F. Lamiraux and L. E. Kavraki, “Positioning of Symmetric and Non-Symmetric Parts Using Radial and Constant Fields:
Computation of All Equilibrium Configurations,” *International Journal of Robotics Research*, vol. 20, no. 8, pp. 635–659, 2001.

Programmable force fields have been used as an abstraction to represent a whole new class of devices that have been proposed for part manipulation. The general idea behind these devices is that a force field is implemented in a plane upon which the part is placed. The forces and torques exerted on the contact surface of the part translate and rotate the part. Manipulation plans for these devices can therefore be considered as strategies for applying a sequence of force fields to bring parts to some desired configuration. Instances of these novel devices are currently implemented using microelectromechanical systems technology, small airjets, vibration, and small motors. Manipulation in this case is sensorless and nonprehensile and promises to address the handling of very small or very fragile parts, such as electronics components, that cannot be handled with conventional pick-and-place robotics techniques. In this paper, the authors consider the problem of bringing a part to a stable equilibrium configuration using force fields. The authors study the combination of a unit radial field with a small constant field. A part placed on the radial field moves toward the origin of the radial field but cannot be oriented due to symmetry. Perturbing the radial field with a constant force field breaks the symmetry and gives rise to a finite number of equilibria. Under certain conditions, there is a unique stable equilibrium configuration. For the case in which these conditions are not fulfilled, the authors provide a comprehensive and unified analysis of the problem that leads to an algorithm to compute all stable equilibrium configurations. The paper contains a detailed discussion on how to implement the algorithm for any part. In the analysis, the authors make extensive use of potential fields. Using the theory of potential fields, the stable equilibrium configurations of a part are equivalent to the local minima of a scalar function. The work presented in this paper leads to the design of a new generation of efficient, open-loop part feeders that can bring a part to a desired orientation from any initial orientation without the need of sensing or a clock.

**Publisher:**
http://dx.doi.org/10.1177/02783640122067589