Dynamically Stable Gait Planning for a Humanoid Robot ... - CiteSeerX

Proceedings of the 2004 IEEE Conference on Robotics, Automation and Mechatronics Singapore, 1-3 December, 2004��

Dynamically Stable Gait Planning for a Humanoid Robot to Climb Sloping Surface Changjiu Zhou1, Pik Kong Yue1, Jun Ni1,2, Shan-Ben Chan2 1

School of Electrical and Electronic Engineering Singapore Polytechnic 500 Dover Road, Singapore 139651 {zhoucj, yue}@sp.edu.sg www.robo-erectus.org 2

Institute of Welding Technology Shanghai Jiao Tong University Shanghai 200030, P. R. China

Abstract—In this paper, we formulate gait synthesis of humanoid biped locomotion as an optimization problem with consideration of some constraints, e.g. zero-moment point (ZMP) constraints for dynamically stable locomotion, internal forces constraints for smooth transition, geometric constraints for walking on an uneven floor, e.g. sloping surface and etc. In the frame of gait synthesis tied with constraint functions, computational learning methods can be incorporated to further improve the gait. The effectiveness of the proposed dynamically stable gait planning and learning approach for humanoid walking on both even floor and sloping surface has been successfully tested on our humanoid soccer robots named RoboErectus, which won first place in the RoboCup 2003 Humanoid League Free Performance competition and got 4 silver awards in the RoboCup Humanoid League 2004. Keywords—Biped gait; Stability of walking; Zero Moment Point; Dynamic walk; Humanoid robots; Climbing slope.

I. INTRODUCTION Humanoid soccer robot league is a new international initiative to foster robotics and AI technologies using soccer games [2, 12]. The Humanoid league (HL) has different challenges from other leagues. The main distinction is that the dynamic stability of the robots needs to be well maintained while the robots are walking, running, kicking and performing other tasks on uneven floor, e.g. ascending and descending stairs [5,6], climbing sloping surface [9] and etc. Furthermore, the humanoid soccer robot will have to coordinate perceptions and biped locomotion, and be robust enough to deal with challenges from other players. Hence, how to generate a dynamically stable gait for the humanoid robots is an important research area for the HL, especially for the new technical challenge – Balancing Challenge, in which humanoid robots must be able to climb sloping surface [12].

0-7803-8645-0/04/$20.00 © 2004 IEEE

The problem of gait planning for humanoid robots is fundamentally different from the path planning for traditional fixed-base manipulator arms due to the inherent characteristics of legged locomotion – unilaterality and underactuation. The humanoid locomotion gait planning method can be classified as two main categories: one is online simplified model based gait generation method; and the other is offline position based gait generation method. There are currently some ways for generating dynamically stable gaits, e.g., heuristic research approach, such as genetic algorithms based gait synthesis; problem optimisation method, such as optimal gradient method; model simplification with iteration, and so on. The main challenges of gait planning and learning include the selection of specific initial conditions, constraint functions and their associated gait parameters. However, finding repeatable gait when the constraint equations involve higher order differential equations still remained unsolved. So, a natural way to solve this problem is to look at numerical methods, e.g. Fourier series expansion and time polynomial functions. One advantage of this technique is that extra constraints can be easily included by adding the coefficients to the polynomials. Disadvantages include the facts that the computing load is high for the humanoid with many degrees of freedom (DOF) and the selection of the polynomials may impose undesirable features to the joint profiles, e.g. oscillation. Another problem is that the gait with many constraints may not be a human-like one. To accomplish this, it is quite natural to attempt using the Human Motion Captured Data (HMCD) to drive the robot. However, some researches show that the HMCD cannot be used directly for a humanoid robot due to kinematic and dynamic inconsistencies between the human subject and the humanoid, which usually require kinematic corrections while calculating the joint angle trajectory. Therefore, how to formulate constraints for humanoid gait planning and learning with consideration of perception-based information of human walking is one of the key challenges. Computation intelligence should play an important role in this field.

341

The rest of this paper is organized as follows. We will briefly present some basic constraints for the dynamically stable gait in Section II. The humanoid soccer robot to be used for the experimental study is introduced in Section III. We demonstrate how to plan a dynamically stable gait for ascending a sloping surface in Section IV. Experimental and simulation results will be elaborated in Section V. Section VI will give concluding remarks and some major technical challenges in this field. II. DYNAMICALLY STABLE GAIT AND CONSTRAINTS Since a humanoid robot tips over easily, it is important to consider stability during planning its gait. Many methods have been proposed for synthesizing walking patterns based on the concept of the zero moment point (ZMP) [1,7,8]. The ZMP is defined as the point on the ground about which the sum of the moments of all the active forces equals zero. If the ZMP is within the convex hull (support polygon) of all contact points between the feet and the ground, the humanoid robot is able to walk dynamically. The humanoid robot is a highly redundant system with many extra DOF. Its gait consists of large number of unknown parameters. This allows us to formulate constraint equations for synthesizing gait. In this paper we formulate an optimization problem to determine the unknown parameters of the gait to achieve dynamic locomotion, i.e. to obtain a good match between the actual and the desired ZMP trajectories as follows.

tf

2

d Pzmp (t ) − Pzmp (t ) dt

Minimize

ti

(1)

subject to the boundary conditions of both p(t ) and p(t ) d at time t i and t f , where Pzmp is the actual ZMP, and Pzmp is the desired ZMP position. Due to the large number of the unknown parameters for the above optimization problem and some requirements of human-like of dynamic walking, we need to specify some constraints. The following are some constraints need to be considered. Stabilization of biped gait (ZMP constraint): The control objective of the humanoid dynamically stable gait can be described as

(

)

Pzmp = x zmp , y zmp ,0 ∈ S

(2)

where ( x zmp , y zmp ,0 ) is the coordinate of the ZMP with respect to O-XYZ. S is the support polygon. Smooth transition constraint: The equation of motion of the centre of the humanoid can be described as mcm acm = f L + f R + mcm g

(3)

Where mcm and acm are the mass of the robot and the acceleration of the COM, respectively. f R and f L represent the ground reaction forces at the right and left foot. During single-support phase, the foot force can be obtained from (3)

as one of f R and f L will be zero. However, during doublesupport phase, only the total of the ground reaction forces is known. Hence, how to resolve the leg reaction forces appropriately to minimize the internal force has to be considered. This can ensure a smooth transition of the internal forces during placement and take-off. Geometrical constraints: The swing limb has to be lifted off the ground at the beginning of the step cycle and has to be landed back at the end of it. Maximum clearance of the swing limb: During swing phase, the foot of the swing limb has to stay clear off the ground to avoid accidental contact. Repeatability of the gait: The requirement for the repeatable gait demands that the initial posture and velocities be identical to those at the end of the step. Continuity of the gait: The horizontal displacements of the hip during the single and double support phases must be continuous. Minimization of the effect of impact: During locomotion, when the swing limb contacts the ground (heel strike), impact occurs, which contacts sudden changes in the joint angular velocities. By keeping the velocity of the foot of the swing limb zero before impact, the sudden jump in the joint angular velocities can be eliminated. III. ROBO-ERECTUS: A FULLY AUTONOMOUS HUMANOID ROBOTS The Robo-Erectus (RE) project (www.robo-erectus.org) aims to develop a low-cost fully-autonomous humanoid platform [11]. We have developed three generations humanoid soccer robots, namely RE40I, RE40II and RE40III. The CAD/CAM Design of the latest model of Robo-Erectus is shown in Fig. 1. The humanoid robot named RE40III, which won second place in Penalty Kick H40 competition is shown in Fig. 2. Our RE humanoid soccer robot has participated in Humanoid League of the RoboCup since 2002, won 2nd place in the Humanoid Walk competition at the RoboCup 2002 and got 1st place in the Humanoid Free Performance competition at the RoboCup 2003, and won 4 second positions in Humanoid Walk, Humanoid Free Performance, Humanoid Penalty Kick H40 and Humanoid Penalty Kick H80 respectively at RoboCup 2004. The configuration of the hierarchical control system for the RE humanoid is given in Fig. 3. We have also implemented fuzzy reinforcement learning to further improve the biped gait [10]. All Robo-Erectus models can be controlled using 3 platforms, namely • PC-based control system • PDA-based control system (Autonomous) • Microcontroller-based control system (Autonomous) Each of the control platforms has its own merits. The PCbased control system is useful for tuning the gait movement of the robot. As the PC-based system is developed using Microsoft Visual Studio, debugging and modification are easy. In addition, the data of each joint can be monitored and analyzed in real time. Microcontroller-base control system is

342

light and can be carried by the robot itself. It is the best control system to implement autonomous control. The PDA-based control system is autonomous and user-friendly. It is easier to reconfigure the PDA-based control system than the microcontroller system. However, PDA-based control system still has its limitation when it comes to electronic interfacing and real-time control.

Figure 1. CAD/CAM design of the latest model of Robo-Erectus

IV. GAIT PLANNING FOR WALKING ALONG SLOPE A. Walking Cycle for a Biped Robot A complete slope walking cycle of biped robots includes two walking phases: a single-support phase and a doublesupport phase. In the single-support phase, one foot stays stationary on the slope while the other foot swings from the rear to the front. At the same time, the hip moves along a relevant trajectory to keep the synchronization of the gait. The double-support phase starts from the forward foot touching the slope and ends with the rear foot leaving the slope. During the double-support phase, the center of gravity (COG) or zero moment point (ZMP) moves from one foot to another foot.

Fig. 4 shows a slope walking cycle starting from the kth step, where Tc is the period for one walking step, Td is the interval of the double-support phase, and k* Tc + Tm is the time when the swing foot reaches its highest point from the slope. To plan a gait for a biped robot to walk along slope, both foot and hip trajectories are generated, and then all joint trajectories are derived by inverse kinematics [3,4]. In this research, we assume that the kth step begins with the heel of the right foot leaving the slop at t = k* Tc , and ends with the heel of the right foot making the first contact with the slope at t = (k+1)* Tc , as shown in Fig. 4.

Figure 2. The newly developed Robo-Erectus RE40III

Figure 4. Walking cycle along slope Path Planning

Gait Synthesizer

Global motion planning

Local motion planning

Joint Controller

Humanoid Robot

Encoders

Range sensors/ gyros/force sensors,...

Vision

Figure 3. The hierarchical control system for RE humanoid robos

B. Foot Trajectories Assuming that Qs is the slope angle, Qb and Q f are the angles of right foot as it leaves and lands on the slope, θ a (t ) denotes the angle of right foot, ( Lao , H ao ) is the position of the best highest point of the swing foot, Ds is the length of one step along slope, L is the distance from the ankle joint of right foot to the origin of the coordinates when robot begins its first step, Lan is the height of the foot, Laf is the length from the ankle joint to the toe, Lab is the length from the ankle joint to the heel, as shown in Fig. 5. ( xa (t ), z a (t ) ) is the coordinate of the ankle position in sagittal plane, the following constraints can be derived.

343

H min + xh (t ) * tgQs , t = k * Tc + 0.5* Td zh (t ) = H max + xh (t ) * tgQs , t = k * Tc + 0.5*(Tc − Td )

H min + xh (t ) * tgQs ,

(9)

t = (k + 1) * Tc + 0.5* Td

We can obtain smooth hip trajectories which satisfy constraints (8) and (9) and some initial and final constraints, using the third-order spline interpolation. V. EXPERIMENTAL RESULTS We have developed a simulation software platform as shown in Fig. 6 to aid the theoretical study of humanoid gait planning. The simulation software is highly graphical. The real-time control is incorporated and the system is able to connect to an actual humanoid robot for testing purpose. Both simulation and experimental studies have been conducted. Fig. 7 shows the stick diagram of humanoid walking on the sloping surface and the experiment setup. It also shows the consecutive walking gait of a biped robot along slope. The whole motion of the biped robot during the single and double support phases can be observed.

Figure 5. Biped robot parameters and coordinate

θ a (t ) =

Qs , Qs − Qb ,

t = k * Tc t = k * Tc + Td

(4)

Qs + Q f , t = (k + 1) * Tc Qs , t = (k + 1) * Tc + Td

θ a (k * Tc ) = 0

(5)

θ a ((k + 1) * Tc + Td ) = 0 xa (k * Tc ) = 0

(6)

xa ((k + 1) * Tc + Td ) = 0 za (k * Tc ) = 0 za ((k + 1) * Tc + Td ) = 0

(7)

From the above via points and initial and final constraints (4)-(7), A1 and A2, smooth foot trajectories can be generated using the third-order spline interpolation, which can guarantee the continuity of the first and the second derivatives. C. Hip Trajectories Assuming that ( xh (t ), zh (t ) ) is the coordinate of the position, xsd and xed are the distances from the hip to ankle joint of the support foot at the start and end of single-support phase respectively, as shown in Fig. 4, following constraints can be derived. L *cos Qs + k * Ds *cos Qs + xed , t = k * Tc xh (t ) = L *cos Qs + (k + 1) * Ds *cos Qs − xsd , t = k * Tc + Td

hip the the the

(8)

L *cos Qs + (k + 1) * Ds *cos Qs + xed , t = (k + 1) * Tc

Assuming that the hip is at the highest point H max from the slope in the middle of the single-support phase, at the lowest point H min from the slope in the middle of the double-support phase, then the following constraints should be considered.

Fig. 8 shows the horizontal displacements of the hip and left ankle and right ankle versus time. Fig. 9 shows the trajectories of the left ankle and the right ankle along the slope which is 10 degrees. It can be seen that all the trajectories are smooth and continuous, i.e. all the velocities are continuous. In Fig. 10, the profiles of both angles of knee joint and ankle joint are given. The angular velocities during the single and double support phases are also shown in Fig. 11. It can be seen that both joint angles and their angular velocities are continuous. However, the angular velocities have some obvious changes when the right foot suddenly touches the ground from single support phase to double support phase. The horizontal displacements of the COG and ZMP during both single and double support phases are illustrated in Fig.12. It can be seen that both COG and ZMP remain in the stable boundary, which ensures stability for both static and dynamic walk when the robot climbs along the slope. In addition, we also make some comparisons on ZMP and the angular velocities of ankle and knee when the slope angle changes. In this experiment, other parameters remain unchanged. Fig. 13 shows the angular velocities of the ankle in different slope angles. Fig. 14 shows the angular velocities of the knee in different slope angles. It can be seen that angular velocities have some obvious changes when the foot transfers from double support phase to single support phase and between the start of single support phase and the time the swing foot achieves the highest point in different slope angles. Fig. 15 shows the horizontal displacements of ZMP in different slope angles. It can be seen that when the slope angle is increased, ZMP fluctuates greatly during the double support phase and the initial part of the single support phase. Once the slope angle exceeds the limit, ZMP will lie out of the stable boundary and the dynamically stable gait cannot achieve while the robot walks along slope.

344

Figure 10. Joint angles of both ankles and knees

Figure 6. Biped robot parameters and coordinate

Figure 11. Angular velocities of both ankles and knees Figure 7. Biped robot parameters and coordinate

Figure 12. Horizontal displacements of COG and ZMP Figure 8. Horizontal displacements of the hip and both ankles

Figure 13. Angular velocities of the ankle in different slope angles Figure 9. Trajectories of both ankles in the sagittal plane

345

ACKNOWLEDGMENT We would like to thank staff and students at the Advanced Robotics and Intelligent Control Center (ARICC) of Singapore Polytechnic for their support in the development of humanoid robots named Robo-Erectus. The research described in this paper was made possible by the jointly support of the Singapore Tote Fund and the Singapore Polytechnic R&D Fund. REFERENCES Figure 14. Angular velocities of the knee in different slope angles

[1]

[2]

[3]

[4] [5] [6] Figure 15. Horizontal displacements of ZMP in different slope angles [7]

VI. CONCLUDING REMARKS In this paper, we formulate gait synthesis of humanoid biped locomotion as an optimization problem with constraints. The effectiveness of the proposed dynamically stable gait planning approach for humanoid walking on a sloping surface has been successfully tested on our humanoid soccer robots named Robo-Erectus. It has demonstrated that the proposed gait planning method can achieve dynamically stable biped locomotion when the robot climbs sloping surface. The future work will be to use computational learning approach to further tune humanoid walking behavior to achieve more stable dynamic gait for humanoids walking in uneven surfaces.

[8]

[9]

[10]

[11]

[12]

xa (t) =

A. Goswami, “Postural stability of biped robots and the foot-rotation indicator (FRI) point,” Int. J. of Robotics Research, Vol.18, No.6, pp.523-533, 1999. H. Kitano and M. Asada, “The RoboCup humanoid challenge as the millennium challenge for advanced robotics,” Advanced Robotics, Vol.13, No.8, pp. 723-736, 2000. Q. Huang, K. Yokoi, S. Kajita, K. Kaneko, H. Arai, N. Koyachi, and K. Tanie, “Planning Walking Patterns for a Biped Robot,” IEEE Trans. Robot. Automat, vol. 17, pp.280-289, 2001. X. Mu and Q. Wu, “Synthesis of a complete sagittal gait cycle for a five-link biped robot,” Robotica, Vol. 21, pp.81-587, 2003. C.-L. Shih, “Gait synthesis for a biped robot”, Robotica, Vol. 15, pp. 599-607, 1997. C.-L Shih, “Ascending and Descending Stairs for a Biped Robot,” IEEE Trans. on Systems, Man, and Cybernetics- Part A: Systems and Humans, Vol. 29, pp. 255-268, 1999. M. Vukobratovic, B. Borovac, D. Surla and D. Stokic, Biped Locomotion: Dynamics, Stability, Control and Application, SpringerVerlag, 1990. M. Vukobratovic and B. Borovac, “Zero-moment point – thirty five years of its life,” Int. J. of Humanoid Robotics, Vol.1, No.1, pp. 157-173, 2004. Y.-F. Zheng and J. Shen, “Gait synthesis for the SD-2 biped robot to climb sloped surface,” IEEE Trans. on Robotics and Automation, Vol. 6, No. 1, pp. 86-96, 1990. C. Zhou and Q. Meng, “Dynamic balance of a biped robot using fuzzy reinforcement learning agents,” Fuzzy Sets and Systems, Vol. 134, No.1, pp.169-187, 2003. C. Zhou and P.K. Yue, “Robo-Erectus: a low cost autonomous humanoid soccer robot,” Advanced Robotics, Vol.18, No.7, pp. 717-720, 2004. C. Zhou, “Rules for the RoboCup humanoid league,” Advanced Robotics, Vol 18, No 7, pp. 721-724, 2004.

L*cos Qs + k * Ds *cos Qs + Lan *sin Qs , L *cos Qs + k * Ds *cos Qs + Lan *sin(Qs − Qb ) + Laf *(cos Qs − cos(Qs − Qb )),

t = k *Tc t = k *Tc + Td

L*cos Qs + k * Ds *cos Qs + Lao *cos Qs ,

t = k *Tc + Tm

(A1)

t = (k +1)*Tc L*cos Qs + (k + 2)* Ds *cos Qs − Lan *sin(Qs + Qf ) − Lab *(cos Qs − cos(Qs − Qb )), L *cos Qs + (k + 2)* Ds *cos Qs − Lan *sin Qs , t = (k +1)*Tc +Td

za (t ) =

Lan *cos Qs − Laf *sin Qs + ( Lan *sin Qs + Laf *cos Qs ) * tgQs + xa (t ) * tgQs ,

t = k * Tc

Lan *cos(Qs − Qb ) − Laf *sin(Qs − Qb ) + ( Lan *sin(Qs − Qb ) + Laf *cos(Qs − Qb )) * tgQs + xa (t ) * tgQs , H ao + xa (t ) * tgQs ,

t = k * Tc + Td t = k * Tc + Tm

Lan *cos(Qs + Q f ) + Lab * sin(Qs + Q f ) + ( Lan *sin(Qs + Q f ) − Lab *cos(Qs + Q f )) * tgQs + xa (t ) * tgQs ,

t = (k + 1) * Tc

Lan *cos Qs + Lab *sin Qs + ( Lan *sin Qs − Lab *cos Qs ) * tgQs + xa (t ) * tgQs ,

t = (k + 1) * Tc + Td

346

(A2)