本研究以上肢復健為例，建立一個含拘束之最佳控制問題，並以非線性規劃法求解。此問題考慮患者關節所受力矩的極限，以及肢體運動的拘束。在基於物理治療師規劃之標準路徑的情況下，經由序列二次規劃計算出最適合患者之上肢復健運動路徑及最佳關節力矩。在標準路徑方面，我們仿照治療師常使用的多項式函數來模擬標準路徑。在關節力矩拘束方面，藉由設定上限值來使關節力矩可以被拘束在一固定範圍內以符合患者的復健需求。此外，我們將量測到的資料搭配 OpenSim 進行肌肉骨骼模型建模並計算出肌肉出力。實驗結果顯示，此系統所評估之關節力矩與肌肉出力有一定可信度，可用於系統性地制定適合患者的復健路徑，並作為治療師評估患者復健成效之依據。

關鍵字：含拘束之最佳控制、序列二次規劃、復健路徑制定、肌肉骨骼建模、OpenSim。

In this study, taking upper limb rehabilitation as an example, an optimal control problem with constraints is established and solved by nonlinear programming method. In particular, the limits of both subjected torques on patient's joints and limb movements are considered. The whole process starts with a standard trajectory initially planned by physiotherapists, and then the most suitable upper limb rehabilitation trajectory and the optimal joint torques of patients are determined through sequential quadratic programming. For the standard trajectory, it is modeled with a polynomial function commonly used by physiotherapists. As for setting the torque constraints, they are limited within a fixed range to meet the rehabilitation needs of patients. In addition, with the measured data, the software, OpenSim, is utilized to construct the musculoskeletal model and simulate the muscle forces. The experimental results show that the simulated joint torques and muscle forces have certain reliability, which can be used to systematically determine a suitable rehabilitation trajectory for patients, and serve as the basis for physiotherapists to evaluate the rehabilitation effects.

Keywords: optimal control with constraints, sequential quadratic programming, rehabilitation trajectory determination, musculoskeletal modeling, OpenSim.

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