State Estimation with Delayed Measurements Considering Uncertainty
of Time Delay
Minyong Choi, Jinwoo Choi, Jonghoon Park and Wan Kyun Chung
Abstract—State estimation problem with time delayed mea-
surements is addressed. In dynamic system with noise, after
taking measurements, it often requires some time until that is
available in a filter. A filter not considering this time delay
cannot be used since a current measurement is related with a
past state. These delayed measurements problem is solved with
augmented state Kalman filter, and uncertainty of the delayed
time is also resolved based on the probability distribution of the
delay. The proposed method is analyzed by a simple example,
and its consistency is verified.
State estimation with noisy measurements is required
in many applications. Researchers have been developed a
number of filtering methods. In order to reduce the effect of
that noise, Extended Kalman Filter (EKF) or particle filter
is widely used.
In general, it is assumed that a measurement is transmitted
to a filter without any time delay. There is, however, a chance
to have the delay between an instant when a measurement
is taken by a sensor and another instant when the measure-
ment is available in the filter. For example, communication
through a network demands a fundamental processing time.
Sometimes, data processing time is required to obtain higher
level information from raw sensor data. This problem that
measurements are available in the filter with delay is referred
as Out Of Sequence Measurement (OOSM) problem . In
addition, uncertainty of the delay should also be resolved.
Even though an average delay is known, it is not an exact
value, and cannot be used in the filter naively.
When there is no time delay, several prediction steps are
carried out using a process model before a measurement
arrives in the filter. Unfortunately, in the OOSM problem,
the predicted state is not meaningful because the delayed
measurement contains information about the past state. In
order to handle it, backward prediction or retrodiction can
be used. The past state can be obtained by applying the
inverse model of the process model to the current state.
The optimal approach of one step delay using the backward
prediction was presented in  and multi step delay was
also solved in . For those cases, linear process model was
supposed to get the inverse model and the optimal solution.
However, if the process model is nonlinear and it is hard to
Minyong Choi and Jinwoo Choi are with the Department of Mechanical
Engineering, Pohang University of Science and Technology, San 31, Pohang,
Jonghoon Park is with SimLab Co., Ltd., Seocho3-Dong, Seocho-Gu,
Seoul, Korea firstname.lastname@example.org
Wan Kyun Chung is with Faculty of Mechanical Engineering, Po-
hang University of Science and Technology, San 31, Pohang, Korea
obtain the inverse model, these methods may require some
Larsen et al.  proposed a method extrapolating the
delayed measurements. As extrapolating the delayed mea-
surements, the current measurement could be calculated and
this value was directly applied to the filtering algorithm. This
approach has low computational cost and does not need any
modification in the filter algorithm. But it might have a weak
point in a dynamic system having fast movements.
A state augmentation for delayed state was also used when
there were several steps of time delay in measurement. After
this augmentation, not only Kalman filter, EKF could be used
directly. Challa et al.  solved the OOSM problem using
this augmented state Kalman filter (ASKF). The uncertainty
of the delay was also resolved by means of Probabilistic
Data Association Filter (PDAF). Merwe et al.  solved
the problem with almost same formulation but sigma-point
Kalman filter was used instead of EKF. In these methods, as
the time delay increases, the number of augmented states also
increases, and the augmentation has a weak point requiring
more memory and computational power. If, however, the
measurement delay shouldn’t be too large, this method is
easy to implement, and be expended the nonlinear models.
Julier and Uhlmann  suggested a method in which
an unknown time delay could be treated by means of the
covariance union algorithm. In this method, the covariance
was estimated conservatively since the worst case, unknown
delay, was considered. But, if the time delay is able to be
modeled as a form of a distribution, it is helpful in estimation
using that quantity considering the uncertainty of the delayed
In vehicle localization, fusion of delayed observations is
also a problem. A practical fusion method for the localization
of an outdoor vehicle is presented in .
In this paper, the state estimation with the delayed mea-
surements, OOSM problem, is considered. Several steps of
the delayed measurement will be treated to reinforce the
current state. The uncertainty of the delayed time will also be
considered using its own distribution because the measured
time delay may have noise in practice. First of all, it is
assumed that the delayed time is small enough to accept
for using augmentation, and augmented state Kalman filter
is adopted for dealing with delays. Then, it is supposed
that a Probability Density Function (PDF) representing the
uncertainty of delay can be modeled. A bundle of updates
are carried out for the delay uncertainty based on the PDF.
Although the proposed method has similar form with Prob-
abilistic Data Association Filter (PDAF), both are different
 Y. Bar-Shalom, H, Chen and M. Mallick, “One-Step Solution for the
Multistep Out-of-Sequence-Measurement Problem in Tracking,” IEEE
Transactions on Aerospace and Electronic Systems, Vol. 40, No. 1,
pp. 27-37, 2004.
 T. D. Larsen, N. A. Andersen, O. Ravn and N. K. Poulsen, “Incorpora-
tion of Time Delayed Measurements in a Discrete-time Kalman Filter,”
Proceedings of the 37th IEEE Conference on Decision & Control, pp.
 S. Challa, R. J. Evans and X. Wang, “A Bayesian solution and its ap-
proximations to out-of-sequence measurement problems,” Information
Fusion, Vol. 4, pp. 185-199, 2003.
 R. Merwe, E. A. Wan and S. J. Julier, “Sigma-Point Kalman Filters for
Nonlinear Estimation and Sensor-Fusion - Applications to Integrated
Navigation -,” AIAA Guidance, Navigation, and Control Conference
and Exhibit, AIAA-2004-5120, 2004.
 S. J. Julier and J. K. Uhlmann, “Fusion of Time Delayed Mea-
surements With Uncertain Time Delays,” Proceedings of the 2005
American Control Conference, pp. 4028-4033, 2005.
 C. Tessier, C. Carious, C. Debain, F. Chausse, R. Chapuis and C.
Rousset, “A Real-Time, Multi-Sensor Architecture for fusion of de-
layed observations: Application to Vehicle Localization,” Proceedings
of the 2006 IEEE Intelligent Transportation Systems Conference, pp.
 Y. Bar-Shalom, X. Li and T. Kirubarajan, Estimation with Applications
to Tracking and Navigation, John Wiley and Sons, NewYork, 2001.
 T. B. Bailey, J. Nieto, J. Guivant, M. Stevens and E. Nebot, “Con-
sistency of the EKF-SLAM Algorithm,” Proceedings of the 2006
IEEE/RSJ International Conference on Intelligent Robots and Systems,
pp. 3562-3568, 2006.