Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Differential game theory for Traffic Flow Modelling

569 views

Published on

Lecture given at the INdAM symposium in Rome, 2017. The lecture shows how you can use differential games to model traffic flows, focussing on pedestrian simulation.

Published in: Science
  • DOWNLOAD FULL BOOKS, INTO AVAILABLE FORMAT ......................................................................................................................... ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... 1.DOWNLOAD FULL. PDF EBOOK here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... 1.DOWNLOAD FULL. EPUB Ebook here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... 1.DOWNLOAD FULL. doc Ebook here { https://tinyurl.com/y3nhqquc } ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... .............. Browse by Genre Available eBooks ......................................................................................................................... Art, Biography, Business, Chick Lit, Children's, Christian, Classics, Comics, Contemporary, Cookbooks, Crime, Ebooks, Fantasy, Fiction, Graphic Novels, Historical Fiction, History, Horror, Humor And Comedy, Manga, Memoir, Music, Mystery, Non Fiction, Paranormal, Philosophy, Poetry, Psychology, Religion, Romance, Science, Science Fiction, Self Help, Suspense, Spirituality, Sports, Thriller, Travel, Young Adult,
       Reply 
    Are you sure you want to  Yes  No
    Your message goes here
  • Hello! Get Your Professional Job-Winning Resume Here - Check our website! https://vk.cc/818RFv
       Reply 
    Are you sure you want to  Yes  No
    Your message goes here

Differential game theory for Traffic Flow Modelling

  1. 1. Modelling (Active) Traffic An Introduction to Applications of Optimal Control and Differential Game Theory
 Prof. dr. Serge Hoogendoorn 1
  2. 2. 2 Pedestrian modeling questions in practise… • Redesign of the Al Mataf Mosque (testing for safety and throughput) • Station design (e.g. Amsterdam Zuid) to show if LOS remains acceptable • Investigating building evacuations (e.g. impact of counterflow WTC 9/11) • Testing crowd management plans during SAIL tallship event in Amsterdam For each of these (and many others) applications, the key question is: “Can we provide predictively valid models that can predict LOS, throughput, and safety pinch points in case of regular or irregular conditions”
  3. 3. “A model is as good as the predictions it provides” • Questionable if from this engineering perspective, ped models are predictively valid (in contrast to models for car traffic) • Why? In our field, DATA is key in the development of theory and models • Pedestrian theory (and cyclists) has suffered from lack of data and has been “assumption rich and data poor” • Some examples of different data collection exercises that we have performed from say 2000 onward 3 Understanding transport begins and ends with data
  4. 4. Let’s start with the pedestrians…
  5. 5. Speed density relation pedestrian flow • Fundamental diagram for pedestrian flow stemming from 2002 experiments • Non-increasing relation between speed and density showing reduction in speed as density increases (or increase in density as speed reduces?) • Strong impact of flow composition (e.g. purpose, age, gender, etc.)
 • Can you interpret the relation? That is, can you explain the shape? 5
  6. 6. 6 Example shared-space region 
 Amsterdam Central Station Empirical facts of self-organisation (2002 experiment) • Bi-directional flow experiment revealed self-organisation of lanes • Process has chaotic features (e.g. equilibrium states formed depend critically on initial and boundary conditions, ill-predictable) • Collected microscopic data allowed for unique quantification self-organised features
  7. 7. 7 Example shared-space region 
 Amsterdam Central Station Empirical facts of self-organisation (2002 experiment) • Bi-directional flow experiment revealed self-organisation of lanes • Process has chaotic features (e.g. equilibrium states formed depend critically on initial and boundary conditions, ill-predictable) • Collected microscopic data allowed for unique quantification self-organised features 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Density = 0.25 to 0.5 Ped/m2 2 0.2 0.4 0.6 Relativefrequency Density = 0.75 to 1 Ped/m2 0 0.2 0.4 0.6 0.8 0.9 1 1.1 1.2 1.3 density (Ped/m2 ) speed(m/s) 1 2 3 4 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Density = 0 to 0.25 Ped/m2 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Density = 0.25 to 0.5 Ped/m2 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Density = 0.5 to 0.75 Ped/m2 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Density = 0.75 to 1 Ped/m2 • Examples show how likelihood of certain number of lanes formed depends on density • Right picture shows how speed-density relation is (to a limited extent) dependent on the number of lanes that is formed
  8. 8. Further experimentation supports and extends our findings • Example shows results by Prof. Seyfried (downloadable via the Juelich Centre website) http://www.fz-juelich.de/ias/jsc/EN/Research/ModellingSimulation/ CivilSecurityTraffic/PedestrianDynamics/Activities/database/
  9. 9. 9 Example shared-space region 
 Amsterdam Central Station Traffic Flow Phenomena • Other self- organised patterns found for other experiments • Example shows formation of diagonal stripes • Other examples include zipper effect, viscous fingering, and faster is slower effect • Self-organisation also occurs in practise…
  10. 10. 10 Example shared-space region 
 Amsterdam Central StationSelf-organisation during LowLands…
  11. 11. Modelling challenge… • To come up with a model (and underlying theory) that can predict observed relations (e.g. speed-density) and phenomena for different situations • In 2003, we opted for differential game theory to model the behaviour of pedestrians that are competing for the use of (scarce) space • Some motivation? - We know that under specific conditions, differential game theory predicts occurrence of (meta-) stable equilibrium state, which (hopefully) are our self- organised patterns… - Moreover, research from the late Seventies and Eighties provides us with evidence that could be used as a basis for a game-theoretical model… • Let us have a brief look at the different behavioural assumptions (the theory) that underly the game theoretical model… 11
  12. 12. Microscopic pedestrian modelling • Main assumption “pedestrian economicus” based on principle of least effort: 
 
 For all available options (accelerating, changing direction, do nothing) he chooses option yielding smallest predicted disutility (predicted effort)
 • When predicting walking effort, he values and combines predicted attributes characterising available options (risk to collide, walking too slow, straying from intended walking path, etc.) 12
  13. 13. Six additional behavioural assumptions… • Pedestrians are feedback-oriented, reconsidering their decisions based on current situation • They anticipate behaviour of others by predicting their walking behaviour according to non-co-operative or co-operative strategies • Their predicting abilities are limited, reflected by discounting effort of their actions over time and space • Pedestrians are anisotropic in that react mainly to stimuli in front of them • They minimise predicted discounted costs resulting from: (a) straying from planned path; (b) vicinity of other peds (+ obstacles); (c) applying control • Pedestrians are more evasive encountering a group than a single pedestrian
  14. 14. The Math… • State equation describes ‘mental model’ peds to predict from 
 
 
 where the state describes the positions and velocities of ped p and his opponents q (e.g. ), and were the control is the acceleration that a pedestrian can apply • Prediction model describes kinematics of pedestrians, e.g. and • Ped p chooses control (accel.) minimising effort for ˙x(t) = f(t, x, u) x(tk) = xksubject to r(t) v(t)x(t) rp(t) u(t) ˙r = v ˙v = u [tk, tk + T) Jp = Z tk+T tk e ⌘s Lp(s, x(s), u(s))ds + e ⌘(tk+T ) p(tk + T, x(tk + T)) t = tk u[tk,tk+T )
  15. 15. Using the assumptions to specify model • Most behavioural assumptions are specified via running cost
 
 
 where:
 
 
 
 and with the proximity cost equal to: Lp(t, x, u) Lp = Lstray p + Lprox p + Laccel p Lstray p = 1 2 (v0 p vp)2 Laccel p = 1 2 u2 p Lprox p = X q2Q e dpq/RP ✓ p + (1 p) 1 + cos ✓pq 2 ◆ and
  16. 16. Using the assumptions to specify model • Most behavioural assumptions are specified via running cost
 
 
 where:
 
 
 
 and with the proximity cost equal to: Lp(t, x, u) Lp = Lstray p + Lprox p + Laccel p Lprox p = X q2Q e dpq/RP ✓ p + (1 p) 1 + cos ✓pq 2 ◆ andLstray p = 1 2 (v0 p vp)2 Straying cost describe the impact of not walking in the desired direction and at the desired speed Laccel p = 1 2 u2 p Acceleration cost describe the cost of applying the control acceleration in long. and lat. direction
  17. 17. Lstray p = 1 2 (v0 p vp)2 Laccel p = 1 2 u2 p Using the assumptions to specify model • Most behavioural assumptions are specified via running cost
 
 
 where:
 
 
 
 and with the proximity cost equal to: Lp(t, x, u) Lp = Lstray p + Lprox p + Laccel p and Proximity cost shows spatial discounting of cost impact using distance Lprox p = X q2Q e dpq/RP ✓ p + (1 p) 1 + cos ✓pq 2 ◆ Impact of ‘groups’ by adding proximity costs over opponents Anisotropy is reflected by making cost dependent on angle ✓pq p q ✓pq rq rp vp dpq = ||rq rp||
  18. 18. Solving the problem? • Problem can be solved using the Minimum Principle of Pontryagin • Without going into details… • Define Hamiltonian function:
 and use it for necessary conditions for optimality of control signal • Next to the state equation + initial conditions, we can derive an equation for the co-states (a.k.a. marginal costs) + terminal condition 
 
 
 and optimality conditions to determine optimal acceleration 18 Hp = e ⌘t Lp + 0 p · f u⇤ [tk,tk+T ) ˙ p = @Hp/@xp p(tk + T) = @ p/@xand u⇤ p = arg min H(t, x, u, p)
  19. 19. Finding solutions? • Assume non-cooperative behaviour: pedestrian p optimises own cost function while begin aware that the opponents will do the same • Novel iterative approach for mixed initial-terminal boundary condition problem can be applied to small test cases due to computational burden • Example shows ‘self-organisation’ crossing flow case: intersection without signals?
  20. 20. Can we come up with a simpler model? • The trick we used: simplify prediction model of pedestrian p • Pedestrian p assumes that opponents do not change their speed or direction (i.e. ) during the prediction period with • Time-invariant infinite horizon discounted cost problem has closed-form solution which is similar to the social-forces model of Helbing: [tk, tk + T)uq = 0 T ! 1 up = v0 p vp ⌧p Ap X q6=p e dpq/Rp ✓ p + (1 p) 1 + cos ✓pq 2 ◆ npq p q ✓pq vp npq Acceleration towards desired velocity Push away from ped q + …
  21. 21. Speed-density relation? • Assume pedestrians walking in straight line • Equilibrium: no acceleration, equal distances R between pedestrians • We can easily determine equilibrium speed for pedestrian q (q > p means q is in front) • Speed-density diagram looks reasonable for positive values of anisotropy factor 21 0 2 4 6 density (P/m) 0 0.5 1 1.5 speed(m/s) 1 speed(m/s) = 0 = 1 = 0.6 ve p = v0 p ⌧pAp X q>p e (q p)d/Rp X q<p e (p q)d/R !
  22. 22. Characteristics of the simplified model • Simple model captures macroscopic characteristics of flows well • Also self-organised phenomena are captured, including dynamic lane formation, formation of diagonal stripes, viscous fingering, etc. • Does model capture ‘faster is slower effect’? • If it does not, what would be needed to include it? Application of differential game theory: • Pedestrians minimise predicted walking cost, due
 to straying from intended path, being too close to 
 others / obstacles and effort, yielding: • Simplified model is similar to Social Forces model of Helbing Face validity? • Model results in reasonable macroscopic flow characteristics • What about self-organisation? 22 Characteristics of the simplified model • Simple model captures some key relations (e.g. speed-density curve) reasonable well! • Also self-organised phenomena are captured, including dynamic lane formation, formation of diagonal stripes, viscous fingering. • Self-organisation depends critically on paramaters and variance: freezing by heating
 • Do you know other features we have not discussed? Would the model be able to reproduce these? • Self-organisation of bi-directional lanes fails if demand becomes too high • Parameters determine threshold
  23. 23. 23 • Faster = slower effect states that the faster people try to get out, the longer an evacuation will take • Different experiments showed substantial reduction in outflow when evacuees rush • Capacity reduction is caused by friction / arc-formation in front of door due to increased pressure
  24. 24. Introducing the Faster is Slower effect • What could you do to reproduce the capacity drop phenomenon / faster = slower effect?
  25. 25. Introducing the Faster is Slower effect • What could you do to reproduce the capacity drop phenomenon / faster = slower effect? • Analogy with squash balls: • Pedestrians are ‘compressible circles’ • Normal force pushing away the other pedestrian • Tangential friction force increasing with reduction of distance of pedestrian centers • Capacity decreases / evacuation time increases with increasing pressure / haste friction normal force
  26. 26. Calibration and validation• Capacity could be reproduced with 4% error 26 empirical simulation 1 2 3 4 0 0.2 0.4 0.6 Number of lanes formed Relativefrequency Some lessons… • Importance of sensitivity analysis to see which parameters are most relevant • Different average parameter values are found when calibrating per situation (bi-dir flow, cross flow, bottleneck) • Reasonable average parameter set was found for all situations • Improvement when including explicit delay • Large differences in microscopic valuesCalibration and validation
  27. 27. Calibration and validation • Novel estimation technique (based on ML) allowed using the microscopic data from experiments to estimate parameter distribution and correlations • Table shows results of estimation procedure for illustration purposes: note that for some parameters, variance is substantial as is the correlation!
  28. 28. 28 Example shared-space region 
 Amsterdam Central Station Fascination with active-mode traffic modelling Amsterdam cycling during rush hour
  29. 29. Versatility of the game-theory approach? • Modelling cyclist behaviour • Approach is very similar to pedestrian modelling, however the kinematics of cyclists is different: • First results of application to cycle flow • Example shows application to mixed pedestrian / cyclist flow (shared space) 29 ˙xp = vp cos ↵p ˙yp = vp sin ↵p ˙vp = ap ˙↵p = !p
  30. 30. Successful shared-space implementation 30 Example shared-space region 
 Amsterdam Central Station Shared-space: interaction between bikes and peds… • Area behind Amsterdam Central Station • Mix between pedestrians going to / coming from station to ferries and crossing pedestrians • Self-organisation yields reasonable flow operations • Modelling using game-theory
  31. 31. Successful shared-space implementation 31 Example simulation results for shared-space situation Shared-space: interaction between bikes and peds… • Area behind Amsterdam Central Station • Mix between pedestrians going to / coming from station to ferries and crossing pedestrians • Modelling using game-theory • Depending on parameter choices, self-organisation occufs…
  32. 32. Use of game theory for C-ACC systems • Use optimal control approach to minimise joint cost of vehicles in platoon by controlling few vehicles • Cost function can reflect efficiency, comfort, emissions: Follower 2 - Human-driven vehicle Follower 1- Cooperative vehicle Leader – Human-driven vehicle s1, Δv1s2, Δv2 J = J(u[t,T ) ) = c1Jsafety +c2Jefficiency +c3Jfuel +...
  33. 33. Use of game theory for C-ACC systems • Differences no control, ACC and C-ACC
  34. 34. Autonomous cars? What about bikes?
  35. 35. Applications of game theory to modelling traffic flows Contributions of talk • Importance of data & examples of typical features of traffic flow • Motivate use of differential game theory • Apply game theory to pedestrian modelling • Show model features and discuss possible further improvements • Calibration and validation issues • Show other applications (cooperative vehicles, bicycle flow and shared space) Work is based on research contributions from 2002 onward with Winnie Daamen, Mario Campanella, Dorine Duives, Meng Wang, and others!
  36. 36. 9th Workshop on the Mathematical Foundations of Traffic @ TU Delft Co-organised by the TU Delft Transport Institute 
 &
 TU Delft Institute for Computational Science and Engineering Dates soon to follow! 36

×