arXiv subjects

James Goppert

Publications and source records attributed to James Goppert.

6 recordsLinked to original sources

arXiv2025

Log-linear Dynamic Inversion for Thrusting Spacecraft on SE2(3)

We demonstrate that the error dynamics of a thrusting spacecraft are nearly group affine on the $SE_2(3)$ Lie group, and the nonlinearity can be bounded, or removed with the application of a dynamic inversion control law. A numerical example validates the results by showing agreement between the error predicted by the log-dynamics and the error obtained from classical integration of trajectories using Newtonian dynamics. The result clarifies how thrusting spacecraft dynamics fit within the invariant systems framework.

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arXiv2025

Log-linear Backstepping control on $SE_2(3)$

Most of the rigid-body systems which evolve on nonlinear Lie groups where Euclidean control designs lose geometric meaning. In this paper, we introduce a log-linear backstepping control law on SE2(3) that preserves full rotational-translational coupling. Leveraging a class of mixed-invariant system, which is a group-affine dynamic model, we derive exact logarithmic error dynamics that are linear in the Lie algebra. The closed-form expressions for the left- and right-Jacobian inverses of SE2(3) are expressed in the paper, which provides us the exact error dynamics without local approximations. A log-linear backstepping control design ensures exponential stability for our error dynamics; since our error dynamics is a block-triangular structure, this allows us to use Linear Matrix Inequality (LMI) formulation or $H_\infty$ gain performance design. This work establishes the exact backstepping framework for a class of mixed-invariant system, providing a geometrically consistent foundation for future Unmanned Aerial Vehicle (UAV) and spacecraft control design.

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arXiv2024

Application of Log-Linear Dynamic Inversion Control to a Multi-rotor

This paper presents an approach that employs log-linearization in Lie group theory and the Newton-Euler equations to derive exact linear error dynamics for a multi-rotor model, and applies this model with a novel log-linear dynamic inversion controller to simplify the nonlinear distortion and enhance the robustness of the log-linearized system. In addition, we utilize Linear Matrix Inequalities (LMIs) to bound the tracking error for the log-linearization in the presence of bounded disturbance input and use the exponential map to compute the invariant set of the nonlinear system in the Lie group. We demonstrate the effectiveness of our method via an illustrative example of a multi-rotor system with a reference trajectory, and the result validates the safety guarantees of the tracking error in the presence of bounded disturbance.

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arXiv2023

A Closed-form Solution for the Strapdown Inertial Navigation Initial Value Problem

Strapdown inertial navigation systems (SINS) are ubiquitious in robotics and engineering since they can estimate a rigid body pose using onboard kinematic measurements without knowledge of the dynamics of the vehicle to which they are attached. While recent work has focused on the closed-form evolution of the estimation error for SINS, which is critical for Kalman filtering, the propagation of the kinematics has received less attention. Runge-Kutta integration approaches have been widely used to solve the initial value problem; however, we show that leveraging the special structure of the SINS problem and viewing it as a mixed-invariant vector field on a Lie group, yields a closed form solution. Our closed form solution is exact given fixed gyroscope and accelerometer measurements over a sampling period, and it is utilizes 12 times less floating point operations compared to a single integration step of a 4th order Runge-Kutta integrator. We believe the wide applicability of this work and the efficiency and accuracy gains warrant general adoption of this algorithm for SINS.

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arXiv2023

Exploiting Sparsity for Localization of Large-Scale Wireless Sensor Networks

Wireless Sensor Network (WSN) localization refers to the problem of determining the position of each of the agents in a WSN using noisy measurement information. In many cases, such as in distance and bearing-based localization, the measurement model is a nonlinear function of the agents' positions, leading to pairwise interconnections between the agents. As the optimal solution for the WSN localization problem is known to be computationally expensive in these cases, an efficient approximation is desired. In this paper, we show that the inherent sparsity in this problem can be exploited to greatly reduce the computational effort of using an Extended Kalman Filter (EKF) for large-scale WSN localization. In the proposed method, which we call the Low-Bandwidth Extended Kalman Filter (LB-EKF), the measurement information matrix is converted into a banded matrix by relabeling (permuting the order of) the vertices of the graph. Using a combination of theoretical analysis and numerical simulations, it is shown that typical WSN configurations (which can be modeled as random geometric graphs) can be localized in a scalable manner using the proposed LB-EKF approach.

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arXiv2022

Log-linear Dynamic Inversion Control with Provable Safety Guarantees in Lie Groups

In this paper, we use the derivative of the exponential map to derive the exact evolution of the logarithm of the tracking error for mixed-invariant systems, a class of systems capable of describing rigid body tracking problems in Lie groups. Additionally, we design a log-linear dynamic inversion-based control law to remove the nonlinearities due to spatial curvature and enhance the robustness of the controller. We apply Linear Matrix Inequalities (LMIs) to bound the tracking error given a bounded disturbance amplified by the distortion matrix and leverage the tracking error bound to create flow pipes. To demonstrate the usefulness of our method, we show its application with Urban Air Mobility (UAM) scenarios using a simplified kinematic aircraft model and polynomial-based path planning methods.

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