# 倒單擺

## 运动方程

### 固定樞紐點

${\displaystyle {\ddot {\theta }}-{g \over \ell }\sin \theta =0}$

${\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta }$

#### 用力矩及轉動慣量來推導

${\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I{\ddot {\theta }}}$

${\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=mg\ell \sin \theta \,\!}$

${\displaystyle I{\ddot {\theta }}=mg\ell \sin \theta \,\!}$

${\displaystyle I=mR^{2}}$

${\displaystyle m\ell ^{2}{\ddot {\theta }}=mg\ell \sin \theta \,\!}$

${\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta }$

### 台車上的倒單擺

#### 拉格朗日方程

${\displaystyle L={\frac {1}{2}}Mv_{1}^{2}+{\frac {1}{2}}mv_{2}^{2}-mg\ell \cos \theta }$

${\displaystyle v_{1}^{2}={\dot {x}}^{2}}$
${\displaystyle v_{2}^{2}=\left({\frac {d}{dt}}{\left(x-\ell \sin \theta \right)}\right)^{2}+\left({\frac {d}{dt}}{\left(\ell \cos \theta \right)}\right)^{2}}$

${\displaystyle v_{2}^{2}={\dot {x}}^{2}-2\ell {\dot {x}}{\dot {\theta }}\cos \theta +\ell ^{2}{\dot {\theta }}^{2}}$

${\displaystyle L={\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}-m\ell {\dot {x}}{\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}-mg\ell \cos \theta }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\partial {L} \over \partial {\dot {x}}}-{\partial {L} \over \partial x}=F}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\partial {L} \over \partial {\dot {\theta }}}-{\partial {L} \over \partial \theta }=0}$

${\displaystyle \left(M+m\right){\ddot {x}}-m\ell {\ddot {\theta }}\cos \theta +m\ell {\dot {\theta }}^{2}\sin \theta =F}$
${\displaystyle \ell {\ddot {\theta }}-g\sin \theta ={\ddot {x}}\cos \theta }$

#### 牛頓第二運動定律

${\displaystyle F-R_{x}=M{\ddot {x}}}$
${\displaystyle F_{N}-R_{y}-Mg=0}$

${\displaystyle {\vec {r}}_{P}=(x-\ell \sin \theta ){\hat {x}}_{I}+\ell \cos \theta {\hat {y}}_{I}}$

${\displaystyle {\vec {a}}_{P/I}=({\ddot {x}}+\ell {\dot {\theta }}^{2}\sin \theta -\ell {\ddot {\theta }}\cos \theta ){\hat {x}}_{I}+(-\ell {\dot {\theta }}^{2}\cos \theta -\ell {\ddot {\theta }}\sin \theta ){\hat {y}}_{I}}$

${\displaystyle R_{x}=m({\ddot {x}}+\ell {\dot {\theta }}^{2}\sin \theta -\ell {\ddot {\theta }}\cos \theta )}$
${\displaystyle R_{y}-mg=m(-\ell {\dot {\theta }}^{2}\cos \theta -\ell {\ddot {\theta }}\sin \theta )}$

${\displaystyle \left(M+m\right){\ddot {x}}-m\ell {\ddot {\theta }}\cos \theta +m\ell {\dot {\theta }}^{2}\sin \theta =F}$

${\displaystyle {\hat {x}}_{B}=\cos \theta {\hat {x}}_{I}+\sin \theta {\hat {y}}_{I}}$

${\displaystyle ({\hat {x}}_{B})^{T}\sum {\vec {F}}=({\hat {x}}_{B})^{T}(R_{x}{\hat {x}}_{I}+R_{y}{\hat {y}}_{I}-mg{\hat {y}}_{I})=({\hat {x}}_{B})^{T}(R_{p}{\hat {y}}_{B}-mg{\hat {y}}_{I})=-mg\sin \theta }$

${\displaystyle R_{p}={\sqrt {R_{x}^{2}+R_{y}^{2}}}}$

${\displaystyle m({\hat {x}}_{B})^{T}({\vec {a}}_{P/I})=m({\ddot {x}}\cos \theta -\ell {\ddot {\theta }})}$

${\displaystyle \ell {\ddot {\theta }}-g\sin \theta ={\ddot {x}}\cos \theta }$

#### 穩定台車倒單擺的方式

1. 若直桿往右傾斜，台車需要往右加速，反之亦然。
2. 台車相對軌道中心的位置${\displaystyle x}$穩定的方式是是將null angle由台車的位置來進行調整，也就是null angle ${\displaystyle =\theta +kx}$，其中${\displaystyle k}$是小的數值。因此桿會輕微的向軌道中心傾斜，若其角度恰好垂直的話，就會穩定在軌道中心。傾斜感測器或是軌道斜率的誤差（本來會造成不穩定的效應）都會變成穩定的位置偏移量，另外增加的偏移量則是為了位置控制。
3. 正常的擺單擺在角頻率為${\displaystyle \omega _{p}={\sqrt {g/\ell }}}$時會共振。為了避免不受控的晃動，樞紐點需要抑制在共振頻率${\displaystyle \omega _{p}}$附近的頻率響應。倒單擺也需要類似的帶拒濾波器才能達到穩定。

## 卡皮察擺

${\displaystyle \left(-\ell \sin \theta ,y+\ell \cos \theta \right)}$

${\displaystyle v^{2}={\dot {y}}^{2}-2\ell {\dot {y}}{\dot {\theta }}\sin \theta +\ell ^{2}{\dot {\theta }}^{2}.}$

${\displaystyle L={\frac {1}{2}}m\left({\dot {y}}^{2}-2\ell {\dot {y}}{\dot {\theta }}\sin \theta +\ell ^{2}{\dot {\theta }}^{2}\right)-mg\left(y+\ell \cos \theta \right)}$

${\displaystyle {\mathrm {d} \over \mathrm {d} t}{\partial {L} \over \partial {\dot {\theta }}}-{\partial {L} \over \partial \theta }=0}$

${\displaystyle \ell {\ddot {\theta }}-{\ddot {y}}\sin \theta =g\sin \theta .}$

y的運動是簡諧運動${\displaystyle y=A\sin \omega t}$，則其微分方程為：

${\displaystyle {\ddot {\theta }}-{g \over \ell }\sin \theta =-{A \over \ell }\omega ^{2}\sin \omega t\sin \theta .}$

## 參考資料

1. ^ C.A. Hamilton Union College Senior Project 1966
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4. ^ 存档副本 (PDF). [2020-06-15]. （原始内容存档 (PDF)于2020-04-15）.
5. Archived copy (PDF). [2012-05-01]. （原始内容 (PDF)存档于2016-03-04）.
6. ^ 存档副本. [2020-06-15]. （原始内容存档于2016-08-27）.
7. ^ The Acrobot and Cart-Pole (PDF). （原始内容存档 (PDF)于2019-08-19）.
8. ^ Cart-Pole Swing-Up. www.cs.huji.ac.il. [2019-08-19]. （原始内容存档于2019-08-19）.
• D. Liberzon Switching in Systems and Control (2003 Springer) pp. 89ff

## 延伸閱讀

• Franklin; et al. (2005). Feedback control of dynamic systems, 5, Prentice Hall. ISBN 0-13-149930-0