Multiplatform Instantaneous GNSS Ambiguity Resolution for Triple- and Quadruple-Antenna Configurations with Constraints

Abstract

Traditionally the relative positioning and attitude determination problem are treated as independent. In this contribution we will investigate the possibilities of using multiantenna (i.e., triple and quadruple) data, not only for attitude determination but also for relative positioning. The methods developed are rigorous and have the additional advantage that they improve ambiguity resolution on the unconstrained baseline(s) and the overall success rate of ambiguity resolution between a number of antennas.

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]: 𝐸(𝑦)=𝐴𝑎+𝐵𝑏,𝐷(𝑦)=𝑄_𝑦, (1) where 𝐸 is the mean or the expected value and 𝐷 is the variance or dispersion of 𝑦. 𝑦 is the vector of “observed minus computed” DD carrier phases and/or code observations of the order 𝑚,𝑎 is the unknown vector of ambiguities of the order 𝑛 expressed in cycles rather than range to maintain their integer character, 𝑏 is the baseline vector, which is unknown for relative navigation applications but for which the length in attitude determination is known, 𝐵 is the geometry matrix containing normalized line-of-sight vectors, that is, a matrix containing DD direction cosines, and 𝐴 is a design matrix that links the data vector to the unknown vector 𝑎. 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 𝑦 is given by the positive definite matrix 𝑄_𝑦 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 ‖⋅⋅‖²_𝑄𝑦=(⋅⋅)^𝑇𝑄_𝑦⁻¹(⋅⋅), from min_{𝑎∈ℤ𝑛,𝑏∈ℝ3}‖𝑦−𝐴𝑎−𝐵𝑏‖²_𝑄𝑦. (2)

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 ‖𝑏‖_𝐼3=𝑙 , where 𝑙 is known. The observation equations then become [16] 𝐸(𝑦)=𝐴𝑎+𝐵𝑏,𝐷(𝑦)=𝑄_𝑦,‖𝑏‖_𝐼3=𝑙,𝑎∈ℤ^𝑛,𝑏∈ℝ³. (3) Then the least squares criterion reads min_{𝑎∈ℤ𝑛,𝑏∈ℝ3,‖𝑏‖𝐼3=𝑙}‖𝑦−𝐴𝑎−𝐵𝑏‖²_𝑄𝑦. (4) 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] min_{𝑎∈ℤ𝑛,𝑏∈ℝ3}‖𝑦−𝐴𝑎−𝐵𝑏‖²_𝑄𝑦=‖𝑒‖̂²_𝑄𝑦+min_{𝑎∈ℤ𝑛,𝑏∈ℝ3}(‖𝑎−𝑎‖²_𝑄𝑎+‖𝑏̂(𝑎)−𝑏‖²_𝑄𝑏̂_(𝑎)), (5) where 𝑒̂=𝑦−𝐴𝑎−𝐵𝑏̂ is the least squares residual of the float solution 𝑎,𝑏̂, and 𝑏̂(𝑎) is the least squares solution for 𝑏, assuming that 𝑎 is known and 𝑄̂𝑏_(𝑎)𝑏̂𝑏̂𝑎=𝑄−𝑄𝑄𝑎⁻¹𝑄𝑎𝑏̂. The last term of (5) can be made zero for any 𝑎. We solve the vector of integer least-squares estimates of the ambiguities 𝑎̌: 𝑎̌=arg(min_{𝑎∈ℤ𝑛}‖𝑎−𝑎‖²_𝑄𝑎), (6) where 𝑎̌ is the vector of integers that minimize the term within the brackets (arg or 𝑎𝑟𝑔𝑢𝑚𝑒𝑛𝑡). A so-called integer search is needed to find 𝑎̌. The search space for this problem is defined as Ψ(𝜒²)={𝑎∈ℤ^𝑛∣‖𝑎−𝑎‖²_𝑄𝑎≤𝜒²}, (7) where 𝜒² is a properly chosen constant. The LAMBDA method is an efficient way to find the minimizer of (6) [29–31].

Once the solution 𝑎̌ has been obtained, the residual (𝑎−𝑎)̌ is used to adjust the float solution 𝑏̂ of the first step, and therefore the final fixed baseline solution is obtained as ̌̂̌̂𝑏=𝑏(𝑎)=𝑏−𝑄𝑏̂𝑎𝑄𝑎⁻¹(𝑎−𝑎)̌.

3.2. Baseline-Constrained LAMBDA Method

The least squares criterion for (4) of the baseline-constrained problem reads as min_{𝑎∈ℤ𝑛,𝑏∈ℝ3,‖𝑏‖=𝑙}‖𝑦−𝐴𝑎−𝐵𝑏‖²_𝑄𝑦=‖𝑒‖̂²_𝑄𝑦+min_{𝑎∈ℤ𝑛}(‖𝑎−𝑎‖²_𝑄𝑎+min_{𝑏∈ℝ3,‖𝑏‖=𝑙}(‖𝑏̂(𝑎)−𝑏‖²_𝑄𝑏̂_(𝑎))). (8) In the constrained approach we will search for the integer least-squares ambiguity vector in the search space: Ψ₁(𝜒²)={𝑎∈ℤ^𝑛∣‖𝑎−𝑎‖²_𝑄𝑎+‖𝑏̂(𝑎)−𝑏̌(𝑎)‖²_𝑄𝑏̂_(𝑎)≤𝜒²}, (9) where 𝑏̌(𝑎) is the fixed solution for 𝑏, assuming that 𝑎 is known: 𝑏̌(𝑎)=arg(min_𝑏∈ℝ^𝑛_{,‖𝑏‖=𝑙}‖𝑏̂(𝑎)−𝑏‖²_𝑄𝑏̂_(𝑎)). 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 Ψ(𝜒²) and store all those that fulfill the inequality: ‖𝑏̂(𝑎)−𝑏̌(𝑎)‖²_𝑄𝑏̂_(𝑎)≤𝜒²−‖𝑎−𝑎‖²_𝑄𝑎. (10) The initial search space is defined as the value 𝜒²₁=‖𝑎−𝑎̌_𝐵‖²_𝑄𝑎 where 𝑎̌_𝐵 is the bootstrapped solution of 𝑎 [15, 29]. This initial value 𝜒²₁ is increased 𝑘 times until the search space Ψ₁(𝜒²_𝑘) is nonempty, using the logic visualized in Figure 1. For every step we enumerate all the integer vectors contained in Ψ₁(𝜒²_𝑘). If the set is nonempty, we pick up the minimizer; otherwise we increase 𝜒²_𝑘 and thus the size of the search space Ψ₁(𝜒²_𝑘).

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 𝑗 (Ant_𝑗) and the antennas 𝑖 (Ant_𝑖) are called baseline 𝑖𝑗 (𝑏_𝑖𝑗). The unconstrained baselines between an antenna at one platform and the antennas onboard another platform are 𝑏₂₃,𝑏₁₃, and 𝑏₂₄ and the constrained baselines are baseline 12 (𝑏₁₂) and baseline 34 (𝑏₃₄) with lengths 𝑙₁₂ and 𝑙₃₄, 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 𝐴 and 𝐵 and the variance-covariance matrix 𝑄_𝑦 are assumed to be identical. We take the ordering of the four antenna pairs such that 𝑦_𝑖𝑗 is the difference of the single-differenced data of antenna 𝑗 minus that of antenna 𝑖.

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 𝑏₁₂ and unconstrained baselines 𝑏₂₃ and 𝑏₁₃, respectively, with common antennas we have the following relationship for the baseline, DD ambiguities, and DD observation vectors: 𝑏₁₃=𝑏₁₂+𝑏₂₃,𝑎₁₃=𝑎₁₂+𝑎₂₃,𝑦₁₃=𝑦₁₂+𝑦₂₃. (11) 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 𝑦₁₂ and 𝑦₂₃, the model becomes 𝐸𝑦₁₂𝑦₂₃=𝑎𝐴𝐵0000𝐴𝐵¹²𝑏₁₂𝑎₂₃𝑏₂₃,𝐷𝑦₁₂𝑦₂₃=1−1100−112𝑄_𝑦01002𝑄_𝑦01002𝑄_𝑦=𝑄−101−101_𝑦−12𝑄_𝑦−12𝑄_𝑦𝑄_𝑦. (12) Note the presence of the nonzero covariance matrix 𝐶(𝑦₁₂,𝑦₂₃)=−(1/2)𝑄_𝑦, which is due to the fact that the DD vectors 𝑦₁₂ and 𝑦₂₃ have an antenna in common.

Applying 𝑃₂=[^1−1/2_−1/21] and the Kronecker product (or symbol) ⊗ gives the following model: 𝐸𝑦₁₂𝑦₂₃=𝐼₂𝑎⊗(𝐴,𝐵)¹²𝑏₁₂𝑎₂₃𝑏₂₃𝑦,𝐷¹²𝑦₂₃=𝑃₂⊗𝑄_𝑦. (13) 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 𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃=𝐼₂⊗((𝐴,𝐵)^𝑇𝑄_𝑦⁻¹(𝐴,𝐵)⁻¹(𝐴,𝐵)^𝑇𝑄_𝑦⁻¹)𝑦₁₂𝑦₂₃,𝐷𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃=𝑃₂⊗(𝐴,𝐵)^𝑇𝑄_𝑦⁻¹(𝐴,𝐵)⁻¹. (14) This shows that 𝑎_𝑖𝑗 and 𝑏̂_𝑖𝑗 are solely determined by the DD vector of the corresponding antenna pair, that is, 𝑦_𝑖𝑗, 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 𝑎_𝑖𝑗 and 𝑏̂_𝑖𝑗 as (𝐴,𝐵)^𝑇𝑄_𝑦⁻¹(𝐴,𝐵)⁻¹=𝑄𝑎𝑄𝑎𝑏̂𝑄̂𝑏𝑎𝑄̂𝑏, (15) then the dispersion of the 3-antenna model can also be written as 𝐷𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃=𝑃₂⊗𝑄𝑎𝑄𝑎𝑏̂𝑄̂𝑏𝑎𝑄̂𝑏 (16) or after reordering 𝐷𝑎₁₂𝑎₂₃𝑏̂₁₂𝑏̂₂₃=𝑃₂⊗𝑄𝑎𝑃₂𝑎𝑏̂𝑃⊗𝑄₂⊗𝑄𝑏̂𝑎𝑃₂⊗𝑄𝑏̂. (17) If one wants to determine 𝑎₁₃ and 𝑏₁₃ from the above results it can be obtained from (see (11)) 𝑎₁₃𝑏̂₁₃=𝐼(1,1)⊗^𝑛00𝐼₃𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃. (18) 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: 𝐷𝑎₁₃𝑏̂₁₃=𝐼(1,1)⊗^𝑛00𝐼₃𝑃₂⊗𝑄𝑎𝑄𝑎𝑏̂𝑄̂𝑏𝑎𝑄̂𝑏×𝐼(1,1)⊗^𝑛00𝐼₃^𝑇=𝑄𝑎𝑄𝑎𝑏̂𝑄̂𝑏𝑎𝑄̂𝑏. (19)

4.2.2. Quadruple-Antenna Configuration

For constrained baselines 𝑏₁₂ and 𝑏₃₄ and unconstrained baselines 𝑏₂₃, 𝑏₁₃, and 𝑏₂₄, respectively, with common antennas we have the following relationship for the baseline, ambiguities, and observation vectors: 𝑏₁₃=𝑏₁₂+𝑏₂₃,𝑎₁₃=𝑎₁₂+𝑎₂₃,𝑦₁₃=𝑦₁₂+𝑦₂₃,𝑏₂₄=𝑏₂₃+𝑏₃₄,𝑎₂₄=𝑎₂₃+𝑎₃₄,𝑦₂₄=𝑦₂₃+𝑦₃₄. (20) 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: 𝐸𝑦₁₂𝑦₂₃𝑦₃₄=𝐼₃𝑎⊗(𝐴,𝐵)¹²𝑏₁₂𝑎₂₃𝑏₂₃𝑎₃₄𝑏₃₄𝑦,𝐷𝑦¹²₂₃𝑦₃₄=𝑃₃⊗𝑄_𝑦 (21) with 𝑃₃=11−20−1211−210−21. (22)

The dispersion of the quadruple-antenna model can again be written as 𝐷𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃𝑎₃₄𝑏̂₃₄=𝑃₃⊗𝑄𝑎𝑄𝑎𝑏̂𝑄̂𝑏𝑎𝑄̂𝑏 (23) or again after reordering 𝐷𝑎₁₂𝑎₂₃𝑎₃₄𝑏̂₁₂𝑏̂₂₃𝑏̂₃₄=𝑃₃⊗𝑄𝑎𝑃₃𝑎𝑏̂𝑃⊗𝑄₃⊗𝑄𝑏̂𝑎𝑃₃⊗𝑄𝑏̂. (24) If one wants to determine 𝑎₁₃,𝑎₂₄ and 𝑏₁₃,𝑏₂₄ from the above results, it can be obtained from 𝑎₁₃𝑏̂₁₃𝑎₂₄𝑏̂₂₄=⊗110𝐼011^𝑛00𝐼₃𝑎₁₂𝑏̂₁₂𝑎₂₃𝑏̂₂₃𝑎₃₄𝑏̂₃₄. (25)

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 𝑏₁₂ is constrained and the baseline 𝑏₂₃ is unconstrained. First we write the sum-of-squares decomposition as 𝑦₁₂−𝐴𝑎₁₂−𝐵𝑏₁₂𝑦₂₃−𝐴𝑎₂₃−𝐵𝑏₂₃²_{𝑃2⊗𝑄𝑦}=𝑒̂₁₂𝑒̂₂₃²_{𝑃2⊗𝑄𝑦}+𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃²_𝑃2⊗𝑄𝑎+𝑏̂₁₂(𝑎₁₂,𝑎₂₃)−𝑏₁₂𝑏̂₂₃(𝑎₁₂,𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎). (26) The ambiguity-constrained baseline solution with variance-covariance matrix is given as 𝑏̂₁₂(𝑎₁₂,𝑎₂₃)𝑏̂₂₃(𝑎₁₂,𝑎₂₃)=𝑏̂₁₂𝑏̂₂₃−(𝑃₂̂𝑏𝑎⊗𝑄)(𝑃₂⊗𝑄𝑎)⁻¹𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃=𝑏̂₁₂𝑏̂₂₃−𝐼₂̂𝑏𝑎⊗𝑄𝑄𝑎⁻¹𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃=𝑏̂₁₂(𝑎₁₂)𝑏̂₂₃(𝑎₂₃),𝐷𝑏̂₁₂(𝑎₁₂,𝑎₂₃)𝑏̂₂₃(𝑎₁₂,𝑎₂₃)=𝑃₂̂𝑏⊗𝑄_(𝑎). (27) Therefore we can conclude that knowledge about 𝑎₁₂ does not improve the conditional baseline 𝑏̂₂₃(𝑎₂₃), and similarly, knowledge about 𝑎₂₃ does not help to improve 𝑏̂₁₂(𝑎₁₂). 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: 𝐹(𝑎₁₂,𝑎₂₃,𝑏₁₂,𝑏₂₃)=min_{𝑎12,𝑎23∈ℤ𝑛,𝑏12,𝑏23∈ℝ3,‖𝑏12‖𝐼3=𝑙}𝑦₁₂−𝐴𝑎₁₂−𝐵𝑏₁₂𝑦₂₃−𝐴𝑎₂₃−𝐵𝑏₂₃²_{𝑃2⊗𝑄𝑦}=𝑒̂₁₂𝑒̂₂₃²_{𝑃2⊗𝑄𝑦}+min_{𝑎12,𝑎23∈ℤ𝑛,𝑏12,𝑏23∈ℝ3,‖𝑏12‖𝐼3=𝑙12}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃²_𝑃2⊗𝑄𝑎+𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎)). (28) The last term on the right-hand side can be rewritten as 𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎)=𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)+𝑏̂₂₃(𝑎₂₃,𝑏₁₂)−𝑏₂₃²_(3/4)𝑄𝑏̂_(𝑎). (29) With the constraint on the baseline 𝑏₁₂ and the ambiguities, the conditional solution of the baseline 𝑏₂₃ becomes 𝑏̂₂₃(𝑎₂₃,𝑏₁₂)=𝑏̂₂₃(𝑎₂₃1)−(−2𝑄̂𝑏_(𝑎)𝑏̂)(𝑄_(𝑎))⁻¹(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂)=𝑏̂₂₃(𝑎₂₃1)+2(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂). (30) The variance for this ambiguity constrained baseline is 𝐷(𝑏̂₂₃(𝑎₂₃,𝑏₁₂))=(3/4)𝑄𝑏̂_(𝑎), and hence the knowledge of the constrained baseline allows us to improve the precision of the ambiguity constrained baseline from 𝑄̂𝑏_(𝑎) to (3/4)𝑄𝑏̂_(𝑎).

The integer least squares solution of (28) then becomes 𝑎̌₁₂𝑎̌₂₃=argmin_{𝑎12,𝑎23∈ℤ𝑛}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃²_𝑃2⊗𝑄𝑎+min_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑏̌)),₁₂=argmin_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎̌₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑏̌),₂₃=𝑏̂₂₃(𝑎̌₂₃,𝑏̌₁₂)=𝑏̂₂₃(𝑎̌₂₃1)+2(𝑏̂₁₂(𝑎̌₁₂)−𝑏̌₁₂); (31) for which the ambiguity vector can also be written as 𝑎̌₁₂𝑎̌₂₃=argmin_{𝑎12,𝑎23∈ℤ𝑛}(‖𝑎₁₂−𝑎₁₂‖²_𝑄𝑎+min_{‖𝑏12‖=𝑙12}(‖𝑏̂₁₂(𝑎₁₂)−𝑏₁₂‖²_𝑄𝑏̂_(𝑎))+‖𝑎₂₃(𝑎₁₂)−𝑎₂₃‖²_(3/4)𝑄𝑎). (32) 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 𝑎₂₃(𝑎₁₂)=𝑎₂₃−(−(1/2)𝑄𝑎)(𝑄𝑎)⁻¹(𝑎₁₂−𝑎₁₂)=𝑎₂₃+(1/2)(𝑎₁₂−𝑎₁₂). 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 𝑏₁₂ in combination with ambiguity vectors for baseline 𝑏₂₃ 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, 𝑏₁₂ and 𝑏₃₄, respectively, on both sides of the ambiguity constrained baseline 𝑏₂₃, we can write 𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝑃3⊗𝑄𝑏̂_(𝑎)=𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)+𝑏̂₂₃(𝑎₂₃,𝑏₁₂,𝑏₃₄)−𝑏₂₃²_(1/2)𝑄𝑏̂_(𝑎)+𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝑄𝑏̂_(𝑎). (33) With the constraint on the baselines 𝑏₁₂ and 𝑏₃₄ and the ambiguities, the conditional solution of the baseline 𝑏₂₃ becomes 𝑏̂₂₃(𝑎₂₃,𝑏₁₂,𝑏₃₄)=𝑏̂₂₃(𝑎₂₃)+12(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂)+12(𝑏̂₃₄(𝑎₃₄)−𝑏₃₄). (34) The second term on the right-hand side of (33) can be made zero for every 𝑎₂₃, and therefore we can write the minimization problem as min_{𝑎12,𝑎23,𝑎34∈ℤ𝑛,𝑏12,𝑏34∈ℝ3,‖𝑏12‖𝐼3=𝑙12,‖𝑏34‖𝐼3=𝑙34}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃𝑎₃₄−𝑎₃₄²_𝑃3⊗𝑄𝑎+𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝐼2⊗𝑄𝑏̂_(𝑎)). (35) The integer least squares solution becomes for the 4-antenna configuration 𝑎̌₁₂𝑎̌₂₃𝑎̌₃₄=argmin_{𝑎12,𝑎23,𝑎34∈ℤ𝑛}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃𝑎₃₄−𝑎₃₄²_𝑃3⊗𝑄𝑎+min_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎))+min_{‖𝑏34‖=𝑙34}(𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝑄𝑏̂_(𝑎)𝑏̌)),₁₂=argmin_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎̌₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑏̌),₃₄=argmin_{‖𝑏34‖=𝑙34}(𝑏̂₃₄(𝑎̌₃₄)−𝑏₃₄²_𝑄𝑏̂_(𝑎)𝑏̌),₂₃=𝑏̂₂₃(𝑎̌₂₃,𝑏̌₁₂,𝑏̌₃₄=𝑏̂₂₃)(𝑎̌₂₃1)+2(𝑏̂₁₂(𝑎̌₁₂)−𝑏̌₁₂1)+2(𝑏̂₃₄(𝑎̌₃₄)−𝑏̌₃₄). (36)(1)(2)(3)(4)(5)(6) Now the variance for this ambiguity-constrained baseline is 𝐷(𝑏̂₂₃(𝑎₂₃,𝑏₁₂,𝑏₃₄))=(1/2)𝑄𝑏̂_(𝑎), and hence the knowledge of 2 constrained baselines, one at each side of the unconstrained baseline, improves the precision of this baseline from 𝑄̂𝑏_(𝑎) to (1/2)𝑄𝑏̂_(𝑎).

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 (𝑏₁₂ and 𝑏₃₄ 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 𝑏₁₂ and 𝑏₂₃ and the unconstrained baseline as 𝑏₃₄ we can write 𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝑃3⊗𝑄𝑏̂_(𝑎)=𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)+𝑏̂₂₃(𝑎₂₃,𝑏₁₂)−𝑏₂₃²_(3/4)𝑄𝑏̂_(𝑎)+𝑏̂₃₄(𝑎₃₄,𝑏₁₂,𝑏₂₃)−𝑏₃₄²_(2/3)𝑄𝑏̂_(𝑎). (37) With the constraint on both the baselines 𝑏₁₂ and the ambiguities, the conditional solution of the baseline 𝑏₂₃ becomes 𝑏̂₂₃(𝑎₂₃,𝑏₁₂)=𝑏̂₂₃(𝑎₂₃)+12(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂). (38) With the constraint on the baselines 𝑏₁₂ and 𝑏₂₃ and the ambiguities, the conditional solution of the baseline 𝑏₃₄ becomes 𝑏̂₃₄(𝑎₃₄,𝑏₁₂,𝑏₂₃)=𝑏̂₃₄(𝑎₃₄)+13(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂)+23(𝑏̂₂₃(𝑎₂₃)−𝑏₂₃). (39) The third term on the right-hand side of (37) can be made zero for every 𝑎₃₄, and therefore we can write the minimization problem, using the decomposition of (29), as min_{𝑎12,𝑎23,𝑎34∈ℤ𝑛,𝑏12,𝑏23∈ℝ3,‖𝑏12‖𝐼3=𝑙12,‖𝑏23‖𝐼3=𝑙23}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃𝑎₃₄−𝑎₃₄²_𝑃3⊗𝑄𝑎+𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎)). (40) The integer least squares solution becomes for this 4-antenna configuration 𝑎̌₁₂𝑎̌₂₃𝑎̌₃₄=argmin_{𝑎12,𝑎23,𝑎34∈ℤ𝑛}(𝑎₁₂−𝑎₁₂𝑎₂₃−𝑎₂₃𝑎₃₄−𝑎₃₄²_𝑃3⊗𝑄𝑎+min_{‖𝑏12‖𝐼3=𝑙12,‖𝑏23‖𝐼3=𝑙23}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎)𝑏̌)),₁₂𝑏̌₂₃=argmin_{‖𝑏12‖𝐼3=𝑙12,‖𝑏23‖𝐼3=𝑙23}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂𝑏̂₂₃(𝑎₂₃)−𝑏₂₃²_𝑃2⊗𝑄𝑏̂_(𝑎)𝑏̌),₃₄=𝑏̂₃₄(𝑎̌₃₄,𝑏̌₁₂,𝑏̌₂₃)=𝑏̂₃₄(𝑎̌₃₄1)+3(𝑏̂₁₂(𝑎̌₁₂)−𝑏̌₁₂2)+3(𝑏̂₂₃(𝑎̌₂₃)−𝑏̌₂₃). (41) Now the variance for this ambiguity constrained baseline is 𝐷(𝑏̂₃₄(𝑎₃₄,𝑏₁₂,𝑏₂₃))=(2/3)𝑄𝑏̂_(𝑎), and hence the knowledge of two constrained baselines, at the same side of the unconstrained baseline, improves the precision of the ambiguity constrained baseline from 𝑄̂𝑏_(𝑎) to (2/3)𝑄𝑏̂_(𝑎). This means that the improvement is larger than for the triple-antenna configuration, as the model becomes stronger due to the two constrained baselines at one side, and less than the quadruple-antenna configuration of the previous section. This is also as expected as the unconstrained baseline is now only constrained at one side; the other side is left free.

Figure 4 

Definition of a different quadruple-antenna configuration (solid arrows indicate baseline with known length).

4.4. Suboptimal Solution of the Fully Integrated Approach

4.4.1. Triple-Antenna Configuration

An approximation of the integer least squares solution as given in Section 4.3.1 can be obtained by first solving the first two terms on the right-hand side of (32) for 𝑎₁₂, and then the third term for 𝑎₂₃, as if the correlation would be absent. The result is a vectorial bootstrapping approach in which we first solve the ambiguity on the constrained baseline and apply the found ambiguity vector in the solution of the unconstrained baseline. This is also the difference with the uncoupled approach from Section 4.5 in which the solutions are found completely independent of each other. This solution, hereafter coined the suboptimal solution, is then given as 𝑎̌₁₂=argmin_{𝑎12∈ℤ𝑛}(‖𝑎₁₂−𝑎₁₂‖²_𝑄𝑎+min_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑎̌)),₂₃=argmin_{𝑎23∈ℤ𝑛}(‖𝑎₂₃(𝑎̌₁₂)−𝑎₂₃‖²_(3/4)𝑄𝑎𝑏̌),₁₂=argmin_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎̌₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑏̌),₂₃=𝑏̂₂₃(𝑎̌₂₃1)+2(𝑏̂₁₂(𝑎̌₁₂)−𝑏̌₁₂). (42a)(42b)(42c)(42d) For this approach in the first step we use the baseline-constrained LAMBDA to estimate the ambiguities of the constrained baseline 𝑏₁₂. In the second step we use standard LAMBDA with 𝑎₂₃(𝑎̌₁₂) and (3/4)𝑄𝑎 on the unconstrained baseline. From (42), and (32), it is expected that the success rate of the integer least squares (optimal) approach is better than the vectorial bootstrapping (suboptimal) approach [37], but because of the low correlation between the two baselines the difference is anticipated to be minimal as discussed in the previous section. This is analyzed using simulated data in Section 5.

Next we will demonstrate that the suboptimal solution is the same as the solution from [13] in which the ambiguity vector for 𝑏₂₃ that minimizes the cost function in the metric of 𝑄𝑎 was found as 𝑎̌₂₃=argmin_{𝑎23∈ℤ𝑛}(‖𝑎₂₃−𝑎₂₃‖²_𝑄𝑎+‖(𝑎₁₃−𝑎₁₂)−𝑎₂₃‖²_𝑄𝑎). (43) We recall that we have obtained in Section 4.3.1 that 𝑎₂₃(𝑎₁₂)=𝑎₂₃+(1/2)(𝑎₁₂−𝑎₁₂), which we can rewrite using 𝑎₁₂=𝑎₁₃−𝑎₂₃, as 𝑎₂₃(𝑎₁₂1)=2(𝑎₂₃+𝑎₁₃−𝑎₁₂). (44) Thus 𝑎₂₃(𝑎₁₂) is the average of 𝑎₂₃ and 𝑎₁₃−𝑎₁₂. Next we introduce the following identity, which is valid for any 𝑢,𝑣,𝑥∈ℝ^𝑛 and arbitrary positive definite matrix 𝑀: ‖12(𝑢+𝑣)−𝑥‖²_𝑀=12‖𝑢−𝑥‖²_𝑀+12‖𝑣−𝑥‖²_𝑀−14‖𝑢−𝑣‖²_𝑀. (45) With this identity we obtain ‖𝑎₂₃(𝑎₁₂)−𝑎₂₃‖²_𝑄𝑎1=‖2(𝑎₂₃+(𝑎₁₃−𝑎₁₂)−𝑎₂₃)‖²_𝑄𝑎=12‖𝑎₂₃−𝑎₂₃‖²_𝑄𝑎+12‖(𝑎₁₃−𝑎₁₂)−𝑎₂₃‖²_𝑄𝑎−14‖𝑎₂₃−(𝑎₁₃−𝑎₁₂)‖²_𝑄𝑎. (46) As the last term on the right-hand side is constant, we obtain the following solution: 𝑎̌₂₃=argmin_{𝑎23∈ℤ𝑛}(‖𝑎₂₃(𝑎̌₁₂)−𝑎₂₃‖²_𝑄𝑎)=argmin_{𝑎23∈ℤ𝑛}(‖𝑎₂₃−𝑎₂₃‖²_𝑄𝑎+‖(𝑎₁₃−𝑎₁₂)−𝑎₂₃‖²_𝑄𝑎). (47a)(47b) The first expression (i.e., (47a)) is the same as (42b) with 𝑄𝑎 instead of (3/4)𝑄𝑎. This difference in scaling will not affect the outcome of the minimization. The second expression (i.e., see (47b)) is the one used in [13].

4.4.2. Quadruple-Antenna Configuration

An approximation of the integer least squares solution as given in Section 4.3.2 can again be obtained by solving the unconstrained baselines, after the constrained baselines are resolved. This would result in three minimizations, one for either 𝑎₁₂ and 𝑎₃₄ and one for 𝑎₂₃, if the correlation would be absent. The result is a vectorial bootstrapping approach in which we first solve the ambiguity on the constrained baselines 𝑏₁₂ and 𝑏₃₄ and apply the found ambiguity vector in the solution of the unconstrained baseline 𝑎₂₃. Hence the suboptimal solution for the 4-antenna configuration is given as 𝑎̌₁₂=argmin_{𝑎12∈ℤ𝑛}(‖𝑎₁₂−𝑎₁₂‖²_𝑄𝑎+min_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑎̌)),₃₄=argmin_{𝑎34∈ℤ𝑛}(‖𝑎₃₄−𝑎₃₄‖²_𝑄𝑎+min_{‖𝑏34‖=𝑙34}(𝑏̂₃₄(𝑎₃₄)−𝑏₃₄²_𝑄𝑏̂_(𝑎)𝑎̌)),₂₃=argmin_{𝑎23∈ℤ𝑛}(‖𝑎₂₃(𝑎̌₁₂,𝑎̌₃₄)−𝑎₂₃‖²_(1/2)𝑄𝑎𝑏̌),₁₂=argmin_{‖𝑏12‖=𝑙12}(𝑏̂₁₂(𝑎̌₁₂)−𝑏₁₂²_𝑄𝑏̂_(𝑎)𝑏̌),₃₄=argmin_{‖𝑏34‖=𝑙34}(𝑏̂₃₄(𝑎̌₃₄)−𝑏₃₄²_𝑄𝑏̂_(𝑎)𝑏̌),₂₃=𝑏̂₂₃(𝑎̌₂₃1)+2(𝑏̂₁₂(𝑎̌₁₂)−𝑏̌₁₂1)+2(𝑏̂₃₄(𝑎̌₃₄)−𝑏̌₃₄). (48)(7)(8)(9)(10)(11)(12)(13) For this approach in the first step we use the baseline-constrained LAMBDA to estimate the ambiguities of the two constrained baselines 𝑏₁₂ and 𝑏₃₄. In the second step we use standard LAMBDA with 𝑎₂₃(𝑎̌₁₂,𝑎̌₃₄) and (1/2)𝑄𝑎 on the unconstrained baseline.

4.5. Uncoupled Approach Using Unconstrained and Constrained Baselines

The simplest way to combine constrained and unconstrained baselines is the uncoupled approach in which the baselines are treated completely independently. This approach provides a lower bound for the empirical success rate of the optimal and suboptimal approaches described in the previous two sections. This follows from the demonstration for these two approaches that the precision of the variance matrix of the ambiguities on the unconstrained baseline, evidently only if the success rate of the constrained baseline is close enough to 1, was improved. For a discussion on lower bounds of the probability of obtaining the correct integer ambiguity vector, we refer to [37].

5. Verification through Simulation

In this section, the introduced approaches are applied, using simulated data, to the most challenging application of single epoch, single frequency ambiguity resolution. We will investigate the experimental or empirical success rate, which depends on the strength of the underlying GNSS model. For analysis of the performance of the described approaches, we compare the true integer ambiguity vector (the “true solution” known in the simulations) and the estimated integer ambiguity vector at every epoch. The empirical success rate is defined as the number of epochs where the obtained integer ambiguity vector was equal to the true integer ambiguity vector divided by the total number of epochs. The baseline length, as long as the atmospheric effects on the GPS observations are negligible (typically if the baseline is shorter than 10 km, see [5]), will not influence the performance of the ambiguity resolution method. In this contribution we will simulate and analyze short baselines; however the results will also apply to longer baselines (between different platforms) as long as the atmospheric influences are small. We will analyze performance both on individual baselines and on the solution of combined baselines (the “overall” solution). The first result is important as the information of the constrained baseline could improve the solution of the unconstrained baseline, and furthermore we would like to confirm that the success rate for the constrained baseline is not changed in the integrated solution compared to the uncoupled solution. The second result is important, as for some applications as described in Section 1.1, we are interested in the estimation of a number of baseline vectors on or between a number of platforms.

5.1. Simulation Setup

In order to investigate the performance of the proposed approaches, we analyze the empirical success rates using simulated data. Table 1 summarizes the conditions of the simulations. Utilizing the VISUAL software [38], based on the location of the receivers and an actual GPS constellation, the design matrices of the model are calculated. In order to obtain good approximations, the number of samples must be sufficiently large [39]. Assuming different levels of noise on the undifferenced phase (from 1 mm to 3 mm) and undifferenced code (from 5 cm to 30 cm) data, a set of 10⁵ data was generated; then each simulation was repeated for different numbers of satellites varying between 5 and 8. Our choice for simulation parameters was to show the improvement of the integrated method compared to uncoupled. The subsets of GPS satellites were selected based on elevation angles.

Table 1 

Simulation specification.

5.2. Simulation Results

In this section we will analyze the proposed integrated approaches. For the suboptimal solution we observed a maximum difference in the empirical success rate with the optimal solution of 0.1%; therefore there is only a marginal difference between the two and we include only the optimal solution's results in this paper.

We analyze the optimal solution using one constrained baseline (as in the triple-antenna configuration described in Section 4.3.1) and two constrained baselines (as in the quadruple-antenna configuration described in Section 4.3.2). The results are presented in three tables. Tables 2, 3, and 4 contain empirical success rates as a function of the number of tracked satellites (𝑁_𝑆𝑉) and the phase and code level noise (𝜎_𝜙,𝜎_𝑝). In Tables 2 and 3, we analyze the success rate on individual baselines, both uncoupled and as part of the optimal solution.

Table 2 

Simulation results: single-frequency, single-epoch success rates for the constrained baseline 𝑏₁₂ using one constrained baseline (C= 1, i.e., two antennas for uncoupled and triple-antenna configuration for the optimal solution).

Table 3 

Simulation results: single-frequency, single-epoch success rates for the unconstrained baseline 𝑏₂₃ using zero (C = 0), one (C = 1, i.e., triple-antenna configuration), or two (C = 2, i.e., quadruple-antenna configuration) constrained baselines.

Table 4 

Simulation results: single-frequency, single-epoch overall success rates for two baselines or three baselines using zero (C = 0), one (C = 1, i.e., triple-antenna configuration), or two (C = 2, i.e., quadruple-antenna configuration) constrained baselines.

In these tables 𝑃(𝑎₁₂=𝑎₁₂) and 𝑃(𝑎̌₁₂=𝑎₁₂) stand for the success rate on the constrained baseline 𝑏₁₂ and 𝑃(𝑎₂₃=𝑎₂₃) and 𝑃(𝑎̌₂₃=𝑎₂₃) for the success rate on the unconstrained baseline 𝑏₂₃ in the uncoupled and optimal solution, respectively. For the unconstrained baseline we will analyze the uncoupled case 𝑃(𝑎₂₃=𝑎₂₃), the case where one constrained baseline is utilized (𝐶=1) and the case where two constrained baselines are utilized (𝐶=2).

In Table 4, we look at the overall empirical success rate, which is the success rate on both baselines (𝑃(𝑎̌₁₂=𝑎₁₂,𝑎̌₂₃=𝑎₂₃)) for the triple-antenna configuration or on all three baselines (𝑃(𝑎̌₁₂=𝑎₁₂,𝑎̌₂₃=𝑎₂₃,𝑎̌₃₄=𝑎₃₄)) for the quadruple-antenna configuration in a combined solution. Again the uncoupled and optimal approaches using one or two constrained baseline(s) are considered.

Uncoupled versus Optimal Solution
In Table 2, we observe that the solution on the constrained baseline, as part of the optimal solution, has the same performance as the uncoupled baseline-constrained solution.
In Table 3, the unconstrained baseline 𝑏₂₃ is presented as uncoupled (𝐶=0) and as part of the optimal solution using a single constrained baseline (𝐶=1) and using two constrained baselines (𝐶=2). As expected, when comparing Tables 2 and 3, the baseline-constrained solution clearly provides much better results than the unconstrained solution. The differences in success rate are particularly pronounced when the strength of the underlying GNSS model becomes weaker (fewer satellites and/or higher measurement noise). According to Table 2 already 5 satellites and a phase standard deviation of 3 mm give a higher than 70% success rate for the constrained solution.
For the unconstrained baseline in Table 3 we observe that the optimal solution has a better performance than uncoupled. The improvement is between 0% and 13% using a single constrained baseline (𝐶=1) and between 0% and 30% using two constrained baselines (𝐶=2), with a larger improvement for weaker GNSS models.
In Table 4 compared to the uncoupled approach (𝐶=0), the improvement of the empirical success rate for the optimal solution is between 0% and 13% if a single constrained baseline (𝐶=1) is used, and between 0% and 29% if two constrained baselines (𝐶=2) are used. Again we observe a larger improvement for weaker GNSS models.

Suboptimal versus Optimal Solution
As already mentioned in the introduction of this section, the suboptimal results are not included in the tables as these were exactly the same as optimal. Nevertheless we would like to report some observations on the difference between the suboptimal and optimal solution. In the suboptimal solution, the constrained baseline is not influenced by the unconstrained baseline; hence the result will be exactly the same as the uncoupled solution. We observed that the optimal solution and the suboptimal solution have almost the same performance for the success rate on individual baselines (i.e., Table 3): the maximum difference was 0.1% in the empirical success rates. Also for the overall success rate (i.e., results in Table 4), the optimal solution and the suboptimal solution have the same performance. This result is expected from the low correlation between the constrained and unconstrained baselines as discussed in Sections 4.3 and 4.4.

Triple versus Quadruple-Antenna Configuration
For the constrained baseline, the success rate for the uncoupled and optimal solution was found to be exactly the same (see Table 2). This table contains simulation results for the optimum solution on the triple-antenna configuration, but the same results are obtained for the quadruple-antenna configuration. The probability of obtaining the correct integer value increases as the precision of 𝑎 improves. For the unconstrained baseline in the quadruple-antenna configuration (𝐶=2 case in Tables 3 and 4) we observe a higher success rate than for the same baseline in the triple antenna configuration (𝐶=1 case in Tables 3 and 4), which is as expected as the variance-covariance matrix is scaled with 1/2 and 3/4, respectively, of the original 𝑄𝑎 of the uncoupled solution. Therefore we can say that the quadruple-antenna configuration is a stronger model than the triple-antenna configuration.

6. Future Work

The basic theory for a three- and four-antenna configuration, situated at two platforms with at most two antennas on a single platform, was developed in this contribution. As future work we will extend the method to be applicable with more antennas. The performance has to be evaluated with realistic noise values for code and carrier observation on moving vehicles including the effects of multipath. Furthermore, we will test the method using data collected in dedicated field experiments of challenging applications as vessel, air-, and spacecraft, where the unconstrained baseline between the platforms will vary rapidly over large distances [5, 34]. In these field tests also the effect of remaining atmospheric delays, signal blocking, and multipath has to be investigated.

7. Conclusions

In this paper we explored methods for the combination of relative positioning and attitude determination for moving platforms, where each platform has multiantennas with known baseline lengths. The objective of this research was to develop a rigorous 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. In order to obtain more insight into the problem we investigated an uncoupled and two integrated strategies (coined the integer least squares or optimal, and vectorial bootstrapping or suboptimal approach) theoretically and experimentally. As was expected from the low correlation between the two baselines, the success rate of the integer least squares approach is similar to the vectorial bootstrapping approach. This was confirmed with simulated data for the single epoch, single frequency application. This is an important result as the suboptimal solution is more computational efficient and in general could be sufficient for the type of applications discussed in this paper. Furthermore we investigated triple- and quadruple-antenna configurations. The unconstrained baseline in the quadruple-antenna configuration provides a higher success rate than on the same unconstrained baseline in the triple-antenna configuration. This is as expected from the developed theory, as in the quadruple-antenna configuration two constrained baselines are placed at both sides of the unconstrained baseline, and in the triple-antenna configuration, only one constrained baseline is aiding the unconstrained baseline. The methods developed are rigorous and have the additional advantage that they improve ambiguity resolution on the unconstrained baseline(s) and the overall success rate of ambiguity resolution between a number of antennas.

Acknowledgments

Part of this work was done by Peter Buist at JAXA-ISAS, Japan, during a research visit with Professor Tatsuaki Hashimoto and the support of his group is acknowledged. The MicroNed-MISAT framework is kindly thanked for their support. Professor Teunissen is the recipient of an Australian Research Council Federation Fellowship (Project no. FF0883188). The research of Sandra Verhagen is supported by the Dutch Technology Foundation STW, Applied Science Division of NWO, and the Technology Program of the Ministry of Economic Affairs. This support is greatly acknowledged.