Formal Foundations of 3D Geometry to Model Robot Manipulators
Résumé
We are interested in the formal specification of safety properties of robot manipulators down to the mathematical physics. To this end, we have been developing a formalization of the mathematics of rigid body transformations in the COQ proof-assistant. It can be used to address the forward kinematics problem, i.e., the computation of the position and orientation of the end-effector of a robot manipulator in terms of the link and joint parameters. Our formalization starts by extending the Mathematical Components library with a new theory for angles and by developing three-dimensional geometry. We use these theories to formalize the foundations of robotics. First, we formalize a comprehensive theory of three-dimensional rotations, including exponentials of skew-symmetric matrices and quaternions. Then, we provide a formalization of the various representations of rigid body transformations: isometries, homogeneous representation, the Denavit-Hartenberg convention, and screw motions. These ingredients make it possible to formalize robot manipulators: we illustrate this aspect by an application to the SCARA robot manipulator.
Mots clés
F41 [Mathematical Logic]: Mechanical theorem proving
I11 [Expressions and Their Representation]: Representations (general and poly- nomial)
I29 [Robotics]: Manipulators
I35 [Computa- tional Geometry and Object Modeling]: Hierarchy and geo- metric transformations Keywords Formal verification
Proof-assistant
Coq
Robot manipulator
3D Geometry
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