Accel curve help needed by math wiz.

T Chap · 2008-10-25 22:02

I have been searching here and google for a while with no luck finding accel formulas that could easily apply (no extra cogs for float) to fix this curve shown below.

Can anyone suggest a quick easy formula to adjust fix the red curve to look more like the blue one?

The idea is, the fix needs to be spread over 4320 loops, and maintain the same start value(0) and end value(MaxSpeed) so that the curve still ends up at the same place at the same time it always did to preserve the profile.

It seems if there is no better way to fix the main curve, then the offset would simply count up to some number, then count back down, possibly even into the negative to retain the same MaxSpeed/Time end point.

Post Edited (Originator) : 10/25/2008 10:31:12 PM GMT

SRLM · 2008-10-26 00:47

It appears to be like the following image:

· attachment.php?attachmentid=56395

This is an image of x² and x^1/2 . The former is on the bottom, and the latter is above. I may be able to come up with a solution on how to convert, but it will take a little while.

Post Edited (SRLM) : 10/26/2008 12:54:24 AM GMT

T Chap · 2008-10-26 01:14

What I did as a temporary workaround that improves the curve:


PUB AccelForward(len)    'len = distance to travel based on 30% new target pos
      aRate := (MinSpeed - MaxSpeed)/Len        '    = 34  at 30%
      dRate := 10 '(MinSpeed - MaxSpeed)/Len  
      a3 := len/2
      posold := pos
      repeat  len                               
        accel   := ((accel - (aRate + A3))  #> MaxSpeed) <# MinSpeed 
        'a3 is an offset using the length (len) / 2,   start subtracting the distance moved from the len/2 
        a3 :=  a3 -  ||(pos-posold)    #> 0                     
        pos := pos  + 1                                                                        
        waitcnt(accel + cnt)
        if newpos < pos
          Quit                                     
      decel := accel

This is crude but effective to put the curve on the top side.

SRLM · 2008-10-26 01:40

Okay. Here's what I have. It's purely a mathematical proof: I leave it up to you to figure out how to use it and tie it into your data.
So, that said, let's begin.

You have two curves:
The given curve: xⁿ = c
The desired curve: x^1/n = a
Where c and a are the values given at point x (the point in your sample list)

''The proof:
 
x[sup]n[/sup] = c                     ''What you have
log[sub]x[/sub]c = n                  ''Convert in order to get n out
1 / n = 1 / log[sub]x[/sub]c          ''Divide both sides by n and log[sub]x[/sub]c
x[sup]1/n[/sup] = x [sup]1 / log[sub]x[/sub]c[/sup] = a        ''Substitute in for 1/n

So to convert, just use x ^{1 / log_xc}· to get the value that you want.

SRLM · 2008-10-26 01:46

Can you give some sample points so that I can test? The above proof is good for converting between a root function and an exponential, but your data might not fit into that range.

I'll give an example of how to use it:

 a = x [sup]1 / log[sub]x[/sub]c[/sup]      ''Your equation to use
 
''To find out the desired data at point .5, and supposing that you measured a value of .25 at that point
 
a = (.5)[sup]1 / log[sub](.5)[/sub](.25)[/sup]
a = .707107

T Chap · 2008-10-26 02:14

Thanks a lot for that SRLM. I will have to put a lot of thought into this to apply it, since the actually numbers used are inverted so to speak, the faster the speed = a smaller number (using waitcnt to set speed between loops). This will have to go on hold till tomorrow but I will let you know what I came up with.

SRLM · 2008-10-26 03:45

Wait: so the graph that you gave me is NOT what you actually measured and want? It's something else? If so, then my solution may or may not work, and you may or may not be able to do some more math to get the equation that you really want.

And, truth be told, I mostly looked at your graph, and skimmed over the rest. Without the full background, it's difficult to make the project in my mind. But, with the graph I can still do the math. [noparse]:)[/noparse]

T Chap · 2008-10-26 21:47

SRLM, The graph I drew was for illustration purposes only.

I have a better idea now, either predefine 10 reference points over the distance of the accel curve, or, define one point as the 'knee' and calculate the rest using a curve/angle value to determine the sharpness/roundness of the curve.

I like the idea much better to be able to semi-graphically define the curve, only in this case the graphic is on paper rather than a GUI.

PUB AccelForward(len)      |   X            'len = ramp up distance to travel based on 30% new target pos  
     'phase 1 =  MinSpeed(=175_000) - MaxSpeed(=25_000) = 150_000   /   
      phaseLen  =   len/10          '   example   distance 4320   /   10 points  = 432 
      
     index := 0
     phasecounter  := 0
     PCindex  := 0
     accel := MinSpeed     '175000  example
    
      AccelPhase[noparse][[/noparse] 0]  := 0                   'predefined curve values  
      AccelPhase[noparse][[/noparse] 1]   :=  20000         
      AccelPhase[noparse][[/noparse] 2]   :=  30000     
      AccelPhase[noparse][[/noparse] 3]   :=  50000
      AccelPhase[noparse][[/noparse] 4]   :=  90000
      AccelPhase[noparse][[/noparse] 5]   :=  110000
      AccelPhase[noparse][[/noparse] 6]   :=  130000
      AccelPhase[noparse][[/noparse] 7]   :=  135000
      AccelPhase[noparse][[/noparse] 8]   :=  140000
      AccelPhase[noparse][[/noparse] 9]   :=  145000
      AccelPhase[noparse][[/noparse] 10] :=   150000 
   
     PhaseOffset[noparse][[/noparse] 1] :=   (AccelPhase[noparse][[/noparse] 1]  - AccelPhase[noparse][[/noparse] 0] )/ phaseLen        'offset value for each iteration 
     PhaseOffset[noparse][[/noparse] 2] :=   (AccelPhase[noparse][[/noparse] 2]  - AccelPhase[noparse][[/noparse] 1] )/ phaseLen      
     PhaseOffset[noparse][[/noparse] 3] :=   (AccelPhase[noparse][[/noparse] 3]  - AccelPhase[noparse][[/noparse] 2] )/ phaseLen    
     PhaseOffset[noparse][[/noparse] 4] :=   (AccelPhase[noparse][[/noparse] 4]  - AccelPhase[noparse][[/noparse] 3] )/ phaseLen      
     PhaseOffset[noparse][[/noparse] 5] :=   (AccelPhase[noparse][[/noparse] 5]  - AccelPhase[noparse][[/noparse] 4] )/ phaseLen
     PhaseOffset[noparse][[/noparse] 6] :=   (AccelPhase[noparse][[/noparse] 6]  - AccelPhase[noparse][[/noparse] 5] )/ phaseLen   
     PhaseOffset[noparse][[/noparse] 7] :=   (AccelPhase[noparse][[/noparse] 7]  - AccelPhase[noparse][[/noparse] 6] )/ phaseLen      
     PhaseOffset[noparse][[/noparse] 8] :=   (AccelPhase[noparse][[/noparse] 8]  - AccelPhase[noparse][[/noparse] 7] )/ phaseLen    
     PhaseOffset[noparse][[/noparse] 9] :=   (AccelPhase[noparse][[/noparse] 9]  - AccelPhase[noparse][[/noparse] 8] )/ phaseLen      
     PhaseOffset[noparse][[/noparse] 0] :=   (AccelPhase[noparse][[/noparse] 10]  - AccelPhase[noparse][[/noparse] 9] )/ phaseLen   

     PhaseChange[noparse][[/noparse] 1]  := phaseLen              'values to indicate when to change to next PhaseOffset value   
     PhaseChange[noparse][[/noparse] 2]  := phaseLen * 2
     PhaseChange[noparse][[/noparse] 3]  := phaseLen * 3
     PhaseChange[noparse][[/noparse] 4]  := phaseLen * 4
     PhaseChange[noparse][[/noparse] 5]  := phaseLen * 5
     PhaseChange[noparse][[/noparse] 6]  := phaseLen * 6
     PhaseChange[noparse][[/noparse] 7]  := phaseLen * 7
     PhaseChange[noparse][[/noparse] 8]  := phaseLen * 8
     PhaseChange[noparse][[/noparse] 9]  := phaseLen * 9
     'PhaseChange[noparse][[/noparse] 10]  := Len                 'doesn't need 10

 {{  accel is derived by MinSpeed - MaxSpeed +  offset .  An example is 175000 - 25000 =  150000 + offset
  the speed of the motor is determined by PWM, PWM is derived from an Error produced by a Profile generating routine 
  below, which increments/decrements a position counter (pos).  A PID calculator compares an 
  encoder value to the position,   and computes a corresponding PWM motor speed to compensate for the Error.  
  Using waitcnt with the Minimum speed  as the starting (slower) speed, amounts are subtracted from the 
  Minimum speed to increase the speed. 
        }} 

      repeat  len                       
        accel   := ((accel  -  PhaseOffset[noparse][[/noparse]Index] ) < # MinSpeed)  #> MaxSpeed  'reduce waitcnt  by Phase[noparse][[/noparse]index]                   
        pos := pos  + 1
        if  phasecounter ==  PhaseChange[noparse][[/noparse] PCindex]    'if phasecounter has reached 1rst phase end, or ex 432  (len/10)... 
              index++                                          'then go to next phaseOffset value                  
              PCindex++                                    'and increment the PhaseChange index
        phasecounter+                                  'increment the phasecounter
        waitcnt(accel + cnt)                         'waitcnt = value to set speed of pulses     
        if newpos < pos                               'if new position has been rec'd, get out and go decel somewhere else
          Quit                                     
      decel := accel       'let decel  value be same value as value that accel left off at

Post Edited (Originator) : 10/26/2008 9:55:11 PM GMT

SRLM · 2008-10-26 22:59

I like your second version better. I'm not sure what you mean by knee, but here's my impression: the first is a series of points, and you extrapolate in a linear fasion to get the data that you want. The second is an equation given by two points and a "curve amount". The easiest thing would be if you could come up with some sort of equation: then it just takes a single line of code to calculate what you want. Otherwise, you're going to have a big ifelse statement (one statement for each predefined point).

T Chap · 2008-10-26 23:18

SRLM said...
The easiest thing would be if you could come up with some sort of equation: then it just takes a single line of code to calculate what you want. Otherwise, you're going to have a big ifelse statement (one statement for each predefined point).

Actually the code above is what is working out fine for 10 predefined points, it take the 10 points and calculates the 10 ranges of offset values, there are no other if statements needed as the counter bumps up the index keeping everything working at the right time.

Thanks for the advice, very helpful.

SRLM · 2008-10-27 02:55

For that though, you need about 30 lines of predefined code. You have to find and decide which point to use as offsets, and your graph will just be a collection of lines. An equation is much more dynamic: you don't need any sort of predefined data. (or at least not as much)

Accel curve help needed by math wiz.

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