## 1 Introduction  
With the development of robot technology and biology, bionic robot technology has achieved considerable progress and many kinds of bionic robots have appeared over the past several decades [1; 2; 3; 4; 5]. A water strider is a small aquatic insect which can stand, glide, and leap on the surface of water. It is mainly comprised of a torso, four supporting legs, and two stroke legs. The legs of the water strider have a hairy hydrophobic structure which can let itself stand and move on the water surface with water tension. The supporting leg receives an upward supporting force on the water surface and the stroke leg is subjected to a forward drag force and an upward lift force [6; 7]. The water strider has many advantages such as highly efficient locomotion, a good stability, quiet movement, and little disturbance to the water surface [8; 9]. Owing to the unique advantage the water strider has of walking on the water surface and the potential applications of bionic robots inspired by water striders in many areas such as water quality monitoring, aquatic search, and rescue [10], tremendous efforts have been devoted to the development of water strider robots in recent years [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Considering that it is relative easy to implement, most of the water strider robots developed imitate the slide movement of the water strider. The first type of water strider robots were designed based on the water surface tension support [9; 10; 13; 14; 15; 16]. The studies on these robots primarily focusedon the selection of the proper hydrophobic materials for the stroke legs and supporting legs to imitate the hydrophobic properties of the water strider. However, due to the low water surface tension, the load capacity of such bionic robots is generally very limited, which in turn greatly restricts their application. Therefore, water strider robots with a much stronger load capacity (e.g., buoyancy-supported water strider robots) have also begun to be considered [17; 18; 19; 20]. This type of the robots uses floats as the support legs and paddles as the stroke legs, which improve the load capacity and practicability of these robots. This article adopts the latter scheme to develop the water strider robot. The current studies on water strider robots mainly concentrate on structural design and movement analysis. However, the robotic counterpart that can generate an ellipse-like spatial stroke trajectory, just as the real water strider does, when moving on the water surface is very limited. The ellipse-like spatial stroke trajectory produced by the water strider via adjusting its joints when rowing with its stroke legs [10; 21] facilitates a larger stroke length for the stroke leg and effectively helps to improve the working efficiency. [PERSON] et al. [10] designed a special actuating leg using a cam-link mechanism which could generate an ellipse-like spatial trajectory similar to that of a real water strider. Due to the limited water surface tension between the supporting legs and the water surface, the load capacity and propulsion provided by the cam-link mechanism were very limited. [PERSON] and [PERSON] [9] superimposed two 1D actuators to create an elliptical path of the actuating leg. The actuating leg was driven by the piezoelectric actuators which is difficult to accurately control and the propulsion generated was small. At present, there is no buoyancy-supported water strider robot capable of an ellipse-like spatial stroke trajectory. The hydrodynamics are important to ascertain the propulsion, resistance, and evaluate the efficiency of the robots. It is also helpful to confirm the speed of the robots. For the hydrodynamic studies on the water strider robots based on the water surface tension support, [PERSON] et al. [14] determined the supporting force of a water strider robot by accumulating the surface tension force and buoyancy of the supporting legs.