Linearizations for mechanical systems in generalized coordinates

We describe an algorithm for calculating the linearization of the dynamics for arbitrary constrained mechanical systems in generalized coordinates without using symbolic equations. Linearizations of dynamics are useful tools for controllability and stability analysis and can be used to generate locally stabilizing controllers for linear and non-linear systems. However, the computational expense for finding linearizations of complex mechanical systems is often cited as a limiting factor that prevents their use. Recent work has introduced new methods of calculating the dynamics of arbitrary mechanical systems in generalized coordinates without deriving large, system-specific equations of motion. This paper extends that approach to calculate the linearizations of the dynamics without using the symbolic equations of motion. Using these ideas, it becomes practical to both simulate, analyze, and control more complex mechanical systems without sacrificing the benefits of generalized coordinates. Furthermore, this method addresses systems with closed kinematic chains, constraints, and external non-conservative forcing. The technique is applied to an example system with a closed kinematic chain and the resulting linearization agrees with results found by symbolically differentiating the full equations of motion.