如何控制機器手臂使其移動沿著預先找出的最佳移動軌跡是一個重要的研究課題，因為這樣可以同時使得該機器手臂減少能量損耗與增加工作效率。本論文利用最佳控制的方式來控制機器手臂模型並與預測控制方法的數值結果做比較。使用預測控制其原因之一為其控制法則中有引入最小預測力量的概念，可使系統達到最小能量輸出。另一個原因是在不需要知道系統模型的情況下可藉由蒐集輸入輸出資料且用系統識別的方式來設計控制器。

由於機器手臂移動的方式約略可分成擺動和高舉，其基本概念相當於單擺及倒單擺模型的行為，因此我們利用單擺及倒單擺的模型得到動態方程式，以其比較兩種控制方法的模擬結果。再把控制方法使用在三軸機器手模型上，並將角度模擬結果放入幾何運動學中觀察手臂軌跡。我們使用時間間隔1毫秒模擬所有的機器手臂模型。在預測控制中，我們將非線性系統的輸入輸出資料使用系統識別後可得線性非時變系統進行控制，而最佳控制則用LQR。兩種控制方法在手臂上的軌跡表現，預測控制比最佳控制可在有限時間內達到目標位置。

Both optimal control and predictive control are used and compared to control the model of a robot arm. The main reason for using the predictive control approach is that we want to attain the optimal performance by predicting the force/torque required to track a desired output trajectory. Another reason is that the predictive control approach enables controller design to be fulfilled in real time by system identification without knowing the system model.

The moving motions of a robot arm can be roughly classified into the swinging motion and the raising motion, and their movement patterns resemble the movement of a pendulum and an inverted pendulum, respectively. Thus we resort to the models of a pendulum and an inverted pendulum for deriving the dynamic equations and to compare the control results of the optimal control approach and the predictive control approach. We then apply the two control approaches to a three-link robot arm. For predictive control, we utilize the input-output data from a nonlinear system to perform system identification and acquire the linear time-invariant system to control our model. For optimal control, we use the linear quadratic regulator design on a linearized model for small motion. We derive the arm trajectories from simulated results for angles, as done in geometric kinematics. All simulations of robot arms are performed on time intervals of 1 millisecond. A comparison of these two trajectories shows that the predictive control approach is superior to the optimal control approach in reaching our desired position in finite time.

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