arXiv subjects

Thomas Beckers

Publications and source records attributed to Thomas Beckers.

35 records · Page 2Linked to original sources

arXiv2021

Learning Rigidity-based Flocking Control with Gaussian Processes

Flocking control of multi-agents system is challenging for agents with partially unknown dynamics. This paper proposes an online learning-based controller to stabilize flocking motion of double-integrator agents with additional unknown nonlinear dynamics by using Gaussian processes (GP). Agents interaction is described by a time-invariant infinitesimally minimally rigid undirected graph. We provide a decentralized control law that exponentially stabilizes the motion of the agents and captures Reynolds boids motion for swarms by using GPs as an online learning-based oracle for the prediction of the unknown dynamics. In particular the presented approach guarantees a probabilistic bounded tracking error with high probability.

eess.SY↗

arXiv2021

Prediction with Approximated Gaussian Process Dynamical Models

The modeling and simulation of dynamical systems is a necessary step for many control approaches. Using classical, parameter-based techniques for modeling of modern systems, e.g., soft robotics or human-robot interaction, is often challenging or even infeasible due to the complexity of the system dynamics. In contrast, data-driven approaches need only a minimum of prior knowledge and scale with the complexity of the system. In particular, Gaussian process dynamical models (GPDMs) provide very promising results for the modeling of complex dynamics. However, the control properties of these GP models are just sparsely researched, which leads to a "blackbox" treatment in modeling and control scenarios. In addition, the sampling of GPDMs for prediction purpose respecting their non-parametric nature results in non-Markovian dynamics making the theoretical analysis challenging. In this article, we present approximated GPDMs which are Markov and analyze their control theoretical properties. Among others, the approximated error is analyzed and conditions for boundedness of the trajectories are provided. The outcomes are illustrated with numerical examples that show the power of the approximated models while the the computational time is significantly reduced.

eess.SY↗

arXiv2021

Geometric Control for Load Transportation with Quadrotor UAVs by Elastic Cables

Groups of unmanned aerial vehicles (UAVs) are increasingly utilized in transportation task as the combined strength allows to increase the maximum payload. However, the resulting mechanical coupling of the UAVs impose new challenges in terms of the tracking control. Thus, we design a geometric trajectory tracking controller for the cooperative task of four quadrotor UAVs carrying and transporting a rigid body, which is attached to the quadrotors via inflexible elastic cables. The elasticity of the cables together with techniques of singular perturbation allows a reduction in the model to that of a similar model with inelastic cables. In this reduced model, we design a controller such that the position and attitude of the load exponentially converges to a given desired trajectory. We then show that this result leads to an uniformly converging tracking error for the original elastic model under some assumptions. Furthermore, under the presence of unstructured disturbances on the system, we show that the error is ultimately bounded with an arbitrarily small bound. Finally, a simulation illustrates the theoretical results.

math.OC↗

arXiv2021

Online learning-based trajectory tracking for underactuated vehicles with uncertain dynamics

Underactuated vehicles have gained much attention in the recent years due to the increasing amount of aerial and underwater vehicles as well as nanosatellites. Trajectory tracking control of these vehicles is a substantial aspect for an increasing range of application domains. However, external disturbances and parts of the internal dynamics are often unknown or very time-consuming to model. To overcome this issue, we present a tracking control law for underactuated rigid-body dynamics using an online learning-based oracle for the prediction of the unknown dynamics. We show that Gaussian process models are of particular interest for the role of the oracle. The presented approach guarantees a bounded tracking error with high probability where the bound is explicitly given. A numerical example highlights the effectiveness of the proposed control law.

eess.SY↗

arXiv2021

The Impact of Data on the Stability of Learning-Based Control- Extended Version

Despite the existence of formal guarantees for learning-based control approaches, the relationship between data and control performance is still poorly understood. In this paper, we propose a Lyapunov-based measure for quantifying the impact of data on the certifiable control performance. By modeling unknown system dynamics through Gaussian processes, we can determine the interrelation between model uncertainty and satisfaction of stability conditions. This allows us to directly asses the impact of data on the provable stationary control performance, and thereby the value of the data for the closed-loop system performance. Our approach is applicable to a wide variety of unknown nonlinear systems that are to be controlled by a generic learning-based control law, and the results obtained in numerical simulations indicate the efficacy of the proposed measure.

eess.SY↗

arXiv2021

Safe Online Learning-based Formation Control of Multi-Agent Systems with Gaussian Processes

Formation control algorithms for multi-agent systems have gained much attention in the recent years due to the increasing amount of mobile and aerial robotic swarms. The design of safe controllers for these vehicles is a substantial aspect for an increasing range of application domains. However, parts of the vehicle's dynamics and external disturbances are often unknown or very time-consuming to model. To overcome this issue, we present a safe formation control law for multiagent systems based on double integrator dynamics by using Gaussian Processes for an online learning of the unknown dynamics. The presented approach guarantees a bounded error to desired formations with high probability, where the bound is explicitly given. A numerical example highlights the effectiveness of the learning-based formation control law.

eess.SY↗

arXiv2021

An Introduction to Gaussian Process Models

Within the past two decades, Gaussian process regression has been increasingly used for modeling dynamical systems due to some beneficial properties such as the bias variance trade-off and the strong connection to Bayesian mathematics. As data-driven method, a Gaussian process is a powerful tool for nonlinear function regression without the need of much prior knowledge. In contrast to most of the other techniques, Gaussian Process modeling provides not only a mean prediction but also a measure for the model fidelity. In this article, we give an introduction to Gaussian processes and its usage in regression tasks of dynamical systems. Try Gaussian process regression yourself: https://gpr.tbeckers.com

eess.SY↗

arXiv2020

Localized active learning of Gaussian process state space models

The performance of learning-based control techniques crucially depends on how effectively the system is explored. While most exploration techniques aim to achieve a globally accurate model, such approaches are generally unsuited for systems with unbounded state spaces. Furthermore, a globally accurate model is not required to achieve good performance in many common control applications, e.g., local stabilization tasks. In this paper, we propose an active learning strategy for Gaussian process state space models that aims to obtain an accurate model on a bounded subset of the state-action space. Our approach aims to maximize the mutual information of the exploration trajectories with respect to a discretization of the region of interest. By employing model predictive control, the proposed technique integrates information collected during exploration and adaptively improves its exploration strategy. To enable computational tractability, we decouple the choice of most informative data points from the model predictive control optimization step. This yields two optimization problems that can be solved in parallel. We apply the proposed method to explore the state space of various dynamical systems and compare our approach to a commonly used entropy-based exploration strategy. In all experiments, our method yields a better model within the region of interest than the entropy-based method.

cs.LG↗

arXiv2019

Closed-loop Model Selection for Kernel-based Models using Bayesian Optimization

Kernel-based nonparametric models have become very attractive for model-based control approaches for nonlinear systems. However, the selection of the kernel and its hyperparameters strongly influences the quality of the learned model. Classically, these hyperparameters are optimized to minimize the prediction error of the model but this process totally neglects its later usage in the control loop. In this work, we present a framework to optimize the kernel and hyperparameters of a kernel-based model directly with respect to the closed-loop performance of the model. Our framework uses Bayesian optimization to iteratively refine the kernel-based model using the observed performance on the actual system until a desired performance is achieved. We demonstrate the proposed approach in a simulation and on a 3-DoF robotic arm.

eess.SY↗

arXiv2019

Keep soft robots soft -- a data-driven based trade-off between feed-forward and feedback control

Tracking control for soft robots is challenging due to uncertainties in the system model and environment. Using high feedback gains to overcome this issue results in an increasing stiffness that clearly destroys the inherent safety property of soft robots. However, accurate models for feed-forward control are often difficult to obtain. In this article, we employ Gaussian Process regression to obtain a data-driven model that is used for the feed-forward compensation of unknown dynamics. The model fidelity is used to adapt the feed-forward and feedback part allowing low feedback gains in regions of high model confidence.

eess.SY↗

arXiv2019

Stable Gaussian Process based Tracking Control of Euler-Lagrange Systems

Perfect tracking control for real-world Euler-Lagrange systems is challenging due to uncertainties in the system model and external disturbances. The magnitude of the tracking error can be reduced either by increasing the feedback gains or improving the model of the system. The latter is clearly preferable as it allows to maintain good tracking performance at low feedback gains. However, accurate models are often difficult to obtain. In this article, we address the problem of stable high-performance tracking control for unknown Euler-Lagrange systems. In particular, we employ Gaussian Process regression to obtain a data-driven model that is used for the feed-forward compensation of unknown dynamics of the system. The model fidelity is used to adapt the feedback gains allowing low feedback gains in state space regions of high model confidence. The proposed control law guarantees a globally bounded tracking error with a specific probability. Simulation studies demonstrate the superiority over state of the art tracking control approaches.

cs.LG↗

arXiv2018

Mean Square Prediction Error of Misspecified Gaussian Process Models

Nonparametric modeling approaches show very promising results in the area of system identification and control. A naturally provided model confidence is highly relevant for system-theoretical considerations to provide guarantees for application scenarios. Gaussian process regression represents one approach which provides such an indicator for the model confidence. However, this measure is only valid if the covariance function and its hyperparameters fit the underlying data generating process. In this paper, we derive an upper bound for the mean square prediction error of misspecified Gaussian process models based on a pseudo-concave optimization problem. We present application scenarios and a simulation to compare the derived upper bound with the true mean square error.

cs.LG↗

arXiv2018

Gaussian Process based Passivation of a Class of Nonlinear Systems with Unknown Dynamics

The paper addresses the problem of passivation of a class of nonlinear systems where the dynamics are unknown. For this purpose, we use the highly flexible, data-driven Gaussian process regression for the identification of the unknown dynamics for feed-forward compensation. The closed loop system of the nonlinear system, the Gaussian process model and a feedback control law is guaranteed to be semi-passive with a specific probability. The predicted variance of the Gaussian process regression is used to bound the model error which additionally allows to specify the state space region where the closed-loop system behaves passive. Finally, the theoretical results are illustrated by a simulation.

eess.SY↗

arXiv2018

Stable Model-based Control with Gaussian Process Regression for Robot Manipulators

Computed-torque control requires a very precise dynamical model of the robot for compensating the manipulator dynamics. This allows reduction of the controller's feedback gains resulting in disturbance attenuation and other advantages. Finding precise models for manipulators is often difficult with parametric approaches, e.g. in the presence of complex friction or flexible links. Therefore, we propose a novel computed-torque control law which consists of a PD feedback and a dynamic feed forward compensation part with Gaussian Processes. For this purpose, the nonparametric Gaussian Process regression infers the difference between an estimated and the true dynamics. In contrast to other approaches, we can guarantee that the tracking error is stochastically bounded. Furthermore, if the number of training points tends to infinity, the tracking error is asymptotically stable in the large. In simulation and with an experiment, we demonstrate the applicability of the proposed control law and that it outperforms classical computed-torque approaches in terms of tracking precision.

eess.SY↗

arXiv2018

Equilibrium Distributions and Stability Analysis of Gaussian Process State Space Models

Gaussian Process State Space Models (GP-SSM) are a data-driven stochastic model class suitable to represent nonlinear dynamics. They have become increasingly popular in non-parametric modeling approaches since they provide not only a prediction of the system behavior but also an accuracy of the prediction. For the application of these models, the analysis of fundamental system properties is required. In this paper, we analyze equilibrium distributions and stability properties of the GP-SSM. The computation of equilibrium distributions is based on the numerical solution of a Fredholm integral equation of the second kind and is suitable for any covariance function. Besides, we show that the GP-SSM with squared exponential covariance function is always mean square bounded and there exists a set which is positive recurrent.

eess.SY↗

arXiv2018

Stable Gaussian Process based Tracking Control of Lagrangian Systems

High performance tracking control can only be achieved if a good model of the dynamics is available. However, such a model is often difficult to obtain from first order physics only. In this paper, we develop a data-driven control law that ensures closed loop stability of Lagrangian systems. For this purpose, we use Gaussian Process regression for the feed-forward compensation of the unknown dynamics of the system. The gains of the feedback part are adapted based on the uncertainty of the learned model. Thus, the feedback gains are kept low as long as the learned model describes the true system sufficiently precisely. We show how to select a suitable gain adaption law that incorporates the uncertainty of the model to guarantee a globally bounded tracking error. A simulation with a robot manipulator demonstrates the efficacy of the proposed control law.

eess.SY↗

arXiv2018

Stability of Gaussian Process State Space Models

Gaussian Process State Space Models (GP-SSMs) are a non-parametric model class suitable to represent nonlinear dynamics. They become increasingly popular in data-driven modeling approaches, i.e. when no first-order physics-based models are available. Although a GP-SSM produces well-behaved approximations and gains increasing popularity, the fundamental system dynamics are just sparsely researched. In this paper, we present stability results for the GP-SSM depending on selected covariance function employing a deterministic point of view as widely done in the literature. The focus is set on the squared exponential function which is one of the most used covariance functions for nonlinear regression. We start with calculations according to the equilibrium points of GP-SSM and continue with conditions for stability.

eess.SY↗