1. Introduction

In this paper we explore methods for the combination of relative positioning and attitude determination for moving platforms, where each platform has multiantennas with known baseline lengths on its own surface and baseline vectors with unknown length to the other platforms. The objective of this research is to develop a method that optimally makes use of all the information available (i.e., the integerness of the ambiguities, the relationship between the ambiguities on the different baselines, and the known baseline length of the constrained baselines) to determine the relative position and orientation of a multiantenna system with unconstrained and constrained baselines. We develop a rigorous integrated method and investigate its ambiguity resolution performance for the unconstrained baselines and the overall success rate of the ambiguity resolution between a number of antennas. The paper begins with a discussion of potential applications and a literature review of previous work that has been done in this field. In Section 2 a general model for unconstrained and constrained baselines is introduced. Section 3 describes the standard methods for ambiguity resolution for unconstrained (e.g., relative navigation) and constrained (e.g., attitude determination) baseline applications. Section 4 introduces three methods for multiantenna ambiguity resolution and describes the methods mathematically for triple and quadruple antenna configurations. In Section 5 the methods are tested using simulated data.

1.1. Applications

1.1.1. Relative Navigation

Currently precise relative navigation using GNSS is under development for a large number of applications on land, on water, in the air, and even in space. The automotive industry shows interest in this application for relative navigation not only between vehicles and reference stations but also between vehicles [1]. Maritime applications, especially inshore relative navigation, require precise and robust methods [2]. Obviously this kind of technique not only is required for a swarm of Unmanned Aerial Vehicles (UAVs) [3, 4] or spacecraft [5] but also could be beneficial for swarms of manned vehicles [6]. Other aircraft applications are aerial refueling as well as, potentially, landing [7]. For relative navigation between aircraft and vessels, landing on aircraft carriers is an important application [8]. If the vehicles have multiple antennas, GNSS could potentially be used for determination of the attitude of the vehicle(s) [9–11]. Traditionally the relative positioning and attitude determination problems are treated as independent. In this contribution we investigate the possibility of using multiantenna data, not only for attitude determination but also to improve the relative positioning.

1.1.2. Absolute and Relative Attitude Determination

Attitude determination using GNSS signals is becoming more and more accepted for real world applications. With 2 antennas/1 baseline, a direction estimate similar to a magnetic compass can be obtained. With 3 antennas/2 baselines, placed at appropriate relative positions, the full attitude can be determined. For some applications we would like to know the relative attitude between two platforms, which also could be provided by GNSS if both platforms have a number of antennas. Examples of these applications are not only aerial refueling, landing on aircraft carriers and rendezvous and docking in space but also formation flying if the elements of the formation have to point in certain directions relative to each other.

1.2. Previous Work

In [12] the use of a quadruple receiver system consisting of two static GPS receivers and two GPS receivers mounted on a single platform was considered for improved On The Fly (OTF) ambiguity resolution with single frequency receivers. The ambiguities between the two static receivers and between the two receivers on the same platform could be determined within a few seconds due to the short and fixed baselines between them. These ambiguities could, in turn, be used as constraints to reduce the number of potential ambiguity solutions for the unconstrained baseline between the static station and the platform and, therefore, to reduce the time to resolution from 810 to about 470 seconds for a configuration without choke rings and from 355 to 180 seconds for a configuration with choke rings. The research used the relationship between the ambiguities but did not model the correlation between the observations at the antennas.

In [13] a system was proposed which provides carrier-based positioning and two axis attitude measurements using three single frequency GPS receivers (i.e., triple-antenna configuration). The aim of this triple-antenna configuration was to increase the success rate of the integer ambiguity resolution process when relative positioning the platform to a base station by utilising knowledge of the integer ambiguities obtained from a constrained baseline in the attitude determination system. The use of baseline length or geometry constraints in the attitude determination environment increased the integer ambiguity success rate. In that paper the knowledge of the integer ambiguities from the attitude determination system is used to reduce the number of candidates during the search for the integer ambiguities arising when the third receiver is included. When these ambiguities are resolved, the unknown baselines between the roving (attitude) receivers and base receiver may be determined and the relative position obtained. The relation between the work of [13] and this paper will be discussed in more detail later.

Also commercial products are starting to use multiantenna data in their relative positioning solutions. One example is the TRIUMPH-4X from JAVAD, which uses quadruple antennas at both the base station and rover to calculate Real Time Kinematic (RTK) solutions, in what they call cluster RTK [14]. As it is a commercial product no details about their processing strategy are available.

2. Modelling

2.1. Model for Unconstrained Baselines

Precise GNSS receivers make use of two types of observations: pseudorange and carrier phase. The pseudorange observations typically have an accuracy of decimeters, whereas carrier phase observations have accuracies up to millimeter level. The double difference (hereafter coined DD) observation equations can be written as a system of linearized observation equations [15]: where is the mean or the expected value and is the variance or dispersion of . is the vector of “observed minus computed” DD carrier phases and/or code observations of the order is the unknown vector of ambiguities of the order expressed in cycles rather than range to maintain their integer character, is the baseline vector, which is unknown for relative navigation applications but for which the length in attitude determination is known, is the geometry matrix containing normalized line-of-sight vectors, that is, a matrix containing DD direction cosines, and is a design matrix that links the data vector to the unknown vector . In this paper the assumption is made that the antennas are close to each other and thus atmospheric effects can be neglected. The variance matrix of is given by the positive definite matrix which is assumed to be known. As explained in [15], the least squares solution of the linear system of observation equations as introduced in (1) is obtained, using , from

2.2. Model for Constrained Baselines

For a baseline-constrained application, as, for example, GNSS-based attitude determination, we can make use of the knowledge that the length of the baseline is known and constant. Hence the baseline-constrained integer ambiguity resolution can make use of the standard GNSS model by adding the length constraint of the baseline , where is known. The observation equations then become [16] Then the least squares criterion reads This least squares problem is coined a Quadratically Constrained Integer Least Squares (QC-ILSs) problem in [17].

3. Ambiguity Resolution

High-precision positioning and attitude determination require the use of the very precise GNSS carrier phase observations, which however are ambiguous by an unknown integer number of cycles. For ambiguity resolution we make use of the LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) method and its recently developed baseline-constrained extension [16]. These methods will briefly be discussed. A large number of ambiguity resolution techniques have been developed for the attitude determination application, as, for example, [18–26]. These are discussed in more detail in [27]. In this publication we focus on the standard and the constrained LAMBDA method but the proposed combination of relative positioning and attitude determination should also work with the other ambiguity resolution techniques.

3.1. The Standard LAMBDA Method

The least squares criterion for the unconstrained problem reads as [15, 28] where is the least squares residual of the float solution , and is the least squares solution for , assuming that is known and . The last term of (5) can be made zero for any . We solve the vector of integer least-squares estimates of the ambiguities : where is the vector of integers that minimize the term within the brackets ( or ). A so-called integer search is needed to find . The search space for this problem is defined as where is a properly chosen constant. The LAMBDA method is an efficient way to find the minimizer of (6) [29–31].

Once the solution has been obtained, the residual is used to adjust the float solution of the first step, and therefore the final fixed baseline solution is obtained as .

3.2. Baseline-Constrained LAMBDA Method

The least squares criterion for (4) of the baseline-constrained problem reads as In the constrained approach we will search for the integer least-squares ambiguity vector in the search space: where is the fixed solution for , assuming that is known: . The method applied in this contribution, and in [27, 32], is referred to as “Expansion approach.” In the Expansion approach, we first use the standard LAMBDA method to collect integer vectors inside the search space and store all those that fulfill the inequality: The initial search space is defined as the value where is the bootstrapped solution of [15, 29]. This initial value is increased times until the search space is nonempty, using the logic visualized in Figure 1. For every step we enumerate all the integer vectors contained in . If the set is nonempty, we pick up the minimizer; otherwise we increase and thus the size of the search space .

Figure 1: Baseline-constrained LAMBDA using the “Expansion approach.”

For completeness we would like to mention that another method, the so-called “Search and Shrink approach,” was developed to solve the same problem [33].

4. Baseline-Constrained Multiantenna Ambiguity Resolution

Precise relative positioning of two moving platforms usually requires dual-frequency phase data, whereas—due to the baseline length constraints—single-frequency phase data may suffice for the precise determination of platform attitudes [5, 27, 32]. These two GNSS problems, relative positioning and attitude determination, are usually treated separately and independent from one another. In this contribution we combine the two into a multiantenna ambiguity resolution problem of which some of the baseline lengths are constrained. Insight in the numerical and statistical properties of these different approaches will be given. First we will introduce a 3- or triple- and 4- or quadruple-antenna configuration, which we will use to investigate the processing strategies theoretically. These triple- and quadruple-antenna configurations are simplified models that represent experiments as described in [5, 27, 34, 35].

4.1. Multibaseline Setup

Consider three or four antennas on two platforms as shown in Figures 2 and 3, respectively. The baselines between antenna () and the antennas () are called baseline (). The unconstrained baselines between an antenna at one platform and the antennas onboard another platform are , and and the constrained baselines are baseline 12 () and baseline 34 () with lengths and , respectively. The antennas are assumed to be sufficiently close, an assumption generally acceptable for the kind of applications discussed in Section 1.1, so that the relative antenna-satellite geometry may be considered the same for all antennas. The design matrices and and the variance-covariance matrix are assumed to be identical. We take the ordering of the four antenna pairs such that is the difference of the single-differenced data of antenna minus that of antenna .

Figure 2: Definition of the triple-antenna configuration (solid arrows indicate baseline with known length).

Figure 3: Definition of the quadruple-antenna configuration (solid arrows indicate baseline with known length).

4.2. Model and Unconstrained Float Solution

4.2.1. Triple-Antenna Configuration

For an integrated approach, we can use the known relationship between constrained and unconstrained baselines. For constrained baseline and unconstrained baselines and , respectively, with common antennas we have the following relationship for the baseline, DD ambiguities, and DD observation vectors: This equation shows that two out of three DD data vectors are sufficient to set up the GNSS model.

For the 3-antenna configuration, if we use and , the model becomes Note the presence of the nonzero covariance matrix , which is due to the fact that the DD vectors and have an antenna in common.

Applying and the Kronecker product (or symbol) gives the following model: For a complete reference on the properties of the Kronecker product we refer to [36]. Now the least squares solution and corresponding variance matrix of the 3-antenna configuration can be given as This shows that and are solely determined by the DD vector of the corresponding antenna pair, that is, , thus parallel processing is possible for the float solution. In Section 4.3, it will be demonstrated that this property is lost once the integer constraints are applied. If we denote the variance-covariance matrix of and as then the dispersion of the 3-antenna model can also be written as or after reordering If one wants to determine and from the above results it can be obtained from (see (11)) Application of the variance propagation law shows that both the integer and baseline solutions on this baseline have the same precision as the integer and baseline solutions at the other baselines:

4.2.2. Quadruple-Antenna Configuration

For constrained baselines and and unconstrained baselines , , and , respectively, with common antennas we have the following relationship for the baseline, ambiguities, and observation vectors: This equation shows that now three out of five double difference data vectors are sufficient to set up the GNSS model.

Using the Kronecker symbol we can write also this model in a more compact form: with

The dispersion of the quadruple-antenna model can again be written as or again after reordering If one wants to determine and from the above results, it can be obtained from

4.3. Optimal Solution of the Fully Integrated Approach

4.3.1. Triple-Antenna Configuration

For the derivation of the integer least squares solution, which is the optimal solution, we use the 3-antenna configuration introduced in Section 4.1, for which the baseline is constrained and the baseline is unconstrained. First we write the sum-of-squares decomposition as The ambiguity-constrained baseline solution with variance-covariance matrix is given as Therefore we can conclude that knowledge about does not improve the conditional baseline , and similarly, knowledge about does not help to improve . This is as expected from (12) assuming that the integers are known.

In order to obtain the unknown parameters we need to solve the following minimization problem: The last term on the right-hand side can be rewritten as With the constraint on the baseline and the ambiguities, the conditional solution of the baseline becomes The variance for this ambiguity constrained baseline is , and hence the knowledge of the constrained baseline allows us to improve the precision of the ambiguity constrained baseline from to .

The integer least squares solution of (28) then becomes for which the ambiguity vector can also be written as The first two terms of the right-hand side of the equation form the ambiguity objective function for the constrained baseline as described in Section 3.2 (see (8)). The third term is due to the correlation between the ambiguities at the two baselines, where . This term contributes to the optimal solution, but because of the low correlation we expect this contribution to be small.

The processing strategy makes use of the steps explained in Sections 3.1 and 3.2 of the standard and the baseline-constrained LAMBDA method. We use the baseline-constrained LAMBDA to enumerate the ambiguities of the constrained baseline in combination with ambiguity vectors for baseline using the correlation between the ambiguities on the two baselines. In the final step we will use (31) to find the integer least squares solution.

4.3.2. Quadruple-Antenna Configuration

For the quadruple-antenna configuration with a constrained baseline, and , respectively, on both sides of the ambiguity constrained baseline , we can write With the constraint on the baselines and and the ambiguities, the conditional solution of the baseline becomes The second term on the right-hand side of (33) can be made zero for every , and therefore we can write the minimization problem as The integer least squares solution becomes for the 4-antenna configuration Now the variance for this ambiguity-constrained baseline is , and hence the knowledge of 2 constrained baselines, one at each side of the unconstrained baseline, improves the precision of this baseline from to .

4.3.3. Intermezzo: Alternative Quadruple-Antenna Configuration

In Section 4.3.2 we have considered a quadruple-antenna configuration, assuming that we have two platforms, each with two antennas. The baseline lengths between the antennas on both platforms were assumed known ( and in Figure 3). For completeness we assume now that we have a quadruple-antenna configuration with 3 antennas on one platform, again with known baseline lengths between the antennas, and 1 antenna on another platform. We will also give a solution for this configuration where both constrained baselines are placed at the same side of the unconstrained baseline (see Figure 4). For a quadruple-antenna configuration with the two constrained baselines and and the unconstrained baseline as we can write With the constraint on both the baselines and the ambiguities, the conditional solution of the baseline becomes With the constraint on the baselines and and the ambiguities, the conditional solution of the baseline becomes The third term on the right-hand side of (37) can be made zero for every , and therefore we can write the minimization problem, using the decomposition of (29), as The integer least squares solution becomes for this 4-antenna configuration