Unmanned cluster topology repair method in high dynamic environment

Abstract

The invention relates to an unmanned cluster topology restoration method in a high dynamic environment, belongs to the technical field of control, and aims to solve the problems that after an unmanned aerial vehicle cluster suffers from large-scale node failure, a network topology structure is damaged, so that the communication efficiency is reduced, and the task execution capability is reduced. According to the method, a network model is constructed, virtual forces such as guiding force, aggregation force, topological force and isolated node boundary force are utilized to guide survival nodes to move, network connectivity is recovered, and a network topological structure is optimized. In addition, through triangulation based on a particle swarm-simulated annealing algorithm, the network coverage is further optimized, and the network performance is improved. According to the method, the stability and reliability of the unmanned aerial vehicle cluster network are effectively improved, the topology repair cost is reduced, the network coverage is improved, and powerful support is provided for application of the unmanned aerial vehicle cluster network in a complex environment.

Description

Technical Field

[0001] The invention belongs to the field of control technology and relates to an unmanned cluster topology repair method in a high dynamic environment. Background Art

[0002] Due to the characteristics of flexible mobility, convenient deployment, low networking cost, and multi-sensor integration, drones have a high degree of autonomy and adaptability. They have been widely used in relay communications, agriculture, logistics, disaster monitoring, environmental protection and monitoring, infrastructure inspection and other fields. At the same time, with the expansion of drone application scenarios and complex and changing application environments, drone cluster networks can achieve collaborative work and information sharing by combining multiple drones into a network. Drone cluster networks have the characteristics of distributed control, self-organizing collaboration, task division and cooperation, strong robustness, and wide coverage. They can achieve more complex and efficient task execution and adapt to various application scenarios and environments.

[0003] In the process of UAV clusters performing tasks, the nodes of UAVs may be damaged due to their own energy limitations or external environment, which will affect the topological structure and task completion ability of the cluster. Therefore, in order to ensure that the UAV cluster can quickly restore the network performance after the topological structure changes and ensure the data transmission and information sharing of the network, a topology repair method for UAV cluster communication network needs to be designed. First, it can improve the reliability and stability of the network. UAV cluster networks are often used in key tasks such as search and rescue, disaster monitoring, etc. that require high network connectivity. Through topology repair, network connection problems caused by failed nodes can be quickly restored to ensure reliable network transmission, so as to better respond to emergencies and protect people's lives. Secondly, topology repair can improve the efficiency and performance of UAV cluster networks. Due to the large scale of UAV cluster networks and frequent topological changes, topology repair can optimize network coverage, extend network life, and optimize data transmission paths, thereby improving overall network performance. Third, topology repair can improve the adaptability of cluster networks. Faced with complex and changing environments and task requirements, through topology repair methods, cluster networks can have the ability to perceive, share information, and make autonomous decisions, quickly adjust network structures, and improve cluster fault tolerance and adaptability.

[0004] On the one hand, the inventors' existing research on topology repair focuses on wireless sensor networks and mobile ad hoc networks. There are relatively few studies on flying ad hoc networks with high mobility, drastic dynamic topology changes, high connectivity requirements, and long link communication distances. Most of the research focuses on formation control, mission planning, and topology construction in specific scenarios. On the other hand, most of the research focuses on repairing static scenarios and special network nodes after failure, and there are fewer studies on dynamic environments and large-scale drone node failures. Therefore, there is a need for a topology repair method for drone cluster networks in a dynamic environment after large-scale node failures to improve the communication efficiency and fault tolerance of drone cluster networks. Summary of the invention

[0005] In view of this, the purpose of the present invention is to provide an unmanned cluster topology repair method in a highly dynamic environment, which can repair the topological connectivity of the drone cluster network in the event of large-scale node failure and optimize the network topology structure.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A method for repairing a drone cluster network topology comprises the following steps:

[0008] S1: Load network parameters;

[0009] S2: Establish communication links between drone nodes and build normal flight and damaged topology models of cluster networks;

[0010] S3: Normal flight phase: Calculate the virtual force of the drone cluster in the normal flight phase based on the predefined path and update the position information;

[0011] S4: Restoring network 1-connectivity phase: Based on the neighborhood information of the surviving nodes, multiple virtual forces are calculated to repair the network topology and restore network connectivity; where network 1-connectivity means that the network connectivity is 1;

[0012] S5: Optimizing network coverage: Optimizing network topology coverage performance based on global network node location information;

[0013] According to the constructed network topology structure, with the goal of repairing the damaged network connectivity and optimizing the network coverage, a UAV network topology repair method based on improved multiple virtual forces is adopted to update the moving direction and speed of the UAV nodes.

[0014] Furthermore, in S1, network parameters loaded include communication range, sensing range and UAV generation location information, UAV predefined trajectory, and initial generation of UAV obeys Poisson disk distribution.

[0015] Furthermore, in S2, a topology model of a normal flight and a damaged network is constructed, and the specific steps are as follows:

[0016] For a drone cluster V = {u1,u2,...u N}, N is the number of clustered drones, and two drone nodes u i and u j The distance between them is d ij , when d ij Less than the communication distance R c There is a bidirectional link between the two nodes, and the entire network is abstracted as a set of nodes V and a set of communication relationships between drones E = {(u i ,u j )|u i ,u j ∈V,d ij ≤R c} network G=(V,E); when the network nodes of the drone cluster are damaged by a large-scale attack during the movement, the network topology structure is divided into G'={G i =(V i ,E i )|i=1,2,...,m}, including m connected sub-regions.

[0017] Further, in S3, the position information of the drone node in the normal flight phase is updated, and the drone node u i Along the predefined path P i ={p i,n =(x i,n ,y i,n )|n=1,2,...,k} move forward to the target area, k is the number of waypoints in the predefined path; node u i The current position is p i,n =(x i,n ,y i,n ), the next moment position is p i,n+1 =(x i,n+1 ,y i,n+1 ), and its guiding force expression is:

[0018]

[0019] k g is the guiding force adjustment coefficient, The unit vector from the current waypoint to the next waypoint, the resultant force acting on the node Calculate the node speed:

[0020]

[0021] Furthermore, in S4, the type and size of the virtual force of the surviving nodes of the network during the network damage phase are designed, specifically including:

[0022] S41: Setting the guiding force: consistent with the guiding force setting described in S3, the surviving node moves forward along the predefined path;

[0023] S42: Setting aggregation force: The node whose neighborhood is attacked is used as a repair node to complete the repair process. The node determines the direction of the initial aggregation force by the failure of other nodes in its neighborhood. The node receives the location information from its neighbors and the one-hop neighbor list maintained by its neighbors to update its own node neighborhood list, including the node's one-hop neighbor list Γ i ={(id j ,pos j )|d ij <R c} and the corresponding two-hop neighbor list I i ={(id j ,pos j ,Γ j )|d ij <R c}; When the network fails, the node will calculate the aggregate position set U based on the particle swarm-simulated annealing algorithm uc ={o1,o2,...,o l}, l is the number of node aggregation positions, and the node aggregation force is defined as:

[0024]

[0025] k c is the cohesion adjustment coefficient, The node disconnected position set is U uc ={o1,o2,...,o l} position vector;

[0026] S43: Setting topological forces: To ensure that the overall topology is roughly stable, topological forces between neighbors are set to maintain the connection relationship between neighbors, including attraction to prevent neighbors from being disconnected and repulsion to prevent nodes from being too close. The specific definition is:

[0027]

[0028] is attractiveness, defined as:

[0029]

[0030] is the repulsive force, defined as:

[0031]

[0032] k aand k r is the topological force adjustment coefficient, α∈(0,1], αR c is the ideal distance to exert attraction on the node, β∈(0,1], βR d is the ideal distance at which a repulsive force is expected to be applied to the node;

[0033] S44: Set isolated node boundary force: An isolated node is a severely damaged node in the neighborhood. It constitutes a connected branch alone and is not affected by the neighborhood topological force. The boundary force is applied to it through the regional boundary to control its deviation from the cluster. The regional boundary is defined as a rectangular area centered on the predefined path of the node at the next moment. The specific definition is:

[0034]

[0035] k b is the boundary force adjustment coefficient;

[0036] S45: Set the resultant force: Restore 1- The resultant force on the node in the connected stage is:

[0037]

[0038] Set k according to the neighbor survival rate of the node c , defines the neighbor survival rate of a node is the number of real-time neighbors of the node, is the number of neighbors of the node at the initial deployment, and the aggregation coefficient is defined as:

[0039]

[0040] Furthermore, in S5, the network coverage is optimized, and the coverage redundancy or void of the network is calculated by triangulation based on the particle swarm-simulated annealing algorithm, and the coverage repair force is applied to optimize the network. The coverage repair force on the node is specifically defined as:

[0041]

[0042] R oj is the radius of the empty circle of the triangle where the node is located after the network triangulation, is the position vector of the center of the empty circle;

[0043] The resultant force on the node in stage S5 is:

[0044] The beneficial effects of the present invention are:

[0045] (1) The present invention proposes a topology repair solution for the situation where large-scale node failures in the drone cluster network cause serious network damage in dynamic scenarios, thereby improving the stability and reliability of the drone cluster network in complex environments.

[0046] (2) The present invention is based on an improved multiple virtual force algorithm to enable UAV nodes to distribute and calculate the moving direction and speed, reduce the topology repair time and node movement cost, and impose restriction forces on isolated nodes to prevent them from leaving the cluster, thereby maintaining the communication performance of the entire UAV cluster network.

[0047] (3) While repairing the network connectivity, the present invention improves the coverage performance of the damaged network by introducing coverage repair force, which helps to optimize the network topology in many aspects.

[0048] Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

[0050] Figure 1 This is a diagram of the implementation steps of the method for repairing the topology of a drone cluster in a high dynamic environment according to the present invention;

[0051] Figure 2 It is a schematic diagram of the topological structure of the network in the normal stage and the damaged stage of the present invention;

[0052] Figure 3 This is a flow chart of a triangulation method based on improved particle swarm-simulated annealing algorithm. DETAILED DESCRIPTION

[0053] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0054] Among them, the drawings are only used for illustrative explanations, and they only represent schematic diagrams rather than actual pictures, and should not be understood as limitations on the present invention. In order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of actual products. For those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0055] The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "front", "rear", etc. indicate the orientation or position relationship, they are based on the orientation or position relationship shown in the drawings, which is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operate in a specific orientation. Therefore, the terms describing the position relationship in the drawings are only used for illustrative purposes and cannot be understood as limiting the present invention. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

[0056] Figure 1 The following is a diagram of the implementation steps of the method for repairing the topology of a drone cluster in a high dynamic environment according to the present invention. The specific steps include:

[0057] S1: Load network parameters;

[0058] S2: Establish communication links between drone nodes and build normal flight and damaged topology models of cluster networks;

[0059] S3: Normal flight phase: Calculate the virtual force of the drone cluster in the normal flight phase based on the predefined path and update the position information;

[0060] S4: Restoring network 1-connectivity phase: Based on the neighborhood information of the surviving nodes, multiple virtual forces are calculated to repair the network topology and restore network connectivity;

[0061] S5: Optimizing network coverage: Optimizing network topology coverage performance based on global network node location information;

[0062] Furthermore, S1 loads the network parameters, including the communication range, sensing range, UAV generation location information, UAV predefined trajectory, and the initial generation of UAVs follows the Poisson disk distribution;

[0063] further, Figure 2 To instantiate the normal flight and damaged topology of the S2 network, the specific construction steps are as follows:

[0064] For a drone cluster V = {u1,u2,...u 60}, assuming that the two drone nodes ui and u j The distance between them is d ij , when d ij Less than the communication distance R c There is a bidirectional link between the two nodes, and the entire network is abstracted as a set of nodes V and a set of communication relationships between drones E = {(u i ,u j )|u i ,u j ∈V,d ij ≤R c} network G=(V,E). When the network nodes of the drone cluster are damaged by a large-scale attack during the movement, the network topology structure is divided into G'={G i =(V i ,E i )|i=1,2,3,4}, including 4 connected sub-regions.

[0065] Furthermore, S3 updates the position information of the drone node during the normal flight phase. i Along the predefined path P i ={p i,n =(x i,n ,y i,n )|n=1,2,...,k} to the target area, k is the number of waypoints in the predefined path. i The current position is p i,n =(x i,n ,y i,n ), the next moment position is p i,n+1 =(x i,n+1 ,y i,n+1 ), and its guiding force expression is:

[0066]

[0067] k g is the guiding force adjustment coefficient, The unit vector from the current waypoint to the next waypoint, the resultant force acting on the node Calculate the node speed:

[0068]

[0069] Furthermore, S4 designs the types and sizes of virtual forces of surviving nodes in the network during the network damage phase, including:

[0070] S41: Setting the guiding force: consistent with the guiding force setting described in S3, the surviving node still moves forward along the predefined path;

[0071] S42: Set aggregation force: For nodes in the neighborhood that have nodes under attack, it will serve as the main repair node to complete the repair process. Since the damage degree of each node's area is inconsistent, the node needs to determine the direction of the initial aggregation force based on the failure of other nodes in its neighborhood. The node will periodically receive location information from its neighbors and the one-hop neighbor list maintained by its neighbors to update its own node neighborhood list, including the node's one-hop neighbor list Γ i ={(id j ,pos j )|d ij <R c} and the corresponding two-hop neighbor list I i ={(id j ,pos j ,Γ j )|d ij <R c}. When the network fails, the nodes will calculate the aggregate position set U based on the particle swarm-simulated annealing algorithm uc ={o1,o2,...,o l}, l is the number of node aggregation positions, and the node aggregation force is defined as:

[0072]

[0073] k c is the cohesion adjustment coefficient, The node disconnected position set is U uc ={o1,o2,...,o l} position vector.

[0074] Figure 3 This is a flow chart of calculating the aggregated position set of surviving nodes based on triangulation of the particle swarm-simulated annealing algorithm of the present invention, and the specific steps are:

[0075] 1) Initialize particle swarm parameters: set the particle swarm size n and the maximum number of iterations T according to the number of drone clusters. max , set the particle initial value x i and speed v i , limiting the initial particle value to one side of the baseline helps the particles quickly approach the optimal solution, the initial value x i Initialization follows:

[0076]

[0077] e1 and e2 are the endpoints of a baseline that has been added to a triangle in the triangulation network, and v is the point that has formed a triangle with the baseline;

[0078] 2) Set the fitness function f(x i ), the specific expression is:

[0079] f(x i ) = g(x i )+mp(x i )

[0080]

[0081] Point o is the baseline and particle x i The center of the outer circle of the triangle, points e1, e2, x i The coordinates of the circle are (x1, y1), (x2, y2), (x3, y3), and the coordinates of the center of the circle (x o ,y o )for:

[0082]

[0083] 3) Initialize the population individual extreme value pbest and the global extreme value gbest;

[0084] 4) Initialize the simulated annealing algorithm parameters: set the initial temperature T, temperature attenuation coefficient α, and initial energy state E=gbest;

[0085] 5) Generate new solutions w', c1', c2' for particle update parameters and calculate E'=max[f(x i ),i=1,2,...,n], let ΔE=E'-E;

[0086] 6) Determine the energy difference ΔE between the new state and the previous state, and accept the new solution with a certain probability p, where p is defined as:

[0087]

[0088] 7) Update particle velocity v i and the particle position x i , and add a random compensation factor during the update to further prevent the particle swarm from being in the local optimum;

[0089] 8) Update the individual extreme value pbest and the global extreme value gbest;

[0090] 9) If the maximum number of iterations is reached, the algorithm terminates, otherwise continue to execute 5)-9);

[0091] After triangulating the neighborhood of the surviving nodes, the nodes can calculate the aggregation position set of the surviving nodes by comparing the side length of the triangle with the node communication range, and then set the direction and size of the virtual aggregation force. The specific instantiation is as follows: let a triangle be ΔABC, A is the main node, B and C are the nodes in its neighborhood, when AB and AC are smaller than the communication range R c When AC<Rc <AB,BC>R c When AC>R c >AB,BC>R c When AC>R c ,AB>R c When the cohesive force is directed toward the center of the circumscribed circle of the triangle (x o ,y o ). Let the set of disconnected locations generated by the node splitting be U uc ={o1,o2,...,o k}.

[0092] S43: Setting topological forces: To ensure that the overall topology is roughly stable, topological forces between neighbors are set to maintain the connection relationship between neighbors, including attraction to prevent neighbors from being disconnected and repulsion to prevent nodes from being too close. The specific definition is:

[0093]

[0094] is attractiveness, defined as:

[0095]

[0096] is the repulsive force, defined as:

[0097]

[0098] k a and k r is the topological force adjustment coefficient, α∈(0,1], αR c is the ideal distance to exert attraction on the node, β∈(0,1], βR d The ideal distance at which a repulsive force is desired for a node.

[0099] S44: Set isolated node boundary force: An isolated node is a severely damaged node in the neighborhood. It constitutes a connected branch alone and is not affected by the neighborhood topological force. The boundary force is applied to it through the regional boundary to control its deviation from the cluster. The regional boundary is defined as a rectangular area centered on the predefined path of the node at the next moment. The specific definition is:

[0100]

[0101] k b is the boundary force adjustment coefficient.

[0102] S45: Set the resultant force: Restore 1- The resultant force on the node in the connected stage is:

[0103]

[0104] Set k according to the neighbor survival rate of the node c , defines the neighbor survival rate of a node is the number of real-time neighbors of the node, is the number of neighbors of the node at the initial deployment, and the aggregation coefficient is defined as:

[0105] k c =w c (1-η i )

[0106]

[0107] Furthermore, S5 optimizes network coverage by Figure 3 The triangulation based on the particle swarm-simulated annealing algorithm is shown to calculate the coverage redundancy or void of the network, and the coverage repair force is applied to optimize the network. The coverage repair force on the node is specifically defined as:

[0108]

[0109] R oj is the radius of the empty circle of the triangle where the node is located after the network triangulation, is the position vector of the center of the empty circle.

[0110] The resultant force on node S5 is:

[0111] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in the scope of the claims of the present invention.

Claims

1. An unmanned cluster topology repair method in a highly dynamic environment, characterized by: The method comprises the following steps: S1: Load network parameters; S2: Establish communication links between drone nodes and build normal flight and damaged topology models of cluster networks; S3: Normal flight phase: Calculate the virtual force of the drone cluster in the normal flight phase based on the predefined path and update the position information; S4: Restoring network 1-connectivity phase: Based on the neighborhood information of the surviving nodes, multiple virtual forces are calculated to repair the network topology and restore network connectivity; where network 1-connectivity means that the network connectivity is 1; S5: Optimizing network coverage: Optimizing network topology coverage performance based on global network node location information; According to the constructed network topology structure, with the goal of repairing the damaged network connectivity and optimizing the network coverage, a UAV network topology repair method based on improved multiple virtual forces is adopted to update the moving direction and speed of the UAV nodes.

2. The unmanned cluster topology repair method in a high dynamic environment according to claim 1, characterized in that: In S1, network parameters are loaded including communication range, sensing range and UAV generation location information, UAV predefined trajectory, and initial generation of UAV follows Poisson disk distribution.

3. The unmanned cluster topology repair method in a high dynamic environment according to claim 2, characterized in that: In S2, a topology model of a normal flight and a damaged network is constructed, and the specific steps are as follows: For a drone cluster V = {u1,u2,...u N }, N is the number of clustered drones, and two drone nodes u i and u j The distance between them is d ij , when d ij Less than the communication distance R c There is a bidirectional link between the two nodes, and the entire network is abstracted as a set of nodes V and a set of communication relationships between drones E = {(u i ,u j )|u i ,u j ∈V,d ij ≤R c } network G=(V,E); when the network nodes of the drone cluster are damaged by a large-scale attack during the movement, the network topology structure is divided into G'={G i =(V i ,E i )|i=1,2,...,m}, including m connected sub-regions.

4. The unmanned cluster topology repair method in a high dynamic environment according to claim 3, characterized in that: In S3, the position information of the drone node in the normal flight phase is updated. i Along the predefined path P i ={p i,n =(x i,n ,y i,n )|n=1,2,...,k} move forward to the target area, k is the number of waypoints in the predefined path; node u i The current position is p i,n =(x i,n ,y i,n ), the next moment position is p i,n+1 =(x i,n+1 ,y i,n+1 ), and its guiding force expression is: k g is the guiding force adjustment coefficient, The unit vector from the current waypoint to the next waypoint, the resultant force acting on the node Calculate the node speed:

5. The unmanned cluster topology repair method in a high dynamic environment according to claim 4, characterized in that: In S4, the type and size of the virtual force of the surviving nodes of the network during the network damage phase are designed, specifically including: S41: Setting the guiding force: consistent with the guiding force setting described in S3, the surviving node moves forward along the predefined path; S42: Setting aggregation force: The node whose neighborhood is attacked is used as a repair node to complete the repair process. The node determines the direction of the initial aggregation force by the failure of other nodes in its neighborhood. The node receives the location information from its neighbors and the one-hop neighbor list maintained by its neighbors to update its own node neighborhood list, including the node's one-hop neighbor list Γ i ={(id j ,pos j )|d ij <R c } and the corresponding two-hop neighbor list I i ={(id j ,pos j ,Γ j )|d ij <R c }; When the network fails, the node will calculate the aggregate position set U based on the particle swarm-simulated annealing algorithm uc ={o1,o2,...,o l }, l is the number of node aggregation positions, and the node aggregation force is defined as: k c is the cohesion adjustment coefficient, The node disconnected position set is U uc ={o1,o2,...,o l } position vector; S43: Setting topological forces: To ensure that the overall topology is roughly stable, topological forces between neighbors are set to maintain the connection relationship between neighbors, including attraction to prevent neighbors from being disconnected and repulsion to prevent nodes from being too close. The specific definition is: is attractiveness, defined as: is the repulsive force, defined as: k a and k r is the topological force adjustment coefficient, α∈(0,1], αR c is the ideal distance to exert attraction on the node, β∈(0,1], βR d is the ideal distance at which a repulsive force is expected to be applied to the node; S44: Set isolated node boundary force: An isolated node is a severely damaged node in the neighborhood. It constitutes a connected branch alone and is not affected by the neighborhood topological force. The boundary force is applied to it through the regional boundary to control its deviation from the cluster. The regional boundary is defined as a rectangular area centered on the predefined path of the node at the next moment. The specific definition is: k b is the boundary force adjustment coefficient; S45: Set the resultant force: Restore 1- The resultant force on the node in the connected stage is: Set k according to the neighbor survival rate of the node c , defines the neighbor survival rate of a node is the number of real-time neighbors of the node, is the number of neighbors of the node at the initial deployment, and the aggregation coefficient is defined as: k c =w c (1st) i ) 6. The unmanned cluster topology repair method in a high dynamic environment according to claim 1, characterized in that: In S5, the network coverage is optimized, the coverage redundancy or void of the network is calculated by triangulation based on the particle swarm-simulated annealing algorithm, and the coverage repair force is applied to optimize the network. The coverage repair force on the node is specifically defined as: R oj is the radius of the empty circle of the triangle where the node is located after the network triangulation, is the position vector of the center of the empty circle; The resultant force on the node in stage S5 is:

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