Heavy load plastic forming robot compliant control method

Abstract

The present application relates to a kind of heavy load plastic forming robot compliance control method, comprising the following steps: S1, the kinematic model of heavy load plastic forming robot is established;S2, the dynamic model of heavy load plastic forming robot is established;S3, the forming load real-time prediction model of heavy load plastic forming robot is established;S4, the compliance control model of heavy load plastic forming robot is established.The present application establishes the forming load real-time prediction model by revealing the approximate linear relationship between normal forming force and contact area, and assumes that the friction forming force parameter is constant in continuous control cycle, can quickly predict stress.The present application constructs the compliance control method of heavy load plastic forming robot by real-time forming load calculation, forming feed speed feedforward and servo motor three-ring control, can effectively improve the control precision of heavy load plastic forming robot.

Description

Technical Field

[0001] The present invention relates to the technical field of plastic forming equipment, and more particularly to a compliant control method for a heavy-load plastic forming robot. Background Art

[0002] Thin-walled, high-ribbed components are common load-bearing structures in aerospace equipment. To achieve optimal performance during the forming process, a novel enveloping forming system using a parallel kinematic mechanism has been developed for machining these components. This system can simultaneously control tool motion and forming force. However, due to the complexity of metal flow and deformation during plastic forming, using a standard spring-damper system to model the interaction between the parallel kinematic mechanism and its environment is inadequate, rendering traditional compliance control methods ineffective and, in turn, compromising the forming accuracy of thin-walled, high-ribbed components. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a compliant control method for a heavy-load plastic forming robot, which can effectively improve the control accuracy of the heavy-load plastic forming robot.

[0004] The technical solution adopted by the present invention to solve its technical problems is: constructing a compliant control method for a heavy-load plastic forming robot, the robot including a machine tool, a guide rail, a fast slider and a six-degree-of-freedom parallel motion mechanism, the parallel motion mechanism including a moving platform and six independent motion branches, which are suspended on the driving slider, each branch being connected in sequence by a servo motor, a ball screw, an upper S-joint, a connecting rod, and a lower S-joint, and fixed to the moving platform, the envelope mold being fixed to the moving platform, the lower mold and the blank being fixed to the machine tool, and the moving platform realizing six-degree-of-freedom spatial forming motion by controlling the six servo motors. The control method comprises the following steps:

[0005] S1. Establish the kinematic model of heavy-load plastic forming robot;

[0006] S2. Establish a dynamic model of a heavy-load plastic forming robot;

[0007] S3. Establish a real-time prediction model for forming load of heavy-load plastic forming robots;

[0008] S4. Establish a compliant control model for heavy-load plastic forming robots.

[0009] According to the above solution, in step S1, the method for establishing the kinematic model of the heavy-load plastic forming robot includes:

[0010] A coordinate system S0 is established at the center of the driving slider plane, a coordinate system S1 is established at the center of the lower mold, a coordinate system S2 is established at the center of the blank placing plane, a coordinate system S3 is established at the center of the envelope mold base surface, and a coordinate system S4 is established at the vertex of the envelope mold; A1-A6 are respectively the center points of the lower S joints, and B1-B6 are respectively the center points of the upper S joints; h s and h e are respectively the distances from the vertex of the envelope mold before and after forming to the center of the blank placing plane; f = [0, 0, 1] T is the feeding direction, i f is the feeding speed, h = i f t is the feeding distance, χ t = [0, 0, i f t] is the moving vector of the vertex of the envelope mold, ψ t = [δcos(ωt), δsin(ωt), 0] is the rotating vector of the envelope mold; M 24 (h) is the coordinate transformation matrix from the coordinate system S4 to S2; in the coordinate system S0, the installation angle, initial position and sliding distance of each driving slider are respectively represented as and r B + s i ; in the coordinate system S3, the installation angle and radius of the upper S joint are respectively represented as and r A ; l g is the length of the connecting rod, and l1, l2 and l3 are respectively the distances between the coordinate systems S0 and S1, S1 and S2 and S3 and S4;

[0011] The coordinate transformation matrix from the coordinate system S3 to S0 is represented by formula (1):

[0012] M 03 = M 02 M 24 M 43 (1)

[0013] In the formula,

[0014] The motion constraint of the heavy load plastic forming robot is represented by formula (2):

[0015]

[0016] According to the above scheme, in the step S2, the method for establishing the dynamics model of the heavy load plastic forming robot comprises:

[0017] A coordinate system S p is established at the mass center of the moving platform, the distance between the coordinate systems S p and S3 is l p , and a coordinate system S4 is established at the mass center of the connecting rod F p and T p are the load force and load moment, respectively, are the forces acting on the upper S-joint and the lower S-joint, is the driving force vector that drives the slider, are the elastic force and damping force on the connecting rod respectively; δ i is the deformation vector of the connecting rod, and its direction is along the connecting rod direction; a i and b i A i and B i The position vector of p =[α p ,β p ,γ p ] T is the Euler angle of the moving platform; p =[x p ,y p ,z p ] T is the position vector of the moving platform, and is obtained from equation (3):

[0018]

[0019] In the sliding direction of the driving slider, the dynamic equation of the driving slider is expressed by formula (4):

[0020]

[0021] Where u s and c s are the friction coefficient and damping coefficient, m s is the mass of the driving slide, are the position, velocity and acceleration of the driving slider respectively;

[0022] The dynamic equation of the connecting rod is expressed by formula (5):

[0023]

[0024] Where m l is the mass of the connecting rod; E is the elastic modulus, c l is the linear damping coefficient of the connecting rod; is the position vector of the connecting rod's center of mass, is the angle vector of the connecting rod; M l and I l are the mass matrix and inertia matrix of the connecting rod, respectively, and are expressed by equation (6):

[0025]

[0026] Where r l is the cross-sectional radius of the connecting rod;

[0027] In addition, the geometric constraint relationship of the connecting rod is expressed by formula (7):

[0028]

[0029] Where R is the transformation matrix based on Euler angles;

[0030] The dynamic equation of the moving platform is expressed by formula (8):

[0031]

[0032] Where m p is the mass of the moving platform, χ p =[x p ,y p ,z p ] is the position vector of the center of mass of the moving platform, ψ p =[α l ,β l ,γ l ] is the angle vector of the moving platform; M p and I p are the mass matrix and inertia matrix of the moving platform, respectively, and are expressed by equation (9):

[0033]

[0034] Where r p and w p are the radius and thickness of the moving platform respectively;

[0035] The geometric constraint relationship of the moving platform is expressed by formula (10):

[0036]

[0037] The overall dynamic model is obtained according to equations (4), (5) and (8).

[0038] According to the above scheme, step S3 includes the following steps:

[0039] S301, perform equivalent contact surface analysis during forming process;

[0040] S302, establishing a normal forming force model;

[0041] S303, establishing a friction forming force model;

[0042] S304, establishing a forming force identification model;

[0043] S305: Establish a forming force prediction model.

[0044] According to the above solution, the method for performing equivalent contact surface analysis during the forming process in step S301 includes:

[0045] The motion of the enveloping die is simplified to rolling along the screw surface, and the contact line is simplified to the intersection of the cone surface and the disk. The equation of the enveloping die and the upper plane of the blank is expressed by formula (11):

[0046]

[0047] Where θ, r, and z are the polar angle, radius, and height of the plane on the blank, respectively;

[0048] Intersecting lines The equation is expressed by formula (12):

[0049]

[0050] In the polar coordinate system, equation (12) is rewritten as:

[0051] ρ=Δh / tanδ / (1-cosθ) (13)

[0052] The polar angles of points P1 and P2 are expressed by equation (14):

[0053] θ n =acos(1-Δh / r n / tanδ),θ w =acos(1-Δh / r w / tanδ) (14)

[0054] To simplify the calculation, the curve Set as an Archimedean spiral curve, and expressed by formula (15):

[0055] θ p (ρ)=(ρ-r n )(θ w -θ n ) / (r w -r n )+θ n (15)

[0056] The contact surface is calculated according to formula (16):

[0057]

[0058] According to the above solution, the method for establishing the normal forming force model in step S302 includes:

[0059] Normal force F p Perpendicular to the contact surface, rolling friction force F fThe direction is along the tangent direction of the blank, and the sliding friction force F τ The direction is the radial direction of the blank; for a selected point, its orthogonal coordinate value is (x, y), and its polar coordinate value is (θ, ρ). The pressure of this point in the three directions is p, f, and τ; the differential volume element ABCDEF is used to calculate the forming force, and the polar angle and polar radius of the element are defined as θ and ρ; the length of AB and the angle of BC are defined as dρ and dθ; the angle of AB is defined as κ, and the lengths of BF and AE are defined as z and dz+z, respectively;

[0060] The normal forming force and tangential friction force acting on the upper surface ABCD are expressed by formula (17):

[0061] pdA=ρdθ / cosκdρ, fdA=fdθ / cosκdρ ​​(17)

[0062] Where dA represents the area of ​​surface ABCD; its geometric relationship is expressed by equation (18):

[0063] tanκ=dz / ρ / dθ (18)

[0064] The normal force along the tangential direction is expressed by equation (19):

[0065] F pt =pdAsinκ=pρdθdρdz / ρdθ=pdρdz (19)

[0066] The friction force in the tangential direction is expressed by formula (20):

[0067] F ft =-fdAcosκ=-fρdθdρ(1+cosκ) / cosκ≈-2fρdθdρ (20)

[0068] Assuming that the pressure of the blank in the Z direction is constant, and defining σ and (σ+dv) as the pressure of the surface ABEF and CDE'F', and defining the areas of the two surfaces as dB and (dB+ddB), respectively, the tangential deformation force is expressed by Equation (21):

[0069]

[0070] When cos(dθ / 2)≈1, equation (21) can be rewritten as equation (22):

[0071] F dt =[σzdρ-(σ+dσ)(z+dz)dρ] (22)

[0072] The force balance equation in the tangential direction is expressed by equation (23):

[0073] F pt +Fft +F dt =pdρdz-2fρdθdρ+σzdρ-(σ+dσ)(z+dz)dρ=0 (23)

[0074] Formula (23) can be rewritten as formula (24):

[0075]

[0076] Based on the theory of plastic deformation, the principal stress is defined as σ θ , σ ρ and p, the plasticity equation is expressed by equation (25):

[0077] σ θ -p=2ν,dσ θ =dp (25)

[0078] Where ν is the plastic deformation of pure shear material, and the final force is expressed by equation (26):

[0079]

[0080] Since dz / z≈0, f=μν, the solution of Equation (26) is expressed by Equation (27):

[0081] p=2fρθ / (h0-h)+p0 (27)

[0082] Where p0 is the initial value of the final force; replace p=p0 and σ θ Substituting σ0 into equation (25), we get equation (28):

[0083] p0=2v+σ0 (28)

[0084] Where σ0 = 2vΔh / (h0-h) is the stress of the two tangential edge elements. The friction stress and friction coefficient are defined as f = μv and μ, respectively. Equation (29) can be obtained:

[0085] p=2uvρθ / (h0-h)+(2v+1)Δh / (h0-h) (29)

[0086] The total forming force is expressed by formula (30):

[0087]

[0088] The equivalent action point of the normal force is expressed by formula (31):

[0089]

[0090] Where, bite angle can be eliminated; the forming force vector and the corresponding action point are expressed by formula (32):

[0091]

[0092] According to the above scheme, in step S303, the friction forming force and its action point are expressed by formula (33):

[0093]

[0094] According to the above scheme, in step S304, the forming force is obtained by real-time identification of formula (34):

[0095]

[0096] Where, F wr and T wr It is the real-time forming force.

[0097] According to the above scheme, in step S305, when the feed speed is Δh', the parameters at the next moment are expressed by formula (35):

[0098]

[0099] When the forward feed speed Δh is given f After that, the forward forming load at the next moment is expressed by formula (36):

[0100]

[0101] According to the above solution, in step S4, the compliant control model includes:

[0102] S401, real-time forming force calculation method;

[0103] A pressure sensor is installed between the driving slider and the ball screw to measure the driving force, and a grating ruler is used to measure the position of the driving slider. Based on the geometric constraint relationship, the real-time transformation matrix is ​​expressed by formula (37):

[0104]

[0105] The actual forming force applied to the blank is expressed by formula (38):

[0106]

[0107] S402, forming feed speed feedback method;

[0108] The input is the real-time position of the driving slider and real-time driving force Define F wd is the required driving force, the current feed speed Δh f From formula (39), we can get:

[0109] F wf (t,Δh f )=F wd (t+Δt)+k F [F wr (t)-F wd (t)] (39)

[0110] Where, F wf (t,Δh d ) is the feed rate Δh f The driving force, F wd (t+Δt) is the required driving force for the next cycle, F wr (t) is the real-time forming force, k F is the gain of the method;

[0111] The load acting on the moving platform is obtained by formula (40):

[0112]

[0113] S403, compliant control method;

[0114] The compliant control system of the heavy-load plastic forming robot consists of forward kinematics dynamics, forming feed rate feedback and six three-loop servo control parts;

[0115] The control rate is expressed by formula (41):

[0116]

[0117] In the formula, the position loop error, speed loop error and current loop error are expressed by:

[0118]

[0119] Where, P n and ψ mg is the number of motor pole pairs and motor flux, T m is the electromagnetic torque of the six motors; K pp , K vp , K vi , K ip and K ii is the control parameter.

[0120] The implementation of the heavy-load plastic forming robot compliance control method of the present invention has the following beneficial effects:

[0121] (1) The present invention considers the change in connecting rod posture caused by the feed distance and the deformation of the connecting rod caused by the forming load, establishes a kinematic and dynamic model, and can achieve precise motion control.

[0122] (2) The present invention reveals the approximate linear relationship between the normal forming force and the contact area, and assumes that the friction forming force parameters are constant within a continuous control cycle, thereby establishing a real-time prediction model for the forming load, which can quickly predict the force.

[0123] (3) The present invention constructs a compliant control method for a heavy-load plastic forming robot through real-time forming load calculation, forming feed speed feedforward and servo motor three-loop control, which can effectively improve the control accuracy of the heavy-load plastic forming robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0124] The present invention will be further described below with reference to the accompanying drawings and embodiments, in which:

[0125] Figure 1 It is the overall configuration and physical diagram of the heavy-load plastic forming robot;

[0126] Figure 2(a)-Figure 2(c) This is a simplified structural diagram of a heavy-load plastic forming robot;

[0127] Figure 3(a) and Figure 3(b) are schematic diagrams of the overall force of the heavy-load plastic forming robot;

[0128] Figure 4 is a schematic diagram of the forming contact surface;

[0129] Figure 5 It is a schematic diagram of forming force and forming moment during the forming process;

[0130] Figure 6 It is a description of the forming force calculation method;

[0131] Figure 7 This is the structural diagram of the forming feed rate feedforward method;

[0132] Figure 8 This is a structural diagram of a compliant control method for a heavy-load plastic forming robot;

[0133] Figure 9(a) and Figure 9(b) are the analysis diagrams of the forming performance of aircraft bevel gears under the compliant control method;

[0134] Figure 10 It is the position error of the driving slide and the control error of the driving slide in the aircraft bevel gear forming process;

[0135] Figure 11 It is the tracking error of different tool point positions in aircraft bevel gear forming;

[0136] Figures 12(a) and 12(b) are analysis diagrams of the forming performance of aircraft window frames under compliant control;

[0137] Figure 13 This is a diagram showing the position of the drive slider and the control error of the drive slider during the aircraft window frame forming process.

[0138] Figure 14 This is a graph of tracking errors at different tool point positions during aircraft window frame forming. DETAILED DESCRIPTION

[0139] In order to have a clearer understanding of the technical features, purposes and effects of the present invention, specific embodiments of the present invention are now described in detail with reference to the accompanying drawings.

[0140] The basic steps of the compliant control method of the heavy-load plastic forming robot of the present invention are as follows:

[0141] S1. Establish the kinematic model of heavy-load plastic forming robot.

[0142] Heavy-duty plastic forming robots such as Figure 1 As shown in the figure, the robot consists of a machine tool, guide rails, a fast slide, and a six-degree-of-freedom parallel kinematic mechanism. The parallel kinematic mechanism consists of a moving platform and six independent kinematic branches suspended from the drive slide. Each branch is connected in sequence by a servo motor, a ball screw, an upper S-joint, a connecting rod, and a lower S-joint, and is fixed to the moving platform. The enveloping die is fixed to the moving platform, while the lower die and blank are fixed to the machine tool. Precise control of the six servo motors enables the moving platform to achieve the pre-designed six-degree-of-freedom spatial forming motion.

[0143] The schematic diagram of the heavy-load plastic forming robot is as follows: Figure 2(a)-Figure 2(c) As shown, coordinate system S0 is established at the center of the driving slider plane, coordinate system S1 is established at the center of the lower mold, coordinate system S2 is established at the center of the blank placement plane, coordinate system S3 is established at the center of the envelope mold base surface, and coordinate system S4 is established at the vertex of the envelope mold. A1~A6 are the center points of the lower S joint respectively, and B1~B6 are the center points of the upper S joint respectively. s and h e f=[0,0,1] is the distance from the vertex of the envelope die to the center of the blank placement plane before and after forming. T is the feed direction, i f is the feed rate, h=i f t is the feed distance, χ t =[0,0,i f t] is the envelope mold vertex movement vector, ψ t =[δcos(ωt),δsin(ωt),0] is the envelope mold rotation vector. M 24 (h) is the coordinate transformation matrix from coordinate system S4 to S2. In the S0 coordinate system, the installation angle, initial position and sliding distance of each driving slider are expressed as s i and r B +s i In the S3 coordinate system, the installation angle and radius of the upper S joint are expressed as and r A . l g is the connecting rod length, l1, l2, l3 are the distances between coordinate systems S0 and S1, S1 and S2, and S3 and S4, respectively.

[0144] The coordinate transformation matrix from coordinate system S3 to S0 is expressed by formula (1):

[0145] M 03 =M 02 M 24 M 43 (1)

[0146] Where,

[0147] Therefore, the motion constraint of the heavy-load plastic forming robot can be expressed by formula (2):

[0148]

[0149] S2. Establish a dynamic model of heavy-load plastic forming robot.

[0150] The force and geometric constraints of the heavy-load plastic forming robot are shown in Figure 3(a) and Figure 3(b). The coordinate system S is established at the center of mass of the moving platform. p , coordinate system S p The distance between S3 is l p , establish a coordinate system at the center of mass of the connecting rod F p and T p are the load force and load moment, are the forces acting on the upper S-joint and the lower S-joint, is the driving force vector that drives the slider, are the elastic force and damping force on the connecting rod respectively. i is the deformation vector of the connecting rod, and its direction is along the connecting rod. i and b i A i and B i Position vector of ψ p =[α p ,β p ,γ p ] T is the Euler angle of the moving platform. p =[x p ,y p ,z p ] T is the position vector of the moving platform, and can be obtained from formula (3):

[0151]

[0152] In the sliding direction of the driving slider, the dynamic equation of the driving slider is expressed by formula (4):

[0153]

[0154] Where u s and c s are the friction coefficient and damping coefficient, m s is the mass of the driving slide, are the position, velocity and acceleration of the driving slider respectively.

[0155] The dynamic equation of the connecting rod is expressed by formula (5):

[0156]

[0157] Where m l is the mass of the connecting rod. E is the elastic modulus, c l is the linear damping coefficient of the connecting rod. is the position vector of the connecting rod's center of mass, is the angle vector of the connecting rod. M l and I l are the mass matrix and inertia matrix of the connecting rod, respectively, and are expressed by equation (6):

[0158]

[0159] Where r l is the cross-sectional radius of the connecting rod.

[0160] In addition, the geometric constraint relationship of the connecting rod is expressed by formula (7):

[0161]

[0162] Where R is the transformation matrix based on Euler angles.

[0163] The dynamic equation of the moving platform is expressed by formula (8):

[0164]

[0165] Where m p is the mass of the moving platform, χ p =[x p ,y p ,z p ] is the position vector of the center of mass of the moving platform, ψ p =[α l ,β l ,γ l ] is the angle vector of the moving platform. M pand I p are the mass matrix and inertia matrix of the moving platform, respectively, and are expressed by equation (9):

[0166]

[0167] Where r p and w p are the radius and thickness of the moving platform respectively.

[0168] The geometric constraint relationship of the moving platform is expressed by formula (10):

[0169]

[0170] In summary, the overall dynamic model can be obtained according to equations (4), (5) and (8). The model includes 6+3*6+3*6+3+3=48 equations. For the inverse dynamics problem, the desired motion trajectory of the moving platform and the load χ p , F p ,T p is a given value, the unknown quantity There are 6+3*6+3*6+6=48. Therefore, the dynamic equation can be solved.

[0171] S3 establishes a real-time prediction model for forming load of heavy-load plastic forming robots

[0172] 1. Analysis of equivalent contact surface during forming process

[0173] The motion of the enveloping die can be simplified to rolling along the screw surface, and the contact line can be simplified to the intersection of the cone surface and the disk, such as Figure 4 As shown. The equation of the envelope die and the upper plane of the blank is expressed by equation (11):

[0174]

[0175] Where θ, r, and z are the polar angle, radius, and height of the plane on the blank, respectively.

[0176] Intersecting lines The equation is expressed by formula (12):

[0177]

[0178] In the polar coordinate system, equation (12) can be rewritten as:

[0179] ρ=Δh / tanδ / (1-cosθ) (13)

[0180] The polar angles of points P1 and P2 are expressed by equation (14):

[0181] θ n=acos(1-Δh / r n / tanδ),θ w =acos(1-Δh / r w / tanδ) (14)

[0182] To simplify the calculation, the curve It can be set as an Archimedean spiral curve and expressed by formula (15):

[0183] θ p (ρ)=(ρ-r n )(θ w -θ n ) / (r w -r n )+θ n (15)

[0184] Therefore, the contact surface can be calculated according to formula (16):

[0185]

[0186] 2. Establish normal forming force model

[0187] like Figure 5 As shown, three forces act on the contact surface, the normal force F p Perpendicular to the contact surface, rolling friction force F f The direction is along the tangent direction of the blank, and the sliding friction force F τ The direction is the radial direction of the blank. Specifically, for a selected point, its orthogonal coordinate value is (x, y) and its polar coordinate value is (θ, ρ), then the pressure at this point in the three directions is p, f, and τ. The differential volume element ABCDEF is used to calculate the forming force, and the polar angle and polar radius of the element are defined as θ and ρ. In the top view, the length of AB and the angle of BC are defined as dρ and dθ. In the unfolded view, the angle of AB is defined as κ, and the lengths of BF and AE are defined as z and dz+z, respectively.

[0188] The normal forming force and tangential friction force acting on the upper surface ABCD are expressed by formula (17):

[0189] pdA=ρdθ / cosκdρ, fdA=fdθ / cosκdρ ​​(17)

[0190] Where dA represents the area of ​​surface ABCD. Its geometric relationship is expressed by equation (18):

[0191] tanκ=dz / ρ / dθ (18)

[0192] The normal force along the tangential direction is expressed by equation (19):

[0193] Fpt = pdAsin κ = pdp dθ dρ dz / pdθ = pdρ dz (19)

[0194] Considering that the value of κ is very small, cos κ ≈ 1, the friction force in the tangential direction is represented by equation (20):

[0195] F ft = -fdAcos κ = -fρdθdρ (1 + cos ≈) / cos κ ≈ -2fρdθdρ (20)

[0196] Assuming that the pressure in the Z direction of the blank is constant, and defining σ and (σ + dσ) as the pressures of the surfaces AB EF and CDE'F', and defining the areas of the two surfaces as dB and (dB + ddB), respectively, the tangential deformation force can be represented by equation (21):

[0197]

[0198] When cos (dθ / 2) ≈ 1, equation (21) can be rewritten as equation (22):

[0199] F dt = [σzdρ - (σ + dσ)(z + dz)dρ] (22)

[0200] The force balance equation in the tangential direction is represented by equation (23):

[0201] F pt +F ft +F dt = pdρ dz - 2fρdθdρ + σzdρ - (σ + dσ)(z + dz)dρ = 0 (23)

[0202] Equation (23) can be rewritten as equation (24):

[0203]

[0204] Based on the plastic deformation theory, the principal stresses are defined as σ θ , σ ρ , and p, and thus the plastic equation is represented by equation (25):

[0205] σ θ - p = 2v, dσ θ = dp (25)

[0206] In equation (25), v is the plastic deformation of a pure shear material. Thus, the final force can be represented by equation (26):

[0207]

[0208] Since dz / z≈0, f=μv, the solution of Equation (26) can be expressed by Equation (27):

[0209] p=2fρθ / (h0-h)+p0 (27)

[0210] Where p0 is the initial value of the final force. θ Substituting σ0 into equation (25), we can obtain equation (28):

[0211] p0=2v+σ0 (28)

[0212] Where σ0 = 2νΔh / (h0-h) is the stress of the two edge elements tangentially. In addition, the friction stress and friction coefficient are defined as f = μν and μ, respectively, and Equation (29) can be obtained:

[0213] p=2uvρθ / (h0-h)+(2ν+1)Δh / (h0-h) (29)

[0214] The total forming force is expressed by formula (30):

[0215]

[0216] The equivalent action point of the normal force can be expressed by formula (31):

[0217]

[0218] Where, bite angle can be eliminated. The forming force vector and the corresponding action point are expressed by formula (32):

[0219]

[0220] 3. Establish friction forming force model

[0221] Since the third component of the force vector and the action point vector are close to zero, the friction forming force and its action point can be expressed by equation (33):

[0222]

[0223] 4. Establishing forming force identification model

[0224] According to the above analysis, the forming force can be obtained by real-time identification of formula (34):

[0225]

[0226] Where, F wr and T wr It is the real-time forming force.

[0227] 5. Establish a forming force prediction model

[0228] Since the control cycle is very small, the change of the system can be regarded as linear. Therefore, when the feed rate is Δh', the parameters at the next moment can be expressed by formula (35):

[0229]

[0230] When the forward feed speed Δh is given f After that, the forward forming load at the next moment is expressed by formula (36):

[0231]

[0232] S4 establishes a compliant control model for heavy-load plastic forming robots

[0233] 1. Real-time forming force calculation method

[0234] like Figure 6 As shown in Figure 3, a pressure sensor is installed between the driving slider and the ball screw to measure the driving force, and a grating ruler is used to measure the position of the driving slider. Based on the geometric constraint relationship, the real-time transformation matrix can be expressed by formula (37):

[0235]

[0236] Since the center of the enveloping die and the center of the blank are almost the same, the actually detected forming force applied to the blank can be expressed by formula (38):

[0237]

[0238] 2. Forming feed speed feedback method

[0239] This method is as Figure 7 As shown, the input is the real-time position of the driving slider and real-time driving force Define F wd is the required driving force, the current feed speed Δh f It can be obtained from formula (39):

[0240] F wf (t,Δh f )=F wd (t+Δt)+k F [F wr (t)-F wd (t)] (39)

[0241] Where, F wf (t,Δh d ) is the feed rate Δh f The driving force, F wd (t+Δt) is the required driving force for the next cycle, Fwr (t) is the real-time forming force, k F is the gain of the method.

[0242] The load acting on the moving platform can be obtained from formula (40):

[0243]

[0244] 3. Soft control method

[0245] Based on the above analysis, a compliant control method for heavy-load plastic forming robots is proposed. Figure 8 As shown in the figure, the control system consists of forward kinematics dynamics, forming feed speed feedback and six three-loop servo control parts.

[0246] The control rate is expressed by formula (41):

[0247]

[0248] In the formula, the position loop error, speed loop error and current loop error can be expressed by:

[0249]

[0250] Where, P n and ψ mg is the number of motor pole pairs and motor flux, T m is the electromagnetic torque of the six motors. K pp , K vp , K vi , K ip and K ii is the control parameter.

[0251] Example

[0252] Based on the desired motion trajectory of the enveloping die, the method provided by this invention can derive a kinematic solution for the heavy-duty plastic forming robot, namely the desired trajectory of the driving slider. This method can also predict the forming force of the heavy-duty plastic forming robot and, combined with kinematics and dynamics, provide real-time feedback within a compliant control method.

[0253] For aircraft bevel gears, to maximize forming efficiency, the required force was set to 500 tons, the target feed distance was set to 28 mm, and the initial and maximum feed rates were both set to 7.5 mm / s. Figure 9(a) shows the real-time feed rate and feed distance, with the blue and red lines representing the real-time feed rate and feed distance, respectively. Figure 9(b) shows the required force and actual forming force, with the blank dashed line and red solid line representing the real-time feed rate. It can be seen that as the feed distance changes from 0 to 28 mm, the feed rate decreases from 7.5 mm / s to zero. Until the 2-second mark, the feed rate remains constant at its maximum value of 7.5 mm / s, and the forming force remains zero. However, after 2 seconds, the forming force gradually increases to nearly 500 tons and then fluctuates around this value, while the feed rate decreases to zero. This indicates that the enveloping die and the blank contact at a feed distance of 15 mm. After contact, the control system stabilizes for approximately 0.1 seconds, and the overshoot of the forming force is approximately 70 tons. The process ends around 9.1 seconds, with the feed distance approaching 28 mm. In the steady-state stage, the forming force error is about 50 tons. Figure 10 The dashed line represents the slider position, and the solid line represents the position error. The left and right y-axes represent the slider position and position error, respectively. The six sub-figures use different colored lines to represent the corresponding slider positions and errors. It can be seen that as the slider reciprocates, the center position decreases with feed. Within the range of -200 to 200 μm, the position error increases with increasing feed distance. Figure 11 The maximum trajectory error and average trajectory error of different tool points in aircraft bevel gear forming are shown, and the maximum trajectory error can reach 90um.

[0254] For aircraft bevel gears, the target feed distance was set to 27 mm, and the initial and maximum feed rates were both set to 7.5 mm / s. The actual feed rate and feed distance are shown in Figure 12(a) as blue and red lines, respectively. Figure 12(b) shows the required and actual forming forces as the dashed and solid red lines, respectively. It can be seen that as the feed distance changes from 0 to 27 mm, the feed rate decreases from 7.5 mm / s to 0. Before reaching a feed distance of 17 mm, the feed rate remains constant at a maximum of 7.5 mm / s, and the forming force remains zero. After reaching a feed distance of 17 mm, the forming force increases to 150 tons, while the feed rate decreases from 2 to 1.5 mm / s. After reaching a feed distance of 20 mm, the required force increases to 350 tons, and the feed rate first increases to 4.4 mm / s and then decreases to 2.5 mm / s. The settling time is approximately 0.7 seconds, and the overshoot of the forming force is approximately 25 tons. After reaching the feed distance of 23 mm, the required force increases to 500 tons, and the feed speed first increases to 4.4 mm / s and then decreases to 0. Figure 13The dashed line represents the position of the drive slider, and the solid line represents the position error. The left and right y-axes represent the position magnitude and position error, respectively. The six sub-graphs use different colored lines to represent the position and error of the corresponding drive slider. It can be seen that as the drive slider reciprocates, the center position decreases with feed. Within the range of -200 to 200 μm, the position error increases with increasing feed. Figure 14 The maximum trajectory error and average trajectory error at each point during the aircraft window frame forming process are shown in Figure 2. The maximum trajectory error can reach 97 μm. It can be found that the compliant control method for the heavy-load plastic forming robot proposed in the present invention can effectively improve the control accuracy of the robot.

[0255] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the guidance of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the present invention and the claims, all of which are protected by the present invention.

Claims

1. A compliant control method for a heavy-load plastic forming robot, the robot comprising a machine tool, a guide rail, a fast slider, and a six-degree-of-freedom parallel kinematic mechanism, the parallel kinematic mechanism comprising a moving platform and six independent motion branches suspended on a driving slider, each branch being connected in sequence by a servo motor, a ball screw, an upper S-joint, a connecting rod, and a lower S-joint, and fixed to the moving platform. The enveloping die is fixed to the moving platform, and the lower die and blank are fixed to the machine tool. By controlling the six servo motors, the moving platform achieves six-degree-of-freedom spatial forming motion, characterized in that: The control method includes the following steps: S1. Establish the kinematic model of heavy-load plastic forming robot; S2. Establish a dynamic model of a heavy-load plastic forming robot; S3. Establish a real-time prediction model for forming load of heavy-load plastic forming robots; S4. Establish a compliant control model for a heavy-load plastic forming robot; Compliant control models include: S401, real-time forming force calculation method; A pressure sensor is installed between the driving slider and the ball screw to measure the driving force, and a grating ruler is used to measure the position of the driving slider. Based on the geometric constraint relationship, the real-time transformation matrix is ​​expressed by formula (37): Establish coordinate system S0 at the center of the driving slider plane, establish coordinate system S1 at the center of the lower mold, establish coordinate system S2 at the center of the blank placement plane, establish coordinate system S3 at the center of the envelope mold base surface, and establish coordinate system S4 at the vertex of the envelope mold; in the S0 coordinate system, the installation angle, initial position and sliding distance of each driving slider are expressed as and r B +s i ; In the S3 coordinate system, the installation angle and radius of the upper S joint are expressed as and r A ; l g is the connecting rod length, l1, l2, l3 are the distances between coordinate systems S0 and S1, S1 and S2, and S3 and S4 respectively; is the coordinate transformation matrix from coordinate system S3 to S0; The actual forming force applied to the blank is expressed by formula (38): Establish a coordinate system S at the center of mass of the moving platform p , coordinate system S p The distance between S3 is l p , establish a coordinate system at the center of mass of the connecting rod F p and T p are load force and load moment respectively; F wr and T wr is the real-time forming force; S402, forming feed speed feedback method; The input is the real-time position of the driving slider and real-time driving force Define F wd is the required driving force, the current feed speed Δh f From formula (39), we can get: F wf (t,Δh f )=F wd (t+Δt)+k F [F wr (t)-F wd (t)] (39) Where, F wf (t,Δh d ) is the feed rate Δh f The driving force, F wd (t+Δt) is the required driving force for the next cycle, F wr (t) is the real-time forming force, k F is the gain of the method; The load acting on the moving platform is obtained by formula (40): S403, compliant control method; The compliant control system of the heavy-load plastic forming robot consists of forward kinematics dynamics, forming feed rate feedback and six three-loop servo control parts; The control rate is expressed by formula (41): In the formula, the position loop error, speed loop error and current loop error are expressed by: Where, P n and ψ mg is the number of motor pole pairs and motor flux, T m is the electromagnetic torque of the six motors; K pp , K vp , K vi , K ip and K ii is the control parameter.

2. The compliant control method for a heavy-load plastic forming robot according to claim 1, characterized in that: In step S1, the method for establishing a kinematic model of a heavy-load plastic forming robot includes: Establish coordinate system S0 at the center of the driving slider plane, establish coordinate system S1 at the center of the lower mold, establish coordinate system S2 at the center of the blank placement plane, establish coordinate system S3 at the center of the envelope mold base surface, and establish coordinate system S4 at the vertex of the envelope mold; A1~A6 are the center points of the lower S joint respectively, and B1~B6 are the center points of the upper S joint respectively; h s and h e are the distances from the vertex of the envelope die to the center of the blank placement plane before and after forming; f = [0,0,1] T is the feed direction, i f is the feed rate, h=i f t is the feed distance, χ t =[0,0,i f t] is the envelope mold vertex movement vector, ψ t =[δcos(ωt),δsin(ωt),0] is the envelope mold rotation vector; M 24 (h) is the coordinate transformation matrix from coordinate system S4 to S2; in the S0 coordinate system, the installation angle, initial position and sliding distance of each driving slider are expressed as and r B +s i ; In the S3 coordinate system, the installation angle and radius of the upper S joint are expressed as and r A ; l g is the connecting rod length, l1, l2, l3 are the distances between coordinate systems S0 and S1, S1 and S2, and S3 and S4 respectively; The coordinate transformation matrix from coordinate system S3 to S0 is expressed by formula (1): M 03 =M 02 M 24 M 43 (1) Where, The motion constraint of the heavy-load plastic forming robot is expressed by formula (2):

3. The compliant control method for a heavy-load plastic forming robot according to claim 2, characterized in that: In step S2, the method for establishing a dynamic model of a heavy-load plastic forming robot includes: Establish a coordinate system S at the center of mass of the moving platform p , coordinate system S p The distance between S3 is l p , establish a coordinate system at the center of mass of the connecting rod F p and T p are the load force and load moment, respectively, and F l i are the forces acting on the upper S-joint and the lower S-joint, is the driving force vector that drives the slider, and are the elastic force and damping force on the connecting rod respectively; δ i is the deformation vector of the connecting rod, and its direction is along the connecting rod direction; a i and b i A i and B i The position vector of p =[α p ,β p ,γ p ] T is the Euler angle of the moving platform; p =[x p ,y p ,z p ] T is the position vector of the moving platform, and is obtained from equation (3): In the sliding direction of the driving slider, the dynamic equation of the driving slider is expressed by formula (4): Where u s and c s are the friction coefficient and damping coefficient, m s is the mass of the driving slide, s i , are the position, velocity and acceleration of the driving slider respectively; The dynamic equation of the connecting rod is expressed by formula (5): Where m l is the mass of the connecting rod; E is the elastic modulus, c l is the linear damping coefficient of the connecting rod; is the position vector of the connecting rod's center of mass, is the angle vector of the connecting rod; M l and I l are the mass matrix and inertia matrix of the connecting rod, respectively, and are expressed by equation (6): Where r l is the cross-sectional radius of the connecting rod; In addition, the geometric constraint relationship of the connecting rod is expressed by formula (7): Where R is the transformation matrix based on Euler angles; The dynamic equation of the moving platform is expressed by formula (8): Where m p is the mass of the moving platform, χ p =[x p ,y p ,z p ] is the position vector of the center of mass of the moving platform, ψ p =[α l ,β l ,γ l ] is the angle vector of the moving platform; M p and I p are the mass matrix and inertia matrix of the moving platform, respectively, and are expressed by equation (9): Where r p and w p are the radius and thickness of the moving platform respectively; The geometric constraint relationship of the moving platform is expressed by formula (10): The overall dynamic model is obtained according to equations (4), (5) and (8).

4. The compliant control method for a heavy-load plastic forming robot according to claim 3, characterized in that: The step S3 comprises the following steps: S301, perform equivalent contact surface analysis during forming process; S302, establishing a normal forming force model; S303, establishing a friction forming force model; S304, establishing a forming force identification model; S305: Establish a forming force prediction model.

5. The compliant control method for a heavy-load plastic forming robot according to claim 4, characterized in that: The method for performing equivalent contact surface analysis during the forming process in step S301 includes: The motion of the enveloping die is simplified to rolling along the screw surface, and the contact line is simplified to the intersection of the cone surface and the disk. The equation of the enveloping die and the upper plane of the blank is expressed by formula (11): Where θ, r, and z are the polar angle, radius, and height of the plane on the blank, respectively; Intersecting lines The equation is expressed by formula (12): In the polar coordinate system, equation (12) is rewritten as: ρ=Δh / tanδ / (1-cosθ) (13) The polar angles of points P1 and P2 are expressed by equation (14): i n =acos(1-Δh / r n / tanδ),θ w =acos(1-Δh / r w / tanδ) (14) To simplify the calculation, the curve Set as an Archimedean spiral curve, and expressed by formula (15): i p (ρ)=(ρ-r n )(θ w -θ n ) / (r w -r n )+θ n (15) The contact surface is calculated according to formula (16):

6. The compliant control method for a heavy-load plastic forming robot according to claim 5, characterized in that: The method for establishing the normal forming force model in step S302 includes: Normal force F p Perpendicular to the contact surface, rolling friction force F f The direction is along the tangent direction of the blank, and the sliding friction force F τ The direction is the radial direction of the blank; for a selected point, its orthogonal coordinate value is (x, y), and its polar coordinate value is (θ, ρ). The pressure of this point in the three directions is p, f, and τ; the differential volume element ABCDEF is used to calculate the forming force, and the polar angle and polar radius of the element are defined as θ and ρ; the length of AB and the angle of BC are defined as dρ and dθ; the angle of AB is defined as κ, and the lengths of BF and AE are defined as z and dz+z, respectively; The normal forming force and tangential friction force acting on the upper surface ABCD are expressed by formula (17): pdA=ρdθ / cosκdρ, fdA=fdθ / cosκdρ ​​(17) Where dA represents the area of ​​surface ABCD; its geometric relationship is expressed by equation (18): tanκ=dz / ρ / dθ (18) The normal force along the tangential direction is expressed by equation (19): F pt =pdAsinκ=pρdθdρdz / ρdθ=pdρdz (19) The friction force in the tangential direction is expressed by formula (20): F ft =-fdAcosκ=-fρdθdρ(1+cosκ) / cosκ≈-2fρdθdρ (20) Assuming that the pressure of the blank in the Z direction is constant, and defining σ and (σ+dσ) as the pressures of the surfaces ABEF and CDE'F', and defining the areas of the two surfaces as dB and (dB+ddB), respectively, the tangential deformation force is expressed by Equation (21): When cos(dθ / 2)≈1, equation (21) can be rewritten as equation (22): F dt =[σzdρ-(σ+dσ)(z+dz)dρ] (22) The force balance equation in the tangential direction is expressed by equation (23): F pt +F ft +F dt =pdρdz-2fρdθdρ+σzdρ-(σ+dσ)(z+dz)dρ=0 (23) Formula (23) can be rewritten as formula (24): Based on the theory of plastic deformation, the principal stress is defined as σ θ , σ ρ and p, the plasticity equation is expressed by equation (25): s θ -p=2ν,dσ θ =dp (25) Where ν is the plastic deformation of pure shear material, and the final force is expressed by equation (26): Since dz / z≈0, f=μν, the solution of Equation (26) is expressed by Equation (27): p=2fρθ / (h0-h)+p0 (27) Where p0 is the initial value of the final force; replace p=p0 and σ θ Substituting σ0 into equation (25), we get equation (28): p0=2ν+σ0 (28) Where σ0 = 2νΔh / (h0-h) is the stress of the two tangential edge elements, and the friction stress and friction coefficient are defined as f = μν and μ, respectively. Equation (29) can be obtained: p=2uvρθ / (h0-h)+(2ν+1)Δh / (h0-h) (29) The total forming force is expressed by formula (30): The equivalent action point of the normal force is expressed by formula (31): Where, bite angle can be eliminated; the forming force vector and the corresponding action point are expressed by formula (32):

7. The compliant control method for a heavy-load plastic forming robot according to claim 6, characterized in that: In step S303, the friction forming force and its action point are expressed by formula (33):

8. The compliant control method for a heavy-load plastic forming robot according to claim 7, characterized in that: In step S304, the forming force is identified in real time by formula (34): Where, F wr and T wr It is the real-time forming force.

9. The compliant control method for a heavy-load plastic forming robot according to claim 8, characterized in that: In step S305, when the feed speed is Δh', the parameters at the next moment are expressed by equation (35): When the forward feed speed Δh is given f After that, the forward forming load at the next moment is expressed by formula (36):

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