Motor Dynamics

Just how good do your motors need to be?

For the sake of this section we will have to make some gross simplifying assumptions. In particular, we shall assume that we can apply constant forces to the mouse and that there are simple, first order frictional effects at work and rolling friction is constant. In practice, motors vary, the available torque is not constant with speed, inertial effects limit how fast we can get motors turning, braking is not the same as accelerating, the floor surface varies quite a lot and gremlins upset everything at random. While these are gross simplifications, they will serve to get some kind of a feel for the problem. You can refine your model to suit your circumstances or you can live with whatever really happens. Like most people, you probably don’t care and will only believe what actually happens when you switch on. If competition is your aim then winning surely is a (the?) goal. In that case, you will need to have some idea of how and where to optimise. For example, is it going to help to put in bigger motors, more batteries, different tyres, software changes…?

Try to remember that many mice don’t even finish and any kind of result may well put you ahead of the rest. At least one UK prize has gone to the only mouse to find the centre while apparently ‘better’ or ‘faster’ mice failed or crashed.

Looking around other pages here, you will see that, for a reasonably competitive mouse, you will need to sustain an average speed, including bends, of about 0.8 m/s or better. Straights, especially long ones could get much faster. Dave Otten’s MITEE mouse III, raced in 1990, boasted a top speed in excess of 3.5 m/s. You probably want to aim for a top speed of at least 1.5 m/s. The control problem starts to get much worse at high speeds. While you can almost certainly keep the wheels turning and drive in a straight line, you need time to respond to your sensors. Speed runs, will use slightly different rules to exploration runs. During exploration, the speed will have to be kept relatively low because you don’t know what is ahead. When the speed run starts, your mouse should know exactly where it is going and can concentrate on not crashing.

The maximum rate of acceleration is constrained by two things – friction between the mouse wheels and the maze floor; and the rotational inertia of the mouse and motor.

Friction is easily dealt with. The mouse exerts a downward force on the maze floor equal to its weight. The weight of the mouse is the result of gravity acting on the mass of the mouse. The coefficient of static friction between mouse wheels and the floor, determines what proportion of that force can be exerted by the wheels to drive the mouse forwards.

Once we have an idea of the coefficient of friction, we can calculate the maximum force available to accelerate the mouse. Typical values for the coefficient of friction are about 0.7 or so giving a maximum acceleration of 0.7g or about 7 m/s/s. Note that the mass of the mouse cancels out in these calculations so that a heavy mouse exerts more downforce but is more massive and so resists acceleration better.

You can immediately see that there is a real advantage in selecting a good tyre if you want sparkling acceleration. Since the maximum accelerating force is a fraction of the downforce, increasing that downforce without increasing mass would be good. We can’t make use of wings like formula 1 cars but at least one mouse has tried using a fan to create low pressure under the mouse.

You can’t go on accelerating for ever. Sooner or later there must come a balance between driving force and friction, at which point your speed will be constant. If you have a motor powerful enough to push you along at a very high speed, you may find you have a problem reliably controlling it at low speed. There may well be a times when you would like your mouse to move at speeds of millimitres per second rather than meters per second. For example during crash recovery or to re-establish some benchmark position within the maze. So, for high positional accuracy, you will want a fast turning motor with a high reduction ratio. For speed, you will want a low ratio. Compromises, ever compromises.

How fast should it go? Here is a table showing the required wheel speed to achieve a given mouse speed for a range of wheel diameters:

Wheel speed (RPM)		Ground Speed (m/s)
		0.5	1.0	1.5	2.0
Wheel diameter	25	380	765	1146	1529
(mm)	35	273	546	819	1092
	45	212	425	637	850
	55	174	347	521	695

The page on gearing has more to say about motor and wheel speeds.