## 1 Introduction  
Implementing the emergency ascent operation is the most critical measure for sailing underwater submarines to survive from emergencies, e.g., flooding, sterplane jam, even escaping from torpedo attack. The conventional operation procedure is to command maximum propeller spin rate and blow the forward ballast tanks to acquire an appropriate pitch angle. Meanwhile, sterplane control could be used to accelerate the pitching process if possible [1]. In this typical scenario, operators concentrate on getting to the surface, while ignore other parameters such as heading, and roll. Nonetheless, both model tests and full-scale trials indicate that an excessive roll, as well as serious roll oscillations, may occur when the submarine emerges through the surface, which threatens its safety. For instance, [PERSON] [2] proved that a large roll angle of 60\({}^{\circ}\) appears before the free-running model getting to the surface. Moreover, [PERSON] [3] also showed how important other variables e.g., yaw and roll etc., are to the stability of the ascent motion, henceforth referred to as \"rising stability\". In the study on emergency rising stability, previous researchers mainly focused on obtaining the hydrodynamic coefficients through physical captive model tests or numerical simulated captive model tests, then combining the 6 degrees of freedom (DOF) motion equations to carry out maneuverability prediction. [PERSON] et al. [4; 5; 6] performed a series of captive model experiments to investigate the stability and control characteristics of submarines, such as straightline towing, rotating arm test etc. [PERSON] et al. [7; 8] conducted the numerical prediction of submarine hydrodynamic coefficients and rotary derivatives using the Computational Fluid Dynamics (CFD) approach based on the standard equations of motion for submarine maneuvers [9]. In addition, [PERSON] [10] carried out a classical stability analysis of a nonpropelled submarine by introducing nonlinear incidence terms to conventional coefficient modeling. [PERSON] pointed out the lateral instability that is caused by excessive buoyancy, and there existed a strong couple effect between roll and yaw according to the experimental results. However, as he acknowledges, his quasi-steady stability analysis did not model the unsteady forces associated with flow separation and vortex shedding [11]. [PERSON] and [PERSON] [12] provided a good description of the mathematical and physical insight for the horizontal plane stability parameters. Furthermore, [PERSON] et al. established a quasi-steady coefficient model for studying roll stability based on experimental data from the wind tunnel tests [13; 14; 15; 16]. It was manifested that the unsteady effects could be approximated with information gleaned from steady state experiments [13]. In his rolling stability analysis, the destabilizing hydrodynamic influence of the sail was considered as the main source of roll instability. Nevertheless, this conclusion cannot explain the early onset of the instability in several diesel-electric submarine trials. Based on his previous work, a 3 DOF stability analysis had been further accomplished, confirming the existence of yaw instability, whereas there were some unexplained trial instabilities remaining. [PERSON] et al. [17] conducted a series of model tests to investigate hydrodynamic coefficients of submarines at high incidence angles, laying the foundation for analyzing rising stability. Given the above information, the stability analysis based on quasi-steady hydrodynamic coefficients has an insufficient capacity of modeling the unsteady viscous effect associated with trailing sail vorticity [3; 10; 13]. Furthermore, these hydrodynamic coefficients obtained from captive model tests or empirical estimates are merely applicable to infinite depth flow fields, without considering the influence of free surface or surfacing motion on the coefficients. Thus, the prediction accuracy of a coefficient-based model needs a further verification.