How do I compute the derivative of the Jacobian with Matlab?

Question

I am studying robotics, and I am trying to write a Matlab code for computing the derivative of the jacobian matrix.

The formula for computing the derivative of the Jacobian is the following:

so it has been done a derivation with respect to time and it has been applied the chain rule.

I know how to do this computation by hand, but it takes really a long time if the Jacobian is large, and moreover if then I have to do some calculations with the derivative of the Jacobian it is hard without the code.

I have found this question here on Stack Exchange Robotics where a code has been posted:

Derivative of a Jacobian matrix

which is similar(it is the same, I copied it but I changed T with q) to:

clear all
clc
syms q1 q2 q3 t;
q1(t) = symfun(sym('q1(t)'), t);
q2(t) = symfun(sym('q2(t)'), t);
q3(t) = symfun(sym('q3(t)'), t);
J11 = -sin(q1(t))(a3cos(q2(t) + q3(t)) + a2*cos(q2(t)))
dJ11dt = diff(J11,t)

but if I use this code I get an error which says:

Error using sym>convertChar (line 1537) Character vectors and strings in the first argument can only specify a variable or number. To evaluate character vectors and strings representing symbolic expressions, use 'str2sym'.

can somebody please help me?

Answer 1

To answer your solution, specifically, all you need to do is:

clear all
clc
syms q1 q2 q3 t a3 a2;

q1(t) = symfun(str2sym('q1(t)'), t);
q2(t) = symfun(str2sym('q2(t)'), t);
q3(t) = symfun(str2sym('q3(t)'), t);

J11 = -sin(q1(t))*(a3*cos(q2(t) + q3(t)) + a2*cos(q2(t)))

dJ11dt = diff(J11,t)

But if you're interested, there is a paper you might want to check that solves for the Jacobian Derivative $\dot{J}(q)$ analytically, (Rhee, J. Y., & Lee, B., 2017). The authors compare the method you presented with their own in terms of computation time. It is significantly faster.

The method they presented is summarized in the following algorithm.

$$ \overline{\underline{\bf{\text{Algorithm 1}}\textrm{ Jacobian and Jacobian Differentiation}}}\\ \begin{array}{lll} 1: & \omega_0^0 \leftarrow\left[0,0,0\right]^T & \\ 2: & z_0^0 \leftarrow\left[0,0,1\right]^T & \\ 3: & \pmb{J}_1 \leftarrow \left[\begin{matrix} z_0^0\times d_n^0 \\ z_0^0\end{matrix}\right] & \\ 4: & \bf{\text{for }} \textrm{i = 2 to n } \bf{\text{do}} & \triangleright \textrm{ Jacobian Computation loop} \\ 5: & \quad \pmb{J}_i = \left[\begin{matrix} z_{i-1}^0 \times d_{i-1,n}^0 \\ z_{i-1}^0\end{matrix}\right] & \\ 6: & \quad \omega_{i-1}^0 \leftarrow w_{i-2}^0 + z_{i-2}^0\dot{q}_{i-1} \\ 7: & \bf{\text{end for}} & \\ 8: & \beta \leftarrow J_{v,i}\dot{q}_i & \\ 9: & \bf{\text{for }} \textrm{i = n to 2 } \bf{\text{do}} & \triangleright \textrm{ Jacobian Differentiation loop}\\ 10: & \quad \dot{z}_{i-1}^0 \leftarrow \omega_{i-1}^0 \times z_{i-1}^0 & \\ 11: & \quad \alpha \leftarrow \left[0,0,0\right]^T & \\ 12: & \quad \bf{\text{for }} \textrm{j = 1 to i-1 } \bf{\text{do}} \\ 13: & \qquad \alpha \leftarrow \alpha + z_{j-1}^0 \times \left(d_n^0 - d_{i-1}^0\right)\dot{q}_j & \\ 14: & \quad \bf{\text{end for}} & \\ 15: & \quad \dot{J}_i = \left[\begin{matrix} \left(\dot{z}_{i-1}^0\right) \times (d_{n}^0- d_{i-1,n}^0) + z_{i-1}^0 \times\left(\alpha + \beta\right) \\ \dot{z}_{i-1}^0 \end{matrix}\right] & \\ 16: & \quad \beta \leftarrow \beta + J_{v,i-1}\dot{q}_{i-1} & \\ 17: & \bf{\text{end for}} & \\ 18: & \dot{J}_1 = \left[\begin{matrix} z_0^0 \times \beta \\ \left[0,0,0\right]^T \end{matrix}\right] & \\ \end{array} \\ $$