scurve trajectory planner

like Aciera said. tilde means the object is in the transformed coordinate system and the numbers of ' show the order of the derivative in respect to curve lengths s. In this case they are from the boundary condition trajectories (lines or arcs) so you get the expression from multiplying equation (26) or (27) with the transformation matrix. Well 26 is easy as 0 stays 0.

Regarding the Frego paper, only the closed form solution needs to start from the straight line, the numerical calculation works for all cases.
However, the paper only gives the recipe how to calculate a 3D clothoid given the starting values. It does not explain how to fit 2 or more of them together to connect 2 points in space with given start and end conditions. So the new paper is the way to go as it also explains this part.

I have been loosely following this for sometime and, I have a sincere question. Would this not be better implemented as a analog NN ?

This problem is difficult to solve because the calculations are based in lots of heavy math and splines where in essence we are attempting to create a plot of points in a 3d cloud. But that's not really optimal in reality.

I wonder could a NN be created to replicate the function of a analog calculator like a Tidal machine that can take the various inputs of a spline, end point, curvature etc and then it would produce via analog calculation the correct output ? Sure seems like it would be a far better solution.

en.wikipedia.org/wiki/Tide-predicting_machine

@Fabian,

Most of the math of Frego is already inside the 2d clothoid library.
There it solves the clothoid G2 problem using 3 connected clothoids.
As later on discussed on github, adding the z component to the xy was proposed by using the Frego 3d paper.

The new paper is for me more understandable.

I have noticed in the new paper, that the 2 gcode segments are transformed to a null plane first.

This same approach i used before when we construct a clothoid when 2 gcode segments share same plane.

Then the 2 gcode segments in 3d space are transformed to xy plane, clothoid is build in xy plane, clothoid + gcode segments are
transformed back to original 3d space position. This worked flawless.

So this part of the paper, we can almost skip. As we know the mathematics for it.

Attached a few files where i tried to implement paper equatations. They may contain wrong approaches.

For now i am stuck at eq.40.

@Collins,
Maybe this is the same as starting with a bspline from segment a to segment b. Then manipulate the spline trajectory
to get a G2 result and apply a clothoid property like
"a curve whose curvature changes linearly with its curve length"

I cannot say anything about your analog predicion machine, as i can't imagine how to appy it to our work.

 

Hi Grotius,

where are you stuck with equation 40? the thetas you already have from the prev equation (39) and the tilded x" y" z" are the transformed derivatives of the starting and end points. If we connect with lines they are all 0 eq. (26). With arcs they are given by eq. (30) + transformation.

@Fabian,

Thanks for your responce.

To me it looks like eq. 40 is all about k, kappa (curvature).

Then x~``traj(s0) for example :
- This is second derivate = curvature, given the ``
- Is already transfomed, given the ~ above the x
- Is for startpoint s0.
- Is for x coordinate.

- It is the curvature kappa value for the transformed startpos?

- I now get it, that a line has no curvature, so value is : 0
- So for a arc, we just put in the curvature value like : "1/Radius" normally.  But they use eq.30 :   -1/r*cos(s/r).....etc.... Why so difficult?



I did a little play around. Nothing seriously.
We can use this little program also to validate other curves.

Setup:
2 red lines, lenght 100mm, are connected by a clothoid at xy ground plane.
The lines have a offset 100mm.
After drawing the clothoid, we know the clothoid total length.

Test:
For the first line we lift the startpoint off the plane, given the z value.
For the second line we lift the endpoint off the plane, given the -z value.
Then both lines are out off plane, they don't share any plane anymore.

Still we are able to project a clothoid on the groundplane.
In the second picture the z value of one end of the line is already z219 and z-219.

On the ground plane i projected the red 2 lines, given the yz plane. (side view)
between the 2 lines is the clothoid lenght for xy. So clothoid is streched out to straight line.
Then another algo growns the xy lenght until the 3d clothoid ends are colinear to the red lines in 3d.