Nonlinear inverse optimization approach for determining the weights of objective function in standing reach tasks

This paper presents a nonlinear inverse optimization approach to determine the weights for the joint displacement function in standing reach tasks. This inverse optimization problem can be formulated as a bi-level highly nonlinear optimization problem. The design variables are the weights of a cost function. The cost function is the weighted summation of the differences between two sets of joint angles (predicted posture and the actual standing reach posture). Constraints include the normalized weights within limits and an inner optimization problem to solve for joint angles (predicted standing reach posture). The weight linear equality constraints, obtained through observations, are also implemented in the formulation to test the method. A 52 degree-of-freedom (DOF) human whole body model is used to study the formulation and visualize the prediction. An in-house motion capture system is used to obtain the actual standing reach posture. A total of 12 subjects (three subjects for each percentile in stature of 5th percentile female, 50th percentile female, 50th percentile male and 95th percentile male) are selected to run the experiment for 30 tasks. Among these subjects one is Turkish, two are Chinese, and the rest subjects are Americans. Three sets of weights for the general standing reach tasks are obtained for the three zones by averaging all weights in each zone for all subjects and all tasks. Based on the obtained sets of weights, the predicted standing reach postures found using the direct optimization-based approach have good correlation with the experimental results. Sensitivity of the formulation has also been investigated in this study. The presented formulation can be used to determine the weights of cost function within any multi-objective optimization (MOO) problems such as any types of posture prediction and motion prediction.