This paper provides a new observer design methodology for invariant systems whose state evolves on a Lie group with outputs in a collection of related homogeneous spaces and where the measurement of system input is corrupted by an unknown constant bias. The key contribution of the paper is to study the combined state and input bias estimation problem in the general setting of Lie groups, a question for which only case studies of specific Lie groups are currently available. We show that any candidate observer (with the same state space dimension as the observed system) results in non-autonomous error dynamics, except in the trivial case where the Lie-group is Abelian. This precludes the application of the standard non-linear observer design methodologies available in the literature and leads us to propose a new design methodology based on employing invariant cost functions and general gain mappings. We provide a rigorous and general stability analysis for the case where the underlying Lie group allows a faithful matrix representation. We demonstrate our theory in the example of rigid body pose estimation and show that the proposed approach unifies two competing pose observers published in prior literature.
Most existing methods for satellite attitude control assume that full knowledge of satellite attitude is available by algebraic manipulation of at least two vector measurements from attitude sensors such as Sun, Earth-horizon, Earth-magnetic or star tracker sensors. Kalman filtering is usually used when only one vector measurement is available, however, asymptotic stability of nonlinear and uncertain dynamics of satellite is not guaranteed in this case. This technical note presents a coupled nonlinear estimator-controller for satellite attitude determination and control by using a 3-axis gyro and a single vector measurement for a fully actuated satellite. We assume that the moment-of-inertia matrix of satellite is unknown. Asymptotic convergence of the satellite attitude and angular velocity to their desired values is proven. Performance of the proposed controller is illustrated in a realistic case study by simulation where a magnetometer is used to provide the single vector measurement.Index Terms-Attitude Determination and control system (ADCS).
In this paper, we provide a general method of state estimation for a class of invariant systems on connected matrix Lie groups where the group velocity measurement is corrupted by an unknown constant bias. The output measurements are given by a collection of actions of a single Lie group on several homogeneous output spaces, a model that applies to a wide range of practical scenarios. The proposed observer consists of a group estimator part, providing an estimate of a bounded state evolving on the Lie group, and a bias estimator part, providing an estimate of the bias in the associated Lie algebra. We employ the gradient of a suitable invariant cost function on the Lie group as an innovation term in the group estimator. We design the bias estimator such that it guarantees uniform local exponential stability of the estimation error dynamics around the zero error state. We propose a systematic methodology for the design of suitable cost functions on Lie groups by lifting invariant cost functions from the homogeneous output spaces. We show that the resulting observer is implementable based on available sensor measurements if the homogeneous output spaces are reductive. As an example, we derive an observer for rigid body attitude using vector and gyro measurements with unknown constant gyro bias.
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