Elmo HARSFEN0602, HARmonica software manual 141, Vt = 3at − t0 2 + 2bt − t0 + c

HARSFEN0602ElmoHARmonicaSoftwareManual PRELIMINARYDRAFT

The user provides, for each motion interval, the boundary positions and speeds. Mathematically, the user provides the following data:

The starting position and speed, denoted by P0 and V0, respectively

•The end position and speed, denoted by PT and VT, respectively.

Let t0 denote the starting time, and let T denote the length of the time interval. The position for t ∈[ 0 , 0 + T] is given by the 3rd order interpolating polynomial:

P(t) = a(t − t0 )3 + b(t − t0 )2 + c(t − t0 ) + d

The speed is given by:

V(t) = 3a(t − t0 )2 + 2b(t − t0 ) + c

The four parameters a, b, c, and d are unknown. They can be solved using the four linear equations P(t0 ) = P0 , Namely d P0 .

V(t0 ) = V0 , Namely c =V0

P(t0 + T) = PT , Namely PT = aT3 + bT2 + cT + d

V(t0 + T) = VT , Namely VT = 3aT2 + 2bT + c

141

Example

In this example, we demonstrate how very few points can accurately describe a smooth and long motion path.

Consider two Amplifiers, driven synchronously do draw an ellipse. One Amplifier drives the x-axis, and the other drives the y-axis.

The long axis of the ellipse is 100000 counts long, and the short axis of the ellipse is 50000 counts long. The entire ellipse is to be traveled within 2.2 seconds.

We planned a PVT motion with a fixed inter-point time of 100millisec. 23 points are enough for the entire ellipse, as seen in the figures below. The motion is planned so that the tangential speed is accelerated to a constant rate, and then decelerated back to zero speed at the end of the ellipse. Near the starting point of the ellipse, the speed is slow – and therefore the PVT points, which are equally spaced in time, are more spatially dense there. The continuous line in the figure below depicts the true ellipse and also the ellipse generated by the Amplifier by interpolating the PVT points.

The original ellipse and the Amplifier interpolation of the PVT points are so close, that they can't be resolved on the plot.