DEVELOPMENT OF A SIX DEGREE OF FREEDOM TRACKED VEHICLE MODEL

Purdy D J

Engineering Systems Department, Defence College of Management and Technology, Cranfield University, Shrivenham,

Swindon, Wiltshire, SN6 7HY, United Kingdom.

Tel: 01793 785352 Fax: 01793 783192 E-mail:d.j.purdy@cranfield.ac.uk

Keywords: Tracked Vehicle, Modelling, Simulation, Handling, Stability. Abstract In this work a six degree of freedom tracked vehicle model is developed, with longitudinal, lateral, bounce, roll, pitch and yaw motions of the hull. A suspension system is incorporated into the model, which allows the forces on the vehicle hull and between the track and ground to be determined. The model is validated using data from a steady state handling trial that was undertaken on a Combat Vehicle Reconnaissance (Tracked) CVR(T). From simulations of the model it is shown that the stability of the vehicle reduces as the speed increases and on low friction surfaces. 1 Introduction The majority of tracked vehicles operated by the military are capable of travelling at relatively high speed, over 80 km/h in some cases. The possibility of this high speed, the nature of the steering system and the inexperience of the drivers results in a number of accidents each year. The size and weight of military tracked vehicles, greater than 60,000 kg for the British Challenger 2 main battle tank, can result in significant amounts of damage and personal injury when an accident does occur.

The vast majority of high-speed military tracked vehicles employ skid steering to change their direction. The mechanics of skid steering are complex and the resulting equations are sufficiently non-linear to prevent linearisation techniques being applied successfully. Thus, the fundamental handling characteristics of tracked vehicle are significantly more difficult to establish theoretically than for wheeled vehicles [1-6]. The reason for this greater complexity is the sliding (skidding) interface between the track and ground, which occurs during turning. To simplify this most of the workers in this field have assumed a hard level surface for the investigation [1-6].

When a skid steer tracked vehicle is in a steady turn the outer track sprocket is rotating faster than the inner. The ratio of the outer to the inner sprocket angular speed is called the steer ratio, n and is analogous to the steer angle of the front wheels of a wheeled vehicle. For modern tracked vehicles, for example the British Challenger 2 and Warrior the steer ratio can be varied infinitely between two limits [7], thus it is

possible to control the steer input to achieve any radius of curvature between straight ahead and an upper limit (tightest turn). With older skid steered vehicle there is commonly only one (and unusually two) fixed steer ratio for each of the gears. Thus at high speed in a particular gear it is possible to initiate a turn, which is too severe for the given conditions. One of the other problems applicable to both old and new vehicle transmissions is that if a bend is approached in too high a gear, then there may be insufficient steer ratio to get round the bend (running wide) requiring one or more downward gear shifts. An introduction to tracked vehicle steering for both the low and high-speed case is given in [4,5].

In the study presented here a six degree of freedom model of a tracked vehicle, Combat Vehicle Reconnaissance (Tracked) CVR(T), is developed. A CVR(T) is a low mass, approximately 10,000 kg, military tracked vehicle shown in Figure 1. This vehicle has five wheel stations, which have torsion bar suspension system utilising trailing arms and dampers on the first and last wheel stations only. Damping on the inner wheel stations being parasitic caused by the rotation of the trailing arms in their bearings.

The steering system employed in the CVR(T) is of a fixed ratio type in each gear, Table 1. The model developed is partially validated against a steady state handling data presented in [4], for third gear up to a lateral acceleration (latac) of about 0.34g.

The model is used to investigate the motion of the hull during cornering on low friction surfaces and at different speeds.

Figure 1. Photograph of CVR(T).

Gear Max Speed m/s# Steer Ratio n Turn Radius* m

1 1.11 7.2 1.71 2 2.50 2.1 3.84 3 3.61 1.6 (1.69◊) 5.33

4 5.83 1.32 8.9 5 10.55 1.18 16.06 6 14.61 1.14 21.28 7 21.66 1.09 33.22

Table 1. CVR(T) steering data, *manufacturer’s data, ◊experimentally determined, #estimated.

Parameter Value Parameter Value

Hull Mass 7938 kg Suspension Stiffness*

70.5 kN/m

Roll Inertia* 2230 kgm2 Suspension Damping*

7.0, 0.3 kNs/m

Pitch Inertia* 10970 kgm2 Distance between wheels

1.7 m

Yaw Inertia* 12006 kgm2 Length of track on ground

2.6 m

Number of Wheel Stations

10 Height of Centre of Mass*

0.9 m

Table 2. Main CVR(T) data, *estimated or assumed. 2 Vehicle Model In this section a brief description of the vehicle model is given. The model has six degrees of freedom, Figure 2, these being; rectilinear motions – longitudinal, lateral and bounce, and rotary motions – roll, pitch and yaw. The track model is based on that given in [4] but has been broken into sections to represent each wheel station. The main assumptions in the model are;

1 Smooth, rigid, level ground. 2 Coulomb friction model, between track and ground. 3 Vehicle centre of mass at plan centre of vehicle. 4 No external forces applied to the vehicle other than

from the tracks on the ground. 5 Uniformly loaded, laterally rigid track.

Figure 2. Vehicle axis system.

A top level simulation diagram of the vehicle model is shown in Figure 3. The transmission model is assumed to consist of a first order lags with time constants of 1.0 second acting between the commands from the driver and the sprocket output. The external forces, which are not used in this investigation, are to allow the affect of forces other than through the tracks to be included, for example; aerodynamic, towing or gravity when on an incline. The next three subsection will describe the models for the; suspension, hull and tracks.

Figure 3. Simulation diagram of the vehicle model.

2.1 Hull Model The hull model, without the suspension system, has been created in ProPac [8,9], which allows a multibody dynamic model to be generated in Mathematica [10] and c-code to be generated. The resulting c-code is then compiled and forms an s-function block within Matlab/Simulink [11]. Using ProPac removes the problem of forming the model from first principles and the difficulties of implementing and debugging it. The model building process allows the user control over the assumptions that are built into the model, i.e. small angles and motions, and thus the resulting model complexity. In this case no simplifying assumptions have been made. The equations for the hull are omitted from the paper due to their complexity.

The external forces acting on the hull and those in the suspension system are handled in Matlab/Simulink using either the inbuilt functions or in the case of the tracks another s-function. Thus the resulting model is a hybrid, using the most appropriate tool for each part. 2.2 Suspension Model The suspension system, for each wheel station, is modelled as shown in Figure 4, which incorporates a spring and damper at each wheel station. The spring rates are the same at each wheel station and only the first and last wheel stations have dampers attached, while parasitic damping is added to the remaining wheel stations, Table 2. In this model zsi and zgi are the displacements of the hull and track at the wheel station, and ksi and csi are the wheel station stiffness and damping. The force in the suspension system is given by; 0 or 0 0si si si si si si si siF k z c z F F F= + ≥ = < (1)

In this equation the force between the track and ground is prevented from going negative. The force in the suspension is applied to both the body and the track model for that wheel station, thus allowing body motion and track shear forces to be determined.

Figure 4. CVR(T) wheel station model.

2.3 Track Model A diagram showing the track ground interface is shown in Figure 5. The axis system for the track is given by; xt, yt and ψt, with xi the distance to a track-link. In this diagram the highlighted track element (link) has longitudinal and lateral sliding velocities given by; (2) and s si

t s s tu u R v v x rω= − = + i

Where Rs and ωs are the sprocket radius and speed of rotation. The longitudinal speed of the track being determined from the output of the transmission. The direction of sliding of the link and its speed are;

1tan and si

i sitt ts

t

v V u vu

α − ⎛ ⎞= =⎜ ⎟

⎝ ⎠2 2s si

t t+ (3)

The sliding motion at the track ground interface is opposed by Coulomb friction, the components of which are; ( ) ( )cos and sini i i i i i

xt t t yt tF F F F tα α= − = − (4)

The vertical force on the track link for this wheel station being derived from the suspension force and the number of links and is given by; zsi

zlisi

FFn

= (5)

Where Fzsi and nsi are the suspension force and number of track-links for the ith wheel station. This process is repeated for all the links in each wheel station and then the force and moments can be determined, these being; , and i i i

xt xt yt yt zt ii i i

ytF F F F M x F= = =∑ ∑ ∑ (6)

Figure 5. Track model.

3 Model Validation In this section the assembled vehicle model is simulated for the CVR(T) and the resulting output is compared to experimental data for the vehicle undertaking a steady state handling trial. The key data for the tracked vehicle simulation is given in Table 2, where assumed or estimated data is highlighted.

The vehicle handling trial was conducted at the Defence College for Management and Technology DCMT, formally the Royal Military College of Science RMCS [4]. The investigation involved driving the vehicle in third gear, Table 1, with the steering system engaged at a range of speeds. These runs produced data for the speed, lateral acceleration and radius of turn for the vehicle. The results of this test and the output from the simulation are plotted in Figure 6. The simulated results used a coefficient of friction between the tracks and the ground of 0.84.

The experimental data show that as the lateral acceleration of the vehicle increases the radius of turn also increases. The initial rate of increase is 2.4 m/g [4], the implication being that the vehicle is initially understeering. The simulated responses show a good correlation with the experimental data, up to where the trial finished. This gives a reasonable level of confidence that the model developed is capable of reproducing the steady state handling characteristics up to 0.34g lateral acceleration. The simulated data then show the vehicle going into oversteer just under 0.5g.

Figure 6. Radius of turn against latac for third gear, o –

experimental data and continuous lines simulation. 4 Model Responses A small sample of the simulated response of the CVR(T) model are presented in this section. The first set of responses are for third gear at the maximum speed given in Table 1 and two values of coefficient of friction between the track and ground. The final response is for sixth gear at two speeds. 4.1 Effect of Friction The cornering response of the vehicle for two different values for the coefficient of friction, 0.84 and 0.17, are shown in Figures 7 to 9. The motion of the vehicle over the ground is displayed in Figure 6, where the effect on the motion of the vehicle is clearly seen. At the higher level of friction the vehicle corners as expected, with the final motion being at a constant radius of about 6.0m. At the reduced level of friction the vehicle goes through a transitional phase, which shows a characteristic loop [6] in the response before settling down to a constant radius of almost 4.0m. This is the onset of an oversteer response with the vehicle starting to “spin out” before settling down to a new smaller radius of turn.

The initial loop shown in the low friction results, Figure 7, corresponds to a rapid increase in the side slip angle of the vehicle, Figure 8. This type of response is expected also and typical of some wheeled vehicles on low friction surfaces. At its peak the side slip angle is just over -70°, thus the vehicle is almost going side ways. The overshoot in the response is virtually 50% indicating a damping ratio of 0.2. The side slip angle then decays sinusoidally to a final value just over -45°. In contrast at the higher level of friction the side slip increases to about -4° without overshoot.

The effect on roll angle is given in Figure 9. At the higher friction value the roll angle increases to just almost -2° again without overshoot. With the lower value of friction the final roll angle is about -0.8°, which is less than the high friction results because the vehicle has rotated around so that it is pointing more towards the centre of turn. Interestingly the roll angle now overshoots its final value by about 40%. This indicates a damping ratio of almost 0.3, thus the overdamped motion at high friction has become underdamped at low values of friction. This reduction in the stability of the vehicle is caused by the coupling together of the motions of the hull and horizontal track forces.

Figure 7. Simulation of the tracked vehicle showing the path

over the ground while cornering in third gear at 3.61 m/s with two levels of friction.

Figure 8. Simulated side slip angle for the tracked vehicle

cornering in third gear at 3.61 m/s with two levels of friction.

Figure 9. Simulated roll angle for the tracked vehicle

cornering in third gear at 3.61 m/s with two levels of friction.

As a tracked vehicle is being driven around a corner and encounters a surface with a reduced friction coefficient, due to water, oil or mud etc. on the roads surface, than it is possible that the driver could lose control even if travelling at a relatively low speed. Anecdotal evidence from drivers of tracked vehicles, suggests that once this type of motion is initiated then there is little they can do until the vehicle comes to rest. 4.2 Effect of Speed In the previous section the effect of reduced friction was considered at low speed, in this part the influence of speed at the higher friction level is briefly examined. A simulation of pitch angle motion in a corner is shown in Figure 10. In this plot the vehicle is in sixth gear when the corner is initiated at 50 and 60% of the maximum speed, Table 1.

In Figure 10, it is seen that the damping (stability) of the hull motion reduces with speed. At 50% maximum speed the damping is relatively low and when increased to 60% the damping approaches zero for this vehicle. While there is some anecdotal evidence for short tracked vehicles becoming “twitchy” at speed, which may be indicated by this study, it is felt that such a rapid reduction in the effective hull motion damping with speed would have been reported by the crews. 4.3 Final Remarks on the Responses The responses at low levels of friction and high speed must be viewed with some caution because they are the result of simulation studies with the model being validated for steady state cornering only [4]. The experimental validation of the results given here could be difficult because of the dangers and cost involved. Bearing this in mind, the simulated responses clearly show that at higher speed and on reduced friction surfaces the vehicle has a tendency to become less stable.

Figure 10. Simulated pitch angle response for 50 and 60%

maximum speed in sixth gear. 5 Conclusion In this work a six degree of freedom tracked vehicle model for a CVR(T) has been developed and validated against steady state handling data. It was shown that at low latac the vehicle initially understeers before going into oversteer at higher latac. It has been demonstrated by simulation that the vehicle becomes less stable at higher speed and on low friction surfaces. References [1] Thai, T. D. and Muro, T., Numerical analysis to predict turning characteristics of rigid suspension tracked vehicles, J. of Terramechanics 36, 1999, pp 183-196.

[2] Merritt, H. E., Some considerations influencing the design of high-speed track-vehicles, The Institution of Automobile Engineers, January 1939, pp 398-429.

[3] Steeds, W., Tracked vehicles – an analysis of the factors involved in steering, Automobile Engineer, April 1950, pp 143-148.

[4] Purdy, D. J. and Wormell, P. J. H., Handling of High-Speed Tracked Vehicles, J. Battlefield Tech. Vol. 6, No. 2, July 2003.

[5] Wormell, P. J. H. and Purdy, D. J., Handling of Tracked Vehicles at Low Speed, J. Battlefield Tech. Vol. 7, No. 1, March 2004.

[6] Kitano, M. and Kuma, M., An analysis of horizontal plane motion of tracked vehicles, J. of Terramechanics, Vol. 44, No. 4, 1977, pp 211-225.

[7] McGuigan, S. J. and Moss, P. J., A Review of Transmission Systems for Tracked Military Vehicles, J. Battlefield Tech. Vol. 1, No. 3, November 1998.

[8] Kwatny, H. G. and Blankenship, G. L, Nonlinear Control and Analytical Mechanics: A Computational Approach, Birkhauser, 2000, ISBN 0-8176-4147-5.

[9] ProPac, Techno-Sciences, Inc, Lanham, MD.

[10] Wolfram Research, Inc., Mathematica, Champaign, Illinois.

[11] Matlab and Simulink, The MathWorks, Inc, Natick, MA, USA.