From e14e79e71fe7f18e75192ae303670aaff51362f3 Mon Sep 17 00:00:00 2001 From: "Jason K. Moore" Date: Mon, 16 Sep 2024 11:34:37 +0200 Subject: [PATCH] Started work on the discussion section. --- main.tex | 161 ++++++++++++++++++++++++++++++++++++------------------- 1 file changed, 106 insertions(+), 55 deletions(-) diff --git a/main.tex b/main.tex index b65f03a..56e38c8 100644 --- a/main.tex +++ b/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 @@ -605,64 +621,89 @@ \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. @@ -670,6 +711,8 @@ \section{Discussion} \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}