arXiv subjects

Armin Lederer

Publications and source records attributed to Armin Lederer.

At least 37 records · Page 2Linked to original sources

arXiv2023

Deep Learning based Uncertainty Decomposition for Real-time Control

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.

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arXiv2023

Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors

Due to the increasing complexity of technical systems, accurate first principle models can often not be obtained. Supervised machine learning can mitigate this issue by inferring models from measurement data. Gaussian process regression is particularly well suited for this purpose due to its high data-efficiency and its explicit uncertainty representation, which allows the derivation of prediction error bounds. These error bounds have been exploited to show tracking accuracy guarantees for a variety of control approaches, but their direct dependency on the training data is generally unclear. We address this issue by deriving a Bayesian prediction error bound for GP regression, which we show to decay with the growth of a novel, kernel-based measure of data density. Based on the prediction error bound, we prove time-varying tracking accuracy guarantees for learned GP models used as feedback compensation of unknown nonlinearities, and show to achieve vanishing tracking error with increasing data density. This enables us to develop an episodic approach for learning Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed. The effectiveness of the derived theory is demonstrated in several simulations.

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arXiv2023

Can Learning Deteriorate Control? Analyzing Computational Delays in Gaussian Process-Based Event-Triggered Online Learning

When the dynamics of systems are unknown, supervised machine learning techniques are commonly employed to infer models from data. Gaussian process (GP) regression is a particularly popular learning method for this purpose due to the existence of prediction error bounds. Moreover, GP models can be efficiently updated online, such that event-triggered online learning strategies can be pursued to ensure specified tracking accuracies. However, existing trigger conditions must be able to be evaluated at arbitrary times, which cannot be achieved in practice due to non-negligible computation times. Therefore, we first derive a delay-aware tracking error bound, which reveals an accuracy-delay trade-off. Based on this result, we propose a novel event trigger for GP-based online learning with computational delays, which we show to offer advantages over offline trained GP models for sufficiently small computation times. Finally, we demonstrate the effectiveness of the proposed event trigger for online learning in simulations.

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arXiv2023

Safe Learning-Based Control of Elastic Joint Robots via Control Barrier Functions

Ensuring safety is of paramount importance in physical human-robot interaction applications. This requires both adherence to safety constraints defined on the system state, as well as guaranteeing compliant behavior of the robot. If the underlying dynamical system is known exactly, the former can be addressed with the help of control barrier functions. The incorporation of elastic actuators in the robot's mechanical design can address the latter requirement. However, this elasticity can increase the complexity of the resulting system, leading to unmodeled dynamics, such that control barrier functions cannot directly ensure safety. In this paper, we mitigate this issue by learning the unknown dynamics using Gaussian process regression. By employing the model in a feedback linearizing control law, the safety conditions resulting from control barrier functions can be robustified to take into account model errors, while remaining feasible. In order to enforce them on-line, we formulate the derived safety conditions in the form of a second-order cone program. We demonstrate our proposed approach with simulations on a two-degree-of-freedom planar robot with elastic joints.

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arXiv2022

Personalized Rehabilitation Robotics based on Online Learning Control

The use of rehabilitation robotics in clinical applications gains increasing importance, due to therapeutic benefits and the ability to alleviate labor-intensive works. However, their practical utility is dependent on the deployment of appropriate control algorithms, which adapt the level of task-assistance according to each individual patient's need. Generally, the required personalization is achieved through manual tuning by clinicians, which is cumbersome and error-prone. In this work we propose a novel online learning control architecture, which is able to personalize the control force at run time to each individual user. To this end, we deploy Gaussian process-based online learning with previously unseen prediction and update rates. Finally, we evaluate our method in an experimental user study, where the learning controller is shown to provide personalized control, while also obtaining safe interaction forces.

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arXiv2022

Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications

Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical settings hinge on the assumption that the kernel hyperparameters are known, which does not apply in general. To mitigate this, we introduce robust Gaussian process uniform error bounds in settings with unknown hyperparameters. Our approach computes a confidence region in the space of hyperparameters, which enables us to obtain a probabilistic upper bound for the model error of a Gaussian process with arbitrary hyperparameters. We do not require to know any bounds for the hyperparameters a priori, which is an assumption commonly found in related work. Instead, we are able to derive bounds from data in an intuitive fashion. We additionally employ the proposed technique to derive performance guarantees for a class of learning-based control problems. Experiments show that the bound performs significantly better than vanilla and fully Bayesian Gaussian processes.

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arXiv2022

Safe Reinforcement Learning via Confidence-Based Filters

Ensuring safety is a crucial challenge when deploying reinforcement learning (RL) to real-world systems. We develop confidence-based safety filters, a control-theoretic approach for certifying state safety constraints for nominal policies learned via standard RL techniques, based on probabilistic dynamics models. Our approach is based on a reformulation of state constraints in terms of cost functions, reducing safety verification to a standard RL task. By exploiting the concept of hallucinating inputs, we extend this formulation to determine a "backup" policy that is safe for the unknown system with high probability. Finally, the nominal policy is minimally adjusted at every time step during a roll-out towards the backup policy, such that safe recovery can be guaranteed afterwards. We provide formal safety guarantees, and empirically demonstrate the effectiveness of our approach.

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arXiv2022

Distributed Bayesian Online Learning for Cooperative Manipulation

For tasks where the dynamics of multiple agents are physically coupled, e.g., in cooperative manipulation, the coordination between the individual agents becomes crucial, which requires exact knowledge of the interaction dynamics. This problem is typically addressed using centralized estimators, which can negatively impact the flexibility and robustness of the overall system. To overcome this shortcoming, we propose a novel distributed learning framework for the exemplary task of cooperative manipulation using Bayesian principles. Using only local state information each agent obtains an estimate of the object dynamics and grasp kinematics. These local estimates are combined using dynamic average consensus. Due to the strong probabilistic foundation of the method, each estimate of the object dynamics and grasp kinematics is accompanied by a measure of uncertainty, which allows to guarantee a bounded prediction error with high probability. Moreover, the Bayesian principles directly allow iterative learning with constant complexity, such that the proposed learning method can be used online in real-time applications. The effectiveness of the approach is demonstrated in a simulated cooperative manipulation task.

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arXiv2022

Diffeomorphically Learning Stable Koopman Operators

System representations inspired by the infinite-dimensional Koopman operator (generator) are increasingly considered for predictive modeling. Due to the operator's linearity, a range of nonlinear systems admit linear predictor representations - allowing for simplified prediction, analysis and control. However, finding meaningful finite-dimensional representations for prediction is difficult as it involves determining features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised problem. In this work, we present Koopmanizing Flows - a novel continuous-time framework for supervised learning of linear predictors for a class of nonlinear dynamics. In our model construction a latent diffeomorphically related linear system unfolds into a linear predictor through the composition with a monomial basis. The lifting, its linear dynamics and state reconstruction are learned simultaneously, while an unconstrained parameterization of Hurwitz matrices ensures asymptotic stability regardless of the operator approximation accuracy. The superior efficacy of Koopmanizing Flows is demonstrated in comparison to a state-of-the-art method on the well-known LASA handwriting benchmark.

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arXiv2022

Networked Online Learning for Control of Safety-Critical Resource-Constrained Systems based on Gaussian Processes

Safety-critical technical systems operating in unknown environments require the ability to quickly adapt their behavior, which can be achieved in control by inferring a model online from the data stream generated during operation. Gaussian process-based learning is particularly well suited for safety-critical applications as it ensures bounded prediction errors. While there exist computationally efficient approximations for online inference, these approaches lack guarantees for the prediction error and have high memory requirements, and are therefore not applicable to safety-critical systems with tight memory constraints. In this work, we propose a novel networked online learning approach based on Gaussian process regression, which addresses the issue of limited local resources by employing remote data management in the cloud. Our approach formally guarantees a bounded tracking error with high probability, which is exploited to identify the most relevant data to achieve a certain control performance. We further propose an effective data transmission scheme between the local system and the cloud taking bandwidth limitations and time delay of the transmission channel into account. The effectiveness of the proposed method is successfully demonstrated in a simulation.

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arXiv2021

Adaptive Low-Pass Filtering using Sliding Window Gaussian Processes

When signals are measured through physical sensors, they are perturbed by noise. To reduce noise, low-pass filters are commonly employed in order to attenuate high frequency components in the incoming signal, regardless if they come from noise or the actual signal. Therefore, low-pass filters must be carefully tuned in order to avoid significant deterioration of the signal. This tuning requires prior knowledge about the signal, which is often not available in applications such as reinforcement learning or learning-based control. In order to overcome this limitation, we propose an adaptive low-pass filter based on Gaussian process regression. By considering a constant window of previous observations, updates and predictions fast enough for real-world filtering applications can be realized. Moreover, the online optimization of hyperparameters leads to an adaptation of the low-pass behavior, such that no prior tuning is necessary. We show that the estimation error of the proposed method is uniformly bounded, and demonstrate the flexibility and efficiency of the approach in several simulations.

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arXiv2021

Inverse Reinforcement Learning: A Control Lyapunov Approach

Inferring the intent of an intelligent agent from demonstrations and subsequently predicting its behavior, is a critical task in many collaborative settings. A common approach to solve this problem is the framework of inverse reinforcement learning (IRL), where the observed agent, e.g., a human demonstrator, is assumed to behave according to an intrinsic cost function that reflects its intent and informs its control actions. In this work, we reformulate the IRL inference problem to learning control Lyapunov functions (CLF) from demonstrations by exploiting the inverse optimality property, which states that every CLF is also a meaningful value function. Moreover, the derived CLF formulation directly guarantees stability of inferred control policies. We show the flexibility of our proposed method by learning from goal-directed movement demonstrations in a continuous environment.

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arXiv2021

Real-Time Regression with Dividing Local Gaussian Processes

The increased demand for online prediction and the growing availability of large data sets drives the need for computationally efficient models. While exact Gaussian process regression shows various favorable theoretical properties (uncertainty estimate, unlimited expressive power), the poor scaling with respect to the training set size prohibits its application in big data regimes in real-time. Therefore, this paper proposes dividing local Gaussian processes, which are a novel, computationally efficient modeling approach based on Gaussian process regression. Due to an iterative, data-driven division of the input space, they achieve a sublinear computational complexity in the total number of training points in practice, while providing excellent predictive distributions. A numerical evaluation on real-world data sets shows their advantages over other state-of-the-art methods in terms of accuracy as well as prediction and update speed.

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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.

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arXiv2021

Distributed Learning Consensus Control for Unknown Nonlinear Multi-Agent Systems based on Gaussian Processes

In this paper, a distributed learning leader-follower consensus protocol based on Gaussian process regression for a class of nonlinear multi-agent systems with unknown dynamics is designed. We propose a distributed learning approach to predict the residual dynamics for each agent. The stability of the consensus protocol using the data-driven model of the dynamics is shown via Lyapunov analysis. The followers ultimately synchronize to the leader with guaranteed error bounds by applying the proposed control law with a high probability. The effectiveness and the applicability of the developed protocol are demonstrated by simulation examples.

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arXiv2021

Uniform Error and Posterior Variance Bounds for Gaussian Process Regression with Application to Safe Control

In application areas where data generation is expensive, Gaussian processes are a preferred supervised learning model due to their high data-efficiency. Particularly in model-based control, Gaussian processes allow the derivation of performance guarantees using probabilistic model error bounds. To make these approaches applicable in practice, two open challenges must be solved i) Existing error bounds rely on prior knowledge, which might not be available for many real-world tasks. (ii) The relationship between training data and the posterior variance, which mainly drives the error bound, is not well understood and prevents the asymptotic analysis. This article addresses these issues by presenting a novel uniform error bound using Lipschitz continuity and an analysis of the posterior variance function for a large class of kernels. Additionally, we show how these results can be used to guarantee safe control of an unknown dynamical system and provide numerical illustration examples.

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arXiv2020

Learning Stable Nonparametric Dynamical Systems with Gaussian Process Regression

Modelling real world systems involving humans such as biological processes for disease treatment or human behavior for robotic rehabilitation is a challenging problem because labeled training data is sparse and expensive, while high prediction accuracy is required from models of these dynamical systems. Due to the high nonlinearity of problems in this area, data-driven approaches gain increasing attention for identifying nonparametric models. In order to increase the prediction performance of these models, abstract prior knowledge such as stability should be included in the learning approach. One of the key challenges is to ensure sufficient flexibility of the models, which is typically limited by the usage of parametric Lyapunov functions to guarantee stability. Therefore, we derive an approach to learn a nonparametric Lyapunov function based on Gaussian process regression from data. Furthermore, we learn a nonparametric Gaussian process state space model from the data and show that it is capable of reproducing observed data exactly. We prove that stabilization of the nominal model based on the nonparametric control Lyapunov function does not modify the behavior of the nominal model at training samples. The flexibility and efficiency of our approach is demonstrated on the benchmark problem of learning handwriting motions from a real world dataset, where our approach achieves almost exact reproduction of the training data.

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arXiv2020

GP3: A Sampling-based Analysis Framework for Gaussian Processes

Although machine learning is increasingly applied in control approaches, only few methods guarantee certifiable safety, which is necessary for real world applications. These approaches typically rely on well-understood learning algorithms, which allow formal theoretical analysis. Gaussian process regression is a prominent example among those methods, which attracts growing attention due to its strong Bayesian foundations. Even though many problems regarding the analysis of Gaussian processes have a similar structure, specific approaches are typically tailored for them individually, without strong focus on computational efficiency. Thereby, the practical applicability and performance of these approaches is limited. In order to overcome this issue, we propose a novel framework called GP3, general purpose computation on graphics processing units for Gaussian processes, which allows to solve many of the existing problems efficiently. By employing interval analysis, local Lipschitz constants are computed in order to extend properties verified on a grid to continuous state spaces. Since the computation is completely parallelizable, the computational benefits of GPU processing are exploited in combination with multi-resolution sampling in order to allow high resolution analysis.

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