Structural Properties of the Spatial Manipulating Systems in Connection with the State and Control Constraints (original) (raw)
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Axioms
This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, certain typical problems in robotics are mathematically formalised and analysed in axiomatic form. The outcomes are sufficiently general that it is possible to discuss the structural properties of robotic manipulation. A generalized theoretical linear model is used, and a thorough analysis is made. The noninteracting nature of this model is also proven through a specific theorem.
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A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnormal ones is also rationalized, showing the existence of a purely kinematical algorithm assigning to each admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. The whole machinery is then applied to constrained variational calculus. The argument provides an interesting revisitation of Pontryagin maximum principle and of the Erdmann-Weierstrass corner conditions, as well as a proof of the classical Lagrange multipliers method and a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system. As a final, highly non-trivial topic, a sufficient condition for the existence of finite deformations with fixed endpoints is explicitly stated and proved.
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Inertial Properties in Robotic Manipulation: An Object-Level Framework
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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
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