Started work on the discussion section.

main.tex

@@ -4,6 +4,8 @@
 \usepackage{booktabs}  % nice tables
 \usepackage[margin=25mm]{geometry}
 \usepackage[natbib=true,style=authoryear]{biblatex}
+% NOTE : this file is automatically generated from Zotero, do not edit
+% manually!
 \addbibresource{references.bib}
 \usepackage{siunitx}  % use for all units
 \usepackage{subcaption}  % for subfigures
@@ -195,6 +197,7 @@ \subsection{Balance Assist Control}
 %
 \begin{align}
   T^\textrm{m}_\delta = -k_{\dot{\phi}}\dot{\phi} = g(v_{stable} - v)\dot{\phi}
+  \label{eq:implemented-controller}
 \end{align}
 %
 where \(v_{\textrm{stable}} = 4.7~\si{\meter\per\second}\) is approximately the
@@ -551,9 +554,10 @@ \section{Results}
 threshold, the probability of falling is significantly lowered with the balance
 assist system on. The skewedness of the probaility curves arrives from the
 interaction effects.\todo{check this statement about skewedness this and think
-about it} Figure~\ref{fig:probablity-10kph} shows the same result for the
-10~\kph trials which has a similar trend of reducing the probability to fall
-with the balance assist system turned on, but the effect is not significant.
+about it} Figure~\ref{fig:probability-10kph} shows the same result for the
+10~\si{\kph} trials which has a similar trend of reducing the probability to
+fall with the balance assist system turned on, but the effect is not
+significant.
 %
 \begin{figure}
   \centering
@@ -577,6 +581,18 @@ \section{Results}
 
 \section{Discussion}
 %
+We have shown that at 6~\si{\kph} the addition of balance assist control
+reduces the chance that a rider will fall when perturbed around the limits of their
+control authority. But this effect diminishes at the higher speed
+scenario of 10~\si{\kph}. We were only able to test these two speed-gain
+scenarios for mostly homogeneous sets of riders within the resources of this
+research project, but additional experimental work could help understand more
+completely the range and limits of the positive effect of the balance system.
+For example, it is possible that simply increasing the controller gain at
+10~\si{\kph} also results in a significant positive effect.
+
+\subsection{Interpretation of the Results}
+%
 The probability that a fall occurs depends on the values of all the independent
 variables in Table~\ref{tab:stat-model-variables}, but we can visualize the
 effect of one or two variables (e.g. Figure~\ref{fig:probability}) to gain
@@ -585,12 +601,12 @@ \section{Discussion}
 \ref{tab:freq-coefs-10} it is important to understand the relationship between
 probability and odds.
 The estimate in Table~\ref{tab:freq-coefs-6} shows that the balance assist
-system halves the odds that a perturbation results in a fall
-(MCiO=0.53)~\todo{Make a variable for MCiO}.
+system halves the odds that a perturbation results in a fall:
+\(e^{\alpha_k}=0.53\).
 This means if the odds are a 1000:1, turning on the balance-assist system
 reduces the odds to 500:1.
-However, the probability that a fall occurs is only reduced from $0.999$ to
-$0.998$ in that case.
+However, in that case the probability that a fall occurs is only reduced from
+$0.999$ to $0.998$.
 If the odds that a fall will occur are smaller, halving the odds has a larger
 influence on the fall probability.
 For example, if the odds that a fall occurs is two, halving it to one reduces
@@ -605,71 +621,98 @@ \section{Discussion}
 the subject's personal threshold between falling or recovery and that large
 perturbations will make you fall regardless of the balance assist's help.
 
-\todo{Consider making a specific value of calcualting the probabilty for a set
-of inputs to the regression model.}
-
-To illustrate the effect of the balance-assist system on fall probability, we will
-give an example of how the data collected during the experiments is used to predict
-fall probability. We use \ref{eq:log-regress} and the coefficients in 
-\ref{tab:freq-coefs-6}. For simplicities sake, the interaction effects are not included.
-Let's assume that the mean angular impulse $\bar{L}$ of all the perturbations applied to 
-a participant is 100~\si{\newton}, and the standard deviation $\sigma^{L}=15$. The centred
-and scaled angular impulse can be calculated by substracting $\bar{L}$ from the applied
-angular impulse $L$, and dividing this by $\sigma^{L}$. The same applies for the 
-perturbation order $c$, initial roll angle $\phi_0$ and initial steer angle $\delta_0$.
-If we take the coefficients estimated for cycling at 6~\si{\kilo\meter\per\hour}, the
-log-odds of falling can be calculated as follows.
-
+To illustrate the effect of the balance-assist system on fall probability, we
+will give an example of how the data collected during the experiments is used
+to predict fall probability. We use \ref{eq:log-regress} and the coefficients
+in \ref{tab:freq-coefs-6}. For simplicities sake, the interaction effects are
+not included.  Let's assume that the mean angular impulse $\bar{L}$ of all the
+perturbations applied to a participant is 100~\si{\newton}, and the standard
+deviation $\sigma^{L}=15$. The centred and scaled angular impulse can be
+calculated by subtracting $\bar{L}$ from the applied angular impulse $L$, and
+dividing this by $\sigma^{L}$. The same applies for the perturbation order $j$,
+initial roll angle $\phi_0$, and initial steer angle $\delta_0$.  If we take
+the coefficients estimated for cycling at 6~\si{\kph}, the log-odds of falling
+can be calculated as follows.
+%
 \begin{align}
-	\log \left( \frac{p_{ij}}{1-p_{ij}} \right) & = \beta + \sum_{k}^{k=0}\alpha_k 
-	\frac{x_{ij}^{k}-\bar{x_{ij}^{k}}} {\sigma^{x^{k}}} 
-	& = -0.29 + 1.69\cdot\frac{110~\si{\newton} - 115\si{\newton}}{15\si{\newton}} 
-	-0.77\cdot\frac{10-20}{11.54} 
-	-0.25\cdot\frac{-6\si{\degree}-2\si{\degree}}{10\si{\degree}} 
-	-0.14\cdot\frac{1\si{\degree}+3\si{\degree}}{5\si{\degree}}
-	-0.64\cdot s
-	& = 1.42 -0.64 \cdot s
+  \log \left( \frac{p_{ij}}{1-p_{ij}} \right)
+  =
+  \beta + \sum_{k}^{k=0}\alpha_k \frac{x_{ij}^{k}-\bar{x_{ij}^{k}}}{\sigma^{x^{k}}}
+  =
+  & -0.29 + 1.69\cdot\frac{110~\si{\newton} - 115\si{\newton}}{15\si{\newton}}
+    -0.77\cdot\frac{10-20}{11.54}  \\
+  & -0.25\cdot\frac{-6\si{\degree}-2\si{\degree}}{10\si{\degree}}
+    -0.14\cdot\frac{1\si{\degree}+3\si{\degree}}{5\si{\degree}} -0.64\cdot s \\
+  =
+  & 1.42 -0.64 \cdot s
 \end{align}
 
-The state of the balance-assist $s$ is a binary variable. If the balance-assist is turned on, 
-the log-odds that a fall occurs are decreased by 0.64. The odds and probability can be calculated:
+The state of the balance-assist $s$ is a binary variable. If the balance-assist
+is turned on, the log-odds that a fall occurs are decreased by 0.64. The odds
+and probability can be calculated:
 
 \begin{align}
-	\frac{p_{ij}}{1-p_{ij}} = e^{1.42-0.64s} = e^{1.42}\cdot e^{-0.64s} = 4.14 \cdot 0.53s
+  \frac{p_{ij}}{1-p_{ij}} = e^{1.42-0.64s} = e^{1.42}\cdot e^{-0.64s} = 4.14 \cdot 0.53s
 \end{align}
 
 \begin{align}
-	p_{ij}^{s=0} = \frac{4.14}{1 + 4.14} = 0.81
+  p_{ij}^{s=0} = \frac{4.14}{1 + 4.14} = 0.81
 \end{align}
 
 \begin{align}
-	p_{ij}^{s=1} = \frac{4.14\cdot0.53}{1 + 4.14\cdot{0.53}]} = 0.69
+  p_{ij}^{s=1} = \frac{4.14\cdot0.53}{1 + 4.14\cdot{0.53}]} = 0.69
 \end{align}
 
-Turning on the balance-assist system reduces the probability that the perturbation results in a 
-fall from 0.81 to 0.69.
+Turning on the balance-assist system reduces the probability that the
+perturbation results in a fall from 0.81 to 0.69.
 
+\subsection{Stability and Human Controlled Plant Dynamics}
+%
+The linear Carvallo-Whipple model indicates that the steer controller
+stabilizes the bicycle-rider system, but this model assumes the rider's hands
+are not connected to the handlebars and that they clamp their body as rigidly
+as possible to the rear frame. In reality, the system's behavior is likely more
+akin to a marginally stable or an easily controllable unstable system due to
+the various un-modeled effects. Our system may not result in a definitely
+stable system, i.e. cannot fall, but having plant eigenvalues with very small
+unstable eigenvalue real parts correlates to ease of control~\citep{Hess2012}.
+
+The controller design we utilize,
+Equation~\ref{eq:implemented-controller}, also increases the weave mode
+frequency by a factor of about three up to about 1~\si{\hertz}. This bandwidth
+is still controllable by the human's neuromuscular system, but may feel
+unnatural as it is more akin to what the steering would feel like at in the
+\SIrange{30}{40}{\kph} speed range. \Citet{Hanakam2023} reported
+dissatisfaction in subjective rider feeling on their similar bicycle to ours
+and this effect to the human-controlled plant dynamics could be connected to
+this.
+
+\subsection{Treadmill Width}
+%
 Angular impulse magnitude has the largest significant effect for predicting
-fall probability, Tables~\ref{tab:freq-coefs-6} and \ref{tab:freq-coefs-10}.
-An increase in angular impulse increases the fall probability both at 6 and
-10~\si{\kph}. At 10~\si{\kph}, the multiplicative change in odds is
-approximately twice as big as at 6~\si{\kph}. Thus, angular impulse is a more
-important predictor at higher speeds compared to lower speeds. The reason for
-this likely has to do with the width of the treadmill and may also be why
-balance assist system did not have a statistically significant effect at
-10~\si{\kph}. As a bicycle travels at higher speeds, the same perturbation
-causes larger lateral deviations. At 10~\si{\kph} almost all falls were due to
-the bicycle exiting the maximum width of the treadmill. If the same experiment
-was performed on an infinite plane, the riders may have recovered from more
-perturbations. At 6~\si{\kph} the riders could often recover in the allotted
-treadmill width. Our results are very much dependent on the two modes of
+fall probability as seen in both Tables~\ref{tab:freq-coefs-6} and
+\ref{tab:freq-coefs-10}. An increase in angular impulse increases the fall
+probability both at 6 and 10~\si{\kph}. At 10~\si{\kph}, the multiplicative
+change in odds is approximately twice as big as at 6~\si{\kph}. Thus, angular
+impulse is a more important predictor at higher speeds compared to lower
+speeds. We posit that this likely has to do with the width of the treadmill and
+that this could also be why balance assist system did not have a statistically
+significant effect at 10~\si{\kph}. As a bicycle travels at higher speeds, the
+same perturbation magnitude causes larger lateral deviations. At 10~\si{\kph}
+almost all falls were due to the bicycle exiting the maximum width of the
+treadmill. If the same experiment was performed on an infinitely wide plane,
+the riders may have recovered from more perturbations. At 6~\si{\kph} the
+riders could often recover in the allotted treadmill width due to the smaller
+lateral deviations. Our results are very much dependent on the two modes of
 falling with use: exit the treadmill width or foot is placed on the belt. Cycle
 paths are a similar width as the treadmill, so rider's are often limited in
 width when recovering from a fall.
 
 \todo[inline]{Could show an impulse response in lateral deviation for different
 speeds.}
 
+\subsection{Learning Effect}
+%
 An increase in the number of perturbation that a participant already
 experienced, decreases the probability that a fall occurs. This is likely due
 to the participants learning how to better recover from the perturbation over
@@ -678,6 +721,8 @@ \section{Discussion}
 learning effect that occurs during the experiment is not strongly dependent on
 the speed.
 
+\subsection{Non-significant Predictors}
+
 Roll and steer angle are not a significant predictor of fall probability,
 neither at 6 or at 10~\si{\kph}. We expected this to have an effect. If you are
 in a rolled and steered state that is far from the upright equilibrium, then a
@@ -689,16 +734,22 @@ \section{Discussion}
 the effect that the roll angle, steer angle, angular impulse, and perturbation
 order have on the probability that a fall will occur.
 
-Difficult to layer this onto the fall statistics, because we don't have the
-details of how people fall. If such data existed we could make estimates on the
-number of falls reduced if everyone road such a bike.
+\subsection{Extrapolation to Natural Falls}
+%
+The positive effect of the balance assist system is coupled to the assumptions
+and experimental scenarios we implemented and there is unfortunately no simple
+way to extrapolate our results to reductions of single-actor crashes we may see
+if such a system were deployed widely to bicyclists. Although, our results do
+indicate that we would see such a reduction, even if only in a class of
+single-actor crashes that most resemble our experimental design. If there were
+more comprehensive and detailed natural data of how people fall we could make
+estimates on the number of falls reduced if everyone road a balance assist
+bicycle.
 
 \section{Conclusion}
 %
 TODO
 
-% NOTE : this file is automatically generated from Zotero, do not edit
-% manually!
 \printbibliography
 
 \end{document}