1. Introduction
Micro Aerial Vehicles (MAVs), which refer to small Unmanned Aerial Vehicles (UAVs), have gained popularity in aerial robotics due to their small size, agility, and versatility. It is possible for MAVs to perform a wide range of tasks, including surveillance, inspection, mapping, and environmental monitoring. However, achieving coordinated missions for MAV formations is a challenging task, as it requires precise and reliable navigation and positioning capabilities. Furthermore, such missions involve complex maneuvers and high synchronization between multiple MAVs, and any errors or inaccuracies in navigation or positioning can result in mission failure or even accidents. Therefore, developing robust and efficient navigation and positioning methods is crucial for enabling MAVs to perform coordinated missions effectively and safely.
Urban or forest environments are renowned for their complexity and heterogeneity, which pose significant challenges to wireless communication systems due to issues such as multipath propagation and shadowing [
1]. In this context, Hermosilla et al. [
2] proposed the use of street-based urban metrics descriptions to quantify various spatial patterns of urban types constructed at different periods, and provided eight different types of urban environments (including urban, residential, and industrial areas) based on their structural characteristics, such as building height distribution or vegetation coverage. These differences may impact wireless signal propagation in different ways. For example, historic urban areas with lower building heights and narrower streets may experience more severe multipath effects, while emerging industrial areas with taller buildings and wider streets may face more severe signal blockage issues. As a result, there are numerous challenges in receiving Global Navigation Satellite System (GNSS) signals in urban or forest environments due to the weak anti-interference ability of GNSS [
3]. Despite some related studies [
4,
5,
6] having reduced the error in GNSS-based positioning systems through multi-information fusion technology, intentional interference may render GNSS unusable in certain specialized fields, such as military applications. Therefore, achieving relative positioning of MAVs in GNSS-denied scenarios has become a potentially fruitful area of research.
The common methods for achieving relative positioning in GNSS-denied environments can be classified into the following three categories: visual positioning [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17], inertial navigation positioning [
18,
19,
20,
21,
22], and radio positioning [
23,
24]. The development status of each category of positioning technology is systematically discussed below, and the advantages and limitations of each technology are analyzed, providing good inspiration for the study of relative positioning methods for MAVs in GNSS-denied environments.
The visual localization methods can be broadly categorized into map-based localization [
7,
8,
9,
10,
11] and map-free localization [
12,
13,
14,
15,
16,
17], depending on whether prior visual maps are utilized. (Note that visual refers to visual information, which is typically collected using sensors such as cameras or laser scanners. Visual maps can be understood as maps constructed based on visual information and used for visual localization and navigation.) In the map-based localization, a pre-constructed visual map was used to aid in localization, and the steps involved in map construction and updating, image retrieval, feature point extraction and matching, and precise localization have been studied extensively [
7,
8,
9,
10]. The main focus of [
7] was the construction of 3D maps, while reference [
8] addressed map quality issues by removing outliers and tracking lanes. Map matching problems were mainly addressed in [
9,
10,
11]. In contrast, map-free visual localization methods do not rely on prior visual maps and instead estimate the pose of the object and surrounding environment. This category can be further divided into Visual Simultaneous Localization and Mapping (VSLAM) [
12,
13,
14,
15] and Structure from Motion (SFM) [
16,
17]. The VSLAM is designed for real-time processing, making it well-suited for applications such as robotics and autonomous vehicles, while SFM prioritizes accuracy and is more appropriate for offline processing applications such as digital reconstruction of scenes [
16]. Specific techniques within these categories have also been put forward. For example, an efficient distributed particle filter (EDPF) was proposed in [
12] to address the difficulty of sampling high-dimensional state spaces in range-only SLAM. Reference [
13] proposed a weight-optimized particle filter-based algorithm for monocular visual SLAM, which aimed to improve the slow environmental interference repair speed of traditional filtering SLAM algorithms. Three-level parallel optimization was adopted in [
14], including the direct method, feature-based method, and pose graph optimization. The method in [
15] involves two stages: the first stage implements a local SLAM process based on filtering techniques, while the second stage utilizes optimization-based techniques for constructing and maintaining a consistent global map of the environment, which includes addressing the loop closure problem. To estimate the state of a MAV, the main filtering technique employed is the Extended Kalman Filter (EKF). In terms of optimization techniques, three methods are used: a local bundle adjustment technique to minimize the total reprojection error, the minimization of the Perspective-n-Point (PnP) problem, and graph optimization to correct the global map. In [
16], 3D point clouds of close-range images were generated using SFM technology. Homologous features between different images were found to determine the shooting direction and position of each image. Finally, reference [
17] used optimization to enhance the convergence rate of SFM by retrieving maximum internal similarity between images. The chosen images and query results were used to reconstruct a three-dimensional scene and estimate relative camera positions with SFM.
The inertial navigation positioning is an autonomous method that provides accurate short-term navigation and positioning without relying on external information or emitting radiation. However, the general positioning system is prone to an error accumulation due to the second-order integration operation [
18,
19,
20,
21,
22]. Various studies have been conducted to address this issue. For instance, a 3D trajectory planning method based on Particle Swarm Optimization-A star (PSO-A*) algorithm was proposed in [
18] to solve the yaw problem of intelligent aircraft. In [
19], the nonlinear error model was considered, and internal measurement information was utilized to correct the nonlinear error of inertial navigation. The zero-velocity detection algorithm was adopted to compensate for the accumulated error of pedestrian inertial navigation [
20,
21]. Additionally, reference [
22] employed Convolutional Neural Networks (CNN) to reduce the error of MEMS IMU sensors.
There are various wireless positioning methods based on radio technology, mainly including infrared positioning [
23], Ultra Wide Band (UWB) positioning [
24], and radio frequency identification (RFID) positioning [
25]. In [
23], the angle of an object equipped with a positioning device was measured using an infrared laser beam emitted by a lighthouse base station, and the position of the object was calculated. An indoor positioning system based on improved adaptive Kalman filter (IAKF) was proposed in [
24] that calculated the distance between the tag and the base stations using the time it took for UWB signals to travel, and then applied a triangulation algorithm to acquire the position of the tag. Similar to [
24], reference [
25] differs mainly in the method of distance measurement, using RFID technology to calculate the distance. In addition, Ref. [
25] mentions that RFID positioning tags can be classified into two types, active and passive, depending on their power requirements. Passive tags mainly involve backscattering communication, and therefore do not require direct power supply, but they have limited communication range compared to active tags. Therefore, RFID technology is mostly used for indoor product tracking. Furthermore, the design, installation, and maintenance of RF navigation systems can be expensive and complex, which may pose challenges for deploying them in certain applications, such as small unmanned aerial vehicles.
In addition, multi-technology fusion is an important means to improve the positioning performance. An important trend in VSLAM is to integrate visual sensor data with other sensor data [
26,
27]. A method was proposed in [
26] to integrate Inertial Measurement Unit (IMU) and dynamic VSLAM, which avoided the static assumption of common SLAM algorithms and solved the problems of fast vehicle motion and insufficient light. This method exhibited higher robustness compared with pure visual dynamic SLAM systems. In [
27], a SLAM autonomous positioning algorithm combining magnetometer, IMU, and monocular camera was proposed to address the initialization instability and drift problem in the visual-inertial SLAM (VI-SLAM) algorithm. In wireless positioning, UWB technology is a potentially productive area of research [
28,
29,
30] in multi-information fusion positioning due to its advantages such as strong penetration ability, low power consumption, small impact of multipath effects, and high positioning accuracy [
31,
32,
33]. In [
28], UWB ranging was used in an indoor positioning scenario, and the position was jointly estimated by fusing the ranging and IMU information through cooperative positioning, which reduced the position drift of Inertial Navigation System (INS). A relative positioning method based on trilateration was proposed for the multi-mobile user mutual positioning scenario [
29], where the UWB ranging and IMU information were fused and integrated into a probabilistic framework for cooperative positioning fault recovery. In [
30], the pedestrian navigation was realized based on IMU and UWB ranging, and a zero-velocity detection algorithm and single anchor point reference were used. Due to the inherent error drift of IMU, relying solely on IMU for inertial navigation positioning is uncommon. Typically, the fusion of multi-sensor information is required to reduce the positioning errors [
26,
27,
28,
29,
30,
31,
32,
33,
34,
35].
Authors have two observations on the existing research on positioning methods.
Motivated by the above facts, this paper investigates a relative positioning method of a two-MAV cooperative formation fusing inter-vehicle distance and IMU information in GNSS-denied environments, which does not require the use of other auxiliary devices such as odometers, except for the IMU and the devices used for ranging. The main contributions of this paper are summarized as follows:
A novel method for relative positioning in GNSS-denied scenarios is proposed based on ranging and IMU information. The method utilizes the EKF algorithm to jointly estimate the relative positions of MAVs, providing continuous, precise, and reliable information for the formation without being affected by the error accumulation problem of IMU. Additionally, the method is capable of achieving relative positioning for an arbitrary number of nodes without requiring an accurate reference anchor. This innovative approach offers notable advantages over existing methods and has great potential for applications in various fields of aerial robotics.
Theoretical derivations of the system observability conditions are presented, along with the specific expressions. The conditions indicate that the system positioning accuracy and reliability can be guaranteed when the flight trajectory of the MAV formation satisfies the observability conditions. Failure to satisfy these conditions may result in decreased positioning accuracy and reliability, as well as a possibility of divergence.
Monte Carlo simulations were conducted, where the correctness of the system observability conditions was verified and the effects of different ranging errors on the positioning accuracy and reliability were investigated. Moreover, the positioning error was reduced by approximately 3.89 times compared to an existing positioning method [
32].
The rest of this paper is organized as follows.
Section 2 describes the system model.
Section 3 presents a detailed description of the relative positioning method.
Section 4 de-rives the system observability conditions and provides the specific expressions.
Section 5 presents simulation results, followed by
Section 6 which concludes this paper.
2. System Model
In this paper, we consider a system model for relative localization of two cooperative MAVs in GNSS-denied environments, as shown in
Figure 1. The definitions of the reference coordinate systems are explained as follows. The global coordinate system is the Earth-fixed North-East-Down (NED) coordinate system, denoted as
n, and assumed to be an inertial frame. The origin of the body-fixed horizontal coordinate system (denoted as
,
i = 1, 2) for MAV
i (
i = 1, 2) is located at its center of gravity, with
x-
y plane and
z-axis paralleling those of
n, respectively, meaning that
is obtained by rotating the
n around its
z-axis by a yaw angle
.
It is worth noting that this paper does not use the typical body-fixed coordinate system (denoted as , i = 1, 2) for the MAV i, which is represented by Euler angles with respect to n. According to the 321-rotation sequence, the corresponding Euler angles are yaw angle , pitch angle , and roll angle , respectively. The reason for using instead of is to simplify the kinematic relationships and minimize the impact of unnecessary factors, such as near-hovering states with small roll and pitch angles.
In the system model, MAV i can measure its own state variables, which include acceleration, angular velocity, and velocity in . Furthermore, by means of wireless communication, MAV i can also obtain velocity and distance information from the other one in the system framework, where two MAVs utilize inter-vehicle distance information to enhance the accuracy and robustness of relative positioning in the system framework.
The relative motion of two MAVs is described in
h. Let
denote the position vector of MAV
i in
n, where the subscript
i refers to the MAV number and the superscript
n indicates the vector is projected onto
n. The projection of the relative position of MAV
k (
k = 1, 2, but
) with respect to MAV
i in
can be expressed as:
where
is the coordinate transformation matrix from
n to
(see Equation (2)). It is only dependent on the yaw angle
of MAV
i, since
x-
y plane and
z-axis of
are parallel to those of
n, respectively.
It can be inferred from Equations (1) and (2) that the z-component of the relative position corresponds to the difference in height (which can be measured by a barometric altimeter) between MAV i and MAV k in n. Therefore, the system model can be simplified from three-dimensional space to a two-dimensional plane in the horizontal direction.
Some of the parameters and symbols used in this section are shown in
Table 1.
4. System Observability Analysis
Considering the nonlinear state vector system shown in Equation (3), the system observability is analyzed by means of Lie derivative. The multiple Lie derivatives are defined as follows:
where
represents the Jacobi matrix of
. Furthermore, the observability matrix of the nonlinear system is:
When the observable matrix
H has full rank, the nonlinear system is considered locally weakly observable [
37].
The first term of the observability matrix
H is equal to:
Since the cross term of the last four rows and columns in the first term of the observability matrix H is the unit matrix, the rank of H cannot be increased by its corresponding observation functions. Therefore, only the distance observation corresponding to the first row in Equation (11), denoted as , needs to be considered.
The first-order Lie derivative corresponding to the distance observation
is equal to:
Calculating the Jacobi matrix of the first-order Lie derivative, the second term of the observability matrix
H can be obtained:
Given that the full rank of the observability matrix
H is seven and the first term
has rank five, calculation of the second-order Lie derivative is necessary. The second-order Lie derivative is given by:
where the simplification is achieved using Equations (19) and (20). The result shows that the yaw rate
of MAV 1 is completely cancelled out.
The third term of the observability matrix
H is the Jacobi matrix of the second-order Lie derivative (see Equation (18)), which can be expressed as:
where the specific expressions of each term are derived simply as follows:
It can be seen from the above simplified results that the yaw rate
of MAV 2 is completely offset. In the event that the combination matrix
A consisting of the Jacobi matrixes of the zero-order, first-order, and second-order Lie derivatives corresponding to the distance observation
has full rank, the rank of observability matrix
H is guaranteed to have full rank, indicating that the nonlinear system is observable.
In light of the above discussion, the observable system needs to satisfy the following condition:
Multiplying each element in the third column of determinant
with the corresponding algebraic cofactor, adding and expanding to calculate, we can obtain the following expression:
Considering the properties of matrix
, when
and Equation (28) satisfies the following inequality (where
m is an arbitrary constant), Equation (27) holds, and the nonlinear system is observable.
While the inequality (29) is not intuitive in revealing the motion constraints of MAVs, it can be used to extract some more obvious conditions (as seen in Equations (30)–(32)), which greatly aid our comprehension of Equation (29).
Equation (30) serves as a prerequisite for Equation (29), which means that the relative position between two MAVs cannot be equal to zero. Equation (31) indicates that the MAVs cannot remain stationary. Both the conditions are obvious. Equation (32) shows that two MAVs cannot fly in parallel unless at least one of the MAVs has a non-zero acceleration, where n is an arbitrary constant.
According to the system observability analysis in this section, we derived the motion conditions that must be satisfied by MAVs (see Equations (29)–(32)), which means if the relative motion between two MAVs in the system violates the observability conditions, the positioning accuracy and reliability of MAVs cannot be guaranteed. In such a case, by actively intervening in the relative motion of MAVs to satisfy the observability conditions, MAVs in the system can still achieve mutual positioning.
Some of the parameters and symbols used in this section are shown in
Table 3.