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 $\tau_j$ .