Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control

Linearity of Koopman operators and simplicity of their estimators coupled with model-reduction capabilities has lead to their great popularity in applications for learning dynamical systems. While nonparametric Koopman operator learning in infinite-dimensional reproducing kernel Hilbert spaces is well understood for autonomous systems, its control system analogues are largely unexplored. Addressing systems with control inputs in a principled manner is crucial for fully data-driven learning of controllers, especially since existing approaches commonly resort to representational heuristics or parametric models of limited expressiveness and scalability. We address the aforementioned challenge by proposing a universal framework via control-affine reproducing kernels that enables direct estimation of a single operator even for control systems. The proposed approach, called control-Koopman operator regression (cKOR), is thus completely analogous to Koopman operator regression of the autonomous case. First in the literature, we present a nonparametric framework for learning Koopman operator representations of nonlinear control-affine systems that does not suffer from the curse of control input dimensionality. This allows for reformulating the infinite-dimensional learning problem in a finite-dimensional space based solely on data without apriori loss of precision due to a restriction to a finite span of functions or inputs as in other approaches. For enabling applications to large-scale control systems, we also enhance the scalability of control-Koopman operator estimators by leveraging random projections (sketching). The efficacy of our novel cKOR approach is demonstrated on both forecasting and control tasks.

Cooperative Learning with Gaussian Processes for Euler-Lagrange Systems Tracking Control under Switching Topologies

This work presents an innovative learning-based approach to tackle the tracking control problem of Euler-Lagrange multi-agent systems with partially unknown dynamics operating under switching communication topologies. The approach leverages a correlation-aware cooperative algorithm framework built upon Gaussian process regression, which adeptly captures inter-agent correlations for uncertainty predictions. A standout feature is its exceptional efficiency in deriving the aggregation weights achieved by circumventing the computationally intensive posterior variance calculations. Through Lyapunov stability analysis, the distributed control law ensures bounded tracking errors with high probability. Simulation experiments validate the protocol's efficacy in effectively managing complex scenarios, establishing it as a promising solution for robust tracking control in multi-agent systems characterized by uncertain dynamics and dynamic communication structures.

Koopman Kernel Regression

Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear dynamical systems. This makes system analysis and long-term predictions simple -- involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.

Variational Integrators and Graph-Based Solvers for Multibody Dynamics in Maximal Coordinates

Multibody dynamics simulators are an important tool in many fields, including learning and control for robotics. However, many existing dynamics simulators suffer from inaccuracies when dealing with constrained mechanical systems due to unsuitable integrators and dissatisfying constraint handling. Variational integrators are numerical discretization methods that can reduce physical inaccuracies when simulating mechanical systems, and formulating the dynamics in maximal coordinates allows for easy and numerically robust incorporation of constraints such as kinematic loops or contacts. Therefore, this article derives a variational integrator for mechanical systems with equality and inequality constraints in maximal coordinates. Additionally, efficient graph-based sparsity-exploiting algorithms for solving the integrator are provided and implemented as an open-source simulator. The evaluation of the simulator shows the improved physical accuracy due to the variational integrator and the advantages of the sparse solvers, while application examples of a walking robot and an exoskeleton with explicit constraints demonstrate the necessity and capabilities of maximal coordinates.

Dext-Gen: Dexterous Grasping in Sparse Reward Environments with Full Orientation Control

Reinforcement learning is a promising method for robotic grasping as it can learn effective reaching and grasping policies in difficult scenarios. However, achieving human-like manipulation capabilities with sophisticated robotic hands is challenging because of the problem's high dimensionality. Although remedies such as reward shaping or expert demonstrations can be employed to overcome this issue, they often lead to oversimplified and biased policies. We present Dext-Gen, a reinforcement learning framework for Dexterous Grasping in sparse reward ENvironments that is applicable to a variety of grippers and learns unbiased and intricate policies. Full orientation control of the gripper and object is achieved through smooth orientation representation. Our approach has reasonable training durations and provides the option to include desired prior knowledge. The effectiveness and adaptability of the framework to different scenarios is demonstrated in simulated experiments.

Towards Data-driven LQR with KoopmanizingFlows

We propose a novel framework for learning linear time-invariant (LTI) models for a class of continuous-time non-autonomous nonlinear dynamics based on a representation of Koopman operators. In general, the operator is infinite-dimensional but, crucially, linear. To utilize it for efficient LTI control, we learn a finite representation of the Koopman operator that is linear in controls while concurrently learning meaningful lifting coordinates. For the latter, we rely on KoopmanizingFlows - a diffeomorphism-based representation of Koopman operators. With such a learned model, we can replace the nonlinear infinite-horizon optimal control problem with quadratic costs to that of a linear quadratic regulator (LQR), facilitating efficacious optimal control for nonlinear systems. The prediction and control efficacy of the proposed method is verified on simulation examples.

Structure-Preserving Learning Using Gaussian Processes and Variational Integrators

Gaussian process regression is often applied for learning unknown systems and specifying the uncertainty of the learned model. When using Gaussian process regression to learn unknown systems, a commonly considered approach consists of learning the residual dynamics after applying some standard discretization, which might however not be appropriate for the system at hand. Variational integrators are a less common yet promising approach to discretization, as they retain physical properties of the underlying system, such as energy conservation or satisfaction of explicit constraints. In this work, we propose the combination of a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty. The simulative evaluation of the proposed method shows desirable energy conservation properties in accordance with the theoretical results and demonstrates the capability of treating constrained dynamical systems.

KoopmanizingFlows: Diffeomorphically Learning Stable Koopman Operators

We propose a novel framework for constructing linear time-invariant (LTI) models for data-driven representations of the Koopman operator for a class of stable nonlinear dynamics. The Koopman operator (generator) lifts a finite-dimensional nonlinear system to a possibly infinite-dimensional linear feature space. To utilize it for modeling, one needs to discover finite-dimensional representations of the Koopman operator. Learning suitable features is challenging, as one needs to learn LTI features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised learning task. For a theoretically well-founded solution to this problem, we propose learning Koopman-invariant coordinates by composing a diffeomorphic learner with a lifted aggregate system of a latent linear model. Using an unconstrained parameterization of stable matrices along with the aforementioned feature construction, we learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum, while ensuring stability regardless of the operator approximation accuracy. We demonstrate the superior efficacy of the proposed method in comparison to a state-of-the-art method on the well-known LASA handwriting dataset.

Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. By learning the conjugacy map between a nonlinear system and its Jacobian linearization through a Normalizing Flow one can guarantee the learned function is a diffeomorphism. Using this diffeomorphism, we construct eigenfunctions of the nonlinear system via the spectral equivalence of conjugate systems - allowing the construction of linear predictors for nonlinear systems. The universality of the diffeomorphism learner leads to the universal approximation of the nonlinear system's Koopman eigenfunctions. The developed method is also safe as it guarantees the model is asymptotically stable regardless of the representation accuracy. To our best knowledge, this is the first work to close the gap between the operator, system and learning theories. The efficacy of our approach is shown through simulation examples.

Linear-Time Contact and Friction Dynamics in Maximal Coordinates using Variational Integrators

Simulation of contact and friction dynamics is an important basis for control- and learning-based algorithms. However, the numerical difficulties of contact interactions pose a challenge for robust and efficient simulators. A maximal-coordinate representation of the dynamics enables efficient solving algorithms, but current methods in maximal coordinates require constraint stabilization schemes. Therefore, we propose an interior-point algorithm for the numerically robust treatment of rigid-body dynamics with contact interactions in maximal coordinates. Additionally, we discretize the dynamics with a variational integrator to prevent constraint drift. Our algorithm achieves linear-time complexity both in the number of contact points and the number of bodies, which is shown theoretically and demonstrated with an implementation. Furthermore, we simulate two robotic systems to highlight the applicability of the proposed algorithm.