2011 IEEE International Conference on Robotics and Automation Shanghai International Conference Center May 9-13, 2011, Shanghai, China
Running and Turning Control of a Quadruped Robot with Compliant Legs in Bounding Gait Xin Wang, Mantian Li, Pengfei Wang, and Lining Sun is controlling hopping height, forward speed and body posture, making complex gaits possible on one-, two-, and four-legged robots [2-4]. Another quadruped robot was developed with articular-joint-type-legs [5] and a Delayed Feedback Control (DFC) system was designed to realize bounding gaits [6, 7]. Utilizing the DFC approach, it shows good self-stabilization property and a slight terrain passing ability. The Stanford University also built a quadruped robot, using direct adaptive fuzzy control, which requires no system modeling but is more computationally expensive, exhibits better response [8, 9]. Using these techniques it has been successful in operating one leg at speeds necessary for a dynamic gallop. The Ohio State University also presents a fuzzy control strategy that manages the complex coupling between the multiple system inputs and outputs to successfully execute high-speed turns over a range of speeds and turning rates [10, 11]. It turns the robot by putting the leg crabwise, not by changing the force. The research of McGill University showed that even compliant movement can be used to stabilize running [12]. They designed a quadruped with compliant legs, and achieved stable running at speeds up to 1.3 m/s by positioning the legs at a desired touchdown angle [13]. However, it is a pity that they didn’t consider the robot dynamics as an energy system, which we believe that the variation of the system energy during running has a significant relationship with properties of energy saving and animal locomotion. The energy approach may advance the robot’s control performance. In this paper, we describe a new compliant quadruped robot and build the bounding gait model of this one. Unlike the McGill’s robot, we discuss the effects of various hip torques and impact angles on robot’s motion states, and build a DFC method to realize the dynamically stable locomotion, under the idea of controlling the energy input. Further to advance the agility, we present a simple turning control method under the energy idea also, which is only using a PD controller to adjust the hip torques. Finally, the running and turning simulation of robot prototype and bounding gait experiment results exhibit the dynamically stable running and turning character with the simple delayed feedback control law. This paper shows the bio-mimetism, simplicity, feasibility and validity of the energy approach. It will help us to understand the animal’s locomotion as a dynamics more essentially. Further, we will deeply discuss the energy saving property and the strategy to make the robot running fast.
Abstract—In this paper, we introduce a quadruped robot designed for bounding gait with only one actuator per compliant leg. Under the analysis of the dynamics model of the robot, a new simple linear running controller using the energy control idea, which requires minimal task level feedback and only controls both the leg torque and ending impact angle, is proposed. It successfully executes fast running from rest till a constant speed and hi-speed turning, both in the prototype simulation and robot experiment. These results contribute to that complex dynamically dexterous tasks may be controlled via simple energy control method and delayed task feedback, which is closer to the animal’s actual locomotion conditions. In the future, we plan to modify this method for reducing the energy expending and make the robot running fast.
I. INTRODUCTION
A
SIDE from the sheer thrill of creating machines that actually run, there are two serious reasons for exploring the use of legs for locomotion. One is mobility; the other is to understand animal locomotion. Animals exhibit impressive performance in adapting difficult rough terrain and demonstrate great versatility. They can reach a much larger fraction of the earth than most of the existing wheeled vehicles. Thus, robots can benefit from the improved mobility and versatility that legs could offer by imitating the animals. Early attempts to implement legged designs resulted in slow moving statically stable systems. However, the superiority of animals’ agility refers to dynamically stable locomotion. A number of quadruped robots, capable of dynamically stable running, have been built with very various physical structures. Unlike statically stable robots, dynamically stable robots can tolerate departures of the center of mass from the support polygon formed by the legs in contact with the ground. The contacts in the stance phase are typically modeled with the spring-loaded inverted pendulum model (SLIP) [1]. Under this basic theory, Raibert’s research revolved around the three-part controller principles, which it
Manuscript received September 11, 2010. This work is supported by National Hi-tech Research and Development Program of China (863 Program, Grant No. 2008AA04Z211), National Natural Science Foundation of China (No. 61005076), Self-Planned Task (NO.SKLR2008011B) of State Key Laboratory of Robotics and System (HIT). Xin Wang is currently a PhD candidate in State Key Laboratory of Robotics and System, Harbin Institute of Technology, China (phone: 86-451-86402217; E-mail:
[email protected]). Mantian Li is currently an associate professor in State Key Laboratory of Robotics and System, Harbin Institute of Technology, China (E-mail:
[email protected]). Pengfei Wang is currently lecturer in State Key Laboratory of Robotics and System, Harbin Institute of Technology, China (E-mail:
[email protected]).. 978-1-61284-385-8/11/$26.00 ©2011 IEEE
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and the torque of each leg with respect to the body. The leg is reduced as a spring-damping massless leg, with only stance or flight phase. The massless legs are considered there are no impacts at touchdown. Also, a toe in contact with the ground is treated as a frictionless pin joint. The assumption of the spring-damping massless legs could derive a simplified mathematical model for the robot, but well exhibit the robot’s kinetic characters [14-16].
II. ROBOT DESIGN The design of robot is an exercise for bounding gait in simplicity shown in Fig.1. It is designed to run straight and to turn as will. Since a bounding gait has the flight phase, the robot should have enough power and ability for dynamic movement. Thus, reducing the complexity components results in rising powerful, minimizing the major sources of failure and increasing the reliability and robustness of the platform.
l
Fig. 1. Photograph of quadruped robot used for exercise.
The robot consists of a spine with four compliant legs. It uses only one single actuator per leg, located at the hip joint, which provides leg rotation in the sagittal plane. Each leg assembly consists of a relative movable part and a hip fixed part, connected by springs and a linear guide to form a compliant joint. Thus each leg has two degrees of freedom (DOF), which are the hip’s actuated rotational DOF and the passive compliant DOF. As the hip joint driving method, we adopt the belt-pulley-system, and each hip joint is driven independently by a Maxon DC-servomotor. The robot’s construction can be departed as two parts, the forelegs and the rear legs. We select a special aluminum profile to connect these two parts just like animal’s spine. The spine could afford a certain extent deflection and elasticity on different directions. This design can absorb the impact cause of the asynchrony of a pair legs’ touchdown, which makes the robot stable on the frontal plane. And during running, the periodic elastic deformation on the sagittal plane can store the energy in legs stance phase, and release the energy when legs flight. These significant effects will be discussed deeply in further work.
Fig. 2. Sagittal plane running model. The variables in this figure are listed in table I.
The robot is designed to run on bounding gait, which uses its fore and rear legs in pairs. It will act on three perpendicularity planes, the sagittal plane, the frontal plane and the transverse plane. But the robot’s locomotion could be researched on these planes individually. Thus, the essentials of the motion take place in the sagittal plane, when it is running straight. And the turning motion takes place on the transverse plane, which we will discuss in section V. The rolling motion has been absorbed by the spine. Hence, we turn our attention to the running and turning analysis. Therefore, the running model can be considered as a three-body dynamic system, see Fig.2, composed of the torso and the fore and rear virtual legs. Actuators control the angle
to Re uc ar h d le Re ow g a lif r n l to eg ff
g le re off Fo lift g le n re dow Fo uch to
III. RUNNING MODEL
g le re off Fo lift g le n re dow Fouch to
to Re uc ar hd le Re ow g a lif r n l to eg ff
When animal is running in bounding gait, the motion can be divided as four phases, triggered by legs touchdown and liftoff events presented in Fig.3, which are foreleg stance phase, rear leg stance phase, double leg stance phase and flight phase. These four phases are changing circularly, as the sequence showing in Fig.3. We always call the clockwise sequence as rotary bounding gait, and counter-clockwise sequence as transverse bounding gait [5].
Fig. 3. Sequences in bounding gait. The bounding gait is composed of four phases and four events. The dark leg represents it in the stance phase, and light one represents the flight phase.
In each phase, the dynamic system equations are different since each of them is characterized by a different constraint set. The double leg stance dynamics is the most complex of them. Here, we give the equations of the double leg stance motion, using a Lagrange equation, which generalized coordinates are the Cartesian variables describing the center of mass position and the main body’s attitude,
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d ⎛ ∂L ⎞ ∂L ∂Δ ∂Π + = , ⎜ ⎟− dt ⎝ ∂q ⎠ ∂q ∂q ∂q
characteristics, which are just like the animal’s locomotion.
(1)
IV. ENERGY CONTROL When the robot is working on bounding gait, the power is injected by hip actuators solely. That means the hip torque is the only energy source of the dynamic system. For this system, if we ignore the damping, it can work independently [17, 18], without any torque input. Thus, the hip torque input is to renew the energy lost caused by system damping, sliding friction or unilateral ground forces during running. If we want to keep or change its locomotion, first, we should control the hip torque input. And the second, it is to adjust the action angle. As we known, the energy input is:
where 1 1 1 L = m ( x 2 + y 2 ) + Iθ 2 − K 2 2 2 , 2 2 ⎡⎣ (l0 − l f ) + (l0 − lr ) ⎤⎦ − mgy
1 2 2 B(l f + lr ) , 2 Π = ϕ f τ f + ϕrτ r . Δ=
In the equations, q = [ x
y θ ] , L is the Lagrangian of T
the system calculated by subtracting the potential energy from the kinetic energy, Δ is Rayleigh dissipation function, Π is the power supplied to the system and the meaning of the variables we used here are shown in Table I. The dynamics equations of other phases can be derived from (1), by removing appropriate terms.
Win = τ ⋅ β
where, Win is the system energy input; τ is the hip torque; β is the action angle. The changing of τ and β could adjust the energy input, then, control the robot locomotion. During the bounding gait, the rear legs are the primary power source, when animal is running [19]. By contraries, the forelegs are only used to support the body offering a comparatively stable state, when rear legs are lifting off. And the control method of forelegs is similar as the rear legs’. Thus, we only focus on the control of the rear legs and don’t discus the foreleg control in this paper. When the robot is working on the stance phase, we give a constant hip torque input, till the leg swept to a special position, which we call it ending impact angle. Then, we give zero torque input, thus, the robot becomes conservative till to the lift off angle. When the legs lift off, we use a position control to ensure the touchdown angle.
TABLE I VARIABLES LIST Symbol
Units
Variable
m I K B x y θ φ γ α β l0 l τ M L Win td end l or r f or r k d
kg kgm2 N/m N/m/s m m rad rad rad rad rad m m Nm Nm m J subscript subscript subscript subscript subscript subscript
Body mass Body inertia Spring stiffness Damping coefficient C horizontal pos. C vertical pos. Body pitch angle Leg relative angle Leg absolute angle Body yaw angle The action angle Leg rest length Leg length Torque at hip Virtual torque on robot body Half body length The system energy input Leg touch down Ending impact Left leg or right leg Foreleg or rear leg The number of steps The desired status
(2)
A. The Effect of Leg Torque Utilizing the bounding model built in the section III, we can analyze the effect of rear leg torque on body status during the rear leg stance phase. The rear leg torque at hip is the main source of the actuator for the whole system. In the analysis, we wipe off the action angle control. As Fig.4 showing, the torque is acting though the whole rear leg stance phase, from the toe touchdown to liftoff, at an initial body status ( x = 1m/s , y = −1m/s , θ = 0.05rad , θ = 0.52rad/s , ϕtd = 0.35rad ). Thus, the influence of β could be ignored. For a SLIP system, the forward velocity decreases first, storing the energy, then, reverts back, releasing the energy. Comparing the forward velocity at the rear leg touchdown and liftoff time, different torque leads the speed to upper or lower than the touchdown one. When hip torque is 25Nm (the markers curve), the forward velocity is equal at touchdown and liftoff time. Fitting the forward velocity at liftoff time to the hip torques, fortunately, the relationship is nearly linearity.
From the running model, it shows that this robot is an under-actuated, highly nonlinear, intermittent dynamic system. And considered the friction constraints and unilateral ground forces will increase the complexity. Thus, the locomotion of such so complex system can not be specified via reference trajectories. So we considered the energy control method described in the next section, which can stabilize periodic motions, resulting in fast running with little task level feedback. And, it also gains some motion 513
Fig. 6. The curves of forward velocity changing with ending impact angle from -10~10˚.
Fig. 4. The curves of forward velocity changing with rear leg torque from 0~50Nm.
Fortunately, the alteration of the ending impact angle has almost no effect on the upward velocity, as Fig.7. That is perfect for simplifying our control method.
Upward velocity y (m/s)
As we found, the forward speed is increasing with the hip torque added. However, the upward velocity would decrease with the torque adding, as Fig.5 showing, and the changing relationship is also linear. Thus, only the torque control can’t effect independently on the forward and upward velocity. We must find another variable to make them self-governed, in order to control the robot status independently.
Fig. 7. The curves of upward velocity changing with ending impact angle from -10~10˚.
C. The Control Method The target of the energy control method is to adjust the forward and upward velocity separately. Base on the analysis above, we can get a linear equation at any body status q to gain a desired velocity:
Fig. 5. The curves of upward velocity changing with rear leg torque from 0~50Nm.
B. The Effect of Ending Impact Angle We detect that the ending impact angle γ end , which is the stop point of the torque action during the rear leg stance phase, can also influence the body status:
⎡ xq ⎤ ⎡ kγx ⎤ ⎡ xd ⎤ ⎡ kτx ⎤ ⎢ y ⎥ = ⎢ y ⎥ τ r + ⎢ ⎥ γ end + ⎢ y ⎥ ⎣ d ⎦ ⎣ kτ ⎦ q ⎣ 0 ⎦q ⎣ q⎦
β = ϕtd − γ end .
where kτx , kτy and kγx are the coefficients of the effects of rear
(3)
(4)
leg torque and ending impact angle; xq and y q is the initial
Actually, the changing of the ending impact angle inflects the action region of the leg torque. The torque is functioning from the touchdown angle till the ending impact angle, and then the hip becomes a passive joint with no driving. Essentially, it is also to control the system energy input. See Fig.6, the forward velocity decreases with the ending impact angle added, at the same initial body status used above. And, the velocity at the liftoff point is nearly linear variation with angle changing.
velocity; xd and y d is the desired velocity. Because these coefficients are uncorrelated, inverse (4), we can get a unique solution of torque and angle input, under a given body status, which lead the system running at an anticipant status. Further more, the body status feedback is not a real time system. We use the previous status gained in flight phase to solve the input values, which is called Delayed Feedback Control (DFC). Thus, for the controller, the previous status is the initial status q 514
< k −1>
. And under this status, we use the desired
τ lr = τ r + τ rc τ rr = τ r − τ rc
velocity as the system input to calculate the output torque and ending impact angle. The equation is: ⎡ τr ⎤ ⎢γ ⎥ ⎣ end ⎦
=K
⎛ ⎡ x ⎤ < d > ⎡ x ⎤ < k −1> ⎞ ⎜⎢ ⎥ − ⎢ ⎥ ⎟ ⎜ ⎣ y ⎦ ⎟ ⎣ y ⎦ ⎝ ⎠
τ rc = kα (α
, −α
< k −1>
) + kdα (α
− α
< k −1>
(6)
)
(5) where the feedback of body yaw angle is also the previous status as in section IV; kα and kdα are the gains of the PD controller. The torque changes of forelegs are the same, only τ fc and gains are not equal.
where K is the coefficient matrix of a fixed initial status; means the desired status and means the number of steps. Using this equation, we can solve the torque and the angle to provide the power that changes the pre-step’s status to the desired one. When the robot starts from rest, the torque input is largest. Till the robot running at a constant speed, the torque varies smaller to afford the energy lost. And the ending impact angle also changes from negative to positive. It is noteworthy that the body status θ is also changing with body velocities. So, the variations of inputs can not be great, in order to ensure the robot will not overturned by the large θ .
VI. PROTOTYPE SIMULATIONS To verify the effectiveness of the control method, the motion of a quadruped robot bounding in the 3D space is simulated. The parameters of the prototype are equal to the real robot, shown in Table II. It is simulating the complex intermittent dynamics of a mechatronic system, which exhibits the friction constraints, unilateral ground forces and other interacts with its environment.
V. TURNING CONTROL
TABLE II
The energy controller could operate the robot running stably in sagittal plane. But on transverse plane locomotion, it has no functions. Thus, we can isolate the turning motion to control it individually on transverse plane. As the Fig.8 shows, we only consider the locomotion in transverse plane.
PARAMETERS OF ROBOT
Parameters
Units
Values
L l0 m I K B
m m kg kgm2 N/m N/m/s
0.21 0.29 12.46 1.296 3600 60
A. Running Simulation In the simulation, the energy control method is implanted to maintain a constant bounding velocity. We get the body status feedback to adjust the hip torques and ending impact angles like section IV discussed. The speed of running is desired to maintain 1m/s. The forward and upward speeds and the rear leg hip torques are shown in Fig.9, which confirms successfully repeating bounding motion. The robot almost runs 7.5m during 2.5~10s. The vertical velocity is varying between -1~1m/s. The period of the bounding gait is 0.26s. As Fig.9 showing, the robot bounding begins from resting state. During 0~2.5s, the robot’s legs are heeled backward, preparing to run. In initiatory cycles, the body velocities and the leg torques undulate acutely. However, after several strides, the robot goes into a stable gait, which the body velocities and the leg torques of stance phase are relatively stable. During this period, the ending impact angle is also changing till stabling at a constant value. When the robot is in stance phase, we control the hip torques, and make the legs sweep a given angle. When the robot is working in flight phase, we use the energy control method to calculate the torques and angles needed during next stride. Before it is touchdown, we fixed the leg relative angle ϕtd = 0.35rad . In the view of Fig.10, we plot the robot body pitch motion on its phase plane. It shows transparently that robot motions
Fig. 8. The turning model of the robot. τ is the hip torque; M represents the virtual torque on robot body; α is the yaw angle between the static coordinate system Ox0y0z0 and the body axes coordinate system Oxyz; l and r are left and right respectively; r and f represent the rear and foreleg.
When it is running straight, the left side torques are equal to the corresponding right side ones. Thus, if the torques are not equal, the robot will turn by the virtual torque M. The M is produced by the imbalance of two side torques. That means if the left toques are greater than right, the robot will turn right and vice versa. But it must be sure that the sum of the rear pair torques and the sum of the fore pair torques are unchanged respectively, in order to keep the forward running. That is because the forward running is controlled by energy method, which is only care of the sum of each pair torques. For example, the changes of rear torque ( τ rc ) is added to one side and reduced on the other side:
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converge to a limit cycle, which indicates stable bounding.
Fig. 9. Stable running graphs. (a) and (b) are the forward and upward speed simulation curves of prototype. (c) is the graph of the rear leg torque during bounding gait
Fig. 11. Turning control simulation graphs. (a) is the curve of body yaw angle. The point in (a) represents the turning point which is the changing of the turning rate and the turning heading. (b) is the graph of robot running path on transverse plane. The point in (b) represents the inflexion of the S-shaped path.
Fig. 10. Phase plot of the body pitch motion limit cycle during running. The dense area of lines in the middle of the picture is the limit cycle of the stable bounding. And the dissociative lines around the cycle represent the starting process.
Fig. 12. Phase plot of the body pitch motion limit cycle during turning. The dense area of lines in the picture is the limit cycle of the stable bounding. And the dissociative lines represent the starting process.
B. Turning Simulation In the turning simulation, we give the turning rate to the controller in order to let the robot turn at a constant angular velocity. The Fig.11a) is the body yaw angle curve during the turning simulation. We control the robot running begin at 2.5s, then turn left from 3s to 5s at the rate of 0.3rad/s. During 5.5s~8.5s, the robot is turned back at the rate of 0.2rad/s. Because the controller uses the feedbacks of previous stride, the turning has some time delay and the slope of the yaw angle has a slight variation. On the transverse plane, the robot runs an S-shaped line as Fig.11b) shows. The curves of Fig.11 confirm that our method could control the robot turning at different curvature radius with one constant forward speed. The Fig.12 is the phase plane diagram of body pitch motion which converges to a limit cycle. That indicates that the robot runs and turns stably. 516
At this point, one must be mentioned that in this simulation we set the robot running at a higher speed, 2m/s. We fixed the leg touchdown angle ϕtd = 0.39rad . Combine the energy control and turning control, we realized the robot running from rest and turning at high speed. Making a comparison of Fig.10 and Fig.12, we could find that the limit cycle in Fig.12 is thicker and more concentrated than Fig.10. That means the faster the robot running the more steady the robot bounding. Although, phase line of the starting process has a larger departure from the limit cycle, it converges more rapidly. And during the simulation, with increasing speed the leg adjustment for stable running becomes less sensitive. This property is also found in Seyfarth’s research on animal and human running criterion [20]. VII. BOUNDING GAIT EXPERIMENT For the sake of realization of robot bounding gait using the
controller above, we developed a sensor and control system for our new robot, as Fig.13 shows. It is equipped with a linear potentiometer per leg for sensing the leg’s length to judge the landing and lifting events. We choose the Crossbow NAV420 as the Attitude & Heading Reference System (AHRS). It is a tri-axis electronic compass combined GPS navigation for sensing the body’s movement status. The control system utilizes ARM7 as the center control processor, and CAN bus for information transfer. Use one dsPIC per motor as a slave processor to control the leg movements.
time. And, there is 1/8 time for fight phase, 1/12 time for double stance phase. We also did the experiment of turning control. Dealing with the attitude data from NAV420, we give a desired body yaw angle curve α = 45 ⋅ [cos(π t / 6) − 1] to form a feedback control. The robot was turning with the forward speed of 0.5m/s. Fig.15 records the variation of body pitch angle and body yaw angle. From 501s to 502s, the robot is under the starting process, and there was not any turning signal input. From 502s, it goes into stable bounding stage, and we began to control the robot turning, as Fig.15b) shows. Comparing these two figures, we can see that both the body pitch angle and the yaw angle have a periodic variation in accordance with the bounding gait cycles. That is because a Delayed Feedback Control is used here, the actual turning angle and the running status feedback have some time delay. The experimental results verify the feasibility and validity of the energy control method.
Fig. 13. Control system.
The running experiments of our new robot were conducted by using the above control method. Fig.14 shows a photographic playback in continual bounding. In the bounding experiment, the sequence of running phases is (1) rear leg stance, (2) double leg stance, (3) foreleg stance, (4) flight, which is just like we hoped.
(1) Rear leg stance
(2) Double leg stance
Fig. 15. Body pitch angle and yaw angle during bounding. The shadow square in (a) is a whole bounding stride. The robot turns right when the yaw angle is negative. In (b) the solid line represents the actual angle, and dashed line represents the desired angle.
(3) Fore leg stance
(4) Flight
VIII. CONCLUSIONS
(5) Rear leg stance
In this paper, we developed a quadruped robot with compliant legs which only one actuator per leg. Based on the study of the dynamics of bounding gait model, we focused on the control of the system energy input. Thus, we proposed a new, simple energy control method to generate a stable bounding gait, which is only changing the primary power source of the whole movement according to a delayed task feedback. Likewise, we present a turning control method, which is only adjusting the hip torques’ balance, to improve the robot’s agility. Under these control method, a running control system using the energy approach is designed to realize bounding gait in robot experiment. The controller
Fig. 14. Bounding experiment.
For the experiment, we controlled the robot bounding at 0.5m/s. The period of bounding is about 0.7s, which is greater than bounding at 1m/s in the simulation, for the speed lower than previous. This tip is just like the animals that the stride frequency decreased with the speed down. And the rear leg stance phase during the experiment takes most time of the whole cycle about 2/3 of one stride, which is conformity with the viewpoint that the rear legs play a leading role during running. By contraries, the foreleg stance phase takes only 1/8 517
realized the bounding running from rest to a constant speed and turning at a high speed, both in prototype simulation and robot experiment. The results show that: 1) the robot’s bounding is a periodic movement; 2) the rear legs play an important role during running; 3) the faster the robot runs the more stable it moves; 4) the stride frequency decrease with the speed down. These results exhibit the bio-mimetism, simplicity, feasibility and validity of the energy approach in quadruped control. It contribute to that complex dynamically dexterous tasks may be controlled via simple energy control method, which is closer to the animal’s actual locomotion conditions. In the future, we plan to modify this method for reducing the energy expending and make the robot running fast.
[15] T. McMahon, “The role of compliance in mammalian running gaits,” J Exp Biol, 1985. 115(1), pp. 263-282. [16] R. Blickhan and R. J. Full, “Similarity in multilegged locomotion: Bouncing like a monopode,” Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioral Physiology, 1993. 173(5), pp. 509-517. [17] I. Poulakakis et al. “On the stable passive dynamics of quadrupedal running” IEEE Int. Conf. on Robotics and Automation (ICRA), Taipei, Taiwan, Sept 2003, pp. 1368-1373. [18] H. Sang-Ho, J. Xin and et al, “Passive running of planar 1/2/4-legged robots,” Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (ICRA), Sendal, Japan, Sept 2004, pp. 3532-3539. [19] L. M. Day and B. C. Jayne, “Interspecific scaling of the morphology and posture of the limbs during the locomotion of cats (Felidae),” Journal of Experimental Biology, 2007. 210(4), pp. 642-654. [20] A. Seyfarth, H. Geyer and et al. “A movement criterion for running,” Journal of Biomechanics, 2002. 35(5), pp. 649-655.
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