## 1 Introduction  
In recent times, piezoelectric driving technology has been widely applied in many engineering fields with requirements of high precision and high speed [1; 2; 3]. This is due to its advantages of simple in structure, high resolution, high response speed, and good environmental adaptability [4; 5]. As a representative of the piezoelectric driving technology, piezoelectric actuators have been successfully employed in aerospace, robotics, medical instruments, and a large number of MEMS devices [6; 7; 8; 9]. In general, piezoelectric actuators can be classified into non-resonant and resonant types according to their working states [10; 11; 12; 13]. The non-resonant piezoelectric actuator operates below its first resonance frequency so that it can achieve high-precision positioning with a nanometer scale [11]. However, this kind of piezoelectric actuator is usually inefficient, and its output speed is limited. Moreover, it produces noise during operations due to the low driving frequency. For instance, [PERSON] et al. proposed a two-dimensional linear piezoelectric stepping platform by applying longitudinal and bending deformation of a piezoelectric actuator [14]. This platform obtained a high resolution of 1 \(\upmu\)m, while its output velocities under no-load condition were 101.7 \(\upmu\)m/s and 124.2 \(\upmu\)m/s in two directions. [PERSON] et al. designed a linear inchworm piezoelectric actuator with a minimum positioning resolution of 7 nm and a maximum output speed of 1.5 mm/s [15]. These cases indicate that the non-resonant piezoelectric actuators are hard to ensure the balance between high speed and high precision. In contrast, the resonant type piezoelectric actuators utilize single vibration mode or coupling of multi vibration modes to generate micro amplitude elliptical motion of their driving foot, and then the macro motion can be achieved due to the friction coupling effect [16, 17]. This accumulation of micro-movement enabled resonant piezoelectric actuators to function with higher output velocity and more considerable movement travel. Additionally, the resonant piezoelectric actuators operate very quietly due to their driving frequency in the ultrasound range. The disadvantage for the resonant piezoelectric actuators is also obvious, which is the serious wear caused by the high frequency vibration. This shortcoming limits the service lifetime of resonant piezoelectric actuators. According to the waveforms, the resonant piezoelectric actuators can be divided into standing wave and traveling wave types. The typical standing wave and traveling wave piezoelectric actuators are the V-shaped linear ultrasonic motor developed by [PERSON] in 1998 [18] and the rotatory ultrasonic motor proposed by [PERSON] in 1983 [19], respectively. To utilize the merits of direct drive and large travel of the resonant piezoelectric actuators, some piezoelectric actuated robots have been emerged. In 2006, [PERSON] et al. developed a miniature-step piezoelectric mobile robot with four electromagnetic legs [20]. A piezoelectric stack was used to deform a rhombic flexible frame to achieve multi degrees of freedom of the robot. In 2013, [PERSON] et al. proposed a novel tracked mobile system driven by a traveling wave piezoelectric actuator with four driving rings, eliminating the bearings, bogie wheels, and reducer mechanisms of traditional tracked systems [21]. In 2014, [PERSON] et al. designed a piezoelectric beam robot by gluing two non-collocated piezoelectric elements on a straight beam to generate traveling wave [22]. In 2016, [PERSON] proposed a piezoelectric mobile micro robot by using a piezoelectric bimorph beam to stimulate the traveling wave of a frame body, which has a small volume of 10 cm\({}^{3}\)[23]. In 2017, [PERSON] et al. developed a piezoelectric miniature legged robot driven by standing wave [24]. At the same time, [PERSON] et al.