models Peter Wagner, DLR 9 July 2018 DLR.de Chart 2 > CF > - - PowerPoint PPT Presentation

Optimization of driver (and vehicle) models based on NDS/ FOT data for simulation models

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 1

Peter Wagner, DLR 9 July 2018

• A driver…
• is a machine…
• that takes objects that might

be on a collision track and estimates an possible action to avoid this

• It does so by changing acceleration

(gas/ brake pedal) and/ or steering.

• Will concentrate today on acceleration

(positive as well as negative, braking)

The Comic

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 2

Sun Y, Wu S, Spence I (2015) The Commingled Division of Visual Attention. PLOS ONE 10(6): e0130611. https://doi.org/10.1371/journal.pone.0130611

• Clearly, a driver converts (at each moment in time 𝑢)
• distance (gap) 𝑕 𝑢 = 𝑌 𝑢 − 𝑦 𝑢 − ℓ,
• own speed 𝑤(𝑢),
• and speed difference Δ𝑤 𝑢 = 𝑊 𝑢 − 𝑤(𝑢)
• into an acceleration 𝑏(𝑢) (positive as well as negative)
• Some people believe, that this

conversion can be quantified into a mathematical function

• 𝑏 = 𝐵(𝑕, 𝑤, Δ𝑤)
• – a map

(dropped argument time 𝑢 for better readability)

A “map” for car-following

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 3

𝒚 𝒖 , 𝒘 𝒖 , 𝒃(𝒖) 𝒀 𝒖 , 𝑾 𝒖 , 𝑩(𝒖) 𝒉 𝒖 = 𝒀 𝒖 − 𝒚 𝒖 − ℓ

• Since 1950, a huge number of 𝐵(𝑕, 𝑤, Δ𝑤)’s have been proposed, I will

work here with just one of them which is easy to understood.

• Such a driving machine wants:

1. To keep a preferred distance, and this is: 𝑕0 = 𝑤𝑈0

• Where 𝑈

0 is the preferred time headway

• (in most countries, there is the recommendation 𝑈

0 ≈ 2 seconds)

2. The speed difference Δ𝑤 = 0 to be zero  then 𝑕 does not change

• Driving machine’s Garden of Eden state: 𝑕 = 𝑕0 and Δ𝑤 = 0

The model(s)

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 4

• Points: data
• Dashed line: 2 s
• Solid: robust fit
• Strong scatter

is typically for these data

• Driver tend to a

preferred gap, but not very strongly

Real NDS data: 𝑕 versus 𝑤

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 5

• Data are filtered: |𝑏| < 0.25 𝑛/𝑡2 and Δ𝑤 < 1𝑙𝑛/ℎ
• Better: look at

distributions, instead of scatterplots.

• There a

tendency to stay near 𝑈0

• (but 𝑈

0 may be 𝑈 0(𝑤))

Real NDS data: again 𝑕 versus 𝑤, now as 𝑞𝑠𝑝𝑐(𝑤, 𝑕)

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 6

• In fact: drivers

want to stay near Δ𝑤 = 0

• (Highest prob

to be there)

Real NDS data: 𝒉 versus 𝚬𝒘

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 7

Helly’s model / a simple AICC AICC = adaptive intelligent cruise control

• Reminder: 𝑕0 = 𝑤𝑈
• If 𝑕 > 𝑕0 (far away)  𝑏 > 0
• If 𝑕 = 𝑕0 (jackpot!)  𝑏 = 0
• If 𝑕 < 𝑕0 (too close)  𝑏 < 0
• In short: 𝑏 = 𝑑1 𝑕 − 𝑤𝑈
• Parameter 𝑑1 scales difference

between actual 𝑕 and preferred distance 𝑕0 = 𝑤𝑈

• (and makes units correct, [𝑕] is

(𝑛)eter and 𝑏 is 𝑛/𝑡2, so [𝑑1] = 1/𝑡2)

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 8

• Reminder: Δ𝑤 = 𝑊 − 𝑤
• If Δ𝑤 > 0 (falling back)  𝑏 > 0
• If Δ𝑤 = 0 (jackpot!)  𝑏 = 0
• If Δ𝑤 < 0 (approaching)  𝑏 < 0
• In short: 𝑏 = c2Δ𝑤
• Parameter 𝑑2 scales the speed

difference and makes the units correct

• 𝑏 = 𝑑1 𝑕 − 𝑤𝑈

0 + c2Δ𝑤

• Again, more complicated models exists.
• This model ignore almost all about perception and driver’s internal

states, it is even in slight contradiction to the data:

• No min/max acceleration
• Continuous control by the driver machine, in any millisecond
• True drivers have a discrete control mechanism
• …
• But: can you really notice these points?
• Let us put it to test!

And so…Helly’s model (~1960)

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 9

• Idea: we have a model, we have data, and this Helly model has three

parameters (𝑑1, 𝑑2, 𝑈

0)  let us fit the model to the data

• In this case, this is very simple: it is nothing but a linear regression

between the dependent variable 𝑏 and the independent variables 𝑕, 𝑤, Δ𝑤

• In NDS data, we have exactly this: for each time-step 𝑢𝑗 (often they are

sampled at 10 Hz) we have a set of data

• (𝑕𝑗, 𝑤𝑗, Δ𝑤𝑗; 𝑏𝑗)
• And a linear model with three parameters:
• 𝑏𝑗 = 𝑑1𝑕𝑗 + 𝑑0𝑤𝑗 + 𝑑2Δ𝑤𝑗
• (𝑑0 = 𝑑1/𝑈

0 is needed in addition)

 So, let us fit this.

Calibration

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 10

𝑏 = 𝑑1 𝑕 − 𝑤𝑈

0 + c2Δ𝑤

lm(formula = acc ~ gap + vel + Dv + 0) Coefficients: Estimate

• Std. Error

t value Pr(>|t|) gap 0.00738 0.00024 31.05 <2e-16 Vel -0.0088 0.00031

• 28.63 <2e-16

Dv 0.08440 0.0023 36.14 <2e-16 Residual standard error: 0.1438 on 6074 degrees of freedom Multiple R-squared: 0.329, Adjusted R-squared: 0.3287 F-statistic: 992.8 on 3 and 6074 DF, p-value: < 2.2e-16

 𝑈

0 = 0.0088/0.00738 = 1.195 s – reasonable number

• The fit is statistically very strong! But what does it mean?

Results I (will show data in a moment)

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 11

𝑏 = 𝑑1 𝑕 − 𝑤𝑈

0 + c2Δ𝑤

• Is this good? Depends: for the acceleration, eventually, for the gaps: nope

Results II

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 12

𝑏 = 𝑑1 𝑕 − 𝑤𝑈

0 + c2Δ𝑤

Green: Data Red: Model

• Model is wrong
• Stochasticity not included, or not modelled correctly (linear model has

white noise) (perception errors for 𝑕, Δ𝑤, …)

• Discrete versus continuous control, but have you watched this?

Not untypical…some “explanations”

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 13

31270 31275 31280 31285 31290 31295 31300 70 80 90 100 110 Time (s) Speed v (km/h)

• Model is wrong
• Stochasticity not included, or not modelled correctly

(perception errors for 𝑕, Δ𝑤, …)

• Discrete versus continuous control, but have you watched this?
• Parameters not constant

Not untypical…some “explanations”

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 14

31270 31275 31280 31285 31290 31295 31300 70 80 90 100 110 Time (s) Speed v (km/h)

• Can be seen by dividing

data-set into pieces

• Blue line: 4 pieces and

fitting each individually

• Fits better, of course
• Can go down to 10 data-

points in a piece – another story

• May be a consequence
• f a model misfit

Current research topic: parameter vary a lot

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 15

Green: Data Red: Model

• Have model(s) with parameters
• Have NDS data
• Have an error measure (this is a long story as well)
• Fit the model to the data, and get error metrics
• Often, just a single number is reported, but: a distribution would be

much better Technicalities:

• Fitting, unfortunately, is very often non-linear
• And, I have tricked you a bit: the independent variables are not

completely independent, which may change the results a bit

That’s it, in a nutshell

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• With all kinds of data
• About 15 of those CF models
• Many different MoE’s (error measures)
• The somewhat surprising result (?): all models are comparable in their

accuracy, despite the fact that they may be very different

• Which is partly due to the fact, that the lead vehicle’s speed is usually

not constant, and therefore most of the behavior is determined by the lead car

Have run many tests

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 17

• Is it good enough? I think not, but: it depends on what you want to do.

10 models to the test

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 18

NDS are a wonderful source of data for

• Developing new –better– driver models (have not talked about this)
• To test and compare existing driver models
• To calibrate/ optimize existing driver models
• Validation is a bit more challenging (this can not be derived from what I

have shown here):

• In a narrow sense, NDS data perfect for doing this
• In a broader sense, validation more often than not have to be done for

the study at hand

• We are still lacking models to link the parameters to “externalities”

(exceptions exists, e.g. maximum speed as function of weather etc)

Summary

> CF > WaP • CF.pptx > 9 July 2018 DLR.de • Chart 19

Thanks for listening

Picture: DLR Berlin at Berlin‘s Long Night of the Sciences