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Thomas Dehaeze 2023-06-28 10:15:15 +02:00
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8
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. 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> <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> <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/). 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> <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 q\_4 &= (k\_1 + k\_2) c
\end{align} \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} \begin{equation}
F\_{f2} = \frac{1}{k\_12 s + c} (q\_1 d + q\_2 j + q\_3 q + q\_4 v) 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. 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>)). 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>)).

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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 - \\(v = 1\\,mm/s\\): the scan velocity
- \\(T\_{s,\text{ctrl}} = 100\\,\mu s\\) the "sampling rate" of the controller - \\(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 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 <fig:jitter_error_example>, right. The errors associated with the "jitter" is shown in Figure [2](#figure--fig:jitter-error-example), right.
```matlab ```matlab
%% Simulation parameters %% Simulation parameters