Robo-Rats Locomotion: Differential

A differential (not to be confused with the differential drive) is a mechanical system of gears that allows force to be divided or combined.  Its typical use is in an automobile drivetrain where it distributes force from the engine to the two drive wheels.  In straight-line driving, force is evenly distributed to the drive wheels, assuming the friction between the wheels and the road is identical (in snow or icy conditions the situation changes, see below for details).  However, when a car is turning the drive wheels must travel different distances--the outside wheel travels farther than the inside wheel.  If a solid axle constrained the drive wheels to turn at identical rates both wheels would slip, causing reduced traction while cornering (and more importantly for Robo-Rats builders, slipping reduces odometry accuracy).  A differential allows the drive wheels to turn at different rates, effectively routing more force to the outer wheel--the wheel with the least resistance gets the most force.  You can think of a differential as a "black box" with three shafts:

The rotational mathematical relationship between the shafts is C = (A + B) / 2, meaning that if C is rotated a full turn, then the amount of rotation experienced by A & B is their sum divided by 2.  In the case of an automobile differential, C is the input shaft from the engine and A & B are the output shafts to the wheels (to be precise, the C shaft is not really a component of the differential itself but I've included it here to simplify the diagraml--for the purposes of discussion assume that there is a 1:1 gear ratio between the C shaft and the differential itself).  The above formula assumes that the rotation direction of A & B is the same as seen from an outside observer, i.e. wheels attached to A & B will rotate in the same direction (from the point of view of the differential, the shafts would be rotating in opposite directions).  The rotation direction of C is somewhat arbitrary, as it is possible to configure it such that either direction will produce the desired rotation of the output wheels, depending on how the shaft is coupled to the differential.  Note that a differential can be used in reverse: if the A & B shafts are rotated, C will be the addition of the rotations at A & B, divided by 2.

Example 1: Assume that wheels are attached to the A & B shafts, and that the rotational resistance of A & B are equal (straight-line motion).  Now if C is rotated by 1 full turn then A & B will rotate by 1 full turn: 1 = (1 + 1) / 2.  In this case, the differential acts like a solid coupling of the A & B shafts.

Example 2: Assume that the A shaft is prevented from rotating and the C shaft is rotated by 1 full turn.  Then the B shaft will rotate 2 full turns: 1 = (0 + 2) / 2.  Note that the amount of torque present at the B shaft will be 1/2 of that in Example 1, since there is an effective 1:2 gear ratio between C and B in this case.  The ability to trade off torque & rotational velocity between the A & B shafts is what makes the differential so useful.

Although the automotive differential solves the problem of force distribution during cornering, it can create other problems. Consider what happens in an automobile if the wheel connected to the B shaft is on ice and the wheel connected to the A shaft is on dry pavement.  Because the friction of the B wheel is zero, all force will be routed to the B wheel, meaning that the wheel on ice will spin and the car will go nowhere.  Because of this problem, some cars are available with a limited slip differential, or LSD.  A limited slip differential limits the maximum amount of force that can be applied to a single wheel.  For example BMW automobiles have 25% LSDs, meaning that at most 75% of the engine force can be routed to one wheel, leaving a minimum of 25% at the other wheel.  This is usually enough to prevent the car from being stuck in the "one wheel on ice" scenario described above.  LSDs usually employ some type of clutch mechanism to limit force distribution.  An enterprising builder has created a Lego limited slip differential.  Note there will be no ice on the Robo-Rats course, so an LSD shouldn't be necessary...

"Real world" differentials are fairly expensive devices (and LSDs are even more so), however there is a nicely done Lego differential available:

The Lego differential has two shafts that correspond to the A & B shafts shown above.  There is no C shaft as shown in the diagram above--force is applied to the differential housing itself (the brown piece in the above photo) which has a set of gear teeth built in at both ends.  The housing is turned by interfacing an external gear to the housing, using either set of gear teeth (only one set can be used at a time, the sets have different numbers of teeth allowing different gear ratios to be used).  As you can see in the photo above, there are three small gears inside the housing.  These gears are not permanently attached to the housing, but held there by the shafts (the black rods in the photo) and by a pin attached to the housing for the central gear.  The gears attached to the shafts are referred to as sun gears and the central gear is called a planet gear (some differentials have two planet gears).  In the case of the Lego differential, the sun & planet gears have the same number of teeth, but this is not usually the case.  There are two different Lego gears that can be used inside the differential housing:

The gear on the left is the original Lego differential gear--it has 14 teeth.  A few years ago, Lego started shipping the gear on the right with its differentials.  This gear has 12 teeth and the teeth are cut in such a way that the gears mesh better than the original gears.  I would suggest that you use the new gears in your differentials--there will be less power lost through friction.  Note that either of these gears can be used as normal beveled gears to route mechanical force at a right angle.

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