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XPS-Q8 Controller
8.2.4
Catmull-Rom Interpolating Splines
To trace a smooth curve that links different predefined trajectory points, the
intermediate points must be calculated following a mathematical model. For the sake of
simplicity, in most cases this is done by a polynomial curve (polynomial interpolation).
For motion systems, the resulting curve should hit all predefined points. This is called
precise interpolation in contrast to approximate interpolation (like Bezier splines),
where the predefined points act only as control points. Within this class of precise
interpolation are:
• Global polynomial interpolation: One polynomial represents the whole trajectory.
Examples are Lagrange polynomials or Newton polynomials.
• Local polynomial interpolation: Each segment that links two consecutive trajectory
points has its own polynomial. The resulting curve is obtained by segment
polynomial concatenation. To limit oscillations inside segments, the polynomial
order is generally limited to 3 or less. This is called spline interpolation. If the
polynomial order is equal to 3, it is called cubic spline interpolation.
The interpolation methods are also classified by the continuity criterion C
interpolating curve has the continuity C
continuous in all its points. The interpolating spline curves generally have C
continuity.
Catmull-Rom splines are a family of local cubic interpolating splines where the
tangent at each point p
the spline. In case of the spline curve tension τ = 1/2 (normal case), the Catmull-Rom
spline is described by the following equation:
Here, p
are the coordinates of the predefined trajectory point in x, y and z (p
i
"u" is the normalized interpolating parameter, varying from 0 (starting at p
(ending at p
).
i+1
Catmull-Rom splines have a C
control and interpolation. Catmull-Rom splines have the advantage of simple
calculation without matrix inversion for on-line calculations, which is a great advantage
for splines with a large number of trajectory points. For this reason, the XPS controller
uses the Catmull-Rom spline interpolation.
8.2.5
Trajectory Elements Arc Length Calculation
Spline contouring at constant speed requires an accurate calculation of the segment's
arc length. The segment's arc length can be expressed as follows:
k
if it and its derivatives up to k-degrees are
is calculated based on the previous p
i
continuity (continuity up to the first derivative), local
1
Figure 31: A Catmull-Rom spline.
97
XPSDocumentation V1.4.x (EDH0301En1060 — 10/17)
Motion Tutorial
k
. An
1
or C
2
and the next point p
on
i-1
i+1
, p
, p
).
xi
yi
zi
) to 1
i
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