# Interpolated Motion

## For many positioning system applications, simple point-to-point moves or constant velocity motion are sufficient. In other applications, more sophisticated movement is required.

Interpolated motion is called for when the path that an object takes through space is important. For example, when a simple point-to-point system moves two axes at any angle other than 45°, the path will be as shown (Figure 18a), with one axis finishing before the other. One simple improvement that simulates interpolated motion is to very finely set the velocity on each axis, so as to have both axes start and stop at about the same time. More sophisticated motion controllers, which offer linear interpolation, can move at an arbitrary angle, and yet have the point-to-point path be a straight line (Figure 18b). More general examples of interpolated motion include circular interpolation, continuous path motion along arbitrary combinations of linear and arc moves (Figure 18c), and smooth, cubic spline motion through a series of points (Figure 18d).

One common application requires that a third theta axis be maintained tangent to the path, as the X and Y axes follow an arbitrary curved path in the X-Y plane. An example of an interpolated motion system with a tangent theta axis would be guiding a knife through material that is being cut; the blade must always face directly into the motion. Another computational problem we encounter consists of three linear axes with bearings at one end, pushing a planar payload in X, Y, and Theta axes. In this case, retaining a fixed (but virtual) “axis of rotation” as the system moves in the X and Y axes requires extensive trigonometric transforms. A similar case involves the need to perform arbitrary Z axis motion with tip and tilt motions using three linear Z axis stages with gimbals supporting a single payload. Another common problem is to take a Linear – Theta stage and use it to perform X-Y motion. Higher order interpolations, of more limited use, include helical, spherical, and elliptical interpolation.

###### Figure 18a – Point-to-Point

###### Figure 18b – Linear Interpolation

###### Figure 18c – Circular Interpolation

###### Figure 18d – Cubic Spline Interpolation

#### Motion Control Handbook

- Abbé Error
- Accuracy in Positioning Systems
- Constant Velocity
- Cosine Error
- Full Coil vs. Half Coil
- High Vacuum Positioning Tables
- Interferometer Feedback Systems
- Interpolated Motion
- Lead Screws and Ball Screws
- Precision Linear Motors
- Limit Sensors
- Limitations of Piezos
- Low Magnetic Field Tables
- Linear Positioning Accuracy
- Mapping
- Microstepping
- Midrange Resonance
- Motion Calculations
- Mounting Issues
- Move and Settle Time
- Positioning Systems Overview
- Repeatability
- Resolution
- Rotary Motor Mount
- Servo Motors
- Slow Down to Speed Up
- Stepper Motor
- Thermal Expansion
- Torque and Force Requirements
- Units of Measure
- Vibration Isolation Systems