Jochen Trumpf

Uwe Helmke, Knut Hüper, Jochen Trumpf

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties.

Jack Henderson, Mohammad Zamani, Robert Mahony et al.

Collaborative Localisation has been studied extensively in recent years as a way to improve pose estimation of unmanned aerial vehicles in challenging environments. However little attention has been paid toward advancing the underlying filter design beyond standard Extended Kalman Filter-based approaches. In this paper, we detail a discrete-time collaborative localisation filter using the deterministic minimum-energy framework. The filter incorporates measurements from an inertial measurement unit and models the effects of sensor bias and gravitational acceleration. We present a simulation based on real-world vehicle trajectories and IMU data that demonstrates how collaborative localisation can improve performance over single-vehicle methods.

Jack Henderson, Mohammad Zamani, Robert Mahony et al.

Accurate localisation of unmanned aerial vehicles is vital for the next generation of automation tasks. This paper proposes a minimum energy filter for velocity-aided pose estimation on the extended special Euclidean group. The approach taken exploits the Lie-group symmetry of the problem to combine Inertial Measurement Unit (IMU) sensor output with landmark measurements into a robust and high performance state estimate. We propose an asynchronous discrete-time implementation to fuse high bandwidth IMU with low bandwidth discrete-time landmark measurements typical of real-world scenarios. The filter's performance is demonstrated by simulation.

Pieter van Goor, Robert Mahony, Tarek Hamel et al.

Visual Simultaneous Localisation and Mapping (VSLAM) is a key enabling technology for small embedded robotic systems such as aerial vehicles. Recent advances in equivariant filter and observer design offer the potential of a new generation of highly robust algorithms with low memory and computation requirements for embedded system applications. This paper studies observer design on the symmetry group proposed in previous work by the authors, in the case where inverse depth measurements are available. Exploiting this symmetry leads to a simple fully non-linear gradient based observer with almost global asymptotic and local exponential stability properties. Simulation experiments verify the observer design, and demonstrate that the proposed observer achieves similar accuracy to the widely used Extended Kalman Filter with significant gains in processing time (linear verses quadratic bounds with respect to number of landmarks) and qualitative improvements in robustness.

Jack Henderson, Jochen Trumpf, Mohammad Zamani

We present a new approach to the cooperative localisation problem by applying the theory of minimum energy filtering. We consider the problem of estimating the pose of a group of mobile robots in an environment where robots can perceive fixed landmarks and neighbouring robots as well as share information with others over a communication channel. Whereas the vast majority of the existing literature applies some variant of a Kalman Filter, we derive a set of filter equations for the global state estimate based on the principle of minimum energy filtering. We show how the filter equations can be decoupled and the calculations distributed among the robots in the network without requiring a central processing node. Finally, we provide a demonstration of the filter's performance in simulation.

Pieter van Goor, Robert Mahony, Tarek Hamel et al.

This paper builds on recent work on Simultaneous Localisation and Mapping (SLAM) in the non-linear observer community, by framing the visual localisation and mapping problem as a continuous-time equivariant observer design problem on the symmetry group of a kinematic system. The state-space is a quotient of the robot pose expressed on SE(3) and multiple copies of real projective space, used to represent both points in space and bearings in a single unified framework. An observer with decoupled Riccati-gains for each landmark is derived and we show that its error system is almost globally asymptotically stable and exponentially stable in-the-large.

Minh-Duc Hua, Jochen Trumpf, Tarek Hamel et al.

This paper presents a new algorithm for online estimation of a sequence of homographies applicable to image sequences obtained from robotic vehicles equipped with vision sensors. The approach taken exploits the underlying Special Linear group structure of the set of homographies along with gyroscope measurements and direct point-feature correspondences between images to develop temporal filter for the homography estimate. Theoretical analysis and experimental results are provided to demonstrate the robustness of the proposed algorithm. The experimental results show excellent performance even in the case of very fast camera motion (relative to frame rate), severe occlusion, and in the presence of specular reflections.

Brian D. O. Anderson, Guodong Shi, Jochen Trumpf

We study continuous-time consensus dynamics for multi-agent systems with undirected switching interaction graphs. We establish a necessary and sufficient condition for exponential asymptotic consensus based on the classical theory of complete observability. The proof is remarkably simple compared to similar results in the literature and the conditions for consensus are mild. This observability-based method can also be applied to the case where negatively weighted edges are present. Additionally, as a by-product of the observability based arguments, we show that the nodes' initial value can be recovered from the signals on the edges up to a shift of the network average.