See also readme_roundabouts

Specification of Simple X Intersections

First, one needs to specify the kind of intersection. A fully fledged
X intersection with 12 origin-destination pairs (4 times right turn,
straight on, and left turn, respectively) will probably be too complex
for the intention of this project: a simple interactive traffic flow
simulator with realistic models  that runs in any browser without
add-ons. A very limited X intersection with only right turns is
already available since these are just four merging roads. A
generalisation of intermediate complexity and moderate programming
effort would be straight-on-only X intersections in three variants
(keeping aside signalized intersections that can be easily added later
on): (i) mainroad crosses secondary road (secondary road with or
without stop signs),  (ii) roads of equal priority where vehicles
coming from the right have the right of way (widespread in Europe),
(iii) 4-way stop X-ings (widespread in the US). 

Type (i) is the easiest to model: Mainroad traffic just passes the
intersection as though no secondary road exists. If there are stop
signs, secondary traffic decelerates to a stop just as it would do
behind a red traffic light (standing virtual vehicle of length zero at
the stop line). As soon as a vehicle on the secondary road has
stopped, a binary decision will be taken at every time step: Move on
(as though the main road does not exist) if there is a sufficiently
large gap, otherwise wait. The definition of a "sufficiently large
gap" is nontrivial and cannot simply reduced to some condition for the
longitudinal accelerations (as would be the case for lane changing and
merging). A simple heuristics would be as follows: 

move on if (C1 AND C2)

C1 (obstruction criterion) is true if the spatial area covered when
crossing the intersection is currently not occupied by any mainroad
vehicle 

C2 (gap criterion) is true if any incoming mainroad vehicle is
sufficiently far away. 


The gap-acceptance criterion C2is more tricky. A plausible ansatz
would be the following: "The lowest value of the anticipated times-to-collision (TTC) should exceed a critical value TTC_crit" (of the order of 3 seconds).

The simplest heuristic assumptions to make the necessary anticipation are the following

(i) the speed v_main of the relevant mainroad vehicle and its actual
distance s_main to the first possible collision point  is known; the
speed is expected to remain unchanged 

(ii) the own acceleration in case of a "go" is the free-flow
acceleration which can be approximated by a constant acceleration with
the IDM parameter value a_IDM 

Since the distance s_cross between the stopping position and the last
possible collision point (where the crossing vehicle leaves the
respective lane of the crossing road) is known, we derive following
explicit rule: 

t_cross=sqrt(2*s_cross/a_IDM)  (1)

s_min=s_main-v_main*t_cross  (2)

TTC=s_min/v_main   (3)

C2=(TTC>TTC_crit)   (4)

This reflects a rather defensive behaviour since the secondary driver
will not start if (TTC<=TTC_crit) AND (TTC+(L_main+w_cross)/v_main<0)
although this means no collision since the approaching vehicle has
already passed when the crossing vehicle intersects with its
trajectory. However, the simple rule (4) is practicable, in most
cases. 


Remarks:


(a) this model has a single parameter TTC_crit and inherits the free
acceleration a_IDM from the longitudinal model. Both the intersection
geometry and the lengths and widths of all cars must be known 

(b) (C1 AND C2) must be satisfied for all lanes of both mainroad directions

(c) with no stop signs (type III), the secondary traffic can cross at
full speed, or at a predefined reduced speed. The stopping line is
then replaced by a fixed "decision point", say, 20-50 m
upstream. Criterium (4) is then checked with an adapted kinematics
(1). If true, accelerate freely, if not, stop at the stopping line and
proceed as in the case of stop signs 

(d) in "right-has -priority" intersections, the roles "mainroad" or
"secondary road w/o stop signs" depends on the considered OD 

(e) right turns can be added independently

(f) left turns can be decided with a similar rule as (4) but the
specifications become more complex and there are many variants.