# Improving Smartphone GNSS Positioning Accuracy Using Inequality Constraints
## 1 Introduction  
In 2016, Google announced that raw Global Navigation Satellite System (GNSS) observations including carrier phase measurements would be accessible to developers, providing more opportunities for the implementation of location-based services (LBS) on smartphones [1]. Extensive studies have been conducted to develop different data processing and positioning algorithms for achieving precise smartphone-based location tracking [2; 3]. Numerous studies have investigated both absolute positioning and relative positioning algorithms for smartphones. [PERSON] et al. initially evaluated the single-point positioning (SPP) performance of the Samsung Galaxy S7 using only GPS single-frequency pseudo-range observations, and the results indicated that the smartphone was capable of providing only meter-level accuracy [4]. To improve code-based positioning performance, carrier smooth code (CSC) or Doppler smooth code (DSC) approaches have been proposed, which take advantage of the high-precision carrier phase measurements available on some smartphones [5]. [PERSON] et al. combined pseudo-range, carrier phase, and Doppler observations using an improved Hatch filter, proposing an algorithm called the three-thresholds and single-difference Hatch filter (TT-SD filter), which achieved sub-meter-level positioning accuracy [6]. [PERSON] et al. proposed an algorithm based on the time-differenced carrier phase (TDCP) filter, which obtained meter-level accuracy [7]. Precise point positioning (PPP) is another important algorithm in absolute positioning, and several studies have investigated its application in smartphones capable of performing carrier phase measurements [8]. These investigations also considered data analysis, observation processing, andmodel modification [9; 10; 11], and the results confirmed that sub-meter-level accuracy in static scenarios and meter-level accuracy in kinematics are achievable on smartphones using the PPP algorithm [12]. In addition to absolute positioning, relative positioning methods are also frequently studied, with real-time kinematics (RTK) being a major approach [13]. In short-baseline cases, atmospheric delay can be eliminated, allowing the RTK algorithm to provide high-precision location estimates [14]. Investigations using different smartphone brands and various use cases have consistently demonstrated the feasibility of achieving high-precision positioning using certain types of smartphone. The aforementioned studies have primarily focused on smartphone positioning that relies on GNSS observations obtained from the smartphone's application programming interface (API). However, considering that smartphones are equipped with multiple sensors, additional measurements can be obtained, which can be used as constraints in positioning algorithms to improve positioning accuracy and stability [15]. Several studies have been conducted to verify the performance of GNSS combined with other sensor information on smartphones to enhance positioning accuracy [16]. For example, it has been demonstrated that combining GNSS and Inertial Navigation System (INS) measurements in a loosely coupled manner can significantly improve smartphone location accuracy [17; 18]. Other observation constraints, such as camera observations, digital map data, and WiFi signals, have also been investigated [19; 20; 21]. In most cases, the additional information is expressed as an equality constraint. However, in some scenarios, it is more appropriate to express the range information as an inequality constraint [22]. For example, a smartphone can compute road direction based on node positions, which are obtained from digital maps, but this information alone may not accurately indicate the actual direction of a vehicle [23]. Thus, expressing the range of direction as an inequality constraint can provide better results.