IROS2019国际学术会议论文集 1541

A Novel Robust Approach for Correspondence Free Extrinsic Calibration Xiao Hu1 Daniel Olesen1 Knudsen Per1 Abstract Extrinsic calibration is a necessary step when using heterogeneous sensors for robotics applications Most existing methods work under the assumption that the prior data correspondence is known Considering data loss and false measurements the correspondence may not be accessible in practice To solve this problem without knowing the corre spondence several probabilistic methods have been proposed However an implicit restriction on input data limits their application Therefore in this paper we propose a more stable correspondence free method with two improvements that can relax the restrictions on inputs and improve the calibration accuracy The fi rst improvement fi nds consistent sets from raw inputs using screw invariants which signifi cantly improve the robustness in case of outliers and data loss A new optimization method on matrix Lie group is proposed as the second improvement which demonstrates better accuracy The experimental results on both numerical and real data show the superiority and robustness of the proposed method I INTRODUCTION Extrinsic calibration for heterogeneous sensors is one of the most fundamental and important tasks for many robotics applications e g sensor fusion and Simultaneous Localiza tion And Mapping SLAM In general extrinsic calibration between two rigidly connected coordinate systems as shown in Fig 1 can be modeled as AX XB A B X are homogeneous matrices in SE 3 which also relates to the hand eye calibration problem The previous work 1 has proved that at least two motion pairs with nonparallel rotation axes are needed to estimate X If the data correspondence is known this problem can be solved with certifi cated global optimality by approach 2 However fi nding pair wise correspondence could be challenging in practice in case of package loss during data transmission or outliers generated by abnormal sensor measurements Therefore several probabilistic methods have been proposed recently to solve the calibration problem without the need to know the correspondence Although the pairwise correspondence is not essential for those methods there is still an implicit assumption that the two sets should be bijective In other words every data in set A must have a corresponding match in set B and vice versa If this assumption does not hold then their performance will drop dramatically which constrains the application of those methods in practice This work was fi nancially supported by the Innovation Fund Denmark through the project Unmanned Aerial Vehicle for high Quality Magnetic Surveying UAV QMS 1Xiao Hu Daniel Olesen and Knudsen Per are with the Na tionalSpaceInstitute TechnicalUniversityofDenmark Elektro vej 2800 Kgs Lyngby Denmark danole space dtu dk xiahaa space dtu dk pk space dtu dk CA y x z CB Ai Ai 1 Bi Bi 1 X X X Fig 1 Extrinsic calibration problem CA CBrepresent the coordinate frames of sensor A and sensor B respectively The corresponding transfor mation matrix from CBto CAis denoted with X The dashed connections are transformation matrices directly measured and dotted connections are transformation matrices to be estimated This paper focuses on solving the calibration problem without knowing the correspondence beforehand By con sidering the restriction on data sets that required by the aforementioned correspondence free approaches we propose a method which can relax the restriction on data sets and improve the accuracy using a new optimization algorithm on matrix Lie group The proposed method starts with a prepro cessing modular to fi nd consistent sets from the raw inputs by computing the consistency using screw invariants After obtaining the consistent sets we carry out the calibration using the proposed optimization approach Compared with previous methods which use linearization this optimization method works directly on special Euclidean group for better accuracy Evaluations on synthetic data and real datasets have been carried out to validate the performance of the proposed method The experimental results show the supe riority and robustness of the proposed method against other correspondence free approaches When compared to state of the art correspondence dependent methods it also shows stable performance The main contributions of our work are 1 we propose a method for consistent set fi nding which can signifi cantly improve the robustness against data loss and outliers 2 we propose a new algorithm using optimization on matrix Lie group which can improve estimation accuracy The remainder of the paper is organized as follows Section II presents related work Section III introduces the notation and mathematical preliminaries used in this paper Section IV explains the details of the proposed approach Section V describes the experiments and results Finally the conclusion is drawn in Section VI II RELATEDWORK The hand eye calibration problem was fi rstly studied in 3 followed by a considerable number of works on hand 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE1 eye calibration Related work is reviewed in this section The fi rst category of methods attempts to solve the hand eye calibration in a decoupled fashion i e fi rstly the rotation estimation and followed by the estimation of the translational part The majority of the earlier proposed approaches 1 3 5 belong to this category Their formulations vary with the parameterizations of the rotational part e g axis and angle unit quaternion etc Decouple rotation from transla tion yields effi cient solutions but at a cost that the rotation estimation errors would propagate to the translational part The second category of methods solves the hand eye calibration jointly by estimating the rotation and translation simultaneously 4 6 8 The introduction of the screw theory in 9 provides a geometric view to analyze the restrictions of the hand eye calibration problem which later motivated the invention of the dual quaternion approach 6 A similar linear formulation was proposed in 7 by using the Kronecker product Due to noise neither the constraint of the unit quaternion q 1 nor the orthogonality constraint of SO 3 can be maintained which requires normalization or orthogonalization To suppress the infl uence of noise iterative nonlinear optimization methods 4 8 have been proposed However iterative approaches rely on good initial values for accuracy and fast convergence while obtaining fa vorable initial values may be diffi cult in practice Therefore global optimization methods arise nowadays which can per form calibration without the request for good initial values e g using Second Order Cone Programming SOCP 10 using multivariate polynomial optimization 11 and by solving the Lagrange dual problem 12 All the aforementioned approaches focus on solving the problem by assuming that the data correspondence is known in advance To deal with motion pairs with unknown cor respondences several stochastic methods 13 15 which we summarize as the third category have been proposed recently The latest work by 14 proposed two novel batch methods that show better performance compared with ap proaches in 13 15 However methods 13 15 share the common assumption that the two sets should be bijective Put differently every data in set A must have a corresponding match in set B and vice versa In case that this assumption breaks their performance will drop dramatically III NOTATION 2 the proposed method is capable of dealing with irregular cases including random outliers and data loss However the proposed method is relatively more time consuming which leaves as a future improvement 050100150200 Number of samples 0 0 1 0 2 0 3 0 4 0 5 0 6 Time s Runtime Comparison B1 B2 BS Fig 5 Runtime comparison the fi gure shows the evaluations of runtime versus the number of measurements Mean runtime of 50 trials is used as the vertical ordinate C Real Datasets In the second experiment we benchmark the perfor mance of BS against correspondence dependent methods on real datasets to evaluate its accuracy and robustness For a fair comparison data correspondence is provided to correspondence dependent methods KR NLQ ATA SOCP GPOLY DUAL In the fi rst comparison the real RGB D data provided by 18 is used The results are 5 TABLE I QUANTITATIVE RESULT ONRGB DDATASET MethodER rad Et m Runtime s KR0 03100 05880 0221 NLQ0 07390 02331 1680 ATA0 02600 01270 5185 SOCP0 10050 12735 8258 GPOLY0 02550 01801 6093 DUAL0 02630 01250 9428 BS0 02800 03440 6671 0 0 2 0 4 0 6 RMSE Rotation KR NLQ ATA SOCP GPOLY DUAL BS 0 0 1 0 2 0 3 RMSE Translation KR NLQ ATA SOCP GPOLY DUAL BS Prime 1 0 0 1 0 2 0 3 RMSE Rotation KR NLQ ATA SOCP GPOLY DUAL BS 0 0 05 0 1 RMSE Translation KR NLQ ATA SOCP GPOLY DUAL BS Prime 2 Fig 6 Evaluations on RMSE of rotation and translation on datasets primesense 1 and primesense 2 For both datasets the RMSE of rotation is shown in the left plot whereas the RMSE of translation is shown in the right plot shown in Table I The rotational and translational errors are computed with respect to the ground truth provided in the dataset Based on Table I GPOLY has the most accurate result for rotation estimation Regard to translation DUAL bypasses others BS shows slightly worse accuracy compared with global optimization methods such as GPOLY DUAL In the second evaluation the datasets primesense 1 and primesense 2 from 19 are used Since no ground truth is provided in those datasets similarly to 19 the Root Mean Squared Error RMSE of the translation and orientation is used for evaluation The boxplots in Fig 6 represents the experimental results For both datasets we can see that except NLQ and SOCP all other methods demonstrate stable calibration performance with slight differences To summarize from the second experiment certain global optimization methods show their superiority in terms of accuracy The proposed correspondence free approach shows stable performance on real datasets but the accuracy is slightly worse than some advanced global optimization meth ods However considering its capability of working under the correspondence free condition we can use the proposed method as the front end for establishing data correspondence and fi nd a good initial value then refi ne the result using advanced global optimization methods for better accuracy VI CONCLUSION In conclusion this paper presents a new and stable correspondence free extrinsic calibration approach We pro pose a consistent set fi nding method that can detect and extract consistent sets from raw inputs which signifi cantly improves the robustness against outliers and missing data Secondly a matrix Lie group based optimization algorithm is applied to estimate mean for better accuracy Experiments have been carried out to evaluate the performance of the proposed method and other state of the art approaches in terms of accuracy stability and runtime The results show the superior performance of the proposed approach over the comparison approaches Future work will be to speed up computation Improve the performance against noise via leveraging soft weighting scheme will also be considered REFERENCES 1 R Y Tsai and R K Lenz A new technique for fully autonomous and effi cient 3d robotics hand eye calibration IEEE Transactions on robotics and automation vol 5 no 3 pp 345 358 1989 2 M Giamou Z Ma V Peretroukhin and J Kelly Certifi ably globally optimal extrinsic calibration from per sensor egomotion IEEE Robotics and Automation Letters 2018 submitted Online Available https arxiv org abs 1809 03554 3 Y C Shiu and S Ahmad Calibration of wrist mounted robotic sensors by solving homogeneous transform equations of the form ax xb IEEE Transactions on robotics and automation vol 5 no 1 pp 16 29 1989 4 R Horaud and F Dornaika Hand eye calibration The 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