Pooya Sekhavat

Optimal Feedback Control: Foundations, Examples, and Experimental Results for a New Approach

Typical optimal feedback controls are nonsmooth functions. Nonsmooth controls raise fundamental theoretical problems on the existence and uniqueness of state trajectories. Many of these problems are frequently addressed in control applications through the concept of a Filippov solution. In recent years, the simpler concept of a π solution has emerged as a practical and powerful means to address these theoretical issues. In this paper, we advance the notion of Caratheodory-π- solutions that stem from the equivalence between closed-loop and feedback trajectories. In recognizing that feedback controls are not necessarily closed-form expressions, we develop a sampling theorem that indicates that the Lipschitz constant of the dynamics is a fundamental sampling frequency. These ideas lead to a new set of foundations for achieving feedback wherein optimality principles are interwoven to achieve stability and system performance, whereas the computation of optimal controls is at the level of first principles. We demonstrate these principles by way of pseudospectral methods because these techniques can generate Caratheodory-π solutions at a sufficiently fast sampling rate even when implemented in a MATLAB® environment running on legacy computer hardware. To facilitate an exposition of the proposed ideas to a wide audience, we introduce the core principles only and relegate the intricate details to numerous recent references. These principles are then applied to generate pseudospectral feedback controls for the slew maneuvering of NPSAT1, a spacecraft conceived, designed, and built at the Naval Postgraduate School and scheduled to be launched in fall 2007.

Low-Thrust, High-Accuracy Trajectory Optimization

J OURNAL OF G UIDANCE , C ONTROL , AND D YNAMICS Vol. 30, No. 4, July–August 2007 Low-Thrust, High-Accuracy Trajectory Optimization I. Michael Ross, ∗ Qi Gong, † and Pooya Sekhavat ‡ Naval Postgraduate School, Monterey, California 93943 DOI: 10.2514/1.23181 Multirevolution, very low-thrust trajectory optimization problems have long been considered difficult problems due to their large time scales and high-frequency responses. By relating this difficulty to the well-known problem of aliasing in information theory, an antialiasing trajectory optimization method is developed. The method is based on Bellman’s principle of optimality and is extremely simple to implement. Appropriate technical conditions are derived for generating candidate optimal solutions to a high accuracy. The proposed method is capable of detecting suboptimality by way of three simple tests. These tests are used for verifying the optimality of a candidate solution without the need for computing costates or other covectors that are necessary in the Pontryagin framework. The tests are universal in the sense that they can be used in conjunction with any numerical method whether or not antialiasing is sought. Several low-thrust example problems are solved to illustrate the proposed ideas. It is shown that the antialiased solutions are, in fact, closed-loop solutions; hence, optimal feedback controls are obtained without recourse to the complexities of the Hamilton–Jacobi theory. Because the proposed method is easy to implement, it can be coded on an onboard computer for practical space guidance. the field to exchange ideas over several workshops. These workshops, held over 2003–2006, further clarified the scope of the problems, and ongoing efforts to address them are described in [12]. From a practical point of view, the goal is to quickly obtain verifiably optimal or near-optimal solutions to finite- and low-thrust problems so that alternative mission concepts can be analyzed I. Introduction C ONTINUOUS-THRUST trajectory optimization problems have served as one of the motivating problems for optimal control theory since its inception [1–4]. The classic problem posed by Moyer and Pinkham [2] is widely discussed in textbooks [1,3,4] and research articles [5–7]. When the continuity of thrust is removed from such problems, the results can be quite dramatic as illustrated in Fig. 1. This trajectory was obtained using recent advances in optimal control techniques and is extensively discussed in [8]. In canonical units, the problem illustrated in Fig. 1 corresponds to doubling the semimajor axis (a 0 1, a f 2), doubling the eccentricity (e 0 0:1, e f 0:2), and rotating the line of apsides by 1 rad. Note that the extremal thrust steering program for minimizing fuel is not tangential over a significant portion of the trajectory. Furthermore, the last burn is a singular control as demonstrated in Fig. 2 by the vanishing of the switching function. Although such finite-thrust problems can be solved quite readily nowadays, it has long been recognized [9–11] that as the thrust authority is reduced, new problems emerge. These well-known challenges chiefly arise as a result of a long flight time measured in terms of the number of orbital revolutions. Consequently, such problems are distinguished from finite-thrust problems as low-thrust problems although the boundary between finite thrust and low thrust is not altogether sharp. Although ad hoc techniques may circumvent some of the low- thrust challenges, it is not quite clear if the solutions generated from such methods are verifiably optimal. As detailed in [8], the engineering feasibility of a space mission is not dictated by trajectory generation, but by optimality. This is because fuel in space is extraordinarily expensive as the cost of a propellant is driven by the routine of space operations, or the lack of it, and not the chemical composition of the fuel. In an effort to circumvent ad hoc techniques to efficiently solve emerging problems in finite- and low-thrust trajectory optimization, NASA brought together leading experts in al Or iti bit In it Fin rb al O Transfer Trajectory Fig. 1 A benchmark minimum-fuel finite-thrust orbit transfer problem. Thrust Acceleration, u s = 0 Switching Function, s Received 13 February 2006; accepted for publication 21 August 2006. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Copies of this paper may be made for personal or internal use, on condition that the copier pay the

Pseudospectral Optimal Control for Military and Industrial Applications

10.00 per- copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/07

Minimum-Time Reorientation of a Rigid Body

10.00 in correspondence with the CCC. Professor, Department of Mechanical and Astronautical Engineering; imross@nps.edu. Associate Fellow AIAA. Research Associate, Department of Mechanical and Astronautical Engineering; qgong@nps.edu. Research Scientist, Department of Mechanical and Astronautical Engineering; psekhava@nps.edu. Singular Control s u time (canonical units) Fig. 2 Extremal thrust acceleration (control) program t7 !u and the corresponding switching function t7 !s for the trajectory shown in Fig. 1.

Pseudospectral Feedback Control: Foundations, Examples and Experimental Results

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.

Autonomous Trajectory Planning Using Real-Time Information Updates

Recent advances in optimal control theory and computation are used to develop minimum-time solutions for the rest-to-rest reorientation of a generic rigid body. It is shown that the differences in the geometry of the inertia ellipsoid and the control space lead to counterintuitive noneigenaxis maneuvers. The optimality of the open-loop solutions are verified by an application of Pontryagins principle as a necessary condition and not as a problem-solving tool. It is shown that the nonsmoothness of the lower Hamiltonian compounds the curse of dimensionality associated in solving the Hamilton-Jacobi-Bellman equations for feedback solutions. These difficulties are circumvented by generating Caratheodory-π trajectories, which are based on the fundamental notion that a closed loop does not necessarily imply closed-form solutions. While demonstrating the successful implementation of the proposed method for practical applications, these closed-loop results reveal yet another counterintuitive phenomenon: a suggestion that parameter uncertainties may aid the optimality of the maneuver rather than hinder it.

A Pseudospectral optimal motion planner for autonomous unmanned vehicles

Suppose optimal open-loop controls could be computed in real time. This implies optimal feedback control. These controls are, typically, nonsmooth. Nonsmooth controls raise fundamental theoretical problems on the existence and uniqueness of feedback solutions. A simple, yet powerful, approach to address these theoretical issues is the concept of a …-solution that is closely linked to the practical implementation of zero-order-hold control sampling. In other words, even traditional feedback controls involve open-loop controls through the process of sampling. In this paper, we advance the notion of

UKF-Based Spacecraft Parameter Estimation Using Optimal Excitation

We present a dynamic optimal control method for autonomous trajectory planning and control of an Unmanned Ground Vehicle (UGV) using real-time information updates. The objective of the UGV is to traverse from an initial start point and reach its goal in minimum time, with maximum robustness, while avoiding both static and dynamic obstacles. This is achieved by deriving the control solution that carries out the initial planning problem while minimizing a cost and satisfying constraints based on the initial global knowledge of the area. To combat the problem of inaccurate global knowledge and a dynamic invironment, the UGV uses its sensors to map the locally detected change in the environment and continuously updates its global map to re-compute a control solution that can achieve an optimal trajectory to the goal. Simulation results illustrate successful implementation of the method in various scenarios.

Unscented Kalman Filtering: NPSAT1 Ground Test Results

This paper presents a pseudospectral (PS) optimal control algorithm for the autonomous motion planning of a fleet of unmanned ground vehicles (UGVs). The UGVs must traverse an obstacle-cluttered environment while maintaining robustness against possible collisions. The generality of the algorithm comes from a binary logic that modifies the cost function for various motion planning modes. Typical scenarios including path following and multi-vehicle pursuit are demonstrated. The proposed framework enables the availability of real-time information to be exploited by real-time reformulation of the optimal control problem combined with real-time computation. This allows the each vehicle to accommodate potential changes in the mission/environment and uncertain conditions. Experimental results are presented to substantiate the utility of the approach on a typical planning scenario.

Closed-Loop Time-Optimal Attitude Maneuvering of Magnetically Actuated Spacecraft

The success or failure of an identification experiment depends heavily on the nature of the system excitation during the parameter estimation process. This paper presents an approach for synthesizing a set of optimized command trajectories to estimate the inertial parameters of a control moment gyro actuated spacecraft using an unscented dual state-parameter Kalman filter. The command trajectories are designed to minimize the condition number of an excitation matrix derived from the system dynamics in order to improve the convergence rate of the parameter estimator and desensitize the estimation process to external disturbances and/or measurement noise. The resulting nonlinear trajectory optimization problem is solved using pseudospectral optimal control. The approach requires no assumptions on the nature of the exciting trajectories, allows physical performance limits to be straightforwardly incorporated as part of the problem formulation, and rapidly generates the exciting trajectories on a typical personal computer. Simulation results demonstrate a typical reduction in parameter convergence time from more than 2800 to less than 250 seconds, when compared to parameter estimation using periodic sinusoidal commands. The efficacy of the approach is also verified f or the case where a priori knowledge of the true values of the system parameters is grossly inaccurate.