Bias estimation for invariant systems on Lie groups with homogeneous outputs

Abstract

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.

Higher Order Nonlinear Complementary Filtering on Lie Groups

“…However, as mentioned in [5], there is no canonical choice for the innovation term for Lie group observers, and as such, its selection must be carefully considered. A method to find suitable innovation terms is considered in [2], where symmetry preserving innovation terms are found via the moving frame method [2], [6]. Alternatively, in [3] and [5] the innovation term is chosen based on the gradient descent direction of a selected invariant cost function and the observer is shown to be almost globally asymptotically stable about the point where the estimated state is equal to the true state.…”

“…In this paper, a nonlinear observer whose system state evolves on a Lie group is considered. The approach taken is similar to previous Lie group observers presented in the literature, including [5], [6], as the innovation is related to the gradient of a cost function. However, while the gradient of the cost function appears directly as the innovation term in [5], in this paper the innovation is based on the output of a linear time-invariant (LTI) system whose input is the gradient of a cost function resolved in a basis of the tangent space.…”

“…Analogous to the classical complementary filter, the proposed method can be understood as a nonlinear complementary filter on a Lie group. Previous observers, including [5] and [6], are analogous to the classical complementary filter with first order sensitivity and complementary sensitivity transfer functions. The introduction of the LTI system in the proposed observer allows for more general and complex higher-order filtering.…”

“…Further, the proposed method is a direct extension of the gradient observer proposed in [5] in that the proposed observer reduces to the gradient observer with the selection of a particular static linear system. As in [6] and [7], the case where velocity measurements are corrupted by constant bias is also considered. However, the solutions given in [6] and [7] are extended in this paper to include harmonic disturbances as well as constant bias.…”

“…As in [6] and [7], the case where velocity measurements are corrupted by constant bias is also considered. However, the solutions given in [6] and [7] are extended in this paper to include harmonic disturbances as well as constant bias. This is done by introducing a disturbance observer that incorporates an internal model of the harmonic disturbances.…”