Journal of Intelligent & Robotic Systems

, Volume 93, Issue 1–2, pp 289–301 | Cite as

Control-enabled Observability and Sensitivity Functions in Visual-Inertial Odometry

  • He BaiEmail author
  • Clark N. Taylor


Visual-inertial odometry (VIO) is an important component in autonomous navigation of Unmanned Aerial Vehicles (UAVs) in GPS-denied or degraded environments. VIO is a nonlinear estimation problem where control inputs, such as acceleration and angular velocity, have significant impact on the estimation performance. In this paper, we examine the effects of controls on the VIO problem. We first propose a sensitivity function that characterizes the relationship between the errors in the control inputs and the state estimation performance. This function depends on the control inputs, which is unique for nonlinear systems since for linear systems, state observability properties are independent of control inputs. We next derive analytical expressions of the sensitivity functions for various VIO scenarios relevant to UAV motions. Using Monte-Carlo simulations, we validate the derived sensitivity functions. We also show an interesting fact that deceleration along the velocity direction yields better estimation performance than acceleration with the same magnitude.


Visual-inertial odometry Unmanned aircraft systems Observability Sensitivity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bai, H., Taylor, C.N.: Control-enabled observability in visual-inertial odometry. In: 2017 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 822–829. (2017)
  2. 2.
    Beard, R.W., McLain, T.W.: Small unmanned aircraft: theory and practice. Princeton University Press, Princeton (2012)CrossRefGoogle Scholar
  3. 3.
    Dellaert, F., Kaess, M.: Square root sam: Simultaneous localization and mapping via square root information smoothing. Int. J. Robot. Res. 25(12), 1181–1203 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22(5), 728–740 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jones, E.S., Soatto, S.: Visual-inertial navigation, mapping and localization: A scalable real-time causal approach. Int. J. Robot. Res. 30(4), 407–430 (2011)CrossRefGoogle Scholar
  6. 6.
    Khalil, H.K.: Nonlinear systems. 2002. ISBN 130673897: 9780130673, 893 (2002)Google Scholar
  7. 7.
    Krener, A.J., Ide, K.: Measures of unobservability. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009., IEEE, pp. 6401–6406 (2009)Google Scholar
  8. 8.
    Li, X.R., Zhao, Z., Jilkov V.P.: Practical measures and test for credibility of an estimator. In: Proceedings of the Workshop on Estimation, Tracking, and Fusion—A Tribute to Yaakov Bar-Shalom, pp. 481–495 (2001)Google Scholar
  9. 9.
    Martinelli, A.: State estimation based on the concept of continuous symmetry and observability analysis: The case of calibration. IEEE Trans. Robot. 27(2), 239–255 (2011)CrossRefGoogle Scholar
  10. 10.
    Martinelli, A: Vision and imu data fusion: Closed-form solutions for attitude, speed, absolute scale, and bias determination. IEEE Trans. Robot. 28(1), 44–60 (2012). CrossRefGoogle Scholar
  11. 11.
    Martinelli, A.: Closed-form solution of visual-inertial structure from motion. Int. J. Comput. Vis. 106(2), 138–152 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mirzaei, F.M., Roumeliotis, S.I.: A kalman filter-based algorithm for imu-camera calibration: Observability analysis and performance evaluation. IEEE Trans. Robot. 24(5), 1143–1156 (2008)CrossRefGoogle Scholar
  13. 13.
    Reif, K., Gunther, S., Yaz, E., Unbehauen, R.: Stochastic stability of the discrete-time extended kalman filter. IEEE Trans. Autom. Control 44(4), 714–728 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rutkowski, A.: The most accurate path from point a to point b is not necessarily a straight line. In: AIAA Guidance, Navigation, and Control Conference, p. 4761 (2012)Google Scholar
  15. 15.
    Taylor, C N, Veth, M J, Raquet, J F, Miller, M M: Comparison of two image and inertial sensor fusion techniques for navigation in unmapped environments. IEEE Trans. Aerosp. Electron. Syst. 47(2), 946–958 (2011)CrossRefGoogle Scholar
  16. 16.
    Titterton, D., Weston, J.L.: Strapdown inertial navigation technology, vol 17. IET, Stevenage (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mechanical and Aerospace EngineeringOklahoma State UniversityStillwaterUSA
  2. 2.Senior Research Electronics Engineer, Sensors DirectorateAir Force Research LabWright-PattersonUSA

Personalised recommendations