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Gradient-like observers for invariant dynamics on a Lie group
C. Lageman, J. Trumpf, R. Mahony
TL;DR
The paper addresses nonlinear observer design for fully measured kinematic systems evolving on finite-dimensional connected Lie groups. It characterizes synchrony and observer factorization into a model-determined internal model and gradient-like innovation, yielding almost-global convergence under Morse-Bott cost assumptions.
Problem
Nonlinear observer design on Lie groups seeks computationally simple state estimators with strong robustness and global-stability guarantees for autonomous robotic systems.
Method
The paper characterizes synchronous invariant systems and designs innovation terms from gradients of invariant or non-invariant cost functions, including a construction procedure for invariant costs.
Results
Under Morse-Bott cost and metric invariance assumptions, observer errors converge to the identity for generic initial conditions, with monotonic cost decrease and locally exponential convergence.
Takeaways & Limitations
The resulting theory provides a practical observer-design methodology for left- or right-invariant Lie-group kinematics with full state and velocity measurements.
Takeaways & Limitations
The innovation term must be chosen carefully to ensure that the observed trajectory is an asymptotically stable limit set of the observer trajectory.
Abstract
from arXiv · showhide
This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialised to systems on Lie groups. For invariant systems this leads to a general factorisation theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.
1 Introduction
The paper develops a general methodology for nonlinear observers on finite-dimensional connected Lie groups, motivated by robust state estimation for autonomous robotic systems. It characterizes observer structure through synchrony and combines system-determined internal models with designed innovation terms.
- Nonlinear Lie-group observers target computationally simple state estimation with strong robustness and global stability guarantees for autonomous robotic systems.
- Compared with nonlinear filters, observers generally require fewer computational resources but provide less information because filters estimate posterior distributions.
- The paper studies observers for kinematic systems evolving on finite-dimensional connected Lie groups with full measurements.
- Its methodology separates a synchronous internal-model term from an innovation term that drives estimates toward the system trajectory despite initialization or measurement errors.
- The resulting theory establishes almost global exponential convergence under mild assumptions and connects gradient-like canonical-error dynamics to the observer structure.
- The paper positions its methodology as a coherent framework for left- or right-invariant kinematics with suitable invariant Morse-Bott costs and full measurements.
2 Notation and problem formulation
The problem is state estimation for Lie-group kinematics using measurements of the state and velocity, with attitude and pose estimation serving as principal robotic examples. The formulation represents tangent vectors through Lie-group translations and treats left- and right-invariant system descriptions as equivalent through the adjoint action.
- The state space is a finite-dimensional connected Lie group G equipped with a Lie algebra and a Riemannian metric, without assuming metric invariance generally.
- The paper considers left-invariant systems with algebra-valued input u and an equivalent right-invariant representation with input v = Ad_X u.
- Observers fuse potentially noisy measurements of the state X and input velocity u or v to estimate X.
- Robotic examples: Attitude estimation models a rigid body on SO(3), with angular velocity represented in the body-fixed frame.
- Robotic examples: The examples discuss exteroceptive measurements for attitude and position, while linear-velocity measurement typically requires fusion of differentiated position and accelerometer data.
- Robotic examples: Full-pose estimation extends the setting to SE(3), combining attitude and translation through the semidirect-product group law (R, p)(S, q) = (RS, p + Rq).
- Scope: The paper demonstrates design principles through attitude and pose examples but does not address detailed sensor selection or noise characterization.
3 Synchrony and error functions
Synchrony is defined by constant evolution of an error function between systems sharing an input. For invariant systems, synchronous errors factor through canonical left or right invariant errors, yielding observer structures whose synchronous terms can be selected independently of the observed state.
- Two systems are E-synchronous when their error E remains constant along trajectories for every admissible input, initial condition, and time.
- The canonical right and left errors are invariant under simultaneous right or left transformations of both system states, respectively.
- Any error function making two invariant systems synchronous factors through a canonical invariant error followed by a smooth map from G to a manifold.
- Synchronous observer structure: The synchronous terms provide the first half of the observer template, while the innovation term must be chosen to make the observed trajectory an asymptotically stable limit set.
- Synchronous observer structure: For right-invariant plants, the corresponding synchronous constraints interchange the roles of the left and right errors.
- Synchronous observer structure: For a left-invariant plant, right synchrony produces left-invariant partner dynamics independent of the observed state, making them suitable as an observer’s internal model.
- Synchronous observer structure: Choosing left synchrony for a left-invariant plant yields partner dynamics depending on the left error and the observed state, which cannot be implemented in a real observer.
4 Internal models and innovation terms
The paper characterizes Lie-group observers as a synchronous internal-model term plus an innovation term, then proposes this decomposition as a general observer-design structure for invariant systems. The innovation must satisfy consistency conditions, while its stability properties require additional careful design.
- Observer decomposition: Any observer containing an internal model can be decomposed into a synchronous term and an innovation term.The synchronous term supplies the internal-model behavior, while the innovation term accounts for the remaining observer dynamics.
- Internal models: The internal model replicates the observed trajectory when supplied with the exact initial condition, state measurements, and input information.This defines the observer's model-reproduction role before innovation is added.
- Synchronous terms: For a left invariant system, the synchronous component has the form ˙ˆX = ˆXw, with w equal to the system input.The corresponding right-invariant construction is obtained analogously for right invariant systems.
- Innovation terms: The innovation term is a smooth tangent-vector correction that vanishes along corresponding system and observer trajectories.Its defining conditions require tangency at the estimate and zero correction on matched trajectories.
- Stability and conditions: The innovation conditions are least restrictive, so α must be designed carefully to make the observed trajectory an asymptotically stable limit set.The stronger condition α(Y,Y,w,t)=0 implies the trajectory-based consistency condition, but the two conditions are generally not equivalent.
- Observer design structure: The proposed left and right observer structures combine the appropriate invariant model term with α, using measurements of the state and input.The resulting observers need not retain the original left or right invariance because α need not be invariant.
5 Gradient observers
The paper designs nonlinear Lie-group observers by using gradients of suitable cost functions as innovation terms, while analyzing the resulting error dynamics. Under invariant cost and metric conditions, the observers converge for generic initial conditions, with locally exponential convergence near the identity; a construction extends the approach to pose estimation on SE(3).
- Innovation design: The proposed observers use the gradient with respect to the first state argument of a smooth cost function as the innovation term.The cost function is non-negative and has global minima on the diagonal, while the product-metric gradient splits by argument.
- Error dynamics: For matched invariant cost functions and Riemannian metrics, the appropriate canonical invariant error has gradient dynamics.For left-invariant systems this applies to the right-invariant error; for right-invariant systems it applies to the left-invariant error.
- Convergence: Under a Morse-Bott cost with a unique global minimum at the identity and no other local minima, both observer errors converge to the identity for generic initial conditions.The relevant cost decreases monotonically to its minimum and convergence is locally exponential near the identity.
- Noise: With suitably bounded input and state noise, the error expressions suggest at least a practical stability result for the corresponding observers.The noise model includes additive driving noise and multiplicative state noise.
- Invariant cost functions: The method provides a construction for right-invariant cost functions by setting f(X,Y)=g(XY^-1), preserving the required Morse-Bott minimum structure when g has suitable properties.This construction addresses the difficulty of obtaining right-invariant costs, especially for non-compact groups.
- SE(3) example: For pose estimation on SE(3), the constructed cost yields an observer whose right-invariant error converges to the identity for generic initial values.The paper states that this SE(3) observer was not previously proposed in the literature, while related prior work used a different position correction term.
6 Gradient-like observers
The section develops gradient-like observers whose innovation terms produce autonomous gradient dynamics for invariant errors, including when the cost function is non-invariant. Under Morse-Bott conditions, the resulting observers achieve almost-global convergence with monotonic cost decrease and local exponential convergence.
- Relaxing cost-function invariance still yields observers whose canonical invariant-error dynamics are gradients of the cost function.The left observer produces gradient dynamics for the canonical right invariant error, while the right observer does so for the canonical left invariant error.
- General observers use a synchronous internal-model term plus a gradient-like innovation term derived from a cost function.For left observers, the innovation uses the canonical right invariant error; for right observers, it uses the canonical left invariant error.
- When the metric and cost function have the appropriate invariance, these observers coincide with the earlier gradient observers.Right invariance applies to the cost function and metric for one observer orientation, and left invariance applies for the other.
- Morse-Bott costs with a global minimum at e and no other local minima give generic convergence of both errors to e.The result applies to both observer orientations and excludes nongeneric initial conditions associated with other critical structures.
- The relevant cost decreases monotonically toward f(e,e), and convergence is locally exponential near e.For the left filter this is f(E_r,e), while for the right filter it is f(E_l,e).
7 Conclusion
The paper presents a coherent theory and practical design methodology for nonlinear observers on Lie groups using invariant or non-invariant Morse-Bott cost functions. Its main scope boundary is the requirement for full measurements of both state and velocity.
- The theory covers invariant kinematics on Lie groups with matched invariant, non-degenerate, Morse-Bott cost functions and full measurements.The key contributions include observer equations and theorems establishing the associated structure and convergence properties.
- The paper provides practical methods for generating invariant cost functions and designing observers when the cost function is non-invariant.The latter methodology is developed for the practical case of non-invariant costs.
- The approach requires full measurement of both state and velocity, while partial state measurements are identified as future work.