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\title{First Look at Rider Biomechanics while Controlling a Bicycle}
\author{Jason~K.~Moore}
\author[Jason~K.~Moore]
{
  Reverend Jason~K.~Moore
}
\institute[Davis]
{
  Sports Biomechanics Lab\\
  Mechanical and Aeronautical Engineering Department\\
  University of California, Davis\\
  \&\\
  Bicycle Dynamics Lab\\
  3mE/PME Engineering Dynamics\\
  Delft University of Technology
}
\date{Thursday October 29, 2009}
\begin{document}

\frame{\titlepage}
\frame
{
	\begin{center}
  	\includegraphics[width=10cm]{delftParking.jpg}
  \end{center}
}

\section[Outline]{}

\frame{\tableofcontents}

\section{Introduction}

\subsection{Handling Qualities}
\frame
{
  \frametitle{The Holy Grail}
	\begin{itemize}
		\item<1-> What are we seeking?
		\begin{itemize}
			\item<2-> To be able to predict the \alert{handling qualities} of a bicycle.
		\end{itemize}
		\item<3-> What is a \alert{handling quality}?
		\begin{itemize}
			\item<4-> A measure that determines the ease and precision with which a rider may complete a given task
		\end{itemize}
	\end{itemize}
}
\frame
{
	\frametitle{But why?}
	Isn't the bicycle perfect the way it is?
  \begin{itemize}
  	\item<2-> The bicycle evolved from 200 years of tinkerers.
  	\item<3-> Alternative designs do not have this luxury.
  	\item<4-> Help shed light on many other human/machine interactions.
  \end{itemize}
	\begin{center}
  	\includegraphics[width=8cm]{Bicycle_evolution-en.pdf}
  \end{center}
}
\frame
{
  \frametitle{Handling qualities road map}
      \begin{columns}[t]
  			\column{5cm}
				\begin{overlayarea}{5cm}{4cm}
				\begin{center}
					\alert<1>{Vehicle Dynamics}\\
					$\downarrow$\\
					\alert<2>{Human Biomechanics}\\
					$\downarrow$\\
					\alert<3>{Manual Control}\\
					$\downarrow$\\
					\alert<4>{Handling qualities}
				\end{center}
				\end{overlayarea}
  			\column{6cm}  
  			\begin{overlayarea}{5cm}{5cm}
  			\begin{figure}[]
    			\begin{center}
    				\includegraphics[width=5cm]<1>{NeimarkFufaev1967.png}
    				\includegraphics[width=5cm]<2>{biomechanics.jpg}
    				\includegraphics[width=5cm]<3>{brain-small.jpg}
    				\includegraphics[width=5cm]<4>{nervousDriver.jpg}
  				\end{center}
  			\end{figure}
  			\end{overlayarea}
  \end{columns}
}
\subsection{What do we know?}
\frame
{
  \frametitle{What do we know about bicycle dynamics?}
  We only confidently know some things about bicycle dynamics:
		\begin{itemize}
		\item<2-> \alert{Fact \# 1}: Some bicycles are stable at various speeds.
		\item<3-> \alert{Fact \# 2}: Steering into the lean will stabilize the bicycle. 
		\item<4-> \alert{Fact \# 3}: To initiate a \structure{right turn} (rightward lean) you have to first steer to the \structure{left}!
	\end{itemize}
}
\note
{
So lets talk about a bicycle's dynamics. There are not many things we can claim about bicycle dynamics, but here are a few:
\begin{enumerate}
	\item Some bicycles are stable at various speeds, if perturbed from its equilibrium it will correct itself.
	\item Steering into the lean will stabilize a bicycle. If you fall to the left you must steer to the left to bring the bicycle back upright. This is similar to balancing a broomstick on your hand. You have to move your hand in the direction of the lean to keep the stick upright.
	\item As a consequence you must always initially turn to the left to go to the right and vice versa.
\end{enumerate}
I will show two videos that demonstrate these phenomena.
}
\frame
{
  \frametitle{Whipple Model Description}
  \begin{columns}
  	\column{0.4\textwidth}
  	\begin{itemize}
  		\item<1-> Four rigid bodies
  		\item<2-> Frictionless revolute joints
  		\item<3-> Knife edge, no-slip wheels
  		\item<4-> Uncontrolled
  		\item<5-> 25 parameters      
  	\end{itemize}
  	\column{0.6\textwidth}
  	\begin{figure}[]
  		\centering
			\includegraphics[width=2.5in]{bicycle_autolev.pdf}
  	\end{figure}
  \end{columns}
}
\note{I have derived the non-linear equations of the Whipple Model. It is made up of four rigid bodies connected by frictionless revolute joints at the wheel axles and the steering axis. The wheels roll without slip and theres is no control input, no-hands riding. A non-minimum set of 25 parameters describe the geometry, mass, mass location and mass distribution.}
\frame
{
  \frametitle{Degrees of Freedom}
	\begin{center}
		\begin{overprint}
  		\only<1>{\alert{8 GCs} - 1 HC - 4 NHCs = 3 DoF}
  		\only<2>{8 GCs \alert{- 1 HC} - 4 NHCs = 3 DoF}
  		\only<3>{8 GCs - 1 HC \alert{- 4 NHCs} = 3 DoF}
  		\only<4>{8 GCs - 1 HC - 4 NHCs \alert{= 3 DoF}}
		\end{overprint}
		\begin{overprint}
  		\only<1>{8 generalized coordinates}
  		\only<2>{1 holonomic constraint}
  		\only<3>{4 nonholonomic constraints}
  		\only<4>{3 degrees of freedom: steer, lean, rear wheel rates}
  	\end{overprint}  
  \end{center}    
  \begin{figure}[]
  	\centering
		\includegraphics[width=6cm]{generalized_coordinates.pdf}
  \end{figure}
}
\note{The configuration of the model is fully described by 8 generalized coordinates. The fact that both wheels have to touch the ground introduces a holonomic kinematic loop constraint. 4 non-holonomic constraints are needed due to the velocity being zero in the horizontal plane  at each wheel. The constraints reduce the model to 3 degrees of freedom. We typically choose the steer, lean and rear wheel rates as the independent generalized speeds.}
\frame
{
  \frametitle{Linearization of the Whipple Model}
  \begin{itemize}
  	\item<1-> Linearized about upright constant forward speed configuration.
  \end{itemize}
  \begin{overprint}
  \onslide<2->
	\begin{beamerboxesrounded}[upper=uppercol,lower=lowercol,shadow=true]{Equations of Motion}
		\begin{center}
			\alert{$\mathbf{M}$}
			$\mathbf{\ddot{q}}+v$
			\alert{$\mathbf{C}_1$}
			$\mathbf{\dot{q}}+[g$
			\alert{$\mathbf{K}_0$}
			$+v^2$
			\alert{$\mathbf{K}_2$}
			$]\mathbf{q}=0$
		\end{center}
	\end{beamerboxesrounded}
	\end{overprint}
  \begin{itemize}	
  	\item<3-> Three modes of motion:
  	\begin{itemize}
  		\item \alert{Weave}: Oscillatory, stable at higher speeds.
  		\item \alert{Capsize}: Stable at lower speeds.
  		\item \alert{Caster}: Always stable.
  	\end{itemize}
  	\item<4-> Average bicycle stable speed range: \alert{11 to 18 $\frac{km}{h}$}(7 to 11 mph).     
  \end{itemize}
}
\note{I linearized my model about an upright constant speed condition, thus reducing the model to 4th order with the states being steer and roll and their rates. The linear equations take this canonical form where the forward speed is now a parameter. The coefficient matrices are combinations of the 25 parameters. The damping and spring coefficient matrices are speed dependent. The model predicts three modes of motion: weave, capsize, and caster. It turns out that typical bicycles have a stable speed range.}
\frame
{
  \frametitle{Eigenvalues vs. Speed}
	\begin{figure}[]
		\centering
		\includegraphics[width=8cm]{0kph.pdf}\\
	\end{figure}
}
\note{After calculating the eigenvalues of the 4th order linear model this graph can be plotted. It is a plot of the eigenvalues versus speed. There are 4 eigenvalues and 3 fundamental modes of motion. If the real part of all of the eigenvalues is negative the model is stable, if at least one is positive the model is unstable. I am now going to show 4 animations to demonstrate the modes of motion.}
\frame
{
  \frametitle{Unstable at 0 km/h (0 mph)}
    \begin{columns}[t]
  		\column{6cm}
  			\begin{figure}[]
  				\centering
  				\includegraphics[width=6cm]{0kph.pdf}\\
  			\end{figure}
  		\column{5cm}  
  			\begin{figure}[]
    			\begin{center}
    				\movie[width=5.00cm,height=4.01cm]{\centering\includegraphics[width=5.04cm]{Bicycle3v0p00.jpg}}{Bicycle3v0p00.avi}
  				\end{center}
  			\end{figure}
  \end{columns}
}
\note{The vertical black line shows the corresponding speed. At zero speed there are two positive real eigenvalues that predict an inverted pendulum type of motion. As we would except the bicycle falls over.}
\frame
{
  \frametitle{Unstable 12.6 km/h (7.9 mph)}
    \begin{columns}[t]
  		\column{6cm}
  			\begin{figure}[]
  				\centering
  				\includegraphics[width=6cm]{12p6kph.pdf}\\
  			\end{figure}
  		\column{5cm}  
  			\begin{figure}[]
    			\begin{center}
    				\movie[width=5.00cm,height=4.01cm]{\centering\includegraphics[width=5.04cm]{Bicycle3v3p50.jpg}}{Bicycle3v3p50.avi}
 				\end{center}
  			\end{figure}
  \end{columns}
}
\note{Once the bicycle speeds up the two negative eigenvalues coalesce into an imaginary pair. The motion is now an oscillatory motion we call weave and is unstable. The motion is a combination of steer and roll where the steer lags the roll. The frequency of the oscillation is dictated by the value of the imaginary part of the eigenvalue.}
\frame
{
  \frametitle{Stable at 17.6 km/h (11.0 mph)}
    \begin{columns}[t]
  		\column{6cm}
  			\begin{figure}[]
  				\centering
  				\includegraphics[width=6cm]{17p6kph.pdf}\\
  			\end{figure}
  		\column{5cm}  
  			\begin{figure}[]
    			\begin{center}
    				\movie[width=5cm,height=4.01cm]{\centering\includegraphics[width=5.04cm]{Bicycle3v4p90.jpg}}{Bicycle3v4p90.avi}
  				\end{center}
  			\end{figure}
  \end{columns}
}
\note{If the bicycle is going just a bit faster, the weave motion now becomes stable and all of the eigenvalues are negative. We see that the weave motion is still present but exponentially decays and the bicycle stabilizes back to the upright position. Unfortunately this animation doesn't make that super clear, but if you watch closely you can see the weave motion.}
\frame
{
  \frametitle{Unstable at 22.7 km/h (14.2 mph)}
    \begin{columns}[t]
  		\column{6cm}
  			\begin{figure}[]
  				\centering
  				\includegraphics[width=6cm]{22p7kph.pdf}\\
  			\end{figure}
  		\column{5cm}  
  			\begin{figure}[]
    			\begin{center}
    				\movie[width=5cm,height=4.01cm]{\centering\includegraphics[width=5.04cm]{Bicycle3v6p30.jpg}}{Bicycle3v6p30.avi}
  				\end{center}
  			\end{figure}
  \end{columns}
}
\note{Finally, at high speeds the models predicts instability again. This time the real eigenvalue we call capsize becomes positive. The bicycle slowly spirals over but the eigenvalue is so small, it is very easy to keep under control by a the rider.}
\frame
{
  \frametitle{Yellow Bicycle}
  \begin{center}
    \movie[width=6.4cm,height=4.8cm]{\centering\includegraphics[width=6.4cm]{Yellowbike2.jpg}}{Yellowbike2.avi}
  \end{center}
}
\frame
{
  \frametitle{Gyrobike}
  \begin{center}
    \movie[width=7.2cm,height=4.8cm]{\centering\includegraphics[width=7.2cm]{New_gyrobike.jpg}}{New_gyrobike.avi}
  \end{center}
}
\frame
{
  \frametitle{Countersteering}
  \begin{center}
    \movie[width=7.2cm,height=4.8cm]{\centering\includegraphics[width=7.2cm]{New_countersteer.jpg}}{New_countersteer.avi}
	\end{center}
}
\frame
{
  \frametitle{Vehicle dynamics and handling}
  \begin{itemize}
  	\item<1-> Is stability beneficial?
		\begin{itemize}
			\item<2-> The Wright Flyer was an unstable aircraft
			\item<2-> Fighter jets are unstable without control
			\item<2-> Race cars are typically on the verge of stability
		\end{itemize}
		\item<3-> \alert{Stability does not necessarily equate to ease of control}
		\item<4-> Are the uncontrolled dynamics and indicator of handling?
		\begin{itemize}
			\item<5-> For aircraft, connections have been found
			\item<5-> Unlike a bicycle, the pilot's motion does not affect the aircraft's dynamics
			\item<5-> Pilot and manual control theory have provided more insight
		\end{itemize}
	\end{itemize}
}
\subsection{What we want to know}
\frame
{
  \frametitle{How do we control the bicycle?}
	\begin{center}
  	\movie[width=6cm,height=4.5cm]{\centering\includegraphics[width=6cm]{MacAskill.jpg}}{MacAskill.avi}\\
  	Danny MacAskill, Pro Trials Rider
  \end{center}
}
\frame
{
  \frametitle{How do we control the bicycle?}
	Obvious control input candidates:
		\begin{itemize}
			\item<2-> Steering
			\item<3-> Upper body lean (especially while riding no-hands)
		\end{itemize}
	Not so obvious candidates:
		\begin{itemize}
			\item<4-> Upper body twist
			\item<5-> Arm movement
			\item<6-> Leg movement
			\item<7-> Shifting on the saddle
		\end{itemize}		
	\onslide<8->\alert{First step}: objectively observe and measure what the rider does
}
\section{Instrumented bicycle}
\subsection{The bicycle}
\frame
{
  \frametitle{The bicycle}
  	\begin{figure}[]
  		\centering
  		\includegraphics[width=8cm]{instruBike.pdf}
  	\end{figure}  
}
\subsection{Experiments}
\frame
{
  \frametitle{Experiments}
    \begin{columns}[t]
  		\column{5.5cm}
  		\begin{itemize}
				\item<1-> Around the town ride
				\item<2-> Treadmill tests: pedaling, no pedaling, no-hands, perturbing, lane change
				\item<3-> Measured bicycle dynamics and observed rider
			\end{itemize}
			\begin{figure}[]
  			\centering
  			\includegraphics[width=5cm]{treadmill.jpg}
  		\end{figure}
  		\column{5.5cm}
    	\begin{figure}[]
  			\begin{center}
    			\movie[width=5cm,height=4cm]{\centering\includegraphics[width=5cm]{arendcycling301505.jpg}}{arendcycling301505.avi}
				\end{center}
			\end{figure}
  \end{columns}
}
\subsection{Conclusions}
\frame
{
	\frametitle{Conclusions}
		\begin{itemize}
			\item<1-> No visual signs of upper body lean
			\item<2-> Steering frequency is dominated by pedaling frequency
			\item<3-> Steering amplitude inversely proportional to the speed
			\item<4-> At low speeds the rider exhibited knee motion
		\end{itemize}
		\onslide<5->\alert{But, no easy way to quantify the rider's movements.}
}
\section{Motion Capture}
\subsection{Experiments}
\frame
{
  \frametitle{Experimental Setup}
  \begin{columns}[t]
  	\column{8cm}
  	\begin{itemize}
  		\item Full body motion capture with active markers
			\item Two different bicycles and three adult male riders
			\item Treadmill tests (pedaling, no pedaling, no-hands, tracking) at different speeds
		\end{itemize}
  	\begin{figure}[]
  		\centering
  		\includegraphics[height=2.5cm]{optotrack.jpg}
  		\includegraphics[height=2.5cm]{jodiBikeMarkers.pdf}
  	\end{figure}
  	\column{3cm}
		\begin{figure}[]
  		\centering
  		\includegraphics[width=2.5cm]{jodiMarkers.pdf}
  	\end{figure}
	\end{columns}	

}
\subsection{Data Processing}
\frame
{
  \frametitle{Principal component analysis}
 	$3\textrm{ riders}\times90\frac{\textrm{runs}}{\textrm{rider}}\times560,000\frac{\textrm{data points}}{\textrm{run}}=$ \alert{$150\cdot10^6\textrm{ data points}$}
	\begin{center}\onslide<2->\structure{Principal Component Analysis!}\end{center}
	\begin{itemize}
		\item<3-> Statistical data reduction technique based on an eigenanalysis of data variance
		\item<4-> Used for face recognition, data compression, characterizing human walking
		\item<5-> Largest eigenvalue corresponds to largest variance in motion
	\end{itemize}
}
\begin{frame}
	\frametitle{PCA in a nutshell}
	\begin{displaymath}
		\mathbf{P}=\left[ \begin{array}{ccccc}
		x_1 & \ldots &  x_j & \ldots & x_n \\
		y_1 & \ldots &  y_j & \ldots & y_n
		\end{array} \right]
	\end{displaymath}
	\pause
	Covariance of the mean subtracted data:
	\begin{displaymath}
		\mathbf{C}=\frac{1}{1-n}(\mathbf{P}-\overline{\mathbf{P}})(\mathbf{P}-\overline{\mathbf{P}})^T
	\end{displaymath}
	\pause
  \begin{displaymath}
		\left[\begin{array}{c}
		x_j\\y_j
		\end{array} \right]=\overline{\mathbf{u}}+a_{1j}\mathbf{v}_1+a_{2j}\mathbf{v}_2
  \end{displaymath}
  $\mathbf{v}_1$ and $\mathbf{v}_2$ are the \alert{eigenvectors} of the covariance matrix, $\mathbf{C}$ 
\end{frame}
\note{In the actual experiments we had 93 time series that describe the location of the markers. Here is a simple example showing one point in two dimensions. Imagine a matrix P that maps the coordinates of a single marker in a planar motion through time. A 2 x 2 symmetric covariance matrix is calculated from the mean subtracted data. Then you can represent any time instance of the marker coordinates with this linear relationship where u bar is the average marker position, the a's are time vary coefficients to two vectors v that are the eigenvectors of the covariance matrix. For larger data sets like the one I collected with 31 markers and 3 coordinates each, the PCA is valuable because only the top 10\% or so of components needs to be retained to describe the motion. The rest can be ignored.}
\frame
{
  \frametitle{Graphical User Interface}
  	\begin{figure}[]
  		\centering
  		\includegraphics[width=9cm]{GUI.pdf}
  	\end{figure}  
}
\frame  
{
  \frametitle{Normal bicycling at 15 km/h}
  \begin{columns}[t]
	\column{5.5cm}
  \begin{figure}[]
  	\begin{center}
    	\movie[width=5.5cm,height=4.4cm,showcontrols]{\centering\includegraphics[width=5.5cm]{3054a.jpg}}{New_3054a.avi}
		\end{center}
	\end{figure}
	\column{5.5cm}
	  \begin{figure}[]
  	\begin{center}
    	\movie[width=5.5cm,height=4.81cm,showcontrols]{\centering\includegraphics[width=5.5cm]{jasonNormal15kph.jpg}}{jasonNormal15kph.avi}
		\end{center}
	\end{figure}
 \end{columns}
}
\subsection{Results}
\frame  
{
  \frametitle{Motions and Groups}
  \begin{columns}[t]
  \column{2.93cm}
	\begin{figure}[]
    \begin{center}
			\includegraphics[height=3cm]{rollSteer.pdf}
			\includegraphics[height=3cm]{yaw.pdf}\\
			Steer-Roll-Yaw Group
    \end{center}
	\end{figure}
	\column{5.14cm}
	\begin{figure}[]
    \begin{center}
			\includegraphics[height=3cm]{pedal.pdf}
			\includegraphics[height=3cm]{lean.pdf}
			\includegraphics[height=3cm]{bend.pdf}
			\includegraphics[width=1.5cm]{twist.pdf}\\
			Pedaling Group
    \end{center}
	\end{figure}
	\column{1.47cm}
	\begin{figure}[]
    \begin{center}
			\includegraphics[height=3cm]{knees.pdf}\\
			Lateral Knee Group
    \end{center}
	\end{figure}
	\column{1.47cm}
	\begin{figure}[]
    \begin{center}
			\includegraphics[height=3cm]{bounce.pdf}\\
			Knee Bounce Group
    \end{center}
	\end{figure}
	\end{columns}
}   
\frame  
{
  \frametitle{Group variance vs speed}
  \begin{figure}[]
  	\begin{center}
			\includegraphics[width=8cm]{pedaling4classes.pdf}
		\end{center}
	\end{figure}
}
\frame  
{
  \frametitle{Normal bicycling at 5 km/h and 25 km/h}
  \begin{figure}[]
  	\begin{center}
    	\movie[width=8.00cm,height=4.64cm]{\centering\includegraphics[width=8.00cm]{63vs59.jpg}}{63vs59.avi}\\
    	\structure{5 km/h} and \structure{25 km/h}
		\end{center}
	\end{figure}
}
\frame  
{
  \frametitle{Steer angle comparisons}
  \begin{columns}[t]
  \column{6cm}
  \begin{figure}[]
  	\begin{center}
			\includegraphics[width=6cm]{steerangle.pdf}
		\end{center}
	\end{figure}
	\column{5cm}
		  \begin{figure}[]
  	\begin{center}
			\includegraphics[width=5cm]{steerangleVsSpeed.pdf}
		\end{center}
	\end{figure}
	\end{columns}
}
\section{Conclusions}
\frame  
{
  \frametitle{Conclusions}
\begin{itemize}[<+->]
\item During normal bicycling the dominant \structure{upper body motions}: lean, bend, twist and bounce,
\alert{are all linked} to the \structure{pedaling motion}.
\item We hypothesize that \structure{lateral control} is mainly done \alert{by steering} since we observed only upper body motion in the pedaling frequency.
\item If \structure{upper body motions} are used for control then this control is in the \structure{pedaling frequency}.
\item When pedaling at \structure{low speed} we observe \alert{lateral knee motions} which are probably also used for control.
\end{itemize}
}
\frame  
{
  \frametitle{Conclusions}
  \begin{columns}[t]
  \column{5.5cm}
  Bicycles present a rich complex and robust system to study
	\begin{itemize}[<+->]
		\item \alert{vehicle dynamics} (rolling contacts, variable stability)
		\item \alert{biomechanics} (human stabilization, locomotion)
		\item \alert{human control} (stabilization + manuveuring)
		\item \alert{handling qualities} (perception, psychology)
	\end{itemize}
	\column{5.5cm}
	  \begin{figure}[]
  	\begin{center}
  	\vspace*{-35pt}
    	\movie[width=5.5cm,height=6.27cm,repeat]{\centering\includegraphics[width=5.5cm]{JasonNoHands10kph.jpg}}{JasonNoHands10kph.avi}
		\end{center}
	\end{figure}
	\end{columns}
}
\end{document}
