Terekhov Alexander V, Pesin Yakov B, Niu Xun, Latash Mark L, Zatsiorsky Vladimir M
Department of Kinesiology, The Pennsylvania State University, University Park, PA 16802, USA.
J Math Biol. 2010 Sep;61(3):423-53. doi: 10.1007/s00285-009-0306-3. Epub 2009 Nov 10.
We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.
我们考虑关于人类动作的各个方面以及不同运动任务，人类动作中正在被优化的是什么这一问题。从数学角度来看，这个问题在于给定未知目标函数达到其最小值时的值，去找出该目标函数。这个问题被称为逆优化问题。到目前为止，解决这个问题的主要方法是试错法，该方法包括引入一个目标函数并检查它如何反映实验数据。使用这种方法，针对相同的运动动作提出了不同的目标函数。在当前论文中，我们专注于具有加性目标函数和线性约束的逆优化问题。此类问题在人类运动科学中很典型。肌肉（或手指）力分配问题就是一个例子。对于此类问题，我们获得了唯一性的充分条件，并提出了一种确定目标函数的方法。为了说明我们的方法，我们分析了抓握任务中手指间的力分配问题。我们从实验数据中估计目标函数，并表明它可以预测施加到被抓握物体上的大范围外力和扭矩下的力分配模式。所得的目标函数是二次的，且具有本质上非零的线性项。
