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content/zettels/feedforward_control.md
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Depending on the physical system to be controlled, several feedforward controllers can be used:
- [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](#org-target--sec-rigid-body-feedforward)
- [Fourth order feedforward](#org-target--sec-fourth-order-feedforward)
- [Model based feedforward](#org-target--sec-model-based-feedforward)
## Rigid Body Feedforward {#rigid-body-feedforward}
<span id="sec-rigid-body-feedforward"></span>
<span class="org-target" id="org-target--sec-rigid-body-feedforward"></span>
Second order trajectory planning: the acceleration and velocity can be bound to wanted values.
Such trajectory is shown in [Figure 1](#figure--fig:feedforward-second-order-trajectory).
Such trajectory is shown in [1](#figure--fig:feedforward-second-order-trajectory).
<a id="figure--fig:feedforward-second-order-trajectory"></a>
@ -36,9 +36,9 @@ F\_{ff} = m a + c v
## Fourth Order Feedforward {#fourth-order-feedforward}
<span id="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](#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>).
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> 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)
@ -103,16 +103,16 @@ 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> 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 <&lambrechts04_trajec>).
## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant}
<span id="sec-model-based-feedforward"></span>
<span class="org-target" id="org-target--sec-model-based-feedforward"></span>
See (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
See <&schmidt20_desig_high_perfor_mechat_third_revis_edition> (Section 4.2.1).
Suppose we have a second order plant (could typically be a piezoelectric stage):
\\[ G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\xi \omega\_0 s + \omega\_0^2} \\]
@ -227,7 +227,4 @@ This can be solved by using **snap feedforward**
## 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>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>
<./biblio/references.bib>