## 1 Introduction  
Compared to traditional welding methods, all-position narrow gap gas metal arc welding (NG-GMAW) has lower heat input, lower material consumption, small thermal deformation, and high productivity. Accordingly, this method has been widely used in diverse industries such as oil and gas and even the nuclear industry to weld large structural parts, pressure vessels, and pipelines [1; 2; 3; 4; 5]. In the all-position narrow gap welding process, when the welding position changes from flat to an overhead position, gravity affects the molten pool in the opposite direction gradually. The molten pool is also affected by other factors such as voltage, current, speed of welding, shielding gas composition, and flow rate. Because of these factors, incomplete fusion is prone to occur in the welding bead, sidewall, and interlayers [2; 6; 7]. Studies show that these defects generally originate from the lack of heat input control and the formation of the poor upper weld bead. Aiming at resolving this problem, numerous investigations have been carried out to improve the quality of narrow gap welding, and different methods, such as angular swing arc [8; 9; 10], rotating arc [1; 11; 12], tandem GMAW [13; 14; 15], laser welding [16; 17; 18], hybrid laser-arc welding [19; 20; 21], and cable-type wire welding [6; 22; 23] have been developed. Although these methods have remarkable achievements, they have high requirements for welding equipment in welding long-distance pipelines in the field environment, which limits the development of automatic welding for pipelines in the field. In order to resolve this shortcoming, vertical oscillation arc welding with relatively fewer requirements on equipment structure was developed, but the stability of the welding process and quality of the weld bead should be further promoted. A review of the literature indicates that mathematical models are powerful tools that can help find the relationship between input and target parameters. [PERSON] [24] adopted the partial factorial design scheme to investigate the influence of input variables and their interactions in determining weld bead shape in NG-GMAW. Response Surface Methodology (RSM) has captured the interest of many researchers and engineers worldwide [25]. [PERSON] et al. [2, 26] used RSM based on central composite design (CCD) to predict and optimize weld bead geometry for angular swing arc NG-GMAW. [PERSON] et al. [27] employed the optimized BP neural network to predict the penetration of asymmetrical fillet root welding in GMAW. [PERSON] et al. [28] developed an empirical relationship to predict the tensile strength of laser beam welded AZ31B magnesium alloy using response surface methodology. [PERSON] et al. [29] proposed a hybrid model to describe the groove sidewall penetration dynamics for the rotational arc NG-GMAW process, which includes a support vector machine (SVM)-based dynamic model and a cubic polynomial regression model to predict the penetration. [PERSON] et al. [30] established the mapping model between weld parameters and process parameters under limited sample conditions based on a four-layer BP neural network optimized with a genetic algorithm. [PERSON] et al. [31] presented an integrated process-performance model, which is composed of a finite element-based process model to calculate the weld area and an artificial neural network-based performance model to predict the weld quality. [PERSON] et al. [7] applied statistical models to study the interactions between welding parameters and geometry of a single bead and provided a guidance trajectory for automatic multi-pass narrow-gap laser welding. [PERSON] et al. [32, 33] optimized laser welding parameters on Mg alloy and its multiple quality characteristics via a Taguchi method-based grey analysis. [PERSON] [34] performed the grey Taguchi analysis and optimized laser parameters in the laser welding of dissimilar materials to increase hardness and reduce HAZ during welding.