The steering angle δ controls the direction of travel of the tractor, it is defined for a single track vehicle model. In this model the left and right tyre of each axle are lumped into a single equivalent tyre. Steering angle δ is defined as the angle in between the center line of the vehicle and a line through the single front wheel in the direction of travel. The bicycle model and the definition of δ is shown in Figure5.2. According to ISO convention, δ is defined to be positive to the left.
Figure 5.2: bicycle model with the definition of δ
CHAPTER 5. OPEN LOOP CONTROL OF THE TRACTOR
The tractor has two front wheels having their own steering angle, δl and δr for the left front wheel and right front wheel respectively. The relation between δl and δr is determined by the geometry of the steering linkage. When taking tight corners, where large steering angles are needed, a relation is needed to minimize tyre slip. The Ackermann steering geometry lets each wheel roll without the tyres having a side slip angle. If a vehicle has Ackermann steering geometry, imaginary lines through the left and right front wheel axles intersect at a point which lies on an imaginary line through the rear axle. This intersection point C is the center of rotation. The Ackermann steering geometry is shown in Figure5.3.
Figure 5.3: Ackermann steering geometry δl and δr can now be defined by:
tan δl= L R −d2f
(5.2)
tan δr= L
R + d2f (5.3)
using the wheelbase of the vehicle L, the corner radius R and the distance in between the front wheels df. These parameters are shown in Figure5.3.
For the two track vehicle assuming a virtual steered wheel positioned at the center line of the vehicle. The steering angle δ can now be defined by:
tan δ = L
R (5.4)
A two track vehicle with 100% Ackermann geometry will steer in the same direction as the bicycle model for a given δ. This has been the motivation to calibrate the steering system of the tractor so that it’s geometry is as close as possible to the Ackermann geometry. The closed loop controllers of [1], [2], [3], [4] and [5] for certain manoeuvres of tractor and trailer have been designed using a single track vehicle model of the tractor and trailer. For these controllers to work on the scaled tractor with minimal adaptation, it should be possible to use δ as an input to steer the scale model.
CHAPTER 5. OPEN LOOP CONTROL OF THE TRACTOR
Another important aspect of the control of the steering system is the relation between the input δ and the front joint value. This relation determines whether δl and δr which are observed at the front wheels correspond to the steering direction as expected for the input δ.
5.3.1 Steering system calibration procedure
To check the Ackermann geometry δl and δr need to be measured. The relation between δl and δr is a result of the steering linkage geometry. The measurements have been done by using a digital leveling instrument. This instrument measures the angle with respect to the horizontal plane. During the measurements the tractor has been placed on its side such that δl and δr at the wheels could be measured with respect to the horizontal plane. The measurement points have been processed in MATLAB.
Using the measured δl and δr the corresponding δ as predicted by Ackermann steering geome-try has been calculated by:
δ = atan(2 tan δltan δr
tan δr+ tan δl
) (5.5)
Equation (5.5) has been derived by rewriting (5.2) and (5.3) and substituting (5.4).
As (5.5) only holds under the assumption of Ackermann geometry, it needs to be checked if the steering system has Ackermann geometry. The measured relation between δl and δrhas therefore been compared to a model of the ideal Ackermann relation given by:
δr= atan( 1
1
tan δl +dLf) (5.6)
Changes have been made to the steering system geometry based on first tests. The geometry can be adapted by changing the length of the tie rod and/or changing the mounting position of the tie rod. Such an adaptation gives the steering system more toe in or more out. Toe out means that the front wheels are further apart on the front side of the front wheels. Toe in is the opposite.
An adaptation made to the tie rod changes the relation between δland δrand therefore results in a reduction or increase in the adherence of the steering system to Ackermann geometry. This is not necessarily the case over the entire range of the steering system. An increase of the adherence to Ackermann geometry in a specific range of δ can result in a reduction in another range. After modifying the steering system, the measurement of δl and δr has been repeated. The process has been repeated three times, after that is has been concluded that changing the geometry again would only give marginal benefits. The measured relation between δland δrcompared to the ideal Ackermann relation, given by (5.6), is shown in Figure 5.4 for the final configuration. It can be concluded that the steering system is close to having Ackermann geometry after the calibration procedure, The RMS error between the ideal δrand the measured δr is 1.5◦.
CHAPTER 5. OPEN LOOP CONTROL OF THE TRACTOR
Figure 5.4: measured vs Ackermann relation, final configuration
The steering system is shown to be close to having Ackermann geometry based on the results of Figure 5.4. Based on the measured δl and δr the corresponding δ has been calculated using (5.5). Together with the front joint value Jvf a polynomial has been fitted. Jvf is the low level control input to the Dynamixel which results from the high level control input δ. This fit is shown in Figure5.5. This polynomial has been fitted through the points formed by the front joint values and the corresponding δ as calculated by (5.5).
Figure 5.5: polynomial fit based on final configuration
CHAPTER 5. OPEN LOOP CONTROL OF THE TRACTOR
The polynomial found by this fit gives the front joint value Jvf as a function of δ as shown by:
Jvf = −1234.0 ∗ δ6− 819.5 ∗ δ5+ 494.7 ∗ δ4
+143.8 ∗ δ3− 203.5 ∗ δ2− 818.0 ∗ δ + 2096.5 (5.7)
Note that in both Figure 5.5 and (5.7) the unit of δ is radians. The reason for using radians, instead of degrees is to ensure that the coefficients of the polynomial have almost the same order of magnitude. When using degrees for δ in the polynomial this is not the case, which can result in floating point errors, which should be avoided.
The polynomial (5.7) is used to transform the desired steer angle δ into a steering motor joint angle request. The degree of the polynomial has been chosen by evaluating the fit error. The effect of the degree of the polynomial on the fit error is shown in Table5.1. For the chosen degree 6 the fit error is within 7 motor units. The motor unit is equal to 0.088◦/step. The root mean squared error for degree 6 is 3 motor units. This gives an RMSE of 0.2640◦ in positioning the servo. As the servo turns over 108.5◦ for the full range of delta from -38◦ to 38◦, this gives a transmission ratio of approximately 1.4. Note that this ratio is an approximation, as it assumes a linear relation between the servo rotation and the change in δ. Using this ratio the RMSE of 0.2640◦in positioning the servo results in an approximate RMSE in positioning δ of 0.19◦.
degree polynomial max fit error (motor units) RMSE (motor units)
1 -105 39
Table 5.1: Effect of polynomial degree on the fit error
The result of the final configuration is shown in Figure5.6. The measured δland δrare plotted against the front joint value. Further the steering angle input δ and the Ackermann δ as calculated by (5.5) are plotted. It can be concluded that these two are almost equal based on a RMSE of 0.6649◦. The polynomial steer model used is therefore able to describe the relation between the input δ and the front joint value up to this accuracy. This means that the observed δ resulting from a steering motor front joint value request, corresponds to the δ expected for Ackermann steering geometry with an RMSE of 0.6649◦. This error is the summation of the error in positioning the servo which results in an approximate RMSE in positioning δ of 0.19◦ and the error caused by free play in the steering system.
CHAPTER 5. OPEN LOOP CONTROL OF THE TRACTOR
Figure 5.6: final wheel angle measurement