Slow locomotion of the three-link systems along a horizontal plane

We continue to investigate the possibility of slow (quasi-static) locomotion of multi-link systems along a horizontal plane owing to changing their configurations. It has been shown in [Figurina T. Yu, 2001] that the quasi-static motion of a two-link system, occurring when the angle between its links varies, is uncontrollable and that the trajectories of the system's vertices are uniquely defined by the initial position of the system. It has been shown in [Chernousko, F.L., 2001] and [Chernousko, F.L., 2000] that, in contrast, both two-link and three-link systems are controllable (i.e. they can be driven to a prescribed position on the plane), when using both fast and slow motions. We state here quasi-static controllability of three-link systems, both star-like and with links connected in series. We investigate a symmetrical star-like three-link system. We show that there exist much possibilities for the quasi-static motion of the system. One can arrange a motion with the central vertex of the system moving along a prescribed broken line on the plane. As an example, we consider two gaits of the three-link system allowing it to move along a straight line and to rotate on a spot. We also consider a star-link three-link system, which contacts the plane only at the three end vertices. This system is an important special case of the star-like three-link system, since it can move along surfaces. We consider its motion only along a horizontal plane and construct a statically stable gait allowing driving the linkage to a desired position on the plane. At the end, we consider a three-link system with links connected in series and prove its quasi-static controllability.