.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/examples/plot_balanceassistv1.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_gallery_examples_plot_balanceassistv1.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_gallery_examples_plot_balanceassistv1.py:


===========================================================
Balance Assist E-Bike with Roll Rate Feedback Steer Control
===========================================================

This example shows how to work with a model that includes a feedback controller
and how to use a simple derivative control with it. The TU Delft Bicycle Lab
developed a bicycle with a steer motor that can be controlled based on sensor
measurements from an inertial measurement unit mounted on the rear frame, a
steer angle sensor, and a speed sensor. The bicycle is based on an e-bike model
from Royal Dutch Gazelle:

.. figure:: https://objects-us-east-1.dream.io/mechmotum/balance-assist-bicycle-400x400.jpg
   :align: center

   Gazelle Grenoble/Arroyo E-Bike modified with a steering motor. Battery in
   the downtube and electronics box on the rear rack.

.. GENERATED FROM PYTHON SOURCE LINES 20-27

.. code-block:: Python

    import numpy as np

    from bicycleparameters.main import Bicycle
    from bicycleparameters.io import remove_uncertainties
    from bicycleparameters.parameter_sets import Meijaard2007ParameterSet
    from bicycleparameters.models import Meijaard2007WithFeedbackModel








.. GENERATED FROM PYTHON SOURCE LINES 28-34

Set Up a Model
==============

First, load the physical parameter measurements of the bicycle from a file
and create a
:class:`~bicycleparameters.parameter_sets.Meijaard2007ParameterSet`.

.. GENERATED FROM PYTHON SOURCE LINES 34-43

.. code-block:: Python


    data_dir = "../../data"

    bicycle = Bicycle("Balanceassistv1", pathToData=data_dir)
    par = remove_uncertainties(bicycle.parameters['Benchmark'])
    par['v'] = 1.0
    par_set = Meijaard2007ParameterSet(par, False)
    par_set





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Found the RawData directory: ../../data/bicycles/Balanceassistv1/RawData
    Looks like you've already got some parameters for Balanceassistv1, use forceRawCalc to recalculate.


.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table><caption>Meijaard2007</caption><tr><th>Variable</th><th>Value</th></tr><tr><td>\(I_{Bxx}\)</td><td>1.120</td></tr><tr><td>\(I_{Bxz}\)</td><td>0.047</td></tr><tr><td>\(I_{Byy}\)</td><td>3.160</td></tr><tr><td>\(I_{Bzz}\)</td><td>2.119</td></tr><tr><td>\(I_{Fxx}\)</td><td>0.100</td></tr><tr><td>\(I_{Fyy}\)</td><td>0.190</td></tr><tr><td>\(I_{Hxx}\)</td><td>0.298</td></tr><tr><td>\(I_{Hxz}\)</td><td>-0.038</td></tr><tr><td>\(I_{Hyy}\)</td><td>0.257</td></tr><tr><td>\(I_{Hzz}\)</td><td>0.057</td></tr><tr><td>\(I_{Rxx}\)</td><td>0.102</td></tr><tr><td>\(I_{Ryy}\)</td><td>0.189</td></tr><tr><td>\(c\)</td><td>0.042</td></tr><tr><td>\(g\)</td><td>9.807</td></tr><tr><td>\(\lambda\)</td><td>0.255</td></tr><tr><td>\(m_B\)</td><td>22.500</td></tr><tr><td>\(m_F\)</td><td>2.235</td></tr><tr><td>\(m_H\)</td><td>4.300</td></tr><tr><td>\(m_R\)</td><td>4.085</td></tr><tr><td>\(r_F\)</td><td>0.352</td></tr><tr><td>\(r_R\)</td><td>0.349</td></tr><tr><td>\(v\)</td><td>1.000</td></tr><tr><td>\(w\)</td><td>1.113</td></tr><tr><td>\(x_B\)</td><td>0.519</td></tr><tr><td>\(x_H\)</td><td>0.921</td></tr><tr><td>\(z_B\)</td><td>-0.521</td></tr><tr><td>\(z_H\)</td><td>-0.860</td></tr></table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 44-46

The following plot depicts the geometry and inertial parameters with the
inertia of the rider included.

.. GENERATED FROM PYTHON SOURCE LINES 46-48

.. code-block:: Python

    par_set.plot_all()




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_001.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <Axes: xlabel='$x$ [m]', ylabel='$z$ [m]'>



.. GENERATED FROM PYTHON SOURCE LINES 49-53

Create a :class:`~bicycleparameters.models.Meijaard2007WithFeedbackModel`.
The parameter set does not include the feedback gain parameters, but
:meth:`~bicycleparameters.parameter_sets.Meijaard2007WithFeedbackParameterSet.to_parameterization`
will be used to convert the parameter set into one with the gains.

.. GENERATED FROM PYTHON SOURCE LINES 53-56

.. code-block:: Python

    model = Meijaard2007WithFeedbackModel(par_set)
    model.parameter_set






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table><caption>Meijaard2007WithFeedback</caption><tr><th>Variable</th><th>Value</th></tr><tr><td>\(I_{Bxx}\)</td><td>1.120</td></tr><tr><td>\(I_{Bxz}\)</td><td>0.047</td></tr><tr><td>\(I_{Byy}\)</td><td>3.160</td></tr><tr><td>\(I_{Bzz}\)</td><td>2.119</td></tr><tr><td>\(I_{Fxx}\)</td><td>0.100</td></tr><tr><td>\(I_{Fyy}\)</td><td>0.190</td></tr><tr><td>\(I_{Hxx}\)</td><td>0.298</td></tr><tr><td>\(I_{Hxz}\)</td><td>-0.038</td></tr><tr><td>\(I_{Hyy}\)</td><td>0.257</td></tr><tr><td>\(I_{Hzz}\)</td><td>0.057</td></tr><tr><td>\(I_{Rxx}\)</td><td>0.102</td></tr><tr><td>\(I_{Ryy}\)</td><td>0.189</td></tr><tr><td>\(c\)</td><td>0.042</td></tr><tr><td>\(g\)</td><td>9.807</td></tr><tr><td>\(k_{T_{\delta}\delta}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\delta}\dot{\delta}}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\delta}\phi}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\delta}\dot{\phi}}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\phi}\delta}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\phi}\dot{\delta}}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\phi}\phi}\)</td><td>0.000</td></tr><tr><td>\(k_{T_{\phi}\dot{\phi}}\)</td><td>0.000</td></tr><tr><td>\(\lambda\)</td><td>0.255</td></tr><tr><td>\(m_B\)</td><td>22.500</td></tr><tr><td>\(m_F\)</td><td>2.235</td></tr><tr><td>\(m_H\)</td><td>4.300</td></tr><tr><td>\(m_R\)</td><td>4.085</td></tr><tr><td>\(r_F\)</td><td>0.352</td></tr><tr><td>\(r_R\)</td><td>0.349</td></tr><tr><td>\(v\)</td><td>1.000</td></tr><tr><td>\(w\)</td><td>1.113</td></tr><tr><td>\(x_B\)</td><td>0.519</td></tr><tr><td>\(x_H\)</td><td>0.921</td></tr><tr><td>\(z_B\)</td><td>-0.521</td></tr><tr><td>\(z_H\)</td><td>-0.860</td></tr></table>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 57-58

The model shows a small self-stable speed range.

.. GENERATED FROM PYTHON SOURCE LINES 58-62

.. code-block:: Python

    speeds = np.linspace(0.0, 10.0, num=501)
    ax = model.plot_eigenvalue_parts(v=speeds)
    ax.set_ylim((-10.0, 10.0))




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_002.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (-10.0, 10.0)



.. GENERATED FROM PYTHON SOURCE LINES 63-70

Add a Rider
===========

If the data files for a rider are present in the data directory, you can add
a rider and the package Yeadon will be used to configure a rider to sit on
the bicycle. You can check if a rider is properly configured by plotting the
geometry which will now include a stick figure depiction of the rider.

.. GENERATED FROM PYTHON SOURCE LINES 70-73

.. code-block:: Python

    bicycle.add_rider('Jason', reCalc=True)
    bicycle.plot_bicycle_geometry(inertiaEllipse=False)




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_003.png
   :alt: Balanceassistv1 Bicycle Geometry
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_003.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    There is no rider on the bicycle, now adding Jason.
    Calculating the human configuration.

    <Figure size 647.214x400 with 1 Axes>



.. GENERATED FROM PYTHON SOURCE LINES 74-76

The inertia representation now reflects the larger inertia of the rear frame
due to the rigid rider addition.

.. GENERATED FROM PYTHON SOURCE LINES 76-81

.. code-block:: Python

    par = remove_uncertainties(bicycle.parameters['Benchmark'])
    par['v'] = 1.0
    par_set = Meijaard2007ParameterSet(par, True)
    par_set.plot_all()




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_004.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_004.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <Axes: xlabel='$x$ [m]', ylabel='$z$ [m]'>



.. GENERATED FROM PYTHON SOURCE LINES 82-83

The self-stable speed range begins at a higher speed and becomes wider.

.. GENERATED FROM PYTHON SOURCE LINES 83-87

.. code-block:: Python

    model = Meijaard2007WithFeedbackModel(par_set)
    ax = model.plot_eigenvalue_parts(v=speeds)
    ax.set_ylim((-10.0, 10.0))




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_005.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_005.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (-10.0, 10.0)



.. GENERATED FROM PYTHON SOURCE LINES 88-94

Add Control
===========

It turns out that controlling the steer torque with a positive feedback on
roll angular rate, the bicycle can be stabilized over a large speed range.
For example, setting :math:`k_{T_\delta \dot{\phi}}=-50` gives this effect:

.. GENERATED FROM PYTHON SOURCE LINES 94-97

.. code-block:: Python

    ax = model.plot_eigenvalue_parts(v=speeds, kTdel_phid=-50.0)
    ax.set_ylim((-10.0, 10.0))




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_006.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_006.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (-10.0, 10.0)



.. GENERATED FROM PYTHON SOURCE LINES 98-100

The eigenvectors at low speed show that the weave mode has a steer dominated
high frequency natural motion. This may not be so favorable.

.. GENERATED FROM PYTHON SOURCE LINES 100-103

.. code-block:: Python

    speed = 2.0
    ax = model.plot_eigenvectors(v=speed, kTdel_phid=-50.0)




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_007.png
   :alt: Eigenvalue: -1.817+10.745j, Eigenvalue: -1.817-10.745j, Eigenvalue: -5.815+0.000j, Eigenvalue: -0.231+0.000j
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_007.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 104-107

You can also gain schedule with respect to speed. A linear variation in the
roll rate gain can make the weave eigenfrequency have lower magnitude than
using simply a constant gain.

.. GENERATED FROM PYTHON SOURCE LINES 107-115

.. code-block:: Python

    vmin, vmin_idx = 1.5, np.argmin(np.abs(speeds - 1.5))
    vmax, vmax_idx = 4.7, np.argmin(np.abs(speeds - 4.7))
    kappa = 10.0
    kphidots = -kappa*(vmax - speeds)
    kphidots[:vmin_idx] = -kappa*(vmax - vmin)/vmin*speeds[:vmin_idx]
    kphidots[vmax_idx:] = 0.0
    model.plot_gains(v=speeds, kTdel_phid=kphidots)




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_008.png
   :alt: $k_{T_{\phi}\phi}$, $k_{T_{\phi}\delta}$, $k_{T_{\phi}\dot{\phi}}$, $k_{T_{\phi}\dot{\delta}}$, $k_{T_{\delta}\phi}$, $k_{T_{\delta}\delta}$, $k_{T_{\delta}\dot{\phi}}$, $k_{T_{\delta}\dot{\delta}}$
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_008.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([[<Axes: title={'center': '$k_{T_{\\phi}\\phi}$'}>,
            <Axes: title={'center': '$k_{T_{\\phi}\\delta}$'}>,
            <Axes: title={'center': '$k_{T_{\\phi}\\dot{\\phi}}$'}>,
            <Axes: title={'center': '$k_{T_{\\phi}\\dot{\\delta}}$'}>],
           [<Axes: title={'center': '$k_{T_{\\delta}\\phi}$'}, xlabel='v'>,
            <Axes: title={'center': '$k_{T_{\\delta}\\delta}$'}, xlabel='v'>,
            <Axes: title={'center': '$k_{T_{\\delta}\\dot{\\phi}}$'}, xlabel='v'>,
            <Axes: title={'center': '$k_{T_{\\delta}\\dot{\\delta}}$'}, xlabel='v'>]],
          dtype=object)



.. GENERATED FROM PYTHON SOURCE LINES 116-117

Using the gain scheduling gives this effect to the dynamics:

.. GENERATED FROM PYTHON SOURCE LINES 117-120

.. code-block:: Python

    ax = model.plot_eigenvalue_parts(v=speeds, kTdel_phid=kphidots)
    ax.set_ylim((-10.0, 10.0))




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_009.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_009.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    (-10.0, 10.0)



.. GENERATED FROM PYTHON SOURCE LINES 121-124

.. code-block:: Python

    kphidot = kphidots[np.argmin(np.abs(speeds - speed))]
    ax = model.plot_eigenvectors(v=speed, kTdel_phid=kphidot)




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_010.png
   :alt: Eigenvalue: -0.786+7.434j, Eigenvalue: -0.786-7.434j, Eigenvalue: -6.242+0.000j, Eigenvalue: -0.457+0.000j
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_010.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 125-127

We can then simulate the system at a specific low speed and see the effect
control has. First, without control:

.. GENERATED FROM PYTHON SOURCE LINES 127-131

.. code-block:: Python

    times = np.linspace(0.0, 2.0, num=201)
    x0 = np.array([0.0, 0.0, 0.5, 0.0])
    ax = model.plot_simulation(times, x0, v=speed)




.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_011.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_011.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 132-133

And with control:

.. GENERATED FROM PYTHON SOURCE LINES 133-135

.. code-block:: Python

    times = np.linspace(0.0, 5.0, num=501)
    ax = model.plot_simulation(times, x0, v=speed, kTdel_phid=kphidot)



.. image-sg:: /gallery/examples/images/sphx_glr_plot_balanceassistv1_012.png
   :alt: plot balanceassistv1
   :srcset: /gallery/examples/images/sphx_glr_plot_balanceassistv1_012.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.867 seconds)


.. _sphx_glr_download_gallery_examples_plot_balanceassistv1.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_balanceassistv1.ipynb <plot_balanceassistv1.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_balanceassistv1.py <plot_balanceassistv1.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_balanceassistv1.zip <plot_balanceassistv1.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_