I have a robot which has two phases of motion. They have null space motion and the underactuated motion. The trajectory planning method for each phase is different. But these motions are integrated together in one trajectory by interpolation. Moreover, the null space motion is calculated first, then the underactuated motion is calculated by using the positions, velocities, and accelerations as boundary conditions for the underactuated motion. In these boundary points, how can I solve the problem of small discontinuity in trajectory for the robot as shown in the attached figure?
Usually, these discontinuities are causing a high torque in the small period.
Regards
Alslamo Aleikom Ahmed
Regards to your problem i think you should force a continuity condition on the second order derivative of the required states. My major is flight trajectory not robotics but the idea here is the same. You need a smooth trajectory so you need to give a concern to the higher derivatives. If you need a smooth V put a constraint on V' and V"
yes it is the same idea. But I'm using position, velocity, and acceleration. But still there is a problem. Because the optimized trajectory match the first point in proper way but the second point doesn't match well.
First of all could you specify your dynamic model ODEs and specify your controls. Secondly 2 consecutive phases. The end of first phase is the start of the second one so continuity in states is already forced. Your problem is in mismatched derivatives of states at the connection point. If you enlarged your dynamic model to contain 2 higher derivatives you will get a 2 phases geometrically smoothed and dynamically smoothed
Hi Ahmed
It looks like you need the torque to be limited ( you did not like the high torque) - then it worth to add explicit constraints on the torque or on the corresponding angular acceleration if it ti simpler. As you describe in your answer you get first point matched ( because you probably put the explicit constraint on the position/velocity/acceleration), but the second point -does not match ( because the system is free to whatever and it returns to the trajectory as soon as it can). Adding the constraint should allow to find the second-phase-trajectory that dose not require so high torque to follow.. Alternatively, you may add the constraint used for executing the planned trajectory. In this case you will just reduce the torque the system is allowed to apply - so slow down the return to the planned trajectory.
First of all, I'd like to thanks, all of you. But, It seems there is a misunderstanding because the constraints on the position, the velocity, and the acceleration are implemented in the two points and this why I'm surprising.
My suggestion is first solve the differential equation for the trajectory state. If the value of the state is zero at any value of the excitation. That means there is a discontinuity in the trajectory.
Suppose: Y(t) = mX(t)+C:P Y: trajectory state: X= excitation
you calculate x(t) for which Y(t) is zero.
I think that you give the system commands that it clearly cannot exactly respond to.
I would pass the trajectory through a model that can do the best that you can expect from the system. However, its output would smooth the corners and so, if you pass this realistic command to the real plant, it would only receive commands that the real system can follow.
This was the basic idea of model following.(I just happen to deal with adaptive model following.)
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