本文根據LQR軌跡跟蹤算法Python/Matlab算法實現_LQRmatrix推導(2)使用代碼實現，進行仿真：
clc
clear all
Kp = 1.0 ;
dt =0.1 ;% [s]
L = 2.9 ;% [m] wheel base of vehicle
Q = eye(3);
% Q(1,1) = 10;
% Q(2,2) =20;
% Q(3,3) = 5;
% Q(4,4) = 10;
%%%%%%%%%% about 27
Q(1,1) = 36;
Q(2,2) =36;
Q(3,3) = 1;
%%%%%%%%%%
R = eye(1)* 200;
max_steer =30 * pi/180; % in rad
target_v =30.0 / 3.6;
cx = 0:0.1:200; % sampling interception from 0 to 100, with step 0.1
for i = 1:500% here we create a original reference line, which the vehicle should always follow when there is no obstacles;
cy(i) = -sin(cx(i)/10)*cx(i)/8;
end
for i = 501: length(cx)
cy(i) = -sin(cx(i)/10)*cx(i)/8; %cy(500);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% here we provide another reference line for testing
% cx = [0.0, 0.0820204315351055, 0.16405076165331595, 0.24610088893773643, 0.32818071197147175, 0.41030012933762705, 0.4924690396193074, 0.5746973413996174, 0.6569949332616624, 0.7393717137885472, 0.8218375815633768, 0.9044024351692563, 0.9870761731892908, 1.0698686942065845, 1.1527898968042436, 1.235849679565372, 1.3190579410730754, 1.4024245799104587, 1.485959494660626, 1.5696725839066838, 1.6535737462317357, 1.7376728802188874, 1.821979884451244, 1.9065046575119102, 1.9912570979839908, 2.076247104450591, 2.161484575494815, 2.24697940969977, 2.3327415056485585, 2.4187807619242876, 2.5051070771100594, 2.5917303497889823, 2.678660478544159, 2.7659073619586954, 2.853480898615696, 2.9413909870982655, 3.0296475259895104, 3.1182604138725343, 3.2072395493304424, 3.29659483094634, 3.3863361573033313, 3.476473426984523, 3.5670165385730184, 3.6579753906519223, 3.749359881804342, 3.84117991061338, 3.9334453756621435, 4.026166175533734, 4.119352208811261, 4.213013374077826, 4.307159569916534, 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% cy = [0.0, -0.037366703007607714, -0.07474321440358046, -0.11213934257628329, -0.1495648959140812, -0.1870296828053393, -0.22454351163842262, -0.2621161908016961, -0.29975752868352495, -0.3374773336722741, -0.37528541415630867, -0.41319157852399363, -0.45120563516369416, -0.48933739246377506, -0.5275966588126015, -0.5659932425985387, -0.6045369522099514, -0.6432375960352048, -0.6821049824626639, -0.721148919880694, -0.7603792166776596, -0.7998056812419262, -0.8394381219618586, -0.8792863472258223, -0.9193601654221818, -0.959669384939302, -1.0002238141655486, -1.0410332614892863, -1.08210753529888, -1.1234564439826953, -1.1650897959290965, -1.207017399526449, -1.2492490631631181, -1.2917945952274683, -1.3346638041078651, -1.3778664981926732, -1.4214124858702577, -1.465311575528984, -1.5095735755572173, -1.5542082943433218, -1.5992255402756625, -1.6446351217426058, -1.6904468471325154, -1.7366705248337566, -1.7833159632346953, -1.8303929707236954, -1.877911355689123, 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6.68746835748119, 6.6751302775518795, 6.657613261458263, 6.634835602206671, 6.606715592803418, 6.573171526254818, 6.534121695567208, 6.48948507667127, 6.439288626224071, 6.383784255864741, 6.323251985344354, 6.257971834413989, 6.188223822824728, 6.114287970327644, 6.03644429667382, 5.95497282161433, 5.870153564900254, 5.782266546282674, 5.691591785512662, 5.598409302341302, 5.502999116519667, 5.405641247798836, 5.306615715929894, 5.20620254066391, 5.10468174175197, 5.002333338945146, 4.89943735199452, 4.796273800651171, 4.693122704666173, 4.590264083790611, 4.487977957775556, 4.386544346372089, 4.286243269331292, 4.187354746404237, 4.090158797342008, 3.9949354418956777, 3.901964699816327, 3.811526590855039, 3.723901134762883, 3.639368351290946, 3.5582082601902973, 3.480700881212022, 3.407126234107199, 3.3377643386269, 3.27289521452221, 3.2127988815442015, 3.1577553594439567, 3.1080446679725546, 3.063946826881071, 3.0257418559205838, 2.9937089792748877, 2.9679835997506907, 2.948391652310318, 2.9347189129743403, 2.926751157763331, 2.924274162697862, 2.9270737037985057, 2.934935557085835, 2.947645498580423, 2.9649893043028404, 2.986752750273661, 3.0127216125134586, 3.0426816670428023, 3.0764186898822676, 3.1137184570524243, 3.1543667445738492, 3.198149328467111, 3.244851984752782, 3.294260489451438, 3.346160618583647, 3.4003381481699866, 3.4565788542310263, 3.514668512787337, 3.574392899859495, 3.635537791468069, 3.697888963633636, 3.761232192376766, 3.8253532537180313, 3.8900379236780003, 3.955071978277253, 4.020241193536361, 4.085331345475892, 4.150128210116423, 4.214417563478519, 4.277985181582762, 4.340616840449722, 4.402098316099968, 4.462215384554076, 4.520753821832612, 4.577499403956158, 4.6322379069452815, 4.684755106820557, 4.734836779602553, 4.782268701311841, 4.826836647969001, 4.868326395594604, 4.906523720209217, 4.941214397833415, 4.972184204487769, 4.999218916192856, 5.022144570218274, 5.0409641291725205, 5.055729140908652, 5.066491154641944, 5.073301719587677, 5.076212384961124, 5.075274699977566, 5.0705402138522775, 5.062060475800537, 5.049887035037623, 5.034071440778809, 5.014665242239377, 4.991719988634599, 4.965287229179759, 4.935418513090126, 4.902165389580982, 4.865579407867605, 4.825712117165269, 4.7826150666892575, 4.73633980565484, 4.686937883277296, 4.634460848771905, 4.5789602513539425, 4.520487640238691, 4.4590945646414175, 4.394832573777406, 4.327753216861931, 4.257908043110272, 4.18534860173771, 4.110126441959512, 4.032293112990962, 3.951900164047335, 3.86899914434391, 3.7836416030959685, 3.695879089518778, 3.605763152827619, 3.5133453422377716, 3.4186772069645106, 3.3218102962231137, 3.222796159228867, 3.1216863451970314, 3.0185324033428924, 2.9133858828817267, 2.8062983330288125, 2.697321302999434, 2.5865063420088523, 2.4739049992723525, 2.359568824005213, 2.2435493654227114, 2.125898172740131, 2.006666795172734, 1.8859067819358044, 1.7636696822446223, 1.6400070453144635, 1.5149704203606134, 1.3886113565983302, 1.2609814032429025, 1.1321321095096089, 1.0021150246137207, 0.8709816977705314, 0.7387836781952948, 0.6055725151032996, 0.4713997577098228, 0.3363169552301395, 0.20037565687954162, 0.06362741187328114, -0.07387623057335091, -0.21208372124507902, -0.35094351092662635, -0.4904040504027045, -0.6304137904580571, -0.7709211818773953, -0.9118746754454454, -1.0532227219469266, -1.194913772166554, -1.336896276889068, -1.4791186868991812, -1.62152945298162, -1.7640770259211065, -1.9067098565023501]
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
p = [cx', cy'];
%計算一階導數
for i = 1:length(cx)-1
pd(i) = (p(i+1,2)-p(i,2))/(p(i+1,1)-p(i,1));
end
pd(end+1) = pd(end);
%計算二階導數
for i =2: length(cx)-1
pdd(i) = (p(i+1,2)-2*p(i,2) + p(i-1,2))/(0.5*(-p(i-1,1)+p(i+1,1)))^2;
end
pdd(1) = pdd(2);
pdd(length(cx)) = pdd(length(cx)-1);
%計算曲率
for i = 1:length(cx)-1
k(i) = (pdd(i))/(1+pd(i)^2)^(1.5);
end
cx= cx';
cy =cy';
cyaw = atan(pd');
% cyaw = [-0.42747397147825394, -0.42750072065132655, -0.42758094489540044, -0.4277145744333646, -0.42790149313092396, -0.42814153873731675, -0.4284345032210795, -0.42878013319963476, -0.4291781304611309, -0.42962815257663806, -0.430129813600478, -0.4306826848561635, -0.4312862958051237, -0.4319401349951135, -0.43264365108494557, -0.4333962539419319, -0.4341973158082054, -0.4350461725318764, -0.4359421248588016, -0.4368844397805786, -0.43787235193423374, -0.43890506504896076, -0.4399817534351663, -0.4411015635110099, -0.4422636153615753, -0.4434670043257879, -0.44471080260618273, -0.4459940608966561, -0.4473158100233627, -0.4486750625939931, -0.4500708146507374, -0.4515020473223469, -0.452967728470822, -0.4544668143283879, -0.4559982511205717, -0.457560976671358, -0.45915392198657834, -0.46077601281187724, -0.4624261711617995, -0.46410331681674805, -0.46580636878477943, -0.4675342467254242, -0.4692858723329443, -0.4710601706766694, -0.47285607149628356, -0.4746725104501654, -0.47650843031511225, -0.4783627821360113, -0.4802345263242414, -0.4821226337038107, -0.484026086504453, -0.48594387930111194, -0.48787501989944665, -0.4898185301671866, -0.49177344681135127, -0.4937388221015239, -0.4957137245395447, -0.49769723947613925, -0.49968846967515207, -0.5016865358261939, -0.503690577006636, -0.5056997510940054, -0.5077132351299432, -0.5097302256369819, -0.511749938889487, -0.5137716111401838, -0.5157944988037568, -0.5178178785990676, -0.5196931149270214, -0.5212243260431461, -0.5224197163289975, -0.5232873261735816, -0.5238339185427856, -0.524065048216695, -0.5239851199517127, -0.5235974365070971, -0.5229042373201364, -0.5219067284762255, -0.5206051044967731, -0.5189985623566975, -0.5170853080421537, -0.5148625558662738, -0.5123265206745073, -0.5094724029903408, -0.5062943670756251, -0.5027855118065777, -0.49893783419608717, -0.4947421853247934, -0.49018821837741167, -0.4852643284170402, -0.47995758346934314, -0.47425364643153906, -0.46813668726977986, -0.46158928492527146, -0.4545923183179331, -0.4471248458215146, -0.43916397259260614, -0.43068470517701307, -0.42165979290260597, -0.41205955571398767, -0.4018516983320383, -0.3910011109578644, -0.3794696572209753, -0.3672159507406615, -0.3541951225846469, -0.34035858314180506, -0.3256537835649078, -0.31002398409323995, -0.293408039362483, -0.2757402143998198, -0.2569500495510399, -0.23696229826460727, -0.2156969686215963, -0.19306950785625335, -0.16899117885754447, -0.14336968857788476, -0.11611013987570126, -0.08711638955650051, -0.0562929045043543, -0.02354721209741704, 0.011206963321640584, 0.048045804465186234, 0.08703071058155766, 0.1282028423989582, 0.1715770293292805, 0.2171352954827804, 0.2648204174206136, 0.3145300919358443, 0.36611243518678976, 0.4193636116084794, 0.47402835394583015, 0.5298039462969809, 0.5863478911541079, 0.643289005215774, 0.7002411717905076, 0.7568185361252836, 0.8118690502380587, 0.8641084724045175, 0.9134969015780486, 0.960059587025011, 1.0038662856881957, 1.0450193867705637, 1.0836435702751392, 1.1198772452847279, 1.1538657621615147, 1.1857562421823056, 1.2156937951233464, 1.2438188751256842, 1.27026553679441, 1.2951603814332422, 1.3186220173456338, 1.3407608921349043, 1.3616793857102756, 1.381472078925828, 1.4002261342177875, 1.418021741613346, 1.4349325966649613, 1.4510263868852697, 1.4663652707516057, 1.481006338868105, 1.4950020508771902, 1.5084006445706022, 1.5212465156536303, 1.5335805679889383, 1.545440535058924, 1.5568612739662109, 1.5678750336361222, 1.5785116990634436, 1.5887990135100851, 1.5987627805478712, 1.6084270477787184, 1.6178142739720436, 1.6269454812500126, 1.635840393834563, 1.6445175647523944, 1.6529944917796306, 1.6612877237992376, 1.6694129586432487, 1.6773851333992504, 1.685218508076781, 1.6929267434543929, 1.7005229738619254, 1.7080198755947573, 1.7154297316071752, 1.7227644930901023, 1.7300358385040377, 1.7372552306108673, 1.744433972028028, 1.751583259815283, 1.7587142395979847, 1.7658380597313836, 1.7729659260183661, 1.7801091575084258, 1.7872792439290976, 1.7944879053332161, 1.801747154587014, 1.8090693633763313, 1.8164673324723377, 1.8239543670758143, 1.8315443581520676, 1.8392518707793255, 1.8470922406647414, 1.855081680137207, 1.8632373951090009, 1.8715777147135213, 1.8801222355794698, 1.8888919829993627, 1.8979095915996806, 1.9071995085302154, 1.9167882226713098, 1.9267045239213823, 1.936979797286335, 1.9476483572613554, 1.9587478288892548, 1.9703195829126208, 1.9824092336227193, 1.9950672093546409, 2.0083494070846104, 2.0223179442338215, 2.0370420225265327, 2.052598920493523, 2.0690751327826553, 2.086567675540389, 2.1051855772731987, 2.125051572999925, 2.1463040149253114, 2.1690990033792454, 2.1936127244444314, 2.2200439511487113, 2.2486166168989628, 2.2795822938387658, 2.3132222925563584, 2.3498489271506036, 2.3898052430761987, 2.433462168992391, 2.4812116264501998, 2.5334536477941385, 2.590575124560417, 2.6529176801580223, 2.7207327634867218, 2.794124005772281, 2.8729807692041156, 2.9569127030418287, 3.0452016965400053, 3.136791572382236, -3.0528526363431867, -2.9588998772229815, -2.866118078467373, -2.775975743675515, -2.689697060846188, -2.608184767719592, -2.532002337116722, -2.4614041798828104, -2.3965048038800147, -2.3397948760409086, -2.291344036566652, -2.249929188652858, -2.2145029415177375, -2.1841871135510753, -2.15825404315402, -2.1361045911182837, -2.117247006290865, -2.1012783168776954, -2.087868670762281, -2.07674849089681, -2.0676981000336987, -2.0605394221252733, -2.0551293927447887, -2.051354762141669, -2.049128030743504, -2.048384309028064, -2.0490789386074084, -2.0511857486905054, -2.0546958525159713, -2.059616912890818, -2.0659728256116554, -2.073803785046152, -2.0831667079739544, -2.0941360000040827, -2.106804653092188, -2.121285661829316, -2.137713738342337, -2.156247287694478, -2.1770705727702664, -2.2003959426511295, -2.226465911379705, -2.2555547413252683, -2.2879689905787797, -2.3240462096482406, -2.364150607796797, -2.4086640629261113, -2.457970378933774, -2.512430353061553, -2.572345306978107, -2.63790775477262, -2.7091404559568084, -2.785693709954596, -2.8629771777058166, -2.9381969335531912, -3.0105047505307554, -3.079236843819787, 3.1392557203024505, 3.0788740583345158, 3.022908226491498, 2.9713293509988272, 2.9240098803318566, 2.880755549938086, 2.841332037315793, 2.805485658729539, 2.772958486701022, 2.743498760169683, 2.7168675954883774, 2.692842944371884, 2.6712215957393837, 2.651819848518777, 2.6344733256498656, 2.6190362691126525, 2.6053805541017656, 2.5933945845903703, 2.582982177799927, 2.574061506732527, 2.566564143762559, 2.5604342309811536, 2.5556277919912787, 2.552112193285384, 2.549865759850427, 2.5488775482819763, 2.5491472807805122, 2.550685444476995, 2.553513562245124, 2.5576646432249412, 2.5631838234373894, 2.570129208795003, 2.5785729340730743, 2.588602451346434, 2.6003220590186795, 2.6138546763318944, 2.6293438557915674, 2.6469560037264426, 2.666882741981156, 2.689343283943847, 2.7145866051993917, 2.7428930491370123, 2.774574803956107, 2.8099744025506865, 2.849460020595651, 2.8920818699119404, 2.936427491384847, 2.982406179787393, 3.02989327954402, 3.0787277146995717, 3.1287126057349637, -3.103567370744916, -3.051999928303144, -3.000050157240052, -2.9480107895802816, -2.8961779597767547, -2.844841379864934, -2.794275039844364, -2.744729158412715, -2.6964239288216736, -2.649545362846886, -2.6042432823839587, -2.560631287824937, -2.5187883747956463, -2.4787617868250718, -2.4405706753745005, -2.4042101741998962, -2.3696555622740223, -2.336866269903388, -2.3057895620788367, -2.276363802585878, -2.2485212577514577, -2.222190439166585, -2.1972980116927356, -2.173770308967378, -2.1515345061653335, -2.1305195014414275, -2.1106565553490393, -2.0918797331909, -2.0741261898580414, -2.057336331021679, -2.041453879044128, -2.0264258669421595, -2.0122025792985068, -1.9987374552102448, -1.9859869651650783, -1.9739104710958324, -1.9624700767129548, -1.9516304734846341, -1.9413587862567534, -1.931624421419231, -1.9223989196776674, -1.9136558148336809, -1.9053704994756007, -1.897520098101366, -1.8900833479118169, -1.8830404873040305, -1.8763731519441875, -1.8700642781943329, -1.8640980135966956, -1.8584596340746704, -1.8531354674846352, -1.8481128231423796, -1.8433799269481883, -1.8389258617424753, -1.834740512537043, -1.8308145162837601, -1.8271392158613986, -1.8237066179815833, -1.8205093547355256, -1.8175406485239551, -1.8147942801330408, -1.8122645597388487, -1.8099463006419043, -1.8078347955515526, -1.805925795257055, -1.8042154895387084, -1.8027004901877028, -1.8013778160180771, -1.8002448797679307, -1.7992994768001862, -1.7985397755256436, -1.7979643094829658, -1.7975719710216365, -1.7973620065449136]
ck = k'
% plot(1:1001, cyaw)
% plot(1:1001, ck)
% plot(1:1001, pd)
pe = 0; pth_e = 0;
i = 1;
x = 0; y = -0.1; yaw = 0; v = 0;
ind =0;
figure
subplot(2,1,1);
plot(cx,cy,'r-')
hold on
while ind < length(cx)
if ind> length(cx)
fprintf('Simuliation finished!\n')
break;
end
[delta,ind,e,th_e] = lqr_steering_control(x,y,v,yaw,cx,cy,cyaw,ck,L,Q,R,dt);
pe =e
pth_e = th_e
if abs(e)> 4
fprintf('mayday mayday!\n')
break;
end
delta
a = PIDcontrol(target_v, v, Kp);
[x,y,yaw,v] = update(x,y,yaw,v, a, delta, dt,L, max_steer);
posx(i) = x;
posy(i) =y;
vel(i) = v;
Delta(i) = delta;
i = i+1;
subplot(2,1,1);
plot(posx(i-1),posy(i-1),'bo')
hold on
subplot(2,1,2);
plot(posx(i-1),Delta(i-1),'.-r')
pause(0.001);
hold on
end
% function"Update" updates vehicle states
function [x, y, yaw, v] = update(x, y, yaw, v, a, delta,dt,L,max_steer)
delta = max(min(max_steer, delta), -max_steer);
x = x + v * cos(yaw) * dt + randn(1)* 0.01;
y = y + v * sin(yaw) * dt + randn(1)* 0.01;
yaw = yaw + v / L * tan(delta) * dt;%remember here the yaw is in rad
v = v + a * dt + randn(1) * 0.1;
end
function [a] = PIDcontrol(target_v, current_v, Kp)
a = Kp * (target_v - current_v);
end
function [angle] = pipi(angle) % the unit of angle is in rad;
if (angle > pi)
angle = angle - 2*pi;
elseif (angle < -pi)
angle = angle + 2*pi;
else
angle = angle;
end
end
function [Xn,iter] = solve_DARE(A,B,Q,R)
X = Q;
maxiter = 150;
epsilon = 0.01;
for i = 1:maxiter
Xn = A' * X * A - A' * X * B * ((R + B' * X * B) \ B') * X * A +Q;
if abs(Xn - X) <= epsilon
% X = Xn;
break;
end
X = Xn;
end
iter = i;
end
function [K,iter] = dlqr (A,B,Q,R)
[X,iter] = solve_DARE(A,B,Q,R);
K = (B' * X * B + R) \ (B' * X * A);
end
function [delta,ind,e,th_e] = lqr_steering_control(x,y,v,yaw,cx,cy,cyaw,ck,L,Q,R,dt)
[ind, e] = calc_target_index(x,y,cx,cy,cyaw);
% e= error_check(x, y, yaw,e,cx,cy,ind)
k =ck(ind);
delta_yaw = v * dt * sin(yaw) / L;
th_e = -pipi(yaw + delta_yaw - cyaw(ind));
% th_e = pipi(yaw - cyaw(ind));
x_e = cx(ind) - x;
y_e = cy(ind) - y;
A = zeros(3,3);
A(1,1) = 1; A(1,3) = -v*dt*sin(yaw); A(2,2) = 1; A(2,2) = 1;...
A(2,3) = v * dt * cos(yaw); A(3,3) = 1;
B= zeros(3,1);
B(3,1) = v*dt/(L * cos(yaw)^2);
[K,iter] = dlqr(A,B,Q,R);
x = zeros(3,1);
x(1,1)=x_e; x(2,1)= y_e; x(3,1) = th_e;
ff = atan(L * (k));
fb = pipi(K * x);
delta = 1*ff + 1*fb;
ff
fb
iter
end
function [ind, error] = calc_target_index(x,y, cx,cy,cyaw)
N = length(cx);
Distance = zeros(N,1);
for i = 1:N
Distance(i) = sqrt((cx(i)-x)^2 + (cy(i)-y)^2);
end
[value, location]= min(Distance);
ind = location
dx1 = cx(ind) - x;
dy1 = cy(ind) - y ;
angle = pipi(cyaw(ind)-atan(dy1/dx1));
heading = cyaw(ind)*180/pi
if y<cy(ind)
error = -value;
else
error = value;
end
% error = value;
end
