## 1 Introduction  
Supercavitation is a hydrodynamic phenomenon in which an underwater body is almost entirely inside the cavity wall (a layer of gas bubbles). Since the density of the gas is much lower than that of water, skin friction drag can be reduced considerably. A supercavitating vehicle is a high-speed vehicle with a cavitator at the nose of the vehicle. Due to the cavitator, a cavity wall is generated containing the body. In this way, skin friction drag becomes almost zero. There are many papers on modelling the system dynamic of supercavitating vehicles, e.g., [1; 2; 3; 4; 5; 6]. The authors of Reference [5] presented a dynamic model that captures the physics of a supercavitating vehicle with a disk cavitator. Based on the dynamic model, Reference [5] presented how to find the trim input so that the vehicle moves straight with a constant velocity. The authors of Reference [3] provided integrated modelling of a supercavitating vehicle from partially cavitating condition at low speed to supercavitation condition at high speed including transition phase. The authors of Reference [4] presented modelling and dynamics for a supercavitating vehicle with no fins. Reference [5] discussed how to optimally design a supercavitating vehicle with a disk cavitator, considering maximum range, maximum turn rate, and high maneuverability. The authors of Reference [6] considered a supercavitating vehicle with a conical cavitator and presented an Linear Quadratic Regulator (LQR) controller for pitch and roll rate control. Many papers have tackled how to control a supercavitating vehicle while maintaining stability. The authors of References [7; 8] considered how to control a supercavitating vehicle as the vehicle moves between a planing and a non-planing condition. References [7; 8] used switching control laws to handle this switching condition. The authors of Reference [9] developed an adaptive control law for a supercavitating vehicle while the vehicle is in the transition phase (partial cavity condition). The authors of Reference [10] presented a control law based on H-infinity optimization so that the pitch angle tracks the reference pitch command while avoiding planing. Our system model is developed based on that in Reference [5]. This paper uses [PERSON]'s model presented in References [5; 11; 12; 13; 14; 15; 16; 17; 18], since this model is known to be accurate considering unsteady cavity. The authors of Reference [5] considered a constant cavitation number. In reality, the cavitation number changes associated to the speed of the vehicle and to the depth of the vehicle. Accordingly, this paper considers the case where the cavitation number changes depending on changing systems. Considering the case where the full cavity is developed, we present control laws to control depth and roll of the vehicle. We designed the control laws based on impulsive inputs. Impulsive inputs are used to change the pitch of the vehicle slightly. This small change is desirable, since a large pitch change can lead to instability due to large planing force. By minimizing the pitch change, we can avoid planing completely. The planing force is large and occurs at dominating frequencies of about 10-50 Hz. Due to this high frequency, very fast and expensive actuators are required to actively stabilize and control the vehicle under planing [19]. Since our control laws avoid planing completely, we can reduce the drag associated to planing force and do not require expensive actuators. Sliding mode controls (SMCs) are well-known for their robustness to uncertain models [20]. This robustness comes from the fact that the SMCs switch between two distinct modes to converge to desired states. Our impulsive control laws are similar to the SMCs in that we switch between two distinct modes to converge to desired states. Consider the case where the vehicle follows a target depth under our impulsive control laws. Since the input of the vehicle switches between two modes, the vehicle depth does not converge to the target depth exactly.