[ODE] More accuracy ???

[Nguyen Binh]

Hi ,

    In moveAndRotateBody (dxBody *b, dReal h) function of step.cpp
    file (ode 0.35 distro) we have :

    ------Begin code fragment
    ......
    
        // handle linear velocity
        for (j = 0; j < 3; j++)
                b->pos[j] += h * b->lvel[j];

    .....
    ------End

    I understand those code update 3D position of the body using first
    order Euler method.In which,
          lvel = pos'
          
    Is it true?

    Look more closely, I found b->facc[j] is 'external force' apply to
    the body along jth axis, so:
        facc =  lvel' = pos''

    That's means we have a "cheap" and more accurate way to solve the
    ODE problem:
        we should update pos with :
        pos(t + h) = pos(t) + h*pos'(t) + h*h/2*pos''(t)
                   = pos(t) + h*lvel(t) + h*h/2*facc(t).

    So, the code is modified to

    ------Begin code fragment
    ......
    
        // handle linear velocity
        dReal h2 = h*h/2;
        for (j = 0; j < 3; j++)
                b->pos[j] += h * b->lvel[j] + h2*b->facc[j];

    .....
    ------End
    So our ODE has O(h3) accuracy compare to just O(h2) with the old
    one.

    How is your idea? Is something wrong with me?
    I have little time to test it, and it works just fine.I mean I see
    no 'wrong' behavior but no 'better' than the old one.

-- 
Best regards,

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   Nguyen Binh
   Software Engineer
   Glass Egg Digital Media
   
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