MODELLING THE DYNAMICS OF A LOAD-HAUL-DUMP VEHICLE

Transcription

1 MODELLING THE DYNAMICS OF A LOAD-HAUL-DUMP VEHICLE B.J. Dragt, F.R. Camisani-Calzolari, an, I.K. Craig Department of Electrical an Electronic Engineering, University of Pretoria, Pretoria, 0002, South Africa. Tel Fax bruce Dragt@tuks.co.za Tel Fax fernano.camisani@eng.up.ac.za Tel Fax icraig@postino.up.ac.za Abstract: This paper provies an overview of the erivation of a ynamic moel of a Loa-Haul-Dump (LHD) vehicle. The moel is erive using Lagrangian Dynamics an makes use of a simplifie tyre moel in an attempt to inclue the effect of basic tyre ynamics. Simulation results obtaine from the vehicle moel are provie an the results iscusse an analyse.copyright c 2005 IFAC Keywors: kinematics, ynamics, LHD vehicle, slip angles, articulate vehicle 1. INTRODUCTION Loa-Haul-ump(LHD) vehicles as shown in figure 1 are the work horses of the moern unergroun trackless mining environment. LHD vehicles are prouce by a number of manufacturers an are available in various ifferent moels using either iesel or electric power an in various sizes. Typically the vehicles vary in length from 8 to 15 metres, weigh between kg with a transportation capacity of up to 25000kg. The LHD vehicle is typically use for the transportation of fragmente ore from the stopes to an ore pass where the ore is transporte by gravity to another hanling point. The LHD an its operator move back an forth along the mine tunnel, which is typically a few hunre metres long, hauling the ore. The more repetitions of this cycle that are complete within a shift the higher the prouction. Therefore for reasons of safety as well as prouctivity it is esirable to automate LHD vehicles. In orer to esign a navigation system for an autonomous vehicle it is necessary to have a vehicle moel that escribes the vehicles position an other vehicle parameters through time. The LHD vehicle poses unique moelling problems in that it s boy consists of two parts connecte together by means of an articulation joint. The front an rear wheel sets are fixe to remain parallel with the vehicle s boy an vehicle steering is achieve by means of hyraulic actuators altering the articulation angle of the vehicle. An articulate vehicle is preferable in the narrow environment of an unergroun mine tunnel because of its higher maneuverability as escribe by Altafini (1999). Altafini (1999) has proven that the articulate vehicle can be moelle by a nonlinear system, with two inputs, namely spee an articulation angle, which is controllable. There are however some ebates as to the complexity of the moel require

2 Fig. 1. An example of a typical LHD. to successfully implement an autonomous vehicle navigation system, (Riley an Corke, 2001), as increase moel complexity places more stringent requirements on the computing power require on the actual vehicle in orer to implement the navigation systems. Fig. 2. Definition of slip angle α. 2. VEHICLE MODELLING The main ifference between the various vehicle moels is whether the moel is base purely on vehicle kinematic geometry or inclues vehicle ynamics. Due to the confine nature of the unergroun mining environment the LHD vehicles usually operate at relatively low spees, typically below 28 km/h. For this reason the path-tracking problem has often been base on the kinematic moel only, (Polotski an Hemami, 1997),(Dragt et al., 2003). This assumption greatly reuces the complexity of the vehicle moel as it is not possible to measure the amount of slip which has occurre uring the vehicles motion. One approach which has been use to inclue the effect of slip is to estimate the amount of slip using an Extene Kalman Filter (EKF),(Scheing et al., 1999). Details on Extene Kalman Filters can be foun in (Brown, 1983). 2.1 Kinematic Moelling In kinematic or low spee moels the motion of a wheele vehicle is etermine only by the pure rolling of the wheels,(genta, 1997). The velocities of the centres of all the wheels are assume to be in the miplane of the wheel, that is there is no ifference between the rotational plane of the tyre an the heaing of the tyre. This means that the slip angles α are infinitesimally small. In this conition the wheels can exert no cornering force to balance the centrifugal force cause by cornering an therefore these moel are only vali at infinitesimally small velocities,(genta, 1997). Figure 2 illustrates the concept of a slip angle which is a ynamic effect which is present in all pneumatic tyres. Figure 3 shows the kinematic vehicle geometry use by Scheing et al. (1999) to erive their vehicle moel base on vehicle s kinematic geometry. Fig. 3. LHD kinematic geometry. Equations (1) is the moel erive by Scheing et al. (1999) from kinematics only. ẋ = V cosφ ẏ = V sinφ φ = V tan( γ 2 ) L (1) where x an y enote the position of the vehicle relative to some fixe global co-orinate frame of reference, an L refers to the istance between the front an rear wheels of the vehicle an the articulation joint which is referre to as the half-length of the vehicle. The angle φ is the orientation of the vehicle with respect to the x- axis also referre to as the heaing an γ is efine as the articulation angle of the vehicle. These equations are base on the assumption that the front an rear wheel velocities of the LHD are ientical an that the articulation angle remains constant. 2.2 Dynamic Moelling In orer to moel the vehicles ynamics it is necessary to take into account the generalise forces acting on the vehicle. Most of the forces acting on the vehicle are epenant on the tyre/roa interaction in a complex nonlinear manner. The tyre forces at the tyre roa interface can be resolve into lateral(cornering) an longituinal(traction) forces at an assume contact point. These forces then etermine the slip angles of each tyre an wheel. It is however not possible to measure these slip angles an they therefore have to be estimate in some manner. Scheing et al. (1999) makes use of an Extene Kalman Filter (EKF) to estimate these slip angles

3 in the vehicle moel which inclues the effect of slip. This is one by incluing only two slip variables (α an β), one for each axle into the kinematic vehicle moel, as illustrate in figure 4. This approach is similar to what is commonly referre to as the single-track or bicycle moel of a wheele vehicle,(gillespie, 1992). Simulation results for this moelling approach can be foun in Dragt et al. (2004). Lagrangian ynamics. The same approach as escribe in (Wells, 1967) an (Chen an Tomizuka, 1997) is followe in this erivation. Fig. 5. Coorinate system use in moel erivation. Fig. 4. LHD kinematic geometry inicate slip angles. This approach oes however have the shortcoming that the moel is epenant on the accuracy of the statistical parameters use to estimate the slip variables in the error moel of the Kalman filter. No etail is given by Scheing et al. (1999) on how these parameters where etermine. An alternative approach, an the one use here, to estimate the slip angles of the tyres is to make use of a characteristic of pneumatic tyres known as the cornering stiffness (C α ). Cornering stiffness is efine as the slope of the lateral force verses slip angle curve evaluate at zero slip angle,(gillespie, 1992). It is possible to etermine the cornering stiffness of a tyre an once this parameter is known, the slip angle of the tyre can be estimate base of the lateral force an vertical loa being applie to the tyre. This approach can be applie to a bicycle or single-track moel as use by Scheing et al. (1999) by using ouble the cornering stiffness so as to account for both wheels attache to a particular axle. Unfortunately these tyre characteristic measurements are not usually carrie out on large tyres as use on an LHD vehicle. Therefore for the purposes of this moel tyre ata for a similar size tyre has been use until the measurements can be carrie out. Alternatively a complex empirical tyre moel can be use to etermine the same parameters shoul more extensive information on the tyres be available. One example of an empirical tyre moel is that of Baraket an Fancher (1989). 2.3 Dynamic Moel Derivation This section provies a summary of the erivation of the LHD vehicle ynamic moel using The coorinate system is efine to characterise the motion on an articulate LHD vehicle. As shown in figure 5, X n, Y n, Z n is the fixe global coorinate system. In applying the Lagrange metho expressions for the kinetic an potential energies are obtaine relative to this coorinate system. X f, Y f, Z f are the coorinate system attache to the front unit of the LHD with Z f position in such a way that it passes through the front units centre of gravity. Similarly X r, Y r, Z r enote the position of the rear unit of the LHD vehicle. The motion of the front unit of the LHD can be escribe by the relative motion of the X f, Y f, Z f coorinate system with respect to the X n, Y n, Z n coorinate system. The motion of the rear unit of the vehicle can be characterise by escribing the articulation angle between the front an rear units, or by escribing the motion of the X r, Y r, Z r coorinate system relative to the X f, Y f, Z f coorinate system. Using this coorinate system a set of variables for the LHD vehicle can be introuce to as follows: x n : position of front unit s C.G. in x-irection on ẋ n : velocity of front unit s C.G. in x-irection on y n : position of front unit s C.G. in x-irection on ẏ n : velocity of front unit s C.G. in x-irection on ɛ 1 : front unit s yaw angle with respect to inertial coorinate system ɛ 1 : front unit s yaw rate with respect to inertial coorinate system ɛ f : articulation angle between front an rear unit of vehicle ɛ f : rate of change of articulation angle between front an rear unit of vehicle The vehicle rotational an translational velocities are then etermine to be as follows for the front an rear units of the LHD vehicle. V CG1/n = x ni n+ y nj n+0 =( x n cosɛ 1 + y n sinɛ 1 )i f +( x n sinɛ 1 + y n cosɛ 1 )j f (2)

4 D 2 D 1 Table 1. Parameters of vehicle moel. h 2 C.G2 H L L H L C.G1 Fig. 6. Schematic iagram of vehicle parameters use in moel. V CG2/n =V CG1/n D 1 t i f 1 t k f D 2 t i r 2 t k r V CG2/n =V CG1/n D 1ɛ 1j f D 2( ɛ 1+ ɛ f )j r (3) These equations are use in eriving the expressions for the kinetic an potential energies necessary to apply Lagrange s equations. It can be shown that the expression for the kinetic energy of the front an rear units of the vehicle are as shown is 5 an 6. The change in potential energy (V ) of the vehicle woul be cause mainly by roll motion in the vehicle suspension as etaile in (Chen an Tomizuka, 1997). However ue to the limite amount of suspension travel an the heavy loas carrie by an LHD vehicle it was ecie to neglect the effects of roll motion. Thus there is no change in potential energy. h 1 V = 0 (4) The kinetic energy of the front unit (T 1 ) of the vehicle can be calculate from it s translational an rotational velocity as show in 5. Parameter m 1 I x1, I y1, I z1 m 2 I x2, I y2, I z2 T w1 T w2 C αf C αr H L L h 1, h 2 D 1, D 2 1, 2 Description Mass of front unit Front unit moment of inertia Mass of rear unit Rear unit moment of inertia Track with of front axle Track with of rear axle Front tyre cornering stiffness Rear tyre cornering stiffness Distance from axle to articulation joint Vehicle wheelbase Heights of centre of gravity (CG) Distance from joint to CG Vertical istance from joint to CG L = F gyn (9) t y n y n L = F gɛ1 (10) t ɛ 1 ɛ 1 L = F gɛf (11) t ɛ f ɛ f where F gxn, F gyn, F gɛ1 an F gɛf refer to the generalise external forces acting on the vehicle. These external forces acting on the vehicle consist of the tyre/roa interface. In orer to calculate the generalise forces, expressions for the generalise forces are erive in terms of longituinal an lateral components of the tyre forces as inicate in figure 7. The full erivation of these forces is too long to be shown here. Fb4 Fa3 Fb1 T 1 = 1 2 m 1V CG ω f I 1 ω f (5) 9 Fa1 similarly the kinetic energy of the rear unit (T 2 ) can be calculate as T 2 = 1 2 m 2V CG ω ri 2 ω r (6) Y n Fb4 X n Fa4 Fb2 9 Fa2 where the V CG1 an V CG2 are the translational velocities an ω f an ω r are the angular velocities of the front an rear unit of the LHD vehicle respectively. The Lagrangian is therefore efine as L = T 1 + T 2 0 (7) Fig. 7. Definition of tyre forces in Cartesian Coorinates. Equations 8, 9, 10,11 can be combine with the expressions erive for the generalise forces an after some manipulation written in the following form. By using Lagrange s equations the following four ynamic equations can be obtaine. Table 1 provies a escription of the parameters use in the vehicle moel an as inicate in figure 6. L = F gxn (8) t x n x n where ( q ) = M 1 h(q, q) + M 1 F g (12) q = x y ɛ 1 ɛ f (13)

5 M is the inertia matrix an F g are the generalise forces acting on the wheels. Equation 12 can be combine with the parameters from the tyre moel to etermine the generalise forces an perform a simulation of the vehicles ynamics by integrating the state variables.the results are shown in the next section. ra/s SIMULATION This section provies simulation results for the erive LHD vehicle moel. The simulations where performe in Matlab an the parameters inicate in table 1 are base on the vehicle parameters of a Sanvik-Tamrock EJC 245 LHD, which has a tramming capacity of 11000Kg,(Sanvik Mining an Construction Global Site, 2003). The simulations were however performe base on the vehicles empty weight an the vehicle is assume to be operating on level terrain with a friction coefficient of 0.7. Parameters which where not available were estimate base on vehicle imensions. Due to a lack of availability of complete tyre ata the simulations were performe using average values of the tyre cornering stiffness which are vali for tyre slip angles of up to 5 egrees. These parameters were use for the entire simulation. The full ynamic tyre moel where parameters vary accoring to loaing an velocities will be implemente when the complete tyre ata are mae available. As the main emphasis of the ynamic moel is on the cornering performance of the LHD vehicle the simulations where carrie out at a constant velocity of 18 km/h. Figure 8 shows a plot of the commane rate of change of steering input against time. At time t = 2 secons the vehicle starts turning to the left once the articulation angle has reache full lock in the one irection the vehicle starts turning back in the other irection. Figure 9 shows the path followe by the vehicle for the rate of change of the steering angle ɛ f shown in figure 8. Figure 10 shows the slip angles for each tyre etermine by the vehicle moel for the motion of the vehicle. As can be seen from figure 10 the slip angles excee 5 egrees towars the en of this motion an thus the cornering forces preicte by the tyre moel lose accuracy. This results in the vehicle moel being biase towars an unersteer characteristic as the lateral forces generate uring cornering will increase with increase slip angles. This is ue to the fact that C α typically increases linearly with increase slip angle up to a certain maximum value at large slip angles an time (s) Fig. 8. Plot of rate of change of articulation angle ( ɛ f ) in raians per secon verses time. meters meters Fig. 9. Plot of vehicle trajectory. egrees Slip angles alpha 1 alpha 2 alpha 3 alpha time (s) Fig. 10. Plot of slip angles α 1, α 2, α 3 an α 4 verses time. the lateral force generate by the ith tyre (F yi ) is given by equation 13,(Genta, 1997). F yi = C α.α i (13) 4. CONCLUSIONS AND FUTURE WORK In conclusion to etermine the cornering forces acting on the vehicle is probably one of the most ifficult tasks. The lateral forces generate by the tyres are entirely epenant on the characteristics of the tyre - primarily the cornering stiffness C α. The results shown in the previous section are thus

6 highly epenant on the parameter C α use in the simulation an are only as accurate as the estimate of this value. The simulations o however show the performance of the vehicle uner the particular operating conition assume. Although this simulation was base on a simplifie tyre moel an certain vehicle parameters ha to be estimate the results appear to be realistic. In future work a more avance an accurate tyre moel will be ae to the vehicle moel as well as the actual values of the estimate vehicle parameters. The moel will also be expane to take into account the effect of pitch on the vehicle ynamics. This moel will then be valiate against tests performe on an LHD vehicle above groun using GPS measurements to recor the vehicles movements. Any necessary tuning of the moel can then be performe. 5. ACKNOWLEDGMENTS This material is base upon work supporte by the National Research Founation uner grant number The authors also wish to thank Dr. Gunter Metzner of De Beers Consoliate Mines for his contribution to this research project. Genta, G. (1997). Motor Vehicle Dynamcis, Moeling an Simulation, Series on avances in Mathematics for Applie Sciences - Vol. 43. Worl Scientific. Lonon. Gillespie, T.D (1992). Funamentals of Vehicle Dynamics. Society of Automotive Engineers, Inc.. Warrenale, PA. Polotski, V. an A. Hemami (1997). Control of articulate vehicle for mining applications: Moeling an laboratory experiments. In: Proceeings of the 1997 IEEE International Conference on Control Applications. Hartfor, CT. Riley, P. an P. Corke (2001). Autonomous control of an unergroun mine vehicle. In: Proceeings 2001 Australian Conference on Robotics an Automation. Syney. Sanvik Mining an Construction Global Site (2003). Last visite: September Scheing, S., G. Dissanayake, E.M. Nebot an H. Durrant-Whyte (1999). An experiment in autonomous navigation of an unergroun mine vehicle. IEEE Transactions on robotics an Automation 15 no.1, Wells, D.A (1967). Schaums s Outline of theory an problems of Lagrangian Dynamics. McGraw-Hill. New York. REFERENCES Altafini, C. (1999). Why to use an articulate vehicle in unergroun mining operations?. In: Proceeings of the 1999 IEEE International Conference on Robotics an Automation. Detroit. Baraket, Z. an P. Fancher (1989). Representation of truck tire properties in braking an hanling stuies: The influence of pavement an tire conitions on frictional characteritics. Technical Report University of Michigan Transportation Research Institute. Brown, R.G (1983). Introuction to Ranom Signal Analysis an Kalman Filtering. John Wiley an Sons. New York. Chen, C. an M. Tomizuka (1997). Moelling an control of articulate vehicles. California PATH research report. Dragt, B.J., I.K. Craig an F.R. Camisani- Calzolari (2003). Autonomous unergroun mine vehicles. In: The 1st African Control Conference (AFCON2003). Cape Town, South Africa. pp Dragt, B.J., I.K. Craig an F.R. Camisani- Calzolari (2004). Moelling of an autonomous unergroun mine vehicle. In: Preprints of The 11th IFAC Symposium on Automation in Mining, Mineral an Metal Processing(MMM2004). Nancy, France.