Controlling complex tasks in robotic systems, such as circular motion for cleaning or following curvy lines, can be dealt with using nonlinear vector fields. This paper introduces a novel approach called the rotational obstacle avoidance method (ROAM) for adapting the initial dynamics when obstacles partially occlude the workspace. ROAM presents a closed-form solution that effectively avoids star-shaped obstacles in spaces of arbitrary dimensions by rotating the initial dynamics toward the tangent space. The algorithm enables navigation within obstacle hulls and can be customized to actively move away from surfaces while guaranteeing the presence of only a single saddle point on the boundary of each obstacle. We introduce a sequence of mappings to extend the approach for general nonlinear dynamics. Moreover, ROAM extends its capabilities to handle multi-obstacle environments and provides the ability to constrain dynamics within a safe tube. By utilizing weighted vector-tree summation, we successfully navigate around general concave obstacles represented as a tree-of-stars. Through experimental evaluation, ROAM demonstrates superior performance in minimizing occurrences of local minima and maintaining similarity to the initial dynamics, outperforming existing approaches in multi-obstacle simulations. Due to its simplicity, the proposed method is highly reactive and can be applied effectively in dynamic environments. This was demonstrated during the collision-free navigation of a 7-degree-of-freedom robot arm around dynamic obstacles.

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.

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.

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. 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.

Indirect trajectory optimization methods such as Differential Dynamic Programming (DDP) have found considerable success when only planning under dynamic feasibility constraints. Meanwhile, nonlinear programming (NLP) has been the state-of-the-art approach when faced with additional constraints (e.g., control bounds, obstacle avoidance). However, a naïve implementation of NLP algorithms, e.g., shooting-based sequential quadratic programming (SQP), may suffer from slow convergence – caused from natural instabilities of the underlying system manifesting as poor numerical stability within the optimization. Re-interpreting the DDP closed-loop rollout policy as a sensitivity-based correction to a second-order search direction, we demonstrate how to compute analogous closedloop policies (i.e., feedback gains) for constrained problems. Our key theoretical result introduces a novel dynamic programmingbased constraint-set recursion that augments the canonical “cost-to-go” backward pass. On the algorithmic front, we develop a hybrid-SQP algorithm incorporating DDP-style closedloop rollouts, enabled via efficient parallelized computation of the feedback gains. Finally, we validate our theoretical and algorithmic contributions on a set of increasingly challenging benchmarks, demonstrating significant improvements in convergence speed over standard open-loop SQP.

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.

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 the closed-loop stability that is rigorously guaranteed and consistently observed for kernel machines.

Working memory has long been thought to arise from sustained spiking/attractor dynamics. However, recent work has suggested that short-term synaptic plasticity (STSP) may help maintain attractor states over gaps in time with little or no spiking. To determine if STSP endows additional functional advantages, we trained artificial recurrent neural networks (RNNs) with and without STSP to perform an object working memory task. We found that RNNs with and without STSP were able to maintain memories despite distractors presented in the middle of the memory delay. However, RNNs with STSP showed activity that was similar to that seen in the cortex of a non-human primate (NHP) performing the same task. By contrast, RNNs without STSP showed activity that was less brain-like. Further, RNNs with STSP were more robust to network degradation than RNNs without STSP. These results show that STSP can not only help maintain working memories, it also makes neural networks more robust and brain-like.

This article presents a closed-form approach to constraining a flow within a given volume and around objects. The flow is guaranteed to converge and to stop at a single fixed point. The obstacle avoidance problem is 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 or obstacles. It asymptotically converges toward 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 enclosed corridors while avoiding static and moving obstacles. It was applied on an autonomous robot (QOLO) in a static complex indoor environment and 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 the presence of a diverse crowd with a nonuniform motion pattern.

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.1

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.

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.

In this work, we consider a group of robots working together to manipulate a rigid object to track a desired trajectory in$\text{SE}(3)$. 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 3-D, 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.

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 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 width 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.

We analyze the effect of synchronization on distributed stochastic gradient algorithms. By exploiting an analogy with dynamical models of biological quorum sensing, where synchronization between agents is induced through communication with a common signal, we quantify how synchronization can significantly reduce the magnitude of the noise felt by the individual distributed agents and their spatial mean. This noise reduction is in turn associated with a reduction in the smoothing of the loss function imposed by the stochastic gradient approximation. Through simulations on model nonconvex objectives, we demonstrate that coupling can stabilize higher noise levels and improve convergence. We provide a convergence analysis for strongly convex functions by deriving a bound on the expected deviation of the spatial mean of the agents from the global minimizer for an algorithm based on quorum sensing, the same algorithm with momentum, and the elastic averaging SGD (EASGD) algorithm. We discuss extensions to new algorithms that allow each agent to broadcast its current measure of success and shape the collective computation accordingly. We supplement our theoretical analysis with numerical experiments on convolutional neural networks trained on the CIFAR-10 data set, where we note a surprising regularizing property of EASGD even when applied to the non-distributed case. This observation suggests alternative second-order in time algorithms for nondistributed optimization that are competitive with momentum methods.

The brain consists of many interconnected networks with time-varying, partially autonomous activity. There are multiple sources of noise and variation yet activity has to eventually converge to a stable, reproducible state (or sequence of states) for its computations to make sense. We approached this problem from a control-theory perspective by applying contraction analysis to recurrent neural networks. This allowed us to find mechanisms for achieving stability in multiple connected networks with biologically realistic dynamics, including synaptic plasticity and time-varying inputs. These mechanisms included inhibitory Hebbian plasticity, excitatory anti-Hebbian plasticity, synaptic sparsity and excitatory-inhibitory balance. Our findings shed light on how stable computations might be achieved despite biological complexity. Crucially, our analysis is not limited to analyzing the stability of fixed geometric objects in state space (e.g points, lines, planes), but rather the stability of state trajectories which may be complex and time-varying.

The brain uses spikes in neural circuits to perform many dynamical computations. The computations are performed with properties such as spiking efficiency, i.e. minimal number of spikes, and robustness to noise. A major obstacle for learning computations in artificial spiking neural networks with such desired biological properties is due to lack of our understanding of how biological spiking neural networks learn computations. Here, we consider the credit assignment problem, i.e. determining the local contribution of each synapse to the network's global output error, for learning nonlinear dynamical computations in a spiking network with the desired properties of biological networks. We approach this problem by fusing the theory of efficient, balanced neural networks (EBN) with nonlinear adaptive control theory to propose a local learning rule. Locality of learning rules are ensured by feeding back into the network its own error, resulting in a learning rule depending solely on presynaptic inputs and error feedbacks. The spiking efficiency and robustness of the network are guaranteed by maintaining a tight excitatory/inhibitory balance, ensuring that each spike represents a local projection of the global output error and minimizes a loss function. The resulting networks can learn to implement complex dynamics with very small numbers of neurons and spikes, exhibit the same spike train variability as observed experimentally, and are extremely robust to noise and neuronal loss.

Wind power is the source of propulsive energy for sailboats and albatrosses. We present the UNAv, an Unmanned Nautical Air-water vehicle, that borrows features from both. It is composed of a glider-type airframe fitted with a vertical wing-sail extending above the center of mass of the system and a vertical surface-piercing hydrofoil keel extending below. The sail and keel are both actuated in pitch about their span-wise axes. Like an albatross, the UNAv is fully streamlined, high lift-to-drag ratio and generates the gravity-cancelling force by means of its airborne wings. Like a sailboat, the UNAv interacts with water and may access the full magnitude of the wind. A trim analysis predicts that a 3.4-meter span, 3 kg system could stay airborne in winds as low as 2.8 m/s (5.5 knots), and travel several times faster than the wind speed. Trim flight requires the ability to fly at extreme low height with the keel immersed in water. For that purpose, a multi-input longitudinal flight controller that leverages fast flap actuation is presented. The flight maneuver is demonstrated experimentally.

Recent advances in perception and planning algorithms have enabled robots to navigate autonomously through unknown, cluttered environments at high-speeds. A key component of these systems is the ability to identify, select, and execute a safe trajectory around obstacles. Many of these systems, however, lack performance guarantees because model uncertainty and external disturbances are ignored when a trajectory is selected for execution. This work leverages results from nonlinear control theory to establish a bound on tracking performance that can be used to select a provably safe trajectory. The Composite Adaptive Sliding Controller (CASC) provides robustness to disturbances and reduces model uncertainty through high-rate parameter estimation. CASC is demonstrated in simulation and hardware to significantly improve the performance of a quadrotor navigating through unknown environments with external disturbances and unknown model parameters.

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.

Finding actions that satisfy the constraints imposed by both external inputs and internal representations is central to decision making. We demonstrate that some important classes of constraint satisfaction problems (CSPs) can be solved by networks composed of homogeneous cooperative-competitive modules that have connectivity similar to motifs observed in the superficial layers of neocortex. The winner-take-all modules are sparsely coupled by programming neurons that embed the constraints onto the otherwise homogeneous modular computational substrate. We show rules that embed any instance of the CSP's planar four-color graph coloring, maximum independent set, and sudoku on this substrate and provide mathematical proofs that guarantee these graph coloring problems will convergence to a solution. The network is composed of nonsaturating linear threshold neurons. Their lack of right saturation allows the overall network to explore the problem space driven through the unstable dynamics generated by recurrent excitation. The direction of exploration is steered by the constraint neurons. While many problems can be solved using only linear inhibitory constraints, network performance on hard problems benefits significantly when these negative constraints are implemented by nonlinear multiplicative inhibition. Overall, our results demonstrate the importance of instability rather than stability in network computation and offer insight into the computational role of dual inhibitory mechanisms in neural circuits.

We present a framework for online generation of robust motion plans for robotic systems with nonlinear dynamics subject to bounded disturbances, control constraints, and online state constraints such as obstacles. In an offline phase, one computes the structure of a feedback controller that can be efficiently implemented online to track any feasible nominal trajectory. The offline phase leverages contraction theory and convex optimization to characterize a fixed-size “tube” that the state is guaranteed to remain within while tracking a nominal trajectory (representing the center of the tube). In the online phase, when the robot is faced with obstacles, a motion planner uses such a tube as a robustness margin for collision checking, yielding nominal trajectories that can be safely executed, i.e., tracked without collisions under disturbances. In contrast to recent work on robust online planning using funnel libraries, our approach is not restricted to a fixed library of maneuvers computed offline and is thus particularly well-suited to applications such as UAV flight in densely cluttered environments where complex maneuvers may be required to reach a goal. We demonstrate our approach through simulations of a 6-state planar quadrotor navigating cluttered environments in the presence of a cross-wind. We also discuss applications of our approach to Tube Model Predictive Control (TMPC) and compare the merits of our method with state-of-the-art nonlinear TMPC techniques.

In recent years, hydrofoils have become ubiquitous and critical components of high-performance surface vehicles. Twenty-meter-long hydrofoil sailing craft are capable of reaching speeds in excess of 45 knots. Hydrofoil dinghies routinely travel faster than the wind and reach speeds up to 30 knots. Besides, in the quest for super-maneuverability, actuated hydrofoils could enable the efficient generation of large forces on demand. However, the control of hydrofoil systems remains challenging, especially in rough seas. With the intent to ultimately enable the design of versatile, small-scale, high-speed, and super-maneuverable surface vehicles, we investigate the problem of controlling the lift force generated by a flexible, surface-piercing hydrofoil traveling at high speed through a random wave field. We present a test platform composed of a rudder-like vertical hydrofoil actuated in pitch. The system is instrumented with velocity, force, and immersion depth sensors. We carry out high-speed field experiments in the presence of naturally occurring waves. The 2 cm chord hydrofoil is successfully controlled with a LTV/feedback linearization controller at speeds ranging from 4 to 10+ m/s.

Various control approaches have been developed for micro/nanomanipulations using optical tweezers. Most existing methods assume that the micro/nanoparticles stay trapped during manipulations, and stochastic perturbations (Brownian motion) are usually ignored for the simplification of model dynamics. However, the trapped particles could escape from the optical traps especially in motion due to several possible reasons: small trapping stiffness, stochastic perturbations, and kinetic energy gained during manipulation. This paper investigates the conditions under which micro/nanoparticles will stay trapped while in motion. The dynamics of the trapped particles subject to stochastic perturbations is analyzed. Dynamic trapping is considered and the maximum manipulation velocity is determined from a probabilistic perspective. A controller with certain velocity bound is proposed, the stability of the system is analysed in presence of stochastic perturbation. Experimental results are presented to show the effectiveness of the proposed control approach.

Previous explanations of computations performed by recurrent networks have focused on symmetrically connected saturating neurons and their convergence toward attractors. Here we analyze the behavior of asymmetrical connected networks of linear threshold neurons, whose positive response is unbounded. We show that, for a wide range of parameters, this asymmetry brings interesting and computationally useful dynamical properties. When driven by input, the network explores potential solutions through highly unstable 'expansion' dynamics. This expansion is steered and constrained by negative divergence of the dynamics, which ensures that the dimensionality of the solution space continues to reduce until an acceptable solution manifold is reached. Then the system contracts stably on this manifold towards its final solution trajectory. The unstable positive feedback and cross inhibition that underlie expansion and divergence are common motifs in molecular and neuronal networks. Therefore we propose that very simple organizational constraints that combine these motifs can lead to spontaneous computation and so to the spontaneous modification of entropy that is characteristic of living systems.

To learn reliable rules that can generalize to novel situations, the brain must be capable of imposing some form of regularization. Here we suggest, through theoretical and computational arguments, that the combination of noise with synchronization provides a plausible mechanism for regularization in the nervous system. The functional role of regularization is considered in a general context in which coupled computational systems receive inputs corrupted by correlated noise. Noise on the inputs is shown to impose regularization, and when synchronization upstream induces time-varying correlations across noise variables, the degree of regularization can be calibrated over time. The resulting qualitative behavior matches experimental data from visual cortex.

In this paper we dicuss 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 methods with training/tuning complexity which is {\em independent} to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper.

Models of cortical neuronal circuits commonly depend on inhibitory feedback to control gain, provide signal normalization, and 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 colocalized 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 colocalization assumption is not valid. In this letter, 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 nonlinear contraction analysis and demonstrate by simulation that distributed WTA subsystems 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 other.

Learning and decision making in the brain are key processes critical to survival, and yet are processes implemented by nonideal biological building blocks that can impose significant error. We explore quantitatively how the brain might cope with this inherent source of error by taking advantage of two ubiquitous mechanisms, redundancy and synchronization. In particular we consider a neural process whose goal is to learn a decision function by implementing a nonlinear gradient dynamics. The dynamics, however, are assumed to be corrupted by perturbations modeling the error, which might be incurred due to limitations of the biology, intrinsic neuronal noise, and imperfect measurements. We show that error, and the associated uncertainty surrounding a learned solution, can be controlled in large part by trading off synchronization strength among multiple redundant neural systems against the noise amplitude. The impact of the coupling between such redundant systems is quantified by the spectrum of the network Laplacian, and we discuss the role of network topology in synchronization and in reducing the effect of noise. We discuss range of situations in which the mechanisms we model arise in brain science and draw attention to experimental evidence suggesting that cortical circuits capable of implementing the computations of interest here can be found on several scales. Finally, simulations comparing theoretical bounds to the relevant empirical quantities show that the theoretical estimates we derive can be tight.

The neocortex has a remarkably uniform neuronal organization, suggesting that common principles of processing are employed throughout its extent. In particular, the patterns of connectivity observed in the superficial layers of the visual cortex are consistent with the recurrent excitation and inhibitory feedback required for cooperative-competitive circuits such as the soft winner-take-all (WTA). WTA circuits offer interesting computational properties such as selective amplification, signal restoration, and decision making. But these properties depend on the signal gain derived from positive feedback, and so there is a critical trade-off between providing feedback strong enough to support the sophisticated computations while maintaining overall circuit stability. The issue of stability is all the more intriguing when one considers that the WTAs are expected to be densely distributed through the superficial layers and that they are at least partially interconnected. We consider how to reason about stability in very large distributed networks of such circuits. We approach this problem by approximating the regular cortical architecture as many interconnected cooperative-competitive modules. We demonstrate that by properly understanding the behavior of this small computational module, one can reason over the stability and convergence of very large networks composed of these modules. We obtain parameter ranges in which the WTA circuit operates in a high-gain regime, is stable, and can be aggregated arbitrarily to form large, stable networks. We use nonlinear contraction theory to establish conditions for stability in the fully nonlinear case and verify these solutions using numerical simulations. The derived bounds allow modes of operation in which the WTA network is multistable and exhibits state-dependent persistent activities. Our approach is sufficiently general to reason systematically about the stability of any network, biological or technological, composed of networks of small modules that express competition through shared inhibition.

THE FUNCTIONAL ROLE OF SYNCHRONIZATION HAS ATTRACTED MUCH INTEREST AND DEBATE: in particular, synchronization may allow distant sites in the brain to communicate and cooperate with each other, and therefore may play a role in temporal binding, in attention or in sensory-motor integration mechanisms. In this article, we study another role for synchronization: the so-called "collective enhancement of precision". We argue, in a full nonlinear dynamical context, that synchronization may help protect interconnected neurons from the influence of random perturbations-intrinsic neuronal noise-which affect all neurons in the nervous system. More precisely, our main contribution is a mathematical proof that, under specific, quantified conditions, the impact of noise on individual interconnected systems and on their spatial mean can essentially be cancelled through synchronization. This property then allows reliable computations to be carried out even in the presence of significant noise (as experimentally found e.g., in retinal ganglion cells in primates). This in turn is key to obtaining meaningful downstream signals, whether in terms of precisely-timed interaction (temporal coding), population coding, or frequency coding. Similar concepts may be applicable to questions of noise and variability in systems biology.

Distributed synchronization is known to occur at several scales in the brain, and has been suggested as playing a key functional role in perceptual grouping. State-of-the-art visual grouping algorithms, however, seem to give comparatively little attention to neural synchronization analogies. Based on the framework of concurrent synchronization of dynamic systems, simple networks of neural oscillators coupled with diffusive connections are proposed to solve visual grouping problems. The same algorithm is shown to achieve promising results on several classical visual grouping problems, including point clustering, contour integration and image segmentation.

Time-frequency representations of audio signals often resemble texture images. This paper derives a simple audio classification algorithm based on treating sound spectrograms as texture images. The algorithm is inspired by an earlier visual classification scheme particularly efficient at classifying textures. While solely based on time-frequency texture features, the algorithm achieves surprisingly good performance in musical instrument classification experiments.

In conventional task-space control problem of robots, a single task-space information is used for the entire task. When the task-space control problem is formulated in image space, this implies that visual feedback is used throughout the movement. While visual feedback is important to improve the endpoint accuracy in presence of uncertainty, the initial movement is primarily ballistic and hence visual feedback is not necessary. The relatively large delay in visual information would also make the visual feedback ineffective for fast initial movements. Due to limited field of view of the camera, it is also difficult to easure that visual feedback can be used for the entire task. Therefore, the task may fail if any of the features is out of view. In this paper, we present a new task-space control strategy that allows the use of dual task-space information in a single controller. We shall show that the proposed task-space controller can transit smoothly from Cartesian-space feedback at the initial stage to vision-space feedback at the end stage when the target is near.

This paper presents a dynamic region following formation control method for a swarm of robots. In this control strategy, a swarm of robots shall move together as a group inside a dynamic region that can rotate or scale to enable the robots to adjust the formation. Various desired shapes can be formed by choosing appropriate functions. Unlike existing formation control methods, the proposed method do not need to have specific identities or orders in the group but yet dynamic formation can be formed for a large group of robots. This enables a swarm of robots to adjust the formation during the course of maneuver. The system is also scalable in the sense that any robot can move into the formation or leave the formation without affecting the other robots. Lyapunov-like function is presented for convergence analysis of the multi-robot systems. Simulation results are presented to illustrate the performance of the proposed controller.

Distributed synchronization is known to occur at several scales in the brain, and has been suggested as playing a key functional role in perceptual grouping. State-of-the-art visual grouping algorithms, however, seem to give comparatively little attention to neural synchronization analogies. Based on the framework of concurrent synchronization of dynamical systems, simple networks of neural oscillators coupled with diffusive connections are proposed to solve visual grouping problems. The key idea is to embed the desired grouping properties in the choice of the diffusive couplings, so that synchronization of oscillators within each group indicates perceptual grouping of the underlying stimulative atoms, while desynchronization between groups corresponds to group segregation. Compared with state-of-the-art approaches, the same algorithm is shown to achieve promising results on several classical visual grouping problems, including point clustering, contour integration, and image segmentation.

Concurrent synchronization is a regime where diverse groups of fully synchronized dynamic systems stably coexist. We study global exponential synchronization and concurrent synchronization in the context of Lagrangian systems control. In a network constructed by adding diffusive couplings to robot manipulators or mobile robots, a decentralized tracking control law globally exponentially synchronizes an arbitrary number of robots, and represents a generalization of the average consensus problem. Exact nonlinear stability guarantees and synchronization conditions are derived by contraction analysis. The proposed decentralized strategy is further extended to adaptive synchronization and partial-state coupling.

We investigate a biologically motivated approach to fast visual classification, directly inspired by the recent work [13]. Specifically, trading-off biological accuracy for computational efficiency, we explore using standard wavelet transforms and patch transforms to parallel the tuning of visual cortex V1 and V4 cells, alternated with max operations to achieve scale and translation invariance. A feature selection procedure is applied during learning to accelerate recognition. We introduce a simple attention-like feedback mechanism, significantly improving recognition and robustness in multiple-object scenes. In experiments, the proposed algorithm achieves or exceeds state-of-the-art performance in object recognition, but also in new applications such as texture classification, satellite image classification, and language identification. Preliminary results on sound classification are shown as well.

In this paper, a region following formation control method for multi-robot systems is proposed. In this control method, the robots move as a group inside a desired region while maintaining a minimum distance among themselves. Various shapes of desired region can be formed by choosing the appropriate objective functions. The robots do not need to have specific identities since the proposed controller does not need specific orders of robots within the group. Therefore, the system is scalable since any robot can come in or go out of the group without affecting the system. Lyapunov-like function is presented for convergence analysis of the multi-robot systems. Simulation results are presented to illustrate the performance of the proposed controller.

A decentralized controller is presented that causes a network of robots to converge to a near optimal sensing configuration, while simultaneously learning the distribution of sensory information in the environment. A consensus (or flocking) term is introduced in the learning law to allow sharing of parameters among neighbors, greatly increasing learning convergence rates. Convergence and consensus is proven using a Lyapunov-type proof. The controller with parameter consensus is shown to perform better than the basic controller in numerical simulations.

In this paper, a vision based tracking controller with adaptation to uncertainty in depth information is presented. Depth uncertainty plays a special role in visual tracking as it appears nonlinearly in the overall Jacobian matrix and hence cannot be adapted together with other uncertain kinematic parameters. We propose a novel parameter update law to update the uncertain parameters of the depth. It is proved that system stability can be guaranteed for the visual tracking task in presence of uncertainties in depth information, robot kinematics and dynamics. Simulation results are presented to illustrate the performance of the proposed controller.

A decentralized, adaptive control law is presented to drive a network of mobile robots to a near-optimal sensing configuration. The control law is adaptive in that it integrates sensor measurements to provide a converging estimate of the distribution of sensory information in the environment. It is decentralized in that it requires only information local to each robot. A Lyapunov-type proof is used to show that the control law causes the network to converge to a near-optimal sensing configuration, and the controller is demonstrated in numerical simulations. This technique suggests a broader application of adaptive control methodologies to decentralized control problems in unknown dynamical environments.

Various forms of animal locomotion have been studied in the biological literature. Neuroscience research suggests the existence of central pattern generators (CPGs), neural networks that generate periodic signals for locomotion. We study simplified modular architectures based on CPGs for robotic applications, and show their global exponential stability using partial contraction analysis. The proposed architectures can reproduce periodic CPG signals for swimming or walking motion of various animals. They can be combined towards increasingly complex behaviors while preserving stability

In many applications of robot manipulators, the end-effector is required to make contact with environment. In these applications, it is necessary to control not only the position but also the interaction force between the robot end-effector and environment. Most research so far on motion and force tracking control has assumed that the kinematics and constraint surface are exactly known. In this paper, we propose a visually-servoed adaptive Jacobian controller for motion and force tracking control with structural uncertainties in kinematics, dynamics and constraint surface. It is shown that uniform ultimate boundedness of the tracking errors can be guaranteed. Simulation results are presented to illustrate the performance of the proposed control law.

A non–linear dynamic system is called contracting if initial conditions are for- gotten exponentially fast, so that all trajectories converge to a single trajectory. We use contraction theory to derive an upper bound for the strength of recurrent connections that guarantees contraction for complex neural networks. Specifi- cally, we apply this theory to a special class of recurrent networks, often called Cooperative Competitive Networks (CCNs), which are an abstract representation of the cooperative-competitive connectivity observed in cortex. This specific type of network is believed to play a major role in shaping cortical responses and se- lecting the relevant signal among distractors and noise. In this paper, we analyze contraction of combined CCNs of linear threshold units and verify the results of our analysis in a hybrid analog/digital VLSI CCN comprising spiking neurons and dynamic synapses.

Most research so far on motion and force tracking control of robots has assumed that the kinematics and dynamics are exactly known. In this paper, we propose an adaptive Jacobian controller for motion and force tracking with uncertainties in kinematics and dynamics. It is shown that the robot end-effector can track the desired position and force trajectories with the uncertain parameters updated online. Simulation results are presented to illustrate the performance of the proposed control law

Joint flexibility is an important factor to consider in the robot control design if high performance is expected for the robot manipulators. The research work on control of rigid-link flexible-joint (RLFJ) robot in the literature has assumed that the kinematics of the robot is known exactly. There have been no results so far that can deal with the kinematics uncertainty in RLFJ robot. In this paper, we present the first study on this problem and propose an adaptive regulation method which can deal with the kinematics uncertainty and uncertainties in both link and motor dynamics of the RLFJ robot system. An observer is designed to avoid the use of acceleration due to the fourth-order overall dynamics. Sufficient conditions are derived to guarantee the asymptotic stability of the closed-loop system. Simulation result illustrates the effectiveness of proposed control method

This paper presents a task-space saturated-proportional, integral and differential (SP-ID) regulation approach for robot manipulators with uncertain kinematics and actuator model. The proposed approach is computationally efficient and easy to implement due to its simple structure. It's interesting to observe that in this paper the simple PID type controller is shown not only capable of compensating unknown gravity force, as has been known for long in robot control literature, but also capable of dealing with uncertainties in robot kinematics and actuator model. Sufficient conditions to guarantee system stability are provided and simulation results are presented to show the performance of proposed control method

Most research so far on motion and force tracking has assumed that the kinematics and dynamics are exactly known. In this paper, we propose an visually-servoed adaptive Jacobian controller for motion and force tracking with uncertainties in kinematics, dynamics and camera model. It is shown that the robot end-effector can track the desired position and force trajectories with the uncertain parameters updated online. Simulation results are presented to illustrate the performance of the proposed control law

Most research so far on trajectory tracking control of robot has assumed that the kinematics of the robot is known exactly. This paper extends our recent work on adaptive Jacobian tracking control by deriving a new algorithm for trajectory tracking of robots with uncertain kinematics and dynamics. The algorithm requires only to measure the end-effector position in visual space, besides the robot’s joint angles and joint velocities. Experimental results are presented to illustrate the performance of the proposed controllers. In the experiments, we demonstrate that the robot’s shadow can be used to control the robot.

Research so far on trajectory tracking control of robot has assumed that the kinematics of the robot is known exactly. In this paper, a new approximate Jacobian adaptive controller is proposed for trajectory tracking of robot with uncertain kinematics and dynamics. It is shown that the robot end effector is able to converge to a desired trajectory with the uncertain kinematics and dynamics parameters being updated online by parameter update laws. Experimental results are presented to illustrate the performance of the proposed controllers.

The richness and complexity of recurrent cortical circuits is an inexhaustible source of inspiration for thinking about high-level biological computation. In past theoretical studies, constraints on the synaptic connection patterns of threshold-linear networks were found that guaranteed bounded network dynamics, convergence to attractive fixed points, and multistability, all fundamental aspects of cortical information processing. However, these conditions were only sufficient, and it remained unclear which were the minimal (necessary) conditions for convergence and multistability. We show that symmetric threshold-linear networks converge to a set of attractive fixed points if and only if the network matrix is copositive. Furthermore, the set of attractive fixed points is nonconnected (the network is multiattractive) if and only if the network matrix is not positive semidefinite. There are permitted sets of neurons that can be coactive at a stable steady state and forbidden sets that cannot. Permitted sets are clustered in the sense that subsets of permitted sets are permitted and supersets of forbidden sets are forbidden. By viewing permitted sets as memories stored in the synaptic connections, we provide a formulation of long-term memory that is more general than the traditional perspective of fixed-point attractor networks. There is a close correspondence between threshold-linear networks and networks defined by the generalized Lotka-Volterra equations.

This paper extends earlier results on stable force reflecting teleoperation in the presence of significant time-delays to the case, frequent in practice, where the transmission delays are themselves varying with time in an unpredictable fashion. It shows that stability can be preserved through the systematic use of specially designed wave-variable filters. The resulting performance of the teleoperation system is illustrated in simulations, and is consistent with reasonable expectations on "ideal" behavior. The results may provide a practical tool for implementing force-reflecting teleoperation over the Internet.

Motivated by fluid analogies, artificial harmonic potentials can eliminate local minima problems in robot path planning. In this paper, simple analytical solutions to planar harmonic potentials are derived using tools from fluid mechanics, and are applied to two-dimensional planning among multiple moving obstacles. These closed-form solutions enable real-time computation to be readily achieved.

Many robotic applications involve interactions with a simple mechanism, such as opening a door or turning a crank. Implementing such tasks using standard controllers may require precise knowledge of the kinematics or result in prohibitively large internal forces. We propose a simple method that learns the shape of the mechanism while in motion and generates little internal forces, in essence following the path of least resistance. The discussion is illustrated experimentally.

Wave variables were originally introduced in the context of time delayed force reflecting teleoperation. Their use provides significant benefits for control, including robustness to arbitrary delays and an inherent hybrid construction, well suited for handling unknown passive environments. They also suggest a new perspective for analysis, presenting information from an alternative viewpoint. In this paper we explore the concept of wave variables in a more general robotic and mechanical setting, leading to alternate control approaches.

We examine the behavior of force reflecting teleoperators which are subjected to large time delays up to several seconds. While stability is guaranteed by the use of passive transmission procedures, such systems can demonstrate particular dynamics which interfere with normal operation. Using the notion of wave variables for the analysis and implementation, and making appropriate design choices, we can construct a telerobot system with consistent and predictable behavior. This follows the design goal of a virtual tool accounting for the implicit limitations imposed by the delay.

The rapid development and formalization of adaptive signal processing algorithms loosely inspired by biological models can be potentially harnessed for use in flexible new learning control algorithms for nonlinear dynamic systems. However, if such controller designs are to be viable in practice, their stability must be guaranteed and their performance quantified. In this paper, the stable adaptive tracking control designs employing “neural” networks, initially presented in Sanner and Slotine (1992), are extended to classes of multivariable mechanical systems, including robot manipulators, and bounds are developed for the magnitude of the asymptotic tracking errors and the rate of convergence to these bounds. This new algorithm permits simultaneous learning and control, without recourse to an initial identification stage, and is distinguished from previous stable adaptive robotic controllers, e.g. (Slotine and Li 1987), by the relative lack of structure assumed in the design of the control law. The required control is simply considered to contain unknown functions of the measured state variables, and adaptive “neural” networks are used to stably determine, in real time, the entire required functional dependence. While computationally more complex than explicitly model-based techniques, the methods developed in this paper may be effectively applied to the control of many physical systems for which the state dependence of the dynamics is reasonably well understood, but the exact functional form of this dependence, or part thereof, is not, such as underwater robotic vehicles and high performance aircraft.

A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems.

It is shown that adaptive extensions of sliding control are effective for precise control of underwater vehicles through experiments with an actual vehicle. Adaptive sliding control permits direct nonlinear control system design, including online parameter estimation. Experimental results are detailed which were obtained from a tethered underwater vehicle equipped with a precision broadband acoustic navigation system and three-degree-of-freedom attitude instrumentation controlled through a real-time network of transputers. Simulation and experimental trials show online determination of vehicle parameters such as mass and nonlinear drag.>

The authors explore performance issues linked to the effective implementation of adaptive manipulator controllers. In particular, they discuss computational implementations of the algorithm directly in Cartesian space, the utilization of kinematic redundancies, and applications to adaptive compliant motion control. The development is illustrated experimentally on a four-degree-of-freedom whole-arm articulated manipulator. It is suggested that the range of application of adaptive tracking controllers may extend well beyond adaptation to grasped loads.>

A framework is proposed for helping the operator of a redundant teleoperator to perform tasks in a 3D environment. The aid is implemented on a graphics workstation. Kinematic control of the redundant teleoperator is performed by specifying end-effector trajectories and using a numerical inverse kinematic algorithm to determine positions of the intermediate links of the teleoperator. Also proposed are graphic displays that aid the operator in the execution of tasks. Experiments with a twelve-degrees-of-freedom vehicle manipulator system that were conducted to compare these suggestions with other methods of kinematic control and display indicate that the proposed framework results in superior performance by reducing both task completion times and collision with obstacles in the environment.>

A method is presented which significantly improves the computational efficiency of time-optimal path-following planning algorithms for robot manipulators with limited actuator torques. Characteristics switching points are identified and characterized analytically as functions of the single parameter defining the position along the path. Limit curves are then constructed from the characteristic switching points, in contrast with the so-called maximum velocity curve approach of existing methods. An efficient algorithm is proposed which utilizes the characteristics of the defined concepts. The algorithm can also account for viscous friction effects and smooth state-dependent actuator bounds. A numerical example, while showing the consistency of the algorithm with the existing techniques, demonstrates its potential for increasing computational efficiency by several orders of magnitude.>

An iterative method of computing the solution of the inverse kinematic problem is developed for redundant systems using the transpose of the Jacobian matrix instead of the pseudoinverse. The solutions may be optimized on a criterion function or on physical constraints, such as obstacle avoidance. Stability and convergence of the method are shown. Although its convergence rate is only about half that of Newton's method, the advantage of the method is that it remains easily tractable close to the singular configurations of the manipulator. A hybrid method combining the Jacobian transpose and Newton's methods is proposed. Results of the application of the method on a 10-link manipulator in 2-D space are shown.>

The theoretical issues linked to the development of indirect adaptive robot controllers are discussed, and some possible solutions are proposed. After a review of the prediction models used for robotic parameter estimation, a variety of parameter estimation methods are discussed under a common framework based on an exact solution approach. A novel indirect adaptive controller structure, which consists of a modified computed torque using parameters obtained from any of the estimators discussed, is presented. It is shown that a critical difficulty in using indirect adaptive control is the necessity to explicitly guarantee that the estimated inertia matrix remains positive definite in the course of adaptation, a requirement avoided by both the direct and the composite adaptive controllers. A practical solution to this difficulty is proposed.>

VSS controller is suited for robotic arms where the robust performances in the presence of parametric variations and disturbances are most important. The practical design of such robust VSS controller is discussed in this paper. The significant issue in the design of VSS is to avoid the chattering caused by the switched input. This problem is solved by introducing the continuous control input instead of switched input. The practical design of such control is obtained by using "Balance Condition". The "Balance Condition" is derived from the careful consideration on the bandwidth of plant dynamics and VSS controller, and gives many benefits in applications of VSS controller to actual systems. This paper shows the validity of balance condition in the robotic arm control. Simulations and experimental results using a position servo system (one degree of freedom robotic arm) are discussed from the practical point of view of robust control. In order to investigate multi-joint cases, simulations with a two-linkage robot arm are provided.

Earlier work (Slotine and Li, 1986) demonstrates that using state feedback to directly modify a manipulator's energy function, rather than its fully expanded dynamics, represents a powerful approach to robot control, and, in particular, yields a simple globally convergent adaptive algorithm for trajectory control problems. An important practical feature of the algorithm is that is does not require measurements or estimates of the manipulator's joint accelerations. This paper rewrites the approach in term of end-effector dynamics, extends it to hybrid motion/force control, and discusses adaptive strategies involving mobile environments, such as the external motion control of an unknown passive mechanism.

Earlier work (Slotine and Li, 1986) exploits the particular structure of manipulator dynamics to develop a simple, globally convergent adaptive algorithm for trajectory control problems. The algorithm does not require measurements or estimates of the manipulator's joint accelerations, nor inversion of the estimated inertia matrix. This paper demonstrates the approach on a high-speed 2 d.o.f. semi-direct-drive robot. It shows that the manipulator mass properties, assumed to be initially unknown, can be precisely estimated within the first half second of a typical run. Similarly, the algorithm allows large loads of unknown mass properties to be precisely manipulated. Further, these experimental results demonstrate that the adaptive controller enjoys essentially the same level of robustness to unmodelled dynamics as a PD controller, yet achieves much better tracking accuracy than either PD or computed-torque schemes. Its superior performance for high speed operations, in the presence of parameter uncertainties, and its relative computational simplicity, make it a attractive option both to address complex industrial tasks, and to simplify high-level programming of more standard operations.

An overall concept and specific system elements for teleoperated vehicles and manipulators are presented. The approach emphasizes continuous, real-time sharing of control between both the human operator and the computer system and is intended for application to the JASON underwater vehicle now in development. As JASON will have extremely high communications bandwidth available through a fiber optic cable, the emphasis will be on aiding and extending the capabilities of the human operator. Specific elements presented include task-resolved motion specification, rule-based inverse kinematics, and robust and adaptive nonlinear tracking control.

Performance enhancement through modeling of high-frequency dynamics (such as structural resonant modes or actuator time-delays) and on-line parameter estimation (of large loads with unknown mass properties, or of characteristics of the environment in compliant motion control) is discussed in the context of sliding control. It is shown that such additions can be easily incorporated in existing controllers and yield predictable performance improvements.

A scheme is presented for the accurate trajectory control of robot manipulators in the presence of model uncertainty and disturbances. Based on the suction control methodology (an extension of sliding mode control), the scheme addresses the following problem: given the extent of parametric uncertainty (such as imprecisions or inertias, geometry, loads) and the frequency range of unmodeled dynamics (such as unmodeled structural modes, neglected time-delays), design a nonlinear feedback controller to achieve optimal tracking performance (in a suitable sense). The methodology is compared with algorithms such as the computed torque method, and is shown to combine in practice improved performance with simpler and more tractable controller designs. Extensions to compliant motion control are also discussed.