658 IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 3, JUNE 2009 Boundary Condition Relaxation Method for Stepwise Pedipulation Planning of Biped Robots Tomomichi Sugihara, Member, IEEE, and Yoshihiko Nakamura, Member, IEEE Abstract—A completely stepwise online pedipulation planning method is proposed. It is an analytical approach based on the general solution of the equation of motion of an approximate dy- namical biped model whose mass is concentrated at the center of mass. A physically feasible referential trajectory with a constraint about the reaction force taken into account is planned only in one interval by relaxing the boundary condition, namely, by admitting a certain level of error between the desired and actually reached states, and discontinuity of zero-moment point at each end of the in- terval. It potentially creates responsive motions that require strong instantaneous acceleration. A semiautomatic continual pedipula- tion planning method is also presented. It generates a referential trajectory of the whole body only from the next desired foot place- ment. The validity of the proposed method is ensured through experiments with a small anthropomorphic robot. Index Terms—Biped robot, legged motion, online motion planning. I. INTRODUCTION E FFICACY of a so-called pattern-based approach for the bipedal motion control, in which robots move in the envi- ronment with a number of degrees of freedom cooperating, has been proven by many previous researches [1]–[7]. It simplifies the problem by breaking it into two subproblems, namely, biped motion planning and feedback control to stabilize the robot in real time. For high-level task executions, path planning is the central issue to avoid collisions in complex environments [8]. Then, a trajectory to track the path is computed as a function of time under dynamical constraints by solving a two-point boundary value problem. In the case of common manipulator controls, jerk minimization, torque rate minimization, etc., work as loose dynamical constraints. In the case of legged robot, however, there works a strong constraint in which no attractive force is available at any contact point with the environment since the robot lacks mechanical connection to the inertial frame. Due to this property, geometry and dynamics are closely linked with each other. If the trajectory is carelessly designed only to satisfy the boundary condition, it does not guarantee physical Manuscript received July 23, 2008; revised November 4, 2008. First pub- lished February 10, 2009; current version published June 5, 2009. This paper was recommended for publication by Associate Editor T. Kanda and Editor W. K. Chung upon evaluation of the reviewers’ comments. This work was supported in part by Category “S” 15100002 of Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, and by “The Kyushu University Research Superstar Program (SSP),” based on the budget of Kyushu University allocated under President’s initiative. T. Sugihara is with the School of Information Science and Electrical Engineer- ing, Kyushu University, Fukuoka 819-0395, Japan (e-mail: zhidao@ieee.org). Y. Nakamura is with the Department of Mechano-Informatics, University of Tokyo, Tokyo 113-8656, Japan (e-mail: nakamura@ynl.t.u-tokyo.ac.jp). Digital Object Identifier 10.1109/TRO.2008.2012336 Fig. 1. Approaches to solve two-point boundary value problem with constraint on the input. (a) If the trajectory is carelessly designed only from the boundary condition, the constraint on the input is not necessarily satisfied. (b) Conven- tional biped motion planning methods assumed the input trajectory is given a priori and sacrificed velocity continuity at both ends. (c) Boundary condition relaxation method strictly satisfies the position–velocity continuity at the initial state. The goal state moderately satisfies the desired condition. Simultaneously, the input is designed to satisfy the constraint. consistency, as shown in Fig. 1(a). This is the primary difficulty of the legged motion planning. Vukobratovi´ c and Juriˇ ci´ c [9] proposed a shuffling motion planning method in which the upper body (a counterweight lever) movement is iteratively computed for given trajectories of the lower extremities in accordance with a criterion that the point of action of the resultant external force (zero-moment point (ZMP) [10]) stays at the tip of the posterior leg. It is an approach in which the ZMP trajectory to satisfy the reaction force constraint is given a priori, and the whole body motion is planned by sacrificing the boundary condition about velocity, as shown in Fig. 1(b). It was augmented by Takanishi et al. [11], Takanishi et al. [1], Nagasaka [3], and Kagami et al. [5] into more generalized forms, and some online implementations of them [12], [13] have been proposed. However, unsatisfied veloc- ity continuity at seams of trajectories limits their applicability to conservative motion. Kajita et al. [7] applied the preview con- trol to an online center-of-mass (COM) trajectory planner with given referential ZMP trajectory in certain length of future du- ration. Since it is optimal regulator by nature, COM necessarily reaches the stationary state above ZMP. A common problem of those methods is that designing the referential ZMP trajectory for tasks to be achieved is not trivial. Another approach is to solve the problem analytically with the piecewise equation of motion of an approximate robot dy- namics model whose mass is concentrated at COM, and a ZMP trajectory function that includes unknown parameters [4], [14], [15]–[17]. Its advantages are that it can deal with velocity con- tinuity, the computational amount and accuracy do not depend on the quantized time step, and the computational cost is less without the necessity of iteration. In the previous methods, the ZMP trajectory function was designed as a zero-order function (constant value) [4], [14], first-order function [16], second-order 1552-3098/$25.00 © 2009 IEEE Authorized licensed use limited to: Central Library M080. Downloaded on June 12, 2009 at 10:36 from IEEE Xplore. Restrictions apply.