Brief paperSpacecraft relative rotation tracking without angular velocity measurements☆
Introduction
Synchronization, coordination and cooperative control are new promising trends within mechanical systems technology. In the space industry, the concept makes the way for new and better applications, such as improved monitoring of the Earth and its surrounding atmosphere, geodesy, deep-space imaging and exploration and even in-orbit spacecraft servicing and maintenance. However, formation flying introduces a control problem with strict and time-varying boundaries on spacecraft reference trajectories, and requires detailed knowledge and tight control of relative distances and velocities for participating spacecraft.
We focus on output-feedback control of relative spacecraft motion; with some exceptions–cf. Bondhus, Pettersen, and Gravdahl (2005) and Krogstad, Gravdahl, and Kristiansen (2005) previous results mainly address the relative translation case. The related problem of attitude output-feedback control without angular velocity measurements of single spacecraft has received a larger interest over the last few years. One approach is to use model-based nonlinear observers to estimate the angular velocity, as suggested in e.g., Caccavale, Natale, and Villani (2003) and Seo and Akella (2007). Other solutions, such as in Akella, Valdivia, and Kotamraju (2005), Caccavale and Villani (1999), Costic, Dawson, de Queiroz, and Kapila (2000), Lizarralde and Wen (1996), Singla, Subbarao, and Junkins (2006), and Subbarao and Akella (2004), employ variations of first-order filters that, if not supplying the controller with the correct angular velocity, at least provides enough information to solve the control problem. For small spacecraft with limited computational resources, this approach is favorable.
For spacecraft formations, some results on output-feedback relative translation control have been extended to 6DOF motion directly; in Pan and Kapila (2001), a nonlinear tracking controller for both translation and rotation was presented, including an adaptation law to account for unknown mass and inertia parameters of the spacecraft. The controller ensures asymptotic convergence of position and velocity errors for all initial states; the proof relies on a standard signal-chasing analysis and Barbalat’s lemma. Based on the latter reference, semi-global asymptotic convergence of relative translation and rotation errors was claimed in Wong, Pan, and Kapila (2005) for an adaptive output-feedback controller using relative position and attitude only. For a thorough review of spacecraft formation control, see Scharf, Hadaegh, and Ploen (2004).
It must be stressed that the qualifier global is often used with an abuse of notation for systems evolving on the rotational sphere. Indeed, global (asymptotic) stability cannot be achieved in general (cf. Bhat and Bernstein (2000)), since there are often more than one equilibrium point in the system’s state space. For instance, in quaternion coordinates as we use in this paper, there exist two equilibria but which correspond to the same physical configuration. Moreover, even if we consider the two equilibria as the same point the term global refers to the whole state space –cf., Hahn (1963). The multiplicity of equilibria can be avoided by using three-parameter representations such as Euler angles or modified Rodriguez parameters (cf., Akella (2001) and Singla et al. (2006)), albeit leading to singularities in the inverse kinematics. This is acceptable for operations where small angle deviations are expected, such as in spacecraft rendezvous and docking operations, but not suitable for general attitude control on the entire sphere.
The ability to perform satisfactory attitude control with few sensors is especially interesting in formations of small spacecraft, to concur with requirements of small mass and reduced communication loads between individual spacecraft. In this paper, we solve the problem of relative rotation tracking control without angular velocity measurements for a leader–follower spacecraft formation. The spacecraft model that we use is expressed in quaternion coordinates hence, enters in the case of study described above. However, a fact that is often neglected and that we use in this paper is that multiple equilibrium cases can be exploited to achieve shorter rotation paths on attitude manoeuvres, by working on different equilibrium points (in the quaternion space) for different manoeuvres.
Our main result extends previous work on attitude control–cf. Caccavale and Villani (1999) and Kristiansen, Loría, Chaillet, and Nicklasson (2006). We assume that only relative attitudes are measured; an approximate-differentiation filter as in Kelly (1993) is used to compensate for the lack of angular velocity measurements. In contrast to the above-cited works, our controller requires less knowledge of the system model through a decreased dependency on the system dynamics, and employs a simpler hence more computationally-favorable filter. We also exploit the quaternion redundancy to achieve shorter rotation path in the transient for the cases where this has an effect on power consumption. Finally, we prove more stringent stability properties of the closed-loop system, both in terms of uniformity and asymptotic convergence. In a strict sense we show that each of the resulting closed-loop systems are uniformly practically asymptotically stable (UPAS).
The rest of the paper is organized as follows: In Section 2 we introduce some notation and definitions. In Section 3 we present the mathematical models of relative-attitude dynamics and kinematics in a leader–follower spacecraft formation. The control solution is presented in Section 4, and simulation results of a system with the derived controller are presented in Section 5. Concluding remarks are given in Section 6.
Section snippets
Mathematical preliminaries
We denote by the solution to the nonlinear differential equation with initial conditions . We denote by the Euclidean norm of a vector and the induced norm of a matrix. The distance to a closed ball of and radius , i.e., is denoted by A continuous function is said to be of class () if it is strictly increasing and . Moreover, is of class () if, in addition, as . A continuous
Relative-attitude model
Notation Reference coordinate frames are denoted by , and we denote by the angular velocity of relative to , referenced in . Matrices representing coordinate transformation between and are denoted .
Problem statement
The control problem is to design a control law that makes the state converge to a time-varying smooth trajectory , satisfying the kinematic relation and under the assumption that only is measurable. 2
A numerical example
In this case-study the orbital perturbations are assumed to be perfectly known or estimated based on measurements. Both spacecraft have moments of inertia . The leader follows an equatorial orbit with a perigee altitude of 250 km and eccentricity , and the leader body and orbit coordinate frames are perfectly aligned at all times. The follower has available continuous torque about all body axes, with a maximum torque of 0.05 Nm. In the simulations, the
Conclusion
We have presented a solution to the problem of tracking relative rotation in a leader–follower spacecraft formation using feedback from relative attitude only, using an approximate-differentiation filter of the attitude error to provide sufficient knowledge about the angular velocity error. The resulting stability properties of the closed-loop systems left by the controller configuration have been derived, and proved to result in a UPAS closed-loop system.
Raymond Kristiansen was born in Finnsnes, Norway, in 1977. He received his B.Sc. degree in Computer Science from Narvik University College (NUC), Norway, in 1998, and his M.Sc. and Ph.D. degrees in Engineering Cybernetics from the Norwegian University of Science & Technology (NTNU), Norway, in 2000 and 2008, respectively. From September through November 2005 he was a graduate scholar visitor at the “Laboratoire de Signaux et Systemes”, Supelec in Paris, France. R. Kristiansen is currently an
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Cited by (0)
Raymond Kristiansen was born in Finnsnes, Norway, in 1977. He received his B.Sc. degree in Computer Science from Narvik University College (NUC), Norway, in 1998, and his M.Sc. and Ph.D. degrees in Engineering Cybernetics from the Norwegian University of Science & Technology (NTNU), Norway, in 2000 and 2008, respectively. From September through November 2005 he was a graduate scholar visitor at the “Laboratoire de Signaux et Systemes”, Supelec in Paris, France. R. Kristiansen is currently an Associate Professor at the Department of Computer Science, Electrical Engineering and Space Technology at Narvik University College (NUC) in Norway. His research interests include modeling and nonlinear control of aerospace systems in general, and spacecraft and helicopters in particular, with a special focus on synchronization and coordinated control in formations. See also http://ansatte.hin.no/rayk.
Antonio Loría was born in Mexico in 1969. He got the B.Sc. degree in Electronic Engineering from the ITESM, Monterrey, Mexico in 1991. He got the M.Sc. and Ph.D. degrees in Control Engg. from the UTC, France in 1993 and Nov. 1996 respectively. From December 1996 through December 1998, he was successively an associate researcher at Univ. of Twente, The Netherlands; NTNU, Norway and the CCEC of the Univ. of California at St Barbara, USA. A. Loria is currently “Directeur de Recherche” (senior researcher), at the French National Centre of Scientific Research (CNRS). He is with the “Laboratoire de Signaux et Systemes”, Supelec, since Dec 2002. His research interests hop amidst mechanical systems, stability theory of dynamical systems, adaptive control, observer design, etc. He serves as associate editor for Systems and Control Letters, Automatica, IEEE Transactions on Automatic Control and is member of the IEEE CSS Conference Editorial Board. See also http://public.lss.supelec.fr/perso/loria.
Antoine Chaillet was born in Douai, France, in 1979. In 2002, he received his B.Sc. degree from ESIEE Amiens, France, and the M.Sc. degree in Control Engineering from Université Paris Sud in 2003. He was an undergraduate scholar visitor to the University of Twente, The Netherlands, and a graduate scholar visitor to the INRIA Sophia Antipolis, France. In July 2006, he received his Ph.D. degree cum laude in Control Theory from Université Paris Sud-L2S: “On stability and robustness of nonlinear time-varying systems — Applications to cascaded systems”. In 2004, A. Chaillet was recipient of a “Marie-Curie” Scholarship to visit Universitá degli Studi di Firenze, Italy. In 2006–2007, he served as a post doc fellow at Centro di Ricerca Piaggio, Pisa, Italy. Since September 2007, he has been serving as an assistant professor at Université Paris Sud, Supélec, EECI. His research interests include stability analysis and stabilization of nonlinear time-varying systems, control of mechanical systems, network controlled systems and synchronization. He is the (co)author of 25 publications. Detailed information and publications are available at www.eeci-institute.eu/index.php?p=chaillet.
Per Johan Nicklasson is a Professor at the Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College (NUC) in Norway. He received his siv.ing. and dr.ing. degrees in Engineering Cybernetics from the Norwegian University of Science & Technology in 1992 and 1996, respectively. He is co-author of Passivity-based Control of Euler–Lagrange Systems (Springer 1998). His research interests include nonlinear control of electromechanical systems.
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The material in this paper was partially presented at American Control Conference, Seattle, WA, June 11–13, 2008. This paper was recommended for publication in revised form by Associate Editor Xiaohua Xia under the direction of Editor Toshiharu Sugie.