Planning and deduction method

Generating new trajectories that conform to the data attributes of the population through generative adversarial network technology, the problem of poor performance of scene matching and interactive scenarios in the existing technology is solved, real-time interactive crowd simulation is realized, and decision-making support is provided for large-scale activity organizations.

CN119475947APending Publication Date: 2025-02-18AEROSPACE SCI & IND GRP INTELLIGENT TECH RES INST CO LTD
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Patent Information

Application Number
CN202311008568.3
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2023-08-10
Publication Date
2025-02-18

AI Technical Summary

Technical Problem

The prior art is difficult to achieve approximate matching in the rule scenarios of simulated characters and the historical data attribute rules in the database, and it is difficult to achieve good results in interactive scenarios (such as collisions, etc.).

Method used

Using a Generative Adversarial Network (GAN) architecture, the generator generates the continuation points of the crowd trajectory and sub-trajectory based on the collected crowd data, and uses the discriminator to distinguish the generated crowd trajectory from the real trajectory until the generated crowd trajectory is confirmed as the final Trajectory, used for real-time crowd simulation.

Benefits of technology

Learn time and spatial attributes through real-world crowd trajectory data, generate new trajectories with similar attributes, and realize real-time interactive crowd simulation, suitable for large-scale event organization decision support.

✦ Generated by Eureka AI based on patent content.

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Abstract

The invention relates to the technical field of real-time simulation, and discloses a planning and deduction method. The method comprises the following steps: collecting crowd data of an activity scene; a generator in a generative adversarial network architecture is utilized, a crowd trajectory pi and continuation points of a sub-trajectory # imgabs0 # are generated according to collected crowd data, the crowd trajectory comprises n pi points # imgabs1 # and sub-trajectories # imgabs2 # which are divided according to a preset time interval delta t, and the sub-trajectories are the sub-trajectories from a point # imgabs3 # to a point # imgabs4 # in the crowd trajectory pi; the continuation point of the sub-track # imgabs5 # is the next point # imgabs7 # after the sub-track # imgabs6 #, a discriminator in the generative adversarial network architecture is used for discriminating the generated crowd track pi and the next point # imgabs8 # from the real crowd track until the next point # imgabs9 # is located outside the region of interest R, and the generated crowd track pi is determined to be the final track; and performing real-time crowd simulation by using the final trajectory.
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Description

Technical Field

[0001] The present invention relates to the field of real-time simulation technology, and in particular to a planning and deduction method. Background Art

[0002] At present, with the rapid improvement of economic levels in various countries, more and more large-scale social activities are being held, including military parades, large-scale sports events, concerts, and event opening and closing ceremonies. These large-scale activities often involve a large number of people, and how to scientifically and efficiently organize a large number of people to complete the event process has become a difficult problem. The use of computer crowd simulation technology can reduce the difficulty of event organization process planning, save a lot of manpower and material resources, and make program planning more intuitive. Therefore, the use of virtual crowd simulation technology to deduce large-scale event organization processes is of great significance for event organization.

[0003] Crowd simulation technology first appeared in a "Boid" model proposed by Reynolds to simulate the group behavior within a flock of birds. Since then, crowd simulation has been continuously studied by scholars and has made great achievements in various fields. Although great progress has been made in the field of crowd simulation at this stage, with the complexity of the social environment and the continuous improvement of computer technology, crowd simulation models in simple scenarios can no longer meet current research needs. Crowd simulation technology is no longer a single computer simulation technology, and it often needs to intersect with disciplines such as psychology and big data.

[0004] The prior art proposes a method for deducing large-scale activities of a large number of people based on a crowd simulation model, wherein the crowd simulation model is divided into a macro model and a micro model. The macro model starts from the global perspective and focuses on the overall movement of the crowd. The micro model starts from the local perspective and focuses on the movement of individuals in the crowd. This method directly obtains the movement trajectory of the crowd based on real-world data such as video surveillance, and stores the input trajectory in a database. Then, when the simulation is running, the environment is divided into different units, and the most matching historical trajectory data is calculated in each unit using an algorithm and applied to the simulation, so as to achieve the optimal result of the global simulation through the optimal matching of different local areas.

[0005] However, although the above existing methods have low simulation calculation costs and can perform optimal matching of different parts through algorithms, if the rule scene of the simulated character is different from the attribute rules of the historical data in the database, it will be difficult to achieve an approximate match, and the simulation task will be greatly different from the actual situation. At the same time, in interactive scenes (such as collisions, etc.), this method is difficult to achieve good results. Summary of the invention

[0006] The present invention provides a planning and deduction method, which can solve the problems in the prior art.

[0007] The present invention provides a planning and deduction method, wherein the method comprises:

[0008] Collect crowd data at the event scene;

[0009] Using the generator in the generative adversarial network architecture, crowd trajectories π and sub-trajectories are generated based on the collected crowd data. The continuation point of the crowd trajectory includes π, including n divided by a predetermined time interval Δt. π Points Subtrack is the midpoint of the crowd trajectory π To point Sub-trajectory, sub-trajectory The continuation point of The next point after

[0010] Using the discriminator in the generative adversarial network architecture, the generated crowd trajectory π and the next point Distinguish from the real crowd trajectory until the next point If it is outside the region of interest R, the generated crowd trajectory π is confirmed to be the final trajectory;

[0011] Use the final trajectories for real-time crowd simulation.

[0012] Preferably, using a generator to generate a crowd trajectory π comprises:

[0013] According to the collected crowd data, the noise vector z is used as the input of the fully connected layer neural network, and the output is the crowd trajectory π represented by a 4-dimensional vector, where the 4-dimensional vector includes the point Coordinates and points coordinate.

[0014] Preferably, the sub-trajectories are generated using a generator The continuation points include:

[0015] Subtrack The noise vector z is used as the input of the long short-term memory network layer that encodes the dynamics of the relevant trajectory. The output result is used as the input of the fully connected layer neural network, and the output is the sub-trajectory represented by a 2D vector The continuation point of , where the 2D vector includes the next point coordinate where θ G The generator parameters.

[0016] Preferably, the discriminator is used to generate the crowd trajectory π and the next point Discrimination with real crowd trajectories includes:

[0017] The entry point of the crowd trajectory π is transformed through a fully connected neural network Normalized to the first scalar in [0,1] The entry point Including the first entry point and the second entry point θ D is the discriminator parameter;

[0018] Sub-trajectories are mapped via fully connected neural networks and long short-term memory networks that encode the dynamics of related trajectories. The planned trajectory in Normalize each point to get the second scalar corresponding to each point

[0019] Using the first scalar The second scalar corresponding to each point Calculate the loss function and determine whether the crowd trajectory π is real based on the loss function.

[0020] Preferably, the loss function is calculated by the following formula:

[0021]

[0022] Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, and Π′ represents the set of crowd trajectories π generated according to different noise vectors z.

[0023] Preferably, the loss function is calculated by the following formula:

[0024]

[0025]

[0026] Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, Π′ represents the set of crowd trajectories π generated according to different noise vectors z, Indicates that the generator is given The next continuation point generated in the case of are real points in the real crowd trajectory.

[0027] Preferably, performing real-time crowd simulation using the final trajectory includes:

[0028] Calculate the attraction point p in the final trajectory att =π(t att ), where t att=min(t+w,T), T is the end time of the crowd trajectory π, the attraction point p att is the point w seconds before π(t), where w is the time window;

[0029] According to the attraction point p att Calculate the preferred speed v pref ;

[0030] Make the agent move at the preferred speed v pref Move along the final trajectory.

[0031] Preferably, the preferred speed v is calculated by the following formula pref :

[0032]

[0033] Where p is the current position of the agent.

[0034] Preferably, in ||v pref ||>s max In the case of v pref Scaling is performed so that ||v pref ||=s max .

[0035] Through the above technical solution, we can learn the time and space attributes of the complete trajectory through the real-world crowd trajectory data, use the generative adversarial network technology to generate new trajectories with similar attributes, and combine the generated trajectories with crowd simulation to conduct multiple rounds of simulation so that the intelligent agents follow these trajectories. Finally, real-time interactive crowd simulation is realized, providing solution support for the decision-making of relevant personnel in the organization of large-scale activities. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The included drawings are used to provide a further understanding of the embodiments of the present invention, which constitute a part of the specification, are used to illustrate the embodiments of the present invention, and together with the text description, explain the principles of the present invention. Obviously, the drawings in the following description are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0037] Figure 1 A flow chart of a planning and deduction method according to an embodiment of the present invention is shown. DETAILED DESCRIPTION

[0038] It should be noted that, in the absence of conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is by no means intended to limit the present invention and its application or use. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0039] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0040] Unless otherwise specifically stated, the relative arrangement of the parts and steps described in these embodiments, numerical expressions and numerical values ​​do not limit the scope of the present invention. At the same time, it should be understood that, for ease of description, the sizes of the various parts shown in the accompanying drawings are not drawn according to the actual proportional relationship. The technology, method and equipment known to ordinary technicians in the relevant field may not be discussed in detail, but in appropriate cases, the technology, method and equipment should be regarded as a part of the authorization specification. In all examples shown and discussed here, any specific value should be interpreted as being merely exemplary, rather than as a limitation. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters represent similar items in the following drawings, so once a certain item is defined in an accompanying drawing, it does not need to be further discussed in subsequent drawings.

[0041] Figure 1 A flow chart of a planning and deduction method according to an embodiment of the present invention is shown.

[0042] like Figure 1 As shown, an embodiment of the present invention provides a planning deduction method, wherein the method includes:

[0043] S100, collecting crowd data at the activity scene;

[0044] S102, using the generator in the generative adversarial network (GAN) architecture, generate crowd trajectory π and sub-trajectory according to the collected crowd data The continuation point of the crowd trajectory includes π, including n divided by a predetermined time interval Δt. π Points Subtrack is the midpoint of the crowd trajectory π To point Sub-trajectory, sub-trajectory The continuation point of The next point after

[0045] S104, using the discriminator in the generative adversarial network architecture, the generated crowd trajectory π and the next point Distinguish from the real crowd trajectory until the next point If it is outside the region of interest R, the generated crowd trajectory π is confirmed to be the final trajectory;

[0046] S106, using the final trajectory to perform real-time crowd simulation.

[0047] In the generative adversarial network, there are two networks for adversarial training. One is the discriminator D, whose goal is to judge as accurately as possible whether a sample comes from real data or is generated by the generator; the other is the generator G, whose goal is to generate samples whose sources cannot be distinguished by the discriminator. Both the generator G and the discriminator D are neural networks, and their parameters are represented by θ. G (generator parameters) and θ D (discriminator parameters). The two networks with opposite goals are constantly trained alternately. When they finally converge, if the discriminator can no longer determine the source of a sample, it is equivalent to the generator being able to generate samples that conform to the real data distribution.

[0048] Through the above technical solution, we can learn the time and space attributes of the complete trajectory through the real-world crowd trajectory data, use the generative adversarial network technology to generate new trajectories with similar attributes, and combine the generated trajectories with crowd simulation to conduct multiple rounds of simulation so that the intelligent agents follow these trajectories. Finally, real-time interactive crowd simulation is realized, providing solution support for the decision-making of relevant personnel in the organization of large-scale activities.

[0049] In the present invention, it can be assumed that the simulation environment is a plane environment, and the virtual people (intelligent agents) therein are modeled as a circle. The crowd trajectory is defined as a mapping π: describes how the agent moves through the environment over a period of T seconds. The trajectory encodes velocity information: it captures the times when the agent accelerates, decelerates, or stops. π Points To separate the trajectory π, that is, each point There is a corresponding timestamp i·Δt. Indicates from point To point Π={π0,π1,…,π N-1} represents a set of n obtained in the real world π The purpose of the generator G is to generate trajectories similar to Π. It is assumed that all generated crowd trajectories start and end at the boundary of the region of interest R, which can be of any shape and any environment.

[0050] According to an embodiment of the present invention, using a generator to generate a crowd trajectory π includes:

[0051] According to the collected crowd data, the noise vector z is used as the input of the fully connected layer neural network, and the output is the crowd trajectory π represented by a 4-dimensional vector, where the 4-dimensional vector includes the point Coordinates and points coordinate.

[0052] According to an embodiment of the present invention, a sub-track is generated using a generator The continuation points include:

[0053] Subtrack The noise vector z is used as the input of the long short-term memory (LSTM) network layer that encodes the dynamics of the relevant trajectory. The output result is used as the input of the fully connected layer neural network, and the output is the sub-trajectory represented by a 2D vector The continuation point of , where the 2D vector includes the next point coordinate where θ G The generator parameters.

[0054] Among them, LSTM is a common recurrent neural network for processing sequence data. The output of this LSTM layer is sent to a fully connected layer, which ultimately produces a 2D vector for the point Coordinates Ideally, this point would be taken from The subsequent unknown distribution may be repeated iteratively to generate the task of sub-trajectory continuation until the newly generated point If it is outside the region of interest R, the trajectory is considered completed.

[0055] According to an embodiment of the present invention, a discriminator is used to compare the generated crowd trajectory π and the next point Discrimination with real crowd trajectories includes:

[0056] The entry point of the crowd trajectory π is transformed through the fully connected layer (FC) neural network Normalized to the first scalar in [0,1] The entry point Including the first entry point and the second entry point (That is, the first two points and ), θ D is the discriminator parameter;

[0057] Sub-trajectories are mapped via fully connected neural networks and long short-term memory networks that encode the dynamics of related trajectories. The planned trajectory in Normalize each point to get the second scalar corresponding to each point

[0058] Using the first scalar The second scalar corresponding to each point Calculate the loss function and determine whether the crowd trajectory π is real based on the loss function.

[0059] In other words, for the discriminator, the discriminator takes the entire real or fake trajectory as input and divides the discrimination task into two tasks with a similar structure as in the generator. For Task 1, i.e. the discrimination entry point part, the fully connected layer will Normalized to the first scalar in [0,1], the first scalar is expressed as The first scalar represents the probability of judging the current entry point as the true trajectory; for task 2, i.e. judging the continuation part, given the predetermined trajectory LSTM and fully connected layers are used for each point Normalize to the second scalar in [0,1] (where k is in the range of 2≤k≤n π ). The second scalar is expressed as That is, the result of the kth point, the second scalar represents the probability of judging the kth point as the true trajectory. π The complete trajectory of points π, the discriminator calculates n π -1 scalars, which together indicate the probability that π is true, and these numbers will be used in the loss function during the training phase.

[0060] During the training phase, each training iteration generates a set of n noise vectors for different noise vectors through the generator. π Then all trajectories (real and fake) are classified by the discriminator. The loss function of the generative adversarial network described in the present invention is the sum of two components. The first component is the n in the set Π of the real crowd trajectories. π The n in the set Π′ of trajectories and generated crowd trajectories π π The entry point of the trajectory is the true success rate, and the second component is the n in the set Π of discriminating the true crowd trajectoryπ The n in the set Π′ of trajectories and generated crowd trajectories π π The continuation point of the trajectory is the actual success rate.

[0061] The first component is expressed as:

[0062]

[0063] The second component is expressed as:

[0064]

[0065] According to one embodiment of the present invention, the loss function is calculated by the following formula:

[0066]

[0067] Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, and Π′ represents the set of crowd trajectories π generated according to different noise vectors z.

[0068] According to an embodiment of the present invention, alternatively, the loss function is calculated by the following formula:

[0069]

[0070] Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, Π′ represents the set of crowd trajectories π generated according to different noise vectors z, Indicates that the generator is given The next continuation point generated in the case of are real points in the real crowd trajectory.

[0071] Specifically, a third component is added in this optional embodiment. For each real crowd trajectory π∈Π, take all valid sub-trajectories of length 5 And let the generator be given , which generates the next continuation point, represented by And calculate its difference with the real point The sum of the Euclidean distances between the real points and the generated points is added to the loss function. This additional component enables the model to converge faster:

[0072]

[0073] In the present invention, a multi-level crowd simulation can be performed, where the calculation of trajectories is separated from the cycle of crowd simulation. This cycle consists of discrete time steps. In each step, new agents may be added, and each agent tries to follow its trajectory while avoiding collisions.

[0074] For the adding agent task, to determine when a new agent should be added to the simulation, an exponential distribution can be used with a parameter λ representing the average time between insertions. This parameter can be obtained from the input dataset to produce similar crowding, but can also be pre-set to other values. Each added agent can follow its own trajectory generated by the generative adversarial network.

[0075] For the trajectory tracking task, in each frame of the simulation cycle, each agent can try to move along its trajectory π while avoiding collisions. The trajectory in the present invention includes a time component that specifies the speed at which the agent should move, and this speed can change over time.

[0076] For example, we can first calculate a preferred speed v pref , the agent will move farther along the crowd trajectory at this speed. This speed can then be input to any existing collision avoidance algorithm to calculate the approach v pref speed while avoiding collisions with other agents.

[0077] According to an embodiment of the present invention, performing real-time crowd simulation using the final trajectory includes:

[0078] Calculate the attraction point p in the final trajectory att =π(t att ), where t att =min(t+w,T), T is the end time of the crowd trajectory π, the attraction point p att is the point that is w seconds before π(t), where w is the time window and t is the time that has passed since the agent joined;

[0079] According to the attraction point p att Calculate the preferred speed v pref ;

[0080] Make the agent move at the preferred speed v pref Move along the final trajectory.

[0081] According to one embodiment of the present invention, the preferred speed v is calculated by the following formula: pref :

[0082]

[0083] Where p is the current position of the agent.

[0084] That is, the preferred speed v pref is the agent's cost t att -t time movement to p att speed.

[0085] According to an embodiment of the present invention, in ||v pref ||>s max In the case of v pref Scaling is performed so that ||v pref ||=s max .

[0086] This prevents the agent from having very high speeds after being blocked for a long time.

[0087] Among them, the time window w and the maximum speed s max You can define how the agent moves along the crowd trajectory. The agent tries to move to a point w seconds ahead along the trajectory π while taking into account the maximum speed.

[0088] In the present invention, the calculated preferred speed v pref Can be used as input to any collision avoidance example.

[0089] It can be seen from the above embodiments that the planning and deduction method described in the above embodiments of the present invention can realize the deduction of the organization process of large-scale activities with massive personnel based on data-driven crowd simulation. The method uses a generative adversarial network to learn the attributes of the input trajectory, and then generates a new trajectory with similar attributes, combined with flexible route tracking that considers time information, for real-time crowd simulation. And it can flexibly create variants of scenes with different densities to meet the actual conditions of different activities.

[0090] In the description of the present invention, it is necessary to understand that the directions or positional relationships indicated by directional words such as "front, back, up, down, left, right", "lateral, vertical, perpendicular, horizontal" and "top, bottom" are usually based on the directions or positional relationships shown in the drawings. They are only for the convenience of describing the present invention and simplifying the description. Unless otherwise specified, these directional words do not indicate or imply that the devices or elements referred to must have a specific direction or be constructed and operated in a specific direction. Therefore, they cannot be understood as limiting the scope of protection of the present invention. The directional words "inside and outside" refer to the inside and outside relative to the contours of each component itself.

[0091] For ease of description, spatially relative terms such as "above", "above", "on the upper surface of", "above", etc. may be used here to describe the spatial positional relationship between a device or feature and other devices or features as shown in the figure. It should be understood that spatially relative terms are intended to include different orientations of the device in use or operation in addition to the orientation described in the figure. For example, if the device in the accompanying drawings is inverted, the device described as "above other devices or structures" or "above other devices or structures" will be positioned as "below other devices or structures" or "below other devices or structures". Thus, the exemplary term "above" can include both "above" and "below". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatially relative descriptions used here are interpreted accordingly.

[0092] In addition, it should be noted that the use of terms such as "first" and "second" to limit components is only for the convenience of distinguishing the corresponding components. If not otherwise stated, the above terms have no special meaning and therefore cannot be understood as limiting the scope of protection of the present invention.

[0093] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A planning deduction method, characterized in that: The method includes: Collect crowd data at the event scene; Using the generator in the generative adversarial network architecture, crowd trajectories π and sub-trajectories are generated based on the collected crowd data. The continuation point of the crowd trajectory includes π, including n divided by a predetermined time interval Δt. π Points Subtrack is the midpoint of the crowd trajectory π To point Sub-trajectory, sub-trajectory The continuation point of The next point after Using the discriminator in the generative adversarial network architecture, the generated crowd trajectory π and the next point Distinguish from the real crowd trajectory until the next point If it is outside the region of interest R, the generated crowd trajectory π is confirmed to be the final trajectory; Use the final trajectories for real-time crowd simulation.

2. The method according to claim 1, characterized in that Using the generator to generate crowd trajectories π includes: According to the collected crowd data, the noise vector z is used as the input of the fully connected layer neural network, and the output is the crowd trajectory π represented by a 4-dimensional vector, where the 4-dimensional vector includes the point Coordinates and points coordinate.

3. The method according to claim 2, characterized in that Generate sub-trajectories using generators The continuation points include: Subtrack The noise vector z is used as the input of the long short-term memory network layer that encodes the dynamics of the relevant trajectory. The output result is used as the input of the fully connected layer neural network, and the output is the sub-trajectory represented by a 2D vector The continuation point of , where the 2D vector includes the next point coordinate where θ G The generator parameters.

4. The method according to claim 3, characterized in that Use the discriminator to generate the crowd trajectory π and the next point Discrimination with real crowd trajectories includes: The entry point of the crowd trajectory π is transformed through a fully connected neural network Normalized to the first scalar in [0,1] The entry point Including the first entry point and the second entry point θ D is the discriminator parameter; Sub-trajectories are mapped via fully connected neural networks and long short-term memory networks that encode the dynamics of related trajectories. The planned trajectory in Normalize each point to get the second scalar corresponding to each point Using the first scalar The second scalar corresponding to each point Calculate the loss function and determine whether the crowd trajectory π is real based on the loss function.

5. The method according to claim 4, characterized in that The loss function is calculated by the following formula: Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, and Π′ represents the set of crowd trajectories π generated according to different noise vectors z.

6. The method according to claim 4, characterized in that The loss function is calculated by the following formula: Among them, V(θ D ,θ G ) represents the loss function, Π represents the set of real crowd trajectories obtained in the real scene, Π′ represents the set of crowd trajectories π generated according to different noise vectors z, Indicates that the generator is given The next continuation point generated in the case of are real points in the real crowd trajectory.

7. The method according to any one of claims 1 to 6, characterized in that Real-time crowd simulation using the final trajectory includes: Calculate the attraction point p in the final trajectory att =π(t att ), where t att =min(t+w,T), T is the end time of the crowd trajectory π, the attraction point p att is the point w seconds before π(t), where w is the time window; According to the attraction point p att Calculate the preferred speed v pref ; Make the agent move at the preferred speed v pref Move along the final trajectory.

8. The method according to claim 7, characterized in that The preferred speed v is calculated by pref : Where p is the current position of the agent.

9. The method according to claim 8, characterized in that In ||v pref ||>s max In the case of v pref Scaling is performed so that ||v pref ||=s max .