An unmanned cluster route planning method based on quantum computing

Abstract

The present application relates to a kind of unmanned cluster route planning method based on quantum computing, comprising: establishing task environment;Unmanned cluster route planning model is established, and is converted into quadratic unconstrained binary optimization problem;Variational quantum algorithm is used to solve planning model;Whether the path corresponding to the solution exists missing or redundancy is judged;Whether the path obtained exists sub-loop is judged, if yes, all nodes contained in sub-loop are extracted, and constraint condition capable of sub-loop elimination is added in planning model;Whether the path obtained exists collision conflict is judged, if yes, all collision conflict conditions are modeled, and variational quantum algorithm is used to solve again, otherwise the result of path planning is output.The method of the present application not only improves the efficiency of path planning, but also reduces the demand for quantum computing physical resources, and has very high application value for the cooperative operation of unmanned cluster.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of swarm path planning, particularly to an unmanned swarm path planning method based on quantum computing. This technical field covers the cross-application of quantum information science, computational optimization theory, and automation control. BACKGROUND

[0002] In the field of modern technology, unmanned swarm path planning technology has attracted much attention due to its significant role in various important application scenarios, such as automated transportation systems, logistics distribution, environmental monitoring, etc. Each individual in the swarm needs efficient and intelligent path planning to improve operational efficiency and safety. Therefore, swarm path planning not only has practical application value but also research significance.

[0003] The problem of unmanned swarm path planning is essentially a complex optimization problem, the core of which is to find the optimal path for individuals in the swarm to meet specific task requirements and constraints. Due to its inherent complexity, the swarm path planning problem is usually classified as an NP-hard problem, which means that as the size of the swarm increases, the computational complexity grows exponentially, and traditional computing methods face great challenges in handling large-scale swarms.

[0004] Quantum computing, as a new computing paradigm, exhibits potential to surpass traditional computing methods in certain problems due to its unique quantum superposition and quantum entanglement characteristics. This feature of quantum computing makes it have acceleration potential in solving certain NP-hard problems, especially in scenarios requiring large-scale parallel computing. Therefore, quantum computing is considered to have the potential to provide new solutions for swarm path planning problems.

[0005] Although quantum computing has great potential in solving swarm path planning problems in theory, relatively few quantum algorithms have been developed for such problems. Existing research has focused on theoretical exploration of quantum algorithms and specific applications, while the development of quantum algorithms for swarm path planning, a specific problem, is still in its infancy. SUMMARY

[0006] The technical problem solved by the present application is that the present application proposes an unmanned swarm path planning method based on quantum computing, aiming to solve the technical problems faced by quantum computing in swarm path planning, particularly how to efficiently utilize limited quantum resources to handle swarm path planning problems involving collision avoidance constraints.

[0007] The solution to the technical problem of the present application is an unmanned swarm path planning method based on quantum computing, comprising the following steps:

[0008] Step 1, establish a task environment, after the individual in the unmanned cluster accepts the mobile task, the number K of individuals participating in the task is determined, the set V of n positions that need to be traversed by the unmanned cluster and the position coordinate information are determined, and the unmanned cluster starts from the starting position;

[0009] Wherein, the set V of positions that need to be traversed by the unmanned cluster is represented as V = {1, 2,..., n}; the starting position and the final arrival position of the individual in the unmanned cluster are recorded as node 0, that is, the individual in the unmanned cluster needs to return to the starting position after completing the mobile task;

[0010] Step 2, establish an unmanned cluster path planning model and convert it into a quadratic unconstrained binary optimization problem;

[0011] Step 3, use the variational quantum algorithm to solve the cluster path planning model in step 2 to obtain a solution;

[0012] Step 4, determine whether the path corresponding to the solution obtained in step 3 is missing or redundant, if so, return to step 3; otherwise, execute the next step;

[0013] Whether the path is missing means that the individual in the unmanned cluster does not completely traverse all the nodes that need to be reached; whether the path is redundant means that there are some nodes that need to be reached are repeatedly reached by the individual in the unmanned cluster 2 times or more than 2 times;

[0014] Step 5, determine whether the path obtained in step 4 has a sub-loop, if so, extract all the nodes contained in the sub-loop, and add a constraint condition capable of eliminating the sub-loop to the unmanned cluster path planning model in step 2, and execute step 3; otherwise, execute the next step;

[0015] The sub-loop means that the individual in the unmanned cluster returns to the starting position without completing the traversal of all the nodes that need to be reached;

[0016] Step 6, determine whether the path obtained in step 5 has a collision conflict, if so, model all collision conflicts, establish a cluster path planning model considering collision avoidance, and use the same variational quantum algorithm as in step 3 to solve; otherwise, output the path planning result, and the calculation is completed;

[0017] Whether the path has a collision conflict means determining whether there is a collision situation in the moving process of the individual in the unmanned cluster.

[0018] Further, the individual in the unmanned cluster in step 1 means an unmanned platform with carrying capacity, including unmanned aerial vehicles, unmanned vehicles and unmanned ships.

[0019] Further, the unmanned cluster path planning model is that the cost between nodes i and j that need to be traversed by the unmanned cluster is defined as w ij ; the set of all starting positions of the unmanned cluster individuals after moving to position i is defined as ; the set of all other nodes that the unmanned cluster individuals can reach from node i is defined as ; x ij is defined as whether the unmanned cluster individual moves from position i to position j: if yes, the value of variable x ij is 1, otherwise 0; the cost function of the unmanned cluster path planning model is constructed according to the above elements as follows:

[0020]

[0021] The constraint condition of the above cost function is as follows:

[0022]

[0023] Further, the conversion into a quadratic unconstrained binary optimization problem is as follows:

[0024] H VRP = H c + H N + H0

[0025] The expressions of H c , H N and H0 are as follows:

[0026]

[0027]

[0028]

[0029] In the above expressions, λ1 and λ2 are two normal numbers, H c represents the total cost of the unmanned cluster path planning, H N and H0 are the constraint conditions of the unmanned cluster path planning model, and H VRP is the cost function of the unmanned cluster path planning model.

[0030] Further, the variational quantum algorithm is that the vector θ is defined as a free parameter in the quantum state evolution process, |ψ(θ)> = U(θ)ψ0> is the final state of the initial state ψ0> after the unit operator U(θ) of the parameter quantum circuit running on the quantum computer, and the cost function C(θ) is defined as follows:

[0031] C(θ) = <ψ(θ)|H|ψ(θ)>

[0032] The quantum circuit consists of n qubits, 1) first need to apply a quantum rotation Y gate to each qubit, 2) then apply quantum CNOT gate to all qubits that satisfy i

[0033] Further, the sub-circuit elimination constraint conditions are as follows:

[0034]

[0035] Wherein, |Q| represents the number of nodes contained in the position node set Q that needs to be reached.

[0036] Further, the modeling of all collision conflicts includes:

[0037] The difference between the individual (i, j) in the unmanned cluster in space and time must be less than the respective minimum safety standard Δ a And Δ t That is: max{A, T} > 0, where A i,t is the position of individual i in the unmanned cluster at time t i , T is the time delay of individual i in the unmanned cluster to reach point , Δ a is the minimum safety standard of the individual in the unmanned cluster in space, Δ t is the minimum safety standard of the individual in the unmanned cluster in time.

[0038] Further, the solution is solved using the same variational quantum algorithm as in step 3, including: after completing the solution, it is necessary to determine whether the solution is infeasible in the actual movement process of the individual in the unmanned cluster.

[0039] Further, the determination of whether the solution is infeasible in the actual movement process of the individual in the unmanned cluster includes: if not, the next step is executed; if yes, add an infeasible solution constraint condition to the solution process and calculate, the infeasible solution constraint condition is:

[0040]

[0041] Where S infea = (s1, s2, …, s n ), s i ∈{0, 1}, i∈{1, 2, …, n} represents the actual infeasible solution, denotes the new solution obtained after adding the infeasible solution constraint condition, Ω0(S infea )={i|s i =0}, Ω1(S infea )={i|s i =1}.

[0042] A computer readable storage medium, the computer readable storage medium stores a computer program, the computer program is executed by the processor to realize the steps of the kind of unmanned cluster path planning method based on quantum calculation.

[0043] The beneficial effects of the present application compared with the prior art are:

[0044] (1) The present application uses variational quantum algorithm as a model solver, realizes the cluster route planning problem containing collision avoidance constraints under limited quantum resources, compared with the traditional algorithm in the prior art, simplifies the problem solving process and improves the solving efficiency under the premise of meeting the collision avoidance requirements.

[0045] (2) The present application optimizes the constraint conditions of the model by introducing infeasible solution constraints, reduces the dependence on quantum bits in the sub-cycle elimination process, even in the worst case, the required number of quantum bits is comparable to the original constraint, which guarantees the accuracy and efficiency of path planning.

[0046] (3) The quantum optimization algorithm proposed in the present application fully utilizes the limited quantum resources through step-by-step optimization, providing a feasible method for solving complex problems, and has high resource utilization and computing efficiency when dealing with large-scale problems. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 The flow chart of the unmanned cluster path planning method based on quantum calculation of the embodiment of the present application;

[0048] Figure 2 The task environment diagram of the simulation case of the unmanned cluster path planning method based on quantum calculation of the embodiment of the present application;

[0049] Figure 3 The quantum calculation algorithm framework used in the embodiment of the present application;

[0050] Figure 4 The result of the unmanned cluster path planning in the embodiment of the present application. DETAILED DESCRIPTION

[0051] With the rapid development of quantum computing technology, its potential in solving complex optimization problems is gradually tapped, especially in scenarios that require processing large-scale data and high-dimensional space search. The present application aims to apply quantum computing technology to the cluster path planning problem, providing a new path planning method for individuals in the cluster, with the goal of reducing energy consumption, improving task execution efficiency and safety. Through quantum algorithms, it is expected to improve the efficiency and accuracy of path planning, thereby promoting technological progress and application expansion in related fields.

[0052] The present application will be further described below in conjunction with the accompanying drawings and examples.

[0053] The present application proposes an unmanned cluster path planning method based on quantum computing, which uses variational quantum algorithms to solve the path planning problem of unmanned clusters. The main process includes constructing an algorithm framework, establishing a task environment, proposing a problem model, designing a quantum computing method and solving.

[0054] 1. Constructing an algorithm framework

[0055] As Figure 1 , the present application introduces quantum computing methods for unmanned cluster path planning problems, considers collision conflict elimination between unmanned cluster individuals, and enables unmanned clusters to complete tasks safely with minimal energy consumption during movement. The specific path planning task process includes:

[0056] Step 1, establish the task environment. After the individuals in the unmanned cluster accept the movement task, determine the number K of individuals participating in the task, and determine the set V of n positions that the unmanned cluster needs to traverse and their position coordinate information, starting from the initial position.

[0057] Step 2, establish the unmanned cluster path planning model and convert it into a quadratic unconstrained binary optimization problem.

[0058] Step 3, use the variational quantum algorithm to solve the cluster path planning model in step 2 to obtain a solution.

[0059] Step 4, determine whether the path corresponding to the solution obtained in step 3 has missing or redundant paths. If so, return to step 3. Otherwise, proceed to the next step.

[0060] The absence of the path means that the unmanned cluster individuals have not completely traversed all the positions of the position nodes that need to be reached.

[0061] The presence of the redundant path means that there are some position nodes that need to be reached by the individuals in the unmanned cluster 2 times or more than 2 times.

[0062] Step 5: Determine whether the path obtained in Step 4 has a sub-loop. If so, extract all nodes contained in the sub-loop and add constraints that can eliminate the sub-loop to the unmanned swarm route planning model (and its corresponding quadratic unconstrained binary optimization problem) in Step 2, and execute Step 3; otherwise, proceed to the next step.

[0063] The sub-loop refers to an individual in an unmanned swarm returning to the starting position without having completed traversing all the required location nodes;

[0064] Step 6: Determine if there are any collision conflicts in the path obtained in Step 5. If so, model all collision conflict scenarios, establish a cluster route planning model that considers collision avoidance, and solve it using the same variable quantum algorithm as in Step 3. If not, output the path planning result and end the calculation.

[0065] The existence of collision conflicts along the path refers to determining whether there are any collisions among individuals in the unmanned swarm during their movement.

[0066] 2. Establish the task environment

[0067] like Figure 2 As shown, an environment is established for executing unmanned swarm path planning tasks. This implementation case sets up two areas, A and B, connected by a river and a bridge. The bridge supports two-way traffic, but only one-way traffic is allowed at any given time. The unmanned swarm in this implementation case consists of unmanned vehicles, which must not enter the river and can only travel between areas A and B via the bridge. There is one unmanned vehicle in area A and one in area B. There are five location nodes in area A and five location nodes in area B that the unmanned vehicles need to reach.

[0068] In this implementation case, two autonomous vehicles need to traverse all required locations without colliding during their movement, and the total distance traveled by the two vehicles to achieve this objective must be minimized. The minimum safe standard distance Δ for the autonomous vehicles is... a Less than or equal to the length of the bridge in this implementation case.

[0069] 3. Propose a problem model

[0070] Let V be the set of locations that the unmanned swarm needs to traverse, denoted as V = {1, 2, ..., 10}; let node 0 be the starting position and the final destination of each individual in the unmanned swarm. Let w be the cost between nodes i and j that the unmanned swarm needs to traverse. ij ;definition Let i be the set of all starting positions that an individual in an unmanned swarm can reach after moving. is the set of all other nodes in the swarm of unmanned vehicles that the individual can reach from node i; define x ij is whether the individual moves from location i to location j in the swarm of unmanned vehicles: if yes, then the variable x ij has a value of 1, otherwise 0; the cost function of the route planning model of the swarm of unmanned vehicles is constructed according to the above elements as

[0071]

[0072] where the constraint condition of the above cost function is:

[0073]

[0074] In the above formula, the mathematical symbol represents "any one", and the mathematical symbol ∈ represents "belongs to".

[0075] The conversion into a quadratic unconstrained binary optimization problem is:

[0076] H VRP = H c + H N + H0,

[0077] where the expressions of H c , H N and H0are respectively:

[0078]

[0079]

[0080]

[0081] In the above expressions, λ1and λ2are two normal numbers, H c represents the total cost of the route planning of the swarm of unmanned vehicles, H N and H0are the constraint conditions of the route planning model of the swarm of unmanned vehicles, and define H VRP is the cost function of the route planning model of the swarm of unmanned vehicles.

[0082] 4. Design a quantum computing method and solve

[0083] As shown in Figure 3 , a quantum computing algorithm is designed and solved. In the figure, the hardware-efficient ansatz is a quantum computing special term, which represents a quantum circuit containing parameters.

[0084] A vector θ is defined as a free parameter in the evolution of the quantum state. |ψ(θ)> = U(θ)ψ0> is the final state of the initial state ψ0> after the unitary operator U(θ) of the parameter quantum circuit running on the quantum computer. The cost function C(θ) is defined as follows:

[0085] C(θ) = <ψ(θ)|H|ψ(θ)>

[0086] The above quantum circuit is composed of n quantum bits, 1) first, a quantum rotation Y gate needs to be applied to each quantum bit, 2) then a quantum CNOT gate is applied to all quantum bits satisfying i < j (requires quantum bit i to control quantum bit j), 3) finally, a quantum rotation Y gate is applied to each quantum bit again, and the last two steps are repeated p times, where p represents the number of layers of the circuit (also known as circuit depth). By continuously optimizing the parameter θ and continuously iterating until the algorithm converges.

[0087] After obtaining the solution of the above calculation process, it is determined that the corresponding path does not have missing or redundant, but there is a sub-loop. Extract all nodes contained in the sub-loop and add the following sub-loop elimination constraint condition:

[0088]

[0089] Where |Q| represents the number of nodes contained in the set of position nodes Q that need to be reached.

[0090] After adding the above sub-loop elimination constraint condition to the unmanned cluster path planning model, the solution is obtained again, and it is found that the path corresponding to the solution has a collision conflict, and the difference between the individual (i, j) in the unmanned cluster in space and time is less than the respective minimum safety standard Δ a and Δ t , that is, max{A, T} > 0, where A i,t is the position of unmanned vehicle i at time t i , and is the time delay of individual i in the unmanned cluster to reach point . After completing the solution, it is further determined whether the solution is feasible in the actual movement of the individual in the unmanned cluster, and if not, the infeasible solution constraint condition is added and calculated during the solution process, where adding the infeasible solution constraint condition means Here S infea = (s1, s2,..., s n ), (s i ∈{0, 1}, i ∈{1, 2,..., n}) represents the actual infeasible solution, represents the new solution obtained after adding the infeasible solution constraint condition, and Ω0(S infea) = {i | s i = 0}, Ω1(S infea ) = {i | s i = 1}. If not, the next step is performed, and the path planning result of the unmanned cluster in this embodiment is obtained as shown in Figure 4 .

[0091] In summary, the cluster path planning method based on quantum computing provided by the present application utilizes the advantages of quantum computing to provide an efficient and innovative solution for the cluster path planning problem.

[0092] The present application provides a computer readable storage medium, which stores computer instructions, when the computer instructions run on a computer, make the computer execute Figure 1 the method.

[0093] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including, but not limited to, disk storage and optical storage, etc.) containing computer-usable program code.

[0094] The present application is described with reference to flowcharts and / or block diagrams according to the methods, devices (systems), and computer program products of the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams can be implemented by computer program instructions. These computer program instructions can be provided to a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to produce a machine, so that the instructions executed by the computer or other programmable data processing devices produce a device that implements the functions specified in the flowcharts and / or block diagrams. Figure 1 flow or multiple flows and / or blocks Figure 1 devices that implement the functions specified in the flowcharts and / or block diagrams.

[0095] These computer program instructions can also be stored in a computer readable memory that can direct the computer or other programmable data processing devices to work in a specific manner, so that the instructions stored in the computer readable memory produce a manufactured product including instruction devices that implement the functions specified in the flowcharts and / or block diagrams. Figure 1 flow or multiple flows and / or blocks Figure 1 devices that implement the functions specified in the flowcharts and / or block diagrams.

[0096] These computer program instructions can also be loaded into a computer or other programmable data processing devices, so that a series of operational steps are performed on the computer or other programmable data processing devices to generate computer-implemented processes, thus the instructions executed on the computer or other programmable data processing devices provide the function of implementing the processes specified in the flowchart Figure 1 one or more flows and / or blocks ​ one or more blocks or steps of the functions specified in the flowchart

[0097] Obviously, those skilled in the art can make various modifications and variations to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application belong to the scope of the claims of the present application and their equivalents, the present application also intends to include these modifications and variations.

[0098] The contents not described in detail in the specification of the present application are the known technology of those skilled in the art.

Claims

1. A method for quantum computing based unmanned swarm path planning, characterized in that, Comprising the following steps: Step 1, establish the task environment, after the individual in the unmanned cluster accepts the mobile task, the number K of individuals participating in the task is determined, the set V of n positions that need to be traversed by the unmanned cluster and the position coordinate information thereof are determined, and the starting position is determined; Wherein, the set V of positions that need to be traversed by the unmanned cluster is represented as V = {1, 2,..., n}; the starting departure position and the final arrival terminal point of the individual in the unmanned cluster are recorded as node 0, that is, the individual in the unmanned cluster needs to return to the starting departure position after completing the mobile task; Step 2, establish an unmanned cluster path planning model and convert it into a quadratic unconstrained binary optimization problem; Said conversion into a quadratic unconstrained binary optimization problem is: H VRP = H c + H N + H0 where H c , H N and H0are expressed respectively as: In the above expression, H c represents the total cost of the unmanned cluster path planning, H N and H0are constraint conditions of the unmanned cluster path planning model, H VRP is the cost function of the unmanned cluster path planning model; w ij is the cost between the nodes i and j that need to be traversed by the unmanned cluster, x ij is whether the individual in the unmanned cluster moves from position i to position j, is the set of all other nodes that the individual in the unmanned cluster can reach from node i, is the set of all starting positions that the individual in the unmanned cluster can reach after moving to position i, λ1and λ2are two normal numbers; Step 3, use the variational quantum algorithm to solve the cluster path planning model in step 2 to obtain a solution; Step 4, determine whether the path corresponding to the solution obtained in step 3 is missing or redundant, if so, return to step 3; otherwise, execute the next step; Said path missing refers to the fact that the individual in the unmanned cluster does not completely traverse all the position nodes that need to be reached; said path redundancy refers to the fact that some position nodes that need to be reached are repeatedly arrived at by the individual in the unmanned cluster 2 times or more than 2 times; Step 5, determine whether the path obtained in step 4 contains a sub-loop, if so, extract all the nodes contained in the sub-loop, add a constraint condition capable of eliminating the sub-loop to the unmanned cluster path planning model in step 2, and execute step 3; otherwise, execute the next step; Said sub-loop refers to the fact that the individual in the unmanned cluster returns to the starting position without completing the traversal of all the position nodes that need to be reached; Step 6, determine whether the path obtained in step 5 contains a collision conflict, if so, model all the collision conflicts, establish a cluster path planning model considering collision avoidance, and use the same variational quantum algorithm as in step 3 to solve it; otherwise, output the path planning result, and the calculation is completed.

2. The method of claim 1, wherein, The individual in the unmanned cluster in step 1 refers to an unmanned platform with carrying capacity, including unmanned aerial vehicles, unmanned vehicles, and unmanned ships.

3. The method of claim 1, wherein, The unmanned cluster path planning model is that the cost between nodes i and j traversed by the unmanned cluster is defined as w ij ; the set of all starting positions of the unmanned cluster is defined as ; the set of all other nodes that the individual in the unmanned cluster can reach from node i is defined as ; x ij is defined as whether the individual in the unmanned cluster moves from position i to position j: if yes, the value of the variable x ij is 1, otherwise 0; and the cost function of the unmanned cluster path planning model is constructed according to the above elements as: Wherein, the constraint condition of the above cost function is:

4. The method of claim 1, wherein, Said variational quantum algorithm is: define the vector theta as a free parameter in the quantum state evolution process, |psi(theta)> = U(theta)|psi0> is the final state of the initial state |psi0> after the unit operator U(theta) of the parameter quantum circuit running on the quantum computer, and the cost function C(theta) is defined as follows: C(theta) = <psi(theta)|H|psi(theta)> Said quantum circuit is composed of n qubits, 1) first, apply a quantum rotation Y gate to each qubit, 2) then apply a quantum CNOT gate to all qubits that satisfy i < j, requiring qubit i to control qubit j, 3) finally, apply a quantum rotation Y gate to each qubit again; repeat the last two steps p times, where p represents the number of layers of the circuit; by constantly optimizing the parameter theta and constantly iterating until the algorithm converges.

5. The method of claim 3, wherein, The constraint condition for eliminating the sub-loop is as follows: Wherein, |Q| represents the number of nodes contained in the position node set Q that needs to be reached.

6. The quantum computing based route planning method for unmanned swarm according to claim 1, wherein, The collision conflict is modeled, including: In an unmanned swarm, the spatial and temporal differences between individuals (i,j) must be less than their respective minimum safety thresholds Δ. a With Δ t That is, max{A,T}>0, where A i,t For individual i in an unmanned swarm, in t i Location at any given moment For individual i in the unmanned swarm, the arrival point The time delay, Δ a Δ is the minimum spatial safety standard for individuals in an unmanned swarm. t This represents the minimum safety standard for individuals in an unmanned swarm over time.

7. The method of claim 1, wherein, The same variational quantum algorithm as in step 3 is used for solving, including: after solving, it is necessary to determine whether the solution is infeasible in the actual movement process of individuals in the unmanned cluster.

8. The quantum computing-based route planning method for unmanned swarm according to claim 7, wherein, The determination of whether the solution is infeasible in the actual movement process of individuals in the unmanned cluster includes: if not, the next step is executed; if yes, an infeasible solution constraint condition is added in the solving process and calculation is performed, and the infeasible solution constraint condition is: Where S infea =(s1,s2,…,s n ),s i ∈{0,1}, i∈{1,2,...,n} represents a practically infeasible solution. This represents the new solution obtained after adding the infeasible solution constraint, Ω0(S infea )={i|s i =0},Ω1(S infea )={i|s i =1}.

9. A computer-readable storage medium storing a computer program, the computer program comprising instructions that, when executed by a computer, cause the computer to perform the method of any one of claims 1 to 8. The computer program is executed by the processor to realize the steps of the method of any one of claims 1-8.

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