Bipedal Walking and Running with Spring-like Biarticular Muscles.pdf

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Bipedal walking and running with spring-like biarticular muscles

Fumiya Iida a,b, ,J¨ rgen Rummel c , Andre´ Seyfarth c

a Computer Science and Artiﬁcial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar Street, Cambridge, MA 02139, USA

b Artiﬁcial Intelligence Laboratory, Department of Informatics, University of Zurich, Andreasstrasse 15, CH-8057 Zurich, Switzerland

c Locomotion Laboratory, Institute of Sport Science, University of Jena Dornburger Strasse 23, 07743 Jena, Germany

Accepted 25 September 2007

Abstract

Compliant elements in the leg musculoskeletal system appear to be important not only for running but also for walking in

human locomotion as shown in the energetics and kinematics studies of spring-mass model. While the spring-mass model assumes

a whole leg as a linear spring, it is still not clear how the compliant elements of muscle–tendon systems behave in a human-like

segmented leg structure. This study presents a minimalistic model of compliant leg structure that exploits dynamics of biarticular

tension springs. In the proposed bipedal model, each leg consists of three leg segments with passive knee and ankle joints that are

constrained by four linear tension springs. We found that biarticular arrangements of the springs that correspond to rectus femoris,

biceps femoris and gastrocnemius in human legs provide self-stabilizing characteristics for both walking and running gaits. Through the

experiments in simulation and a real-world robotic platform, we show how behavioral characteristics of the proposed model agree

with basic patterns of human locomotion including joint kinematics and ground reaction force, which could not be explained in the

previous models.

Keywords: Bipedal walking and running; Compliant leg; Biarticular springs; Stability; Legged robot

1. Introduction

1997; Full and Koditschek, 1999; Dickinson et al.,

2000; Srinivasan and Ruina, 2006 ). Among other results,

a recent important discovery is that a theoretical model,

the so-called spring-mass model, explains not only the

dynamics of human running ( Blickhan, 1989; McMahon

and Cheng, 1990; Farley and Gonza´ lez, 1996; Seyfarth

et al., 2002 ), but also that of walking ( Geyer et al.,

2006 ). The importance of this research progress on the

compliant leg models lies in the fact that, on the one

hand, the dynamics of human walking can be better

explained (see the next section for more details), and on the

other, a single biped model can explain both walking and

running gaits.

While the spring-mass model generally assumes a whole

leg as a linear spring, it is still not clear how the elastic

components of muscle–tendon systems behave in a human-

like segmented leg structure. This study presents a mini-

malistic model of compliant leg structure that exploits

dynamics of biarticular tension springs. In the proposed

bipedal model, each leg consists of three leg segments with

The model of ballistic walking was proposed a few

decades ago inspired from the observation of relatively

low muscle activities during the swing leg of human

walking ( Mochon and McMahon, 1980 ). Since then, there

have been a number of studies investigating minimalistic

walking models ( McGeer, 1990; Garcia et al., 1998;

Collins et al., 2001 ), and they inspired for the construction

and demonstrations of robotic platforms ( Collins et al.,

2005 ).

Although stiff legs are generally assumed in these

models, a number of biomechanics studies of human

locomotion reported the roles of compliant elements in

animals’ leg structures ( Cavagna et al., 1977; Alexander,

Corresponding author. Tel.: +1 617 324 9136; fax: +1 617 253 0778.

E-mail addresses: iida@csail.mit.edu (F. Iida) ,

andre.seyfarth@uni-jena.de (J. Rummel) ,

juergen.rummel@uni-jena.de (A. Seyfarth) .

doi: 10.1016/j.jbiomech.2007.09.033

Please cite this article as: Iida, F., et al., Bipedal walking and running with spring-like biarticular muscles. Journal of Biomechanics (2007), doi: 10.1016/

j.jbiomech.2007.09.033

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passive knee and ankle joints that are constrained by

four linear tension springs. Biarticular arrangements of the

springs provide both self-stabilizing and energy efﬁcient

characteristics for both walking and running gaits. In

particular, we focus on three biarticular tension springs,

corresponding to rectus femoris (RF), biceps femoris (BF) and

gastrocnemius (GAS) in human legs that play signiﬁcant roles

in the stability and segmental organization of both gaits.

The model is analyzed in simulation and in a real-world

robotic platform, and we compare the behavior with that of

human locomotion.

2. Walking and running in human locomotion

In order to characterize the nature of human walking

and running, we ﬁrst analyze the joint trajectories and

ground reaction force (GRF). Hereto, the subject was

asked to walk and run on a treadmill operated at a

constant velocity of 2m/s. The locomotion patterns were

recorded by motion capture system (six Qualisys motion

capture units; sampling frequency of 240Hz) and two force

plates measuring the GRF at each foot. We used 27

tracking points attached to the human subject, from which

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2000

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Time (sec)

Fig. 1. Time-series trajectories of human locomotion: (a) walking and (b) running both at 2m/s. The trajectories indicate hip joint angle (y Hip ), vertical

movement of body (y), knee joint angle (y Kne ), ankle joint angle (y Ank ) and vertical GRF (from top to bottom ﬁgures) of 15 steps which are aligned by the

stance phase. The stance phase is indicated by gray rectangle areas in the ﬁgures. The coordinate system of the measurement follows the deﬁnition in

Fig. 2 (a).

Please cite this article as: Iida, F., et al., Bipedal walking and running with spring-like biarticular muscles. Journal of Biomechanics (2007), doi: 10.1016/

j.jbiomech.2007.09.033

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ﬁve points are used to analyze angular movements of the

hip, knee and ankle joints. Fig. 1 shows the joint

trajectories and vertical GRF of walking and running

during 15 steps which are aligned with respect to the stance

phase.

A set of basic characteristics of human gaits are shown in

these ﬁgures, which are generally agreed in biomechanics.

Firstly, walking and running dynamics can be clearly

distinguished by observing the vertical GRF: while the

vertical GRF exhibits two peaks in a stance phase during

walking, there is only one peak in running ( Keller et al.,

1996 ). Secondly, the vertical body excursion during

walking increases toward the middle of the stance phase,

whereas it decreases in running ( Pandy, 2003; Geyer et al.,

2006 ). And thirdly, in walking, the knee and ankle joints of

the stance leg show ﬂexion ( Saunders et al., 1953;

McMahon, 1984 ). It is important to note that the ballistic

walking models are generally not capable of reproducing

some of these aspects of human walking dynamics. For

example, the ﬁxed knee joint in the stance leg cannot

generate dynamic angular movement ( Lee and Farley,

1998 ), and accordingly, the vertical GRF generally exhibits

only one peak in ballistic walking ( Pandy, 2003 ).

Based on these basic observations of human locomotion,

in the following sections, we investigate a bipedal locomo-

tion model that generates both walking and running

dynamics. To account for the discrepancy between human

locomotion and the existing models, we implement the

following dynamic elements to the model. Firstly, we

employ a simple sinusoidal oscillation in the control of hip

joint. Secondly, instead of ﬁxating the passive knee and

ankle joints during stance phase as in the ballistic walking

model, we constrain the joints by linear tension springs. By

having the compliant stance legs, the system is able to

reduce ground impact force at touchdown, on the one

hand, and to increase the locomotion stability against the

irregularity in the foot–ground interaction. And ﬁnally, we

constrain the movements of the trunk to the horizontal and

vertical directions only in order to avoid the complexity

derived from the rotational movements.

3. Bipedal locomotion model with compliant legs

The bipedal model investigated in this study consists

of seven limb segments (three segments in each leg and

one body segment), two motors at the hip joints, four

passive knee and ankle joints and eight linear tension

springs ( Fig. 2 ). Two ground contact points are deﬁned in

each foot segment.

The conﬁguration of springs are determined such that

they can constrain the passive joints for natural locomotion

behavior, and support the body weight of the entire system.

In each leg, three springs are connected over two joints

(i.e. two springs attached between the hip and the shank

and one between the thigh and the heel). These springs

correspond to biarticular muscles, rectus femoris (RF: hip

joint ﬂexor and knee joint extensor), biceps femoris

(BF: hip joint extensor and knee joint ﬂexor) and

Fig. 2. (a) Bipedal locomotion model with compliant legs (only one of the two legs is shown in this ﬁgure). The model consists of an actuated hip joint

(denoted by a circle with a cross) and three limb segments which are connected through two passive hinge joints (open circles). The segment mass is deﬁned

at the center of each segment. The dashed lines represent the tension springs (S 11 : BF, S 12 : TA, S 21 : GAS and S 22 : RF), and two ground contact points are

deﬁned in the foot segment (G 1 and G 2 ). The design parameters used in this study are speciﬁed in Appendix A. (b) Photograph of the biped robot. Each leg

of this robot consists of a hip joint controlled by a servomotor and three leg segments which are connected through two passive joints. Four tension springs

are attached to the segments and rubber materials are implemented at the two ground contact points of the foot segment.

Please cite this article as: Iida, F., et al., Bipedal walking and running with spring-like biarticular muscles. Journal of Biomechanics (2007), doi: 10.1016/

j.jbiomech.2007.09.033

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F. Iida et al. / Journal of Biomechanics ] (]]]]) ]]]–]]]

gastrocnemius (GAS: knee joint ﬂexor and ankle joint

extensor) in human legs. Additionally, a monoarticular

spring, corresponding to the tibialis anterior (TA: ankle

joint ﬂexor), is implemented. The model parameters are

described as M ¼½LWS , which consists of the indivi-

dual segment lengths L ¼½l 1 ... l 11 , and weight parameters

W ¼½w 1 w 2 w 3 w 4 . As shown in Appendix A, we set

the relative proportion of these segment length and

weight parameters as close as those of humans while

considering the mechanical constraints for robotic

implementation such as spring attachment. Three para-

meters are then used to characterize each spring

(spring constant K ij , intrinsic damping factor D ij and rest

length N ij ) as follows:

speciﬁed limit determined by the sliding and stiction

friction coefﬁcients, m slide and m stick , respectively.

G yi ¼ ajy ci j 3 ð1 b y ci Þ,

ð5Þ

m slide G yi

x ci

j x ci j

if F xci 4m stick G yi

x ci

j x ci j ;

G xi ¼

ð6Þ

F xci

otherwise;

where x ci and y ci denote the horizontal velocity and the

vertical distance of the contact point i from the ground

surface, respectively. F xci represents the computed force at

the foot contact point i. We used the following parameters

to simulate the ground interaction: a ¼2:5 10 5 N=m 3 ,

b ¼ 3:3s=m, m slide ¼ 0:6 and m stick ¼ 0:7.

In parallel, the model was implemented to a robotic

platform as shown in Fig. 2 (b). This robot consists of

passive knee and ankle joints, and two servomotors

(Conrad HS-9454) at the hip joints as in the simulation

model. We used four tension springs and rubber material at

the two ground contact points in each foot segment in

order to gain sufﬁcient ground friction and to minimize

impact force at touch down. A supporting boom was

attached to the body segment in order to restrict roll

and pitch of the upper body segment. The same

control parameters were used to conduct a set of

experiments. Since this robot is not able to change the

spring parameters, we tuned the parameters before each

experiment.

S ¼½S 11 S 12 S 21 S 22

K 11 K 12 K 21 K 22

D 11 D 12 D 21 D 22

N 11 N 12 N 21 N 22

ð1Þ

These springs S 11 , S 12 , S 21 and S 22 correspond to BF,

TA, GAS and RF, respectively ( Fig. 2 ). The force

generated in these tension springs F ¼½F 11 F 12 F 21 F 22

are calculated as

(

K ij ðx ij N ij ÞD ij x ij ; x ij 4N ij ;

F ij ¼

otherwise;

(2)

where x ij denotes the length of the spring S ij .

This model requires only three control parameters in hip

joint actuation: C ¼½ABo, amplitude, offset angle and

frequency, respectively. These parameters determine a

simple oscillation of the hip joints as follows:

4. Dynamics of walking and running

The dynamics of the proposed model is analyzed in

terms of the time-series data of system variables, vertical

body movement (y), knee and ankle joints (y Knee and

y Ankle ), vertical GRF (G y1 þ G y2 ) and a vector of forces

generated in the springs (F). The GRF of one leg is deﬁned

as a sum of GRF in two contact points of the foot ( Fig. 2 ).

With the spring and control parameters S walk and C walk

(see Appendix A), the model exhibits stable walking gait as

shown in Figs. 3 (a) and (c) and 4 (a). 1 The behavior of

each joint shows the similarity to those of human walking

( Fig. 1 (a)). More speciﬁcally, in Fig. 3 (a), the knee joint

starts slightly ﬂexing at the beginning of stance phase

(0.09 s), extending and ﬂexing again toward the end

(0.27 s). The ankle joint extends at the end of stance phase

which results in ground clearance for the subsequent swing

phase. Note that the spring S 12 (TA) generates small force

during the swing phase which stabilizes the ankle joint at

the angle required for ground clearance.

For a running gait, the spring and control parameters

are set to S run and C run in which the spring constants K ij

and the motor oscillation frequency o are set to

signiﬁcantly larger values than those of walking. In

addition, the rest length of spring N 11 (BF) is set to a

y HipR ðtÞ¼A sinð2potÞþB,

ð3Þ

y HipL ðtÞ¼A sinð2pot þ pÞþB.

ð4Þ

While the leg segmentation is similar to that of humans,

the size of this model is scaled down as shown in

Appendix A in order to facilitate the real-world imple-

mentation to the robotic platform. And for the sake of

simplicity, this model is restricted to motions within a

plane, thus no rotational movement (roll or pitch) of the

body segment is considered. In the following simulation

and robot experiments, all of the parameters S and C were

determined by considering the geometric constraints

explained in Section 5.

For the simulation experiments, we implemented the

model to Matlab (The Mathworks Inc.) together with the

SimMechanics toolbox. A level ground surface with a

physically realistic interaction model is deﬁned based on

Gerritsen et al. (1995) . The vertical GRFs are approxi-

mated by nonlinear spring–damper interaction, and the

horizontal forces are calculated by a sliding–stiction model.

The model switches between sliding and stiction when the

velocity of the foot becomes lower or higher than the

1 See also the video clip of the simulation and robot experiments

(Appendix A).

Please cite this article as: Iida, F., et al., Bipedal walking and running with spring-like biarticular muscles. Journal of Biomechanics (2007), doi: 10.1016/

j.jbiomech.2007.09.033

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F. Iida et al. / Journal of Biomechanics ] (]]]]) ]]]–]]]

time (sec) = 0.0

0.09

0.18

0.27

0.36

0.45

time (sec) = 0.0

0.07

0.13

0.20

0.26

0.33

Fig. 3. One cycle behavior of the model in simulation: (a) walking and (b) running. Black and gray leg segments represent the right and left legs,

respectively, and gray areas depict the stance phase of the right leg. The cycle time is set to 0.45 s (o ¼ 2:2Hz) for walking, and 0.33 s (o ¼ 3:0Hz) for

running. The ﬂight phase of running is approximately 0.06 s before and after the stance phase (see also Fig. 4 (b)). Time-series photographs of the biped

robotic platform during (c) walking and (d) running. A high-speed camera was used to record the experiments (Basler A602 fc: resolution 656 490 pixels,

frame rate 100 fps). The interval between two pictures is approximately 10ms. See also the video clip in Appendix A.

shorter length for antagonistic torque equilibrium at the

knee joint. As shown in Figs. 3(b) and (d) and 4(b) , the

running gait exhibits clear ﬂight phases (around 0.07 and

0.26 s). By comparing the simulation results with those of

human ( Figs. 1(b) and 4(b) ), the knee and ankle joints

show similar behavioral patterns. For example, the knee

joint exhibits multiple peaks in a cycle (0.10 and 0.20 s),

and the ankle joint ﬂexes and extends in the stance phase.

The contrast between two gaits, that is similar to human

locomotion, can be observed in the vertical body excursion

and the vertical GRF ( Fig. 4 ). Namely, this model exhibits

the maximum peak of vertical body movement in the

stance phase in walking (0.12 s in Fig. 4 (a)), while the

minimum peak in running (0.13 s). In addition, vertical

GRF shows multiple humps during walking while there is a

large bell curve in running.

The roles of biarticular spring arrangement can be

identiﬁed further by comparing with the model that has

only monoarticular arrangement of the springs by setting

l 9;10;11 ¼ 0. With the spring and control parameters S walk ,

S run , C walk and C run (see Appendix A), the model with

monoarticular spring arrangement can also perform

periodic gait patterns as shown in Fig. 5 . By comparing

with Fig. 4 , however, there are a few salient differences.

Firstly, because the hip motor torque and GRF do not

directly inﬂuence the knee and ankle joints through the

springs, the model with monoarticular springs exhibits less

joint dynamics especially in running. Namely the knee and

ankle joints show signiﬁcantly smaller ﬂuctuation in both

stance and swing phases. Secondly, the monoarticular

spring arrangement often induces locomotion instability

originated in kinematic singularity of the knee joint (i.e. the

joint angle exceeds 180 ). For this reason, the knee joint

angle needs to be maintained at a relatively lower angle

with smaller dynamics, although the biarticular arrange-

ment allows the legs to extend up to 180 .

Fig. 6 shows the kinematics and GRF during 10

steps of the robot walking and running. In general, the

Please cite this article as: Iida, F., et al., Bipedal walking and running with spring-like biarticular muscles. Journal of Biomechanics (2007), doi: 10.1016/

j.jbiomech.2007.09.033

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