A review of pseudospectral optimal control: From theory to flight

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

Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jackson's theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.

https://www.elissarglobal.com/wp-content/uploads/2012/05/Spectral-Algorithm-for-Pseudospectral-Methods-in-Optimal-Control.pdf

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Recently, the Legendre pseudospectral (PS) method migrated from theory to ∞ight application onboard the International Space Station for performing a flnite-horizon, zeropropellant maneuver. A small technical modiflcation to the Legendre PS method is necessary to manage the limiting conditions at inflnity for inflnite-horizon optimal control problems. Motivated by these technicalities, the concept of primal-dual weighted interpolation, introduced earlier by the authors, is used to articulate a unifled theory for all PS methods for optimal control. This theory illuminates the previously hidden fact of the unit weight function implicit in the the Legendre PS method based on Legendre-Gauss-Lobatto points. The unifled framework also reveals why this Legendre PS method is the most appropriate method for solving flnite-horizon optimal control problems with arbitrary boundary conditions. This conclusion is borne by a proper deflnition of orthogonality needed to generate convergent approximations in Hilbert spaces. Special boundary conditions are needed to ensure the convergence of the Legendre PS method based on the Legendre-Gauss-Radau (LGR) and the Legendre-Gauss (LG) points. These facts are illustrated by simple examples and counter examples which reveal when and why PS methods based on LGR and LG points fail. A new kind of consistency in the primal-dual weight functions allows us to generate dual maps (such as Hamiltonians, adjoints etc) without resorting to solving di‐cult two-point boundary-value problems. These concepts are encapsulated in a unifled Covector Mapping Theorem.

/pdf/advances-in-pseudospectral-methods-for-optimal-control-4rl425qwgf.pdf

Advances in Pseudospectral Methods for Optimal Control

During the last decade, pseudospectral methods for optimal control, the focus of this tutorial session, have been rapidly developed as a powerful tool to enable new applications that were previously considered impossible due to the complicated nature of these problems. The purpose of this tutorial section is to introduce this advanced technology to a wider community of control system engineering. We bring in experts of pseudospectral methods from academia, industry, and military and DoD to present topics covering a large spectrum of pseudospectral methods, including the theoretical foundation, numerical techniques of pseudospectral optimal control, and military/industry applications.

/pdf/pseudospectral-optimal-control-for-military-and-industrial-45qtoon1iq.pdf

Pseudospectral Optimal Control for Military and Industrial Applications

Program to Optimize Simulated Trajectories II (POST2) is used as a basis for an end-to-end descent and landing trajectory simulation that is essential in determining design and integration capability and system performance of the lunar descent and landing system and environment models for the Autonomous Landing and Hazard Avoidance Technology (ALHAT) project. The POST2 simulation provides a six degree-of-freedom capability necessary to test, design and operate a descent and landing system for successful lunar landing. This paper presents advances in the development and model-implementation of the POST2 simulation, as well as preliminary system performance analysis, used for the testing and evaluation of ALHAT project system models.

/pdf/advances-in-post2-end-to-end-descent-and-landing-simulation-2yr2aaavse.pdf

Advances in POST2 End-to-End Descent and Landing Simulation for the ALHAT Project

Airdrop technology is a vital Department of Defense (DoD) capability that supports rapid deployment of war fighters and supplies. Consequently, the Army has sponsored development of gliding, steerable airdrop systems that can be deployed from high altitudes, with large offset, carrying small through large pay loads. The goal was to enable payload delivery within 100 meters of the target. Under this effort, Draper Laboratory developed modular guidance, navigation, and control (GN&C) software to precision guide ram-air parafoils using a combination of Global Positioning System (GPS) and inertial navigation system (INS) data. A high fidelity simulator was constructed to evaluate the expected performance of the Draper software. Also, in conjunction with NASA, flight tests with an 88 sq. ft. parafoil and a 170 pound payload were performed to evaluate the GN&C system performance under real flight conditions. A number of GN&C system design refinements were formulated after review of initial flight test results that ultimately enabled a payload delivery accuracy of about 50 meters. This paper summarizes the motivation for precision guided airdrop systems, reviews the Draper GPS/INS based GN&C for ram-air parafoils, and presents both simulation and flight test results.

Precision guided airdrop system flight test results

All Apollo landings were performed by the crew, manually commanding the lunar module (LM) attitude and rate-of-descent (ROD). In future missions, the astronauts will again need manual control of the flight path and attitude. Crew interaction mechanisms have been proposed to re-designate a landing aimpoint during the approach phase. However, manual control modes, crew control capability, and resulting vehicle performance during terminal descent have not been thoroughly investigated. A rate-control attitude hold (RCAH) mode was ultimately used in the LM for lateral flight and descent rate was controlled incrementally (P66), although other modes were considered and evaluated. These modes, as well as others proposed during Apollo, are reviewed and discussed in terms of their applicability to Altair. The ALHAT guidance, navigation, and control (GNC) algorithms for autonomous precision lunar landing were modified to include two manual flight control modes: RCAH with incremental ROD (A66) and incremental lateral velocity control with incremental ROD (A68) using Altair LDAC1-Delta vehicle parameters. Crew interactions with the ALHAT GNC system are described throughout the mission phases from lunar orbit to touchdown, focusing on manual control of flight path and attitude during terminal descent. These ALHAT manual attitude control modes are described and vehicle performance is discussed in terms of its estimated impact on handling quality ratings.

Design and analysis of lunar lander manual control modes

Precision airdrop from high altitude is important for a range of missions while limiting risk to carrier aircraft. Since 1998 the New World Vistas program has sponsored development of a Draper Laboratory onboard Precision Aerial Delivery Planner and a Planning Systems, Inc. WindPADS wind/density field estimator to address these mission requirements. The resulting system derives an airdrop Computed Aerial Release Point on-board the carrier aircraft while in-flight to the Drop Zone (DZ). This enables adaptation to DZ changes while accurately accounting for current environment data. Reference trajectories for low-glide airdrop systems are also generated en route to the DZ. Two Ethernet-connected laptop personal computers are being used for the Planner, with its embedded airdrop simulation, and for WindPADS. The Planner PC also has access to data from the carrier aircraft 1553 data bus. Interfaces between the PCs and to the carrier aircraft have been successfully tested. Data is being collected by the Army to improve the Planner's

Thomas Fill

Papers

Advances in POST2 End-to-End Descent and Landing Simulation for the ALHAT Project

Precision guided airdrop system flight test results

Design and analysis of lunar lander manual control modes

Status of an on-board pc-based airdrop planner demonstration

Constrained trajectory optimization for lunar landing