# Design and Impedance Control of a Traveling Wave Ultrasonic Motor with a Spherical Rotor
## 1 Introduction  
An ultrasonic motor (USM) is one of the potential suitable actuators for haptic interfaces due to its advantages, such as a fast response, high positioning accuracy, power self-locking, large torque/weight ratio, good electromagnetic compatibility, low speed and large torque, easy miniaturization, etc. As a special actuator, the USM is well known for its high positioning accuracy as well as the flexibility of its multi-degree-of-freedom (MDOF) structure design. To some extent, a USM has advantages over a DC motor, which is mostly adopted in MDOF haptic interactions. If applying a DC motor to a haptic interface, it is necessary to use multiple actuators combined with a complex transmission mechanism. This will cause problems, such as a bulky construction, accumulative transmission clearance, and complicated control system, which will bring certain difficulties to the rapid and accurate reproduction of the force feedback. Although some MDOF electromagnetic motors have been proposed, the applications are limited due to their complex internal structure, large moment of inertia, and difficulty in machining [1; 2]. For a USM, because of its flexible structure and friction driving mode, it is more convenient to realize a multidirectional driving motion in a prototype. In recent years,various USMs have been proposed for MDOF driving applications, and the most typical structure is a spherical ultrasonic motor [3; 4; 5; 6; 7; 8; 9; 10; 11].  
However, because of the unique driving principle, a USM has different characteristics from a DC motor when using it for haptic interaction applications. For a DC motor, the relationship between the output torque and control variable is certain, but it is not very clear for the USM due to its highly nonlinear dynamics. In previous years, many researchers have explored various prototypes and control methods for USMs used in haptic interfaces. Some of them use an indirect structure to achieve the force feedback [12; 13] and some others use a model-based torque control method to achieve the force feedback of USM [14; 15; 16]. In fact, the motion of the USM is generated by the mechanical vibration of the stator and the friction drive between the rotor and stator. The no-load speed of the USM can be controlled in three ways: voltage amplitude modulation, frequency modulation, and phase difference modulation. The action mechanism of the former two is mainly used to change the amplitude of particles on the stator surface, and the larger the amplitude is, the faster the motor speed will be. The latter one is used to adjust the speed by changing the shape of the elliptical trace of particles on the stator surface. Although these control strategies are simple, the force interaction requires frequent changes in motor output speed and torque, which in turn affect the stator vibration state. For example, a torque applied to the shaft of a motor changes the amplitude of the vibration, and a stick-slip friction contact between the stator and rotor will cause a chattering phenomenon when it is at a low speed [17]. These characteristics are difficult to predict because they depend on the amplitude, speed, and output torque.  
In this study, a traveling wave ultrasonic motor (TWUM) with a spherical rotor was designed and fabricated for a haptic interface application. The expression and influencing factors of the output torque of the rotor are deduced. A mechanical characteristics testbed of the motor was built, and the behaviors under various parameters were tested. The force feedback method based on impedance control was studied for the TWUM with a spherical rotor through the virtual linear spring and virtual wall experiments. Also, a direct torque control experiment was conducted to contrast with the proposed method.