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Update Content - 2023-06-28

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Thomas Dehaeze 2023-06-28 10:15:15 +02:00
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content/zettels/feedforward_control.md
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@ -20,7 +20,7 @@ Depending on the physical system to be controlled, several feedforward controlle
Second order trajectory planning: the acceleration and velocity can be bound to wanted values.
Such trajectory is shown in Figure <fig:feedforward_second_order_trajectory>.
Such trajectory is shown in Figure [1](#figure--fig:feedforward-second-order-trajectory).
<a id="figure--fig:feedforward-second-order-trajectory"></a>
@ -38,7 +38,7 @@ F\_{ff} = m a + c v
<span class="org-target" id="org-target--sec-fourth-order-feedforward"></span>
The main advantage of "fourth order feedforward" is that it takes into account the flexibility in the system (one resonance between the actuation point and the measurement point, see Figure <fig:feedforward_double_mass_system>).
The main advantage of "fourth order feedforward" is that it takes into account the flexibility in the system (one resonance between the actuation point and the measurement point, see Figure [2](#figure--fig:feedforward-double-mass-system)).
This can lead to better results than second order trajectory planning as demonstrated [here](https://www.20sim.com/control-engineering/snap-feedforward/).
<a id="figure--fig:feedforward-double-mass-system"></a>
@ -76,7 +76,7 @@ q\_3 &= (m\_1 + m\_2)c + k\_1 k\_2 + (k\_1 + k\_2) k\_{12} \\\\
q\_4 &= (k\_1 + k\_2) c
\end{align}
This means that if a fourth-order trajectory for \\(x\_2\\) is used, the feedforward architecture shown in Figure <fig:feedforward_fourth_order_feedforward_architecture> can be used:
This means that if a fourth-order trajectory for \\(x\_2\\) is used, the feedforward architecture shown in Figure [3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used:
\begin{equation}
F\_{f2} = \frac{1}{k\_12 s + c} (q\_1 d + q\_2 j + q\_3 q + q\_4 v)
@ -103,7 +103,7 @@ q\_4 &= c\_1 k
and \\(s\\) the snap, \\(j\\) the jerk, \\(a\\) the acceleration and \\(v\\) the velocity.
The same architecture shown in Figure <fig:feedforward_fourth_order_feedforward_architecture> can be used.
The same architecture shown in Figure [3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used.
In order to implement a fourth order trajectory, look at [this](https://www.mathworks.com/matlabcentral/fileexchange/16352-advanced-setpoints-for-motion-systems) nice implementation in Simulink of fourth-order trajectory planning (see also (<a href="#citeproc_bib_item_1">Lambrechts, Boerlage, and Steinbuch 2004</a>)).

4
content/zettels/position_jitter_due_to_asynchronous_acquisition.md
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@ -31,8 +31,8 @@ Let's choose the following parameters:
- \\(v = 1\\,mm/s\\): the scan velocity
- \\(T\_{s,\text{ctrl}} = 100\\,\mu s\\) the "sampling rate" of the controller
The encoder position as well as the stored value on the PEPU and the position used in the controller are shown in Figure <fig:jitter_error_example>, left.
The errors associated with the "jitter" is shown in Figure <fig:jitter_error_example>, right.
The encoder position as well as the stored value on the PEPU and the position used in the controller are shown in Figure [2](#figure--fig:jitter-error-example), left.
The errors associated with the "jitter" is shown in Figure [2](#figure--fig:jitter-error-example), right.
```matlab
%% Simulation parameters