# Simple Degree-of-Freedom Modeling of the Random Fluctuation Arising in Human-Bicycle Balance
## 1 Introduction  
Bicycles provide a useful means of short-distance transportation, and their utilization is expected to contribute to building a healthy and environmentally friendly society [1]. However, at least on Japanese roads, bicycle transportation is not always necessarily safe due to collisions with automobiles. The Japan National Police Agency reported that over 83% of bicycle accidents in Japan in the last ten years have involved automobiles [2]. To avoid such accidents, autonomous vehicle technology will play an important role; if it can predict bicycle motions, the resulting self-driving cars may reduce such accidents. For this purpose, accurate simulation models of bicycle motions are required, and they should be provided as simply as possible for the potential use of electronic control units in self-driving cars.  
Bicycle motions with human riders in traffic seem to be broadly classified into two types: voluntary and involuntary. The former comprises purposeful motions such as right and left turning at a street intersection. Google [3] reported that their sensors can detect a cyclists' hand signals as an indication of an intention to make a turn or shift over. The latter comprises unconscious motions such as human fluctuated balance motions, which have already been found universally in human quiet standing [4; 5; 6], human stick balancing [7; 8], human visuomotor tracking [9; 10; 11], and so on.  
In this study, we propose a simple stochastic model that allows us to simulate the latter type of bicycle motion, i.e., involuntary fluctuated human-bicycle balance motions. To this end, we have conducted an experiment in which each human participant rides a bicycle on bicycle-trainer rollers, allowing it to move without rolling or yawing constraints. During this experiment, we measured the bicycle's rolling motion using a three-dimensional motion sensor attached to it. The measuredtime-series of the rolling motion are characterized by their probability density functions (PDFs). Next, we designed our proposed human-bicycle model as a simple pendulum mechanism controlled by our human controller model, which was successfully used in our previous study [11] to simulate random human fluctuations during a visuomotor tracking task. We then identified the model parameters based on the measured PDFs as training data, using particle swarm optimization (PSO) to minimize the Kolmogorov-Smirnov (KS) distance between the measured PDFs and those simulated by our proposed human-bicycle model. The results show that our proposed model successfully reproduces the measured PDFs with fitnesses of over 98.7%. Furthermore, we conducted a statistical hypothesis test called the KS test [12; 13] on our results to check their stochastic reliability, showing that the measured time series were much better supported by our model than the Gaussian distribution.  
Our stochastic human-modeling approach sharply contrasts with other studies in the fields of autonomous or unmanned bicycle-control systems [14; 15; 16; 17; 18; 19; 20; 21; 22] because their models have been deterministic and not designed to have randomly fluctuating terms. Our approach also differs from Google's study on voluntary bicycle motions, as mentioned above [3]. Although there has been another study on the stochastic modeling of bicycle fluctuated motion [23], it addressed large-scale bicycle running paths, unlike our study, which deal with small fluctuations.  
The rest of the paper is structured as follows: Section 2 describes the experimental test of human-bicycle balance motions. In Section 3, our proposed stochastic model of these motions is presented. In Sections 4 and 5, we describe the method of parameter identification and the identification results are presented with the KS testing results. Section 6 concludes our study.