Planning method for equidistant spiral motion with width by taking dragon stool as prototype

Abstract

The invention relates to the technical field of equidistant spiral motion, in particular to a planning method for equidistant spiral motion with width by taking a stool dragon as a prototype, which comprises the following steps of: 1, establishing a generalized equidistant spiral motion model, and determining a motion track and calculating the speed; 2, the model in the step 1 is optimized, the coordinate range of the bench where collision occurs is determined, and a strategy for avoiding collision is planned; 3, on the basis of the models in the step 1 and the step 2, calculating the minimum pitch of the U-turn space; step 4, optimizing a turn-round path; and 5, calculating the maximum traveling speed. According to the method, the real-time motion trail of the equidistant spiral motion with the width such as the bench dragon can be analyzed, and the research result can be used for damage prediction in nature such as tornado and typhoon.

Description

Technical Field

[0001] The invention relates to the technical field of isometric spiral motion, and in particular to a planning method for isometric spiral motion with a width taking a bench dragon as a prototype. Background Art

[0002] In traditional Chinese culture, the "stool dragon" is a distinctive folk art form, especially popular in the Jiangnan region. It is usually a long dragon dance performed by multiple people in festivals and celebrations. With its unique structure and dynamic beauty, the bench dragon not only shows the cultural heritage of the Chinese nation, but also contains rich mechanical principles and spatial geometry knowledge. The present invention proposes a new method, which uses the bench dragon as a prototype and combines the motion characteristics of the equidistant spiral with width to conduct research.

[0003] An equidistant spiral is a special mathematical curve that keeps a constant angle between the ray starting from the origin and the spiral. An equidistant spiral with width takes the width variation into further consideration and can be found in many natural phenomena and technical applications, such as the growth pattern of snail shells and the eye structure of hurricanes. The present invention aims to explore the potential application of equidistant spirals with width in the performance of bench dragons through an in-depth study of their motion characteristics. In addition, the research results can also be extended to other technical fields that require simulation of complex curved motion, such as industrial automation and robot trajectory planning, and have important theoretical value and practical significance. Summary of the invention

[0004] The purpose of the present invention is to solve the shortcomings existing in the prior art, and a planning method for equidistant spiral motion with width based on the bench dragon as the prototype is proposed. With the size of the bench, the turning space distance, the pitch, etc. as constraints, a dynamic spiral trajectory tracking simulation optimization model is established. The Euler method, the improved Euler method, the simulated annealing algorithm, the sparrow search algorithm and the locust optimization algorithm are comprehensively used to finally obtain the optimal travel path based on the equidistant spiral motion of the bench dragon.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A planning method for a width equidistant spiral motion based on a bench dragon prototype comprises the following steps:

[0007] Step 1: Establish a generalized equidistant spiral motion model to determine the motion trajectory and calculate the speed;

[0008] Step 2: Optimize the model in step 1, determine the coordinate range of the bench where the collision occurs, and plan a strategy to avoid the collision;

[0009] Step 3: Based on the models of steps 1 and 2, calculate the minimum pitch of the turning space;

[0010] Step 4: Optimize the U-turn path;

[0011] Step 5: Calculate the maximum travel speed.

[0012] Preferably, in step 1, a motion equation in polar coordinates is established: first, the motion trajectory of the dragon dance team is converted into a polar coordinate system, and a position equation of the handle in front of the dragon head is established, and the radial distance r and polar angle (the angle formed by the line connecting the handle and the origin of the coordinates and the positive direction of the x-axis) in polar coordinates are used to describe the position of the dragon dance team;

[0013] The polar coordinate equation of an equidistant spiral is:

[0014] r=α+bθ (1)

[0015] Where α is the radial pitch from the center of the spiral to the initial point, b is the pitch divided by 2π, and the equation for converting polar coordinates to rectangular coordinates is:

[0016]

[0017] Then the speed and coordinates of the front and rear handles of the bench dragon were implemented to further improve the model;

[0018] First, establish the polar coordinate equation:

[0019]

[0020] Where p is the fixed pitch; n0 is the initial number of turns of the faucet;

[0021] By linking the polar coordinate system with the rectangular coordinate system, the coordinate position of the faucet can be expressed as:

[0022]

[0023] Then, the motion model of the first section of the dragon body is derived based on the dragon head motion equation:

[0024]

[0025] Since the distance between the front and rear handles of the bench is constant, the relationship is obtained based on the distance formula between two points:

[0026] (x1-x head ) 2 +(y1-y head ) 2 = l head 2 (7)

[0027] Set up the nonlinear equation:

[0028]

[0029] Solve the equation to get θ1;

[0030] Among them l head The length of the faucet plate minus the distance d between the center of the two holes and the nearest plate head;

[0031] Restrictions:

[0032]

[0033] where θ head is the polar angle of the faucet front handle (i.e. the angle formed by the line connecting the faucet front handle and the origin of the polar coordinate system and the x-axis);

[0034] The motion model of the dragon's front handle and the dragon's tail handle:

[0035]

[0036] The nonlinear equation is established in the same way:

[0037]

[0038] Among them l i It is the length of the dragon body of the i-th section minus the distance between the center of the two holes and the nearest board head, i = 2, 3, 4, ..., 223, when i = 223, it is the handle behind the dragon tail;

[0039] Restrictions:

[0040]

[0041] The Euler method is used to solve the ordinary differential equation of the position of the front handle of the dragon head at each time point, and then the improved Euler method is used to solve the ordinary differential equation of the position of the front handle of each section of the dragon body and the rear handle of the dragon tail.

[0042] Preferably, in step 2, coordinate representation and collision detection: by finding out the coordinates of the four vertices of the dragon head bench and calculating the coordinates of the handle of the dragon body bench closest thereto, to determine the potential collision area;

[0043] System of equations:

[0044]

[0045] where α i is the inclination angle of the perpendicular bisector of the line segment formed by the front and rear handles of the i-th dragon body, β i is the inclination angle of the straight line where the front and rear handles of the i-th dragon body are located, L i is the length of the i-th dragon body bench;

[0046] Substituting the known parameters corresponding to the four vertices of the dragon body into equation (15) to obtain the four equations, the four coordinates obtained by solving the combined equations are the four vertices of the dragon body of the ith section, which are recorded as: (x i1 ,y i1 )、(x i2 ,y i2 )、(x i3 ,y i3 )、(x i4 ,y i4 );

[0047] Then the area where this section of the dragon body is located is:

[0048] {(x,y)|min{x ij}≤x≤max{x ij},min{y ij}≤y≤max{y ij}}j=1,2,3,4(16)

[0049] When the four vertices of the dragon head appear within the range of the dragon body for the first time, the dragon dance team enters the stop time, which is recorded as t 停 ;

[0050] Collision moment prediction: The linear multi-step method is used to solve the model and predict the moment when the dragon dance team stops moving.

[0051] Preferably, in step 3, an objective function is established to represent the minimum pitch required for the front handle of the faucet to enter the turning space;

[0052] Under the model conditions of step 1 and step 2, the expression of the pitch is obtained as follows:

[0053]

[0054] Where p is the pitch of the equidistant spiral motion of the bench dragon;

[0055] The objective function to be solved is:

[0056] minP(18)

[0057] The constraints are:

[0058]

[0059] Optimization algorithm solution: Use simulated annealing or sparrow search algorithm to optimize the objective function and obtain the exact value of the minimum pitch.

[0060] Preferably, in step 4, geometric relationship analysis: analyzing the geometric relationship of each arc segment in the U-turn path to ensure the smoothness of the U-turn path;

[0061] The movement trajectory of the bench dragon in the turning space is simplified and shown through geometric exploration;

[0062] Total length of the U-turn curve is a fixed value;

[0063] Where D is the diameter of the turning space; since the positions of the tangents of the large arc and the small arc to the turning range are fixed, The size of is fixed, that is, R is a fixed value

[0064] Establish model and algorithm solution: Establish a model of the U-turn path and use the locust optimization algorithm to calculate the position and speed of each handle within a specific time.

[0065] Preferably, in step 5, a constraint condition is added: on the basis of the overall moving model of the bench dragon in step 2 and the minimum non-collision model in step 3, a constraint condition is added that the speed of each handle does not exceed the maximum safe speed;

[0066] Traversal calculation: Use the traversal method to search the speed of the faucet handle in steps of 0.1 to find the maximum travel speed that meets the constraints.

[0067] By adopting the above technical scheme: in order to achieve the research purpose, combined with the influencing factors such as the width and pitch of the actual spiral, a dynamic spiral trajectory tracking simulation optimization model is established, and then the simulated annealing method, sparrow search algorithm, etc. are used to solve it. Both algorithms are heuristic optimization algorithms with unique search mechanisms and global search capabilities. They can find the optimal solution in the entire solution space rather than being limited to a local space. At the same time, both can be adjusted according to specific problems to optimize the search efficiency, and finally the motion trajectory characteristics of a spiral with equidistant width are obtained.

[0068] Compared with the prior art, the present invention has the following beneficial effects:

[0069] 1. The present invention utilizes the motion equation under polar coordinates, combined with numerical solutions such as the improved Euler method and the linear multi-step method, to accurately predict the position and speed of the bench dragon during the performance; this method can not only improve the safety of the performance, but also enhance the viewing experience, because more complex performance movements can be designed without worrying about collisions.

[0070] 2. The present invention can effectively prevent accidental collisions during performances by determining the end time (i.e., the collision time) of the dragon dance team's movement; by calculating the coordinates of the four vertices of the dragon head bench and comparing them with the coordinates of the outer dragon body benches, once a possibility of collision is found, the dragon dance team will stop moving, thereby protecting the personal safety of the performers.

[0071] 3. The present invention calculates the minimum pitch required for the front handle of the faucet to turn smoothly through optimization algorithms such as simulated annealing and sparrow search algorithm, so that complex turning actions can be completed in a limited space, improving the utilization rate of the performance venue and making the performance more compact and smooth.

[0072] 4. The present invention determines the maximum travel speed of the faucet under the premise of ensuring that the speed of each handle does not exceed a fixed value through a traversal calculation method, which not only ensures the stability of the performance, but also improves the performance efficiency, making the performance more exciting. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 Schematic diagrams of two possible collision situations in the embodiments of the present invention, wherein (a) shows the outer vertex of the front section of the dragon head bench colliding with the outer dragon body bench; (b) shows the outer vertex of the rear end of the dragon head bench colliding with the outer dragon body bench;

[0074] Figure 2 Schematic diagram of movement in the turning space in an embodiment of the present invention. DETAILED DESCRIPTION

[0075] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings, so that those skilled in the art can better understand the advantages and features of the present invention, thereby making a clearer definition of the protection scope of the present invention. The embodiments described in the present invention are only a part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present invention.

[0076] A planning method for a width equidistant spiral motion based on a bench dragon prototype comprises the following steps:

[0077] Step 1: Establish a generalized equidistant spiral motion model to determine the motion trajectory and calculate the speed;

[0078] Step 2: Optimize the model in step 1, determine the coordinate range of the bench where the collision occurs, and plan a strategy to avoid the collision;

[0079] Step 3: Based on the models of steps 1 and 2, calculate the minimum pitch of the turning space;

[0080] Step 4: Optimize the U-turn path;

[0081] Step 5: Calculate the maximum travel speed.

[0082] Specifically, in step 1, the motion equation in polar coordinates is established: first, the motion trajectory of the dragon dance team is converted into the polar coordinate system, and the position equation of the handle in front of the dragon head is established. The radial distance r and the polar angle (the angle formed by the line connecting the handle and the origin of the coordinates and the positive direction of the x-axis) in the polar coordinates are used to describe the position of the dragon dance team;

[0083] The polar coordinate equation of an equidistant spiral is:

[0084] r=α+bθ (1)

[0085] Where α is the radial pitch from the center of the spiral to the initial point, b is the pitch divided by 2π, and the equation for converting polar coordinates to rectangular coordinates is:

[0086]

[0087] Then the speed and coordinates of the front and rear handles of the bench dragon were implemented to further improve the model;

[0088] First, establish the polar coordinate equation:

[0089]

[0090] Where p is the fixed pitch; n0 is the initial number of turns of the faucet;

[0091] By linking the polar coordinate system with the rectangular coordinate system, the coordinate position of the faucet can be expressed as:

[0092]

[0093] Then, the motion model of the first section of the dragon body is derived based on the dragon head motion equation:

[0094]

[0095] Since the distance between the front and rear handles of the bench is constant, the relationship is obtained based on the distance formula between two points:

[0096] (x1-x head ) 2 +(y1-y head ) 2 = l head 2 (7)

[0097] Set up the nonlinear equation:

[0098]

[0099] Solve the equation to get θ1;

[0100] Among them l head The length of the faucet plate minus the distance d between the center of the two holes and the nearest plate head;

[0101] Restrictions:

[0102]

[0103] where θ head is the polar angle of the faucet front handle (i.e. the angle formed by the line connecting the faucet front handle and the origin of the polar coordinate system and the x-axis);

[0104] The motion model of the dragon's front handle and the dragon's tail handle:

[0105]

[0106] The nonlinear equation is established in the same way:

[0107]

[0108] Among them l i It is the length of the dragon body of the i-th section minus the distance between the center of the two holes and the nearest board head, i = 2, 3, 4, ..., 223, when i = 223, it is the handle behind the dragon tail;

[0109] Restrictions:

[0110]

[0111] The Euler method is used to solve the ordinary differential equation of the position of the front handle of the dragon head at each time point, and then the improved Euler method is used to solve the ordinary differential equation of the position of the front handle of each section of the dragon body and the rear handle of the dragon tail.

[0112] Establish the motion speed model of each handle:

[0113] By discretizing the time t, we gradually calculate the position and speed of each section of the dragon body and tail for each time t, calculate the position of the dragon head, and then deduce the position of the dragon body and tail based on the position of the dragon head. The speed of each section of the bench is calculated by numerical differentiation, and we get:

[0114]

[0115] The positions of the handles of the dragon head, dragon body and dragon tail are calculated, and the velocity differential equations of each handle are written using the difference method. Then, the idea of ​​limit differential is used to replace the differential with the differential, and the velocity differential equation is rewritten into a differential equation. The Euler formula is then used to solve the speed of the dragon head, the front handles of the dragon body and the rear handle of the dragon tail.

[0116] Euler method to solve the position: Based on the equation of motion in polar coordinates, the Euler method is used to solve the position of the faucet front handle to obtain the position coordinates within a given time interval.

[0117] Improved Euler method to solve the positions of other handles: Use the improved Euler method to further solve the positions of the handles in front of the dragon's body and the handles behind the dragon's tail to reduce the error.

[0118] Speed ​​calculation: Based on the position data, the speed of each handle is calculated using the difference method and converted into a differential equation to further improve the accuracy.

[0119] Specifically, in step 2, coordinate representation and collision detection: by finding the coordinates of the four vertices of the dragon head bench and calculating the coordinates of the handle of the dragon body bench closest to them, the potential collision area is determined;

[0120] System of equations:

[0121]

[0122] where α i is the inclination angle of the perpendicular bisector of the line segment formed by the front and rear handles of the i-th dragon body, β i is the inclination angle of the straight line where the front and rear handles of the i-th dragon body are located, L i is the length of the i-th dragon body bench;

[0123] Substitute the known parameters corresponding to the four vertices of the dragon body into the four equations obtained by equation (15). The four coordinates obtained by solving the four equations are the four vertices of the dragon body of the ith section, which are recorded as: (x i1 ,y i1 )、(x i2 ,y i2 )、(x i3 ,y i3 )、(x i4 ,y i4 );

[0124] Then the area where this section of the dragon body is located is:

[0125] {(x,y)|min{x ij}≤x≤max{x ij},min{y ij}≤y≤max{y ij}}j=1,2,3,4(17)

[0126] When the four vertices of the dragon head appear within the range of the dragon body for the first time, the dragon dance team enters the stop time, which is recorded as t 停 ;

[0127] Collision moment prediction: The linear multi-step method is used to solve the model and predict the moment when the dragon dance team stops moving.

[0128] Figure 1(a) and (b) respectively demonstrate two possible collision situations, where (a) shows the outer vertex of the front section of the dragon head bench colliding with the bench of the outer dragon body; (b) shows the outer vertex of the rear section of the dragon head bench colliding with the bench of the outer dragon body.

[0129] Specifically, in step 3, an objective function is established to represent the minimum pitch required for the front handle of the faucet to enter the turning space;

[0130] Under the model conditions of step 1 and step 2, the expression of the pitch is obtained as follows:

[0131]

[0132] Where p is the pitch p of the equidistant spiral motion of the bench dragon;

[0133] The objective function to be solved is:

[0134] minP(19)

[0135] The constraints are:

[0136]

[0137] Optimization algorithm solution: Use simulated annealing or sparrow search algorithm to optimize the objective function and obtain the exact value of the minimum pitch.

[0138] Specifically, in step 4, geometric relationship analysis: analyzing the geometric relationship of each arc segment in the U-turn path to ensure the smoothness of the U-turn path;

[0139] The movement trajectory of the bench dragon in the turning space is simplified and shown through geometric exploration;

[0140] Total length of the U-turn curve is a fixed value;

[0141] Where D is the diameter of the turning space; since the positions of the tangents of the large arc and the small arc to the turning range are fixed, The size of is fixed, that is, R is a fixed value

[0142] Establish model and algorithm solution: Establish a U-turn path model and use the locust optimization algorithm to calculate the position and speed of each handle within a specific time. Figure 2 shown.

[0143] Specifically, in step 5, a constraint condition is added: based on the overall moving model of the bench dragon in step 2 and the minimum non-collision model in step 3, a constraint condition is added that the speed of each handle does not exceed the maximum safe speed;

[0144] Traversal calculation: Use the traversal method to search the speed of the faucet handle in steps of 0.1 to find the maximum travel speed that meets the constraints.

[0145] In summary, the present invention constructs the motion trajectory equation of the bench dragon when it coils in and out and turns around, takes the size of the bench, the turning space distance, the pitch, etc. as constraints, establishes a dynamic spiral trajectory tracking simulation optimization model, and comprehensively uses the Euler method, the improved Euler method, the simulated annealing algorithm, the sparrow search algorithm, the locust optimization algorithm, etc., to solve the problems of accurate motion trajectory prediction, collision avoidance strategies, turning space optimization, and maximum travel speed determination in turn, and obtains the optimal travel path of the bench dragon.

[0146] The description and practice disclosed in the present invention are easy to think and understand for ordinary technicians in the technical field, and several improvements and modifications can be made without departing from the principles of the present invention. Therefore, modifications or improvements made without departing from the spirit of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A planning method for equidistant spiral motion with width based on the bench dragon, characterized in that: The steps include: Step 1: Establish a generalized equidistant spiral motion model to determine the motion trajectory and calculate the speed; Step 2: Optimize the model in step 1, determine the coordinate range of the bench where the collision occurs, and plan a strategy to avoid the collision; Step 3: Based on the models of steps 1 and 2, calculate the minimum pitch of the turning space; Step 4: Optimize the U-turn path; Step 5: Calculate the maximum travel speed.

2. According to claim 1, a method for planning a spiral motion with equal width based on a bench dragon as a prototype, characterized in that: In step 1, establish the motion equation in polar coordinates: first, convert the motion trajectory of the dragon dance team into the polar coordinate system, establish the position equation of the front handle of the dragon head, and use the radial distance r and polar angle in polar coordinates to describe the position of the dragon dance team; The polar coordinate equation of an equidistant spiral is: r=α+bθ (1) Where α is the radial pitch from the center of the spiral to the initial point, b is the pitch divided by 2π, and the equation for converting polar coordinates to rectangular coordinates is: Then the speed and coordinates of the front and rear handles of the bench dragon were implemented to further improve the model; First, establish the polar coordinate equation: Where p is the fixed pitch; n0 is the initial number of turns of the faucet; By linking the polar coordinate system with the rectangular coordinate system, the coordinate position of the faucet is expressed as: Then, the motion model of the first section of the dragon body is derived based on the dragon head motion equation: Each parameter in this equation corresponds to the position parameter of the faucet front handle; Since the distance between the front and rear handles of the bench is constant, the relationship is obtained based on the distance formula between two points: (x1-x head ) 2 +(y1-y head ) 2 =l head 2 (7) Set up the nonlinear equation: Solve the equation to get θ1; Among them l head The length of the faucet plate minus the distance d between the center of the two holes and the nearest plate head; Restrictions: where θ head It is the polar angle of the faucet’s front handle; The motion model of the dragon's front handle and the dragon's tail handle: The nonlinear equation is established in the same way: Among them l i It is the length of the dragon body of the i-th section minus the distance between the center of the two holes and the nearest board head, i = 2, 3, 4, ..., 223, when i = 223, it is the handle behind the dragon tail; Restrictions: The Euler method is used to solve the ordinary differential equation of the position of the front handle of the dragon head at each time point, and then the improved Euler method is used to solve the ordinary differential equation of the position of the front handle of each section of the dragon body and the rear handle of the dragon tail.

3. The method for planning a spiral motion with equal width based on a bench dragon as a prototype according to claim 1, characterized in that: In step 2, coordinate representation and collision detection: by finding the coordinates of the four vertices of the dragon head bench and calculating the coordinates of the handle of the dragon body bench closest to them, the potential collision area is determined; System of equations: where α i is the inclination angle of the perpendicular bisector of the line segment formed by the front and rear handles of the i-th dragon body, β i is the inclination angle of the straight line where the front and rear handles of the i-th dragon body are located, L i is the length of the i-th dragon body bench, k i and k t is the unknown coefficient in the equation; Substituting the known parameters corresponding to the four vertices of the dragon body into equation (15) to obtain the four equations, the four coordinates obtained by solving the combined equations are the four vertices of the dragon body of the ith section, which are recorded as: (x i1 ,y i1 )、(x i2 ,y i2 )、(x i3 ,y i3 )、(x i4 ,y i4 ); Then the area where this section of the dragon body is located is: {(x,y)|min{x ij }≤x≤max{x ij },min{y ij }≤y≤max{y ij }}j=1,2,3,4(16) When the four vertices of the dragon head appear within the range of the dragon body for the first time, the dragon dance team enters the stop time, which is recorded as t 停 ; Collision moment prediction: The linear multi-step method is used to solve the model and predict the moment when the dragon dance team stops moving.

4. The method for planning a spiral motion with equal width based on a bench dragon as a prototype according to claim 1, characterized in that: In step 3, an objective function is established to represent the minimum pitch required for the front handle of the faucet to enter the turning space; Under the model conditions of step 1 and step 2, the expression of the pitch is obtained as follows: Where P is the pitch of the equidistant spiral motion of the bench dragon; The objective function to be solved is: minP(18) The constraints are: Optimization algorithm solution: Use simulated annealing or sparrow search algorithm to optimize the objective function and obtain the exact value of the minimum pitch.

5. The method for planning a spiral motion with equal width based on the bench dragon as the prototype according to claim 1 is characterized in that: In step 4, geometric relationship analysis: analyzing the geometric relationship of each arc segment in the U-turn path to ensure the smoothness of the U-turn path; The movement trajectory of the bench dragon in the turning space is simplified and shown through geometric exploration; Total length of the U-turn curve is a fixed value; Where D is the diameter of the turning space; since the positions of the tangents of the large arc and the small arc to the turning range are fixed, The size of is fixed, that is, R is a fixed value Establish model and algorithm solution: Establish a model of the U-turn path and use the locust optimization algorithm to calculate the position and speed of each handle within a specific time.

6. The method for planning a spiral motion with equal width based on a bench dragon as a prototype according to claim 1, characterized in that: In step 5, add constraints: based on the model of the overall movement of the bench dragon in step 2 and the minimum non-collision model and equation in step 3, add the constraint that the speed of each handle does not exceed the maximum safe speed; Traversal calculation: Use the traversal method to search the speed of the faucet handle in steps of 0.1 to find the maximum travel speed that meets the constraints.

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