The derivation of the Jacobian of a two link arm.

Introduction to Robotics Mechanics and Control

Two-link-manipulator

small notatiom notes before continuing:

  • , 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 is the position and orientation of then end-effector with respect to the base frame. The Jacobian is the partial derivative of with respect to the joint angles.

Each row of 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 .

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

Jacobian summary

end-effector velocity:

: invserse of the Jacobian is required. If there are singularities (loss of full rank) the inverse cannot be commputed.

: Cartesian forces can be converted to torques without having to inverse the Jacobian.