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

DLR.de • Chart 1 > CF > WaP • CF.pptx > 9 July 2018 Optimization of driver (and vehicle) models based on NDS/ FOT data for simulation models Peter Wagner, DLR 9 July 2018

DLR.de • Chart 2 > CF > WaP • CF.pptx > 9 July 2018 The Comic • A driver… • i s a machine… • that takes objects that might be on a collision track and estimates an possible action to avoid this 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 • It does so by changing acceleration (gas/ brake pedal) and/ or steering. • Will concentrate today on acceleration (positive as well as negative, braking)

DLR.de • Chart 3 > CF > WaP • CF.pptx > 9 July 2018 A “map” for car -following 𝒚 𝒖 , 𝒘 𝒖 , 𝒃(𝒖) 𝒉 𝒖 = 𝒀 𝒖 − 𝒚 𝒖 − ℓ 𝒀 𝒖 , 𝑾 𝒖 , 𝑩(𝒖) • 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)

DLR.de • Chart 4 > CF > WaP • CF.pptx > 9 July 2018 The model(s) • 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 = 𝑤𝑈 0 • Where 𝑈 0 is the preferred time headway • (in most countries, there is the recommendation 𝑈 0 ≈ 2 seconds) The speed difference Δ𝑤 = 0 to be zero  then 𝑕 does not change 2. • Driving machine’s Garden of Eden state: 𝑕 = 𝑕 0 and Δ𝑤 = 0

DLR.de • Chart 5 > CF > WaP • CF.pptx > 9 July 2018 Real NDS data: 𝑕 versus 𝑤 • 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

DLR.de • Chart 6 > CF > WaP • CF.pptx > 9 July 2018 Real NDS data: again 𝑕 versus 𝑤 , now as 𝑞𝑠𝑝𝑐(𝑤, 𝑕) • Data are filtered: |𝑏| < 0.25 𝑛/𝑡 2 and Δ𝑤 < 1𝑙𝑛/ℎ • Better: look at distributions, instead of scatterplots. • There a tendency to stay near 𝑈 0 • (but 𝑈 0 may be 𝑈 0 (𝑤) )

DLR.de • Chart 7 > CF > WaP • CF.pptx > 9 July 2018 Real NDS data: 𝒉 versus 𝚬𝒘 • In fact: drivers want to stay near Δ𝑤 = 0 • (Highest prob to be there)

DLR.de • Chart 8 > CF > WaP • CF.pptx > 9 July 2018 Helly’s model / a simple AICC AICC = adaptive intelligent cruise control • Reminder: 𝑕 0 = 𝑤𝑈 • Reminder: Δ𝑤 = 𝑊 − 𝑤 0 • If 𝑕 > 𝑕 0 (far away)  𝑏 > 0 • If Δ𝑤 > 0 (falling back)  𝑏 > 0 • If 𝑕 = 𝑕 0 (jackpot!)  𝑏 = 0 • If Δ𝑤 = 0 (jackpot!)  𝑏 = 0 • If 𝑕 < 𝑕 0 (too close)  𝑏 < 0 • If Δ𝑤 < 0 (approaching)  𝑏 < 0 • In short: 𝑏 = 𝑑 1 𝑕 − 𝑤𝑈 • In short: 𝑏 = c 2 Δ𝑤 0 • Parameter 𝑑 1 scales difference • Parameter 𝑑 2 scales the speed between actual 𝑕 and preferred difference and makes the units distance 𝑕 0 = 𝑤𝑈 0 correct • (and makes units correct, [𝑕] is ( 𝑛 )eter and 𝑏 is 𝑛/𝑡 2 , so [𝑑 1 ] = 1/𝑡 2 )

DLR.de • Chart 9 > CF > WaP • CF.pptx > 9 July 2018 And so…Helly’s model (~1960) • 𝑏 = 𝑑 1 𝑕 − 𝑤𝑈 0 + c 2 Δ𝑤 • 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!

𝑏 = 𝑑 1 𝑕 − 𝑤𝑈 0 + c 2 Δ𝑤 DLR.de • Chart 10 > CF > WaP • CF.pptx > 9 July 2018 Calibration • Idea: we have a model, we have data, and this Helly model has three parameters (𝑑 1 , 𝑑 2 , 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 𝑢 𝑗 (often they are sampled at 10 Hz) we have a set of data • (𝑕 𝑗 , 𝑤 𝑗 , Δ𝑤 𝑗 ; 𝑏 𝑗 ) • And a linear model with three parameters: • 𝑏 𝑗 = 𝑑 1 𝑕 𝑗 + 𝑑 0 𝑤 𝑗 + 𝑑 2 Δ𝑤 𝑗 • ( 𝑑 0 = 𝑑 1 /𝑈 0 is needed in addition)  So, let us fit this.

𝑏 = 𝑑 1 𝑕 − 𝑤𝑈 0 + c 2 Δ𝑤 DLR.de • Chart 11 > CF > WaP • CF.pptx > 9 July 2018 Results I (will show data in a moment) 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?

𝑏 = 𝑑 1 𝑕 − 𝑤𝑈 0 + c 2 Δ𝑤 DLR.de • Chart 12 > CF > WaP • CF.pptx > 9 July 2018 Results II • Is this good? Depends: for the acceleration, eventually, for the gaps: nope Green: Data Red: Model

DLR.de • Chart 13 > CF > WaP • CF.pptx > 9 July 2018 Not untypical…some “explanations” • 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? 110 Speed v (km/h) 100 90 80 70 31270 31275 31280 31285 31290 31295 31300 Time (s)

DLR.de • Chart 14 > CF > WaP • CF.pptx > 9 July 2018 Not untypical…some “explanations” • 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 110 Speed v (km/h) 100 90 80 70 31270 31275 31280 31285 31290 31295 31300 Time (s)

DLR.de • Chart 15 > CF > WaP • CF.pptx > 9 July 2018 Current research topic: parameter vary a lot • Can be seen by dividing Green: Data data-set into pieces Red: Model • 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 of a model misfit

DLR.de • Chart 16 > CF > WaP • CF.pptx > 9 July 2018 That’s it, in a nutshell • 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

DLR.de • Chart 17 > CF > WaP • CF.pptx > 9 July 2018 Have run many tests • 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

DLR.de • Chart 18 > CF > WaP • CF.pptx > 9 July 2018 10 models to the test • Is it good enough? I think not, but: it depends on what you want to do.

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

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