装载机前端工作机构建模及仿真

ORIGINAL ARTICLECompound mechanism modeling of wheel loaderfront-end kinematics for advance engineering simulationYing Li &Wenyuan liu &Samuel FrimpongReceived:23September 2014/Accepted:23November 2014/Published online:6December 2014#Springer-Verlag London 2014Abstract Compound closed-loop mechanism methodology is proposed to model the wheel loader front-end kinematics for simulating the mechanism motion in a dynamic environ-ment.For the numerical modeling purpose,the mechanism is considered as consisting of three vector loops connected by the joint angles or displacements.Each vector loop is formu-lated as time-dependent ordinary differential equations (ODEs).The simulations cover the numerical solutions of the forward and reverse kinematics differential equations.The consistency error check approach is employed to validate the simulation.A case study is presented to demonstrate the reverse kinematics simulation of the compound mechanism model in Simulink environment,which includes the user-defined program,numerical solution,and error analysis.The displacement-time variations of two driving cylinders are obtained with the inputs of the lift and rotation bucket accel-erations.The simulation is validated with the numerical anal-ysis of the error results.This research provides a significant method to simulate the wheel loader motion for wide-ranging engineering application.Keywords Wheel loader .Compound mechanism modeling .Closed vector loop .Kinematics simulation .Error analysis1IntroductionThe wheel loaders are the heavy mining and construc-tion equipment,which function as loading,lifting,transporting,and discharging materials as displayed in Fig.1a –d ,respectively.Figure 2illustrates the main structural components of the wheel loader front-end attachment.It has a front-mounted bucket and a link-age controlled by two mechanisms:one is the lift mechanism (links 1–3)driven the hydraulic cylinder 3for lifting arm 4,and the other is the rotation mecha-nism (links 8–15)driven by hydraulic cylinder 8for rotating bucket 7[1].Kinematics model of wheel loader allows engineers to design dynamics control system to achieve the operation automation,and kine-matics simulation of wheel loader allows engineers to examine dynamic machine performance to achieve the design optimization.The detailed literature survey of the modeling and simulation of the mining equipment front-end attach-ment is presented in our previous publications [2–4].The kinematics modeling methods regarding the front-end attachments of mining equipment are focused on the closed or open kinematics loop [2–8].The closed-loop modeling method based on the theory of machin-ery kinematics [9]is applied to the kinematics modeling of the simple mechanism with four or less links.The open-loop modeling method based on the theory of robot kinematics [10]is applied to the kinematics modeling of the complicated mechanism with five or more links.The simulation methods are condensed on the forward and reverse kinematics simulation.The for-ward simulation is to find a trajectory of the bucket for the given motions (displacements,velocities,and accel-erations)of driving links,whereas the reverse simula-tion is to solve the required displacements,velocities,Y .Li (*)North University of China,Taiyuan,Shanxi,China 030051e-mail:liyinglzh@W.liuUniversity of Texas-UTMB,Galveston,TX,USA S.FrimpongUniversity of Missouri-Rolla,Rolla,MO,USAInt J Adv Manuf Technol (2015)78:341–349DOI 10.1007/s00170-014-6640-7and accelerations of the driving links for a given tra-jectory of the bucket.A typical wheel loader consists of multi-links as described in Fig.2,and the linkage motions are intertwined with other links as indicated in Fig.1.In our recent academic research [3],a wheel loader front-end is modeled as a seven-bar mechanical manipulator via open-loop concepts.The forward kinematics method is employed to formulate the kinematics equations in Cartesian coordinates for kinematics simulation.This model is the graduate student level understanding of kinematics and may not be accepted by all engineers.For the research purpose,a novel closed kinematics loop model of the cable shovel is developed based on kinematics of machinery in our lab [2].The kinematics model of the cable shovel is modeled as a seven-bar linkage with four degrees of freedom,which comprises the upper attachment and the dumping mechanisms.The motion equation of each mechanism is given by attaching Cartesian coordinates to schematic diagram of the kinematics model.A virtual prototype of the cable shovel is built,which allows the visualization of(a) Loading (b) LiftingFig.1A typical operation process of wheelloaderFig.2Main structure components of a wheel loader front-end attachment.1front vehicle body,2crank 3lift cylinder,4lift arm,5arm link,6bellcrank,7bucket,8bucket cylinder,9rotating link,10crank,11frame,12rotating link,13connecting rod,14crank,and 15framea three-dimensional motion of general mechanical sys-tem and the prediction of interference among the struc-tural components of the cable shovel.The numerical results of displacement,velocity,and acceleration can be used to evaluate the kinematics performance of cable shovel for its optimization.This study provides a solid foundation for further kinematics modeling and simula-tion of wheel loader in the engineering applications.For the engineering application purpose,the closed kinematics loop models,which are developed base on undergraduate student course [1,11],are proposed to simulate wheel loader motions.However,there are mas-sive links in the front-end attachment,so the link kine-matics model cannot be established in one closed loop.This study outlines the compound closed-loop mecha-nisms method,which can allow engineers to develop the wheel loader kinematics model with multi-simple mechanisms.The research include the following:first,the wheel loader front-end is modeled as a compound mechanism with three simple mechanisms;second,the vector loop equations of each mechanism are developed according to the geometric relations among the machinery links;third,the forward and reverse kinematics simulations are described;and fourth,an example is presented for demonstrating the method applications.2Kinematics modeling of the front-end compound mechanismFigures 3,4,and 5show the vector loop schematic diagrams of the wheel loader front-end attachment with local coordinate systems.The front-end attachment is regarded as a compound mechanism including a lift mechanism (slider-crank mechanism)with links 1–3and a rotation mechanism,which consists of a slider-crank mechanism with links 8–11and a four-bar mecha-nism with links 12–15.These three mechanisms form three vector loops 1–3,respectively [1].If each vector loop has N links and each link can associate a vector,the magnitude of this vector r i is the length of the link,whereas the direction of this vector is along the link.i and r i are the velocity and acceleration of link i ,respectively.The angle θi is defined as counterclockwise positive from the positive x -direction and drawn to the tail of each vector.ωi and αi are defined as the angular velocity and acceleration for link i ,respectively.The mathematical representation for each vector loop is given as Eq.1,where the sum of all vectors in the loop must be equal to zero [11].X i ¼1N r i ¼0ð1ÞIf each vector can be represented in Eq.2,Eq.1can be written as the loop-closure Eq.3.r i ¼r i cos θi ;sin θiÀÁTð2ÞX i ¼1N r i ¼X i ¼1N r i cos θi ;sin θi ðÞT ¼0ð3ÞThe loop-closure equations of each loop are established as follows.Figure 3displays a vector loop 1diagram of the slider-crank mechanism,which is attached to the front vehicle body by two revolute joints O 1and O 2.The local coordinate system O 2(X 2,Y 2,Z 2)is affixed to the link 1at joint O 2with the X 2axis along O 2O 1.The lift motion of the bucket can be obtained by hydraulic cylinder (link 3)rotating lift arm.The frame is link 1,the crank is link 2,and the connecting rod is link 3.Equation 3can be written as x and y coordinates (Eqs.4–5).r i cos θ1þr 2cos θ2þr 3cos θ3¼0ð4Þr 1sin θ1þr 2sin θ2þr 3sin θ3¼0ð5ÞThe velocity matrix Eq.6can be derived by taking the firsttime derivative of Eqs.4–5.−r 3sin θ3cos θ3r 3cos θ3sin θ3 !ω3r 3!¼r 2ω2sin θ2−r 2ω2cos θ2 !ð6ÞThe acceleration matrix Eq.7can be obtained by taking the second derivative of Eqs.4–5.−r 3sin θ3cos θ3r 3cos θ3sin θ3!α3r 3!¼2r 3ωsin θ3þr 3ω23cos θ3þr 2α2sin θ2þr 2ω22cos θ2−2r 3ω3cos θ3þr 3ω23sin θ3−r 2α2cos θ2þr 2ω12sin θ2"#ð7ÞFigure 4illustrates a vector loop 2diagram of the slider-crank mechanism,which is attached to the front vehicle bodyby two revolute joints O 2and O 3.The local coordinate system O 3(X 3,Y 3,Z 3)is assigned to the link 11at point O 3,with the X 3axis along O 3O 2.The link 9motion can be given by hydraulic cylinder (link 8)and lift arm (link 10).The frame is link 11,the cranks are links 9and 10,and the connecting rod is link 8.Equation 3can be broken down into x and y coordinates (Eqs.8–9).r 8cos θ8þr 9cos θ9þr 10cos θ10þr 11cos θ11¼0ð8Þr 8sin θ8þr 9sin θ9þr 10sin θ10þr 11sin θ11¼0ð9ÞThe velocity matrix Eq.10can be derived by taking the first derivative of Eqs.8–9for loop 2.−r 8sin θ8cos θ8r 8cos θ8sin θ8!ω8r 8!¼r 9ω9sin θ9þr 10ω10sin θ10r 9ω9cos θ9−r 10ω10cos θ10!ð10ÞThe acceleration matrix Eq.11can be given by taking the second derivative of Eqs.8–9.−r 8sin θ8cos θ8r 8cos θ8sin θ8!α8r 8!¼2r 8ω8sin θ8þr 8ω28cos θ8þr 9ω29cos θ9þr 9α9sin θ9þr 10ω210cos θ10þr 10α10sin θ10−2r 8ω8cos θ8þr 8ω28sin θ8þr 9ω29sin θ9−r 9α9cos θ9þr 10ω210sin θ10−r 10α10cos θ10"#ð11ÞFigure 5indicates a vector loop 3diagram of the four-barmechanism,which is attached to the lift arm by two revolute joints O 4and O 5.The local coordinate system O 4(X 4,Y 4,Z 4)is assigned to the lift arm at joint O 4,with the X 4axis along O 4O 5.The rotating motion of the bucket can be generated by rotating link 9.The rotating motion of the bucket can be generated by rotating link 12.The dynamic frame is link 15,the cranks are links 12and 14,and the connecting rod is link 13.Equation 3can become x and y coordinates (Eqs.12–13).r 13cos θ13þr 14cos θ14þr 12cos θ12þr 15cos θ15¼0ð12Þr 13sin θ13þr 14sin θ14þr 12sin θ12þr 15sin θ15¼0ð13ÞThe matrix velocity Eq.14can be derived by taking the first time derivative of Eqs.12–13for loop 3.r 12sin θ12r 13sin θ13r 12cos θ12r 13cos θ13 !ω12ω13 !−r 14ω14sin θ14−r 14ω14cos θ14!ð14ÞThe acceleration matrix can be expressed as Eq.15by taking the second derivative of Eqs.12–13.local coordinatesystemFig.4Vector loop 2diagram of the slider-crank mechanism with local coordinate systemr12sinθ12r13sinθ13 r12cosθ12r13cosθ13!α12α13!¼−r13ω213cosθ13−r14α14sinθ14−r14ω214sinθ14−r12ω212cosθ12r13ω213sinθ13−r14α14cosθ14þr14ω214sinθ12þr12ω212sinθ12!ð15ÞThree loop equations are combined in the assembled form through mathematical relationships to capture the front-end system motion characteristics.Therefore,the combined math-ematical model is developed as time-dependent ordinary dif-ferential equations(ODEs)and ready to be solved numerically in kinematics simulation.3Kinematics simulation of the front-end compound mechanismFigure6describes a schematic diagram of compound mech-anism used for the kinematics simulation of the wheel loader front-end attachment.Two problems of forward kinematics and inverse kinematics are solved as discussed below. Forward kinematics simulation For the given displacement-time variables of the two hydraulic cylinders,r3(t)and r8(t), the bucket position-time variables,θ2andθ14,are yielded by the solutions of Eqs.4,5,8,9,12,and13through loops1–3.&In loop1,the slider-crank mechanism initial position is defined by the link lengths of r1,r2,and r3=r3(0)and angles ofθ1,θ2=θ2(0),andθ3=θ3(0).For a given input displacement of the hydraulic cylinder(link3),r3=r3(t), the two link angles ofθ2andθ3,are found by solving Eqs.4–5.&In loop2,the slider-crank mechanism initial position is defined by the link lengths of r8=r8(0),r9,r10,and r11and angles ofθ8=θ8(0),θ9=θ9(0),θ10=θ10(0),andθ11.One of the solutions of Eqs.4–5,θ2,becomes one of inputsθ10 into loop2.When other one is the displacement of thehydraulic cylinder(link8),r8=r8(t),the solution of Eqs.8–9produces the link9motionθ9.&In loop3,the four-bar mechanism initial position is de-fined by the link lengths of r12,r13,r14,and r15and angles ofθ12=θ12(0),θ13=θ13(0),θ14=θ14(0),andθ15.When the solutions from the loops1and2,θ15(θ2),andθ12(θ9), are used as inputs,θ13andθ14are given by solving Eqs.12–13.Therefore,it needs two independent input motions to op-erate the bucket.This means the displacements r3(t)and r8(t) have to be provided by the two hydraulic cylinders3and8 (see Fig.2),respectively.Reverse kinematics simulation For the given position-time variables of the bucket,θ2andθ14,the displacement-time variables of two hydraulic cylinders,r3(t)and r8(t),are yielded by the solutions of equations from15to4through loop3to loop1.&In loop3,the four-bar mechanism initial position is de-fined by the link lengths r12,r13,r14,and r15and angles of θ12=θ12(0),θ13=θ13(0),θ14=θ14(0),andθ15.If the motion of bucket,θ13andθ14,is given,the angular displacementsθ12(θ9)andθ15(θ2)are yielded by solving Eqs.12–13.The solutions of Eqs.14–15product the angular velocitiesω12(ω9)andω15(ω2),andα12(α9)accel-eration andα15(α2),respectively.&In loop2,the slider-crank mechanism initial position is defined by the link lengths of r8=r8(0),r9,r10,and r11, and angles ofθ8=θ8(0),θ9=θ9(0),θ10=θ10(0),andθ11.As the solutions from loop3,θ9andθ10(θ2),are asinputs, coordinate systemthe solutions of Eqs.8–11product the hydraulic cylinder (link 8)displacement r 8,velocity 8,and acceleration r 8,respectively.&In loop 1,the slider-crank mechanism initial position is defined by the link lengths r 1,r 2,and r 3=r 3(0)and angles of θ1,θ2=θ2(0),and θ3=θ3(0).When the solutions from loop 3,θ2,becomes an input into loop 1,the solutions of Eqs.4–7product the hydraulic cylinder (link 3)displace-ment r 3,velocity 3,and acceleration r 8,respectively.Therefore,it needs two independent bucket motion param-eters to solve the compound mechanism driving variables,which is provided by the two hydraulic cylinders 3and 8(see Fig.2),respectively.4Simulation validation of the front-end compound mechanismThe kinematics simulation of the front-end compound mech-anism described in “section 3”is validated using the consis-tency check approach [11].No matter what kind of simulation it is,the inputs for the forward kinematics or the reverse kinematics simulations are the motion parameters of the two input links;the remaining velocities and accelerations of the linkages are solved by deriving the vector loop equations;and then these variables are integrated to compute the displace-ments of two hydraulic cylinders,r 3and r 8,or the bucket positions,θ2and θ14.An error is defined as difference between the resulting position and original position as describedinFig.6Compound mechanism diagram of the wheel loader front-endattachmentFig.7Simulink diagram for kinematics simulation of wheel loader front-end compound mechanismEq.16.E ¼X i ¼1N r i ¼X i ¼1N r i cos θi ;r i sin θi ðÞTð16ÞFor loop 1,the error Eq.16can be expressed as x and y coordinates (Eqs.17–18).e x 1¼r 1cos θ1þr 2cos θ2þr 3cos θ3ð17Þe y 1¼r 1sin θ1þr 2sin θ2þr 3sin θ3ð18ÞFor loop 2,the error Eq.16can be described as x and y coordinates (Eqs.19–20).e x 2¼r 8cos θ8þr 9cos θ9þr 10cos θ10þr 11cos θ11ð19Þe y 2¼r 8sin θ8þr 9sin θ9þr 19sin θ10þr 11sin θ11ð20ÞFor loop3,the error Eq.16can be indicated as x and y coordinates (Eqs.21–22).e x 3¼r 13cos θ13þr 14cos θ14þr 12cos θ12þr 15cos θ15ð21Þe y 3¼r 13sin θ13þr 14sin θ14þr 12sin θ12þr 15sin 15ð22ÞThe norm of the three vector Eqs.16–22generates the root-mean-square error of the compound mechanism as showed in Eq.23.e ¼e 2x 1þe 2y 112þe 2x 2þe 2y 2 12þe 2x 3þe 2y 312ð23ÞOnce the acceptance tolerance limit is defined,the simula-tion error can be checked within the time ually,theerror tolerance is caused by the model structure and numerical solution in the coding program.If the numerical integration does not converge,lower the acceptance tolerance limit is suggested.If the error is within the acceptance allowance limit,the simulation is validated.5Case study of the front-end compound mechanism A case study is performed for indicating the reverse kinemat-ics modeling,simulation,and error check of the wheel loader front-end compound mechanism in the MATLAB/Simulink software environment.The simulation model is developed using Simulink blocks and user-defined program blocks.The simulation involves the numerical solution of the ODEs that describe the time-varying responses of the front-end compound mechanism.The displacement-time results of two driving cylinders are obtained with the inputs of the lift and rotation bucket motions.The dynamic simulation errors are examined by analyzing the error-time and error-cylinder relationships.Simulation model Figure 7illustrates a Simulink block dia-gram model used to simulate the motion of a wheel loader front-end compound mechanism.The model embodies three user-defined program blocks to solve a series of the time-dependent ODEs derived in vector loops 1–3.The numerical integration blocks are employed for calculating velocities and displacements by integrating accelerations.Gain block sums the consistency errors of three loop equations.Simulation configuration parameters are set as Runge-Kutta ODE Solver,automatic step-size control and 18-s run time.The appropriate kinematics parameters such as the link geometric properties and initial conditions are addressed in Table 1.The reverse kinematics simulation refers to five steps:(a)input the angular accelerations of the lift arm and rotation bucket,(b)repeatedly solve the three loop ODEs through time steps,(c)integrate the resultant accelerations and velocities at each iteration step,(d)output the displacement values of two hydraulic cylindersTable 1Structural kinematics parameters and initial conditionsLength (m)Angle (°)Loop 1r 1r 2r 3(0)θ1θ2(0)θ3(0)1.9940.925 2.177−18088.57334.86Loop 2r 8(0)r 9r 10r 11θ8(0)θ9(0)θ10(0)θ112.4 1.0 2.6090180−67.38−180Loop 3r 12r 13r 14r 15θ12(0)θ13(0)θ14(0)θ151.01.10.5211.25838.318−78.25461.347−180over simulation time,(e)and check the simulation root-mean-square errors to ensure accurate solutions.Simulation inputs Figure8shows that the simulation inputs during the operating time18s with9s for lifting and9s for rotating motions.Figure8a indicates the angular acceleration of the lift arm variation with time relative to the O2(X2,Y2,Z2) coordinate system(see Fig.3).The angular acceleration jumps from0to0.077rad/s2at the0th second,linearly decreases to −0.078rad/s2from0to9s,sharply returns to0rad/s2at the 9th second,and then stays at the constant0rad/s2from9to 18s.The minimum angular acceleration of−0.078rad/s2 occurs at the9th second while the maximum of0.077rad/s2 occurs at the0th second.Figure8b depicts the angular accel-eration of the rotation bucket variation with time relative to the O4(X4,Y4,Z4)coordinate system(see Fig.5).The angular acceleration keeps a constant0rad/s2during the first9s,rises from0to0.13rad/s2at the9th second,linearly decreases from 0.13to−0.13rad/s2with time during the last9s and then jumps back from−0.13to0rad/s2at the18th second.The minimum angular acceleration of−0.13rad/s2occurs at the 18th second while the maximum of0.13rad/s2occurs at the 9th second.These inputs indicate that the arm mechanism motion is performed during the first9s while the bucket rotating motion is performed during the last9s.Simulation results Figure9shows that the simulation outputs during the operating time18s with9s for lifting and9s for rotating motions.The displacement of the lift cylinder varia-tion with time relative to the O2(X2,Y2,Z2)coordinate system (see Fig.3)is given in Fig.9a.The displacement non-linearly increases from a minimum of2.18m at the beginning to a maximum of2.82m at9s,and then stays at a constant of 2.82m from9to18s.The displacement within the last time segment(9to18s)does not change with time because the lifting motion is completed at the end of the9th second. Figure9b depicts the displacement of the rotation bucket cylinder variation with time relative to the O4(X4,Y4,Z4) coordinate system(see Fig.5).The displacement keeps a constant maximum of2.4m during the first9s,non-linearly decreases from2.4m to a minimum of2.2m during9to14s, and non-linearly increases from2.2to2.25m during the last 4s.The displacement within the first time segment(0to9s) does not change with time,because the bucket rotating motion starts at the9th second.Two output results show that the cylinder displacements vary smoothly with time during the operation.Simulation validation Figure10indicates that the root-mean-square error check during the simulation time18s with9s for lifting and9s for rotating motions.Figure10a gives theerror Fig.9Two cylinder displacement outputs with time:a lift cylinder displacement and b rotation bucket cylinderdisplacementFig.8Two link angular acceleration inputs with time:a lift arm angular acceleration and b rotation bucket angular accelerationvariation with time and Fig.10b displays the error variation with displacements of the lift cylinder and rotation bucket cylinder.The results show that the simulation error is quiet small and nearly constant of6.19×10−3m over time.In this validation,the acceptance tolerance limit is suggested as 10−2m considering the geometric model error and initial condition measurement errors[3].The resulting error is ac-ceptable,which is less than10−2m.Therefore,the resulting the positions of two hydraulic cylinders satisfy the original three vector loop equations throughout simulation time.The reverse kinematics simulation of the front-end compound mechanism is validated.6ConclusionsCompound mechanism methodology has been proposed to model the wheel loader front-end kinematics for efficient engi-neering simulation.The front-end mechanism is decomposed into three simple mechanisms,which forms three closed kine-matics chains.The mathematical modeling technology of the closed vector loop is employed to derive kinematics ODEs for each simple mechanism.The three loops are related by the joint kinematics parameters of the angles or displacements between adjacent links.The forward and reverse kinematics is used to simulate the compound mechanism model by the numerical solutions of differential equations.The consistency check meth-od is applied to validate the simulation.An example is given to indicate the reverse kinematics simulation of the compound mechanism model in Simulink environment.The study involves the user-defined program, numerical integration,dynamic simulation,and error check. The position problems of two driving cylinders are solved with the inputs of the lift and rotation bucket accelerations. 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