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HomeMatlabPath Planning for Formulation Pupil Driverless Vehicles Utilizing Delaunay Triangulation

Path Planning for Formulation Pupil Driverless Vehicles Utilizing Delaunay Triangulation


On this weblog, Veer Alakshendra will present how one can develop a primary path planning algorithm for Formulation Pupil Driverless competitions.
Earlier than we get began, we simply wish to point out that you would be able to run this code in your browser or can obtain the whole dwell script utilizing the buttons on the backside proper nook.

Introduction

Numerous Formulation Pupil competitions have launched the driverless class, the place the purpose for the groups is to design and construct an autonomous car that may compete in several disciplines. On this script, now we have demonstrated the steps to plan a path via a racing monitor utilizing Delaunay triangulation. The appliance is analogous to the primary lap path planning of the Formulation Pupil Driverless competitors to plan the trail via the coordinates of the detected cones.
Please observe that the Delaunay triangulation is simply one of many strategies for planning a path for Formulation Pupil Driverless competitions. You may as well attempt to develop a sampling-based planner like RRT, RRT*, and so on, or every other customized algorithm that finest fulfills your necessities. To develop such planners utilizing MATLAB, please take a look at the features listed on the movement planning webpage.
Determine 1

What’s Delaunay Triangulation?

First, allow us to briefly attempt to perceive Delaunay triangulations. The basic property is the Delaunay criterion. The criterion says that for a set of factors in 2-D, a Delaunay triangulation of those factors ensures the circumcircle related to every triangle incorporates no different level in its inside. Within the determine under, the circumcircle related to T2 is empty. It doesn’t include a degree in its inside. Therefore, this triangulation is a Delaunay triangulation.
Within the algorithm under, now we have used this property to create a path utilizing the detected cones as vertices.
Determine 2

Methodology

Determine 3 exhibits the methodology now we have carried out to plan the trail via the cones. To grasp the algorithm, allow us to undergo the code.
Determine 3

Step 1: Create 2-D Delaunay triangulation

  • Load cone coordinates
As a primary step, we are going to load the x and y coordinates of the internal and outer cones. It’s assumed that the notion algorithm is detecting the yellow and blue cones. As some of the frequent approaches in Formulation Pupil competitions, you need to use the YOLO community to detect cones. For reference, you’ll be able to watch this video to learn to design and practice a YOLO community in MATLAB.
clc
clear
innerConePosition = [6.49447171658257,41.7389113024907;8.49189149682204,41.8037451937836;10.4848751821667,41.8690573815958;12.4735170408320,41.9319164105607;14.4579005366844,41.9894100214277;16.4380855350346,42.0386448286094;22.3534294484539,42.1081444836209;24.3165071700586,42.0957886616701;26.2748937812757,42.0609961552005;28.2282468521326,42.0009961057027;30.1761149294924,41.9130441723441;32.0795946202318,41.7896079706255;33.8817199327800,41.5914171121172;37.1238045770298,40.8042280456036;38.5108000329817,40.1631834270328;40.7498148915033,38.2980872410971;41.6428012725483,37.0572740958520;42.3684952713925,35.6541128117207;42.9237412798850,34.1220862330776;43.4562377484922,30.9418382209873;43.4041825643156,29.3682424552239;43.1310900802234,27.8454335309325;42.6373380962399,26.4081334491868;41.9213606895600,25.0723633391567;39.8493389129556,22.6733478226331;37.1330772214184,20.7325660900255;35.6438750732877,19.9947555768891;34.0970198095490,19.4366654165469;32.5117013027651,19.0709696852617;30.9105163807122,18.9081142458350;29.3050197645280,18.9545691109581;27.6851829730976,19.2045278218877;24.4759113345748,20.2570787967781;21.3685688158133,21.9377743574784;19.8072187451783,22.9775182612717;18.2040616100214,24.1054677217171;16.5291821840059,25.2778069272720;14.7490275765791,26.4446189166734;10.7566166245341,28.5122919557571;8.59035703212379,29.2671114310357;6.36970146209175,29.8034933598624;4.11753580708921,30.1252372241127;-0.405973075502234,30.1500578730274;-2.75740378439869,29.8083496207198;-7.53032772904185,27.7630095130657;-9.55882345130782,25.8360602858122;-10.9967405848897,23.4446538263345;-11.7530941810305,20.7728147412547;-11.7945553145768,18.0098876247875;-11.1377067999357,15.3398589997259;-9.84329815385971,12.9293912425976;-4.31864858110937,8.32717777382664;-2.64821833398598,7.33684919790624;-1.23396167898668,6.38669110355496;-0.139159724307982,5.41310565569044;0.583196113153409,4.42232005043196;1.00846697640521,3.25999992647677;1.42452204766312,-0.235409796458984;1.70769766109080,-2.50157264279400;2.43395748729283,-5.02746251429141;5.90118769712182,-9.36241405660719;8.13219227914984,-10.6723062142501;10.4108490166236,-11.5249176387577;14.7174578544944,-12.4162331505645;16.6611999625517,-12.7135418322201;18.4669199253036,-13.0527675263258;20.2513584750154,-13.4902268953088;22.0275043214267,-14.0208096878619;23.7938485314227,-14.6351269687361;29.0380553533840,-16.8864873288738;30.7681622622371,-17.7392626196140;32.3760783979903,-18.6039808584518;33.7897740653103,-19.5166383512928;34.9667330396653,-20.5090458088496;35.8614855279756,-21.5808775016001;36.4612858521334,-22.7451308643890;36.8711477182864,-25.4557910211687;36.6696611457761,-26.8930663715963;35.5175477601575,-29.6867414698392;34.6369025552692,-30.9011171932826;33.5888282300982,-31.9233335254203;32.4026659456202,-32.7184009884738;29.6211021726631,-33.5886007841515;27.9849204584717,-33.7170193136260;26.2183096377485,-33.6848022646400;24.3309596262912,-33.5465676173721;22.3208062313579,-33.3726612899257;18.1328018280879,-33.1964401661521;14.1429922166079,-33.1731958647648;12.2313503871593,-33.1266878316603;10.3778348140314,-33.0132052855498;8.58829435522285,-32.8040580663346;6.84064584789227,-32.4725374962639;5.07009944482550,-32.0172508962124;3.27322533222869,-31.4640545504384;-0.418687595522242,-30.1796696447230;-4.09627739578414,-28.8238238024923;-5.89202160786470,-28.1122858460715;-7.65255868890126,-27.3645667343373;-9.37355826729420,-26.5706223212060;-11.0772549160090,-25.7164049091686;-14.4775165017317,-23.8562440887552;-16.1852335110450,-22.8738429891042;-17.9066984601430,-21.8724503885825;-19.6280899733090,-20.8717543910881;-22.9283852622260,-18.8697018891971;-24.4790807692059,-17.8277052050408;-27.3051265922114,-15.5801090683244;-28.5448479313762,-14.3530174024786;-29.6530218737847,-13.0404842531107;-30.6444406510768,-11.6234683085559;-31.5169602294234,-10.1146356600596;-32.2692264908853,-8.52739205631154;-32.9016624434539,-6.87420998418598;-34.1160057737752,-1.60803776182190;-34.3261608207142,0.237322459171913;-34.4624174228984,2.11880932640591;-34.5407037273032,4.03273612643633;-34.5779532171168,5.97700580252438;-34.5920061096721,7.95066706685753;-34.6203010935088,11.9628384547546;-34.6463516562825,13.9656473217268;-34.6751545298116,15.9616504117489;-34.7230350194217,19.9334733431980;-34.7332354577762,21.9093507078716;-34.7284338451120,23.8784778261844;-34.6466389854161,27.7320065787716;-34.4887912178893,29.4532110956616;-34.1674004338218,30.9833511541837;-33.6523388115164,32.2835000565933;-32.9540435069214,33.3001054449054;-32.0153881487289,34.0792118997941;-28.7867224485036,35.7027232116937;-26.8737074947600,36.7067299514829;-21.7623826032159,39.5958663210330;-20.1418625665089,40.3549987041733;-18.5286281071084,40.9596629960629;-16.9269162480936,41.3805845888608;-15.2947586791386,41.6060043323431;-13.5312107482484,41.6777684016896;-9.64289482747188,41.5853268011948;-7.61112976471737,41.5421552074769;-5.58318659783131,41.5226049040385;-1.53976115651896,41.5427010208205;0.475431094342800,41.5765527022834;2.48619653317858,41.6224387797260;6.49447171658257,41.7389113024907;8.49189149682204,41.8037451937836;10.4848751821667,41.8690573815958;12.4735170408320,41.9319164105607;14.4579005366844,41.9894100214277] % load internal cone x and y coordinates
outerConePosition = [8.29483356036796,47.8005083348189;10.2903642978411,47.8659036790991;12.2903853701921,47.9291209919643;14.2949817462715,47.9871977358879;16.3042155724446,48.0371512121282;18.3181128601480,48.0759783968909;24.3872198920862,48.0953719564451;26.4188343006777,48.0592693339474;28.4542241392916,47.9967391176812;30.4929111587893,47.9046750145358;32.5715480310539,47.7694057800486;34.7354354110013,47.5303707137414;36.9671887094624,47.1090924863221;41.4608084515852,45.3878796220444;43.5486168233166,43.9466679046951;46.7637769841785,40.1838709296144;47.8711744070693,38.0458741563760;48.6759026786991,35.8287318442088;49.2095764033813,33.5308461962448;49.3717906596030,28.7456240961468;48.9402528124274,26.3442245678724;48.1379348685719,24.0115869446911;46.9718766730536,21.8331831495909;45.5183908761479,19.8937545099424;42.0344522843090,16.7521854290723;38.0024261140352,14.4777602906678;35.7974585974047,13.6826661179064;33.4965697064231,13.1523520909686;31.1302557069132,12.9121393769316;28.7495405480103,12.9803375419597;26.4275819814214,13.3378047377686;24.1935407090117,13.9452300781406;20.0475993937396,15.7534500265274;16.3951857815830,18.0421326577611;14.7344934919394,19.2103582158950;13.1401972893467,20.3265665388672;11.5833537070008,21.3477072094266;10.0428032559345,22.2341839570762;6.90006888859156,23.5101221144020;5.24310276950825,23.9102111346615;3.54614319085073,24.1525066527486;1.82453456285617,24.2384953941222;-1.49646987010121,23.9423419780574;-2.88611819700968,23.5188503619890;-4.87663538345929,22.0841121388762;-5.48860547878733,21.0654842962416;-5.81567263287249,19.9085084532591;-5.83349192339180,18.6923268134878;-5.53917605118359,17.4977407053060;-4.95012865665813,16.4016948398736;-4.05317915721677,15.4107620877673;0.492121533127250,12.4494087837712;2.38653784056378,11.1712478467585;4.26363974073903,9.48929957533922;5.87263326747036,7.25460792254726;6.83628597561633,4.68706884998182;7.22868549746256,2.31823546262226;7.60968340642873,-1.42149659832866;7.97509056184070,-2.72619237583490;8.54802041473498,-3.68009075055991;10.6601317831479,-5.23084300440234;12.1167992046948,-5.77254995615783;13.7805173778787,-6.17261534649334;17.6231089179688,-6.79114948102549;19.7388585053217,-7.18913620702706;21.8298950982966,-7.70159820373082;23.8771264994103,-8.31301696223576;25.8797524901441,-9.00938215804746;27.8385154732156,-9.77726113305095;33.4828656272925,-12.3885256946890;35.3884273243004,-13.4149776838374;37.3243202944926,-14.6682383106842;39.1922962154652,-16.2493960759111;40.8455371260767,-18.2403341823979;42.0705518955054,-20.6153105774461;42.7346589897432,-23.1596769590206;42.5147079384368,-28.2478460553850;41.7538036771420,-30.6144579632713;39.1831183440142,-34.8167170198948;37.3770559671547,-36.5762176430613;35.2523671945937,-37.9984770098583;32.8791324240351,-38.9934826446380;28.1270516052592,-39.7153356388220;25.9046030197368,-39.6765956654354;23.8121031317075,-39.5240911796868;21.8464792201864,-39.3538830619781;19.9742827992146,-39.2418187116909;16.0937484438592,-39.1847191943436;11.9928246424488,-39.1219447460144;9.86352410414552,-38.9911216862446;7.69014987529461,-38.7364552625087;5.51782683234722,-38.3249002543397;3.42382206323132,-37.7869797271760;1.40682879137184,-37.1663842455962;-0.541821138011257,-36.5027177844325;-4.34015621146816,-35.1410426555394;-8.16436000121084,-33.6653461038145;-10.0767778404464,-32.8530238495275;-11.9792230934156,-31.9752972248347;-13.8428086034457,-31.0410375539532;-15.6623928384561,-30.0682796038662;-19.1958077261686,-28.0638760232219;-20.9238240938196,-27.0586776214187;-22.6524765196268,-26.0537507263292;-24.4077080359509,-25.0144494209715;-27.9360780875981,-22.7317004681533;-29.6698538901663,-21.4381474644069;-32.9453564203185,-18.4316844689044;-34.4107565227371,-16.6961594315364;-35.7065080013589,-14.8445665903860;-36.8313109009813,-12.8998963192641;-37.7878917266941,-10.8820330087871;-38.5811746376266,-8.80892992160400;-39.2178699842561,-6.69652639898087;-40.3011807339499,-0.309611756649864;-40.4533406202231,1.78890139416343;-40.5382789183978,3.86217286202336;-40.5775538214583,5.90777727960811;-40.5919497511314,7.92466131148779;-40.6014196538181,9.91275631233656;-40.6457557289961,13.8810850824754;-40.6745334498720,15.8753221224371;-40.7017394704418,17.8763494567535;-40.7332264185658,21.8989357934230;-40.7282871101318,23.9204396925825;-40.7033369152928,25.9485603009070;-40.4291604069450,30.2970202551981;-39.9260632680985,32.6679289532538;-38.9665731582284,35.0689826553948;-37.4022554152914,37.3266934320522;-35.2924298638551,39.1052438930590;-33.2056190082383,40.2549711124138;-29.7935826692359,41.9483259837501;-28.1179159337869,42.9113732230407;-22.4882559681602,45.8771756304123;-20.3655683666973,46.6715497668858;-18.1093389106216,47.2629205744315;-15.7909195473899,47.5854545071348;-13.5681633690504,47.6776546092619;-11.4644254110510,47.6457593795914;-7.51988574761303,47.5414613781388;-5.55737846449435,47.5225493988029;-3.59058902924480,47.5236739806379;0.355197826404633,47.5753479114301;2.33404449770947,47.6205092826547;4.31684977705784,47.6749328204622;8.29483356036796,47.8005083348189;10.2903642978411,47.8659036790991;12.2903853701921,47.9291209919643;14.2949817462715,47.9871977358879;16.3042155724446,48.0371512121282] % load outer cone x and y coordinates
  • Preprocess the info
After loading the info, now we have merged the internal and outer coordinates with alternate coordinates (Determine 4). This step will be sure that the enter to the perform delaunayTriangulation is a matrix whose columns are the x-coordinates, and y-coordinates of the triangulation factors.
Determine 4
[m,nc] = measurement(innerConePosition); % measurement of the internal/outer cone positions information
P = zeros(2*m,nc); % provoke a P matrix consisting of internal and outer coordinates
P(1:2:2*m,:) = innerConePosition;
P(2:2:2*m,:) = outerConePosition; % merge the internal and outer coordinates with alternate values
xp = []; % create an empty numeric xp vector to retailer the deliberate x coordinates after every iteration
yp = []; % create an empty numeric yp vector to retailer the deliberate y coordinates after every iteration
  • Kind triangles
In actual situations, the sensors mounted on the car will detect solely a sure variety of yellow and blue cones whereas going via the race monitor. Therefore, now we have carried out a for loop that enables the code to create Delaunay triangulation objects for each nth interval of the cone place. For instance, if n=4, the Delaunay triangulation shall be created based mostly on the coordinates of 4 cones. The picture under illustrates the process.
Determine 5
interv = 10; % interval
for i = interv:interv:2*m
DT = delaunayTriangulation(P(((abs((i-1)-interv)):i),:)); % create Delaunay triangulation for abs((i-1)-interv)):i factors
Pl = DT.Factors; % coordinates of abs((i-1)-interv)):i vertices
Cl = DT.ConnectivityList; % triangulation connectivity matrix
[mc,nc] = measurement(Pl); % measurement
determine(1) % plot delaunay triangulations
triplot(DT,‘okay’)
grid on
ax = gca;
ax.GridColor = [0, 0, 0]; % [R, G, B]
xlabel(‘x(m)’)
ylabel(‘y (m)’)
set(gca,‘Coloration’,‘#EEEEEE’)
title(‘Delaunay Triangulation’)
maintain on

Step 2: Take away exterior triangles

  • Outline constraints
Whereas performing triangulation, the coordinates of the internal and outer cones are sure to create triangles exterior the boundary of the monitor. For example, Determine 6 exhibits one of many circumstances the place the outside triangle is fashioned.
Determine 6
As these triangles can generate a mistaken path, now we have eliminated them by imposing constraints, C. These constraints are the vertex IDs of constrained edges, specified as a 2-column matrix. Every row of C corresponds to a constrained edge and incorporates two IDs:
C(j,1) is the ID of the vertex at first of an edge.
C(j,2) is the ID of the vertex at finish of the sting.
For example, Determine 7 exhibits the vertex IDs of the constrained edges the place the matrix C = [2 1;1 3;3 5;5 6;2 4;4 6].
Determine 7
So now now we have outlined the constraints for the internal and outer boundaries.
% internal and outer constraints when the interval is even
if rem(interv,2) == 0
cIn = [2 1;(1:2:mc-3)’ (3:2:(mc))’; (mc-1) mc];
cOut = [(2:2:(mc-2))’ (4:2:mc)’];
else
% internal and outer constraints when the interval is odd
cIn = [2 1;(1:2:mc-2)’ (3:2:(mc))’; (mc-1) mc];
cOut = [(2:2:(mc-2))’ (4:2:mc)’];
finish
C = [cIn;cOut]; % create a matrix connecting the constraint boundaries
  • Create Delaunay triangulation with constraints
As soon as now we have outlined the constraints, now we have used the delaunayTriangulation object to create 2-D Delaunay triangulations.
TR = delaunayTriangulation(Pl,C); % Delaunay triangulation with constraints
Earlier than we transfer to the following step, you will need to introduce you to the ‘Connectivity Listing.’ This property shall be used within the subsequent steps to create a brand new triangulation by excluding the outside triangles.
As per the documentation, the triangulation connectivity record is a matrix with the next traits:
  • Every aspect in DT.ConnectivityList is a vertex ID.
  • Every row represents a triangle or tetrahedron within the triangulation.
  • Every row variety of DT.ConnectivityList is a triangle or tetrahedron ID.
For instance, in Determine 8 the weather of the primary row [2 1 3] signify the vertices of the primary triangle.
Determine 8
Now that you just understood the that means of the connection record, allow us to output the connectivity record of TR.
TRC = TR.ConnectivityList; % triangulation connectivity matrix
Upon getting listed the connectivity matrix, we have to take away the rows that assemble exterior triangles. Determine 9 exhibits that the second row [1 5 3] represents an exterior triangle.
Determine 9
With the delaunayTriangulation object, you’ll be able to carry out a wide range of topological and geometric queries. For our case, now we have used the article perform isInterior which returns a column vector of logical values that point out whether or not the triangles are inside a bounded geometric area. The ith triangle within the triangulation is taken into account to be contained in the area if the ith logical flag is true, in any other case, it’s exterior. For instance, as proven in Determine 10, the outside triangle is assigned to the logical worth 0 or false.
Determine 10
TL = isInterior(TR); % logical values that point out whether or not the triangles are contained in the bounded area
TC = TR.ConnectivityList(TL,:); % triangulation connectivity matrix

From the earlier step, now we have obtained a brand new connectivity matrix that doesn’t include the outside triangles. So now on this step, now we have used the up to date connectivity matrix TC to create 2-D triangulation utilizing the factors in matrix Pl.
[~,pt] = type(sum(TC,2)); % non-obligatory step. The rows of connectivity matrix are organized in ascending sum of rows…
% This ensures that the triangles are related in progressive order.
TS = TC(pt,:); % connectivity matrix based mostly on ascending sum of rows
TO = triangulation(TS,Pl); % create triangulations based mostly on sorted connectivity matrix
determine(2) % plot delaunay triangulations
triplot(TO,‘okay’)
grid on
ax = gca;
ax.GridColor = [0, 0, 0]; % [R, G, B]
xlabel(‘x(m)’)
ylabel(‘y (m)’)
set(gca,‘Coloration’,‘#EEEEEE’)
title(‘Delaunay Triangulation with out Outliers’)
maintain on

Step 3: Discover midpoints of inner edges

As soon as now we have eliminated the outliers, the following step is easy. We simply must compute the midpoint of the inner edges.
Determine 11
xPo = TO.Factors(:,1);
yPo = TO.Factors(:,2);
E = edges(TO); % triangulation edges
iseven = rem(E, 2) == 0; % neglect boundary edges
Eeven = E(any(iseven,2),:);
isodd = rem(Eeven,2) ~=0;
Eodd = Eeven(any(isodd,2),:);
xmp = ((xPo((Eodd(:,1))) + xPo((Eodd(:,2))))/2); % x coordinate midpoints
ymp = ((yPo((Eodd(:,1))) + yPo((Eodd(:,2))))/2); % y coordinate midpoints
Pmp = [xmp ymp]; % midpoint coordinates

Step 4: Interpolate midpoints

Lastly, to acquire a easy path now we have carried out interpolation.
Determine 12
distancematrix = squareform(pdist(Pmp));
distancesteps = zeros(size(Pmp)-1,1);
for j = 2:size(Pmp)
distancesteps(j-1,1) = distancematrix(j,j-1);
finish
totalDistance = sum(distancesteps); % complete distance travelled
distbp = cumsum([0; distancesteps]); % distance for every waypoint
gradbp = linspace(0,totalDistance,100);
xq = interp1(distbp,xmp,gradbp,‘spline’); % interpolate x coordinates
yq = interp1(distbp,ymp,gradbp,‘spline’); % interpolate y coordinates
xp = [xp xq]; % retailer obtained x midpoints after every iteration
yp = [yp yq]; % retailer obtained y midpoints after every iteration
Plot outcomes
determine(3)
% subplot
pos1 = [0.1 0.15 0.5 0.7];
subplot(‘Place’,pos1)
pathPlanPlot(innerConePosition,outerConePosition,P,DT,TO,xmp,ymp,cIn,cOut,xq,yq)
title([‘Path planning based on constrained Delaunay’ newline ‘ triangulation’])
% subplot
pos2 = [0.7 0.15 0.25 0.7];
subplot(‘Place’,pos2)
pathPlanPlot(innerConePosition,outerConePosition,P,DT,TO,xmp,ymp,cIn,cOut,xq,yq)
xlim([min(min(xPo(1:2:(mc-1)),xPo(2:2:mc))) max(max(xPo(1:2:(mc-1)),xPo(2:2:mc)))])
ylim([min(min(yPo(1:2:(mc-1)),yPo(2:2:mc))) max(max(yPo(1:2:(mc-1)),yPo(2:2:mc)))])
finish
h = legend(‘yCone’,‘bCone’,‘begin’,‘midpoint’,‘inner edges’,
‘internal boundary’,‘outer boundary’,‘deliberate path’);
Pp = [xp’ yp’]; % concatenated deliberate path
pathPlanningGIF.gif
Determine 13

What subsequent?

So, the algorithm solely computes the trail via the cones. Nevertheless, in Formulation Pupil Driverless competitions, the car must concurrently plan and monitor the trail within the first lap. Therefore as a subsequent job, you’ll be able to attempt to implement a trajectory monitoring controller. Here’s a tutorial that exhibits the way to implement trajectory monitoring controllers in MATLAB and Simulink: Simulating Trajectory Monitoring Controllers for Driverless Vehicles.
Additional, in case you are to generate an optimized raceline please be happy to take a look at this GitHub repository from Gautam Shetty: Raceline Optimization.
Additionally, in case of any queries associated to this weblog please be happy to achieve out to us at racinglounge@mathworks.com.
perform y = pathPlanPlot(innerConePosition,outerConePosition,P,DT,TO,xmp,ymp,cIn,cOut,xq,yq) %perform to animate the plot
plot(innerConePosition(:,1),innerConePosition(:,2),‘.y’,‘MarkerFaceColor’,‘y’)
maintain on
plot(outerConePosition(:,1),outerConePosition(:,2),‘.b’,‘MarkerFaceColor’,‘b’)
plot(P(1,1),P(1,2),‘|’,‘MarkerEdgeColor’,‘#77AC30’,‘MarkerSize’,15, ‘LineWidth’,5)
grid on
ax = gca;
ax.GridColor = [0, 0, 0]; % [R, G, B]
xlabel(‘x(m)’)
ylabel(‘y (m)’)
set(gca,‘Coloration’,‘#EEEEEE’)
maintain on
plot(xmp,ymp,‘*okay’)
drawnow
maintain on
triplot(TO,‘Coloration’,‘#0072BD’)
drawnow
maintain on
plot(DT.Factors(cOut’,1),DT.Factors(cOut’,2),
‘Coloration’,‘#7E2F8E’,‘LineWidth’,2)
plot(DT.Factors(cIn’,1),DT.Factors(cIn’,2),
‘Coloration’,‘#7E2F8E’,‘LineWidth’,2)
drawnow
maintain on
plot(xq,yq,‘Coloration’,‘#D95319’,‘LineWidth’,3)
drawnow
finish
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