Here's a planar assembly of blocks on a table. We'd like to know if this assembly will stand or fall.

To test if standing is a possible solution, for each block we can write a vector static balance equation, where F_ext is the external force acting on the block, in this case gravity, and the contact wrenches on each block must balance the gravitational force. The F_i are the wrenches corresponding to friction cone edges, and the k_i are nonnegative coefficients.

For the left block, we can write the static balance equation as shown here. There are 8 friction cone edges acting on the left block: F_1 through F_4 from the table and F_5 through F_8 from the top block. The total contact wrench is in the positive span of these 8 wrenches.

For the right block, we can write another vector static balance equation. The 8 friction cone edges acting on the right block are labeled F_9 through F_16.

Finally, for the keystone block at the top of the arch, the 8 friction cone edges acting on it are minus F_5 through minus F_12. Since the keystone block must apply a wrench to the left block that is opposite the wrench that the left block applies to the keystone, the coefficients k_5 through k_8 are the same as those we used in our analysis of the left block. Similarly, the coefficients k_9 through k_12 are the same as those we used in our analysis of the right block.

Counting the coefficients and the constraints, we have 16 nonnegative coefficients to satisfy 9 wrench-balance equations. If a linear constraint satisfaction solver finds a set of nonnegative coefficients satisfying the equations, then standing up is a feasible solution for the arch.

Recall the problem of transporting a waiter's tray from the beginning of this chapter.

We could formulate a dynamic version of the arch stability problem, and ask if the arch stays standing as its support surface moves.

In this case, the equation for each rigid body is written as you see here. If we plug in the twist V and the acceleration V-dot of the tray and we can still find positive coefficients k_i satisfying the 9 equations, then the assembly can stay assembled during the motion. If any of the coefficients k_i has to become negative to satisfy the equations, then the assembly must be glued together to keep from collapsing.

If the assembly were in 3 dimensions, nothing changes about the analysis except that we would approximate the quadratic friction cones as polyhedral cones.

This concludes Chapter 12 on grasping and manipulation. In this chapter you learned about the kinematics of contact constraints, the forces that can be applied through a contact, and how to use this information to analyze different kinds of manipulation problems.