A Discrete Variational Integrator for Optimal Control Problems on SO(3)

An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and,...

We reconsider the variational integration of optimal control problems for mechanical systems based on a direct discretization of the Lagrange-d'Alembert principle. This approach yields discrete dynamical constraints which by construction preserve important structural properties of the system, like the evolution of the momentum maps or the energy behavior. Here, we employ higher order quadrature rules based on polynomial collocation. The resulting variational time discretizati...

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in ...

In this work, we consider optimal control problems for mechanical systems on vector spaces with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term and we allow linear coordinate changes in the configuration space. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a va...

We analyze discrete optimal control problems and their connection with back propagation and deep learning. We consider in particular the symmetric representation of the discrete rigid body equations developed via optimal control analysis and optimal flows on adjoint orbits

In this paper, we develop a novel method for deriving a global optimal control strategy for stochastic attitude kinematics on the special orthogonal group SO(3). We first introduce a stochastic Lie-Hamilton-Jacobi-Bellman (SL-HJB) equation on SO(3), which theoretically provides an optimality condition for the global optimal control strategy of the stochastic attitude kinematics. Then we propose a novel numerical method, the Successive Wigner-Galerkin Approximation (SWGA) meth...

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: $SE\left( 3\right) $ and $SO\left( 3\right) \times \mathbb{R}^{3}$. Since the performance of the numerical integration...

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 $L\colon T^{(k)}Q\to\mathbb{R}$ with $k\geq 1$, the resulting discrete equations define a generally implicit numerical integrator algorithm on $T^{(k-1)}Q\times T^{(k-1)}Q$ ...

This paper provides new results for a tracking control of the attitude dynamics of a rigid body. Both of the attitude dynamics and the proposed control system are globally expressed on the special orthogonal group, to avoid complexities and ambiguities associated with other attitude representations such as Euler angles or quaternions. By selecting an attitude error function carefully, we show that the proposed control system guarantees a desirable tracking performance uniform...

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of ske...