Calculation method of stool dragon mathematical model

Abstract

The invention discloses a stool dragon mathematical model calculation method, which comprises the following steps of S1, establishing a stool dragon motion mathematical model; s2, on the basis of the model in S1, setting an algorithm under the condition that stools are stopped due to mutual collision and cannot continue to enter, and solving; the method has the advantages that the variable step size search algorithm based on dichotomy is adopted, the optimal solution can be efficiently approached in the global range, fine optimization can be achieved in the local area, and in the model building process, the optimal solution can be accurately obtained, so that the optimal solution can be accurately obtained. Double mapping is constructed by using an idea of converting a curve into a straight to obtain a correlation between variables, an iterative solution idea is provided, and a vector method is used for participating in modeling, so that the calculation amount is reduced, and the programming difficulty is reduced.

Description

Technical Field

[0001] The invention relates to the technical field of calculation methods, and in particular to a calculation method for a mathematical model of a bench dragon. Background Art

[0002] "Bench Dragon", also known as "Panlong", is a traditional folk cultural activity in Zhejiang and Fujian. People connect dozens or even hundreds of benches end to end to form a winding bench dragon. When coiling the dragon, the dragon head leads the way, and the dragon body and tail follow and coil, forming a disc shape as a whole.

[0003] Generally speaking, under the premise that the dragon dance team can move in and out freely, the smaller the area required for the dragon and the faster the moving speed, the better the viewing experience. In order to achieve the above purpose, this application proposes to use optimization algorithms to solve related problems related to the dragon dance team's movement. Summary of the invention

[0004] The purpose of the present invention is to solve the shortcomings in the prior art and to propose a calculation method for the mathematical model of a bench dragon.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] A method for calculating a mathematical model of a bench dragon comprises the following steps:

[0007] S1: Establish the mathematical model of bench dragon movement;

[0008] S2: Based on the model of S1, set up an algorithm and solve the problem when the benches cannot continue to be placed in each other due to collision;

[0009] S3: Set the algorithm for setting the minimum pitch of the dragon's front handle along the spiral to the boundary of the turning space and solve it;

[0010] S4: Set the algorithm for the minimum turning length curve when turning the dragon's front handle and solve it;

[0011] S5: Set the algorithm for the maximum travel speed of the faucet and solve it.

[0012] Preferably, when establishing the model in S1, in order to facilitate the establishment of the model, we regard each bench in motion as a line segment. First, according to the Archimedean spiral equation, the mapping relationship between the rotation angle θ and the time t is established; then, according to the movement speed and initial position of the faucet, the position of the faucet at different times is obtained, and considering the connection relationship between the benches, the positions of the various parts of the faucet and the tail are gradually obtained through the position of the faucet; then, the derivative of the position with respect to time is used to obtain the speed of each part; finally, we use the binary search method with variable step size to solve the equation.

[0013] Preferably, in S2, according to the length, width and connection mode between adjacent benches, a spatial geometric relationship is established when a collision occurs, and then the collision condition is determined. In this process, we use Schmidt orthogonalization on the vector between the judgment point and the geometric center of the bench, and the collision condition can be quickly determined by comparing the modulus of the two components of the vector with the length of the corresponding side of the rectangle; combining the collision condition and the equation model in problem one, the variable step size bisection method is used for iteration to obtain the moment when the dragon dance team collides.

[0014] Preferably, in S2, according to the process of the dragon dance freely turning into the boundary of the turning space, the properties of the spiral are analyzed in combination with the knowledge of differential geometry, and then a relationship model between the pitch and the spiral is established; then, a single-objective optimization model is established with the minimum Archimedean pitch as the optimization objective function and no collision as the constraint condition; for each pitch, the boundary conditions are brought in, and after the time t is calculated, collision detection is iterated to determine whether the boundary conditions are met.

[0015] Preferably, in S4, we first establish the relationship between the radius and the center of the two arcs, and further obtain the total arc length expression of the S-curve using the arc length formula; then, taking the shortest total arc length as the objective function and the relevant geometric relationship as the constraint condition, a single-objective programming model is established to adjust the U-turn path. In this process, the constraint conditions are integrated and processed and combined with the vector method to obtain a nonlinear optimization model, which is traversed and solved using a numerical method to obtain the minimum U-turn curve length.

[0016] Preferably, in S5, based on the travel route in S4, a single-objective planning model needs to be established with the maximum dragon head travel speed as the optimization target and the upper limit of the speed of the dragon dance team at each moment as a constraint condition; a variable step-size algorithm based on binary search is used to search for the maximum dragon head travel speed. If the speed of the dragon dance exceeds the upper limit at a certain moment in the search process, the iteration is stopped immediately and returned to the previous iteration to solve for the maximum travel speed of the dragon head.

[0017] The beneficial effects of the present invention are: the present invention provides a method for calculating the mathematical model of the bench dragon. 1. When solving the model, we adopt a variable step-size search algorithm based on binary search, which can not only efficiently approximate the optimal solution in the global range, but also perform fine optimization in the local area.

[0018] 2. In the establishment of the model in this paper, a double mapping is constructed by using the idea of ​​turning curves into straight lines to obtain the correlation between variables and provide an iterative solution idea.

[0019] 3. Use vector methods to participate in modeling, reduce the amount of calculation and reduce the difficulty of programming. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1It is a basic structural schematic diagram of the present invention;

[0021] Figure 2 It is the algorithm flow chart of the present invention;

[0022] Figure 3 It is a simulation diagram of the movement of the bench dragon at the collision moment of the present invention;

[0023] Figure 4 It is a diagram for proving the equality of central angles of a circle of the present invention;

[0024] Figure 5 It is a schematic diagram of Schmidt orthogonalization of the present invention. DETAILED DESCRIPTION

[0025] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions 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. Based on the embodiments of 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.

[0026] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as a limitation on the present invention.

[0027] like Figure 1 - Figure 5 As shown, the present invention provides a method for calculating the mathematical model of a bench dragon, comprising the following steps:

[0028] S1: Establish the mathematical model of bench dragon movement;

[0029] S2: Based on the model of S1, set up an algorithm and solve the problem when the benches cannot continue to be placed in each other due to collision;

[0030] S3: Set the algorithm for setting the minimum pitch of the dragon's front handle along the spiral to the boundary of the turning space and solve it;

[0031] S4: Set the algorithm for the minimum turning length curve when turning the dragon's front handle and solve it;

[0032] S5: Set the algorithm for the maximum travel speed of the faucet and solve it.

[0033] When the model is established in S1, for the convenience of model establishment, we regard each bench in motion as a line segment. First, according to the Archimedean spiral equation, the mapping relationship between the rotation angle θ and the time t is established; then, according to the movement speed and initial position of the faucet, the position of the faucet at different times is obtained. Considering the connection relationship between the benches, the positions of the faucet and the tail are gradually obtained through the position of the faucet; then, the derivative of the position with respect to time is used to obtain the speed of each part; finally, we use the variable step size bisection method to solve the equation;

[0034] The Archimedean spiral is a spiral curve whose shape expands at equal distances in each rotation period, so it is called an "equidistant spiral". The polar coordinate equation of the Archimedean equidistant spiral can be expressed as:

[0035] ρ=α+βθ.

[0036] Where α is the distance from the starting point to the origin of the coordinate system, and β is the distance between two adjacent curves, thus obtaining the spiral equation in the Cartesian coordinate system:

[0037] x=(α+βθ)cos(θ)y=(α+βθ)sin(θ)

[0038] (x(θ), y(θ)) are the position coordinates of any point on the equidistant spiral;

[0039] When the pitch of the equidistant spiral is 55 cm, we can get α=0, Where d represents the pitch.

[0040] When the continuous function is in [α, β], the formula is used:

[0041]

[0042] Get the length of the curve;

[0043] s i As the upper limit of integration, the initial value θ=0 as the lower limit of integration, the formula is used:

[0044]

[0045] Get the travel distance of the faucet front handle;

[0046]

[0047] The s in the formula i ,θ i Represents the position of the faucet handle at each moment.

[0048] The formula to calculate the distance of the faucet handle is:

[0049] s i =s t -vt, (3) Where v is the speed of the dragon's front handle, which is 1 m / s. Substitute θ0 = 32π into formula (1) and solve for the total length of the spiral s. t .

[0050] According to the mapping properties, we can know that:

[0051] s i =f(θ i ). (4)

[0052] θ i =f -1 (s i ). (5)

[0053] The distance between the front handles of the bench dragon is a fixed value, and we can know that:

[0054]

[0055] Among them l j,j+1 Represents the Euclidean distance between the jth dragon front handle and the j+1th dragon front handle. They represent the position components on the front handle of the j-th or j+1-th dragon at time i respectively.

[0056] The speed of the faucet and each part at a time point is calculated using the formula:

[0057]

[0058] in Represents the weight of the j-th dragon's front handle in the Cartesian coordinate system at time i. They represent the position components on the j-th or j+1-th dragon's front handle at time i respectively. Δt is the time interval, which is 1s in S1.

[0059] A variable step-size algorithm based on binary search is used.

[0060] Step 1: Determine a more appropriate change interval and set the initial step length k to half of the original step length.

[0061] Step 2: Determine whether the current parameter M(i) needs to change in the opposite direction. If so, the change step size is halved. If not, the change direction is changed with the current step size.

[0062] Step 3: During the iteration process, the algorithm continuously updates the iteration point until the distance between the point and the optimal target point is less than the set threshold, then stops the iteration and considers that the optimal solution has been reached.

[0063] Determine the initial conditions:

[0064]

[0065] According to formulas (2)-(5), the differential equation can be obtained to calculate the position coordinates of the faucet at each moment:

[0066]

[0067] The relationship between adjacent dragon front handles is expressed using a difference equation:

[0068]

[0069] According to (7)-(9), the differential equation can be obtained to calculate the speed of each dragon's front handle:

[0070]

[0071] According to the above formula, the positions and speeds of the faucet front handle, the 1st, 51st, 101st, 151st, 201st dragon body front handles behind the faucet, and the dragon tail rear handle are calculated at 0s, 60s, 120s, 180s, 240s, and 300s, as shown in the following figure:

[0072]

[0073]

[0074] In S2, according to the length, width and connection mode of the benches, the spatial geometric relationship when the collision occurs is established, and then the collision condition is determined. In this process, we use Schmidt orthogonalization on the vector between the judgment point and the geometric center of the bench. By comparing the modulus of the two components of the vector with the length of the corresponding side of the rectangle, the collision condition can be quickly determined; combining the collision condition and the equation model in problem 1, the variable step size binary search method is used for iteration to obtain the moment when the dragon dance team collides; Figure 1 As shown, we know that D is a vertex of the other rectangle, a and b are the length and width of the rectangle respectively. Perform Schmidt orthogonalization. The unit vectors in the two directions are have:

[0075]

[0076] It can be known exist The projection in the direction is:

[0077]

[0078] Knowable exist The projection in the direction is:

[0079]

[0080] Combining formulas (9) and (10), we can find the vertices of the rectangle Calculate, the calculation formula is:

[0081]

[0082] If the situation where no collision occurs is constrained, the calculation formula is:

[0083]

[0084] Use a variable-step search based on binary search to find the jth front handle coordinates Calculate the coordinates of j+1 front handles The corresponding speed is obtained by solving formula (10).

[0085] Step 1: Use the above calculation formula, substitute time, and find out the position of each faucet.

[0086] Step 2: The position P of the jth handle and the j+1th handle j With P j+1 ,pass Find the geometric center.

[0087] Step 3: Use v=P j+1 -P j Find a direction vector and then use Make it unit, and then use w j =(-v y , v x ) find the value perpendicular to ν j The unit vector of .

[0088] step4. Through the long side length a and the short side length b, Find the four vertices and record all the vertices of all the wooden boards.

[0089] step5. i and Substitute it into the collision judgment. If a collision occurs, terminate the algorithm, return the collision time, and output the intersection mark; otherwise, after traversing all elements, output the non-intersection mark.

[0090] The following table shows the positions and velocities of some handlebars when a collision occurs after substituting into the formula:

[0091]

[0092] In S2, according to the process of the dragon dance freely entering the boundary of the turning space, the properties of the spiral are analyzed in combination with the knowledge of differential geometry, and then a relationship model between the pitch and the spiral is established; then, the minimum Archimedean pitch is used as the optimization objective function, and no collision is used as a constraint condition to establish a single-objective optimization model; for each pitch, the boundary conditions are brought in, and after the time t is calculated, collision detection is iterated to determine whether the boundary conditions are met; the minimum Archimedean pitch is used as the optimization objective function

[0093] min d (16)

[0094] The U-turn space is a circular area with a diameter of 9m and the center of the spiral. When the dragon dance moves to the boundary of the U-turn space, certain restrictions are met:

[0095]

[0096] Where θ e is the angle of rotation when the dragon dance moves to the boundary, ρ is the polar diameter at this time, which is 4.5 m. From formula (4), we can know:

[0097] s0=f(θ0),s e =f(θ e ), (18)

[0098] s e is the displacement of the dragon dance when it moves to the boundary. The change of speed and time can be known from formula (3):

[0099] s e -s0=vt. (19)

[0100] The constraint condition of no collision can be calculated by formula (15). The dragon dance team freely enters and exits in a clockwise direction, which shows that:

[0101]

[0102] The above formulas are synthesized to establish a single-objective optimization model with the minimum pitch d as the goal:

[0103] min d

[0104]

[0105] In S4, we first establish the relationship between the radius and the center of the two arcs, and use the arc length formula to further obtain the total arc length expression of the S-curve; then, with the shortest total arc length as the objective function and the relevant geometric relationship as the constraint condition, a single-objective planning model is established to adjust the U-turn path. In this process, the constraint conditions are integrated and combined with the vector method to obtain a nonlinear optimization model, which is traversed and solved using a numerical method to obtain the minimum U-turn curve length;

[0106] Assume that the turning path is an S-shaped curve formed by two tangent arcs. The radius of the former arc is twice that of the latter arc, and it is tangent to both the winding-in and winding-out spirals.

[0107] Assume that the point of tangency between the spiral and the turning area is P = (x, y), and solve the equation of the spiral and the equation of the circle simultaneously:

[0108] x 2 +y 2 =r 2 , It can be known that the coordinates of this point are The tangent unit vector is The normal unit vector is Where r is the turning point;

[0109] In order to reduce the length of the U-turn curve as much as possible, assuming that the lengths of the first and second sections of the S-curve are L1 and L2 respectively, we can get:

[0110] min L1+L2 (20) The first and second segments of the S-curve have only one circle. Assuming that the centers of the circles are (x1, y1) and (x2, y2), the corresponding center angles are θ1 and θ2, and the radius lengths are R1 and R2, we can know that:

[0111] L1+L2=R1θ1+R2θ2 (21)

[0112] So we know that the equations of circle 1 and circle 2 are:

[0113]

[0114] Considering that the exit spiral and the entry spiral are centrally symmetric about the center, the coordinates of the tangent point between the exit spiral and the turning area are From geometry we know that the tangent unit vector is And its Parallel, the normal unit vector is And its Parallel; vector addition and subtraction of the two tangent points can reveal the coordinates of the two circle centers:

[0115]

[0116] According to the distance relationship between the center of the circle (x1, y1) and (x2, y2), we can know that:

[0117]

[0118] R1=2R2 (27)

[0119] The tangent points of the arc all satisfy the arc equation:

[0120]

[0121] The above formulas are synthesized to establish a single-objective optimization model:

[0122] min L1+L2

[0123]

[0124] Furthermore, we can substitute the nonlinear equation:

[0125]

[0126] By solving this nonlinear equation, we can obtain all the geometric parameters of this problem.

[0127] The speed of the handle in front of the dragon head is always maintained at 1m / s. Taking the turning start time as zero, from -100s to 100s, the positions and speeds of the handle in front of the dragon head, the handles in front of the 1st, 51st, 101st, 151st, 201st sections of the dragon body behind the dragon head, and the handle behind the dragon tail are calculated as follows:

[0128]

[0129]

[0130] In S5, based on the route in S4, a single-objective planning model is needed to be established with the maximum dragon head speed as the optimization target and the upper limit of the dragon dance team's speed at each moment as the constraint condition; a variable step-size algorithm based on binary search is used to search for the maximum dragon head speed. If the speed of the dragon dance exceeds the upper limit at a certain moment during the search process, the iteration is immediately stopped and returned to the previous iteration to solve the maximum dragon head speed; the formula for setting the maximum dragon head speed is:

[0131] max v (30)

[0132] Calculate with the above formula, if the speed of each handle of the dragon dance team does not exceed 2m / s, we can know:

[0133]

[0134] It can be known that the maximum speed of the faucet is 1.6871m / s.

[0135]

[0136] Working principle: When solving the model, a variable step search algorithm based on binary search is adopted, which can not only efficiently approximate the optimal solution in the global range, but also optimize the local area finely. In the establishment of the model in this article, a double mapping is constructed to obtain the correlation between variables by using the idea of ​​turning curves into straight lines, providing iterative solution ideas, and using vector methods to participate in modeling, reducing the amount of calculation and reducing the difficulty of programming.

[0137] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and descriptions are only preferred examples of the present invention and are not intended to limit the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A method for calculating a bench dragon mathematical model, characterized in that: The following steps are included: S1: Establish the mathematical model of bench dragon movement; S2: Based on the model of S1, set up an algorithm and solve the problem when the benches cannot continue to be placed in each other due to collision; S3: Set the algorithm for setting the minimum pitch of the dragon's front handle along the spiral to the boundary of the turning space and solve it; S4: Set the algorithm for the minimum turning length curve when turning the dragon's front handle and solve it; S5: Set the algorithm for the maximum travel speed of the faucet and solve it.

2. The method for calculating the mathematical model of a bench dragon according to claim 1, characterized in that: When establishing the model in S1, in order to facilitate the establishment of the model, we regard each bench in motion as a line segment. First, based on the Archimedean spiral equation, we establish the mapping relationship between the rotation angle θ and the time t; then, based on the movement speed and initial position of the faucet, we obtain the position of the faucet at different times. Taking into account the connection relationship between the benches, we gradually obtain the positions of the various parts of the faucet and the tail through the position of the faucet; then, we use the derivative of the position with respect to time to obtain the speed of each part; finally, we use the variable step size bisection method to solve the equation.

3. The method for calculating the mathematical model of a bench dragon according to claim 2, characterized in that: In S2, according to the length, width and connection mode of the benches, the spatial geometric relationship when the collision occurs is established, and then the collision condition is determined. In this process, we use Schmidt orthogonalization on the vector between the judgment point and the geometric center of the bench. By comparing the modulus of the two components of the vector with the length of the corresponding side of the rectangle, the collision condition can be quickly determined; combined with the collision condition and the equation model in problem 1, the variable step size bisection method is used for iteration to obtain the moment when the dragon dance team collides.

4. The method for calculating the mathematical model of a bench dragon according to claim 3, characterized in that: In S3, according to the process of the dragon freely spinning into the boundary of the turning space, the properties of the spiral are analyzed in combination with the knowledge of differential geometry, and then a relationship model between the pitch and the spiral is established; then, a single-objective optimization model is established with the minimum Archimedean pitch as the optimization objective function and no collision as the constraint condition; for each pitch, the boundary conditions are brought in, and after the time t is calculated, collision detection is iterated to determine whether the boundary conditions are met.

5. The method for calculating the mathematical model of a bench dragon according to claim 4, characterized in that: In S4, we first establish the relationship between the radius and the center of the two arcs, and use the arc length formula to further obtain the total arc length expression of the S-curve; then, with the shortest total arc length as the objective function and the relevant geometric relationship as the constraint condition, a single-objective programming model is established to adjust the U-turn path. In this process, the constraint conditions are integrated and combined with the vector method to obtain a nonlinear optimization model, which is traversed and solved using a numerical method to obtain the minimum U-turn curve length.

6. The method for calculating the mathematical model of a bench dragon according to claim 5, characterized in that: In S5, based on the travel route in S4, it is necessary to establish a single-objective planning model with the maximum dragon head travel speed as the optimization target and the upper limit of the dragon dance team's speed at each moment as a constraint condition; use a variable step-size algorithm based on binary search to search for the maximum dragon head travel speed. If the speed of the dragon dance exceeds the upper limit at a certain moment in the search process, stop the iteration immediately and return to the previous iteration to solve for the maximum dragon head travel speed.

Images