PRESENTADOS POR ORGANISMOS Y ADMINISTRACIONES

The third case is an attempt to acknowledge that it is difficult for a machine to learn and understand human behaviour, but it is plausible that a human driver will develop an understanding of what the pedal feedback controller is aiming for and may well cooperate with it. Figure 4-3 illustrates this framework in block diagram form.

Figure 4-3 – One-sided cooperative control block diagram

Start again with the pair of prediction equations from ( 4.1 ) and ( 4.2 ): 𝐙one_{(k) = 𝚿}coop_{𝐱(k) + 𝚯}one_𝐔

1(k) + 𝛀coop𝐔2(k) ( 4.31 ) Where 𝒁one(𝑘) = [𝐙1(𝑘) 𝐙2(𝑘)] and 𝚯 one _{= [}𝚯1 0] Notice that in place of 𝚯2, we have a zero matrix instead.

4.3 One-sided Cooperative Feedback 137 The feedback controller is unaware of the driver, so the feedback controller gains can be calculated as the driver model was in 3.2:

𝐔2(k) = 𝐊2fullone 𝛆2one(k) ( 4.32 )

𝐊_2fullone = [𝐒𝐐2𝚯2 𝐒𝐑2

] \ [𝐒𝐐2

0 ] ( 4.33 ) By applying the receding horizon approach, the control law becomes:

f2opt(k) = 𝐊2one{ 𝐱(k) 𝐓2(k)} ( 4.34 ) where 𝐊₂one_{= [−𝐊} 2full one _{(1, : )𝚿} 2 𝐊2fullone (1, : )]

The cost function for the model driver is the same as that used in the cooperative case V₁one_{(k) = ‖𝐙}one_{(k) − 𝐓}coop_(k)‖

𝐐coop 2 _{+ ‖𝐔} 1(k)‖_𝐑 1 coop 2 _{( 4.35 )}

and hence the controller can be derived in the same way as the cooperative controllers in 4.2 were. The tracking error is defined as

𝛆₁one_{(k) = 𝐓}coop_{(k) − 𝚿}coop_{(k)𝐱(k) − 𝛀}coop_𝐔

2(k) ( 4.36 )

So the cost function is

V1one(k) = ‖𝚯one𝐔1(k) − 𝛆one1 (k)‖𝐐2coop+ ‖𝐔1(k)‖_𝐑

1 coop 2 _{( 4.37 )} Alternatively expressed as V₁one_{(k) = ‖}𝐒𝐐coop{𝚯 one_𝐔 1(k) − 𝛆1one(k)} 𝐒_R 1 coop𝐔₁(k) ‖ 2 ( 4.38 ) To minimise the cost function:

𝐔1(k) = 𝐊1fullone 𝛆1one(k) ( 4.39 )

where

𝐊_1fullone = [𝐒Qone𝚯

one

𝐒R₁one ] \ [

𝐒Qone

0 ] Substituting into ( 50 ) into ( 53 ):

𝐔1(k) = [−𝐊1fullone 𝚿coop 𝐊1fullone ] [

𝐱(k)

𝐓coop(k)] − 𝐊1fullone𝛀coop𝐔2(k) ( 4.40 )

and then substituting ( 46 ) in

𝐔1(k) = [−𝐊1fullone𝚿coop 𝐊1fullone − 𝐊1one𝛀coop𝐊2fullone] {

𝐱(k) 𝐓coop_(k)

𝐓2(k)

} ( 4.41 ) Following Wang (2015), by dividing 𝐊1fullone into two components 𝐊1full1one and 𝐊1full2one that correspond

138 Pedal Feedback f1opt(k) = [𝐊s1one 𝐊p1one 𝐊p2one] {

𝐱(k)

𝐓coop_(k)} ( 4.42 )

where

𝐊s1one = −𝐊1fullone 𝚿coop+ 𝐊1fullone𝛀coop𝐊2fullone 𝚿2

𝐊p1one= 𝐊1full1one

𝐊s2one= 𝐊1full2one − 𝐊1fullone 𝛀coop𝐊2fullone

Finally, using the receding horizon method, the control law can be condensed to f1opt(k) = 𝐊1one{ 𝐱(k) 𝐓coop(k)} ( 4.43 ) where 𝐊₁one _{= [𝐊} s1 one_{(1, : ) 𝐊} p1 one_{(1, : ) 𝐊} p2 one_{(1, : )]}

4.4 Driving Experiments and Data Analysis

With two frameworks in place for the modelling of the driver’s interaction with pedal feedback, a series of experiments were designed and performed to identify and validate the models.

4.4.1 Experiment Design

The fixed base driving simulator was used once again, and a full description is in Section 3.4. In this experiment, however, the pedal motor was powered, and exerted feedback forces on the pedal.

For the drive-cycle following task, the feedback pedal controller was given speed error weighting, 𝑞2𝑠= 80 and pedal position weighting, 𝑞2𝜙= 50 to provide a feedback pedal controller

driving style more focussed on the reduction in pedal displacements than the human drivers measured in Chapter 3. The car-following feedback pedal controller was given speed error weighting 𝑞2𝑠 = 80, pedal position weighting 𝑞2𝜙= 50 and following distance error weighting

𝑞2𝑑= 40 (for the same reason as the drive cycle cost function weightings) and a target following

distance of 2.5s THW plus a constant 5m offset to avoid the risk of collisions at low speeds. In both the drive cycle and car following scenario, the feedback pedal controllers generate their previews using a constant acceleration assumption – the controller identifies the current target acceleration (either demanded drive cycle acceleration, or acceleration of the vehicle in front) and assumes this remains constant throughout the preview horizon, enabling the calculation of demanded speeds. For the human driver in the drive-cycle scenario, an accurate preview of upcoming speed demands is available. For the car following scenario, the human driver, like the feedback controller, has to make assumptions on how the target vehicle will behave.

4.4 Driving Experiments and Data Analysis 139

4.4.2 Participants

The same nine participants were selected for this experiment as the experiments in 3.4, so all drivers had already completed the unassisted tasks.

4.4.3 Procedure

Once again, the experiment consisted of two scenarios.

In the first scenario, drive cycle following, a four minute segment of the Modified Millbrook Suburban Drive Cycle for HGVs was set as the target speed. The driver was given a practice run through to familiarise themselves with the behaviour of the vehicle, the dynamics of the pedal and the characteristics of the feedback. Drivers were asked to follow the target speed. The test was then completed three times, to improve the reliability of the results.

In the second scenario, car following, the speed of the target vehicle was set as the Modified Millbrook Suburban Drive Cycle for HGVs. Again, the driver was allowed a familiarisation run to get used to the virtual world and the behaviour of the feedback controller. Drivers were asked to follow the vehicle in front at a safe distance, as they normally would on the road. Again, once the familiarisation run was completed, the test was repeated three times.

When all drivers had completed the experiment, the data was analysed and differences in driver behaviours compared.

4.4.4 Drive Cycle Results

Figure 4-4 illustrates the recorded mean speed error profiles from the drivers. Generally, the drivers maintained a speed quite close to the target speed profile, as they were assisted by the pedal feedback. From further analysis, however, it is noted that the higher errors tended to occur after high accelerations. At lower acceleration rates, the drivers were better at anticipating the future speed demand. At speed peaks, the human drivers tended to keep accelerating until the maximum speed was reached, and then coast down until they reached the target speed again.

The human drivers appear to be more consistently closer to the target with the aid of the pedal feedback device. Speed errors in this scenario mostly remain within the range −1.0 ms−1 to 0.7 ms−1_{, whereas in the unassisted case of Chapter 3 (Figure 3-21), the speed errors spread within}

the range −1.5 ms−1 to 1.0 ms−1. The pedal feedback has contributed to a 40% drop in the range of speed errors.

140 Pedal Feedback

Figure 4-4 – Mean speed error against time for the drive cycle following task

The introduction of pedal feedback has resulted in several high peaks in pedal position over the drive cycle - Figure 4-5. The peaks are more numerous and of greater magnitude than those that were observed in the unassisted case (Figure 3-22). This suggests that the feedback controller is very sensitive to changes in acceleration, and that when it detects one (its previews operate on a constant acceleration assumption, so the controllers are unable to anticipate it) the controller reacts very quickly, resulting in the pedal position peaks. Although not very clear in the figure, the driver’s intermittent control is still apparent when they are assisted by the pedal feedback.

4.4 Driving Experiments and Data Analysis 141

Figure 4-5 – Mean pedal position against time for the drive cycle following task

By comparing the driver pedal forces for the unassisted and assisted cases, (Figure 4-6 with Figures 3-23 show the forces for the full drive cycle), it is noted that the magnitude of the forces from the driver decreases by approximately 50%, suggesting that the pedal feedback has helped decrease speed errors, whilst keeping effort on the driver’s part, low. Note that, as mentioned in Chapter 3, there is some concern about the validity of pedal forces at low pedal displacements, caused by a small clip on the pedal. The driver pedal force does not appear in the fitting approaches described in later chapters, as the pedal displacement and feedback force are sufficient.

There is little variation in feedback force across the drivers. A spread of approximately 4N of feedback force covers the range of drivers – Figure 4-7. The feedback force remains near constant during constant accelerations.

It is interesting to note that the addition of pedal feedback has greatly reduced the RMS pedal forces for the drivers, suggesting the workload has decreased, but that the RMS speed errors have not significantly decreased - Figure 4-8 compared with Figure 3-24. This is especially the case for Driver 9, which has switched from one the highest RMS pedal forces in the unassisted case, to the lowest RMS pedal force in this instance. This implies that the pedal feedback has had a big impact on the effort Driver 9 puts into the control of the vehicle.

142 Pedal Feedback

Figure 4-6 – Mean driver pedal force against time for the drive cycle following task

4.4 Driving Experiments and Data Analysis 143

Figure 4-8 – Cost diagram for the drive cycle scenario with feedback – stars (non-professionals) and diamonds (professionals) illustrate data from the assisted drivers. Their original data points

from the unassisted case are illustrated with circles (non-professionals) and squares (professionals). Colours are unchanged.

As a final consideration, the mean vehicle speed for each driver is plotted with one standard deviation of the individual’s three runs shaded either side in Figure 4-9. Once again, this clearly demonstrates that the standard deviations within each driver is similar to the standard deviation across drivers in this scenario, meaning that there is not a statistically significant difference between most of the drivers across most of the test. It is interesting to note that the addition of pedal feedback has greatly reduced the standard deviations in vehicle speed compared with the unassisted case in Figure 3-25.

144 Pedal Feedback

Figure 4-9 - Driver mean vehicle speeds with one driver standard deviation shaded either side

4.4.5 Car-Following Results

The mean speed profiles for the assisted drivers in the car-following scenario are displayed in Figure 4-10. All drivers remain quite close to the speed of the target vehicle, but it is noted that Drivers 1 and 9 drift further from the target vehicle speed than other drivers.

Now observing the pedal position in Figure 4-11, it can be seen that the addition of pedal feedback has smoothed the pedal position curve when compared to Figures 3-26. This is due to the fact that the pedal feedback assesses and acts at every time step of 0.02s, whereas the driver, when implementing intermittent control, can go for much longer (up to a second) without updating. The addition of pedal feedback has also reduced the peaks in pedal position. The unassisted driver would frequently depress the pedal to 60% and occasionally up to 80%. The assisted driver, on the other hand only occasionally depresses above 60%. The reduced pedal positions are an indicator of improved fuel usage.

4.4 Driving Experiments and Data Analysis 145

Figure 4-10 – Mean vehicle speed against time for the car-following task

146 Pedal Feedback As already mentioned, the speed profile for Driver 1 looks different to the other drivers’. This is matched by the driver’s pedal force (Figure 4-12). The other drivers maintain forces in the 0-35N bracket, whereas Driver 1 is exerting forces up to 45N. This implies that the driver is not prepared to accept the guidance of the feedback controller and therefore tries to oppose it. The pedal feedback force (Figure 4-13) for Driver 1 is also significantly higher than the other drivers.

The result of this conflict between Driver 1 and the feedback controller is a vehicle that has poor vehicle following performance (Figure 4-14 and Figure 4-15) whilst having a highly fluctuating pedal position and vehicle speed, which are likely to have negative effects on vehicle fuel consumption. The other drivers on the other hand have utilised the guidance from the feedback controller to adopt a more consistent vehicle following strategy, and now accurately follow the target vehicle at a 2.5s THW gap.

Unlike the drive cycle scenario, the addition of pedal feedback here has increased both the drivers’ RMS pedal forces and RMS following distance errors in some cases - Figure 4-16. This is because the RMS following distance error is based on the target estimated from the driver in the unassisted case, as explained in 3.4.5. The pedal feedback guides the driver towards a different THW target, resulting in greater errors relative to the driver’s original target. This measure is used, however, as a direct comparison with the unassisted case in Chapter 3. The RMS pedal forces also dramatically increases for Driver 1 as the driver tries and oppose the pedal feedback to remain close to their original target.

4.4 Driving Experiments and Data Analysis 147

Figure 4-13 – Mean feedback force against time for the car-following task

148 Pedal Feedback

4.4 Driving Experiments and Data Analysis 149

Figure 4-16 – Cost diagram for the car-following scenario with feedback – stars (non- professionals) and diamonds (professionals) illustrate data from the assisted drivers. Their original

data points from the unassisted case are illustrated with circles (non-professionals) and squares (professionals). Colours are unchanged.

Finally, the following distance standard deviations are shaded alongside the means for each driver in Figure 4-17. It is very noticeable that the addition of pedal feedback has not only greatly reduced the following distances for a few drivers (mainly Drivers 4 and 9), but has also greatly reduced the standard deviation in following distance (compare with Figure 3-32). The feedback has enabled the drivers to be much more consistent in their following distance behaviour.

150 Pedal Feedback

Figure 4-17 - Driver mean following distances with one driver standard deviation shaded either side