arXiv subjects

Matthew R. Kirchner

Publications and source records attributed to Matthew R. Kirchner.

12 recordsLinked to original sources

arXiv2026

A Data-Driven Algorithm for Model-Free Control Synthesis

Presented is an algorithm to synthesize the optimal infinite-horizon LQR feedback controller for continuous-time systems. The algorithm does not require knowledge of the system dynamics but instead uses only a finite-length sampling of input-output data. A necessary condition that relates the optimal LQR gain to any arbitrary solution trajectory of the system is presented. An algorithm using this necessary condition, based on constrained optimization, is developed that estimates the LQR gain matrix. In addition to calculating the standard feedback gain matrix, a feedforward gain can be found to implement a reference tracking controller. This paper presents a theoretical justification for the method and shows several examples, including a validation test flight on a real scale aircraft with unknown dynamics.

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arXiv2026

Human as an Actuator Dynamic Model Identification

This paper presents a method for estimating parameters that form a general model for human pilot response for specific tasks. The human model is essential for the dynamic analysis of piloted vehicles. Data are generated on a simulator with multiple trials being incorporated to find the single model that best describes the data. The model is found entirely in the time domain by constructing a constrained optimization problem. This optimization problem implicitly represents the state of the underlying system, making it robust to natural variation in human responses. It is demonstrated by estimating the human response model for a position control task with a quadcopter drone.

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arXiv2025

A Model-Free Data-Driven Algorithm for Continuous-Time Control

Presented is an algorithm to synthesize an infinite-horizon LQR optimal feedback controller for continuous-time systems. The algorithm does not require knowledge of the system dynamics, but instead uses only a finite-length sampling of (possibly suboptimal) input-output data. The algorithm is based on a constrained optimization problem that enforces a necessary condition on the dynamics of the optimal value function along an arbitrary trajectory. This paper presents the derivation as well as shows examples applied to both linear and nonlinear systems inspired by air vehicles.

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arXiv2023

Trajectories for the Optimal Collection of Information

We study a scenario where an aircraft has multiple heterogeneous sensors collecting measurements to track a target vehicle of unknown location. The measurements are sampled along the flight path and our goals to optimize sensor placement to minimize estimation error. We select as a metric the Fisher Information Matrix (FIM), as "minimizing" the inverse of the FIM is required to achieve small estimation error. We propose to generate the optimal path from the Hamilton-Jacobi (HJ) partial differential equation (PDE) as it is the necessary and sufficient condition for optimality. A traditional method of lines (MOL) approach, based on a spatial grid, lends itself well to the highly non-linear and non-convex structure of the problem induced by the FIM matrix. However, the sensor placement problem results in a state space dimension that renders a naive MOL approach intractable. We present a new hybrid approach, whereby we decompose the state space into two parts: a smaller subspace that still uses a grid and takes advantage of the robustness to non-linearities and non-convexities, and the remaining state space that can by found efficiently from a system of ODEs, avoiding formation of a spatial grid.

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arXiv2022

The Impact of Message Passing in Agent-Based Submodular Maximization

This paper considers a set of sensors, which as a group are tasked with taking measurements of the environment and sending a small subset of the measurements to a centralized data fusion center, where the measurements will be used to estimate the overall state of the environment. The sensors' goal is to send the most informative set of measurements so that the estimate is as accurate as possible. This problem is formulated as a submodular maximization problem, for which there exists a well-studied greedy algorithm, where each sensor sequentially chooses a set of measurements from its own local set, and communicates its decision to the future sensors in the sequence. In this work, sensors can additionally share measurements with one another, in order to augment the decision set of each sensor. We explore how this increase in communication can be exploited to improve the results of the nominal greedy algorithm. Specifically, we show that this measurement passing can improve the quality of the resulting measurement set by up to a factor of $n+1$, where $n$ is the number of sensors.

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arXiv2021

A Hamilton-Jacobi Formulation for Optimal Coordination of Heterogeneous Multiple Vehicle Systems

We present a method for optimal coordination of multiple vehicle teams when multiple endpoint configurations are equally desirable, such as seen in the autonomous assembly of formation flight. The individual vehicles' positions in the formation are not assigned a priori and a key challenge is to find the optimal configuration assignment along with the optimal control and trajectory. Commonly, assignment and trajectory planning problems are solved separately. We introduce a new multi-vehicle coordination paradigm, where the optimal goal assignment and optimal vehicle trajectories are found simultaneously from a viscosity solution of a single Hamilton-Jacobi (HJ) partial differential equation (PDE), which provides a necessary and sufficient condition for global optimality. Intrinsic in this approach is that individual vehicle dynamic models need not be the same, and therefore can be applied to heterogeneous systems. Numerical methods to solve the HJ equation have historically relied on a discrete grid of the solution space and exhibits exponential scaling with system dimension, preventing their applicability to multiple vehicle systems. By utilizing a generalization of the Hopf formula, we avoid the use of grids and present a method that exhibits polynomial scaling in the number of vehicles.

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arXiv2020

Reachability as a Unifying Framework for Computing Helicopter Safe Operating Conditions and Autonomous Emergency Landing

We present a numeric method to compute the safe operating flight conditions for a helicopter such that we can ensure a safe landing in the event of a partial or total engine failure. The unsafe operating region is the complement of the backwards reachable tube, which can be found as the sub-zero level set of the viscosity solution of a Hamilton-Jacobi (HJ) equation. Traditionally, numerical methods used to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension, which is problematic for the high-fidelity dynamics models required for accurate helicopter modeling. We avoid the use of spatial grids by formulating a trajectory optimization problem, where the solution at each initial condition can be computed in a computationally efficient manner. The proposed method is shown to compute an autonomous landing trajectory from any operating condition, even in non-cruise flight conditions.

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arXiv2020

Heterogeneous Measurement Selection for Vehicle Tracking using Submodular Optimization

We study a scenario where a group of agents, each with multiple heterogeneous sensors are collecting measurements of a vehicle and the measurements are transmitted over a communication channel to a centralized node for processing. The communication channel presents an information-transfer bottleneck as the sensors collect measurements at a much higher rate than what is feasible to transmit over the communication channel. In order to minimize the estimation error at the centralized node, only a carefully selected subset of measurements should be transmitted. We propose to select measurements based on the Fisher information matrix (FIM), as "minimizing" the inverse of the FIM is required to achieve small estimation error. Selecting measurements based on the FIM leads to a combinatorial optimization problem. However, when the criteria used to select measurements is both monotone and submodular it allows the use of a greedy algorithm that is guaranteed to be within $1-1/e\approx63\%$ of the optimum and has the critical benefit of quadratic computational complexity. To illustrate this concept, we derive the FIM criterion for different sensor types to which we apply FIM-based measurement selection. The criteria considered include the time-of-arrival and Doppler shift of passively received radio transmissions as well as detected key-points in camera images.

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arXiv2019

A Level Set Approach to Online Sensing and Trajectory Optimization with Time Delays

Presented is a method to compute certain classes of Hamilton-Jacobi equations that result from optimal control and trajectory generation problems with time delays. Many robotic control and trajectory problems have limited information of the operating environment a priori and must continually perform online trajectory optimization in real time after collecting measurements. The sensing and optimization can induce a significant time delay, and must be accounted for when computing the trajectory. This paper utilizes the generalized Hopf formula, which avoids the exponential dimensional scaling typical of other numerical methods for computing solutions to the Hamilton-Jacobi equation. We present as an example a robot that incrementally predicts a communication channel from measurements as it travels. As part of this example, we introduce a seemingly new generalization of a non-parametric formulation of robotic communication channel estimation. New communication measurements are used to improve the channel estimate and online trajectory optimization with time-delay compensation is performed.

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arXiv2018

Automatic Thresholding of SIFT Descriptors

We introduce a method to perform automatic thresholding of SIFT descriptors that improves matching performance by at least 15.9% on the Oxford image matching benchmark. The method uses a contrario methodology to determine a unique bin magnitude threshold. This is done by building a generative uniform background model for descriptors and determining when bin magnitudes have reached a sufficient level. The presented method, called meaningful clamping, contrasts from the current SIFT implementation by efficiently computing a clamping threshold that is unique for every descriptor.

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arXiv2018

A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems

Presented is a method for efficient computation of the Hamilton-Jacobi (HJ) equation for time-optimal control problems using the generalized Hopf formula. Typically, numerical methods to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension. The generalized Hopf formula avoids the use of grids and numerical gradients by formulating an unconstrained convex optimization problem. The solution at each point is completely independent, and allows a massively parallel implementation if solutions at multiple points are desired. This work presents a primal-dual method for efficient numeric solution and presents how the resulting optimal trajectory can be generated directly from the solution of the Hopf formula, without further optimization. Examples presented have execution times on the order of milliseconds and experiments show computation scales approximately polynomial in dimension with very small high-order coefficients.

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arXiv2017

Time-Optimal Collaborative Guidance Using the Generalized Hopf Formula

Presented is a new method for calculating the time-optimal guidance control for a multiple vehicle pursuit-evasion system. A joint differential game of k pursuing vehicles relative to the evader is constructed, and a Hamilton-Jacobi-Isaacs (HJI) equation that describes the evolution of the value function is formulated. The value function is built such that the terminal cost is the squared distance from the boundary of the terminal surface. Additionally, all vehicles are assumed to have bounded controls. Typically, a joint state space constructed in this way would have too large a dimension to be solved with existing grid-based approaches. The value function is computed efficiently in high-dimensional space, without a discrete grid, using the generalized Hopf formula. The optimal time-to-reach is iteratively solved, and the optimal control is inferred from the gradient of the value function.

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