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Thomas Dehaeze 2024-12-17 11:37:04 +01:00
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content/zettels/feedforward_control.md
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@ -9,9 +9,9 @@ Tags
Depending on the physical system to be controlled, several feedforward controllers can be used:
- [sec-fourth_order_feedforward](#sec-fourth_order_feedforward)
- [sec-model_based_feedforward](#sec-model_based_feedforward)
- [sec-rigid-body-feedforward](#sec-rigid-body-feedforward)
- [sec-fourth-order-feedforward](#sec-fourth-order-feedforward)
- [sec-model-based-feedforward](#sec-model-based-feedforward)
## Rigid Body Feedforward {#rigid-body-feedforward}
@ -38,7 +38,7 @@ F\_{ff} = m a + c v
<span id="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](#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 [fig-feedforward-double-mass-system](#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](#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 [fig-feedforward-fourth-order-feedforward-architecture](#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](#fig-feedforward_fourth_order_feedforward_architecture) can be used.
The same architecture shown in Figure [fig-feedforward-fourth-order-feedforward-architecture](#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>)).