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Wrench Analysis of Grasps. Wrench Analysis. Analysis based on forces that can be exerted by the fingers. Multiples of forces. Resist external forces (k 1 ,k 2 >0). Forces on a Particle. f 1. −k 1 f 1. f 2. −(k 1 f 1 +k 2 f 2 ). f 1. Positive Bases.
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Wrench Analysis Analysis • based on forces that can be exerted by the fingers.
Multiples of forces Resist external forces (k1,k2>0) Forces on a Particle f1 −k1f1 f2 −(k1f1+k2f2) f1
Positive Bases f1,f2,f3 can resist all external forces if they form a positive basis: • any vector f can be written as -f= k1f1+k2f2+k3f3,for some k1,k2,k3≥0. • k1f1+k2f2+k3f3 = O,for some k1,k2,k3>0(convex hull of f1,f2,f3 has O in its interior) f2 f3 f1
Forces on a Part • Point p of application of force f matters, expressed by moment p x f. • Force f and moment p x f encoded in wrench:Wrench space is 3D for 2D parts and 6D for 3D parts.
Wrench Analysis Force closure: wrenches w1,…,wk induced by fingers should be able to resist any external wrench, so w1,…,wk form a positive basis for wrench space, so convex hull of w1,…,wk has O in its interior. (4 wrenches necessary in 2D, 7 wrenches necessary in 3D.)
Example Triangle with corners (0,0), (8,0) and (4,4), held by four point contacts p1,…,p4. p4=(5,3) p3=(2,2) p2=(7,0) p1=(1,0)
Example p4=(5,3) p3=(2,2) p2=(7,0) p1=(1,0)
Example • Origin lies in the interior of the convex hull of w1,w2,w3,w4 Force closure w2 w1 w4 w3
