Jochen Trumpf

Gradient-Like Observers for Invariant Dynamics on a Lie Group

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 specialized to systems on Lie groups. For invariant systems this leads to a general factorization 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.

Frequency, Temperature and Salinity Variation of the Permittivity of Seawater

With the emergence of unmanned marine robots, underwater communication systems have received much attention in recent years. To successfully develop radio wave based communication solutions, it is essential to understand properties of electromagnetic wave transmission in seawater. These properties are determined by the frequency variation of the permittivity of seawater. Existing models for the permittivity of saline water are empirical ones that best fit experimental data. We propose a physically realistic model, similar to the one used in plasma physics, for the variation of the dielectric constant of water with varying frequencies and salinities. Our model is in excellent agreement with existing empirical fits for frequencies between 1 and 256 GHz. We use this model to study the propagation of electromagnetic waves in seawater. We explain that large propagation distances would be possible at MHz frequencies if the conductivity of seawater decreases at small field strengths due to the hydrogen bonding of water molecules. However, we were unable to experimentally verify any reduction in the conductivity of seawater

L1 rotation averaging using the Weiszfeld algorithm

We consider the problem of rotation averaging under the L1 norm. This problem is related to the classic Fermat-Weber problem for finding the geometric median of a set of points in IRn. We apply the classical Weiszfeld algorithm to this problem, adapting it iteratively in tangent spaces of SO(3) to obtain a provably convergent algorithm for finding the L1 mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L1 mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L2 norm (the usual averaging process). We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L1 optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L2 averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set.

Nonlinear attitude observers on SO(3) for complementary and compatible measurements: A theoretical study

This paper considers the question of designing an attitude observer exploiting the structure of the Special Orthogonal Group SO(3) for both inertial and body-fixedframe measurements. We consider measurements from a minimal sensor suite, typically a rate gyroscope along with several measurements of inertial and/or body-fixed vector directions. We propose fully nonlinear pose observers based directly on the vector measurements, allowing a mix of both inertial and body-fixed measurements.We provide a comprehensive stability analysis of the observer error dynamics. We show that even with a single vector measurement it is often possible to recover asymptotically stable observer error dynamics.

Newton-like methods for numerical optimization on manifolds

Many problems in signal processing require the numerical optimization of a cost function, which is defined on a smooth manifold. Especially, orthogonally or unitarily constrained optimization problems tend to occur in signal processing tasks involving subspaces. In this paper we consider Newton-like methods for solving these types of problems. Under the assumption that the parameterization of the manifold is linked to so-called Riemannian normal coordinates our algorithms can be considered as intrinsic Newton methods. Moreover, if there is not such a relationship, we still can prove local quadratic convergence to a critical point of the cost function by means of analysis on manifolds. Our approach is demonstrated by a detailed example, i.e., computing the dominant eigenspace of a real symmetric matrix.

A nonlinear observer for 6 DOF pose estimation from inertial and bearing measurements

This paper considers the problem of estimating pose from inertial and bearing-only vision measurements. We present a non-linear observer that evolves directly on the Special Euclidean group SE(3) from inertial measurements and bearing measurements, such as provided by a visual system tracking known landmarks. Local asymptotic convergence of the observer is proved. The observer is computationally simple and its gains are easy to tune. Simulation results demonstrate robustness to measurement noise and initial conditions.

Rotation averaging with application to camera-rig calibration

We present a method for calibrating the rotation between two cameras in a camera rig in the case of non-overlapping fields of view and in a globally consistent manner. First, rotation averaging strategies are discussed and an L1-optimal rotation averaging algorithm is presented which is more robust than the L2-optimal mean and the direct least squares mean. Second, we alternate between rotation averaging across several views and conjugate rotation averaging to achieve a global solution. Various experiments both on synthetic data and a real camera rig are conducted to evaluate the performance of the proposed algorithm. Experimental results suggest that the proposed algorithm realizes global consistency and a high precision estimate.

Minimum-Energy Filtering for Attitude Estimation

In this work, we study minimum-energy filtering for attitude kinematics with vectorial measurements using Mortensens approach. The exact form of a minimum-energy attitude observer is derived and is shown to depend on the Hessian of the value function of an associated optimal control problem. A suitably chosen matrix representation of the Hessian operator leads to a Riccati equation that approximates a minimum-energy attitude filter. An extended version of the proposed approximate filter is included for a situation where there is slowly time-varying bias in the gyro measurements. A unit quaternion version of the proposed filter is derived and shown to outperform the multiplicative extended Kalman filter (MEKF) for situations with large initialization errors or large measurement errors.

Analysis of Non-Linear Attitude Observers for Time-Varying Reference Measurements

This paper provides a comprehensive stability analysis of a suite of nonlinear attitude observers that have been developed over the last few years. The observers considered are based on vectorial measurements of an a priori known reference direction. By treating the reference direction and the measurement in the same analysis framework, and allowing time-variation of either, we are able to define general persistency of excitation criteria that incorporate and generalize convergence criteria used in prior work. A key outcome is conditions that ensure almost global asymptotic and local exponential stability of attitude observers based on a single vector measurement as long as the excitation conditions are met on the reference and system trajectory. The approach generalizes stability results provided in prior work, based on rank conditions, that required at least two or more vector measurements.

Observer design on the Special Euclidean group SE(3)

This paper proposes a nonlinear pose observer designed directly on the Lie group structure of the Special Euclidean group SE(3). We use a gradient-based observer design approach and ensure that the derived observer innovation can be implemented from position measurements. We prove local exponential stability of the error and instability of the non-zero critical points. Simulations indicate that the observer is indeed almost globally stable as would be expected.