273 research outputs found
Formalization of Robot Collision Detection Method based on Conformal Geometric Algebra
Cooperative robots can significantly assist people in their productive
activities, improving the quality of their works. Collision detection is vital
to ensure the safe and stable operation of cooperative robots in productive
activities. As an advanced geometric language, conformal geometric algebra can
simplify the construction of the robot collision model and the calculation of
collision distance. Compared with the formal method based on conformal
geometric algebra, the traditional method may have some defects which are
difficult to find in the modelling and calculation. We use the formal method
based on conformal geometric algebra to study the collision detection problem
of cooperative robots. This paper builds formal models of geometric primitives
and the robot body based on the conformal geometric algebra library in HOL
Light. We analyse the shortest distance between geometric primitives and prove
their collision determination conditions. Based on the above contents, we
construct a formal verification framework for the robot collision detection
method. By the end of this paper, we apply the proposed framework to collision
detection between two single-arm industrial cooperative robots. The flexibility
and reliability of the proposed framework are verified by constructing a
general collision model and a special collision model for two single-arm
industrial cooperative robots
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps
This paper presents a formalized proof of a discrete form of the Jordan Curve
Theorem. It is based on a hypermap model of planar subdivisions, formal
specifications and proofs assisted by the Coq system. Fundamental properties
are proven by structural or noetherian induction: Genus Theorem, Euler's
Formula, constructive planarity criteria. A notion of ring of faces is
inductively defined and a Jordan Curve Theorem is stated and proven for any
planar hypermap
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