Removed :name: directives as advised by Henry

copter/source/docs/systemid-mode-operation.rst

@@ -104,12 +104,12 @@ main loop frequency divided by a sub-sample factor.
 The next figure depicts the :ref:`SID_AXIS<SID_AXIS>` injection points in yellow and some details of the most inner PID loops when :ref:`ATC_RATE_FF_ENAB<ATC_RATE_FF_ENAB>` == 0.
 
 .. image:: ../images/ArduCopter4.1-PID-loops_FF_dis.png
-:name: fig-ArduCopter4.1-PID-loops_FF_dis
+fig-ArduCopter4.1-PID-loops_FF_dis
 
 The next figure depicts the :ref:`SID_AXIS<SID_AXIS>` injection points in yellow and some details of the most inner PID loops when :ref:`ATC_RATE_FF_ENAB<ATC_RATE_FF_ENAB>` == 1.
 
 .. image:: ../images/ArduCopter4.1-PID-loops_FF_en.png
-:name: fig-ArduCopter4.1-PID-loops_FF_en
+fig-ArduCopter4.1-PID-loops_FF_en
 
 
 Identification of a Multicopter
@@ -129,13 +129,13 @@ This ensures that the waveforms at the inputs of the plant model are mostly chir
 With the default controller parameters, which are quite low, the magnitude of the rate controller outputs are relatively low.
 
 .. image:: ../images/ControlSystemDiagram.png
-:name: fig-ctrl-sys-ardupilot
+fig-ctrl-sys-ardupilot
 
 Model Structure
 ---------------
 
 .. image:: ../images/bodyDiagramQuad.PNG
-:name: fig-body-diagram
+fig-body-diagram
 
 The quadcopter model is based on its equations of motion.
 Looking at the quadcopter as a rigid body, it has 6 degrees of freedom, consisting of three translational and three rotational motions.
@@ -145,21 +145,21 @@ Its axes are represented by the vectors (:math:`b_{x}`, :math:`b_{y}`, :math:`b_
 The equations of motion within the body-fixed frame can be expressed as
 
 .. image:: ../images/EquationsOfMotion.PNG
-:name: fig-eq-motion
+fig-eq-motion
 
 where :math:`u`, :math:`v` and :math:`w` are the velocities in :math:`b_{x}`, :math:`b_{y}`, :math:`b_{z}` direction respectively, whereas :math:`p`, :math:`q`,
 :math:`r` represent the angular velocities about the :math:`x`, :math:`y` and :math:`z` axis of the body-frame.
 The variables :math:`X`, :math:`Y`, :math:`Z` and :math:`L`, :math:`M`, :math:`N` represent forces and torques acting on the airframe due to thrust :math:`T_i` and reaction torque :math:`Q_i` generated by the propellers as well as external forces and moments caused by air resistance and wind. [#f2]_
 :math:`\phi` and :math:`\theta` are the Euler angles, that describe the orientation of the copter within the earth-frame. Thrust and reaction torque can be calculated by the following equations
 
 .. image:: ../images/EquationsThrustReactionTorque.PNG
-:name: fig-eq-thrust-torque
+fig-eq-thrust-torque
 
 with :math:`\omega_i` being the propellers angular velocity and :math:`C_T` as well as :math:`C_Q` standing for constants dependent on the propeller geometry.
 Based on these two quantities, the created forces and torques by the propulsion system acting on the copter's airframe for an X-configuration can be written as
 
 .. image:: ../images/EquationForceTorqueAllocation.PNG
-:name: fig-eq-force-torque-prop
+fig-eq-force-torque-prop
 
 where :math:`d` represents the arm length between the respective propeller and the airframe's center of gravity, while :math:`\varphi_i` stands for the angle of the propeller's arm in regard to the :math:`b_x` axis of the copter. [#f3]_
 :math:`\Gamma` is called the allocation matrix.
@@ -171,28 +171,28 @@ This assumes that the parameter :ref:`MOT_THST_EXPO<MOT_THST_EXPO>` has been :re
 As an example, the lateral velocity :math:`u` can then be expressed by its stationary condition :math:`u_0` and its perturbation :math:`\Delta u`
 
 .. image:: ../images/SmallPertU.PNG
-:name: fig-eq-small-pert-u
+fig-eq-small-pert-u
 
 While hovering, the stationary condition of every state and input quantity is 0, since there is no translational or rotational motion of the vehicle. [#f4]_
 The equations of motion can then be simplified to
 
 .. image:: ../images/equationsOfMotionLin.PNG
-:name: fig-eq-motion-lin
+fig-eq-motion-lin
 
 This linearization removes the interdependencies between the pitch, roll and yaw motions of the copter, thereby yielding three decoupled systems.
 Next, the perturbations in each force and torque are modeled to obtain the influences of these system inputs on the state variables.
 By using a Taylor series expansion, the forces and torques are described as a linear function of the disturbances in the state and input variables that they depend on.
 The coefficients of each state variable are the so-called stability or control derivatives of the respective force and torque. [#f4]_ [#f5]_
 
 .. image:: ../images/perturbedForcesTorques.PNG
-:name: fig-eq-pert-forces-torques
+fig-eq-pert-forces-torques
 
 By inserting these descriptions of forces and torques, the linearized equations of motion are rewritten in a state-space representation,
 with the control forces :math:`F_{c}` and torques :math:`M_{c}` generated by the propellers being the model inputs.
 Since only perturbed quantities are contained in the equations, the :math:`\Delta` indicating the small perturbations is left out.
 
 .. image:: ../images/equationsOfMotionLinPerturbation.PNG
-:name: fig-eq-motion-lin
+fig-eq-motion-lin
 
 The actual control inputs of the system are the rate controller outputs as shown in the :ref:`control diagram<fig-ctrl-sys-ardupilot>`.
 The motor mixer converts the controller outputs to thrust demands for each motor.
@@ -207,13 +207,13 @@ The acceleration or deceleration of the motor leads to an additional, dynamic re
 The transfer function can then be written as
 
 .. image:: ../images/motorModel.PNG
-:name: fig-eq-motor-model
+fig-eq-motor-model
 
 The thrust demands of the motors are calculated from the controller outputs by inverting the :ref:`allocation matrix<fig-eq-force-
 torque-prop>` [#f9]_
 
 .. image:: ../images/ctrlOutputsToThrstCmds.PNG
-:name: fig-eq-ctrlout-thrstCmds
+fig-eq-ctrlout-thrstCmds
 
 As a consequence, the inverted matrix above and the :ref:`allocation matrix<fig-eq-force-torque-prop>` cancel each other out,
 thereby leaving the following four separated control paths: heave, roll, pitch and yaw.
@@ -223,7 +223,7 @@ To model this in a state-space representation, control forces and torques are ad
 The state-space models for the roll, pitch and yaw axis are formulated as
 
 .. image:: ../images/axisModels.PNG
-:name: fig-eq-axis-models
+fig-eq-axis-models
 
 The used transfer function model of each axis is derived from the state-space models above through Laplace-Transformation and
 by solving for the angular velocities :math:`p`, :math:`q` and :math:`r` as the outputs of each model.
@@ -232,7 +232,7 @@ Also, dead time terms are added to account for delays, that were not regarded in
 These three transfer functions represent the model structures used in the identification for the pitch, roll and yaw axes.
 
 .. image:: ../images/axisModelsTf.PNG
-:name: fig-eq-axis-models-Tf
+fig-eq-axis-models-Tf
 
 
 Example of Identification Process
@@ -299,15 +299,15 @@ The reason for the sweep attenuation is the amplified controller output due to t
 Although attenuated, the system excitation is still large enough to obtain a reliable frequency response with a sufficiently high coherence as shown in the following paragraph.
 
 .. image:: ../images/rollSweepPlantInput.png
-:name: fig-sweep-rll
+fig-sweep-rll
 
 
 .. image:: ../images/pitchSweepPlantInput.png
-:name: fig-sweep-pit
+fig-sweep-pit
 
 
 .. image:: ../images/yawSweepPlantInput.png
-:name: fig-sweep-yaw
+fig-sweep-yaw
 
 The frequency response of each axis is obtained through spectral analysis of the flight data.
 Only test flights with a sufficient coherence between input and output are used for the system identification.
@@ -316,35 +316,35 @@ The following diagrams show the data-based frequency responses of all three axes
 The bottom plot shows the coherence between input and output which quantifies the linearity between input and output.
 
 .. image:: ../images/bodeDataRll.png
-:name: fig-bode-data-rll
+fig-bode-data-rll
 
 
 .. image:: ../images/bodeDataPit.png
-:name: fig-bode-data-pit
+fig-bode-data-pit
 
 
 .. image:: ../images/bodeDataYaw.png
-:name: fig-bode-data-yaw
+fig-bode-data-yaw
 
 The composite frequency responses are used to determine the parameters of the :ref:`transfer function models<fig-eq-axis-models-Tf>`.
 The parameters of the plant model transfer functions are optimized to maximize their fit to the collected real-world data frequency responses.
 The result is shown in the following three figures.
 
 .. image:: ../images/bodeTfRll.png
-:name: fig-bode-data-rll
+fig-bode-data-rll
 
 
 .. image:: ../images/bodeTfPit.png
-:name: fig-bode-data-pit
+fig-bode-data-pit
 
 
 .. image:: ../images/bodeTfYaw.png
-:name: fig-bode-data-yaw
+fig-bode-data-yaw
 
 The derived transfer function models are as follows:
 
 .. image:: ../images/identifiedAxisModelsTf.PNG
-:name: fig-identified-models
+fig-identified-models
 
 Example of Identification Results and Model Verification
 --------------------------------------------------------
@@ -364,43 +364,43 @@ As seen in the following three figure, each model is capable of reproducing the
 Roll:
 
 .. image:: ../images/modelValidationSweepRoll.png
-:name: fig-val-sweep-rll
+fig-val-sweep-rll
 
 Pitch:
 
 .. image:: ../images/modelValidationSweepPitch.png
-:name: fig-val-sweep-pit
+fig-val-sweep-pit
 
 Yaw:
 
 .. image:: ../images/modelValidationSweepYaw.png
-:name: fig-val-sweep-yaw
+fig-val-sweep-yaw
 
 Since the frequency-sweeps are used for the identification, it is important to test the models against another test signal to check for their robustness.
 A widely recommended verification signal is the so called doublet maneuver that is basically a double step [#f8]_.
 For time-domain validation, a similar signal is used as the input of the stabilize controller.
 As shown in the following figure for the roll axis, it consists of two consecutive doublet maneuvers.
 
 .. image:: ../images/modelValidationDoublet.png
-:name: fig-val-doublet
+fig-val-doublet
 
 The next three figures contain the angular rates of the axis models compared to the measured ones during the validation flights.
 Due to the high fitting between model outputs and measured angular rates, the fidelity of the models is seen as satisfactory.
 
 Roll:
 
 .. image:: ../images/modelValidationRollAng.png
-:name: fig-val-doublet-rll
+fig-val-doublet-rll
 
 Pitch:
 
 .. image:: ../images/modelValidationPitchAng.png
-:name: fig-val-doublet-pit
+fig-val-doublet-pit
 
 Yaw:
 
 .. image:: ../images/modelValidationYaw.png
-:name: fig-val-doublet-yaw
+fig-val-doublet-yaw
 
 
 .. rubric:: References