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Originally Posted by Ether
Once you know the equation of the cubic connecting the starting point and the destination, you can compute the radius of curvature (and thus the required rate of rotation) at any point along the curve.
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Attached is an example for starting point = (0,0) with a slope of 1/6
and a finish point = (3,9) with a slope of 2.
The blue line is the path and the red line is the reciprocal of the radius at each point along the curve.
If you download and install Maxima, you can play with the endpoints and see the curve that results.
Quote:
Originally Posted by ewhitman
you need a 4th order polynomial (y= ax^3+bx^2+cx+d)
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For the record, that's a third-order polynomial (cubic).
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you can't get a semi-circle
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semi-circle would be a turn of pi. The OP limited the turn to pi/2.
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polynomials can give you weird behavior in some situations... The math is more complex, but I would recommend looking into splines as a more versatile tool for path planning.
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If your starting and ending points are such that a single cubic results in a less-than-desirable curve, you can use intermediate waypoints to get to your destination (i.e. cubic splines). I suspect that in this application for many (most?) cases a single cubic will be satisfactory.