Anton H. J. de Ruiter

Adaptive Spacecraft Attitude Control with Actuator Saturation

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.

Distributed finite-time velocity-free attitude coordination control for spacecraft formations

In this paper, the finite-time velocity-free attitude coordination control for spacecraft formation flying under an undirected communication graph is addressed. A finite-time observer is introduced to obtain an accurate estimation of unmeasurable angular velocity and a decentralized finite-time observer is employed to estimate the angular acceleration of the virtual leader. With the application of the finite-time observer, the decentralized finite-time observer, and the homogeneous method, a continuous distributed finite-time attitude coordination control law is designed for a group of spacecraft without requiring angular velocity measurements. A rigorous proof shows that semi-global finite-time stability of the overall closed-loop system can be achieved and the proposed velocity-free control law guarantees a group of spacecraft to simultaneously track a common time-varying reference attitude in finite time even when the reference attitude is available only to a subset of the group members. The performance of the control scheme derived here is illustrated through numerical simulations.

Differential Drag as a Means of Spacecraft Formation Control

This paper investigates the feasibility of using differential drag as a means of nano-satellite formation control. Differential drag is caused when the ballistic coefficients of the spacecraft in a formation are not equal. The magnitude of differential drag depends on the difference in ballistic coefficients and also the altitude of the spacecraft formation. AGIs Satellite Tool Kit (STK) is used initially to assess the magnitude of drifts caused due to differential drag for different altitudes. This information is then used to show that it is feasible to use differential drag for spacecraft formation control. A simple PID controller is then implemented that adjusts the cross- sectional areas of the satellites such that the energies of the orbits remain equal. Results are presented that show that the control law can maintain the formation separation with reasonable accuracy.

Magnetic Attitude Control of a Flexible Satellite

1 M.A.Sc. Candidate, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Student Member AIAA. 2 Assistant Professor, Ryerson University, Department of Aerospace Engineering, 350 Victoria Street, Toronto, ON, Canada, M5B 2K3, and Senior Member AIAA. 3 Assistant Professor, McGill University, Department of Mechanical Engineering and Centre for Intelligent Machines, 817 Sherbrooke Street West, Montreal, QC, Canada, H3A 0C3, and Member AIAA. 4 Associate Professor, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Senior Member AIAA. 5 Professor, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Associate Fellow AIAA. 6 Research Scientist, Canadian Space Agency, Space Science and Technology, 6767 route de l’Aeroport, St.Hubert,QC, J3Y 8Y9, and AIAA member.

Distributed attitude synchronization control for a group of flexible spacecraft using only attitude measurements

This paper considers the problem of attitude synchronization control of a group of flexible spacecraft under an undirected communication graph and in the absence of measurements of both modal variables and spacecraft angular velocities. To solve this problem, a nonlinear observer is introduced to estimate the unmeasurable modal variables and spacecraft angular velocities. Then, the backstepping technique is used to design the control law. The stability of the overall closed-loop system is guaranteed by the Lyapunov approach together with Barbalats Lemma. The performance of the control scheme derived here is illustrated through numerical simulations.

Spacecraft Attitude Tracking with Guaranteed Performance Bounds

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

Continuous-time norm-constrained Kalman filtering

This paper considers continuous-time state estimation when part of the state estimate or the entire state estimate is norm-constrained. In the former case continuous-time state estimation is considered by posing a constrained optimization problem. The optimization problem can be broken up into two separate optimization problems, one which solves for the optimal observer gain associated with the unconstrained state estimates, while the other solves for the optimal observer gain associated with the constrained state estimates. The optimal constrained state estimate is found by projecting the time derivative of an unconstrained estimate onto the tangent space associated with the norm constraint. The special case where the entire state estimate is norm-constrained is briefly discussed. The utility of the filtering results developed are highlighted through a spacecraft attitude estimation example. Numerical simulation results are included.

Magnetic Control of Dual-Spin and Bias-Momentum Spacecraft

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.

Observer-Based Adaptive Spacecraft Attitude Control With Guaranteed Performance Bounds

This technical note considers observer-based adaptive attitude tracking of fully actuated spacecraft with known bounds on the disturbance torques acting on the spacecraft. Given any attitude and angular velocity observer with known ultimate bounds on the estimation errors, sequences of successively less conservative ultimate bounds on the attitude and angular velocity tracking errors are obtained.

Linear-Matrix-Inequality-Based Solution to Wahba’s Problem

WAHBA’S problem was introduced in 1965 by Grace Wahba [1] and is an important problem in aerospace engineering that typically involves finding an optimal rotation to fit a series of vector measurements. There have been many different methods developed to solve Wahba’s problem, both directly in terms of the rotation matrix [2,3] and in terms of the unit quaternion [4], the most famous method being QUEST [5]. A good survey of the different methods may be found in [6] and the references therein. Recently, a useful generalization of Wahba’s problem has been made, allowing the determination of both attitude and body rate using a time history of vector measurements (see [7–9]). However, this Note does not examine this problem and treats only Wahba’s original problem. This Note presents a new characterization of the solution to Wahba’s problem, directly in terms of the rotation matrix. It is shown that, under a mild condition (that is satisfied in many practical applications), Wahba’s problem may be recast as a convex linear matrix inequality (LMI) optimization problem. This opens the door to a whole new class of solvers for Wahba’s problem. This is accomplished by relaxing the nonconvex special orthogonal group [SO 3 ] constraint on the rotation matrix to a convex LMI constraint. This constraint relaxation approach has applications beyond the solution of Wahba’s problem and can potentially be useful for other optimization problems involving vehicle attitude, such as guidance and control problems. The remainder of the Note is organized as follows. Section II presents an overview ofWahba’s problem and its solution in terms of the singular value decomposition (SVD) [3]. Section III demonstrates that, under a mild condition, the Wahba problem may be recast as a LMI problem, leading to an identical solution, and conditions under which this mild condition is satisfied are investigated. Section IV presents a pair of numerical examples comparing the LMI-based solution to existing well-established solutions to Wahba’s problem. Section V contains concluding remarks. The appendix contains a technical mathematical result, which is used in the Note. II. Wahba’s Problem and Solution