OPTIMAL ATTITUDE-CONTROL OF A RIGID-BODY

Being mainly interested in the control of satellites, we investigate t he problem of maneuvering a rigid body from a given initial attitude t o 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 cla ssical calculus of variations approach to optimal control problems wit hout control constraints, but formulated for control problems on arbit rary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state spa ce G = SO(3), thereby completely circumventing the inevitable difficul ties (singularities and ambiguities) associated with the use of parame ters such as Euler angles or quaternions. The angular velocities omega (k) about the body's principal axes are used as (unbounded) control va riables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectory t --> g(t) to a trajectory on T*G which is then revealed a s an integral curve of a certain time-invariant Hamiltonian vector fie ld. Next, the calculus of Poisson brackets is applied to derive a syst em of differential equations for the optimal angular velocities t --> omega(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 u nknown initial values omega(k)(t(0)) can be found by a shooting proce dure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-v alue problem. In fact, our approach completely avoids the explicit int roduction of costate (or adjoint) variables and yields a differential equation for the control variables rather than one for the adjoint var iables. This has the consequence that only variables with a clear phys ical significance (namely angular velocities) are involved for which g ood a priori estimates of the initial values are available.