Spacecraft Attitude Tracking with Guaranteed Performance Bounds

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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