Automatic Differentiation in Automatic Generation of the Linearized Equations of Motion

Abstract

Abstract In the ongoing search for mathematically efficient methods of predicting the motion of vehicle and other multibody systems, and presenting the associated results, one of the avenues of continued interest is the linearization of the equations of motion. While linearization can potentially result in reduced fidelity in the model, the benefits in computational speed often make it the pragmatic choice. Linearization techniques are also useful in modal and stability analysis, model order reduction, and state and input estimation. This paper explores the application of automatic differentiation to the generation of the linearized equations of motion. Automatic differentiation allows one to numerically evaluate the derivative of any function, with no prior knowledge of the differential relationship to other functions. It exploits the fact that every computer program must evaluate every function using only elementary arithmetic operations. Using automatic differentiation, derivatives of arbitrary order can be computed, accurately to working precision, with minimal additional computational cost over the evaluation of the base function. There are several freely available software libraries that implement automatic differentiation in modern computing languages. In the paper, several example multibody systems are analyzed, and the computation times of the stiffness matrix are compared using direct evaluation and automatic differentiation. The results show that automatic differentiation can be surprisingly competitive in terms of computational efficiency.