Amit K. Sanyal

Rigid-Body Attitude Control

Rigid-body attitude control is motivated by aerospace applications that involve attitude maneuvers or attitude stabilization. The set of attitudes of a rigid body is the set of 3 X 3 orthogonal matrices whose determinant is one. This set is the configuration space of rigid-body attitude motion; however, this configuration space is not Euclidean. Since the set of attitudes is not a Euclidean space, attitude control is typically studied using various attitude parameterizations. Motivated by the desire to represent attitude both globally and uniquely in the analysis of rigid-body rotational motion, this article uses orthogonal matrices exclusively to represent attitude and to develop results on rigid-body attitude control. An advantage of using orthogonal matrices is that these control results, which include open-loop attitude control maneuvers and stabilization using continuous feedback control, do not require reinterpretation on the set of attitudes viewed as orthogonal matrices. The main objec tive of this article is to demonstrate how to characterize properties of attitude control systems for arbitrary attitude maneuvers without using attitude parameterizations.

Inertia-Free Spacecraft Attitude Tracking with Disturbance Rejection and Almost Global Stabilization

We derive a continuous nonlinear control law for spacecraft attitude tracking of arbitrary continuously differentiable attitude trajectories based on rotation matrices. This formulation provides almost global stabilizability, that is, Lyapunov stability of the desired equilibrium of the error system as well as convergence from all initial states except for a subset for which the complement is open and dense. This controller thus overcomes the unwinding phenomenon associated with continuous controllers based on attitude representations, such as quaternions, that are not bijective and without resorting to discontinuous switching. The controller requires no inertiainformation,noinformation onconstant-disturbance torques,andonlyfrequencyinformation forsinusoidal disturbance torques. For slew maneuvers (that is, maneuvers with a setpoint command in the absence of disturbances), the controller specializes to a continuous, nonlinear, proportional–derivative-type, almost globally stabilizing controller, in which casethe torque inputs can be arbitrarily bounded a priori. For arbitrary maneuvers, we present an approximate saturation technique for bounding the control torques.

Global optimal attitude estimation using uncertainty ellipsoids

A deterministic attitude estimation problem for a rigid body in a potential field, with bounded attitude and angular velocity measurement errors is considered. An attitude estimation algorithm that globally minimizes the attitude estimation error is obtained. Assuming that the initial attitude, the initial angular velocity and measurement noise lie within given ellipsoidal bounds, an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained. The center of the uncertainty ellipsoid provides point estimates, and the size of the uncertainty ellipsoid measures the accuracy of the estimates. The point estimates and the uncertainty ellipsoids are propagated using a Lie group variational integrator and its linearization, respectively. The attitude and angular velocity estimates are optimal in the sense that the sizes of the uncertainty ellipsoids are minimized.

Dynamics and control of a 3D pendulum

New pendulum models are introduced and studied. The pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force and control forces and moments. Several different pendulum models are developed to analyze properties of the uncontrolled pendulum. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Rigid pendulum and multi-body pendulum control problems are proposed. The 3D pendulum models provide a rich source of examples for nonlinear dynamics and control, some of which are similar to simpler pendulum models and some of which are completely new.

An Almost Global Tracking Control Scheme for Maneuverable Autonomous Vehicles and its Discretization

This technical note treats the challenging control problem of tracking a desired continuous trajectory for a maneuverable autonomous vehicle in the presence of gravity, buoyancy and fluid dynamic forces and moments. A realistic dynamics model that applies to maneuverable vehicles moving in 3-D Euclidean space is used for obtaining this control scheme. While applications of this control scheme include autonomous aerial and underwater vehicles, we focus on an autonomous underwater vehicle (AUV) application because of its richer, more nonlinearly coupled, dynamics. The desired trajectory and trajectory tracking errors are globally characterized in the nonlinear state space. Almost global asymptotic stability to the desired trajectory in the nonlinear state space is demonstrated both analytically and through numerical simulations.

Asymptotic Tracking Control for Spacecraft Formation Flying with Decentralized Collision Avoidance

This paper presents a tracking control scheme for spacecraft formation flying with a decentralized collision-avoidance scheme, using a virtual leader state trajectory. The configuration space for a spacecraft is the Lie group SE(3), which is the set of positions and orientations in three-dimensional Euclidean space. A virtual leader trajectory, in the form of attitude and orbital motion of a virtual satellite, is generated offline. Each spacecraft tracks a desired relative configuration with respect to the virtual leader in an autonomous manner, to achieve the desired formation. The relative configuration between a spacecraft and the virtual leader is described in terms of exponential coordinates on SE(3). A continuous-time feedback tracking control scheme is designed using these exponential coordinates and the relative velocities. A Lyapunov analysis guarantees that the spacecraft asymptotically converge to their desired state trajectories. This tracking control scheme is combined with a decentralized co...

Almost Global Robust Attitude Tracking Control of Spacecraft in Gravity

In this paper, we treat the practical problem of tracking the attitude and angular velocity of a spacecraft in the presence of gravity and disturbance moments. Autonomous trajectory tracking in the presence of disturbance moments is a challenging problem for robotic spacecraft, besides autonomous aerial and ground vehicles. The approach used here achieves almost global stable trajectory tracking by using a globally defined dynamics model that includes a moment on the vehicle created by a gravity potential and a disturbance moment that vanishes when the required angular velocity to be tracked is zero. The feedback control law is also globally defined. In the presence of the particular type of disturbance moments considered, this control law achieves almost global asymptotic tracking. Lyapunov-type methods on the nonlinear space of rigid body rotations are used to analyze the properties of the closed loop system and show near global stability and asymptotic convergence to the desired attitude and angular velocity trajectory. The treatment in this paper utilizes concepts from geometric mechanics to treat the dynamics of the feedback system in a global manner.

Optimal Attitude Estimation and Filtering Without Using Local Coordinates Part I: Uncontrolled and Deterministic Attitude Dynamics

Most existing algorithms for attitude estimation of mechanical systems use generalized coordinates to represent the group of rigid body rotations. Generalized coordinate representations of the group of rotations have some associated problems. While minimal (focal) coordinate representations exhibit kinematic singularities for large rotations, the quaternion representation requires satisfaction of an extra constraint. This paper treats the attitude estimation and filtering problem as a deterministic optimization problem, without using generalized coordinates, in the framework of geometric mechanics. An attitude estimation algorithm and filters are developed, that minimize the attitude and angular velocity estimation errors from noisy measurements. For filter propagation, the attitude kinematics and deterministic dynamics equations (Eulers equations) for a body in an attitude-dependent potential are used. Vector attitude measurements, with or without angular velocity measurements, are used for attitude and angular velocity estimation

Finite-time control for spacecraft body-fixed hovering over an asteroid

A finite-time control scheme for autonomous body-fixed hovering of a rigid spacecraft over a tumbling asteroid is presented. The relative configuration between the spacecraft and asteroid is described in terms of exponential coordinates on the Lie group SE(3), which is the configuration space for the spacecraft. With a Lyapunov stability analysis, the finite-time convergence of the proposed control scheme for the closed-loop system is proved. Numerical simulations validate the performance of the proposed control scheme.

Stability and Stabilization of Relative Equilibria of Dumbbell Bodies in Central Gravity

A dumbbell-shaped rigid body can be used to represent certain large spacecraft or asteroids with bimodal mass distributions. Such a dumbbell body is modeled as two identical mass particles connected by a rigid, massless link. Equations of motion for the five degrees of freedom of the dumbbell body in a central gravitational field are obtained. The equations of motion characterize three orbit degrees of freedom, two attitude degrees of freedom, and the coupling between them. The system has a continuous symmetry due to a cyclic variable associated with the angle of right ascension of the dumbbell body. Reduction with respect to this symmetry gives a reduced system with four degrees of freedom. Relative equilibria, corresponding to circular orbits, are obtained from these reduced equations of motion; the stability of these relative equilibria is assessed. It is shown that unstable relative equilibria can be stabilized by suitable attitude feedback control of the dumbbell. Nomenclature er = unit vector along local vertical (radial) direction ex = unit vector along longitudinal axis of dumbbell ey, ez = orthogonal unit vectors spanning plane perpendicular to dumbbell axis eλ