# Mechanical Compass Notes

## One Differential

### The Powered Heading Neglector

Also known as the common auto-mobile. A differential allows the vehicle to be driven through a pair of wheels and vary its heading without wheel slip. Although the movements of the wheels contain heading information, the motor sees only distance.

### The Powered Distance Neglector

Imagine an auto-mobile. Now imagine that the shaft between one wheel and the differential is replaced with an inverter (built with three crown gears or a blocked differential). Now the differential allows the vehicle to roll freely forwards and back and, assuming the other wheels are replaced with a castor, the motor drives the heading. We might call the thing an auto-rotile, a vehicle that rotates under its own power.

### The Passive Heading Reflector

The auto-rotile described above seems quite useless, but viewed as a passive device we have a sort of mechanical compass. If the motor is replaced by an counter, the mechanism maps the combined movements of its two wheels to a single heading reference neglecting any distance covered.

If the counter is geared to a pointer, rotating in the horizontal plane, in a carefully chosen ratio, the pointer reflects the heading information so as to negate any change in heading and remain pointing in the same direction as the vehicle is moved about.

Such devices go back at least to the time of Huang Di when the construction of a South Pointing Chariot was documented [www.chinatown-online.co.uk/pages/culture/legends/huangdi.html], but like the wheel the South Pointing Chariot has been reinvented many times. See Harry Siebert's South Pointing Chariots page for a careful discussion of various designs.

Note that the so-called South Pointing Chariot actually indicates relative rather than absolute heading. If it is set pointing South it continues, ideally, to point South so long as the ground is flat and its wheels do not lose contact or slip. Like a compass, the device provides a heading reference. Unlike a compass the reference accumulates error. A large error - introduced, say, by upsetting the chariot - is forever included in its reckoning.

### Driving the Heading Reflector

While drawings of the South Pointing Chariot show a sort of hand cart, it can be turned into an auto-mobile by powering the wheels and adding a castor for balance. The wheels must be powered independently, like a wheel chair - although a South Pointing Chariot with a south pointing seat would make an odd sort of wheel chair indeed [www.exploratorium.edu/exhibit_services/exhibits/s/south_cart.html]. Note that a counter indicates the total change in heading - let's call it a rotometer - whereas the pointer on the South Pointing Chariot does not record the number of complete rotations. A rotometer is very useful in Lego robotics where the number of sensors is limited [ www.convict.lu/Jeunes/NSWEMain.htm].

## Two Differentials

### The Powered Heading Reflector

With a second differential, a single motor can power the cart through the same wheels that measure the heading or two motors can be used to drive distance and heading independently. In Lego robotics, this can be useful for driving in a straight line without feed back (simply switch the heading motor off), but for many manoeuvres this makes less efficient use of the motors.

### The Mechanical Dead Reckoner

As many robot designs feature two, independently powered wheels and are expected to dead reckon their position, the pair of differentials makes a useful preprocessor. Moreover, the device is continuous and so puts off the vexing question of discretization.

### A Cute Design

If two differentials are placed in series between a pair of coaxial wheels and the means (the shaft that is the mean of the other two) connected by an inverter, the shaft between the differentials gives distance and either mean (or the crown of the inverter) gives heading. With a little care the crown of the inverter may be geared to a pointer to obtain a South Pointing Chariot, which I prefer to call a mechanical compass.

## Some Mathematics

Whereas naval dead reckoning typically proceeds from measurements of heading (made with a compass) and distance (derived from speed and elapsed time) to deduce one's position, wheeled vehicles must first transform wheel movements into rotation and distance. We may then transform these into position information, although for some applications total rotation and total distance are enough.

If we work in speeds, rather than positions, the basic relationships are quite simple. Let L and R be the rotational speeds of the left and right wheels. The forward speed of the vehicle s is then the mean of the forward speeds of the two wheels, s = (wL + wR)/2, where w is the wheel radius. The speed of rotation of the vehicle r (change in heading per change in time) is propotional to the difference between the forward speeds of the two wheels and is given by (wL - wR)/2a, where a is the half the axle length. To simplify the presentation, we may choose units and construct the vehicle so as to make w and a equal to 1. the relationship between the four speeds is then described by two equations, 2s = L + R and 2r = L - R. These may transformed into the equations L = s + r (add and divide by 2) and R = s - r (substract and divide by two), which describe the same relationship.

### Infinitesimal Actions - Lie and Otherwise

In studies of continuous actions, Sophus Lie found it useful to consider spaces of infinitesimal actions. When the parameters of the continuous action form a group, so that actions may be undone, the space of infinitesimal actions carries extra structure, known as a Lie bracket, and is now known as a Lie algebra.

Here the movements of the wheels act on the position of the vehicle. The vector space of infinitesimal actions is two dimensional. The pair of differentials sets up an isomorphism between two presentations of this space, one generated by the independent movement of the two wheels and one generated by changes in heading and forward motion.

Do these spaces have a Lie bracket? A Lie bracket requires the difference between A followed by B and B followed by A, where A and B are any two infinitesimal actions, to be given by a single infinitesimal action C. When C exists as a function of A and B it is written [A,B]. Suppose the left wheel moves forward a small amount followed by the right and compare this with the reverse where the right wheel moves first. The difference in final position is a small displacement perpendicular to the wheels, precisely the movement the vehicle can never make. So this system does not have Lie bracket: it is not Lienear. This is not such a terrible word as systems that do have a Lie bracket share some of the convenient properties of linear systems - OK, it is a very terrible word, but what a pity.

### Holonomy

A pair of coaxial wheels is a standard example of a nonholonomic system known as a skate, an object with a heading that, although it may change heading, can only move in the direction of its heading. Skates include ice skates, roller skates and many robots.

A. Eppendahl, 2002 - 2011