Study On Dynamics Of Non-smooth Rigid Body System Based On Udwadia-Kalaba Method

The non-smooth rigid body system has the characteristics of nonlinearity,discontinuity and variable topological structure.The dynamics problem of non-smooth rigid body system considering friction and collision has always been a hot and difficult problem in the field of multi-body system dynamics.Different from the general Lagrangian method for constrained systems,the Udwadia-Kalaba(U-K)method decouples the generalized acceleration and the generalized constraint force of the system.The generalized constraint force of the system is directly obtained by solving the explicit equation,and then the generalized acceleration is obtained.At present,the application of U-K method mainly focuses on the dynamics and control of smooth system.The purpose of this paper is to study the U-K method for the dynamics of non-smooth rigid body systems,and to provide a new research tool and idea for the dynamics and control of nonsmooth multibody systems.The main work of this paper is as follows:(1)The U-K method of constrained system is briefly introduced,the modeling theory and modeling process are described,the dynamic simulation is carried out through a specific example,and the simulation results are analyzed.Then,the detailed process of modifying the U-K method in the dynamics of non-ideal constrained systems with Coulomb sliding friction is described,and the significance of the modification is discussed and analyzed.(2)Based on the modified U-K method,the dynamics of non-smooth rigid body system with unilateral constraint is studied.The dynamics model of rigid body system with unilateral constraint and corresponding numerical solution scheme are established under the condition of considering the states change of sliding viscous collision.The dynamics simulation of two examples shows that compared with the original form of UK method,the modified U-K method can better reflect the actual physical characteristics,which verifies the correctness of the modified U-K method and its feasibility in solving the dynamics problem of unilateral constrained non-smooth rigid body system.(3)Based on the modified U-K method,the dynamics of the non-smooth rigid body system with bilateral constraints is studied.The dynamics equation of the system with friction bilateral constraints is established.With the help of the smooth Newton algorithm,the absolute value problem of the unknown constraint force in this mathematical model is studied,and the corresponding numerical solution scheme is given.Two examples with bilateral constraints are simulated to show the effectiveness of the proposed method.(4)The modified U-K method is applied to the trajectory tracking control of manipulator with friction.A new control strategy of the manipulator is presented.The trajectory tracking control is transformed into constraints.The constraint force generated by the constraint equation is applied to the unconstrained dynamics equation,and the constraint torque corresponding to the joint angle is calculated,which is the driving torque required for the manipulator to meet the control requirements.The simulation of a two joint manipulator shows the effectiveness of the control strategy,which provides a new research idea for this kind of control problem.