41:12·C.Wang et al.. difference between the two positions(i.e.,lo and lg)at antenna A is derived based on Eq.(1)as: △9b,）=8s6,A)-（6.A=4 ATAT.+mod2元 (4) △8r,小=9pt,A-8,,A=4rA.l-lA.dl+omod2元 入 where .measures the Euclidean distance and @(B)calculates the phase offset due to the change of the tag orientation.The phase deviation Ode of each tag could be canceled during the above calculation.Theoretically,if the distance between rotation center O and RFID reader antenna A,AOl,is much larger than IT6.0Ol,ATb,Bl- ATb,o is approximated to the distance change Axb along the collinear direction as follows: A=moOI-cos)≈hmBl-haol=(A06-6圆+wherek=0±1. (5) Similarly,we also derive the change on the distance between the red tag Ax,and antenna A as: A,=E,oOa-cosm≈hr,fl-hrdl=(A8c,-6+k号where=0,士1 (6) 2π To remove the phase offset resulted from the changes of tag orientation @(B),we combine the phase changes with respect to the two tags as: Too1-osm≈(A9b.-A9+)where k=0士1 (7) 2x Therefore,given the distance between the two tagsIT.oTr.ol,deviation angle B could be obtained based on the PDT△0b,A)-△(r,A). Once the deviation angle B with respect to one RFID antenna is obtained,the possible postures of the limb form a conical surface around the collinear direction,where the apex of the cone is rotation center O.Figure 7(b) presents a sample for the estimation.Given the deviation angles with respect to two antennas located at two different positions,we can determine the posture of the limbs by finding the common places of the two conical surfaces.Since we use only two antennas,there is an ambiguity on the mirror side with respect to the line between the two antennas. We validate the feasibility of using PDT for the deviation angle estimation via the following two basic experiments.1)In the first experiment,we vary the distance of |AO from 150cm to 300cm,and in the meanwhile spin the line lo from 0 to 90 as shown Figure 7(a).The red tag is always fixed at O(Tr.oO=0)during the movement.The actual and theoretical PDT variations of different AO distances presented in Figure 7(c)are closely overlapping.It indicates that the estimation of the deviation angle via PDT is independent of the distance between the tag and antenna.2)For the second experiment,before spinning line lo around O in a similar way as above,we change the location of rotation center O by varying the distance Tr.oO from Ocm to 10cm.As shown in Figure 7(d),the actual and theoretical PDT variations are still similar to each other,so the effectiveness of using PDT for the limb orientation estimation is verified.In addition,since all the tags on the same part of limb(ie., the upper arm or lower arm)have the same orientation,the phase offset due to the tag orientation is canceled through the PDT calculation and will not affect the final estimation of the deviation angle. 4.3.2 AoA-based Orientation Refinement.Since the resolution of the 3D Limb Orientation Estimation,reflected as the range of possible PDT,is determined by the relative distance between two tags,it is difficult to achieve higher resolution by increasing the distance between tags,due to the limited length of the human arm.Besides, the PDT is capable of estimating the deviation angle related to the transmitting direction of the antenna,but can hardly differentiate orientations with the same deviation angle.Therefore,we utilize an AoA-based method [35] to refine the orientation estimation.As shown in Figure 8,we use a simple case to show the basic idea of using Proceedings of the ACM on Interactive,Mobile,Wearable and Ubiquitous Technologies,Vol.2,No.1,Article 41.Publication date:March 2018.41:12 • C. Wang et al. difference between the two positions (i.e., l0 and lβ ) at antenna A is derived based on Eq. (1) as: ∆θ(b,A) = θβ (b,A) − θ0(b,A) = 4π |ATb, β | − |ATb,0 | λ + ω(β) mod 2π, ∆θ(r,A) = θβ (r,A) − θ0(r,A) = 4π |ATr, β | − |ATr,0 | λ + ω(β) mod 2π, (4) where | · | measures the Euclidean distance and ω(β) calculates the phase offset due to the change of the tag orientation. The phase deviation θdev of each tag could be canceled during the above calculation. Theoretically, if the distance between rotation center O and RFID reader antenna A, |AO|, is much larger than |Tb,0O|, |ATb, β | − |ATb,0 | is approximated to the distance change ∆xb along the collinear direction as follows: ∆xb = |Tb,0O|(1 − cos β) ≈ |ATb, β | − |ATb,0 | = ( ∆θ(b,A) − ω(β) 2π + k) λ 2 where k = 0, ±1, · · · . (5) Similarly, we also derive the change on the distance between the red tag ∆xr and antenna A as: ∆xr = |Tr,0O|(1 − cos β) ≈ |ATr, β | − |ATr,0 | = ( ∆θ(r,A) − ω(β) 2π + k) λ 2 where k = 0, ±1, · · · . (6) To remove the phase offset resulted from the changes of tag orientation ω(β), we combine the phase changes with respect to the two tags as: |Tb,0Tr,0 |(1 − cos β) ≈ (∆θ(b,A) − ∆θ(r,A) 2π + k) λ 2 where k = 0, ±1, · · · . (7) Therefore, given the distance between the two tags |Tb,0Tr,0 |, deviation angle β could be obtained based on the PDT ∆θ(b,A) − ∆θ(r,A). Once the deviation angle β with respect to one RFID antenna is obtained, the possible postures of the limb form a conical surface around the collinear direction, where the apex of the cone is rotation center O. Figure 7(b) presents a sample for the estimation. Given the deviation angles with respect to two antennas located at two different positions, we can determine the posture of the limbs by finding the common places of the two conical surfaces. Since we use only two antennas, there is an ambiguity on the mirror side with respect to the line between the two antennas. We validate the feasibility of using PDT for the deviation angle estimation via the following two basic experiments. 1) In the first experiment, we vary the distance of |AO| from 150cm to 300cm, and in the meanwhile spin the line l0 from 0 ◦ to 90◦ as shown Figure 7(a). The red tag is always fixed at O (|Tr,0O| = 0) during the movement. The actual and theoretical PDT variations of different AO distances presented in Figure 7(c) are closely overlapping. It indicates that the estimation of the deviation angle via PDT is independent of the distance between the tag and antenna. 2) For the second experiment, before spinning line l0 around O in a similar way as above, we change the location of rotation center O by varying the distance |Tr,0O| from 0cm to 10cm. As shown in Figure 7(d), the actual and theoretical PDT variations are still similar to each other, so the effectiveness of using PDT for the limb orientation estimation is verified. In addition, since all the tags on the same part of limb (i.e., the upper arm or lower arm) have the same orientation, the phase offset due to the tag orientation is canceled through the PDT calculation and will not affect the final estimation of the deviation angle. 4.3.2 AoA-based Orientation Refinement. Since the resolution of the 3D Limb Orientation Estimation, reflected as the range of possible PDT, is determined by the relative distance between two tags, it is difficult to achieve higher resolution by increasing the distance between tags, due to the limited length of the human arm. Besides, the PDT is capable of estimating the deviation angle related to the transmitting direction of the antenna, but can hardly differentiate orientations with the same deviation angle. Therefore, we utilize an AoA-based method [35] to refine the orientation estimation. As shown in Figure 8, we use a simple case to show the basic idea of using Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies, Vol. 2, No. 1, Article 41. Publication date: March 2018