Equivariant Filter Design for Discrete-time systems (2209.04965v1)

Abstract: The kinematics of many nonlinear control systems, especially in the robotics field, admit a transitive Lie-group symmetry, which is useful in high performance observer design. The recently proposed equivariant filter (EqF) exploits equivariance to generate high performance filters for a wide range of real-world systems. However, existing work on the equivariant filter, and equivariance of control systems in general, is based on a continuous-time formulation. In this paper, we first present the equivariant structure of a discrete-time system. We then use this to propose a discrete-time version of the equivariant filter. A novelty of the approach is that the geometry of the symmetry group naturally appears as parallel transport in the reset step of the filter. Preliminary results for linear second order kinematics with separate bearing and range measurements indicate that the discrete EqF significantly outperforms both a discretized version of the continuous EqF and a classical discrete EKF.

• The paper presents a discrete-time equivariant filter that extends continuous observer design using Lie-group symmetry.
• It integrates geometric techniques, such as parallel transport in the reset step, to maintain covariance consistency and improve convergence.
• Numeric simulations on a kinematic system demonstrate significantly enhanced transient response and asymptotic performance compared to classical EKF.

Equivariant Filter Design for Discrete-time Systems

This paper presents a significant advancement in the design of observers for discrete-time systems characterized by state symmetry and formulated on homogeneous spaces. The central contribution is the introduction of an Equivariant Filter (EqF) that exploits the transitive Lie-group symmetry inherent in many nonlinear control systems, particularly in robotics, to construct high-performance filters. Previous work in this area has largely focused on continuous-time systems, and the authors address this gap by developing a discrete-time version of the EqF.

Core Contributions

The paper's focus is on discrete-time systems on manifolds that allow for a transitive group action by a Lie group, resulting in equivariance. The key technical advancement is the discrete-time extension of the EqF, which was previously restricted to continuous systems. The authors exhibit how the geometry of symmetry groups naturally manifests as parallel transport in the filter's reset step, innovatively integrating geometric considerations directly into the observer design.

This discrete-time EqF significantly outperforms a straightforward discretization of the continuous-time EqF and a classical discrete Extended Kalman Filter (EKF) in numeric simulations. The improvement is attributed to the careful attention given to the parallel transport of the covariance during the reset step, maintaining consistency with the natural symmetries of the system.

Numerical Insights

Strong numerical results are provided through simulations of a linear second-order kinematic system in three-dimensional space with separate bearing and range measurements. These results underpin several claims, demonstrating the discrete EqF's superior performance in the transient response and significant enhancements in asymptotic behavior compared to both the continuous-time EqF and the classical EKF.

The simulation results highlighted:

• The benefit of including a reset covariance step in the discrete EqF when compared to a variant without this feature. This adjustment makes a marked difference in early filter convergence, where initial errors are significant.
• The discrete-time EqF shows enhanced asymptotic error convergence relative to the other filters evaluated, affirming the theoretical advantages posited by the authors.

Theoretical and Practical Implications

The implications of this work are multi-faceted. Theoretically, it extends the framework of equivariant systems to discrete-time scenarios, providing a robust grounding for further exploration of symmetry-based observer design beyond continuous domains. Practically, this approach benefits various applications in robotics and control where systems are naturally described in discrete time, including scenarios without straightforward or stable continuous-time equivalents.

Future Directions

The work lays a foundation for further research in extending equivariance principles to broader classes of discrete systems and investigating more complex manifolds and symmetry groups. One area for future exploration is the development of more comprehensive methodologies for the algebraic manipulation of the geometric structures encountered in these systems. Additionally, the incorporation of other affine connections and deeper exploration of their impact on system performance presents an intriguing opportunity for theoretical advancement.

Overall, this paper adds substantial value to the field of observer design for discrete systems, offering a well-grounded methodology that balances theoretical innovation with practical performance improvements. It opens avenues for the deployment of symmetry-based techniques in real-world applications, especially within robotics, where discrete-time models are prevalent.