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content/zettels/feedforward_control.md
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@ -11,17 +11,14 @@ Below, the "References" heading will be auto-inserted.
Depending on the physical system to be controlled, several feedforward controllers can be used: Depending on the physical system to be controlled, several feedforward controllers can be used:
- - [sec:fourth_order_feedforward](#sec:fourth_order_feedforward)
- <sec:fourth_order_feedforward> - [sec:model_based_feedforward](#sec:model_based_feedforward)
- <sec:model_based_feedforward> - [sec:rigid-body-feedforward](#sec:rigid-body-feedforward)
- <sec:rigid-body-feedforward>
(<a href="#citeproc_bib_item_1">Boerlage et al. 2003</a>)
## 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. 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} ## 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/). 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>
@ -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 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 [fig:feedforward_fourth_order_feedforward_architecture](#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)
@ -108,14 +105,14 @@ 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 [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>)). 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} ## 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). 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" >}} {{< 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"> <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> </div>