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content/zettels/feedforward_control.md
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@ -9,19 +9,21 @@ Tags
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)
- [sec:rigid-body-feedforward](#sec:rigid-body-feedforward)
-
- <sec:fourth_order_feedforward>
- <sec:model_based_feedforward>
- <sec:rigid-body-feedforward>
<&boerlage03_model>
## Rigid Body Feedforward {#sec: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 [Figure 1](#figure--fig:feedforward-second-order-trajectory).
<a id="figure--fig:feedforward-second-order-trajectory"></a>
@ -37,9 +39,9 @@ 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 [2](#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>).
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>
@ -77,7 +79,7 @@ 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 [3](#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> 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)
@ -104,14 +106,14 @@ 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 [3](#figure--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 <&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 <&schmidt20_desig_high_perfor_mechat_third_revis_edition> (Section 4.2.1).