Thomas Beckers

Thomas Beckers, Sandra Hirche

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.

Kaiyuan Tan, Peilun Li, Jun Wang et al.

The ability to predict trajectories of surrounding agents and obstacles is a crucial component in many robotic applications. Data-driven approaches are commonly adopted for state prediction in scenarios where the underlying dynamics are unknown. However, the performance, reliability, and uncertainty of data-driven predictors become compromised when encountering out-of-distribution observations relative to the training data. In this paper, we introduce a Plug-and-Play Physics-Informed Machine Learning (PnP-PIML) framework to address this challenge. Our method employs conformal prediction to identify outlier dynamics and, in that case, switches from a nominal predictor to a physics-consistent model, namely distributed Port-Hamiltonian systems (dPHS). We leverage Gaussian processes to model the energy function of the dPHS, enabling not only the learning of system dynamics but also the quantification of predictive uncertainty through its Bayesian nature. In this way, the proposed framework produces reliable physics-informed predictions even for the out-of-distribution scenarios.

Kaiyuan Tan, Peilun Li, Thomas Beckers

Modeling the dynamics of flexible objects has become an emerging topic in the community as these objects become more present in many applications, e.g., soft robotics. Due to the properties of flexible materials, the movements of soft objects are often highly nonlinear and, thus, complex to predict. Data-driven approaches seem promising for modeling those complex dynamics but often neglect basic physical principles, which consequently makes them untrustworthy and limits generalization. To address this problem, we propose a physics-constrained learning method that combines powerful learning tools and reliable physical models. Our method leverages the data collected from observations by sending them into a Gaussian process that is physically constrained by a distributed Port-Hamiltonian model. Based on the Bayesian nature of the Gaussian process, we not only learn the dynamics of the system, but also enable uncertainty quantification. Furthermore, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

Thomas Beckers, Jacob Seidman, Paris Perdikaris et al.

Data-driven approaches achieve remarkable results for the modeling of complex dynamics based on collected data. However, these models often neglect basic physical principles which determine the behavior of any real-world system. This omission is unfavorable in two ways: The models are not as data-efficient as they could be by incorporating physical prior knowledge, and the model itself might not be physically correct. We propose Gaussian Process Port-Hamiltonian systems (GP-PHS) as a physics-informed Bayesian learning approach with uncertainty quantification. The Bayesian nature of GP-PHS uses collected data to form a distribution over all possible Hamiltonians instead of a single point estimate. Due to the underlying physics model, a GP-PHS generates passive systems with respect to designated inputs and outputs. Further, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

Thomas Beckers, Qirui Wu, George J. Pappas

Variational autoencoders allow to learn a lower-dimensional latent space based on high-dimensional input/output data. Using video clips as input data, the encoder may be used to describe the movement of an object in the video without ground truth data (unsupervised learning). Even though the object's dynamics is typically based on first principles, this prior knowledge is mostly ignored in the existing literature. Thus, we propose a physics-enhanced variational autoencoder that places a physical-enhanced Gaussian process prior on the latent dynamics to improve the efficiency of the variational autoencoder and to allow physically correct predictions. The physical prior knowledge expressed as linear dynamical system is here reflected by the Green's function and included in the kernel function of the Gaussian process. The benefits of the proposed approach are highlighted in a simulation with an oscillating particle.

Armin Lederer, Alexandre Capone, Thomas Beckers et al.

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.

Neha Das, Jonas Umlauft, Armin Lederer et al.

Data-driven control in unknown environments requires a clear understanding of the involved uncertainties for ensuring safety and efficient exploration. While aleatoric uncertainty that arises from measurement noise can often be explicitly modeled given a parametric description, it can be harder to model epistemic uncertainty, which describes the presence or absence of training data. The latter can be particularly useful for implementing exploratory control strategies when system dynamics are unknown. We propose a novel method for detecting the absence of training data using deep learning, which gives a continuous valued scalar output between $0$ (indicating low uncertainty) and $1$ (indicating high uncertainty). We utilize this detector as a proxy for epistemic uncertainty and show its advantages over existing approaches on synthetic and real-world datasets. Our approach can be directly combined with aleatoric uncertainty estimates and allows for uncertainty estimation in real-time as the inference is sample-free unlike existing approaches for uncertainty modeling. We further demonstrate the practicality of this uncertainty estimate in deploying online data-efficient control on a simulated quadcopter acted upon by an unknown disturbance model.

Thomas Beckers, Leonardo Colombo, Sandra Hirche et al.

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.

Thomas Beckers, Sandra Hirche

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.

Alexandre Capone, Jonas Umlauft, Thomas Beckers et al.

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.

Thomas Beckers, Sandra Hirche

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.

Thomas Beckers, Jonas Umlauft, Sandra Hirche

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.