Moving direction y Fig.10.The distance differences are different for Fig.11.Package stacking Fig.12.Determining package stacking in the 3D space the equal tag spacing along different directions VII.DETERMINE PACKAGE STACKING ↑Z ↑2 FOR MULTIPLE PACKAGES PP A.Limitations of the 1-dimensional Mobile Scanning for Ps Pa P7 P8 3 P Determining Package Stacking When we determine the package orientation,we derive y the indicators for the relative positions among the tags on a single package.As the geometry relationships of these tags are (a)along the X axis (b)along the Y axis known,we can combine these indicators (perpendicular dis- tance and perpendicular point)from different tags to estimate the indicators of the package's center point.Then.similarly. we compare the indicators of the center points of different P6 P7 Ps Pa packages to determine their stacking situation.Note that.when we determine the package orientation,we only need to perform 1-dimensional scanning,but it may not support the package (c)split plane stacking determination due to the 3-DoF in the 3D space.As (d)split space Fig.13.Illustration of determining the packages'order in the 3D space shown in Fig.11,suppose the antenna is above all the packages and is in front of the packages along the X axis,it moves along Combining their relative positions along the two directions. the y axis.It is easy to determine the packages'orders along the 3D space is divided into many pieces.For each piece,two the Y axis by referring to their perpendicular points,but it dimensions are fixed,the packages in it are lining up,so we may be a problem to determine their orders in the XZ plane. can use the perpendicular distances of the packages'centers If the packages line up,that is,the packages are along the to determine their orders in the piece.Thus,all the packages X axis or along the Z axis,as the left two cases shown in relative positions in the 3D space are determined. Fig.11,we can determine their orders along their lining up Taking the scene in Fig.12 for example,there are eight direction by comparing their perpendicular distances of their packages in total (we use the packages'centers to represent centers.For instance,the perpendicular distance of package the corresponding packages).The dash lines are parallel to A should be smaller than others,so package A is ahead of the different axes respectively.The antenna is above all tags. other packages along the X/Z axis.If not,however,as the It performs a 2-dimensional mobile scanning along the X 2 x 2 package stacking in Fig.11,we cannot identify the and Y axes.All of the tags are always at the same side orders of the packages exactly along the X and Z axes at the against the antenna plane.For the scanning along the X axis, same time only through the 1-dimensional mobile scanning. based on the perpendicular points of the packages'centers, To solve this problem,our solution is to perform one more the packages can be split into two sets:[P2,P3,P6,Pi mobile scanning along the orthogonal direction of the previous and [P,P,P,Ps).In each set,the X coordinates of the scanning direction,so as to limit the number of the free packages are the same,as they share the same perpendicular dimensions in the 3D space.Note that,the more times of the points projected to the X axis.Similarly,based on the mobile mobile scanning is,the much more the cost will be,thus we scanning along the Y axis,the packages can also be split into adopt the mobile scanning twice as the least needed times. two sets:[P5,P6,P7,Ps}and [P1,P2,P3,P}.Combining these two split results,we have four sets then:[P,P, B.Determine Package Stacking with a 2-dimensional Mobile (P2,P,(Ps,Ps}and {P6,P}(in Fig.13).In each set Scanning the packages share the same coordinates along the X and Y Through once mobile scanning,after determining the pack- axes.Then,we just need to determine the packages'orders age orientation for each single package,we derive the indica- along the Z axis.By referring to the perpendicular distances tors of the packages'centers for the relative positions along of the packages'centers,we can identify the packages'orders the scanning direction,so we know their relative orders along in each piece.As we determine the relative positions of each that direction.Similarly,by performing the other scanning piece,and the packages'relative positions in each piece,the along the orthogonal direction of the previous one,we can stacking situation of these packages are determined,the 3D get the packages'relative positions along the new direction. reconstruction for multiple packages is done.ο݀௩ ο݀௛ ܶଵ ܶଶ ܶଷ Fig. 10. The distance differences are different for the equal tag spacing along different directions ܣ ܤ ܥ ܦ ܣ ܤ ܥ ܦ ܣ ܤ ܥ ܦ    Fig. 11. Package stacking ‘‹‰ †‹‡…‹‘    ଼ܲ ܲଵ ܲ ܲ଺ ହ ܲସ ܲଶ ܲଷ ܲ଻ Fig. 12. Determining package stacking in the 3D space VII. DETERMINE PACKAGE STACKING FOR MULTIPLE PACKAGES A. Limitations of the 1-dimensional Mobile Scanning for Determining Package Stacking When we determine the package orientation, we derive the indicators for the relative positions among the tags on a single package. As the geometry relationships of these tags are known, we can combine these indicators (perpendicular dis￾tance and perpendicular point) from different tags to estimate the indicators of the package’s center point. Then, similarly, we compare the indicators of the center points of different packages to determine their stacking situation. Note that, when we determine the package orientation, we only need to perform 1-dimensional scanning, but it may not support the package stacking determination due to the 3-DoF in the 3D space. As shown in Fig. 11, suppose the antenna is above all the packages and is in front of the packages along the X axis, it moves along the Y axis. It is easy to determine the packages’ orders along the Y axis by referring to their perpendicular points, but it may be a problem to determine their orders in the XZ plane. If the packages line up , that is, the packages are along the X axis or along the Z axis, as the left two cases shown in Fig. 11, we can determine their orders along their lining up direction by comparing their perpendicular distances of their centers. For instance, the perpendicular distance of package A should be smaller than others, so package A is ahead of other packages along the X/Z axis. If not, however, as the 2 × 2 package stacking in Fig. 11, we cannot identify the orders of the packages exactly along the X and Z axes at the same time only through the 1-dimensional mobile scanning. To solve this problem, our solution is to perform one more mobile scanning along the orthogonal direction of the previous scanning direction, so as to limit the number of the free dimensions in the 3D space. Note that, the more times of the mobile scanning is, the much more the cost will be, thus we adopt the mobile scanning twice as the least needed times. B. Determine Package Stacking with a 2-dimensional Mobile Scanning Through once mobile scanning, after determining the pack￾age orientation for each single package, we derive the indica￾tors of the packages’ centers for the relative positions along the scanning direction, so we know their relative orders along that direction. Similarly, by performing the other scanning along the orthogonal direction of the previous one, we can get the packages’ relative positions along the new direction.   ܲଶ ܲଷ ܲ଺ ܲ଻ ܲଵ ܲସ ܲହ ଼ܲ ଶݔ ଵݔ (a) along the X axis   ܲହ ܲ଺ ܲ଻ ଼ܲ ܲଵ ܲଶ ܲଷ ܲସ ଶݕ ଵݕ (b) along the Y axis   ଵݕ ଻ܲ ଺ܲ ݕଶ ܲଶ ܲଷ ܲଵ ܲସ ܲହ ଼ܲ ଶݔ ଵݔ (c) split plane    (d) split space Fig. 13. Illustration of determining the packages’ order in the 3D space Combining their relative positions along the two directions, the 3D space is divided into many pieces. For each piece, two dimensions are fixed, the packages in it are lining up, so we can use the perpendicular distances of the packages’ centers to determine their orders in the piece. Thus, all the packages’ relative positions in the 3D space are determined. Taking the scene in Fig. 12 for example, there are eight packages in total (we use the packages’ centers to represent the corresponding packages). The dash lines are parallel to the different axes respectively. The antenna is above all tags. It performs a 2-dimensional mobile scanning along the X and Y axes. All of the tags are always at the same side against the antenna plane. For the scanning along the X axis, based on the perpendicular points of the packages’ centers, the packages can be split into two sets: {P2, P3, P6, P7} and {P1, P4, P5, P8}. In each set, the X coordinates of the packages are the same, as they share the same perpendicular points projected to the X axis. Similarly, based on the mobile scanning along the Y axis, the packages can also be split into two sets: {P5, P6, P7, P8} and {P1, P2, P3, P4}. Combining these two split results, we have four sets then: {P1, P4}, {P2, P3}, {P5, P8} and {P6, P7} (in Fig. 13). In each set, the packages share the same coordinates along the X and Y axes. Then, we just need to determine the packages’ orders along the Z axis. By referring to the perpendicular distances of the packages’ centers, we can identify the packages’ orders in each piece. As we determine the relative positions of each piece, and the packages’ relative positions in each piece, the stacking situation of these packages are determined, the 3D reconstruction for multiple packages is done.