Added inverse dynamics correlation plots.

main.tex

@@ -250,11 +250,19 @@ \section*{Results}
 %
 The formulation above allows one to choose a variety of combinations of
 potential sensor, $\mathbf{s}(t)$, and control, $\mathbf{m}(t)$, signals to
-generate a controller.
+generate a controller. For this analyses we focus on a planar controller that
+involves the hip, knee, and ankle joints. The angle and angular rate of each
+joint are available as sensors and a torque at each joint are available as the
+actuators. Further more the formulation allows for different types of control
+that isolate sensors from actuators:
 
 \begin{description}
   \item[Joint Isolated Control] The torque generated by a joint is only
     affected by the sensors that give information about that joint.
+  \item[Side Isolated Control] The torque generated at a joint is only affected
+    by the sensors on the same side of the body.
+  \item[Full Control] The torque genderated by a joint is affected by all of
+    the available sensors.
 \end{description}
 
 \subsection*{Example Result}
@@ -290,10 +298,72 @@ \subsection*{Example Result}
   \end{center}
 \end{figure}
 %
-\subsection*{$\mathbf{m}^*$ accounts for most of the variation}
-%
 \subsection*{Random variations predict random gains}
 %
+If the variation in the sensor and actuator measurements is purely random, the
+identified gains should be random. To verify this we follow this procedure:
+
+\begin{itemize}
+  \item Select the necessary marker coordinates and ground reaction load
+    measurments for a single gait cycle from normal walking.
+  \item Compute the inverse dynamics from sensor and actuator measurements.
+  \item Fit a Fourier series to each time series so that the cycle can be
+    perfectly cyclic.
+  \item Form a time series of 500 gait cycles based on the fitted data.
+  \item Add normal Gaussian noise to each sensor and actuator measurement.
+  \item Identify a joint isolated controller from this artificially created
+    data.
+\end{itemize}
+
+The identified gains show no apparent correlation.
+
+\todo{Add a gain plot here showing this. The results are shown here:
+http://nbviewer.ipython.org/github/csu-hmc/gait-control-direct-id-paper/blob/master/notebooks/artificial_variations.ipynb}
+
+\subsection*{Correlations due to inverse dynamics bias gains}
+%
+The measurement noise present in the marker coordinates is propogated into both
+the sensor values (joint angles and angular rates) and the actuator values
+(joint torques) through the inverse kinematic and inverse dynamic computations,
+thus the sensors and actuator measurement estimates are correlated simply by
+this. We exposed this correlation by the following procedure:
+
+\begin{itemize}
+  \item Select the necessary marker coordinates and ground reaction load
+    measurments for a single gait cycle from normal walking.
+  \item Fit a Fourier series to each time series so that the cycle can be
+    perfectly cyclic.
+  \item Form a time series of 500 gait cycles based on the fitted data.
+  \item Add normal Gaussian noise to each measurement.
+  \item Compute the inverse dynamics from the noisy data to produce the sensor
+    and actuator measurements.
+  \item Identify a joint isolated controller from this artificially created
+    data.
+\end{itemize}
+
+The correlation based entirely on the random noise is shown in the example gain
+plot in Figure~\ref{fig:inverse-dynamics=correlation-gains}.
+%
+\begin{figure}
+  \begin{center}
+    \includegraphics{figures/example-inverse-dynamics-correlation-gains.pdf}
+    \caption{Gait phase percent scheduled gains for right (blue) and left (red) legs.}
+    \label{fig:inverse-dynamics=correlation-gains}
+  \end{center}
+\end{figure}
+
+The proportional gains show a clear bias that is strikingly similar to a
+proportionally scaled vertical ground reaction force.
+
+\subsection*{$\mathbf{m}^*$ accounts for most of the variation}
+%
+Ideally the variation in the gait cycles is sufficient to identify the
+acutation due to feedback encapsulated by the gait matrix. If the feedback
+contribution is assumed to be zero, then $\mathbf{m}^*$ can be identified
+alone. If the VAF decreases significantly with $\mathbf{K}(\phi}=0$ then one
+can assume that the feedback terms are not just contributing to overfitting the
+data.
+
 \subsection*{Similar gains are predicted from both unperturbed and perturbed}
 %
 \subsection*{Increasing the number of sensors for each control increases vaf}

src/example_inverse_dynamics_correlation_plot.py

@@ -0,0 +1,46 @@
+#!/usr/bin/env python
+
+
+# standard library
+import os
+
+# external
+from scipy.constants import golden
+import matplotlib.pyplot as plt
+
+# local
+import utils
+
+paths = utils.config_paths()
+
+params = {'backend': 'ps',
+          'axes.labelsize': 8,
+          'axes.titlesize': 8,
+          'font.size': 10,
+          'legend.fontsize': 6,
+          'xtick.labelsize': 6,
+          'ytick.labelsize': 6,
+          'text.usetex': True,
+          'font.family': 'serif',
+          'font.serif': ['Computer Modern'],
+          'figure.figsize': (6.0, 6.0 / golden),
+          }
+
+plt.rcParams.update(params)
+
+trial_number = '019'
+event = 'Artificial Data'
+structure = 'joint isolated'
+
+trial = utils.Trial(trial_number)
+trial.remove_precomputed_data()
+trial.identify_controller(event, structure)
+
+fig, axes = trial.plot_joint_isolated_gains(event, structure)
+plt.tight_layout()
+filename = 'example-inverse-dynamics-correlation-gains.pdf'
+fig.savefig(os.path.join(paths['figures_dir'], filename))
+
+fig, axes = trial.plot_validation(event, structure)
+filename = 'example-inverse-dynamics-correlation-fit.pdf'
+fig.savefig(os.path.join(paths['figures_dir'], filename))