TRAJECTORY PLANNING
T. Bajd and M. Mihelj
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory planning
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.
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 so
called trapezoidal
velocity profile.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory between two points
• Symmetric trajectory
• Acceleration phase
and
and
• Constant velocity phase
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory between two points
• End of acceleration phase
• After rearrangement
• Duration of acceleration
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory interpolation between points
• Acceleration phase
• Constant velocity phase
• Deceleration phase
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
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.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
• Via points
• Reached by the robot in time intervals
• Interpolation functions
• Polynomial coefficients to be determined
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
• Velocities in linear segments
• Computation of coefficients
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
• Acceleration in a via point
• Time of motion through the via point
• Computation of coefficients
• Velocity at
at
equals
equals
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
and velocity
Interpolation by use of via points
• Considering
we obtain
• Sum of the above equations leads to
• Difference of the above equations leads to
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
• Considering continuity of position at instant
• and
• leads to
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Interpolation by use of via points
• 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.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Conditions in the initial point
The lighter curve
represents the
trajectory without
correction (position
error can be noticed),
while the darker curve
shows the corrected
trajectory
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory corrections
• Small position changes are assumed in interval
therefore
• and
• leading to
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory corrections
• Duration of acceleration in the first point on the
trajectory equals
• or
• where the initial acceleration equals
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Trajectory corrections
• Similar correction is applied also for the last point
• and
• leading to
• where
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010