## 2 Linear and Rotational Doppler Effect on Hand Gesture Motion
(2) Wrist motion from flexion to extension: As shown in Figure 2b, the hand moved from the position with the palm away from the radar to that with the palm toward the radar. Initially, the wrist was flexed. Then, regarding the angular displacement, the hand underwent acceleration, constant speed, and then deceleration stages. Finally, the wrist was extended. The pivot region, which can be considered as the wrist joint, possesses  
Figure 2: Hand gestures: (**a**) Push motion. (**b**) Wrist from flexion to extension. (**c**) Pinch out motion. (**d**) Hand rotation motion. (**e**) Used ultrasonic emitter and receiver. zero velocity during the entire motion, thus resulting in a zero Doppler frequency shift. The largest Doppler frequency shift was contributed by the largest value of the hand velocity at a certain scatterer point on the hand multiplying the unit direction vector from that scatterer to the radar. (3) Pinch out: This motion is mainly performed by the thumb and index finger. In the initial state, the front ends of the index finger and thumb contacted each other. The remaining fingers were clenched. The motion is illustrated in Figure 2c. The front ends of the two fingers gradually opened up along with the forward movement of the hand. The thumb and index finger underwent acceleration from the initial position and then decelerated to a stop position, thus completing this motion. The index finger rotated in a clockwise direction, and the thumb rotated in a counterclockwise direction. These two motions provide a negative Doppler frequency shift with the magnitude gradually increasing, reaching the maximum, and then decreasing to zero. Meanwhile, the remaining fingers formed a partial fist and underwent acceleration and deceleration phases. This caused a similar Doppler frequency shift as the push motion, but covered a smaller scattering area compared with the push motion. (4) Hand rotation: The motion of hand rotation is similar to the wrist motion from flexion to extension, with the major difference being the axis of rotation. Regarding hand rotation, the hand was aligned with the lower arm, and all the fingers closed to form a plane with the palm (Figure 2d). The axis of rotation was the centerline of the lower arm. The rotation motion started from the palm toward the radar and rotated to make the palm face away from the radar. The hand motion underwent two phases of acceleration and deceleration. The axis of rotation had zero velocity, and thus contributed zero frequency shift. During the motion, the part of the hand from the axis of rotation to the little finger edge provided a positive Doppler frequency shift, followed by a negative Doppler frequency shift. Meanwhile, the rest of the hand produced a negative Doppler frequency shift, followed by a small positive Doppler frequency shift. Hand gesture can be served as an important tool for human interaction. Compared to existing interfaces, hand gestures have the advantages of being intuitive and easy to use. The investigated four different kinds of hand gestures which could be applied to human-robot interaction. The importance of the characterization of these hand gestures are as following: the push motion commands the robot arm to move forward, the wrist motion from flexion to extension asks the robot grip to make a right turn, the pinch out motion instructs the gripper of robot to open, and the hand rotation motion requests the robot body/platform to turn left (Figure 3). Figure 3: The potential robot operating application of the proposed hand gesture recognition.