Intrinsic Optimal Control for Mechanical Systems on Lie Group

Abstract

The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the HamiltonJacobi-Bellman equation. For a special case on SO(3), the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.

Real-Time Model Predictive Control of Rigid Body Motion via Discretization Using the Cayley Map

“…The special Euclidean group SE(3) is a semidirect product of SO (3) and R 3 , and an element of SE(3) represents a pair of a rotation matrix and a position vector in 3-dimensional space. From the definition, the pose (position and orientation) of a…”

“…Rigid body kinematics and dynamics are the fundamentals of the modeling, control, and measurement of aerial robots, spacecraft, robot manipulators, etc. The rigid body motion is described with a dynamical model in the special Euclidean group SE (3). The dynamics must be constrained on the group, and thus, we need to consider specific control laws to achieve regulation and tracking.…”

“…The dynamics must be constrained on the group, and thus, we need to consider specific control laws to achieve regulation and tracking. For example, [1] presents coordinatefree geometric PD controllers on SE (3) and its subgroup SO (3), the special orthogonal group, based on the metric structure of the groups. Lee et al [2] study a trajectory tracking controller of a quadrotor aerial vehicle on SE (3) using an intrinsic attitude tracking error defined on SO (3).…”

“…Liu et al [3] show an analytic solution of a quadratic optimal control problem in a compact Lie group such as SO(3) by using the logarithm map. The authors of this paper have presented an analytic optimal control method for SE (3) considering a quadratic objective function with specific weight matrices [4]. It also shows an application of the control of a fully actuated hexarotor aerial vehicle ( Fig.…”

“…In Section II, we first introduce a continuous-time model of a rigid body dynamical system. Then, structure-preserving geometric integrators based on the exponential map and the Cayley map for SE (3) are explained. Based on the Cayley map, we present a discrete-time model of the rigid body dynamics in Section III.…”