# American Institute of Mathematical Sciences

doi: 10.3934/jcd.2021026
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## Motion tomography via occupation kernels

 1 Oak Ridge National Laboratory, Computer Science and Mathematics Division 2 Oklahoma State University, School of Mechanical and Aerospace Engineering 3 Oregon State University, Collaborative Robotics and Intelligent Systems Institute 4 University of South Florida, Department of Mathematics and Statistics

* Corresponding author: Benjamin P. Russo

§ This work was partially completed while the author was at Farmingdale State College SUNY.
Research supported by AFOSR Awards FA9550-20-1-0127 and FA9550-21-1-0134.
Research supported by NSF grants ECCS-2027976, ECCS-2027999 and NRI 2.0 - 1925147.
This work was partially completed while the author was at the University of Michigan.
** The following YouTube playlist discusses the contents of this manuscript: https://www.youtube.com/playlist?list=PLldiDnQu2phuBWvO40eiqNve6-_GOdSsz.
Notice: This manuscript has been authored, in part, by UT-Battelle, LLC, under contract DEAC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

Received  February 2021 Revised  October 2021 Early access January 2022

The goal of motion tomography is to recover a description of a vector flow field using measurements along the trajectory of a sensing unit. In this paper, we develop a predictor corrector algorithm designed to recover vector flow fields from trajectory data with the use of occupation kernels developed by Rosenfeld et al. [9,10]. Specifically, we use the occupation kernels as an adaptive basis; that is, the trajectories defining our occupation kernels are iteratively updated to improve the estimation in the next stage. Initial estimates are established, then under mild assumptions, such as relatively straight trajectories, convergence is proven using the Contraction Mapping Theorem. We then compare the developed method with the established method by Chang et al. [5] by defining a set of error metrics. We found that for simulated data, where a ground truth is available, our method offers a marked improvement over [5]. For a real-world example, where ground truth is not available, our results are similar results to the established method.

Citation: Benjamin P. Russo, Rushikesh Kamalapurkar, Dongsik Chang, Joel A. Rosenfeld. Motion tomography via occupation kernels. Journal of Computational Dynamics, doi: 10.3934/jcd.2021026
##### References:
 [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.1090/S0002-9947-1950-0051437-7.  Google Scholar [2] A. S. Aweiss, B. D. Owens, J. Rios, J. R. Homola and C. P. Mohlenbrink, Unmanned aircraft systems (UAS) traffic management (UTM) national campaign Ⅱ, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Kissimmee, FL, 2018. doi: 10.2514/6.2018-1727.  Google Scholar [3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/coll/048.  Google Scholar [4] R. L. Burden and J. D. Faires, Numerical Analysis, $7^{th}$ edition, Brooks/Cole, 2001. Google Scholar [5] D. Chang, W. Wu, C. R. Edwards and F. Zhang, Motion tomography: Mapping flow fields using autonomous underwater vehicles, Internat. J. Robotics Res., 36 (2017), 320-336.  doi: 10.1177/0278364917698747.  Google Scholar [6] R. E. Moore, Computational Functional Analysis, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985.  Google Scholar [7] V. I. Paulsen and M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316219232.  Google Scholar [8] J. Petrich, C. A. Woolsey and D. J. Stilwell, Planar flow model identification for improved navigation of small AUVs, Ocean Engrg., 36 (2009), 119-131.  doi: 10.1016/j.oceaneng.2008.10.002.  Google Scholar [9] J. A. Rosenfeld, R. Kamalapurkar, L. F. Gruss and T. T. Johnson, Dynamic mode decomposition for continuous time systems with the Liouville operator, preprint, J. Nonlinear Sci., 32 (2022).  doi: 10.1007/s00332-021-09746-w.  Google Scholar [10] J. A. Rosenfeld, B. Russo, R. Kamalapurkar and T. T. Johnson, The occupation kernel method for nonlinear system identification, preprint, arXiv: 1909.11792. Google Scholar [11] V. Stepanyan and K. S. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Grapevine, TX, 2017. doi: 10.2514/6.2017-0670.  Google Scholar [12] W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.  Google Scholar [13] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511617539.  Google Scholar [14] W. Wu, D. Chang and F. Zhang, Glider CT: Reconstructing flow fields from predicted motion of underwater gliders, Proceedings of the Eighth ACM International Conference on Underwater Networks and Systems, (2013), 1-8.  doi: 10.1145/2532378.2532403.  Google Scholar

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##### References:
 [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.1090/S0002-9947-1950-0051437-7.  Google Scholar [2] A. S. Aweiss, B. D. Owens, J. Rios, J. R. Homola and C. P. Mohlenbrink, Unmanned aircraft systems (UAS) traffic management (UTM) national campaign Ⅱ, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Kissimmee, FL, 2018. doi: 10.2514/6.2018-1727.  Google Scholar [3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/coll/048.  Google Scholar [4] R. L. Burden and J. D. Faires, Numerical Analysis, $7^{th}$ edition, Brooks/Cole, 2001. Google Scholar [5] D. Chang, W. Wu, C. R. Edwards and F. Zhang, Motion tomography: Mapping flow fields using autonomous underwater vehicles, Internat. J. Robotics Res., 36 (2017), 320-336.  doi: 10.1177/0278364917698747.  Google Scholar [6] R. E. Moore, Computational Functional Analysis, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985.  Google Scholar [7] V. I. Paulsen and M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316219232.  Google Scholar [8] J. Petrich, C. A. Woolsey and D. J. Stilwell, Planar flow model identification for improved navigation of small AUVs, Ocean Engrg., 36 (2009), 119-131.  doi: 10.1016/j.oceaneng.2008.10.002.  Google Scholar [9] J. A. Rosenfeld, R. Kamalapurkar, L. F. Gruss and T. T. Johnson, Dynamic mode decomposition for continuous time systems with the Liouville operator, preprint, J. Nonlinear Sci., 32 (2022).  doi: 10.1007/s00332-021-09746-w.  Google Scholar [10] J. A. Rosenfeld, B. Russo, R. Kamalapurkar and T. T. Johnson, The occupation kernel method for nonlinear system identification, preprint, arXiv: 1909.11792. Google Scholar [11] V. Stepanyan and K. S. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Inf. Syst. - AIAA Infotech @ Aerosp., Grapevine, TX, 2017. doi: 10.2514/6.2017-0670.  Google Scholar [12] W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182, Readings in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.  Google Scholar [13] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511617539.  Google Scholar [14] W. Wu, D. Chang and F. Zhang, Glider CT: Reconstructing flow fields from predicted motion of underwater gliders, Proceedings of the Eighth ACM International Conference on Underwater Networks and Systems, (2013), 1-8.  doi: 10.1145/2532378.2532403.  Google Scholar
Algorithm 1, used for estimation of the vector field in (14). The true trajectories are calculated via RK4 over the time frame $[0, 1]$. Using Gaussian RBFs with a kernel width of 1, we performed $10$ iterations of Algorithm 1
Calculated initial trajectories and output of Algorithm 1 for 5, 10, and 20 iterations on Gliderpalooza data
] for estimation of the vector field in (14)">Figure 3.  A comparison of Algorithm 1 and the method of [5] for estimation of the vector field in (14)
To demonstrate universality of the occupation kernel basis, Algorithm 1 is used to estimate three different flow fields. The flow field in (14) (Flow field 1), a linear flow field (Flow field 2), and a constant flow field (Flow field 3). The plot shows the average estimation error, as defined in 16, plotted against the number of iterates of Algorithm 1 for the three flow fields
] (blue arrows)">Figure 5.  Simultaneous plots for estimation of the unknown flow field in the Gliderpalooza experiment using Algorithm 1 (green arrows) and the method of [5] (blue arrows)
Algorithm 1, along with the technique in [5] are used to estimate the vector field in (14). The table shows the maximum and the average estimation error, as defined in (15) and (16), respectively, for the two methods
 Method in [5] Algorithm 1 Max Error 1.1849 0.25321 Mean Error 0.51549 0.025642
 Method in [5] Algorithm 1 Max Error 1.1849 0.25321 Mean Error 0.51549 0.025642
Norm difference statistics between estimates of the Gliderpalooza flow field, approximated using Algorithm 1 and the method in [5]
 max norm difference 0.14877 mean norm difference 0.0088628 variance 0.0001869
 max norm difference 0.14877 mean norm difference 0.0088628 variance 0.0001869
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