Transport
Plan
Plan de Systems
de Formación
Formación (40h)
(40h)
Lecture 2. Trajectories
Learning
• Learning goals: Competences to estimate, describe and characterize how
vehicles, people and goods overcome distance; competence to infer kinematic
variables of one or multiple vehicles from a space-time diagram; to plan
transport services in linear infrastructures to maximize the service efficiency or
to guarantee safety conditions; competence to rebuild traffic crashes.
• Previous knowledge: Time and space means of a solid (Continuum
mechanics), vehicular flow and density concepts in roadways (Roadways and
Railways)
• Exercises and cases: Shared trips among different travellers,
acceleration lanes, percentage of heavy vehicles based on aerial photography,
design of sidetracks in single-track railways, management of airport runways,
linear infrastructure capacity.
1
Trajectories (s-t) + queues (N-t)
Transport operations:
• Trajectories (in motion –
overcome the distance)
– Speed, acceleration, etc.
s
• Queues (waiting – overcome
the time)
̶ Waiting time, maximum
queue and average, etc.
• Dual analysis of motion
N
Q
t
t 3
Definition
• Determination of the position of a vehicle (x) in relation to a
reference point during the time (t).
• Geometrical place of the vehicle position (space) respect to
the time: function x(t). Graphic representation of x(t) in the
plane (t, x) is a curve named trajectory.
• Applications:
– Minimum travel time between stations in a railway line for a
maximum speed.
– Determine the initial speed of a car from the braking marks on its
tires.
– Length of runways and location of exits.
– Length of acceleration / deceleration lanes.
– Etc.
2
Diagrams s-t
x
TRAJECTORY
Instantaneous speed
v(t) = dx(t)/dt
1
Instantaneous acceleration
2 a(t) = dv(t)/dt = d2x(t)/dt2
3
* t
(*) does not represent a real physical trajectory
In case uniform motion, v(t) = v(constant) and a(t) = 0(constant):
In case uniformly accelerated motion, a(t) = a(constant):
Interpretation of trajectories (1/2)
s t Acceleration
(recovering
Deceleration speed)
(intersection,
pedestrian
crossing, bump)
Free
speed
Vehicle running at a
lower speed and with
a longer spacing than
the previous vehicles
3
Interpretation of trajectories (2/2)
¿If we record a video
in a section of a road,
what do we see?
Construction of Trajectories
A) by recording times of vehicle passing through fixed sections (elevators,
pedestrians, automobiles, etc.).
B) by photographs in sequenced areas in time or by time ranges in systems
operated by scheduled in a circuit (buses with navigation systems).
C) by recording of the overtaking time of vehicles to an observer in motion at a
constant speed.
D) by continuous “tracking” with GPS (in vehicles such as cars, trucks, urban
buses via SAE or private smart phones) or bluetooth (roads).
4
Diagrams (s,t) in traffic
Real trajectories in
a highway
Traffic jams in highways
10
5
Comparison of PT modes
11
Bus bunching
TRAJECTORIES D'AUTOBUSOS
4500,00 bus 1
4000,00 bus 2
3500,00 bus 3
TEMPS (seg.)
3000,00 bus 4
2500,00 bus 5
2000,00 bus 6
1500,00 bus 7
1000,00 bus 8
500,00 bus 9
0,00 bus 10
0 2 4 6 8 10 12 bus 11
PARADES bus 12
12
6
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
13
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
14
7
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
15
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
16
8
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
17
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
18
9
Bus bunching
AUTOBUSES PAREADOS
2000
1800
1600
1400
1200
Tiem po (s )
1000
800
600
400
200
0
0 5 10 15 20 25
Paradas
19
Trajectories PV and PT
Trajectòries
4.000
3.800
3.600
3.400
TRAJECTORIES B
TRAJECTORY PV OF BUSES
3.200
H
3.000
2.800
A
2.600 E
2.400
G
Posició
2.200
2.000 C
1.800 I
1.600
H2
1.400
1.200 A2
1.000
J
800
600
B2
400
200
0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200
Temps
20
10
Trajectories PV and PT
TRAJECTORIES
TRAJECTORY PV OF BUSES
21
Micro and Macro traffic variables
Micro: h, headway
Macro: q, flow
i=1
Micro: s, spacing
Macro: k, density
j=1
22
11
Time and Space means
Time-mean speed
vi : instantaneous speed at a
specific location during a time T
Space-mean speed
vj : instantaneous speed at a
specific time along a length L
One stationary observer sees two fast vehicles
per one slow vehicle, but a picture shows two
slow vehicles and one fast vehicle!!
Wardrop relationship
23
Average speed of a group of vehicles
Example 1. Three family of vehicles crossing a highway section with
the same flow and speeds v1 = 30 km/h, v2 = 60 km/h and v3 = 60
km/h.
Example 2. Circular highway of length 2 km with 3 vehicles running
at speeds 100, 120 and 140 km/h respectively.
Generalization for the calculation of speed means:
24
12
Fundamental Traffic Equation
x
Stationary:
- Constant speed
- Constant headway
L n, # vehicles - Constant spacing
Identical values of variables at any point of
the diagram (x,t)
t
T
Stationary traffic:
Fundamental equation:
25
Group of vehicles
2 families of vehicles
Ex: Highway with one lane for light vehicles and one lane
for heavy vheicles
Assumption: Uniform headway and constant spacing
between vehicles of the same family (stationary traffic)
hl vl sl l 1,2
Additive magnitudes
ql kl vl l 1,2
q ql ; k kl l 1,2
l l
Fundamental Traffic Equation: q k vl (kl /k ) vs k l 1,2
l
ql kl
l 1,2
q k
26
13
Railway station capacity
• Capacity of railway stations with high frequency:
ht t g t s t a t d
1
C
ht
Maximum time admitted between
tg front and back between two
consecutive trains
ts Time of getting on / off
passengers
ta td Times of acceleration and
deceleration
27
Train line capacity (1)
Critical factors: - Line length
- Running speed
- Number and location of sidetracks
- Maintenance time
- Modelo de explotación adoptado
Mantenimiento
Figueres
120
110
100
90
80
Recorrido, km
Apartaderos
70
60
50
40
30
20
10
Mollet 0
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Horario, horas 28
14
Train line capacity (2)
Figueres
- L= 122km,
v=80 km/h,
4 trenes
Recorrido
Hmin=9min consecutivos
Mollet
Horario
•Figueres
•Recorrido
14 trenes
consecutivos
Mollet
•Horario 29
Airport runway capacity (1)
6-10 nm
50 nm
30
15
Airport runway capacity (2)
Type of Max. take-off Num. of Approaching Classification
aircraft weight, MTOW (kgs) engines speed (Kts) according to capacity
A Mono
<7000 100 Light (L)
B Multi
C 7000-136000 Multi 120 Medium (M)
D >136000 Multi 140 Heavy (P)
31
Airport runway capacity (3)
1. Minimum distance between two consecutive take-offs:
SPAIN USA
Aircraft follower
T>1 min if routes diverge more than 45º
Aircraft leader
T>2 min if routes are parallel but vi –v i+1>70 kt L M P
L 90s 60s 60s
T>3 min other cases
M 120s 60s 60s
P 120s 60s 60s
2. Minimum separation between two consecutive landings:
USA
Aircraft follower
Aircraft leader
L M P
L 102s 77s 77s
M 150s 90s 90s
P 210s 144s 108s
32
16
Airport runway capacity (4)
3. Minimum distance between take-off and landing:
• Standard occupancy time of a runway with fast exit is 1 min (50 sec)
• For a take-off preceding a landing: Dda=2 nm
33
Airport runway capacity (5)
4. Development of analytic solutions:
– Case i: vi≤vj
tij
vj
– Case ii: vi≥vj
1 1
tij
vj v v
j i
– CAPACITY
1
PIJ Fraction of events that an aircraft of characteristics j is
PIJ TIJ preceded by other of characteristics i
I ,J TIJ Time between events characterized for the succession ij
34
17
Connection btw trajectories and cumulative diagrams
35
18
