Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 557from rest, one can reach any configuration from a given initial configuration.However, as a traditional control system, it is not controllablebecause of conservation of angular momentum. If one asks for the states(i.e., configurations and velocities) reachable from configurations with zeroinitial velocity, one finds that not all states are reachable. This is a consequenceof the fact that angular momentum is conserved, even with inputs.Thus if one starts with zero momentum, the momentum will remain zero(this is what enables one to treat the system as nonholonomic). Nevertheless,all configurations are accessible. This suggests that the question ofcontrollability is different depending on whether one is interested in configurationsor states. We will be mainly interested in reachable configurations.Considering the system with just one of the two possible inputforces is also interesting. In the case where we are just allowed to useF 2 , the possible motions are quite simple; one can only move the ball onthe leg back and forth. With just the force F 1 available, things are abit more complicated. But, for example, one can still say that no matterhow you apply the force, the ball with never move ‘inwards’ [Lewis (1995);Lewis and Murray (1997)].Fig. 4.5A simple robotic leg (see text for explanation).In general, simple mechanical control systems are characterized by:• An nD configuration manifold M;• A Riemannian metric g on M;• A potential energy function V on M; and• m linearly independent 1−forms, F 1 , ..., F m on M (input forces; e.g.,in the case of the simple robotic leg, F 1 = dθ − dψ and F 2 = dr).When we say these systems are not amenable to liberalization–basedmethods, we mean that their liberalizations at zero velocity are not controllable,and that they are not feedback linearizable. This makes simple