arXiv subjects

William Clark

Publications and source records attributed to William Clark.

At least 37 records · Page 2Linked to original sources

arXiv2022

Optimality of Zeno Executions in Hybrid Systems

A unique feature of hybrid dynamical systems (systems whose evolution is subject to both continuous- and discrete-time laws) is Zeno trajectories. Usually these trajectories are avoided as they can cause incorrect numerical results as the problem becomes ill-conditioned. However, these are difficult to justifiably avoid as determining when and where they occur is a non-trivial task. It turns out that in optimal control problems, not only can they not be avoided, but are sometimes required in synthesizing the solutions. This work explores the pedagogical example of the bouncing ball to demonstrate the importance of "Zeno control executions."

math.OC↗

arXiv2022

Optimal Control of Nonholonomic Systems via Magnetic Fields

Geometric optimal control utilizes tools from differential geometry to analyze the structure of a problem to determine the control and state trajectories to reach a desired outcome while minimizing some cost function. For a controlled mechanical system, the control usually manifests as an external force which, if conservative, can be added to the Hamiltonian. In this work, we focus on mechanical systems with controls added to the symplectic form rather than the Hamiltonian. In practice, this translates to controlling the magnetic field for an electrically charged system. We develop a basic theory deriving necessary conditions for optimality of such a system subjected to nonholonomic constraints. We consider the representative example of a magnetically charged Chaplygin Sleigh, whose resulting optimal control problem is completely integrable.

math.OC↗

arXiv2022

Lie Algebraic Cost Function Design for Control on Lie Groups

This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching $π$ in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).

math.OC↗

arXiv2022

How do we walk? Using hybrid holonomy to approximate non-holonomic systems

Why do we move forward when we walk? Our legs undergo periodic motion and thus possess no net change in position; however, our bodies do possess a net change in position and we are propelled forward. From a geometric perspective, this phenomenon of periodic input producing non-periodic output is holonomy. To obtain non-zero holonomy and propel forward, we must alternate which leg is in contact with the ground; a non-zero net motion can be obtained by concatenating arcs that would individually produce no net motion. We develop a framework for computing the holonomy group of hybrid systems and analyze their behavior in the limit as the number of impacts goes to infinity.

math.DG↗

arXiv2022

Orientation Control of the Bouncing Ball

Control of a hybrid dynamical system can manifest in one of two main ways: either through the continuous or the discrete dynamics. An example of controls influencing the continuous dynamics is legged locomotion, where the joints are actuated but the location and nature of the impacts are uncontrolled. In contrast, an example of discrete control would be in tennis; the player can only influence the trajectory of the ball through striking it. This work examines the latter case with two key emphases. The first is that controls manifest through changing the location of the guard (as opposed to changing only the reset). The second is that the location of the guard is described by "external variables" while the goal is to control "internal variables." As a simple test of this theory, orientation control of a bouncing ball is explored; the ball is only controlled during impacts which are exclusively position-dependent.

math.OC↗

arXiv2022

Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno

Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the "boundary identity property" along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top.

math.DS↗

arXiv2021

Optimal Target Shape for LiDAR Pose Estimation

Targets are essential in problems such as object tracking in cluttered or textureless environments, camera (and multi-sensor) calibration tasks, and simultaneous localization and mapping (SLAM). Target shapes for these tasks typically are symmetric (square, rectangular, or circular) and work well for structured, dense sensor data such as pixel arrays (i.e., image). However, symmetric shapes lead to pose ambiguity when using sparse sensor data such as LiDAR point clouds and suffer from the quantization uncertainty of the LiDAR. This paper introduces the concept of optimizing target shape to remove pose ambiguity for LiDAR point clouds. A target is designed to induce large gradients at edge points under rotation and translation relative to the LiDAR to ameliorate the quantization uncertainty associated with point cloud sparseness. Moreover, given a target shape, we present a means that leverages the target's geometry to estimate the target's vertices while globally estimating the pose. Both the simulation and the experimental results (verified by a motion capture system) confirm that by using the optimal shape and the global solver, we achieve centimeter error in translation and a few degrees in rotation even when a partially illuminated target is placed 30 meters away. All the implementations and datasets are available at https://github.com/UMich-BipedLab/optimal_shape_global_pose_estimation.

cs.CV↗

arXiv2021

A Geometric Approach to Optimal Control of Hybrid and Impulsive Systems

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory is applied to these systems and a hybrid version of Pontryagin's maximum principle is presented. This hybrid maximum principle is presented to emphasize its geometric nature which makes its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach is that Zeno behavior can be strongly controlled for "generic" control problems. Moreover, when the underlying control system is a mechanical impact system, additional structure is present which can be exploited and is thus explored. Multiple examples are presented for both mechanical and non-mechanical systems.

math.OC↗

arXiv2021

Nonparametric Continuous Sensor Registration

This paper develops a new mathematical framework that enables nonparametric joint semantic and geometric representation of continuous functions using data. The joint embedding is modeled by representing the processes in a reproducing kernel Hilbert space. The functions can be defined on arbitrary smooth manifolds where the action of a Lie group aligns them. The continuous functions allow the registration to be independent of a specific signal resolution. The framework is fully analytical with a closed-form derivation of the Riemannian gradient and Hessian. We study a more specialized but widely used case where the Lie group acts on functions isometrically. We solve the problem by maximizing the inner product between two functions defined over data, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Low-dimensional cases are derived with numerical examples to show the generality of the proposed framework. The high-dimensional derivation for the special Euclidean group acting on the Euclidean space showcases the point cloud registration and bird's-eye view map registration abilities. An implementation of this framework for RGB-D cameras outperforms the state-of-the-art robust visual odometry and performs well in texture and structure-scarce environments.

math.OC↗

arXiv2020

Asymmetric Private Set Intersection with Applications to Contact Tracing and Private Vertical Federated Machine Learning

We present a multi-language, cross-platform, open-source library for asymmetric private set intersection (PSI) and PSI-Cardinality (PSI-C). Our protocol combines traditional DDH-based PSI and PSI-C protocols with compression based on Bloom filters that helps reduce communication in the asymmetric setting. Currently, our library supports C++, C, Go, WebAssembly, JavaScript, Python, and Rust, and runs on both traditional hardware (x86) and browser targets. We further apply our library to two use cases: (i) a privacy-preserving contact tracing protocol that is compatible with existing approaches, but improves their privacy guarantees, and (ii) privacy-preserving machine learning on vertically partitioned data.

cs.CR↗

arXiv2020

A New Framework for Registration of Semantic Point Clouds from Stereo and RGB-D Cameras

This paper reports on a novel nonparametric rigid point cloud registration framework that jointly integrates geometric and semantic measurements such as color or semantic labels into the alignment process and does not require explicit data association. The point clouds are represented as nonparametric functions in a reproducible kernel Hilbert space. The alignment problem is formulated as maximizing the inner product between two functions, essentially a sum of weighted kernels, each of which exploits the local geometric and semantic features. As a result of the continuous models, analytical gradients can be computed, and a local solution can be obtained by optimization over the rigid body transformation group. Besides, we present a new point cloud alignment metric that is intrinsic to the proposed framework and takes into account geometric and semantic information. The evaluations using publicly available stereo and RGB-D datasets show that the proposed method outperforms state-of-the-art outdoor and indoor frame-to-frame registration methods. An open-source GPU implementation is also provided.

cs.CV↗

arXiv2020

Density of Gabor Systems Via the Short Time Fourier Transform

We apply a new approach to the study of the density of Gabor systems, and obtain a simple and straightforward proof of Ramanathan and Steger's well known result regarding the density of Gabor frames and Gabor Riesz sequences. Moreover, this point of view allows us to extend this result in several directions. The approach we use was first observed by A. Olevskii and the third author in their study of exponential systems, here we develop and simplify it further.

math.CA↗

arXiv2020

A Path-Dependent Variational Framework for Incremental Information Gathering

Information gathered along a path is inherently submodular; the incremental amount of information gained along a path decreases due to redundant observations. In addition to submodularity, the incremental amount of information gained is a function of not only the current state but also the entire history as well. This paper presents the construction of the first-order necessary optimality conditions for memory (history-dependent) Lagrangians. Path-dependent problems frequently appear in robotics and artificial intelligence, where the state such as a map is partially observable, and information can only be obtained along a trajectory by local sensing. Robotic exploration and environmental monitoring has numerous real-world applications and can be formulated using the proposed approach.

cs.RO↗

arXiv2020

Demonstration of a Reconfigurable Entangled Radiofrequency-Photonic Sensor Network

Quantum metrology takes advantage of nonclassical resources such as entanglement to achieve a sensitivity level below the standard quantum limit. To date, almost all quantum-metrology demonstrations are restricted to improving the measurement performance at a single sensor, but a plethora of applications require multiple sensors that work jointly to tackle distributed sensing problems. Here, we propose and experimentally demonstrate a reconfigurable sensor network empowered by continuous-variable (CV) multipartite entanglement. Our experiment establishes a connection between the entanglement structure and the achievable quantum advantage in different distributed sensing problems. The demonstrated entangled sensor network is composed of three sensor nodes each equipped with an electro-optic transducer for the detection of radiofrequency (RF) signals. By properly tailoring the CV multipartite entangled states, the entangled sensor network can be reconfigured to maximize the quantum advantage in distributed RF sensing problems such as measuring the angle of arrival of an RF field. The rich physics of CV multipartite entanglement unveiled by our work would open a new avenue for distributed quantum sensing and would lead to applications in ultrasensitive positioning, navigation, and timing.

quant-ph↗

arXiv2019

A New Dynamic Model of a Two-Wheeled Two-Flexible-Beam Inverted Pendulum Robot

Two-wheeled inverted pendulum robots are designed for self-balancing and they have remarkable advantages. In this paper, a new configuration and consequently dynamic model of one specific robot is presented and its dynamic behavior is analyzed. In this model, two cantilever beams are on the two-wheeled base and they are excited by voltages to the attached piezoelectric actuators. The mathematical model of this system is obtained using the extended Hamilton's Principle. The results show that the governing equations of motion are highly nonlinear and contain several coupled terms. These complex equations are solved numerically and the natural frequencies of the system are extracted. The system is then simulated in both lateral and horizontal plan movements. This paper proposes a new model of two-wheeled two-flexible-beam inverted pendulum Robot and investigates its complex dynamic; however, the derived equations will be validated experimentally and a suitable control strategy will be applied to the system to make it fully automated and more applicable in the future works.

math.DS↗

arXiv2019

Adaptive Continuous Visual Odometry from RGB-D Images

In this paper, we extend the recently developed continuous visual odometry framework for RGB-D cameras to an adaptive framework via online hyperparameter learning. We focus on the case of isotropic kernels with a scalar as the length-scale. In practice and as expected, the length-scale has remarkable impacts on the performance of the original framework. Previously it was handled using a fixed set of conditions within the solver to reduce the length-scale as the algorithm reaches a local minimum. We automate this process by a greedy gradient descent step at each iteration to find the next-best length-scale. Furthermore, to handle failure cases in the gradient descent step where the gradient is not well-behaved, such as the absence of structure or texture in the scene, we use a search interval for the length-scale and guide it gradually toward the smaller values. This latter strategy reverts the adaptive framework to the original setup. The experimental evaluations using publicly available RGB-D benchmarks show the proposed adaptive continuous visual odometry outperforms the original framework and the current state-of-the-art. We also make the software for the developed algorithm publicly available.

cs.RO↗

arXiv2019

The Bouncing Penny and Nonholonomic Impacts

The evolution of a Lagrangian mechanical system is variational. Likewise, when dealing with a hybrid Lagrangian system (a system with discontinuous impacts), the impacts can also be described by variations. These variational impacts are given by the so-called Weierstrass-Erdmann corner conditions. Therefore, hybrid Lagrangian systems can be completely understood by variational principles. Unlike typical (unconstrained / holonomic) Lagrangian systems, nonholonomically constrained Lagrangian systems are not variational. However, by using the Lagrange-d'Alembert principle, nonholonomic systems can be described as projections of variational systems. This paper works out the analogous version of the Weierstrass-Erdmann corner conditions for nonholonomic systems and examines the billiard problem with a rolling disk.

math.DS↗

arXiv2019

Continuous Direct Sparse Visual Odometry from RGB-D Images

This paper reports on a novel formulation and evaluation of visual odometry from RGB-D images. Assuming a static scene, the developed theoretical framework generalizes the widely used direct energy formulation (photometric error minimization) technique for obtaining a rigid body transformation that aligns two overlapping RGB-D images to a continuous formulation. The continuity is achieved through functional treatment of the problem and representing the process models over RGB-D images in a reproducing kernel Hilbert space; consequently, the registration is not limited to the specific image resolution and the framework is fully analytical with a closed-form derivation of the gradient. We solve the problem by maximizing the inner product between two functions defined over RGB-D images, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Energy-based approaches have been extremely successful and the developed framework in this paper shares many of their desired properties such as the parallel structure on both CPUs and GPUs, sparsity, semi-dense tracking, avoiding explicit data association which is computationally expensive, and possible extensions to the simultaneous localization and mapping frameworks. The evaluations on experimental data and comparison with the equivalent energy-based formulation of the problem confirm the effectiveness of the proposed technique, especially, when the lack of structure and texture in the environment is evident.

cs.RO↗