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.