Roll angle estimation for smart munitions under GPS jamming environment

Transcription

1 Proceedings of the 17th World Congress he International Federation of Automatic Control Roll angle estimation for smart munitions under GPS jamming environment Han Sung Lee*. Heeoung Park*. KwangJin Kim**. Jang Gu Lee*. Chan Gook Park**. *School of Electrical Engineering and Computer Science, Seoul National Universit, Seoul , South Korea, ( **School of Mechanical and Aerospace Engineering, Seoul National Universit, Seoul , South Korea, ( Astract: A smart munition revolves ~3Hz when it flights. Before guidance starts, the roll estimation of a smart munition is an important factor to improve the navigation performance. he roll estimation algorithm is alread known using IMU and GPS. But GPS can t e availale as a measurement under GPS jamming environment. his paper explains how to estimate the roll angle of a smart munition under jamming environment using IMU, earth magnetism measurements. he sstem and measurement models of the extended Kalman filter are designed for GPS jamming environment. Under GPS jamming environment, the proposed method shows etter performance than the previous method Monte Carlo simulation. 1. INRODUCION A smart munition is a low cost militar weapon with a guidance sstem. When it is fired from an artiller or tank, it flights the inertial force efore guidance starts. hen a guidance usuall starts after its height is a maximum value. A smart munition revolves ~3Hz in flight. So it is difficult to know the roll angle of it. he roll angle estimation is ver important factor to reduce the navigation error of a smart munition. o estimate the roll angle of a smart munition, IMU(Inertial Measurement Unit) and GPS(Gloal Positioning Sstem) can e used(lucia, 1995). he performance analsis of a MEMS(Micro Electro Mechanical Sstem) device under high gravit environment is alread studied(.gordon, 3). And the performance analsis of a MEMS device under high spin condition is alread studied(bradford, 1998). A low cost MEMS groscope can t measure the angular rate of a smart munition. So another method is needed to know the roll angle. A roll angle is related to a pitch rate and aw rate(lucia, 1995). If the pitch rate and aw rate of a smart munition are given, we can calculate the roll angle from it. he pitch rate and aw rate of a smart munition can e otained from the GPS velocit of the smart munition. In chapter, the extended Kalman filter(ekf) with the GPS measurement is introduced, which is alread known previous paper(heeoung, 7). Under GPS jamming environment, a pitch rate and aw rate can t e otained. In this case, we can assume that the pitch rate and aw rate are constant which is calculated fall free trajector of a smart munition. But this assumption makes the roll error increased. o reduce roll error, other measurement is needed in the EKF. he earth magnetism is a good measurement to reduce the roll error. In Chapter 3, this paper explains the roll estimation method using the earth magnetism as a measurement. In general, the flight distance of a smart munition is short within 1Km. Within this area, the earth magnetism is hardl changed. So we can assume that the earth magnetism is constant. o represent the earth magnetism with respect to the navigation frame, an earth magnetism frame is defined. After the transformation etween two coordinates is explained, the sstem and measurement models are designed. In chapter 4, we explain three simulation results, which are the previous method without GPS jamming, the previous method with GPS jamming and the proposed method with GPS jamming.. he EKF design using the GPS measurement.1 he roll angle from a pitch rate and aw rate he attitude of a groscope is aligned with the od frame. he X axis output of a groscope can t measure its roll rate ecause the roll rate is usuall larger than Hz. If the pitch rate and aw rate are given from GPS velocit, he roll angle is calculated the following equations(lucia, 1995). ω cos sin φ φ & θ ω sinφ cosφ z ψ& ω : axis output of a groscope aligned with od frame ωz : Z axis output of a groscope aligned with od frame φ : roll angle of a smart munition & θ : pitch rate of a smart munition ψ& : aw rate of a smart munition (1) /8/$. 8 IFAC /876-5-KR-11.41

2 From the equation 1, the following equation holds. & θω + ψω & cosφ & θ + ψ& z ψω & & θωz sinφ & θ + ψ& Figure 1 shows the roll angle related to the pitch rate and aw rate of a smart munition when the roll angle is 9 degrees. In this case, he axis of a groscope measures the aw rate of the smart munition and the Z axis measures its negative pitch rate. his result is coincident with the equation 1. he equation 1 can e explained the concept of a coordinate rotation for an angles as well as 9 degrees. () (3) ψ& Δψ (7) Δ t. he EKF design using the GPS measurement he EKF using GPS measurement is known previous paper(heeoung, 7). he state variales of this filter are the roll rate( p ), initial roll( φ ), axis ias of the groscope( ω ) and Z axis ias of the groscope( ω ). ias he sstem equation is 4 4 identit matrix. It is assumed that all state variales are constant regardless of time(lucia, 1995). he sstem equation is as follows. x& ( k + 1) F( k) x( k) + w( k) (8) ias z Z od ω z Roll 9deg where w( k): sstem noise xk ( ) ias ias p φ ω ω z pitch rate of the shell X shell X od Fk ( ) I Z shell ω od aw rate of the shell shell he equations (), (3) are used to drive the measurement equations. We can also use the groscope ias as a measurement. he are otained the following equations. Fig. 1. he roll angle related to the pitch rate and aw rate he pitch rate and aw rate of a smart munition can e otained GPS data. he pitch angle and aw angle of the smart munition are calculated the equations 4, 5 and their rates are calculated the equations 6, 7(Lucia, 1995). ω N ias ω (9) 1 N ω N z ias ωz (1) 1 N tanθ V V D N + VE VE tanψ (5) V N In the equations (4), (5), V N, V E, V D are the GPS velocit of a smart munition, which is represented in the navigation frame. & Δθ θ (6) Δ t (4) B appling perturation method to the equations (), (3), (9), (1), the measurement equations are as follows. δ zk ( + 1) H( k) δ xk ( ) + vk ( ) (11) where v measurement noise 1 ( ˆ ) ( ˆ ω ω z δzk ( ) δ sin φ( t) δ cos φ( t) ) δ δ k k ψ& & θ tcos( pt+ φ) cos( pt+ φ) & θ + ψ& & θ + ψ& & θ ψ& H1( k) tsin( pt+ φ) sin( pt+ φ) & θ + ψ& & θ + ψ&

3 he extended Kalman filter is given as follows(e.w. Kamen, 1999). ime propagation x ( k) F( k 1) x( k1) P ( k) F( k 1) P( k1) F ( k 1) + Q( k1) Measurement update ( ) ( ) Kk ( ) P( kh ) ( k) HkP ( ) ( kh ) ( k) + Rk ( ) 1 Pk ( ) IKkHk ( ) ( ) P( k) x( k) x ( k) + K( k) z( k) h( k, x ( k)) ( ) 1 (1) degrees(down). his means that the aw(ψ ) is -6.5 degrees and the pitch(θ ) is -53 degrees in coordinate transformation. We think these values as constant ecause the trajector of a smart munition is within 1Km. 3. he EKF design using the earth magnetism measurement 3.1 he earth magnetism expressed in the navigation frame he earth magnetism consists two parts. One is declination, the other is inclination. he magnetic declination is defined as the angle etween magnetic north and geographic north. he magnetic inclination is defined as the angle etween magnetic vector and the tangential plane of the horizon at the position where is measured. In figure and figure 3, the axes of the navigation frame( Xn, n, Z ) are defined as geographic north, east, down n respectivel. In figure, the attitude of a magnetic sensor is exactl aligned with the od frame( X,, Z ). he magnetic sensor output consists of the values measured earth magnetism in the od frame. We normalize magnetic sensor output dividing its magnitude. he normalized magnetic sensor output can e expressed with respect to the navigation frame coordinate transformation. he equation (13) represents these relations. he matrix C n is a coordinate transform matrix from the od frame to the navigation frame. In figure 3, we define the earth magnetism frame ( Xem, em, Z ). he X axis of the earth magnetism frame is the em direction of earth magnetic vector. he, Z axis are determined the coordinate transformation from the navigation frame to the earth magnetism frame. In this coordinate transformation, the aw is declination, the pitch is inclination and the roll is zero. In other words, the earth magnetism frame can e aligned with the navigation frame coordinate transformation. he normalized magnetic sensor output expressed in the earth magnetism frame is, which is theoretical value. he equation (14) represents these relations. In this equation, the matrix m Cn the coordinate transform matrix from the navigation frame to the earth magnetism frame. is Fig.. Navigation frame and od frame Fig. 3. Navigation frame and earth magnetism frame X n X X n n C1CC 3 C Z Z Z n where cosψ sinψ cosθ sinθ 1 C1 sinψ cosψ, C 1, C 3 cosφ sinφ 1 sinθ cosθ sinφ cosφ (13) In Seoul(the capital cit of Korea) the magnetic declination is 6.5 degrees(west) and the magnetic inclination is

4 1 X em em Z em cos β sin β cos( α) sin( α) X n 1 sin( α) cos( α) n sin β cos β 1 Z n X n m C n n Z n (14) In figure and figure 3, these two normalized magnetic outputs expressed the navigation frame must e same value. In the equation (13), [ Xn, n, Z n] is sustituted into the equation (14), the following equation holds. We will use this equation (15) as a measurement equation in the EKF. ([ Xm, m, Zm] ) H( k) p C p 3 m C C1 Cn ([ Xm, m, Zm] ) H3( k) φ C 3 m C C1 Cn φ ([ Xm, m, Zm] ) H4( k) θ C m C3 C1 Cn θ ([ Xm, m, Zm] ) H5( k) ψ C1 m C3 C Cn ψ (18) (19) () (1) X m 1 n m m ( C ) ( C n ) Z m (15) 3. he EKF design using an earth magnetism measurement If GPS signal can t e availale ecause of GPS jamming, we can use the magnetic sensor output as a measurement to reduce the navigation error. he state variales are as follows. 4. Simulation In this chapter, the previous method and the proposed method are simulated. Figure 4 shows the simulation trajector of a smart munition. he start point is (,,), which is marked circle. Its initial velocit is 1m/s and total flight time is 18.6 seconds. he initial pitch and aw are 35 degrees and degrees respectivel. he initial roll and its roll rate are random valuales, whose ranges are given tale 1. In this trajector, we make the groscope data, GPS data and magnetic sensor data, which have noise. he repetition of Monte Carlo simulation is 1 and we calculate the mean of the asolute roll error. B this value, we evaluate the performance of three roll estimation methods. ias ias xk ( ) ω ω p φ θ ψ (16) z he sstem equation is 6 6 identit matrix as the equation (8). It is assumed that all state variales are constant regardless of time. he measurement equation is as follows. δ zk ( + 1) Hk ( ) δxk ( ) + ν( k) (17) Hk ( ) H ( k) 1 4 O3 H( k) H3( k) H4( k) H5( k) O Fig. 4. he trajector of a smart munition In the equation (17), H 1 is given the equation (11). From H to H 5, all components of these matrices are otained partial derivative in the equation (15) as follows. he detail conditions in this simulation are given in tale 1. 95

5 ale 1. Simulation condition Characteristics Exit velocit Initial roll Initial pitch Initial aw Roll rate Groscope sampling rate Groscope ias Rate noise densit Scale factor error Non-orthogonal angle Value 1 m/s ~36 deg (random) 35 deg deg 1~3 Hz (random) 1 Hz 15 deg/h.5 deg/s/ Hz 1 ppm 3 deg 4. he simulation with a constrained GPS measurement under GPS jamming environment If GPS jamming occurs, we can t use GPS signal as a measurement. If the initial pitch angle is 35 degrees and flight time is 18.6 seconds. he angle variation is 7 degrees, so the pitch rate is approximated -.64deg/s. We fixed the pitch rate and aw rate as -.64deg/s and deg/s respectivel. In figure 6, efore the GPS jamming time(at 38.7 second), roll error trajector is the same as that in section 4.1. But after jamming time, the roll error increases. So another measurement is needed to reduce this error. Magnetic sensor noise densit GPS velocit noise GPS position noise GPS sampling rate Monte Carlo simulation.1 m/s 1 m 1 Hz 5 (repetition) 4.1 he simulation with a GPS measurement In the equations (), (3), the pitch rate, aw rate can e calculated GPS data. he simulation result using the GPS measurement is shown in figure 5. he dot line is the mean of the asolute roll error and upper line is its standard deviation. Within 1 seconds, the trajector has a ig error ecause the EKF can t know the initial roll and its rate. In real case, it is impossile to know the initial roll and its rate ecause the smart munition revolves ver fast at firing. In figure 5, the EKF estimates the roll error within 1 deg after 3 seconds. Fig. 6. Roll estimation error using a constrained GPS measurement under jamming environment 4.3 he simulation with an earth magnetism measurement under jamming environment In this section, we use the earth magnetism as a measurement under GPS jamming environment. We fixed the pitch rate and aw rate same reason in section 4.. In figure 7, under GPS jamming time, we use the earth magnetism measurement in the EKF and the navigation performance is improved. he simulation result shows ver good performance. he roll estimation is stailized within short time using earth magnetic sensor. he roll error converges to. deg within 3 seconds after GPS jamming in figure 7. Fig. 5. Roll estimation error using onl a GPS measurement 953

6 Fig. 7. Roll estimation error using an earth magnetism measurement under jamming environment Emedded GPS/Micro-Mechanical Inertial Navigation Sstem, MS hesis, MI, Camridge, Massachusetts.. Gordon Brown (3). Harsh Militar Environments and Microelectro mechanical (MEMS) Devices, Sensors, 3. Proceedings of IEEE, -4, Oct. 3 pp itterton, D.H and J.L. Weston (1997). Strapdown Inertial Navigation echnolog, Peter Peregrinus Ltd., London, U.K. Heeoung Park, KwnagJin Kim, Jang Gu Lee, Chan Gook Park (7), Roll Angle Estimation for Smart Munitions, IFAC Smposium on Automatic Control in Aerospace, oulouse, France. 5. CONCLUSIONS he previous roll estimation algorithm of a smart munition is introduced. Based on this knowledge, the roll estimation algorithm using the earth magnetism is suggested. We drive sstem and measurement equations for the EKF. Without GPS jamming, previous method is 1 deg within seconds. But under GPS jamming environment, the previous method shows that the roll estimation error is increased. o reduce this roll estimation error, the earth magnetism is used as a measurement. Under GPS jamming environment, we didn t use GPS data. We fixed the pitch rate and aw rate as constant using the initial velocit and flight time of a smart munition. he simulation result of the proposed method shows good performance. he roll estimation error converges to. deg within seconds under GPS jamming environment. ACKNOWLEDGMEN his stud is accomplished as a fundamental research project (ADD-6-7-1) of Defence Acquisition Program Administration(DAPA) he authors wish to thank Agenc for Defense Development, Brain Korea 1, Automatic and Sstem Research Institute, and Automatic Control Research Center. REFERENCES Bradford S. (1998). Davis. Using low-cost MEMS accelerometers and groscopes as strapdown IMUs on rolling projectiles, Position Location and Navigation Smposium IEEE 1998, -3 April 1998 pp E.W. Kamen and J.K. Su, Introduction to Optimal Estimation, Springer-Verlag Ltd., London, U. K., Lucia, D.J. (1995). Estimation of the Local Vertical State for a Guided Munition Shell with an 954