Geometric and Numerical Foundations of Movements

In previous works we reanalyzed the kinematics of hand movements and locomotion, and suggested that several geometries are used conjointly by the brain for according the shape and the duration along trajectories; this was done in collaboration with Tamar Flash and her collaborators [10, 64, 67], and with Quang-Cuong Pham [79]. The variety of geometries which were implied in this process, were associated to sub-groups of the affine group of a plane: full affine, equi-affine and Euclidean. Other studies have shown how the above geometries constrain the production of the movements [92], or began to use the affine geometry in Robotics [80]. In this article, we propose to use a new variety of geometries which extends the preceding series in another direction, to cover wider contexts and more complex movements, like prehension, initiation of walking, locomotion, navigation, imagined motion. The new spaces adapted to those geometries have no points; they come from topos theory, which is an extension of set theory replacing sets by fields and graphs of dynamics. Any given topos generates a variety of different geometries, which can be mixed as in the preceding studies. Such geometries take into account efforts, forces and dynamics; they do not neglect them aside as does traditional geometry. In this preliminary report we indicate the simplest characteristics of spaces which underly the above examples. The hypothesis is also that these spaces are implemented in different, although overlapping, central nervous system networks in the brain, corresponding to the different action spaces mentioned above. Here, as for the known classical geometries, the most concrete suggestion concerns the timing of movement: we predict that different components of the controlled system are using different intrinsic time courses, and that the mapping between these different internal durations is an important part of the dynamic under geometrical control. This reminds us of a well known psychological observation, for instance that time in imagination does not flow as ordinary clocks time, but this also suggests that reaching an object with the hand has its own time, or that equilibrium control in walking works within a specific time, which is different from the walking trajectory displacement time.

In this chapter, we present a mathematical theory of human movement vigor. At the core of the theory is the concept of the cost of time. According to it, natural movement cannot be too slow because the passage of time entails a cost which makes slow moves undesirable. Within this framework, an inverse methodology is available to reliably and robustly characterize how the brain penalizes time from experimental motion data. Yet, a general theory of human movement pace should not only account for the self-selected speed but should also include situations where slow or fast speed instructions are given by an experimenter or required by a task. In particular, the limit case of a “maximal speed” instruction is linked to Fitts’s law, i.e. the speed/accuracy trade-off. This chapter first summarizes the cost of time theory and the procedure used for its accurate identification. Then, the case of slow/fast movements is investigated but changing the duration of goal-directed movements can be done in various ways in this framework. Here we show that only one strategy seems plausible to account for both slow/fast and self-paced reaching movements. By relying upon a free-time optimal control formulation of the motor planning problem, this chapter provides a comprehensive treatment of the linear-quadratic case for single degree of freedom arm movements but the principles are easily extendable to multijoint and/or artificial systems.

If some resources of a robot are redundant with respect to a given objective, they can be used to address other, additional objectives. Since the amount of resources required to realize a given objective can vary, depending on the situation, this gives rise to a limited form of decision making, when assigning resources to different objectives according to the situation. Such decision making emerges in case of conflicts between objectives, and these conflicts appear to be situations of linear dependency and, ultimately, singularity of the solutions. Using an elementary model of a mobile manipulator robot with two degrees of freedom, we show how standard resolution schemes behave unexpectedly and inefficiently in such situations. We propose then as a remedy to introduce carefully tuned artificial conflicts, in the form of a trust region.

Exciting recent developments at the interface of optimization and control have shown that several fundamental problems in dynamics and control, such as stability, collision avoidance, robust performance, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization. In this chapter, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance guarantees as well as fundamental limitations. The topics covered include the concepts of “dsos and sdsos optimization”, path-complete and non-monotonic Lyapunov functions, and some lower bounds and complexity results for Lyapunov analysis of polynomial vector fields and hybrid systems. In each case, our relevant papers are tersely surveyed and the challenges/opportunities that lie ahead are stated.

We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows us to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton–Jacobi–Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.

Recent advances in distributed control, coupled with an exponential growth in data gathering capabilities, have made feasible a wide range of applications with potential to profoundly impact society, from safer self-aware environments and smart cities to enhanced model-based medical therapies. Yet, achieving this vision requires addressing the challenge of handling large amounts of very high dimensional data. In this chapter, we provide a tutorial showing how to exploit the inherent sparsity of the data, which is present in a large class of identification problems, to overcome the “curse of dimensionality”. The concepts presented here extend traditional ideas from machine learning linking big data and sparsity, to challenging dynamic settings. In particular, we explore the connections between system identification and information extraction from large data sets, using as an example human activity analysis from video data.

In this paper, we discuss numerical foundations and computational results for inverse optimal control of human locomotion based on human motion capture data. The task of inverse optimal control is to identify the precise underlying objective function that is optimized in an observed motion. The presented methods can cope with partial and imprecise measurements of the state variables which is typically the case for motion capture recordings. We investigate human walking and running motions on different levels of detail and consequently different underlying models which all have their own motivation depending on the question asked. Whole-body models are used to explore the mechanisms of motions on joint level, while simple models describing the subject as a single entity can be used to describe overall locomotion behavior. At an intermediate level, template models describe some relative motions of bodies while maintaining simplicity and computational efficiency. Results will be presented for all model types and different walking tasks. We also show for some of them how the identified objective functions can be used to generate new waking motions for humanoid robots in novel scenarios.

Traditional industrial robot controllers are typically dedicated to a specific task, while humans always interact with new objects yielding unknown interaction forces and instability. In this chapter, we examine the neuromechanics of such contact tasks. We develop a model of the necessary adaptation of force, mechanical impedance and planned trajectory for stable and efficient interaction with rigid or compliant surfaces of different structures. Simulations demonstrate that this model can be used as a novel adaptive robot controller yielding versatile control in representative interactive tasks such as cutting, drilling and haptic exploration, where the robot acquires a model of the geometry and structure of the surface along which it is moving.

This chapter provides a theoretical perspective on action and the control of movement from the point of view of the free-energy principle. This variational principle offers an explanation for neuronal activity and ensuing behavior that is formulated in terms of dynamical systems and attracting sets. We will see that the free-energy principle emerges when considering the ensemble dynamics of biological systems like ourselves. When we look closely what this principle implies for the behavior of systems like the brain, one finds a fairly straightforward explanation for many aspects of action and perception; in particular, their (approximately Bayesian) optimality. Within the Bayesian brain framework, the ensuing dynamics can be separated into those serving perceptual inference, learning and behavior. Variational principles play a key role in what follows; both in understanding the nature of self-organizing systems but also in explaining the adaptive nature of neuronal dynamics and plasticity in terms of optimization—and the process theories that mediate optimal inference and motor control. A special focus of this chapter is the pre-eminent role of heteroclinic cycles in providing deep and dynamic (generative) models of the sensorium; particularly the sensations that we generate ourselves through action. In what follows, we will briefly rehearse the basic theory and illustrate its implications using simulations of action (handwriting)—and its observation.

The modeling and online-generation of human-like body motion is a central topic in computer graphics and robotics. The analysis of the coordination structure of complex body movements in humans helps to develop flexible technical algorithms for movement synthesis. This chapter summarizes work that uses learned structured representations for the synthesis of complex human-like body movements in real-time. This work follows two different general approaches. The first one is to learn spatio-temporal movement primitives from human kinematic data, and to derive from this Dynamic Movement Primitives (DMPs), which are modeled by nonlinear dynamical systems. Such dynamical primitives are then coupled and embedded into networks that generate complex human-like behaviors online, as self-organized solutions of the underlying dynamics. The flexibility of this approach is demonstrated by synthesizing complex coordinated movements of single agents and crowds. We demonstrate that Contraction Theory provides an appropriate framework for the design of the stability properties of such complex composite systems. In addition, we demonstrate how such primitive-based movement representations can be embedded into a model-based predictive control architecture for the humanoid robot HRP-2. Using the primitive-based trajectory synthesis algorithm for fast online planning of full-body movements, we were able to realize flexibly adapting human-like multi-step sequences, which are coordinated with goal-directed reaching movements. The resulting architecture realizes fast online planing of multi-step sequences, at the same time ensuring dynamic balance during walking and the feasibility of the movements for the robot. The computation of such dynamically feasible multi-step sequences using state-of-the-art optimal control approaches would take hours, while our method works in real-time. The second presented framework for the online synthesis of complex body motion is based on the learning of hierarchical probabilistic generative models, where we exploit Bayesian machine learning approaches for nonlinear dimensionality reduction and the modeling of dynamical systems. Combining Gaussian Process Latent Variable Models (GPLVMs) and Gaussian Process Dynamical Models (GPDMs), we learned models for the interactive movements of two humans. In order to build an online reactive agent with controlled emotional style, we replaced the state variables of one actor by measurements obtained by real-time motion capture from a user and determined the most probable state of the interaction partner using Bayesian model inversion. The proposed method results in highly believable human-like reactive body motion.

Humans out-perform contemporary robots despite vastly slower ‘wetware’ (e.g. neurons) and ‘hardware’ (e.g. muscles). The basis of human sensory-motor performance appears to be quite different from that of robots. Human haptic perception is not compatible with Riemannian geometry, the foundation of classical mechanics and robot control. Instead, evidence suggests that human control is based on dynamic primitives, which enable highly dynamic behavior with minimal high-level supervision and intervention. Motion primitives include submovements (discrete actions) and oscillations (rhythmic behavior). Adding mechanical impedance as a class of dynamic primitives facilitates controlling physical interaction. Both motion and interaction primitives may be combined by re-purposing the classical equivalent electric circuit and extending it to a nonlinear equivalent network. It highlights the contrast between the dynamics of physical systems and the dynamics of computation and information processing. Choosing appropriate task-specific impedance may be cast as a stochastic optimization problem, though its solution remains challenging. The composability of dynamic primitives, including mechanical impedances, enables complex tasks, including multi-limb coordination, to be treated as a composite of simpler tasks, each represented by an equivalent network. The most useful form of nonlinear equivalent network requires the interactive dynamics to respond to deviations from the motion that would occur without interaction. That suggests some form of underlying geometric structure but which geometry is induced by a composition of motion and interactive dynamic primitives? Answering that question might pave the way to achieve superior robot control and seamless human-robot collaboration.

How do humans control their actions and interactions with the physical world? How do we learn to throw a ball or drink a glass of wine without spilling? Compared to robots human dexterity remains astonishing, especially as slow neural transmission and high levels of noise seem to plague the biological system. What are human control strategies that skillfully navigate, overcome, and even exploit these disadvantages? To gain insight we propose an approach that centers on how task dynamics constrain and enable (inter-)actions. Agnostic about details of the controller, we start with a physical model of the task that permits full understanding of the solution space. Rendering the task in a virtual environment, we examine how humans learn solutions that meet complex task demands. Central to numerous skills is redundancy that allows exploration and exploitation of subsets of solutions. We hypothesize that humans seek solutions that are stable to perturbations to make their intrinsic noise matter less. With fewer corrections necessary, the system is less afflicted by long delays in the feedback loop. Three experimental paradigms exemplify our approach: throwing a ball to a target, rhythmic bouncing of a ball, and carrying a complex object. For the throwing task, results show that actors are sensitive to the error-tolerance afforded by the task. In rhythmic ball bouncing, subjects exploit the dynamic stability of the paddle-ball system. When manipulating a “glass of wine”, subjects learn strategies that make the hand-object interactions more predictable. These findings set the stage for developing propositions about the controller: We posit that complex actions are generated with dynamic primitives, modules with attractor stability that are less sensitive to delays and noise in the neuro-mechanical system.

The control and planning of interaction forces is fundamental for locomotion and manipulation tasks since it is through the interaction with the environment that a robot can walk forward or manipulate objects. In this chapter we present a control and planning strategy focused on the control of interaction forces to generate multi-contact whole-body behaviors. Centered around the robot momentum dynamics, our approach consists of a hierarchical inverse dynamics controller that treats the control of the robot’s momentum as a contact force task and a trajectory optimization algorithm that can generate desired whole-body motions, momentum and desired contact forces for multiple contacts. Experimental results demonstrate the capabilities of the approach on a humanoid robot.

Controlling complex robotic platforms is a challenging task, especially in designs with high levels of kinematic redundancy. Novel variable stiffness actuators (VSAs) have recently demonstrated the possibility of achieving energetically more efficient and safer behaviour by allowing the ability to simultaneously modulate the output torque and stiffness while adding further levels of actuation redundancy. An optimal control approach has been demonstrated as an effective method for such a complex actuation mechanism in order to devise a control strategy that simultaneously provides optimal control commands and time-varying stiffness profiles. However, traditional optimal control formulations have typically focused on optimisation of the tasks over a predetermined time horizon with smooth, continuous plant dynamics. In this chapter, we address the optimal control problem of robotic systems with VSAs for the challenging domain of switching dynamics and discontinuous state transition arising from interactions with an environment. First, we present a systematic methodology to simultaneously optimise control commands, time-varying stiffness profiles as well as the optimal switching instances and total movement duration based on a time-based switching hybrid dynamics formulation. We demonstrate the effectiveness of our approach on the control of a brachiating robot with a VSA considering multi-phase swing-up and locomotion tasks as an illustrative application of our proposed method in order to exploit the benefits of the VSA and intrinsic dynamics of the system. Then, to address the issue of model discrepancies in model-based optimal control, we extend the proposed framework by incorporating an adaptive learning algorithm. This performs continuous data-driven adjustments to the dynamics model while re-planning optimal policies that reflect this adaptation. We show that this augmented approach is able to handle a range of model discrepancies in both simulations and hardware experiments.