Impulsive Control for Anthropomorphic Biped
Many authors study the dynamics of two-legged walk theoretically. They are, for instance, Beletsky, Berbjuk, Bolotin, Farnsworth, Frank, Gubina, Hemami, Larin, Morecki, Moreinis, Novozhilov, Schiehlen, Seireg, Townsend, Vukobratovich, Zackiorski.
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Impulsive Control for Anthropomorphic Biped
IMPULSIVE CONTROL FOR ANTHROPOMORPHIC BIPED
A.M. Formalsky Moscow State University, Moscow, Russia
Introduction Many authors study the dynamics of two-legged walk theoretically. They are, for instance, Beletsky, Berbjuk, Bolotin, Farnsworth, Frank, Gubina, Hemami, Larin, Morecki, Moreinis, Novozhilov, Schiehlen, seireg, Townsend, Vukobratovich, Zackiorski. Most researchers investigate the dynamics of two-legged walk using the inverse or semi-inverse approach. They prescribe the motion of the biped, fully or partly, and then, using the motion equations, find the forces applied to the biped, evaluate the consumed energy. Several control algorithms for dynamic walk of two- and one-legged mechanisms have been designed and tested experimentally (Kato et aI, Katoh and Mori, Miura and shimoyama, Raibert, Furusho and Mashubushi, Furusho and Sano, Zheng and sias, McGeer, Kajita et al). In the present theoretical investigation direct method of biped locomotion design is used. Biped and its motion equations Hodel description. We study the mathematical model of biped walk in sagittal plane. The investigated model of biped contains the torso and two identical legs. Each leg consists of the thigh and the shin (Fig. 1). All these five links are massive and absolutely rigid. We neglect the friction in the hip and knee joints considering them as ideal. Note, that in human joints the friction is very small. A seven-link model with massless feet added is considered a well. We describe the position of the considered biped mechanism in the plane XY by seven generalized coordinates: x, y, ~, al' a 2 , ~l' ~2 (Fig. 1). A. Morecki et al. (eds.), Theory and Practice of Robots and Manipulators © Springer-Verlag Wien 1995 1
388
A.M. Fonnalsky
Let q1 and q2 be the torques of the forces acting between the trunk and thighs, u 1 and u 2 be the torques of the forces in the knee joints Band D, TIl and TI2 be the torques in the ankle joints A and E, R1 (R Ix , R 1y ) and R2 (R 2x , R 2y ) be the forces applied to the leg tips A and E (Fig. 2). Similarly to the human walk, the walk modeled here consists of alternating phases of single and double support. In the double-support phase both legs are on the bearing surface (on the ground), the forces R1 and R2 are the support reactions, and they are non-zero. During the single-support motion, one of reactions R 1 , R2 is zero.
Equations of planar motion of five-link biped. The equations of the planar motion of the described biped mechanism can be obtained by the second Lagrange method. Omitting the intermediate calculations, we write the equations in the matrix form ·2 B(Z)Z· + gF sinz ( 1) + D(z) z C(z)Q I I where Z
sinz I
= x, y, 1/1,
0, 1,
0:
1'
sinl/l, sino:
1'
0:
2 ' (31 ' (32 sino:
2 '
sin(31 ' sin(32
·2 Z I
Q
Here
the
u 2 , q1' q2' TI 1 , TI 2 , asterisk
denotes
R lx , R ly , R 2x , R 2y
transpose.
The
•
symmetrical,
positive
definite matrix of kinetic energy B(z) is of the size (7x7), matrices F, D(z) and C(z) are of the
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