Although the classical tools of nonlinear control theory provide powerful tools for analysing observer and filter design, existing theory lacks a cohesive understanding tailored to the principles of symmetry. Symmetry is a fundamental property of physical systems that applies to a wide range of system models in modern robotics, computer vision, and navigation systems. To understand symmetry it is necessary to understand the role of velocity and state measurement for a kinematic system at a fundamental level. This, in turn, leads to an even more fundamental question about what is meant by a kinematic model in the context of control systems theory.

We consider observer synthesis using Lyapunov design principles for fundamental systems such as the special orthogonal group, the special linear group, and the special linear group. We begin with cost functions on the outputs that can be realized from available measurements. By imposing invariance on the costs, we can lift these costs to a non-degenerate cost in the error coordinates. For an actual design problem, the simplest approach is to undertake a direct Lyapunov design process, and we provide examples to demonstrate how this can be done. More generally, we show how the cost can be used to define an equivariant gradient innovation once an invariant metric on the Lie group is defined. This construction leads to a gradient flow in the error coordinates that is straightforward to analyze for stability. We also provide the intuition and the main formulas required to extend the proposed design methodology to an observer that also estimates either an unknown constant bias offset in the measured velocity or a time-varying bias based on the internal model principle. Finally, we provide experimental evidence of the performance of the observer design methodology on different useful examples.

The classic extended Kalman filter and its variants are not well-suited to dealing with states in non-Euclidean spaces, such as Lie groups. Various estimators that intrinsically consider the constraints of the states have been developed, mainly by adapting the Kalman filter equations on the Lie groups. A different path can be taken by leveraging the Bayesian interpretation of the Kalman filter. The concentrated Gaussian distribution on Lie groups can be seen as an equivalent of the normal distribution in Euclidean space because it minimizes the entropy. This parallel paves the way to developing new Bayesian estimators on Lie groups by assuming that the posterior distribution is a concentrated Gaussian distribution and then adjusting its first and second-order moments. Such an approach facilitates the evaluation of the uncertainty in the estimates. In this framework, we present different filters, depending on whether a discrete or continuous-time state evolution is considered, and based on different approximation techniques (extended or iterated Kalman filtering).

Regarding confidence in estimates, we compute posterior Cramér-Rao bounds, indicative of minimum achievable error. However, these bounds are not directly applicable to Lie groups. To address this, we introduce a novel performance bound tailored to the intrinsic error criterion of Lie groups.

This talk discusses challenges in inertial state estimation algorithms and presents a solution using the tangent group GTG : SE2(3) ⋉ se2(3) as Lie group symmetries. While this symmetry was extended for multi-sensor-aided inertial systems, the focus here is on visual-inertial navigation systems. The Multi-State Constraint Equivariant Filter (MSCEqF) is introduced as an equivariant filter with self-calibration capabilities, addressing IMU biases and providing autonomous navigation error dynamics. The MSCEqF demonstrates superior accuracy, robustness, and consistency performance compared to existing solutions. Real-world scenario evaluations showcase its effectiveness against expected and unexpected errors, and its real-time capabilities are demonstrated in closed-loop control of a resource-constrained aerial platform.

The bundle framework for nonlinear observer design on a manifold with a Lie group action has been relatively unexplored. The group action decomposes the manifold into a quotient structure and an orbit space, and the problem of observer design for the entire system can likewise be decomposed into a design over the orbit (group) space and a design over the quotient space. The emphasis, almost always, is on the observer design in the orbit space. Since many observer design problems are found in mechanical or aerospace systems where the second-order dynamics rule the roost, and additional structures such as the affine connection come into play, it is worth our while to examine such a mathematical structure and its implications on observer design.

Lie theory has been heavily applied to various problems in control and robotics since at least the 1970s. The success of the complementary filter and invariant EKF for attitude estimation in the mid-2000s has driven renewed interest in estimating systems with Lie symmetry. Group-affine systems are a broad class of systems on homogeneous spaces for which the dynamics may be decomposed into left-invariant and normalizing vector fields of the symmetry group. We discuss the relationship between this class of systems and the notion of a synchronous observer, that is, an observer for which the observer error function is governed only by the chosen correction function and is stationary when the correction function is nullified. We discuss the inertial navigation problem using this perspective and, through it, demonstrate the power of the synchrony property by describing the first known solutions to feature almost global asymptotic stability.

Inertial navigation systems (INS) are important in operating autonomous aerial vehicles. They combine measurements from translational motion sensors (accelerometers) and rotational motion sensors (gyroscopes) to track a vehicle's position, velocity, and orientation with respect to a reference frame. Inertial navigation systems are usually aided by other sensors, which provide full or partial position information expressed in an inertial or body frame.  Examples of such sensors are those that provide full position measurements (e.g., GPS), range measurements (e.g., Ultra-Wide Band (UWB) sensors), stereo or monocular bearing measurements (e.g., vision system), or altitude measurements (altimeter).

We propose universal nonlinear observers capable of estimating the extended pose of the vehicle (position, linear velocity, and attitude) using an IMU and the available position information. The innovation terms of the proposed observers are derived via direct inspection of the geometric estimation errors,  which are different from the standard linear errors used in classical EKF-based filters. Unlike the existing local Kalman-type filters, our proposed nonlinear observers have strong stability results (e.g., almost semi-global asymptotic/exponential stability). Detailed uniform observability analysis has been conducted for different types of measurements, and sufficient conditions have been derived.

To establish connections and explore the ever-present duality between control and observer design, we discuss the problem of designing a geometric trajectory tracking controller for slung-load transportation using a quadrotor. The feedback laws are defined as direct functions of the system's internal state represented in its natural space, namely SO(3) for attitude and the 2-sphere for the cable carrying the suspended. A direct Lyapunov design is applied to provide stability guarantees and a large region of attraction for which a conservative estimate can be obtained. Experimental results showcase the performance of the controller design in a demanding scenario involving large but controlled cable angles and accelerations to cross a small window frame.