Path Control: Linear and Near-Linear Solutions

1. Path Control: Linear and Near-Linear Solutions Slide Set 9: ME 4135 R. Lindeke, PhD

2. Linear Path Control • Sometimes thought of as Cartesian Control • It is based on the idea of Transitions between consecutive required geometries • These transitions are based on the solution of a Drive Matrix: The matrices T(r), Ra(r), and Ro(r) are the translation, rotation wrt Z and rotation wrt Y in transitioning from Pinitial to Pfinal

3. Developing the Drive Matrix: • Given: P1 is (n1, o1, a1, d1) • And: P2 is (n2, o2, a2, d2) • Then: *Derivation is found in Paul’s Reference: pgs 139 - 151

4. Cartesian Control • It is used when very exact interaction is required • It ‘guaranties’ accurate tool placement at all times • It is typically used in time dependent solutions – like interaction while a product is moving

5. Cartesian Control NOTE: On the conveyor, the H-frame is a time dependent pose in C (conveyor space)

6. Cartesian Control • We desire to attach the “Quality Tag” to the part as it moves by the robot station • Requires that the part and robot tool must be in exact contact throughout the attachment process • This becomes a ‘Time-based’ Mapping problem

7. Cartesian Control • At Time 1 (P1): • At Time 2 (P2):

8. Cartesian Control • Using these two (time-dependent) Poses, we can build the desired drive matrix • We can compute the accuracy of the path then as a series of changes to the three control vectors: a, o and d • These are updated in real time

9. Cartesian Control • Problems that can result (and must be accounted for): • Intermediate points that are unreachable – After we compute the initial and final points (that prove to be reachable as individuals), we request the tracking of a, o and d vectors but they exceed joint capabilities or require positions outside the work envelope during the driving action • In certain situations where only certain solutions are possible for the robot, like being near singularities, the desired linear velocity may require very high joint velocities – exceeding capabilities – and the path actually followed will deviate from the one desired as the joints run at their velocity limits

10. Near Cartesian (Joint Interpolated) Control • This is a semi-precise control method developed as a compromise between full-Cartesian and point-to-point motion • Basically it is used when a process needs to be held within a ‘band’ about an ideal linear path – for example during painting or bar-code scanning • The path is designed to ‘track’ the work as it moves and maintains no more than a given “focal distance” separation between the tool and work surface • It is a path that is close to the target path at all poses but exact only at a few!

11. Joint-Interpolated Control • Step 1: determine the desired path • Step 2: Compute the tolerable error and the number of points (‘VIAs’) needed to maintain tool–to–work distances • Step 3:Compute IKS’s at each of the VIAs • Step 4: Determine “Move time” for each segment:

12. Joint-Interpolated Control • Step 5: Divide the Tseg into ‘m’ equal time intervals:

13. Joint-Interpolated Control • Step 6: For each joint, determine angular distance during each time segment tseg: • Step 7: at the beginning of the nth step over a path, joint i servo control receives a target point:

14. Joint-Interpolated Control • Implementing this method begins with determination of the distance between and ultimately the number of Via Points needed • This is (really!) a simple trigonometry problem based on the offset distance and error tolerance () at ‘closest approach’

15. Joint-Interpolated Control -- Model 2 2 1 Note: R = R1 = R2 Robot Base

16. Joint-Interpolated Control -- Model • Notice line #1, #2 and R1 form a right triangle R (= R1) is taken at point of closest approach between the robot and part! #2 is half the distance between Via Points!

17. Lets try one: • A Part 6m long moves by a stationary robot on a conveyor moving at 0.04mps (counter flowing compared to painting direction) If we desire that the robot complete its spraying in 15 seconds • Then, the robot must travel 5.4 m to spray the side of the part nearest it since the part moves during the painting operation. • At closest approach, the robot is 1.5 m from the part and needs to have its sprayer 20 cm ( 5 cm) from the part. • From this data, the R value is: 1.5 - .20 + .05 = 1.35 m

18. Lets try one: • Distance between Via’s is found using: • Therefore: The number of Via’s • = 5.4/1.020 = 5.29 so round up to 6 • plus the initial point = 6 + 1 = 7

19. Follow-up What if we equally space the Via’s? • Distance between is: 5.4/6 = .9m (rather than 1.020m) • What is actual Error band? • Here we see it is: 3.86cm • Typically, we find that Joint Interpolated solutions provide better than required (or expected) process control!