A review of pseudospectral optimal control: From theory to flight

To increase the science return of future missions to Mars and to enable sample return missions, the accuracy with which a lander can be deliverer to the Martian surface must be improved by orders of magnitude. The prior work developed a convex-optimization-based minimum-fuel powered-descent guidance algorithm. In this paper, this convex-optimization-based approach is extended to handle the case when no feasible trajectory to the target exists. In this case, the objective is to generate the minimum-landing-error trajectory, which is the trajectory that minimizes the distance to the prescribed target while using the available fuel optimally. This problem is inherently a nonconvex optimal control problem due to a nonzero lower bound on the magnitude of the feasible thrust vector. It is first proven that an optimal solution of a convex relaxation of the problem is also optimal for the original nonconvex problem, which is referred to as a lossless convexification of the original nonconvex problem. Then it is shown that the minimum-landing-error trajectory generation problem can be posed as a convex optimization problem and solved to global optimality with known bounds on convergence time. This makes the approach amenable to onboard implementation for real-time applications.

Minimum-Landing-Error Powered-Descent Guidance for Mars Landing Using Convex Optimization

A class of computational methods for solving a wide variety of optimal control problems is presented; these problems include nonsmooth, nonlinear, switched optimal control problems, as well as standard multiphase problems. Methods are based on pseudospectral approximations of the differential constraints that are assumed to be given in the form of controlled differential inclusions including the usual vector field and differential-algebraic forms. Discontinuities and switches in states, controls, cost functional, dynamic constraints, and various other mappings associated with the generalized Bolza problem are allowed by the concept of pseudospectral (PS) knots. Information across switches and corners is passed in the form of discrete event conditions localized at the PS knots. The optimal control problem is approximated to a structured sparse mathematical programming problem. The discretized problem is solved using off-the-shelf solvers that include sequential quadratic programming and interior point methods. Two examples that demonstrate the concept of hard and soft knots are presented.

Pseudospectral Knotting Methods for Solving Nonsmooth Optimal Control Problems

Planetary soft landing is one of the benchmark problems of optimal control theory and is gaining renewed interest due to the increased focus on the exploration of planets in the solar system, such as Mars. The soft landing problem with all relevant constraints can be posed as a finite-horizon optimal control problem with state and control constraints. The real-time generation of fuel-optimal paths to a prescribed location on a planet's surface is a challenging problem due to the constraints on the fuel, the control inputs, and the states. The main difficulty in solving this constrained problem is the existence of nonconvex constraints on the control input, which are due to a nonzero lower bound on the control input magnitude and a nonconvex constraint on its direction. This paper introduces a convexification of the control constraints that is proven to be lossless; i.e., an optimal solution of the soft landing problem can be obtained via solution of the proposed convex relaxation of the problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain optimal solutions of the original nonconvex optimal control problem.

Lossless Convexification of Nonconvex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem

In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.

/pdf/connections-between-the-covector-mapping-theorem-and-1poeo9t2fp.pdf

Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control

A guidance law which minimizes the commanded acceleration along with the (weighted) final time is developed. This guidance law is a linear function of the states (relative to the landing point) and a nonlinear function of the time-to-go. The time-togo is obtained as a solution to a quartic equation which is solved analytically. The advantage of this guidance law is that it does not involve any iterations whatsoever. It is the exact solution to the two-point boundary-value problem associated with the first variation necessary conditions. It also satisfies the second variation necessary conditions for a minimum. An example of a lunar landing is given to demonstrate the optimality of this guidance law.

An optimal guidance law for planetary landing

With the progressive sophistication of future missions, it has become increasingly apparent that a new framework is necessary for efficient planning, analysis, and optimization of various concepts of operations (CONOPS). In recognizing that CONOPS involve categorical variables, we propose a hybrid optimal control framework that mathematically formalizes such problems. Hybrid optimal control theory extends ordinary optimal control theory by including categorical variables in the problem formulation. The proposed formalism frees mission planners to focus on high-level decision making by automating and optimizing the details of the inner loops. The eventual goal of this formalism is to develop efficient tools and techniques to support the objective of increasing autonomy for future systems. In using the pseudospectral knotting method to solve hybrid optimal control problems, we generate a mixed-variable programming (MVP) problem. A simple, feasible integer programming subproblem is identified that reduces the combinatorial complexity of solving the MVP. In addition to developing the framework using various examples from aerospace engineering, we provide details for a two-agent benchmark problem associated with a multiagent launch system. The entire process is illustrated with a numerical example.

https://www.elissarglobal.com/wp-content/uploads/2012/03/Hybrid-Optimal-Control-Framework-for-Mission-Planning.pdf

Hybrid Optimal Control Framework for Mission Planning

Autonomous rendezvous and docking of a spacecraft with a cooperative target vehicle is critical for a wide array of future mission applications, and flash light detection and ranging sensors are one of the most promising sensors for achieving this task. This paper presents an end-to-end assessment of how these sensors may be used for relative navigation. Within a unified framework, detailed discussions are provided on the topics of navigation architecture, flash light detection and ranging sensors measurements and image processing, reflector identification, and the estimation of relative position and attitude.

Cooperative Relative Navigation of Spacecraft Using Flash Light Detection and Ranging Sensors

This study introduces novel algorithms and the underlying mathematics to process photographs of planetary illuminated bodies and use them for navigation purposes. The goal is to accurately estimate the observer-to-body relative position in inertial coordinates. The main motivation is to provide autonomous navigation capabilities to spacecrafts by observing a planet or a moon. This is needed, for example, in human-rated vehicles in order to provide navigation capabilities in a loss-of-communications scenario. The algorithm is derived for the general case of a triaxial ellipsoid that is observed bounded by an elliptical cone. The orientation of the elliptical cone reference frame is obtained by eigenanalysis, and the offset between the elliptical cone axis and the body center direction as well as the equation of the terminator are quantified. The main contribution of this paper is in the image-processing approach adopted to derive centroid and distance to the body. This is done by selecting a set of pixels ...

Christopher D'Souza

Papers

An optimal guidance law for planetary landing

Hybrid Optimal Control Framework for Mission Planning

Cooperative Relative Navigation of Spacecraft Using Flash Light Detection and Ranging Sensors

Image Processing of Illuminated Ellipsoid

A Fast Approach to Multi-Stage Launch Vehicle Trajectory Optimization