IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL.XX,NO.XX,2020 a phase-based model for the rigid motion tracking in Sec- Central Beam Region tion 4,and propose a tracking solution based on the motion 180H model in Section 5.Then,we build an RSSI-based model 160 for the touch gesture detection in Section 6,and propose the corresponding solution to realize the absolute touch position detection robustly and accurately in Section 7. 120 4 RIGID MOTION TRACKING 1005 -100-80 -60 -40 -20 0 20 100 In this section,we propose the definition of linear region and Fig.3.Phase contours of RF-signals non-linear region based on phase contour variations.Then, we illustrate the relationship between the tag movement and phase variation in different regions.Further,we model the rigid motion to decompose the translation and rotation. Linear Region 0.6x0.6m2 4.1 Linear Region and Non-linear Region In RFID systems,the RF phase is a common attribute of the Non-linear Region wireless signal,ranging from 0 to 2m.It is very sensitive to the tag-antenna distance.Suppose the distance between the tag and the antenna is d,so the signal traverses a distance Fig.4.Effective scanning area:linear region vs non-linear region of 2d in the backscatter communication.Then,the phase provided by the antenna can be expressed as: of antenna Ay smaller than a certain value,we think this position belongs to the central beam of Ay.Also,as shown 0= ×2d+4 mod 2. (1) in Fig.3,the width of the central beam changes with the perpendicular distance.The larger the perpendicular where A is the wavelength,u represents the phase offset distance is,the wider the central beam is at that distance,the caused by the diversity of hardware characteristics.Accord- central beam region is trapezoidal.Therefore,taking Fig.4, ing to the phase expression in Eq.(1),besides the diversity if setting the small displacement as 5cm,the phase different term,the phase value mainly depends on the distance threshold as 0.4 radians,for the position with the distance between the tag and the antenna.Therefore,the phase of 1.2m from two antennas,the size of the linear region is contours should form concentric circles with the antenna about 0.6 x 0.6m2,centered at that position.Apart from the at the center in an ideal situation.We thus conduct an linear region,the phase variations in the other scanning area experiment to validate the above hypothesis.We build a 2D are not linear to the displacements along either the X-axis coordinate system according to the parallel direction (X- or Y-axis,they depend on the exact tag position instead. axis)and perpendicular direction (Y-axis)of the antenna, Hence,we denote the other area as the non-linear region. and set the origin (0,0)at the center of the antenna.Then, we collect phase values in a rectangle space in front of the 4.2 Rigid Transformation antenna,ranging from -100cm to 100cm along the X-axis During the continuous movement of an object,its position and from 100cm to 180cm along the Y-axis,the step is 5cm. and orientation are changing all the time.For a rigid body, The collected phase values are plotted in Fig.3.Based on such change of the position and orientation in the 2D the experiment results,we have the following observation: space can be defined by the rigid transformation R,S, Observation 1:The phase contours are very close to concen- where R is a 2 x 2 rotation matrix and S is a 2 x 1 tric circles with the antenna at the center.Besides,in the central translation matrix.Here,the rotation means a circular move- beam region marked with blue lines in Fig.3,the phase contours ment that the device rotates around a rotation center,and are almost parallel to each other and stretching along the X-axis. the translation means a linear movement that every point That is,in this region,the phase can be regarded as linearly related of the device moves with the same displacement.As the to the perpendicular distance from the tag to the antenna plane. continuous movement of an object consists of a series of As shown in Fig.4,assume that two antennas Ar and instant movements at different time,we denote the instant Ay are deployed in a mutually orthogonal manner and movement as the micro-movement,each micro-movement can separated with a fairly large distance.Then,according to be expressed with the rotation and translation.Thus,we can Observation 1,in the intersection area of the central beams use the rigid transformation to depict the micro-movement. of the two antennas,the displacement of a tag along the By attaching a tag array on an object,it is possible to X-axis and Y-axis should be linear to the phase variations track the rigid transformation based on the movement of received by the antenna Ar and Au,respectively.We thus each tag in the array.Note that,different from the rigid body, denote this intersection region as the linear region.The size i.e.,the tagged object,the tag attached on the object actually of the linear region depends on two factors:the central beam represents a single point of the object,so its movement can region of each antenna,and the perpendicular distance to be regarded as the particle movement,which only has the each antenna.Specifically,the central beam region relies on translation rather than the rotation.E.g.,assume an object the tolerance of the small displacement along the horizontal is attached with a tag array with the layout of rectangle, direction for the antenna.E.g.,if the small displacement as shown in Fig.5,the tags are denoted as solid points on along the X-axis at one position incurs the phase difference the rectangle.For any micro-movement in the continuousIEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. XX, NO. XX, 2020 4 a phase-based model for the rigid motion tracking in Sec￾tion 4, and propose a tracking solution based on the motion model in Section 5. Then, we build an RSSI-based model for the touch gesture detection in Section 6, and propose the corresponding solution to realize the absolute touch position detection robustly and accurately in Section 7. 4 RIGID MOTION TRACKING In this section, we propose the definition of linear region and non-linear region based on phase contour variations. Then, we illustrate the relationship between the tag movement and phase variation in different regions. Further, we model the rigid motion to decompose the translation and rotation. 4.1 Linear Region and Non-linear Region In RFID systems, the RF phase is a common attribute of the wireless signal, ranging from 0 to 2π. It is very sensitive to the tag-antenna distance. Suppose the distance between the tag and the antenna is d, so the signal traverses a distance of 2d in the backscatter communication. Then, the phase provided by the antenna can be expressed as: θ =  2π λ × 2d + µ  mod 2π, (1) where λ is the wavelength, µ represents the phase offset caused by the diversity of hardware characteristics. Accord￾ing to the phase expression in Eq. (1), besides the diversity term, the phase value mainly depends on the distance between the tag and the antenna. Therefore, the phase contours should form concentric circles with the antenna at the center in an ideal situation. We thus conduct an experiment to validate the above hypothesis. We build a 2D coordinate system according to the parallel direction (X￾axis) and perpendicular direction (Y -axis) of the antenna, and set the origin (0, 0) at the center of the antenna. Then, we collect phase values in a rectangle space in front of the antenna, ranging from −100cm to 100cm along the X-axis and from 100cm to 180cm along the Y -axis, the step is 5cm. The collected phase values are plotted in Fig. 3. Based on the experiment results, we have the following observation: Observation 1: The phase contours are very close to concen￾tric circles with the antenna at the center. Besides, in the central beam region marked with blue lines in Fig. 3, the phase contours are almost parallel to each other and stretching along the X-axis. That is, in this region, the phase can be regarded as linearly related to the perpendicular distance from the tag to the antenna plane. As shown in Fig. 4, assume that two antennas Ax and Ay are deployed in a mutually orthogonal manner and separated with a fairly large distance. Then, according to Observation 1, in the intersection area of the central beams of the two antennas, the displacement of a tag along the X-axis and Y -axis should be linear to the phase variations received by the antenna Ax and Ay, respectively. We thus denote this intersection region as the linear region. The size of the linear region depends on two factors: the central beam region of each antenna, and the perpendicular distance to each antenna. Specifically, the central beam region relies on the tolerance of the small displacement along the horizontal direction for the antenna. E.g., if the small displacement along the X-axis at one position incurs the phase difference -100 -80 -60 -40 -20 0 20 40 60 80 100 100 120 140 160 180 1 2 3 4 5 6 Central Beam Region Fig. 3. Phase contours of RF-signals � � �! Linear Region 1.2� 1.2� 0.6×0.6�! Non-linear Region �" � Fig. 4. Effective scanning area: linear region vs non-linear region of antenna Ay smaller than a certain value, we think this position belongs to the central beam of Ay. Also, as shown in Fig. 3, the width of the central beam changes with the perpendicular distance. The larger the perpendicular distance is, the wider the central beam is at that distance, the central beam region is trapezoidal. Therefore, taking Fig. 4, if setting the small displacement as 5cm, the phase different threshold as 0.4 radians, for the position with the distance of 1.2m from two antennas, the size of the linear region is about 0.6 × 0.6m2 , centered at that position. Apart from the linear region, the phase variations in the other scanning area are not linear to the displacements along either the X-axis or Y -axis, they depend on the exact tag position instead. Hence, we denote the other area as the non-linear region. 4.2 Rigid Transformation During the continuous movement of an object, its position and orientation are changing all the time. For a rigid body, such change of the position and orientation in the 2D space can be defined by the rigid transformation R, S , where R is a 2 × 2 rotation matrix and S is a 2 × 1 translation matrix. Here, the rotation means a circular move￾ment that the device rotates around a rotation center, and the translation means a linear movement that every point of the device moves with the same displacement. As the continuous movement of an object consists of a series of instant movements at different time, we denote the instant movement as the micro-movement, each micro-movement can be expressed with the rotation and translation. Thus, we can use the rigid transformation to depict the micro-movement. By attaching a tag array on an object, it is possible to track the rigid transformation based on the movement of each tag in the array. Note that, different from the rigid body, i.e., the tagged object, the tag attached on the object actually represents a single point of the object, so its movement can be regarded as the particle movement, which only has the translation rather than the rotation. E.g., assume an object is attached with a tag array with the layout of rectangle, as shown in Fig. 5, the tags are denoted as solid points on the rectangle. For any micro-movement in the continuous