Paper note: Robust Jacobian Estimation for Uncalibrated Visual Servoing

This paper propose a robust Jacobian estimation for uncalibrated visual servoing. For the term "uncalibrated", authors refer to model-free and nonparametric Jacobian (i.e. non-analytic form).
To estimate the Jacobian, a Broyden rank-one secant update has been proposed which requires a good initial guess. Farahmand et al. propose local least-squares (LLS) estimation to utilize the memory of visual-motor data. This method estimated the Jacobian of any point in the workspace directly from raw visual-motor data in a close neighborhood (K-NN) of the point under consideration.
The following equation consider the Jacobian estimation as a minimization problem:

It is pointed out that the least-squares (LS) estimator is not robust to outliers as its weight function assign weight equally to all data including outliers. The L1-norm is more robust but both have the least possible breakdown point (BDP). The BDP refers to  the smallest proportion of incorrect samples that the estimator can tolerate before they arbitrarily affect the model fitting. In other words, BDP of an estimator is a measure of its resistance to outliers. The maximum BDP is 50% where outliers and inliers have equal amount. In the paper, two other M-estimators with a redescending  influence function, Tukey’s Biweight (BW) function and Geman-McClure (GM) estimator, is investigated. The  function for GM estimator is written as:

For other type of M-estimators, refer to this link. Note that the formulation of the GM estimator in the paper is a bit different from the link, more investigation needed.



The scale parameter σ quantifies how the probability distribution is spread. For example, variance is a measure of scale for the normal distribution. To estimate a scale for a M-estimator, the paper uses Median Absolute Deviation (MAD), which has the highest possible BDP of 50% and a bounded influence function and is computationally efficient, regardless its low Gaussian efficiency (37%).
More info about MAD and other way to measure scale of data see: link

A common method to solve (6) is the Iteratively Reweighted Least Squares (IRLS) that is widely used as an efficient implementation of robust M-estimator in nonlinear optimization domains. The IRLS used in the paper for the Jacobian estimation is shown as:

The algorithm presented in the paper can be summarized as follows:

A.1 Initialize visual-motor memory
A.2 Determine neighbors: K-NN
A.3 Estimate initial scale: Use MAD to find initial measure of scale σ
A.4 Find initial weights: Initialize weight matrix W0 according to the found norm and scale.
A.5 Estimate the Jacobian: Use JACOBIANESTIRLS
A.6 Update control signal
A.7 Update memory: The new visual-motor pair is added to the memory for later use P = P +1 (All pairs are kept?)
A.8 Goto step A.2










Reference:
[1] A. Shademan, A. M. Farahmand, and M. Jägersand, “Robust Jacobian estimation for uncalibrated visual servoing,” Proc. - IEEE Int. Conf. Robot. Autom., pp. 5564–5569, 2010.