Optimal Attitude Control of a Rigid Body

摘要

Being mainly interested in the control of satellites, we investigate the problem of maneuvering a rigid body from a given initial attitude to a desired final attitude at a specified end time in such a way that a cost functional measuring the overall angular velocity is minimized. This problem is solved by applying a recent technique of Jurdjevic in geometric control theory. Essentially, this technique is just the classical calculus of variations approach to optimal control problems without control constraints, but formulated for control problems on arbitrary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state space G=SO(3), thereby completely circumventing the inevitable difficulties (singularities and ambiguities) associated with the use of parameters such as Euler angles or quaternions. The angular velocities wk about the body's principal axes are used as (unbounded) control variables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectory t→g*(t) to a trajectory on T*G which is then revealed as an integral curve of a certain time-invariant Hamiltonian vector field. Next, the calculus of Poisson brackets is applied to derive a system of differential equations for the optimal angular velocities t→w*k(t); once these are known the controlling torques which need to be applied are determined by Euler's equations. In special cases an analytical solution in closed form can be obtained. In general, the unknown initial values w*k(t0) can be found by a shooting procedure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-value problem. In fact, our approach completely avoids the explicit introduction of costate (or odjoint) variables and yields a differential equation for the control variables rather than one for the adjoint variables. This has the consequence that only variables with a clear physical significance (namely angular velocities) are involved for which good a priori estimates of the initial values are available.