# Simple Degree-of-Freedom Modeling of the Random Fluctuation Arising in Human-Bicycle Balance
## 2 Human-Bicycle Balance Experiment
### Experimental Data  
During each trial, the time series of the measurement vector,  
\[\mathbf{x}(t)=(x_{1}(t),x_{2}(t))^{T}:=(\theta(t),\dot{\theta}(t))^{T} \tag{1}\]  
(hereafter, \((\cdot)^{T}\) denotes a transpose), was obtained by the motion sensor. Here, \(\theta\) [rad] is the roll angle from the vertical line to the bicycle's vertical axis, and \(\dot{\theta}:=d\theta/dt\) [rad/s] is the corresponding angular velocity. Figure 2 schematically shows the definition of the roll angle in the front view of the bicycle. The bicycle's vertical axis was nominally determined based on the direction of the bicycle frame. The motion sensor was initially calibrated to output \(\theta=0\) when the bicycle's vertical axis was parallel to the vertical line. Therefore, \(\theta=0\) does not indicate the upright equilibrium of the unmanned bicycle. Nevertheless, since the resulting calibration setting was commonly maintained for all participants and trials, the obtained datasets can be compared with each other.  
Throughout the experiment, \(\mathbf{x}(t)\) were stored in the computer in the following form:  
\[\{\mathbf{x}_{hum}^{(s,n)}(t_{0}),\cdots,\mathbf{x}_{hum}^{(s,n)}(t_{i}),\cdots,\mathbf{x}_ {hum}^{(s,n)}(t_{1-1})\},\quad i=0,\cdots,I-1,\ s=1,\cdots,S,\ n=1,\cdots,N, \tag{2}\]  
where \(t_{i}:=i\Delta t\) [s] is a discrete time with a sampling period of \(\Delta t\) [s], \(I\) is the length of the time series, \(s\) and \(S\) are an index and the number of participants, respectively, and \(n\) and \(N\) are an index and the number of trials, respectively.  
For this study, we chose \(\Delta t=10^{-2}\) s and \(I=18,\!001\) to obtain the physical data length \((I-1)\Delta t=180\) s. The number of participants was \(S=8\), and the number of trials undertaken by each participant was \(N=5\).  
Figure 3 shows the measured time series for \((s,n)=(1,1)\), i.e., for the first participant's first trial. The result clearly exhibits fluctuations specific to human balancing motions in which large-amplitude spikes intermittently arise among the moderate-amplitude fluctuation process, which has already been recognized as temporal intermittency in the field of nonlinear physics [6; 7; 8]. Thus, we obtained the time series for all \(s=1,\cdots,S\) and \(n=1,\cdots,N\).  
Figure 2: Schematic front view of the bicycle during the experiment.  
As shown in the upper graph of Figure 3, the vibrational center of \(\theta(t)\) is shifted from the origin, \(\theta=0\). This is mainly because the calibrated \(\theta=0\) does not indicate the vertical equilibrium of the bicycle, as mentioned above. We statistically evaluate these shifts by identifying them as the mean roll angles given by  
\[E^{(s)}[\theta]:=\frac{1}{N\times I}\sum_{n=1}^{N}\sum_{i=0}^{I-1}x_{1}^{(s,n)} (t_{i}),\quad s=1,\cdots,S, \tag{3}\]  
which is the temporal average of the \(s\)th participant's \(\theta(t)\), further averaged over all of his trials. The resulting values are listed in Table 1 with standard deviations given by  
\[SD^{(s)}[\theta]=\sqrt{V^{(s)}[\theta]},\quad V^{(s)}[\theta]:=\frac{1}{N \times I+1}\sum_{n=1}^{N}\sum_{i=0}^{I-1}\left(x_{1}^{(s,n)}(t_{i})-E^{(s)}[ \theta]\right)^{2},\quad s=1,\cdots,S. \tag{4}\]  
Viewing the average over all participants' values, \(E^{(s)}[\theta]\) takes about \(1.75\times 10^{-2}\), indicating that the actual equilibrium angle of the bicycle was approximately at \(\theta\approx 1.75\times 10^{-2}\) rad (or \(1.01^{\circ}\)) in our setup. In addition, each \(E^{(s)}[\theta]\) value is slightly different due to the respective riding forms of the participants.