Elmo HARSFEN0602, HARmonica software manual 153, PT Motion What Is PT, Interpolation Mathematics

2T

P(k + 1) − P(k)

T

P(k) − P(k −1)

T

P(k + 1) − P(k −1)

HARSFEN0602ElmoHARmonicaSoftwareManual PRELIMINARYDRAFT

12.1.7PT Motion

12.1.7.1What Is PT

PT stands for Position-Time.

In a PT motion, the user specifies sequence of absolute positions to be visited by the Amplifier with equal time spaces. The time space must be an integer multiple of the Amplifier sampling time. Between the user specified positions, the Amplifier interpolates smooth motion.

The position specifications are absolute.

Interpolation Mathematics

PT implements a 3rd order interpolation between the position data points provided by the user. Let T = m ⋅ Ts

Where:

Ts is the sampling time of the position controller. The parameter WS[29] reads Ts . T is the sampling time of the PT trajectory

m(The system parameter MP[4]) is the integer parameter relating Ts and T.

For the case m 1, no interpolation is required.

For m 1, there=are sampling instances of the position controller for which the path command must be interpolated>. We use a 3rd order polynomial interpolation.

The user provides the position points P(k), k = 1...N .

The Amplifier calculates the speeds V(k), k = 1...N for the points P(k), k = 1...N as follows: If k is an ordinary point inside the path: V(k) =

If k is the first programmed point in the path: V(k) =

If k is the last programmed point in the path: V(k) =

For each motion interval, we have four requirements to satisfy:

Start position.

End position.

Start speed.

•End speed.

These four requirements exactly suffice to solve the interpolating 3rd order interpolating polynomial.

12.1.7.2Example

Consider the position controller reference signal P(t) = sin(2π ⋅10t) where t is the time in seconds. The controller has a position sampling time of 200 microseconds.

The path is programmed with a data point once per MP[4]=50 controller sampling-times (10msec). The set of PT reference points are below depicted in circles.

The path interpolated by the Amplifier is shown as a solid line.

153