### Generalities

This is a relatively technical post. Its purpose is mainly to teach myself why the Jacobian transpose is so useful when doing inverse kinematics.

We are going to solve the following problem:

We have a mechanism with effectors applying forces whose components are given by , with going from 1 to . The required power is provided by a series of torques, whose components are called , where goes from 1 to . Get the required values of as a function of .

### Jacobian

Let’s call the angular coordinate associated with the torque components and the normal (“linear”) coordinate of the effector associated with the force component . In most useful cases, the positions of the effectors can be expressed as a function of the angular coordinates, . The Jacobian will be the linearization of this relationship around some point ,

.

### Virtual work

If we can ignore inertia forces (either because we are dealing with a purely static problem or because inertia is negligible), we can get

,

where is the infinitesimal linear displacement associated with the force component and is the infinitesimal angular displacement associated with the torque component .

### Putting things together

As the previous expression uses infinitesimal movements, we can use the Jacobian to relate the linear displacements to the angular ones:

.

If we replace that result in the virtual work equation,

,

and we do some rearrangements, we get an expression with infinitesimal angular displacements in both sides:

.

As the infinitesimal angular displacements are arbitrary, their factors should match:

.

By representing this equation in matrix form,

,

we see how we arrive naturally to the Jacobian transpose.