Jacobian calculation of symbolic variables which are function of other variables.

Hi everyone,

I am trying to find the jacobian for a transformation matrix. I am using symbolic variables (T, l, m, n). Each of these variables are function of 4 others variables (delta1 delta2 delta3 and delta4), as:

syms u v w p q r phi theta psi x y z;
syms delta1 delta2 delta3 delta4;
% Aerodynamics
V = sqrt(u^2 + v^2 + w^2);
q_bar = (1/2) * rho * V^2;


m = (-1.35* Kf * delta1^2) + (1.35* Kf * delta2^2) + (1.35*K * Kf * delta3^2) + (-1.35* Kf * delta4^2);
l = (0.904* Kf *delta1^2) + (-0.904* Kf *delta2^2) + (0.904* Kf *delta3^2) + (-0.904* Kf *delta4^2);
n = (Km * delta1^2) + (Km * delta2^2) - (Km * delta3^2) - (Km * delta4^2);

T1= (Kf * delta1^2);
T2= (Kf * delta2^2);
T3= (Kf * delta3^2);
T4= (Kf * delta4^2);

T= T1 + T2 + T3 + T4;


phi_dot = p + tan(theta) * (q * sin(phi) + r * cos(phi));
theta_dot = q * cos(phi) - r * sin(phi);
psi_dot = (q * sin(phi) + r * cos(phi)) / cos(theta);

x_dot = cos(psi)*cos(theta)*u + (cos(psi)*sin(theta)*sin(phi) - sin(psi)*cos(phi))*v + (cos(psi)*sin(theta)*cos(phi) + sin(psi)*sin(phi))*w;
y_dot = (sin(psi)*cos(theta))*u + (sin(psi)*sin(theta)*sin(phi) + cos(psi)*cos(phi))*v + (sin(psi)*sin(theta)*cos(phi) - cos(psi)*sin(phi))*w;
z_dot = -sin(theta)*u + cos(theta)*sin(phi)*v + cos(theta)*cos(phi)*w;

f_x = - mass*g * sin(theta);
f_y = mass*g * sin(phi) * cos(theta);
f_z = mass*g * cos(phi) * cos(theta) - T  ;

u_dot = r*v - q*w + (1/mass) * (f_x);
v_dot = p*w - r*u + (1/mass) * (f_y);
w_dot = q*u - p*v + (1/mass) * (f_z);

p_dot = gam(1)*p*q - gam(2)*q*r + gam(3)*l + gam(4)*n;
q_dot = gam(5)*p*r - gam(6)*(p^2 - r^2) + (1/J_yy) * m;
r_dot = gam(7)*p*q - gam(1)*q*r + gam(4)*l + gam(8)*n;

% Collect dynamics
f = [    x_dot;
         y_dot;
         z_dot;
         phi_dot;
         theta_dot;
         psi_dot;
         u_dot;
         v_dot;
         w_dot;
         p_dot;
         q_dot;
         r_dot];

jacobian(f,[T l m n]);

So when calculating jacobian(f,[T l m n]) , i have the error:

"Invalid argument at position 2. Argument must be a variable, a symfun without a formula, or a symfun whose formula is a variable."

Can someone please give me a solution to the problem ?

Answer 1

[Steven Lord]

I tried running your code but there are several variables that are not defined: rho, Kf, K, Km, mass, g, gam, J_yy. Specifying dummy values for those variables, let's look at your f variable and see if it's functions of T, l, m, and n.

rho = 42;
Kf = 999;
K = 17;
Km = 18;
mass = 12345;
g = 9.8;
gam = 9:-1:1;
J_yy = 343455;

syms u v w p q r phi theta psi x y z;
syms delta1 delta2 delta3 delta4;
% Aerodynamics
V = sqrt(u^2 + v^2 + w^2);
q_bar = (1/2) * rho * V^2;
m = (-1.35* Kf * delta1^2) + (1.35* Kf * delta2^2) + (1.35*K * Kf * delta3^2) + (-1.35* Kf * delta4^2);
l = (0.904* Kf *delta1^2) + (-0.904* Kf *delta2^2) + (0.904* Kf *delta3^2) + (-0.904* Kf *delta4^2);
n = (Km * delta1^2) + (Km * delta2^2) - (Km * delta3^2) - (Km * delta4^2);
T1= (Kf * delta1^2);
T2= (Kf * delta2^2);
T3= (Kf * delta3^2);
T4= (Kf * delta4^2);
T= T1 + T2 + T3 + T4;
phi_dot = p + tan(theta) * (q * sin(phi) + r * cos(phi));
theta_dot = q * cos(phi) - r * sin(phi);
psi_dot = (q * sin(phi) + r * cos(phi)) / cos(theta);
x_dot = cos(psi)*cos(theta)*u + (cos(psi)*sin(theta)*sin(phi) - sin(psi)*cos(phi))*v + (cos(psi)*sin(theta)*cos(phi) + sin(psi)*sin(phi))*w;
y_dot = (sin(psi)*cos(theta))*u + (sin(psi)*sin(theta)*sin(phi) + cos(psi)*cos(phi))*v + (sin(psi)*sin(theta)*cos(phi) - cos(psi)*sin(phi))*w;
z_dot = -sin(theta)*u + cos(theta)*sin(phi)*v + cos(theta)*cos(phi)*w;
f_x = - mass*g * sin(theta);
f_y = mass*g * sin(phi) * cos(theta);
f_z = mass*g * cos(phi) * cos(theta) - T  ;
u_dot = r*v - q*w + (1/mass) * (f_x);
v_dot = p*w - r*u + (1/mass) * (f_y);
w_dot = q*u - p*v + (1/mass) * (f_z);
p_dot = gam(1)*p*q - gam(2)*q*r + gam(3)*l + gam(4)*n;
q_dot = gam(5)*p*r - gam(6)*(p^2 - r^2) + (1/J_yy) * m;
r_dot = gam(7)*p*q - gam(1)*q*r + gam(4)*l + gam(8)*n;
% Collect dynamics
f = [    x_dot;
         y_dot;
         z_dot;
         phi_dot;
         theta_dot;
         psi_dot;
         u_dot;
         v_dot;
         w_dot;
         p_dot;
         q_dot;
         r_dot]

f = 

I don't see variables named T, l, m, or n in the array f. You have defined expressions in T, l, m, and n but you cannot use those as the second input to the jacobian function. Take a look at T:

T

T = 

What exactly would you expect the derivative of f(10) with respect to T to be?

f(10)

ans = 

[A]

Sorry, i forgot to share the constants (rho, g , ...) because it was defined in another script, here are my values:

rho = 1.225;
Kf = 0.0121;
K = 1;
Km = 3.5e-04;
mass = 60;
g = 9.8;
gam = [0.1752 0.7835 0.365 0.0036 0.65 0.2042 0.2204 0.025];
J_yy = 16.53;

the array f includes variables which contains the variables T,l,m,n so technically, it contains these variables.

My objective is to linearize my non-linear system to obtain a state space representation, with this code:

vars = [x y z u v w p q r phi theta psi T l m n];
trim = [1000 1000 -1000 0 0 0 0 0 0 0 0 0 mass*g 0 0 0];
A_lon = double(subs(jacobian(f,[x y z phi theta psi u v w p q r]),vars,trim));
 % B_lon = double(subs(jacobian(f,[delta_t_mr1 delta_t_mr2 delta_t_mr3 delta_t_mr4]),vars,trim));
 B_lon = double(subs(jacobian(f,[T l m n]),vars,trim));

I get the A matrix without any problem. But when comming to the B matrix, i get the error.

I actually want (T, l, m, n) as inputs for my system. That's why, i don't want to use the variables (delta1 delta2 delta3 delta4) in the B jacobian.

[Steven Lord]

the array f includes variables which contains the variables T,l,m,n so technically, it contains these variables.

Which element of f contains the character T?

The array f was created using the expressions stored in the variable T, but that's not the same as saying that f contains the variable T.

To repeat my question from the end of my answer, what exactly would you expect the derivative of f(10) with respect to T to be? [You can use the values of the constants from your comment instead of the dummy values I added to generate f(10), but show us the values of T, f(10), and the specific value you expect to have.]

[A]

The derivative of f(10) with respect to T should be -1/mass, which should give -0.016667. That's what I am trying to verify with my script.