Enhancement of boundary condition relaxation method for 3D hopping motion planning of biped robots

Boundary condition relaxation (BCR) method proposed by the authors (2005) is enhanced to enable 3D hopping motion plannings from arbitrary initial conditions. The original BCR has an advantage that stepwise legged motion planning is realized in online by accepting an error from the desired goal state of the center of mass (COM) and discontinuity of the trajectory of the zero-moment point (ZMP). The main difficulty of the enhancement lies on that heterogeneous piecewise equations of motions have to be handled at once, and seamless conditioning about the angular-momentum conservation and the body attitude before/after contact phase changing have to be achieved. For the former issue, multiple boundary conditions of piecewise differential equations are set up and solved in the same way with the original BCR. They are based on an approximately mass-concentrated biped model, so that contact state transition and severe time constraints are dealt with at low computational cost. Vertical-horizontal interference of COM trajectory is also taken into account by applying numerical solution of the initial value problem of differential equations. For the latter, a Jacobian-based inverse kinematics with a continuously-varying weight-blending in accordance with the shift of the contact state is presented.