Compliance in Robot Legs
Jonathan Hurst
Outline
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Introduction
 What is the long-term goal of this work?
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Background, motivation
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What is the intent of this presentation?
Running: Spring Loaded Inverted Pendulum (SLIP)
Why are real springs important?
Future work
Current Research
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Hardware!
Simulation and Control (in collaboration with Joel Chestnutt)
Future work
Introduction
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The long-term goal is to build a bipedal robot that
can walk, run, jump, hop on one foot up stairs,
recover from a stumble, and generally behave in a
dynamically stable manner
The goal of this presentation is to convince the
listener of the following:
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Series compliance is essential for a successful running
robot
Physically varying the stiffness of this series compliance is
useful
Running
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Animals
 Compliant elements in
limbs, used for energy
storage
 Energy consumption is
lower than work output
The motion of the center of
mass of a running animal is
similar to that of a pogo
stick, and is common to all
animals [Blickhan and Full, 93]
Running
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Running is loosely defined
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Aerial phase
Energy transfer
The Spring Loaded Inverted
Pendulum (SLIP) model [Schwind
and Koditschek, 97] closely
approximates the motion of a
running animal’s center of mass
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Assumes no leg dynamics at all
during flight
Assumes lossless, steady state,
cyclical running gait
Assumes point mass ballistic
dynamics for mass
Ideal, lossless
model
SLIP
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Control inputs:
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Gait parameters at steady
state [schwind, kod, 97]:
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Leg Touchdown Angle, q
Leg Stiffness, K
Spring rest position, X
Leg + Ground Stiffness
Leg Length at the bottom of
stance phase
Leg angular velocity at the
bottom of stance
OR
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Stride Length
Hopping Height
Leg + Ground Stiffness
SLIP: Observations of Animals
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Animals maintain a relatively constant stride length,
and change leg stiffness for these reasons:
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Changing ground stiffness
Different speeds within a gait
Changing gravity or payload
Ground stiffness changes are a bigger problem for
bigger animals[Ferris and Farley, 97]
SLIP: stiffness adjustment vs. mass
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From experimental
observations, leg stiffness
scales with animal body
mass[Farley, Glasheen,
McMahon, 93]:
Springs in series add as
inverses:
Ground stiffness changes
significantly for different
terrain types
The lower the leg stiffness,
the less global stiffness is
affected by changing
ground stiffness
SLIP:
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Observations of animal behavior gives us
hints, not proofs
Do we really need a physical spring, or is
spring-like behavior achievable without one?
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Springs are needed for energetic reasons
Springs are needed for power output reasons
Springs are needed for bandwidth reasons
Energetics
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Energy consumption should be
minimized when designing and
building a running robot
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Tether-free
Large payload capacity
Long battery life
Natural dynamics affect energy
consumption
Mimicking the control model
(SLIP) with the system’s
natural dynamics is a good
idea. So far, every running
robot has used physical series
springs.
Energetics: CMU Bowleg
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70% spring restitution
Mass distribution:
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0.8% spring
5% batteries
20% entire mechanism
80% ballast
Used a spring hanging from the
ceiling to simulate operation in
0.35G
Tensioned leg spring during flight
If a slightly larger motor replaced
some ballast weight, the Bowleg
could hop in 1G, but not without the
spring
Energetics: ARL Monopod
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The most energy-efficient
legged robot
Running speed of 4.5 km/h
Total power expenditure of
48W
10.5 Joules of energy
exerted by leg motor in
each hop, for 135J of
mechanical work
Energetics
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A 4kg robot hopping 0.5m high yields a flight
phase of 0.632 seconds
Assume stance and flight are symmetrical:
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Robot with series spring and 70% restitution:
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Constant force of 40N
Work output of 20J
Power output of 32W
Constant force of 40N
Work output of 6J by the motor, 14J by the
spring
Power output of 3.8W by the motor, 28.8W by
the spring
Violating the assumption of constant force
spring only enhances the difference, favoring
the series-spring method
Power Considerations
Bandwidth Considerations
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Reflected rotor inertia dominates the natural
dynamics
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Inertia is proportional to the square of the gear
reduction
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Given the following values:
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Gear reduction = 16 rev/m
Rotor inertia = 0.00134 kg-m2
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Reflected inertia of the motor is equivalent to leg
mass of 13.5 kg
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Kinetic energy in leg momentum is lost as an
inelastic collision with the ground (a highfrequency input)
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For a 30kg robot, much of the energy will be
lost in an inelastic collision, and cannot be
recovered through the electric motor
Summary of the facts so far:
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Animals have leg compliance
SLIP
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Stride Length
Hopping Height
Leg + Ground Stiffness
Animals physically vary leg
stiffness
Series springs are important:
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Bandwidth
Power
Energy
Further Research
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I think variable stiffness
is important for a
human-scale legged
robot
The extent to which
physically variable
stiffness is important
should be calculable
•Can’t make the stride length
longer
•Can’t lower hopping height
•Stiffness is the only thing
left!
Current Research
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Actuator with physically variable compliance
2-DOF device, 1-DOF actuator
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Motor 1: spring set point
Motor 2: cable tension=spring stiffness
Mechanism Design
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Cable drive
Lightweight – about 3 kg
Fiberglass springs for high
energy density
Spiral pulleys impart
nonlinearity to spring
function
Electric motors allow for
precise control
Very low friction on the
“leg” side of the springs
Mechanical Model
Motor Position
time
Leg Position
time
Control
Control
Performance
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We created a plot of
comparative max force
against frequency.
Peak spring force is measured on two
models:
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The dynamic simulation, with physically
realistic spring adjustment limits and the
controller on M1
An idealized simulation, with no spring
adjustment limits and M1 held stationary
X2 is forced to a sine function, cycling
from 1 to 100 Hz
If the Bode plot for the dynamic
simulation were divided by the Bode plot
for the idealized simulation, this would be
the result.
Frequency-Magnitude plots
Frequency-Magnitude Plots
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Physical adjustment is limited
to 10 kN/m
Two discrepancies are
apparent:
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0.78 is the difference between
f=kx, described by the software
controller, and the polynomial
fit of our physical spring
function
0.6 is the difference between
the peak forces of the natural
dynamics of the two systems
System validation
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We built a simulation of
a runner with the full
dynamic model of the
actuator built in – so it’s
almost a SLIP
Raibert-style controller
commands leg angle,
energy insertion for a
SLIP
Future Work
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Show analytically how bandwidth is affected by the
various parameters and situations of the actuator
Calculate the required range of variable stiffness,
and rate of change
Put a hip on this thing, make it hop
Research and implement controllers for hopping
height, stride length, speed on a step-to-step basis
Working with a team, build and control a running
biped that can hop on one foot up stairs