# Inverse kinematics and the Jacobian transpose

### 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 $f_i$, with $i$ going from 1 to $N$. The required power is provided by a series of torques, whose components are called $\tau_j$, where $j$ goes from 1 to $M$. Get the required values of $\tau_j$ as a function of $f_i$.

### Jacobian

Let’s call $\theta_j$ the angular coordinate associated with the torque components $\tau_j$ and $x_i$ the normal (“linear”) coordinate of the effector associated with the force component $f_i$. In most useful cases, the positions of the effectors can be expressed as a function of the angular coordinates, $x_i(\theta_j)$. The Jacobian will be the linearization of this relationship around some point $\theta_j^0$,

$\displaystyle J_{ij}(\theta_j^0) = \left. \frac{\partial x_i}{\partial \theta_j} \right|_{\theta_j=\theta_j^0}$.

### 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

$\displaystyle \sum_{i=1}^N f_i \delta x_i = \sum_{j=1}^M \tau_j \delta \theta_j$,

where $\delta x_i$ is the infinitesimal linear displacement associated with the force component $f_i$ and $\delta \theta_j$ is the infinitesimal angular displacement associated with the torque component $\tau_j$.

### Putting things together

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

$\displaystyle \delta x_i = \sum_{j=1}^M J_{ij} \delta \theta_j$.

If we replace that result in the virtual work equation,

$\displaystyle \sum_{i=1}^N f_i \left(\sum_{j=1}^M J_{ij} \delta \theta_j\right) = \sum_{j=1}^M \tau_j \delta \theta_j$,

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

$\displaystyle \sum_{j=1}^M \left(\sum_{i=1}^N f_i J_{ij}\right) \delta \theta_j = \sum_{j=1}^M \tau_j \delta \theta_j$.

As the infinitesimal angular displacements $\delta \theta_j$ are arbitrary, their factors should match:

$\displaystyle \sum_{i=1}^N f_i J_{ij} = \tau_j$.

By representing this equation in matrix form,

$\displaystyle \boldsymbol{\tau} = \mathbf{J}^T \mathbf{f}$,

we see how we arrive naturally to the Jacobian transpose.