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Unformatted text preview: Kinematic transformation of mechanical behavior Neville Hogan Generalized coordinates are fundamental If we assume that a linkage may accurately be described as a collection of linked rigid bodies, their generalized coordinates are a fundamental requirement for any model of mechanical behavior. However, to describe functional behavior it will often be necessary to express mechanical behavior as it appears in a different frame of reference, for example, the Cartesian coordinates of the endpoint. Knowledge of the geometry relating the two frames is sufficient to transform mechanical behavior, but care is required. Transformation to endpoint coordinates Express the kinematic equations relating endpoint coordinates to generalized coordinates. ( ) θ L x ~ = The relations between incremental displacements, velocities, forces and momenta are obtained by differentiating and using power continuity. ( ) θ θ J θ θ L x d d d ~ ~ = ∂ ∂ = ( ) ω θ J v ~ = ( ) f θ J τ t ~ = ( ) p θ J η t ~ = Note that transformation of motion variables (displacement, velocity) is always well defined from generalized coordinates to any other coordinates. Conversely, the transformation of force variables (force, momentum) is always welldefined to generalized coordinates from any other coordinates. In general, the inverse of these transformations may not be welldefined. Inertia Inertia relates momentum and velocity. ( ) ω θ I η = If every rigid body in the linkage has nonzero mass, the inertia tensor is positivedefinite and its inverse exists. To define stored kinetic energy, the inverse of this relation is required, the causallypreferred form for an inertia. ( ) η θ I ω 1 − = Transformation to endpoint coordinates is a straightforward matter of substitution. p M v 1 − = x ( ) ( ) ( ) p θ J θ I θ J v t ~ ~ 1 − = page 1 ( ) ( ) ( ) ( ) t x θ J θ I θ J θ M ~ ~ 1 1 − − = The inverse inertia at the endpoint can always be defined. Due to linkage geometry it varies with endpoint position and linkage configuration. In generalized coordinates, inertia (and hence inverse inertia) is positive definite. In endpoint coordinates, inverse inertia is only positive semidefinite (strictly nonnegative); in some configurations it may lose rank. In those cases the endpoint inertia approaches infinity—force may be may lose rank....
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 Fall '06
 NevilleHogan
 Jacobian, endpoint coordinates

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