Symmetry Reduction,Stability Analysis And Control For Nonholonomic Systems

Nonholonomic constraints widely exist in various vehicle systems and sports,and have applications in robotics,aerospace and other fields.Due to the nonintegrability of constraints,the motion of a nonholonomic system does not have the general property of making the Hamilton’s action a stationary value under constraints,which brings a series of mathematical difficulties to dynamics modeling,stability analysis and control.This thesisxestablishes a theoretical framework of the structures and stability of relative equilibria for a class of Voronets systems,and then focuses on two classical nonholonomic systems,the rattleback and bicycle,and carries out systematic research on symmetry reduction,stability analysis and control.The main work are as follows:(1)A theoretical framework of the structures and stability of relative equilibria for a class of Voronets systems is established.The possible local structures of relative equilibria under regularity conditions are revealed,and the intersection structure of the static and dynamic equilibria manifolds is discovered for the first time.The basic properties of the Jacobian matrix of the reduced dynamic system are studied,and the stability criteria for relative equilibria are given based on the center manifold theorem.(2)A strict mathematical description of contact constraints between the rattleback whose bottom shape is an arbitrary convex surface and the horizontal plane is given,and a reduced dynamic system is established by combining the symmetry of the rattleback system and the Voronets equations.Using the theoretical framework of the relative equilibria of Voronets systems,the relative equilibria and their stability of the rattleback system are studied,a necessary condition for the rattleback parameters that there are other nontrivial relative equilibria near a trivial relative equilibrium is obtained,and the chiral behavior of the rattleback’s rotation motion is explained.Numerical verification is carried out by taking a semi-ellipsoid as an example.(3)Both the holonomic and nonholonomic constraint equations for the Whipple bicycle on a horizontal ground are established,and a proper description of the bicycle’s symmetry is given.Some properties of the coefficients of the Voronets equations are revealed and a reduced dynamic system is established.Under the theoretical framework of the relative equilibria of Voronets systems,the relative equilibria of the bicycle system on the horizontal ground are studied,and a strict statement of the nonlinear stability of relative equilibria is given for the first time.(4)The constraint equations and reduced dynamic system of bicycle on a surface of revolution are established for the first time.Using the theoretical framework of the relative equilibria of Voronets systems,the relative equilibria and their stability of the bicycle system on the surface of revolution are investigated.Taking the paraboloid of revolution as an example,the numerical results are provided,and the variation of the structures of the relative equilibria manifolds with respect to the coefficient of the paraboloid of revolution is revealed.(5)A linear servo-constrained control law is proposed for bicycle,and a complete twodimensional nonlinear model for bicycle’s controlled motion is established.Based on the theoretical analysis and numerical research on the stability and bifurcation behaviors of the relative equilibria,the ranges of control parameters for realizing the bicycle’s uniform straight motion and uniform circular motion are obtained,and the mechanisms of the STF and CST phenomena in bicycle riding are revealed.Considering the measurement error of the gyroscope,a modified control law is proposed,and a three-dimensional nonlinear model for bicycle’s controlled motion is built.Through the stability analysis of the relative equilibria,the bicycle’s uniform straight motion and uniform circular motion under the interference of measurement error are achieved,and it is found that the bicycle has a supercritical Hopf bifurcation behavior under some control parameters.The theoretical results are verified by using the self-developed bicycle experimental system.Using the basic concepts of geometric mechanics and combining theoretical analysis,numerical computation and experimental verification,this thesis studies the basic properties of the relative equilibria of a class of Voronets systems,and fully demonstrates the rich nonlinear dynamic behavior of this class of systems by taking the rattleback and bicycle as examples.The research work of this thesis is of great significance to the development of the basic theory of nonholonomic systems and its applications in robotics,aerospace and other fields.