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matlab/index.org
@ -65,7 +65,7 @@
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- $k$: Actuator's Stiffness [N/m]
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- $m$: Payload's mass [kg]
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- $\omega_0 = \sqrt{\frac{k}{m}}$: Resonance of the (non-rotating) mass-spring system [rad/s]
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- $\omega_r = \dot{\theta}$: rotation speed [rad/s]
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- $\Omega = \dot{\theta}$: rotation speed [rad/s]
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#+name: fig:rotating_xy_platform
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@ -141,7 +141,7 @@ Without the coupling terms, each equation is the equation of a one degree of fre
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Thus, the term $- m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
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** Constant Rotating Speed
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To simplify, let's consider a constant rotating speed $\dot{\theta} = \omega_r$ and thus $\ddot{\theta} = 0$.
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To simplify, let's consider a constant rotating speed $\dot{\theta} = \Omega$ and thus $\ddot{\theta} = 0$.
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#+NAME: eq:coupledplant
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\begin{equation}
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@ -154,13 +154,15 @@ To simplify, let's consider a constant rotating speed $\dot{\theta} = \omega_r$
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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# Explain each term
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#+NAME: eq:coupled_plant
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\begin{equation}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
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\frac{\frac{1}{k}}{\left( \frac{s^2}{{\omega_0}^2} + (1 - \frac{{\omega_r}^2}{{\omega_0}^2}) \right)^2 + \left( 2 \frac{{\omega_r} s}{{\omega_0}^2} \right)^2}
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\frac{\frac{1}{k}}{\left( \frac{s^2}{{\omega_0}^2} + (1 - \frac{{\Omega}^2}{{\omega_0}^2}) \right)^2 + \left( 2 \frac{{\Omega} s}{{\omega_0}^2} \right)^2}
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\begin{bmatrix}
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\frac{s^2}{{\omega_0}^2} + 1 - \frac{{\omega_r}^2}{{\omega_0}^2} & 2 \frac{\omega_r s}{{\omega_0}^2} \\
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-2 \frac{\omega_r s}{{\omega_0}^2} & \frac{s^2}{{\omega_0}^2} + 1 - \frac{{\omega_r}^2}{{\omega_0}^2} \\
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\frac{s^2}{{\omega_0}^2} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} & 2 \frac{\Omega s}{{\omega_0}^2} \\
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-2 \frac{\Omega s}{{\omega_0}^2} & \frac{s^2}{{\omega_0}^2} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
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\end{bmatrix}
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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