Is there any need to guarantee the duration of one planned trajectory is an integer times rather than a decimal times of interpolation period Ti

Question

Take double S of velocity profile for example where the seven segments exist. Given the desired displacement s, the constraints jmax,amax, and vmax, after the computation I found that each duration which it takes respectively to reach amax,vmax,s are not the integer times of the period e.g. Ti=2 ms .

This picture is from Trajectory Planning for Automatic Machines and Robots.

I expect this would result in some problem when processed in CPU discretely. This would exceed the amax when round up the Tj1 to integer times of period or not reach the amax when round down. The similar problem also apply to vmax and s. This would not satisfy the accuracy of displacement.

Is there some solution or relevant investigation about this quesiton?

I thought up a solution by assigning the duration which equals the rounded up time as new constraints to resolve for the new jmax after the original computation.

Answer 1

You're mentioning a period without going into any detail about it, so I'm not positive what you're asking about, but I'm guessing your concern is about a computation period - the inverse of the computation frequency. Is this right? That you mean it's possible to overshoot or undershoot a desired position because the time step required to achieve the target position doesn't fall on integer increments of the controller's sample time?

If this is what you're asking for, you're technically correct in that you're almost certainly guaranteed to not hit the position exactly correctly, but you're also describing open-loop control, where you're going to drive the system without regard to positioning error.

You can reduce the effect of the timing mismatch by reducing the effective mismatch, either by reducing the controller period (by increasing the computing frequency) or by reducing the speeds and accelerations.

There are a whole host of issues that are going to come up like this any time you talk about needing precision with open-loop control, for example: your ability to accelerate and decelerate will depend on the motor speed regulator, which in turn will need to be tuned to the exact vehicle weight; motor speed will depend on gearing and wheel radii; motor acceleration will depend on motor torque, which in turn will depend on the exact winding resistance and system voltage; etc.

Your (in)ability to drive the motor at the exact acceleration curve is probably going to have a much larger impact on ultimate accuracy than any timing issues. The only way to achieve the exact position is to use position feedback and build a controller to drive the positioning error to zero.