Showing papers by "N.H. McClamroch published in 2007"

Global Attitude Estimation using Single Direction Measurements

Abstract: A deterministic attitude estimator for a rigid body under an attitude dependent potential is studied using small error assumptions. This estimator requires only a single direction measurement to a known reference point at each measurement instant. The measurement cannot completely determine the attitude, but an attitude estimation scheme based on this measurement is developed; a feasible set compatible with the measurement is described and it is combined with an attitude dynamics model to obtain an attitude estimate. The attitude is globally represented by a rotation matrix, and the uncertainties are described by ellipsoidal sets. A numerical example for a spacecraft in a circular orbit is presented.

Optimal Attitude Control for a Rigid Body with Symmetry

Abstract: Optimal control problems are formulated and efficient computational procedures are proposed for attitude dynamics of a rigid body with symmetry. The rigid body is assumed to act under a gravitational potential and under a structured control moment that respects the symmetry. The symmetry in the attitude dynamics system yields a conserved quantity, and it causes a fundamental singularity in the optimal control problem. The key feature of this paper is its use of computational procedures that are guaranteed to avoid the numerical ill-conditioning that originates from this symmetry. It also preserves the geometry of the attitude dynamics. The theoretical basis for the computational procedures is summarized, and examples of optimal attitude maneuvers for a 3D pendulum are presented.

Attitude stabilization of the inverted 3D pendulum on TSO(3) with control saturation

Abstract: This paper treats the asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold of the 3D pendulum, taking into account saturation of the control moment. Control saturation is accommodated by a novel feedback structure that is based on the special geometric features of the 3D pendulum, namely that the attitude representation lies in the compact manifold SO(3). This attitude stabilization problem is solved by use of Lyapunov methods applied to closed loop dynamics that evolve on the tangent bundle TSO(3). The construction of a Lyapunov function for the closed-loop, that explicitly involves saturation, exemplifies a method to handle saturation for attitude stabilization on SO(3) when potential forces exist. The controller provides freedom to influence the local dynamics of the closed loop near the specified equilibrium and the global dynamics away from the equilibrium, as well as some freedom to shape the manifold of solutions that do not converge to the specified equilibrium.

Propagation of uncertainty in rigid body attitude flows

Abstract: Motivated by attitude control and attitude estimation problems for a rigid body, computational methods are proposed to propagate uncertainties in the angular velocity and the attitude. Uncertainties in the angular velocity and attitude are described in terms of ellipsoidal sets that are propagated through this nonlinear attitude flow. Computational methods are proposed, one method based on a local linearization of the attitude flow and two methods based on propagation of a small (unscented) sample selected from the initial uncertainty ellipsoid. Each of these computational methods is constructed using a Lie group variational integrator, viewed as a discretization of the attitude dynamics. Computational results are obtained that indicate: (1) the strongly nonlinear attitude flow characteristics; and (2) the limitations of each of these methods, and indeed any method, in providing effective global bounds on the nonlinear attitude flow.

Stabilization of a Specified Equilibrium in the Inverted Equilibrium Manifold of the 3D Pendulum

Abstract: This paper treats the asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold of the 3D pendulum. This attitude stabilization problem is solved by use of Lyapunov methods applied to closed loop dynamics that evolve on the tangent bundle TSO(3). A smooth controller is proposed that achieves almost global asymptotic stabilization of the specified equilibrium; the controller provides freedom to influence the local dynamics of the closed loop near the specified equilibrium as well as some freedom to shape the manifold of solutions that do not converge to the specified equilibrium.

Optimal motion planning for dual-spacecraft interferometry

Abstract: We address the question of design and optimal control of a class of dual-spacecraft interferometric imaging formations. The first main contribution is that we combine two ideas introduced separately in the literature and propose a maneuver that offers improved imaging performance. We then formulate an optimal control problem to minimize fuel consumption and maximize image quality by minimizing the relative speed, which is proportional to the signal-to-noise ratio (SNR) of the reconstructed image. We show that the necessary conditions are also sufficient and that the resulting optimal control is unique. Finally, we apply a continuation method to solve for the unique optimal trajectory.

A combinatorial optimal control problem for spacecraft formation reconfiguration

Abstract: We consider a spacecraft formation reconfiguration problem in the case of identical spacecraft. This introduces in the optimal reconfiguration problem a permutational degree of freedom, in addition to the choice of individual spacecraft trajectories. We approach this non-convex optimization problem using a coupled combinatorial and continuous optimization framework, in which the inner loop consists of computing the costs associated with a particular assignment by using a geometrically exact and numerically efficient discrete optimal control method based on Lie group variational integrators. In the outer optimization loop, combinatorial optimization techniques are used to determine the optimal assignments based on the costs computed in the inner loop. The proposed method is demonstrated on the optimal reconfiguration problem for 5 identical spacecraft to go from an inline configuration to one equally spaced on a circle.

3D Pendulum Experimental Setup for Earth-based Testing of the Attitude Dynamics of an Orbiting Spacecraft

Abstract: A 3D pendulum implemented using a triaxial air bearing system is proposed for Earth-based testing of orbiting spacecraft attitude dynamics and closed loop systems. This proposal is based on prior research on attitude dynamics and control of orbiting spacecraft, attitude dynamics and control of the 3D pendulum, and an experimental implementation of the 3D pendulum in our laboratory, referred to as the triaxial attitude control testbed (TACT). These research themes are integrated to assess the strengths and weaknesses of such an Earth-based testbed for spacecraft attitude dynamics and control hardware and software components. Several different cases, based on the importance of orbital effects and gravitational effects, are analyzed.