Jean-Jacques Slotine

Spencer M. Richards, Navid Azizan, Jean-Jacques Slotine et al. · mit

Real-time adaptation is imperative to the control of robots operating in complex, dynamic environments. Adaptive control laws can endow even nonlinear systems with good trajectory tracking performance, provided that any uncertain dynamics terms are linearly parameterizable with known nonlinear features. However, it is often difficult to specify such features a priori, such as for aerodynamic disturbances on rotorcraft or interaction forces between a manipulator arm and various objects. In this paper, we turn to data-driven modeling with neural networks to learn, offline from past data, an adaptive controller with an internal parametric model of these nonlinear features. Our key insight is that we can better prepare the controller for deployment with control-oriented meta-learning of features in closed-loop simulation, rather than regression-oriented meta-learning of features to fit input-output data. Specifically, we meta-learn the adaptive controller with closed-loop tracking simulation as the base-learner and the average tracking error as the meta-objective. With both fully-actuated and underactuated nonlinear planar rotorcraft subject to wind, we demonstrate that our adaptive controller outperforms other controllers trained with regression-oriented meta-learning when deployed in closed-loop for trajectory tracking control.

Carlos Esteves, Jean-Jacques Slotine, Ameesh Makadia

Spherical CNNs generalize CNNs to functions on the sphere, by using spherical convolutions as the main linear operation. The most accurate and efficient way to compute spherical convolutions is in the spectral domain (via the convolution theorem), which is still costlier than the usual planar convolutions. For this reason, applications of spherical CNNs have so far been limited to small problems that can be approached with low model capacity. In this work, we show how spherical CNNs can be scaled for much larger problems. To achieve this, we make critical improvements including novel variants of common model components, an implementation of core operations to exploit hardware accelerator characteristics, and application-specific input representations that exploit the properties of our model. Experiments show our larger spherical CNNs reach state-of-the-art on several targets of the QM9 molecular benchmark, which was previously dominated by equivariant graph neural networks, and achieve competitive performance on multiple weather forecasting tasks. Our code is available at https://github.com/google-research/spherical-cnn.

Spencer M. Richards, Jean-Jacques Slotine, Navid Azizan et al. · mit

Even for known nonlinear dynamical systems, feedback controller synthesis is a difficult problem that often requires leveraging the particular structure of the dynamics to induce a stable closed-loop system. For general nonlinear models, including those fit to data, there may not be enough known structure to reliably synthesize a stabilizing feedback controller. In this paper, we discuss a state-dependent nonlinear tracking controller formulation based on a state-dependent Riccati equation for general nonlinear control-affine systems. This formulation depends on a nonlinear factorization of the system of vector fields defining the control-affine dynamics, which always exists under mild smoothness assumptions. We propose a method for learning this factorization from a finite set of data. On a variety of simulated nonlinear dynamical systems, we empirically demonstrate the efficacy of learned versions of this controller in stable trajectory tracking. Alongside our learning method, we evaluate recent ideas in jointly learning a controller and stabilizability certificate for known dynamical systems; we show experimentally that such methods can be frail in comparison.

Márton Pósfai, Yang-Yu Liu, Jean-Jacques Slotine et al.

A dynamical system is controllable if by imposing appropriate external signals on a subset of its nodes, it can be driven from any initial state to any desired state in finite time. Here we study the impact of various network characteristics on the minimal number of driver nodes required to control a network. We find that clustering and modularity have no discernible impact, but the symmetries of the underlying matching problem can produce linear, quadratic or no dependence on degree correlation coefficients, depending on the nature of the underlying correlations. The results are supported by numerical simulations and help narrow the observed gap between the predicted and the observed number of driver nodes in real networks.

Ian R. Manchester, Jean-Jacques E. Slotine

We introduce the concept of a control contraction metric, extending contraction analysis to constructive nonlinear control design. We derive sufficient conditions for exponential stabilizability of all trajectories of a nonlinear control system. The conditions have a simple geometrical interpretation, can be written as a convex feasibility problem, and are invariant under coordinate changes. We show that these conditions are necessary and sufficient for feedback linearizable systems, and also derive novel convex criteria for exponential stabilization of a nonlinear submanifold of state space. We illustrate the benefits of convexity by constructing a controller for an unstable polynomial system that combines local optimality and global stability, using a metric found via sum-of-squares programming.

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.

Marco Tulio Angulo, Jean-Jacques Slotine

In many large systems, such as those encountered in biology or economics, the dynamics are nonlinear and are only known very coarsely. It is often the case, however, that the signs (excitation or inhibition) of individual interactions are known. This paper extends to nonlinear systems the classical criteria of linear sign stability introduced in the 70's, yielding simple sufficient conditions to determine stability using only the sign patterns of the interactions.

Silvere Bonnabel, Jean-Jacques Slotine

The contraction properties of the Extended Kalman Filter, viewed as a deterministic observer for nonlinear systems, are analyzed. This yields new conditions under which exponential convergence of the state error can be guaranteed. As contraction analysis studies the evolution of an infinitesimal discrepancy between neighboring trajectories, and thus stems from a differential framework, the sufficient convergence conditions are different from the ones that previously appeared in the literature, which were derived in a Lyapunov framework. This article sheds another light on the theoretical properties of this popular observer.

Ian R. Manchester, Jean-Jacques E. Slotine

Finding separable certificates of stability is important for tractability of analysis methods for large-scale networked systems. In this paper we consider the question of when a nonlinear system which is contracting, i.e. all solutions are exponentially stable, can have that property verified by a separable metric. Making use of recent results in the theory of positive linear systems and separable Lyapunov functions, we prove several new results showing when this is possible, and discuss the application of to nonlinear distributed control design via convex optimization.

Tal Shnitzer, Ronen Talmon, Jean-Jacques Slotine

In this paper, we propose a non-parametric method for state estimation of high-dimensional nonlinear stochastic dynamical systems, which evolve according to gradient flows with isotropic diffusion. We combine diffusion maps, a manifold learning technique, with a linear Kalman filter and with concepts from Koopman operator theory. More concretely, using diffusion maps, we construct data-driven virtual state coordinates, which linearize the system model. Based on these coordinates, we devise a data-driven framework for state estimation using the Kalman filter. We demonstrate the strengths of our method with respect to both parametric and non-parametric algorithms in three tracking problems. In particular, applying the approach to actual recordings of hippocampal neural activity in rodents directly yields a representation of the position of the animals. We show that the proposed method outperforms competing non-parametric algorithms in the examined stochastic problem formulations. Additionally, we obtain results comparable to classical parametric algorithms, which, in contrast to our method, are equipped with model knowledge.

Ian R. Manchester, Jean-Jacques E. Slotine

This paper addresses the problems of stabilization, robust control, and observer design for nonlinear systems. We build upon recently a proposed method based on contraction theory and convex optimization, extending the class of systems to which it is applicable. We prove converse results for mechanical systems and feedback-linearizable systems. Next we consider robust control, and give a simple construction of a controller guaranteeing an L2-gain condition, and discuss connections to nonlinear H-infinity control. Finally, we discuss a "duality" result between nonlinear stabilization problems and observer construction, in the process constructing globally stable reduced-order observers for a class of nonlinear systems.

Winfried Lohmiller, Philipp Gassert, Jean-Jacques Slotine

While much progress has been achieved over the last decades in neuro-inspired machine learning, there are still fundamental theoretical problems in gradient-based learning using combinations of neurons. These problems, such as saddle points and suboptimal plateaus of the cost function, can lead in theory and practice to failures of learning. In addition, the discrete step size selection of the gradient is problematic since too large steps can lead to instability and too small steps slow down the learning. This paper describes an alternative discrete MinMax learning approach for continuous piece-wise linear functions. Global exponential convergence of the algorithm is established using Contraction Theory with Inequality Constraints, which is extended from the continuous to the discrete case in this paper: The parametrization of each linear function piece is, in contrast to deep learning, linear in the proposed MinMax network. This allows a linear regression stability proof as long as measurements do not transit from one linear region to its neighbouring linear region. The step size of the discrete gradient descent is Lagrangian limited orthogonal to the edge of two neighbouring linear functions. It will be shown that this Lagrangian step limitation does not decrease the convergence of the unconstrained system dynamics in contrast to a step size limitation in the direction of the gradient. We show that the convergence rate of a constrained piece-wise linear function learning is equivalent to the exponential convergence rates of the individual local linear regions.

Michaela Ennis, Leo Kozachkov, Jean-Jacques Slotine

To push forward the important emerging research field surrounding multi-area recurrent neural networks (RNNs), we expand theoretically and empirically on the provably stable RNNs of RNNs introduced by Kozachkov et al. in "RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks". We prove relaxed stability conditions for salient special cases of this architecture, most notably for a global workspace modular structure. We then demonstrate empirical success for Global Workspace Sparse Combo Nets with a small number of trainable parameters, not only through strong overall test performance but also greater resilience to removal of individual subnetworks. These empirical results for the global workspace inter-area topology are contingent on stability preservation, highlighting the relevance of our theoretical work for enabling modular RNN success. Further, by exploring sparsity in the connectivity structure between different subnetwork modules more broadly, we improve the state of the art performance for stable RNNs on benchmark sequence processing tasks, thus underscoring the general utility of specialized graph structures for multi-area RNNs.

Bing Song, Jean-Jacques Slotine, Quang-Cuong Pham

We propose a novel way to integrate control techniques with reinforcement learning (RL) for stability, robustness, and generalization: leveraging contraction theory to realize modularity in neural control, which ensures that combining stable subsystems can automatically preserve the stability. We realize such modularity via signal composition and dynamic decomposition. Signal composition creates the latent space, within which RL applies to maximizing rewards. Dynamic decomposition is realized by coordinate transformation that creates an auxiliary space, within which the latent signals are coupled in the way that their combination can preserve stability provided each signal, that is, each subsystem, has stable self-feedbacks. Leveraging modularity, the nonlinear stability problem is deconstructed into algebraically solvable ones, the stability of the subsystems in the auxiliary space, yielding linear constraints on the input gradients of control networks that can be as simple as switching the signs of network weights. This minimally invasive method for stability allows arguably easy integration into the modular neural architectures in machine learning, like hierarchical RL, and improves their performance. We demonstrate in simulation the necessity and the effectiveness of our method: the necessity for robustness and generalization, and the effectiveness in improving hierarchical RL for manipulation learning.

Anand Srinivasan, Jean-Jacques Slotine

Recently, the vanishing-step-size limit of the Sinkhorn algorithm at finite regularization parameter $\varepsilon$ was shown to be a mirror descent in the space of probability measures. We give $L^2$ contraction criteria in two time-dependent metrics induced by the mirror Hessian, which reduce to the coercivity of certain conditional expectation operators. We then give an exact identity for the entropy production rate of the Sinkhorn flow, which was previously known only to be nonpositive. Examining this rate shows that the standard semigroup analysis of diffusion processes extends systematically to the Sinkhorn flow. We show that the flow induces a reversible Markov dynamics on the target marginal as an Onsager gradient flow. We define the Dirichlet form associated to its (nonlocal) infinitesimal generator, prove a Poincaré inequality for it, and show that the spectral gap is strictly positive along the Sinkhorn flow whenever $\varepsilon > 0$. Lastly, we show that the entropy decay is exponential if and only if a logarithmic Sobolev inequality (LSI) holds. We give for illustration two immediate practical use-cases for the Sinkhorn LSI: as a design principle for the latent space in which generative models are trained, and as a stopping heuristic for discrete-time algorithms.

Leo Kozachkov, Patrick M. Wensing, Jean-Jacques Slotine

We prove that Riemannian contraction in a supervised learning setting implies generalization. Specifically, we show that if an optimizer is contracting in some Riemannian metric with rate $λ> 0$, it is uniformly algorithmically stable with rate $\mathcal{O}(1/λn)$, where $n$ is the number of labelled examples in the training set. The results hold for stochastic and deterministic optimization, in both continuous and discrete-time, for convex and non-convex loss surfaces. The associated generalization bounds reduce to well-known results in the particular case of gradient descent over convex or strongly convex loss surfaces. They can be shown to be optimal in certain linear settings, such as kernel ridge regression under gradient flow.

Thomas T. C. K. Zhang, Stephen Tu, Nicholas M. Boffi et al.

Motivated by bridging the simulation to reality gap in the context of safety-critical systems, we consider learning adversarially robust stability certificates for unknown nonlinear dynamical systems. In line with approaches from robust control, we consider additive and Lipschitz bounded adversaries that perturb the system dynamics. We show that under suitable assumptions of incremental stability on the underlying system, the statistical cost of learning an adversarial stability certificate is equivalent, up to constant factors, to that of learning a nominal stability certificate. Our results hinge on novel bounds for the Rademacher complexity of the resulting adversarial loss class, which may be of independent interest. To the best of our knowledge, this is the first characterization of sample-complexity bounds when performing adversarial learning over data generated by a dynamical system. We further provide a practical algorithm for approximating the adversarial training algorithm, and validate our findings on a damped pendulum example.

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.

Leo Kozachkov, Michaela Ennis, Jean-Jacques Slotine

Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits applications to very specific classes of systems. We introduce a novel nonparametric adaptive algorithm that estimates an infinite-dimensional density over parameters online to learn an unknown dynamics in a reproducing kernel Hilbert space. Surprisingly, the resulting control input admits an analytical expression that enables its implementation despite its underlying infinite-dimensional structure. While this adaptive input is rich and expressive - subsuming, for example, traditional linear parameterizations - its computational complexity grows linearly with time, making it comparatively more expensive than its parametric counterparts. Leveraging the theory of random Fourier features, we provide an efficient randomized implementation that recovers the complexity of classical parametric methods while provably retaining the expressivity of the nonparametric input. In particular, our explicit bounds only depend polynomially on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. As an illustration of the method, we demonstrate the ability of the randomized approximation algorithm to learn a predictive model of a 60-dimensional system consisting of ten point masses interacting through Newtonian gravitation. By reinterpretation as a gradient flow on a specific loss, we conclude with a natural extension of our kernel-based adaptive algorithms to deep neural networks. We show empirically that the extra expressivity afforded by deep representations can lead to improved performance at the expense of closed-loop stability that is rigorously guaranteed and consistently observed for kernel machines.

Lukas Huber, Jean-Jacques Slotine, Aude Billard

This paper presents a closed-form approach to constrain a flow within a given volume and around objects. The flow is guaranteed to converge and to stop at a single fixed point. We show that the obstacle avoidance problem can be inverted to enforce that the flow remains enclosed within a volume defined by a polygonal surface. We formally guarantee that such a flow will never contact the boundaries of the enclosing volume and obstacles, and will asymptotically converge towards an attractor. We further create smooth motion fields around obstacles with edges (e.g. tables). Both obstacles and enclosures may be time-varying, i.e. moving, expanding and shrinking. The technique enables a robot to navigate within an enclosed corridor while avoiding static and moving obstacles. It was applied on an autonomous robot (QOLO) in a static complex indoor environment, and also tested in simulations with dense crowds. The final proof of concept was performed in an outdoor environment in Lausanne. The QOLO-robot successfully traversed a marketplace in the center of town in presence of a diverse crowd with a non-uniform motion pattern.

Brett T. Lopez, Jean-Jacques E. Slotine

The stable combination of optimal feedback policies with online learning is studied in a new control-theoretic framework for uncertain nonlinear systems. The framework can be systematically used in transfer learning and sim-to-real applications, where an optimal policy learned for a nominal system needs to remain effective in the presence of significant variations in parameters. Given unknown parameters within a bounded range, the resulting adaptive control laws guarantee convergence of the closed-loop system to the state of zero cost. Online adjustment of the learning rate is used as a key stability mechanism, and preserves certainty equivalence when designing optimal policies without assuming uncertainty to be within the control range. The approach is illustrated on the familiar mountain car problem, where it yields near-optimal performance despite the presence of parametric model uncertainty.

Heng Yang, Chris Doran, Jean-Jacques Slotine

We study the problem of aligning two sets of 3D geometric primitives given known correspondences. Our first contribution is to show that this primitive alignment framework unifies five perception problems including point cloud registration, primitive (mesh) registration, category-level 3D registration, absolution pose estimation (APE), and category-level APE. Our second contribution is to propose DynAMical Pose estimation (DAMP), the first general and practical algorithm to solve primitive alignment problem by simulating rigid body dynamics arising from virtual springs and damping, where the springs span the shortest distances between corresponding primitives. We evaluate DAMP in simulated and real datasets across all five problems, and demonstrate (i) DAMP always converges to the globally optimal solution in the first three problems with 3D-3D correspondences; (ii) although DAMP sometimes converges to suboptimal solutions in the last two problems with 2D-3D correspondences, using a scheme for escaping local minima, DAMP always succeeds. Our third contribution is to demystify the surprising empirical performance of DAMP and formally prove a global convergence result in the case of point cloud registration by charactering local stability of the equilibrium points of the underlying dynamical system.

Spencer M. Richards, Navid Azizan, Jean-Jacques Slotine et al.

Real-time adaptation is imperative to the control of robots operating in complex, dynamic environments. Adaptive control laws can endow even nonlinear systems with good trajectory tracking performance, provided that any uncertain dynamics terms are linearly parameterizable with known nonlinear features. However, it is often difficult to specify such features a priori, such as for aerodynamic disturbances on rotorcraft or interaction forces between a manipulator arm and various objects. In this paper, we turn to data-driven modeling with neural networks to learn, offline from past data, an adaptive controller with an internal parametric model of these nonlinear features. Our key insight is that we can better prepare the controller for deployment with control-oriented meta-learning of features in closed-loop simulation, rather than regression-oriented meta-learning of features to fit input-output data. Specifically, we meta-learn the adaptive controller with closed-loop tracking simulation as the base-learner and the average tracking error as the meta-objective. With a nonlinear planar rotorcraft subject to wind, we demonstrate that our adaptive controller outperforms other controllers trained with regression-oriented meta-learning when deployed in closed-loop for trajectory tracking control.

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.

Brett T. Lopez, Jean-Jacques E. Slotine

This work develops a new direct adaptive control framework that extends the certainty equivalence principle to general nonlinear systems with unmatched model uncertainties. The approach adjusts the rate of adaptation online to eliminate the effects of parameter estimation transients on closed-loop stability. The method can be immediately combined with a previously designed or learned feedback policy if a corresponding model-parameterized Lyapunov function or contraction metric is known. Simulation results of various nonlinear systems with unmatched uncertainties demonstrates the approach.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

We study the problem of adaptively controlling a known discrete-time nonlinear system subject to unmodeled disturbances. We prove the first finite-time regret bounds for adaptive nonlinear control with matched uncertainty in the stochastic setting, showing that the regret suffered by certainty equivalence adaptive control, compared to an oracle controller with perfect knowledge of the unmodeled disturbances, is upper bounded by $\widetilde{O}(\sqrt{T})$ in expectation. Furthermore, we show that when the input is subject to a $k$ timestep delay, the regret degrades to $\widetilde{O}(k \sqrt{T})$. Our analysis draws connections between classical stability notions in nonlinear control theory (Lyapunov stability and contraction theory) and modern regret analysis from online convex optimization. The use of stability theory allows us to analyze the challenging infinite-horizon single trajectory setting.

Brett T. Lopez, Jean-Jacques E. Slotine

This work proposes a quaternion-based sliding variable that describes exponentially convergent error dynamics for any forward complete desired attitude trajectory. The proposed sliding variable directly operates on the non-Euclidean space formed by quaternions and explicitly handles the double covering property to enable global attitude tracking when used in feedback. In-depth analysis of the sliding variable is provided and compared to others in the literature. Several feedback controllers including nonlinear PD, robust, and adaptive sliding control are then derived. Simulation results of a rigid body with uncertain dynamics demonstrate the effectiveness and superiority of the approach.

Nicholas M. Boffi, Stephen Tu, Nikolai Matni et al.

Many existing tools in nonlinear control theory for establishing stability or safety of a dynamical system can be distilled to the construction of a certificate function that guarantees a desired property. However, algorithms for synthesizing certificate functions typically require a closed-form analytical expression of the underlying dynamics, which rules out their use on many modern robotic platforms. To circumvent this issue, we develop algorithms for learning certificate functions only from trajectory data. We establish bounds on the generalization error - the probability that a certificate will not certify a new, unseen trajectory - when learning from trajectories, and we convert such generalization error bounds into global stability guarantees. We demonstrate empirically that certificates for complex dynamics can be efficiently learned, and that the learned certificates can be used for downstream tasks such as adaptive control.

Krzysztof Choromanski, Jared Quincy Davis, Valerii Likhosherstov et al.

We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the orthogonal group O(d). This nested system of two flows, where the parameter-flow is constrained to lie on the compact manifold, provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem which is intrinsically related to training deep neural network architectures such as Neural ODEs. Consequently, it leads to better downstream models, as we show on the example of training reinforcement learning policies with evolution strategies, and in the supervised learning setting, by comparing with previous SOTA baselines. We provide strong convergence results for our proposed mechanism that are independent of the depth of the network, supporting our empirical studies. Our results show an intriguing connection between the theory of deep neural networks and the field of matrix flows on compact manifolds.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These observations have important implications for modern deep learning but remain poorly understood due to the difficulty of the associated nonconvex optimization problem. Towards an understanding of this phenomenon, we analyze a family of pseudogradient methods for learning generalized linear models under the square loss - a simplified problem containing both nonlinearity in the model parameters and nonconvexity of the optimization which admits a single neuron as a special case. We prove non-asymptotic bounds on the generalization error that sharply characterize how the interplay between the optimization geometry and the feature space geometry sets the out-of-sample performance of the learned model. Experimentally, selecting the optimization geometry as suggested by our theory leads to improved performance in generalized linear model estimation problems such as nonlinear and nonconvex variants of sparse vector recovery and low-rank matrix sensing.

Preston Culbertson, Jean-Jacques E. Slotine, Mac Schwager

In this work, we consider a group of robots working together to manipulate a rigid object to track a desired trajectory in $SE(3)$. The robots do not know the mass or friction properties of the object, or where they are attached to the object. They can, however, access a common state measurement, either from one robot broadcasting its measurements to the team, or by all robots communicating and averaging their state measurements to estimate the state of their centroid. To solve this problem, we propose a decentralized adaptive control scheme wherein each agent maintains and adapts its own estimate of the object parameters in order to track a reference trajectory. We present an analysis of the controller's behavior, and show that all closed-loop signals remain bounded, and that the system trajectory will almost always (except for initial conditions on a set of measure zero) converge to the desired trajectory. We study the proposed controller's performance using numerical simulations of a manipulation task in 3D, as well as hardware experiments which demonstrate our algorithm on a planar manipulation task. These studies, taken together, demonstrate the effectiveness of the proposed controller even in the presence of numerous unmodeled effects, such as discretization errors and complex frictional interactions.

Jared Quincy Davis, Krzysztof Choromanski, Jake Varley et al.

Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient, constant memory backpropagation. Neural ODEs are universal approximators only when they are non-autonomous, that is, the dynamics depends explicitly on time. We propose a novel family of Neural ODEs with time-varying weights, where time-dependence is non-parametric, and the smoothness of weight trajectories can be explicitly controlled to allow a tradeoff between expressiveness and efficiency. Using this enhanced expressiveness, we outperform previous Neural ODE variants in both speed and representational capacity, ultimately outperforming standard ResNet and CNN models on select image classification and video prediction tasks.

Nicholas M. Boffi, Jean-Jacques E. Slotine

Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors -- out of the many that will achieve perfect tracking or prediction -- for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian. We show that its Euler Lagrange equations lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.

Jozef Bucko, Yulia Sandamirskaya, Jean-Jacques Slotine

Analytical approach to SLAM problem was introduced in the recent years. In our work we investigate the method numerically with the motivation of using the algorithm in a real hardware experiments. We perform a robustness test of the algorithm and apply it to the robotic hardware in two different setups. In one we try to recover a map of the environment using bearing angle measurements and radial distance measurements. The another setup utilizes only bearing angle information.

Sumeet Singh, Spencer M. Richards, Vikas Sindhwani et al.

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in Reproducing Kernel Hilbert Spaces, we formulate stabilizable dynamics learning as a functional optimization with convex objective and bi-convex functional constraints. Under a mild structural assumption and relaxation of the functional constraints to sampling-based constraints, we derive the optimal solution with a modified Representer theorem. Finally, we utilize random matrix feature approximations to reduce the dimensionality of the search parameters and formulate an iterative convex optimization algorithm that jointly fits the dynamics functions and searches for a certificate of stabilizability. We validate the proposed algorithm in simulation for a planar quadrotor, and on a quadrotor hardware testbed emulating planar dynamics. We verify, both in simulation and on hardware, significantly improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when learning from small supervised datasets. The results support the conjecture that the use of stabilizability constraints as a form of regularization can help prune the hypothesis space in a manner that is tailored to the downstream task of trajectory generation and feedback control, resulting in models that are not only dramatically better conditioned, but also data efficient.

Sumeet Singh, Vikas Sindhwani, Jean-Jacques E. Slotine et al.

We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key idea is to develop a new control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, which guarantees that the learned system can be accompanied by a robust controller capable of stabilizing any open-loop trajectory that the system may generate. By leveraging tools from contraction theory, statistical learning, and convex optimization, we provide a general and tractable semi-supervised algorithm to learn stabilizable dynamics, which can be applied to complex underactuated systems. We validated the proposed algorithm on a simulated planar quadrotor system and observed notably improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when using a small number of demonstration examples. The results presented illustrate the need to infuse standard model-based reinforcement learning algorithms with concepts drawn from nonlinear control theory for improved reliability.

Winfried Lohmiller, Philipp Gassert, Jean-Jacques Slotine

We discuss technical results on learning function approximations using piecewise-linear basis functions, and analyze their stability and convergence using nonlinear contraction theory.

Patrick M. Wensing, Günter Niemeyer, Jean-Jacques E. Slotine

This paper presents an algorithm to geometrically characterize inertial parameter identifiability for an articulated robot. The geometric approach tests identifiability across the infinite space of configurations using only a finite set of conditions and without approximation. It can be applied to general open-chain kinematic trees ranging from industrial manipulators to legged robots, and it is the first solution for this broad set of systems that is provably correct. The high-level operation of the algorithm is based on a key observation: Undetectable changes in inertial parameters can be represented as sequences of inertial transfers across the joints. Drawing on the exponential parameterization of rigid-body kinematics, undetectable inertial transfers are analyzed in terms of observability from linear systems theory. This analysis can be applied recursively, and lends an overall complexity of $O(N)$ to characterize parameter identifiability for a system of $N$ bodies. Matlab source code for the new algorithm is provided.

Patrick M. Wensing, Jean-Jacques E. Slotine

This paper studies collaboration through the cloud in the context of cooperative adaptive control for robot manipulators. We first consider the case of multiple robots manipulating a common object through synchronous centralized update laws to identify unknown inertial parameters. Through this development, we introduce a notion of Collective Sufficient Richness, wherein parameter convergence can be enabled through teamwork in the group. The introduction of this property and the analysis of stable adaptive controllers that benefit from it constitute the main new contributions of this work. Building on this original example, we then consider decentralized update laws, time-varying network topologies, and the influence of communication delays on this process. Perhaps surprisingly, these nonidealized networked conditions inherit the same benefits of convergence being determined through collective effects for the group. Simple simulations of a planar manipulator identifying an unknown load are provided to illustrate the central idea and benefits of Collective Sufficient Richness.

Feng Tan, Winfried Lohmiller, Jean-Jacques Slotine

This paper solves the classical problem of simultaneous localization and mapping (SLAM) in a fashion which avoids linearized approximations altogether. Based on creating virtual synthetic measurements, the algorithm uses a linear time- varying (LTV) Kalman observer, bypassing errors and approximations brought by the linearization process in traditional extended Kalman filtering (EKF) SLAM. Convergence rates of the algorithm are established using contraction analysis. Different combinations of sensor information can be exploited, such as bearing measurements, range measurements, optical flow, or time-to-contact. As illustrated in simulations, the proposed algorithm can solve SLAM problems in both 2D and 3D scenarios with guaranteed convergence rates in a full nonlinear context.

Marco Tulio Angulo, Yang-Yu Liu, Jean-Jacques Slotine

Network motifs are overrepresented interconnection patterns found in real-world networks. What functional advantages may they offer for building complex systems? We show that most network motifs emerge from interconnections patterns that best exploit the intrinsic stability characteristics of individual nodes. This feature is observed at different scales in a network, from nodes to modules, suggesting an efficient mechanism to stably build complex systems.

Ian R. Manchester, Jean-Jacques E. Slotine

In this paper we introduce the concept of universal stabilizability: the condition that every solution of a nonlinear system can be globally stabilized. We give sufficient conditions in terms of the existence of a control contraction metric, which can be found by solving a pointwise linear matrix inequality. Extensions to approximate optimal control are straightforward. The conditions we give are necessary and sufficient for linear systems and certain classes of nonlinear systems, and have interesting connections to the theory of control Lyapunov functions.

Feng Tan, Jean-Jacques Slotine

Quorum sensing is a decentralized biological process, through which a community of cells with no global awareness coordinate their functional behaviors based solely on cell-medium interactions and local decisions. This paper draws inspirations from quorum sensing and colony competition to derive a new algorithm for data clustering. The algorithm treats each data as a single cell, and uses knowledge of local connectivity to cluster cells into multiple colonies simultaneously. It simulates auto-inducers secretion in quorum sensing to tune the influence radius for each cell. At the same time, sparsely distributed core cells spread their influences to form colonies, and interactions between colonies eventually determine each cell's identity. The algorithm has the flexibility to analyze not only static but also time-varying data, which surpasses the capacity of many existing algorithms. Its stability and convergence properties are established. The algorithm is tested on several applications, including both synthetic and real benchmarks data sets, alleles clustering, community detection, image segmentation. In particular, the algorithm's distinctive capability to deal with time-varying data allows us to experiment it on novel applications such as robotic swarms grouping and switching model identification. We believe that the algorithm's promising performance would stimulate many more exciting applications.

Ian R. Manchester, Jean-Jacques E. Slotine

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems.

Youssef Mroueh, Tomaso Poggio, Lorenzo Rosasco et al.

In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this setting it is possible to derive the first provably consistent regularized method with training/tuning complexity which is independent to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper.

Patrick Bechon, Jean-Jacques Slotine

With the advent of inexpensive simple humanoid robots, new classes of robotic questions can be considered experimentally. One of these is collective behavior of groups of humanoid robots, and in particular robot synchronization and swarming. The goal of this work is to robustly synchronize a group of humanoid robots, and to demonstrate the approach experimentally on a choreography of 8 robots. We aim to be robust to network latencies, and to allow robots to join or leave the group at any time (for example a fallen robot should be able to stand up to rejoin the choreography). Contraction theory is used to allow each robot in the group to synchronize to a common virtual oscillator, and quorum sensing strategies are exploited to fit within the available bandwidth. The humanoids used are Nao's, developed by Aldebaran Robotics.

Ueli Rutishauser, Jean-Jacques Slotine, Rodney J. Douglas

Models of cortical neuronal circuits commonly depend on inhibitory feedback to control gain, provide signal normalization, and to selectively amplify signals using winner-take-all (WTA) dynamics. Such models generally assume that excitatory and inhibitory neurons are able to interact easily, because their axons and dendrites are co-localized in the same small volume. However, quantitative neuroanatomical studies of the dimensions of axonal and dendritic trees of neurons in the neocortex show that this co-localization assumption is not valid. In this paper we describe a simple modification to the WTA circuit design that permits the effects of distributed inhibitory neurons to be coupled through synchronization, and so allows a single WTA to be distributed widely in cortical space, well beyond the arborization of any single inhibitory neuron, and even across different cortical areas. We prove by non-linear contraction analysis, and demonstrate by simulation that distributed WTA sub-systems combined by such inhibitory synchrony are inherently stable. We show analytically that synchronization is substantially faster than winner selection. This circuit mechanism allows networks of independent WTAs to fully or partially compete with each other.