Add some analysis
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@ -109,7 +109,7 @@ We obtain:
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+ \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
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\end{align*}
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By injecting the previous result into the Lagrangian equation [[eq:lagrangian_eq_inertial]], we obtain:
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By injecting the previous result into the Lagrangian equation, we obtain:
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\begin{align*}
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m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
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-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
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@ -137,7 +137,7 @@ We can then subtract and add the previous equations to obtain the following equa
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\end{align*}
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#+end_important
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** Analysis
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** TODO Analysis
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We obtain two differential equations that are coupled through:
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- *Euler forces*: $m d_v \ddot{\theta}$
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- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
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@ -145,12 +145,24 @@ We obtain two differential equations that are coupled through:
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Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k-d_u m\dot{\theta}^2$.
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Thus, the term $-d_u m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
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*** Stiff actuators
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Let's say we use stiff actuators such that $m \ddot{d_u} + (k - m\dot{\theta}^2) d_u \approx k d_u$.
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Let's suppose that $F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \approx F_u$.
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Then we obtain $d_u = \frac{F_u}{k}$ that we can re inject in the other equation to obtain:
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\[ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\frac{\dot{F_u}}{k}\dot{\theta} - m \frac{F_u}{k}\ddot{\theta} \]
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*** Negative Stiffness
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If $\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}$, then the negative spring effect is negligible and $k - m\dot{\theta}^2 \approx k$.
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* Analytical Computation of forces for the NASS
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For the NASS, the Euler forces should be less of a problem as $\ddot{\theta}$ should be very small when conducting an experiment.
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First we will determine the value for Euler and Coriolis forces during regular experiment.
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** Euler and Coriolis forces
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** Parameters
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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Let's define the parameters for the NASS.
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#+begin_src matlab :exports code :results silent
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@ -166,6 +178,9 @@ Let's define the parameters for the NASS.
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ddot = 0.2; % [m/s]
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#+end_src
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** Euler and Coriolis forces
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First we will determine the value for Euler and Coriolis forces during regular experiment.
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#+begin_src matlab :exports none :results silent
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Felight = mlight*d*wdot;
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Feheavy = mheavy*d*wdot;
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@ -188,25 +203,25 @@ We then compute the corresponding values of the Coriolis and Euler forces, and t
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| Coriolis | 44.0 N | 1.8 N |
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| Euler | 3.5 N | 8.5 N |
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** Spring Softening Effect
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** Negative Spring Effect
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#+begin_src matlab :exports none :results silent
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Klight = mlight*d*wdot^2;
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Kheavy = mheavy*d*wdot^2;
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#+end_src
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The values for the spring softening effect are displayed in table [[tab:spring_softening]].
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The values for the negative spring effect are displayed in table [[tab:negative_spring]].
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This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
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#+begin_src matlab :results value table :exports results :post addhdr(*this*)
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ans = sprintf(' | Light | Heavy | \n Spring Soft. | %.1f N/m | %.1f N/m', Klight, Kheavy)
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ans = sprintf(' | Light | Heavy | \n Neg. Spring | %.1f N/m | %.1f N/m', Klight, Kheavy)
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#+end_src
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#+NAME: tab:spring_softening
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#+CAPTION: Spring Softening effect
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#+NAME: tab:negative_spring
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#+CAPTION: Negative Spring effect
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#+RESULTS:
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| | Light | Heavy |
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|--------------+---------+---------|
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| Spring Soft. | 3.5 N/m | 8.5 N/m |
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|-------------+---------+---------|
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| Neg. Spring | 3.5 N/m | 8.5 N/m |
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* Control Strategies
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<<sec:control_strategies>>
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@ -229,11 +244,10 @@ The block diagram is shown on figure [[fig:control_measure_fixed_2dof]].
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The loop gain is then $L = G(\theta) K J(\theta)$.
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*** QUESTION Is the loop gain is changing with the angle ?
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Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
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One question we wish to answer is: is $G(\theta) J(\theta) = G(\theta_0) J(\theta_0)$?
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** Measurement in the rotating frame
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Let's consider that the measurement is in the rotating reference frame.
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Let's consider that the measurement is made in the rotating reference frame.
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The corresponding block diagram is shown figure [[fig:control_measure_rotating_2dof]]
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@ -245,6 +259,11 @@ The loop gain is $L = G K$.
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* Effect of the rotating Speed
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<<sec:effect_rot_speed>>
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
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** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
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@ -262,7 +281,14 @@ The loop gain is $L = G K$.
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#+NAME: matlab-init
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#+BEGIN_SRC matlab :results none :exports none
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clear; close all; clc;
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%% Add path with some functions
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addpath('./src/');
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%% Intialize Laplace variable
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s = tf('s');
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%% Initialize ans with org-babel
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ans = 0;
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#+END_SRC
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