Sensitivity analysis represents a powerful tool for the optimization of multibody system dynamics. The performance of a gradient-based optimization algorithm is strongly tied to the dynamic and the sensitivity formulations considered. The accuracy and efficiency are critical to any optimization problem, thus they are key factors in the selection of the dynamic and sensitivity analysis approaches used to compute an objective function gradient. Semi-recursive methods usually outperform global methods in terms of computational time, even though they involve sometimes demanding recursive procedures. Semi-recursive methods are well suited to be combined with different constraints enforcement schemes as the augmented Lagrangian index-3 formulation with velocity and acceleration projections (ALI3-P), taking advantage of the robustness, accurate fulfillment of constraint equations and the low computational burden. The sensitivity analysis of the semi-recursive ALI3-P formulation is studied in this document by means of the direct differentiation method. As a result, a semi-recursive ALI3-P sensitivity formulation is developed for an arbitrary reference point selection, and then two particular versions are unfolded and implemented in the general purpose multibody library MBSLIM, using as reference point the center of mass (RTdyn0) or the global origin of coordinates (RTdyn1). Besides, the detailed derivatives of the recursive terms are provided, which will be useful not only for the direct sensitivity formulation presented herein, but also for other sensitivity formulations relying on the same recursive expressions. The implementation has been tested in two numerical experiments, a five-bar benchmark problem and a buggy vehicle.

灵敏度分析是优化多体系统动力学的强大工具。基于梯度的优化算法的性能与所考虑的动态和灵敏度公式密切相关。准确性和效率对于任何优化问题都至关重要，因此它们是选择用于计算目标函数梯度的动态和灵敏度分析方法的关键因素。半递归方法在计算时间方面通常优于全局方法，尽管它们有时涉及要求较高的递归过程。半递归方法非常适合与不同的约束执行方案相结合，如具有速度和加速度投影的增强拉格朗日指数 3 公式（ALI3-P），利用鲁棒性，约束方程的精确满足和低计算负担。本文通过直接微分法研究了半递归 ALI3-P 公式的敏感性分析。因此，开发了半递归 ALI3-P 灵敏度公式，用于任意参考点选择，然后在通用多体库 MBSLIM 中展开并实现两个特定版本，使用质心 (RTdyn0) 作为参考点或全局坐标原点 (RTdyn1)。此外，还提供了递归项的详细导数，这不仅对于本文提出的直接灵敏度公式有用，而且对于依赖相同递归表达式的其他灵敏度公式也有用。该实现已在两个数值实验中进行了测试，