4.1 Testing control rules for the SUV

The literature on control engineering is outside the scope of this study, but it has been reported [78, p.214] that orthodox methods find the problem of bicycle control very difficult. It would appear that there have not been any published claims to have discovered supposed rules that humans might use for riding bicycles, nor rules that describe specifically human bicycle riding.

Logically, after deriving a supposed human or human-like control rule for the SUV, it would need testing to assess its success in use. In fact, a program to test control rules was written first, because it followed on naturally from previous pole-and-cart work. Initially, a pole-balancing simulation program written by Michael Bain of the Turing Institute was adapted, with help from him, to simulate the SUV rather than the pole and cart (this was the author's first experience of C).

The present author alone then continued to adapt and modify the program, adding a routine to display a grid of lines as it would appear to a rider of the SUV, so that some visual impression of the performance of the rules was available. Since this was implemented using black and white, on a Sun workstation without any special graphics facilities, the quality of the graphics was rudimentary. An alternative drawing routine plotted the position of the vehicle on an X-Y grid, so that one could see the time development of the path traced out by the vehicle. The source code for this program, written in C, amounted to around three hundred lines.

4.1.1 Description of the operation of the test program

In this program, the SUV is represented by a state vector of seven components:

x	the X-coordinate on the ground
y	the Y-coordinate on the ground
phi	the angle of the direction of motion (yaw)
theta	the angle of inclination from the vertical (roll)
omega	(i.e. d theta / dt) the rate of change of the roll
alpha	the angle of the handlebars
v	the speed along the ground (constant)

The simulation equations, devised by the present author, are not based on an exact analysis of the system, but incorporate a number of linear approximations valid for small angles and small time steps. At each time step Delta t (0.02s) the following calculations are performed:

where s stands for distance travelled along the path taken, t stands for time (their conventional meanings), ks is a constant related to the angle of the front fork and the wheelbase, and += has the same meaning as in C, i.e., add the RHS to the LHS. Various values for the physical parameters were tried out, based on an adult mountain bike and a child's bike. A good set of adult bicycle parameters were:

At each time step, a decision was made about how to change the handlebar angle. As in the case of pole-balancing, this was done in a number of variant ways. The closest in spirit to pole balancing was for control along a straight road. For this arrangement, we define desired quantities in the form y0 (for y), and discrepancies between y and y0 in the form y'. In this case, y0 is the value of y that defines the desired path of the SUV. The calculation then becomes:

where Lwb is the wheelbase length. The discrepancy in the handlebar angle, alpha', then dictates which of five set increments will be added to the current handlebar angle. The constants k1 ... k4 were set by intuitively guided trial and error. Values which gave good results were:

This means in practice that discrepancies in roll rate (omega) are dealt with most quickly, followed by roll (theta), yaw (phi), and position (y).

It may be instructive to compare this strategy with a qualitative strategy for riding in a circle. In these calculations (which, for the sake of clarity, omit some implementation details), r is the distance from the centre of the circle, r0 is the desired value of r, r' is the discrepancy of r from its desired value, r dot is the rate of change of r, and theta omega and alpha are as before.

As before, the discrepancy in the handlebar angle, alpha' dictates which of five set increments will be added to the current handlebar angle.

Again, the constant values were experimented with intuitively. Values of the Q constants that gave good results were:

4.1.2 Results and discussion

The quantitative rules, such as the ones described above for the straight road case, gave very smooth and accurate performance, which, looking at the display, did not look like what one would expect from human control. In contrast, the qualitative rules, such as the ones most recently discussed, give more erratic behaviour. This behaviour is still too regular to look very human, but it certainly looks more like human behaviour than the quantitative rules. We have intuitions about what a natural human performance of bicycle-riding looks like, and this is something that is much less clear for pole-balancing.

The program allowed the initial setting, via command line parameters, of the angle of roll, the speed, and, for some configurations, the desired angle of roll. Setting different initial values of roll ensured that the course followed was not a special case dependent on initial conditions. Variations of speed proved particularly interesting. For the adult bike configuration, the interdependencies were clearly complex, but around 1m/s even the quantitative rule control was inclined to fall over. The child's bike configuration could, on the whole, tolerate lower speeds than the adult bike. Though these results were not explored in great detail, it is reassuring that they are qualitatively similar to the results that one would expect of human control.

Things may be learnt from these tests.

1. Straightforward control strategies were able to balance the simulation. The more qualitative variations on these strategies were able to produce behaviour more like human than the more quantitative ones.
2. This encourages us to look for fairly simple rules characterising human control of a similar system, which may have some similar features.