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Cubic Fit for Two Given Positions |
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To begin with, let us consider a single revolute joint's motion between any two specified points i and j. At these two points, the position and velocity are prescribed viz.,  . These could be the initial and final positions and so we may require starting from rest and coming to rest at the final position i.e.  . On the other hand, both these points could be intermediate or via points and so  . |
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The specified  and the tangents  are indicated in Fig.12.2.1. We consider a local time fame ( ,0  T) for the motion between the points i, j and wish to find a smooth curve () satisfying these end conditions indicated by the dotted line in the figure. Since there are four constraints, we can fit a cubic spline. Let, |
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(12.2.1) |
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(12.2.2) |
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Substituting the end conditions, |
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(12.2.3) |
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(12.2.4) |
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(12.2.5) |
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(12.2.6) |
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We can solve these four equations,(12.2.3 to 12.2.6) to get the four coefficients c0 – c3 . Solving, we will obtain the coefficients as, |
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(12.2.7) |
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(12.2.8) |
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(12.2.9) |
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(12.2.10) |
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