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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: 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: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>
<&boerlage03_model>
## Rigid Body Feedforward {#sec:rigid-body-feedforward} ## 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. 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> <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} ## 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/). 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>
@ -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 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 [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} \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)
@ -104,14 +106,14 @@ 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 [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>). 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} ## 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). See <&schmidt20_desig_high_perfor_mechat_third_revis_edition> (Section 4.2.1).