Jacobian

Synonyms

Singularities

Definition

A robot manipulator’s Jacobian matrix maps the linear or first-order differential relationship between its joint and Cartesian spaces, i.e., it relates the manipulator joint rates to velocities in the Cartesian space. It also relates the joint torques to end-effector forces and moments.

Overview

Serial Manipulator Jacobian

Let the position and orientation of the end-effector of a serial chain manipulator be given by $\mathbf {x} \in \mathbb {R}^M$, which can be described as a vector function of the set of joint variables $\mathbf {q} \in \mathbb {R}^N$ as:

$$\displaystyle \begin{aligned} \mathbf{x} = \mathbf{x}(\mathbf{q}) \end{aligned} $$

(1)

Then, the Jacobian form of the above expression is given by:

$$\displaystyle \begin{aligned} \dot{\mathbf{x}} = \mathbf{J} \dot{\mathbf{q}} \end{aligned} $$

(2)

where J is the M × N Jacobian matrix. Individual elements of J consist of:

$$\displaystyle \begin{aligned} {\mathbf{J}}_{ij} = \frac{\partial x_i}{...