# Kinematics with Product-of-Exponentials terminology: not forward, not inverse

I'm working on a kinematic estimation algorithm, and am trying to come up with the right terminology.

I'm using the Product-of-Exponentials formulation for kinematic chains (and ignoring position for now):

$$R = e^{\omega_1 \theta_1}e^{\omega_2 \theta_2}...e^{\omega_n \theta_n}R_0$$

where:
$$R \in SO(3)$$: orientation of end effector
$$\theta_{1..n}$$: joint coordinates
$$\omega_{1..n} \in so(3)$$: joint screw axes

Forward Kinematics: Solve for $$R$$ given $$\omega_{1..n}, \theta_{1..n}$$
Inverse Kinematics: Solve for $$\theta_{1..n}$$ given $$R, \omega_{1..n}$$
?___?: Solve for $$\omega_{1..n}$$ given $$R, \theta_{1..n}$$

My question is, what phrase fills that blank? I've been using the PoE formula for so long that I am starting to forget the DH paradigm but I'd imagine there's an equivalent with DH params too.

The literature I'm comparing against calls this "system identification" but I think that's because they're treating it like a dynamical system...I am treating it like a standard robot kinematics problem.

Is there an agreed-upon term for this process?

• \omega is the so(3) matrix and \theta is the joint coordinate? Shouldn't there be an SO(3) matrix on the end, the orientation when all \theta are zero? Commented Apr 27, 2019 at 22:14
• @PeterCorke correct, I had omitted for brevity, just edited for clarity ;) Commented Apr 28, 2019 at 2:26