| The problem is that a straight line tangent to a circular arc
has only first order geometric continuity. That is, the curvature
undergoes a discontinuous jump, and this is felt as a sudden
force.
Any curve which joins the circular arc and the straight line which
has 3-rd order geometric continuity could smooth out the jump
in acceleration. A cubic curve in the shape of a spiral is the
minimum order polynomial curve which would accomplish this.
You could also think of it in terms of Fourier analysis - the idea
would be to have no harmonics below a certain order in the path
by "low pass filtering" the path.
Here's a possible way. Suppose you approach a circle of radius
1 from below at a speed of w units per second, meeting it at (1,0)
at t = 0, and thereafter follow the circle with velocity w radians per
second.
then
x(t) = 1, y(t) = wt t < 0
x(t) = cos(wt), y(t) = sin(wt) t >= 0
x'(t) = 0, y'(t) = w t < 0
x'(t) = -w*sin(wt), y'(t) = w*cos(wt) t >= 0
x''(t) = 0, y''(t) = 0 t < 0
x''(t) = -w^2*cos(wt), y''(t) = -w^2*sin(wt) t >= 0
You can see a jump in the second x derivative at t = 0; simply
specify an acceleration that smoothly joins the discontinuity
over some time interval [-tau,0] so that the path still meets the circle.
It could even be a simple linear ramp in x acceleration... upon
integration this would give a cubic curve.
- Jim
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