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Turning an articulated truck on a spreadsheet

Author - Jim McGovern

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Steering control

With the aid of power steering it is possible to change the steering angle without any forward movement. It is more usual however (and much better for the tyres) to use the steering wheel to change the steering angle progressively as the truck moves. The analysis described here did not take dynamic effects into account and was based on finite displacements of the tractor’s reference point (the centre of the tractor’s rear axle group). For the purpose of defining and plotting the path travelled, a distance of 0.5 metres was a sufficiently small travel increment. However, still smaller sub-increments were used in the calculations to ensure high accuracy in the coordinates of each new plotted position along the path. In establishing the path of the articulated truck the steering lock was used (as a percentage value ranging from –100% to +100%). This corresponded to the amount by which a driver would have rotated the steering wheel. In the calculation sheet, provision was made for any arbitrary steering lock change at any path position, as well as for any arbitrary steering lock change from the previous position to the current position. Table 3 illustrates some of the path data from the Paths sheet. For the purpose of the example described here, the highest rate of steering-lock change with distance was taken as 20% per 0.5 metres, or 40% per metre of travel. Whenever the steering lock was zero, the tractor would travel in a straight line, and so the turning radius was infinite. To avoid displaying extremely large values for the turning radius, it was useful to use the inverse turning radius instead (zero when the tractor was travelling in a straight line). Keeping track of the position and direction of the tractor was a matter of cumulative addition and trigonometry.

Modelling of the paths traced out by the principal points on the trailer depended on being able to calculate how the position of the centre of the rear axle group varied as the towing king pin moved. The trailer was a trailing link attached to the king pin of the tractor and had two degrees of freedom: it could slide (or, more correctly, roll) in the direction of its centre-line, while also rotating about the centre of the rear axle group. Where the tractor travelled in a straight line, the centre of the rear axle group would typically have followed a curve. Of course, the tractor sometimes travelled in a curve and this complicated the path of the trailing link still further. This type of trailing-link motion was best modelled by considering a succession of very small increments of the overall movement, and this was the approach that was used.

For the example that is shown in Figure 4 an articulated truck was to leave a test lane and do a U-turn while taking the minimum space to the front and side of the test lane exit. At the start of the manoeuvre the articulated truck was in line with the testing lane and the tractor had advanced straight ahead until the centre of the rear axle group was level with the exit from the lane. As an immediate right turn would quickly bring the trailer body into contact with the right gate, the tractor first turned toward the left and then quickly changed to a full lock to the right. The tractor continued at full lock for some distance until its direction came close to the desired exit direction. The lock was then quickly reduced to zero so that the exit was in exactly the desired direction. A human driver would achieve this readily from his or her experience of truck driving. In the spreadsheet, the maximum steering change rate of 40% lock per metre was used, and the fine adjustment of the exit angle was achieved by starting and finishing with a lower absolute steering change rate. By some trial and error the desired exit direction of the tractor was achieved. Table 4 details the final steering adjustments from the Paths sheet of the workbook. The path distance between the successive positions was 0.5 metres. Just before the final adjustment the steering lock was –100%. This was reduced progressively to 0% in such a way that the truck left with the desired direction of 270º. The steering changes were made as the truck travelled, and therefore the values in the row entitled ‘Steering lock increment at this point’ remained at zero.

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