Kinematics of Planar Elastic Chains
This Demonstration shows an -link planar kinematic chain with a fixed base in which each joint is a linearly elastic torsional spring. The shape of this chain in static equilibrium can be represented as the solution of a discrete-time optimal control problem, with boundary conditions that vary with the position and orientation of the last link. McCarthy and Bretl proved that the set of all solutions to this problem is a smooth manifold that can be parameterized by a single chart. We have added gravitational force to this model.
Given a fixed position for the left end of the chain (shown in blue), equilibrium configurations can be obtained by minimizing the equation of potential energy. In this minimization process, we obtain costate vectors (similar to Lagrange multipliers) and we find that there is a diffeomorphism between the initial costate vectors and the space of nonsingular equilibrium configurations . The initial constants can also be interpreted as the force and torque of the first (yellow) link on the initial (blue) link.
 Z. McCarthy and T. Bretl, "Mechanics and Manipulation of Planar Elastic Kinematic Chains," in Proceedings of the 2012 IEEE International Conference on Robotics and Automation (ICRA12), St. Paul, Minn., IEEE, 2012 pp. 2798–2805.