Stabilization Study On A Class Of Nonholonomic Mechanical Control Systems

The nonholonomic systems are the nonlinear systems with nonholonomic constraints, which represent a large class of mechanical systems and have very broad applications in real life, such as pure rolling bail without sliding movement, ice skates sliding on the ice, flexible robots, satellites running in space and space shuttles. These systems have a common characteristic: the constraint equations of such systems include a nonintegrable constraint equations with a differential ferm at least.Because of the presence of nonholonomic constraints, such systems don’t satisfy the Brockett necessarv conditions of stabilization, i.e. there exists no smooth time-invariant state feedback controller, that stabilize or asymptotically stabilize the system on the equilibrium point. Therefore, to carry out a complete system stabilization control study has important theoretical value and practical application prospects.During the last two decades, the stabilization problem of the nonholonomic systems have received more researchs and a series of fruitful results. But most of these studies are carrried on the frist-order nonholonomic constraint systems, and less on the second-order nonholonomic constraint systems.The main work of this paper is on the stabilization of the second-order chained systems as follows. where x=[x1, x2,?·?, xn]T is states of the system, u=[u1, u2]T is control input.Firstly, the background knowledge and the research of the nonholonomic systems are introduced,especially for the second-order chained systems.Secondly, smooth time-varying exponential control law is proposed to stabilize globally a class of nonholonomic systems that are transformed into the second order canonical chained form systems. All the states of the system converge to the desired equilibrium point exponentially. The obtained result is applied to stabilize a3-link planar manipulator with a free joint. The simulation results show that part of states and controls decrease obviously and local characteristics are better than the previous works.Finally, globally continuous K-exponential control law is proposed to stabilize a class of nonholonomic systems that are transformed into the second order canonical chained form systems by using an exponential decay term related to the initial state values. All the states of the system converge to the desired equilibrium point exponentially. The obtained result was applied to stabilize a rigid body with horizontal motion.