Driver model drive-cycle task

An optimisation algorithm is used to fit the driver model cost function weightings to the recorded data. The Matlab fmincon function is used to find the cost function weightings that minimise the sum of squared speed errors and pedal displacement errors when the driver model is applied to a drive cycle following task. This is called the fitting function:

3.5 Model Identification 109 Γ𝐷𝐶(𝑞1𝑠, 𝑞1𝜙, 𝑞1𝜙̇) = 1 2|∑𝑡=240𝑡=0 (𝜙𝑃𝑒𝑑𝑎𝑙𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑− 𝜙𝑃𝑒𝑑𝑎𝑙𝑀𝑜𝑑𝑒𝑙𝑙𝑒𝑑)2|₀ ∑ (𝜙𝑃𝑒𝑑𝑎𝑙𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑡=240 𝑡=0 − 𝜙𝑃𝑒𝑑𝑎𝑙𝑀𝑜𝑑𝑒𝑙𝑙𝑒𝑑)2 + 1 2|∑𝑡=240𝑡=0 (𝑣𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑− 𝑣𝑚𝑜𝑑𝑒𝑙𝑙𝑒𝑑)2|₀ ∑ (𝑣𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑− 𝑣𝑚𝑜𝑑𝑒𝑙𝑙𝑒𝑑)2 𝑡=240 𝑡=0 ( 3.46 ) where |… |0 denotes a term to be assessed with all 𝑞 values set to zero (i.e. the only component of

the driver cost function is pedal force, so the model will respond by keeping the vehicle stationary throughout each task). The component weightings are such that the value of Γ𝐷𝐶 will be 1 for the

case of all 𝑞 values set to zero. It is worth noting that there are two cost functions in this process: the driver cost function, 𝑉1, which determines how the driver model behaves, and the fitting

function, Γ𝐷𝐶, which determines how the driver model is fitted to the data.

For the optimisation, each driver cost function weighting is constrained to be positive and the pedal force weighting, 𝑟1, is set to 1. In order to assess the significance of each term in the

driver’s cost function, different cost function combinations were assessed to see the effect on the fitting function. The results are summarised below in Table 3-4. The numbers included here are the average values over all drivers.

Table 3-4 – Average fitting function values for different driver cost functions. A dash indicates that the corresponding 𝑞 value was not included in that particular cost function.

𝑞1𝑠 𝑞1𝜙 𝑞1𝜙̇ Γ𝐷𝐶

4.3 × 104 2.3 × 10−1 4.1 × 108 0.0135 3.8 × 102 5.0 × 10−4 - 0.0134

6.5 × 106 _{- 4.2 × 10}8 _0.0135

3.8 × 102 - - 0.0134

It can be seen from Table 3-4 that there is little difference in the fitting function value if neither of the pedal displacement or pedal speed weightings are included. As extra weightings add extra complexity to the driver model, it is proposed to adopt the simplest cost function assessed, the cost function that includes only speed errors, 𝑞1𝑠, and pedal force, 𝑟1. As the pedal force weighting,

𝑟1, is set to a value of 1, the value of speed error weighting, 𝑞1𝑠, is hence the relative weighting on

110 Driver Model A sensitivity study is completed to assess how the fitting function values vary with speed error weighting, 𝑞1𝑠, for Driver 2: Figure 3-33. In this figure the components of the fitting function

are also included. It can be seen that at low values of speed error weighting, 𝑞1𝑠, the Fitting

Function, Γ𝐷𝐶, approaches 1. This is where the vehicle remains stationary and the difference

between measured and modelled forces are large as the model exerts zero force. There is a critical weighting at which point the modelled vehicle is able to accelerate from stationary, and this is detected from the kink in the speed curve. At high speed error weightings, 𝑞1𝑠, the speed errors

reach near zero as both modelled and measured drivers are close to the speed target, but there is some steady pedal force error required. As the speed error weighting, 𝑞1𝑠, increases, the modelled

pedal force increases to achieve smaller speed errors relative to the target, but this results in greater speed and pedal force errors relative to the measured driver.

The resulting curve (Figure 3-33) identifies a clear lower bound to the value of speed error weighting, 𝑞1𝑠 at approximately 𝑞1𝑠= 5. A minimum is found at approximately 𝑞1𝑠= 300.

Examining how the driver’s cost function summed over the task varies with speed error weighting, 𝑞1𝑠, as well (Figure 3-34), reveals that the driver’s cost also varies little above the minimum

detected from the fitting function technique. This suggests that the fitting function technique is a good method for finding a lower bound to the value of speed error weighting, 𝑞1𝑠, and that

exceeding the lower bound has little in the way of impact on driver cost function performance. Similar conclusions are drawn from the other drivers, although their sensitivity studies are not included here for conciseness.

3.5 Model Identification 111

Figure 3-33 – Fitting function sensitivity study for the drive cycle task

Figure 3-34 –Summed driver cost function sensitivity study for drive cycle task

Γ𝐷𝐶

𝑞1𝑠

112 Driver Model

The resulting cost function weightings for all nine drivers are displayed in Figure 3-35. The fitting function values are included in Figure 3-36 as an indicator of goodness of fit.

It can be seen that the driver model had a best fit with Driver 9, and the worst fit with Driver 7. Driver 7’s speed profile is dominated by large overshoots when acceleration demands change – a behaviour that the driver model will not replicate because it demonstrates a better anticipation of future speed demands. The poor fit is also exaggerated because the driver model does not account for the driver’s imperfect perception of speed and limited understanding of the vehicle dynamics. Driver 9 on the other hand has much smaller overshoots and achieves speeds closer to the target. The driver model is able to recreate this behaviour much more closely.

3.5 Model Identification 113

Figure 3-36 – Total driver-model driver squared errors for Drive Cycle scenario

Figure 3-37, Figure 3-38 and Figure 3-39 compare the recorded speeds, pedal positions and pedal forces of Driver 9 going through the drive cycle scenario with the modelled Driver 9. (Driver 9 is selected as the best agreement between human driver and model driver occurs for this driver). The speed profiles of the two compare very well, with the most significant difference being the lack of small overshoots in the model driver’s performance. The model driver sticks very close to the target speed profile, except for rounding off speed troughs and peaks where accelerations change sharply.

The modelled pedal position and pedal forces differ slightly more from the recorded values. In the pedal position case, as the human driver is driving at a slightly different speed and acceleration to the model driver, the demanded torque from the engine is slightly different. As the vehicle models are identical in both the driving simulator and the model case, this means that different pedal positions are required. These differences are passed onto the pedal forces as well. However, as discussed in Section 3.1.2, the linear pedal model does not represent the actual pedal dynamics completely accurately. It is also expected that the human driver also introduces significant noise to the system that is not included in the driver model. The result is that the modelled pedal positions and forces are much smoother than the recorded data from the simulator.

The high pedal force and displacement witnessed at the start of the drive cycle are explained by the one non-linearity programmed into the driving simulator – the constant opposition forces

114 Driver Model (rolling resistance and gradient component of mass) drop to zero at zero speed. In the driver model, this is not the case as the full model is linear.

Figure 3-37 – Measured human driver and modelled driver speed profile for Driver 9

3.5 Model Identification 115

Figure 3-39 - Measured human driver and modelled driver pedal force for Driver 9

Further analysis of the driver model is completed through a parameter study examining how by varying the cost function weightings, differences in driver model behaviour can be seen. Figure 3-40 demonstrates the trade-off between different costs for different driving styles. The driver model is given a drive-cycle following task using the modified Millbrook cycle and the RMS pedal force and RMS speed error are calculated.

Looking at the RMS speed error and RMS pedal force plot (Figure 3-40), several key features of the shape are noted. Firstly, generally, as the weighting on speed error (𝑞1𝑠) increases,

116 Driver Model

Figure 3-40 – Trade off graph for drive-cycle scenario

The horizontal line at approximately 16 ms−1 in the speed errors plot is a result of the weightings meaning that the pedal is not depressed enough to overcome the drag forces and accelerate the vehicle. As the vehicle cannot accelerate, the RMS speed error is therefore equal to the RMS value of the drive-cycle speeds – a constant value.

At some critical speed error weighting, approximately 𝑞1𝑠= 1.2, the driver model is able

accelerate the vehicle from stationary, and the corresponding RMS speed error decreases. There is a limit to this behaviour, however, as to keep reducing the speed errors further, high accelerations of the vehicle are required. For high accelerations, high pedal displacements are required, and hence high pedal forces. This results in a higher RMS pedal force for the drive cycle.

It is interesting to compare the data points from the individual drivers with the curve for the driver model. Although the human drivers all have a similar RMS speed error to each other, their RMS pedal forces vary more, and it is this variation that draws the individuals away from the driver model plot.