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Very good explained - thank you - I've bookmarked your post. Perhaps it could be also for me useful sometime in future.
In fact I enjoyd this discussion very much.
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Yes, I know it very well ... but also (from another Forum where I'm very active) I knew how useful help (and also a discussion) can be.
Mostly it is not the first idea, which guides to the goal, but most often the discussion does ...
Along these lines ...
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You should edit/Update your question with actual code and explain what is actual problem.
So that everyone can see it without digging in discussion.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Why not use some sort of cubic spline interpolation? This gives you smooth acceleration, as well as smooth velocity.
In your case, you would have "control points" (or knots) consisting of points xi through which the robot arm must pass at times ti. For three-dimensional movement, you would have a vector Xi, and a separate interpolation for each ordinate.
This gives you a useful way of describing the constraints:
- For N "control points", you have N-1 cubic functions.
- Each function is of the form f(x) = ax^3 + bx^2 +cx + d
- The 1st derivative of each function is the velocity f'(x) = 3ax^2 + 2bx + c, and it will be continuous (no abrupt velocity changes)
- The 2nd derivative of each function is the acceleration f''(x) = 6ax + 2bx, and it will be continuous (no jerks)
- f0(t0) = x0
- f0'(t0) = v0
- fN-1(tN) = xN
- fN-1'(tN) = vN
Calculating the spline is a fairly straightforward operation.
If you have an important point to make, don't try to be subtle or clever. Use a pile driver. Hit the point once. Then come back and hit it again. Then hit it a third time - a tremendous whack.
--Winston Churchill
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Hi Daniel,
Thank you for your suggestion.
A Spline is perfect to join movements and, as you say it is a good option.
In fact I would like to have a controlled acceleration and having all the job done till now I would like to keep it. This is the last week I spend with that... if it is not working, then I'll have to switch to other methods and the spline would be an option.
Thank you for your comment! 
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Working now! thank you for your comment Daniel, I've updated the original post to explain the solution.
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Glad I could help.
If you have an important point to make, don't try to be subtle or clever. Use a pile driver. Hit the point once. Then come back and hit it again. Then hit it a third time - a tremendous whack.
--Winston Churchill
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Joan M wrote: I've tried to make my own which works perfectly when the values allow the motor to reach the maximum speed "plateau" but it is not working well when is not possible to achieve this speed
First calc if motor will reach plateau or not.
if not reach the plateau:
do shorter acceleration and deceleration in order to create a plateau at lower speed.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Hi ppolymorphe,
Thank you for your comment.
What you suggest changes the acceleration... I would like to change the speed and find the lowest permissible speed without changing the acceleration.
The idea is to keep the same dynamics for all the movements.
Doing what you suggest would end with a near to infinite acceleration in really short movements... reducing the speed would keep movements soft and get the same result...
But I've not been able to understand why I can't find it directly, if you take a look at the code snippet I've pasted into the thread with Ralf you will see that at the end, after a first approximation, I'm searching for a suitable speed decreasing it by one (terrible).
Thank you for your post.
Joan.
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Joan M wrote: What you suggest changes the acceleration... I would like to change the speed and find the lowest permissible speed without changing the acceleration.
I suggest to change the acceleration duration, not the rate.
In a car, if it takes 60 secondes to accelerate from 0 to 100km/h, then plateau and decelerate.
it will take 15 secondes to reach 25km/h, 30 secondes to reach 50km/h, 45 secondes to reach 75km/h.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Of course, Patrice, you are right ... because the acceleration (or the deceleration) represents the angle of a right-angled triangle.
The Trapezoid consists of 3 parts : a triangle for the acceleration, a rectangle and finally a triangle for the deceleration.
There are 2 Problems in this discussion :
1st : the distance between both points don't allow to reach the max. speed with the given acceleration. In this case we get only 2 triangles with a reduced peak - but all the time the same dynamic
2nd : the movement consists of 2 parts with different max.-speed. In this case we get a triangle for the acceleration, a rectangle, a triangle for the 2nd acceleration or deceleration, the 2nd rectangle and finally again triangle for the deceleration.
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Hi ppolymorphe,
First post updated with the real problem explained, a piece of code and the new question.
If you could take a look at it...
Thank you very much for your time! 
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Code looks not so bad.
Can you give example values of a test that work and one that fail.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Of course Patrice:
Notice that the variables I've written that start with u_ are values entered by the user and the ones that start with c_ are values calculated by the algorithm.
This set of parameters doesn't reach the user vmax and require 37 iterations (ugliest code in the world) to fins the right one:
u_vmax 3000 mm/s
u_v0 2200 mm/s <--------- the important different one
u_vf 0 mm/s
u_a 2000 mm/s^2
u_x0 3000 mm
u_xf 6000 mm
c_dist 3000 mm
c_vmax 2901,250001 mm
This set of parameters works without reaching the vmax as the dynamics doesn't allow the motor to reach it:
u_vmax 3000 mm/s
u_v0 0 mm/s <--------- the important different one
u_vf 0 mm/s
u_a 2000 mm/s^2
u_x0 3000 mm
u_xf 6000 mm
c_dist 3000 mm
c_vmax 2449,489743 mm
This set of parameters work as the vmax entered by the user is reached and we have some part of the movement at constant speed:
u_vmax 3000 mm/s
u_v0 2600 mm/s <--------- the important different one
u_vf 0 mm/s
u_a 2000 mm/s^2
u_x0 3000 mm
u_xf 6000 mm
c_dist 3000 mm
c_vmax 3000 mm
Thank you!
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You should change
LimitIterations = 1000
to
LimitIterations = vmax
because for very short moves, you will need more than 1000 iterations to get the answer.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Yes, but this is not acceptable...
That code will run in a very small processor in a realtime deterministic system and I can't afford spending this amount of time in each cycle.
That's why I need to find the vmax calculating and not by decreasing the value little by little...
I can see two possible approaches:
A) graphical approach with trigonometry trying to find the intersection point of the two straight lines in a time/velocity graph.
B) trying to find the solution via quadratic equations.
In case none of them work, then, at least I would implement a sort of binary search to minimize the searching time for vmax, but I will work to avoid that.
Thank you for your suggestion though. 
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c) dichotomy search
Expect 12 iterations for a vmax between 0 and 4000.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Yes, that would be an option, but I would prefer to solve it mathematically and with only one cycle.
Anyway, that binary (or dichotomy) search would do it more or less well.
Usually the dichotomy search is used to find exact values, in this case I should search for two states in each cycle:
vmax + 1 is not OK
vmax is ok
But that's the last resort if all the other calculations fail.
Thank you!
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Solved... updated the initial post to show the solution.
Thank you Patrice!
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I have to write a program that, given an array of n numbers, finds the longest contiguous subarray whose minimum is the first element of the subarray and maximum the last element of the subarray. For example, with [5, 18, 2, 12, 4, 7, 13, -2, -5, 12] the answer should be [2, 12, 4, 7, 13] (2 <= of all the elements of the subarray, 13 >= of all the elements of the subarray). If possible, the time complexity in the worst case has to be equal to O(n). If it isn't possible, could you explain me why? I already tried to find the minimums and then the maximums, but doesn't work with arrays like [-1,12,1,2,3,4]. I also tried to split the array when there is an element followed by a smaller element and then merge the subarrays, but it doesn't work for arrays like [2,15,6,10,3,20].
modified 5-Feb-17 6:11am.
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Member 10566089 wrote: I have to write a program that
OK, go ahead then.
So what exactly is your question? What do you need help with?
The difficult we do right away...
...the impossible takes slightly longer.
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I can't find an algorithm that solves this problem in O(n). Do you have any idea?
modified 5-Feb-17 13:06pm.
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Member 10566089 wrote: I can't find an algorithm that solves this problem in O(n). Do you have any idea?
You have not been told it is possible, you are asked if it is possible or not. If you don't find an algorithm, the answer is 'no'.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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Show what you have done so far. We will not write it for you.
Patrice
“Everything should be made as simple as possible, but no simpler.” Albert Einstein
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