Assignment of Inertial Technology 《惯性技术》作业Autumn 2015Assignment 1: 2-DOF response simulationA 2-DOF gyro is subjected to a sinusoidal torque with amplitude of 4 g.cm and frequency of 10 Hz along its outer ring axis. The angular moment of its rotor is 10000 g.cm/s , and the angular inertias in its equatorial plane are both 4 g.cm/s 2. Please simulate the response of the gyro within 1 second, and present whatever you can discover or confirm from the result.In this simulation, we are going to discuss the response of a 2-DOF gyro to sinusoidal torque input. According to the transfer function of the 2-DOF gyro, the outputs can be expressed as:12222()()()()yx y x y x y J Hs M s M s J J s H s J J s H α=-++ 12222()()()()x yx x y x y J Hs M s M s J J s H s J J s H β=+++ The original system transfer function is a 2-input, 2-output coupling system. But the given input only exists one input, we can treat the system as 2 separate FIFO systemAs a consequence, we can establish the block diagram of the system in simulink in Fig 1.1. Substitute the parameters into the system and input, then we have the input signals as follow: 0,4sin(20)y ox M M t π==Then the inverse Laplace transform of the output equals the response of the gyro in timedomain as follows:0222200020222200()sin sin ()()()cos cos ()()ox a oxa x a x a ox ox a ox a a a a a M M t t t J J M M M t t t H H H ωαωωωωωωωωωβωωωωωωωω=-+--=+---Fig 1.1 The block diagram of the system in simulinkAnd the simulation results in time domain within 1 second are shown in the follwing pictures. Fig1.2 is the the output of outer ring ()t α. Fig1.3 is the output of inner ring ()t β. Fig1.4 is the trajectory of 2-DOF gyro ’s response to sinusoidal input. As we can see from the Fig1.3,there are obvious sawtooth wave in the output of the inner ring. It ’s a unexpected phenomenon in my original theoretical analysis.Fig1.2 The output of inner ring ()t βFig 1.3 The output of outer ring ()t αI believe the sawtooth wave is caused by the nutation. For the frequency of the nutation is obtained as 010*******/3974e H rad s Hz J ω===≈, which is far higher than the frequencyof the applied sinusoidal torque , namely 0a ωω.Fig 1.4 Trajectory of 2-DOF gyro ’s response to sinusoidal inputThe trajectory of 2-DOF gyro ’s response to sinusoidal input are shown in Fig1.4. As we can see, it ’s coupling of X and Y channel scope output. The overall shape is an ellipse, which is not perfect for there are so many sawteeth on the top of it.Note that the major axis of ellipse is in the direction of the forced procession, amplitude of which is 0ox M H ω, whereas the minor axis is in the direction of the torsion spring effects,with amplitude ox aM H ω.The nutation components are much smaller than that of the forced vibration, which can be eliminated to get the clear static response.22002220()sin sin ()cos (1cos )()ox oxaa x a ox ox oxa a a a a aM M t t t J H M M M t t t H H H αωωωωωωβωωωωωωω≈≈-≈-=--To prove it, we eliminate the effects of the nutation namely the quadratic term in the denominator and get Fig 1.5, which is a perfect ellipse. We can conclude that when input to the 2-DOF gyro is sinusoidal torque, the gyro will do an ellipse conical pendulum as a static response, including procession and the torsion spring effects, together with a high-frequency vibration as the dynamic response.Fig 1.5 Trajectory of the gyro’s response without nutationAssignment 2: Single-axis INS simulation2.1 description of the problemA magnetic levitation train is being tested along a track running north-south. It first accelerates and then cruises at a constant speed. Onboard is a single-axis platform INS, working in the way described by the courseware of Unit 5: Basic problems of INS. The motion information and Earth parameters are shown in table 2-1, and the possible error sources are shown in Table2-2.Fig2.1 The sketch map of the single-axis INS problemYou are asked to simulate the operation of the INS within 10,000 seconds, and investigate, first one by one and then altogether, the impact of these error sources on the performance of the INS.The block diagram in the courseware might be of some help. However, there lurks an inconspicuous error, which you have to correct before you can obtain reasonable results.There are one core relevant formula, to get the specific form of its solution, we should substitute the unknown parameters.(1)()c N a y A K y ga =∆++-Firstly, the input signal is accelerometer of the platform, and the velocity of the platform is the integration of the acceleration.0/p y y ydty Rω=+=-⎰The acceleration along Yp may contains two parts:cos gsin g yp f y y ααα=-≈-When accelerometer errors are concerned, the output of accelerometer will be:(1)N a yp N a K f A =++When gyro errors concerned:'(1)p g p K ωωε=++Only αis unknown:0[(1)]tg p t tp t K dt dtααωεϕϕω=+++-∆∆=⎰⎰And the reference block diagram and simulink block diagram are as follows in Fig2.2, Fig2.3. There is a small fault in the reference block, which is that the sign of the marked add operation should be positive instead of negative.The results of the simulation are shown in Fig 2.4 to Fig 2.13.Fig2.2 The reference block diagram in the courseware(unrectified)Fig2.3 The simulink block diagram for Assignment 22.2 results and analysis of the problemFig.2.4 real acceleration,velocity and displacement output without error sourcesKFig2.5 position bias when only accelerometer scale factor error exists0.0001aFig2.6 position bias output when only gyro scale factor error exists 0.0001g K =Fig2.7 position bias when only acceleromete bias error exists 20.0002/N A m s ∆=Fig2.8 position bias output when only initial velocity error exists 200.01/y m s ∆=Fig2.9 position bias output when only initial position error exists 010y m ∆=Fig2.10 position bias output bias when only gyro bias error exists 0.000000024240.00681/3/h rad s ε=︒=Fig2.11 position bias output when only initial platform misalignment angle exists0.000012120''345radα==Fig2.12 output considering all the error sourcesFig2.13 position bias output considering all error sourcesAs we can see in the above simulation results, if there is no error we can navigate the train ’s motion correctly, which comes from north to the south as shown in Fig2.4, beginning with an constant acceleration within 60 seconds then cruises at a constant speed, approximately 65 m/s. However, the situation will change a lot when different errors put into the simulation. The initial position error 010y m ∆= effects least as Fig.2.8, for this error doesn ’t enter into the closed loop and it won ’t influence the iterative process. The position bias is constant and can be negligible.In the second case, when the accelerometer scale factor error exists, 0.0001a K =, as shown in Fig2.5, the result are stable and almost accurate, the position bias is a sinusoidal output.So it is with the accelerometer bias error situation, 20.0002/N A m s ∆=, in Fig2.7, the initialvelocity error, 200.01/y m s ∆= in Fig2.9, and the initial platform misalignment angle,05''α=, in Fig2.11. However, the influence degrees of the different factors are not in the samemagnitude. The accelerometer scale factor influences the least with magnitude of 5, then the initial velocity larger magnitude of 8, and the accelerometer bias magnitude of 25. The influence of the initial platform misalignment angle is much more significant with a magnitude of 150. All the navigation bias in the second kind case is sinusoidal, which means they ’re limited and negligible as time passes by.In the third case, such as the gyro scale factor error situation, 0.0001g K =, in Fig2.6, and the gyro bias error,0.01/h ε=︒, results in Fig2.10, effects the most significant, the trajectory of the navigation disvergence accumulated as time goes. The position bias is a combination of sinusoidal signal and ramp signal. They also show that the longitudinal and distance errors resulted from gyro drifts are not convergent in time. It means the errors in the gyroscope do most harm to our navigation. And due to the significant influence of the gyro bias errors and the gyroscope scale factor error, results considering all the error sources disverge, and the navigationposition of the motion will be away from the real motion after a enough long time, as shown in Fig2.10. The gyro bias error is the most significant effect factor of all errors. By the time of 10000s, it has reaches 1600m, and it’s nearly the quantity of the position bias considering all error sources. Through contrasting all the results, We can conclude that the gyro bias error is the main component of the whole position bias.Impression of the Whole simulation experimentThrough contrasting all the results, We can conclude that the gyro bias error is the main component of the whole position bias, and the the gyro bias or the drift error do most harm to our navigation. So it is a must for us to weaken or eliminate it anyway. In spite of all the disadvantages discussed above, the INS still shows us a relatively accurate results of single-axis navigation.Assignment 3: SINS simulation3.1 Task descriptionA missile equipped with SINS is initially at the position of 46o NL and 123 o EL, stationary on a launch pad. Three gyros, GX, GY, GZ, and three accelerometers, AX, AY, AZ, are installed along the axes Xb, Yb, Zb of its body frame respectively.Case 1: stationary testThe body frame of the missile initially coincides with the geographical frame, as shown in the figure, with its pitching axis Xb pointing to the east, rolling axis Yb to the north, and azimuth axis Zb upward. Then the body of the missile is made to rotate in 3 steps:(1) -22o around Xb(2) 78o around Yb(3) –16o around ZbFig 3.1 Introduction to assignment 3After that, the body of the missile stops rotating. You are required to compute the final outputs of the three accelerometers on the missile, using quaternion and ignoring the device errors. It is known that the magnitude of gravity acceleration is g = 9.8m/s2.Case 2: flight tes tInitially, the missile is stationary on the launch pad, 400m above the sea level. Its rolling axis is vertical up,and its pitching axis is to the east. Then the missile is fired up. The outputs of the gyros and accelerometers are both pulse numbers. Each gyro pulse is an angular increment of 0.01arc-sec, and each accelerometer pulse is 1e-7g, with g =9.8m/s2. The gyro output frequency is 200Hz, and the accelerometer’s is 10Hz. The outputs of the gyros and accelerometers within 1315s are stored in a MA TLAB data file named imu.mat, containing matrices gm of 263000× 3 and am of 13150× 3 respectively. The format of the data is as shown in the tables, with 10 rows of each matrix selected. Each row represents the outputs of the type of sensors at each sampling time. The Earth can be seen as an ideal sphere, with radius 6371.00km and spinning rate 7.292× 10-5 rad/s, The errors of the sensors are ignored, so is the effect of height on the magnitude of gravity. The outputs of the gyros are to be integrated every 0.005s. The rotation of the geographical frame is to be updated every 0.1s, so are the velocities and positions of the missile.You are required to:(1) compute the final attitude quaternion, longitude, latitude, height, and east, north, vertical velocities of the missile.(2) compute the total horizontal distance traveled by the missile.(3) draw the latitude-versus-longitude trajectory of the missile, with horizontal longitude axis.(4) draw the curve of the height of the missile, with horizontal time axis.Fig 3.2 simplified navigation algorithm for SINS 3.2Procedure code3.2.1 Sub function code：quaternion multiply code:function [q1]=quml(q1,q2);lm=q1(1);p1=q1(2);p2=q1(3);p3=q1(4);q1=[lm -p1 -p2 -p3;p1 lm -p3 p2;p2 p3 lm -p1;p3 -p2 p1 lm]*q2;endquaternion inversion code:function [qni] =qinv(q)q(1)=q(1);q(2)=-q(2);q(3)=-q(3);q(4)=-q(4);qni=q;end3.2.2case1 DCM algorithm：function ans11cz=[cos(-22/180*pi) sin(-22/180*pi) 0 ;-sin(-22/180*pi) cos(-22/180*pi) 0;0 0 1]; %The third rotation DCMcx=[1 0 0 ;0 cos(78/180*pi) sin(78/180*pi);0 -sin(78/180*pi) cos(78/180*pi)]; %The first rotation DCMcy=[cos(-16/180*pi) 0 -sin(-16/180*pi);0 1 0;sin(-16/180*pi) 0 cos(-16/180*pi)]; %The second rotation DCM A=cz*cy*cx*[0;0;-9.8]end3.2.3case1 quaternion algorithm：function ans12g=[0;0;0;-9.8];q1=[cos(-11/180*pi);sin(-11/180*pi);0;0]; %The first rotation quaternionq2=[cos(39/180*pi);0;sin(39/180*pi);0]; %The second rotation quaternionq3=[cos(-8/180*pi);0;0;-sin(-8/180*pi)]; %The third rotation quaternionr=quml(q1,q2); %call the quaternion multiplication subfunction q=quml(r,q3);P1=[q(1) q(2) q(3) q(4);-q(2) q(1) q(4) -q(3);-q(3) -q(4) q(1) q(2);-q(4) q(3) -q(2) q(1)];P2=[q(1) -q(2) -q(3) -q(4);q(2) q(1) q(4) -q(3);q(3) -q(4) q(1) q(2);q(4) q(3) -q(2) q(1)];P=P1*P2;gn=P*g;gn=gn(2:4)end3.2.4 case2 SINS quaternion algorithm code：function ans2clc;clear;%parameters initializing：T=0.005;K=13150;R=6371000; %radius of earthwE=7.292*10^(-5); %spinning rate of earthQ=zeros(4, 263001) ;%quaternion matrix initializinglongitude=zeros(1,13151);latitude=zeros(1,13151);H=zeros(1,13151); %altitude matrixQ(:,1)=[cos(45/180*pi);-sin(45/180*pi);0;0]; % initial quaternion longitude(1)=123; %initial longitudelatitude(1)=46; % initial latitudeH(1)=400; %initial altitudelength=0;g=9.8;vE = zeros(1,13151); %eastern velocityvN = zeros(1,13151); %northern velocityvH = zeros(1,13151); %upward velocityvE(1)=0;vN(1)=0;vH(1)=0;load imu.mat %data loading%main calculation sectionfor N=1:Kq1=zeros(4,11);q1(:,1)=Q(:,N);for n=1:20 % Attitude iteration wx=0.01/(3600*180)*pi*gm((N-1)*10+n,1); % Angle incrementwy=0.01/(3600*180)*pi*gm((N-1)*10+n,2);wz=0.01/(3600*180)*pi*gm((N-1)*10+n,3);w=[wx,wy,wz]';normw=norm(w); % Norm calculationW=[0,-w(1),-w(2),-w(3);w(1),0,w(3),-w(2);w(2),-w(3),0,w(1);w(3),w(2),-w(1),0];I=eye(4);S=1/2-normw^2/48;C=1-normw^2/8;q1(:,n+1)=(C*I+S*W)*q1(:,n);Q(:,N+1)=q1(:,n+1);endWE=-vN(N)/R; % rotational angular velocity component of a geographic coordinate systemWN=vE(N)/R+wE*cos(latitude(N)/180*pi);WH=vE(N)/R*tan(latitude(N)/180*pi)+wE*sin(latitude(N)/180*pi);attitude=[WE,WN,WH]'*T; %correction of the quaternion by updating the rotation of geographic coordinatenormattitude=norm(attitude);e=attitude/normattitude;QG=[cos(normattitude/2);sin(normattitude/2)*e];Q(:,N+1)=quml(qinv(QG),Q(:,N+1));fx=1e-7*9.8*am(N,1); %specific force measured by accelerometerfy=1e-7*9.8*am(N,2);fz=1e-7*9.8*am(N,3);Fb=[fx fy fz]';F=quml(Q(:,N+1),quml([0;Fb],qinv(Q(:,N+1)))); %The specific force is decomposed into geographic coordinate system.FE(N)=F(2);FN(N)=F(3);FU(N)=F(4);%calculate the velocity of the vehicle:VED(N)=FE(N)+vE(N)*vN(N)/R*tan(latitude(N)/180*pi)-(vE(N)/R+2*wE*cos(latitude(N)/ 180*pi))*vH(N)+2*vN(N)*wE*sin(latitude(N)/180*pi);VND(N)=FN(N)-2*vE(N)*wE*sin(latitude(N)/180*pi)-vE(N)*vE(N)/R*tan(latitude(N)/180 *pi)-vN(N)*vH(N)/R;VUD(N)=FU(N)+2*vE(N)*wE*cos(latitude(N)/180*pi)+(vE(N)^2+vN(N)^2)/R-g;%integration and get the relative velocity of vehicle:vE(N+1)=VED(N)*T+vE(N);vN(N+1)=VND(N)*T+vN(N);vH(N+1)=VUD(N)*T+vH(N);% integration and get the position of vehicle:longitude(N+1)=vE(N)/(R*cos(latitude(N)/180*pi))*T/pi*180+longitude(N);latitude(N+1)=vN(N)/R*T/pi*180+latitude(N);H(N+1)=vH(N)*T+H(N);length=sqrt((vE(N))^2+(vN(N))^2)+length;endfigure(1) %picture the longitude-latitude curve of the motion within 1315 secondstitle(' longitude-latitude ');hold ongrid onplot(longitude,latitude);figure(2) % picture the altitude curve of the motion within 1315 secondstitle('altitude');hold ongrid onplot(0:13150,H);3.3Simulation outputs and results analysisIn case 1, the DCM algorithm and quaternion results are the same as follow:A =7.53165.9788-1.8892And the results suggest the solution of DCM and quaternion is equivalence in this problem.In case 2, the latitude-versus-longitude trajectory of the missile(with horizontal longitude axis) and the altitude curve of the missile are shown as follows in Fig 3.3 and Fig 3.4.Fig3.3 the latitude-versus-longitude trajectory of the missile(with horizontal longitude axis)Fig3.4the altitude curve of the missileImpression of the Assignment 3In this assignment, we simulate the missile navigation situation in a real problem, as the problem we have done before. I did this based on my own original code, but to my surprise, it’s harder than I have expected. For the detailed thoughts for my program I have forgotten. I have to recheck the code line by line, But I am still troubled in correcting the initial parameters for many times. It has taught me a lesson, which is never to be egotistical for your ability to memory and the work you have done or understood.。