numerical solution of differential equations

[Ether]

Unless I was careless with the algebra (it happens when I'm tired),
Steve's catapult can be modeled with an ODE of the form

θ'' = k₁ + k₂∙cos(θ) + k₃∙θ'

Attached is an Octave script that uses Octave's built-in ODE solver "lsode"
to numerically integrate arbitrary ODEs of the form x'' = f(t,x,x')

[sspoldi]

Hillbilly solution:
one forward Euler integration, one backward Euler integration, less typing and good enough for government work.

Surprising how often that works...

Cheers,
Steve.

P.S. The equation (for the catapult) should be something like θ" = K1∙(K2 - θ'), θ is just along for the ride.

[GeeTwo]

Originally Posted by sspoldi

Hillbilly solution:
one forward Euler integration, one backward Euler integration, less typing and good enough for government work.

Surprising how often that works...

That is:

x'[n+1]  =  x'[n] + dt*x''[n]   //backward looking
x[n+1]   =  x[n]  + dt*x'[n+1]  //forward looking
x''[n+1] = -x[n+1]

It looks OK on amplitude, but overpredicted the resonant frequency by .. almost half a part per thousand. Certainly good enough for FRC.

Originally Posted by sspoldi

P.S. The equation (for the catapult) should be something like θ" = K1∙(K2 - θ'), θ is just along for the ride.

The cosθ term is gravity acting on the boulder (and lever arm).

[sspoldi]

Originally Posted by GeeTwo

The cosθ term is gravity acting on the boulder (and lever arm).

Yea, I'd like to say we just ignore stuff like that, but sometimes it's a real factor. In 2014 we had a hammer with a 3 pound head on a 1 foot arm, gravity definitely made a difference.

Since we typically don't have a lot of time (who does), I like to get the kids to do a simple model up front, and then we do some system id and fit the actual robot behavior to a model. This way we can tune control systems quickly, and it gives them a chance to do some data based optimization in addition to a little physics up front.

Cheers,
Steve.

[Ether]

Originally Posted by sspoldi

one forward Euler integration, one backward Euler integration

V_n+1 = V_n + A_n∙dt;
X_n+1 = X_n + V_n+1∙dt;

Maybe provides some insight: the above is algebraically equivalent to

V_n+1 = V_n + A_n∙dt;
X_n+1 = X_n + dt∙(V_n+V_n+1)/2 + ½∙A_n∙dt²

[GeeTwo]

Originally Posted by Ether

Maybe provides some insight: the above is algebraically equivalent to

V_n+1 = V_n + A_n∙dt;
X_n+1 = X_n + dt∙(V_n+V_n+1)/2 + ½∙A_n∙dt²

..so counting the constant acceleration term twice in the position calculation mostly offsets not counting the jerk in the velocity calculation..at least in this case.

[Ether]

Originally Posted by Ether

θ'' = k₁ + k₂∙cos(θ) + k₃∙θ'

derivation of k₁ k₂ k₃