https://studywolf.wordpress.com/2013/09/02/robot-control-jacobians-velocity-and-force/

Jacobian defines a dynamic relationship between two different representations of a system. It is the equivalent of gradient (which is for vector to scalar functions), but for vector-to-vector functions.

Since gradient has a scalar values, we only need to to take the partial derivative of the scalar value with respect to the components

e.g. if we define a function $f$ where $f(x_1, x_2, x_3)$ is a scalar

Then

$$

\nabla f = [\frac{\partial f(x_1, x_2, x_3)}{x_1}, \frac{\partial f(x_1, x_2, x_3)}{x_2}, \frac{\partial f(x_1, x_2, x_3)}{x_1}] $$

However, Jacobians are used for vector-to-vector functions so they look like this

$$ J = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} && \frac{\partial f_1}{\partial x_2} && \frac{\partial f_1}{\partial x_3} && ...\\\\ \frac{\partial f_2}{\partial x_1} && \frac{\partial f_2}{\partial x_2} && \frac{\partial f_2}{\partial x_3} && ...\\\\ \frac{\partial f_3}{\partial x_1} && \frac{\partial f_3}{\partial x_2} && \frac{\partial f_3}{\partial x_3} && ... \end{pmatrix} $$

In robotics, the Jacobian is usually referring to the how the movement of the elements of $q$ causes movement of the elements of $x$, or you can think of it as a transform matrix for velocity

$$ J = \frac{\partial x}{\partial q} $$

Applying chain rule and rearranging terms, we get

$$ \frac{\partial x}{\partial t} = J \frac{\partial q}{\partial t} $$

This shows that the end-effector velocity is equal to the Jacobian multiplied by joint angle velocity. We want to find the equivalent end-effector forces (x, y, z) by applying joint torques (q), and this relationship helps us find the forward kinematics.

However, we actually want forces and not velocities.

Energy equivalence

$$ W = \int F^Tvdt $$

$$ P = \frac{W}{t} $$

$$ P = \frac{W}{t} = F^T \frac{d}{t} = F^Tv $$

Due to energy equivalence, work is performed at the same rate.