Structure-Preserving Algorithms For Dynamic Systems With Constraints And Its Optimal Control Problem

Dynamic systems with constraints and its optimal control problem widely exist in the fields of robotics,aerospace and automatic control.Their state or motions satisfy constraints caused by physical laws or mathematical characteristics.Since the constraint equations contain important state informations of the corresponding system,they should be satisfied as much as possible in the numerical solution.In recent years,the geometric properties and qualitative analysis of mechanical systems have attracted more and more attention,how to make numerical solutions maintain inherent geometric properties of the system while ensuring high accuracy has become a very important topic.Structure-preserving algorithms aim at reflecting the inherent properties of the system in a most realistic way.They have great advantages in terms of improving the accuracy of the algorithm and maintaining the invariance of the system.Due to the high accuracy and good numerical stability,especially the huge advantage in long-time simulation and robustness,structure-preserving algorithms have become an important tool in the field of scientific computing and engineering.In this thesis,we focus on holonomic constrained dynamic systems,nonholonomic constrained dynamic systems,and optimal control systems with holonomic constraints,aiming at constructing high-order structure-preserving algorithms while satisfying constraint equations with high precision.The main research work covers the following aspects:(1)In Lagrangian system,symplectic algorithms with high-order are proposed for holonomic constrained dynamic systems based on the Hamilton's principle.In order to improve the accuracy,the augmented action of the variational principle is divided into two parts,unconstrained and constrained,which are integrated separately using different quadrature rules.Lagrangian polynomials that passing through given interpolation points are used to approximate the generalized coordinates and Lagrange multipliers,and the constraint equations are strictly satisfied at the constraint points.Then 8 algorithms are constructed by choosing different interpolation points and quadrature rules.The symplecticity of the algorithms are proved analytically,and the influence of various factors on the accuracy of the algorithms is discussed by numerical examples.It shows that symplectic algorithms with high-order can be easily obtained based on the proposed method,and the algorithms satisfy the holonomic constraints with high precision,and provide good performance for long-time integration.(2)In Lagrangian system,a modified Lagrange-d'Alembert principle is presented,and then high-order symmetric algorithms are proposed for dynamic systems with non-ho]onomic constraints based on this principle.The modified Lagrange-d'Alembert principle is obtained by adding an augmented term to the Lagrange-d'Alembert principle,the augmented term allows the modified variational principle to derive equations of motion and nonholonomic constraints simultaneously.Then,based on the Lagrange-d'Alembert principle and the modified Lagrange-d' Alembert principle,two classes of high-order symmetric algorithms are constructed.For algorithms based on the Lagrange-d'Alembert principle,as the variational principle cannot derive the constraint equations,constraint points are needed to discretize the constraint equation.For algorithms based on the modified Lagrange-d'Alembert principle,as the modified variational principle can derive equations of motion and nonholonomic constraints simultaneously,the constraint points are no longer needed.The symmetric property of algorithms is discussed analytically and the high-order convergence is discussed numerically.Moreover,numerical results show that algorithms based on the modified Lagrange-d'Alembert principle have better numerical performances than algorithms based on the Lagrange-d' Alembert principle.(3)Using dual variables,high-order structure-preserving algorithms for holonomic and nonholonomic constrained dynamic systems are constructed in Hamiltonian system.Compared with the Lagrangian system,the Hamiltonian system is more essential and more important to some extent as all conservative systems can be expressed under the Hamiltonian system.High-order symplectic algorithms of the holonomic constrained system and high-order symmetry algorithms of the nonholonomic constrained system are constructed based on the Hamilton's principle represented by dual variables and the Lagrange-d'Alembert principle represented by dual variables,respectively.The discretization of the variational principle under the Hamiltonian system also depends on three factors,i.e.,interpolation points,quadrature rules and constraint points.But different from the construction under the Lagrangian system,the construction under the Hamiltonian system not only discretizes the displacements and the Lagrange multipliers but also discretizes the momentums by Lagrange polynomials.Algorithms with different order of the displacements,momentums and Lagrange multipliers are disccussed in order to find the best combination which has the highest efficiency and accuracy.The high-order convergence,good long-time simulation ability and high-precision properties of the structure-preserving algorithms for the holonomic and nonholonomic constrained systems are verified numerically.(4)A new augmented cost function and its corresponding dual variable variational principle is constructed for optimal control problem of holonomic constrained dynamic systems,then high-order symplectic algorithms are established based on the proposed variational principle and taken the state variable at right end and the costate variable at left end as independent variables.On the basis of the classical Lagrange type of cost function,augmented cost function which involves the motion equations and the constraint equations is established using the Lagrangian multiplier method.The dual variable variational principle corresponding to this augmented cost function can derive optimal solution that containing holonomic constraints.By discretizing this variational principle and choosing the state variable at right end and the costate variable at left end as independent variables,we put forword high-order symplectic algorithms which can satisfy the holonomic constraints with machine precision.Symplectic property of the continuous system and the numerical algorithms are proved analytically,and the high-order convergence of the algorithms is verified by numerical examples.The application on the flying formation and control problem for multiple unmanned aerial vehicles is implemented.