Detection algorithm for separation and merging of drone swarm based on envelope characteristics of photoelectric detection

Abstract

The present application relates to the field of unmanned aerial vehicle group detection, in order to solve the technical problems such as low detection resolution of the existing detection method and inability to carry out overall situation estimation, and proposes a detection algorithm for group separation and combination based on the envelope characteristics of photoelectric detection unmanned aerial vehicle group: a single frame image of the unmanned aerial vehicle group target is collected, the cluster center coordinates, group elliptical envelope edge coordinates, group elliptical envelope principal axis direction, major axis length, minor axis length and eccentricity of each unmanned aerial vehicle group are extracted; the central trajectory motion direction angle, the two edge trajectory motion direction angle, the angle of the two edge trajectories relative to the central trajectory motion direction, the eccentricity difference of each group of three frames, and the distance between the centers of the groups are calculated respectively; the above data is used as input for separation or combination judgment processing, and the combined key frame and the separated key frame are extracted, so that the grouping or combination detection of the unmanned aerial vehicle group target is completed.

Description

Technical Field

[0001] This invention relates to the field of unmanned aerial vehicle (UAV) swarm detection, and more specifically to a detection algorithm for swarm separation and merging based on the envelope characteristics of UAV swarms detected by photoelectric sensors. Background Technology

[0002] Swarm target tracking technology is a crucial research area in machine vision, with wide applications in both military and civilian sectors. As drone applications become more widespread and diverse, simply detecting multiple drones within the field of view is no longer sufficient to meet the demands of swarm-based applications and combat scenarios. Real-time tracking of swarm targets and acquiring spatial situational awareness of multiple drone groups to meet practical application needs is the main future development trend of swarm target tracking technology.

[0003] Group target tracking establishes temporal correlations between states at different times to form a continuous spatiotemporal track, including track initiation, maintenance, termination, separation, and merging; among these, group target separation and merging detection is a key process feature that reflects the overall situation of the group targets.

[0004] Currently, there are two main methods for detecting the separation and merging of swarm targets: one is to consider the swarm's structural model and use an evolutionary graph network model to detect the overall situation of the swarm targets for distinguishable targets; the other is to use information such as position, velocity direction, and trajectory history, based on methods such as clustering, pattern space, and PDA to detect the separation and merging of swarm targets. For low-altitude, slow-flying UAV swarms with high dispersion density, the blind zone of low-altitude flight is large, and the overall characteristics of the swarm are not easy to detect, track, and identify. Both of the above detection methods use radar signals as input. Due to the insufficient detection resolution of radar for low-altitude, slow-flying, and small UAVs, it can only detect individual targets in dense UAV swarm targets and cannot form an overall situation estimate of the swarm targets, nor can it detect key frames of changes in the movement trend of swarm targets in the image sequence. Therefore, it is urgent to find a detection method that can image swarm targets in real time, detect and track individual targets in the field of view, and can associate and manage the state estimates of multiple frames based on video information, distinguish the emergence, splitting, and merging of trajectories, and form an overall situation estimate of the swarm targets. Summary of the Invention

[0005] The purpose of this invention is to address the technical problems of existing swarm target separation and merging detection methods that use radar signals as input. These methods are insufficient in detection resolution for low-altitude, slow-flying UAV swarms with high dispersion density and large blind zones. They can only detect individual targets and cannot form an overall situational estimate of the swarm targets, nor can they detect key frames of changes in the movement trend of the swarm targets in the image sequence. Therefore, this invention proposes a swarm separation and merging detection algorithm based on the envelope characteristics of photoelectric detection UAV swarms.

[0006] The design concept of this invention is as follows: During the target tracking process of a UAV swarm, multiple swarm motion trajectories are established by acquiring swarm envelope characteristic parameters from a sequence of single-frame images of the UAV swarm target. Separation and merging detection are performed in the swarm target tracking process by jointly using parameters such as swarm motion trajectory direction, eccentricity, and swarm cluster center distance, providing accurate real-time swarm situation parameter estimation for anti-UAV operations. By identifying the key frames for separation or merging, the trajectory splitting or merging can be determined, providing accurate criteria for swarm target trajectory updates and improving the stability of swarm target tracking.

[0007] Using photoelectric detection observation data as input, the system detects the separation and merging of group targets by analyzing information such as position, velocity direction, and trajectory history; it also detects key frames in video sequences that show changes in the movement trend of group targets, providing prior information for correcting data association and tracking results.

[0008] To achieve the above objectives, the technical solution proposed by this invention is as follows:

[0009] A detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms is characterized by the following steps:

[0010] S1. Acquire single-frame images of the UAV swarm target using a photoelectric detector, and extract the cluster center coordinates (Ti_x0, Ti_y0) and the edge coordinates (Ti_x1, Ti_y1) and (Ti_x2, Ti_y2) of each UAV swarm Ti in the single-frame image. Obtain the principal axis direction b1, major axis length λ1, minor axis length λ2, and eccentricity of the elliptical envelope.

[0011] S2. Calculate the angle Ti_θ0 of the center trajectory motion direction, the angles Ti_θ1 and Ti_θ2 of the two edge trajectories motion direction, and the angle Ti_Δθ(t) of the two edge trajectories relative to the center trajectory motion direction at time t for each group in a single frame image.

[0012] Calculate the difference in eccentricity Ti_Δe(t) between the three frames before and after each group at time t;

[0013] Calculate the pairwise center distance Tab_Δd(t) of the group at time t;

[0014] S3. Using the angle between the movement directions of each trajectory of the group target, the difference in eccentricity, and the distance between the centers of each pair of groups obtained in step S2 as input, perform separation or merging judgment processing, extract merging key frames and separation key frames, correct the group target tracking parameters and group situation estimation parameters, thereby completing the grouping and merging detection of UAV group targets.

[0015] Furthermore, step S2 specifically includes:

[0016] S2.1 The angle Ti_θ0 of the center trajectory motion direction of each group is calculated by the following formula:

[0017]

[0018] The formulas for calculating the angles Ti_θ1 and Ti_θ2 between the movement directions of the edge trajectory are as follows:

[0019]

[0020]

[0021] The formula for calculating the angle Ti_Δθ(t) between the two edge trajectories and the direction of motion of the center trajectory is as follows:

[0022] Ti_Δθ(t)=|Ti_θ0(t)-Ti_θ1(t)|+|Ti_θ0(t)-Ti_θ2(t)|

[0023] Where (Ti_x0(t), Ti_y0(t)) and (Ti_x0(t-1), Ti_y0(t-1)) are the center trajectory positions of the UAV swarm Ti at time t and t-1, respectively; (Ti_x1(t), Ti_y1(t)) and (Ti_x2(t), Ti_y2(t)) are the two edge trajectory positions of the group elliptical envelope of the UAV swarm Ti at time t; (Ti_x1(t-1), Ti_y1(t-1)) and (Ti_x2(t-1), Ti_y2(t-1)) are the two edge trajectory positions of the group elliptical envelope of the UAV swarm Ti at time t-1; Ti_θ0(t) is the angle between the motion directions of the center trajectory of the UAV swarm Ti at time t; and Ti_θ1(t) and Ti_θ2(t) are the angle between the motion directions of the two edge trajectories of the group elliptical envelope at time t.

[0024] S2.2 The formula for calculating the eccentricity difference Ti_Δe(t) of each group across three consecutive frames is as follows:

[0025] Ti_Δe(t)=Ti_e(t)-Ti_e(t-3)

[0026] Where: Ti_e(t) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t, and Ti_e(t-3) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t-3.

[0027] S2.3, The formula for calculating the pairwise center distance Tab_Δd(t) of the group at time t is:

[0028]

[0029] Where: (Ta_x0(t), Ta_y0(t)) is the center trajectory position of group a at time t, and (Tb_x0(t), Tb_y0(t)) is the center trajectory position of group b at time t.

[0030] Furthermore, step S3 specifically includes: group separation detection and group merging detection;

[0031] The group separation detection includes the following steps:

[0032] S3A.1. The calculated angle Ti_Δθ(t) between the two edge trajectories and the center trajectory at time t in step S2.1 is compared with the set separation angle threshold θ. th Compare; if Ti_Δθ(t)≥θ th Then proceed to step S3A.2; if Ti Δθ (t)<θ th If the frame is not separated, return to step S2 and proceed to the next frame for processing.

[0033] S3A.2. The calculated result of the eccentricity difference Ti_Δe(t) of each group in the three frames before and after each group in step S2.2 is compared with the set eccentricity threshold e. th Compare; if Ti_Δe(t)≥e th If the separation occurs at time t, the image frame at that time is considered the separation initiation keyframe Ti_N(t), thus completing the separation detection and determination of the UAV swarm target, confirming the addition of a new swarm target tracking trajectory, and updating the tracking filter parameters; if Ti Δe (t)<e th If the frame is not separated, return to step S2 and proceed to the next frame for processing.

[0034] The group merging detection includes the following steps:

[0035] S3B.1. Compare the calculated result of the pairwise center distance Tab_Δd(t) of the group at time t in step S2.3 with the set group center distance threshold d. th Compare, if Tab Δd (t)≤d th If Tab Δd (t)>d th If the frame is not merged, return to step S2 and proceed to the next frame for processing.

[0036] S3B.2. Compare the calculated result of the merged trajectory center angle threshold Tab_Δθ(t) in step S2.4 with the set group merged trajectory center angle threshold ab_θ. thCompare; if five consecutive frames satisfy Tab_Δθ(t)≤ab_θ th If the merging of group a and group b occurs at time t, the image frame at that time is the merging start keyframe Tab_N(t), thus completing the merging detection and determination of group a and group b, merging the two trajectories into one trajectory, and updating the tracking filter parameters; if one of the five consecutive frames does not satisfy Tab_Δθ(t)≤ab_θ th If the frame is not merged, it is determined that no merging has occurred, and the process returns to step S2 to proceed to the next frame for processing.

[0037] Furthermore, in step S3B.2, the threshold value of the included angle between the centers of the group merging trajectories is ab_θ. th Determined by the following formula:

[0038]

[0039] in:

[0040] (Ta_x0(t), Ta_y0(t)) represents the position of the center trajectory of group a at time t, and (Ta_x0(t-1), Ta_y0(t-1)) represents the position of the center trajectory of group a at time t-1;

[0041] (Tb_x0(t), Tb_y0(t)) represents the position of the center trajectory of group b at time t, and (Tb_x0(t-1), Tb_y0(t-1)) represents the position of the center trajectory of group b at time t-1.

[0042] The beneficial effects of this invention are:

[0043] [1] The detection algorithm for group separation and merging based on the envelope characteristics of photoelectric detection UAV swarms in this invention can solve the problems of large blind spots, slow flight and high dispersion density of UAV swarms at low altitudes, and inability to identify the overall characteristics of the swarm. By acquiring key frames for group target separation and merging, it provides accurate prior information for correcting data association and tracking results, realizes stable tracking of group targets, and effectively improves the detection accuracy of group separation and merging. It can not only perform real-time imaging of group targets, but also detect and track individual targets in the field of view. It can also associate and manage the state estimation of multiple frames based on video information, distinguish the emergence, splitting and merging of trajectories, and form an overall situational estimation of group targets, which effectively improves the stability and reliability of group target tracking.

[0044] [2] The present invention can track group targets through two-dimensional observation data. It can effectively estimate the movement status of group targets through a simple and effective algorithm, eliminate misjudgment of the overall expansion of group targets and the intersection of group targets, and effectively improve the detection accuracy of group separation and merging.

[0045] [3] By associating and managing the multi-frame state estimation of group targets, the present invention distinguishes between trajectory splitting and merging, and can obtain the overall situational distribution estimation and group correlation of group targets, providing monitoring personnel with intuitive and visual data results, enabling UAV group to achieve intuitive and visual observation and situational estimation. The invention uses dual judgment criteria to detect group targets, realizes independent estimation of group separation and group merging, and effectively improves the detection accuracy of group targets. Attached Figure Description

[0046] Figure 1 This is a schematic diagram illustrating the group target separation determination in an embodiment of the detection algorithm for group separation and merging based on the envelope characteristics of photoelectric detection UAV swarms of the present invention;

[0047] Figure 2 This is a schematic diagram illustrating the group target merging determination in an embodiment of the present invention;

[0048] Figure 3 These are simulation results of keyframe detection for group target separation in this embodiment of the invention;

[0049] Figure 4 The above are simulation results of keyframe detection for merging group targets in an embodiment of the present invention. Detailed Implementation

[0050] A detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms includes the following steps:

[0051] S1. Acquire single-frame images of the UAV swarm target using a photoelectric detector, and extract the cluster center coordinates (Ti_x0, Ti_y0) and the edge coordinates (Ti_x1, Ti_y1) and (Ti_x2, Ti_y2) of each UAV swarm Ti in the single-frame image. Obtain the principal axis direction b1, major axis length λ1, minor axis length λ2, and eccentricity of the elliptical envelope.

[0052] S2. Calculate the angle Ti_θ0 of the center trajectory motion direction, the angles Ti_θ1 and Ti_θ2 of the two edge trajectories motion direction, and the angle Ti_Δθ(t) of the two edge trajectories relative to the center trajectory motion direction at time t for each group in a single frame image.

[0053] Calculate the difference in eccentricity Ti_Δe(t) between the three frames before and after each group at time t;

[0054] Calculate the pairwise center distance Tab_Δd(t) of the group at time t;

[0055] S2.1 The angle Ti_θ0 between the center trajectory directions of each group is calculated using the following formula:

[0056]

[0057] The formulas for calculating the angles Ti_θ1 and Ti_θ2 between the motion directions of the edge trajectory are:

[0058]

[0059]

[0060] The formula for calculating the angle Ti_Δθ(t) between the directions of motion of the two edge trajectories relative to the center trajectory is as follows:

[0061] Ti_Δθ(t)=|Ti_θ0(t)-Ti_θ1(t)|+|Ti_θ0(t)-Ti_θ2(t)|

[0062] Where (Ti_x0(t), Ti_y0(t)) and (Ti_x0(t-1), Ti_y0(t-1)) are the center trajectory positions of the UAV swarm Ti at time t and t-1, respectively; (Ti_x1(t), Ti_y1(t)) and (Ti_x2(t), Ti_y2(t)) are the two edge trajectory positions of the elliptical envelope of the UAV swarm Ti at time t; (Ti_x1(t-1), Ti_y1(t-1)) and (Ti_x2(t-1), Ti_y2(t-1)) are the two edge trajectory positions of the elliptical envelope of the UAV swarm Ti at time t-1; Ti_θ0(t) is the angle between the motion directions of the center trajectory of the UAV swarm Ti at time t; and Ti_θ1(t) and Ti_θ2(t) are the angle between the motion directions of the two edge trajectories of the elliptical envelope of the UAV swarm at time t.

[0063] S2.2 The formula for calculating the eccentricity difference Ti_Δe(t) of each group across three frames is as follows:

[0064] Ti_Δe(t)=Ti_e(t)-Ti_e(t-3)

[0065] Where: Ti_e(t) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t, and Ti_e(t-3) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t-3.

[0066] S2.3, The formula for calculating the pairwise center distance Tab_Δd(t) of the group at time t is:

[0067]

[0068] Where: (Ta_x0(t), Ta_y0(t)) is the center trajectory position of group a at time t, and (Tb_x0(t), Tb_y0(t)) is the center trajectory position of group b at time t.

[0069] S3. The angle between the movement directions of each trajectory of the group target, the difference in eccentricity, and the distance between the centers of each pair of groups obtained in step S2 are used as inputs for separation or merging judgment processing. Merging key frames and separation key frames are extracted, and the group target tracking parameters and group situation estimation parameters are corrected, thereby completing the grouping and merging detection of UAV group targets.

[0070] The process of determining whether to separate or merge groups specifically includes group separation detection and group merging detection.

[0071] Group separation detection includes the following steps:

[0072] S3A.1, such as Figure 1 As shown, the calculated angle Ti_Δθ(t) between the two edge trajectories and the center trajectory at time t in step S2.1 is compared with the set separation angle threshold θ. th Compare; if Ti_Δθ(t)≥θ th Then proceed to step S3A.2; if Ti Δθ (t)<θ th If the frame is not separated, return to step S2 and proceed to the next frame for processing.

[0073] S3A.2, such as Figure 3 As shown, the calculated result of the eccentricity difference Ti_Δe(t) of each group in the three frames before and after in step S2.2 is compared with the set eccentricity threshold e. th Compare; if Ti_Δe(t)≥e th If the separation occurs at time t, the image frame at that time is considered the separation initiation keyframe Ti_N(t), thus completing the separation detection and determination of the UAV swarm target, confirming the addition of a new swarm target tracking trajectory, and updating the tracking filter parameters; if Ti Δe (t)<e th If the frame is not separated, return to step S2 and proceed to the next frame for processing.

[0074] Group merge detection includes the following steps:

[0075] S3B.1, such as Figure 2 As shown, the calculation result of the pairwise distance Tab_Δd(t) between the group centers at time t in step S2.3 is compared with the set group center distance threshold d. th Compare, if Tab Δd (t)≤d th If Tab Δd (t)>d th If the frame is not merged, return to step S2 and proceed to the next frame for processing.

[0076] S3B.2, such as Figure 4 As shown, the calculation result of the merged trajectory center angle threshold Tab_Δθ(t) in step S2.4 is compared with the set group merged trajectory center angle threshold ab_θ. th Compare; if five consecutive frames satisfy Tab_Δθ(t)≤ab_θ th If the merging of group a and group b occurs at time t, the image frame at that time is the merging start keyframe Tab_N(t), thus completing the merging detection and determination of group a and group b, merging the two trajectories into one trajectory, and updating the tracking filter parameters; if one of the five consecutive frames does not satisfy Tab_Δθ(t)≤ab_θ th If the frame is not merged, it is determined that no merging has occurred, and the process returns to step S2 to proceed to the next frame for processing.

[0077] Among them, the threshold of the included angle between the centers of the group merging trajectories is ab_θ. th Determined by the following formula:

[0078]

[0079] in:

[0080] (Ta_x0(t), Ta_y0(t)) represents the position of the center trajectory of group a at time t, and (Ta_x0(t-1), Ta_y0(t-1)) represents the position of the center trajectory of group a at time t-1;

[0081] (Tb_x0(t), Tb_y0(t)) represents the position of the center trajectory of group b at time t, and (Tb_x0(t-1), Tb_y0(t-1)) represents the position of the center trajectory of group b at time t-1.

Claims

1. A detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms, characterized in that, Includes the following steps: S1. Acquire single-frame images of the UAV swarm target using a photoelectric detector, and extract the cluster center coordinates (Ti_x0, Ti_y0) and the edge coordinates (Ti_x1, Ti_y1) and (Ti_x2, Ti_y2) of each UAV swarm Ti in the single-frame image. Obtain the principal axis direction b1, major axis length λ1, minor axis length λ2, and eccentricity of the elliptical envelope. S2. Calculate the angle Ti_θ0 of the center trajectory motion direction, the angles Ti_θ1 and Ti_θ2 of the two edge trajectories motion direction, and the angle Ti_Δθ(t) of the two edge trajectories relative to the center trajectory motion direction at time t for each group in a single frame image. Calculate the difference in eccentricity Ti_Δe(t) between the three frames before and after each group at time t; Calculate the pairwise center distance Tab_Δd(t) of the group at time t; S3. Using the angle between the movement directions of each trajectory of the group target, the difference in eccentricity, and the distance between the centers of each pair of groups obtained in step S2 as input, perform separation or merging judgment processing, extract merging key frames and separation key frames, correct the group target tracking parameters and group situation estimation parameters, thereby completing the grouping and merging detection of UAV group targets.

2. The detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms according to claim 1, characterized in that, Step S2 is as follows: S2.1 The angle Ti_θ0 of the center trajectory motion direction of each group is calculated by the following formula: The formulas for calculating the angles Ti_θ1 and Ti_θ2 between the movement directions of the edge trajectory are as follows: The formula for calculating the angle Ti_Δθ(t) between the two edge trajectories and the direction of motion of the center trajectory is as follows: Ti_Δθ(t)=|Ti_θ0(t)-Ti_θ1(t)|+|Ti_θ0(t)-Ti_θ2(t)| Where (Ti_x0(t), Ti_y0(t)) and (Ti_x0(t-1), Ti_y0(t-1)) are the center trajectory positions of the UAV swarm Ti at time t and t-1, respectively; (Ti_x1(t), Ti_y1(t)) and (Ti_x2(t), Ti_y2(t)) are the two edge trajectory positions of the elliptical envelope of the UAV swarm Ti at time t; (Ti_x1(t-1), Ti_y1(t-1)) and (Ti_x2(t-1), Ti_y2(t-1)) are the two edge trajectory positions of the elliptical envelope of the UAV swarm Ti at time t-1; Ti_θ0(t) is the angle between the motion directions of the center trajectory of the UAV swarm Ti at time t; and Ti_θ1(t) and Ti_θ2(t) are the angle between the motion directions of the two edge trajectories of the elliptical envelope of the UAV swarm at time t. S2.2 The formula for calculating the eccentricity difference Ti_Δe(t) of each group across three consecutive frames is as follows: Ti_Δe(t)=Ti_e(t)-Ti_e(t-3) Where: Ti_e(t) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t, and Ti_e(t-3) is the eccentricity of the group ellipse shape of the UAV swarm Ti at time t-3. S2.3, The formula for calculating the pairwise center distance Tab_Δd(t) of the group at time t is: Where: (Ta_x0(t), Ta_y0(t)) is the center trajectory position of group a at time t, and (Tb_x0(t), Tb_y0(t)) is the center trajectory position of group b at time t.

3. The detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms according to claim 2, characterized in that, Step S3 specifically includes: group separation detection and group merging detection; The group separation detection includes the following steps: S3A.

1. The calculated angle Ti_Δθ(t) between the two edge trajectories and the center trajectory at time t in step S2.1 is compared with the set separation angle threshold θ. th Compare; if Ti_Δθ(t)≥θ th Then proceed to step S3A.2; if Ti Δθ (t)<θ th If the frame is not separated, return to step S2 and proceed to the next frame for processing. S3A.

2. The calculated result of the eccentricity difference Ti_Δe(t) of each group in the three frames before and after each group in step S2.2 is compared with the set eccentricity threshold e. th Compare; if Ti_Δe(t)≥e th If the separation occurs at time t, the image frame at that time is considered the separation initiation keyframe Ti_N(t), thus completing the separation detection and determination of the UAV swarm target, confirming the addition of a new swarm target tracking trajectory, and updating the tracking filter parameters; if Ti Δe (t)<e th If the frame is not separated, return to step S2 and proceed to the next frame for processing. The group merging detection includes the following steps: S3B.

1. Compare the calculated result of the pairwise center distance Tab_Δd(t) of the group at time t in step S2.3 with the set group center distance threshold d. th Compare, if Tab Δd (t)≤d th If Tab Δd (t)>d th If the frame is not merged, return to step S2 and proceed to the next frame for processing. S3B.

2. Compare the calculated result of the merged trajectory center angle threshold Tab_Δθ(t) in step S2.4 with the set group merged trajectory center angle threshold ab_θ. th Compare; if five consecutive frames satisfy Tab_Δθ(t)≤ab_θ th If the merging of group a and group b occurs at time t, the image frame at that time is the merging start keyframe Tab_N(t), thus completing the merging detection and determination of group a and group b, merging the two trajectories into one trajectory, and updating the tracking filter parameters; if one of the five consecutive frames does not satisfy Tab_Δθ(t)≤ab_θ th If the frame is not merged, it is determined that no merging has occurred, and the process returns to step S2 to proceed to the next frame for processing.

4. The detection algorithm for swarm separation and merging based on the envelope characteristics of photoelectric detection UAV swarms according to claim 3, characterized in that: In step S3B.2, the threshold ab_θ of the included angle between the centers of the group merging trajectories th Determined by the following formula: in: (Ta_x0(t), Ta_y0(t)) represents the position of the center trajectory of group a at time t, and (Ta_x0(t-1), Ta_y0(t-1)) represents the position of the center trajectory of group a at time t-1; (Tb_x0(t), Tb_y0(t)) represents the position of the center trajectory of group b at time t, and (Tb_x0(t-1), Tb_y0(t-1)) represents the position of the center trajectory of group b at time t-1.

Images