Image 8.32: Ramps with smoothing: Red = actual speed; grey = actual position
8.3.6 Interpolation
If there is no analytical description available for a function, and only individual points
are known instead, it will not be possible to evaluate the function at just any point.
Image 8.33: Known points
ID No.: 0842.26B.5-01 Date: 09.2020
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8 Motion profile
By using a suitable interpolation method, the function’s values between known
points can be estimated. This is termed an interpolation problem. There are a
number of solutions to the problem; the user must select the appropriate functions.
Depending on the functions chosen, a different interpolant is obtained.
Interpolation is a kind of approximation: the function under analysis is precisely
reproduced by the interpolation function at the interpolation points and at the
remaining points is at least approximated. The quality of approximation depends on
the method chosen. In order to estimate it, additional information above the function f
is required. This information is usually obtained naturally even if f is unknown:
boundedness, continuity, and differentiability can frequently be assumed.
8.3.6.1 Linear interpolation
Image 8.34: Linear interpolation
Here two given datum points f
0
and f
1
are connected by a line. To n+1 differing
datum point pairs there is exactly one n-th order interpolation polynomial, which
matches at the specified interpolation points.
