Spline Interpolation in Real-Time Applications using Three Control Points

Abstract

Spline interpolation is widely used in many different applications like computer graphics, animations and robotics. Many of these applications are run in real-time with constraints on computational complexity, thus fueling the need for computational inexpensive, real-time, continuous and loop-free data interpolation techniques. Often Catmull-Rom splines are used, which use four control-points: the two points between which to interpolate as well as the point directly before and the one directly after. If interpolating over time, this last point will ly in the future. However, in real-time applications future values may not be known in advance, meaning that Catmull-Rom splines are not applicable. In this paper we introduce another family of interpolation splines (dubbed Three-Point-Splines) which show the same characteristics as Catmull-Rom, but which use only three control-points, omitting the one "in the future". Therefore they can generate smooth interpolation curves even in applications which do not have knowledge of future points, without the need for more computational complex methods. The generated curves are more rigid than Catmull-Rom, and because of that the Three-Point-Splines will not generate self-intersections within an interpolated curve segment, a property that has to be introduced to Catmull-Rom by careful parameterization. Thus, the Three-Point-Splines allow for greater freedom in parameterization, and can therefore be adapted to the application at hand, e.g. to a requested curvature or limitations on acceleration/deceleration. We will also show a method that allows to change the control-points during an ongoing interpolation, both with Thee-Point-Splines as well as with Catmull-Rom splines.

Task Space Trajectory Planning for Robot Manipulators to Follow 3-D Curved Contours

“…There was no interest in approximation in this study. Consequently, the planned trajectory meets certain requirements [24]. The first requirement was that it should be at least C 1 continuous.…”

“…Several studies have focused on trajectory planning using smooth functions to connect multiple knots [17][18][19][20][21][22][23][24][25][26]. There are two types of trajectory planning, namely, interpolation and approximation.…”

“…The cubic spline in references [23,24] provides a simple method to generate a smooth robot trajectory. Here, the piecewise-cubic trajectory, which passes through the waypoints and satisfies velocity continuity can be determined.…”

“…However, one setback of the cubic spline is its computational limitation because it needs additional processes to compute intermediate velocities in order to find the function of the curve [2,23]. A cubic spline interpolation with three control points was then proposed in reference [24] to overcome the Catmull-Rom spline using four points. They applied the proposed method to limited cases and compared it with the method using the Catmull-Rom.…”

“…To apply to industrial robots, closed-form solutions are required to assign a position, velocity, and acceleration to them at each sampling time. Consequently, previous studies used other methods such as B-spline, cubic spline, trigonometric spline, and Bezier spline to plan a trajectory for multi-points [17][18][19][20][21][22][23][24][25][26]. The spline trajectory planning method can be divided into trajectory planning in the joint space and trajectory planning in the task space (Cartesian coordinate space).…”