Manuscript:
Robust and efficient walking with spring-like legs
∗†
Juergen Rummel, Yvonne Blum and Andre Seyfarth
Lauflabor Locomotion Laboratory, University of Jena,
Dornburger Strasse 23, 07743 Jena, Germany
juergen.rummel@uni-jena.de, andre.seyfarth@uni-jena.de
Abstract
1 Introduction
An interesting subject in engineering sciences is the
field of bipedal robots, which is inspired by human locomotion. Most of the robots can walk on
flat ground and a few of them can even run with
short aerial phases. While it seems that walking
on two legs is naturally internalized in humans, it
is a challenge to implement this gait into bipedal
robots. One way of implementing walking into an
artificial system could be done by copying anthropomorphic data and reproducing human-like leg kinematics (Zielińska 2009, Wehner and Bennewitz 2009).
A second possibility is to simplify the motion and,
thus, simplify the system, which also leads to a walking gait as demonstrated with passive dynamic walkers (McGeer 1990). Here, the leg is straight during the stance phase which causes the lifting of the
body toward midstance. A third idea of copying human walking is to reproduce leg dynamics with corresponding centre of mass kinematics. Hereto, the
ground reaction forces of single legs with a double
humped vertical force is common (Alexander and
Jayes 1978). The simplest model, which is able to
represent such leg dynamics and centre of mass motion of human walking, is the bipedal spring-mass
model (Geyer et al. 2006). Although, human legs
are very complex in structure and neural control, it
seems that human legs can generate a simple springlike behaviour during stance phase in walking at preferred walking speeds (Lipfert 2010), as already described for bouncing gaits (Blickhan 1989, McMahon
and Cheng 1990).
Spring-like leg behaviour in locomotion can be
produced by technical springs in the leg (de Lasa
and Buehler 2001, Iida et al. 2008, Seyfarth et al.
2009), but also by simulating elasticity with actuators
The development of bipedal walking robots is inspired
by human walking. A way of implementing walking
could be done by mimicking human leg dynamics. A
fundamental model, representing human leg dynamics during walking and running, is the bipedal springmass model which is the basis for this paper. The aim
of this study is the identification of leg parameters
leading to a compromise between robustness and energy efficiency in walking. It is found that, compared
to asymmetric walking, symmetric walking with flatter angles of attack reveals such a compromise. With
increasing leg stiffness, energy efficiency increases
continuously. However, robustness has a maximum
at moderate leg stiffness and decreases slightly with
increasing stiffness. Hence, an adjustable leg compliance would be preferred, which is adaptable to the environment. If the ground is even, a high leg stiffness
leads to energy efficient walking. However, if external
perturbations are expected, e.g. when the robot walks
on uneven terrain, the leg should be softer and the angle of attack flatter. In case of underactuated robots
with constant physical springs, the leg stiffness should
be larger than e
k = 14 in order to use the most robust
gait. Soft legs, however, lack in both, robustness and
efficiency.
Keywords: legged locomotion; bipedal spring-mass
model; SLIP; leg stiffness; periodic gaits; stability;
basin of attraction; cost of transport.
∗ Bioinspiration
& Biomimetics, Vol. 5, No. 4, 046004 (13pp),
December 2010.
† DOI: 10.1088/1748-3182/5/4/046004
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
1
METHODS
Robust and efficient walking with spring-like legs
(Seyfarth et al. 2007). In human and animal legs, a
simple linear spring does not exist, but a spring-like
function is probably generated by properly adjusted
muscle activation during stance (Geyer et al. 2003).
Furthermore, compliance of legs is gained due to the
coupling of muscles with passive elastic structures,
i.e. the tendons.
One criterion for successful walking is robustness
against perturbations such that the risk to fall is low.
Geyer et al. (2006) have shown that in walking compliant legs provide stabilizing effects and the parameter ranges of stable walking decrease with increasing
leg stiffness. However, stability was only measured
for small perturbations as frequently done for nonlinear systems. So far, it is not proven if compliant legs
are robust against larger perturbations. We assume
that, similar to the parameter range, robustness increases when the leg is more compliant.
The second important criterion for selecting walking gait patterns is energy efficiency to, for instance,
save battery power during locomotion. The most efficient leg configuration in walking occurs when the
leg is not compressed during stance (Srinivasan and
Ruina 2006). This was impressively shown by robots
based on the idea of passive dynamic walking (Collins
et al. 2005). However, such a stiff leg behaviour results in extensive forces at touch down and lift off
(Srinivasan and Ruina 2006), which increases the risk
of damage. By contrast, compliant legs in walking
systems enable the reduction of impact forces but
may require more energy, since each time the leg is
loaded dissipation occurs. The dissipative energy loss
requires a compensation by actuators. Hence, it can
be supposed that the more compliant a leg is, the
more mechanical work has to be provided.
The two main topics of this study are robustness
and energy efficiency in walking. It seems that both
criteria are opposing optimization targets in adjusting leg configurations. In this study, we compare
both criteria in walking with spring-like legs and identify leg configurations that could compromise both.
Therefore, we (1) identify and describe human-like
walking gaits characterized by step-to-step periodicity, (2) select stable patterns, and (3) investigate
them concerning robustness and efficiency.
leg 1
events
touch down
lift off
leg 2
leg 1
VLO1
VLO0
velocity
angle θ
L = L0
center of mass
trajectory (x,y)
point
mass
m
leg 2
2
rest length L0
leg stiffness k
y
FP2
FP1
angle of
αTD attack
x
single
support
double
support
single
support
double
support
Figure 1: The bipedal spring-mass model for walking.
The simulations start at the instant of Vertical Leg Orientation (VLO) during single support phase. For the
second step of leg 1, an additional reference line for
L = L0 is shown to imagine how the leg is compressed
during stance.
F directed from their foot point rFP at the ground
to the point mass r. During swing phase, the swingleg does not affect the system dynamics as the leg is
massless and it generates no leg force. The equation
of motion is defined in the sagittal plane
m r̈ = F1 + F2 + m g
(1)
T
with the gravitational acceleration g = [0, g] and
g = −9.81m s−2 . The force of leg 1 during stance is
F1 = k
L0
− 1 (r − rFP1 ) .
|r − rFP1 |
(2)
We detect the transition from stance to swing
phase when the leg force decreases to zero. The transition from swing to stance occurs when the landing
condition yTD = L0 sin(αTD ) is fulfilled while the vertical velocity ẏ is negative. The parameter αTD is the
angle of attack or touch down angle of the leg.
For two identical legs, the model is described by the
following five parameters: body mass m, rest length
of the leg L0 , leg stiffness k, angle of attack αTD ,
and the system energy E. To compare the model
and its results with other bipedal walking systems
2 Methods
of different sizes, we use dimensional analysis. Herewith, the number of parameters is reduced to three
2.1 Bipedal Spring-Mass Model
(Geyer et al. 2006), i.e. the dimensionless leg stiffk = k L0 /(m g), the angle of attack αTD , and
The basis for this study is the bipedal spring-mass ness e
e = E/(L0 m g).
model (figure 1), which describes the action of the the dimensionless system energy E
stance leg by a linear spring of rest length L0 and leg For the simulations presented in this study, we apply
stiffness k. The body is represented by a point mass the individual parameters based on a human subject
T
m located at r = [x, y] . The legs generate forces having a body mass of m = 80kg and a leg length
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
2
2
METHODS
Robust and efficient walking with spring-like legs
of L0 = 1m. The simulation model is implemented
in Matlab/Simulink (The MathWorks Inc., Natick, MA, USA). We use an integrated Runge-Kutta
variable step integrator (ode45) with absolute and
relative error tolerance of 10−11 .
period-1, i.e. each walking step is equal to the previous step, can be found. To locate periodic patterns or
rather fixed points in the map, we compute the map
with a relatively fine grid. A fixed point could appear
when the deviations yi+1 − y0 and θi+1 − θ0 change
their sign between two grid points. Then, we locate
the according fixed point in the map by applying a
two-dimensional Newton-Raphson algorithm. A fixed
point is identified when both deviations |yi+1 − y0 |
and |θi+1 − θ0 | are less than 10−9 m and 10−9 rad, respectively. If one or more fixed points are known, we
change one system parameter a little bit and compute
the next fixed points. To accelerate the computation,
we use S∗ of the first fixed points as initial guesses
for the Newton-Raphson algorithm.
In the next step, we prove stability of the identified walking solutions. Local stability of a periodic solution is estimated by calculating the effect
of small perturbations on the system’s motion. If
the perturbation is reduced after one step, the periodic gait pattern is indicated as stable limit-cycle. To
compute the effect of small perturbations on the system, a linear approximation is applied [Si+1 − S∗ ] =
J (S∗ ) [Si − S∗ ], where J (S∗ ) is the Jacobian matrix. The eigenvalues of the Jacobian are λ1 and
λ2 . If the magnitudes of all eigenvalues are less than
one, the gait pattern is locally stable (Guckenheimer
and Holmes 1983), also denoted as orbitally stable
(Dingwell and Kang 2007).
Each stable walking solution has a basin of attraction within the space of initial conditions. The area
of this basin of attraction ABOA is a measure for robustness of the corresponding stable solution. Dependent on the choice of the simulation’s starting point,
the number and type of initial conditions can vary.
Hence, the size and dimensional unit of ABOA varies
too. However, the qualitative results are not affected,
i.e. the relationship of the area with respect to parameter variations remains unchanged. In this study,
the space of initial conditions is defined by the instant
of VLO with y0 and θ0 as variables. Since the model
is energetically conservative, we make sure that the
basin of attraction is calculated for constant energy.
With initial conditions that change the system energy, e.g. the horizontal velocity ẋ0 , one cannot estimate stability or robustness of a single periodic solution because for each change in energy another solution might exist and be attractive.
To estimate the basin of attraction, its boundary is computed by (1) a method based on nonlinear dynamics theory, and (2) an intuitive approach.
The first method requires a second fixed point in the
Poincaré map, representing a saddle point. A saddle point is an unstable fixed point, but there exists
one line of initial conditions that are attractive for the
saddle point, namely its stable manifold. At the same
2.2 System Analysis
The bipedal spring-mass model is capable of running
and walking. Given the three dimensionless model
parameters (Section 2.1), the motion is completely
described by the initial conditions, i.e. the position
T
T
r0 = [x0 , y0 ] and the velocity ṙ = [ẋ0 , ẏ0 ] of the
centre of mass. We choose the initial conditions such
that one leg has ground contact and the other leg
is in swing-phase. Simulations start when the supporting leg is oriented vertically (x0 = xFP1 ) and a
single step is completed when the same situation occurs with the counter leg (x = xFP2 ). We call this
situation Vertical Leg Orientation (VLO) (Rummel
et al. 2010).
With the VLO as starting point, we can reduce
the number of independent initial conditions. At the
instant of VLO, the horizontal position is zero with
respect to the actual foot point. Since the model
is energetically conservative, the magnitude of the
initial velocity |ṙ| depends on system parameters and
on initial height y0 :
2
|ṙ0 | =
2
m
k
2
E − m g y0 − (L0 − y0 ) .
2
(3)
The initial velocity is separated into two components,
the horizontal and the vertical velocity, and its direction can be expressed by the velocity angle
ẏ0
.
(4)
ẋ0
With this definition, the number of independent initial conditions is reduced to two, the height y0 and
the velocity angle θ0 at VLO.
A walking motion of a legged system is typically
an oscillation, which can be analyzed using Poincaré
return maps. The system’s state S at predefined
Poincaré sections is investigated depending on the
previous step. The selection of the Poincaré section
has no influence on system dynamics and periodic solutions (Poincaré 1892). Without loss of generality,
we use the instant of VLO as Poincaré section, where
T
the system state is defined as Si = [yi , θi ] , with i indicating the step number. The Poincaré map of two
subsequent steps is Si+1 = F (Si ). We first identify
periodic walking patterns, also known as limit-cycle
trajectories. A periodic gait pattern corresponds to
a fixed point in the Poincaré map S∗ = F (S∗ ). With
a single-step Poincaré map, only walking patterns of
θ0 = arctan
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
3
1
1.5
y [m]
0
0.5
1
1.5
vGRF [body weight]
0
y [m]
1
0
0
0.5
1
1.5
y [m]
1
0
0
0.5
1
1.5
1
0
0
0.2
0.4
0.6
power P [kW]
0.5
vGRF [body weight]
0
double
double
support single support
support
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
power P [kW]
1.5
2
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0.873
2
1
0
0
0.2
0.4
0.574
2
1
0
0
0.2
0.4
0.664
power P [kW]
1
1
E
~
E = 1.05
αTD = 74.4 deg
0.5
y [m]
0
D
~
E = 1.05
αTD = 69 deg
0
1
C
~
E = 1.05
αTD = 72 deg
LO
vGRF [body weight]
0
B
~
E = 1.035
αTD = 75 deg
VLO
y [m]
~
E = 1.05
αTD = 69 deg
1 TD
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
power P [kW]
A
power P [kW]
Robust and efficient walking with spring-like legs
vGRF [body weight]
METHODS
vGRF [body weight]
2
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
2
1
0
0
0.2
0.4
0.6 0.773
2
1
0
0
0.2
x [m]
0.4
0.6 0.773
time [s]
TD LOC
0
0.2
TDC LO
0.4
0.6
0.873
WPOS
WNEG
0
0
0.2
0.2
0
0.2
0
0.2
0.4
0.4
0.4
0.574
0.664
0.6 0.773
0.4
0.6 0.773
time [s]
Figure 2: Representative examples of periodic walking based on the bipedal spring-mass model. The examples A,
B, and E are stable walking patterns; C and D are unstable. The left column contains reduced stick figures with
the events touch down (TD), vertical leg orientation (VLO) and lift off (LO) of one step. In the middle column,
the vertical ground reaction forces (vGRF) of both legs are shown for a complete stride. Mechanical power and
mechanical work (area below the power curve) are shown in the right column. The leg stiffness is e
k = 20 for all
walking patterns and the individual parameters are printed at the left-most position.
time, this stable manifold is the boundary of the sta- sion by refining the underlying grid in the boundary’s
ble fixed point’s basin of attraction (Guckenheimer surrounding region.
and Holmes 1983). Background and details of this
Beside robustness, we investigate energetic effimethod are explained in Appendix A.
ciency of periodic walking patterns. The energy effiSometimes, the saddle point does not exist or ciency is estimated with an opposing gait parameter,
e
the stable manifold cannot be completely calcu- the dimensionless work-based cost of transport C.
lated. In such cases we apply the second, more intu- For a periodic gait, the mechanically defined workitive approach, i.e. a steps-to-fall method (Seyfarth based cost of transport is
et al. 2002), to estimate the basin of attraction. We
!
discretize both initial conditions and simulate the
Z T
Z T
2
+
−
model for each combination. A combination of initial
e=
[P1 ] dt −
[P1 ] dt
(5)
C
mgd
conditions is located within the basin of attraction if
0
0
a minimum number of 50 walking steps is reached.
We further make sure that the motion converges to- where d is the length of a complete stride (two steps),
wards the stable fixed point. The boundary of the and T is the stride time. The power of leg 1 is
basin of attraction is estimated with a higher preci- P1 = F1 (t) L̇1 (t), with L̇1 being the lengthening rate
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
4
METHODS
Robust and efficient walking with spring-like legs
.07
~
E=
1
~ .06
E=
1.0
5
~
E=
1.0
4
~
E=
1.0
3
~
E=
1.0
2
~
E=
1.0
1
1
~
E=
1
~
E=
1
.08
2
0.99
1
0.98
height at VLO y0 [m]
y TD
0.99
height at VLO y0 [m]
~
E = 1.06
A
0.97
B
0.96
~
E = 1.02
~
E = 1.01
~
E = 1.00
0.95
~
~ E = 1.035
E = 1.04
~
E = 1.05
0.97
E
D
0.96
0.95
0.94
~
E = 1.037
~
E = 1.04
~
E = 1.05
0.93
0.92
C
0.91
60
tB
65
tC
yT
D
0.94
75
70
angle of attack αTD [deg]
65
75
70
angle of attack αTD [deg]
80
85
Figure 3: Initial conditions (thick grey and black lines)
of symmetric walking dependent on angle of attack αTD
e and constant leg stiffness
and selected system energies E
e
k = 20. Each point on a thick line represents the initial
VLO height of a periodic gait. A black line highlights
stable walking solutions in contrast to unstable solutions
(thick grey lines). The three selected initial conditions
A, B, and C correspond to the examples in figure 2. Circles illustrate transcritical bifurcation points where the
asymmetric walking patterns intersect symmetric solutions. These same points are shown in figure 4. The
diamond at αTD = 72.4 deg and y0 = 0.968m stands for
a unique bifurcation within symmetric walking solutions
(see Appendix C). The thin dotted line represents the
centre of mass height yTD at the instant of touch down.
The inset shows the separation of the symmetric gaits
(black lines) explained in Appendix B. The abbreviations tA, tB, and tC stand for walking patterns of type
A, B, and C, respectively.
+
30
velocity angle at VLO θ0 [deg]
60
80
85
(a)
tA
0.93
y TD
0.98
E
tE
~
E = 1.05
~
E = 1.04
symmetry axis
10
0
~
E = 1.037
-10
tD
-20
-30
D
60
65
transcritical
bifurcation
points
75
70
angle of attack αTD [deg]
80
85
(b)
Figure 4: Map of initial conditions (thick grey and black
lines) of asymmetric walking dependent on angle of ate for a given leg stiffness
tack αTD and system energy E
e
k = 20. The initial conditions y0 and θ0 are separated
into two panels (a) and (b), respectively. Similar to figure 3, each point on a thick line represents a fixed point
in the VLO return map, which corresponds to a periodic walking pattern. Stable solutions are indicated as
black line. Small circles in (a) show transcritical bifurcation points, i.e. points where the chains of asymmetric
walking solutions intersect those of symmetric walking
solutions. These bifurcation points and those in figure
3 are the same. The selected initial conditions D and
E correspond to the walking examples in figure 2 and
both are the representatives for the gait types tD and
tE, respectively (see Appendix B). The thin dotted line
in panel (a) illustrates the height yTD at touch down
that depends on the angle of attack.
−
of this leg. The notations [P ] and [P ] are rectifications separating positive and negative power, respectively. To calculate the cost of transport, 5 uses the
work of one leg, since the second leg exhibits the same
behaviour in periodic gaits with step-to-step periodicity. The work of the second leg is considered by a
multiplication of two.
With a simplified gait model it is not possible
to calculate the specific energetic cost of transport
(Collins et al. 2005) or specific resistance (Gabrielli
and von Kármán 1950), which are based on the energy input for the whole system. In contrast, with
e we estimate the
the work-based cost of transport C
mechanical requirements to achieve a periodic gait.
In a real robot, the energetic cost of transport could
be lower than the work-based cost of transport if the
most part of the mechanical work is done passively
by physical springs. The cost of transport in the sim-
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
~
E = 1.06
20
plified manner, as used in this study, is applicable as
long as we use it for comparisons within one model
(Ruina et al. 2005).
Using the bipedal spring-mass model as template,
energy expenditure can be determined for the stance
phase, but hardly for the swing phase of a leg. As the
leg is assumed to be massless, a swing forward does
not require mechanical work. Hence, the analyses
5
RESULTS
swing duration tSW [s]
2
0.05
1.8
A
1.4
C
0.15
1.2
0.4
5
0.2
0.5
1
3
3 Results
tC
0.1
1.6
0.
regarding energy efficiency would be incomplete, i.e.
the internal work needed to protract a real leg with
mass is not predicted by the model. Therefore, we
calculate the swing duration (time between take off
and touch down of one leg), which is necessary for
periodic walking. If the predicted swing duration is
too short, a real robot would spend too much energy
to move the leg, or more important, the gait pattern
would be hardly feasible for the robot.
Robust and efficient walking with spring-like legs
average velocity vx [m s-1]
3
0.8
0.2
B
tB
tA
0.6
0.4
0.2
The bipedal spring-mass model shows several kinds of
0
75
60
65
70
80
85
55
periodic walking patterns, which can be distinguished
angle of attack αTD [deg]
by the number of peaks in the vertical ground reaction force (Rummel et al. 2009). At the beginning
of this section, we describe human-like walking solu- Figure 5: Average speed and swing time (isolines) of
symmetric walking patterns. Regions of stable walking
tions, characterized by two peaks in the vertical force.
solutions are highlighted as dark grey areas. The seThen, we select stable walking solutions, which we
lected examples A, B, and C correspond to the walking
analyze with respect to robustness and efficiency.
patterns in figure 2. The regions of symmetric walking
are categorized into three types of patterns, i.e. tA, tB,
and tC (see Appendix B). The regions tA and tB are
disconnected by transcritical bifurcation points shown
as a circle at αTD = 71.6 deg and vx = 1.13m s−1 . The
dashed line indicates the boundary of tC.
3.1 Periodic Walking
Figure 2 shows five representative walking examples.
In most cases (e.g. patterns A, B, D and E), these
patterns have double-humped ground reaction forces,
but we also observe a transformation from doublehumped to single-humped forces (pattern C).
The examples A, B and C are symmetric walking
patterns, with the centre of mass trajectory being
mirrored with respect to the vertical axis at midstance. Here, midstance occurs at the same instant
as the VLO. Beside symmetric gaits, indicated by a
velocity angle θ0 = 0, there also exist asymmetric
walking solutions, e.g. patterns D and E. Here, the
velocity angle θ0 is different from zero (figure 4(b))
and the ground reaction force has two peaks of different maxima.
We categorize and abbreviate walking patterns in
Appendix B. Patterns associated with example A are
called ’patterns of type A’ (tA) (see also figures 3 and
4).
Figure 3 shows a map of initial conditions for periodic and symmetric walking gaits at a constant leg
stiffness (e
k = 20). The VLO height is always above
the height at touch down yTD , which means that in
symmetric walking the centre of mass is always lifted
during midstance. The greater the distance between
y0 and yTD for a given angle of attack, the larger is
the vertical motion of the system. This behaviour increases the cost of transport even if the walking speed
does not change. In most cases, we observe that this
distance y0 − yTD increases with increasing system
energy. Hence, the added energy is mainly shifted
into potential energy and less into horizontal veloc-
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
ity. Taking pattern A as an example, for the energy
e = 1.05 the average speed is 1.203m s−1 with cost of
E
e = 0.242. An increase about ∆E
e = 0.01
transport C
gains the speed to 1.228m s−1 , while a complete exertion of the added system energy into kinetic energy
would lead to 1.282m s−1 . The cost of transport ine = 0.306. This behaviour,
creases about 26% to C
e with
the gain of vertical excursion and increase of C
increasing energy, is more pronounced for walking at
steeper angles of attack, e.g. for tB, and asymmetric
walking tD and tE (figure 4(a)). The lowest power,
respectively the lowest work-based cost of transport,
is required when the centre of mass shows almost no
vertical motion, as observable in pattern C in figure 2.
Here, the VLO height, and therefore the vertical motion, decreases with increasing system energy (figure
3). The average velocity is more affected by changes
of the system energy. The speed of walking pattern
e = 0.01 inC is 1.248m s−1 . A small change of ∆E
−1
creases the walking speed to 1.325m s , which means
that the added energy is completely transformed into
kinetic energy, while the potential energy is reduced.
3.2 Gait Selection
Within three of the five walking types, i.e. tA, tB and
tE, local stability is found (thick lines in the figures 3
and 4), which is indicated by eigenvalues |λ| less than
6
3
RESULTS
Robust and efficient walking with spring-like legs
one.
The region of stable walking inside tB exists
e =
for a small range of system energies, i.e. E
e =
[1.0259, 1.0392], and for a given energy, e.g. E
1.035, the corresponding range of angles of attack is
small (αTD = [74.2, 75.5] deg). Moreover, the stable
walking patterns of tB require short swing durations
tSW < 0.18s compared to tA (tSW > 0.22s) (figure
5), and tE (tSW ≈ 0.221s) . Therefore, we neglect
tB in further analyses. The remaining stable walking
patterns of tA and tE span a relatively large range
in angle of attack since they are connected at the
transcritical bifurcation points (see Appendix B). For
e = 1.05 the range of angles of attack
example, at E
is αTD = [68.0, 71.6] deg for stable walking of tA.
This range is enlarged about ∆αTD = 4.5 deg by the
asymmetric walking tE.
By exploring the maps of initial conditions for varying leg stiffness, we observed a shift for the system
energy dependent on leg stiffness. This shift is pronounced for tE, as for e
k = 5 and 35 the corree = [0.933, 1.036] and
sponding energy ranges are E
e
E = [1.051, 1.065], respectively. For largely varying
leg stiffness the energy ranges do not overlap. For
the analyses, we take the shift of system energy into
account. Therefore, we manually select for each stiffness one representative system energy, i.e. the energy where the αTD -range of asymmetric patterns tE
is maximal. For the stiffness range e
k = [5, 35] this
energy is then fitted with the logarithmic function
transcritical bifurcation is an exception of this trend.
Here, the eigenvalues strongly increase until the bifurcation point is reached, where the maximum eigenvalue is one. his behaviour is visible in the inset of
figure 6(a). A solution with max |λ| = 1 is neutrally
stable (Guckenheimer and Holmes 1983), or rather,
it is neither stable nor unstable. This case requires
further analyses to uncover its stabilizing properties
and will be elucidated in the paragraph after next.
centre of the αTD -range. The neighbourhood of a
Energy efficiency is evaluated using an inversely
The stabilizing behaviour of a walking solution
with respect to larger perturbations can be quantified
by the size of the basin of attraction. In this reduced
model, the basin of attraction is a two-dimensional
area within the space of initial conditions y0 and
θ0 , and its size is illustrated in figure 6(b). The
area ABOA of the basin of attraction is small for soft
legs (e
k < 8) compared to legs with medium stiffness
(e
k = [10, 20]). We observe two maxima of ABOA , one
for symmetric (tA) and one for asymmetric walking
(tE). The maximum size of the basin of attraction for
symmetric walking is at e
k = 12.9, and for asymmetric
walking at e
k = 10.1. With further increasing leg stiffness, the area of the basin of attraction slightly decreases. This reduction can partially be explained by
the shift in angle of attack dependent on leg stiffness.
The angle of attack influences the touch down height
yTD = L0 sin αTD , and the touch down height limits
the basin of attraction in case of negative velocity angles θ0 . This is shown in figure 7(a) at αTD = 70, 72
and 74 deg, where the basin of attraction is cut at the
height yTD for θ0 < 0. Below this height, touch down
e e
E(
k) = −0.054 log10 (e
k)2 + 0.189 log10 (e
k) + 0.898,
does not occur and the model falls down immediately.
(6) Again, the touch down height depends on α
TD and
which increases steadily within the mentioned stiff- this is shifted to higher values with increasing stiffness range. It represents the tendency to walk with ness. Furthermore, we observe that the decrease of
higher centre of mass trajectories for stiffer legs. A
e
BOA with increasing k is caused by a decrease of
Therefore, with increasing leg stiffness the potential the range of velocity angles θ , as shown for selected
0
energy increases and consequently the total energy basins of attraction in figure 6(b) (insets).
too.
At the centrally aligned transcritical bifurcations,
shown in figure 6(b), the basin of attraction reaches
3.3 Characteristics of Stable Walking
a minimum in size. Moreover, the fixed point in the
Figure 6 shows the parameter region and charac- return map is closely aligned to the boundary of the
teristics of stable period-1 walking. The lowest leg basin of attraction. This is illustrated in figure 7(a)
stiffness for stable walking is e
k ≈ 5.6. The range at αTD = 72 deg. Here, the system is very sensiin angle of attack is maximal for e
k ≈ 12.8 with tive against perturbations that lead to VLO heights
∗
∆αTD = 9.6 deg, and slightly decreases with increas- larger than y (VLO height of the fixed point). The
extreme cases are the neutrally stable transcritical
ing e
k.
Figure 6(a) describes the stabilizing behaviour bifurcation points, indicated by max |λ| = 1. These
when small perturbations occur. The smaller the fixed points lie on the boundary of their own basin. If
∗
maximum eigenvalues max |λ|, the faster such a per- perturbations occur where y0 is less than y , the systurbation is compensated. The maximum eigenvalues tem is able to continue walking, while conditions with
∗
of all walking patterns are larger than 0.75, and for y0 > y lead to unstable behaviour. This means, the
soft legs (e
k < 10) they are larger than 0.85. For system has a one-sided robustness at the transcritical
a given leg stiffness, max |λ| decreases towards the bifurcation point.
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
7
RESULTS
Robust and efficient walking with spring-like legs
-3
maximum eigenvalue max| λ |
area of the basin of attraction ABOA [10 m rad]
35
1.0
5
20
10
y0 [m]
20
20
15
50
0.
9
15
10
78
20
76
0 0.7
θ0 [rad]
30
74
αTD [deg]
25
~
leg stiffness k
72
0.9
-0.7
0.8
70
30
0.8
~
leg stiffness k
25
1
0.9
0.8
0.7
local stability
30
max| λ |
transcritical
bifurcation
5
0.9
10
5
50
10
1.0
55
60
75
65
70
angle of attack αTD [deg]
80
5
50
85
55
60
75
65
70
angle of attack αTD [deg]
(a)
30
transcritical
bifurcations
0.2
25
25
0.3
~
leg stiffness k
~
leg stiffness k
85
swing duration tSW [s]
35
30
25
0.
20
0.35
0.3
15
0.2
symmetric
walking
tA
20
0.25
15
5
0.4
10
0.5
0.3
0.3
10
0.6
5
50
80
(b)
~
work-based cost of transport C
35
30
35
40
3
55
60
75
65
70
angle of attack αTD [deg]
80
85
(c)
5
50
55
60
75
65
70
angle of attack αTD [deg]
asymmetric
walking
tE
80
85
(d)
Figure 6: Parameter range and gait characteristics of stable walking solutions. The grey-scale in all graphs is chosen
such that a dark colour indicates favourable values. The maximum eigenvalue in (a) reflects the compensation of
small perturbations. The inset in (a) shows max |λ| in detail for e
k = 25. The area of the basin of attraction in (b)
is a measure for robustness against larger perturbations in the state space (y0 , θ0 ). The insets in (b) show examples
of basins of attraction with the stable fixed point (black dot), and the touch down height yTD (horizontal lines).
The ranges of the insets are θ0 = [−0.7, 0.7]rad and y0 = [0.9, 1]m, for the horizontal and vertical axis, respectively.
Energy efficiency of a periodic gait is estimated by the work-based cost of transport (c). Graph (d) shows the swing
duration required to continue the corresponding walking gait. For these simulations, the system energy depends on
leg stiffness using 6 and the walking speed is between 1.0 and 1.2 m s−1 .
related measure, i.e. the cost of transport based on
the the total mechanical work done during stance.
This is visualized in figure 6(c) for stable walking. In
general, the cost of transport decreases with increasing leg stiffness. For a constant stiffness, the cost
of transport increases with increasing angle of attack
(see also the grey highlighted lines in figure 7(b)). As
the symmetric walking patterns tA exist at flatter angles compared to asymmetric walking tE, symmetric
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
walking is more efficient. Furthermore, we observe
that stable walking has lower costs of transport than
unstable walking, as exemplarily shown in figure 7(b).
e as used in
Selecting one parameter set, e.g. e
k and E
figure 7(b) and αTD = 70 deg, there exist two individual walking patterns, one of tA and one of tD. The
stable pattern tA is more efficient than the unstable
pattern tD. The patterns tC are an exception with
costs less than the half of the most efficient stable
8
4
DISCUSSION
Robust and efficient walking with spring-like legs
height at VLO y0 [m]
1
transcritical
bifurcation
point
tB
0.98
tA
0.96
0.94
tC
tD
0.92
40
tE
20
0
velocity angle
θ0 [deg]
-20
-40
67
68
69
70
71
72
73
74
unstable
tB
75
77
76
angle of attack αTD [deg]
~
work-based cost of transport C
(a)
0.5
0.4
transcritical
bifurcation point
tD
tE
unstable
0.3
0.2
stable
stable
tA
tC
0.1
0
67
68
69
70
71
72
73
angle of attack αTD [deg]
unstable
74
75
76
77
(b)
Figure 7: The graph (a) shows five typical basins of attraction (white transparent regions) in the state space (y0 , θ0 )
in combination with the periodic solutions. These are illustrated as dots on the two-dimensional maps and thick
lines in the three-dimensional graph. The dotted lines represent asymmetric walking where θ0 6= 0, and solid lines
stand for symmetric walking solutions (θ0 = 0). The circles ◦ on the two-dimensional maps represent the stable
fixed points. A diamond indicates a saddle-point whose stable manifold provides the boundary of the basin of
attraction. A filled dot • stands for another unstable fixed point. The graph (b) shows the work-based cost of
transport based of the periodic walking solutions corresponding to (a). The grey highlighted lines represent the
e = 1.051 which was selected based on 6.
stable walking patterns. The constant parameters are e
k = 20, and E
gait’s cost. Compared to other walking gaits with
four peaks in the mechanical power, walking with tC
has only two peaks (figure 2). Hence, the leg is less
stressed when tC is used for locomotion.
decrease of tSW with increasing stiffness is connected
with the increase of stride frequency, as typical for
springs.
The work-based cost of transport consider the
stance phase, but not swing-phase requirements. To
complete the analyses regarding efficiency, we investigate the swing duration tSW shown in figure 6(d).
The swing duration decreases with increasing leg stiffness in case of asymmetric walking solutions tE. Here,
the swing duration is independent on the angle of attack. An opposing behaviour is found for symmetric
walking, where tSW decreases with increasing αTD .
Due to the shift of the angle of attack with increasing
leg stiffness, the swing duration of symmetric walking decreases with increasing leg stiffness too. The
4 Discussion
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
In this paper, we investigate the interplay of robustness and efficiency of walking based on a bipedal
spring-mass model. Walking systems with compliant legs (1) demonstrate a variety of periodic walking
solutions with different gait characteristics, (2) provide a large range of leg stiffness and angle of attack
adjustments leading to stable walking, (3) are most
robust at medium stiffness, (4) provide efficient locomotion at high stiffness, and (5) facilitate longer
swing-phases for softer legs, which reduces the inter-
9
4
DISCUSSION
Robust and efficient walking with spring-like legs
nal work to swing the leg forward.
Based on the model predictions, we found that soft
legs (e
k < 10) provide poor stability and energy efficiency (figure 6). Hence, for a given gait (tA or tE),
we expect a minimum leg stiffness being required for
stable and efficient walking. For a constant leg stiffness, we found that, compared to symmetric walking
tA, the stable asymmetric walking tE requires higher
costs of transport (figure 6(c) and 7(b)) and shorter
swing durations (figure 6(d)). Therefore, symmetric walking with flatter angles of attack allows for a
compromise between efficiency and robustness. The
minimum leg stiffness is then e
k ≈ 14 with a relatively
large basin of attraction.
The angle of attack is a control parameter which
is online adjustable during locomotion. By contrast,
the leg stiffness could be constant in the robot dependent on the robot design. In order to choose the leg
parameter e
k and the corresponding αTD , one has to
take the general robot design into account. For this,
we categorize robots into three classes, i.e. underactuated robots with fixed leg stiffness based on simple springs (e.g. Skipper II (Hyon and Emura 2005),
and PDR400 (Owaki et al. 2010)), actuated robots
with adaptable or simulated compliance (e.g. MABEL (Sreenath et al. 2010) and humanoid robots
like Reem-B (Tellez et al. 2008) or LOLA (Lohmeier
et al. 2009)), and passive dynamic walkers with stiff
legs (e.g. Denise (Wisse et al. 2007)). Here, we focus
on robots with physical or simulated springs.
For underactuated robots with fixed leg stiffness
a robust walking pattern is preferable. This can be
achieved with a medium stiffness, i.e. e
k = [15, 25]
with αTD = [65, 70] deg. As the predicted range with
respect to angle of attack is also relatively large for
medium stiffnesses, the leg control could then also be
relaxed. In robots with partially or fully actuated
legs, the mechanical work is mainly executed by actuators. Therefore, energy efficiency plays a more important role than underlying robustness, as the latter
performance criteria could be enhanced by a higherlevel leg controller. Hence, actuated robots with
simulated compliance or adaptable spring stiffness
should implement a higher leg stiffness, e.g. e
k > 30,
to keep the cost of transport low. As these robots
can change the leg stiffness during locomotion, they
should adapt it dependent on the environment. If
they walk on even ground, e.g. on the laboratory
floor or a sidewalk, where no obstacles are expected,
they could walk with a high leg stiffness, for instance
e
k = 35 which enables for energy efficient locomotion.
If the ground is less even or obstacles are expected,
leg stiffness should be adapted to lower values, e.g.
e
k = 20 with αTD = 68 deg.
When leg stiffness and angle of attack are selected
for the robot, the remaining parameter is the system
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
energy. System energy is mainly a model parameter,
however, it strongly relates to the VLO height, or
rather to the vertical motion of the walker. The lower
the energy inside the possible range, the lower is the
vertical motion which reduces the cost of transport,
respectively gains efficiency of walking.
In this study, we focus on robustness against larger
perturbations, when local stability is provided. A
comparison between both quantified gait characteristics, i.e. area of the basin of attraction and maximum eigenvalue, leads in case of symmetric walking
tA to opposing results. The basin of attraction is
largest when the eigenvalue is almost one. Here, the
walker can deal with normal internal and external
disturbances but it would hardly show a real periodic gait. Investigations on passive dynamic walker
models reveal similar findings. For example, Garcia et al. (1998) mentioned that some basins of attraction of stable chaotic walking were larger than
that of stable period-1 walking, and Su and Dingwell
(2007) found stabilizing behaviour for larger perturbations although the periodic walking solution was
locally unstable.
The symmetric gait tA, as the preferred gait, has
the largest basins of attraction at the margin of the
stable parameter range with respect to angle of attack. An angle of attack chosen a bit too flat does
not mean that the walker will fall down immediately.
There still exist periodic walking solutions and domains in the state space where the robot is able to
continue walking for ten steps and more before it fails.
Hence, a higher-level leg controller gains time to compensate system perturbations.
Humans have a spring-like leg function, respectively an almost linear force-length relationship,
when they walk at their preferred walking speed
(Lipfert 2010). When humans walk faster, a springlike leg function is no longer observed and the vertical ground reaction force changes to an asymmetric
pattern with first a higher and then a lower maximum. Although, the force-length relationship is no
longer described by a simple spring, the force pattern
is qualitatively predicted by the bipedal spring-mass
model, i.e. by the gait tE (example E in figure 2).
This gait is found at steeper angles of attack compared to stable symmetric walking tA. The choice of
a steeper angle of attack has the benefit that after
touch down the risk of sliding is reduced. If walking
systems would keep the gait symmetric, as typical for
tA, the angle of attack has to be flatter with increasing speed (figure 5). In asymmetric walking, the leg
angle at take off is flatter than the angle of attack.
This flatter angle increases the risk of sliding at the
end of the stance phase. At that time the body is
already supported by the leading leg, hence, a sliding
at late stance would hardly increase the risk of fall. In
10
4
DISCUSSION
Robust and efficient walking with spring-like legs
conclusion, a change from symmetric to asymmetric
walking at speeds higher than the preferred walking
speed may help to support a safe ground contact during walking.
Compared to the stable walking tA and tE, the
most efficient gait with respect to work-based cost of
transport, is the gait tC (figure 7(b)). This is caused
by higher walking speeds and negligible vertical oscillations of the centre of mass, which reduces the mechanical work. Our simulation results stay in contrast
to experimental results on human walking, applying
a similar walking gait, but having higher metabolic
costs of transport than normal walking (Ortega and
Farley 2005). This indicates that the internal work,
mainly the work required to swing the leg forward, is
an important factor that contributes to the observed
high metabolic cost of transport. This internal work
depends on leg mass and inertia, swing time and the
angle swept during stance phase. The swing time is
predicted to be very short in tC, hence, the internal
work to swing the leg forward would be large. However, the internal work cannot be measured with a
template, but requires a more realistic model including leg and segment masses. Due to the short swing
time, it would be hardly feasible to implement tC in
a bipedal robots. In order to deal with short swing
durations one could implement a rotating leg like the
bipedal RHex with S-shaped legs (Neville et al. 2006),
or the compliant legged spoke-like walker, which is
described by Asano (2010). Another possibility is to
implement more than two legs, like in quadrupeds,
to safely support the body when one leg is off the
ground.
Focusing on human-like walking, represented by
tA, the predicted swing duration is still relatively
short with values between 0.2 and 0.3s. Based on
human experiments, a mechanism was revealed that
could help to achieve short swing-phases, i.e. the catapult mechanism at the ankle joint (Lipfert 2010). At
the end of the stance phase, the pre-stretched Achilles
tendon pushes the lower leg forward. This mechanism
takes advantage of leg segmentation and passive elastic structures. An implementation of such a mechanism into robot legs should be possible in the near
future.
Regarding swing duration, a surprising behaviour
is found, namely the independence of swing duration on angle of attack for asymmetric walking (figure
6(d)). In the case of asymmetric walking tE and constant leg stiffness, the leg angle swept during stance
phase remains constant for all αTD , and the spatiotemporal parameters (e.g. average speed, duty factor,
stride time) are not affected. This behaviour could
be useful for robots. Let us assume, a robot walks
symmetrically (tA) with a flatter angle of attack and
long swing durations. Now, a disturbance occurs such
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
that αTD has to be adjusted towards a steeper angle.
This means that the required swing duration, which
is shorter for steeper angles of attack in tA, decreases
and the swing leg needs to be more accelerated. As
the swing duration in tA tends to zero with steeper
αTD , the asymmetric patterns tE become attractive.
They keep the required spatio-temporal parameters,
e.g. swing duration, at one level such that the robot
can continue walking. This could also be understood
as a kind of robustness in compliant legged walkers.
In the case of symmetric walking tA, a clear dependency of swing duration on angle of attack is observed. For a given leg stiffness, the swing duration
increases with flatter αTD . At the same time, the
duty factor (= stance duration/(swing duration +
stance duration)) decreases. Hence, with flatter angles of attack, the swing phases and stance phases are
prolonged with similar durations. This means that in
a robot a simple scheme for the hip control could be
implemented, e.g. a sinusoidal swing of the thigh
as used in the JenaWalkers (Iida et al. 2008, Seyfarth et al. 2009). The energy efficiency could be
improved if the oscillation frequency is close to the
natural frequency of the swing leg (Mochon and
McMahon 1980).
In this paper, the bipedal spring-mass model serves
as a conceptual framework for investigating potential
walking gaits. Due to the reduced description of leg
function in the template, it enables to identify general leg strategies, e.g. the adjustment of leg compliance and leg angle at touch down for robust and
efficient walking. While some fundamental insights
into walking with compliant legs can be given, the
complete requirements to robots can hardly be stated
since the model is limited in the number of parameters and state variables. However, this reduction has
the advantage of directly calculating possible walking patterns. The outcomes of this study regarding
periodic walking solutions could be used as starting
points in models of higher complexity or in robots.
In future studies, the limitations of the model need
to be addressed as for instance the influence of leg
mass and leg segmentation on stability in walking
and which swing-leg strategies could enhance robustness. For this, it is required to carefully increase the
complexity of the models to focus on specific questions while using the bipedal spring-mass model as
the underlying template.
Acknowledgments
This study was supported by the German Research
Foundation (DFG, SE1042/1 and SE1042/7). The
authors thank H. Moritz Maus and Christian Rode
for discussions about discrete maps and Monica A.
11
4
DISCUSSION
Robust and efficient walking with spring-like legs
35
30
S0 S-1
S-2
S-3
stable manifold
SU*
0.989
S-4
fixed point
(saddle point)
BASIN OF
ATTRACTION
0.988
fixed point
(stable)
-1
1
2
3
0
velocity angle at VLO θ0 [deg]
saddle points
are among tD
10
saddle points
are among tB
4
5
50
Figure 8: Detailed map of initial conditions S0 =
[y0 , θ0 ]T corresponding to figure 7(a) with αTD = 72 deg.
The grey area represents the basin of attraction of the
stable fixed point S∗S . The boundary of the basin of attraction is the stable manifold (thick black line) of the
saddle point S∗U , which is an unstable fixed point. The
grey points on the stable manifold are examples for timereverse iterations on the manifold (see Appendix A for
details). The arrows visualize the direction of how the
systems state S at VLO change after one walking step
when S0 is at the origin of an arrow.
... method 2
20
15
SS*
-2
basin of attraction
computed with ...
... method 1
25
~
leg stiffness k
height at VLO y0 [m]
0.99
55
60
75
65
70
angle of attack αTD [deg]
80
85
Figure 9: The dark grey region in the parameter space
shows where method 1 (the nonlinear dynamics approach explained in Appendix A) is applied compared
to the light grey region where method 2 (the straight
forward steps-to-fall method described in section 2.2) is
used. Beside a stable fixed point, method 1 requires another fixed point, namely a saddle point. On the left side
of the transcritical bifurcation (thick black line), the saddle points are among the asymmetric walking patterns
tD and on the right side they are part of the symmetric
patterns tB. If the saddle point does not exist, method
2 has to be used.
Daley for motivations considering cost of transport.
The authors further acknowledge Susanne W. Lipfert,
Christophe Maufroy, Andreas Merker, and an anonystable manifold, it has a destabilizing characteristic,
mous reviewer for helpful comments.
i.e. if we start with a small deviation away from the
manifold, the system diverges from the manifold. A
manifold always separates two parts. In our map,
Appendix A
the stable manifold separates the basin of attraction
For calculating the boundary of the basin of attrac- of the stable fixed point from other unstable regions
tion, we use for five sevenths of all cases (figure 9) a (figure 8).
As mentioned, the stable manifold is an unstable
nonlinear dynamics approach, which is briefly mentioned in section 2.2. Here, we explain this method line inside the space of initial conditions - in a forward simulation. By simulating the model backwards
and the background in detail.
At first, we detect all fixed points in the return in time, stability properties are reversed. This means
map for a given parameter set. In many cases two that the stable fixed point becomes unstable and the
or three fixed points can be found. If one of them is manifold has now stabilizing effects. Now, the sadidentified as a stable fixed point (all eigenvalues of the dle point is the source of the manifold and the latter
Jacobian matrix are less than one), we check if one effect can be used to compute the manifold.
In the neighbourhood of the saddle point, the staof the other fixed points represents a saddle point.
A saddle point is characterized by (1) real eigenval- ble manifold is linear. The first part of the stable
ues, (2) one eigenvalue is less than one, and (3) the manifold can be determined from the saddle point’s
other eigenvalue is larger than one. Since the largest Jacobian matrix, whose eigenvector, which correeigenvalue is larger than one, this fixed point is an sponds to the lower eigenvalue, indicates the direction
unstable fixed point. A property of a saddle point of the stable manifold. We select one point on the
is that there exists one way within the space of ini- detected line as starting point for a backward simutial condition for which the saddle point is attractive. lation, e.g. S0 in figure 8. After one walking step, we
This way is called stable manifold. In other words, get the system variables at VLO S−1 which is a point
if we start simulations of the model on this mani- on the stable manifold and another initial condition
fold, the system follows the manifold and converges for the next iteration step. By continuing this proto the saddle point. Although, this manifold is called cess we get some points that lie on the manifold and
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
12
4
DISCUSSION
Robust and efficient walking with spring-like legs
by varying the initial conditions S0 we increase the
precision of the resultant line. In this study, the iteration of the manifold was optimized using the method
of Hobson (1993), which often reduces the computational effort but yet generating more precise results.
The stable manifold consist of two parts, on the left
and right side of the saddle point in figure 8. Hence,
we have to compute the stable manifold a second time
but now starting at the other side of the saddle point.
Now, we explain how the time-reverse simulation
is implemented the bipedal spring-mass model. In
a forward simulation, the model has piecewise holonomic constraints, i.e. the footpoints at positions
rFP1 and rFP2 , which are set when touch down of
the associated leg occurs (except for the simulation
start at VLO). In the case of time-reverse simulations (we further use the index R), these constraints
must not change during a trial. Hence, a time-reverse
simulation can be done as a single step only, which is
from VLOi to the previous VLOi−1 . The footpoint of
T
leg 1 at simulation start is defined as rFP1R = [0, 0]
on even ground. The position of the second footT
point is then rFP2R = [xFP2R , 0] where xFP2R is an
expected value defined in advance. During the single step simulation, the system has a lift-off angle
αLO1R at leg 1, and a touch-down angle αTD2R at leg
2. Both angles have a mirrored meaning in forward
simulations. As mentioned in Section 2.1, the touchdown angle αTD is a constant parameter in forward
simulations. Hence, it is necessary that its equivalent in time-reverse simulations, that is αLO1R , has
the same value. Therefore, the horizontal position of
leg 2, xFP2R , is optimized as long as the condition
αLO1R = αTD is fulfilled. A one-dimensional Newton
algorithm is applied to find the according value for
xFP2R .
Appendix B
In this appendix, we categorize the identified walking patterns. The asymmetric walking patterns can
be distinguished by their velocity angles θ0 . The examples D and E in figure 2 have negative and positive
velocity angles, respectively. With this, we call the
walking patterns with θ0 < 0 patterns of ’type D’
(tD) and those with θ0 > 0 patterns of ’type E’ (tE).
Following a line of initial conditions that corresponds to periodic walking tD in figure 4, we observe
that this line crosses the symmetry axis (θ0 = 0) and
afterwards, the line corresponds to tE. This crossing
is a transcritical bifurcation because the bifurcation
point is also part of the symmetric walking patterns
shown in figure 3. An example of this crossing or
transcritical bifurcation is shown in figure 7(a). A
typical property of a transcritical bifurcation is the
Bioinspir. Biomim. 5(4): 046004 (13pp), 2010.
change of local stability from one line of fixed points
to another while the transcritical bifurcation point is
neutrally stable (one eigenvalue is one). In our model,
stability switches from symmetric walking to asymmetric walking tE. We use the transcritical bifurcation with its switching behaviour to separate symmetric walking patterns. At angles of attack flatter than
that of the transcritical bifurcation point we locate
walking patterns that belong to example A (figures 2
and 3), which we call tA (patterns of type A). Patterns of type B (tB) exist at the other side of the
transcritical bifurcation point.
In the case of symmetric walking patterns we distinguish another gait type, i.e. those who belong to
example C (figure 2). For separating these patterns,
we use an unique bifurcation (see Appendix C). The
bifurcation point is indicated by a diamond in figure 3. The walking solutions identified for energies
e > 1.0392 and VLO heights y0 < 0.968m are simE
ilar to example C, therefore, we abbreviate them as
tC (patterns of type C).
Appendix C
In the previous section, a unique bifurcation was used
to separate symmetric walking patterns. As there
exist several bifurcations within the map of walking
solutions, we briefly describe those which appear for
symmetric walking in this appendix. For investigating bifurcations, a map of initial conditions belonging
to fixed points in the return map is used. Figure 10(a)
is such an example map including two types of bifure = 1.037
cations. For the selected system energy E
two regions with fixed points exist, i.e. one at flatter
and one at steeper angles of attack. We begin the
analysis at αTD = 73.0 deg where, for this energy, no
fixed point is found. With an increased angle of attack to αTD = 73.61 deg, a single fixed point appears.
This point is called a limit-point or saddle-node bifurcation point due to a furcation with increasing αTD
. Here, we identify two different fixed points which
means that two different kinds of walking patterns
e A
exist for identical system parameters (e
k, αTD , E).
similar structure in the map can be identified for flatter angles of attack. Here, the limit-point bifurcation
is found at αTD = 71.64 deg and the furcation occurs
for decreasing αTD .
With an increasing system energy, we observe that
the distance between both limit-points decreases as
indicated by the grey arrows in figure 10(a). At
e = 1.039188) they converge to a sinone energy (E
gle point. For this system energy and variable angle
of attack, the mentioned point represents a transcritical bifurcation point within the map of symmetric
walking solutions. It is a crossing of two lines of fixed
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E = 1.042
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0.97
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0.96
unique transcritical
bifurcation point
limit-point
0.95
69
72
73
71
angle of attack αTD [deg]
70
74
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Figure 10: Map of initial conditions y0 for periodic solutions of symmetric walking (θ0 = 0). The solutions in
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