Hiroyasu Tsukamoto

Ting-Wei Hsu, Hiroyasu Tsukamoto

We present a true-dynamics-agnostic, statistically rigorous framework for establishing exponential stability and safety guarantees of closed-loop, data-driven nonlinear control. Central to our approach is the novel concept of conformal robustness, which robustifies the Lyapunov and zeroing barrier certificates of data-driven dynamical systems against model prediction uncertainties using conformal prediction. It quantifies these uncertainties by leveraging rank statistics of prediction scores over system trajectories, without assuming any specific underlying structure of the prediction model or distribution of the uncertainties. With the quantified uncertainty information, we further construct the conformally robust control Lyapunov function (CR-CLF) and control barrier function (CR-CBF), data-driven counterparts of the CLF and CBF, for fully data-driven control with statistical guarantees of finite-horizon exponential stability and safety. The performance of the proposed concept is validated in numerical simulations with four benchmark nonlinear control problems.

Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques E. Slotine

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit the systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is, therefore, to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

Benjamin P. S. Donitz, Declan Mages, Hiroyasu Tsukamoto et al.

Interstellar objects (ISOs) represent a compelling and under-explored category of celestial bodies, providing physical laboratories to understand the formation of our solar system and probe the composition and properties of material formed in exoplanetary systems. In this work, we investigate existing approaches to designing successful flyby missions to ISOs, including a deep learning-driven guidance and control algorithm for ISOs traveling at velocities over 60 km/s. We have generated spacecraft trajectories to a series of synthetic representative ISOs, simulating a ground campaign to observe the target and resolve its state, thereby determining the cruise and close approach delta-Vs required for the encounter. We discuss the accessibility of and mission design to ISOs with varying characteristics, with special focuses on 1) state covariance estimation throughout the cruise, 2) handoffs from traditional navigation approaches to novel autonomous navigation for fast flyby regimes, and 3) overall recommendations about preparing for the future in situ exploration of these targets. The lessons learned also apply to the fast flyby of other small bodies, e.g., long-period comets and potentially hazardous asteroids, which also require tactical responses with similar characteristics.

Hiroyasu Tsukamoto, Benjamin Rivière, Changrak Choi et al.

The key innovation of our analytical method, CaRT, lies in establishing a new hierarchical, distributed architecture to guarantee the safety and robustness of a given learning-based motion planning policy. First, in a nominal setting, the analytical form of our CaRT safety filter formally ensures safe maneuvers of nonlinear multi-agent systems, optimally with minimal deviation from the learning-based policy. Second, in off-nominal settings, the analytical form of our CaRT robust filter optimally tracks the certified safe trajectory, generated by the previous layer in the hierarchy, the CaRT safety filter. We show using contraction theory that CaRT guarantees safety and the exponential boundedness of the trajectory tracking error, even under the presence of deterministic and stochastic disturbance. Also, the hierarchical nature of CaRT enables enhancing its robustness for safety just by its superior tracking to the certified safe trajectory, thereby making it suitable for off-nominal scenarios with large disturbances. This is a major distinction from conventional safety function-driven approaches, where the robustness originates from the stability of a safe set, which could pull the system over-conservatively to the interior of the safe set. Our log-barrier formulation in CaRT allows for its distributed implementation in multi-agent settings. We demonstrate the effectiveness of CaRT in several examples of nonlinear motion planning and control problems, including optimal, multi-spacecraft reconfiguration.

Hiroyasu Tsukamoto, Soon-Jo Chung, Yashwanth Kumar Nakka et al.

Interstellar objects (ISOs) are likely representatives of primitive materials invaluable in understanding exoplanetary star systems. Due to their poorly constrained orbits with generally high inclinations and relative velocities, however, exploring ISOs with conventional human-in-the-loop approaches is significantly challenging. This paper presents Neural-Rendezvous -- a deep learning-based guidance and control framework for encountering fast-moving objects, including ISOs, robustly, accurately, and autonomously in real time. It uses pointwise minimum norm tracking control on top of a guidance policy modeled by a spectrally-normalized deep neural network, where its hyperparameters are tuned with a loss function directly penalizing the MPC state trajectory tracking error. We show that Neural-Rendezvous provides a high probability exponential bound on the expected spacecraft delivery error, the proof of which leverages stochastic incremental stability analysis. In particular, it is used to construct a non-negative function with a supermartingale property, explicitly accounting for the ISO state uncertainty and the local nature of nonlinear state estimation guarantees. In numerical simulations, Neural-Rendezvous is demonstrated to satisfy the expected error bound for 100 ISO candidates. This performance is also empirically validated using our spacecraft simulator and in high-conflict and distributed UAV swarm reconfiguration with up to 20 UAVs.

Koki Hirano, Hiroyasu Tsukamoto

We propose a fully data-driven, Koopman-based framework for statistically robust control of discrete-time nonlinear systems with linear embeddings. Establishing a connection between the Koopman operator and contraction theory, it offers distribution-free probabilistic bounds on the state tracking error under Koopman modeling uncertainty. Conformal prediction is employed here to rigorously derive a bound on the state-dependent modeling uncertainty throughout the trajectory, ensuring safety and robustness without assuming a specific error prediction structure or distribution. Unlike prior approaches that merely combine conformal prediction with Koopman-based control in an open-loop setting, our method establishes a closed-loop control architecture with formal guarantees that explicitly account for both forward and inverse modeling errors. Also, by expressing the tracking error bound in terms of the control parameters and the modeling errors, our framework offers a quantitative means to formally enhance the performance of arbitrary Koopman-based control. We validate our method both in numerical simulations with the Dubins car and in real-world experiments with a highly nonlinear flapping-wing drone. The results demonstrate that our method indeed provides formal safety guarantees while maintaining accurate tracking performance under Koopman modeling uncertainty.

Rihan Aaron D'Silva, Hiroyasu Tsukamoto

This paper presents novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction. Our framework employs conformal inference to generate coverage-based confidence sets for the closed-loop dynamics around arbitrary reference trajectories, by constructing a joint nonconformity score to quantify both the validity of contraction (i.e., incremental stability) conditions and the impact of external stochastic disturbance on the closed-loop dynamics, without any distributional assumptions. Via appropriate constraint tightening, chance constraints can be reformulated into tractable, statistically valid deterministic constraints on the reference trajectories. This enables a formal pathway to leverage and validate learning-based motion planners and controllers, such as those with neural contraction metrics, in safety-critical real-world applications. Notably, our statistical guarantees are non-diverging and can be computed with finite samples of the underlying uncertainty, without overly conservative structural priors. We demonstrate our approach in motion planning problems for designing safe, dynamically feasible trajectories in both numerical simulation and hardware experiments.

Manav Vora, Gokul Puthumanaillam, Hiroyasu Tsukamoto et al.

Communication can improve coordination in partially observed multi-agent reinforcement learning (MARL), but learning \emph{when} and \emph{who} to communicate with requires choosing among many possible sender-recipient pairs, and the effect of any single message on future reward is hard to isolate. We introduce \textbf{SCoUT} (\textbf{S}calable \textbf{Co}mmunication via \textbf{U}tility-guided \textbf{T}emporal grouping), which addresses both these challenges via temporal and agent abstraction within traditional MARL. During training, SCoUT resamples \textit{soft} agent groups every \(K\) environment steps (macro-steps) via Gumbel-Softmax; these groups are latent clusters that induce an affinity used as a differentiable prior over recipients. Using the same assignments, a group-aware critic predicts values for each agent group and maps them to per-agent baselines through the same soft assignments, reducing critic complexity and variance. Each agent is trained with a three-headed policy: environment action, send decision, and recipient selection. To obtain precise communication learning signals, we derive counterfactual communication advantages by analytically removing each sender's contribution from the recipient's aggregated messages. This counterfactual computation enables precise credit assignment for both send and recipient-selection decisions. At execution time, all centralized training components are discarded and only the per-agent policy is run, preserving decentralized execution. Project website, videos and code: \hyperlink{https://scout-comm.github.io/}{https://scout-comm.github.io/}

Minjae Cho, Hiroyasu Tsukamoto, Huy Trong Tran

Control contraction metrics (CCMs) provide a framework to co-synthesize a controller and a corresponding contraction metric -- a positive-definite Riemannian metric under which a closed-loop system is guaranteed to be incrementally exponentially stable. However, the synthesized controller only ensures that all the trajectories of the system converge to one single trajectory and, as such, does not impose any notion of optimality across an entire trajectory. Furthermore, constructing CCMs requires a known dynamics model and non-trivial effort in solving an infinite-dimensional convex feasibility problem, which limits its scalability to complex systems featuring high dimensionality with uncertainty. To address these issues, we propose to integrate CCMs into reinforcement learning (RL), where CCMs provide dynamics-informed feedback for learning control policies that minimize cumulative tracking error under unknown dynamics. We show that our algorithm, called contraction actor-critic (CAC), formally enhances the capability of CCMs to provide a set of contracting policies with the long-term optimality of RL in a fully automated setting. Given a pre-trained dynamics model, CAC simultaneously learns a contraction metric generator (CMG) -- which generates a contraction metric -- and uses an actor-critic algorithm to learn an optimal tracking policy guided by that metric. We demonstrate the effectiveness of our algorithm relative to established baselines through extensive empirical studies, including simulated and real-world robot experiments, and provide a theoretical rationale for incorporating contraction theory into RL.

Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques Slotine et al.

This paper presents a theoretical overview of a Neural Contraction Metric (NCM): a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, the existence of which is a necessary and sufficient condition for incremental exponential stability of non-autonomous nonlinear system trajectories. Its innovation lies in providing formal robustness guarantees for learning-based control frameworks, utilizing contraction theory as an analytical tool to study the nonlinear stability of learned systems via convex optimization. In particular, we rigorously show in this paper that, by regarding modeling errors of the learning schemes as external disturbances, the NCM control is capable of obtaining an explicit bound on the distance between a time-varying target trajectory and perturbed solution trajectories, which exponentially decreases with time even under the presence of deterministic and stochastic perturbation. These useful features permit simultaneous synthesis of a contraction metric and associated control law by a neural network, thereby enabling real-time computable and probably robust learning-based control for general control-affine nonlinear systems.

Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques E. Slotine

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit the systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is, therefore, to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques Slotine

Adaptive control is subject to stability and performance issues when a learned model is used to enhance its performance. This paper thus presents a deep learning-based adaptive control framework for nonlinear systems with multiplicatively-separable parametrization, called adaptive Neural Contraction Metric (aNCM). The aNCM approximates real-time optimization for computing a differential Lyapunov function and a corresponding stabilizing adaptive control law by using a Deep Neural Network (DNN). The use of DNNs permits real-time implementation of the control law and broad applicability to a variety of nonlinear systems with parametric and nonparametric uncertainties. We show using contraction theory that the aNCM ensures exponential boundedness of the distance between the target and controlled trajectories in the presence of parametric uncertainties of the model, learning errors caused by aNCM approximation, and external disturbances. Its superiority to the existing robust and adaptive control methods is demonstrated using a cart-pole balancing model.

Hiroyasu Tsukamoto, Soon-Jo Chung

This paper presents Learning-based Autonomous Guidance with RObustness and Stability guarantees (LAG-ROS), which provides machine learning-based nonlinear motion planners with formal robustness and stability guarantees, by designing a differential Lyapunov function using contraction theory. LAG-ROS utilizes a neural network to model a robust tracking controller independently of a target trajectory, for which we show that the Euclidean distance between the target and controlled trajectories is exponentially bounded linearly in the learning error, even under the existence of bounded external disturbances. We also present a convex optimization approach that minimizes the steady-state bound of the tracking error to construct the robust control law for neural network training. In numerical simulations, it is demonstrated that the proposed method indeed possesses superior properties of robustness and nonlinear stability resulting from contraction theory, whilst retaining the computational efficiency of existing learning-based motion planners.

Hiroyasu Tsukamoto, Soon-Jo Chung, Jean-Jacques E. Slotine

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic neural contraction metric, as illustrated in simulation results.

Hiroyasu Tsukamoto, Soon-Jo Chung

This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems.

Hiroyasu Tsukamoto, Soon-Jo Chung

This paper presents ConVex optimization-based Stochastic steady-state Tracking Error Minimization (CV-STEM), a new state feedback control framework for a class of Ito stochastic nonlinear systems and Lagrangian systems. Its innovation lies in computing the control input by an optimal contraction metric, which greedily minimizes an upper bound of the steady-state mean squared tracking error of the system trajectories. Although the problem of minimizing the bound is non-convex, its equivalent convex formulation is proposed utilizing state-dependent coefficient parameterizations of the nonlinear system equation. It is shown using stochastic incremental contraction analysis that the CV-STEM provides a sufficient guarantee for exponential boundedness of the error for all time with L2-robustness properties. For the sake of its sampling-based implementation, we present discrete-time stochastic contraction analysis with respect to a state- and time-dependent metric along with its explicit connection to continuous-time cases. We validate the superiority of the CV-STEM to PID, H-infinity, and baseline nonlinear controllers for spacecraft attitude control and synchronization problems.