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The development of a computational adaptive optimal controller for regulation and tracking of spacecraft attitude with unknown moment of inertia is the subject of this paper. The control technique accomplishes this by sequentially prioritizing higher goals of stabilization, optimization and identification. The proposed controller assumes measurement of angular velocities, and control torques. Conditions are derived that ensure asymptotic stability of the states. The proposed methodology uses an integral reinforcement learning based approach to drive the system to a linear quadratic optimal control solution. A novel identification method is derived that follows from the optimal controller formulation. Simulations of a spinning satellite with unknown moment of inertia are considered which shows successful optimal regulation and tracking of angular velocities along with accurate identification of unknown inertia.

a = positive parameter atx aty atz T = linear acceleration due to the tether tension, m · s−2 ax ay az T = linear acceleration of the gripper’s thruster force, m · s−2 C k = matrix in coordinated desaturation controller d = position vector of the capture position, m Fl = tether tension, N I = inertia matrix of the combination, kg · m I0 = nominal value of combination’s inertia matrix, kg · m Kξ = positive-definite design matrix k2 = positive-definite design matrix m = mass of tethered space robot–target combination, kg Oxlylzl = space tether frame Oxpypzp = space platform orbital frame Oxtytzt = combination orbital frame Ox 0 t y 0 t z 0 t = combination body frame P = positive-definite design matrix R = transformationmatrix from platform orbital frame to combination body frame S k = constant positive weighting matrix Tl = tether control torque, Nm x y z T = centroid position of the combination in the platform orbital frame, m ΔI = inertia matrix uncertainty, kg · m e = positive parameter λ k = Lagrange multiplier λL = upper bound of disturbance λL = estimation values of disturbance μ = positive design parameter ξ = state of the auxiliary design system σ = modified Rodrigues parameters σd = desired modified Rodrigues parameters τ k = vector of optimal thruster force and tether tension τc = control torque of the combination, N · m τd = disturbing torques, N · m τt = control torque of the thruster, N · m τl = control torque of the tether, N · m ω = absolute angular velocity of the combination, rad · s−1 ωd = desired angular velocity of the combination, rad · s−1

This paper proposes a nonlinear adaptive position and attitude-tracking controller for satellite proximity operations between a target and a chaser satellite. The controller requires no information about the mass and inertia matrix of the chaser satellite and takes into account the gravitational acceleration, the gravity-gradient torque, the perturbing acceleration due to Earth’s oblateness, and constant (but otherwise unknown) disturbance forces and torques. Sufficient conditions to identify the mass and inertia matrix of the chaser satellite are also given. The controller is shown to ensure almost global asymptotical stability of the translational and rotational position and velocity tracking errors. Unit dual quaternions are used to simultaneously represent the absolute and relative attitude and position of the target and chaser satellites. The analogies between quaternions and dual quaternions are explored in the development of the controller.

M ODEL uncertainties and measurement errors are very important factors that practical spacecraft attitude-control systems are subject to. Adaptive control is a well-known method for dealing with modeling uncertainty [1–3]. Recently, new control laws have been obtained that treat disturbances and model uncertainties [4–8]. References [4,5] deal with the attitude regulation problem only. References [6,7] both present globally convergent control laws for the attitude tracking problem when bounds on the spacecraft inertiamatrix and the disturbances are known. The advantage of these approaches is that the form of the disturbance need not be known, only the bound. In [8], the inertia matrix and linearly parameterizable disturbances are estimated adaptively. On the other hand, all of the aforementioned works are based on the availability of perfect measurements. Reference [9] explicitly considers measurement errors and studies performance, given bounds on the measurement error in the context of a model reference adaptive controller. This is a very important issue that any practical control system design must address. In particular, guaranteed performance bounds will be very useful for the control system design if performance specifications are given. There are well-established techniques to obtain these bounds for linear systems, however, they are generally lacking for the more general nonlinear case ([9] being an exception). In practice, extensive simulation-basedMonteCarlo analyses are used to determine closedloop performance,which can be quite time-consuming, particularly if they are used to determine suitable control gains. In this Note, nonadaptive and adaptive attitude tracking are considered. Guaranteed analytical performance bounds are obtained in the presence of model uncertainties and measurement errors. The bounds can be useful for attitude-control system designers to assist in gain selection given steady-state performance specifications, thus reducing the need for time-consuming Monte Carlo analyses. TheNote is organized as follows. First, a result on the filtered error from [6] is generalized. It is shown that if the filtered error is ultimately upper bounded with known bound, then the attitude and body-rate errors are also ultimately upper bounded. Subsequently, making use of this result together with sequential Lyapunovtype analyses, bounds on the steady-state tracking errors are derived when bounded model uncertainties and measurement errors are present. II. Mathematical Preliminaries In this Note, the vector and matrix norms used are kxk xx p and kXk λmax XX p (where λmax · denotes the maximum eigenvalue), respectively. The identity matrix will be denoted by 1. We will denote the unit quaternion by q; q4 , where q ∈ R is the vector part of the quaternion, and q4 ∈ R is the scalar part. Associated with a vector a ax ay az T ∈ R is the matrix

This paper examines simultaneous attitude control and momentum-wheel management of dual-spin and biasmomentum spacecraft using magnetic actuation. Transformations of variables are presented, leading to the derivation of control laws that yield proven stability and asymptotic convergence under appropriate assumptions on theEarth’smagneticfield. The results remain valid in the presence ofmagnetic torquer saturation. Furthermore, it is shown that for a spacecraft equipped with three orthogonal magnetic torquers, the control laws are tolerant to the failure of a single magnetic torquer in the case of a dual-spin spacecraft and tolerant to the failure of two magnetic torquers in the case of a bias-momentum spacecraft, provided that the remainingmagnetic torquer does not generate its dipolemoment parallel to themomentum-wheel spin axis.Additionally, the stability analyses show that the control laws remain stabilizing under the effects of control quantization. The theoretical results rely on the assumption that the spacecraft principal and body axes coincide. Robustness to uncertainties in the spacecraft inertia matrix and to disturbance torques are demonstrated with a numerical example.

The problem of rigid-body attitude tracking is examined for the case in which there is an unknown gyro bias, and vectorattitudemeasurementsareusedthatnecessitateestimationoftheattitudequaternion.Anewadaptivecontrol scheme is proposed that meets the attitude tracking control objective of forcing the attitude and angular velocity tracking errors to zero by a suitable reparameterization of the unknown gyro-bias parameters. In addition, it is shown that the bias and the quaternion estimates converge to their corresponding true values, irrespective of the nature of the underlying attitude reference trajectory. Controllers are introduced for the cases in which the body’s inertia matrix is both known and unknown. Convergence is shown through a Lyapunov-like stability analysis, and performance of the controllers is shown through simulation. Furthermore, simulations are shown with noise on the available measurements and with slowly varying gyro bias to highlight the performance benefits of the proposed control scheme. HE problem of rigid-body/spacecraft attitude tracking has been studied extensively with varying complexities. Awide range of controller and observer–controller combinations have been developed for different problem formulations. The difference between the formulations usually lies with either presence of unmodeled dynamics/disturbances or the availability and accuracy of certain measurements relating to the state or parameters of the rigid body. In thispaper,weextendontheexistingliteraturetoincludethescenario in which the measured angular velocity has an unknown constant bias and the attitude (represented by a quaternion) is estimated through a set of vector measurements. In addition, we widen the scope of the problem to include inertia matrix uncertainties.

P RACTICAL spacecraft attitude control systems must operate in the presence of disturbances, modeling errors, and actuator limitations. These issues have been the subject of much research interest. Adaptive control, where the unknown system parameters are estimated adaptively, is one of the proposed approaches for dealing with modeling uncertainty (see, for example, [1–4]). Both [1,2] deal with the attitude tracking problem, but they do not treat disturbances or actuator saturation. Reference [3] also deals with the tracking problem; it includes actuator saturation but not disturbances. Reference [4] includes bounded disturbances, but it does not treat actuator saturation, and it only deals with the attitude regulator problem. Recently, new control laws have been obtained that treat both disturbances and actuator limitations simultaneously [5–8]. References [5,6] deal with the attitude regulation problem only. References [7,8] treat the attitude tracking problem, and both present globally convergent control laws, given bounds on the spacecraft inertia matrix and the disturbances. The advantage of these approaches is that the form of the disturbance need not be known, only the bound. On the other hand, these approaches have no ability to learn the system model, which could be a useful feature if the attitude motion is to be optimized. This Note shows that, when an adaptive attitude control law based on the form given in [2] is appropriately designed, any linearly parameterizable disturbances can be accommodated; the closed-loop system is stable, with asymptotic tracking in the presence of actuator saturation. The unknown system parameters are learned adaptively.

use of control in one axis only. To show the feasibility of such a concept, the technique of single-input/single-output feedback linearization using the Lie derivative method is employed. The problem is solved for different numbers of jets and for different configurations of the inertia matrix. In the case of an axisymmetric spacecraft, the closed-loop stability is analyzed through the zero dynamics of the internal dynamics. In the cases in which the inertia matrix has cross products of inertia different from zero, the internal dynamics becomes practically intractable. In these cases, only an assessment of closed-loop stability coming from the analysis of different simulation cases is provided. The results show that for the proposed concept and control method to be feasible, a specific geometric layout of the actuators is required for each inertia configuration and number of jets employed in the system.

The attitude trajectory tracking control problem for a rigid spacecraft with external disturbances is addressed using the robust inverse optimal-control method. The proposed feedback-control law is optimal with respect to a meaningful cost functional involving tracking errors, control efforts, and extended disturbances, and the associated Lyapunov function satisfies a Hamilton‐Jacobi‐Isaacs partial differential equation. The controller is H∞ optimal with respect to extended disturbances. The performance limitation of the inverse optimal feedback controller is analyzed and guidelines for the selection of the controller gains are established. Numerical simulations are performed to demonstrate the effectiveness of the proposed control algorithm and the tuning guidelines. Nomenclature � A� = induced 2-norm of the matrix A ∈ R n × n , � A �= √ [λmax(A T A)] A T = transpose of A |a| = Euclidean norm of the vectora ∈ R n , |a |= √ (a T a) d, ˆ d =e xternal disturbance and extended disturbance, respectively J = inertia matrix of the spacecraft, J = J T > 0 L f V = Lie derivative of the Lyapunov function V (x) with respect to f (x), L f V = ∂ V (x) ∂ x f (x)

An adaptive attitude control law is presented that realizes linear closed-loop dynamics in the attitude error vector. The modified Rodrigues parameters (MRPs) are used as the kinematic variables since they are nonsingular for all possible rotations. The desired linear closed-loop dynamics can be of either PD or PID form. Only a crude estimate of the moment of inertia matrix is assumed to be known. An open-loop nonlinear control law is presented which yields linear closed-loop dynamics in terms of the MRPs. An adaptive control law is developed which asymptotically enforces these desired linear closed-loop dynamics in the presence of large inertia and external disturbance model errors. Since the unforced closed-loop dynamics are nominally linear, standard linear control methodologies can be employed to satisfy design requirements. The adaptive control law is shown to track the desired linear performance asymptotically without requiring a priori knowledge of either the inertia matrix or external disturbance.

A correction to the previously published article "New Form of Kane's Equations of Motion for Constrained Systems" is presented. Misuse of the transformation matrix between time rates of change of the generalized coordinates and generalized speeds (sometimes called motion variables) resulted in a false conclusion concerning the symmetry of the generalized inertia matrix. The generalized inertia matrix (sometimes referred to as the mass matrix) is in fact symmetric and usually positive definite when one forms nonminimal Kane's equations for holonomic or simple nonholonomic systems, systems subject to nonlinear nonholonomic constraints, and holonomic or simple nonholonomic systems subject to impulsive constraints according to Refs. 1, 2, and 3, respectively. The mass matrix is of course symmetric when one forms minimal equations for holonomic or simple nonholonomic systems using Kane s method as set forth in Ref. 4.

A novel method is presented to carry out near-minimum-time maneuvers of a spacecraft of unknown inertia. The formulation assumes the presence of three orthogonal reaction wheels located near the system mass center and oriented arbitrarily with respect to the spacecraft principal axes. Modie ed Rodrigues parameters along with their shadow set are used as the primary attitude coordinates. Open-loop maneuver laws are designed while solving the equations of motion by inversedynamics approach. This approach permits theapproximateimposition of the maximum saturation torque constraint. The torque proe les are near-bang-bang, with the instantaneous switches replaced by cubicsplinesof specie edduration.An adaptivetrackingcontrollawisdeveloped to determine perturbations to the nominal open-loop torque commands that will ensure the actual motion to follow the nominal motion in the presence of uncertainty in the inertia matrix and errors in the initial attitude. Global stability of the overall closed-loop controller is proved analytically and demonstrated by numerical simulations.

A sliding-mode control (SMC) algorithm is derived and applied to quaternion-based spacecraft attitude tracking maneuvers. Based on some interesting properties related to the spacecraft model, a class of linear sliding manifolds is selected. Significantly, a Lyapunov function is introduced in the SMC design, which can avoid the inverse of the inertia matrix and thus simplify the controller design. To improve the transient response before reaching the sliding manifold, the smoothing model-reference sliding-mode control (SMRSMC) is further developed, which requires well-estimated initial conditions. Simulation results are included to demonstrate the usefulness of the SMRSMC method.

This paper presents stability issues of second-order multidimensional linear time-varying systems. The second-order form is a result of modeling a class of dynamical systems directly using their physical parameters (e.g., inertia matrix). Three sufficient conditions for stability and two for instability were derived. Lyapunov's second method is first used to derive a sufficient stable condition which gives a range of variation for each parameter matrix. A new analysis method is used to derive a stable and an unstable condition. This method gives conditions on the integral of certain quantities. In this way, unlike conditions imposed by a typical Lyapunov analysis, instantaneous values of these quantities do not directly affect the proof of stability.

Nonlinear feedback linearization is used to develop an analytical control law which performs both spacecraft attitude control and Control Moment Gyroscope (CMG) momentum management. The nonlinear control law is expressed in terms of the spacecraft attitude, angular rates, and total stored CMG momentum, and makes no small angle, small rates, or negligible cross product of inertia approximations. Through the development of the nonlinear control law a nonlinear transformation is introduced that raises the spacecraft dynamic system to a companion form. The nonlinear transformation is not global and generates singularities in the control law. An expression that defines all of the singularities in the control law is developed in terms of the spacecraft inertia matrix and the attitude of the spacecraft. The behavior of the singularity expression is investigated. Simulation results show the behavior of the nonlinear control law as the singularities are approached. (Author)

Real-time algorithms that estimate the mass-property parameters commonly used in spacecraft control laws are developed based upon a stochastic estimation viewpoint. The elements of the inverse inertia matrix and the center-of-mass location vector are estimated from noisy measurements of the angular velocity using a secondorder filter, while estimates of the mass reciprocal are generated from noisy linear velocity measurements using a Kalman filter. Simulation results show that the rate of convergence of each estimate depends strongly upon the particular maneuver being performed, but that the mass properties can be estimated to within 1% error.