CN111477035B - Low-altitude navigation network geometric structure generation method oriented to safety distance constraint - Google Patents
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Abstract
The invention discloses a low-altitude navigation network geometric structure generation method oriented to safe distance constraint, which comprises the following steps: establishing a general model of a navigation network; and (3) generating an airway structure: on the basis of generating basic geometric structures of the air routes, determining basic structural parameters of each air route, and calculating to obtain safety distance constraint conditions and corresponding isolation belt widths of all the air routes by taking the distance between unmanned aerial vehicles positioned in different air routes as a target to be larger than an obstacle avoidance distance; and (3) generating an intersection structure: on the basis of generating a basic geometric structure of the intersection, calculating to obtain safety distance constraint conditions and corresponding radiuses of all intersections and updating the widths of isolation belts of part of the air routes by taking the distance between unmanned aerial vehicles positioned in the intersection as a target to be larger than an obstacle avoidance distance. The method can automatically generate a navigation network model consisting of a plurality of geometric structures, reduces the pressure of a low-altitude traffic control system, and is favorable for safe and orderly operation of air traffic.
Description
Technical Field
The invention relates to the technical field of unmanned aerial vehicle navigation network management, in particular to a low-altitude navigation network geometric structure generation method oriented to safety distance constraint.
Background
Due to the characteristics of low cost, convenience in taking off and landing, strong maneuverability and the like, the unmanned aerial vehicle capable of taking off and landing vertically is widely applied to the practical fields of logistics transportation, cruise monitoring, agricultural plant protection and the like in a low-altitude airspace. However, low-altitude airspaces are susceptible to magnetic interference, GPS interference, weather, and complex terrain. A large amount of low-altitude unmanned-machine disordered flight can bring harm to ground facilities, public safety, aerial manned aircrafts and the like. The design can take off and land unmanned aerial vehicle's low latitude navigation network perpendicularly, makes unmanned aerial vehicle people fly in the navigation network of planning. Therefore, the pressure of the low-altitude traffic control system can be relieved from a strategic aspect, and the safe and orderly operation of air traffic is facilitated. With the further popularization of unmanned aerial vehicle application, the traditional dotted line road network is increasingly not suitable. Therefore, it is very important and meaningful to research a low-altitude navigation network route and intersection geometric structure design method oriented to safety distance constraint.
Disclosure of Invention
The invention aims to provide a safety distance constraint-oriented low-altitude navigation network geometric structure generation method, wherein the geometric structure of a navigation network is not an abstract point or line any more, but a navigation network model consisting of a plurality of geometric structures is automatically generated by combining airspace characteristics and obstacle avoidance distances of unmanned aerial vehicles, and then the obstacle avoidance distances among the unmanned aerial vehicles in different navigation channels are ensured to be kept as an optimization target, and relevant parameters of the navigation network model are optimally adjusted. The invention can reduce the pressure of the low-altitude traffic control system from the strategic aspect and is also beneficial to the safe and orderly operation of air traffic.
To achieve the above object, with reference to fig. 1, the present invention provides a method for generating a low-altitude navigation network geometry oriented to a safe distance constraint, where the method includes:
s1, establishing a navigation network general model, wherein the navigation network general model comprises intersections, navigation paths, routes and a navigation network;
the intersection refers to the intersection or end point of each route, is in a cylindrical shape and is represented by V in the graph theory;
the air route is an unmanned aerial vehicle flight pipeline which is established based on a base station and is connected with each intersection and is represented by E in the graph theory;
the route is a route from a flying point to a landing point of the unmanned aerial vehicle and consists of related channels;
the navigation network is an unmanned aerial vehicle traffic network which is formed by the navigation paths and intersections in one airspace and is represented by a directed graph G (V, E);
s2, generating a route structure: on the basis of generating basic geometric structures of the air routes, determining basic structural parameters of each air route, introducing obstacle avoidance distances, analyzing characteristics of all the air routes, and calculating to obtain safety distance constraint conditions and corresponding isolation belt widths of all the air routes by aiming at the fact that the distances between unmanned aerial vehicles positioned in different air routes are larger than the obstacle avoidance distances;
s3, generating an intersection structure: on the basis of generating basic geometric structures of the intersections, analyzing the characteristics of all the intersections, calculating to obtain the safety distance constraint conditions and the corresponding radiuses of all the intersections by taking the distance between unmanned aerial vehicles positioned in the intersections as a target to be larger than the obstacle avoidance distance, and updating the widths of the isolation belts of part of the air routes.
As a preferable example of the intersection, the intersection comprises an airport and a free flight airspace.
As a preferred example, the waterway comprises at least two channels separated by independent isolation belts.
As a preferred example, the generating the route structure includes the following steps:
s21, generating basic geometric structure of air route, and setting symbolDenotes the ith route, i 1,2Width 2rAW+rIS>0, high hAWA cuboid shape > 0 comprising three parts: one longWidth rIS>0, high hAWCuboid isolation belt greater than 0Two opposite-direction driving channels separated by the isolation beltAndline segment [ p ]s,i,pe,i]Represents the vertical centerline of the side of the ith airway, wherein,respectively representing the geometric centers of the left side surface and the right side surface of the ith airway;
s22, introducing an obstacle avoidance distance ra,ra>0;
S23, analyzing the characteristics of all the air routes, and calculating the safety distance constraint conditions and the corresponding widths of the isolation belts of all the air routes by taking the distance between the unmanned aerial vehicles in different air routes as the purpose that the distance is larger than the obstacle avoidance distance:
s231, defining a set S1And S2The distance between them is as follows:
s232, judging whether the unmanned aerial vehicles in the different channels belong to different channels of the same route or different channels of different routes, if so, entering the step S233, otherwise, entering the step S234;
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaAnd the width r of the isolation belt of the output airwayISIt satisfies the safe distance constraint:
rIS>ra;
s234, for the ith routeAnd jth airwayWherein i ≠ j, the design route satisfies the safety distance constraint:
namely:
in the formula, hAWIs the height of the airway, rAWIs the width of the channel, rISWidth of isolation strip, ps,i,pe,iRespectively, the geometric centers of the left and right sides of the ith airway.
As a preferred example thereof, the generating of the intersection structure includes the steps of:
s31, generating an intersection basic geometry, wherein for the xth intersection, x is 1,2xAt which the routes meet and the centre lines of these routesAll intersect at a point oIT,xI.e. by
Wherein x isjIs a pointer, symbol, of the air route number indexRepresents the x-th intersection, which is a center point at oIT,xRadius rIT,xHigh h isIT≥hAWThe cylindrical body of (a) is,
s32, analyzing the characteristics of all crossroads and determining the number M of routes connected with the xth crossroadxIntersections are divided into two categories: when M isxWhen the intersection is 2, the intersection is defined as a circular intersection; when M isx> 2, the intersection is defined as a direct intersection;
s33, aiming at the fact that the distance between unmanned aerial vehicles in the intersection is larger than the obstacle avoidance distance, calculating to obtain the safety distance constraint conditions and the corresponding radiuses of all intersections, and updating the widths of the isolation belts of part of the roads:
wherein
Namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaNumber of routes M connecting to the intersectionxParameter r of connected routesIS,rAW,hAW,Radius r of output intersectionIT,x,ra,hAW,rAW,rISAre all constants;
for a direct junction, the direct junction is designed to satisfy a safety distance constraint:
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaNumber of routes M connecting to the intersectionxParameters of connected routesOutputting the width r of the isolation belt of the air route connected with the crossroadIS。
Compared with the prior art, the technical scheme of the invention has the following remarkable beneficial effects:
(1) the geometric structure of the navigation network is not an abstract point or line any more, but a navigation network model consisting of a plurality of geometric structures is automatically generated by combining the airspace characteristics and the obstacle avoidance distance of the unmanned aerial vehicle, so that the pressure of a low-altitude traffic control system can be relieved from a strategic level, and the safe and orderly operation of air traffic is facilitated.
(2) The method for automatically optimizing the model parameters of the airway network is provided, when an airway changes or a new airway is increased, the relevant structural parameters of the airway network model are automatically adjusted, and the method has strong applicability and expansibility for continuously and rapidly developing unmanned aerial vehicle technology.
It should be understood that all combinations of the foregoing concepts and additional concepts described in greater detail below can be considered as part of the inventive subject matter of this disclosure unless such concepts are mutually inconsistent. In addition, all combinations of claimed subject matter are considered a part of the presently disclosed subject matter.
The foregoing and other aspects, embodiments and features of the present teachings can be more fully understood from the following description taken in conjunction with the accompanying drawings. Additional aspects of the present invention, such as features and/or advantages of exemplary embodiments, will be apparent from the description which follows, or may be learned by practice of specific embodiments in accordance with the teachings of the present invention.
Drawings
The drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. Embodiments of various aspects of the present invention will now be described, by way of example, with reference to the accompanying drawings, in which:
FIG. 1 is a flow chart of a method for generating a low-altitude navigation network geometry facing a safe distance constraint according to the invention.
Fig. 2 is a three-dimensional schematic diagram of the unmanned aerial vehicle navigation network of the present invention.
FIG. 3 is a schematic view of a model of the airway network of the present invention.
FIG. 4 is a schematic representation of the airway structure of the present invention, hAWIs the height of the airway, rAWIs the width of the channel, rISWidth of isolation strip, ps,i,pe,iRespectively, the geometric centers of the left and right sides of the ith airway.
FIG. 5 is a schematic view of the unmanned aerial vehicle of the present invention flying in opposite directions in the same flight path, raThe obstacle avoidance distance between the unmanned aerial vehicles.
Fig. 6 is a schematic view of the unmanned aerial vehicle of the present invention flying in opposite directions in different air paths.
FIG. 7 is a cross-road structure diagram of the present invention, oIT,xIs the center of the x-th intersection, rIT,xRadius of the x-th intersection, hIT,xIs the height of the xth intersection.
FIG. 8 is a schematic view of the intersection of the present invention with the route at the x-th position1Each route x2And x2The paths meeting, their respective centre line segmentsAndintersect at a point oIT,x。rIT,xRadius of the x-th intersection, oIT,xWhich is the center of the xth intersection.Are respectively the x-th1The right/left lane boundary of the individual airway and the midpoint of the intersection segment to the side of the xth intersection.Is a line segmentAndthe intersection point of (a).Is a line segmentAndthe angle of,is a line segmentAndthe included angle of (a).
FIG. 9 is a schematic view of the invention showing unmanned planes flying in opposite directions at a direct junction, xth1Individual airway and x2Each route meets at the x-th intersection oIT,xIs the center of the intersection.Are respectively the x-th1Isolation belt for individual airwayWith the intersection of the x-th intersection middle cross section.Are respectively the x-th2Isolation belt for individual airwayWith the intersection of the x-th intersection middle cross section. t is a pointTo line segmentIs used for the foot drop.
Fig. 10 is a map of the results of the aircraft network based on MATLAB display of the present invention, where the cylinder is the intersection, the cuboid is the airway, and the numbers 1,2, … 12 are the airway labels.
Detailed Description
In order to better understand the technical content of the present invention, specific embodiments are described below with reference to the accompanying drawings.
Detailed description of the preferred embodiment
The invention provides a low-altitude navigation network design method based on a vertical take-off and landing unmanned aerial vehicle, and provides a new method for establishing a low-altitude navigation network of an unmanned aerial vehicle. The specific implementation steps are as follows:
step 1: general model establishment for navigation network
Like a flight corridor in civil aviation, an abstract unmanned plane route is expanded into an unmanned plane pipeline, so that a three-dimensional schematic diagram of an unmanned plane route network shown in fig. 2 is established. The intersection refers to an intersection or an end point of each route, and may be an airport or a free flight airspace. Usually, the unmanned aerial vehicle needs to go to the next route through the intersection, and therefore the cylinder that designs to have a certain volume provides the space that turns to for unmanned aerial vehicle. All M intersections are represented by V in the graph theory. The air route refers to unmanned aerial vehicle flight pipelines which are established based on the base station and are connected with intersections. The routes herein are straight routes and have width and height limitations, all N routes being denoted by E in graph theory. The channel is an unmanned aerial vehicle flight channel which is connected with each intersection and has a specified passing direction. An interior of a waterway may be subdivided into a plurality of lanes, similar to lanes on a highway. The route refers to a route from a flying point to a landing point of the unmanned aerial vehicle. It consists of related channels, similar to navigation routes. The navigation network is an unmanned aerial vehicle traffic network which is formed by a navigation path and an intersection together in one airspace, and is represented by a directed graph G ═ V, E.
The three-dimensional route network shown in fig. 2 is further represented as a route network model as shown in fig. 3. On the basis of the general model, in order to ensure the flight safety of the unmanned aerial vehicle, the unmanned aerial vehicle is only required to enter the following three areas: a channel functioning similarly to a lane, an intersection formed by at least two straight routes, and a take-off and landing airport.
Step 2: structural design of air route
2.1 basic geometric Structure design of airway
A model of the airway with the median as shown in figure 4 was designed. SymbolDenotes the ith route, i 1,2, N, is a long routeWidth 2rAW+rIS>0, high hAWA cuboid > 0. The method comprises three parts in total: one longWidth rIS>0, high hAWCuboid isolation belt greater than 0Two opposite-direction driving channels separated by the isolation beltAndas indicated by the dashed line in fig. 4, line segment ps,i,pe,i]Representing the vertical centerline of the side of the ith airway. Wherein the content of the first and second substances,respectively, the geometric centers of the left and right sides of the ith airway.
2.2 geometric design of airway under safety distance constraint
Introducing a concept raAnd (4) indicating the obstacle avoidance distance between the unmanned planes. Keeping distance r between unmanned aerial vehicles designed to be located in different channelsaAnd if the distance is more than 0, the unmanned aerial vehicles in different channels do not need to avoid each other. The different channels mainly comprise two cases, wherein the different channels in the first case refer to channels in the same route; in the second case, different lanes refer to lanes in different routes. To facilitate subsequent deductions, a set S is defined1And S2The distance between them is as follows:
in the first case, as shown in FIG. 5, for the ith routeDesigning an airway to meet the safety distance constraint:
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaAnd the width r of the isolation belt of the output airwayISIt satisfies the safe distance constraint:
rIS>ra。 (3)
as shown in fig. 6Two cases, for the ith flight routeAnd jth airwayWherein i ≠ j, the design route satisfies the safety distance constraint:
namely:
specifically, the formula is realized by adjusting the radius of the intersection in step 3.
And step 3: intersection structure design
3.1 basic geometry design of crossroad
For the xth intersection, x is 1,2xAt which the routes meet and the centre lines of these routesAll intersect at a point oIT,xNamely:
wherein x isjIs a pointer to the route number index. Thereby designing an intersection model as shown in fig. 7. SymbolRepresents the x-th intersection, which is a center point at oIT,xRadius rIT,xHigh h isIT≥hAWThe cylindrical body of (a) is,hIT,xis the height of the xth intersection.
3.2 design of geometry of crossroad under constraint of safe distance
As shown in fig. 8, for the xth intersectionOf roads meeting at the intersectionAre all parallel to the ground plane and are located at the same height. Wherein j is 1, …, Mx. According to the number M of the routes connected with the xth intersectionxIntersections are classified into two categories. In the first class M x2, for example, 1 st, 7 th, 8 th and 9 th intersections in fig. 3, which are collectively referred to as roundabout intersections; m in the second classxAnd 2, such as 2 nd, 3 rd, 4 th, 5 th and 6 th intersections in the figure 3, which are collectively called direct intersections.
In the first case, as shown in FIG. 8, for the xth intersectionDesigning a circular intersection to meet safety distance constraints:
wherein:
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaNumber of routes M connecting to the intersectionxConnected to the air routeParameter rIS,rAW,hAW,Radius r of output intersectionIT,xThereby supporting formulas and equations.
From the formulara,hAW,rAW,rISAre all constants and follow the formula as rIT,xIncrease of (2)Therefore, if rIT,xIf it is large enough, r must be found according to the formulaIT,xThe solution of (1). But to avoid rIT,xToo large, it should be notedThe value of (A) is not to be too small.
As in the second case shown in fig. 9, it is assumed that the unmanned aerial vehicle is passing in opposite directions within a virtual channel in the direct intersection. Except for designing the xth intersectionThe radius of the first and second electrodes satisfies the formula, and the first and second electrodes are further designed andthe connected routes satisfy the safety distance constraint:
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaNumber of routes M connecting to the intersectionxParameters of connected routesOutput and crossRoad isolation belt width r connected at crossroadISThereby supporting the virtual channel to satisfy the formula.
Detailed description of the invention
Taking a local low-altitude airspace with the length of 500m, the width of 400m and the height of 20m as an example, the method designs and displays results of a navigation network in the low-altitude airspace based on MATLAB. The simulation and calculation processes are carried out on a computer with main frequency of 3.70Ghz and internal memory of 32.0GB and MATLABR2018b under a Win10 professional operating system. The method comprises the following specific implementation steps:
the method comprises the following steps: general model establishment for navigation network
The construction of a generic model of the navigation network requires 2 parameters:and V ═ I1,2,I1,3,4,I2,5,6,I3,5,7,8,I4,7,9,I8,9,10,I11,12,I6,11,I10,12And obtaining a directed graph G (V, E) containing M9 intersections and N12 routes.
Step two: structural design of air route
The routes are designed according to the basic structure shown in FIG. 4, and a parameter h is set for each of the routes EAW=16m,rAW=16m,ra=4m。
According to a formula and a safety distance constraint condition shown by the formula, the isolation bandwidth value of any air route can be obtained as rIS=4.2m。
Step three: intersection structure design
Intersections were designed according to the basic structure shown in fig. 7, and the parameter h was set for each intersection in VITThe intersection center is set to be o at 16mIT,1=[100,0,30]T,oIT,2=[300,0,30]T,oIT,3=[0,150,30]T,oIT,4=[250,150,30]T,oIT,5=[400,150,30]T,oIT,6=[300,300,30]T,oIT,7=[100,300,30]T,oIT,8=[50,225,30]T,oIT,9=[200,300,30]T。
The M-9 intersections are classified into two types according to the number of connecting routes. For the first type of roundabout I1,3,4,I2,5,6,I3,5,7,8,I4,7,9,I8,9,10Obtaining the radius r of the corresponding intersection according to the safety distance constraint condition shown by the formulaIT,2=52.80m,rIT,3=50.22m,rIT,4=41.43m,rIT,5=49.60m,rIT,652.80 m. For the second type direct connection intersection I1,2,I11,12,I6,11,I10,12Obtaining the radius r of the corresponding intersection according to the safety distance constraint condition shown by the formulaIT,1=31.04m,rIT,7=31.04m,rIT,8=28.71m,rIT,928.71 m; then according to the safety distance constraint condition shown by the formula, the width of the isolation belt is updated to be rIS=4.764m。
Finally, the designed route network is drawn by using the 3D drawing function of MATLAB (as shown in fig. 10).
In this disclosure, aspects of the present invention are described with reference to the accompanying drawings, in which a number of illustrative embodiments are shown. Embodiments of the present disclosure are not necessarily defined to include all aspects of the invention. It should be appreciated that the various concepts and embodiments described above, as well as those described in greater detail below, may be implemented in any of numerous ways, as the disclosed concepts and embodiments are not limited to any one implementation. In addition, some aspects of the present disclosure may be used alone, or in any suitable combination with other aspects of the present disclosure.
Although the present invention has been described with reference to the preferred embodiments, it is not intended to be limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, the protection scope of the present invention should be determined by the appended claims.
Claims (5)
1. A low-altitude navigation network geometric structure generation method oriented to safe distance constraint is characterized by comprising the following steps:
s1, establishing a navigation network general model, wherein the navigation network general model comprises intersections, navigation paths, routes and a navigation network;
the intersection refers to the intersection or end point of each route, is in a cylindrical shape and is represented by V in the graph theory;
the air route is an unmanned aerial vehicle flight pipeline which is established based on a base station and is connected with each intersection and is represented by E in the graph theory;
the route is a route from a flying point to a landing point of the unmanned aerial vehicle and consists of related channels;
the navigation network is an unmanned aerial vehicle traffic network which is formed by the navigation paths and intersections in one airspace and is represented by a directed graph G (V, E);
s2, generating a route structure: on the basis of generating basic geometric structures of the air routes, determining basic structural parameters of each air route, introducing obstacle avoidance distances, analyzing characteristics of all the air routes, and calculating to obtain safety distance constraint conditions and corresponding isolation belt widths of all the air routes by aiming at the fact that the distances between unmanned aerial vehicles positioned in different air routes are larger than the obstacle avoidance distances;
s3, generating an intersection structure: analyzing the characteristics of all intersections on the basis of generating basic geometric structures of the intersections, calculating safety distance constraint conditions and corresponding radiuses of all the intersections by taking the distance between unmanned aerial vehicles positioned in the intersections as a target to be larger than an obstacle avoidance distance, and updating the widths of isolation belts of part of the air routes;
wherein, one route is internally subdivided into a plurality of routes; the airway is provided with an isolation belt which separates two opposite driving airways;
the generating of the airway structure comprises the following steps:
s21, generating basic geometric structure of air route, and setting symbolDenotes the ith route, i 1,2Width 2rAW+rISGreater than 0, high hAWA cuboid shape > 0 comprising three parts: one longWidth rISGreater than 0, high hAWCuboid isolation belt greater than 0Two opposite-direction driving channels separated by the isolation beltAndline segment [ p ]s,i,pe,i]Represents the vertical centerline of the side of the ith airway, where ps,i,Respectively representing the geometric centers of the left side surface and the right side surface of the ith airway;
s22, introducing an obstacle avoidance distance ra,ra>0;
S23, analyzing the characteristics of all the air routes, and calculating the safety distance constraint conditions and the corresponding widths of the isolation belts of all the air routes by taking the distance between the unmanned aerial vehicles in different air routes as the purpose that the distance is larger than the obstacle avoidance distance:
s231, defining a set S1And S2The distance between them is as follows:
s232, judging whether the unmanned aerial vehicles in the different channels belong to different channels of the same route or different channels of different routes, if so, entering the step S233, otherwise, entering the step S234;
namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaAnd the width r of the isolation belt of the output airwayISIt satisfies the safe distance constraint:
rIS>ra;
s234, for the ith routeAnd jth airwayWherein i ≠ j, the design route satisfies the safety distance constraint:
namely:
in the formula, hAWIs the height of the airway, rAWIs the width of the channel, rISWidth of isolation strip, ps,i,pe,iRespectively, the geometric centers of the left and right sides of the ith airway.
2. The safe-distance-constraint-oriented low-altitude navigation network geometry generation method as claimed in claim 1, wherein the intersection comprises both an airport and a free-flight airspace.
3. The safe-distance-constraint-oriented low-altitude navigation network geometry generation method as claimed in claim 1, wherein the navigation path comprises at least two navigation paths separated by independent isolation belts.
4. The method for generating a low-altitude navigation network geometry oriented to safe distance constraint according to claim 1, characterized in that the method for generating the intersection structure comprises the following steps:
s31, generating an intersection basic geometry, wherein for the xth intersection, x is 1,2xAt which the routes meet and the centre lines of these routesAll intersect at a point oIT,xI.e. by
Wherein x isjIs a pointer, symbol, of the air route number indexRepresents the x-th intersection, which is a center point at oIT,xRadius rIT,xHigh h isIT≥hAWThe cylindrical body of (a) is,
s32, analyzing the characteristics of all crossroads and determining the number M of routes connected with the xth crossroadxIntersections are divided into two categories: when M isxWhen the intersection is 2, the intersection is defined as a circular intersection; when M isx> 2, the intersection is defined as a direct intersection;
s33, aiming at the fact that the distance between unmanned aerial vehicles in the intersection is larger than the obstacle avoidance distance, calculating to obtain the safety distance constraint conditions and the corresponding radiuses of all intersections, and updating the widths of the isolation belts of part of the roads:
wherein
Namely by inputting the obstacle avoidance distance r between the unmanned aerial vehiclesaNumber of routes M connecting to the intersectionxParameter r of connected routesIS,rAW,hAW,Radius r of output intersectionIT,x,ra,hAW,rAW,rISAre all constants;
for a direct junction, the direct junction is designed to satisfy a safety distance constraint:
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