Workspace estimation and path planning.pdf
- 1. Basics of Trajectory Planning
- 2. Introduction In the previous topics, we studied the kinematics and dynamics of robots. This means that, using the equations of motion of the robot, we can determine where the robot will be if we have the joint variables or we can determine what the joint variables must be in order to place the robot at a desired position and orientation with a desired velocity. Path and trajectory planning relates to the way a robot is moved from one location to another in a controlled manner.
- 3. Introduction... In trajectory planning, we will study the sequence of movements that must be made to create a controlled movement between motion segments, whether in straight-line motions or sequential motions. Path and trajectory planning requires the use of both kinematics and dynamics of robots. In practice, precise motion requirements are so intensive that approximations are always necessary.
- 4. Path Vs Trajectory ❖ A path is defined as the collection of a sequence of configurations a robot makes to go from one place to another without regard to the timing of these configurations. ❖ A trajectory is related to the timing at which each part of the path must be attained. ❖ Therefore, the points at which the robot may be on a path and on a trajectory at a given time may be different, even if the robot traverses the same points.
- 5. Path Vs Trajectory... In Figure 5.1, if a robot goes from point (configuration) A to point B to point C, the sequence of the configurations between A and B and C constitutes a path.
- 6. Path Vs Trajectory... A trajectory is related to the timing at which each part of the path must be attained. As a result, regardless of when points B and C in Figure 5.1 are reached, the path is the same, whereas depending on how fast each portion of the path is traversed, the trajectory may differ.
- 7. Therefore, the points at which the robot may be on a path and on a trajectory at a given time may be different, even if the robot traverses the same points. On a trajectory, depending on the velocities and accelerations, points B and C may be reached at different times, creating different trajectories.
- 8. Joint-Space Vs Cartesian-Space descriptions Consider a 6-axis robot at a point A in space, which is directed to move to another point B. Using the inverse kinematic equations of the robot, we may calculate the total joint displacements the robot needs to makes to get to the new location. The joint values thus calculated can be used by the controller to drive the robot joints to their new values and, consequently move the robot arm to its new position. The description of the motion to be made by the robot by its joint values is called joint-space description.
- 9. Joint-Space Vs Cartesian-Space descriptions Consider again robot move from A and B in a straight line. Divide the line into small portions as shown in figure and move robot through all intermediate points. We use inverse kinematic equations, and controller is directed to drive the robot to those values. The Cartesian-space description is much more computationally intensive than the joint-space description, but yields a controlled and known path. • Both joint-space and Cartesian-space descriptions are very useful and are used in industry.
- 10. Joint-Space Vs Cartesian-Space descriptions Sequential motions of a robot to follow a straight line
- 11. Joint-Space Vs Cartesian-Space descriptions ❖ Both joint-space and Cartesian-space descriptions are very useful and are used in industry. ❖ Cartesian-space trajectories are very easy to visualize. ❖ Since the trajectories are in the common Cartesian space in which we all operate, it is easy to visualize what the end effector’s trajectory must be. ❖ Cartesian-space trajectories are computationally expensive and require a faster processing time for similar resolution than joint-space trajectories. ❖ Additionally, although it is easy to visualize the trajectory, it is difficult to visually ensure that singularities will not occur.
- 12. Joint-Space Vs Cartesian-Space descriptions Cartesian-space trajectory problems. In (a), the trajectory specified in Cartesian coordinates may force the robot to run into itself; in (b), the trajectory may require a sudden change in the joint angles.
- 14. Basics of Trajectory Planning Case 1 To move the robot from point A to point B. The configuration of the robot at point A is shown, with 𝛼 =20° and 𝜷=30° Suppose it has been calculated that in order for the robot to be at point B, it must be at 𝛼 =40° and 𝜷=80° Also suppose that both joints of the robot can move at the maximum rate of 10 degrees/sec.
- 15. Basics of Trajectory Planning... Case1: Solution One way to move the robot from point A to B is to run both joints at their maximum angular velocities. This means that at the end of the second time interval, the lower link of the robot will have finished its motion, while the upper link continues for another three seconds The trajectory of the end of the robot is also shown. As indicated, the path is irregular, and the distances traveled by the robot’s end are not uniform.
- 16. Basics of Trajectory Planning... Joint-space, non-normalized movements of a 2-DOF robot
- 17. Basics of Trajectory Planning... Case 2: Now suppose the motions of both joints of the robot are normalized such that the joint with smaller motion will move proportionally slower so that both joints will start and stop their motion simultaneously. In this case, both joints move at different speeds, but move continuously together. • 𝛂 changes 4 degrees/second while 𝞫 changes 10 degrees/second.
- 18. Basics of Trajectory Planning... Joint-space, normalized movements of a robot with 2 DOF
- 19. Basics of Trajectory Planning... ❖ Notice that the segments of the movement are much more similar to each other than before, but the path is still irregular (and different from the previous case). ❖ Both of these cases were planned in joint-space as we were only concerned with the values of the joints, not the location of the end of the mechanism. ❖ The only calculation needed was the joint values for the destination and, in the second case, normalization of the joint velocities
- 20. Basics of Trajectory Planning... ❖ Now suppose we want the robot’s hand to follow a known path between points A and B, say, in a straight line. ❖ The simplest solution would be to draw a line between points A and B, divide the line into, say, 5 segments, and solve for necessary angles 𝛼 and 𝜷 at each point. ❖ This is called interpolation between points A and B. ❖ Notice that in this case, the path is a straight line, but the joint angles are not uniformly changing.
- 22. Basics of Trajectory Planning... ❖ Although the resulting motion is a straight (and consequently, known) trajectory, it is necessary to solve for the joint values at each point. ❖ Obviously, many more points must be calculated for better accuracy; with so few segments the robot will not exactly follow the lines at each segment. ❖ This trajectory is in Cartesian-space since all segments of the motion must be calculated based on the information expressed in a Cartesian frame. ❖ In this case, it is assumed that the robot’s actuators are strong enough to provide the large forces necessary to accelerate and decelerate the joints as needed.
- 23. Basics of Trajectory Planning... Assume robot is to go from A to B and then C. Blended path reduces the stress on the robot and requires less energy. Due to blending, the robot will go through a different point B1 and not B (fig a). To exactly through B,then specify different point B’’ (fig b).
- 24. Basics of Trajectory Planning... An alternative scheme for ensuring that the robot will go through a specified point during blending of motion segments. Two via points C and D are picked such that point B will fall on the straight-line section of the segment ensuring that the robot will pass through point B.