Three problems in robotics

Three rather different problems in robotics are studied using the same technique from screw theory. The first problem concerns systems of springs. The potential function is differentiated in the direction of an arbitrary screw to find the equilibrium position. The second problem is almost identical in terms of the computations; the least-squares solution to the problem of finding the rigid motion undergone by a body given only data about points on the body is sought. In the third problem the Jacobian of a Stewart platform is found. Again, this is achieved by differentiating with respect to a screw. Furthermore, second-order properties of the first two problems are studied. The Hessian of second derivatives is computed, and hence the stability properties of the equilibrium positions of the spring system are found.