CHAPTER 2 SENSOR DATA SIMULATION: A KINEMATIC APPROACH

Transcription

1 27 CHAPTER 2 SENSOR DATA SIMULATION: A KINEMATIC APPROACH 2.1 INTRODUCTION The standard technique of generating sensor data for navigation is the dynamic approach. As revealed in the literature (John Blakelock 1991), a forward approach is dealt which computes the sensor data by solving the system equations systematically involving all the transfer functions in the system. However some non-linearities of the system creep into the simulated data. Another common approach towards obtaining the sensor data is to use data from real flight trials. Here also inadvertently some real time fluctuations get reflected in the data. Such fluctuations become undesirable in several applications of stand-alone navigation system studies like transfer alignment and hybrid navigation systems. In case of transfer alignment (Bar-Itzhack and Mallove 1980), there is no scope of using the dynamic approach to simulate sensor data since the vehicle is not accompanied with guidance and control loop. Performance of hybrid navigation systems must be studied in isolation. For these applications, a kinematic approach was developed towards simulation of sensor data (Partha Pratim Adhikari et al 2002). This chapter presents the kinematic approach towards proper simulation of angular rate and acceleration data for a space vehicle from a given trajectory involving the six degrees of freedom (Partha Pratim Adhikari et al 2002) as shown in Figure 2.1. This can facilitate an integrated approach for studying navigation system performance by decoupling the navigation

2 28 from guidance and control loops. The present problem of simulating the angular rate and acceleration data from a given trajectory consists of two phases. First, from geometry of the given trajectory, Euler angles such as pitch and yaw can be computed. Roll can be computed from banking concept. With this Euler angle information (roll, pitch and yaw), body rates are computed. Gyro data can be derived from the body rates. Secondly, the same attitude information is used to solve the reverse Strapdown Inertial Navigation System (SDINS) equation for generating the accelerometer data. The steps involved in the sensor data generation algorithm are provided in the block diagram given in Figure 2.1. Flight segment Latitude, longitude, altitude A flight trajectory with respect to FLLF T x e y e z e ECEF V xe V xe V xe Geodetic Frame L,l, h Calculation of S xy,,, Calculation of,. NED Acceleration DCM from ECEF to VLLF Calculation of pt, qt, rt, Computation Velocity Earth rate and Transport rate Body angular rate Computation of Body acceleration Figure 2.1 Block Diagram for sensor data simulation for a given trajectory

3 29 To compute the sensor data from a given trajectory, the trajectory is transformed with respect to various reference frames like Earth Centered Earth Fixed (ECEF), Fixed Locally Local Frame (FLLF) and Variable Locally Local Frame (VLLF) which refers to the North East Down (NED) Frame. Finally the sensor data is calculated at the Body Frame as body rates and body accelerations. 2.2 GENERATION OF ANGULAR RATES AND ACCELERATIONS FROM A GIVEN TRAJECTORY X-Plane is the world's most comprehensive flight simulator and the most realistic flight simulator available for personal computers. In settings menu, data output window is used to output about 124 flight data to the cockpit display, a graphical output, a disk file, or even the internet via UDP (User Datagram Protocol). The trajectory is generated by flying the aircraft in X-Plane flight simulator and the corresponding latitude, longitude and altitude data are stored to a disk file. Figure 2.2 shows the 3D plot of the obtained trajectory. Figure 2.2 Generated trajectory using X-Plane

4 30 Various trajectories are obtained using X-Plane and Flight gear simulation software for the validation of the sensor data generation algorithm. Figure 2.2 is the plot of the latitude, longitude and altitude data stored to a disk file during the trial flight using X-plane Body Angular Rate Generation Algorithm Angular rate computation is carried out for an aircraft moving along a six degree of freedom trajectory generated with respect to fixed locally level NWV frame i.e. X along north (N), Y along west (W) and Z along vertical (V). Orientation of body frame is: X b along tail to nose, Y b along right wing, Z b along vertical to X b Y b plane and passing through center of gravity. Consider the two successive points A and B from the given trajectory as shown in Figure 2.3. Here, roll is assumed to be absent. Let S xy be the projection of the path on XY plane, between two successive instants. Figure 2.3 Calculation of pitch and yaw angle between two successive points Pitch is calculated from the angle made by the path i.e. line joining AB with XY plane at any particular instant and yaw from the angle made by projection of the trajectory on the XY plane i.e. line joining AO with x axis at

5 31 any particular instant. For simple geometry shown in Figure 2.3, these parameters can be evaluated as follows: 2 2 1/2 S x x y y (2.1) xy tan z1 z 2 /Sxy (2.2) (2.3) 1 tan ((y2 y 1) / (x 2 x 1)) where (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) represent the two consecutive sampling points A and B with respect to time and (, ) denote the pitch angle and yaw angle respectively. Roll () is calculated using banking concept based on the Equations ( ) as shown in Figure m R mv T (2.4) Lsin mv T and Lcos mg (2.5) tan V g 1 T (2.6) Where V T is tangential velocity; L is lift vector; m is mass of the aircraft; R is radius of gyration; g is acceleration due to gravity. - Figure 2.4 Geometry of an aircraft in banking turn

6 32 The incremental pitch, yaw and roll angles are obtained by taking the differences of successive pitch, yaw and roll is given as follows: 1. If (new angle old angle) is negative and abs (new angle old angle) is greater than 90º then angle º) angle = (new angle old 2. If (new angle old angle) is positive and abs (new angle old angle) is greater than 90º then angle 360 º) angle = (new angle old The body rates p, q and r of vehicle are related to the time rate of Euler angles, giving rise to the following expressions: pt sin (2.7) q t cos cossin (2.8) r t sin coscos (2.9) Gyro is an inertial sensor, hence, along with the body rates computed above, resolved components (in body frame) of earth rate () is to be added. Hence, gyro output with sampling time Equation (2.10). t is given by T b pt,qt,rt C n t (2.10) where body frame. b C n is the Direction Cosine Matrix (DCM) from navigation frame to

7 Body Acceleration Data Generation Algorithm In order to obtain the acceleration data in body frame from the simulated trajectory with respect to fixed locally leveled frame at launch point the following steps are necessary: A. Generation of acceleration, velocity and position in a variable locally leveled frame (VLLF) and B. Generation of body frame acceleration data: A. Generation of acceleration, velocity, position in variable locally leveled frame (VLLF) The objective is to find acceleration with respect to variable locally level frame, varying with local latitude and longitude. This is achieved in three steps: Step1: DCM C b n and a translation matrix are developed to transform the position co-ordinates from fixed locally level frame (FLLF) (x,y,z) to Earth Centered Earth Fixed (ECEF) frame (x e, y e, z e ). x x R h cos cos y C y R h cos sin z z R h sin e e e f e 0 0 (2.11) sin cos sin sin cos e f 0 0 C sin cos 0 cos cos cos sin sin T (2.12) where o, 0,h0 being initial latitude, longitude, height and R is the radius of the earth.

8 34 Step2: The ECEF position is transformed to geodetic frame (latitude, longitude, height) by using standard transformation Equations ( ) h e e e R x y z (2.13) h Rh R (2.14) Where Rh is the height of the aircraft from the center of earth. (2.15) = tan -1 e z 2 2 e e x +y 1 y tan e xe (2.16) v e Step3: Another DCM ( C varying with, and h) is developed to transform the acceleration from ECEF frame to moving or variable locally level frame (North- East-Up) corresponding to latitudes, longitudes obtained and this yields acceleration in navigation frame. Thus velocity is computed in locally leveled frame. B. Generation of body frame acceleration data Now, the basic navigation equation is : n an C b ab 2 vg (2.17)

9 35 where a n is the acceleration vector in navigation frame, a b is the acceleration vector in body frame, v is velocity vector in navigation frame and g is gravity vector resolved in navigation frame. From this, body frame acceleration data (a b ) is computed given by the following Equation (2.18) n 1 b b b a C a 2 vg (2.18) 2.3 RESULTS AND DISCUSSION Simulation of Angular Rates and Accelerations for Various Trajectories Figures 2.5 and 2.6 are the generated angular rates about roll, pitch and yaw axes and accelerations along the aircraft axes corresponding to the trajectory shown in Figure 2.2. Figures 2.5 and 2.6 are obtained using the sensor data generation algorithm explained in Figure 2.1. Yaw rate deg/sec Pitch rate deg/sec Roll rate deg/sec Figure 2.5 Generated body angular rates

10 36 Vertical Acceleration (Z) m/sec 2 Lateral Acceleration (Y) m/sec 2 Nose Acceleration (X) m/sec 2 Figure 2.6 Generated body accelerations Due to the non-availability of the real signal, the data from the flight simulation software X-plane has been used for comparison and validation. Figures 2.5 and 2.6 shows that the angular rate about roll, pitch and yaw axes and accelerations along the aircraft axes are generated corresponding to the trajectory generated using X-plane software COMPARISON OF SIMULATED AND REAL DATA Navigation system studies can be conducted in simulated data mode and real data mode. Simulated data mode is important for tuning and validation of system models. Raw sensor data are simulated corresponding to the trajectory generated using X-plane. The sensor data generation using the kinematic approach is carried out and also validated by comparing simulated sensor data with that of the real sensor data. The validation responses are shown in Figure 2.8 and Figure 2.9 for the trajectory shown in Figure 2.7.

11 37 6 DOF Trajectory Altitude Km Altitude m Latitude latitude deg Longitude longitude deg deg Figure DOF Trajectory in 3D view Figure 2.8 Comparison of simulated and real angular rates

12 38 Figure 2.9 Comparison of simulated and real accelerations Figures 2.8 and 2.9 clearly tell us that the real sensor data is noisy when compared to the ideal sensor data generated by the data generation algorithm. 2.4 CONCLUSION This chapter has dealt with the sensor data generation algorithm for the trajectories obtained from Flight Gear / X-plane flight simulator and the Figures 2.8 and 2.9 clearly tell us that the real sensor data is noisy when compared to the ideal sensor data generated by the data generation algorithm. Therefore the error modeling of inertial sensors is required to make the simulated data resemble the true sensor data. The modeling of various inertial sensor errors and the simulation of SDINS are being discussed in the next chapter.