Motion Profile

Motion Profile 是一種開環系統，可以客製化機器移動的加加速度，加速度，與速度，以達到順滑且精準的移動。

影片範例：

Trapezoidal Motion Profile

特點：

• 加速度和減速度（acceleration）固定，不考慮加加速度（jerk）

Image referenced from https://www.motioncontroltips.com/what-is-a-motion-profile/

At acceleration phase:

$$v = at$$

$$x = \frac{1}{2} at^2$$

At decceleration phase:

$$v = v_0 - at$$

$$x = x_0 +vt - \frac{1}{2} at^2$$

Sigmoid Motion Profile

特點：

• 設定最大加加速度為常數（jerk），加速度（acceleration）隨加加速度（jerk）變化

Image referenced from https://www.semanticscholar.org/paper/S-curve-profile-switching-method-using-fuzzy-system-Lee-Kang/efa3dde6006ccc4a5f149723202c831daca25551/figure/0

At increasing acceleration phase: $0 \le t \le T_1$

$$a_1(t) = jt$$

$$v_1(t) = \frac{1}{2} jt^2$$

$$x_1(t) = \frac{1}{6} jt^3$$

At constant acceleration phase: $T_1 \le t \le T_2$

$$v_{2}\left(t\right)\ =\ v_{1}\left(T_{1}\right)+\ a_{max}\left(t-T_{1}\right)$$

$$x_2(t) =\frac{1}{2}a_{max}\left(t-T_{1}\right)^{2}+v_{1}\left(T_{1}\right)\left(t-T_{1}\right)+\frac{1}{6}j\left(T_{1}\right)^{3}$$

At decreasing acceleration phase: $T_2 \le t \le T_3$

$$a_{3}\left(t\right)=a_{2}\left(T_{2}\right)-j\left(t-T_{2}\right)$$

$$v_{3}\left(t\right)\ =\ v_{2}\left(T_{2}\right)-\ \frac{1}{2}j\left(t-T_{2}\right)^{2}+a_{2}\left(T_{2}\right)\cdot\left(t-T_{2}\right)$$

$$x_3(t)=-\frac{1}{6}j\left(t-T_{2}\right)^{3}+\frac{1}{2}a_{2}\left(T_{2}\right)\cdot\left(t-T_{2}\right)^{2}+v_{2}\left(T_{2}\right)\left(t-T_{2}\right)+x_{2}(T_{2})$$

At constant velocity phase: $T_3 \le t \le T_4$

$$x_4(t)=v_{3}\left(T_{3}\right)\ \left(t-T_{3}\right)+x_{3}\left(T_{3}\right)$$

At increasing deceleration phase: $T_4 \le t \le T_5$

$$a_{5}\left(t\right)=-j\left(t-T_{4}\right)$$

$$v_{5}\left(t\right)=v_{4}\left(t\right)+a_{4}\left(t\right)\left(t-T_{4}\right)-\frac{1}{2}j\left(t-T_{4}\right)^{2}$$

$$x_5(t)=-\frac{1}{6}j\left(t-T_{4}\right)^{3}+v_{4}\left(t\right)\left(t-T_{4}\right)+x_{4}\left(T_{4}\right)$$

At constant deceleration phase: $T_5 \le t \le T_6$

$$v_{6}\left(t\right)=-a_{max}\left(t-T_{5}\right)+v_{5}\left(T_{5}\right)$$

$$x_6(t)=-\frac{1}{2}a_{max}\left(t-T_{5}\right)^{2}+v_{5}\left(T_{5}\right)\left(t-T_{5}\right)+x_{5}\left(T_{5}\right)$$

At decreasing deceleration phase: $T_6 \le t \le T_7$

$$a_{7}\left(t\right)=-a_{max}+j\left(t-T_{6}\right)$$

$$v_{7}\left(t\right)=v_{6}\left(T_{6}\right)+a_{6}\left(T_{6}\right)\cdot\left(t-T_{6}\right)+\frac{1}{2}j\left(t-T_{6}\right)^{2}$$

$$x_7(t)=\frac{1}{6}j\left(t-T_{6}\right)^{3}+\frac{1}{2}a_{6}\left(T_{6}\right)\left(t-T_{6}\right)^{2}+v_{6}\left(T_{6}\right)\left(t-T_{6}\right)+x_{6}\left(T_{6}\right)$$