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Thomas Dehaeze 2024-12-17 11:28:29 +01:00
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Depending on the physical system to be controlled, several feedforward controllers can be used:
-
- <sec:fourth_order_feedforward>
- <sec:model_based_feedforward>
- <sec:rigid-body-feedforward>
(<a href="#citeproc_bib_item_1">Boerlage et al. 2003</a>)
- [sec:fourth_order_feedforward](#sec:fourth_order_feedforward)
- [sec:model_based_feedforward](#sec:model_based_feedforward)
- [sec:rigid-body-feedforward](#sec:rigid-body-feedforward)
## Rigid Body Feedforward {#sec:rigid-body-feedforward}
## Rigid Body Feedforward {#rigid-body-feedforward}
<span class="org-target" id="org-target--sec-rigid-body-feedforward"></span>
<span id="sec-rigid-body-feedforward"></span>
Second order trajectory planning: the acceleration and velocity can be bound to wanted values.
@ -41,9 +38,9 @@ F\_{ff} = m a + c v
## Fourth Order Feedforward {#fourth-order-feedforward}
<span class="org-target" id="org-target--sec-fourth-order-feedforward"></span>
<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>).
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>
@ -81,7 +78,7 @@ 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 [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)
@ -108,14 +105,14 @@ 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 [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>)).
## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant}
<span class="org-target" id="org-target--sec-model-based-feedforward"></span>
<span id="sec-model-based-feedforward"></span>
See (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
@ -229,8 +226,10 @@ This can be solved by using **snap feedforward**
{{< figure src="/ox-hugo/feedforward_schematic_snap.png" >}}
## References
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Boerlage, M., M. Steinbuch, P. Lambrechts, and M. van de Wal. 2003. “Model-Based Feedforward for Motion Systems.” In <i>Proceedings of 2003 Ieee Conference on Control Applications, 2003. Cca 2003.</i> <a href="https://doi.org/10.1109/cca.2003.1223174">https://doi.org/10.1109/cca.2003.1223174</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lambrechts, P., M. Boerlage, and M. Steinbuch. 2004. “Trajectory Planning and Feedforward Design for High Performance Motion Systems.” In <i>Proceedings of the 2004 American Control Conference</i>. doi:<a href="https://doi.org/10.23919/acc.2004.1384042">10.23919/acc.2004.1384042</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
</div>