CN105159137B - A kind of series parallel type automobile electrophoretic coating conveyor structure dynamic modeling method - Google Patents
A kind of series parallel type automobile electrophoretic coating conveyor structure dynamic modeling method Download PDFInfo
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Abstract
The invention discloses a kind of dynamic modeling methods of series parallel type automobile electrophoretic coating conveyor structure, the symmetrical structure feature of mechanism is made full use of first, pass through the speed of each passive joint in the method analysis institution of analytic geometry, acceleration, then it introduces spinor theory and obtains the diarthrodial speed of each master of mechanism, acceleration, secondly, the kinetics equation of principle of virtual work set up mechanism Screw is applied on this basis, each active joint axial driving force of conveying mechanism of control can be directly realized by by being finally calculated, complete the structure of the kinetic model of achievable high performance control.Analytic geometry method, spinor theory are combined by the present invention with the principle of virtual work, it solves the problems, such as to need to be coordinately transformed because not having coordinate invariance when the modeling of spatial complex mechanism dynamic, computational complexity is reduced, while dynamic modeling method proposed by the present invention is relatively simple, form is neat, is easy to sequencing realization.
Description
Technical Field
The invention relates to a dynamic modeling method of a series-parallel automobile electrophoretic coating conveying mechanism, belonging to the field of hybrid/parallel robots.
Background
With the rapid development of the automobile manufacturing industry, automobile painting is developed from workshop type to modern industrial painting suitable for mass flow production. In the coating process, the automobile body electrophoretic coating mechanical conveying system penetrates through the whole process of an automobile coating production line and is a main artery of the coating production line. At present, a novel series-parallel type automobile electrophoretic coating conveying mechanism is proposed. The series-parallel automobile electrophoretic coating conveying mechanism has the advantages of a parallel mechanism and a series mechanism, and has the characteristics of large rigidity-weight ratio, small accumulated error, compact structure, strong bearing capacity and large working space. However, due to the adoption of the hybrid mechanism, from the control perspective, the novel conveying mechanism is a nonlinear and strongly-coupled multi-input multi-output system, and when the mechanism moves at a higher speed, the dynamic characteristics of the mechanism can greatly influence the motion control precision, so that in order to ensure that the conveying mechanism can stably, reliably and accurately operate under various working conditions and in various coating and conveying processes, a dynamic control method needs to be researched and designed aiming at the conveying mechanism control system.
When the dynamic control method is adopted, a dynamic model of the conveying mechanism needs to be established. The invention provides a method for establishing a novel conveying mechanism dynamic model by combining an analytic geometry method, a momentum theory and a virtual work principle. Compared with other dynamics modeling methods, such as a Newton-Euler equation method and a Lagrange method, the modeling method has obvious geometric characteristics, is low in computational complexity, neat and concise in expression form and easy to realize in a programmed manner, and the obtained dynamics equation is relatively simple and is convenient for efficiently realizing the high-performance control of the series-parallel automobile electrophoretic coating conveying mechanism.
The method provided by the invention is not reported in the same or similar methods after looking up literature data.
Disclosure of Invention
The invention aims to provide a dynamic modeling method for high-performance control, which aims to simplify the modeling process of a series-parallel automobile electrophoretic coating conveying mechanism and facilitate the more efficient implementation of dynamic control.
The technical scheme adopted by the invention is as follows: firstly, the symmetrical structure characteristics of the mechanism are fully utilized, the speed and the acceleration of each passive joint in the mechanism are analyzed by an analytic geometry method, then the speed and the acceleration of each active joint of the mechanism are obtained by introducing a momentum theory, on the basis, a dynamic equation in a momentum form of the mechanism is established by applying a virtual work principle, and further, the axial driving force of each active joint of the mechanism which can directly realize control is calculated.
The technical scheme of the invention comprises the following steps:
1) establishing a basic coordinate system { B } - { O-XYZ }, determining coordinates (xn, yn, zn) T of the center point of each kinematic pair of the conveying mechanism under the basic coordinate system { B }, and calculating the motion rotation of each joint of the mechanism;
2) obtaining the position relation of the passive joints of each branched chain of the mechanism according to the geometric relation, and determining the speed and the acceleration of the passive joints by derivation;
3) obtaining the speed and the acceleration of the active joint of each branched chain of the mechanism by adopting a momentum theory and combining the speed and the acceleration of the passive joint, and obtaining a speed Jacobian matrix J of the mechanism;
4) calculating the speed rotation, the acceleration rotation and the force rotation of each branched chain and the middle connecting assembly of the mechanism relative to a reference point;
5) and (3) obtaining a kinetic equation by utilizing a virtual work principle, thereby calculating a system generalized driving force vector tau of the mechanism, and obtaining an axial driving force vector Q of each driving joint of the mechanism by utilizing conversion of a Jacobian matrix J.
The invention has the advantages and positive effects that:
the invention mainly relates to a novel dynamic modeling method of a series-parallel automobile electrophoretic coating conveying mechanism, which can be used for designing and realizing high-performance control of the conveying mechanism based on a dynamic model, and provides a dynamic modeling method combining an analytic geometry method, a momentum theory and a virtual work principle, and has the advantages that:
(1) the method has the characteristics of coordinate invariance, neat and simple expression form, low calculation complexity, easy programming realization and the like;
(2) the whole system can be regarded as a whole, all counter forces do not need to be solved, and the obtained kinetic equation is relatively simple and is more beneficial to realizing control.
(3) Meanwhile, the dynamic modeling method provided by the invention is simple, neat in form and easy to realize in a programmed manner.
(4) Experiments prove that the driving force/moment change curve of each branched chain driving joint is obtained by adopting ADAMS virtual prototype simulation. Compared with an MATLAB simulation result, the average relative error of the driving force/moment of each branched chain obtained by ADAMS simulation is 9.7%, and the dynamic characteristics of the driving force/moment of each branched chain and the moment of each branched chain have high consistency, so that the accuracy and the reliability of the novel automobile electrophoretic coating conveying mechanism dynamic model established by combining an analytic geometry method, a momentum theory and a virtual work principle are shown.
Drawings
FIG. 1 is a schematic diagram of a series-parallel automotive electrophoretic coating conveying mechanism according to an embodiment of the present invention;
fig. 2 is a schematic structural diagram of a novel automobile electrophoretic coating conveying mechanism, wherein: FIG. 2(a) is a schematic diagram of the first, second, third, and fourth branched structures; FIG. 2(b) is a schematic diagram of the structures of fifth and sixth branched chains;
FIG. 3 is a schematic structural view of a first plane multi-bar mechanism in the novel automobile electrophoretic coating conveying mechanism structure;
FIG. 4 is a graph of the driving force/torque of each active joint of the novel automobile electrophoretic coating conveying mechanism obtained by MATLAB software simulation along with time; wherein: FIG. 4(a) is a graph of the driving force of the first, second, third and fourth branched active joints of the mechanism changing with time, and FIG. 4(b) is a graph of the driving torque of the fifth and sixth branched active joints of the mechanism changing with time;
FIG. 5 is a graph of the driving force/torque of each branched active joint versus time obtained by ADAMS virtual prototype simulation, wherein: fig. 5(a) is a graph showing the change of the driving force of the first, second, third and fourth branched chain braking joints of the mechanism along with time, and fig. 5(b) is a graph showing the change of the driving torque of the fifth and sixth branched chain active joints of the mechanism along with time.
In the figure: 1-a first driver, 2-a first speed reducer, 3-a first guide rail, 4-a first lead screw, 5-a first nut, 6-a first rotating pair, 7-a first lead screw seat, 8-a second driver, 9-a second speed reducer, 10-a second guide rail, 11-a second lead screw, 12-a second nut, 13-a second rotating pair, 14-a second lead screw seat, 15-a third rotating pair, 16-a third driver, 17-a driving wheel, 18-a transmission belt, 19-a driven wheel, 20-an intermediate connecting rod, 21-a walking driver, 22-a guide wheel, 23, 24-a walking wheel, 25-a base and 26-a guide rail.
Detailed Description
The technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
The invention provides a novel series-parallel automobile electrophoretic coating conveying mechanism dynamics modeling method, which comprises the following steps:
firstly, establishing a basic coordinate system { B } - { O-XYZ }, and determining coordinates (x) of the central point of each kinematic pair of the conveying mechanism under the basic coordinate system { B }, wherein the coordinates are as followsTi，yTi，zTi)T(units are m), wherein xTiIs the value of the coordinate x, y of the kinematic pair center point under the base coordinate system { B }TiIs the value of the coordinate y, z, of the kinematic pair center point in the base coordinate system BTiThe kinematic pair center point is the value of coordinate z under the base coordinate system { B }. Then according to the momentum theory, calculating the motion momentum $ of each joint of each branched chain under a basic coordinate system { Bij(i denotes the ith branch of the mechanism and j denotes the jth joint of the corresponding branch).
Secondly, obtaining the position relation of the passive joint of each branched chain of the mechanism according to the geometric constraint relation, and determining the speed and the acceleration of the passive joint by derivation:
in the formulas (1) and (2),(rad/s) is the speed of the jth passive joint of the ith branched chain;(unit is rad/s)2) The acceleration of the jth passive joint of the ith branched chain is obtained; q is a system generalized coordinate; gijIs a first order motion influence coefficient; hijIs a second order motion influence coefficient.
Thirdly, by utilizing the momentum theory, because the first branched chain and the second branched chain form a closed chain system, an equation is obtained:
in the formulas (3) and (4),(the unit is m/s) is the active joint speed in the first branched chain and the second branched chain respectively;(the units are each m/s2) Respectively the acceleration of the active joint in the first branched chain and the second branched chain; $ fit for the treatment of diabetesRi(i-1, 2) is an additional term,wherein [,]indicating a curly bracket operation.
Equations can be obtained by other branched chains, so that the speed and the acceleration of the active joint of each branched chain are obtained through calculation:
in the formulas (5) and (6), i represents the ith branched chain of the mechanism;(m/s) is the active joint velocity in the ith branch;(unit is m/s)2) The acceleration of the active joint in the ith branched chain is obtained;in the form of a system's generalized velocity vector,is a system generalized acceleration vector; j. the design is a squareiIs a first order motion influence coefficient, HiIs a second order motion influence coefficient.
And combining the analysis to obtain a mechanism speed Jacobian matrix J.
Fourthly, calculating the speed rotation and the acceleration rotation of each branched chain and the intermediate connecting rod assembly of the conveying mechanism relative to the corresponding reference point by using a rotation theory, and obtaining an inertia force calculation formula in a rotation form on the basis according to the Dalnbell principle, thereby obtaining the external force rotation of each branched chain and the intermediate connecting assembly of the conveying mechanism relative to the corresponding reference point.
And calculating the velocity momentum V and the force momentum F of each branched chain of the system. According to the momentum theory, the velocity momentum and the acceleration momentum of the ith branched chain in the mechanism by taking the tail end of the branched chain as a reference point under a basic coordinate system are as follows:
in the formulae (7) and (8), ViThe velocity vector of the ith branched chain taking the tail end of the branched chain as a reference point under a basic coordinate system; a. theiThe acceleration momentum of the ith branched chain taking the tail end of the branched chain as a reference point under a basic coordinate system;(rad/s) is the velocity of the jth passive joint of the ith branch.
Assuming that the mass of the rigid body is m (unit is kg), r is a vector of a point O on the rigid body pointing to a centroid C, and according to the darenbell principle, the moment of inertia of the rigid body with O as a reference point can be expressed as:
in the formula, VO、AORespectively, velocity momentum, acceleration momentum of rigid body using O as reference point, d (A)O) Is AOEven vector of (A), p (A)O) Is AOThe other is analogized in the same way; i isOIs an inertia matrix of a rigid body under a basic coordinate system relative to a point O,wherein,is an inertia matrix R under a rigid body mass center coordinate system (a coordinate system which takes the rigid body mass center as an origin and is established by coordinate axes along the main axis direction of the rigid body)OAs coordinates of the centroid of a rigid bodyAn attitude transformation matrix tied to the base coordinate system.
Calculating according to the formula (9) to obtain the force rotation quantity of the intermediate connecting rod component of the mechanism as follows:
in the formula, Vp、ApRespectively, the speed rotation and the acceleration rotation of the mechanism intermediate connecting rod component by taking P as a reference point, d (A)P) Is APEven vector of (A), p (A)P) Is APThe other is analogized in the same way; i isPIs an inertial matrix of the mechanism intermediate link rod assembly with respect to point P in the base coordinate system,wherein,is an inertia matrix R under the coordinate system of the center of mass of the middle connecting rod component (the coordinate system which takes the center of mass of the middle connecting rod component as the origin and is established by coordinate axes along the main axis direction of the rigid body)PAnd (3) an attitude transformation matrix from the center-of-mass coordinate system of the middle connecting rod assembly to the basic coordinate system.
And (3) calculating according to the formula (9) to obtain the force rotation quantity of each branched chain component of the mechanism as follows:
in the formula, Vi、AiThe speed rotation and the acceleration rotation of the ith branched chain component of the mechanism are respectively; i isiIs an inertia matrix of the ith branched chain component of the mechanism relative to a reference point under a basic coordinate system,wherein,is an inertia matrix R under the i-th branched chain component mass center coordinate system (a coordinate system which takes the i-th branched chain component mass center of the mechanism as the original point and is established by coordinate axes along the main axis direction of a rigid body)iAnd (4) an attitude transformation matrix from the i-th branched chain assembly centroid coordinate system to the basic coordinate system.
Fifthly, by utilizing the conclusion that the result of reciprocal product operation of the motion momentum and the force momentum in the momentum theory shows the instantaneous power of the force in the motion direction, the momentum theory and the virtual work principle are tightly combined to obtain a kinetic equation of the momentum form of the conveying mechanism, so that a system generalized driving force vector tau of the conveying mechanism is obtained, and an axial driving force vector Q of each driving joint of the conveying mechanism is obtained by utilizing the Jacobian matrix J for conversion.
Obtaining a kinetic equation by utilizing a virtual work principle;
in the formula, FPFor the amount of force rotation of the intermediate link rod assembly of the mechanism, FiThe force rotation quantity of the ith branched chain component of the mechanism; vPSpeed rotation, V, of the mechanism intermediate link rod assembly with P as reference pointiThe speed rotation of the ith branched chain component of the mechanism; tau is an equivalent generalized force vector; omicron represents the reciprocal product.
The equivalent generalized force vector and the axial driving force vector have the following relationship:
τ＝JTQ (13)
in the formula, tau is an equivalent generalized force vector; q is the axial driving force vector of the driving joint of the mechanism; j is the mechanism velocity Jacobian matrix.
The axial driving force of the driving joint of the mechanism can be obtained:
Q＝(JT)+τ
MATLAB and ADAMS software are used for respectively simulating the established kinetic model, and the correctness and reliability of the kinetic modeling method provided by the invention are verified through comparison of simulation results.
Example (b):
the mechanism shown in fig. 1 is a series-parallel automobile electrophoretic coating conveying mechanism, and the mechanism comprises a traveling mechanism and a lifting turnover mechanism. The walking mechanism realizes the walking and transporting functions of the conveying mechanism through the matching of the walking driver 21, the guide wheel 22, the walking wheel 23, the walking wheel 24 and the guide rail 26. The frame of the lifting turnover mechanism is a moving part of the walking mechanism, and the lifting turnover mechanism mainly comprises two groups of same plane multi-rod mechanisms. The plane multi-rod mechanism comprises three branches, wherein the first branch comprises: the device comprises a first driver 1, a first speed reducer 2, a first guide rail 3, a first lead screw 4, a first nut 5, a first rotating pair 6 and a first lead screw seat 7; the first driver 1 is fixedly arranged on the first speed reducer 2, the first guide rail 3, the first lead screw seat 7 and the first speed reducer 2 are fixed with each other, one end of the first lead screw 4 is driven by the first driver 1 through the first speed reducer 2, meanwhile, the other end of the first lead screw 4 is supported on the first lead screw seat 7, the first nut 5 is arranged on a base 25 of the travelling mechanism through a first revolute pair 6, the first nut 5 and the first lead screw 4 form a screw pair, and meanwhile, the first nut 5 and the first guide rail 3 form a translation pair; the second branch comprises: the second driver 8, the second reducer 9, the second guide rail 10, the second lead screw 11, the second nut 12, the second revolute pair 13 and the second lead screw base 14; the second driver 8 is fixedly arranged on the second speed reducer 9, the second guide rail 10, the second lead screw seat 11 and the second speed reducer 9 are fixed with each other, one end of the second lead screw 11 is driven by the second driver 8 through the second speed reducer 9, meanwhile, the other end of the second lead screw 11 is supported on the second lead screw seat 14, the second nut 12 is arranged on a base 25 of the travelling mechanism through a second revolute pair 13, the second nut 12 and the second lead screw 11 form a screw pair, and meanwhile, the second nut 12 and the second guide rail form a translation pair; the first branch and the second branch are connected through a third revolute pair 15. The third branch comprises: a third driver 16, a driving wheel 17, a transmission belt 18 and a driven wheel 19; the third driver 16 is fixedly installed on the second speed reducer 9, the driving wheel 17 is installed on the second speed reducer 9 through a rotating pair, the driven wheel 19 is installed on the second screw seat 14 through a rotating pair, and the driving wheel 17 driven by the third driver 16 drives the driven wheel 19 to rotate through the transmission belt 18. The two groups of plane multi-rod mechanisms are connected through a middle connecting rod 20, and an automobile body is fixed on the middle connecting rod 20 through a fixing frame.
The method for modeling the dynamics of the novel series-parallel automotive electrophoretic coating conveying mechanism provided by the invention is further described by taking the novel series-parallel automotive electrophoretic coating conveying mechanism as shown in fig. 2 as an example, with reference to fig. 1. The mechanism mainly comprises a walking mechanism and a lifting turnover mechanism. The walking mechanism part and the lifting turnover mechanism part are mutually independent, wherein the lifting turnover mechanism is a main mechanism of the mechanism, and the embodiment mainly aims at the lifting turnover mechanism part. The lifting turnover mechanism is provided with 6 branched chains, wherein the first, second, third and fourth chains are the same PRR (P represents a moving pair, and R represents a rotating pair) kinematic chain, and the fifth and sixth branched chains are the same R kinematic chain. The first, second, third and fourth branch chains are all provided with lead screws, and the lead screws are driven by an alternating current servo motor to realize linear movement through nut matching. And the driving wheel in the fifth branched chain and the sixth branched chain drives the driven wheel to rotate through the transmission belt, and the driving wheel is directly driven by the overturning driving motor. Specific parameters of the series-parallel automobile electrophoretic coating conveying mechanism are shown in table 1.
TABLE 1 parameters of novel series-parallel automobile electrophoretic coating conveying mechanism
The specific implementation of the modeling method of the invention is as follows:
1. as shown in fig. 2, a base coordinate system { B } - { O-XYZ }, with an origin O located atB1And B2The Z axis is perpendicular to the base feature plane and the Y axis is along P1P2in the direction, because the parallel mechanism movable platform only has three degrees of freedom of movement along the direction X, Y and rotation around the Y axis, the position and posture parameters of the middle connecting rod are selected to be q ═ x, z, β]T(x and z are m, respectively, and β is rad).
From the analysis of FIG. 2, it can be seen that the hinge point B between the nut and the base in the first, second, third and fourth branchesi(i ═ 1,2, 3, 4) coordinates in the basic coordinate system and the hinge points P between the two ends of the intermediate connecting rod and the respective branchesi(i ═ 1, 2) the coordinates in the base coordinate system are:
P1＝(x 0 z)T、P2＝(x h z)T
wherein L is1(m) is the distance between the center points of the revolute pairs connected with the first branched chain and the second branched chain and the base; l is2(m) is the distance between the center points of the revolute pairs connected with the third branched chain and the fourth branched chain and the base; h (in m) is the length of the intermediate connecting rod.
Because the first plane multi-rod mechanism composed of the first, second and fifth branched chains is symmetrical to the second plane multi-rod mechanism composed of the third, fourth and sixth branched chains, in order to simplify the analysis, only the first plane multi-rod mechanism is subjected to the kinematic analysis, while the kinematic problem of the second plane multi-rod mechanism is directly obtained from the symmetrical relation, and the schematic diagram of the first plane multi-rod mechanism is shown in fig. 3.
According to the rotation theory, under the basic coordinate system, the KP rotation of each branched chain of the first planar multi-bar mechanism can be expressed as:
S5＝($51＝(0 1 0 -z 0 x)T)
wherein S isi(i ═ 1,2, 5) are the first, second, and fifth branch KP helicities, respectively; $ represents the corresponding amount of motion rotation;
2. at theta1i(unit is rad) denotes ∠ OB in FIG. 3iP1(i＝1，2)，θ2i(unit is rad) denotes ∠ OP in FIG. 21Bi(i is 1, 2), and the position relation of each branched chain passive joint of the first plane multi-bar mechanism is obtained by the geometrical relation as follows:
and (3) obtaining the speed and the acceleration of the first and second branch chain passive joints by using the derivation of the formulas (14) and (15):
wherein,(unit is rad/s) is the angular speed of a rotating pair connecting the nut and the base in the ith branched chain;(unit is rad/s) is the angular speed of a rotating pair connecting the tail end of the lead screw and the middle connecting rod in the ith branched chain;(unit is rad/s)2) The angular acceleration of a rotation pair connecting the nut and the base in the ith branched chain is respectively measured;(unit is rad/s)2) The angular acceleration of a rotating pair connecting the tail end of a lead screw and the middle connecting rod in the ith branched chain is respectively; g1i＝(Gi1Gi20)，H1iFor the second-order motion-influencing coefficients, is the generalized speed of the central point of the middle connecting rod,
3. since the first branch chain and the second branch chain form a closed chain system, P1Is the common end of the first branch chain and the second branch chainThen, according to the theory of the rotation quantity, an equation is obtained:
in the formulae (17) and (18),(the unit is m/s) respectively represents the linear moving speed of the lead screw in the first branched chain and the second branched chain; (the units are each m/s2) Linear moving acceleration of the lead screw in the first branched chain and the second branched chain respectively; $ fit for the treatment of diabetesR1、$R2In order to add the items to the list,wherein [,]indicating a curly bracket operation.
Solving the following equations (16) and (17):
wherein, Ji＝(Ji1Ji20)T，i＝1，2， HiIs a second order motion-influencing coefficient matrix and is expressed as
Wherein
For the fifth branch, there are
In the formulae (18) and (19),(rad/s) represents the driving wheel angular velocity in the fifth branch;(unit is rad/s)2) Representing the driving wheel angular acceleration in the fifth branch.
The speeds of all joints in the second plane multi-rod mechanism are obtained by the symmetrical relation. The analysis is combined to obtain a velocity Jacobian matrix of the mechanism driving joint as follows:
J＝[J1J2J1J2J5J5]T
4. the moving parts of the mechanism are divided into an intermediate connecting rod component and a branched chain component, wherein the intermediate connecting rod component comprises an intermediate linkThe connecting rod and a fixed bracket fixedly connected with the connecting rod, and the central point P of the middle connecting rod is a reference point of the component. According to the movement form of each component in the first, second, third and fourth branched chains, the first, second, third and fourth branched chain components are decomposed into branched chain body components and screw rod components, and B is controlledi(i ═ 1,2, 3, 4) is the reference point for the branched chain body assembly and the lead screw assembly. The velocity spin V and force spin F are calculated.
The instantaneous speed momentum, the instantaneous acceleration momentum and the force momentum of the middle connecting rod assembly with P as a reference point are respectively as follows:
wherein m isp(in kg) is the mass of the intermediate connecting rod assembly; g is a gravity coefficient vector, g ═ 009.8067T；rP＝(-rcsinβ 0 -rccosβ)TVector pointing to the center of mass C of the intermediate link assembly for point P, rc(in m) is the distance P from the center of mass C of the intermediate link assembly; i ispIs an inertial matrix of the intermediate link assembly in the base coordinate system.
Using a kinematic model to characterize the first and second branched chain body components in BiThe instantaneous speed momentum, the instantaneous acceleration momentum and the force momentum of the reference point (i is 1, 2) are respectively:
wherein m isi(in kg) are the masses of the ith branched assembly, respectively; r isLi(in m) are each Bi(i ═ 1, 2) a vector pointing to the centroid of the ith branched chain body assembly; i isLiRespectively, the inertia matrixes of the ith branched chain body assembly under the basic coordinate system.
The instantaneous speed momentum, the instantaneous acceleration momentum and the force momentum of the first branched chain lead screw assembly and the second branched chain lead screw assembly when rotating around the axis of the first branched chain lead screw assembly and the second branched chain lead screw assembly are respectively as follows:
wherein m isSi(kg) is the mass of the screw in the ith branched chain body component; s (unit is m) is a lead screw lead; i isSiAnd (i is 1 and 2) are inertia matrixes of the screw rod in the ith branched chain body assembly under a basic coordinate system respectively.
For the fifth branch chain of the lifting turnover mechanism, the instantaneous speed momentum, the instantaneous acceleration momentum and the force momentum with the central point of the driving wheel as a reference point are respectively as follows:
wherein,rdathe unit is m, which is a vector from the center point of the driven wheel to the center point of the driving wheel; m is5(in kg) is the mass of the fifth branched chain body component; i isL5Is an inertia matrix of the fifth branched chain body component under the basic coordinate system.
5. By τ ═ (τ)1τ2τ3)TRepresenting the generalized equivalent of the mechanism, the kinetic equation is derived from the principles of virtual work in conjunction with the above analysis:
the axial driving force of the lifting turnover mechanism can be obtained through a combined type (20):
Q＝(JT)+τ (21)
wherein Q ═ F1F2F3F4τ5τ6)TIn which F is1、F2、F3、F4(the unit is N) respectively represents the axial driving force of the first, second, third and fourth branched chain active joints of the mechanism, and tau5、τ6(the unit is N.m) respectively represents the axial driving torque of the fifth branch chain driving joint and the sixth branch chain driving joint of the mechanism, (J)T)+Is a matrix JTPseudo-inverse of (J)T)+＝J(JTJ)-1。
And (3) setting an expected motion track equation of the conveying mechanism as shown in a formula (22), and obtaining the motion speed and the acceleration of the mechanism by calculating a first derivative and a second derivative of time through the track equation. A diagram of the change of the driving force/moment of each active joint of the novel automobile electrophoretic coating conveying mechanism along with time is obtained by simulating by using MATLAB software and is shown in figure 4.
In order to further verify the correctness and reliability of the established dynamic model, an ADAMS virtual prototype is adopted for simulation, and a driving force/moment change curve of each branched chain driving joint is obtained and is shown in figure 5. Compared with an MATLAB simulation result, the average relative error of the driving force/moment of each branched chain obtained by ADAMS simulation is 9.7%, and the dynamic characteristics of the driving force/moment of each branched chain and the moment of each branched chain have high consistency, so that the accuracy and the reliability of the novel automobile electrophoretic coating conveying mechanism dynamic model established by combining an analytic geometry method, a momentum theory and a virtual work principle are shown.
To sum up, the invention discloses a dynamic modeling method of a series-parallel automobile electrophoretic coating conveying mechanism, which comprises the steps of firstly, fully utilizing the symmetrical structural characteristics of the mechanism, analyzing the speed and the acceleration of each passive joint in the mechanism by a method of analyzing geometry, then, introducing a momentum theory to obtain the speed and the acceleration of each active joint of the mechanism, secondly, establishing a dynamic equation of a mechanism momentum form by applying a virtual work principle on the basis, and finally, calculating to obtain the axial driving force of each active joint of the conveying mechanism, which can directly realize control, so as to complete the construction of a dynamic model capable of realizing high-performance control. The invention combines the analytic geometry method, the momentum theory and the virtual work principle, solves the problem that the coordinate transformation is needed because the spatial complex mechanism dynamics modeling does not have coordinate invariance, reduces the computational complexity, and simultaneously, the dynamics modeling method provided by the invention is simpler, neat in form and easy to realize by programming.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
Claims (5)
1. A dynamic modeling method of a series-parallel automobile electrophoretic coating conveying mechanism is characterized by comprising the following steps:
1) establishing a three-dimensional coordinate system { B } - { O-XYZ }, and determining the coordinates (x) of the central point of each kinematic pair of the conveying mechanism in the three-dimensional coordinate system { B }, wherein the central point of each kinematic pair is located in the three-dimensional coordinate system { B }, and the central point of each kinematic pair is located in the three-n,yn,zn)TCalculating the motion rotation amount of each joint of the conveying mechanism;
2) obtaining the position relation of the driven joints of the branched chains of the conveying mechanism according to the geometric relation, and determining the speed and the acceleration of the driven joints of the conveying mechanism by derivation;
3) describing instantaneous motion of the tail end of each branched chain of the conveying mechanism relative to a three-dimensional coordinate system by using a momentum motion equation in a momentum theory, so as to respectively obtain a speed equation and an acceleration equation by combining the structural characteristics that each branched chain of the conveying mechanism has a common tail end, then solving the equations to obtain the speed and the acceleration of the driving joint of each branched chain of the conveying mechanism, and obtaining a speed Jacobian matrix J of the conveying mechanism;
4) calculating the speed rotation and the acceleration rotation of each branched chain and the intermediate connecting rod assembly of the conveying mechanism relative to the corresponding reference point by using a rotation theory, and obtaining an inertia force calculation formula in a rotation form on the basis according to the Dalnbell principle so as to obtain the force rotation of each branched chain and the intermediate connecting rod assembly of the conveying mechanism relative to the corresponding reference point;
5) the method is characterized in that the conclusion that the instantaneous power of the inertia force in the motion direction is expressed by the result of reciprocal product operation of the motion momentum and the force momentum in the momentum theory is utilized, the momentum theory and the virtual work principle are tightly combined to obtain a kinetic equation of the momentum form of the conveying mechanism, so that a system generalized driving force vector tau of the conveying mechanism is obtained, and an axial driving force vector Q of each driving joint of the conveying mechanism is obtained by conversion of a Jacobian matrix J.
2. The dynamics modeling method of the series-parallel automobile electrophoretic coating conveying mechanism according to claim 1, characterized in that: in the step 1), a three-dimensional coordinate system { B } ═ O-XYZ } is established, and an origin O is located at a hinge point B between the nut and the base in the first branch chain1And a hinge point B of the nut and the base in the second branch chain2The Z axis is perpendicular to the base feature plane and the Y axis is along P1P2the directions P1 and P2 are hinge points of two ends of the middle connecting rod and each branched chain, and the parallel mechanism moving platform of the conveying mechanism only has three degrees of freedom of movement along the X, Y direction and rotation around the Y axis, so the posture parameter of the middle connecting rod is selected to be q ═ x, z, β]Tx and z are x-axis and z-axis three-dimensional coordinate system coordinates respectively, the units of x and z are m respectively, β is the angle of the parallel mechanism moving platform rotating around the Y axis, and the unit of β is rad.
3. The dynamics modeling method of the series-parallel automobile electrophoretic coating conveying mechanism according to claim 1, characterized in that: and 2) determining the speed and the acceleration of each passive joint of the conveying mechanism by using a geometric analysis method in the step 2).
4. The dynamics modeling method of the series-parallel automobile electrophoretic coating conveying mechanism according to claim 1, characterized in that: the screw amount of the middle connecting rod assembly in the step 4) relative to the corresponding reference point is as follows:
in the formula, Vp、ApRespectively, the speed rotation and the acceleration rotation of the mechanism intermediate connecting rod component by taking P as a reference point, d (A)P) Is APEven vector of rpVector pointing to the center of mass C of the intermediate link rod assembly for point P, P (A)P) Is APThe other is analogized in the same way; i isPIs an inertial matrix of the mechanism intermediate link rod assembly in relation to point P in a three-dimensional coordinate system,wherein,is an inertia matrix R under the coordinate system of the center of mass of the middle connecting rod component (the coordinate system which takes the center of mass of the middle connecting rod component as the origin and is established by coordinate axes along the main axis direction of the rigid body)PAn attitude transformation matrix from a center-of-mass coordinate system of the middle connecting rod assembly to a three-dimensional coordinate system;
the force rotation amount of each branched chain component of the conveying mechanism corresponding to the reference point is as follows:
in the formula, Vi、AiRespectively the speed rotation and the acceleration rotation r of the ith branched chain component of the mechanismiVector pointing to the center of mass C of the intermediate link rod assembly for point P, IiIs an inertia matrix of the ith branched chain component of the mechanism relative to a reference point under a three-dimensional coordinate system,wherein,is an inertia matrix under the mechanism ith branched chain component mass center coordinate system, the mass center coordinate system is a coordinate system which is established by taking the mechanism ith branched chain component mass center as an original point and coordinate axes along the main shaft direction of a rigid body, RiAnd (4) an attitude transformation matrix from the i-th branched chain assembly centroid coordinate system to the three-dimensional coordinate system.
5. The dynamics modeling method of the series-parallel automobile electrophoretic coating conveying mechanism according to claim 1, characterized in that: in the step 5), the kinetic equation of the spiral quantity form of the conveying mechanism is as follows:
in the formula, FpFor the amount of rotation of the intermediate link assembly about the corresponding reference point, VpThe speed rotation quantity of the mechanism intermediate connecting rod component by taking p as a reference point and the pose parameter of q as the intermediate connecting rod
The axial driving force vector of the lifting turnover mechanism of the conveying mechanism can be obtained by combining the following formula:
Q＝(JT)+τ
wherein Q ═ F1F2F3F4τ5τ6)TIn which F is1、F2、F3、F4Respectively represents the axial driving force, tau, of the first, second, third and fourth branch chain driving joints of the mechanism5、τ6Respectively represents the axial driving torque of the fifth branch chain driving joint and the sixth branch chain driving joint of the mechanism, (J)T)+Is a matrix JTPseudo-inverse of (J)T)+＝J(JTJ)-1，VLiB denotes a first, second branched chain-mer componentiI 1,2, as the instantaneous speed rotation of the reference point, FLiIs a first branched chain body component and a second branched chain body component in the BiI 1,2, the amount of rotation of the reference point, VSiThe instantaneous speed rotation amount of the first and second branched-chain screw rod components when rotating around the axes thereof is FSiThe first and second branched-chain screw components rotate around their own axesL5A fifth branch chain of the lifting turnover mechanism of the conveying mechanism, and an instantaneous speed rotation quantity taking the central point of the driving wheel as a reference point, FL5The fifth branch chain of the lifting turnover mechanism of the conveying mechanism takes the center point of the driving wheel as the force rotation amount of the reference point, wherein B1 is the hinge point of the nut and the base in the first branch chain, and B2 is the hinge point of the nut and the base in the second branch chain.
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