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the geometry of perfect parking
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The Geometry of Perfect Parking by Simon R. Blackburn

Blackburn's formula does this by sketching the arc of your car's turning capability into a full circle, then using the center of the circle to create the right-angle triangles Pythagoras loved.

That's a lot of work just to tell you if you have enough space for an easy park. And it doesn't tell you how to do the parking. That's something you have to learn by doing, which is how most people figure out whether they have enough space to park in the first place. Devlin says that behind all that guessing, math is at work.

"Mathematics gives you a way of understanding in detail what people have learned to do simply by practice and expertise," he says.

"In fact, when we practice something, be it on the athletic field or in an automobile, we are becoming very good mathematicians at doing a particular kind of operation," Devlin says. "But usually we don't call it mathematics — and we certainly don't give people a pass on the math test because they can park their car."

How to Park Perfectly

 

 

 

Parallel Steering

Figure 7 Basic configuration used for parallel steering

Figure 8
Green = Inner radius
Blue = Outer radius
Red = Steering angle

A car-like steering configuration in which the wheels used for turning a vehicle or robot (typically the front wheels) are kept in parallel by a connecting rod (called a "tie rod") regardless of the steering angle. Due to its relative simplicity, it is often used for robots that have a car-like wheel pattern.


As long as the robot is traveling in a straight line there are no drawbacks to this configuration. However, as can be seen in figure 8, when turning the wheels are each traversing a different circumference. The wheel on the inside of the turn follows a path with a tighter radius than does the wheel on the outside of the turn.


This results in:

· both wheels experiencing an increase in friction due to having to follow a path to which they are not properly aligned

· an increase in energy being required to make the turn

· extra stress being placed upon the wheels and motors

· excessive ware on the surface of the wheels.


Fortunately there is a relatively simple way to deal with this problem, which is described in the next section.

 

Ackerman Steering

Figure 9 Basic Ackermann steering configuration

Figure 10
Green = Inner radius
Blue = Outer radius
Red = Steering angle

Ackermann steering (named for its inventor Rudolph Ackermann) solves an inherent problem with parallel steering. That problem being undesirable friction and stress being placed upon the wheels during a turn. This is a result of the fact that the wheel on the inside of the turn is traversing a tighter radius than the wheel on the outside of the turn. Unfortunately a parallel steering configuration does not properly aligned the wheels to the curve they need to follow.

As show in figure 9, a simple approximation to perfect Ackermann steering geometry may be generated by angling the steering arms inward so that the linkage pivot points lie on a line drawn between the steering kingpins and the center of the rear axle. The linkage pivot points are joined by a rigid bar called the “tie rod”.

As shown in figure 10, using this type of configuration causes the wheel on the inside of the turn to be angled more acutely than the wheel on the outside of the turn. A perfect Ackermann angle will insure that at any steering angle, both wheels will be properly aligned to trace out the necessary radius on each side of the turn. This of course results in a minimum of friction and stress on the wheels.

This has only been a cursory description of the Ackermann steering geometry. It should be pointed out that sometimes there are reasons why a perfect Ackermann angle may not be wanted. Instead what is called a positive Ackermann, or a negative Ackermann angle may be more desirable. It is recommended that the reader check out some of the external references listed below to learn more about these alternative configurations.