Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Abstract

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian L : T (k) Q → R with k ≥ 1, the resulting discrete equations define a generally implicit numerical integrator algorithm on T (k−1) Q × T (k−1) Q that approximates the flow of the higher-order Euler-Lagrange equations for L. The algorithm equations are called higher-order discrete Euler-Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map.We construct an exact discrete Lagrangian L e d using the locally unique solution of the higher-order Euler-Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of L e d , we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem. Contents 24 References 24Mathematics Subject Classification (2010): 70G45, 70Hxx, 49J15.

Geometric Control for Load Transportation with Quadrotor UAVs by Elastic Cables

“…In addition, we are currently working to add uncertainties in order to further explore the robustness of the proposed controller. We also plan to study the construction of force variational integrators in optimal control problems, in a similar fashion to [35] and [36], dynamic interpolation problems [37], and obstacle avoidance problems [38] for the cooperative task between quadrotors UAVs presented in this paper.…”

Riemannian cubics close to geodesics at the boundaries

“…is the matrix of system (15) and R(V, V ) is given by (16). Denote the matrix −R(V, V ) by A, the matrix vv t by D and the constant −K v 2 by η, so that A = ηI n + KD.…”

“…In optimal control applications such as rigid body control in robotics, spacecraft control in aeronautics, quantum control in quantum information processing, 3D animation in computer graphics or regression schemes for computational anatomy in medical imaging, the problem of path planning requires the initial and final path position and velocity to be prescribed. Developing geometric variational integrators for optimal control problems is a subject of growing interest, especially challenging when boundary conditions are imposed (see, for instance, [16] and the recent work of Bloch and collaborators in [9]).…”

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

“…Our focus is on the discretization procedure and the mathematical tools that it involves, whereas the numerical behavior is planned to be explored in further works. However, regarding the local truncation error and stability (which will be still ensured thanks to the symplecticity of the HO variational integrators), discussed in the last paragraphs, the construction of L E d for HO systems has been developed in [21]. In this reference the reader can find some numerical tests and more details on numerical aspects.…”