bounded deviation for straight line motion

Question

I came across the paper(link given below) which discusses about bounded deviation joint path for straight line motion.

Planning and Execution of Straight Line Manipulator Trajectories (RH Taylor) https://pdfs.semanticscholar.org/e01a/58608f4e68f31c7b9e7cdbddceae645727bb.pdf

In this method, the assumption is that the maximum deviation happens at or near the midpoint between the start and end point.

1) Is this assumption true in all cases? 2) Even if the assumption may not be true, will resulting trajectory be a straight line if this method is used for trajectory planning?

I hope someone shed some light on this. Thank you.

Answer 1

1) Is this assumption true in all cases?

I think it is pretty difficult to conclude that the assumption is true in all cases, as it depends on so many factors such as robot geometries. However, as the sampled points are closer to one another, it is reasonable to assume that the maximum deviation happens near the midpoint (as can be seen from Figure 3 in the paper you mentioned).

2) Even if the assumption may not be true, will resulting trajectory be a straight line if this method is used for trajectory planning?

If you are referring to the second method discussed in the paper (bounded deviation joint paths), the resulting trajectory is not a straight line in the Cartesian space and that does not really depend on the validity of the assumption. The assumption is stated just for them to have a reasonable method to select more intermediate points if the particular trajectory segment exceeds the deviation bound. The generated trajectory will (only) follow the straight line closely enough that it does not deviates from the real straight line more than the specified maximum deviation allowed.

Answer 2

I don't know about the algorithms for a lot of industrial manipulators, but I suspect that the two main approaches cited by Taylor: that of velocity control by Whitney, and that of rapid position computational updates by Paul, are still primary approaches taken. Note that the computing power to calculate these inverse solutions is magnitudes higher with today's robots compared with 1979. The drawbacks of Paul's approach using 100ms updates are likely eliminated since we can update more than 10 times as quickly now. The important innovations over the past two decades, in my opinion, are related to incorporating advanced dynamic models into manipulator control. We kind of take for granted the ability to compute positions or velocities much more rapidly than is required for accurate kinematics.