A Geometrical Setting for the Newtonian Mechanics of Robots

Nonlinear Dynamics, 2013

This work is devoted to deriving and investigating conditions for the correct application of Newton's law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton's law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton's law on

Journal of Robotic Systems, 1994

Using a Riemannian metric in the special Euclidean group we define a kinematic metric on the space of kinematics of robot manipulators. The metric can be used as an instrument in the kinematic design and performance evaluation of robot manipulators.

International Journal of Robotics Research, 1999

The set of rigid body motions forms the Lie group SE(3), the special Euclidean group in three dimensions. In this paper we investigate Riemannian metrics and a ne connections on SE(3) that are suited for kinematic analysis and robot trajectory planning. In the rst part of the paper, we study metrics whose geodesics are screw motions. We prove that no Riemannian metric can have such geodesics and we show that metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannian metrics. In the second part of the paper we investigate a ne connections which through the covariant derivative give the expression for the acceleration of a rigid body that agrees with the expression used in kinematics. We prove that there is a unique symmetric connection with this property. Further, we show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positive de nite constant metric on the group of translations.

Nonlinear Dynamics, 2016

This study is focused on a class of discrete mechanical systems subject to equality motion constraints involving time and acatastatic terms. In addition, their original configuration manifold possesses time-dependent geometric properties. The emphasis is placed on a proper application of Newton's law of motion. A key step is to consider the corresponding event manifold, whose dimension is bigger by one than the configuration manifold, since a temporal coordinate is added to the original set of spatial coordinates. Then, its geometric properties are determined and Newton's law is applied on it, when no motion constraints exist. Next, the way of introducing time dependence in the geometric properties of the configuration manifold through a coordinate transformation in the event manifold is investigated and clarified. Moreover, similar time effects introduced through the motion constraints are also examined. Based on these and application of foliation theory, a geometric definition of a scleronomic manifold is then provided, accompanied by a set of coordinate invariant conditions. The analysis is completed by deriving an appropriate set of equations of motion on the original configuration manifold, when additional constraints are imposed. These equations appear as a system of second-order ordinary differential equations. Finally, the analytical findings are enhanced

arXiv (Cornell University), 2023

In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In the first part of the paper, we review how quasi-static mechanical manipulation tasks can be naturally described via the so-called force-space, i.e. the cotangent bundle of the configuration space, and its Lagrangian submanifolds. Then, via a second order analysis, we derive the control Hessian of total energy. As this is not necessarily positive-definite, from an optimal control perspective, we propose the use of the squared-Hessian, also motivated by insights derived from both mechanics (Gauss' Principle) and biology (Separation Principle). In the second part of the paper, we apply such methods to the problem of an elastically-driven, inverted pendulum. Despite its apparent simplicity, this example is representative of an important class of robotic manipulation problems for which we show how a smooth elastic potential can be derived by regularizing mechanical contact. We then show how graph theory can be used to connect each numerical solution to 'nearby' ones, with weights derived from the very metric introduced in the first part of the paper.

2001

Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its metric connection with mechanical structures produces some features which are absent in the case of general (non-Riemannian) manifolds. The aim of present paper is to discuss these features and develop special language for describing Newtonian, Lagrangian, and Hamiltonian dynamical systems in Riemannian manifolds.

American Journal of Mechanical Engineering, 2014

In this article are described the theoretical basics of geometric mechanics and differential geometry. In the introductory part of article, basic notions are explained that frequently occurring in the concept of geometric mechanics. It contains the basic building blocks that are used to create the configuration spaces. Further were described groups applied to the kinematic description of structure and to control in area of robotics and the fiber bundle that represents the configuration space for mechanical systems. The last part deals with the actions, such as the left and right action, lifted action and adjoint action.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1995

fur mechanische Systemr mii Zwangsbedingungen. Dies ,jiihrt zu einem geometrischen Prinzip der analytischen Mechanik. This paper provides a geometrical interpretation of the explicit equations of motion f o r constrained mechanical systems. ThiJ leads to a geometric principle of analytical mechanics. MSC (1991): 70835, 70B05, 70G05, 70505

A framework for the analysis and control of manipulator systems with respect to the dynamic behavior of their end-effectors is developed. First, issues related to the description of end-effector tasks that involve constrained motion and active force control are discussed. The fundamentals of the operational space formulation are then presented, and the unified approach for motion and force control is developed. The extension of this formulation to redundant manipulator systems is also presented, constructing the end-effector equations of motion and describing their behavior with respect to joint forces. These results are used in the development of a new and systematic approach for dealing with the problems arising at kinematic singularities. At a singular configuration, the manipulator is treated as a mechanism that is redundant with respect to the motion of the end-effector in the subspace of operational space orthogonal to the singular direction.

IEEE Transactions on Robotics and Automation, 2000

This paper presents a configuration manifold (C-manifold) embedding model for robot dynamic systems analysis and control algorithms development from a geometrical and topological perspective. The concepts of C-manifolds and their isometric embeddings are introduced, and the explicit forms of their representations are then developed. For an open serial-chain robotic system, a topological equivalence, i.e., a diffeomorphism between its combined C-manifold and the minimum embeddable C-manifold is found and demonstrated to be useful for dynamic model reduction. The study further shows that kinematics of a dynamic system determines the topology of its C-manifold so that the kinematics becomes a structure of the dynamics. By taking advantage of adaptive control, developing a kinematic model is shown to be sufficient for dynamic control purpose. Furthermore, we discover that the entire dynamic model of a robot can be significantly reduced, and the lower bound of the model reduction is a subsystem with the minimum embeddable C-manifold in the sense of topology. The paper also gives examples to illustrate the procedure of determining their C-manifold embedding models. One of the examples is simulated in computer to verify its trajectory-tracking adaptive control process.