Exponential Stabilization for Caplygin Systems Based on A Simplified Rank Condition

The paper investigates the global ρ -exponential stabilizability of nonholonomic Caplygin systems. A novel decomposition of state is given first. When systems are linear in certain state variables, a simple and easily verified rank condition can be proposed to guarantee the global ρ -exponential stabilizability of Caplygin systems. In our design, all parameters can be explicitly determined from the constraint function. Moreover, an interesting coordinate transformation can be used to change a Caplygin system into another one so that the proposed criterion can be applied to various situations. For an important class of Caplygin systems, the rank condition is further reduced to some conditions relating to the degree and non-zero property of the lowest polynomials of constraint function. Several interesting examples including of the knife-edge, the extended power-form, the rolling wheel and hopping robot systems are shown that they can be exponentially stabilized by an easy test.