How to get the jacobian of a symbolic vector?

Hi,

The following code uses diff(X) and jacobian(r,X) in order to get velocity and accelaration of a point. However, this error appears when I try run:

"Error using sym/jacobian (line 44)
The second argument must be a vector of variables."

t is a variable (time), Phi, Theta and R are functions of t, X is a vector using Phi, Theta and R,(position of the point in cylindrical coordenates) r is function of Phi, Theta and R (position of a point in the cartesian coordenates).

syms t
Phi=sym('Phi(t)');
Theta=sym('Theta(t)');
R=sym('R(t)');
X = [Phi, Theta, R].'
dXdt=diff(X);                  
d2Xdt2=diff(dXdT);               
r = [R*cos(Phi)*sin(Theta), R*sin(Phi)*sin(Theta), R*cos(Theta)].' %position
Ht = jacobian (r,X)              %jacobian
vt=Ht*dXdt                       %velocity
at=Ht*d2Xdt2+diff(Ht,t)*dXdt     %acceleration

=========================================================

It's strange, because the "jacobian(r,X)" works very well in the following code:

syms Phi Theta R
X = [Phi Theta R].';
r = [R*cos(Phi)*sin(Theta), R*sin(Phi)*sin(Theta), R*cos(Theta)].'
jacobian(r,X)

In this case, I can't get dXdt, but when Phi, Theta and R are functions of t, I can't get the jacobian :/

[John D'Errico]

Use the "{} Code" button to format your code. Otherwise, it shows as an unreadable mess.

[Marlon Saveri Silva]

Thanks John, it's edited.

Answer 1

[John D'Errico]

Note that in your code, you have the lines...

dXdt=diff(X);                  
d2Xdt2=diff(dXdT);

MATLAB gets upset here, because you can't type. It gives the error message:

Undefined function or variable 'dXdT'.

You never defined dXdT, only dXdt.

Next, the error you get reflects a problem in your code. You have defined the FUNCTIONS, Theta, Phi and R, as parametric functions of t. They are not variables. The only variable in this is t.

[Marlon Saveri Silva]

Does it mean that's impossible get the jacobian when Theta, Phi and R are parametric functions?

To build the jacobian of a function r(r1, r2, r3) relation to X(x1,x2,x3), I need values like: diff(r1,x1). Since x1 is a parametric function Phi(t), it's impossible get diff(r1,Phi(t))?

diff(R*cos(Phi)*sin(Theta), Phi(t))

[Marlon Saveri Silva]

Well, if its not possible, the code below solve my problem.

%This code is used to check position, velocity and acceleration of a point in the cartesian coordinates system from the cylindrical coordinates system 
%Obs.: when I use "_t", It means not parametric anymore, but with variables changed by its functions of "t". It was necessary because Matlab can not recognize "jacobian(r,X)" when X = X(Psi(t), Theta(t), R(t)). Therefore, it's necessary write Psi(t), Theta(t) and R(t) in a specific part of the code (after get the jacobian matrix).

clear
clc
syms t Psi Theta R
x = [Psi, Theta, R].';              %position of a point in the cylindrical coordinates system
r = [R*cos(Psi)*sin(Theta), R*sin(Psi)*sin(Theta), R*cos(Theta)].'; %position of this point in the cartesian system
Ht = jacobian (r,x);                %jacobian matrix

%Define Psi, Theta and R functions of t
Psi_t=10;
Theta_t=3*t^2;
R_t=2*t;        

%Get diff(Ht,t)
Ht_t=subs(Ht,Theta, Theta_t);
Ht_t=subs(Ht_t,R,R_t);
Ht_t=subs(Ht_t,Psi, Psi_t);
d_Ht_t=diff(Ht_t,t);                   

%Get dx/dt and d²x/dt²
dxdt= [diff(Psi_t,t) diff(Theta_t,t) diff(R_t,t)].';
d2xdt2= [diff(dxdt(1),t) diff(dxdt(2),t) diff(dxdt(3),t)].';

%Velocity Equation
vt=Ht_t*dxdt;

%Acceleration Equation
at=Ht_t*d2xdt2+d_Ht_t*dxdt