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

In this paper we present a general framework that allows one to study discretization of certain dynamical systems. This generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent bundles and cotangent bundles respectively. In particular we show how to obtain a large class of discrete algorithms using this geometric approach. We give new geometric insight into the Newmark model for example and we give a direct discrete formulation of the Hamilt...

In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned Runge-Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and St\"{o}rmer-Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a d...

This paper contributes towards the development of motion tracking algorithms for time-critical applications, proposing an infrastructure for solving dynamically the inverse kinematics of highly articulate systems such as humans. We present a method based on the integration of differential kinematics using distance measurement on SO(3) for which the convergence is proved using Lyapunov analysis. An experimental scenario, where the motion of a human subject is tracked in static...

Adaptive tracking control for rigid body dynamics is of critical importance in control and robotics, particularly for addressing uncertainties or variations in system model parameters. However, most existing adaptive control methods are designed for systems with states in vector spaces, often neglecting the manifold constraints inherent to robotic systems. In this work, we propose a novel Lie-algebra-based adaptive control method that leverages the intrinsic relationship betw...

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of va...

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimension...

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover express...

In this article we present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We val...

Since they were introduced in the 1990s, Lie group integrators have become a method of choice in many application areas. These include multibody dynamics, shape analysis, data science, image registration and biophysical simulations. Two important classes of intrinsic Lie group integrators are the Runge--Kutta--Munthe--Kaas methods and the commutator free Lie group integrators. We give a short introduction to these classes of methods. The Hamiltonian framework is attractive ...

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov's problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction $\boldsymbol{\xi}$. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov's problem for the rotation group $SO(3)$. W...