# An Improved Single-Epoch Attitude Determination Method for Low-Cost Single-Frequency GNSS Receivers
## 1 Introduction  
With the continuous development of sensor technology, multi-source sensor fusion technology is increasingly used for navigation and positioning services in land vehicles and other applications with low-cost sensors. This requires not only accurate position and velocity information, but also the vehicle attitude, to ensure every single sensor's normal operation [1, 2, 3, 4]. As an independent satellite-based positioning technology, GNSS is often used as an external reference for other technologies. In the field of position and velocity estimation, some scholars have studied the problem of random models of observations and used low-cost receivers to obtain more reliable results, such as Android smartphones [5, 6]. As long as the phase ambiguity is correctly fixed, high-precision attitude information can be obtained [7]. In recent years, GNSS attitude determination has developed rapidly [8, 9, 10]. However, due to the accuracy limitations of low-cost sensors, most previous researches mainly focused on the use of other sensors, with low-precision GNSS receivers, to design an attitude output module [11, 12, 13]. Even when considering multi-sensor fusion, the loosely coupled technology is more widely adopted and used in practical applications. GNSS often prefers independent resolution, so it is particularly necessary to explore the independent attitude determination using low-cost GNSS receivers. The most critical problem of GNSS attitude determination is how to fix the ambiguity quickly and accurately. For land vehicles, complex environments will cause frequent carrier phases to lose lock and multipath errors. In order to avoid the phase observation cycle slip issue, the land vehicle's attitude determination is often carried out as a single epoch method. However, the application of low-cost antennas and receivers will make the pseudorange observations easily affected by the multipath, and the correspondent ambiguity fixing success rate becomes quite low [14]. Therefore, more robust methods are demanded to resolve the ambiguity fixing problem under this condition. The most commonly used GNSS ambiguity fixing method is the integer least-squares method, namely, the LAMBDA method. In order to improve the ambiguity success rate, the constrained LAMBDA (C-LAMBDA) method has been proposed. This method extends the ambiguity objective function by incorporating the baseline length and its variance into the ambiguity search process [15; 16]. CLAMBDA has been widely used for applications with low-cost receivers. For example, [PERSON] et al. conducted both simulated and real baseline experiments [17]. In addition to the baseline length constraint, [PERSON] et al. also incorporated the attitude information provided by inertial and magnetic field sensors into the ambiguity search range, in a special extended LAMBDA manner [18]. Given the unreliability of float ambiguity solutions, some scholars tried to obtain a more reliable fixed ambiguity solution, by using the interval search method [19; 20; 21]. [PERSON] et al. selected the CLAMBDA method and MC-LAMBDA method for analysis in basic principle, equivalency, and advantages and disadvantages; it shows that all prior information is fully utilized, but the computational burden is increased [22]. To sum up, many previous studies have shown the effectiveness of CLAMBDA and MCLAMBDA [23]. However, there are few studies on multi-baseline attitude determination with CLAMBDA or MC-LAMBDA, in the context of low-cost receivers and complex environments. Many in-field problems related to actual observing environments still remain unresolved, such as urban or woody environments. For land vehicles, a single baseline can only determine two attitude angles in two degrees of freedom. An additional baseline vector is required to derive attitude information in the direction of the other degree of freedom. Similar to CLAMBDA, the multivariate constrained LAMBDA, MC-LAMBDA, is developed for multi-baseline ambiguity fixing.