Probabilistic double guarantee kidnapping detection in SLAM
Tian and Ma Robot. Biomim.
Probabilistic double guarantee kidnapping detection in SLAM
Yang Tian 0
Shugen Ma 0 1
0 Department of Robotics, Ritsumeikan University , Shiga 525-8577 , Japan
1 Department of Electrical Engineering and Automation, Tianjin University , Tianjin 300072 , China
For determining whether kidnapping has happened and which type of kidnapping it is while a robot performs autonomous tasks in an unknown environment, a double guarantee kidnapping detection (DGKD) method has been proposed. The good performance of DGKD in a relative small environment is shown. However, a limitation of DGKD is found in a large-scale environment by our recent work. In order to increase the adaptability of DGKD in a large-scale environment, an improved method called probabilistic double guarantee kidnapping detection is proposed in this paper to combine probability of features' positions and the robot's posture. Simulation results demonstrate the validity and accuracy of the proposed method.
Kidnapping detection; Simultaneous localization and mapping; Autonomous mobile robots
Different fields like factories, hospitals and houses
require mobile robots to navigate autonomously and to
perform tasks by themselves. In this situation, robots
should be able to make a map of the environment and
recognize its posture (position and orientation) in this
map [1–3]. Simultaneous localization and mapping
(SLAM) is a fundamental technique that can provide the
required information to mobile robots [4, 5]. In SLAM, a
robot incrementally builds a consistent map of the
environment while simultaneously determining its posture
within this map. Many algorithms have been proposed
to be implemented in a number of different autonomous
mobile robots ranging from indoor and outdoor robots
to underwater and airborne vehicles. The methods
existing today allow the problem to be considered as solved,
but some issues still need to be studied.
Some SLAM methods using current sensor and
odometry data are based on pose tracking, which is a
localization method for detecting the location of the mobile robot
based on a given initial robot posture. Starting from this
point, the robot posture is recognized by continuously
tracking the robot’s path. If a well-tracked robot is suddenly
moved to somewhere else without being told, the problem
is called kidnapping. The autonomous robot should detect
this problem in real time; otherwise, failures or faults are
caused by the effects from kidnapping. Thus, the robot may
execute a wrong action using the wrong information, even
hurting humans, environment or damaging itself.
Especially in SLAM process, there are two
different types of situations that may occur if kidnapping
happens, which are shown in Fig. 1. When the mobile
robot is doing the SLAM process, the environment will
be mapped with features. If kidnapping happened, like
in situation 1, the mobile robot will be kidnapped to a
previously explored area. Then, the mobile robot can
recognize its posture in the global coordinates with the
known map which was created by SLAM. In situation 2,
since the robot is kidnapped to a new area, the true
location cannot be estimated by the existing map. The mobile
robot thus needs to create a new SLAM process to
estimate its position in a new global coordinates. Either
situation needs kidnapping detection to identify whether
kidnapping has happened, because the existing
kidnapping recovery method is impossible to be executed all the
time if the kidnapping belongs to situation 2.
While kidnapping happens in SLAM, the original
information including states of the robot and the map
generated by SLAM will be affected without a
kidnapping detection, as shown in Fig. 2. First, the robot creates
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Fig. 1 Two kidnapping situations in SLAM which includes the robot
kidnapped to explored area or unexplored area
Fig. 2 Information state during kidnapping in SLAM. When the SLAM
is performing, the information about the robot and the environment
is building; since the kidnapping will cause the information incorrect,
correct information should be rebuilt by suitable methods with
the extra incorrect information, such as a wrong
position of the robot and new features. Since this incorrect
information cannot refer to original global coordinates,
the mobile robot cannot utilize this information to
navigate autonomously. Second, with some classic SLAM
algorithms such as EKF-SLAM  and FastSLAM [7, 8],
the original information built before kidnapping is also
deformed. This is the point that is different in known
map kidnapping problem. Since the state of posture of
robot and all features are correlated, there is influence on
all information with the incorrect information. Therefore,
the original information will be totally deformed
without a timely kidnapping detection. For recovering from
kidnapping, the mobile robot should rebuild
information to locate itself. In this case, the efficiency of SLAM
would be significantly reduced, especially in a large-scale
In our previous research , we have proposed a
double guarantee kidnapping detection (DGKD) in SLAM.
It constructs a double guarantee to judge whether
kidnapping has happened and which type of kidnapping it
is. However, DGKD has its own limitation in the
largescale environment. In this paper, an improved method
‘probabilistic double guarantee kidnapping detection
(P-DGKD)’ is proposed to maintain similar ability of
kidnapping detection in the large-scale environment. In the
next section, the related work and limitation of DGKD
will be described.
Most of the literature focusing on the pose estimation
has concentrated on the pose tracking problem. Some
approaches explicitly dealing with sensing, model and
movement uncertainty have appeared . Common to
these approaches is that they use a probabilistic
formulation to represent and update the pose of the robot. This
has the advantage of enabling them to handle uncertainty
in a natural and convenient manner. These approaches,
also known as Markovian methods, use a spatially
discretized representation of the environment where each
cell holds the probability that the robot occupies the area
represented by the cell. They use a two-step procedure to
update this representation, namely (using their
nomenclature) the ‘move’ step where the fact that the robot moves
is accounted for by shifting probability mass between cells
according to the robot movement, and the ‘sense’ step
where Bayesian updating is used to incorporate new
evidence stemming from a feature/map comparison. In
general, the ‘sense’ step concentrates probability mass in some
areas and the ‘move’ step disperses it. This ’blurring’ is due
to the fact that the probability mass is not only shifted, but
also smeared to account for robot movement inaccuracies.
For the global localization to ’converge,’ it is important that
the evidence achieved in the ‘sense’ step more than
compensates for the additional pose uncertainty introduced by
the ‘move’ step. This fact stresses the importance of having
an efficient movement/sensing strategy, i.e., to do active
sensing, since moving randomly in general does guarantee
gaining evidence efficiently enough. However, they do the
re-localization (global localization) all the time, regardless
of whether kidnapping has happened or not. For
maintaining validity, these methods need to be executed at high
frequency. It results in wasting lots of computational cost,
which is inefficient and is not well suited for SLAM.
On the other hand, some efficient methods to solve the
kidnapping detection have been proposed recently. Physical
methods that use incorporate sensors to measure whether
kidnapping has happened or not can be applied simply and
directly, e.g., barometer , accelerometer  and switch
. However, there are limitations existing in these
methods. First, additional sensors are required to be mounted on
the robot. Therefore, the reliability of these methods cannot
be ensured while the sensors’ state is abnormal. Second,
each of the sensors can detect only a specific type of
kidnapping. Third, the robots equipped with low power source
cannot adopt these methods. Mathematical methods only
utilize inherent sensors to observe abnormal situations.
Compared to the physical methods, they can be used in the
robots that have proprioceptive and exteroceptive sensors
to locate themselves. Using entropy of location probabilities
, the robot can detect kidnapping with the given
information. However, it cannot be applied in SLAM where the
information of the map is unknown. Metric-based
detection  that can be utilized in unknown environment has
a good performance in detecting the kidnapping. However,
in some specific situations, it may fail to evaluate correctly
if kidnapping has happened, such as the robot is kidnapped
into a similar place in unexplored area.
Comparing with the other studies, two new processes
are added to execute the evaluation while SLAM is
working in DGKD procedure. They have the double
guarantees to judge whether kidnapping has happened and
which type of kidnapping it is. Figure 3 shows overall
workflow of DGKD. The method increases the reliability
of kidnapping detection that prevents the information
from deforming. In addition, a more reasonable metric
has been introduced to avoid a misjudgment in a specific
situation and a convenient method has been proposed
to determine reasonable thresholds for the metrics on a
real-time condition. However, DGKD has its own
limitation in a relatively large-scale environment. To show the
limitation of DGKD, simulation has been done in a
relatively large-scale environment without kidnapping
happening under the conditions shown in Table 1.
Table 1 Simulation condition
The size of the robot in our simulation is shown in
Fig. 4. The map and non-kidnapping simulation are
shown in Fig. 5. As indicated in Fig. 5a, ‘RPF’ denotes
the real positions of the features. ‘Wpoint’ denotes the
waypoint and ‘Wpath’ denotes the path connected with
waypoints. The robot needs to drive itself toward each
waypoint by the shortest path. The scale of the
environment is relatively large according to the size of the robot
and the range of the sensor. The scale of the map is about
11 times larger than the robot size and 15 times larger
than the range of the sensor. Figure 5b shows the
nonkidnapping SLAM progress with the map. ‘APR’ denotes
the actual position of the robot. ‘EPF’ and ‘ECE’ represent
the estimated position of the feature and its covariance
ellipses. In non-kidnapping situation, the EPF is near
the RPF and the distance between them becomes larger
as time step passed. Figure 6 shows the response of
metrics and their adapted thresholds. In non-kidnapping
situation, the value of metrics Qp and Qs should be lower
than the threshold Tp1 and Ts, where Qp and Qs belong to
front-check process and verification process separately.
However, some values of the metrics are beyond
thresholds when time step increased, as shown in Fig. 6. This is
because the uncertainty of the state has been increased
by the process of SLAM, which is its own problem in
the SLAM process. When the mobile robot is working
in the large-scale environment without loop closure, the
Fig. 3 Overall workflow of the DGKD. It embeds two new processes
in the ordinary SLAM processes to construct double guarantee
Fig. 4 Dimensions of the robot
Fig. 5 Map and EKF-SLAM. a Map for simulation. b Result of EKF-SLAM without kidnapping
Fig. 6 Non-kidnapping with DGKD. a Response of the metric Qp. b Response of the metric Qs
probability of false alarm is increased in DGKD by this
increased uncertainty, which means that the total
performance of DGKD is decreased. For keeping similar
performance of DGKD in the large-scale environment, an
improved method called probabilistic double guarantee
kidnapping detection (P-DGKD) is described in the next
Probabilistic double guarantee kidnapping
In this paper, we assume that the robot works in
2-dimensional space and the observation can be
measured all the time. The robot’s state is described by the
vector Xr = [xr , yr , φr ]T, in which (xr , yr ) represents the
position and φr represents the orientation of a frame
X (k + 1|k) =
where (Xr (k|k), Xm(k|k)) is the state at the time step k,
u(k) indicates the control measurement at time step k,
wr (k) is the process noise assumed to be white Gaussian
with zero mean, and its covariance matrix is denoted as
Q. The function f and F depends on the robot model.
Prediction of the state covariance matrix P(k + 1|k) is
Vi(k) = Zi(k + 1|k + 1) − Zi(k + 1|k)
Si(k) = Ri(k + 1|k + 1) + Ri(k + 1|k)
and N represents the number of the overlapped observed
features between sequential time step k and k + 1. Zi
denotes the observation of ith overlapped feature. Qo is
attached to the robot. The state of features is denoted by
Xm = [XmT1, XmT2, · · · ]T, in which Xmi = [xmi, ymi]T
represents the position of the feature i in the global
coordinates. Xmi is given by
where (Lxi,L yi) represents the position of feature i
referred to the local coordinates frame attached on the
robot. Therefore, the state vector is X = [XrT , XmT ]T ,
which contains both the robot state Xr and the feature
In predicting process in SLAM, the predicted state
X (k + 1|k) = [XrT (k + 1|k), XmT (k + 1|k)]T at time steps
k is given by
X (k + 1|k) = F (Xr (k|k), u(k)) + w(k)
P(k + 1|k) = ∇FX P(k|k)∇FXT + Q
where ∇FX is the Jacobian of F with respect to X
evaluated at X(k|k), P(k|k) denotes the state covariance matrix
at time step k.
The observations Z(k + 1|k) that are obtained from the
state X (k + 1|k) at the time step k are given by
Z(k + 1|k) = H (X (k + 1|k)) + v(k)
where H defines the nonlinear coordinates
transformation from the state to the observation Z(k + 1|k). The
observation noise v(k) is assumed to be white Gaussian
with zero mean and uncorrelated the process noise wr (k).
In observation process, the Z(k + 1|k + 1) is measured
from the actual environment, and its covariance matrix
is denoted by R(k + 1|k + 1). To compare observations
in sequential time steps, the kidnapping can be detected
Update the state and the associated covariance matrix
using the observation Z(k + 1|k + 1),
X (k + 1|k + 1) = X (k + 1|k) + K (k + 1)[Z(k + 1|k + 1)
− H (X (k + 1|k))]
P(k + 1|k + 1) = P(k + 1|k)
− K (k + 1)S(k + 1)K (k + 1)T
Qs is given by
Wi(k) = Zi(k + 1|k + 1) − Zi(k|k)
Mi(k) = Ri(k + 1|k + 1) + Ri(k|k)
K (k + 1) = P(k + 1|k)∇HXT S(k + 1)−1
S(k + 1) = ∇HX P(k + 1|k)∇HXT + R(k + 1|k + 1) (7)
and ∇HX is the Jacobian of H with respect to X evaluated
at X (k + 1|k).
The main structure and workflow of P-DGKD are same
with DGKD. The main difference with DGKD is the
metrics. In P-DGKD, the uncertainty of state is combined to
the metrics. New improved metrics have the same name
as DGKD and are used in original processes separately.
With the root-mean-square, Qp is given by
Dmi(k) = Xmi(k + 1|k + 1) − Xmi(k + 1|k)
Omi(k) = Pmi(k + 1|k + 1) + Pmi(k|k)
and M denotes the number of the overlapped features
between sequential time step k and k + 1.
Comparing with DGKD, P-DGKD adds the
associated covariance matrix into metrics. This modification
can decrease the effect of the increasing uncertainty in
SLAM process. It can decrease the rate of false alarm in
the whole detection reports. It causes that the whole
performance of P-DGKD is better than DGKD. About the
thresholds of metrics, the method proposed in DGKD is
used in P-DGKD. Detection and classification condition
are the same as that in DGKD.
Simulations were conducted to investigate the feasibility
and accuracy of the proposed method. We implemented
the described method using MATLAB in a personal
computer (CPU: 3.40GHz Intel Core i5, Memory: 8 GB DDR3).
The source code is based on EKF-SLAM in SLAM
package of Tim Bailey. We modified and added our method in
it. Other basic parameters for simulations are same to our
previous research of DGKD . The map of simulation has
been shown in Fig. 5a. We did simulations in a no actual
kidnapping situation during the whole SLAM process. The
results of metrics of Qp and Qs are shown in Fig. 7.
In non-kidnapping situation, the value of metric Qp is
less than the first threshold Tp1, and metric Qs is lower
than the threshold Ts. Comparing with the simulation
results of DGKD shown in Fig. 6, the value of both
metrics is below the thresholds after time steps increased.
This means that, although the uncertainty of the state
increased by the SLAM process, influence in kidnapping
detection has been decreased.
For getting the whole performance of P-DGKD, we also
did tests in kidnapping situation during SLAM process.
The kidnapping is a man-made movement to the robot
to simulate the real kidnapping that the robot is moved
by the human in the environment. Then, we can judge
whether the report of P-DGKD is a true alarm or a false
alarm or no alarm. The result of the metrics where
kidnapping happened at time step 600 is shown in Fig. 8.
Before the kidnapping, the value of metric Qp is lower
than the first threshold Tp1. When kidnapping happens in
the 600th time step, Qp is larger than the second
threshold Tp2. Moreover, the value of the metric Qs also exceeds
the threshold Ts. It means P-DGKD can detect
kidnapping correctly in this situation.
Several other simulations were conducted to show the
performance of P-DGKD and DGKD. Different
simulations were conducted to verify the performance under
the kidnapping situation and the non-kidnapping
situation. In each simulation, one-time kidnapping event is
set randomly from step 500 to time step 800. The results
processed by ROC are shown in Table 2. About report
of kidnapping in DGKD and P-DGKD, the true-positive
rate is the fraction of detected kidnapping out of the total
number of actual kidnapping events, and the false-positive
rate is the fraction of non-kidnapping time steps that is
incorrectly detected out of the total number of actual
nonkidnapping time steps. Comparing with DGKD, the
falsepositive rate is decreased in P-DGKD, which means that
the possibility of the false alarm is decreased. At the same
time, the true-positive rate of P-DGKD is also decreased
a little comparing with DGKD. That means although the
rate of false alarm is decreased, P-DGKD is not as sensitive
as DGKD dealing with the real kidnapping. Totally
considering, the performance of P-DGKD is better than DGKD.
In this paper, we have presented a probabilistic double
guarantee kidnapping detection in SLAM. Compared
with our proposed method DGKD, P-DGKD is more
suitable for the large-scale environment. In P-DGKD,
a probabilistic formulation is used to add uncertainty
of robot posture and features in the metrics which can
judge whether kidnapping happened. Using the proposed
method, the kidnapping event can be detected accurately
and robustly. Simulation result shows the validity and
feasibility of the proposed method.
In future studies, we plan to detect the kidnapping of
the robot in the case that the kidnapping happens over
Fig. 7 Non-kidnapping with P-DGKD. a Response of the metric Qp. b Response of the metric Qs
Fig. 8 Kidnapping with P-DGKD. a Response of the metric Qp. b Response of the metric Qs
Table 2 Operating characteristics of detection
a long time. Our proposed method can well solve the
problem for a short-time kidnapping event. However, if
the kidnapping happens in a long time, such as a human
carrying the robot for a long distance, or the robot slips
all the time in a specific area, the method introduced in
this paper could result in a failure. Further verification of
DGKD in a real environment is also our future work.
YT proposed the method and drafted the manuscript. SM supervised the
study. Both authors read and approved the final manuscript.
Both authors declare that they have no competing interests.
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