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I have two degrees of freedom robot with a camera mounted on the tool center point where the system is modeled as

$^wT_c = {^wT_z}(\theta_1){^zT_m}(\theta_2) {^mT_c}(\textbf{r})$, $\textbf{r} \in SO3$

I have to calibrate mount to camera transformation ${^mT_c}$ without the robot joint encoder value which means robot joint angles are parameters to estimate as well.

I assume that the following values are given.

The rotation axis of each joint ${^wT_z}(\theta_1){^zT_m}(\theta_2)$ is known.

The world reference frame to camera $^wT_c$ is also known from the camera observation.

The rotation axis of each motor in the camera frame $\textbf{c}_{1,2}$ and in the world frame $\textbf{v}_{1,2}$ is known.

The constraints that I have tried to estimate $\textbf{r}$ are

$\textbf{e}_{w_i} = {^wT_c}({^wT_z}(\theta_{1_i}){^zT_m}(\theta_2) {^mT_c}(\textbf{r}))^{-1}$

$\textbf{e}_{v1} = {^wT_z}(\theta_{1_i})_{[1:3,2]}-\textbf{v}_1$

$\textbf{e}_{v2} = ({^wT_z}(\theta_{1_i}){^zT_m}(\theta_{2_i}))_{[1:3,3]}-\textbf{v}_2$

$\textbf{e}_{c1} = {^wT_z}(\theta_{1_i}){^zT_m}(\theta_{2_i}) {^mT_c}(\textbf{r})\textbf{c}_1-\textbf{v}_1$

$\textbf{e}_{c2} = {^wT_z}(\theta_{1_i}){^zT_m}(\theta_{2_i}) {^mT_c}(\textbf{r})\textbf{c}_2-\textbf{v}_2$

The result of the optimization with simulation data is a bit weird. The estimation of $\theta_{1_i}$ are correct but not $\theta_{2_i}$. $\theta_{2_i}$ are slightly different from the ground truth values. As a result, $\textbf{r}$ is slightly different from the ground truth value. The residual of the constraints drops to the reasonable level which was around 5.6179e-17 after 7 iterations.

So, I guess this is an underconstrained problem but I am not sure what to add to fix the problem. Any idea or thought will be appreciated.

Note: Since the robot joint angles are not known it is not possible to use AX=XB kind of closed-form solution.

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    $\begingroup$ "around 5.6179e-17 after 7 iterations" That sounds really suspicious. No real world optimization I saw (especially including cameras) ever got that close to a perfect zero. I don't know what kind of optimization you do, but 7 iterations also sounds a bit quick. $\endgroup$ – FooTheBar Jun 13 '19 at 9:49
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    $\begingroup$ Thanks for the comment. I think that is normal for a simulated input. The simulated dataset is without error. I am just using LM. $\endgroup$ – C.O Park Jun 13 '19 at 11:17
  • $\begingroup$ Actually, I just found out that the absolute value of θ2i is not measurable. Only the relative angles will be found by the optimization. The real rotation offset added to θ2i will be mixed in the estimated $\textbf{r}$. So, I need to add a home position sensor to properly estimate θ2i. $\endgroup$ – C.O Park Jun 13 '19 at 11:22

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