Pe oPa (a)Translation (b)Rotation (c)Translation with rotation Fig.6.Micro-movement decomposition the constant diversity term,the phase variation A caused by C.Rigid Transformation Decomposition s is as follows: As aforementioned,during the continuous moving process △0= 2T×2△d 2T×2 scosT- (2)of the rigid body.the micro-movement can be defined by the rigid transformation including the rotation and translation. Meanwhile,as 1.s=llsl cos,according to Eq.(2). Meanwhile,the tag movement can be regarded as the particle 1 movement only with the translation.Therefore,we investigate 阿s 4T (3)the relationship between the tag movement and the rigid transformation of the tagged object,i.e.,translation,rotation Note that,is a normalized vector of 1,it depends on the and translation with rotation,respectively. position ofp relative to A.Assume s=(△z,△y,而= 1)Translation:The translation means a linear movement (,y),then,according to Eq.(3), that every point of the device moves with the same displace- x1△x+l△y= 0. ment.Suppose a rigid body is attached with a tag array T, when the center of the rigid body translates from position (4) x+7=1. Ps to position Pe,each tag Ti in the tag array has the same translation Then,to compute the tag movement s=(△z,△y)）according (=[)Let and be the coordinates of tag Ti when the rigid body is at position to the phase variations,we investigate their relationships in P and P,respectively,then: the linear region and non-linear region,respectively. 1)Tag Movement in the Linear Region:In the linear Ti,e Ti,s +S. (7) region,the phase variations detected from the two orthogonally Vi.e Vi.s deployed antennas are linear to the tag's moving distances Fig.6(a)shows an example of the translation when the rigid along the two orthogonal axes,respectively.E.g.,as shown body is attached with a rectangle tag array. in Fig.3,antenna A detects the phase variation of the tag 2)Rotation:The rotation means a circular movement that movement along the X-axis,whereas antenna Ay detects the phase variations of the tag movement along the Y-axis.Let the device rotates around a rotation center.Suppose a rigid body is attached with a tag array T.when the rigid body Ar and A0y be the phase variations from antenna A and Ay, rotates around a rotation center Pa by the angle of o,all respectively,so the tag movement s is computed as follows: the tags should have the same rotation angle.Specifically,let △x 「△0 andT be the coordinates of tag T when 六△， (5) the rigid body starts rotation and ends rotation,respectively, 2)Tag Movement in the Non-linear Region:In the non- let (a,ya)be the coordinates of rotation center P,then linear region,since the corresponding phase variations are Ti,e-Ta 工i,s-a not linear to the tag movement,we need to figure out their =R (8) yi,e-Ya yi,s-Ya relationship according to the geometric property.Given the phase variations A and A0y respectively collected from the -sin a where R is a rotation matrix two orthogonally deployed antennas A and Ay,according to sin a cosa representing Eg.(4),we have: the counter-clockwise rotation of angle a.Fig.6(b)shows an example of the rotation when the rigid body is attached with 2△x十功△y= a rectangle tag array. 4π (6) 3)Translation with Rotation:According to the definition 9y x,△x+功，△y=4 of the rigid transformation,any arbitrary rigid body motion can be decomposed into the combination of the rotation and where (and denote the normalized vector translation.Suppose a rigid body is attached with a tag array for the polar axis AP from the antenna A and Au.respec- T,when the center of the rigid body translates from the tively.Therefore,as long as the starting position of movement position P to the position Pe,the rigid body also rotates s,i.e,P,is known,the values of (L)and ()can around a local rotation center Pa by the angle of a,the be figured out.Then,by solving the linear equations in Eq.(6), local rotation center has the same translation as the rigid we can directly compute[△x,△f. body as well.Without loss of generality,we can model the௬ݏ ܲ௦ ܲ௘ ௫ݏ ܺ ܻ ܱ (a) Translation ܺ ܻ ܲ௘ ܲ௦ ߙ ܲ௔ ܱ (b) Rotation ܲ௘ ௬ݏ ௫ݏ ܲ௦(ܲ௔) ߙ ܺ ܻ ܱ (c) Translation with rotation Fig. 6. Micro-movement decomposition the constant diversity term, the phase variation ∆θ caused by s is as follows: ∆θ = 2π λ × 2∆d = 2π λ × 2 ksk cos γ. (2) Meanwhile, as l · s = klk · ksk cos γ, according to Eq.(2), l klk · s = λ 4π ∆θ. (3) Note that, l klk is a normalized vector of l, it depends on the position of P relative to A. Assume s = h∆x, ∆yi, l klk = hxl , yli, then, according to Eq.(3),    xl∆x + yl∆y = λ 4π ∆θ, x 2 l + y 2 l = 1. (4) Then, to compute the tag movement s = h∆x, ∆yi according to the phase variations, we investigate their relationships in the linear region and non-linear region, respectively. 1) Tag Movement in the Linear Region: In the linear region, the phase variations detected from the two orthogonally deployed antennas are linear to the tag’s moving distances along the two orthogonal axes, respectively. E.g., as shown in Fig.3, antenna Ax detects the phase variation of the tag movement along the X-axis, whereas antenna Ay detects the phase variations of the tag movement along the Y -axis. Let ∆θx and ∆θy be the phase variations from antenna Ax and Ay, respectively, so the tag movement s is computed as follows:  ∆x ∆y  =  λ 4π ∆θx λ 4π ∆θy  . (5) 2) Tag Movement in the Non-linear Region: In the non￾linear region, since the corresponding phase variations are not linear to the tag movement, we need to figure out their relationship according to the geometric property. Given the phase variations ∆θx and ∆θy respectively collected from the two orthogonally deployed antennas Ax and Ay, according to Eq.(4), we have:    xlx∆x + ylx∆y = λ 4π ∆θx, xly∆x + yly∆y = λ 4π ∆θy, (6) where hxlx , ylx i and xly , yly  denote the normalized vector for the polar axis AP from the antenna Ax and Ay, respec￾tively. Therefore, as long as the starting position of movement s , i.e, P, is known, the values of hxlx , ylx i and xly , yly  can be figured out. Then, by solving the linear equations in Eq.(6), we can directly compute [∆x, ∆y] T . C. Rigid Transformation Decomposition As aforementioned, during the continuous moving process of the rigid body, the micro-movement can be defined by the rigid transformation including the rotation and translation. Meanwhile, the tag movement can be regarded as the particle movement only with the translation. Therefore, we investigate the relationship between the tag movement and the rigid transformation of the tagged object, i.e., translation, rotation and translation with rotation, respectively. 1) Translation: The translation means a linear movement that every point of the device moves with the same displace￾ment. Suppose a rigid body is attached with a tag array T, when the center of the rigid body translates from position Ps to position Pe, each tag Ti in the tag array has the same translation  S = sx, sy T  . Let [xi,s, yi,s] T and [xi,e, yi,e] T be the coordinates of tag Ti when the rigid body is at position Ps and Pe, respectively, then:  xi,e yi,e =  xi,s yi,s + S. (7) Fig.6(a) shows an example of the translation when the rigid body is attached with a rectangle tag array. 2) Rotation: The rotation means a circular movement that the device rotates around a rotation center. Suppose a rigid body is attached with a tag array T, when the rigid body rotates around a rotation center Pa by the angle of α, all the tags should have the same rotation angle. Specifically, let [xi,s, yi,s] T and [xi,e, yi,e] T be the coordinates of tag Ti when the rigid body starts rotation and ends rotation, respectively, let (xa, ya) be the coordinates of rotation center Pa, then  xi,e − xa yi,e − ya  = R  xi,s − xa yi,s − ya  , (8) where R is a rotation matrix  cos α − sin α sin α cos α  , representing the counter-clockwise rotation of angle α. Fig.6(b) shows an example of the rotation when the rigid body is attached with a rectangle tag array. 3) Translation with Rotation: According to the definition of the rigid transformation, any arbitrary rigid body motion can be decomposed into the combination of the rotation and translation. Suppose a rigid body is attached with a tag array T, when the center of the rigid body translates from the position Ps to the position Pe, the rigid body also rotates around a local rotation center Pa by the angle of α, the local rotation center has the same translation as the rigid body as well. Without loss of generality, we can model the