lecturenotes1.20100204.4b6af98e20d6f6.00568087

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Coordinate charts De f nition 1 A coordinate chart ( C , ξ ) is i) a subset C of allowable con f gurations of a mechanism and ii) a one-to-one vector-valued function ξ from C to an open subset of R n , for some integer n .Thein te ge r n is the number of geometric degrees of freedom of the mechanism. De f nition 2 The components of the vector-valued function ξ are called coordinate functions or generalized coordinates . De f nition 3 A coordinate chart ( C , ξ ) is said to be global if C contains all allowable con f gurations. De f nition 4 Consider two coordinate charts ( C 1 , ξ 1 ) and ( C 2 , ξ 2 ) on a collection of allowable con f gurations of a mechanism, such that C 1 C 2 6 = .T h e n ,a coordinate transformation from ξ 1 to ξ 2 is a one-to-one relationship ξ 2 = f ( ξ 1 ) for some function f de f ned on ξ 1 ( C 1 C 2 ) . For every con f guration C C 1 C 2 , ξ 2 ( C )= f ( ξ 1 ( C )) and ξ 1 ( C )= f 1 (
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