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The Physics of Active Modes

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Talk given at the kick-off of the ERC MAGnUM PhD week on the ALLEGRO program. The talk gives both an overview of ALLEGRO and then focusses more on active mode traffic operations.

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The Physics of Active Modes

  1. 1. unrAvelLing sLow modE travelinG and tRaffic With innOvative data to a new transportation and traffic theory for pedestrians and bicycles 1
  2. 2. The battle for urban space With increased densification of cities, where can we find the space for performing those functions that the city was actually built for? It is clear that active modes can play a major role in making cities liveable! Car 50 km/h, driver only Car Parked Tram 50 passangers Cyclist 15 km/h Bicycle Parked Pedestrian Walking Pedestrian Standing140 m2 20 m2 7 m2 5 m2 2 m2 2 m2 0.5 m2
  3. 3. Towards greener, healthier, more liveable cities…
 
 How active mode friendly is your city? What makes people in your city walk or cycle (or not!) instead of using car
 Can pedestrians and cyclists find their way easily through the city?
 Can your city / transfer hub deal with large numbers of people?
 Is your active mode infrastructure (roads, control) well designed?

  4. 4. Our central proposition… Science has not yet delivered adequate tools (empirical insights, theory, models, guidelines) to support planners, designers, and traffic managers… The ‘science of active mode mobility’ has been hampered by lack of data!
  5. 5. Unique large-scale cycling experiments (pilot) Large-scale experiment in May 2018 in AHOY Revealed preference route choice data Collaboration with MoBike, and The Student Hotel Innovations in data collection for active modes Active mode monitoring dashboard incl. Social Data Short-run and long-run household travel dynamics MPN longitudinal survey active mode “specials”
  6. 6. Active Mode 
 UML Engineering Applications Transportation & Traffic Theory for Active Modes in Cities Data collection and fusion toolbox Social-media data analytics AM-UML app Simulation platform Walking and Cycling Behaviour Traffic Flow Operations Route and Mode Choice and Scheduling Theory Planning anddesign guidelines Organisation of large-scale events Data Insights Tools Models Impacts Network Knowledge Acquisition (learning) Factors determining route choice Real-timepersonalised guidance
  7. 7. Active Mode 
 UML Engineering Applications Transportation & Traffic Theory for Active Modes in Cities Data collection and fusion toolbox Social-media data analytics AM-UML app Simulation platform Walking and Cycling Behaviour Traffic Flow Operations Route and Mode Choice and Scheduling Theory Planning anddesign guidelines Organisation of large-scale events Data Insights Tools Models Impacts Network Knowledge Acquisition (learning) Factors determining route choice Real-timepersonalised guidance
  8. 8. Active Mode Physics Data, phenomena, modelling and solutions
 Prof. dr. Serge Hoogendoorn 8
  9. 9. Complexity of pedestrian flow
  10. 10. An average cycling day in Amsterdam… Understanding and modelling require access to data… Which techniques are available?
  11. 11. Advanced SP and Simulators Field observations Controlled experiments Social Data Crawling
  12. 12. 0 2 4 6 8 10 Effort ValidityControllability Field observations Controlled Experiments Advanced SP and Simulators Social Data Crawling Trade-offs in data collection • Selection of data collection approach is trade-off between different factors (e.g. effort, controllability, data validity) • In general strive for optimal mix between different data collection techniques • Nevertheless, for young research fields with limited prior knowledge, advantage of controlled conditions are compelling… 12 Ease data collection Controllability Validity
  13. 13. Walker and cycling experiments From experiments to theory and models
  14. 14. A bit of theory… • Pedestrians moving with speed v (in m/s) need space A (in m2) to move • The faster one walks, the more space one needs: • Density = number of pedestrians / square meter: • We can thus express the density as a function of speed: • Equally, we can express speed as a function of density… 14 A(v) = A0 + γ ⋅ v ρ = 1/A ρ = ρ(v) = 1 A(v) = 1 A0 + γ ⋅ v ρ
  15. 15. • We have:
 
 • Given that speed can
 not be larger than 
 maximum speed, 
 we can rewrite: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Speedv Density k A bit of theory… 15 ρ(v) = 1 A0 + γ ⋅ v ρ(0) = 1 A0 = ρjam v(ρ) = { v0 , 1 γ ( 1 ρ − A0)} − ρ
  16. 16. F 0 5 10 15 20 25 30 0 200 400 600 Speedv(km/h) Density k (bike/km) • Do-It-Yourself bicycle experiment revealing fundamental diagram for bicycle flows • Single file assumption relaxed with recently performed experiments • Fundamental diagram also exist for bicycle flows
  17. 17. Bi-directional flows Self-organisation of bi-directional lanes
  18. 18. Which fundamental diagram is bi-directional flow (the other is uni-directional)? How many lanes are formed in a bi-directional flow (4 m wide) 3 4 s formed 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 .75 Ped/m2 0.6 cy 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 formedRelativefrequency Density = 0.75 to 1 Ped/m2 Figure (b) shows the data of the bi- directional flow experiment Clearly, bi-directional flows are very efficient! (a) (b)
  19. 19. !19 Example shared-space region 
 Amsterdam Central Station Other forms of self-organisation? Many other forms of self-organisation are found in pedestrian flow Diagonal stripes in crossing flows, zipper effect in bottlenecks, viscous fingering when group of pedestrians move through crowd, etc.
  20. 20. Self-organisation does not occur under all circumstances

 Love Parade Duisburg
  21. 21. Understanding by game-theoretic modelling • Main assumption “pedestrian economicus” based on principle of least effort: 
 
 For all available options (accelerating, changing direction, do nothing) she chooses option yielding smallest predicted effort (i.e. predicting behaviour of others)
 • Under specific conditions, the game-theoretic setting yields emergence of Nash equilibrium situations in which no pedestrian can unilaterally improve her situation 21
  22. 22. Game-theoretic pedestrian flow model 22 • Considering the following effort or cost components: - Straying from desired direction and speed - Walking close to or colliding with other pedestrians - Frequently slowing down and accelerating • Using a very simple prediction model for behaviour of others: i j Acceleration towards desired velocity Push away from ped j + … 1. Introduction This memo aims at connecting the microscopic modelling principles underlying the cial-forces model to identify a macroscopic flow model capturing interactions amongst edestrians. To this end, we use the anisotropic version of the social-forces model pre- nted by Helbing to derive equilibrium relations for the speed and the direction, given e desired walking speed and direction, and the speed and direction changes due to teractions. 2. Microscopic foundations We start with the anisotropic model of Helbing that describes the acceleration of edestrian i as influence by opponents j: ) ~ai = ~v0 i ~vi ⌧i Ai X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ here Rij denotes the distance between pedestrians i and j, ~nij the unit vector pointing om pedestrian i to j; ij denotes the angle between the direction of i and the postion j; ~vi denotes the velocity. The other terms are all parameters of the model, that will ⃗n ij ⃗v j ϕij
  23. 23. 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? 23 Characteristics of NOMAD • Simple model captures some key relations (e.g. speed- density curve) reasonable well! • All self-organised phenomena are captured, including dynamic lane formation, formation of diagonal stripes, viscous fingering • Playing around with model input and parameters allows us to understand conditions for self-organisation better
  24. 24. Game-theoretic pedestrian flow model • Breakdown probability demands on many factors, including: - Demand levels (see figure) - Variability in desired walking speeds (see figure: low (-), medium (-), high (-) - Variability in physical size (limited) - Level of anticipation / delayed response • Calibration reveals substantial heterogeneity in parameters (and correlation) • No empirical basis for threshold values formed motivation for CrowdLimits experiment @TUDelft in May 2018 1.2 1.4 1.6 1.8 2.01.0 0 1 Demand (P/s) Breakdownprob. Parameter Mean CoV Free speed (m/s) 1.34 0.23 Relaxation time (s) 0.74 0.23 Interaction strength (m/s2) 11.33 0.64 Interaction radius (m) 0.35 0.11 Reaction time (s) 0.28 0.07
  25. 25. CrowdLimits Testing the limits of self-organisation
  26. 26. CrowdLimits Testing the limits of self-organisation
  27. 27. Bicycle and mixed flows Using game theory to model bicycle and mixed flows and understanding conditions for self-organisation
  28. 28. Graphical explanation… • Data collection for modelling and capacity estimation • 25 scenarios (overtaking, merging, crossing, …) • Structures in and upstream b-n determine capacity • Discovery of capacity drop phenomenon for cycle flows Bottleneck width (m) Capacityflow(cyc/s)
  29. 29. Modelling waiting positions • Capacity and flow operations is determined by way queue is formed • Discretisation of area using diamonds (representing bicycle shape) • Estimation of discrete choice model to predict waiting location • Waiting location choice determined by various attributes (distance from stop line, distance to others, distance curb) • Use approach for other important processes (waiting passengers on platform) 29
  30. 30. Active mode traffic management Modelling for real-time prediction and control applications
  31. 31. Modelling for estimation and prediction • NOMAD / Social-forces model as starting point: • Equilibrium relation stemming from model (ai = 0): • Interpret density as the ‘probability’ of a pedestrian being present, which gives a macroscopic equilibrium relation (expected velocity), which equals:
 
 
 31 the desired walking speed and direction, and the speed and direction changes due to interactions. 2. Microscopic foundations We start with the anisotropic model of Helbing that describes the acceleration of pedestrian i as influence by opponents j: (1) ~ai = ~v0 i ~vi ⌧i Ai X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ where Rij denotes the distance between pedestrians i and j, ~nij the unit vector pointing from pedestrian i to j; ij denotes the angle between the direction of i and the postion of j; ~vi denotes the velocity. The other terms are all parameters of the model, that will be introduced later. In assuming equilibrium conditions, we generally have ~ai = 0. The speed / direction for which this occurs is given by: (2) ~vi = ~v0 i ⌧iAi X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ Let us now make the transition to macroscopic interaction modelling. Let ⇢(t, ~x) pedestrian i as influence by opponents j: (1) ~ai = ~v0 i ~vi ⌧i Ai X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ where Rij denotes the distance between pedestrians i and j, ~nij the unit vector pointing from pedestrian i to j; ij denotes the angle between the direction of i and the postion of j; ~vi denotes the velocity. The other terms are all parameters of the model, that will be introduced later. In assuming equilibrium conditions, we generally have ~ai = 0. The speed / direction for which this occurs is given by: (2) ~vi = ~v0 i ⌧iAi X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ Let us now make the transition to macroscopic interaction modelling. Let ⇢(t, ~x) denote the density, to be interpreted as the probability that a pedestrian is present on location ~x at time instant t. Let us assume that all parameters are the same for all pedestrian in the flow, e.g. ⌧i = ⌧. We then get: (3) ~v = ~v0 (~x) ⌧A ZZ exp ✓ ||~y ~x|| ◆ ✓ + (1 ) 1 + cos xy(~v) ◆ ~y ~x ⇢(t, ~y)d~y (1) ~ai = ~vi ~vi ⌧i Ai X j exp Rij Bi · ~nij · i + (1 i) 1 + cos ij 2 where Rij denotes the distance between pedestrians i and j, ~nij the unit vector pointing from pedestrian i to j; ij denotes the angle between the direction of i and the postion of j; ~vi denotes the velocity. The other terms are all parameters of the model, that will be introduced later. In assuming equilibrium conditions, we generally have ~ai = 0. The speed / direction for which this occurs is given by: (2) ~vi = ~v0 i ⌧iAi X j exp  Rij Bi · ~nij · ✓ i + (1 i) 1 + cos ij 2 ◆ Let us now make the transition to macroscopic interaction modelling. Let ⇢(t, ~x) denote the density, to be interpreted as the probability that a pedestrian is present on location ~x at time instant t. Let us assume that all parameters are the same for all pedestrian in the flow, e.g. ⌧i = ⌧. We then get: (3) ~v = ~v0 (~x) ⌧A ZZ ~y2⌦(~x) exp ✓ ||~y ~x|| B ◆ ✓ + (1 ) 1 + cos xy(~v) 2 ◆ ~y ~x ||~y ~x|| ⇢(t, ~y)d~y Here, ⌦(~x) denotes the area around the considered point ~x for which we determine the Microscopic models are great for off-line assessment, but too slow for real-time applications…. Can we come up with a macroscopic version?
  32. 32. Modelling for estimation and prediction • Taylor series approximation:
 
 
 yields a closed-form expression for the equilibrium velocity , which is given by the equilibrium speed and direction: • Equilibrium speed V shows that speed reduces with density / density gradient • Equilibrium direction is function of desired walking direction and density gradient (pedestrians move away from dense areas) • Completing model by including ped. conservation: 32 ! v = ! e ⋅V 2 SERGE P. HOOGENDOORN From this expression, we can find both the equilibrium speed and the equilibrium direc- tion, which in turn can be used in the macroscopic model. We can think of approximating this expression, by using the following linear approx- imation of the density around ~x: (5) ⇢(t, ~y) = ⇢(t, ~x) + (~y ~x) · r⇢(t, ~x) + O(||~y ~x||2 ) Using this expression into Eq. (3) yields: (6) ~v = ~v0 (~x) ~↵(~v)⇢(t, ~x) (~v)r⇢(t, ~x) with ↵(~v) and (~v) defined respectively by: (7) ~↵(~v) = ⌧A ZZ ~y2⌦(~x) exp ✓ ||~y ~x|| B ◆ ✓ + (1 ) 1 + cos xy(~v) 2 ◆ ~y ~x ||~y ~x|| d~y and (8) (~v) = ⌧A ZZ ~y2⌦(~x) exp ✓ ||~y ~x|| B ◆ ✓ + (1 ) 1 + cos xy(~v) 2 ◆ ||~y ~x||d~y FROM MICROSCOPIC TO MACROSCOPIC INTERACTION MODELING 3 , we see that for ~↵, we find: ~↵(~v) = ↵0 · ~v ||~v|| ermine this directly from the integrals?) (6), with ~v = ~e · V we can derive: V = ||~v0 0 · r⇢|| ↵0⇢ ~e = ~v0 0 · r⇢ V + ↵0⇢ = ~v0 0 · r⇢ ||~v0 0 · r⇢|| he direction does not depend on ↵0, which implies that the magnitude of tself has no e↵ect on the direction, while the gradient of the density does direction. FROM MICROSCOPIC TO MACROSCOPIC INTERACTION MODEL Furthermore, we see that for ~↵, we find: (10) ~↵(~v) = ↵0 · ~v ||~v|| (Can we determine this directly from the integrals?) From Eq. (6), with ~v = ~e · V we can derive: (11) V = ||~v0 0 · r⇢|| ↵0⇢ and (12) ~e = ~v0 0 · r⇢ V + ↵0⇢ = ~v0 0 · r⇢ ||~v0 0 · r⇢|| Note that the direction does not depend on ↵0, which implies that t the density itself has no e↵ect on the direction, while the gradient of influence the direction. 2.1. Homogeneous flow conditions. Note that in case of homogen i.e. r⇢ = ~0, Eq. (11) simplifies to (13) V = ||~v0|| ↵0⇢ = V 0 ↵0⇢ and @⇢ @t + r · (⇢ · ~v) = r s <latexit sha1_base64="YZ7rjNHP+BleFepPZE4KeXcaJSM=">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</latexit><latexit sha1_base64="YZ7rjNHP+BleFepPZE4KeXcaJSM=">AAACNHicbVDLSgMxFM34rPVVdenmYhEUscyIYF0IBTcuXChYFTql3EnTNphJhiRTKEN/yo3/4UoXLlTc+g2mD0SrBwKHc87l5p4oEdxY33/2pqZnZufmcwv5xaXlldXC2vq1UammrEqVUPo2QsMEl6xquRXsNtEM40iwm+judODfdJk2XMkr20tYPca25C1O0TqpUTjPwgS15Sgg1B0FoXJp+NZsH/YglBgJhJA2lYWdUWzIwy6j0N2FE9CwDwYahaJf8oeAvyQYkyIZ46JReAybiqYxk5YKNKYW+ImtZ4PlVLB+PkwNS5DeYZvVHJUYM1PPhlf3YdspTWgp7Z60MFR/TmQYG9OLI5eM0XbMpDcQ//NqqW2V6xmXSWqZpKNFrdSVoWBQITS5ZtSKniNINXd/BdpBjdS6ovOuhGDy5L+kelA6LgWXh8VKedxGjmySLbJDAnJEKuSMXJAqoeSePJFX8uY9eC/eu/cxik5545kN8gve5xfF5akO</latexit><latexit sha1_base64="YZ7rjNHP+BleFepPZE4KeXcaJSM=">AAACNHicbVDLSgMxFM34rPVVdenmYhEUscyIYF0IBTcuXChYFTql3EnTNphJhiRTKEN/yo3/4UoXLlTc+g2mD0SrBwKHc87l5p4oEdxY33/2pqZnZufmcwv5xaXlldXC2vq1UammrEqVUPo2QsMEl6xquRXsNtEM40iwm+judODfdJk2XMkr20tYPca25C1O0TqpUTjPwgS15Sgg1B0FoXJp+NZsH/YglBgJhJA2lYWdUWzIwy6j0N2FE9CwDwYahaJf8oeAvyQYkyIZ46JReAybiqYxk5YKNKYW+ImtZ4PlVLB+PkwNS5DeYZvVHJUYM1PPhlf3YdspTWgp7Z60MFR/TmQYG9OLI5eM0XbMpDcQ//NqqW2V6xmXSWqZpKNFrdSVoWBQITS5ZtSKniNINXd/BdpBjdS6ovOuhGDy5L+kelA6LgWXh8VKedxGjmySLbJDAnJEKuSMXJAqoeSePJFX8uY9eC/eu/cxik5545kN8gve5xfF5akO</latexit><latexit sha1_base64="YZ7rjNHP+BleFepPZE4KeXcaJSM=">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</latexit> ⃗e
  33. 33. 33 Macroscopic model yields plausible results…
 • First macroscopic model able to reproduce self-organised patterns (lane formation, diagonal stripes) • Self-organisation breaks downs in case of overloading • Continuum model inherits properties of the microscopic model underlying it • Forms solid basis for real-time prediction module • First trials in model-based optimisation and use of model for state-estimation are promising
  34. 34. Centraal Metro Station access to concourse gate line escalator stairway railway line control area (PI) 0 20 40 60 80 100 m Flurin H¨anseler (TU Delft) 20 Model Predictive Crowd Control (MPCC) 34 • Case: controlling turnstiles in Amsterdam Central Station • In the MPCC framework, the macroscopic model is used to compute predictions given the current state and the control signal • The controller iteratively determines the control 
 signal that optimised the predicted objective function Controller: Crowd dynamics Crowd Dynamics Model Optimizer Objective Function Demand Prediction Model predicted state performance control signal estimated state predicted demand historical data timetable, schedule optimal control signal Controller
  35. 35. 1. Monitoring Microscopic data is collected via video-based sensors, and combined with smartcard data Smart station and MPC 2. Estimation Based on data, current state is estimated and used as initial state for prediction 3. Prediction & optimisation Optimal control signal is computed, yield a 10% decrease in crowding cost
  36. 36. GPS tracks Routes based on Instagram Crowdedness Social-media activiteit Micro-posts related to ‘crowdedness’ From – To relations (WiFi + cam)Socio-demographics (Instagram / Twitter) 0 2000 4000 6000 8000 10000 12000 6 8 10 12 14 16 18 Ruijterkade In Out Total 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 11 12 13 14 15 16 17 18 19 Dichtheden)Veemkade dichtheid2(ped/m2) Scaling up! • Combination of mixed datasources (counting cameras, Wifi, social data, GPS, apps, etc.) • Reliable picture of current situation in at event site by fusion of data sources • Sentiment analysis • Operational web- based dashboard • Development of efficient models for prediction
  37. 37. Cooperative Bicycle Control • Application of model- based stochastic control • Optimise trade-off between missing green phase and consuming energy for traffic responsive control • Inform rider using on-board device (alt. road-side sign) • Chance to catch green phase +65% • Energy consumption -30%
  38. 38. Large scale applications Monitoring, estimation and prediction and MPC at the city scale Possibilities of area-based models?
  39. 39. Recall: Fundamental Diagram for pedestrians 39 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Speedv Density k 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Flowrateq Density k v(ρ) = v0 , 1 γ ( 1 ρ − 1 ρjam ) − q(ρ) = ρ ⋅ v0 , 1 γ ( 1 − ρ ρjam ) − q = ρ ⋅ v ρ ρ
  40. 40. Analytical derivation of P-MFD • Suppose that we have an area that we partition into subareas • For each subarea, flow operations are described by Greenshields FD: • Then we can easily show that for the entire area we have the P-MFD: • where denotes the spatial variation in density Area U U Ui qi(t) = Q(⇢i) = v0 ⇢i (1 ⇢i/⇢jam) U ¯q(t) = Q(¯⇢(t)) (v0 /⇢jam) · 2 (t) 2 (t) = 1 m P m(⇢i ¯⇢)2 Ui ⇢i(t) *) Illustration only: we consider walking pedestrians Also for other FDs, we can show that the P- MFD exist! It is given by the FD with a correction due to spatial variation!
  41. 41. Multi-scale modelling for large areas • Coarse modelling of network flow operations, where dynamics 
 of (sub-)area are described via P-MFD: where • Requires specification of spatial variation; preliminary data analysis points towards: • Approach is equivalent to macroscopic model presented before development of multi-scale simulation approach dni dt = P j fji(t) Fi(ni(t), i(t)) fji(t) = ji(t) · Fj(nj(t), j(t)) (¯⇢, ˙¯⇢) = 0.277 · ¯⇢ 0.039 · ˙¯⇢ n1(t) n2(t) f21(t) F1(t)
  42. 42. Closing Relevance for Sustainable Urban Mobility Need for investing in active mode mobility
  43. 43. Closing remarks • Lecture provided insight into the physics of active modes (empirics, modelling, applications) allowing efficient design and control active mode infrastructure • Only part of the puzzle! • ALLEGRO also provides insight into why people choice to walk or cycle (top figure), or to understand which are the determinants for route choice (bottom table) • Predict impact of policy interventions Determinant Influence Distance Negative Distance in morning peak hour Negative  (stronger than other moments) # Intersections / km Negative % Separate cycle paths No / slightly positive Overlap of routes Positive Rain No Daylight No
  44. 44. The battle for urban space Should NOT a battle of the modes! Rethinking the optimal mobility mix and the role of active modes In the end, the key question is: what type of city do you want to live in and which mobility mix best suits that desire! Car 50 km/h, driver only Car Parked Tram 50 passangers Cyclist 15 km/h Bicycle Parked Pedestrian Walking Pedestrian Standing140 m2 20 m2 7 m2 5 m2 2 m2 2 m2 0.5 m2
  45. 45. Beyond the autonomous car…

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