Efficient Procedures for Form-Closure Grasp Planning and Fixture Layout Design

Illustration of the form-closure condition: −wd+1 lying in the interior of the convex hull of w1−wd+1,w2−wd+1,…,wd−wd+1,0. The convex hull (the gray area) results from the convex cone of w1−wd+1,w2−wd+1,…,wd−wd+1 with apex at 0 truncated by their affine hull.

The orthogonal projection pi of 0 on the affine hull Fi of w1,w2,…,wi and wd+1. The dashed line denotes the affine hull of w1,w2,…,wi. The convex hull Si of w1,w2,…,wi,wd+1 is the gray area bounded by thick lines. (a) pi is in the relative interior of Si; that is, sum(xi)<1 and min(xi)>0. (b) View of (a) by shifting the origin of the reference frame to wd+1. (c) sum(xi)>1 and min(xi)>0. (d) sum(xi)<1 and min(xi)<0.

Search for the face of Si on which the orthogonal projection of 0 is the closest to 0 itself. The origin 0 can be projected onto the relative interiors of several faces of Si, such as the one-dimensional face given by {w1,wd+1} or {w2,wd+1} and the two-dimensional face given by {w1,w2,wd+1} or {w1,wi,wd+1}. Among them, the face determined by {w1,wi,wd+1} is adopted, since the distance between 0 and its orthogonal projection on the face is the smallest.

Illustration of how to select a substituting candidate wrench. To increase hi∗, a candidate contact wrench in the different half space from Sd should be adopted to replace a contact wrench on the closest facet.

Contact positions on an ellipse. The dots represent the candidate contact positions, which are numbered counterclockwise from the right triangle. The initial position for the fourth contact to start Procedure 1 is selected at (a) the seventh candidate or (b) the 15th candidate, as circled in the figure. The contact positions attained by Procedures 1 and 2 are marked with four triangles and four squares, respectively. The quadrangles shading from gray to black show the iterations of Procedure 2 converging to the optimal results.

The values of h versus the iteration number in running Procedure 2. The numbers in the legend designate where the fourth contact is initialized for starting Procedure 1.

The simplices of the contact wrenches corresponding to the case of Fig. 5. The balls depict the candidate contact wrenches. The four numbers are the indices of candidates. (a) The tetrahedron represents the simplex produced by Procedure 1. (b) The tetrahedrons whose edges from gray to black and finally become cylinders show the convergence of wrench simplices in Procedure 2 to an optimum, which seems to be the largest simplex that can be found in the candidate contact wrenches.

The wrench simplices yielded by (a) Procedure 1 and (b) Procedure 2 in the case of Fig. 5. Although the optimal simplex in (b) herein is smaller than the one in Fig. 7, the minimum distance h from the wrench origin to the facets is increased remarkably by Procedure 2 in comparison with (a). In (b), the origin coincides with the centroid of the simplex.

The required CPU times for running the procedures versus the number of candidate points

Contact positions on a U-pipe. The circles and balls portray the contact positions computed by Procedures 1 and 2, respectively. Final contacts on the rear of the pipe are indicated by arrows. (a) The best optimal contact positions in trials. (b) The worst optimal contact positions in trials.

Multiple trials of the procedures with different initial points. The circles and upward triangles denote the values of h for the results of Procedure 1 and those optimized sequentially by Procedure 2, respectively.

The values of h versus the iteration number in running Procedure 2. Each number in the legend indicates an initial position of the seventh contact for starting Procedure 1.

The simplices of the contact wrenches corresponding to the case of Fig. 1. Either 6D wrench simplex is projected onto 3D force subspaces (a) or (c) and 3D moment subspaces (b) or (d). (a) and (b) depict the force and moment components of the wrench simplex determined by Procedure 1, while (c) and (d) depict those resulting from Procedure 2. The balls denote the force or moment components of the candidate contact wrenches.

Viewing the 6D wrench simplices corresponding to the case of Fig. 1 from their force or moment component. (a) and (c) Force component; (b) and (d) moment component. (a) and (b) Result of Procedure 1; (c) and (d) result of Procedure 2.

The required CPU times for running the procedures versus the number of candidate points