1. Using Parametric Curves to Describe Motions
Motion Curves
Using Parametric Curves to Describe Motions

2. Outline Trajectories vs. Paths (5 minutes)
Important features of trajectories (5 minutes)
Single axis construction(15 minutes)
Multi-axis construction (10 minutes)
Limitations (10 minutes)

3. Trajectories vs. Paths So far the emphasis was on a curve’s shape
A parametric curve however is much more expressive, and can be used to describe a dynamic process, like motion
We call this motion description a trajectory

4. Trajectories vs. Paths
To describe a shape/path we used the entire set of points that are the projection of the domain [a, b] by the curve c(t)
To describe a motion we will treat c(t) as the position at time t

5. Trajectories vs. Paths The difference? Parametrization matters
c(t) in [0, 1] and c(2t) [0, 0.5] are no longer equivalent, they would describe the same path/shape, but different motions
Regularity is no longer assumed (we can stop, we can go back)

6. Important features of trajectories
Minimal domain (minimal time) [0, T]
Explicitly bound derivatives
Edge (start/end) conditions

7. Important features of trajectories
Explicitly bound derivatives
For velocity and acceleration the need is clear
Generally accepted that Jerk (x’’’) limits have benefits (especially when human interaction is involved)
Some works show instances where higher limits reduce position error, and vibrations, but it is still an open question as to the extent higher limits are useful

8. Single axis construction
The expected solution is Bang-zero-bang
highest derivative is maximum, minimum or zero
This result is related to Pontryagin's minimum (or maximum) principle
PONTRYAGIN, Lev Semenovich. Mathematical theory of optimal processes. CRC Press, 1987.‏

9. Single axis construction
As a consequence the solution will be the mth integration of a piecewise constant function

10. Single axis construction
M = 1

11. Single axis construction
M = 2

12. Single axis construction
An observation: the general structure can be described recursively

13. Single axis construction
The structure can also be constructed recursively

14. Single axis construction
The algorithm (extremely shortened version)
Guess we will cruise at a certain velocity
(recursively) Construct a velocity solution from the initial conditions to the cruising velocity (order m-1)
(recursively) Construct a velocity solution from the cruising velocity to the final conditions (order m-1)
Calculate the difference between the distance travelled by the two velocity curves and the required distance (∆) add a cruising phase so that ∆ is eliminated
If the cruising phase has positive time try guessing a higher speed, if it has a negative time guess a lower cruising speed

15. Single axis construction
Calculating ∆ in step 4 is done by polynomial integration
In step 5 we can terminate if the cruising phase is short enough (sufficient accuracy is achieved)
Step 5 is basically a binary search for the best cruising velocity

16. Single axis construction
The terminating step of the recursion is when we get to a low enough m, and we can solve the problem directly.
m=1 is trivial
m=2 can also be solved (quadratic equations)
m=3 has some algorithms
m > 3 other solutions assume start/end at rest

17. Single axis construction
The choice of m depends on the application
Higher m is smoother and easier to follow, lower m is faster

18. Single axis construction
Asymmetric bounds

19. Single axis construction
Non monotonous trajectory

20. Single axis construction
Sample calculation times

21. Multi-axis construction
Multi axis trajectories are constructed by synchronizing multiple single-axis trajectories

22. Multi-axis construction
Synchronization is done by slowing down faster trajectories
This is trivial for trajectories start/end at rest
For trajectories that start and end in motion a modification of the algorithm is needed
We add a parameter Tmax and if a solution is quicker than this we guess a slower speed

23. Multi-axis construction
The (shortened) multi axis algorithm is:
Tmax is set to 0
Solve all single axis trajectories with Tmax
If all trajectories finish at close enough times we’re done, otherwise set Tmax to the slowest time, and go to 2
The domain of valid motion times is not necessarily continuous

24. Multi-axis construction
Sample trajectories (or rather their paths)
T1 = 0.743
T2 = 0.701
T3 = 0.683
T4 = 0.620

25. Multi axis construction
The smoothness of the path and smoothness of the trajectory are not the same

26. Multi-axis construction
A More complex Motion

27. Multi-axis construction
Another robot

28. Multi-axis construction
Trajectories for the last example

29. Limitations
We assumed that ∆ is a monotonous function of the cruising velocity
This is not always true

30. Multi-axis construction
Runtime in general is 2n-1 times the single axis trajectory calculation time (n being the number of axes/dimensions)

31. Limitations We assumed there is a constant velocity phase
not always true even for rest-to-rest motion

32. Demo
Show Demo