By Armando Carlos de Pina Filho

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2437–2442. , Inaba, M. & Inoue, H. (2002). Online generation of humanoid walking motion based on a fast generation method of motion pattern that follows desired ZMP, IEEE/RSJ 2002 International Conference on Intelligent Robots and System 3. , Dip, G. & Meng-Hwee, C. (2007). Disturbance rejection by online ZMP compensation, Robotica 26(1): 9–17. , Nakamura, Y. & Inoue, H. (2002). Real-time humanoid motion generation through ZMP manipulation based on inverted pendulum control, Proceedings of 2002 IEEE International Conference on Robotics and Automation 2.

The following assumptions are made in deriving the equations of motion. • The biped is modeled only in the lateral plane; the motion in the longitudinal plane is neglected. • The impact of the swing leg at the end of each step is infinitesimally small. • The impact of the swing leg is plastic. , the swing leg becomes the stance leg, and vice versa. • The angular momentum is conserved during impact. With these assumptions, the governing equation of the biped can be derived for the swing phase as follows.

Calculate PZ MP by using (11). θ LA − θ RA . 2 (26) 24 Biped Robots 5. Compute the generalized force τφ according to (13). 6. Distribute the generalized force τφ to each joint torque τ so as to satisfy (14). Namely, T = (J T (Θ ))∗ τφ + ( I − J T (Θ )(J T (Θ ))∗ )p. (27) Here, ∗ denotes the generalized inverse matrix, and p is an arbitrary 4-dimensional vector. 2 for the calculation of J(Θ ). 7. Output the joint torque by using the actuators. 4 Switching from double- to single-support phase According to control law (13), the ZMP position is shifted to the side of the next supporting leg by following Pd .

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