# 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? – Peter Corke Apr 27 at 22:14
• @PeterCorke correct, I had omitted for brevity, just edited for clarity ;) – Eric Peltola Apr 28 at 2:26