Enhancement of Boundary Condition Relaxation Method for D Hopping Motion Planning of Biped Robots University oftokyo Tomomichi Sugihara and Yoshihiko Nakamura Abstract Boundary Condition Relaxation method[], which realizes stepwise legged motion planning by accepting an error from the desired goal state of the center of mass and discontinuity of the zero-moment point, is enhanced to enable hopping motion planning from arbitrary initial conditions. The main issues of the enhancement are how to utilize intermittent reaction force by solving piecewise equation of motion of the mass-concentrated model, and how to condition the angular-momentum conservation and body attitude control seamlessly by a variable weighted inverse kinematics. Key Words: Biped robot, Hopping motion, Legged motion planning, Weighted inverse kinematics. Raibert [] Hyon [] [] [] [] [] Nagasaka [] [9] [] [] []. m(p + g) =f () m p =[xyz] T g =[g] T f =[f x f y f z ] T
z supporting phase aerial phase supporting phase p G z z T z z quardic curve parabolic curve quardic curve f, n (A)Mass distributed model f, n f p Z (B)Mass concentrated model Fig. Mass distributed VS mass concentrated model Fig. B (p p Z ) f = () p Z =[x Z y Z z Z ] T ZMP[] z Z p Z S(t) () S(t) z = z Z Eq. x =! (x x Z ) () y =! (y y Z ) () z = f z m g ()! r z + g z z Z : () Eq. x Z y Z f z p Eq. m(p + g) = () Eq. t T p() = [x y z ] T _p() = [v x v y v z ] T p(t ) = [x T y T z T ] T _p(t )=[v xt v yt v zt ] T p(t) x Z (t) y Z (t) f z (t) T T T t lift-off touch-down Fig. Phase transition in hopping motion Eq. =tt Fig. T T T T T T T z(t )=z z(t )=z T t T z(t) z(t) = g(t T )(t T )+ z (t T ) z (t T ) T T (9) _z(t) =gt + g(t + T )+ z z T T () T T _z(t )=v z g(t T )+ z z T T () _z(t )=v z g(t T )+ z z T T () z(t) =g t T z() = z _z() = v z z(t )= z _z(t )=v z z(t )=g T t T z(t )=z _z(t )=v z z(t )=g z(t )=z T _z(t )=v zt z(t) Eq.! Eq. t T T t T!!() =!!(T )=! T x Z (t) y Z (t) Eq. Eq. x
ZMP x Z (t) = x Z (x Z x Z ) e!t ( t T ) ( ) (T <t<t ) x ZT (x ZT x Z ) e! T (tt ) (T t T ) () > = Eq. x(t) = C e!t + C e!t + x Z x Z e!t + x Z ( t T ) x + v (t T ) (T <t<t ) C e! T (tt ) + C e! T (tt ) + x ZT x Z e! T (tt ) + x ZT (T t T ) () C C x v C C _x(t) =! C e!t C e!t x Z x Z e!t ( t T ) v (T <t<t )! T C e! T (tt ) C e! T (tt ) x ZT x Z e! T (tt ) (T t T ) () C C O O C C O O O O C C O O C C, C C C B! ; C c x Z c x Z = + ; C x B x B x () x T + ; C ; B + T T! T ; e!t ; e! T (TT ) C c C x x v x! ; C c ; x Z x Z C ; x x v x Z ; x T x T v xt! T ; x Z x Z x ZT x x T x Eq. ZMP Eq. Eq. x x T d x T x Z x Z Eq. d x Z d x Z x Z x Z ZMP x c c Eq. ~C ~C = + ~x T () = ~ C ( + ~xt ) ()! (T T ) +! T + C C x Z x v C C x ZT ; x ( +) d x Z v + d x Z! ( +) d x Z d x Z d x Z d x Z d x Z d x Z ; ~x T x T Eq. x Z x ZT x Z = D + D x T () x ZT ( d ) T Q ( d )! minimum subject to D = s (QP)
[x Z x ZT x T v T =! T ] T d d x Z d x ZT d x T d v T =! T T w supporting phase aerial phase supporting phase w ml A Q diagfq i g (i = ; q i > ) D h D i ; s D (QP) = d QD T (DQD T ) (D d s) (9) c c x d x Z d x ZT t = t = T q q Eq. T T ZMP x z (t)! ZMP ZMP ZMP x z (t) z(t) Eq. Runge-Kutta. p U p U = J U () J = () J w Ω T T T t Fig. Variable weight in accordance with contact phase t = T T J A = d L () d L J A J A NX 9 mi (p i p) J Gi + R i i I i i J i i= () N m i i p i i J Gi i R i i i I i i i i J i i i Fig. w w A w = w A = ml T t ( t T ) T (T t T ) t T (T t T ) T T t ( t T ) T (T t T ) T t (T t T ) T T () () l ml l [] Newton=Raphson
CPU:Intel PentiumM GHz RAM:MB [s]. Name: UT-:magnum height: [mm] weight:. [kg] Number of joints: ( for arms, for legs) Fig. External view and specications of the robot. Fig. [ : ] T [ : ] T [ :] T T =: T = : T = : z = :9 z = : z T = : x Z = x Z = x Z =: x Z =: =: d x T =: h K =: QP Q diagf; ; :; :g Fig.(A) ZMP x (B) z (C) z Runge-Kutta-Gill x [mm] Fig.(D) ZMP (C) (E) Fig. i) ii) ZMP iii) iv), (S) ( ) [],.., pp. {,. [] Marc H. Raibert, H. Benjamin Brown Jr., and Michael Chepponis. Experiments in Balance with a D One-Legged Hopping Machine. The International Journal of Robotics Research, Vol., No., pp. {9, 9. [] S. H. Hyon and T. Mita. Development ofabiologically Inspired Hopping Robot { \Kenken. In Proceedings of the IEEE International Conference on Robotics & Automation, pp. 9{ 99,. [],.., pp. {, 999. [],,,,.. 9, pp. {,. [],,,.., Vol., No., pp. {,.
9 Fig. A hopping-forward motion by a miniature humanoid robot [m]. [m]... original referential COM position -. modified referential COM position referential feet position referential ZMP position -..... [s] (A) COG, feet and ZMP in x-axis [m]...... [m/s ] vertical position of COM vertical position of feet.... [s] (B) COG and feet in z-axis - - vertical acceleration of COM -.... [s] [deg] (C) COG acceleration in z-axis -. COM locus referential ZMP ZMP locus.... [s] (D) ZMP locus by inverse dynamics commanded trunk pitch angle referential trunk pitch angle.... [s] (E) pitch angle of trunk Fig. Planned trajectory of COG, foot and ZMP [].. PhD thesis,,. [] Ken'ichiro Nagasaka, Yoshihiro Kuroki, Shin'ya Suzuki, Yoshihiro Itoh, and Jin'ichi Yamaguchi. Integrated Motion Control for Walking, Jumping andrunningonasmallbipedalentertainment Robot. In Proceedings of the IEEE International Conference onrobotics and Automation, pp. 9{9,. [9] Tomomichi Sugihara and Yoshihiko Nakamura. Contact Phase Invariant Control for Humanoid Robot based on Variable Impedant Inverted Pendulum Model. In Proceedings of the IEEE International Conference on Robotics & Automation, pp. {,. [],,.. IFToMM, pp. {,. [],.., Vol., No., pp. {9, 9. [] M. Vukobratovic and J. Stepanenko. On the Stability of Anthropomorphic Systems. Mathematical Biosciences, Vol., No., pp. {, 9.