Method for obtaining a linkage mechanism layout for a specified motion trajectory

Abstract

The present application belongs to the field of mechanical structure optimization, and particularly relates to a method for obtaining a linkage mechanism layout with a specified motion trajectory, which comprises the following steps: step 1, establishing a linkage structure coordinate system to obtain an initial linkage mechanism layout; step 2, obtaining the length, density and target number of each rod; step 3, determining the functional relationship of the motion points and force points; step 4, establishing a topological model of the linkage mechanism layout; step 5, obtaining the coordinates of each point by using the gradient descent method for Q positions respectively; step 6, determining the sensitivity; step 7, obtaining the rod density by using the MMA algorithm; step 8, cyclic judgment; and step 9, obtaining the linkage mechanism layout. The present application proposes a topological model of the linkage mechanism layout, and combines the gradient descent method and the MMA algorithm in the model solving process, thereby simplifying the model solving process and making the linkage structure obtained according to the present application have good motion precision and reliability.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of mechanical structure optimization, and particularly relates to a method for obtaining a layout of a linkage mechanism for a specified motion trajectory. BACKGROUND

[0002] Topology optimization design of linkage mechanisms plays a crucial role in modern mechanical engineering, and its far-reaching significance not only lies in improving the overall performance of mechanical systems, but also in accurately achieving the control of specified motion trajectories, which is particularly important for aircraft, a highly complex and precise means of transportation. Linkage mechanisms, as key components for connecting and transmitting motion and force, play a crucial role in aircraft design. Through reasonable layout and design, they can ensure that each component of the aircraft moves accurately according to the predetermined trajectory and angle, thereby realizing a series of complex actions such as take-off, landing, turning, and climbing. In the flight control system of an aircraft, linkage mechanisms are widely used. For example, in the deflection control of an aircraft wing, the deflection angle of the wing directly affects the lift, drag, and flight stability of the aircraft, and is a key link in flight control. Traditional wing deflection control relies on complex hydraulic systems or electric systems, but these systems often have problems such as heavy weight, high energy consumption, and complex maintenance. Through topology optimization of linkage mechanisms, the structure can be lightweight, energy consumption can be reduced, and the reliability and durability of the system can be improved while ensuring the accuracy and stability of wing deflection.

[0003] Topology optimization design of linkage mechanisms is essentially an optimization method that maximizes structural efficiency by adjusting material distribution and structural shape under the premise of meeting specific performance requirements. In the design of aircraft wing deflection control, topology optimization can help designers accurately calculate the optimal layout and shape of linkage mechanisms, enabling the linkage mechanisms to achieve the most accurate deflection angle control with the least material consumption and lowest energy consumption during wing deflection. This optimization not only improves the accuracy and stability of wing deflection control, but also significantly reduces the overall weight and energy consumption of the aircraft, improving the flight performance and fuel consumption of the aircraft. In addition, topology optimization of linkage mechanisms has broad application prospects in other key components of aircraft. For example, in the landing gear system of an aircraft, through reasonable linkage mechanism design, accurate folding and unfolding of the landing gear during take-off and landing can be achieved, improving the take-off and landing efficiency of the aircraft while reducing the weight and complexity of the landing gear system. In the rudder control system of an aircraft, topology optimization of linkage mechanisms can also achieve accurate deflection control of the rudder, improving the maneuverability and flight stability of the aircraft.

[0004] Topological optimization of linkage mechanisms is crucial in aircraft design. Through rational linkage layout and design, the specified motion trajectory of aircraft components can be precisely controlled, improving flight performance and fuel consumption. Therefore, this paper investigates a method for obtaining a linkage mechanism layout that satisfies a specified motion trajectory. Based on dynamic equations and the method of steepest descent, the sensitivity of the target to the design variables is derived, resulting in a mechanism layout that satisfies the specified trajectory. This method can be applied to designs that achieve specified displacements, ensuring motion accuracy. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the present invention provides a method for obtaining a linkage mechanism layout with a specified motion trajectory, enabling the linkage mechanism to follow the prescribed route. The method first plans the target motion curve based on actual operating conditions. Then, using topology optimization, the method determines the target region, the topological model for the linkage mechanism layout, and the necessary design variables. Finally, the method optimizes the linkage mechanism layout by combining gradient descent with the MMA algorithm (Method of Moving Asymptotes), ultimately achieving a structure that meets the requirements.

[0006] To achieve the above object, the present invention discloses a method for obtaining a linkage mechanism layout of a specified motion trajectory, which comprises the following steps:

[0007] Step 1: Establish the linkage structure coordinate system and obtain the initial linkage mechanism layout;

[0008] Determine the area range of the linkage mechanism in its initial state and establish the linkage structure coordinate system; establish a mesh structure within the area range, and each point in the mesh structure is used as a vertex of the rod. There are a total of W points, forming a point set D = {D1, D2, ..., D w}, all points are connected in pairs as the initial linkage layout, the initial linkage layout includes N e root pole;

[0009] Step 2: Get the rod length, rod density and target number of rods for each rod;

[0010] Select any two points D from the point set D in step 1 i and D j , the obtained rod length l ij , i≠j; initial rod density ρ ij is the set value, 0.2≤v ij ≤0.5, target number N is the set value, 2≤N≤N e ;

[0011] Step 3: Determine the functional relationship between the moving point and the force point;

[0012] Step 4: Establishing a topological model of the linkage layout;

[0013] The topological model of the linkage layout is:

[0014]

[0015] 0≤ρ ij ≤1 (6)

[0016] where q is a position of the force point on the predetermined trajectory, p q目标 is the position coordinate of the motion point when the force point is at position q obtained according to the function relationship between the motion point and the force point in Step 3, p q is the position coordinate of the motion point when the force point is at position q obtained according to the determined linkage; obj q is the minimum change of the rod length when q is the position of the force point on the predetermined trajectory; W is the number of points in the point set D, ρ ij is the rod density between two points in D i and D j , L ij is the rod length between D i and D j obtained according to the current linkage layout, i < j, D i , D j ∈ D, l ij is the rod length between D i and D j obtained according to Step 2; N is a preset target number of rods, and ε1 is a set value;

[0017] Step 5: Obtaining the coordinates of each point by using the gradient descent method for the Q positions respectively;

[0018] Selecting Q positions on the predetermined trajectory of the force point, for a position q, specifying the corresponding horizontal coordinate or vertical coordinate position of the motion point at this time, using the inner solving formula to apply the gradient descent method to obtain the coordinates (XY) q of the rod vertexes obtained by using the current linkage layout when the force point moves to the specified position q.

[0019] Step 6: Determining the sensitivity;

[0020] Deriving the sensitivity with respect to the rod density according to the target function c;

[0021] Step 7: Obtaining the rod density by using the MMA algorithm;

[0022] Taking the rod density ρ ij as the design variable of the MMA algorithm, taking the target function c obtained according to the coordinates in Step 5 as the target function of the MMA, and taking the sensitivity with respect to the rod density ρ ijOptimization is performed, in which the second constraint condition and the third constraint condition need to be satisfied;

[0023] Step 8: loop judgment;

[0024] If the target function c is less than a predetermined value or the number of loops reaches a maximum value, the loop is stopped, and step 9 is performed; if neither condition is met, the rod density p obtained in the current step 7 is used as the initial value of the next loop, and the loop is restarted. ij Step 5 is returned to, at which time step 5 uses the updated rod density p ij A new loop is started;

[0025] Step 9: obtain the linkage layout;

[0026] According to the rod density p obtained in the current MMA ij , when the rod density p ij ≈1 indicates that a rod exists between D i and D j , and when the rod density p ij ≈0 indicates that a rod does not exist between D i and D j , and the linkage layout is obtained according to the rod density p ij .

[0027] Preferably, the initial rod density p ij in step 2 is set to 0.3.

[0028] Preferably, the rod length in step 2 is obtained according to the following formula:

[0029]

[0030] where l ij is the rod length between D i and D j , (x i , y i ) is the coordinate of point D i , (x j , y j ) is the coordinate of point D j , and i≠j.

[0031] Preferably, the function relationship for determining the motion point and the force point in step 3 is as follows:

[0032]

[0033] where (x 受力 , y 受力 ) is the coordinate value of the force point, (x 运动 , y 运动 ) is the coordinate value of the corresponding motion point; f1() and f2() are obtained by fitting.

[0034] Preferably, in step 5, the inner layer solving formula is:

[0035]

[0036] Wherein, obj q is the rod length difference obtained according to the rod density ρ ij when the displacement of the force point to the coordinate position q, W is the number of points in the point set D, ρ ij is the rod density of the rod between D i and D j , L ij is the rod length between D i and D j determined after the gradient descent method, i < j, D i , D j ∈ D, l ij is the rod length between D i and D j obtained according to step 2, n is the iteration number, g n is the gradient at the n th iteration, λ n is the learning rate at the n th iteration, i.e. the step size in the gradient descent method, XY n is a vector composed of the coordinates of the points in the set D after the n th iteration, The coordinates in the initial XY 0 are the coordinates of the point set D in step 1, except that the coordinates of the force point and the moving point are modified to the coordinates after the displacement.

[0037] Preferably, the gradient descent method is an adaptive gradient descent algorithm, and the adaptive step size adjustment formula is as follows:

[0038]

[0039] Wherein, g n is the gradient at the n th iteration, r n represents the cumulative sum of the squares of all gradients up to the n th iteration, and ε is a set value to prevent the denominator from being zero.

[0040] Preferably, the sensitivity in step 6 is:

[0041]

[0042] Wherein, ρ ij is the density of the rod, there are Ne variables in total, and the sensitivity is a 1*Ne vector obtained by multiplying the vector with .

[0043] Compared with the prior art, the present application has the following beneficial effects:

[0044] (1) The present application proposes a topological model of a linkage mechanism layout, which has the advantages of simplicity, universality and easy solution.

[0045] (2) In the model solution process, the gradient descent method and the MMA algorithm are combined to simplify the model solution process.

[0046] (3) The linkage structure obtained by the method of the present application ensures that the motion point moves along the specified trajectory, and has good motion accuracy and reliability. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 Flowchart of the method for obtaining a linkage mechanism layout of a specified motion trajectory

[0048] Figure 2 Schematic diagram of the region range and vertex in Example 1;

[0049] Figure 3 Schematic diagram of the initial structure of the linkage structure in Example 1;

[0050] Figure 4 Schematic diagram of the final structure of the linkage structure in Example 1;

[0051] Figure 5 Error diagram of the motion trajectory and the target trajectory of the linkage structure in Example 1. DETAILED DESCRIPTION

[0052] The exemplary embodiments, features and aspects of the present application will be described in detail below with reference to the accompanying drawings. The same reference numerals in the drawings represent functionally identical or similar elements. Although various aspects of the embodiments are shown in the drawings, the drawings are not necessarily drawn to scale unless specifically indicated.

[0053] The present application provides a method for obtaining a linkage mechanism layout of a specified motion trajectory, as shown in Figure 1 which comprises the following steps:

[0054] Step 1: Establish a linkage structure coordinate system to obtain an initial linkage mechanism layout.

[0055] Determine the region range of the initial state of the linkage mechanism, the region range is a rectangle, and a vertex of the region range is taken as the origin, and one edge passing through the vertex is taken as the X-axis and the other edge is taken as the Y-axis to establish a linkage structure coordinate system.

[0056] In the connecting rod structure coordinate system, points are taken in sequence on the X-axis and Y-axis within the region from the origin with the set minimum length of the rod. When the distance between the point and the edge of the region is less than the set minimum length of the rod, the point taking ends, and the intersection of the region and the axis is taken as the last point. At this time, there are A points on the X-axis including the origin, and B points on the Y-axis including the origin, where A and B are both positive integers. Based on the points on the X-axis and the Y-axis, a mesh structure is formed within the region after extending parallel to the X-axis and the Y-axis. There are a total of W = A × B intersections within the region, recorded as point set D, D = {D1, D2, ..., D w}, the coordinates of D1 are (x1, y1), the coordinates of D2 are (x2, y2), ..., D W The coordinates of (x W ,y W ).

[0057] Each point in the mesh structure is regarded as the vertex of the rod, and all points are connected in pairs to form the initial linkage layout. At this time, the initial linkage layout includes N e Strokes, total strokes .

[0058] Step 2: Get the rod length, rod density and target number of rods for each rod.

[0059] Select any two points D from the point set D in step 1 i and D j , the obtained rod length is:

[0060]

[0061] Among them, l ij D i and D j The length of the rod between (x i ,y i ) is point D i The coordinates of (x j ,y j ) is point D j 's coordinates, and i≠j.

[0062] Initialize D i and D j Rod density ρ between two points ij ,ρ ij The value of D i and D j The probability of the existence of a rod between two points is when the rod density ρ ij =1 indicates D i and D j A rod exists between two points when the rod density ρ ij =0 means Di and D j There is no rod between the two points. The rod density ρ of all rods at the beginning is ij are all the same and can be set to 0.2≤ρ ij ≤0.5, preferably ρ ij =0.3.

[0063] The total number of rods in the initial linkage layout is N e , so we need to calculate N e The rod lengths are initialized for each rod density. Get the preset target number of rods N, 2≤N≤N e .

[0064] Step 3: Determine the functional relationship between the moving point and the force point.

[0065] According to actual needs, in the initial linkage mechanism layout, select the force point P that applies force and the motion point M that moves according to the specified trajectory, move the force point P to the specified coordinate position, and obtain the corresponding force point P coordinate position. According to the coordinate values ​​of multiple force points P (x 受力 ,y 受力 ) and the corresponding coordinate value of the moving point M (x 运动 ,y 运动 ) and obtain the functional relationship between the coordinates of the force point and the coordinates of the motion point:

[0066]

[0067] Among them, (x 受力 ,y 受力 ) is the coordinate value of the force point P, (x 运动 ,y 运动 ) is the coordinate value of the corresponding moving point M, and f1() and f2() are obtained by fitting.

[0068] Step 4: Establish a topological model of the linkage mechanism layout.

[0069] The goal of the linkage layout is to ensure that when the force-bearing point moves along a predetermined trajectory, the resulting moving point also moves along the preset target trajectory. Therefore, by selecting Q positions on the predetermined trajectory of the force-bearing point, and using the functional relationship between the moving point and the force-bearing point in step 3, the positions of the corresponding moving points are obtained as target positions. The actual position reached by the moving point of the linkage is compared with the target position to form the objective function.

[0070] In order to obtain the layout of the linkage mechanism, three constraints are included: the first is that the rod length changes minimally when the force point moves, the second is the limit on the number of rods, and the third is that the rod density can only be between [0, 1].

[0071] According to the objective function and constraints, the topological model of the linkage mechanism layout is obtained as follows:

[0072]

[0073] 0≤ρ ij ≤1 (6)

[0074] wherein, formula (3) is a target function, formula (4) is a first constraint condition, formula (5) is a second constraint condition, and formula (6) is a third constraint condition; q is one position of a force point on a predetermined trajectory, and Q positions are selected, p q目标 is a position coordinate of a motion point when the force point is located at the q position, which is obtained according to a function relationship between the motion point and the force point in step 3, p q is a position coordinate of the motion point when the force point is located at the q position, which is obtained according to a determined linkage mechanism; obj q is the minimum change of the rod length when q is the q position of the force point on the predetermined trajectory; W is the number of points in the point set D, and ρ ij is the rod density of the rod between D i and D j , L ij is the rod length between D i and D j obtained according to the current linkage mechanism layout, i < j, D i , D j ∈ D, l ij is the rod length between D i and D j obtained according to step 2; N is a preset target rod number, and ε1 is a set value, because the optimization process needs to be closest to N, a certain range needs to be given, and ε1 is a set minimum value.

[0075] According to the analysis of the topological model of the linkage mechanism layout, in order to obtain the linkage mechanism layout, the overall process is divided into inner layer operation and outer layer optimization. The inner layer operation adopts the gradient descent method to obtain, when a specified displacement is applied, the coordinates of each point after the rod displacement caused by the application of all the rods and the corresponding rod density. The outer layer optimization adopts the MMA algorithm to realize the optimization of the rod density ρ ij .

[0076] Step 5: The gradient descent method is used to obtain the coordinates of each point for the Q positions respectively.

[0077] When the displacement of the specified force point is to the position q, the corresponding horizontal coordinate or vertical coordinate position of the specified motion point at this time is specified. It needs to be noted that the position of the motion point cannot be fixed at this time, and a space amount needs to be left for subsequent optimization, and then the inner layer solving formula is used to apply the gradient descent method to obtain the coordinates of each rod vertex caused by the application of the current linkage mechanism layout when the displacement of the force point is to the specified position. The inner layer solving formula is:

[0078]

[0079] XY n+1 = XY n - λ n o n (8)

[0080]

[0081] wherein, obj q is the rod length difference obtained according to the rod density p ij when the force point is moved to the coordinate position q, W is the number of points in the point set D, p ij is the rod density of the rod between two points D i and D j , L ij is the rod length between D i and D j determined after the gradient descent method, i < j, D i , D j ∈ D, l ij is the rod length between the initial D i and D j obtained according to step 2, n is the iteration number, g n is the gradient at the n-th iteration, λ n is the learning rate at the n-th iteration, i.e. the step size in the gradient descent method, XY n is a vector composed of the coordinates of the points in the set D after the n-th iteration, , the coordinates in the initial XY 0 are the coordinates of the point set D in step 1, only the coordinates of the force point and the moving point are modified to the coordinates after moving.

[0082] The gradient descent method can be preferably an adaptive gradient descent algorithm. The adaptive gradient descent algorithm has the feature that in the early stage, the step size is large, which can quickly converge, and in the later stage, the step size is small, which can more finely adjust the model parameters. The adaptive step size adjustment formula is as follows:

[0083]

[0084] wherein, g n is the gradient at the n-th iteration, r n represents the cumulative sum of the squares of all gradients up to the n-th iteration, and ε is a set value, which is usually a very small number, to prevent the denominator from being zero.

[0085] According to formulas (7)-(9), the gradient descent method is used to obtain obj qThe value of XY at the minimum point, using gradient descent method for Q positions, then there are Q sets of values of XY, so the value of XY can be represented as (XY) q , at this time the value of (XY) q is (XY) q , and the value of obj ij is obj

[0086] For Q positions, respectively, using gradient descent method, the value of obj q at the minimum point when the force point moves to coordinate position q is (XY) q , and the value of obj

[0087] Step 6: Determine the sensitivity.

[0088] According to the target function c, the sensitivity of the rod density is obtained, but because the target function c does not directly include the solution variable p ij , therefore the sensitivity needs to be obtained by implicit method, and the sensitivity is:

[0089]

[0090] , (XY) q is a 1*2W vector, that is, (XY) q contains a set of values of x, y coordinates of W points when the force point moves to coordinate position q, p ij is the density of the rod, and there are Ne variables in total, so the value of p at each position is: , because c is a set of relationships including XY coordinates of Q positions, therefore q the value of (XY) at the qth time needs to be brought in to obtain, that is, , then the sum of the formulas obtained at the Q positions is: , and the size of this vector is 1*2W. Therefore, the sensitivity is: , which is a 1*Ne vector obtained by multiplying the vector with

[0091] Step 7: Obtain the rod density by using MMA algorithm.

[0092] The optimization of the rod density uses MMA algorithm, taking the rod density p ij as the design variable of the MMA algorithm, taking the target function c obtained according to the coordinates in step 5 as the target function of the MMA, and optimizing the rod density p ij according to the sensitivity, and in the optimization process, the second constraint condition and the third constraint condition also need to be met. The optimized rod density p ijBecause there are N e bars in common, the optimized N e bar density is obtained.

[0093] Step 8: Loop judgment.

[0094] If the objective function c is less than a predetermined value ε2 or the number of loops reaches a maximum value, the loop is stopped, and step 9 is executed; if neither condition is met, the bar density ρ ij is returned to step 5, which uses the updated bar density ρ ij to start a new loop and increments the number of loops by 1 as the new number of loops, with the initial value of the number of loops being 0.

[0095] Step 9: Obtain the linkage mechanism layout.

[0096] According to the bar density ρ ij obtained from the current MMA, when the bar density ρ ij = 1, it indicates that a bar exists between D i and D j ; when the bar density ρ ij = 0, it indicates that a bar does not exist between D i and D j ; and according to the bar density ρ ij , the linkage mechanism layout is obtained. However, because only an approximation of the objective function is used in the actual optimization process, the bar density ρ ij cannot be only 0 or 1, so in practice, when the bar density ρ ij ≈ 1, it indicates that a bar exists between D i and D j ; when the bar density ρ ij ≈ 0, it indicates that a bar does not exist between D i and D j ; and according to the bar density ρ ij , the linkage mechanism layout is obtained.

[0097] Example 1

[0098] This example is to obtain a linkage mechanism layout that can walk in a straight line.

[0099] Step 1: Establish a linkage structure coordinate system to obtain an initial linkage mechanism layout.

[0100] First, determine the initial state of the linkage mechanism as a rectangular area of ​​30cm*20cm. With one vertex of the area as the origin, one side passing through this vertex as the X-axis, and the other side as the Y-axis, establish the linkage structure coordinate system. The minimum length of the rod is 10cm. Therefore, the X-axis includes 4 points (0, 1, 2, and 3), and the Y-axis includes 3 points (0, 1, and 2). Through the points on the X and Y axes, a network structure is formed within the area, including a total of 12 points, as shown in the figure. Figure 2 As shown, each point in the mesh structure is regarded as the vertex of the rod, and all points are connected in pairs as the initial linkage mechanism layout, as shown in Figure 3 As shown. At this time, the initial linkage layout includes a total number of rods

[0101] For the convenience of description in this embodiment, the side length of the area range is set to an integer multiple of the minimum length of the rod. In practice, the area range may be 35cm*23cm, then the coordinate values ​​of the points on the X-axis are 0, 1, 2 and 3.5 respectively, and the coordinate values ​​of the points on the Y-axis are 0, 1 and 2.3 respectively.

[0102] Step 2: Get the rod length, rod density and target number of rods for each rod.

[0103] The total number of rods N obtained in step 1 e And the endpoint coordinates of each rod, calculate the rod length of each rod, the target number N is preset to N = 5. Set the initial rod density ρ of each rod ij are all 0.3, where ρ ij D i

[0104] and D j Rod density of rods between two points.

[0105] Step 3: Determine the functional relationship between the moving point and the force point.

[0106] In this embodiment, in the connecting rod structure coordinate system, the point with coordinates (0, 2) is selected, that is, Figure 3 Point 1 in the figure is used as the force point. The force point moves downward along the Y axis and the point with coordinates (3, 1) is selected. Figure 3 Point 11 in the figure is used as the moving point, and the moving point moves in a horizontal straight line.

[0107] Therefore, initially, when the force point P is at (0, 2), the moving point M is at (3, 1). When the force point P moves to (0, 1.5), the moving point M is at (3.6, 1). When the receiving point P moves to (0, 1) at its maximum, the displacement point is (4.2, 1). The functional relationship between the coordinates of the force point and the coordinates of the moving point is obtained by fitting:

[0108]

[0109] Step 4: Establishing the topological model of the linkage layout.

[0110] The goal of the linkage layout is that when the force point moves along the predetermined trajectory, the moving point also moves along the preset target trajectory. Therefore, by selecting Q positions on the predetermined trajectory of the force point, the positions of the corresponding moving points are obtained as target positions according to the function relationship between the moving point and the force point in Step 3, and the positions reached by the actual moving points of the linkage are compared with the target positions as the target function.

[0111] In order to obtain the linkage layout, there are three constraints, the first is that the length of the rod changes the least when the force point moves, the second is the rod number limit, and the third is that the rod density can only be between [0, 1].

[0112] In this embodiment, 5 positions are selected on the predetermined trajectory of the force point, and therefore the topological model of the linkage layout obtained according to the target function and the constraints is specifically:

[0113]

[0114] 0≤ρ ij ≤1 (6)

[0115] Wherein, formula (3) is the target function, formula (4) is the first constraint condition, formula (5) is the second constraint condition, and formula (6) is the third constraint condition; q is a position of the force point on the predetermined trajectory, and a total of 5 positions are selected, p q目标 is the position coordinate of the moving point when the force point is located at the q position obtained according to the function relationship between the moving point and the force point in Step 3, p q is the position coordinate of the moving point when the force point is located at the q position obtained according to the determined linkage; obj q is the minimum change of the rod length when q is the q position of the force point on the predetermined trajectory; there are a total of 12 points in the point set D, ρ ij is the rod density between D i and D j , L ij is the rod length between D i and D j obtained according to the current linkage layout, i<j, D i , D j ∈D, l ij is the rod length between D i and D j obtained according to Step 2; the preset target rod number is 5, and ε1 is set to 0.02.

[0116] Step 5: The gradient descent method is used to obtain the coordinates of each point for the Q positions respectively.

[0117] When the displacement of the force point is specified to position q, the corresponding horizontal coordinate or vertical coordinate position of the motion point at this time is specified. Note that the position of the motion point at this time cannot be fixed, and a certain amount of space must be left for subsequent optimization. Then, the inner layer solving formula is used to apply the gradient descent method to obtain the coordinates of the rod vertex when the displacement of the force point is specified to the specified position using the current linkage mechanism layout. The inner layer solving formula is:

[0118]

[0119] XY n+1 = XY n - λ n g n (8)

[0120]

[0121] According to the coordinates of the point set D in step 1, and the coordinates of the determined force point and motion point, the initial XY 0 = [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0], because in this embodiment, the y value of point 11 is a determined value, and the coordinate values of the x and y axes of the motion point can only have one set as a fixed value, so the y value is selected as the determined value and the x value is unchanged. The adaptive gradient descent algorithm is used to obtain the XY value when obj1 is minimized. According to the XY value, the value of L ij is obtained. When step 5 is first executed, ρ ij are all set to the initial value of 0.3.

[0122] For the 5 positions, this method is used respectively to obtain the XY q value coordinates when the displacement of the 5 force points to 5 different coordinate positions minimizes obj q , so that the first constraint condition is satisfied.

[0123] Step 6: Determine the sensitivity.

[0124] The sensitivity formula is obtained by taking the derivative of the objective function c with respect to the rod density;

[0125]

[0126] where (XY) q is a 1*24 vector, and in this embodiment there are 12 points, so there are 24 coordinate values to form (XY) q , ρ ij is the density of the rod, and there are Ne = 66 variables in total, so there are 5 positions for each position. For 24*66 matrix, since c is a set of relations containing XY coordinates of Q positions, thus The qth(XY) q value is needed to be put into the equation to get Then the Q positions are summed up by the equation, which is The vector size is 1*24, and thus the objective function is The multiplication of and the multiplication of two matrices is a 1*66 vector.

[0127] Step 7: Obtain the rod density by using the MMA algorithm.

[0128] Optimize the rod density by using the MMA algorithm, and the rod density ρ ij is taken as the design variable of the MMA algorithm. Since there are 66 rods, there are 66 design variables. The objective function c obtained according to the coordinates in step 5 is taken as the objective function of the MMA, and the rod density ρ ij is optimized according to the sensitivity. In the optimization process, the second constraint condition and the third constraint condition also need to be met.

[0129] Step 8: Loop judgment.

[0130] Judge whether the objective function c is less than a predetermined value ε2=10 -3 or the number of loops 1000 reaches the maximum value. If it reaches, stop the loop and execute step 10; if neither of them is met, use the rod density ρ ij obtained by the previous MMA to return to step 5 to continue the loop, and add 1 to the number of loops as the new number of loops. The initial value of the number of loops is 0.

[0131] Step 9: Obtain the layout of the linkage mechanism.

[0132] According to the rod density ρ ij obtained by the current MMA, when the rod density ρ ij ≈1, it means that the rod exists between D i and D j . When the rod density ρ ij ≈0, it means that the rod does not exist between D i and D j . According to the rod density ρ ij , the layout of the linkage mechanism is obtained. According to the rod density ρ ij , the layout of the linkage mechanism is obtained, as shown in Figure 4 .

[0133] In this embodiment, the relative error between the motion trajectory of the motion point and the target trajectory is measured according to the obtained layout of the linkage mechanism, and the result is shown in Figure 5As shown, the horizontal coordinate is the displacement of the force point on the Y axis, and the relative error h is controlled below 0.1.

[0134] The above-described embodiments are only to describe the preferred embodiments of the present application, and are not intended to limit the scope of the present application. Without departing from the design spirit of the present application, various modifications and improvements to the technical solutions of the present application made by those skilled in the art shall fall within the protection scope determined by the claims of the present application.

Claims

1. A method for obtaining a linkage layout for a specified motion trajectory, characterized by: It includes the following steps: Step 1: Establish the linkage structure coordinate system and obtain the initial linkage mechanism layout; Determine the area range of the linkage mechanism at the initial state and establish the linkage structure coordinate system; establish a mesh structure within the area range, and each point in the mesh structure is used as the vertex of the rod. There are a total of W points, forming a point set ={ }, all points are connected in pairs as the initial linkage layout, the initial linkage layout includes root pole; Step 2: Get the rod length, rod density and target number of rods for each rod; Gather from step 1 Pick any two points and , the obtained rod length , ; Initial rod density is the set value, , the target number N is the set value, ; Step 3: Determine the functional relationship between the moving point and the force point; Step 4: Establish a topological model of the linkage mechanism layout; The topological model of the linkage mechanism layout is: （3）； （4）； + （5）； （6）； Where q is the position of the force point on the predetermined trajectory, is the coordinate of the moving point when the force point is at position q, obtained according to the functional relationship between the moving point and the force point in step 3. is the coordinate of the moving point when the force point is at position q, obtained based on the determined linkage mechanism; When the displacement of the force point to the coordinate position q is calculated based on the rod density The obtained rod length difference; W is the point set The number of points in , for and The rod density of the rods between two points, is obtained based on the current linkage layout and The length of the rod between , is the initial and The length of the rod between; N is the preset target number of rods, is the set value; Step 5: Use gradient descent method to obtain the coordinates of each point at Q positions; Select Q positions on the predetermined trajectory of the force point. For position q, specify the corresponding horizontal or vertical coordinate position of the moving point at this time. Use the inner layer solution formula to apply the gradient descent method to obtain the coordinates of the rod vertices when the force point moves to the specified position q. Apply the current linkage mechanism layout. ; Step 6: Determine sensitivity; The sensitivity is obtained by differentiating the rod density according to the objective function c; Step 7: Use the MMA algorithm to obtain the rod density; The rod density As the design variable of the MMA algorithm, the objective function c obtained according to the coordinates of step 5 is used as the objective function of MMA, and the sensitivity to the rod density is During the optimization process, the second constraint of formula (5) and the third constraint of formula (6) need to be satisfied. Step 8: Loop judgment; Determine whether the objective function c is less than the predetermined value or the number of cycles reaches the maximum value. If so, stop the loop and execute step 9. If neither is satisfied, use the rod density obtained in step 7. Return to step 5, which now uses the updated rod density Start a new cycle; Step 9: Get the linkage layout; Rod density according to current MMA , when the rod density 1 o'clock and A rod exists between two points when the rod density 0 o'clock and There is no rod between the two points, according to the rod density Get the linkage layout.

2. The method for obtaining a linkage layout of a specified motion trajectory according to claim 1, wherein: In step 2, the initial rod density Set the value to 0.

3.

3. The method for obtaining a linkage layout of a specified motion trajectory according to claim 1, wherein: In step 2, the rod length is obtained according to the following formula: （1）； in, for and The length of the rod between for point The coordinates of for point The coordinates of .

4. The method for obtaining a linkage layout of a specified motion trajectory according to claim 1, wherein: The functional relationship between the moving point and the force point in step 3 is: （2）； in, is the coordinate value of the force point, is the coordinate value of the corresponding moving point; and Obtained by fitting.

5. The method for obtaining a linkage mechanism layout of a specified motion trajectory according to claim 1, wherein: In step 5, the inner layer solution formula is: (7) ； (8) ； （9）； Where n is the number of iterations, is the gradient at the nth iteration, is the learning rate at the nth iteration, that is, the step size in the gradient descent method, After the nth iteration, the set The vector formed by the coordinates of the midpoint, ,initial The coordinates in are the set of points in step 1 The coordinates of the force point and the movement point are simply modified to the coordinates after movement.

6. The method for obtaining a linkage mechanism layout of a specified motion trajectory according to claim 5, characterized in that: The gradient descent method is an adaptive gradient descent algorithm, and the adaptive step size adjustment formula is as follows: (10) ； in, represents the cumulative sum of all squared gradients up to the nth iteration, is a set value to prevent the denominator from being zero.

7. The method for obtaining a linkage mechanism layout of a specified motion trajectory according to claim 1, wherein: The sensitivity in step 6 is: （11）； Among them, the sensitivity is and One obtained by multiplying vectors vector.

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