## 2 Structure Design, Working Process, and Dynamic Model Analysis  
The proposed 2-DOF linear PIDM is shown in Figure 1a and is made up of two orthogonal PAs, a stage, and a slider. According to present studies, the 2-DOF linear PIDM can be regarded as two spring-mass-damper systems. The simplified system of the PIDM is shown in Figure 1b, where \(m_{px}\) is the equivalent mass of the PA and stage in the \(x\)-axis, \(x_{p}\) is the distance of the PA in the \(x\)-axis, \(x_{s}\) is the distance of the slider in the \(x\)-axis, \(k_{x}\) is the stiffness of the PA and stage in the \(x\)-axis, \(c_{x}\) is the damper of the PA and stage in the \(x\)-axis, \(F_{px}(t)\) is the equivalent driving force of the PA in the \(x\)-axis, \(f_{x}\) is the friction force in the \(x\)-axis, \(m_{xx}\) is the mass of the slider in the \(x\)-axis, \(m_{py}\) is the equivalent mass of the PA and stage in the \(y\)-axis, \(y_{p}\) is the distance of the PA in the \(y\)-axis, \(y_{s}\) is the distance of the slider in the \(y\)-axis \(k_{y}\) is the stiffness of the PA and stage in the \(y\)-axis, \(c_{y}\) is the damper of the PA and stage in the \(y\)-axis, \(F_{py}(t)\) is the equivalent driving force of the PA in the \(y\)-axis, \(f_{y}\) is the friction force in the \(y\)-axis, and \(m_{sy}\) is the mass of the slider in the \(y\)-axis [32; 33]. Figure 1: A two-degrees-of-freedom (2-DOF) linear piezoelectric impact drive mechanism (PIDM): (**a**) Schematic diagram; (**b**) Simplified system. Kinematics equations of the 2-DOF orthogonal motions in the parallel 2-DOF PIDM can be extended from the 1-DOF PIDM model [34] and expressed as  
\[\left\{ \begin{array}{l}m_{px}\ddot{x}_{p}=F_{px}(t)-c_{x}\dot{x}_{p}-k_{x}x_{p}- f_{x}\\ m_{ss}\ddot{x}_{s}=f_{x}\\ F_{px}(t)=\delta_{x}V_{x}(t)\end{array} \right., \tag{1}\]  
\[\left\{ \begin{array}{l}m_{py}\ddot{y}_{p}=F_{py}(t)-c_{y}\dot{y}_{p}-k_{y}y_{ p}-f_{y}\\ m_{yy}\ddot{y}_{s}=f_{y}\\ F_{py}(t)=\delta_{y}V_{y}(t)\end{array} \right., \tag{2}\]  
where the subscript \(x\) and \(y\) represent the two directions of motion and \(\delta\) is the conversion coefficient between \(F_{p}(t)\) and driving voltage \(V(t)\). When the \(x\)-axis and \(y\)-axis motions of the 2-DOF PIDM are driven independently, there is no coupled motion and the control is very simple. Taking the \(x\)-axis motion of 2-DOF PIDM as an example, the working principle is shown in Figure 2. One working cycle can be divided into three parts: (1) initial state: all components of the structure are still and they stay in their initial positions; (2) stick state: with the slow rise of the driving voltage, the PA extends to a length of \(x_{1}\) and the stage moves along the PA with the distance of \(x_{1}\) in a low speed as a result of the static friction force; (3) slip state: with the sudden decrease of the driving voltage, the PA contracts back to its original position quickly. The stage is adhered to the PA, so the stage moves back to its original position with a high speed. Because the slider moves by the friction force between the contact surface of the stage and slider, the speed of slider is less than the speed of the stage. Thus, the slider cannot catch the stage and it moves back with a small displacement of \(x_{2}\). After the working cycle, the slider moves forward with a stepping displacement of \(x_{1}\)-\(x_{2}\). With repeated working cycles, the slider will move forward continuously. When the driving voltage rises rapidly and falls down slowly, the slider will move backward [35]. The working principle of the \(y\)-axis is similar to the \(x\)-axis motion when the 2-DOF PIDMs are driven independently. Because the two motions share a common friction interface, the coupled motion will exist when the two PAs are working together. The study of friction distribution and the value in the \(x\)-axis and \(y\)-axis of a 2-DOF PIDM is very import.