Design and Investigation of a Sandwich-Type Two-Degree-of-Freedom Ultrasonic Motor for Robotic Arms(https://doi.org/10.63386/617727)

Zhen Wang^1*, Zhiwei Zhao¹, Tenglin Liang¹

 ¹School of Mechanical Engineering, Xi’an University of Science and Technology, Xi’an, Shaanxi Province 710054, China

*Corresponding author: Zhen Wang

Email: 22205016039@stu.xust.edu.cn

Zhiwei Zhao

Email: 22205224136@stu.xust.edu.cn

Tenglin Liang

 Email: 22205016021@stu.xust.edu.cn

Abstract: In this paper, a novel two-degree-of-freedom ultrasonic motor is designed. By employing the principles of longitudinal-bending and bending-bending composite vibrations, the basic structure and dimensions of the ultrasonic motor are determined. A finite element model was established to investigate the effects of various structural parameters on the resonance frequency through simulation analysis. This ultimately enabled the superposition of multiple modes. Elliptical motion trajectories corresponding to different composite motions were acquired via transient analysis. An experimental platform is set up to test the output performance of the prototype. Under the longitudinal-bending composite vibration mode, the optimal working frequency is 40.00 kHz. At an excitation voltage of 150 Vp-p, the velocity reaches 330.56 mm/s, and under an output force of 35 N, the velocity is 40.10 mm/s. Under the bending-bending composite vibration mode, the optimal frequency is 40.60 kHz, with a velocity of 317.06 mm/s, and under an output force of 30 N, the velocity is 66.85 mm/s. A closed-loop control system is established, and based on PID control, taking the bending-bending vibration mode as an example, the maximum error in sin displacement signal tracking is less than 0.1 mm. The ultrasonic motor achieves two-degree-of-freedom motion, characterized by high speed and high precision, providing significant support for the research on ultrasonic motor-driven robotic arms.

Keywords: two-degree-of-freedom, ultrasonic motor, frequency degeneracy, finite element analysis

1. Introduction

With the increasing complexity and spatial constraints of working environments, robotic arms are transitioning from fixed-point operations to intelligent autonomous tasks, which imposes higher demands on their motion flexibility and environmental adaptability [1-3]. Leveraging their advantages of multiple degrees of freedom and large workspace, robotic arms hold significant application value in fields such as confined space exploration and medical surgery [4-6]. However, traditional electromagnetic drive systems require the introduction of independent gear mechanisms for each additional degree of freedom, leading to increased volume, structural complexity, and reduced robustness, which severely limits their working capability in confined spaces.

Ultrasonic motors, based on the principle of multi-physical field coupling resonance, achieve electromechanical energy conversion through vibration mode control. They offer advantages such as no electromagnetic interference, high thrust density, and micrometer-level motion precision [7-9]. Their compact structure and direct-drive characteristics can effectively replace electromagnetic systems and have demonstrated unique value in fields such as precision manufacturing and biomedical applications [10-12]. In recent years, the development of new materials (such as relaxor ferroelectric single crystals) and structural innovations (such as longitudinal-bending composite modes) has significantly enhanced dynamic response characteristics and environmental adaptability [13-14], providing new pathways to break through the bottleneck of multi-degree-of-freedom integration.

Sashida pioneered the standing wave-type ultrasonic motor in 1980, utilizing a Langevin structure to output a torque of 0.25 N•m at 27.8 kHz, laying the foundation for standing wave drive theory[15]. To overcome the limitation of unidirectional movement, Kuribayashi proposed a double eccentric orthogonal structure in 1998, achieving bidirectional control through the coupling of in-phase/anti-phase modes and increasing the no-load speed to 3.5 mm/s[16]. In 2001, Cheol-Ho innovatively combined first-order longitudinal vibration with second-order bending vibration modes, enabling the motor to reach a no-load speed of 0.47 m/s and a maximum output force of 92 N, significantly enhancing practical performance[17]. Multi-degree-of-freedom drive became a research focus in the 21st century. In 2009, Mashimo designed a three-stator space drive system, utilizing a 120° symmetrically distributed ring stator to achieve torque vector addition, outputting a torque of 3.45 N•m at 58 kHz, surpassing traditional two-degree-of-freedom constraints[18]. Concurrently, British scholar Shafik developed a patch-type longitudinal-bending composite motor, utilizing second-order bending and first-order longitudinal vibrations to generate elliptical trajectories, promoting miniaturization development[19].

With the continuous optimization of ultrasonic motor structures, multimodal composite ultrasonic motors have gradually been investigated by scholars in succession. In 2010, Cheol-Ho proposed a rectangular groove topology, reducing the motor weight to 1.6 g through modal decoupling and achieving a driving speed of 1.12 m/s[20]. Tanoue developed a quadruped linear motor in 2019, employing d31 mode piezoelectric elements to achieve clamping-driving coordinated control, enabling a 20 mm stator to achieve a high-speed motion of 258 mm/s under no-load conditions[21]. These innovations mark a technological leap for ultrasonic motors from single-mode to multi-degree-of-freedom integration and from macro to micro-nano scale development.

As an intelligent material motor, ultrasonic motors possess unique advantages, such as no electromagnetic interference, rapid response, high precision, no need for reducers, and high energy density[22-25]. They can directly drive the joints of robotic arms. Research on rotating ultrasonic motors using traveling wave technology is relatively mature and comprehensive, and has been preliminarily applied in robotic arms and aerospace fields[26-27]. Ye designed a ring-beam piezoelectric motor for a small high-precision robotic arm, generating a planar traveling wave on the ring to drive the robotic arm[28]. Izuhara and colleagues developed a micro robotic finger driven by an innovative micro linear ultrasonic motor, which is a key component of a micro robotic hand[29].

Despite some progress in the development of Sandwich-type ultrasonic motors, their application in robotic arms remains limited. This paper introduces a laminated dual-degree-of-freedom ultrasonic motor that can directly drive a robotic arm using a dual-degree-of-freedom ultrasonic motor. Subsequently, Section 2 outlines the configuration and working principles of the two-degree-of-freedom ultrasonic motor. In Section 3, a finite element model of the motor is established, multiple modes are combined through finite element analysis, and elliptical motion trajectories of various composite motions are extracted using transient analysis. In Section 4, an experimental prototype is manufactured to test the output characteristics, and precise positioning control experiments are conducted. Finally, conclusions are drawn in Section 5.

2. The Configuration and Working Principle of two-degree-of-freedom ultrasonic motor

To achieve the two-degree-of-freedom motion of the ultrasonic motor, this paper first proposes the basic structure of the ultrasonic motor. The structure of the ultrasonic motor is illustrated in Figure 1, consisting primarily of a front cover, a rear cover, and piezoelectric ceramics sandwiched between them. Positive and negative copper electrodes are placed between each piezoelectric ceramic piece, and insulating tape is wound around the screw to prevent direct contact with the metal base. The ceramic pieces are clamped between the front and rear covers and operate under compressive stress.

Figure 1. Structure of ultrasonic motor

To facilitate the realization of two-degree-of-freedom motion, the driving foot of the ultrasonic motor adopts a cylindrical structure, and the conical section of the front cover can amplify the vibration amplitude. As shown in Figure 2(b), the rear cover features a double-sided symmetrical screw hole design, which provides a rigid connection with external structures through bolts, offering stable elastic constraints for the motor. Sixteen piezoelectric ceramics are evenly arranged between the front and rear covers, and during assembly, a threaded preload force is applied to ensure optimal operation. In Figure 2(a), the driving foot is pre-machined into a hexagonal structure for torque application to ensure that the front cover reaches the specified preload force, and post-assembly, it is precision-machined to the final dimensions.

Figure 2. (a) Schematic diagram of front cover of ultrasonic motor (b) Schematic diagram of rear cover of ultrasonic motor

In Figure 3, the sixteen piezoelectric ceramics are divided into two groups, with the blue and green pieces placed in order as shown, allowing for the excitation of different vibration modes by applying different excitation signals to different areas of the piezoelectric ceramic pieces.

Figure 3. Quantity and arrangement of piezoelectric ceramics

Figure 4 illustrates the polarization method of the piezoelectric ceramics. The structure adopts an alternating polarization arrangement, with adjacent ceramic pieces having opposite polarization directions. The piezoelectric ceramics are polarized alternately, each ceramic piece being evenly divided into four 90° fan-shaped zones, with the polarization direction along the thickness. The upper half of odd-numbered ceramic pieces has two zones polarized negatively and the lower half has two zones polarized positively; the even-numbered ceramic pieces are polarized oppositely.

Figure 4. Polarization direction of piezoelectric ceramic pieces

When excitation voltages with a phase difference of π/2 of sine and cosine are applied to zones A and C and zones B and D, bending vibration modes are excited in the OX and OY directions, respectively, and the desired elliptical motion trajectory is generated through the synthesis of the two-phase vibrations. By applying sine signals to zones A and B and sine signals with a phase difference of π to zones C and D of the two groups of ceramics closer to the rear cover in the two-degree-of-freedom ultrasonic motor, and applying the same sine signals to the remaining 12 zones of A, B, C, and D, the ultrasonic motor can achieve bending (OY) while extending/retracting (OZ) , ultimately synthesizing an elliptical motion trajectory. As shown in Figure 5, the ultrasonic motor primarily relies on the two elliptical motion trajectories in the XOY plane and the YOZ plane driven by the driving foot to achieve linear and rotational movements of the robotic arm.

Figure 5. Motion trajectory of driving foot for two-degree-of-freedom ultrasonic motor

The linear degree of freedom of the two-degree-of-freedom ultrasonic motor designed in this study is achieved through the synthesis of longitudinal and bending vibrations to produce elliptical motion, as shown in Figure 6. After applying the incentive voltage according to the method mentioned above. At the beginning of the signal, the first two groups of zones A and B elongate, while zones C and D also elongate due to their opposite polarities. The rest six groups see zones A and B elongating while zones C and D shorten, resulting in the overall extension and downward bending of the motor. When reaching π, zones A and B of the first two ceramic pieces shorten, along with zones C and D; the subsequent six groups exhibit the reverse pattern, leading to the overall contraction and upward bending of the motor, completing the elliptical motion in the YOZ plane to achieve linear freedom.

Figure 6. Composite vibration mode diagram for linear degree of freedom of ultrasonic motor

The rotational degree of freedom utilizes two different directions of bending vibrations to synthesize elliptical motion trajectories, as illustrated in Figure 7. All ceramic pieces have sine signals applied to zones A and C, and cosine signals applied to zones B and D. At the signal’s onset, zone A elongates while zone C shortens; zone B elongates while zone D shortens. At π/2, zones A and C maintain their motion, while zone B shortens and zone D elongates. At π, zone A shortens while zone C elongates; zones B and D maintain their motion. At 3π/2, zones A and C continue their motion, while zone B elongates and zone D shortens. Completing one cycle results in one elliptical motion in the XOY plane, achieving rotational freedom.

Figure 7. Composite vibration mode diagram for rotational degree of freedom of ultrasonic motor

To facilitate the ultrasonic motor’s use on a robotic arm, a basic structure for the ultrasonic motor and robotic arm has been preliminarily designed. The preliminary configuration of the robotic arm, including the main body of the arm, a sliding mechanism mounted on the arm’s main body to allow joints to slide along the arm, and the ultrasonic motor installed on the sliding mechanism, is shown in Figure 8.

Figure 8. Conceptual design diagram of robotic arm and ultrasonic motor

3. Finite Element Analysis of Two-Degree-of-Freedom Ultrasonic Motor

The ultrasonic motor proposed in this study is a sandwich-type composite vibration motor with excellent output performance. However, achieving multiple vibration combinations requires frequency degeneracy. This motor forms elliptical motion trajectories of the driving foot through the combination of two different vibration modes. Using ANSYS finite element analysis software for simulation, appropriate operational modes are selected and transient simulations are conducted to fully simulate the vibration characteristics of the ultrasonic motor.

Optimization designs are implemented for different functional components of the sandwich-type ultrasonic motor: high-strength, high elastic modulus 45 steel is chosen for the bolts and rear cover to meet structural pre-tightening requirements; the front cover assembly utilizes 2A12 aluminum alloy for energy transmission and vibration amplification, providing efficient energy transfer and low loss characteristics. The materials of the rear cover, piezoelectric ceramics, and front cover are respectively 45 steel, PZT-4, and 2A12 aluminum alloy, with their performance parameters listed in Table 1.

Table 1: Ultrasonic Motor Material Parameter Table

Material	Density (kg/m³)	Poisson’s Ratio	Young’s Modulus (GPa)
45 Steel	7800	0.30	206
Aluminum Alloy (2A12)	2810	0.33	72
PZT-4	7600	0.32	76

In the finite element modeling process, reasonable simplifications were made for the copper electrode pieces and insulating tape. The thickness of the copper electrode pieces is only 0.2 mm, with minimal impact on the overall modal characteristics. Neglecting them during modeling can significantly reduce computation time. The insulating tape primarily serves for electrical isolation, with a small mass fraction, and its mechanical effects can be disregarded. The final finite element model was constructed using SOLID227 high-order elements with a volume of 1 mm³, and the structural parameters are detailed in Table 2.

Table 2: Ultrasonic Motor Dimensional Parameter Table

Structure	Value (mm)		Structure	Value (mm)
Piezoelectric Ceramic Diameter	30	Piezoelectric Ceramic Center Hole Diameter		14
Connecting Bolt Diameter	12	Connecting Bolt Length		28
Front Cover Conical Section Length	20	Front Cover Cylindrical Section Length		10
Driving Foot Diameter	7.5	Rear Cover Width		80
Rear Cover Rectangular Plate Thickness	2	Rear Cover Rectangular Plate Width		50
Rear Cover Cylindrical Section Length	7

Figure 9. Finite element model of ultrasonic motor

According to the principles of the longitudinal-bending composite two-degree-of-freedom ultrasonic motor, linear freedom is attained by utilizing the bending mode of the driving foot in the OY direction and the longitudinal vibration mode in the OZ direction (The resonant frequencies are close and there are no other vibration modes between the two.). Rotational freedom is achieved by employing the bending mode of the driving foot in the OY direction and the bending mode in the OX direction. Modal analysis has revealed three key vibration modes: Mode A (OX direction bending vibration at 40004 Hz), Mode B (OY direction bending vibration at 41610 Hz), and Mode C (OZ direction longitudinal vibration at 40795 Hz), illustrated in Figures 10, 11, and 12.

Figure 10. Initial mode shape A diagram (40004 Hz)	Figure 11.Initial mode shape B diagram (41610 Hz)

Figure 12. Initial mode shape C diagram (40795 Hz)

To achieve an elliptical motion trajectory driven by the foot, it is necessary to degenerate the resonance frequencies of mode A, mode B, and mode C, meaning the differences in their characteristic frequencies should be within a certain range, and the resonance frequency of mode B should be located between those of mode A and mode C. However, simulation results indicates that mode B’s resonant frequency does not fall between modes A and C, necessitating structural parameter adjustments for the ultrasonic motor.

Analysis depicted in Figure 13 illustrates the varied impact of structural size alterations on mode A (OX direction bending vibration) resonant frequency. The driving foot’s size exerts the most significant influence due to its impact on the conical section’s small end diameter, crucial for vibration amplification. The rectangular plate primarily provides support and has minimal impact on the resonant frequency. Control within a 15% range is advised for changes in the outer diameter of the piezoelectric ceramic to prevent distortion of mode A’s vibration mode. Both the rear cover cylindrical segment and the front cover cylindrical segment exhibit a consistent impact on mode A’s resonant frequency. Therefore, adjusting the resonant frequency of mode A is primarily achieved by modifying the rear cover cylindrical segment, the front cover cylindrical segment, and the conical section.

Figure 13. Resonance frequency parameter sensitivity curve of mode A

According to the sensitivity analysis of different structural sizes on Mode B (bending vibration in the OY direction) shown in Figure 14, the impact is similar to Mode A, but with opposite effects for the front cover cylindrical segment and the conical section. The driving foot diameter, front cover cylindrical segment, and front cover conical section have a significant influence on Mode B and can be adjusted by modifying the front cover cylindrical segment and conical section to achieve frequency degeneracy for Mode B.

Figure 14. Sensitivity curve of mode B resonant frequency parameters

As depicted in Figure 15, the effects of various structural sizes on Mode C (longitudinal vibration in the OZ direction) indicate that the overall length of the ultrasonic motor significantly affects the resonant frequency. Increasing the rear cover cylindrical segment, front cover cylindrical segment, and front cover conical section all lead to a decrease in the resonant frequency of Mode C because the OZ direction vibration is closely related to the motor’s length.

Figure 15. Sensitivity curve of mode C resonant frequency parameters

Based on the information provided, the optimization of key structural parameters of the ultrasonic motor is essential to achieve resonant frequencies for different modes and avoid vibration mode distortion. Mainly, achieving mode degeneracy is done through adjusting the rear cover cylindrical segment, front cover cylindrical segment, and conical section. Combining the sensitivity analysis results of the parameters with the feasibility of the manufacturing process, the critical structural parameters of the ultrasonic motor are optimized. The adjusted structural dimensions are presented in Table 3.

Table 3: Final Dimensional Parameters of the Ultrasonic Motor

Structure	Value (mm)		Structure	Value (mm)
Piezoelectric Ceramic Diameter	30	Piezoelectric Ceramic Center Hole Diameter		14
Connecting Bolt Diameter	12	Connecting Bolt Length		28
Front Cover Conical Section Length	19	Front Cover Cylindrical Section Length		11
Driving Foot Diameter	7.5	Rear Cover Width		80
Rear Cover Rectangular Plate Thickness	2	Rear Cover Rectangular Plate Width		50
Rear Cover Cylindrical Section Length	10

By performing modal analysis to select suitable modes and their corresponding frequencies, and comparing them with other vibration modes, the OZ directional vibration mode with a frequency of 40.83 kHz was chosen (this frequency enables the ultrasonic motor to generate longitudinal vibrations in the OZ direction, while also being close to the frequencies of other vibrations).

Figure 16. OZ directional longitudinal vibration mode

The bending vibration mode along the OY direction with a frequency of 40.95 kHz was selected, with a frequency difference of 0.3%. The bending vibration mode along the OY direction is illustrated in Figure 17.

Figure 17. OY directional longitudinal vibration mode	Figure 18. OX directional longitudinal vibration mode

To achieve elliptical motion in the XOY plane, it is necessary to select a driving mode in the OY direction with bending vibration and an OX direction bending mode that is adjacent in order to the OY direction bending mode. After careful selection, two modes were chosen with frequencies of 41.24 kHz (as shown in Figure 18) and 40.95 kHz (as shown in Figure 17). The frequency deviation between the two modes is 0.7%, attributed to the incomplete symmetry in the structure and mesh.

To obtain the motion trajectory on the driving foot of the ultrasonic motor, a transient analysis was conducted on the established finite element model. Following the excitation method described earlier, two signals were applied to the corresponding nodes of the finite element model, with frequencies of 40.90 kHz (located between the resonant frequencies of the two modes), a voltage of 150 Vp-p, 100 cycles. Theoretically, the driving foot should generate an elliptical actuation trajectory.

The displacement map of the YOZ plane for the final cycle of a 100-cycle run was extracted (as shown in Figure 19). The YOZ plane displacement map indicates that the displacements along the OZ and OY directions stabilize at the micron level, demonstrating the motor’s capability to achieve micron-level motion, with all three points on the driving foot moving in the same direction. When the phase difference of the applied voltage signal shifts from 90° to -90°, the motor can reverse its motion. The transient analysis confirms that the elliptical trajectory of the driving foot serves the dual functions of driving and clamping, validating the design’s rationality.

Figure 19. YOZ plane displacement map

The displacement signals of 150 Vp-p were applied to the piezoelectric ceramic plates in the respective manner. Both signals had a frequency of 41.10 kHz, comprised 100 cycles. Figure 20 illustrates the displacement curves of three points in the XOY plane.

Figure 20. XOY plane displacement map

The trajectory analysis results depicted in Figure 20 indicate that the motion of the points on the driving surface in the XOY plane exhibits a distinct elliptical trajectory. This observation aligns with the analysis of the operational principles of the ultrasonic motor, validating that the motor can achieve elliptical motion trajectory synthesis by combining two orthogonal bending vibrations.

4. Experiment Evaluation of two-degree-of-freedom ultrasonic motor

To evaluate the output performance of the ultrasonic motor, experimental prototypes were fabricated based on the dimensions of the two-degree-of-freedom ultrasonic motor determined through ANSYS finite element analysis. In the finite element analysis, the piezoelectric ceramic plates were electrodeless; however, to ensure correct signal application and avoid interference, 0.2 mm thick electrode plates were placed. Theoretically, each side of the piezoelectric ceramic plate requires electrodes, but by using the rear end cap as the grounding electrode, one electrode plate could be reduced, making a total of 16 electrode plates with a combined thickness of 3.2 mm. According to parameter sensitivity analysis, the front end cap had opposite and low sensitivity effects on the A and B modes and minimal impact on the C mode. Therefore, a cylindrical segment was chosen for length compensation on the front end cap, adjusted to 7.8 mm to minimize its impact on the overall experiment. The fabricated experimental prototype is depicted in Figure 21.

Figure 21. Experimental prototype image

The impedance characteristics of the ultrasonic motor were analyzed using an impedance analyzer (ZX80A). The resonance frequencies of the longitudinal-bending composite vibration mode were 40.05 kHz and 40.12 kHz, with a difference of 0.07 kHz. During testing, the electrode connections of the first two sets of ceramic plates in the A, B areas (or C, D areas) and the rear six sets in the A, B, C, D areas were connected to the positive pole, and the negative pole was connected to the negative terminal of the impedance analyzer (as shown in Figure 22). Due to the symmetrical structure of the ultrasonic motor, the impedance analysis results of the two vibration modes were similar. Only one set of images was displayed, with errors arising from uneven epoxy thickness during assembly and machining dimensional deviations.

Figure 22. Impedance analysis of longitudinal-bending composite vibration

The resonance frequencies of the longitudinal-bending composite vibration mode were 40.82 kHz and 40.76 kHz, with a difference of 0.08 kHz. During testing, the electrode connections of all ceramic plates in the A, C areas (or B, D areas) were connected to the positive pole, following the same connection method as the longitudinal-bending composite vibration (as shown in Figure 23). Deviations between the measured resonance frequencies and the finite element simulation results were primarily due to the electrode structure simplification in the finite element model and the influence of dimensional tolerances and assembly stresses during the machining process.

Figure 23. Bending-bending composite impedance analysis

The motion characteristics of the two-degree-of-freedom ultrasonic motor are significantly influenced by the input signal parameters. By adjusting the parameters of the signal generator to change the frequency and amplitude of the input signal, different modal frequencies can be excited, thereby altering the motion curve and characteristics of the ultrasonic motor. In this section, the frequency-speed and voltage-speed curves of the ultrasonic motor were experimentally tested. The experimental setup included the two-degree-of-freedom ultrasonic motor, power amplifier, signal generator, and guide rail, as depicted in Figure 24.

Figure 24. Experimental platform for longitudinal-bending composite

The preloading force was maintained at 140N during the experiment, with two-phase signals set at an effective value of 150 Vp-p. Speed measurements were taken every 200 Hz within a frequency range of 39 kHz to 41 kHz. The speed-frequency relationship curve for the longitudinal-bending composite vibration is illustrated in Figure 25. Within the range of 39.0 kHz to 40.00 kHz, the speed increased with frequency; however, in the range of 40.00 kHz to 41.0 kHz, the speed decreased with higher frequencies. Under 150 Vp-p excitation, the maximum speed of the longitudinal-bending composite vibration reached 330.56 mm/s. Analysis indicates that 40.00 kHz is the optimal operating frequency for the two-degree-of-freedom ultrasonic motor’s longitudinal-bending composite motion.

Figure 25. Longitudinal-bending composite frequency-speed curve

Based on the optimal operating frequency of 40.00 kHz, experiments were conducted to measure the output speed at different voltages. Starting from 70 Vp-p, voltage was increased in increments of 10 Vp-p up to 200 Vp-p, with corresponding speed measurements taken. The results showed that the output speed of the ultrasonic motor increased with higher voltages, demonstrating a positive correlation between voltage and speed. At 200 Vp-p, the rail speed reached 449.69 mm/s. Conversely, when the voltage was below 70 Vp-p, the output speed was 0 mm/s. This behavior is attributed to the ultrasonic motor’s coupling based on orthogonal composite vibration modes, enabling the driving foot to move along an elliptical trajectory. When the excitation voltage is below 70 Vp-p, the vertical vibration amplitude of the driving foot significantly decreases, preventing effective detachment even in the upper half cycle of the elliptical trajectory despite applying preload force, resulting in a stalled state.

Figure 26. Longitudinal-bending composite voltage-speed curve

The characteristics relationship curve between output force and speed of the ultrasonic motor is depicted in Figure 27. Experimental results indicate that as the output force increases, the output speed of the ultrasonic motor shows a declining trend. Even under the condition of a significant output force of 35 N, the ultrasonic motor can maintain a motion speed of 40.10 mm/s, demonstrating excellent output force capability of the developed motor.

Figure 27. Longitudinal-bending composite force-speed curve

To test the rotational degree of freedom, a bending composite vibration test platform was constructed as shown in Figure 28. While a photoelectric encoder can detect rotational motion, a linear output measurement method was employed for accurate evaluation of the output characteristics under this mode. Based on the kinematic conversion principle, rotational motion was transformed into linear motion measurement using the relationship v=ω•r (where v represents linear speed, ω is angular speed, and r is the radius of rotation), simplifying the measurement process and enhancing the reliability of data collection.

Figure 28. Bending-bending composite vibration test platform

Figure 29 illustrates the characteristic curve of motor speed varying with frequency under 150 Vp-p. The experiment shows that the system reaches a maximum output speed of 317.06 mm/s when the driving frequency is 40.60 kHz, with a significant speed decrease as the frequency deviates from this value. The error between the resonant frequency measured by the impedance analyzer and the actual operating frequency is less than 0.2%, indicating good consistency between the two.

Figure 29. Bending-bending composite frequency-speed curve

As shown in Figure 30, at a fixed driving frequency of 40.60 kHz, the relationship between excitation voltage and output speed reveals that the system is in a dead zone with zero output speed when the voltage is below 90 Vp-p. Within the range of 90-200 Vp-p, the speed rapidly increases from 0 mm/s to 472.14 mm/s.

Figure 30. Bending-bending composite voltage-speed curve

Figure 31 presents the output force-motion speed characteristic curve based on the elliptical actuating trajectory. Under the conditions of a driving frequency of 40.60 kHz and 150 Vp-p, the system can achieve a maximum output force of 35 N, with a corresponding motion speed of 22.33 mm/s. In the low output force range (0-15 N), the system exhibits good operational stability, maintaining a relatively high output speed.

Figure 31. Bending-bending composite vibration output force-speed curve

Through comprehensive experimental evaluation, this study systematically assessed the performance of the ultrasonic motor under different operating modes. The experiments demonstrate that adjusting the input frequency and excitation voltage effectively regulates the motor’s output characteristics. The motor combines high-speed motion and large output force advantages, successfully achieving two-degree-of-freedom motion switching: generating rotational motion under the bending-bending composite vibration mode and achieving linear motion under the longitudinal-bending vibration mode. This validation confirms the design rationality and provides a basis for its application in the field of multi-degree-of-freedom precision drives.

To achieve precise positioning of the ultrasonic motor, a corresponding PID control platform was established. By utilizing a PID control strategy to compensate for dead zone voltage and setting a tolerance at the target position, closed-loop displacement control of the ultrasonic motor was realized. The control block diagram is depicted in Figure 32. As the control processes for longitudinal-bending composite and bending-bending composite are similar, only the bending-bending composite vibration example was used for closed-loop control experiments.

Figure 32. Precision positioning control flowchart

The experimental results in the preceding sections indicate a significant correlation between the output speed of the ultrasonic motor and the frequency and amplitude of the excitation signal. Moreover, the speed exhibits a good linear relationship with the excitation voltage amplitude, providing an ideal variable for speed closed-loop control. Therefore, this study opts to achieve precision positioning by adjusting the excitation voltage amplitude. Based on the maximum output capability of the signal generator (10 Vp-p), a voltage amplitude limit was set to ensure that the control signal remains within a safe range, guaranteeing the safe operation of the equipment.

Figure 33. Sinusoidal displacement function tracking chart

In the closed-loop control experiments, the optimal operating frequency of the bending composite vibration was selected to ensure maximum dynamic response capability. Closed-loop control tests were conducted using 0.5 Hz and 1 Hz sinusoidal signals with 20 mm amplitude as the target displacement. The results, as shown in Figure 33, indicate highly consistent measured displacement with the target displacement, and smooth control. Real-time error analysis reveals that at angular frequencies of 0.5 Hz and 1 Hz, the maximum tracking errors are 0.096 mm and 0.073 mm, respectively, with non-reversal point errors less than 0.05 mm. The larger reversal point error is attributed to the inconsistent characteristics of forward and reverse movements of the ultrasonic motor. The experimental results validate that the closed-loop control system can achieve high-precision trajectory tracking at different frequencies.

5. 5. Conclusion

The article addresses the driving requirements of a robotic arm and designs two types of ultrasonic motors with elliptical motion trajectories, namely longitudinal-bending composite and bending composite, to meet these needs. These motors achieve two composite vibration trajectories through specific arrangements and excitation signals. A finite element model was established using ANSYS to optimize structural dimensions through parameter sensitivity analysis. The resonant frequencies of the three-axis vibration modes were determined to be 40.83 kHz, 40.95 kHz, and 41.24 kHz with frequency deviations less than 5%, achieving frequency degeneracy.Transient analysis revealed that under 150 Vp-p, the end displacement reached the micron level, successfully synthesizing planar elliptical motion trajectories.

In the experiments, prototypes were fabricated, and the resonant frequencies were measured to be 40.05 kHz for the longitudinal-bending mode and 40.76 kHz for the bending mode. At the optimal operating frequencies, the unloaded speed for the longitudinal-bending mode was 330.56 mm/s, with a speed of 40.10 mm/s at 30 N output force. For the bending mode, the maximum unloaded speed was 317.06 mm/s, with a speed of 66.85 mm/s at 30 N output force. Utilizing PID control, the maximum tracking error of the sin displacement signal in the bending vibration mode was less than 0.1 mm.

This ultrasonic motor exhibits characteristics of high speed, high output force, and precise control at the sub-micron level. It can directly drive hyper-redundant robotic arms without the need for reduction devices, laying the foundation for performance optimization and control strategy design in multi-degree-of-freedom precision drive systems.

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