The derivation of the Jacobian of a two link arm.

Introduction to Robotics Mechanics and Control

small notatiom notes before continuing:

• $c_1 = cos(\theta_1)$, the letter indicates if it is sin or cos and the number corresponds to the joint angle.

The kinematic chain of the two-link-manipulator:

The above kinematic equation is a function:

which maps from joint to cartesian space, where $Y$ is the position and orientation of then end-effector with respect to the base frame. The Jacobian is the partial derivative of $F$ with respect to the joint angles.

Each row of $F$ corresponds to one non-linear equation and for each row the partial derivative with respect to $\theta$ is taken. As there are 4 rows there are 4 equations of the following form:

where $i = 1, 2, 3, 4$.

The superscript index is for the frame and subscript is for the link.

## Jacobian summary

end-effector velocity: $v = \begin{bmatrix} \upsilon \\ \omega \end{bmatrix} \in (6 \times 1)$

$\dot{\theta} = J^{-1}(\theta)\,v$: invserse of the Jacobian is required. If there are singularities (loss of full rank) the inverse cannot be commputed.

$\tau = J^{T}(\theta)\,\mathcal{F}$: Cartesian forces can be converted to torques without having to inverse the Jacobian.