## 3 The Kinematics Analysis of the Stroke Mechanism  
The mechanical structure of the stroke mechanism provides a hardware basis for the movement of the paddle. However, the way in which the stepper motor and the linear stepper motor cooperate with each other to make the stroke mechanism generate an ellipse-like spatial trajectory still needs to be analyzed. The kinematics of the stroke mechanism also influenced the hydrodynamic modeling of the robot, as seen in the next section. Although we have already analyzed the kinematics of the stroke mechanism in our previous work [25], here we use a relatively easier method to analyze its kinematics in more detail. The stroke mechanism is a linear guide and rotation coupled mechanism which could be simplified as a double joint mechanism with two degrees of freedom. The spatial position of a point on the paddle with reference to the nonmoving part of the robot itself is determined by the angle and position between the joints (i.e., the linear stepper motor and the stepper motor). Therefore, two Cartesian coordinate systems were established to describe the position of any point on the paddle relative to the torso of the robot which is simpler than the method of using three coordinate systems in the conference paper. A local coordinate system was built on the paddle to describe the coordinate value of any point on the paddle. A global coordinate system was built on the base plate to depict the movement of the local coordinate system. By a homogeneous coordinate transformation [26], the coordinates of any point on the paddle in the local coordinate system can be converted to the global coordinate system. The center of the upper surface of the base plane \(o\) was set as the origin of the global coordinate system labeled coordinate system O (Figure 3). The paddle was simplified as a face in this study since it was very thin. The intersection of the rotation axis and the top edge of the paddle \(q\) was set as the origin of the local coordinate system, labeled coordinate system A. The two origins overlapped with an offset of \(a\) in the horizontal direction and \(b\) in the vertical direction. Since points on the paddle rotate around the same axis (i.e., the axis of the stepper motor) and move along the linear guide mechanism, the movement trajectories of the points are similar. For any point \(M\) on the paddle, its position in the local coordinate system A is constant and is expressed as (0, \(y_{0}\), \(z_{0}\), 1) and the position of point \(M\) in the global coordinate O is (0, \(y_{0}\) + \(a\), \(-b-z_{0}\), 1). When the paddle rotates and moves up or down, the position of the point \(P\) in the global coordinate system O is given by  
\[M_{\text{O}}=(y_{0}\sin\theta,y_{0}\cos\theta+a,-b-b_{1}-z_{0},1) \tag{1}\]  
where \(b_{1}\) is the displacement of the paddle in the vertical direction and \(b_{1}=\int\omega_{1}P_{1}dt\), \(\omega_{1}\) is the angular velocity of the linear stepper motor, \(P_{1}\) is the pitch of the screw (i.e., 3 mm), \(t\) is running time, \(\theta\) is rotational angle of the paddle and \(\theta=\int\omega_{2}dt\), \(\omega_{2}\) is the angular velocity of the stepper motor with the reducer. By replacing \(b_{1}\) with \(\int\omega_{1}P_{1}dt\), and replacing \(\theta\) with \(\int\omega_{2}dt\), Equation (1) can be rewritten as  
\[P_{\text{O}}=(y_{0}\sin(\int\omega_{2}dt),y_{0}\cos(\int\omega_{2}dt)+a,-b- \int\omega_{1}P_{1}dt-z_{0},1) \tag{2}\]According to Equation (2), the paddle can produce many different ellipse-like spatial trajectories such as a water strider by controlling the velocities of the stepper motor and the linear stepper motor. It is difficult to study all ellipse-like spatial trajectories, so this article selects a trajectory for research based on the robot structure and hydrodynamics of the paddle which allows the paddle to generate as much power as possible. In the vertical direction, considering the length of the screw in the linear stepper motor, the maximum displacement of the paddle \(b_{1}\) was set as 4.8 cm.