Autonomous vehicles require control systems that ensure robust, stable, and high performance response. The most popular and common control designs depend on full knowledge of the internal state of the system, and a fundamental component of the whole system design is a separate state observer (for a deterministic system model) or filtering algorithm (for a stochastic system model) whose role is to provide state estimates to the control algorithm. The system's overall reliability depends crucially on this algorithm's performance, robustness, and stability. Typically, observers and filters must be an order of magnitude faster, more stable and robust than the dependent control design.

The Extended Kalman Filter (EKF) is a mainstream sensor fusion algorithm. In its classical form, however, it has many drawbacks in real robotics applications. It lacks strong convergence results and often requires careful tuning to work. The reality is that the EKF in poorly conditioned local coordinates and with non-Gaussian noise often displays very poor robustness and diverges in the presence of mild disturbances or poor initialisation.

Recent work on `invariant' or `equivariant' geometric observers has shown that there are canonical choices of coordinates for a whole class of robotics problems for which EKF inspired algorithms can be developed that overcome the inherent issues in classical EKF. The key idea is to exploit the rich geometric structure of the state space (symmetries, group structure, etc.) to develop algorithms enjoying good convergence properties and low computational cost. Significant progress has been made in the last ten years in the design of state estimation for attitude and pose, inertial navigation systems, tracking of homographies, and velocity aided attitude estimation or example, by exploiting properties of symmetry. These state estimators are becoming recognised as the state-of-the-art algorithms in industry and academia, motivating a recognition of the emerging field of invariant and equivariant systems theory.

This workshop focuses primarily on the key foundations of invariant and equivariant observer design frameworks. It offers a practical introduction to the field, emphasizing the application of Equivariant Systems Theory in systems and control. Participants will gain valuable insight into how to design observers and filters by leveraging the symmetry properties of systems. The objective is to demystify the equivariant observer design framework by applying sound engineering principles. Throughout the workshop, we will examine various case studies drawn from aerial robotic applications, including attitude estimation, velocity-aided attitude estimation, pose estimation, homography estimation, SLAM, and visual odometry. Through these examples, participants will develop a deep understanding of the potential of exploiting symmetries in filtering and state estimation within real-world robotic contexts.

The theoretical framework developed will provide the systems and control and robotics communities and companies with new and powerful tools ensuring robustness and reliability of key perception and control algorithms that solve challenging new problems and improve solutions considered previously.