Turning of a Legged Robot via a Switching Max-Plus Linear System

Abstract

Legged locomotion is an example of a discrete event system (DES) with only synchronization and no concurrency. This particular kind of DES can be framed in max-plus algebra, which consists of maximization and addition as its basic operations. Utilizing the max-plus operations between matrices, max-plus linear (MPL) system is constructed.

Switching max-plus linear (SMPL) system is a class of MPL system where the element of the system matrices can be changed by a switching function. In modeling legged locomotion as an SMPL system, the touchdown and lift-off time instant of every leg is set as the states. The SMPL system is then used to schedule when each leg should touchdown and lift-off the ground. The swing period of a leg starts with lift-off and ends with touchdown. The stance period of a leg starts with touchdown and ends with lift-off. The SMPL system then incorporates the gait and the desired swing and stance period, to schedule the touchdown and lift-off time instant of every leg.

The legged robot which is considered in is called Zebro. Zebro is a robot with six half-circular legs, developed in TU Delft. In this report, the trajectory-following SMPL system for Zebro is proposed. With the trajectory-following SMPL system, a Zebro is able to move forward in all directions by means of the legs scheduling. To turn, Zebro needs to have different linear velocity on the left and right side. While the Zebro is turning with the trajectory-following SMPL system, the stance period of the inner legs are longer by tp seconds compared to the outer legs. On the other hand, the swing period inner legs are shorter by tp seconds compared the outer legs.

The motion of the Zebro is predicted by the switching kinematic algorithm in this report. Based on the relation between coordinate frames, first Jacobian matrices of the legs are formed. Each jacobian matrix describes the relationship between the motion of a leg and the motion of Zebro. The jacobian matrices are combined in the sensed forward solution, to predict the motion of Zebro if there are several legs on stance period. The switching kinematic algorithm will detect which legs are on the stance period, and the change of angle of the respective legs. The switching kinematic algorithm then forms a suitable sensed forward solution.

The results of the simulation show that the Zebro can turn by the trajectory-following SMPL system. In general, if the tp is increased, the turning radius of Zebro is smaller. However, the Zebro does not always turn to the intended direction. To predict the trajectory of Zebro, a switching kinematic algorithm can be used. The switching kinematic algorithm can predict the trajectory of Zebro if a certain lower boundary of tp is exceeded. The Zebro can also turn to the intended direction if that lower boundary is exceeded. The results of the simulation still cannot be verified by the results of the implementation.