Problem with diff(f, diff())

Hi there,

I want to differentiate a long equasion L with respect to thetaAdot and ran into a problem. I managed to break it down to the following:

Example 1:

syms x a
f(x, a)=3*x+2*a^2;
df=diff(f, a)

ans=4a -> perfectly fine

Example 2:

syms x a adot
adot=diff(a);
f(x, adot)=3*x+2*adot^2;
df=diff(f, adot)
-> Error

It seems MATLAB has a problem with the derivative but I cannot figure out why? Can you help me with this?

Thank you in advance!

Answer 1

[Walter Roberson]

Your first problem is that you did not declare a to be a symbolic function. diff() applied to a constant variable is going to yield 1, not a placeholder derivative.

syms a(x) adot
adot = diff(a);

but then you have

f(x, adot)=3*x+2*adot^2;

adot is now a function. It is not possible to define a function like f with a parameter that is a function. It is valid to define

f(x) = 3 * x + 2*adot(x)^2

Then you have

df=diff(f, adot)

which attempts take the derivative of f with respect to a function. diff() can only take derivatives with respect to variables. There is functionalDerivative() in newer versions of MATLAB, but the best it would be able to do would be to take the derivative with respect to the function a

>> functionalDerivative(f,a)
ans(x) =
-4*diff(a(x), x, x)

Perhaps you would prefer,

   syms AD
   f(x, AD) = subs(3 * x + 2*adot(x)^2, adot, AD)
   subs(diff(f, AD), AD, adot)

The problem with this is that it assumes that adot and x are independent of each other, which is not the case.

[Backtobasics]

Thank you very much for your explanation! I could transfer it to my example but for some reason, the "real" task still leads to errors. So here's what it is all about: I have to differentiate this equation first wrt to thetaAdot and then the result again wrt t.

My code now is

syms t theta0(t) thetaA(t) thetaB(t) x(t) y(t) m2 m1 m0 L2 L1 L0
thetaAdot=diff(thetaA, t); 
thetaBdot=diff(thetaB, t); 
theta0dot=diff(theta0, t);
xdot=diff(x, t); ydot=diff(y, t);

T=0.5*m2*(xdot^2+ydot^2)+0.5*m1*((L2-0.5*L1)^2*theta0dot^2+xdot^2+xdot^2+2*(L2-0.5*L1)*(ydot*cos(theta0)-xdot*sin(theta0)*theta0dot))+0.5*m0*((L0^2/4)*(thetaAdot+thetaBdot)^2+xdot^2+ydot^2+L0*(ydot*cos(thetaA+theta0)-xdot*sin(thetaA+theta0))*(thetaAdot+theta0dot))+0.5*m0*(L0^2*(thetaAdot+theta0dot)^2+0.25*(thetaBdot+thetaAdot+theta0dot)+0.5*(thetaAdot+theta0dot)*(thetaBdot+thetaAdot+theta0dot)*cos(thetaB)+xdot^2+ydot+2*L0(ydot*cos(thetaA+theta0)-xdot*sin(thetaA+theta0))*(thetaAdot+theta0dot)+L0*(ydot*cos(thetaB+thetaA+theta0)-xdot*sin(thetaB+thetaA+theta0))*(thetaBdot*thetaAdot+theta0dot));
D=functionalDerivative(T, thetaA);

I'm relatively new to MATLAB and I just cannot figure out where I make the mistake(s). I hope you can help me?

[Walter Roberson]

Search your code for

2*L0(ydot*cos(thetaA+theta0)

Notice you missed the * between the L0 and what follows.

[Backtobasics]

Oh dear, thank you very much! I think I checked it a dozen times and didn't see this mistake! Now it works perfectly fine, thank you so so much! :)