## Observers for Lagrangian Systems with Unilateral Constraints

This page is dedicated to simulation results obtained by implementing a certain class of observers for Lagrangian systems with unilateral constraints. The state estimators used for this purpose are based on the results developed in the following paper:
• A. Tanwani, B. Brogliato, and C. Prier. Observer Design for Unilaterally Constrained Lagrangian Systems: A Passivity-Based Approach. IEEE Trans. Automatic Control, 61(9):2386-2401, 2016.
• A. Tanwani, B. Brogliato, and C. Prieur. Passivity-based observer design for a class of Lagrangian systems with perfect unilateral constraints. Proceedings of 52nd IEEE Conf. on Decision and Control, Florence, Italy, 2013.

The simulations of the nonsmooth dynamical systems studied in our work have baeen carried out using the software SICONOS, which is being developed by Team BipOp at INRIA, Grenoble. The animations have been carried out using Python libraries.

Example 1: We consider a 3 ball chain moving in 1-dimension and their motion is constrained by two walls, one on either side of the chain. The coefficient of restitution is set to 1, and the external force is set to zero. The simulation results corresponding to different initial conditions appear in the following videos:
Example 2: We consider a ball moving within a nonlinear convex billiard. In the first case, the external force is the gravity which is pulling the ball downwards, and coefficient of restitution equals 0.9, which eventually results in Zeno phenomenon. In the second case, we set the external force to be zero, and let coefficient of restitution equal 1, so that the impact behavior in the corners could be seen.
Example 3: We next consider the motion of a ball within a hyperbolic billiard so that the admissible set for the position of the particle is nonconvex, and defined by nonlinear gap functions. The coefficient of restitution is set to $e = 1$, and the external forces are set to zero. Because of the chaotic behavior in such billiards, the results are highly sensitive to numerical precision. In our calculations we have linearized the boundary of the constraint set locally so that algorithms from convex optimization could be used to compute projection on the admissible set.