Evidence for Composite Cost Functions in Arm Movement Planning: An Inverse Optimal Control Approach

An important issue in motor control is understanding the basic principles underlying the accomplishment of natural movements. According to optimal control theory, the problem can be stated in these terms: what cost function do we optimize to coordinate the many more degrees of freedom than necessary to fulfill a specific motor goal? This question has not received a final answer yet, since what is optimized partly depends on the requirements of the task. Many cost functions were proposed in the past, and most of them were found to be in agreement with experimental data. Therefore, the actual principles on which the brain relies to achieve a certain motor behavior are still unclear. Existing results might suggest that movements are not the results of the minimization of single but rather of composite cost functions. In order to better clarify this last point, we consider an innovative experimental paradigm characterized by arm reaching with target redundancy. Within this framework, we make use of an inverse optimal control technique to automatically infer the (combination of) optimality criteria that best fit the experimental data. Results show that the subjects exhibited a consistent behavior during each experimental condition, even though the target point was not prescribed in advance. Inverse and direct optimal control together reveal that the average arm trajectories were best replicated when optimizing the combination of two cost functions, nominally a mix between the absolute work of torques and the integrated squared joint acceleration. Our results thus support the cost combination hypothesis and demonstrate that the recorded movements were closely linked to the combination of two complementary functions related to mechanical energy expenditure and joint-level smoothness.运动控制领域的核心议题之一，是阐明自然运动完成背后的基本原理。根据最优控制理论（optimal control theory），该问题可被表述为：为达成特定运动目标，我们需要协调远超实际所需的自由度，此时我们会优化何种代价函数？这一问题迄今尚无定论，因为所优化的对象在一定程度上取决于任务的具体要求。过往已有诸多代价函数被提出，且其中多数均与实验数据相符。因此，大脑用以实现特定运动行为的实际原理仍不明晰。现有研究结果表明，运动并非单一代价函数最小化的产物，而是复合代价函数共同作用的结果。为进一步阐明这一观点，我们采用了一种创新性的实验范式：在目标存在冗余的条件下开展手臂伸手动作实验。在此框架下，我们运用逆最优控制技术（inverse optimal control technique），自动推断出最契合实验数据的最优准则（组合）。实验结果显示，即便未预先指定目标点位，受试者在各实验条件下仍表现出一致的行为模式。结合逆最优控制与直接最优控制分析可知，当优化两类代价函数的组合时，我们能最佳复现受试者的平均手臂运动轨迹——具体而言，该组合近似于力矩绝对功与积分平方关节加速度的混合形式。综上，我们的研究结果支持了代价组合假说，并证实所记录的运动与两类互补函数的组合密切相关：这两类函数分别对应机械能耗与关节层面的运动平滑性。