# On the Controllability of Lagrangian Systems by Active Constraints

## TL;DR

This paper investigates the controllability of Lagrangian mechanical systems using active, frictionless constraints, introducing a simplified differential inclusion model and demonstrating approximation and exact reachability results.

## Contribution

It introduces a simplified differential inclusion model for controlled Lagrangian systems and proves approximation and reachability properties of the original system.

## Key findings

- Trajectories of the differential inclusion can approximate those of the original system.
- Under stronger assumptions, the system can reach the same terminal point.
- The approach provides a new way to analyze controllability via active constraints.

## Abstract

We consider a mechanical system which is controlled by means of moving constraints. Namely, we assume that some of the coordinates can be directly assigned as functions of time by means of frictionless constraints. This leads to a system of ODE's whose right hand side depends quadratically on the time derivative of the control. In this paper we introduce a simplified dynamics, described by a differential inclusion. We prove that every trajectory of the differential inclusion can be uniformly approximated by a trajectory of the original system, on a sufficiently large time interval, starting at rest. Under a somewhat stronger assumption, we show this second trajectory reaches exactly the same terminal point.