T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

TRAJECTORY PLANNING

T. Bajd and M. Mihelj

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

The aim of trajectory planning is to generate the reference inputs to the robot control system, which will ensure that the robot end-effector will follow the desired trajectory.

Trajectory planning

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Trajectory between two points

With industrial manipulators moving between two points we most often select the socalled trapezoidal velocity profile.

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Symmetric trajectory

• Acceleration phase and

and

• Constant velocity phase

Trajectory between two points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• End of acceleration phase

• After rearrangement

• Duration of acceleration

Trajectory between two points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Acceleration phase

• Constant velocity phase

• Deceleration phase

Trajectory interpolation between points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Trajectory consists of a sequence of linear segments describing the movements between two via points and parabolic segments representing the transitions through the via points.

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Via points

• Reached by the robot in time intervals

• Interpolation functions

• Polynomial coefficients to be determined

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Velocities in linear segments

• Computation of coefficients

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Acceleration in a via point

• Time of motion through the via point

• Computation of coefficients

• Velocity at equals and velocity at equals

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Considering we obtain

• Sum of the above equations leads to

• Difference of the above equations leads to

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Considering continuity of position at instant

• and

• leads to

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• In via point the trajectory deviates from the specified via point value by

• The first and the last point on the trajectory are different from other via points and need to be addressed specifically. Namely, the initial and final robot velocity is usually zero and no deviations are allowed.

Interpolation by use of via points

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

The lighter curve represents the trajectory without correction (position error can be noticed), while the darker curve shows the corrected trajectory

Conditions in the initial point

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Small position changes are assumed in interval therefore

• and

• leading to

Trajectory corrections

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Duration of acceleration in the first point on the trajectory equals

• or

• where the initial acceleration equals

Trajectory corrections

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Similar correction is applied also for the last point

• and

• leading to

• where

Trajectory corrections