Say we want to move an object to a particular point in the shortest amount of time, given a maximum acceleration. It follows that the object must accelerate constantly until it comes to a point where it must decellerate constantly in order to come to a stop at the destination. Looking at this problem we have a known starting velocity, a known final velocity, a known total distance, and a known acceleration. There is a kinematics equation that uses all of those values:

v**2 = v0**2 + 2 * a * d

Unfortunately, that's only one acceleration, but our problem actually consists of two parts: an acceleration and a decelleration. At the end of the object's acceleration it reaches a maximum velocity, the final velocity for that part. The maximum velocity then becomes the starting velocity for the second part. Therefore we have two equations:

v_max**2 = v0**2 + 2 * a * d1

vf**2 = v_max**2 + 2 * -a * d2

This leads to two more unknowns: d1 and d2, the distances travelled during the first and second parts, respectively. However, the total of d1 and d2 is known as the distance from the start to the destination. Therefore, if we solve for d1 and d2 and add the results, it should equal the total distance.

d1 = (v_max**2 - v0**2) / (2 * a)

d2 = (vf**2 - v_max**2) / (2 * -a)

d = d1 + d2

The maximum velocity is still unknown, though, so let's sub in the formulas for d1 and d2:

d = (v_max**2 - v0**2) / (2 * a) + (vf**2 - v_max**2) / (2 * -a)

... simply it:

d = (2 * v_max**2 - v0**2 - vf**2) / (2 * a)

... and solve for the maximum velocity:

v_max = sqrt ((2.0 * a * d + v0**2 + vf**2) / 2.0)

Now, we simply have the object accelerate to that velocity and then decellerate back down to the final velocity. In the end, the object should be at the destination. Of course, now that the maximum velocity is a known value, one can now calculate d1 and use that value to determine how long the object should accelerate.

## Saturday, June 20, 2009

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