards knee. Note
also that the backwards knee gait separates periods of high torque from those of significant
angular displacement, whereas the forwards knee gait exhibits considerable hip excursion
during periods of high torque. Because absolute work corresponds with the area between
this curve and the horizontal axis (shaded for forwards gait), it is apparent that the hip
actuator of the stance leg performs far less absolute work for the backwards knee gait.
will be offset by increased effort at the knee, we might still expect net savings for backwards
running because, all else being equal,
" reciprocating motion of the distal joint, and thus only the lower leg, may require less
absolute work due to the reduced reciprocating mass, and
" application of a force due to torque at the distal joint only could reduce the integral
of torque squared due to the smaller moment arm of the lower leg.
While simplified reasoning like this can yield erroneous conclusions, the data in Figures
6-17, 6-18, 6-19, 6-20 show that the improved cost of transport for backwards running can
indeed be attributed to the reduced effort of the hip, despite increased effort at the knees.
This trend is true for nearly all designs randomly sampled from the design space, which
suggests that the trend is inherent to the direction of the knee and does not depend on
106
120
-
Forwards Knee
--------
100-
s Knee
80-
~600
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (s)
Figure 6-16: Trlajectory of hip torque of stance leg plotted against time for the example
gaits, of Figures 6-7 and 6-9. Both peak torque and total time are much smaller in the case
of the backwards knee. Note also that periods of high torque are of much shorter duration
for the backwards knee gait, whereas the forwards knee gait exhibits considerable torque for
almost the entire period. Because the integral of torque squared corresponds with the area
between the square of this curve and the horizontal axis, it is apparent that the hip actuator
of the stance leg contributes far less to the torque squared integral for the backwards knee
gait.
the running speed or any other aspect of the robot design. A simple explanation for the
unique ability of the backwards knee to avoid hip use is motivated by Figures 6-22 and
6-21: without motion of the hip, only the backwards knee is able to support the propulsion
and protraction required of a running gait. On the other hand, a machine with a forwards
knee requires hip motion for protraction and propulsion, and thus tends to incur the higher
absolute work and torque squared costs associated with it.
Future Work
6.4.2
There are two shortcomings of the model used which might be addressed in future work,
namely:
1. A more realistic contact model between the foot and the ground would obey a friction
cone and prevent the foot from pulling on the ground. Also, the model should include
107
Gait
SForward Knee
Knee Gait
SBackward
70
1601
450
0
d 40
30
20
.10
Robot Design (each cohumn)
work
Figure 6-17: Rank of hip absolute work cost of transport, that is, absolute mechanical
of
traveled,
performed by the hip motors normalized by the product of weight and distance
knee
running gaits for each robot design. There is a marked tendency for the backwards
backwards
for
hip
the
at
efficiency
superior
gaits to have lower numbered ranks, suggesting
gaits.
80
N Forward Knee Gait
70
0 Backward Knee Gait
60
11161
0 50
4-
0
40
30
20
20
10
Robot Design (each column)
robot
Figure 6-18: Rank of knee absolute work cost of transport of running gaits for each
numbered
lower
have
to
gaits
knee
forwards
the
for
design. There is a marked tendency
ranks, suggesting superior efficiency at the knee for forwards gaits.
108
80
* Forward Knee Gait
* Backward Knee Gait
S70
0
i
60
50
0
40
0
S30
20
0
Robot Design (each column)
Figure 6-19: Rank of hip torque squared cost of transport, that is, the time integral of
torque squared performed by the hip motors normalized by the product of weight and
distance traveled, of running gaits for each robot design. There is a marked tendency for
the backwards knee gaits to have lower numbered ranks, suggesting superior efficiency at
the hip for backwards gaits.
80
M
70 -
Forward Knee Gait
Backward Knee Gait
0
S60
0
0
U
50
40
S30
20
10
Robot Design (each column)
Figure 6-20: Rank of knee torque squared cost of transport of running gaits for each robot
design. There is a marked tendency for the forwards knee gaits to have lower numbered
ranks, suggesting superior efficiency at the knee for forwards gaits.
109
Forward Knee
Collapses a
Backward Knee
Propels a
*
Figure 6-21: Propulsion advantage of the backwards knee. With the hip joint locked, only
the backwards knee supports propulsion of the running machine; a running machine with a
forwards knee would collapse.
Direction of Running
O
Backward Knee
Lifts and Prot racts
Forward Knee
Lifts and Retracts
Figure 6-22: Protraction advantage of the backwards knee. With the hip joint locked,
only the backwards knee can lift from the ground and prepare for another step; a running
machine with a forwards knee requires significant hip motion to complete the running cycle.
110
a hard stop, rather than optimization constraints, to prevent knee hyperextension.
New techniques for trajectory optimization through contact [91] could be used to
address both of these items and produce even more realistic results.
2. Optimization constraints included only for numerical tractability (bounds on maximum angular velocities, maximum torques, etc...)
should be even more generous
to prevent them from becoming active and affecting results. Also, other constraints
(minimum and maximum step length and time, periodicity of torque profiles, etc...),
although not believed to significantly affect results, should be improved in order to
behave as intended. Indeed, this work is already under way, and results from the
improved experiment, albeit with fewer data points, indicate no change to the conclusions of the study. Additional constraints on actuator speed/torque relationship, such
as the linear speed-torque curve of an electric motor, should be implemented.
Additionally, the model could be extended to include a separate foot segment and/or allow
the joints to bend in both directions. It would also be very interesting to replace actuator
constraints with those representative of muscles in order to gain insight on the effect of knee
direction on the efficiency of biological systems.
6.5
Conclusion
The results demonstrate that under the conditions tested, robots with backwards knees
exhibited a strong tendency to be more efficient, in terms of absolute work and torque
squared cost of transport, than robots with forwards knees over a broad range of parameters
and running speeds. If running efficiency is of primary concern to a robot designer, and the
designer must choose either the forwards or backwards knee as the nominal running style
in preliminary design phases, then this evidence supports the choice of a backwards knee.
More generally, principles extracted using the methods implemented in this chapter can
be applied to engineering designs to save time and monetary resources and improve design
performance.
111
Part IV
Concluding Discussion
112
In my thesis, I have demonstrated the importance of principle extraction in the biomimetic
design process, outlined the difficulties of isolating design principles from biological examples, and detailed a framework under which to approach principle extraction to overcome
these difficulties.
Furthermore, I have demonstrated the effectiveness of the framework
through several case studies, including a simple example to illustrate the implementation
of the framework in Chapter 5, and a very sophisticated example in Chapter 6 that shows
the power of the framework to tackle difficult problems. Here, I assess the utility of the
framework and suggest areas of improvement.
Recall that the answer to Question 4 is that for q percent of the researcher-defined design
space a particular treatment improves the output of interest relative to the control. This
does not necessarily imply that an engineer who has designed a machine that falls within
the design space has a q percent chance of improving the output of interest by applying
the treatment rather than the control.
This is because designing a machine is not the
same as random sampling of a machine from the researcher's design space, and we cannot
assume that a design process will converge to all designs with the same probability density
of random sampling. Nonetheless, when q is found to be especially high
(~ 0%),
(a
100%) or low
the designer might still decide that in the absence of other information and lack
of resources to perform more focused study that the better design decision would be to
implement the treatment or not as recommended by the general study. When q is neither
high nor low
(~
50%), this suggests to the designer that more information is needed to
make a decision. The answer to Question 5, the average improvement due to the treatment,
should be used in conjunction: it will suggest whether the treatment is very important or
not. Together, these results can be sufficient to proceed with design, or to suggest the
importance of performing a more focused study.
I recognize that the answers to Questions 4 and 5 do not provide all the information a
designer might need to optimize a design. More detailed understanding of the coupled effect
of several input variables on several output variables can be ascertained by performing more
extensive experimentation using a spatially structured experimental design (e.g. factorial,
Latin square, etc...) and regression analysis.
Also, ANOVA or other techniques should be utilized when comparing multiple treat113
ments or performing post-hoc testing on the data to avoid 'false alarms', that is, Type I
error. Bayesian statistics and Gaussian processes may be useful for making inferences about
the location of global minima given many local minima. Alternatively, cluster recognition
algorithms or other techniques might be useful to objectively assign a single scalar for the
output values of multiple local minima for simpler comparison among treatments. New
research on sampling from manifolds
[31]
may allow some variables currently classified by
necessity as decision variables to be treated as sampled variables instead, improving the
generality of results.
Previously I defined design principles as 'relationships between input design parameters
and output objectives and constraints applicable to many systems', and the framework
presented here is formulated with such a definition in mind. I recognize that this may
not be a universal definition of the phrase 'design principle', but I would argue that there
are benefits in attempting to view the principle extraction process from this perspective.
Thinking about biological and engineering design in terms of functions, inputs, and outputs
is quite general, and is also very useful because it helps frame them in the construct of
mathematics, which provides a vast array of tools for the researcher and designer. Also,
while the framework is not a complete, step-by-step procedure, this is not unlike many
elements of design. There is currently no substitute for human understanding, creativity,
and ingenuity. The importance of scientific understanding of the biological system through
experimental biology has already been highlighted as essential to investigating the right
question and proposing a fruitful model.
Nonetheless, the framework highlights some of the important decisions that the researcher must make and suggests tools for verifying the researcher's hypotheses using Monte
Carlo computer experiments and optimization. Overall, the framework guards against logical fallacies which would otherwise plague principle extraction studies, namely false analogy and hasty generalization, in the hope that future principle extraction research will
produce reliable conclusions to expedite the design process and advance the capabilities of
biologically-inspired design.
114
Appendix A
Difficulties of Principle Extraction
Many challenges inhibit the extraction of biologically-inspired design principles for application to engineering design. It is useful to examine some of the most common challenges and
recommended solutions before considering the general framework in Chapter 2.
A.1
Domain Transfer Difficulties
Experimental biologists have shown that for both humans [84] and quadrupeds [116], adding
distal mass to the limbs increases the energetic cost of locomotion more than adding the
same mass proximally. Can we transfer a principle, that increasing leg distal mass reduces
robot locomotion efficiency, from these biological studies to benefit robot design? While it
is quite plausible, perhaps even intuitive, that this principle is valid for robots, I argue that
this principle should not be assumed true for legged robots, nor can elementary physical
arguments adequately justify its application to legged robot design. The physics of legged
locomotion are complex , and the typical morphologies, actuators, and control systems
of robots are quite different from their biological counterparts, so direct transfer of this
principle from experimental biology to engineering design is very difficult to support.
This biological principle, however, is of great value in suggesting a hypothesis to be
explored, or a question to be answered, in the engineering domain. That is, considering the
morphologies, actuators, and control systems of robots, we can test whether this principle
holds with analytical models, computer simulation, and/or physical experiments. Therefore,
115
while immediate, unsupported transfer of principles from the biological domain to the engineering domain is difficult or impossible to justify, we circumvent this challenge by using
the biological principle to inspire hypotheses to test for engineering systems rather than as
evidence for a particular conclusion.
A.2
Hypothesis Testing Challenges and Solutions
The process of testing hypotheses for engineering systems presents its own set of challenges.
Typically, analytical studies are impeded by the complexity of the system; principles can
be deduced only for the most heavily simplified cases.
Physical testing of engineering
systems offer the greatest guarantees for the accuracy of conclusions, but constructing
and testing subject machines can be prohibitively time consuming and expensive. When
systems governed by relatively well-understood physics can be modeled to simulate behavior
quickly and inexpensively, computer experiments are often the method of choice for testing
hypotheses.
I focus on in silico testing for the framework because of its ability to treat complex systems with relatively low resource requirements. Indeed, many researchers have attempted
to employ computers to answer biologically-inspired questions of the form 'What is the
effect of <single input> on <output>?', the answers to which would serve as principles to
guide robot design. Unfortunately, even with the power of computer simulation, there are
difficulties with answering questions of this form. Many of these are commonly dealt with in
fields with longer histories in rigorous inductive study, but it seems that many biomimetic
principle studies either do not attempt to generalize conclusions, or suffer from inductive
fallacies, such as false analogy and hasty generalization. Therefore we continue with additional challenges of bio-inspired principle extraction related to the inductive process and
propose solutions.
A.2.1
Multiple Inputs
Although we may seek the the effect of a particular input of interest, say leg center of mass
location, on one output, such as a measure of energetic efficiency, a legged robot of a given
116
architecture has many physical parameters other than leg center of mass. Over the wide
range of values that a designer might select for the other inputs, it is uncertain a priori
how the relationship we seek is affected. It is possible that the effects of the many inputs
are coupled, and the desired relationship of interest may change as the other inputs are
varied. How, then, can we extract a general principle applicable to any particular robot
design despite the variety of possible designs?
Some studies neglect this variability and instead answer a question for a particular
example:
Question 1 What is the effect of <input of interest> on <output>, assuming all other
inputs are fixed?
Unfortunately, the conclusions of studies which fix the values of all inputs other than the one
of interest are only known to be valid under the particular conditions tested. Rationalizing
that the conclusions are valid for a more general class of systems may be as difficult as
justifying the direct transfer of a principle from biology to engineering. Alone, such a study
may be useful for the design of a particular system, but may not be help other researcher
design their own systems.
Another approach is, for each level of the input of interest, to optimize all other inputs
to achieve the minimal or maximum value of some objective, such as the output of interest.
In other words, the question is changed to
Question 2 What is the effect of <input> on the optimal value of <output>?
This can be rationalized when the purpose of the study is to aid a designer in achieving some
design objective related to the output of the system, and thus it is reasonable to expect
that a designer would choose values for the other inputs that would also help achieve that
objective, and so I provide a case study to introduce this idea in Chapter 3. Still, while
this might provide information of more general interest than a study with an arbitrarily
fixed combination of input values, it is uncertain whether the conclusions will be accurate
for researchers designing robots that are sub-optimal with respect to the chosen output.
Some researchers have used parameter sweeps to study the desired relationship over a
range of values of the other inputs. That is, they answer the question
117
Question 3 What is the effect of each <input> on the <output>, for any given combination of inputs?
An advantage of this approach is that it can be a straightforward way to gain insight
about the full multiple-input coupling relationships of the system, and so an example of
such a study is included in Chapter 4. While this certainly increases the generality of
any conclusions, the computational expense of parameter sweeps increases exponentially
(O(MN)) as the number of unique inputs N increases and the number of levels M of each
input remains constant. If the model is simple enough to permit full parameter sweeps, it
may be too simple for predictive study of a physical system as complex as a legged robot,
as the results of Chapter 3 suggest.
To overcome all of these difficulties, we suggest the statistical approach commonly used
in many disciplines, such as experimental biology, but rare in legged robot literature.
In the field of experimental design, it is common to select a value of the input of interest
as a 'control' and refer to other values of the input as different 'treatments'. The set of all
possible combinations of the other inputs may be referred to as the 'population'. Then it
may be possible to learn enough about the relationship of interest for design purposes by
answering either of the following questions:
Question 4 What is the effect of the treatment on the percentage of the population with
improved <output>, relative to the control?
Question 5 What is the effect of the treatment on the average (across the population)
change in <output>, relative to the control?
An example study of this form is included in Chapter 5. While answering these questions
does not provide detailed information about the coupling relationships among the inputs,
the conclusions can still be applicable to the full class of system under study and at far
lower computational cost than parameter sweeps. These questions can be answered using
Monte Carlo methods [75], that is, sampling at random from the population, measuring the
effect of the treatment for these samples, and using the results to make an inference about
the entire population. Critically, we can also quantify our uncertainty in the conclusion
118
with statistical tests, such as the binomial test [15] or a resampling method [75], and be
sure to collect sufficient data to build confidence in the generality of the result.
Note that there is an important difference between many physical and computer experiments and the type described here. Usually, the population being sampled is well defined:
in a poll to predict the results of a political election, the population may be the (finite)
set of all eligible voters. In our case, the population is likely an infinite set of engineering
systems that may not be defined. The experimenter may have to assume a definition, and
this definition can have significant influence on the answer to Question 4 and Question 5. I
consider population definition in greater detail in Chapter 2.
A.2.2
Constrained Inputs
Sometimes constraints add relationships among the inputs such that the values of the inputs
cannot be chosen independently. For example, given a robot with a finite set of control
parameters, only particular combinations of control parameters will allow the robot to walk.
For other combinations the robot will fall, and the values of outputs, such as locomotion
speed, will be undefined. In this case it may be impossible to answer Question 1 even if
we wish: we cannot significantly vary a control parameter of interest independently of the
others and measure the resulting performance at each value because the robot will fall.
This may not present such a challenge if, given a value of the input of interest, the
constraints are satisfied by a unique combination of the remaining inputs. In this case,
when we change the control parameter of interest, we could tune the remaining control
parameters to the unique values that enable walking, and evaluate the output.
Yet this is not always the case; the family of constraint solutions is not always onedimensional. If we change the control parameter of interest, we must re-tune the remaining
control parameters to enable walking, but there may be an infinitude of possible solutions.
For which of these combinations of control parameters should the output be evaluated?
We might like to randomly sample from the solution space of control parameters repeatedly, evaluate the output for each, and answer either Question 4 or Question 5. However,
the presence of constraints may complicate the random sampling process considerably, as
is considered in Chapter 2.
119
The compromise is to use constrained optimization techniques to solve for a set of control
parameters which optimize the output and answer Question 2. While the applicability of the
results to a designer depend on the extent to which his/her design satisfies this assumption
of optimality, presumably the designer intends to improve the output of interest if he/she
is interested in applying the principle in question.
A.2.3
Function Inputs
Sometimes system inputs are not naturally represented as scalar or vector quantities; some
inputs may be functions of space, such as the shape of a component, or time and state, such
as a control signal. In this case, the input-output relationship in question is not a function,
but a functional (a scalar-valued function of a function [68]). How is it possible to design
an experiment to determine the relationship between the input of interest and the output
of interest that is valid for the variety of input functions a designer might choose?
Function inputs present no added difficulty if we choose to answer Question 1 by evaluating the relationship for a particular, pre-determined input function, but again the conclusions of these approaches suffer from lack of generality. In using Monte Carlo methods to
answer Question 4 or Question 5, how do we define the population of functions from which
to sample, and how do we sample from it?
Certain input functions can be parametrized, and sometimes the range of reasonable
values for the parameters is apparent. It would then be possible to define the distributions
for these parameters and randomly (or exhaustively) sample parameter values from the
distributions. Unfortunately, input functions are not always simple to parametrize. An
important input of a legged robot that must be chosen by the designer is the controller,
which may be a very complicated function of time and the robot's state. Many different
forms of controllers are in common use, many of which cannot be parametrized in any
simple way. In this case it may not be possible to sample a controller at random, because
the population is difficult to define.
In such cases, it may be most relevant to use the calculus of variations or optimal control
theory [68] to optimize the input functions to minimize or maximize the output of interest.
While this limits the applicability of the results, in many cases, it is indeed reasonable to
120
expect that other designers would also choose an input function that would tend to optimize
the objective, and so this choice is justified. Implementation of optimization for this purpose
is considered in greater detail in Chapter 2.
121
122
Appendix B
Mathematical Preliminaries
In order to make this thesis more accessible and the ideas within more useful, I present
summaries of some of the key mathematical preliminaries in a self-contained format.
B.1
Computational Dynamics
The framework for principle extraction relies on computer experiments for verifying proposed hypotheses. Essential to the computer experiment is the modeling and simulation of
the physics of the system. In the field of legged robots, under which context this work developed, the most relevant physics are known as rigid body dynamics, which can be derived
entirely from the principle of conservation of angular momentum and the assumption that
bodies do not deform under load. The background knowledge assumed here is familiarity
with the mathematics of vectors and their derivatives with application to kinematics. Because the dynamics of a wide class of legged robotics is well described in the sagittal plane,
we restrict our attention to computational dynamics in two dimensions.
B.1.1
Equations of Motion
Consider a system of rigid bodies numbered i E [1, 2, . .. , NB] in the plane, each with mass
mi and moment of inertia with respect to its center of mass
I, with P holonomic, scalar
constraints among them, forming an open loop kinematic chain. The dynamic state of
the system may be expressed in terms of a set of M = 3NB - P independent generalized
123
coordinates q = [qi, q2,
...
, qM4 and their first derivatives with respect to time. The state
of each body is described in terms of q via the position of the center of mass with respect
to some reference point fixed in an inertial frame (e.g. the Earth) ri = ri(q), the velocity
Vi=
, the angle measured positive counter-clockwise with respect to some fixed reference
(typically horizontal)
(i(q),
and the angular velocity wi =-
L
The kinetic energy of rigid body i is
212
(B.1)
miV + 1jjW?
Ti =
The gravitational potential energy of a rigid body in a field of constant gravitational
acceleration g is
(B.2)
Vf = -mig - ri.
Prismatic springs numbered
j
E [1, 2,..., Nk] of constant stiffness kj and rest length
l
between any two points rj and r produce equal and opposite forces between the two points
of magnitude
Fk = kj
1
where
.
r - r
3\
- 1
3
,
(B.3)
:1,
denotes the (euclidean) 2-norm. Such an element posesses an elastic potential
energy
V= k
Torsion springs numbered k E [1,2,
between any lines at angles
g
rk - r
...
, N,] of constant stiffness
( B.4)
2
nk
and rest angle 00
0a and Ob produce equal and opposite torques between the two
lines of magnitude
T - 0a - 00 .(B.5)
Such an element posesses an elastic potential energy
b-0a-0 2
1
Vk' = 2Kk (o - k -0)
(B.6)
Other conservative forces and torques can be included by defining additional potential
energy terms V 0 .
124
Then the total kinetic energy is
T=
Ti,
(B.7)
the total potential energy is
Gravity
Prismatic Springs
V = 9 V? +
Z vk
Torsion Springs
EVr
+
Other Conservative Elements
+
VI
,
k
i
(B.8)
and the Lagrangian is L = T - V.
Non-conservative forces and torques cannot be expressed as potential fields and are taken
into account through generalized forces, each corresponding with a generalized coordinate.
The generalized force corresponding with generalized coordinate q, due to forces Ff (numbered
f
E [1, 2,... , NF]) acting at points rf and torques T (numbered t E [1, 2,
...
NT])
,
acting on bodies it is
QM
Ff - r
=
+E
a,
f
t
Tt
am
.
(B.9)
Then the equations of motion are given in terms of the generalized coordinates by
dO8L
dt a9r
B.1.2
OL
Oqm
d Qm = 0
(B.10)
Impact Equations
Collisions occur when the distance between two points
Ct = rt - ra
(B.11)
- or
(B.12)
or angle between two lines
C'a =
vanish. In this case, algebraic equations are needed to determine the state after impact.
Here we assume, without loss of generality, that the same generalized coordinates are used
before and after impact. This is possible even when different generalized coordinates are
125
used for the continuous phases before and after impact because the two sets of generalized
coordinates can always be transformed into a common (potentially redundant) set prior to
performing collision calculations.
When rigid bodies collide, the collision is modeled as instantaneous. Thus, positions and
orientations of the bodies before and after impact are the same, and so the values of the
generalized coordinates after impact q+ are equal to the values of generalized coordinates
before impact q-.
The collisions are also assumed to be perfectly inelastic; i.e., the points or lines that
collide are assumed to remain in contact after the collision. Thus the conditions
dC I
=
dC=
0
(B.13)
0
(B.14)
dt tt+
apply.
Integrating Equation B.10 from the instant before impact t- to the instant after impact
t+ [40] yields
O9L
Oqm
where
+ 9dt
interval and
Qm
-
t-t+
L
94
Om,
(B.15)
-
= 0 has vanished because the integrand is finite over an infinitessimal
is the generalized impulse.
At points of collision, pairs of equal and opposite translational (force) impulses F act.
Likewise, rotational (moment) impulses t act between colliding lines. In a problem without
closed loop kinematic chains, these are the typically the only places where unbalanced
generalized impulses act, and therefore,
QM F i9qm + T 9qm
(B.16)
Equations B.13, B.14, and B.15 are solved simultaneously for the unknown translational
and rotational impulses and generalized velocities after impact.
126
B.1.3
Numerical Integration
Together, the continuous dynamic equations of B.10, collision conditions such as Equations
B. 11 and B. 12, and discrete dynamic equations B. 15 constitute a hybrid dynamic system
in which phases of continuous motion are interrupted by discrete transition events. The
continuous dynamic equations given by Equation B.10 combined with initial conditions
qm(t = 0) constitute an initial value problem that is usually impossible to solve analytically.
Such problems are typically solved approximately using a numerical integration scheme that
'propagates' the solution forward iteratively through successive points in time ti by solving
an equation
l(zi, ui, zi+1, At) = 0,
(B.17)
where zi ~~z(ti) approximates the instantaneous state of the system, ui = u(ti, zi) represents
the instantaneous control forces and torques, zj+1 ~ z(ti+1) approximates the state at the
next time point, and At = ti+1 -ti a the step between time points. Thus, the hybrid system
is solved by propagating the continuous dynamics forward until a collision is detected, that
is, an event condition is satisfied. Numerical integration stops, the impact equations are
solved for the state after impact, and numerical integration (of a potentially different set of
continuous dynamics) resumes.
The continuous dynamic equations given by Equation B.10 are initially of the form
h(q, 4,4, u) = 0,
(B.18)
but in order to be integrated numerically they typically must be converted into first-order
form
i=f(z)
(B.19)
in order to use any of several numerical integration algorithms, such the routines in the
MATLAB
ODE suite [113].
This can be accomplished by first rearranging Equation B.18 into in the form
.M(q, 4)4 = g(q, 4),7
127
(B.20)
where M E RM""
is a matrix and q E RM is a vector of the generalized coordinates. This
is possible because the equations are linear in the terms of the accelerations
4. Thus, the
procedure of transforming B.18 into B.20 can be automated, as
M(q, 4)
=
g(q, d, u) =
q)4..
M(q, 4)q
(B.21)
,
a4
-
h(q, 4, 4, u).
(B.22)
The equations can then be arranged into the required first-order form by making the
substitutions
q
Z =(B.23)
Note that it may be impractical or inaccurate to substitute
which case
t
=
M-1g symbolically, in
4 may instead be evaluated by solving the linear Equation B.20 numerically at
each instant ti.
B.2
Optimal Control
The objective of optimal control is to design a controller which influences a dynamic system
such that some cost functional of the state and control trajectories is minimized (or equivalently, that some objective functional is maximized). We will restrict our attention here to
trajectory optimization; that is, designing an open-loop control trajectory that minimizes a
cost functional.
The optimal control problems we are interested in can be expressed as in [99]:
Given a set of P phases (where p E [1,...
, P]),
minimize the cost functional
P
J=
+
<b(p)(x(
(t0), to, )x((tf), t(, qB)2
P-1
t(P)(B.24)
fL(P)(x(P (t), u(P)(t), t, q(P))dt
(P)
128
subject to the dynamic constraints
xc(P
= f (P (x(P, u(P, t, q(P),
(B.25)
the inequality path constraints
in
C)(x((t) ud (t), t, q))
C
5 CP)
(B.26)
Wfq):5 #(P
(B.27)
the boundary conditions
#p) < #(P (x(P)(to), t(,P
),
,P
and the linkage constraints
L2~~(i
where xd() (t),
u (t), q(,
(S)
, (PI),
x
U(tO), t(P0,
(P-')
:5 Lis
(B.28)
and t are, respectively, the state, control, static parameters, and
time in phase p, L is the number of pairs of phases to be linked, and p' , p' are the "left"
and "right" phase numbers of linkage s
e
[1,..., L], respectively.
For example, the optimal control problem for a legged robot may be composed of two
phases (P = 2): a flight phase (p = 1) with dynamics f(1), and a stance phase (p = 2) with
dynamics f(2). There would be two linkage conditions (L = 2): the first (s = 1) between
flight (pj = 1) and stance (pr = 2) corresponding with the impact dynamics of touchdown
L(1),
and the second (s = 2) between stance (p2 = 2) and flight (p2 = 1) corresponding
with takeoff dynamics L(2).
The boundary condition for flight
#(1)
would include some
condition on final state x((tf) such that the stance foot would be in contact with the
ground (touchdown), and the boundary condition for stance 0(2) would include conditions
on the final flight state x(2)(tf) and control U(2) (tf) such that the force between the foot
and the ground would be zero (takeoff). A path constraint during stance C(2) might require
that the swing foot remain above ground height. If the objective were to maximize average
speed, then the 4p(2) term of the cost functional would be the negative of the horizontal
position component of x(2)(tf) divided by the final time tf. It is important to note that
129
while this form of optimal control problem is quite general, it is not obviously more general
or complex than necessary for work with legged robot control; in this relatively simple
example almost all features of the general form have been utilized.
Also note that the representation of a given problem in this form is not unique, and in
a few cases the problems described in this thesis were not treated in the most obvious way.
For example, while the example above consisted of two phases, a single step of running can
also be treated with three phases (P = 3): apex of flight to touchdown (p = 1), touchdown
to takeoff (p = 2), and takeoff to apex of flight (p = 3). An advantage of this formulation
is that the vertical velocity at the beginning and end of the step must be zero because the
runner is at apex. Of course, there is also a choice over the definition of apex; in the case
studies I use the apex of the hip joint rather than apex of the center of mass. Such choices
affect the way the problem is solved, and selection of the best option may require trial and
error.
B.2.1
Direct/Indirect Methods
There are two main branches of solution methods for this type of problem. The branch
known as indirect methods begins by deriving the necessary conditions for optimality using
techniques extended from the calculus of variations, which take the form of a boundary
value problem. These boundary value problems are usually difficult or impossible to solve
analytically, and so the boundary value problem is usually discretized and solved numerically. Direct methods begin by discretizing the optimal control problem from the outset,
resulting in a nonlinear programming (NLP) problem. We will only consider direct methods because of their relative simplicity, as they do not require the derivation of optimality
conditions, and because of the tendency for the resulting NLP problems to be be easier
to solve computationally than the boundary value problems of indirect methods, especially
without any prior knowledge of the solution. Direct methods are further classified by the
manner in which the optimal control problem is discretized.
130
B.2.2
Discretization
Single Shooting
In perhaps the most conceptually simple approach, single shooting, the NLP problem is
to search for the initial conditions of the first phase x(l)(to), static parameters q, and a
finite number of parameters describing the continuous control trajectory u that yield the
minimal value of the objective functional while satisfying the constraints. The dynamic
constraints and most linkage constraints or phase boundary conditions are automatically
satisfied because the solver takes a 'shot' by propagating the dynamics numerically, using
a technique like those described in B.1.3, from the initial conditions at the initial time of
the first phase to the final conditions at the end of the final phase. Note, however, that
linkage constraints relating the end of the final phase to the beginning of the first phase are
not automatically satisfied, such as when the motion is required to be periodic. Likewise,
boundary conditions involving the beginning of the first phase or the end of the final phase
are not automatically satisfied, as when there are constraints on the initial conditions. Path
constraints are usually taken into account by ensuring that the conditions are satisfied at a
finite number of times. Often, the control trajectory is specified using parameters to describe
zeroth- or first-order holds, but other parameterization methods, such as representing the
control trajectory as a sum of sinusoids or polynomials, is also possible.
Multiple Shooting
Multiple shooting is the same as single shooting except that the trajectory is broken up
into multiple segments, single shooting is performed for each, and constraints are added to
ensure that the final conditions of a trajectory segment are the same as the initial conditions
of the following segment. The most natural points at which to break the trajectories might
be at the linkage between phases, but finer segmentation may also be desirable.
Collocation
Collocation takes the central idea of multiple shooting to the limit that trajectory segmentation is as fine as the time discretization of numerical integration. That is, rather than
131
relying on a separate numerical integration scheme to satisfy the dynamics, the constraints
imposed by the dynamics are added to the nonlinear programming problem. Now, the NLP
problem is to search not only for the initial conditions x(1 )(to), static parameters q, and
a finite number of parameters describing the continuous control trajectory u, but also a
finite number of parameters describing the continuous state trajectory x. A very simple
collocation scheme, for instance, might let the NLP solver select decision variables x(ti) at
a finite number of times ti, but require that the dynamics be enforced according to the
'backward Euler' approximation by constraints x(ti) = x(ti_1) + Atf(x(ti), u(ti)), where f
are the dynamics; i.e., the time derivatives of the states.
Pseudospectral Method
The particular collocation scheme used in the work, a pseu-
dospectral method, is described in detail in [10, 99], but the essential ideas for how the
continuous dynamics are included as constraints in the NLP are presented here for completeness.
We begin by selecting, for each phase, N discrete collocation points ti, i
e
{1, 2, ...
, N}
in time at which the solution is to satisfy the dynamics exactly. Corresponding with each
ti we construct a Lagrange interpolating polynomial Li of order N - 1 such that
1
i=j
Li(t) = Jij =
.
10
(B.29)
i 7 j
A formula for the Lagrange polynomial [117] is
N
Li(t)
H=1,i
t - t- tj
(B.30)
We can construct polynomials to approximate the trajectory of the state derivative globally,
that is, over the entire interval, according to
N
Li(t)fi,
XN(t) =
i=1
132
(B.31)
where fi = f(ti, x(ti), u(ti), p). By construction, the approximate state derivative equals the
dynamics at the collocation points,
kN(ti) =
f(ti, x(ti), u(ti), p), and
XN(t)
approximates
the true *(t) elsewhere. Then the approximate solution to the differential equation is:
t
t
1
XNt
to
N(t)dT
+
= x(to)
N
E
Li(T)f idr
(B.32)
to i= 1
Only the Lagrange polynomials remain under the integral sign, that is,
N
XN(t)
(Jt
= x(to) +
I
Li(T)dr) fi.
(B.33)
If we define
Aji=
Li()dT,
(B.34)
to
then for each collocation point tj the relationship
N
Aji fi.
xN (tj) = x(to) +
(B.35)
i=1
must hold. We can combine these relationships for all tj into a single matrix equation
XN = X 0 + AF(XN)
(B.36)
where
" the jth row of XN is xN(tj) expressed as a row vector,
" every row of Xo is x(to) expressed as a row vector,
* A is an integration approximation matrix defined by Equation B.34, and
" the ith row of F(XN) is fi expressed as a row vector.
This is the essential equation of the integral form of a pseudospectral method, in which
'pseudo' refers to the use of collocation (as opposed to tau or Galerkin formulations), and
'spectral' refers to the exponential convergence to the solution of sufficiently smooth problems as N is increased [117].
In summary, Lagrange polynomials are used to construct interpolating polynomials that
133
exactly satisfy the differential equations at collocation points, and thus approximate the
trajectory of the state time-derivatives (dynamics) over the interval. These interpolating
polynomials are integrated exactly, and to the NLP we add the constraint Equation B.36,
which requires that the integrals of the polynomials equal the states at each collocation
point.
Implementation details, such as the choice of Gauss points to improve accuracy and
avoid the Runge phenomenon [42], the differential form of the method, the division of each
phase into multiple segments, and other important parts of the method are explained in
detail in references [10, 99].
B.2.3
Nonlinear Programming
Direct methods discretize optimal control programs into problems of the form
min J(z)
(B.37)
Z
sAt.
g(Z) < go
where z are the decision variables, J is a scalar-valued cost function, and Z is the set
of z that satisfy the constraints. Such problems are solved numerically by any of several
nonlinear programming algorithms. The simplest method, gradient descent, is summarized
here to motivate the importance of derivative computation for nonlinear programming.
A differentiable function J(z) can be approximated near an expansion point zo by the
first-order truncation of its Taylor series expansion,
J(zo + Az) ~ J(zo) + VJ - Az.
(B.38)
Within that small region, the change AJ in the function due to a change Az in the argument
is
AJ = J(zo + Az) - J(zo) ~ VJ - Az =
IVJJiAzI cos 0,
(B.39)
where 0 is the angle between the two vectors. The function decreases most rapidly when
cos 0
=
-1,
which requires that 0 = wr (or equivalent), and so the direction of steepest
134
descent is -VJ.
This suggests an algorithm for finding an argument which minimizes the value of J: from
a guess zo, iteratively improve the guess by subtracting a component along the direction of
steepest descent, so that
where
(B.40)
a VJz=z;,
zi+1 = zj -
a is a step length that depends on the particular implementation of the algorithm.
The algorithm terminates when
IVJ
-: 0, which suggests that a local minimum is reached.
While this algorithm's utility is limited to well-behaved functions without constraints
on z, the use of the gradient VJ = 4 is common to all nonlinear programming algorithms.
In constrained problems, the jacobian of the constraints !
is also used. Thus, the success
of nonlinear programming algorithms depends on the accurate and efficient evaluation of
partial derivatives.
B.2.4
Derivative Evaluation
Analytic
When the cost function and constraints are known symbolically and consist of expressions
that can be evaluated in closed form, it is possible to perform differentiation analytically and
evaluate the resulting expressions for the derivatives as necessary. However, there are many
situations when this is not possible, such as when the cost function includes an integral
that cannot be evaluated in closed form, or when the constraints can be evaluated in code
but are not known symbolically. In such cases, there are a variety of other techniques for
evaluating derivates, each with benefits and limitations.
Finite Difference
The most obvious, and perhaps most generally applicable, way of evaluating derivatives is
to approximate them with finite differences. From the Taylor series expansion of a scalarvalued function f(x) of a single variable x,
df
f(Xo + AX) = f(Xo) + Ax
135
1
+ 2
d2f2
dX
..
(B.41)
For small changes Ax about the expansion point xo, the function is well approximated by
the first-order truncation of the series
f(xo + Ax) a f(xo) + Ax dx
(B.42)
x=x0
Rearranging yields an approximation for the derivative,
df
-x
f(xo + Ax) - f(xo)
~=OA
(B.43)
Similarly, for a multi-valued function of several inputs, the partial derivative of the ith
output with respect to the jth input is
dfi
dx
f (xO + Axj) - f (Xo)
~
(B.44)
While this first order approximation is very easy to evaluate in a wide variety of situations,
the step size Ax is a parameter that must be chosen by the user, and the accuracy of the
approximation depends on this choice. Moreover, there is a tradeoff in the choice of Ax
which limits the accuracy to which the approximation can be made: a small Ax is necessary
to make the first-order approximation valid and avoid error due to the truncation of the
Taylor series, but the smaller the Ax, the greater the error incurred due to finite precision
arithmetic roundoff error.
Complex Step
A similar, but more accurate, technique can be used when the function being evaluated is
analytic and complex arithmetic is permitted [76]. Consider the first-order truncation of
the Taylor Series expansion of an analytic function
f(z) about a real expansion point zo for
a small step along the imaginary axis Az = hi
f(zo + hi) ; f(zo) + hi df
dz Z=ZO
136
(B.45)
Rearranging,
* df
_
dz Z=zO
f(zo + hi) - f(zo)
h
(B.46)
Taking the imaginary part of both sides yields an approximation for the derivative we seek,
df I
=
(f(zo + hi))
= Im
dz Z=zO
h
-
,
(B.47)
where Im(f(zo)) = 0 has vanished because functions of interest are typically real valued
for real-valued arguments.
The strength of this approximation is that the value of the
function at the expansion point itself is not subtracted, and thus the method avoids the
roundoff error of finite precision arithmetic, allowing the derivative to be approximated to
near machine precision using very small steps h.
Automatic (Algorithmic) Differentiation
Not all functions are analytic, however, and it is not possible to perform complex arithmetic
efficiently in all circumstances. Automatic, or algorithmic, differentiation [47] is a way of
evaluating derivatives exactly (i.e. to machine precision) directly from computer code. It
stems from the fact that computer code can be treated as a series of nested functions: the
output of an operation in code is the input to a successive operation of code. After predefining the analytical derivative of each individual operation, such as the cosine or scalar
multiplication operation, in terms of its arguments, the derivative of a long and complicated
sequence of such operations can be evaluated using the chain rule. Intlab [104] implements
algorithmic differentiation for code involving most standard MATLAB functions, and this
is typically the method of choice when a collocation scheme is used to discretize the optimal control problem. However, when numerical integration must be performed for function
evaluation, as for the single- or multiple-shooting discretization methods, algorithmic differentiation is inconvenient due to the iterative nature of the the integration algorithms.
Indeed, Intlab algorithmic differentiation does not work when the code involves calls to
built-in MATLAB numerical integration algorithms [113]. In such cases, yet another method
of estimating derivatives may be preferred.
137
Derivatives of Solutions to Initial Value Problems
Consider the continuous dynamics of one phase of a hybrid system defined by
5(t) = fi(x(t), u(t)),
0 < t < ti,
(B.48)
starting from the initial conditions xo; note that the notation is simplified from the original
definition of the optimal control problem. Assume an open-loop controller of the form
u(t) = u(t, a)
(B.49)
where a contains controller parameters ul, U2, ... ui and tf that parameterize an open loop
control trajectory. The variable a may also contain initial conditions xO for the problem
considered here. With relatively minor extensions, a may contain other variables, such as
constants appearing in the dynamics or those that parameterize a feedback controller, but
these cases will not be considered explicitly. When the transition condition
< (x(t)) = 0
(B.50)
is satisfied at t = t-, the system state is written x-. The map
x+
X(X-)
+
(B.51)
defines the discrete dynamics to the state x+ at the beginning of the next phase. This map
may be defined implicitly by a transition function
T(x+, x-) = 0.
(B.52)
After the transition, the system may evolve according to dynamics of the second phase,
k(t) = f2(x(t), u(t)),
from the initial state x+.
138
ti < t < t2 ,
(B.53)
The sensitivities, or derivatives, of the final state of the system with respect to changes
in parameters
a are essential if optimal control is to be performed using single shooting.
However, the sensitivities cannot be calculated analytically in general; for the hybrid dynamics of legged robots we must resort to numerical evaluation. The limited accuracy of
finite differences may be insufficient, especially if the discrete dynamics are very sensitive;
the complex step approximation is not applicable when the solution is not analytic; and
automatic differentiation cannot readily be used for all numerical integration routines, such
as the
MATLAB
ODE suite [113]. In such cases, one of two main methods may be used:
adjoint sensitivity analysis [19], and forward sensitivity analysis [9][53].
Only the latter
shall be covered here.
The following method has been assembled from [119], with extensions from [120], assistance from personal communication with Hongkai Dai, and ideas for clarification from [41].
It is written here in hope of treating hybrid systems more explicitly than [119] and [120]
and being more instructive than [41]. The original article on sensitivity analysis of hybrid
systems, [100], is an excellent reference.
Sensitivity During Initial Phase
Suppose we wish to determine the sensitivity ax(t)
on the interval 0 <t < ti. Differentiating the initial condition yields:
0xX(t)t)
a t=to
x
(B.54)
a
The evolution of this initial sensitivity is found by taking the derivative of Equation B.48,
d Ox
dt oDa
>()
f Ox +
9x 09a
F.(t) P(t)
f
u(B.55)
9U 09a
F.(t) Q(t)
where
" Fx(t) and F.(t) are found analytically by differentiation of the dynamics and evaluating at t
" Q(t) is found analytically by differentiating the expression for the open loop controller
and evaluating at t
139
* P(t) defined by this differential equation can be integrated numerically alongside the
dynamics from initial condition P(O) defined by Equation B.54.
Solution of Equation B.55 yields the desired sensitivity as a function of time until time ti.
If the integration end time ti depends on a, then this effect on the sensitivy must be taken
into account. This is most easily seen from the solution of Equation B.48,
t'
x(ti) = xo +
fi(x(t), u(t))dt.
(B.56)
The chain rule and differentiation under the integral sign yield
Ox(ti)
xo
Oa
Oa
+fi (x(t 1), u(ti))
+ lfa
0
The term fi (x(ti), u(ti)) 1
X
Da
(B.57)
+ aaudt,
'
a
au
is the additional term when ti depends on a. In the case
where ti depends explicitly on a, say ti = tf, 2L is simple to evaluate analytically. When
tj depends implicitly on a, say because a influences the time ti
surface is reached, 't
=
t- at which a switching
can still be evaluated. Taking the derivative of Equation B.50 with
respect to a yields
19# ax
09# =- 49 Ox + -x
6t at- = 0.
(B.58)
Solving yields
at-
-9_
-
ax
a
(B.59)
Ox t t=t-
Sensitivity After Impact
Differentiation of Equation B.51 yields
Ox+
Ox+ 0x-+ - =
+ N .(B.60)
We know
-
by evaluating Equation B.57 at ti = t-, the time of transition. If the discrete
dynamics are defined explicitly, then
are define implicitly, we can find
y
is easy to evaluate. Even if the discrete dynamics
by differentiating Equation B.52 with respect to x140
as
0T
0T ax+
(B.61)
=0
+
and solving for
O(B.62)
- -
t+ = t~
Integrating Equation B.53 from t
Sensitivity During a Subsequent Phase
and x = x+ after transition yields
x(t2) = X+ ++
(B.63)
tf2(x(t), u(t))dt.
Ift2
The chain rule and differentiation under the integral sign yield
ax(t2 )
=
Ox
at+
+
a
f2(X, ,U(t ))
&t2
+f2 (X(t2), 0~2)) 5-(B.64)
/t2 +it+
f
+ f au
x
+
x aa
-
dt.
Du 1a
The -f2 (x+, u(t+)) L term is new. It is necessary because of the dependence of t+ on
a, that is,
. Note that this indicates that the new initial condition for
which the integral can be numerically solved is
ax+
Other Sensitivities
- f2
-
from
(x(ti), u(ti)) & rather than simply
The preceding paragraphs outlined the procedure for calculating the
sensitivity of the state x to the parameters a. This can be directly applied, say, for determining the gradients of a constraint function for a gradient-based nonlinear programming
technique. An equivalent procedure can be used to find other sensitivities.
For example, suppose for the previous system a measure of cost is defined
J(ti) =
j
g(x, u)dt,
and we wish to determine the gradient of the cost J with respect to parameters
141
(B.65)
a. Similar
to Equation B.57,
=a g
(x, u(ti)) --
(B.66)
tI ag Ox
+1
x 5a +
(t)
G.
g
ud
u--
P(t)
G.
dt'
Q(t)
where
* G.(t) and Gu(t) are found analytically by differentiation of the instantaneous cost
g(x(t), u(t)),
* Q(t) and P(t) are identical to those defined in Equation B.57.
If a transition occurs at ti = t- = t+ and cost continues to acrue after the transition, then
we can define the cost
j
J(t 2 ) = J(t+) +
g(x, u)dt.
(B.67)
We continue integration after the transition and the sensitivity of the final cost sensitivity
is
OJ(t2 )
aa
J(ti)
+
t
=Jt)a~j
- g (x+,
u(ti)) - 1
Oa
Oa
__
at2
Da--
+9 (x(t2), U(t2))
l2 ag
+j
Computational Considerations
L
x
jf
(B.68)
agandt.
+
+u
a
From a computational standpoint, note that integra-
tion of the equations for the dynamics of the gradients such as Equation B.55 and B.66 can
be done in parallel with integration of the dynamics of the state. The state and the relevant
gradients may be combined in an augmented state vector to be integrated according to
augmented dynamics from an augmented initial condition. In MATLAB, the ode45() function requires that the state be a column vector, so jacobian matrices such as P should be
reshaped as necessary. Upon satisfaction of a transition condition, Equation B.52, detected
using events, the augmented state vector can be broken apart and processed to determine
the effects of the transition according to Equation B.61, then reassembled with proper
142
re-initialization, and integrated again according to Equation B.64.
B.3
Statistics and Design of Experiments
As the method of principle extraction will be to form a hypothesis and then test that
hypothesis using computer experiments, some background in the statistical methods of
Design of Experiments is essential. In particular, this section is intended to be a primer
sufficient for posing and answering Questions 4 and 5 presented in Section A.2.1.
B.3.1
Binomial Distribution
Consider a binary experiment, that is, one with only two possible outcomes, which might
be denoted yes/no, true/false, success/failure, etc....
If the probability of success of any
particular trial is a constant p, and a sequence of N independent trials are conducted, then
the binomial distribution B(p, N) describes the probability of achieving k E [0, N] successes
over the entire experiment.
A simple but illustrative question that can be answered using the binomial distribution
is "What is the probability of observing exactly one 'heads' in two tosses of a fair coin?"
Flipping a coin can be considered as a binary experiment with only two possible outcomes,
heads and tails. A fair coin has equal chances of obtaining heads or tails on an independent
flip, so we can determine the probability by listing all the different, equally likely, possible
outcomes of the repeated trials, counting the number of possibilities which contain exactly
one 'heads', and dividing by the total number of possibilities. As shown in Table B.3.1, there
are two possibilities with exactly one 'heads', and there are four equally likely possibilities
in total, and thus the probability of observing exactly one 'heads' in two tosses is 0.5.
Alternatively, the question could be answered by referring to the probability mass function of the binomial distribution [15], which gives the probability of observing k successes
in a sequence of n independent trials, each with probability of success p:
f 5 (k, n,p)
(
(=
W
pk(i _ P)n-k = k!(n
k
k! (n - k)!
(1
r
P),-k
rh
(B.69)
While the given example is probably easier to solve by enumeration rather than this formula,
143
Possibility #
Flip 1
Flip 2
# of Heads
1
Tails
Tails
0
2
Tails
Heads
1
3
Heads
Tails
1
4
Heads
Heads
2
Table B.1: Possible outcomes of two coin flips
the formula's utility is apparent for large n and p $ 0.5.
B.3.2
Hypothesis Testing
Now that we can determine the probability of achieving a given number of heads under the
hypothesis that the coin is fair, we can use this information to help us analyze the results of
an experiment to assess whether the coin is indeed fair. Suppose we perform an experiment
by flipping a coin 20 times. Intuitively, if the coin flips heads and tails approximately the
same number of times, this would not be evidence to reject our hypothesis that the coin
is fair. On the other hand, if the coin flips either almost all heads or almost all tails, we
would be tempted to reject our hypothesis in favor of the alternative that the coin is not
fair. Using our understanding of the binomial distribution which governs the flip of the
coin, we can make our intuitive notions more precise.
Say that the coin flips heads 19 times out of 20. What is the probability of an outcome
equally or more extreme, that is, equally or even more unlikely, occuring under the hypothesis that the coin is fair? Since we would expect the coin to flip heads about 10 times,
1
head would have been an equally extreme outcome as 19 heads, and 0 heads or 20 heads
would have been even more extreme outcomes. We add up the probabilities of each of these
possibilities to get the total probability of an outcome equally or more extreme than the
one observed:
P = fB(0, 20,0.5) + fB(1, 20,0.5) + fB(19, 20,0.5) + fB( 2 0, 20,0.5) ~ 0.000040
(B.70)
The fact that we observed such an extreme outcome, which is very unlikely to occur under
144
the hypothesis that the coin is fair, suggests that the coin is not fair.
On the other hand, suppose the coin flipped heads 6 times out of 20. The probability
of a fair coin flipping an outcome equally or more extreme than that observed-either 0-6
heads or 14-20 heads-is approximately P = .115. In this case, we might attribute the
extreme outcome to chance rather than a biased coin; we do not have strong evidence to
reject the hypothesis that the coin is fair.
In this example, the hypothesis that the coin is fair is called the "null hypothesis", and
the alternative that the coin is not fair is called the "alternative hypothesis". Typically,
we choose some threshold value such as a = 0.05 by which to judge the "significance" of
the result. If the probability of an outcome as or more extreme than that observed is less
than
a, then we reject the null hypothesis at the a level of significance. This idea, called
hypothesis testing, is a central concept in statistics for experimentation.
B.3.3
Confidence Intervals
The same idea can be extended further to infer a range of values for p*, the true probability
that the coin lands on heads, for which the observation would not be unlikely. Again suppose
that the coin had flipped heads 19 times out of 20, but now suppose that our null hypothesis
is that p = 0.6. Intuitively, if the coin had flipped heads approximately pN = 12 times,
this would not be evidence to reject our null hypothesis. On the other hand, had the coin
flipped any number of heads very different from pN
=
12, we would consider rejecting the
null hypthesis. To make these intuitive notions more precise, previously we asked 'what is
the probability of an outcome equally or more extreme occuring under the null hypothesis'.
This was reasonable before, because there were ranges of possible outcomes, 0-1 and 19-20,
both above and below the expected outcome of 10 that were equally unlikely to occur and
were also equally far away ('9 heads away') from the the expected outcome. Because the
binomial distribution B(N = 20, p = 0.6) is discrete and asymmetric, however, we can no
longer pose the question in this way. For instance, limits of the ranges 0-5 and 19-20 are
equally far from the expected value pN = 12 ('7 heads away'), but the probabilities of
landing in these two ranges are not the same; they are not equally unlikely. Indeed, there is
no range of integer values below the expected value that has the same probabillity as 19-20.
145
To preserve the idea that outcomes on either side of the expected value should have equal
weight as evidence for rejecting the null hypothesis, the accepted practice is to instead ask
'What is the probability of the observed event (19 heads) or an event more extreme event
in the same direction (20 heads) occuring under the null hypothesis' and to compare this to
the threshold
}a instead
is P ;:: 0.0005240 <
=
of
a. Because the probability of observing either 19 or 20 heads
0.025, we would still reject the null hypthesis.
Imagine sliding the null hypthesis for the value of p further and further toward the
observed proportion of heads. What is the lowest value p = p, for which hypothesis testing
would fail to reject the null hypothesis? Likewise, consider if the null hypothesis began at
the upper extreme of p = 1 (which can surely be rejected if we observe only 19/20 heads)
and were to slide downwards. What is the highest value p = pu for which the data does
not suggest we reject the null hypothesis? We call the range of values [p1,pJ] for which the
null hypothesis is not inconsistent with the observed data a 'confidence interval' in which
we believe the actual probability of heads p* lies. In fact, it can be shown that if the
experiment is repeated, the confidence interval will contain the actual probability of heads
p* with probability 1 -
B.3.4
a > 0.95 [1], and thus it is called a '95% confidence interval'.
Monte Carlo Integral Estimation
The binomial distribution can be used to help understand much more than the toss of a
coin. Suppose that a dart player with no length measurement implement wishes to estimate
the area of the triple ring relative to the area of the bullseye on the dartboard. Fortunately,
the dart player is so poor that his throws are equally likely to hit any particular point on
the wall on which the dartboard lies. Because the throws follow a uniform distribution
over the surface of the wall, the probability that a throw will fall within any given region
on the wall is proportional to the region's area. Then of the darts that happen to fall
in either the bullseye or the triple ring, the probability of a dart landing in the bullseye
region is equal to the area of the bullseye relative to the area of the bullseye plus the triple
ring. This suggests another application of the binary distribution: ignoring all darts that
fall neither in the bullseye nor triple ring, we are left with an experiment with only two
possible outcomes; we can compare bullseye to heads on a biased coin and the triple ring
146
to tails. The probability of bullseye on any particular trial is a constant p proportional
to the bullseye's relative area. If a sequence of N throws hitting either the bullseye or
triple ring (not counting throws that fall outside those regions) are made, then the binomial
distribution B(p, N) describes the probability of achieving k E [0, N] bullseyes over the
entire experiment. Our dart player could perform such an experiment and thus calculate a
confidence interval for the relative area of the bullseye.
More abstractly, similar procedures can be used to estimate integrals much more general
than a spatial area.
Such an approximation can be extremely useful when the integral
involves a very complicated, high dimensional region. The class of techniques based on this
principle is called 'Monte Carlo' after the Monte Carlo casino, presumably because of the
importance of statistics in games of chance.
B.3.5
Other Distributions
Suppose now that the dart player is interested in the expected (average, or mean) value of
his dart throws that hit the board. Given that his throws are uniformly distributed, the
mean value would be be equivalent to the integral of the function in Figure B-1 normalized
by the area of the board. Without any knowledge of that function or the relative areas on
the dartboard, however, the player could throw many darts, record the value of each throw,
and take the mean (not counting those that missed the board). This 'sample mean' would
converge to the true mean as the number of throws approached infinity.
However, with only a finite number of samples (throws), the dart player should estimate
the uncertainty in his approximation. As in the previous example, the dart player seeks to
generate a confidence interval with probability 1 - a of containing the true mean. However,
this typically requires understanding (or assumptions) about the reference distribution from
which the data was sampled. He has no knowledge of the relative areas of the different
regions on the board, so he cannot calculate the probability of observing a given sample
mean; the form of the reference distribution is unknown.
In such situations, it is still
possible to estimate a confidence interval using a class of techniques known as 'resampling'.
The essential idea is to use the observed data in place of a reference distribution, and
repeatedly re-sample from the observed data to generate a set of means that might also
147
Figure B-1: The 'dartboard function': to every point on the plane, we assign the point
value associated with the corresponding region of the dartboard. Outside the boundaries
of the board, the function is undefined; these points are not counted in the calculation of
the mean.
have been observed. Intuitively, if all of these 'resampled' means are clustered, then we
can infer that there is little uncertaintly in our knowledge of the mean.
On the other
hand, if the distribution of the resampled means is dispersed, then we know that there is
significant uncertainty in our knowledge of the mean. The resampling method known as
'the bootstrap' [32, 36, 33] is particularly simple to implement: from a sample of N data
points, (re-)sample N data points with replacement M times, and calculate the mean of
each of the M 'bootstrap samples'. An estimate for the 1 - a confidence interval is the
interval between the a and 1 - a percentiles of the bootstrap sample means [33].
148
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