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Theory and Experimentation Acta Astronautica 122 (2016) 114–136 Contents lists available at ScienceDirect Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro Suboptimal LQR-based spacecraft full motion control: Theory and experimentation Leone Guarnaccia b,1, Riccardo Bevilacqua a,n, Stefano P. Pastorelli b,2 a Department of Mechanical and Aerospace Engineering, University of Florida 308 MAE-A building, P.O. box 116250, Gainesville, FL 32611-6250, United States b Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy article info abstract Article history: This work introduces a real time suboptimal control algorithm for six-degree-of-freedom Received 19 January 2015 spacecraft maneuvering based on a State-Dependent-Algebraic-Riccati-Equation (SDARE) Received in revised form approach and real-time linearization of the equations of motion. The control strategy is 18 November 2015 sub-optimal since the gains of the linear quadratic regulator (LQR) are re-computed at Accepted 18 January 2016 each sample time. The cost function of the proposed controller has been compared with Available online 2 February 2016 the one obtained via a general purpose optimal control software, showing, on average, an increase in control effort of approximately 15%, compensated by real-time implement- ability. Lastly, the paper presents experimental tests on a hardware-in-the-loop six- Keywords: Spacecraft degree-of-freedom spacecraft simulator, designed for testing new guidance, navigation, Optimal control and control algorithms for nano-satellites in a one-g laboratory environment. The tests Linear quadratic regulator show the real-time feasibility of the proposed approach. Six-degree-of-freedom & 2016 The Authors. Published by Elsevier Ltd. on behalf of IAA. This is an open access Real-time article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Experimental test 1. Introduction As spacecraft technology has evolved, robust and effi- cient automated control has become an essential mission Abbreviation: ADAMUS, Advanced Autonomous Multiple Spacecraft; capability. Over the last fifty years, in fact, worldwide APF, artificial potential function; AS, attitude stage; BP, balancing plat- aerospace research environments have addressed their form; CPU, central processing unit; DARPA, Defense Advanced Research work to the optimization of guidance, navigation and Projects Agency; DCM, direction cosine matrix; EKF, extended Kalman filter; GDC, guidance dynamics corporation; IBPS, intelligent battery and control performances, also paying attention to the pro- power system; IPOPT, interior point optimizer; KKT, Karush–Kuhn– pellant consumption and time-to-launch costs. Tucker; LGR, Legendre Gauss Radau; LKF, linear Kalman filter; LQE, linear It is clear, then, how high efficiency controls, ensuring quadratic estimator; LQR, linear quadratic regulator; LVLH, local vertical local horizontal; MIMO, multi-input multi-output; MP, moving platform; both position accuracy and propellant optimization, have NLP, nonlinear programming; ONR, Office of Naval Research; PWM, pulse become a critical issue in the control systems scope and width modulator; RTAI, Real-Time Application Interface; S/C, spacecraft; specifically in the aerospace sector. SNOPT, Sparse Nonlinear OPTimizer; TS, translational stage The spacecraft six degrees of freedom optimal control n Corresponding author. Tel.: þ1 352 392 6230. E-mail addresses: [email protected] (L. Guarnaccia), problem has been widely analyzed in the literature, lately bevilr@ufl.edu (R. Bevilacqua), [email protected] (S.P. Pastorelli). focusing on spacecraft relative motion, usually requiring 1 Present address: 48 Via Salvatore Quasimodo, Acicatena numerical methods [1]. Spacecraft formation and the opti- 95022, Italy. 2 Politecnico di Torino, Department of Mechanical and Aerospace mization of relative maneuvers are becoming increasingly Engineering, Room 6939. important topics of investigation. This is due to the benefits http://dx.doi.org/10.1016/j.actaastro.2016.01.016 0094-5765/& 2016 The Authors. Published by Elsevier Ltd. on behalf of IAA. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). L. Guarnaccia et al. / Acta Astronautica 122 (2016) 114–136 115 Nomenclature m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffispacecraft simulator mass (kg) μ = 3 n ¼ È R0 generic orbital rate (rad/s) 0mÂn zero matrix with dimensions m, n ω angular velocity vector (rad/s) A state matrix (Jacobian) ωx; ωy; ωz angular velocity vector components (rad/s) Arot state matrix rotational contribute P solution of the Riccati differential equation Atransl state matrix translational contribute P position cost (–) B input matrix (Jacobian) p thrusters inclination with respect to the ima- Btransl input matrix translational contribute ginary square circumscribed to the attitude Brot input matrix rotational contribute stage basis (p ¼ cos ð451Þ¼ sin ð451Þ) C output matrix Q state weighting matrix (mixed dimensions to D feedforward matrix generate dimensionless cost) d thrusters moment arm with respect to the Q transl state weighting matrix translational center of rotation of the AS [d¼0.32 m] contribute E DCMB direction cosine matrix from the body (B) to Q rot state weighting matrix rotational contribute the inertial reference frame (E) [23] R input weighting matrix (mixed dimensions to – F propellant cost ( ) generate dimensionless cost) F generalized force vector (F; M) R0 generic orbit radius (m) F nominal thrust (F ¼0.3 N) ω t t kinθ_ kinematics matrix relating the time derivative Fthrust force generated by the onboard thrusters (N) of the Euler angles with the angular velocity F LQR optimal generalized force, output of the LQR [23] Simulink block ρ additional gain influencing the input weight- 3 À 1 À 2 G universal gravitational constant (m kg s ) ing matrix (–) H thruster distribution matrix or θ Euler angles vector (rad) mapping matrix θx; θy; θz Euler angles (rad) Hf force term of the thruster distribution matrix UÃ generic LQR optimal solution H Ucont normalized continuous thrust vector (N) Hm torque term of the thruster distribution matrix Ucost net propellant expenditure (–) H (m) u10 normalized binary thrust vector (–) ImÂn identity matrix with dimensions m, n u dimensionless parameter with magnitude 2 a J inertia matrix (kg m ) correspondent to the nominal thrust J global maneuver cost (–) vx; vy; vz linear velocity vector components (m/s) K Kalman gain X generalized state vector (x; x_; θ; ω) Kpos, Krot additional dimensionless gains characterizing Xdes desired generalized state the adaptive tuning, applied to the state Xerr actual error vector weighting matrix position and angular x position vector (m) terms only x_ linear velocity vector (m/s) M angular momentum vector (N m) x; y; z position vector components (m) M È Earth mass (kg) μ μ Y output vector È Earth gravitational parameter ¼ GM È (m3 s À 2) in cost, responsiveness, and flexibility of a multi-spacecraft pushing the envelope with regards to fast computation of system versus the classical monolithic satellite. practical optimal/sub-optimal trajectories. Particularly new is the use of continuous on–off engines, The recent numerical approaches to the optimal control appearing on small spacecraft. This control constraint adds problem could be in short classified in two main categories: new complexity to finding the optimal solution. In fact, most of the literature on spacecraft optimal control assumes that 1. the indirect methods which employ the calculus of the thrust can be finely modulated, partially to mitigate the variations to obtain the first-order optimality conditions aforementioned problem. Unfortunately, this is not the case [6], where the resulting boundary-value problem is with real engines, which are usually limited to some sus- sometimes impossible or time-consuming to solve; tained value for thrust. As a partial response to this problem, 2. direct methods which approximate the trajectory via a new methodology has been presented in [2] with the aim parameterization, and transform the cost functional into to control spacecraft rendezvous maneuvers assuming multi- a cost function [7–9]. level continuous thrusters and impulsive thrusters on the same vehicle. Furthermore, very recently, the interests of the These second methods appear to be a viable tool for Department of Defense have been focusing on time/pro- real-time spacecraft optimal control. However, the major pellant optimal rendezvous and capture maneuvers of a non- issues with direct methods relate to the difficulties of cooperative target satellite [3–5]. This research has been defining parameters to represent feasible trajectories. The 116 L. Guarnaccia et al. / Acta Astronautica 122 (2016) 114–136 chosen parameterization may lead, for example, to unsa- was sub-optimized in real-time through re-computation of tisfactory final conditions, or, it may satisfy the final con- the LQR at each sample time, while the APF performed ditions but violate some of the constraints on the controls. collision avoidance and a high level decisional logic. Adding penalties to the cost function may help, but it There is a gap that has not been addressed directly in provides no guarantee of improvement as in [5,9]. Also, the past literature: real-time techniques for optimal con- the trajectory computation must be as fast as possible, trol of both attitude and position. This is partly due to the since it is iterated by an outer
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