Randomly Added False Constraints



To validate our robust pose graph optimization technique, the Manhattan 3500 data set will be utilized. As an initial test, the original data set is corrupted by randomly adding erroneous range constraints. An example of a corrupted pose graph is provide in the right-hand-side of Fig. 1, where the red lines represent erroneous constraints. For this evaluation, up to 1500 false constraints (i.e., over 20\% of the total constraints) are added to the original pose graph.



Figure 1 :: Manhattan 3500 dataset. (Left-hand-side) Un-corrupted graph. (Right-hand-side) Pose-graph corrupted by erroneous constraints



Utilizing the corrupted Manhattan dataset, the proposed methodology is evaluated along side two commonly used approaches. To quantify the accuracy of the optimzier, the median of the residual sum of squares (RSOS) of the $\it{X-Y}$ positioning error is reported, as in Fig. 2. From Fig. 2, it can be seen that the max-mixtures approach, with a pre-defined mixture model, performs considerable worse than the switchable constraint and clustering technique as the number of erroneous constraints increases. Additionally, it should be noted that both switchable constraints and the clustering technique stay relatively constant, with respect to the median RSOS error, as the number of false constraints is increased; however; the clustering optimization technique provides a smaller bias.



Figure 2 :: Accuracy of the estimated measurement covariance model, with respect to the Frobenius norm, as a function of the number of erroneous constraints



With it shown that the proposed method is as robust other state-of-the-art optimization techniques, the discussion can proceed to the principle benefit of the proposed approach, which is that accurate knowledge of the $a-priori$ measurement covariance is not required. To expand upon this idea, the pose graph thats corrupted by 1500 erroneous constraints in greater detail.

First, we can extract the residuals from the optimized graph and visually evaluate the performance of the estimated covariance. A scatter plot of the residuals is provided in the left-hand side of Fig. 3, where the black cluster represent the inlier distribution and the red scatter represents the residuals of the erroneous constraints. On the right-hand side of Fig. 3, we can see our estimated inlier covariance encapsulating the inlier residuals.



Figure 3 :: Estimated measurement inlier distribution for the Manhattan 3500 data set corrupted by 1500 erroneous constraints. An identity $a-priori$ measurement covariance was provided.



We can extend the analysis of the optimizer’s ability to accurately estimate the measurement covariance model by evaluating the performance as the number of erroneous constraints varies. To quantify the accuracy of the estimated covariance model, the Frobenius norm of the difference between the true covariance matrix and the estimated matrix is utilized, as depicted in Fig. 4. From Fig. 4, it should be noted that the error in the estimated covariance distribution is relatively flat with respect to the number of erroneous constraints.



Figure 4 :: Accuracy of the estimated measurement covariance model, with respect to the Frobenius norm, as a function of the number of erroneous constraints