Compliant control method for heavy-load plastic forming robot

Abstract

The invention relates to a compliance control method for a heavy-load plastic forming robot. The compliance control method comprises the following steps: S1, establishing a kinematic model of the heavy-load plastic forming robot; s2, establishing a dynamic model of the heavy-load plastic forming robot; s3, a forming load real-time prediction model of the heavy-load plastic forming robot is established; and S4, establishing a compliance control model of the heavy-load plastic forming robot. According to the method, the approximate linear relation between the normal forming force and the contact area is disclosed, it is assumed that the friction forming force parameter in the continuous control period is constant, the forming load real-time prediction model is established, and stress can be rapidly predicted. Through real-time forming load calculation, forming feeding speed feedforward and servo motor three-loop control, the compliance control method for the heavy-load plastic forming robot is constructed, and the control precision of the heavy-load plastic forming robot can be effectively improved.

Description

Technical Field

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

[0002] Thin-walled high-rib components are commonly used load-bearing structures in aerospace equipment. In order to achieve the best performance of thin-walled high-rib components during the forming process, a new envelope forming equipment using a parallel motion mechanism was developed for processing thin-walled high-rib components. The equipment can simultaneously control tool movement and forming force. However, due to the complexity of metal flow and deformation during plastic forming, it is not enough to use a standard spring-damper system to model the interaction between the parallel motion mechanism and its environment, making the traditional flexibility control method invalid, which in turn affects the forming accuracy of thin-walled high-rib 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 the technical problem is: construct a compliant control method for a heavy-load plastic forming robot, the robot includes a machine tool, a guide rail, a fast slider and a six-degree-of-freedom parallel motion mechanism, the parallel motion mechanism includes a moving platform and six independent motion branches, and is suspended on a driving slider, each branch is connected by a servo motor, a ball screw, an upper S joint, a connecting rod, and a lower S joint in sequence, and is fixed on the moving platform, the envelope mold is fixed on the moving platform, the lower mold and the blank are fixed on the machine tool, and the moving platform realizes six-degree-of-freedom spatial forming motion by controlling the six servo motors. The control method includes the following steps:

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

[0006] S2. Establish the dynamic model of 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 scheme, in step S1, the method for establishing the kinematic model of the heavy-load plastic forming robot includes:

[0010] 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, B1~B6 are the center points of the upper S joint; 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 length of the connecting rod, l1, l2, l3 are the distances between the coordinate systems S0 and S1, S1 and S2, and S3 and S4 respectively;

[0011] The coordinate transformation matrix from coordinate system S3 to S0 is expressed 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 expressed by formula (2):

[0015]

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

[0017] Establish the 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 connecting rod's center of mass 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 equation (4):

[0020]

[0021] In the formula, 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 equation (5):

[0023]

[0024] In the formula, 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] In the formula, 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 equation (8):

[0031]

[0032] In the formula, 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] In the formula, 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 kinetic 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 scheme, the method for performing equivalent contact surface analysis in 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 plane on the billet is expressed by equation (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 It is set as an Archimedean spiral curve and is 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 scheme, 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 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), its polar coordinate value is (θ, ρ), and the pressure of this point in 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 equation (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 tangent 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 equation (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), 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 =σ0Substitute into formula (25), and we get formula (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, and 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 equation (31):

[0089]

[0090] In the formula, the 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] In the formula, 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 equation (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 scheme, 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 equation (37):

[0104]

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

[0106]

[0107] S402, forming feed speed feedback method;

[0108] Input is the real-time position of the driving slider and real-time driving force Definition 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] In the formula, 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 speed 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 takes into account 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 establishes a real-time prediction model for forming load by revealing the approximate linear relationship between the normal forming force and the contact area and assuming that the friction forming force parameters are constant within a continuous control cycle, 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 schematic diagram of the heavy-duty plastic forming robot;

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

[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 It is the structural diagram of the forming feed amount feedforward method;

[0132] Figure 8 It 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] Fig.10 It is the position and control error of the driving slide in the aircraft bevel gear forming process;

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

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

[0137] Fig.13 It is the driving slide position and driving slide control error diagram in aircraft window frame forming;

[0138] Fig.14 It is a tracking error diagram of different tool point positions in 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, a guide rail, a fast slider and a six-degree-of-freedom parallel kinematic mechanism. The parallel kinematic mechanism includes a moving platform and six independent motion branches, which are suspended on the driving slider. Each branch is connected by a servo motor, a ball screw, an upper S joint, a connecting rod, and a lower S joint in turn, and is fixed on the moving platform. The enveloping mold is fixed on the moving platform, and the lower mold and the blank are fixed on the machine tool. Through the precise control of the six servo motors, the moving platform can realize the pre-designed six-degree-of-freedom spatial forming motion.

[0143] The schematic diagram of the heavy-duty 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, and B1~B6 are the center points of the upper S joint. 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 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 . 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] In the formula,

[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 a 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 connecting rod's center of mass 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. 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 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 can be obtained by equation (3):

[0151]

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

[0153]

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

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

[0156]

[0157] In the formula, 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] In the formula, 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 equation (8):

[0164]

[0165] In the formula, 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. pand I p are the mass matrix and inertia matrix of the moving platform, respectively, and are expressed by equation (9):

[0166]

[0167] In the formula, 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 expected 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-duty plastic forming robots

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

[0173] The enveloping die motion 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 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 of this point in 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 equation (17):

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

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

[0191] tanκ=dz / ρ / dθ (18)

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

[0193] Fpt =pdAsinκ=pρdθdρdz / ρdθ=pdρdz (19)

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

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

[0196] Assuming that the pressure of the blank in the Z direction is constant, and defining σ and (σ+dσ) as the pressure of the surfaces ABEF and CDE'F', and defining the areas of the two surfaces as dB and (dB+ddB), the tangential deformation force can be expressed 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 expressed by equation (23):

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

[0202] Formula (23) can be rewritten as formula (24):

[0203]

[0204] Based on the theory of plastic deformation, the principal stress is defined as σ θ , σ ρ and p, so the plasticity equation is expressed by equation (25):

[0205] σ θ -p=2v,dσ θ =dp (25)

[0206] Where v is the plastic deformation of pure shear resistant material. Therefore, the final force can be expressed 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 =σ0 Substituting into equation (25), we can get equation (28):

[0211] p0=2v+σ0 (28)

[0212] In the formula, σ0 = 2νΔh / (h0-h) is the stress of the two edge elements in the tangential direction. In addition, the friction stress and friction coefficient are defined as f = μν and μ, 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] In the formula, the bite angle can be eliminated. The forming force vector and the corresponding action point are expressed by equation (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 identified in real time by formula (34):

[0225]

[0226] In the formula, 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 speed 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 Establishment of Compliant Control Model for Heavy-Load Plastic Forming Robot

[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 equation (38):

[0237]

[0238] 2. Forming feed speed feedback method

[0239] This method is Figure 7 As shown, the input is the real-time position of the driving slider and real-time driving force Definition 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] In the formula, 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 by formula (40):

[0243]

[0244] 3. Soft control method

[0245] Based on the above analysis, a compliant control method for a heavy-load plastic forming robot is proposed. Figure 8 As shown, 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 expected motion trajectory of the envelope die, the kinematic solution of the heavy-duty plastic forming robot, i.e., the expected trajectory of the driving slider, can be obtained according to the method provided by the present invention. According to the method provided by the present invention, the predicted value of the forming force of the heavy-duty plastic forming robot can be obtained, and combined with kinematics and dynamics, real-time feedback can be provided in the compliant control method.

[0253] For aircraft bevel gears, in order to maximize the forming efficiency, the required force is set to 500 tons, the target feed distance is set to 28 mm, and the initial and maximum feed rates are both set to 7.5 mm / s. The real-time feed rate and feed distance are represented by the blue and red lines in Figure 9(a), respectively, and the required force and actual forming force are represented by the blank dashed line and red solid line in Figure 9(b). It can be found that when the feed distance changes from 0 to 28 mm, the feed rate decreases from 7.5 mm / s to zero. Before the 2nd second mark, the feed rate remains unchanged at the 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 envelope die and the blank are in contact at a feed distance of 15 mm. The stabilization time of the control system after contact is about 0.1 seconds, and the overshoot of the forming force is about 70 tons. The process ends at about 9.1 seconds, and the feed distance is close to 28 mm. In the steady-state stage, the forming force error is about 50 tons. Fig.10 In the figure, the dotted line indicates the position of the slider, the solid line indicates the position error, the left and right y axes indicate the size of the slider position and position error, and the six sub-figures indicate the position and error of the corresponding slider with lines of different colors. It can be found that the slider moves back and forth, and the center position decreases with the feed. In the range of -200 to 200um, the position error increases with the increase of the feed distance. Fig.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 is set to 27 mm, and the initial and maximum feed speeds are both set to 7.5 mm / s. The real-time feed speed and feed distance are shown in Figure 12(a) with blue and red lines, respectively. Figure 12(b) shows the required and actual forming forces with blank dashed lines and red solid lines. It can be found that when the feed distance changes from 0 to 27 mm, the feed speed decreases from 7.5 mm / s to 0. Before reaching the feed distance of 17 mm, the feed speed remains unchanged at the maximum value of 7.5 mm / s, and the forming force remains zero. After reaching the feed distance of 17 mm, the forming force increases to 150 tons, while the feed speed decreases from 2 to 1.5 mm / s. After reaching the feed distance of 20 mm, the required force increases to 350 tons, and the feed speed first increases to 4.4 mm / s and then decreases to 2.5 mm / s. The settling time is about 0.7 seconds, and the overshoot of the forming force is about 25 tons. After reaching a 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. Fig.13As shown in the figure, the dotted line represents the position of the driving slider, the solid line represents the position error, the left and right y axes represent the position size and position error respectively, and the six sub-figures use lines of different colors to represent the position and error of the corresponding driving slider. It can be found that the driving slider moves back and forth, and the center position decreases with the feed. In the range of -200 to 200um, the position error increases with the increase of feed. Fig.14 The maximum trajectory error and average trajectory error of each point in the aircraft window frame forming process are shown in Figure 1. The maximum trajectory error can reach 97um. It can be found that the compliant control method of 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 modes, which are merely illustrative rather than 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 within the protection of 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 motion mechanism, the parallel motion mechanism comprising a moving platform and six independent motion branches, and 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 on the moving platform, the envelope mold being fixed on the moving platform, the lower mold and the blank being fixed on the machine tool, the moving platform realizing six-degree-of-freedom spatial forming motion by controlling the six servo motors, characterized in that: The control method comprises the following steps: S1. Establish the kinematic model of heavy-load plastic forming robot; S2. Establish the dynamic model of 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 heavy-load plastic forming robots.

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 the kinematic model of the heavy-load plastic forming robot includes: 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, B1~B6 are the center points of the upper S joint; 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 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 length of the connecting rod, l1, l2, l3 are the distances between the 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 equation (1): M 03 =M 02 M 24 M 43 (1) In the formula, 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 heavy-load plastic forming robot dynamics model includes: Establish the 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 connecting rod's center of mass 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 equation (4): In the formula, 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 equation (5): In the formula, 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): In the formula, 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 equation (8): In the formula, 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): In the formula, 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 kinetic 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 in 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 plane on the billet is expressed by equation (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 It is set as an Archimedean spiral curve and is 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 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), its polar coordinate value is (θ, ρ), and the pressure of this point in 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 equation (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 tangent direction is expressed by equation (19): F pt =pdA sinκ=pρdθdρdz / ρdθ=pdρdz (19) The friction force in the tangential direction is expressed by equation (20): F ft =-fdA cosκ=-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 pressure of the surface ABEF and CDE'F', and defining the areas of the two surfaces as dB and (dB+ddB), 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=μv, 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 =σ0Substitute into formula (25), and we get formula (28): p0=2ν+σ0 (28) 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, and 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 equation (31): In the formula, the 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 equation (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 equation (34): In the formula, 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):

10. The compliant control method for a heavy-load plastic forming robot according to claim 9, characterized in that: In step S4, the compliance control model includes: 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 equation (37): The actual detected forming force applied to the blank is expressed by formula (38): S402, forming feed speed feedback method; Input is the real-time position of the driving slider and real-time driving force Definition 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) In the formula, 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 speed 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.

Images