This document discusses trajectory planning and generation for robot manipulators. It covers specifying trajectories in joint space or Cartesian space, generating smooth trajectories with continuous velocities and accelerations, and using polynomial curves for interpolation between initial, final, and intermediate points. As an example, it presents a problem to generate a cubic trajectory for the first joint of a 3R robot from 30 to 60 degrees in 3 seconds, passing through 55 degrees at 2 seconds, with given initial, final, and intermediate velocities.

1. Planning of Manipulator Trajectories, Control and Kinematics

2. Trajectory planning and generation
Trajectory : Time history of position, velocity and acceleration of either
actuated joints or the end-effector of the robot
Issues in trajectory planning and generation:
1. Ease and flexibility of planning – how the trajectory can be specified
by the robot operator.
2. Efficient representation of the trajectory in computer.
3. Generation of the desired trajectory at run time.

3. How the trajectory can be specified…
1. Robot operator specify the desired time history of a single or a group
of joints -- Joint space scheme
1. Robot operator simply state the desired time history of the end
effector or the tool of the manipulator – Cartesian space scheme

4. Planned and Generated trajectory:
• Should be a smooth function of time – derivatives w.r.t time, up to a
desired order, must exist and be smooth.
• Robot operator specifies the initial and the final positions of a joint/end
effector.
• sometimes intermediate points, the desired velocities and/ accelerations
also be specified
• The robot control computer and the trajectory planning algorithm should
generate the exact path velocity and acceleration profiles between start,
intermediate and end points

6. Joint space scheme
• Fundamental Task is to find the appropriate curve between the
given initial and final values.
• There are many possible smooth curves.
• Choosing the smooth curve between two points called interpolation
• Restricted with simple polynomial curve, they are sufficient for the
purpose of trajectory planning in robotics manipulators.

15. Problem -2
The first joint of a 3R robot is to rotate from 30degree to 60degree in 3
sec. It is also required that the joint is at 55degree at t=2 sec. The initial
and final angular velocities are +10 deg/sec and -30 deg/sec. The
velocity at the intermediate via point is -10deg/sec. Find the equations
represent the continuity in the velocity and acceleration and draw the
cubic trajectory with via point.