Real-time multibody application for tree harvester truck simulator

Original Article

Real-time multibody application for treeharvester truck simulator

Mohamad Ezral Baharudin1, Asko Rouvinen2,Pasi Korkealaakso2 and Aki Mikkola1

Abstract

A real-time simulator for a tree harvester has been developed for training in more effective vehicle and cutter operation

and tree management. The equations of motion of the constrained mechanical system of the tree harvester are

expressed using a recursive formulation. The hydraulic actuator modelling of the harvester is based on lumped fluid

theory, in which the hydraulic circuit is divided into discrete volumes where pressures are assumed to be distributed

equally, while pressure wave propagation in pipes and hoses is assumed to be negligible. For modelling purposes, valves

are broken up into a number of adjustable restrictors, which can be modelled separately. The contact model used

comprises two parts: collision detection and response. Collision detection identifies whether, when and where moving

bodies may come in contact. Collision response prevents penetration when contact occurs and identifies how it should

behave after collision. A penalty method is used in this study to establish object collision events. The major achievement

of this study is combining these three modelling methods in the application of a real-time simulator.

Keywords

Multibody dynamics, recursive method, hydraulic actuator, harvester truck, contact modelling

Date received: 27 August 2013; accepted: 3 January 2014

Introduction

As computer simulation tools become more sophisti-cated, the option of virtually building and analysingcomplex machine systems becomes an increasinglyattractive alternative to physical prototyping. If thesimulation is used for operator training, the simula-tion system should be able to present the workingenvironment and machine functions accurately togive the operator a true-to-life experience. A numberof tree harvester training simulators have been devel-oped and are already being marketed by well-knowncompanies. These include, for example, the JohnDeere Forestry Machine Simulator, the ValmetKomatsu Forestry Simulator and the CreanexTraining Simulator.

In the past, tree harvester simulators required agreat deal of space and highly skilled (expensive)teams to manage set-up and maintenance.1

However, with the increasing sophistication of com-puter technology and growing knowledge of multi-body systems, machine simulators are becomingsimpler and more user friendly. In practice, commonreal-time machine simulation should consist of thedescription of several subsystems such as the machine

mechanics, the actuators and the control system. Thedevelopment of a comprehensive computer-based treeharvester simulator is claimed to have started in theyear 1997. The simulation models involved two majorsub-models: the kinematic of four degrees of freedommanipulator and collision/contact detection of con-tact between the log and ground and between themanipulator and log.2 The dynamic models ofmachines become complicated when the manipulatorsare attached to a moving base. The moving base in theiterative Newton–Euler dynamic model can bedescribed by employing a symbolic approach in theimplementation.3

On the other hand, a conventional linear graphmethod can also be implemented for the models of

Proc IMechE Part K:

J Multi-body Dynamics

2014, Vol. 228(2) 182–198

! IMechE 2014

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DOI: 10.1177/1464419314521182

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1Department of Mechanical Engineering, Lappeenranta University of

Technology, Finland2MeVEA Ltd, Finland

Corresponding author:

Mohamad Ezral Baharudin, Department of Mechanical Engineering,

Lappeenranta University of Technology, Skinnarilankatu 34,

Lappeenranta 53850, Finland.

Email: [email protected]

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the mechanical structure and hydraulic actuators of aharvester machine manipulator.4 In the method, eachsingle component is identified together with its consti-tutive equations. Linear graphs for each subsystemprovide detailed variables and can be interconnectedwith other subsystems if they share variables.However, three-dimensional models produced usingthis approach result in less efficient dynamic model-ling than models produced using a recursive multi-body formulation.5,6 To accelerate the computingtime, it also possible to reduce the electro-hydraulicactuator model where the dynamics of fluid lines areeliminated and modelled as a purely resistive net-work.7,8 The hydraulic system can also be modelledusing a simple computing algorithm and self-tuningmodel reference adaptive control.9

Another almost similar study on developing thedynamics model and simulation of a heavy machinemanipulator deals with cherry pickers where thesystem consists of a truck (base), two flexible armsand a passenger basket. The system is characterisedby the two-mode vibrations of a zero vibration inputshaper and an extra-insensitive input shaper that sig-nificantly reduces the motion-induced oscillation ofthe system.10,11 The dynamic behaviour of themachine suspension can be modelled using an active,semi-active or passive suspension system solution, asexplained in the study of Grott et al.12

In conclusion, many factors may affect the comput-ing time in order for it to approach real-time applica-tion, such as the method of implemented dynamics,the integration time step, the numerical integrationalgorithm, the CPU type, the source code structure,the computing processing method and so forth13 andthe key to success in developing the harvester simula-tor is the simulation models of manipulators, vehiclemotion due to unmade surface, cutting and loadingthe log.14

The primary objective of the work described in thisarticle is to introduce a new and accurate simulationapproach that closely models the performance of acomplex machine system in a real environment andwith representative feedback. This approach wasimplemented to develop a tree harvester simulator,and the modelling of dynamics of the harvester’smechanics, hydraulic actuators and contact phenom-ena are discussed in depth in the following sections.The mechanical system was modelled dynamicallyusing a semi-recursive formulation and solved via par-allel computing techniques. The equations of motionwere developed recursively for an open loop (chaintopology) rigid body system. The hydraulic actuatorswere modelled using lumped fluid theory, in which thehydraulic circuit volume is divided into discrete vol-umes, and pressures are assumed to be distributedequally. The penalty method was applied to definemodel collision and contact. The tires-ground contactis modelled using the LuGre friction model. Theintroduced simulation approach, applied to producereal-time simulation models for a tree harvester, isdescribed in ‘Numerical example’.

Mechanical model of tree harvester

Mechanical structure of the tree harvester is modelledusing multibody system dynamics approach. Theapproach can be applied to dynamic analysis of sys-tems consisting of bodies that are constrained to moverelative to one another using kinematic connections.The system may have a tree or chain structure (openloop) or a structure with closed links of bodies (closedloop). In the chain structure system, the link betweenany two arbitrary bodies in the system is unique. Asan example, a tree harvester truck can be simplifiedinto a chain structure multibody system, as shown inFigure 1.

Figure 1. Tree harvester illustrated as multiple rigid bodies.

Baharudin et al. 183



The tree harvesting mechanism, which excludes thevehicle body, comprises five bodies. Bodies 1, 2, 3, 4and 5 are connected to the preceding bodies withrevolute joints. The vehicle body B0 is designatedthe base body. Body B1 is attached to body B0 by arevolute joint q1. The fourth joint q4 consists of tworevolute joints – one vertical and one horizontal.Joints q2, q3 and q5 are revolute joints. The mechan-ical system has six degrees of freedom. Because move-ment is possible in both the horizontal and verticaldirections, the motion of the mechanism is threedimensional.

Recursive formalism

The recursive formulation for this study derives fromthe equations of motion for a constrained mechanicalsystem. The kinematic properties such as position,velocity and acceleration are developed based on therelative coordinates between contiguous bodies con-nected by a joint.15 Hooker and Margulies16 proposedthis formulation to the dynamic analysis of satellitesand found the computational costs to increase linearlywith the number of bodies. This algorithm has beenused and extended by several researchers and hasrecently been generalised to improve its implementa-tion and efficiency.17–20 The dynamics equation in arecursive formulation is written in terms of the sys-tem’s degrees of freedom and, typically, is writtenwith a lower dimensionality than those in an aug-mented formulation.

As shown in Figure 1, bodies are connected withjoints. Each body number can be represented as Bn.Preceding bodies are designated Bn�1. The arrangedset of body numbers is called a ‘body connectionarray’.21 Joint numbers are named before body num-bers. The revolute, prismatic, cylindrical, translationalor spherical joints can consist of single or multipleaxes.

Kinematics. Relative motion between neighbouringbodies and constraints are the two main aspectsof recursive kinematics used to generate thetotal system matrices and solve the equations ofmotion for the multibody system. Figure 2 shows anelementary system of two bodies interconnected bya joint.

The orientation relationship of contiguous bodiesis obtained by sequentially transforming from thebody reference frame on body B0 to the body refer-ence frame on body Bn. The system of bodies is con-sidered an open chain and point 0 is the globalreference frame. The relative kinematics of positionrn for each body can be described with respect tothe global reference as follows22

rn ¼ rn�1 þ sn�1 þ dn ð1Þ

where dn is the joint relative displacement vector frompoint n� 10 to n, and sn–1 is a vector from point n� 1to n� 10.

The orthogonal rotation matrix An�1 is obtainedfrom the preceding body Bn–1. The velocities can bedetermined from the time derivative of equation (1)

vn ¼ _rn�1 þ ~un�1sn�1 þ _dn ð2Þ

un ¼ un�1 þ ud ð3Þ

where matrix ~un�1 is the skew-symmetric matrix ofthe cross-product ~un�1sn�1 � un�1 � sn�1. Vector ud

is the relative angular velocity of the joint. The accel-eration equations can be obtained by differentiatingequations (2) and (3) as follows

€rn ¼ €rn�1 þ _~un�1sn�1 þ ~un�1 ~un�1sn�1 þ €dn ð4Þ

_un ¼ _un�1 þ _ud ð5Þ

In equations (1) to (5), the terms dn, Ad and ud arefunctions of relative joint position, the joint rotationmatrix and joint angular velocity, respectively. Thefunctions of velocity and acceleration for eachcommon type of joint (prismatic, spherical, etc.) canbe obtained by deriving expressions as suggested byAvello et al.23

Equation of motion. The equations of motion can bedeveloped from the principle of virtual power, as pro-posed by Avello et al.23 and Haug.24 In Figure 2, ifposition vector rn is shifted to the centre of gravity ofthe body becoming un, un is the velocity of body n andNb is the number of contiguous bodies. The equationof motion can be written in matrix form as follows

�Nb

n¼1�_uTn �u

Tn

� � mnI 0

0 Jn

� �€un

_un

� ��

þ0

~unJnun

� ��

Fn

Tn

� ��¼ 0 ð6Þ

Figure 2. Relationship of contiguous bodies.

184 Proc IMechE Part K: J Multi-body Dynamics 228(2)



where �_un and �un are virtual translational and angu-lar velocities, matrix Jn is inertia (obtained fromAnJnA

Tn ), mn is body mass and Fn and Tn are external

forces and torques. The equations of motion can besimplified as

�_qT M€qþ C�Qð Þ ¼ 0 ð7Þ

From equation (7), the dependent virtual velocities�_q should be eliminated to obtain the final equationwith a small dimension. Therefore, the vector of inde-pendent virtual velocities �_z with Nf number ofdegrees of freedom is introduced. As mentioned pre-viously, _q has a dimension of 6Nb, which covers thetranslational and angular velocities of each body.However, the angular position cannot be obtainedby integrating _q, which necessitates the introductionof the new variable pq to represent the dependent pos-ition. The new variable can be defined by separatelydefining the translational and orientation as suggestedin the Cartesian coordinate approach25 or other suit-able methods. In this case, _q and _pq can be representedas follows

_pq ¼ Eq _q ð8Þ

_q ¼ Fq _pq ð9Þ

In the preceding equations, Eq is the velocity trans-formation matrix and Fq is the inverse velocity trans-formation matrix. Similar to _q, _z cannot be integratedto obtain a position, which necessitates introducingthe position vector pz and the velocity transformationmatrix Ez with the size (Nf�Nf)

_pz ¼ Ez _z ð10Þ

In this particular equation, pq can be represented as afunction of pz and time as follows

pq ¼ pqðpz, tÞ ð11Þ

By taking the time derivative of equation (11), thefinal _q can be obtained as

_q ¼ R_zþ b ð12Þ

where

R ¼ Fq

@pq@pz

Ez and b ¼ Fq

@pq@t

The acceleration equation can be formulated bytaking the time derivative of equation (12)

€q ¼ R€zþ _R_zþ _b ð13Þ

Substituting equation (13) into the original equa-tion (7) leads to these expressions

�_zTðRTMR€zþRTM _R_zþRTM_bþRTC�RTQÞ ¼ 0

ð14Þ

Equation (14) can be reconstructed in simpler formas follows

RTMR€z ¼ �RTM _R_z� RTM_b� RTCþ RTQ

¼ RTðQ� CÞ � RTMð _R_zþ _bÞ

ð15Þ

Computing matrix R. Here, it is clearly shown that vel-ocity transformation matrix R plays an importantrole. Consequently, it is important to pay attentionto solving the matrix R that affects each body. Forexample, referring to the simple analogy of Figure 1and assuming that all joints have one degree of free-dom, the total size of matrix R is (30� 5), where thecolumns of matrix R correspond to each joint in thesystem. If body B3 has been chosen to identifymatrix R3, the columns corresponding to joints 4and 5 have all zero elements and only the 1, 2 and3 joints need be computed. Therefore, matrix R3 is a(6� 3) matrix R1

3R23R

33

� . The superscripts represent

the joint number. Each submatrix is a (6� 1) vectorand the first three components of the vector areobtained from equation (2). The last three compo-nents come from equation (3). From here, it can begeneralized that Rn can be formed intoR1

nR2n . . .RNq

n

� with a size of (6�Nd), where Nd

and Nq are the number of degrees of freedom andnumber of joints from body n to the vehicle body,respectively. The size of the sub-matrix Rj

n is (6�Nj),where Nj is the number of degrees of freedom atjoint j.

Numerical algorithm for open loop system. The system ofdifferential algebraic equations has been trans-formed into ordinary differential equations by inte-grating variables €z and _pz. In general, the algorithmis derived for generating the real-time simulation foran open loop system. The topology of the systemshould be defined first by labelling all of the bodiesand joints. The algorithm to recursively computethe dynamics of the open loop system is startedby assigning the initial values for positions pz andvelocity _z at t¼ 0. Then, by solving the positionproblem, rn, qn and An can be obtained.Subsequently, the velocities _rn, !n and _R_zþ _b

�are computed as a velocity-dependent acceleration.With the help of a parallel computing loop fromn¼ 1:Nb:

1. Compute Mn, Cn, Qn and finally Rn

2. Compute RTn Qn � Cnð Þ and RT

nMTn

3. Compute RTnMn

�Rn and RT

nMn

�_R_zþ _b �

.

By using equation (25), the acceleration of €z can beobtained. Compute _pz ¼ Ez _z and by integrating€z, _pz �

, _z, pz �

tþ�tcan be obtained. In this algorithm,




the parallel computing loop requires the most com-puting costs and may cause problems if not handledaccurately.

Modelling of actuators

In this study, hydraulic actuators, the assumed driversfor the tree harvester systems, are modelled usinglumped fluid theory, which divides a hydraulic circuitinto discrete volumes with hydraulic pressure distrib-uted equally. Valves are modelled using a semi-empirical approach that makes use of flow parametersthat can be obtained, in many cases, directly frommanufacturer catalogues. Usually, a hydraulicsystem has a high nominal frequency response and,therefore, the calculation time step must be short toproduce reliable results. In this approach, pressurewave propagation in pipes and hoses is assumed tobe negligible.26 The hydraulic pressure in eachhydraulic volume i can be described as follows

_pl ¼Bei

Vi

Xncj¼1

Qij ð16Þ

where Bei is the effective bulk modulus that defines theflexibility of the hydraulics, Qij is the outgoing orincoming flow rate and nc is the total number ofhydraulic components, all related to volume i, Vi.The effective bulk modulus can be calculated as

Bei ¼1

1BoilþPnc

j¼1Vj

ViBjþPnh

k¼1 QijVk

ViBk

ð17Þ

In equation (17), nh is the total number of pipesand hoses related to volume i. The bulk modulus ofoil, Boil, accounts for the amount of non-dissolving airin the oil. It is a function of pressure and the max-imum value is typically Boilmax¼ 1.6� 109Pa. Thebulk moduli Bj and Bk of components j and k arealso dependent on the component type.

The magnitude of volume Vi is solved based on thecombined volume of pipes, hoses and other hydrauliccomponents related to the volume as follows

Vi ¼Xncj¼1

Vj þXnhk¼1

Vk ð18Þ

where the volume Vj of a single component dependson the types of the component. The bulk modulus Bj

of component j is also dependent on the componenttype.

Modelling of valves

For modelling purposes, a valve is assumed to consistof several adjustable restrictor valves that can each be

modelled separately.27 Figure 3 shows an example of4/3 directional valve where flow rate at each port (PA,PB, PT and PP) can be calculated. With small pressuredifferences (< 1 bar), the flow over the restrictor isthought to be laminar and with larger differences, itis thought to be turbulent. When using the semi-empiric modelling method, the flow over the restrictorcan be written as follows

Q ¼ CvUffiffiffiffiffidp

pð19Þ

where Cv is the flow rate constant defining the size ofthe valve, U is a variable that defines the spool orpoppet position and dp is the pressure differencebetween pressure ports. For a number of valvetypes, the variable U can be defined using a first-order differential equation

_U ¼Uref �U

�ð20Þ

where Uref is the spool reference position and � is thetime constant describing the dynamics of the valvespool.

Modelling of hydraulic cylinders

A hydraulic cylinder can be modelled using thedimensions of the cylinder and the pressure obtainedfrom equation (16). The volumes of hydraulic cylinderchambers are solved as a function of cylinder strokeas follows

VjA ¼ xAA

VjB ¼ ðl� xÞAB

ð21Þ

where AA is the area of the cylinder piston side and AB

is the area on the cylinder piston rod side, x is thestroke of the cylinder and l is the cylinder maximumstroke. The motion of the hydraulic cylinder producesa flow rate Q to the hydraulic volume i as follows

QjA ¼ � _xAA

QjB ¼ _xAB

ð22Þ

The stroke velocity of the cylinder is _x, as depictedin Figure 4.

Figure 3. Direction valve.




The force produced by the hydraulic cylinder Fs

can be defined as

Fs ¼ p1A1 � p2A2 � F� ð23Þ

where F� is the total friction force of the cylinder andp1 and p2 are pressures acting in the cylinder cham-bers. Friction force is a function of pressures, cylinderefficiency � and velocity. The friction force can bedescribed in a simple case as follows

F� ¼ ð p1A1 � p2A2Þð1� �Þ f ð _xÞ ð24Þ

The velocity-dependent friction coefficient, f ð _xÞ,can be described using a spline curve, as shown inFigure 5.

In this particular model, the maximum speed of thehydraulic cylinder is 1ms–1, as shown in Figure 5.Friction is a physical process of shear between twosliding surfaces as a function of velocity. In the fric-tion model used, the coefficient decreases at a lowvelocity and increases proportionally when the vel-ocity increases after a specific velocity.28 The min-imum velocity-dependent friction coefficient is 0.2 at

0.1ms–1. Hydraulic cylinder end damping is modelledusing impact force as follows

FImin ¼ kðx� xminÞ þ c _x STEP ð _x, � 1e� 3, 0, 1, 0Þ,

x5 xmin

FImax ¼ kðx� xmaxÞ þ c _x STEP ð _x, 0, 1e� 3, 0, 1Þ,

x4 xmax ð25Þ

where k is the end damping spring constant and c isthe damping coefficient. The STEP function that usesa cubic polynomial to fit between two states is used inthe formulation code in order to avoid the instantan-eously contact force.29

Collision and contact modelling

The contact model is important and must be con-sidered when simulating more than one body. Itsfunction is to prohibit the interpenetration ofbodies. The contact algorithm determines when andwhere contact occurs and calculates the contactingelements.

In practice, there are two steps in contact model-ling: contact detection and collision response.30

Contact detection is the first and most importantstep. It models the moving bodies and determineswhen and where a potential collision will occur. Theaccuracy of the results depends on the geometry of thecontacting bodies. Contact response prevents penetra-tion between bodies by analysing the contact force ofcolliding bodies based on their geometric properties,relative velocities and material makeup. However,interbody penetration between rapidly movingbodies may occur between times steps if excessivetime steps are being used.31

There are several ways to define contact in multi-body systems, including the penalty technique, analyt-ical method and impulse method.32–34 In this study,contacts between bodies are defined using the penaltymethod, which allows small penetrations betweenbodies, and a temporal spring damper is addedbetween the collision points.35–37 Since it permitsinterpenetration,38 the penalty method is also knownas the surface compliance or soft contact method. Thegeneral algorithm for contact/collision models is illu-strated by the state transition diagram in Figure 6.

In Figure 6, d and _d are the distance and velocitybetween two objects, while FN and FT are normal andtangent forces. The idea of this algorithm is to alwayscompare the distance between two bodies and a colli-sion occurs when the distance d is equal to or less than0. Then, the collision response model will be imple-mented. The contact force is applied to the model asan external force.

In this study, the object-oriented bounding box(OOBB) model is used for collision detection andthe penalty method is applied as a collision responseFigure 5. The velocity-dependent friction co-efficient.

Figure 4. Hydraulic cylinder transforms hydraulic pressure

into a mechanical force.




between the harvester head and log. Contact betweentires and the ground is modelled using the LuGremodel as a friction model.

Collision detection

The collision between bodies with different geometriesis detected using the bounding volume (BV) approach.This method uses detection trees to determine whetheror not proximate bodies are intersecting. The BVapproach uses simple bounding volumes such asspheres and boxes to encapsulate the more complexbody geometries. The simplicity and tightness of thebounding box affect the efficiency of collision detec-tion. There are a few types of bounding methodsavailable for collision detection, such as boundingspheres,39 an axis-aligned bounding box,40 a dis-crete-orientation polytope41 and an oriented bound-ing box.42

In this study, collision detection is accomplishedby employing the OOBB approach. This approachoffers the minimum rectangular solid embodying theobject and along the direction of axis. The tightnessof OOBB can significantly reduce the number ofbounding boxes involved in intersection detection.If the object has variety of shapes, the OOBB canbe divided into small boxes that are connected toeach other, that is, an OOBB tree. Each box canbe treated separately for collision detection.However, the higher the number of OOBBs in thehierarchy, the greater the computing costs.The separating axis theorem is used to test whetherthe two OOBBs on an axis either intersect or not.For a three-dimensional case, this theorem offers 15potential separating axes, which need to be

examined, and if overlapping occurs on everysingle separating axis, the boxes intersect. Theseare the three local axes of the first OOBB, thethree local axes of the second and the nine cross-products of an axis of the first OOBB with an axisof the second. If the boxes do not intersect, the givenaxis is called as a separating axis. Figure 7 shows anOOBB collision detection approach in a two-dimen-sional case between harvester head and log.

In Figure 7, A1, A2, B1 and B2 are normalised axesof boxes A and B while a1, a2, b1 and b2 are the radiiof boxes A and B. L is a normalised direction and T isthe distance from the centres of A and B. The formu-lations for pA and pB are

pA ¼ a1A1Li þ a2A2Li

pB ¼ b1B1Li þ b2B2Li

ð26Þ

Boxes A and B do not overlap in a two-dimensionalcase if T.L> pAþ pB. For a three-dimensional case, all15 separating axes must be tested and the two OBBsare separated if

T � Li 4 pAi þ pBi i ¼ 1 : 15 ð27Þ

Under certain conditions, collision detection needsspecial treatment, for example, when dealing withparticles. Even though this collision detectionmethod will affect computing costs, especially whena large number of particles are involved, it willproduce good results for extended simulation. In gen-eral, particle-based collision detection can be dividedinto several steps and involves different approachessuch as the neighbour search method or the forcemodel.43

Figure 6. State transition diagram track of contact phase between two bodies.




Collision response

In practice, it is difficult to avoid body penetrationeven when using smaller time steps in the collisionprediction algorithm. Using smaller time steps reducespenetration occurrences but increases computingtime. Hence, there is a limit to how small the timesteps can be made. This limitation may result ininaccurate simulation results. Therefore, applyingthe penalty method could produce an acceptableresult in the real-time environment. The penaltymethod can be categorised into two types: thenormal force methods and the geometry-basedmodel. The normal force method deals with thesingle force model and will account for the entire col-lision. The geometry-based model can be divided intosingle collision point models, multiple collision pointmodels and volumetric models.44–48 In this particularstudy, the geometry-based model (single point)was chosen.

The kinematics of collision points can be describedusing a global reference frame. Figure 8 gives anexample of collision direction before and during acollision.

In Figure 8, Ft and Fn are the tangential normalforce at the collision point. Distance d can be writtenfrom the basic global reference frame approach asfollows

rPA¼ rA þ AA �uA

rPB¼ rB þ AB �uB

ð28Þ

d ¼ rPB� rPA

¼ rB þ AB �uB � rA � AA �uA ð29Þ

where AA and AB are a rotation matrix of bodies Aand B in three dimensions and �u¼ [x y z]T is a relativeposition vector of bodies A and B. Knowing that the

relative velocity is the time derivative of d, the relativevelocity is

vAB ¼ _rB þ ~uB �uB þAB_�uB � _rA � ~uA �uA �AA

_�uA

¼ _rB � ~�uBuB þHB _qB � _rA þ ~�uAuA �HA _qA

ð30Þ

As mentioned earlier, a spring and damper areadded in the penalty method in order to define thecontact forces when a small penetration is allowedbetween contacting bodies. Basically, the contactforce at the contact point can be formulated as

F ¼ �Kx� Cðn � vABÞ ð31Þ

where n is the contact normal, x is the penetrationdepth, K is the coefficient of elasticity and C is thedamping factor. The dot product of relative velocityvAB and contact normal n is the component of therelative velocity in the direction of the collisionnormal. The choice of value for the two coefficientsK and C is vital and may result in a range of desiredcollision response types.

Tire modelling

The harvester truck will manoeuvre on the unmadesurfaces/off road and each movement will affect thedynamic behaviour of the system. There are differentapproaches to model the contact between the tires andground and the most realistic model should considerthe contact patch interaction between the tires andsoil using finite element method and multibodydynamics.49,50 However, simulating the detail dimen-sions of the contact patch between the tires andground surface/soil will increase the computing

(a) (b)

Figure 7. Object Oriented Bounding Box (OOBB): (a) not intersected and (b) intersected.




costs, which is not desirable for the real-time simula-tion. Therefore, in order to meet the real-time criter-ion, the tires and ground are assumed to be simplerigid bodies and the profile of the ground can be con-structed depending on the required environment (off-road profile).

For the implementation of this particular tree har-vester simulator, the tire of the truck is presumed as aseries of discs, as shown in Figure 9(a), whereas thetypical forces involved are longitudinal force Fx, lat-eral force Fy and vertical force Fz. In this simulation,the contact between tire and ground is assumed ascontact between two rigid bodies. Figure 9(b) showsthe friction model with punctual contact, whereas Fn

is the normal force, F is the friction force and v is therelative velocity between the bodies.

The lumped LuGre model is used to model thecontact between tire and ground. The LuGre frictionmodel is an enhancement of the Dahl model with theStribeck effect added. This model captures many

important aspects of friction, such a stiction, theStribeck effect, stick slip, zero slip displacement andhysteresis.51–54 The contact between tire and ground ismodelled as two rigid bodies that contact through anelastic bristle at the microscopic level, as shown inFigure 9(c). When contact occurs, the tangentialforce will affect the bristles by deflecting like springsand create the friction force.55,56 Whena large force isimposed during the contact, some of the bristles coulddeflect excessively and start to slip. When the tirerotates, the bristle will deflect with respect to timeand the deflection of the bristles denoted by z isderived by defining the normalised longitudinal as

z ¼�1 � �0�0

ð32Þ

where �1 is the distance from the deformed bristlepoint to point q while �0 is the distance from an

(b)(a)

Figure 8. Collision between bodies A and B.

(a) (b) (c)

Figure 9. LuGre tire model.




undeformed bristle point to point q. The averagedeflection of bristles is the time derivative of z as

dz

dt¼ v�

�0 vj j

gðvÞz ð33Þ

where v is the relative velocity between two slidingsurfaces. This shows that the deflection is relative tothe integral of the relative velocity. The function g(friction) must always be positive and depends on

material properties, lubricants, temperature andother factors. The friction force generated from thebending bristles is

F ¼ �0zþ �1dz

dtþ �2v

� �Fn ð34Þ

where �0 is the longitudinal lumped stiffness, �1 is thelongitudinal lumped damping coefficient and �2 is vis-cous relative damping. The Stribeck effect occurs

Figure 11. Topology of tree harvester model.

Figure 10. Tree harvester connecting components.




when the typical friction g(v) decreases with increasingvelocity v for low velocities. Therefore, function g(v),which describes the Stribeck sliding friction forceeffect is

gðvÞ ¼ Fc þ ðFs � FcÞe� v

vs

1=2ð35Þ

where Fc is Coulomb friction, Fs is normalized staticfriction and vs is the Stribeck relative velocity.

Numerical example

The following numerical example is based on a typicaltree harvester, as shown in Figure 10. The simulationmodel of the harvester consists of a number of bodiesand is modelled using a recursive method. Figure 11shows the simplified topology of the harvester used asa reference when implementing the recursive method.The models include several force components such asbooms, hydraulics and log cutters (harvester head).

The mechanics of a tree harvester can be con-sidered as a set of rigid bodies with mechanicaljoints limiting the relative motion of the coupledbodies. Configuration changes with time based onthe forces and motions applied to the harvester’s com-ponents. Kinematic models consider system bodymotions without taking into account the forces that

Figure 13. Control valve signal and reference control signal.

Figure 12. Simplified description of outer boom hydraulic

system.

Table 1. Summary of model.

Name Number

Number of bodies 25

Number of joints 29

Number of joint coordinates 31

Number of closed loop constraint equations 7

Degrees of freedom 23

State variables 177




produce the motion. A kinematic model shows howthe bodies move within the system. It is solved usinglinear and non-linear systems of equations. However,a dynamic model must be used to reveal the move-ment of the bodies and the realistic responses of thesystem to applied and resultant forces. To develop the

kinematic and dynamic model simulations, the struc-ture of the machine needs to be identified early so thatall simulation procedures are based on a commonstructure.

Figure 11 is the topology of the tree harvestermodel, which shows the structures are connected to

Figure 15. Pressure rates at the inlet (A and AC) and outlet (B) of the cylinder.

Figure 14. Flow rates through the valve during work cycle.




each other through joints from the bogie to the har-vester head. There are two major types of joints in thisstructure: the revolute joint and the cylindrical joint.This structure consists of open loop and closed looplinks. For the closed loop link, the cut joint is intro-duced and the penalty approach must be added as aconstraint equation. For this particular tree harvestertopology, the model structure can be summarised asin Table 1.

Since the model under investigation is used intraining simulators, the hydraulic model example forthis study is a simplified version of the actual hydraul-ics for the outer boom, as shown in Figure 12.

A hydraulic circuit consists of the hydraulic cylin-der(s), the pressure-compensated proportional direc-tional valve(s), the pressure relief valve(s) and thepump(s). Figures 13 and 14 give examples of asimple work cycle. Figure 13 shows the valve controlsignal and spool opening. Figure 14 shows the flowrates through the outer boom circuit valve. Thepressure rates of the cylinder volumes are shown inFigure 15.

The collision simulation between the harvesterhead and ground is shown in Figure 16, where theharvester head is pressed against the ground andthen the boom swing motion is added. The initial

Figure 16. Collision force between harvester head and ground.




condition is that the boom points to the x-direction. Itis lowered until the harvester head collides with theground. The boom is swung to the left and the pres-sure towards the ground is constant.

The simulation was carried out using the MeVEAreal-time simulation environment, which offers thepossibility of offline simulation for a more detailedmodel or real-time simulation and visualisation forsimplified models. The environment is compatiblewith the MeVEA Full Mission Solution, whichoffers a motion platform and visualisation environ-ment combined with a user interface that includes aseat for the operator, joysticks, and pedal and so forthconfigured specifically for the application.57

Conclusions

In real-time simulation, a machine must be consideredas a coupled system that consists of mechanical com-ponents and actuators. This study introduced a generalsimulation approach that can be applied to the real-time simulation of a hydraulically driven harvester.The introduced approach was based on the recursivemethod coupled with lumped fluid theory for the mod-elling of hydraulic circuits and the penalty method forcontact modelling. The introduced simulationapproach was applied to create real-time simulationmodels of a tree harvester. The simulation model ofthe harvester was installed and tested in a real-timesimulation environment consisting of a visualisationdisplay, a motion platform and an I/O interface. Thereal-time simulation environment merged the operatorvirtually with the simulated tree harvester.

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

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Appendix

Notation

a1, a2 radii of box Ab1, b2 radii of box BA1, A2 normalized axes of box AAA area behind the hydraulic pistonAA rotation matrix of body AAB area a front of hydraulic pistonAB rotation matrix of body BAn area of cylinder chamberB1, B2 normalized axes of box BBei effective bulk modulusBj bulk modulus of component jBk bulk modulus of component kBn body with number of nBoil bulk modulus of oilc damping coefficientC damping factorC total inertiaCv flow rate constantdn joint relative displacement vector€dn joint relative acceleration_dn joint relative velocitydp pressure different between pressure

portsEq velocity transformation matrixEz new velocity transformation matrixf ð _xÞ velocity-dependent friction coefficientFc Coulomb friction forceFlmin, Flmax hydraulic cylinder end dampingFn external force at body nFn normal forceFq inverse velocity transformation matrixFs normalized static frictionFs total force produced by the cylinderFm total friction force of the cylinderg(v) Stribeck sliding friction force effectI identity matrixJn inertia of body nk end damping spring constantK coefficient of elasticity

l cylinder maximum strokeL normalized directionmn mass matrix of body nM total mass matrixn contact normalnc number of hydraulic componentnh number of hosesNb number of contiguous bodies_pl hydraulic pressurepn pressure acting in the cylinder

chamber_pq dependent velocitiespz new position vector_pz new velocity€q accelerationqn generalized coordinate of joint nQ flow over restrictorQ total forcesQjA flow rate behind the cylinder pistonQjB flow rate a front of cylinder pistonQij outgoing or incoming flow ratern relative kinematics of position for

body n_rn relative kinematics of velocity for

body n€rn acceleration of body nR projection matrixsn–1 vector from point n–1 to n–1’T distance from centreT timeTn torque at body n�uA relative position vector of body A�uB relative position vector of body Bun new shifted position to the centre of

gravity of rn€un acceleration of body body nU variable of spool or poppet position_U velocity of position displacementUref spool reference positionvn velocity at joint nvs Stribeck relative velocityvAB dot product of relative velocityVi total hydraulic volumeVj volume of single componentVjA volume behind the hydraulic pistonVjB volume a front of hydraulic pistonVk volume of single hosex displacement of cylinder stroke_x stroke velocity of the cylinderz deflection of the bristles

�un virtual angular velocities�_un virtual translational�_q dependent virtual velocities�0 distance from an undeformed bristle

point to origin point�1 distance from the deformed bristle

point to origin point� cylinder efficiency




�0 longitudinal lumped stiffness�1 longitudinal lumped damping

coefficient�2 viscous relative damping� time constant describing dynamic of

valve spoolud relative angular velocity of the joint

_un relative angular acceleration of body n_ud relative angular acceleration of the joint~un skew-symmetric matrix of the cross-

product_~un�1 differentiating of skew-symmetric

matrix of body n–1




Real-time multibody application for tree harvester truck simulator

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