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Concept# Rigid body dynamics

Résumé

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.
The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law (kinetics) or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation

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Human beings like all organisms, are subject to a variety of diseases. Musculoskeletal diseases such as arthritis, that affect our muscles and bones, are particularly debilitating in that they considerably limit our ability to interact with our environment. The symptoms of arthritis are joint pain and loss of movement. There is deterioration of the cartilage in our articulations. The precise determination of the underlying cause of the deterioration is a challenging task. It is believed that it is caused by excessive force in the joints due to inappropriate muscle forces. Since only forces in muscles just beneath the skin can be measured, the force hypothesis remains unproven. Musculoskeletal models are essential in analysing musculoskeletal diseases because they address the lack of information on the forces involved. Such models are used to estimate muscle and joint reaction forces. Determining the key elements in a musculoskeletal model to assess its quality raises several challenges. In this thesis, a musculoskeletal model of the shoulder is presented. The model is governed by the laws of rigid-body mechanics and is similar to a model of a cable-driven mechanism. The model contains both the kinematic and dynamic aspects of the shoulder. Applying the theory of rigid body mechanics requires a certain level of rigour to ensure compatibility between the kinematic and dynamic parts of the model. Therefore, a considerable part of the thesis is devoted to presenting the details of the model's construction. The model is designed specifically for estimating muscle and joint-reaction forces in quasi-static and dynamic situations. The muscle-force estimation problem is defined as a nonlinear program and solved in this thesis using a two-step approach. In a first step, the kinematics is constructed and inverse dynamics is used to estimate the associated joint torques. In a second step, the nonlinear program is solved using null-space optimisation. An initial solution to the estimation problem is obtained by taking a pseudo-inverse of the moment-arms matrix. The solution is then corrected using the matrices null-space to satisfy the constraints. This approach redefines the estimation problem as a quadratic program and considerably reduces the time required to find a solution. Once the muscle-forces are estimated, the joint reaction forces are deduced from the dynamic model. Muscle and joint-reaction forces are compared to other results from the literature.

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in so(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.

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Kinematic couplings are used when two rigid bodies need to be repeatedly and accurately positioned with respect to each other. They allow for sub-micron positioning repeatability by suppressing play and reducing strains in the bodies. Typical applications are lens mounts, work piece mounts and docking interfaces for astrophysics, semiconductor and metrology applications. This thesis generalizes the well-known concept of two-body kinematic couplings to three-body kinematic mounts. The goal of the thesis is: To pave the way for high precision assembly using kinematic mounts by providing an exhaustive catalogue of all twobody and three-body kinematic mounts and to test key configurations experimentally. The main contributions of this thesis are: - State of the art survey of essential knowledge in the field of kinematic couplings. - Rigorous problem statement for the design of two-body and three-body kinematic mounts. - Rigorous limitation of the scope of research to three-body kinematic mounts whose contact points lie exclusively on three convergent orthogonal lines and whose constraint lines are parallel to these lines. - An exhaustive catalogue of three-body kinematic mounts consisting of seven configurations in 3D and nine configurations in 2D. - An exhaustive set of four conditions satisfied by three-body 3-dimensional kinematic mounts. - An exhaustive set of seven conditions satisfied by three-body 2-dimensional kinematic mounts. - Realization of a two-body kinematic mount and a three-body kinematic mount in metal, and precise measurement of their positioning accuracy on a 3D coordinate measurement machine at the Swiss Federal Institute of Metrology. Positioning error of 0.2 microns and 5 microradian achieved with two-body kinematic mounts. Positioning error of 1 micron and 50 microradian achieved with three-body kinematic mounts. - Realization of three-body kinematic mounts in Silicon by Deep Reactive Ion Etching processes (DRIE) and experimental measurement of their positioning error. - Physical implementation of nesting forces and assembly methods allowing for the physical construction of kinematic mounts. - Physical realizations in robotics, optics and aerospace using our new kinematic mounts.