Turning an articulated truck on a spreadsheet
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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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