The power dissipation method and kinematic reducibility of multiple-model robotic systems

This paper develops a formal connection between the power dissipation method (PDM) and Lagrangian mechanics, with specific application to robotic systems. Such a connection is necessary for understanding how some of the successes in motion planning and stabilization for smooth kinematic robotic systems can be extended to systems with frictional interactions and overconstrained systems. We establish this connection using the idea of a multiple-model system, and then show that multiple-model systems arise naturally in a number of instances, including those arising in cases traditionally addressed using the PDM. We then give necessary and sufficient conditions for a dynamic multiple-model system to be reducible to a kinematic multiple-model system. We use this result to show that solutions to the PDM are actually kinematic reductions of solutions to the Euler-Lagrange equations. We are particularly motivated by mechanical systems undergoing multiple intermittent frictional contacts, such as distributed manipulators, overconstrained wheeled vehicles, and objects that are manipulated by grasping or pushing. Examples illustrate how these results can provide insight into the analysis and control of physical systems

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

On least-cost path for realistic simulation of human motion

We are interested in "human-like" automatic motion simulation with applications in ergonomics. The apparent redundancy of the humanoid wrt its explicit tasks leads to the problem of choosing a plausible movement in the framework of redundant kinematics. Some results have been obtained in the human motion literature for reach motion that involves the position of the hands. We discuss these results and a motion generation scheme associated. When orientation is also explicitly required, very few works are available and even the methods for analysis are not defined. We discuss the choice for metrics adapted to the orientation, and also the problems encountered in defining a proper metric in both position and orientation. Motion capture and simulations are provided in both cases. The main goals of this paper are: to provide a survey on human motion features at task level for both position and orientation, to propose a kinematic control scheme based on these features, to define properly the error between motion capture and automatic motion simulation

Computational Modeling, Visualization, and Control of 2-D and 3-D Grasping under Rolling Contacts

This chapter presents a computational methodology for modeling 2-dimensional grasping of a 2-D object by a pair of multi-joint robot fingers under rolling contact constraints. Rolling contact constraints are expressed in a geometric interpretation of motion expressed with the aid of arclength parameters of the fingertips and object contours with an arbitrary geometry. Motions of grasping and object manipulation are expressed by orbits that are a solution to the Euler-Lagrange equation of motion of the fingers/object system together with a set of first-order differential equations that update arclength parameters. This methodology is then extended to mathematical modeling of 3-dimensional grasping of an object with an arbitrary shape. Based upon the mathematical model of 2-D grasping, a computational scheme for construction of numerical simulators of motion under rolling contacts with an arbitrary geometry is presented, together with preliminary simulation results. The chapter is composed of the following three parts. Part 1 Modeling and Control of 2-D Grasping under Rolling Contacts between Arbitrary Smooth Contours Authors: S. Arimoto and M. Yoshida Part 2 Simulation of 2-D Grasping under Physical Interaction of Rolling between Arbitrary Smooth Contour Curves Authors: M. Yoshida and S. Arimoto Part 3 Modeling of 3-D Grasping under Rolling Contacts between Arbitrary Smooth Surfaces Authors: S. Arimoto, M. Sekimoto, and M. Yoshid

Programming by Demonstration on Riemannian Manifolds

This thesis presents a Riemannian approach to Programming by Demonstration (PbD). It generalizes an existing PbD method from Euclidean manifolds to Riemannian manifolds. In this abstract, we review the objectives, methods and contributions of the presented approach. OBJECTIVES PbD aims at providing a user-friendly method for skill transfer between human and robot. It enables a user to teach a robot new tasks using few demonstrations. In order to surpass simple record-and-replay, methods for PbD need to \u2018understand\u2019 what to imitate; they need to extract the functional goals of a task from the demonstration data. This is typically achieved through the application of statisticalmethods. The variety of data encountered in robotics is large. Typical manipulation tasks involve position, orientation, stiffness, force and torque data. These data are not solely Euclidean. Instead, they originate from a variety of manifolds, curved spaces that are only locally Euclidean. Elementary operations, such as summation, are not defined on manifolds. Consequently, standard statistical methods are not well suited to analyze demonstration data that originate fromnon-Euclidean manifolds. In order to effectively extract what-to-imitate, methods for PbD should take into account the underlying geometry of the demonstration manifold; they should be geometry-aware. Successful task execution does not solely depend on the control of individual task variables. By controlling variables individually, a task might fail when one is perturbed and the others do not respond. Task execution also relies on couplings among task variables. These couplings describe functional relations which are often called synergies. In order to understand what-to-imitate, PbDmethods should be able to extract and encode synergies; they should be synergetic. In unstructured environments, it is unlikely that tasks are found in the same scenario twice. The circumstances under which a task is executed\u2014the task context\u2014are more likely to differ each time it is executed. Task context does not only vary during task execution, it also varies while learning and recognizing tasks. To be effective, a robot should be able to learn, recognize and synthesize skills in a variety of familiar and unfamiliar contexts; this can be achieved when its skill representation is context-adaptive. THE RIEMANNIAN APPROACH In this thesis, we present a skill representation that is geometry-aware, synergetic and context-adaptive. The presented method is probabilistic; it assumes that demonstrations are samples from an unknown probability distribution. This distribution is approximated using a Riemannian GaussianMixtureModel (GMM). Instead of using the \u2018standard\u2019 Euclidean Gaussian, we rely on the Riemannian Gaussian\u2014 a distribution akin the Gaussian, but defined on a Riemannian manifold. A Riev mannian manifold is a manifold\u2014a curved space which is locally Euclidean\u2014that provides a notion of distance. This notion is essential for statistical methods as such methods rely on a distance measure. Examples of Riemannian manifolds in robotics are: the Euclidean spacewhich is used for spatial data, forces or torques; the spherical manifolds, which can be used for orientation data defined as unit quaternions; and Symmetric Positive Definite (SPD) manifolds, which can be used to represent stiffness and manipulability. The Riemannian Gaussian is intrinsically geometry-aware. Its definition is based on the geometry of the manifold, and therefore takes into account the manifold curvature. In robotics, the manifold structure is often known beforehand. In the case of PbD, it follows from the structure of the demonstration data. Like the Gaussian distribution, the Riemannian Gaussian is defined by a mean and covariance. The covariance describes the variance and correlation among the state variables. These can be interpreted as local functional couplings among state variables: synergies. This makes the Riemannian Gaussian synergetic. Furthermore, information encoded in multiple Riemannian Gaussians can be fused using the Riemannian product of Gaussians. This feature allows us to construct a probabilistic context-adaptive task representation. CONTRIBUTIONS In particular, this thesis presents a generalization of existing methods of PbD, namely GMM-GMR and TP-GMM. This generalization involves the definition ofMaximum Likelihood Estimate (MLE), Gaussian conditioning and Gaussian product for the Riemannian Gaussian, and the definition of ExpectationMaximization (EM) and GaussianMixture Regression (GMR) for the Riemannian GMM. In this generalization, we contributed by proposing to use parallel transport for Gaussian conditioning. Furthermore, we presented a unified approach to solve the aforementioned operations using aGauss-Newton algorithm. We demonstrated how synergies, encoded in a Riemannian Gaussian, can be transformed into synergetic control policies using standard methods for LinearQuadratic Regulator (LQR). This is achieved by formulating the LQR problem in a (Euclidean) tangent space of the Riemannian manifold. Finally, we demonstrated how the contextadaptive Task-Parameterized Gaussian Mixture Model (TP-GMM) can be used for context inference\u2014the ability to extract context from demonstration data of known tasks. Our approach is the first attempt of context inference in the light of TP-GMM. Although effective, we showed that it requires further improvements in terms of speed and reliability. The efficacy of the Riemannian approach is demonstrated in a variety of scenarios. In shared control, the Riemannian Gaussian is used to represent control intentions of a human operator and an assistive system. Doing so, the properties of the Gaussian can be employed to mix their control intentions. This yields shared-control systems that continuously re-evaluate and assign control authority based on input confidence. The context-adaptive TP-GMMis demonstrated in a Pick & Place task with changing pick and place locations, a box-taping task with changing box sizes, and a trajectory tracking task typically found in industr

On geodesic paths and least-cost motions for human-like tasks

We are interested in ”human-like” automatic mo- tion generation. The apparent redundancy of the humanoid wrt its explicit tasks lead to the problem of choosing a plausible movement in the framework of redundant kinematics. Some results have been obtained in the human motion literature for reach motion that involves the position of the hands. We discuss these results and a motion generation scheme associated. When orientation is also explicitly required, very few works are available and even the methods for analysis are not defined. We discuss the choice for metrics adapted to the orientation, and also the problems encountered in defining a proper metric in both position and orientation. Motion capture and simulations are provided in both cases. The main goals of this paper are : - to provide a survey on human motion features at task level for both position and orientation, - to propose a kinematic control scheme based on these features - to define properly the error between motion capture and automatic motion simulation