Ballistic Walking Design via Impulsive Control

Abstract

Mathematical models of a biped and quadruped walking are considered. The planar five-link biped consists of a one-link trunk and two identical two-link legs. In the single support motion swing phase, the biped has five degrees of freedom and is described by a system of nonlinear ordinary differential equations of the 10th order. These equations are written in a visible matrix form. A seven-link planar biped with massless feet is also considered. In this paper, the swing motion of the biped is assumed a ballistic passive one. There are no active torques in the interlink joints during the single support motion—only the gravity force and ground reaction forces are applied to the biped. The problem of design of ballistic swing motion is reduced to the boundary-value problem for the system of nonlinear differential equations with given initial and final configurations and the duration of the half-step. It is assumed that there is no friction in the interlink joints note that the friction in the human joints is very small. Therefore, in the ballistic swing motion the complete energy of the system kinetic energy plus potential one is conserved and the system has the energy integral. Due to this fact some properties of symmetry of ballistic motions are proved. Linearized model can be reduced to the canonical Jordan form. Then the linear boundary-value problem can be solved analytically. Using numerical investigations of the linear model we have animated the biped walking, which occurs similar to the human gait: the transferring leg moves over the support, the legs bend with knees forward, and the trunk makes one vibration during one half-step. All these features have not been prescribed beforehand in the statement of the problem. Iteration process is used to solve the nonlinear boundary-value problem. For some cases, several solutions of this nonlinear problem are found numerically. The symmetry properties of the ballistic motions help to find numerically the solutions of this complex nonlinear problem. The ballistic motion is also designed numerically for the three-dimensional biped model with 6, 8, 9, and finally with 11 degrees of freedom. The double-support phase is assumed an instantaneous one. During this phase there is a collision of the transferring swing leg and the support. Active impulsive torques are applied in the interlink joints at this instant. These impulsive torques and ground reaction forces are described by delta functions of Dirac. Thus, with this impulsive control, most of the efforts are applied in the double-support phase. Formulas for the energy cost of impulsive control actions have been found. Some problems of optimal distribution of the impulsive actions between the joints are discussed. A planar seven-link model of biped with arms is considered. This model consists of a one-link trunk, two identical one-link arms, and two identical two-link legs with point feet. Ballistic gait of this biped model is studied. The goal of this study is to find the optimal amplitude of the arms swinging, minimizing the energy consumption. We show numerically the existence of the optimal amplitude of the arms swinging. Ballistic walking of a quadruped is investigated too. Three types of quadruped gaits, bound, amble, and trot, are considered. For each kind of the gait, ballistic locomotion is designed and energy consumption is evaluated.