A secant approximation method for nonlinear constraints of a redundant drive system

Abstract

The application provides a secant approximation method for nonlinear constraints of a redundant drive system, and belongs to the technical field of dynamic control distribution of the redundant drive system. The method comprises the following steps: firstly, according to the control input model of the redundant drive system with any pair of constraint components being nonlinear constraints, a closed region formed by the intersection of a rectangle and an ellipse in a geometric plane is obtained; then, after the closed region is divided into the union of the rectangle and the ellipse triangle, the approximation result of the closed region is obtained by performing the approximation of the rectangle and the triangle combination on the ellipse triangle, so that the linear approximation of the pair of nonlinear constraint components is realized. The application jointly uses the triangle and the rectangle to perform the approximation on the region surrounded by the nonlinear constraints, converts the nonlinear constraints into multiple linear constraints, converts the nonlinear constraint set into a linear constraint set, effectively solves the problem that the control reachable set cannot be determined due to the nonlinear constraint relationship between the actuators in the redundant drive system, and is helpful to realize the real-time control of the redundant drive system.

Description

Technical Field

[0001] This invention belongs to the field of dynamic control and allocation technology for redundant drive systems, and specifically relates to a secant approximation method for nonlinear constraints of redundant drive systems. Background Technology

[0002] Control assignment is responsible for distributing the desired system control vector to each redundant actuator for execution. Direct assignment based on the control reachability set is an important method for control assignment. The calculation of the control reachability set involves determining the boundary of the system control reachability vector that can be reached by all actuators operating simultaneously, given the known variation range of each actuator, thereby revealing the control capability of the redundant drive system. The calculation of the control reachability set is the foundation of the direct assignment method.

[0003] The control reachability set of a parallel redundant drive system can be mathematically represented as:

[0004] Φ={v|v=B·u,u∈Ω} (1-1)

[0005] In equation (1-1), u is the control vector, representing the control input of the redundant drive system, u = (u1, ..., u2) / (u3, ..., u4) / (u5, ..., u6) / (u7, ..., u8) / (u9 ... m ) T Where T is the matrix transpose symbol, and u is the i-th control component. i u represents the control action of the i-th actuator, 1 ≤ i ≤ m, where m is the number of actuators. imin ≤u i ≤u imax ,u imin u is the minimum value of the control action of the i-th actuator. imax The maximum value of the control action of the i-th actuator; each u i There are often linear or nonlinear constraints between them; Ω is the control set, Ω={u}; v is the control reachability vector of the redundant drive system, v=(v1,...,v m ) T , represents the control output of the redundant drive system, where v j Let Φ be the j-th control reachable component, 1≤j≤n, where n is the dimension of the control reachable vector, n<m; Φ is the control reachable set; and B is the control efficiency matrix with n rows and m columns.

[0006] The physical meaning expressed by the above equation (1-1) is: given the set Ω of control vectors consisting of m control inputs of a redundant drive system, how to determine the set Φ of control reachable vectors consisting of n control outputs through the control efficiency matrix B.

[0007] The patent "Method for Determining the Control Reachable Set of a Redundant Drive System under Multiple Pairs of Linear Constraint Control Components" (Patent No.: ZL201911405939.5) solves the problem of determining the control reachable set when there are linear constraints between the control actions of multiple pairs of actuators.

[0008] However, for redundant drive systems with nonlinear constraints between actuators, such as four-wheel independent drive-independent braking-independent steering vehicles, there is currently no effective method to determine their control reachable set.

[0009] For control sets formed by control vectors with nonlinear constraints, if the nonlinear constraints can be approximated, transforming the nonlinear inequality constraints into multiple linear constraints, and the nonlinear problem into a linear problem, and then mature methods can be used to calculate the control reachable set, the difficulty of the problem can be effectively reduced. Summary of the Invention

[0010] The purpose of this invention is to overcome the shortcomings of existing technologies and propose a secant approximation method for nonlinear constraints in redundant drive systems. This invention addresses the linear approximation problem in redundant drive systems where the geometry formed by nonlinear constraints can be divided into a combination of rectangles and elliptical triangles. It combines triangles and rectangles to approximate the region enclosed by the nonlinear constraints, transforming the nonlinear constraints into multiple linear constraints and converting the set of nonlinear constraints into a set of linear constraints. This effectively solves the problem in current redundant drive systems where nonlinear constraint relationships exist between actuators, making it impossible to determine their control reachable set, thus facilitating real-time control of redundant drive systems.

[0011] This invention proposes a secant approximation method for nonlinear constraints in redundant drive systems, comprising:

[0012] 1) By constructing a control input model of a redundant drive system with any pair of nonlinear constraint components, the closed region formed by the intersection of rectangles and ellipses in the geometric plane corresponding to the model is obtained;

[0013] The control input model of the redundant drive system, where any pair of constraint components are nonlinear constraints, is expressed as follows:

[0014]

[0015] In the formula, u1 and u2 represent two control actions in a pair of nonlinear constraint components, -a≤u1≤a represents the range of the control action corresponding to u1, -b≤u2≤b represents the range of the control action corresponding to u2, and u imin u is the minimum value of the current control action of the i-th actuator. imax The maximum value of the current control action of the i-th actuator, i = 1, 2, -a ≤ u1min <u 1max ≤a,-b≤u 2min <u 2max ≤b;

[0016] Then, the ellipse in the closed figure has a major semi-axis length of a and a minor semi-axis length of b; the rectangle has a length of u. 1max -u 1min Width is u 2max -u 2min ;

[0017] 2) Based on the closed region obtained in step 1), draw perpendicular lines from the intersection of the rectangle and the ellipse to the major axis and minor axis of the ellipse respectively, dividing the closed region into the union of rectangles and elliptical triangles. Place the rectangles in the union into an initially empty set W, and place the elliptical triangles into an initially empty set Y.

[0018] The elliptical triangle is a figure formed by the two legs of a right triangle and the elliptical arc connecting the two vertices of the hypotenuse of the right triangle.

[0019] 3) Approximate the elliptical triangle obtained in step 2) by combining rectangles and triangles; the specific steps are as follows:

[0020] 3-1) In set Y, choose any elliptical triangle denoted as M1P1N. The intersection of the two legs of this elliptical triangle is M1, the other endpoint of the leg perpendicular to the major axis of the ellipse is N, and the other endpoint of the leg perpendicular to the minor axis of the ellipse is P1. Let the coordinates of points M1, P1, and N be... (x n ,y n );

[0021] 3-2) Let i = 1, and construct an initially empty set denoted as Γ;

[0022] 3-3) To find the elliptic arc P after one approximation i The endpoint P of the next elliptic arc to be approximated on N. i+1 Calculate the secant slope k i :

[0023] When the elliptic triangle M i P i When N is in the first or second quadrant, solve the equation shown in equation (2) to obtain... slope k i Among them, the elliptical triangle M i P i The intersection of the two legs of the right triangle N is M. i The other endpoint of the right-angled leg perpendicular to the major axis is N, and the other endpoint of the right-angled leg perpendicular to the minor axis is P.i ;

[0024]

[0025] Where e is the preset error coefficient, Let P be the point i x-coordinate Let P be the point i The ordinate;

[0026] When the elliptic triangle M i P i When N is in the third or fourth quadrant, solve the equation shown in equation (3) to obtain... slope k i ;

[0027]

[0028] 3-4) By calculating P i+1 Find the coordinates of the elliptic arc P after one approximation. i The endpoint of the next elliptical arc to be approximated on N; the specific steps are as follows:

[0029] 3-4-1) The real root k obtained in step 3-3) i Let k be the number of times. ij j = 1, ..., τ, where τ is the real root k i The number of elements, τ≤4; let l=1;

[0030] 3-4-2) Let k i =k il , will k i Substitute the equations into the system shown below:

[0031]

[0032] Solving for point P yields the solution. i+1 coordinates

[0033] 3-4-3) Determine the result of step 3-4-2):

[0034] If point P i+1 In elliptic arc P i If N is on the line, then point P is on the line. i+1 To complete an approximation of the elliptic arc P i Find the endpoint of the next elliptical arc to be approximated on N, and proceed to steps 3-5);

[0035] Otherwise, proceed to step 3-4-4);

[0036] 3-4-4) Determination: If l = τ, then set the vertices as N and P.i M i The triangle is placed into set Γ. The elliptical triangle M1P1N is approximated. Proceed to step 3-6). If l < τ, let l = l + 1, and then return to step 3-4-2.

[0037] 3-5) Judgment:

[0038] like or Then let the vertices be N and P. i M i Place the triangle into set Γ and proceed to steps 3-6);

[0039] Otherwise, point P. i+1 To line segment P i M i Draw a perpendicular line, and denote the foot of the perpendicular as T. i ; Self-point P i+1 To line segment NM i Draw a perpendicular line and denote the foot of the perpendicular as M. i+1 According to P i+1 The coordinates of the point determine triangle T i P i P i+1 Rectangle M i T i P i+1 M i+1 Triangle T i P i P i+1 and rectangle M i T i P i+1 M i+1 Add it to set Γ, then let i = i + 1, return to step 3-3), and continue to approximate the updated elliptic triangle;

[0040] 3-6) All rectangles and triangles in set Γ are approximations of the elliptic triangle M1P1N. Put all rectangles and triangles in set Γ into set W, remove the elliptic triangle M1P1N from set Y, and proceed to step 4).

[0041] 4) Determination: If set Y is empty, then all rectangles and triangles in set W constitute the approximation result of the closed region obtained in step 1); otherwise, return to step 3-1).

[0042] In one specific embodiment of the present invention, the method further includes:

[0043] Repeat steps 1)-4) until the approximation results for all nonlinear constraint component pairs of the redundant drive system are obtained, at which point the approximation is complete.

[0044] Features and beneficial effects of the present invention:

[0045] 1. This invention fully utilizes the high-precision approximation capability of secant lines to elliptical arcs, resulting in higher approximation efficiency.

[0046] 2. The present invention transforms nonlinear constraints into multiple linear constraints, and transforms a set of nonlinear constraints into a set of linear constraints. This method can effectively solve the problem of being unable to determine the control reachable set in a class of problems where there are nonlinear constraint relationships between actuators in redundant drive systems.

[0047] 3. This invention can be used to evaluate the control capabilities of parallel configuration systems with redundant drive characteristics and multiple pairs of nonlinear constraint control components, such as advanced satellites, aircraft, ships, automobiles, and parallel robots. It can provide a basis for system control allocation and be used for fault-tolerant control of systems after the failure of some actuators. Attached Figure Description

[0048] Figure 1 This is an overall flowchart of a secant approximation method for nonlinear constraints of a redundant drive system according to an embodiment of the present invention.

[0049] Figure 2 This is a schematic diagram of a closed region formed by the intersection of a rectangle and an ellipse in a specific embodiment of the present invention.

[0050] Figure 3 This is a schematic diagram of a closed region formed by the intersection of a rectangle and an ellipse in a specific embodiment of the present invention.

[0051] Figure 4 This is a schematic diagram of elliptical triangular secant approximation in a specific embodiment of the present invention.

[0052] Figure 5 This is a schematic diagram of nonlinear constraint secant approximation of the active wheels of a redundant drive vehicle in a specific embodiment of the present invention. Detailed Implementation

[0053] This invention proposes a secant approximation method for nonlinear constraints of redundant drive systems, which is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0054] This invention proposes a secant approximation method for nonlinear constraints in redundant drive systems. The overall process is as follows: Figure 1 As shown, it includes the following steps:

[0055] 1) Construct a control input model for a redundant drive system with any pair of nonlinear constraint components, and obtain the closed region corresponding to the model in the geometric plane.

[0056] In a specific embodiment of the present invention, the control input model expression of the redundant drive system in which any pair of constraint components are nonlinear constraint components is as follows:

[0057]

[0058] In the formula, u1 and u2 represent two control actions in a pair of nonlinear constraint components, -a≤u1≤a represents the range of the control action corresponding to u1, -b≤u2≤b represents the range of the control action corresponding to u2, and u imin u is the minimum value of the current control action of the i-th actuator. imax The maximum value of the current control action of the i-th actuator, i = 1, 2, -a ≤ u 1min <u 1max ≤a,-b≤u 2min <u 2max ≤b. The model described by equation (1) is geometrically represented as a closed region formed by the intersection of a rectangle and an ellipse.

[0059] Figure 2 This is a schematic diagram of a closed region formed by the intersection of a rectangle and an ellipse in a specific embodiment of the present invention. The ellipse is centered at the origin, with a major axis of length *a* and a minor axis of length *b*. The lower left vertex of the rectangle is R1, with a length of *u*. 1max -u 1min Width is u 2max -u 2min Because of u imin and u imax The value is different at different times, therefore the position of the rectangle is different at different times.

[0060] The problem of secant approximation of nonlinear constraints is: how to approximate the geometric figure formed by the nonlinear constraints using a combination of rectangles and triangles to achieve a linear approximation of the nonlinear constraints with higher accuracy and fewer figures.

[0061] 2) Divide the closed region of step 1) into several combinations of rectangles and elliptical triangles.

[0062] In this embodiment, an elliptical triangle is defined as a figure bounded by the two legs of a right triangle and an elliptical arc connecting the two vertices of the hypotenuse of the right triangle. A schematic diagram of an elliptical triangle is shown below. Figure 2 As shown. In Figure 2 In the diagram, the figure formed by connecting line segment S1S6, elliptical arc S6S5, and line segment S5S1 end to end is an elliptical triangle.

[0063] Draw perpendicular lines from the intersection of the rectangle and the ellipse to the major and minor axes of the ellipse, respectively. Divide the closed region formed by the intersection of the rectangle and the ellipse into the union of the rectangle and the elliptical triangle. Place the rectangle obtained from the union into an initially empty set W, and the elliptical triangle into an initially empty set Y. Figure 3 This is a schematic diagram of a closed region formed by the intersection of a rectangle and an ellipse in a specific embodiment of the present invention. (See diagram below.) Figure 3 As shown, the rectangle D1D9D obtained according to equation (1) 10 The region formed by the intersection of D6 and the ellipse is a closed region enclosed by line segment D1D2, elliptical arc D2D7, elliptical arc D7D5, line segment D5D6, line segment D6D8, and line segment D8D1. (This is followed by a reference to rectangle D1D9D.) 10 Draw perpendicular lines from the intersection points D2 and D5 of the rectangle and the ellipse to the major and minor axes of the ellipse, respectively. This divides the region formed by the intersection of the rectangle and the ellipse into rectangles D1D2D3D8, D4D5D6D8 and elliptical triangles D3D2D7, D4D7D5. It should be noted that, in cases such as... Figure 3 In the example shown, the area formed by the intersection of the rectangle and the ellipse is divided into 2 rectangles and 2 elliptical triangles; however, in other cases, the number of rectangles and elliptical triangles divided into different closed areas may be different.

[0064] 3) Approximate the elliptical triangle obtained in step 2) by combining rectangles and triangles. The specific steps are as follows:

[0065] 3-1) Select any elliptical triangle from set Y, denoted as M1P1N; wherein, in a specific embodiment of the present invention, the elliptical triangle secant approximation diagram is as follows. Figure 4 As shown. Figure 4 In the triangle, the intersection of the two right-angled sides is M1, the other endpoint of the right-angled side perpendicular to the major axis of the ellipse is N, and the other endpoint of the right-angled side perpendicular to the minor axis of the ellipse is P1. Let the coordinates of points M1, P1, and N be... (x n ,y n The major semi-axis of the ellipse is a, and the minor semi-axis is b. All of the above data are known.

[0066] 3-2) Let i = 1, and construct an initially empty set denoted as Γ.

[0067] 3-3) To find the elliptic arc P after one approximation i The endpoint P of the next elliptic arc to be approximated on N. i+1 Calculate the secant slope k i .

[0068] When the elliptic triangle M i P iWhen N is in the first or second quadrant, solving the equation shown in equation (2) yields the following result. slope k i Among them, the elliptical triangle M i P i The intersection of the two legs of the right triangle N is M. i The other endpoint of the right-angled leg perpendicular to the major axis is N, and the other endpoint of the right-angled leg perpendicular to the minor axis is P. i .because The slope of ...

[0069]

[0070] Where e is a preset error coefficient, and in this embodiment, 0 < e ≤ 0.05. Let P be the point i x-coordinate Let P be the point i The ordinate.

[0071] When the elliptic triangle M i P i When N is in the third or fourth quadrant, solving the equation shown in equation (3) yields the following result. slope k i ;because The slope of ...

[0072]

[0073] It should be noted that equations (2) and (3) in this embodiment are derived from the following process:

[0074] In elliptic arc P i Find a point on N, and denote it as P. i+1 P i+1 The coordinates are as For P i+1 x-coordinate For P i+1 The ordinate of the line. Draw a secant line. In elliptic arc P i P i+1 Find some R on the top i R i The coordinates are as For R i x-coordinate For R i The ordinate of the line. From this point towards the secant. Draw a perpendicular line, and denote the foot of the perpendicular as Q. i Qi The coordinates are as For Q i x-coordinate For Q i The ordinate of the line is . Then the secant line... slope

[0075] Denote the perpendicular segment Length is make We can obtain:

[0076]

[0077] right Seeking information about The derivative of, and let We can obtain:

[0078]

[0079]

[0080] Will Substitution And order When the elliptic triangle is in the first or second quadrant, the following equation can be obtained:

[0081] When the elliptic triangle is in the third or fourth quadrant, the following equation can be obtained:

[0082]

[0083] 3-4) By calculating P i+1 Find the coordinates of the elliptic arc P after one approximation. i The endpoints of the next elliptical arc to be approximated on N. The specific steps are as follows:

[0084] 3-4-1) Step 3-3) Calculated k i Not unique, k will be a real root i Let k be the number of times. ij j = 1, ..., τ, where τ is the real root k i The number of elements, τ≤4; let l=1;

[0085] 3-4-2) Let k i =k il , will k i Substitute the equations into the system shown below:

[0086]

[0087] Solving for point P yields the solution. i+1 coordinates

[0088] 3-4-3) Determine the result of step 3-4-2): If point P i+1 In elliptic arc P i If N is on the line, then point P is on the line. i+1 To complete an approximation of the elliptic arc P i Find the endpoint of the next elliptical arc to be approximated on N, and proceed to steps 3-5);

[0089] Otherwise, proceed to step 3-4-4);

[0090] 3-4-4) Determination: If l = τ, then set the vertices as N and P. i M i The triangle is placed into set Γ. The elliptical triangle M1P1N is approximated. Proceed to step 3-6). If l < τ, let l = l + 1, and then return to step 3-4-2.

[0091] 3-5) Judgment:

[0092] like or Then let the vertices be N and P. i M i Place the triangle into set Γ and proceed to steps 3-6).

[0093] Otherwise, point P. i+1 To line segment P i M i Draw a perpendicular line, and denote the foot of the perpendicular as T. i ; Self-point P i+1 To line segment NM i Draw a perpendicular line and denote the foot of the perpendicular as M. i+1 According to P i+1 The coordinates of the point can determine triangle T. i P i P i+1 Rectangle M i T i P i+1 M i+1 Triangle T i P i P i+1 and rectangle M i T i P i+1 M i+1 Add it to set Γ, then let i = i + 1, and return to step 3-3) to continue approximating the updated elliptic triangle.

[0094] 3-6) All rectangles and triangles in set Γ are approximations of the elliptic triangle M1P1N. Add all rectangles and triangles in set Γ to set W, and remove the elliptic triangle M1P1N from set Y. Proceed to step 4);

[0095] 4) Determination: If set Y is empty, then all rectangles and triangles in set W constitute the approximation result of the closed region formed by the intersection of the rectangle and the ellipse corresponding to the pair of nonlinear constraints obtained in step 1); otherwise, return to step 3-1).

[0096] Furthermore, the method of the present invention also includes:

[0097] 5) Repeat steps 1)-4) until the approximation results of all nonlinear constraint component pairs of the redundant drive system are obtained, and the approximation is complete.

[0098] In this embodiment, after obtaining the approximation result of the nonlinear constraint, the linear constraint control reachable set algorithm is called to calculate the control reachable set subset corresponding to each rectangular and triangular constraint in the approximation result. After the calculation is completed, the union of all control reachable set subsets is taken to obtain the control reachable set corresponding to the nonlinear constraint, so as to realize the control of the redundant drive system.

[0099] The method of the present invention will be further described below with reference to specific embodiments.

[0100] Example

[0101] In one specific embodiment of the present invention, a secant approximation is performed on the nonlinear constraints of the active wheels of the redundant drive vehicle.

[0102] For a single driving wheel, when it is in a combined driving / braking-steering condition, it will generate a longitudinal tire force F. x and lateral force F y Longitudinal force F x and lateral force F y These are two actuators that obey an elliptic nonlinear relationship. Figure 5 This is a schematic diagram of the nonlinear constraint secant approximation of the active wheels of a redundant drive vehicle in a specific embodiment of the present invention. Figure 5 The ellipse in the figure represents the longitudinal force F that the tire can provide under certain loads, road conditions, tire pressure, tire slip ratio, and tire slip angle. x and lateral force F y Minimum and maximum values ​​(-25kN≤F) x ≤25kN, -20kN≤F y ≤20kN), in this embodiment a=25, b=20, then F x and F y There is an elliptic nonlinear relationship between them. (Translated by Gu Bailiang et al., BOSCH Automotive Engineering Handbook, Beijing Institute of Technology Press, 2nd edition, February 2004). Figure 5 In the diagram, the intersection of rectangle FGHI and the ellipse forms a graph composed of elliptical arc E1CB1, line segment E1F, line segment FG, and line segment GB1, representing the longitudinal force F that the tire can provide at a specific moment. x and lateral force F y The range (at this time, -10kN≤F) x ≤7kN, 17.5kN≤F y ≤20kN, and ).

[0103] Requirement: Perform a linear approximation of the closed region formed by the intersection of rectangle FGHI and ellipse.

[0104] In this embodiment, the secant approximation method for nonlinear constraints of a redundant drive system includes the following steps: 1) Constructing a control input model of a redundant drive system with any pair of nonlinear constraint components, and obtaining the closed region corresponding to the model in the geometric plane.

[0105] This embodiment aims to approximate the region formed by the intersection of a rectangle and an ellipse using a combination of rectangles and triangles, that is... Figure 5 The area enclosed by line segment E1F, line segment FG, line segment GB1, arc B1C, and arc CE1. Figure 5 In the diagram, the four vertices of the rectangle are FGHI, with coordinates (-10, 17.5), (7, 17.5), (7, 20), and (-10, 20). The intersection points of the rectangle and the ellipse are B1, C, and E1, with coordinates (7, 19.2), (0, 20), and (-10, 18.33), respectively. T is the intersection point of side FG of the rectangle and the minor axis of the ellipse, with coordinates (0, 17.5).

[0106] 2) Divide the closed region of step 1) into the union of several rectangles and elliptical triangles.

[0107] In this embodiment, the region formed by the intersection of the rectangle and the ellipse is the area enclosed by line segments E1F, FG, GB1, arc B1C, and CE1. A perpendicular line is drawn from point E1 to the major axis of the ellipse, with the foot of the perpendicular at D1. A perpendicular line is also drawn from point B1 to the minor axis of the ellipse, with the foot of the perpendicular at A1. Therefore, the region formed by the intersection of the rectangle and the ellipse is the union of rectangles FTD1E1, TGB1A1, elliptical triangles A1B1C, and D1E1C. Rectangles FTD1E1 and TGB1A1 are placed in set W1, and elliptical triangles A1B1C and D1E1C are placed in set Y1.

[0108] 3) Approximate the elliptical triangle obtained in step 2) by combining rectangles and triangles. The specific steps are as follows:

[0109] 3-1) In set Y1, select the elliptical triangle A1B1C. The coordinates of points A1, B1, and C are (0, 19.2), (7, 19.2), and (0, 20) respectively. The major semi-axis of the ellipse is a = 25, and the minor semi-axis is b = 20. Points A1, B1, and C correspond to points M1, P1, and N in the algorithm steps. Therefore, x n =0,y n =20;

[0110] 3-2) Let i = 1, and construct an initially empty set denoted as Γ1;

[0111] 3-3) Since the elliptic triangle A1B1C is in the first quadrant, solve the equation:

[0112]

[0113] The two real roots obtained are -0.1148 and -0.3603. The error coefficient is e, which is 0.01 in this embodiment.

[0114] 3-4) By solving for the coordinates of point P2, find the endpoints of the next elliptical arc to be approximated on the elliptical arc P1N after one approximation. The specific steps are as follows:

[0115] 3-4-1) The two real roots obtained in step 3-3) are denoted as k respectively. 11 k 12 k 11 = -0.1148, k 12 =--0.3603;

[0116] 3-4-2) Let k1 = k 11 Substitute k1 into the following system of equations:

[0117]

[0118] The coordinates of point P2 are obtained by solving the problem. Proceed to step 3-4-3);

[0119] 3-4-3) Since point P2 lies on elliptical arc P1N, proceed to step 3-5);

[0120] 3-5) Proceed to steps 3-6);

[0121] 3-6) Place the triangle with vertices A1, B1, and C into Γ1. All rectangles and triangles in Γ1 approximate the elliptic triangle A1B1C. Place all rectangles and triangles in Γ1 into set W1, and remove the elliptic triangle A1B1C from set Y1. Proceed to step 4);

[0122] 4) If set Y1 is not empty, then return to step 3-1.

[0123] 3-1) In set Y1, select the elliptical triangle L1E1C. The coordinates of points L1, E1, and C are (0, 18.33), (-10, 18.33), and (0, 20) respectively. The major semi-axis of the ellipse is a = 25, and the minor semi-axis is b = 20. Points L1, E1, and C correspond to points M1, P1, and N in the algorithm steps. Therefore, x n =0,y n =20.

[0124] 3-2) Let i = 1, and construct an initially empty set denoted as Γ2;

[0125] 3-3) Since the elliptic triangle L1E1C is in the second quadrant, solve the equation:

[0126]

[0127] The two real roots are 0.2223 and 0.4898.

[0128] 3-4) By solving for the coordinates of point P2, find the endpoints of the next elliptical arc to be approximated on the elliptical arc P1N after one approximation. The specific steps are as follows:

[0129] 3-4-1) The two real roots obtained in step 3-3) are denoted as k respectively. 11 k 12 k 11 =0.2223, k 12 =0.4898;

[0130] 3-4-2) Let k1 = k 11 Substitute k1 into the following system of equations:

[0131]

[0132] The coordinates of point P2 are obtained by solving the problem. Proceed to step 3-4-3);

[0133] 3-4-3) Since point P2 lies on elliptical arc P1N, proceed to step 3-5);

[0134] 3-5) Proceed to steps 3-6);

[0135] 3-6) Place the triangle with vertices L1, E1, and C into Γ2. All rectangles and triangles in Γ2 approximate the elliptic triangle L1E1C. Place all rectangles and triangles in Γ2 into set W1, and remove the elliptic triangle L1E1C from set Y1. Proceed to step 4);

[0136] 4) When set Y1 is empty, the algorithm ends. The figure formed by all rectangles and triangles in W1 is a linear approximation of the closed region formed by the intersection of rectangles and ellipses.

[0137] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A secant approximation method for nonlinear constraints in a redundant drive system, characterized in that, include: 1) By constructing a control input model of a redundant drive system with any pair of nonlinear constraint components, the closed region formed by the intersection of rectangles and ellipses in the geometric plane corresponding to the model is obtained; The control input model of the redundant drive system, where any pair of constraint components are nonlinear constraints, is expressed as follows: In the formula, u1 and u2 represent two control actions in a pair of nonlinear constraint components, -a≤u1≤a represents the range of the control action corresponding to u1, -b≤u2≤b represents the range of the control action corresponding to u2, and u imin u is the minimum value of the current control action of the i-th actuator. imax The maximum value of the current control action of the i-th actuator, i = 1, 2, -a ≤ u 1min <u 1max ≤a,-b≤u 2min <u 2max ≤b; Then, the ellipse in the closed figure has a major semi-axis length of a and a minor semi-axis length of b; the rectangle has a length of u. 1max -u 1min Width is u 2max -u 2min ; 2) Based on the closed region obtained in step 1), draw perpendicular lines from the intersection of the rectangle and the ellipse to the major axis and minor axis of the ellipse respectively, dividing the closed region into the union of rectangles and elliptical triangles. Place the rectangles in the union into an initially empty set W, and place the elliptical triangles into an initially empty set Y. The elliptical triangle is a figure formed by the two legs of a right triangle and the elliptical arc connecting the two vertices of the hypotenuse of the right triangle. 3) Approximate the elliptical triangle obtained in step 2) by combining rectangles and triangles; the specific steps are as follows: 3-1) In set Y, choose any elliptical triangle denoted as M1P1N. The intersection of the two legs of this elliptical triangle is M1, the other endpoint of the leg perpendicular to the major axis of the ellipse is N, and the other endpoint of the leg perpendicular to the minor axis of the ellipse is P1. Let the coordinates of points M1, P1, and N be... (x n ,y n ); 3-2) Let i = 1, and construct an initially empty set denoted as Γ; 3-3) To find the elliptic arc P after one approximation i The endpoint P of the next elliptic arc to be approximated on N. i+1 Calculate the secant slope k i : When the elliptic triangle M i P i When N is in the first or second quadrant, solve the equation shown in equation (2) to obtain... slope k i Among them, the elliptical triangle M i P i The intersection of the two legs of the right triangle N is M. i The other endpoint of the right-angled leg perpendicular to the major axis is N, and the other endpoint of the right-angled leg perpendicular to the minor axis is P. i ; Where e is the preset error coefficient, Let P be the point i x-coordinate Let P be the point i The ordinate; When the elliptic triangle M i P i When N is in the third or fourth quadrant, solve the equation shown in equation (3) to obtain... slope k i ; 3-4) By calculating P i+1 Find the coordinates of the elliptic arc P after one approximation. i The endpoint of the next elliptical arc to be approximated on N; the specific steps are as follows: 3-4-1) The real root k obtained in step 3-3) i Let k be the number of times. ij j = 1, ..., τ, where τ is the real root k i The number of elements, τ≤4; let l=1; 3-4-2) Let k i =k il , will k i Substitute the equations into the system shown below: Solving for point P yields the solution. i+1 coordinates 3-4-3) Determine the result of step 3-4-2): If point P i+1 In elliptic arc P i If N is on the line, then point P is on the line. i+1 To complete an approximation of the elliptic arc P i Find the endpoint of the next elliptical arc to be approximated on N, and proceed to steps 3-5); Otherwise, proceed to step 3-4-4); 3-4-4) Determination: If l = τ, then set the vertices as N and P. i M i The triangle is placed into set Γ. The elliptical triangle M1P1N is approximated. Proceed to step 3-6). If l < τ, let l = l + 1, and then return to step 3-4-2. 3-5) Judgment: like or Then let the vertices be N and P. i M i Place the triangle into set Γ and proceed to steps 3-6); Otherwise, point P. i+1 To line segment P i M i Draw a perpendicular line, and denote the foot of the perpendicular as T. i ; Self-point P i+1 To line segment NM i Draw a perpendicular line and denote the foot of the perpendicular as M. i+1 According to P i+1 The coordinates of the point determine triangle T i P i P i+1 Rectangle M i T i P i+1 M i+1 Triangle T i P i P i+1 and rectangle M i T i P i+1 M i+1 Add it to set Γ, then let i = i + 1, return to step 3-3), and continue to approximate the updated elliptic triangle; 3-6) All rectangles and triangles in set Γ are approximations of the elliptic triangle M1P1N. Put all rectangles and triangles in set Γ into set W, remove the elliptic triangle M1P1N from set Y, and proceed to step 4). 4) Determination: If set Y is empty, then all rectangles and triangles in set W constitute the approximation result of the closed region obtained in step 1); otherwise, return to step 3-1).

2. The method as described in claim 1, characterized in that, The method further includes: Repeat steps 1)-4) until the approximation results for all nonlinear constraint component pairs of the redundant drive system are obtained, at which point the approximation is complete.

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