Unscented Optimal Control for Orbital and Proximity Operations in an Uncertain Environment: A New Zermelo Problem

Abstract

Orbital and proximity operations in an uncertain environment are critical technology requirements for NASA's asteroid retrieval mission and the OSIRIS-REx program. One approach for establishing orbits around a body with an uncertain gravity field is to use the concept of operations employed in the NEAR-Shoemaker mission. Such operations may not be easily or cost-effectively portable to highly irregularly-shaped asteroids that may be in a tumbling state. In this paper, we propose the concept of unscented optimal control to design safe guidance commands that can produce orbits with guaranteed safety margins. Unscented optimal control is based on combining the concept of the unscented transform with standard optimal control to produce a new approach for enabling an open-loop management of navigational, gravitational and other uncertainties. Many space operations problems in an uncertain environment can be conceptualized as Zermelo-type problems with uncertain currents. Under this backdrop, a collection of uncertain Zermelo problems is discussed using unscented optimal control. Results show dramatic improvements in performance including the case of near-zero targeting errors under uncertainty.

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

“…Given that x 0 is an uncertain parameter, the simplest formulation is to set μ x 0 to some fixed value, where μ x 0 is the mean of x 0 . Such a formulation is used in [8,9] in one instantiation of a Riemann-Stieltjes formulation called unscented optimal control. Generalizing along this direction, we may set μ x 0 to be a decision variable that must be selected from some given set X 0 μ ⊆ R N x .…”

Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites

“…The gimbal angle uncertainties have Normal distributions centered on the nominal initial conditions, each with variance σ 2 . The uncertainty is propagated via copies of fx; u; t (see [5][6][7] for details) so that the expectations on the CMG singularity index may be computed over the optimal trajectory. Additional requirements on the conservation of angular momentum and the quaternion norm for each copy of fx; u; t are embodied in the additional constraints hx; u; t, given in (11).…”

“…In fact, as will be seen later, the condition SΔt ≪ S min can occur and cause the steering to become singular. Rather than reserving a larger margin on control authority by increasing the value of S min (this can increase maneuver times to the point of operational infeasibility), we employ the newly developed framework of Riemann-Stieltjes optimal control [5][6][7] and implement a tychastic constraint on SΔt.…”

“…One approach is to solve the problem using pseudospectral methods [3] and Monte Carlo sampling, but a very large number of samples (greater than 1000) are generally required. To reduce the dimension of the problem, it is better to use sigma points [6,7] or hyper-pseudospectral (HS) points [5], r r 0 1 ; r 0 2 ; : : : ; r 0 N T at t t 0 , which, along with a set of associated weights, w w 1 ; w 2 ; : : : ; w N T , provide an efficient cubature formula for approximating Riemann-Stieltjes integrals of the form P N i1 w i Er i . To determine the appropriate number of discretization points for solving (11), the first n moments of the singularity index PD may be computed.…”

“…The approach thus provides a margin for feedback, which is calculable in a statistical sense. The resulting nonstandard optimal control problems are solved using the RiemannStieltjes optimal control framework [5][6][7]. Experimental results from a series of tests performed on Honeywell's momentum control system testbed are presented for both nominal and off-nominal operating conditions to illustrate the feasibility of implementing the approach on a practical CMG attitude control system.…”