=Paper=
{{Paper
|id=Vol-3919/short23
|storemode=property
|title=Trunk-Mounted PDR System Based on Inverted Pendulum Model
|pdfUrl=https://ceur-ws.org/Vol-3919/short23.pdf
|volume=Vol-3919
|authors=Lei Cao,Wenchao Zhang,Dongyan Wei,Hong Yuan
|dblpUrl=https://dblp.org/rec/conf/ipin/CaoZWY24
}}
==Trunk-Mounted PDR System Based on Inverted Pendulum Model==
https://ceur-ws.org/Vol-3919/short23.pdf
Trunk-Mounted PDR System Based on Inverted
Pendulum Model
Lei Cao1,2, Wenchao Zhang 1,∗, Dongyan Wei1 and Hong Yuan1
1
Aerospace Information Research Institute, Chinese Academy of Sciences, No.9 South Dengzhuang Road, Haidian District,
Beijing 10009, China
2
School of Electronic, Electrical and Communication Engineering , University of Chinese Academy of Sciences, No.1 Yanqihu
East Road, Huairou District, Beijing, 101408, China
Abstract
The Pedestrian dead reckoning (PDR) using IMU mounted on trunk (e.g., back or chest), which is called
trunk-mounted PDR, based on the inertial navigation system (INS) has better positioning performance due
to the use of INS mechanization. However, the positioning error would accumulate rapidly because of the
high noise level of low-cost inertial measurement unit (IMU). Existing trunk-mounted PDR are almost based
on the assumption that the lateral and vertical velocity are zero, the same with the IMU is affixed to the
vehicle, called nonholonomic constraint (NHC). However, the human body is a non-stationary platform
with swaying motion observed in pedestrian movement, which does not align with the NHC. In this paper,
the pedestrian movement pattern is modeled as an inverted pendulum model (IPM), that one point is the
foot on the ground and the other one is the IMU mounted on the trunk, with the length of the pendulum is
the distance from the IMU to the ground. At the same time, a trunk-mounted PDR based on IPM is proposed,
which contains velocity measurement based on IPM (IPM-V) and distance increasement measurement based
on IPM (IPM-D). Based on IPM-V, lateral speed is calculated using angular rate from gyroscope and length
of the pendulum without assuming zero value, while lateral and vertical distances remain at zero within a
single step cycle depending on IPM-D. Experimental findings demonstrate that the proposed method offers
enhanced positioning accuracy and robustness compared to existing trunk-mounted PDR methods, with an
80% improvement in positioning accuracy.
Keywords
Pedestrian navigation, inertial navigation systems (INS), inverted pendulum mode(IPM).1
1. Introduction
The demands for navigation and positioning have increased rapidly in people’s daily lives[1]. While
the global navigation satellite system (GNSS) effectively fulfills these requirements in outdoor
settings, it faces limitations in indoor environments where satellite signals are obstructed. Various
methods such as Bluetooth[2], Wi-Fi[3], UWB[4], and magnetic field matching [5]are commonly
employed for indoor navigation. However, these techniques are infrastructure- or database-based,
necessitating significant financial investment and human resources prior to the operational
functionality of the system. Pedestrian dead reckoning (PDR) is an approach utilizing inertial
navigation systems (INS) for indoor positioning and navigation. This autonomous method relies
solely on data from the inertia measurement unit (IMU), which includes accelerometers and
gyroscopes, to determine position without the need of external information or devices[6].
Consequently, PDR is considered a cost-effective and practical solution to indoor navigation
compared to techniques that depend on infrastructure or databases.
Current PDR primarily relies on the integration of inertial navigation systems, specifically the
Foot-mounted PDR which utilizes an IMU embedded in shoes. With high frequency of sample, Foot-
mounted PDR has the ability to provide continuous 3-dimension position in real world space. Indeed,
Proceedings of the Work-in-Progress Papers at the 14th International Conference on Indoor Positioning and Indoor
Navigation (IPIN-WiP 2024)，October 14- 17, 2024, Hong Kong,China
∗
Corresponding author.
caolei22@mails.uacs.ac.cn (Lei Cao),zhangwenchao@aoe.ac.cn (Wenchao Zhang),weidy@aircas.ac.cn (Dongyan Wei),
yuanhong@aircas.ac.cn (Hong Yuan)
© 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
the errors will accumulate rapidly because of measurement noise when using a built-in Micro-Electro
Mechanical System (MEMS) IMU without any other measurement[7]. To address this issue, some
methods such as zero-velocity update technology (ZUPT)[8] and heuristic drift elimination (HDE)[9]
are proposed. However, the requirement of special shoes to assure the device could operate normally
is a notable constraint on Foot-mounted PDR[10].
Some researchers have taken attention to the potential of mounting IMU on trunk(e.g., waist,
back, chest) to promote PDR to broader applications. [11] proposes an inverted pendulum based on
a waist-mounted IMU to estimates step length; [12] integrates heading from madgwick algorithm,
step length from the step model and an efficient map-matching algorithm based on particle filtering
using a chest-mounted IMU; [13] estimates step length by Weinberg model and heading by a
quaternion-based derivation of the explicit complementary filter based on a head-mounted IMU
without rotation when operating. The scenario where a mobile phone is placed in a jacket pocket
can be compared to affixing an IMU on the trunk of an individual. In this situation, it is assumed that
the mobile phone has zero lateral velocity[14], which is similar to nonholonomic constraint (NHC),
then the INS result is used to fuse with other information like magnetic field[15].However, while an
individual moving, a distinct sway is evident, suggesting that the hypothesis does not match the real
observations.
This paper introduces a trunk-mounted PDR based on only one IMU, using the inverted pendulum
model (IPM) derived from pedestrian movement patterns. This system contains measurement of
velocity based on IPM (IPM-V) and measurement of distance increasement based on IPM (IPM-D).
Based on IPM-V, lateral speed can be computed by the date from the gyroscope. Based on IPM-D,
lateral and vertical distance are zero in one step cycle. The proposed system only relies on an IMU
consisted by accelerometers and gyroscopes, without additional devices. The experiment results
show the proposed method exhibits superior accuracy and robustness compared to existing methods
with nearly 80% improvement.
The rest of the paper is organized as follows. Section 2 introduces the principles of trunk-PDR
based on IPM including Extended Kalman Filter (EKF) and the IPM derived from pedestrian moving
patterns. In Section 0, experimental analyses are carried out to evaluate the proposed approach
against conventional methods, focusing on the positioning accuracy. Section 4 summarizes this paper
and looks forward to future research.
2. Proposed method
2.1. Coordinate systems
There are three primary coordinate systems in this paper: the IMU coordinate system (b), the human
coordinate system (h), and the local horizontal coordinate system (n). The b-frame represents the
IMU coordinate system, with its origin at the center of the IMU and the x-y-z directions indicating
forward-right-down. The h-frame represents the human body coordinate system, sharing the same
origin as the b-frame, with x-axis pointing towards the pedestrian's front, y-axis pointing to the right,
and z-axis forming a right-handed orthogonal system with x-axis and y-axis. The n-frame represents
the local horizontal coordinate system, with x-axis pointing towards virtual north, y-axis towards
virtual east, and z-axis pointing downward.
2.2. System overview
The proposed algorithm workflow is depicted in Figure 1. The IMU provides angular rate and specific
force inputs to the INS mechanization module for the computation of the IMU's navigation states,
including position, velocity, and attitude (PVA). Concurrently, the IPM module utilizes the gyroscope
angular rate with the rotation radius to produce lateral velocity measurements and zero increments
in lateral and vertical directions. At the same time, the forward velocity module utilizes specific force
data to detect steps and estimate step length. The Extended Kalman Filter (EKF) module then fuses
these measurements to improve the system's accuracy and reliability.
IMU INS
Mechanization
IPM
Accelerometer
Lateral E
Velocity K
PVA
F
Zero Distance
Increasement
Gyroscope
Forward
Velocity
Figure 1: Architecture of the trunk-mounted PDR system based on IPM
2.3. INS mechanization
The traditional INS mechanization is a typical inertial navigation algorithm with very rigorous
theoretical logic. As the MEMS-IMU has a high level of noise, some items like earth rotation can be
disregarded. therefore, the formulation of simplified INS mechanization for the MEMS-IMU is below
[16]:
n
rk rk 1 vk dt
n n
n n kb vkb
vk vk 1 Cb ,k vk g dt
n b n
(1)
2
Cbn, k Cbn,k 1 I kb k 1 k
b b
12
Where k means the sample time; dt is the sample interval; rkn and vkn are the position and
n
velocity in the n-frame at the k-th epoch; Cb,k donates the direction matrix from the b-frame to the
n-frame at the k-th epoch. vk fk ba,k dt and k k bg ,k dt donate velocity
b b b b
increasement and angle increasement in the b-frame, respectively. f kb is the specific force. kb is the
angular rate. b a , k and bg , k are the bias of the acceleration and the gyroscope, respectively. g n is the
location gravity vector in the n-frame, and is the cross-product form of the vector.
2.4. Extended Kalman Filter
In this article, the 15-dimensional error state based EKF is chosen to integrate the IMU information
and virtual measurement.
T
X r n v n ba bg (2)
Where r n is the position error in the n-frame; v n is the velocity error in the n-frame; is the
attitude error; ba and bg are errors of bias of the accelerometer and the gyroscope, respectively,
which are modeled as a first-order Markov process.
2.5. Measurement based on IPM
In this paper, the IMU is mounted on the back as an example to analyze the data from the IMU and
model the movement when a pedestrian moving. Existing trunk-mounted PDR methods almost
depend on the assumption that the lateral velocity is zero. However, as shown in Figure 2, the output
of the accelerometer on the y-axis in the h-frame cannot be disregarded because of its notable
strength and repetitive characteristics.
Acceleration(m/s 2 )
Figure 2: The data of accelometer on x-axis and y-axis in the h-fram during individual moving .
Hence, as shown in Figure 3, a pedestrian's gait involves a rhythmic alternation of the left and
right feet swinging and resting while moving. To maintain balance, the pedestrian's center of gravity
shifts to the right when the left foot swings forward and the right foot is planted on the ground. This
process is mirrored when the right foot swings forward and the left foot is grounded. As a result, the
IPM is proposed to describe the pattern of an individual’s movement. The trajectory of the IMU
mounted on the trunk follows a curve, represented by a red point in Figure 3.
𝑥
IMU
𝑦
𝑧
Figure 3: The IPM while the pedestrian moving
2.5.1. Measurement of velocity on the basis of IPM(IPM-V)
When the IMU is firmly affixed to the trunk, it will exhibit lateral movement in accordance with the
body's motion. The dynamics of the IMU can be modeled as an inverted pendulum. The foot, in
contact with the ground, serves as the pivot point, and the IMU traces a curved path. Consequently,
the tiny displacement of the curve can be articulated as follows, as illustrated in Figure 4, when
viewed from the xh direction of Figure 3:
ds d l (3)
Where d s is the tiny displacement of the IMU; d is the tiny angle through which the lever
rotates; l is the arm of the inverted pendulum, which is nearly the height from IMU to the ground.
By deducing the above equation and projecting the vector to the h-frame, the lateral velocity
obtained from the IPM can be expressed as:
v yh Cbhnb
b
, xl (4)
Where vhy is the velocity of y-axis in the h-frame; C bh is the translation matrix form the b-frame
to the h-frame, which is confirmed by the mounting angle. nb, x is the angular rate from the b frame
b
to the n frame. Considering that the MEMS IMU cannot measure the earth rotation for its high noise
level, hence, nbb ibb .
The forward velocity is calculated by step-model, which includes detecting steps and estimating
step length[17], and the vertical velocity is zero the in h-frame. According to the introduction to
measurement of velocity above, the measurement of velocity is:
SL
vh [ Cbhibb , x l 0] v h v (5)
dt
Where vh means the measurement of velocity in the h-frame; vh is the truth of velocity vector.
SL is the step length and dt is the time of step interval at n-th step. SL divided by dt is the forward
velocity. v is the noise of measurement of velocity. Given that the velocity update occurs within
the h-frame, it is necessary to convert the velocity of the state from the n-frame to the h-frame.
Therefore, the velocity from the INS mechanization in the n-frame from can be expressed as:
vˆ h C h Cˆ b vˆ n
b n
C C ( I )(v n v n )
h
b
b
n
v C C v C C (v )
h h
b
b
n
n h
b
b
n
n
Where I is the Identity matrix; is the attitude error; is the Skew-symmetric matrix of the
vector. Therefore, the measurement of error states can be written as:
zv vˆ h vh
v h Cbh Cnb v n Cbh Cnb (v n ) (v h ev ) (7)
Cbh Cnb v n Cbh Cnb (v n ) v
Hence, the measurement transition matrix is:
H k 013 Cbh，, k Cnb, k Cbh，, k Cnb, k vkn 013 013 (8)
IMU 𝑦
𝑧
l
Figure 4: Lateral velocity on the basis of IPM.
2.5.2. Measurement of distance increments on the basis of IPM (IPM-D)
Simultaneously, the movement of the IMU and feet is depicted in Figure 5 (seeing Figure 3 from the
zh direction). The illustration shows that when both feet are in contact with the ground, the IMU is
positioned centrally between the feet. Consequently, during a single step cycle, the lateral and
vertical distances of the IMU are zero, based on the IPM. As Figure 5 shows, the right foot moves
forward from k-1 to k, and in this step, the lateral distance of the IMU is zero. This establishes a
motion constraint where there is no change in the y-axis and z-axis coordinates in the h-frame.
In a single step cycle, assuming that the velocity is linear over a short period, the distance of
measurement and the INS mechanization can be expressed as [18]:
s h skh
k
1
2 v v dt
i k N 1
h
i 1 i
h
(9)
[s 0 0]T s
1 h
vˆk 1 vˆkh dt
tk
sˆkh vˆ h (t )dt vˆkh1/2 dt (10)
tk 1 2
k
1
sˆ h sˆkh 2 vˆ vˆ dt
i k N 1
h
i 1
h
i (11)
Where: vkh and vˆkh denote the velocity determined by the step-model and INS in the k-th in the h-
frame, respectively. i means the sample time in a single step cycle. Then,
k
z s sˆ h s h sˆ s
i k N 1
h
k
h
k s
1 k
vi 1 vi dt s
2 i k N 1
(12)
1 k
H i 1 xi 1 H i xi dt s
2 i k N 1
x k N k 1/k N x k (13)
Where, k / k N is the state transition matrix from tk N to t k , and N is the sample times in a single
step cycle. Finally, in a single step cycle, the measurement transition matrix is:
k
1 1
H s H k N k 1/ k N dt H i k1/ i dt H k dt (14)
2 i k N 1 2
Figure 5: Schema of IPM overlooked
3. Experiment and results
3.1. Test description
Figure 6 illustrates the positional relationship of the sensors worn by the tester, including the back
and heel. The experimental area comprises the standard office indoor environment and the outdoor
environment. The MTw IMU device from Xsens is used to collect data with 100Hz and its
specification is shown in Table 1. Five methods are used to handle the data form the IMU. They are:
1)IPM-V: This method uses the IPM-V to constrain the accumulated error from INS mechanization;
2)IPM-D: This method uses the IPM-D to constrain the accumulated error from INS
mechanization;
3)IPM-VD: This method uses the IPM-V and IPM-D both;
4)NHC-PDR: This method is state-of-the-art with the assumption that lateral velocity is zero all
the time[19];
5)Foot-PDR: This method uses the IMU mounted on the foot with basic zero-velocity update
(ZUPT) and zero-integrated heading rate (ZIHR), without additional observation such as
environment information.
The step detection and step length estimation methods in IPM-V, IPM-D, IPM-VD and NHC-PDR
are the same. FOOT-PDR uses the data from IMU mounted on the foot; IPM-V, IPM-D, IPM-VD and
NHC-PDR use the data from the IMU mounted on the back. Since FOOT-PDR is relative positioning
methods, use the angle between the tenth step and reference line to correct the test trajectory. This
procedure enables the conformity of the trajectory of the five methods.
Table 1
The specification of the MTw IMU
Sensor Types Accelerometer Gyroscope
Range (of scales) ±160m/s 2
±1200deg/s
Linearity 0.2% 0.1%
Stability - 20deg/hour
Noise 0.003m/s /Hz
2 1/2
0.05deg/s/Hz1/2
Figure 6: The positional relationship between Trunk-mounted IMU and Foot-mounted IMU.
3.2. Indoor test
The indoor trajectory is 54.4 m in length with two 90-degree turns. This test contains 10 tests for
each method. The end position error for each method, along with its average and variance, is
presented in Table 2 and Figure 7 to assess the accuracy of the position estimation in meters. In Table
2, the best and second-best results among the five methods in each trial are highlighted in red and
blue, respectively. According to Table 2, IPM-VD and IPM-V achieve the best and second-best results,
outperforming NHC-PDR and FOOT-PDR.
GroundTruth
20 IPM-V
IPM-D
IPM-VD
15 NHC-PDR
Foot-PDR
10
Error(m)
y/m
5
0
-5
-10
0 5 10 15 20 25 30 35 40 45
x/m
(a) (b)
Figure 7: The results of indoor test: (a) The estimated trajectories of indoor test in 10 tests; (b) End
point position error of the indoor test trajectories in 10 tests
Table 2
The End Point Position Error ot Five Methods in 10 Tests
test 1 2 3 4 5 6 7 8 9 10 Mean Variance
IPM-V 0.590 2.834 1.038 1.438 0.808 1.796 1.026 1.517 1.378 2.947 1.537 0.571
IPM-D 2.635 0.549 3.353 3.866 1.413 4.133 1.242 3.194 0.800 3.067 2.425 1.548
IPM-VD 0.957 2.606 0.769 1.037 0.140 1.257 0.058 1.418 0.989 1.511 1.074 0.471
NHC-PDR 3.823 3.764 3.919 4.792 3.174 0.885 0.673 2.013 1.883 1.409 2.634 1.864
FOOT-PDR 2.177 2.639 4.532 3.430 5.961 3.180 5.811 5.831 3.833 2.985 4.038 1.798
3.3. Outdoor test
The outdoor test area is shown in Figure 8, which is a rectangle of 400m in length. Five trajectories
derived from five methods and ground truth obtained from GNSS are depicted in Figure 8. In Table
3, the closing errors and positioning errors of the second turning of the five trajectories. The best
and second-best results among the five methods in each trial are also highlighted in red and blue,
respectively. The closing error of IPM-VD is 3.231m, which is 79% and 82% better than NHC-PDR
and FOOT-PDR, respectively. Considering that the results of the INS mechanization would neutralize
each other in opposing directions, the position error of the second turn, which is the farthest point
from the starting point on the entire test track, is used to evaluate the performance of the five
methods. The positioning error of the second turning of IPM-VD is 2.540m, which is 77% and 87%
better than NHC-PDR and FOOT-PDR, respectively.
The reason is that NHC-PDR assuming that lateral velocity is zero, does not align with an
individual's movement pattern, while FOOT-PDR lacks the capability to determine direction. The
proposed approach imposes restrictions on lateral and vertical velocity or distance, thereby
improving the accuracy of direction estimation.
Figure 8: The estimated trajectories of outdoor test
Table 3
The Position Error of Five Methods in Outdoor Test
test IPM-V IPM-D IPM-VD NHC-PDR FOOT-PDR
closing error/m 8.907 7.704 3.231 15.146 17.733
positioning error of the second
3.711 7.456 2.540 11.2639 19.1625
turning/m
4. Conclusion
This paper presents trunk-mounted PDR based on IPM to use only an IMU to fulfill pedestrian
positioning. The proposed method constraints the accumulated errors in the INS mechanization with
IPM derived from pedestrian movement patterns， which matches the real observations with the
lateral velocity caused by tiny slosh and the lateral and vertical distance are zero in one step cycle in
h-frame, which called IPM-V and IPM-D. The results of the experiments suggest that the proposed
methods demonstrate superior position accuracy compared to current methods, showing an
approximate 80% improvement in positioning accuracy in the outdoor test. Trunk-PDR based on IPM
provides a simple way to install and has huge potential to promote to smartphone-based pedestrian
navigation when the smartphone is put in a jacket pocket. In the future, Trunk-INS is expected to
integrate with additional techniques employed in foot-INS systems to enhance positional accuracy.
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