Single GPS antenna attitude determination of a fixed wing aircraft aided with aircraft aerodynamics

Transcription

1 AIAA Guidance, Navigation, and Control Conference and Exhibit August 5, San Francisco, California AIAA Single GPS antenna attitude determination of a fixed wing aircraft aided with aircraft aerodynamics K.P.A. Lievens,J.A. Mulder and P. Chu Department of Control and Simulation, Delft University of Technology, The Netherlands The satellite-based Global Positioning Systems (GPS) creates a new era for navigation, surveying and geodesy. Precise airborne and shipboard navigation, precise static geodetic positioning over baselines from a few meters up to thousands of kilometers, and kinematic positioning are just a few among numerous applications of GPS. In addition to providing position and velocity information, the GPS system can also be used to estimate the attitude parameters of the platform. It has the potential of replacing, to some extent, some sophisticated and expensive attitude sensors such as Inertial Navigation Systems (INS). Recently, a considerable amount of effort has been directed at developing low-cost systems for attitude determination. Methods to determine the attitude of the aircraft using a known wind vector and angle of attack and a single antenna GPS receiver already exist. In this colloquium a new method for attitude determination using a single GPS receiver aided with an aerodynamic model and a Kalman filter will be presented. Using no additional information but a single GPS receiver the complete attitude of the aircraft and the wind vector will be estimated. Results from simulation and real aircraft log files will be shown. When the flight path is adapted to the attitude determination algorithm for wind estimation the aircraft is required to fly a turn at a constant velocity, accuracies of a few degrees for all attitude angles can be obtained. The faster the update rate of the GPS receiver, the more accurate the method becomes at estimating the faster dynamics of the aircraft. In the future this method could prove a useful addition to the cockpit as a backup attitude determination system for General Aviation, or as primary attitude determination system for use in Unmanned Aerial ehicles. I. Introduction The GPS system has already widely been used for attitude determination ranging from integrated systems and multiple GPS antenna systems to single antenna GPS systems. The single GPS antenna is developed for navigation, but can provide additional information that can be used to estimate the attitude of the vehicle. Using carrier to noise measurements the direction of the antenna relative to the satellite can be determined. Or using a kinematic point mass model accelerations can be calculated from the velocity estimations made from the phase and doppler measurements from the GPS, see Szarmes. 1 These accelerations can then be used to estimate pseudo-euler angles as done by Johnson. With a known wind vector and a known angle of attack, and neglecting the slip, an estimation can be made of the true euler angles as done by Kornfeld 3, 4. This paper presents a new method that estimates the true euler angles and the angle of attack and the slip angle, using only a single GPS antenna and a aerodynamic model of the aircraft. As the wind and airspeed remains unknown, the aircraft is required to fly a turn at constant airspeed from time to time to be able to estimate the wind from GPS ground speed measurements, to make a rough estimation of these velocities. It could be used as a backup system This method is tested in simulation and using log files from a GPS and a reference system recorded in real flights. PhD Student, Delft University of Technology, Faculty of Aerospace Engineering, Department of Control and Simulation, Aerospace Software and Technologies Institute, Kluyverweg 1, 69 HS, Delft. Prof, Delft University of Technology, Faculty of Aerospace Engineering, Department of Control and Simulation, Kluyverweg 1, 69 HS, Delft. Doctor, Delft University of Technology, Faculty of Aerospace Engineering, Department of Control and Simulation, Kluyverweg 1, 69 HS, Delft. 1 of 14 Copyright 5 by the author(s). Published by the, Inc., with permission.

2 II. An observability analysis Is an aircraft equiped with a single GPS antenna an observable system? This question can be answered in theory by making a linearization of this system. And by calculating the observable, controllable and stable subspaces. Combining these subspaces will make clear which states of the system are observable and which not. A linear system is designed using a linear state space model of an aircraft. And then adding a linear model of a GPS. The state is further expanded with a wind vector which will determine the difference between the GPS ground speed and airspeed used by the linear model. A. A linear aircraft model The linear model of the aircraft is made as described by Mulder. 5 The equations of motion of the aircraft are derived from Newton s third law. In these equations we consider the mass of the aircraft to be constant. A second assumption is, that the aircraft may be considered a rigid body during the motion to be considered. So, the effects of elastic deformations are neglected for simplicity. We can schematize the aircraft, all aerodynamic forces must be equal to the kinematic forces due to the different velocities and accelerations. The results of this statement can be seen in equation (1). F x = W sin θ + X = m( u + qw rv) F y = +W cos θ sin ϕ + Y = m( v + ru pw) F z = +W cos θ cos ϕ + Z = m(ẇ + pv qu) M x = L = I x ṗ + (I z I y )qr J xz (ṙ + pq) M y = M = I y q + (I x I z )rp J xz (p r ) M z = N = I z ṙ + (I y I x )pq J xz (ṗ + rq) (1) These equations here derived are based on the following restrictive assumptions, some mentioned already before. They can be summarized as follows. The aircraft s mass is constant in the time interval during which the motion is studied The aircraft is a rigid body in the motion under consideration The mass-distribution of the aircraft is symmetric relative to the X B OZ B -plane The rotation of the earth in space, as well as the curvature of the earth s surface are negligible Due to these assumptions, no coupling exists between the symmetric and the asymmetric aircraft motions. The equations of motion of an aircraft may be used in several different ways. Most commonly, a disturbance function is given, such as a time-dependant control surface deflection or a gust velocity. The response of the aircraft to such a disturbance is then calculated. This is the way the model will be used in this report. Equations (1) become after linearizing around u =, w =, α = and θ = γ : Symmetric motions W cos θ θ + X u u + X w w + X q q + X δe δ e =m u W sin θ θ + Z u u + Z w w + Zẇẇ + Z q q + Z δe δ e =m(ẇ q ) M u u + M w w + Mẇẇ + M q q + M δe δ e =I y q θ =q () of 14

3 Asymmetric motions W cos θ ϕ + Y v v + Y v v + Y p p + Y r r + Y δa δ a + Y δr δ r =m( v + r ) L v v + L p p + L r r + L δa δ a + L δr δ r =I x ṗ J xz ṙ N v v + N v v + N p p + N r r + N δa δ a + N δr δ r =I z ṙ J xz ṗ ψ = r cos θ ϕ =p + r tan θ (3) These equations will now be added to each other to form one big state space equation which holds both symmetric and asymmetric equations of motion. The different equations derived in the foregoing can be applied directly to the calculation of the symmetric and asymmetric aircraft motions about a given condition of steady, straight, symmetric flight as a response to a given disturbance. ery often, however, the equations are used in a non-dimensional form. The reason for this practice is that the aerodynamic forces and moments in the equations are usually in non-dimensional coefficients. Equations () then become: Symmetric motions C Xu µ c D c C Xα C Z C Xq C Zu C Zα + (C Z α µ c )D c C X C Zq + µ c D c 1 C mu C mα + C m α D c C mq µ c KY D c = C Xδ e C Zδ e C mδ e δ e (4) Asymmetric motions C Yβ + (C Y β µ b )D b C L C Yp C Yr 4µ b 1 1 C lβ C lp 4µ b KX D b C lr + 4µ b K XZ D b C nβ + C n βd b C np + 4µ b K XZ D b C nr 4µ b KZ D b = C Yδa C Yδa C lδ a C lδ a C nδa C nδa [ δ a δ r (5) ] These equations of motion are transformed to the state-space form as in, ẋ = Ax + Bu (6) This form can be obtained by rearranging equation (4) and (5). The result is, Symmetric motion û α θ q c = x u x α x θ z u z α z θ z q c m u m α m θ m q û α θ q c + x δe x δt z δe m δe [ δ e δ T ] (7) 3 of 14

4 x... z... m... u C X u C Z u Cm C mu +C α Z u µc C Z α c µ c c µ c C Z α c µ c KY α C X α C Z α Cm C mα +C α Z α µc C Z α c µ c c µ c C Z α c µ cky θ q Cm C Z c µ c c C X µ c C Z α c C α X µc C Z α µ c KY C X q µ c+c Z q µc+c Z C mq +C q m α µc C Z α c µ c c µ c C Z α c µ c KY (8) δ e C X δ e C Z δ e Cm C mδ e +C α Z δ e µc C Z α c µ c c µ c C Z α c µ c KY δ T C X T c µ c Asymmetric motion β ϕ ṗb ṙb = y β y ϕ y p y r b l β l p l r n β n p n r β ϕ pb rb + y δr l δa n δa l δr n δr [ δ a δ r ] (9) y... l... n... β C Y β b µ b b C lβ K Z +Cn β K XZ 4µ b (K X K Z K XZ ) b C lβ K XZ +C nβ K X 4µ b (K X K Z K XZ ) ϕ b C L µ b p r C Y p b µ b b C Y r 4µ b b µ b b C l p K Z +Cn p K XZ 4µ b (K X K Z K XZ ) b C lr K Z +Cn r K XZ 4µ b (K X K Z K XZ ) b C lp K XZ +C n p K X 4µ b (K X K Z K XZ ) C lr K XZ +C nr K X 4µ b (K X K Z K XZ ) (1) δ a C Y δ a C lδ a K Z +C n K δ XZ a C lδa K XZ +C nδ a K X b µ b b 4µ b (KX K Z K XZ ) b 4µ b (KX K Z K XZ ) δ r C Y δr C lδr K Z +Cn δr K XZ C lδ r K XZ+C nδr K X b µ b b 4µ b (KX K Z K XZ ) b 4µ b (KX K Z K XZ ) These equations will now be added to each other to form one big state space equation which holds both 4 of 14

5 symmetric and asymmetric equations of motion. û α θ q c β ϕ ṗb ṙb = x u x α x θ z u z α z θ z q c m u m α m θ m q y β y ϕ y p y r b l β l p l r n β n p n r û α θ q c β ϕ pb rb + x δe x δt z δe m δe y δr l δa l δr n δa n δr δ e δ T δ a δ r (11) B. A linear GPS model The output of the GPS-system will contain the position and the speed and direction in the earth frame of reference. We have to find a linear relation between the state of the aircraft, which is in body frame of reference, and the output of the GPS, which is in earth frame of reference. This is impossible if we do not ad some variables to the state of the aircraft. To obtain the complete state space equations, we still need to find the C-matrix, the matrix that relates the state to the output. We will now try to find this matrix. One part of this equation will be in the body frame of reference, and the other in the earth frame of reference. The aerodynamics of the aircraft are dependent of body frame angles and speeds, and the output of the GPS is in the earth frame of reference. The transformation matrix L EB is as follows: L EB = cos θ sin ϕ sin θ cos ψ cos ϕ sin ψ cos ϕ sin θ cos ψ + sin ϕ sin ψ cos θ sin ϕ sin θ sin ψ + cos ϕ cos ψ cos ϕ sin θ sin ψ sin ϕ cos ψ sin θ sin ϕ cos θ cos ϕ cos θ The geographic position coordinates of the origin of F B (i.e. center of gravity) with respect to F E satisfy the following relations: E = L EB B + W Where W denotes the components of the atmospheric wind along the axes of F E. This relation describes almost perfectly the relation between the state and the output of the GPS. The only problem are the variables v and w, they are not part of the state of the aircraft. But they can be related to the variable of the state with the following equations: v = u tan(β) and w = u tan(α) When we write this out we get : ẋ E ẏ E = (u cos (θ) + (u tan (β) sin (φ) + u tan (α) cos (φ)) sin (θ)) cos (ψ) (u tan (β) cos (φ) u tan (α) sin (φ)) sin (ψ) + W xe = (u cos (θ) + (u tan (β) sin (φ) + u tan (α) cos (φ)) sin (θ)) sin (ψ) + (u tan (β) cos (φ) u tan (α) sin (φ)) cos (ψ) + W ye ż E = u sin (θ) (u tan (β) sin (φ) + u tan (α) cos (φ)) cos (θ) + W ze (1) These equations are rearranged and linearized into: ẋ E ẏ E ż E = flx E u flx E α flx E θ flx E β flx E ϕ flx E ψ fly E u fly E α fly E θ fly E β fly E ϕ fly E ψ flz E u flz E α flz E θ flz E β flz E ϕ flz E ψ du dα dθ dβ dϕ dψ + W xe W ye W ze (13) Where flx Eu stands for the first derivative with respect to u of the component of along the X E axis, etc. This linearization will be different for each set of euler angles, and therefore all possible combinations should be considered before making a conclusion about the observability of the non-linear system. 5 of 14

6 C. The complete linear model Now we can compose the complete state space equations: u α θ q β ϕ ṗ ṙ ψ ẋ E ẏ E ż E Ẇx E Ẇy E Ẇz E = xu xα x θ zu zα z θ zq c mu mα m θ mq y β yϕ yp yr b l β lp lr n β np nr 1 flx E u flx E α flx E θ flx E β flx E ϕ flx E ψ 1 fly E u fly E α fly E θ fly E β fly E ϕ fly E ψ 1 flz E u flz E α flz E θ flz E β flz E ϕ flz E ψ 1 u α θ q β ϕ p r ψ x E y E z E Wx E Wy E Wz E + x δ e x δt z δ e cm δ e y δ r bl δa bl δr bn δ a bn δ r δe δ T δa δr (14) χ x E y E z E ẋ E ẏ E ż E = flx E u flx E α flx E θ flx E β flx E ϕ flx E ψ 1 fly E u fly E α fly E θ fly E β fly E ϕ fly E ψ 1 flz E u flz E α flz E θ flz E β flz E ϕ flz E ψ 1 u α θ q β ϕ p r ψ x E y E z E Wx E Wy E Wz E (15) D. Observability analysis 1. Stability Stability of a state space system can be checked by calculating the eigenvalues of the A matrix. It turns out that the GPS and wind are asymptotically stable.. Controllability The controllability of a linear system (A, B, C, D) can be characterized by the matrices A and B. This procedure is described in mathematical systems theory. 6 The controllability matrix R is equal to [ B AB... A n 1 B ]. The controllability matrix of this system does not have full ranks, as the wind is of course uncontrollable. But all other states are in the controllable subspace. 3. Observability The observability of a linear system (A, B, C, D) can be characterized by the matrices A and C. This procedure is described in mathematical systems theory. 6 The observability matrix W is equal to: [ C CA... CA n 1] T. The observability matrix has full ranks for each possible set of euler angles. 6 of 14

7 4. Conclusion As the observability matrix has full ranks it should be possible to build an observer for such a system. The unstable states that are controllable should not be a problem, but the wind is uncontrollable and unstable. A problem can be expected there. III. The observer algorithm A. An overview The core element of the algorithm is the unscented Kalman filter which tries to estimate a state consisting of the following elements : {u, α, θ, q, β, ϕ, p, r, x E, y E, z E, W xe, W ye, W ze }. This is done by combining measurements with a prediction made with a model of the system. The GPS receiver provides three direct measurements of the fourteen states. Using different methods provided by Kornfeld 3, 4 a first estimate of a few states is made (as described in section III C), these are then used as measurements of the state by the Kalman filter. By comparing the ground velocity with a previous opposite ground velocity from a log file, the wind is estimated and this estimate is also given to the Kalman filter. The measurements from the GPS receiver get a smaller covariance than the covariance of the estimates of the other states, as they are more likely to contain more errors. The current state is also predicted using a model (see III B) that has the the previous state and the pilot inputs as input. These two state estimates (prediction and measurements ) are then combined using a weighted average into one best estimate of the state. Figure 1 shows a schematic of the whole process of state estimation. Prediction Model Q GPS Position and velocity First estimate Measurement R + First estimate R Unscented Kalman Filter Best estimated state Figure 1. A schematic of the attitude determination algorithm B. The model The model used in the filter is a combination of the same linear aircraft model as described earlier in section II A, and the non-linear GPS equations (1). Coefficients of the Cessna Citation I trimmed at cruise speed are used. It uses the pilots inputs to calculate the new aircraft state with an aerodynamic model. And using the last state euler angles, new euler angles are calculated. The new aircraft state and euler angles are then used to calculate the new GPS state using the the GPS equations (1). C. The measurements The measurements feed to the filter can be divided into two groups : The measurements from the GPS, consisting of the position, the speed and the heading. The estimates of the euler angles made from the speed and accelerations obtained from the GPS. It is possible, with some simplifications, to make an estimate of almost every state using only ground speed. The roll 7 of 14

8 This method to calculate the roll angle is a commonly used by other attitude determination algorithms. It is based on the fact pilots are trained to make controlled turns, which means that when making a turn the total acceleration should point in the Z BODY direction. The aircraft should fly at constant speed, a xbody = and should not change altitude,ż NED =. In such a case the turn rate of the controlled turn can be calculated by differentiating the track angle, Ω = ψ. Once Ω is found, the additional acceleration due to the circle a turn and the radius of the turn R can be calculated with Ω = g R and a turn = RΩ These two combined give the following equation: a turn = g Ω When the additional accelerations due to flying a circle are known the roll angle can be calculated with ϕ = arctan a turn g = arctan Ω g. A lot of assumptions and simplifications had to be made to calculate this angle, which would questions its usability. But in reality this method give a fairly good estimate of the roll angle, and improved the performance of the filter a great deal. An estimation of the pitch A not so accurate estimate can be made of θ can be made by using the pitch. We know that the pitch : θ = γ + α. We can simplify this equation assuming the angle of attack is around degrees, which is the case in most of the flight: θ = γ + α, with α deg. This estimate will almost never be correct, only in cases when α is degrees, but the error will never be more then 1 degrees. The filter will be able to make a better estimate taking this into account. The covariance of this measurement will be set to 1deg An Indication of the true airspeed The aerodynamic model can give an indication of the attitude angles of the aircraft, and will give an indication of how the true air speed changed due to a elevator deflection. But it does not tell us anything about the changes in true air speed due to the throttle position. Therefore a measurement of the true airspeed could help. The error in this estimation will be as big as the wind. Thus, it will be added with a covariance of 4m/s An estimation of the angular velocities In this section you will find a way to determine the angular velocities p,q and r of the aircraft about the body axes using the rates of change of the three attitude angles ψ, θ and ϕ. Because the three attitude angles are not known the estimates of them will be used. That is ψ χ, θ γ +α and ϕ arctan Ω g as described earlier. These attitude angles are then differentiated to obtain ψ, θ and varphi. The angular velocities about the aircraft body axes are obtained by adding the components of ψ, θ and ϕ along each of the body axes, p = ψ sin θ, q = ψ cos θ sin ϕ + θ cos ϕ and r = ψ cos θ cos ϕ θ sin ϕ. This estimate is a derivative of four other estimates which will make it a very noisy and un reliably estimate. But when using a butter filter to smoothen the noise and when taken into account that there may be a large error on this estimate, it becomes a useful addition to the filter. The wind using the average of opposite speeds We are not searching for small turbulence but the larger wind directions and speeds which are much more constant. Therefore we can permit us a slower update rate. Every minute the algorithm will make the average of the logged speeds in eight wind directions (North North-East East South-East South South-West West North-West). Making averages decreases the effect of changes of the true air speed of the aircraft. From these eight average speeds in the different wind directions, four new averages of opposite directions can be made. These these four averages are estimates of the wind, which can in turn be averaged to one best estimate of the wind. This method will be used by the attitude determination algorithm, and has to be calculated ones every time step. The covariance of this measurement will be set to 1m/s. 8 of 14

9 D. The Unscented Kalman filter The Unscented Kalman filter will compare the measurements from with the prediction from the aerodynamic model from section to calculate an estimate for the states described. First the model is used to calculate an prediction of the state. The model is a function that uses the previous state x k 1 and the control inputs u to calculate the state x k a certain time step t later(equation 18). But because the previous state is uncertain, all we have is an estimate x k 1 and a covariance P, one should calculate the next state using all possible states within that covariance. This would take to much computer power, and it is not completely necessarily. This filter uses sigma points, which are carefully chosen points on the border of the covariance. For example when one wants to estimate the position of a vehicle, using an estimate of the previous position and a covariance. Then this covariance would be a circle around the estimated position, and the only certainty would be that the vehicle s previous position was some where in that circle. Then the sigma points would simply be on top, above, under, left and right of the estimated position on that circle. Calculating the next state from these five sigma points gives an almost as good indication of the next state as calculating the next state from all possible points in that circle, but takes only a fraction of that in calculating time. Now in stead of having two states as in the position of the vehicle and five sigma points, we have the 17 states. They will provide 35 sigma points (equation 17). From these 35 sigma points a next state is calculated using the aerodynamic model (equation 18). This will give us 35 estimates of the next state with 35 covariances. Or one mean prediction and one mean covariance, which are calculated using a weighted average(equation 19). Out of all 35 sigma points a prediction of the measurement can be calculated (equation 1). With these 35 predicted measurements a mean predicted measurement and covariance is calculated using that same weighted average(equation ). With these predicted state and predicted measurement, the difference between the state and the measurement is estimated. This difference is then used to calculate the best estimate of the state using the real measurements. And it is also used to calculate the covariance of that best estimate of the state(equation 3). This story described exactly what is done by the filter in the matlab programs used to determine the attitude of the Cessna Citation in this report. A clearer explanation of the unscented Kalman filter can be found in. 7 Table 1. Initialize with: Calculate Sigma points: The UKF Algorithm ˆx =x P =P (16) Xk 1 a = [ˆx a k 1 ˆx a k 1 ± sqrt(l + λ)pk 1 a ] (17) Time update: Xk k 1 x =F [ Xk 1, x Xk 1 v ] ˆx k P k = L i= = L i= (18) W (m) i X x i,k k 1 (19) [ ] [ ] T W (c) i Xi,k k 1 x ˆx k Xi,k k 1 x ˆx k [ ] Y k k 1 =H Xk k 1 x, Xx k 1 ŷ k = L i= Measurement update: L Pỹk ỹ k = P xk y k = i= L i= κ =P xk y k P 1 ỹ k ỹ k () (1) W (m) i Y i,k k 1 () W (c) [ i Yi,k k 1 ŷ ] [ k Yi,k k 1 ŷ ] T k W (c) [ i Xi,k k 1 ˆx ] [ k Yi,k k 1 ŷ ] T k ˆx k =ˆx k + κ ( y k ŷ k ) P k =P k κp ỹ k ỹ k κ T (3) [ where x a = x T v T n ] T T,X a = [ ] (X x ) T (X v ) T (X n ) T T, λ = composite scaling parameter, L = dimension of augmented state, P v = process noise covariance, P v = measurement noise covariance and W i = weights. 9 of 14

10 I. Results A. The results in simulation 1. The non-linear model of the Cessna Citation I The model used in these simulations is a model that was developed in DASMAT, state and witch used the syntax of MATLAB stuur input p q Transport wens attitude wens attitude state r 4. The model was then rewritten to Delay vtas attitude_pid_controller alpha navigation_pid_controller beta phi MATLAB 6 by Alexander eldhuijzen, theta psi he In1 1 xe as can be seen in his Master Thesis report. 8 Another version of this model ye Turbulence gain he_dot Out6 alpha_dot beta_dot In Turb gain aircraft output M outputs ias nz gamma_a Subsystem Actuators Input Inputs chi_a exists in a program extension file (.dll), gamma_a_dot chi_a_dot Wind/Turb wind gamma_k External Turb chi_k Actuators this has been developed by Erwin Kipperman. 9 1 da gamma_k_dot Cessna Citation 5 chi_k_dot Non Lineair 6 DOF model bankangle g InCo de dr External Turbulance In1 dte In 3 df In this simulation a wind model was 4 In3 plac C In4 5 6 In5 used, it uses a wind direction and wind External Turbulence In6 7 8 In7 In8 velocity, and next to this constant wind turbulence can be added. Explanations Figure. The Cessna Citation I model in simulink on this wind model can be found in the Master Thesis report by Alexander eldhuijzen. 8 The wind direction was zet to 45deg, and the wind velocity to 1m/s. lat_gps lon_gps alt_gps u_gps v_gps w_gps gamma psi da de dr u_tas alpha theta q beta phi p r v_wind track_wind flight_log.mat To File. The selected flight path A small PID controller was developed to make the model follow a certain trajectory. In this simulation the aircraft will perform some turns alternated by a straight forward flight. The turns are to test the wind estimation tool, and the alternately straight pieces are to show how well the algorithm can use the information obtained from one turn. The navigation controller is designed to keep the same height and a pulse function controls the desired roll angle..5 x trajectory The results of the estimations of the attitude The symmetrical states The results of the estimates of the symmetrical states are very good, especially for the angle of attack α, the pitch θ and the angular velocity around the X B -axis q. When comparing the results from the simulation with or without turbulence, you can not see any difference in the results of the filter. The symmetrical states are effected less then the asymmetrical states, especially the pitch and the true airspeed. But the filter can still follow every movement of the aircraft x 1 4 Figure 3. The trajectory flown in simulation The asymmetrical states The asymmetrical states are affected more by the turbulence then the symmetrical states, this has its effect on the filter. When the states fluctuate too much, the filter can not follow and smoothes the results. It sees the rapid movements of the aircraft due to the turbulence as noise and filters them out. As the filter stays stable, it does not diverge from the reference, this should not be a problem. It may even be advantageous to have a smoothed reading for some high frequency controllers who may become unstable when trying to follow the fluctuating signal. 4. Conclusion The results in simulation without turbulence are very good, in spite of the difference in the models used. The simulations were made with a non-linear model of the Cessna Citation I and the algorithm uses a linear model trimmed for cruise of the Cessna Citation I. Due to these different models, one can see that the estimation has a few errors. Especially when the aircraft diverges from cruise condition with large angles of roll. These manoeuvres are not modelled in the linear model and therefore badly estimated. 1 of 14

11 Another cause of error in the estimation is due to the slow update rate of the GPS. Especially when there is turbulence added to the simulation, the aircraft starts making all kinds of fast manoeuvres, which are not seen by a slow GPS. When we change the update frequency of the GPS to 1 Hz the estimations errors are diminished greatly, and turbulence is no longer insurmountable. B. The results in real flight 1. The flight data The flight data used is a result of the recording of a flight of the faculty s flying laboratory, the Cessna Citation II. On the 6nd of may 3, the Cessna Citation II (C55) PH-LAB made a flight that had as primary objective: manoeuvres to perform sensor calibration and aerodynamic model identification. The airplane took off from Schiphol and landed back on Schiphol (The Netherlands). The data was recorded by an ADAM computer, using FTIS (designed by Marco Soijer) as data-collecting system software. The GPS data comes from a single antenna GPS receiver with a rate of 1 hertz. This was used as input for the filter. The output is then compared to the reference measured by different systems: The pitch and roll come from a device called TARSYN. And the angular velocities from an IMU (with fiber optic gyroscopes). The true airspeed is measured by a pitot tube. And the angle of attack and slip are measured by a vane on top of a boom on top of the nose of the aircraft. This data has been post processed by Joao Oliveira.. A single turn In this section you we will look deeper into a small part of the complete flight. To test the effectiveness of the algorithm it is now tested on a single turn. The algorithm has had a complete reset prior to this turn, which means that there is no initial information on wind or true airspeed or any other state, except for the data received from the GPS. Before starting this turn, the filter had a complete reset. This means that the filter had no estimation of the wind or any other state, except for the ones measured by the GPS. This experiment with a single turn shows us that the algorithm works for most of the states, even with no estimate of the wind. The estimation of the wind and the true airspeed only becomes liable after making half a turn. But the most important state variables needed by an autopilot to guide plane, such as the angle of attack, the pitch, the roll angle and the angle of slip, are directly avaleble. With the knowledge of these angles, the aircraft could be guided through a turn by a autopilot. Then the true airspeed and wind would become known. The flight path The chosen piece from the complete flight log is a single turn. This is done to show the impact of a turn on the wind estimation, and on the estimates of the states. The symmetrical states The estimate of the true airspeed is the worst estimates of all the estimates mode for this turn. As the estimate of the wind does not improve enough, the estimate of the true airspeed remains inaccurate. The angle of attack is an estimate made solely by the Kalman filter, there are no first estimates for this state. And yet the Kalman filter manages to describe an almost exact copy of the angle of attack vane installed in the Cessna Citation II. The biggest difference between the measurements by the vane and the estimates by the Kalman filter is less than.degrees. The first estimate of the pitch was made by assuming it would be almost equal to the vertical path angle plus a mean angle of attack. This estimate has now been improved by the Kalman filter, with use of the estimation of the angle of attack. The results can be seen in figure 5(c) x 14 trajectory Figure 4. The trajectory The asymmetrical states The angle of slip is a state that has no first estimation, it is solely estimated by the Kalman filter. And yet it became a very nice estimation of the angle of slip, with a maximum difference with the slip vane mounted on the Cessna Citation II of 3.5degrees.From the ground speeds a track angle can be calculated, using this calculated track angle a change of track angle can be calculated, and using this change of track angle and the ground speed an estimate of the roll can be made. A second 11 of 14

12 estimate is then made using the Kalman filter. There were no differences seen between the first and the second estimate.the results can be seen in figure 5(d).. Conclusion In this report a filter was designed on the idea that an aircraft with no other sensor but a single GPS receiver is observable. This was proven in theory for a linearized model. A Kalman filter was designed to solve this problem in simulation and later on in reality. The Kalman filter uses the control inputs for the aircraft, and the position and velocity data from the single GPS-receiver. The control inputs are used in a linearized aerodynamic model of the aircraft to help predict the next state. And the position and velocity data are used as measurements, to help the prediction to stay aligned. The velocity data from the GPS-receiver are also used to calculated additional estimates, these are then added to the Kalman filter as measurements of lesser value. These additional estimates help the Kalman filter to stay aligned, and not divert from the reality. The results are very good, they have only minor errors compared to the reality of the simulation, or compared to the measurement equipment onboard of the Cessna Citation II. As this algorithm uses a very simple model of the aircraft, it is a very fast algorithm. This means that it will be easily applicable in a small computer, which can be carried by small aircrafts. The estimates of the attitude of the Cessna Citation II are comparable to the measurements of the measurement equipment of the aircraft. All the expensive and heavy onboard measurement equipment can be matched using a simple aircraft model, a single GPS-receiver, and a small computer. Which is off course a great advantage for small aircraft such as UA s. The algorithm has been tested in a simulation of a Cessna Citation I and tested on the flight log of a real live flight of a Cessna Citation II, without making any changes to the algorithm. From this, we can conclude that it is a fairly robust filter. Due to the slow update frequency of the GPS, which is in most cases only 1 Hertz, some fast dynamic movements of the aircraft are not shown with the algorithm. Only a GPS with higher update frequency could decrease this problem. In the slower dynamic cases the algorithm has a precision witch is comparable to standard flight measurements tools used in the GA community. When operating in the fast dynamics region the accuracy decreases, by a factor of. This paper proves the potential of an algorithm that is capable of estimating the state of an aircraft only equipped with a single GPS antenna receiver. The results can be improved by using a more accurate model of the aircraft. For example a non-linear model, of multiple linear models depending the flight case. The update rate of the GPS receiver imposes a direct limit on the system as to estimating fast movements. Especially with turbulence or highly manoeuvrable aircrafts (such as small UA s) this will impose a problem. This problem can be seen in figure 5(c) and 5(d), at 5 seconds a fast manoeuvre is measured by the reference measurements equipment and is not visible in the filter results. The only way to overcome this is by increasing the update rate of the GPS receiver. I. Recommendations The results from the attitude determination algorithm could be improved by looking at the following topics: The GPS receiver The results could be greatly improved, especially for faster manoeuvres, by using a higher update frequency for the GPS receiver. At 1 Hertz some manoeuvres, such as turbulence, are simply invisible. Having a higher update of the measurements would also decrease the covariance of the estimate. The aircraft model. In this report the model used by the Kalman filter for dynamic aiding in the attitude determination was a linear aerodynamic model of a Cessna Citation I trimmed for cruise flight at 18m/s. This simplification was done to speed up calculations, and was not altered as the first estimates were satisfactory. This model was then expanded with a some non-linear equations which incorporated the euler angles which state the difference between the body (used by the aerodynamic model) and earth (used by the GPS receiver) frame of reference. This means that the aerodynamic model is not influenced by the wind. This could be improved by using a completely non-linear model which uses the wind in its aerodynamic calculations. But this will slow down the calculation, a trade off will have to be made. 1 of 14

13 Better performance in different flight cases. The results of the attitude determination for take of and landing are useless. To improve these estimates the model could trimmed around different flight cases, so that the filter can switch from one model to an other according to the flight case. This may cause large glitches in the prediction of the measurement which may cause instabilities in the Kalman Filter. Design of a compatible controller. Designing a controller specially tuned for the data received from this attitude determination algorithm. This controller could take in to account that the fast dynamics are not visible, and that when the speed measurements are not accurate anymore a turn should be made to improve the wind estimation. Tests on a small UA. This case is ideal for flight tests on a small UA. The hardware is reduced to the bare minimum for a UA, a small computer and a single antenna GPS receiver. The algorithm has been optimized for speed, which makes it almost ready to implement in a small processor. References 1 M.Szarmes, S. and G.Lachapelle, DGPS High Accuracy Aircraft elocity Determination Using Doppler Measurements, Tech. rep., The University of Calgary, E.N.Johnson, S.Fontaine, and A.D.Kahn, Minimum Complexity Uninhabited Air ehicle Guidance And Flight Control System, Tech. rep., School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia, 1. 3 R.P.Kornfeld, R. and J.J.Deyst, Single-Antenna GPS-Based Aircraft Attitude Determination, Journal of the Institute of Navigation, ol. 45, No. Spring, 1998, pp R.P.Kornfeld, R.J.Hansman, J. D. K. A. and E.M.Walker, Applications of Global Positioning System elocity-based Attitude Information, Journal of Guidance, Control and Dynamics, ol. 4, No. 5, September-October 1, pp J.A.Mulder, W.H.J.J. van Staveren, J. v. d.., Flight Dynamics, Delft University of Technology, Faculty of Aerospace Engineering,. 6 G.J.Olsder, J. v. d. W., Mathematical Systems Theory, Delft University of Technology, Information Technology and Systems, nd ed. 7 Eric A. Wan, R. v. d. M., The Unscented Kalman Filter for Nonlinear Estimation, Tech. rep., Oregon Graduate Institute of Science and Technology, NW Walder Rd, Beaverton, Oregon eldhuijzen, A., Integrating flight path oriented control with the tunnel-in-the-sky display, Master s thesis, Delft University of Technology, november. 9 Kipperman, E., Object-oriented aircraft modeling. implementing flight dynamics in Dymola, Master s thesis, Delft University of Technology, october of 14

14 95 true airspeed [m/s] 3.5 alpha [deg].6 beta [deg] 35 phi [deg] theta [deg] (a) symmetric simulation q [deg/s] p [deg/s] (b) asymmetric simulatoin r [deg/s] tas [m/s] theta [deg] alpha [deg] q [deg/s] beta [deg] p [deg/s] 1 phi [deg] r [deg/s] (c) symmetric real flight (d) asymmetric real flight Figure 5. result) The states of the aircraft estimated by the Kalman filter (red is the reference, blue is the filter 14 of 14