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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 end-point. Knowledge of the geometry relating the two frames is sufficient to transform mechanical behavior, but care is required. Transformation to end-point coordinates Express the kinematic equations relating end-point 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 well-defined to generalized coordinates from any other coordinates. In general, the inverse of these transformations may not be well-defined. Inertia Inertia relates momentum and velocity. ( ) ω θ I η = If every rigid body in the linkage has non-zero mass, the inertia tensor is positive-definite and its inverse exists. To define stored kinetic energy, the inverse of this relation is required, the causally-preferred form for an inertia. ( ) η θ I ω 1 − = Transformation to end-point 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 end-point can always be defined. Due to linkage geometry it varies with end-point position and linkage configuration. In generalized coordinates, inertia (and hence inverse inertia) is positive definite. In end-point coordinates, inverse inertia is only positive semi-definite (strictly non-negative); in some configurations it may lose rank. In those cases the end-point inertia approaches infinity—force may be may lose rank....
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- Fall '06
- Jacobian, end-point coordinates