Influence of Dynamics and Trajectory on Integrated GPS/INS Navigation Performance

Transcription

1 Journal o Global Positioning Systems (23) Vol. 2 o. 2 : Inluence o ynamics and Trajectory on Integrated GPS/IS aigation Perormance J. Wang H.K. Lee S. Hewitson and Hyung-Keun Lee The Uniersity o ew South Wales Sydney SW 252 Australia Receied: 2 oember 23 / Accepted: 28 ecember 23 Abstract. The integrated GPS/IS system has become an indispensable tool or proiding precise and continuous position elocity and attitude inormation or many positioning and naigation applications. Thereore it is important to gain insights into the characteristics o the integrated GPS/IS system perormance particularly their relationships with key operational actors such as the trajectory and dynamics. Such knowledge can be used to improe the quality o positioning and naigation results rom integrated GPS/IS systems. In order to analyse the inluence o ehicle dynamics and trajectory simulation and ield tests hae been carried out in this research. The test results show that the ehicle dynamic changes signiicantly aect the Kalman ilter initialisation time and estimation perormance depending on the system operational enironments. Key words: Integrated GPS/IS system aigation Vehicle dynamics and trajectory 1 Introduction Global Positioning System (GPS) and Inertial aigation System (IS) hae complementary operational characteristics (e.g Saage 2). It is well known that IS can proide a complete set o naigation parameters with a short-term stability due to the error characteristics o its sensor components. Thus the accuracy o a standalone IS deteriorates ery rapidly with time which can be compensated by GPS. Integrating GPS with IS can arguably leerage the best o each component system. The adantages o GPS/IS integration relatie to either GPS or IS only are reported to be a high data rate o complete naigation solutions (e.g. position elocity and attitude) with a consistent long-term accuracy improed aailability smoother trajectories and greater integrity (Farrell and Barth 1998; Greenspan 1996). Hence these systems hae been used or a wide range o applications or instance aerial photogrammetry and graimetry mobile mapping ehicle naigation guidance and control (see e.g. Beely et al. 2; a et al. 1997; Grejner-Breinska et al. 1998b; Kwon and Jekeli 21; Wang et al. 23). Integrated GPS/IS can be implemented using a Kalman ilter in dierent modes such as loosely tightly and ultra-tightly coupled. In these integration modes the IS sensor error states together with all naigation error states and other unknown parameters o interest are estimated using a dynamic model and GPS measurements such as oppler pseudo-ranges and/or carrier phases. It has been reported rom the literature that the ehicle dynamic and trajectory changes can improe the Kalman ilter estimation perormance (e.g. Bar-Ithack & Port 198; Port & Bar-Ithack 1981; Hong et al. 22; Wang et al. 23). The improement can be described by the act that the error model becomes time-arying nature that enhances the ilter obserability. Hence it is necessary to get insights into the characteristics o the integrated GPS/IS positioning and naigation perormance particularly their relationships with the ehicle trajectory and dynamic changes. This paper will study how the ehicle dynamics and trajectory changes inluence on the perormance o an integrated GPS/IS system through both simulation and real data analyses. The optimisation o trajectories and dynamics during system initialisation and operational naigation mode and in the eent o GPS signal blockages will be discussed. 2 rror Model and Coariance Analysis 2.1 Strapdown IS (SIS) error model In order to study the behaiour o an inertial naigation system an appropriate presentation error model is necessary. The description o the IS error propagation

2 11 Journal o Global Positioning Systems using a linearised error model has been widely used to derie the characteristics o IS error behaiour. A number o the dierent models can be ound in the literature. Among them the Bar-Ithack and Berman s mode (Bar-Ithack and Berman 1988) is adopted in this research. As the model is designed or Gimballed IS modiication is made through including the coordinate transormation matri between body and naigation rames to drie the Strapdown IS (SIS) model. ote that a local leel coordinate system (: orth ast own) is used as a naigation rame. Hence the SIS error model can be described as ollows: & A + w w ( Q ) & r I & F21 Fj C & ψ & F & ε 44 C F 55 r ψ ε w w + w w w where I and are the third-order identity and ero matrices; C is the coordinate transormation matri between the body and naigation rames; w are all ero-mean Gaussian white noise ectors; r r ε r ψ ε (1) ψ and ε are respectiely the position elocity attitude accelerometer and gyro measurement error ectors. The error states included in these ectors are as ollows: r [ δr δ r δr ] T (2a) [ δ δ δ ] T (2b) [ δψ δψ δψ ] T ψ (2c) [ δ y δ ] [ δε δε δε ] T T δ (2d) ε y (2e) where is accelerometer bias and ε is gyro drit. Both two accelerometer bias and gyro drit are modeled as irst order Gaussian-Marko processes here. The details o A matri are gien below: where with & λ cos L L& & λ sin L ( ω + & λ) where cos L L& ( ω + & λ) sin L ( 2ω + & λ) cos L L& ( 2ω + & λ) sin L (3) (4) (5) y F j C (6a) y g g 2g F21 diag- - (6b) Rn Rn Rn [-ξ -ξ -ξ ] [ β β β ] F 44 diag (6c) F diag (6d) 55 in which ω is the arth rate ector; L and λ respectiely denotes latitude and longitude; y and are the measurements sensed by an accelerometer; g is the graity alue; R n represents the

3 Wang et al.: Inluence o ynamics and Trajectory on Integrated GPS/IS 111 radius o the parallel curature; ξ and β are 1/(correlation time) o accelerometer bias and gyro drit processes respectiely. 2.2 Coariance analysis Coariance analysis is a common tool to proide numerical time histories depicting the accuracy o a gien coniguration in terms o the coariance o its associated error state ector (Saage 2). Hence the analysis can be used to ealuate the perormance o the suboptimal ilter that operates in a real world enironment and can be utilied as a basic design tool during the synthesis and test o the suboptimal coniguration which is typically based on a simpliied error state dynamic/measurement model. Formulation o linear coariance equation with respect to a system or which eedback the optimally estimated state ector or control reset is (Saage 2; Maybeck 1979): P c k e ( + ) { I K H } P { I K H } + K RK T k T k k T (7) A coariance simulation was carried out to inestigate comprehensie GPS/SIS system obserability (e.g perormance) due to the act that the error coariance o Kalman ilter is one o the indices to check the degree o obserability (Ham and Brown 1983). A trajectory used in the simulation comprised our segments: a constant elocity manoeuring or 6 seconds an accelerating with.167 m/sec 2 or 1 seconds a 9 degree turning with angular elocity o 1.8 deg/sec and a constant elocity moing or 1 seconds. Figure 1 shows the coariance simulation results or twele states in the integration ilter ecept or three position states which are directly obserable rom the measurements. The igure indicates that heading error horiontal accelerometer bias and ertical gyro drit hae poor estimation perormance (e.g. poor obserability) during the irst constant elocity manoeuring when compared with other states. Howeer it can also be seen rom the igure that the estimation perormance o these our states is improed when the ehicle dynamics are changed (e.g. acceleration and 9 angle turn). where c denotes the application o control resets; e is the application o estimation resets; denotes the eedback matri In this study a suboptimal ilter consisting o 17-sates is designed to estimate the IS naigation and sensor errors and eedback them to the IS input (control reset) with respect to a designed real world system (true) model comprising 71 error states. ote that all the analyses will be conducted using a GPS/IS integration system based on GPS pseudo-range obserations and a tactical-grade SIS (5 deg/h 5µg). 3 Simulation Study In order to inestigate the inluence o ehicle dynamics and trajectory on the GPS/IS integration system perormance a serious o coariance simulation analyses are carried out. 3.1 ect o ehicle dynamics on ilter estimation The obserability o a linear system represents the possibility o determining the state ariables using the inormation on the input and the output o a system. One o the reasons or considering the obserability o a dynamic system is the need to determine the eiciency o a Kalman ilter that estimates the states o that system. Fig. 1 Coariance simulation results or our dynamic segments In order to study how dierent ehicle trajectories and dynamics aect ilter estimation perormance urther simulations analyses were conducted with respect to the states that hae poor obserability. In these analyses our dierent trajectories (deined as Circle Line Rectangle S-turn ) were considered. Their plane trajectories and elocity changes are depicted in Fig. 2. ote that the irst two segments (e.g. stationary mode or 3 seconds and accelerating with.45 m/sec 2 or 4 seconds) are commonly considered in all the trajectory generations then the ehicle moes according to the

4 112 Journal o Global Positioning Systems characteristics o each trajectory. Hence all the simulation analyses were carried out or 36 seconds. When taking a look at Figure 2 the characteristics o each trajectory can be described as: - Circle: the heading angle is continuously changed along clock-wise direction; - Line: moing with the constant heading and orth elocity (e.g. degree and 18 m/sec respectiely); - Rectangle: a 9 degree turn is made eery 5 seconds; - S-turn: the ehicle turns are continuously made along clock-wise and counter clock-wise directions. orthing (Km) 1.5 Circle Line Rectangle 2 Velocity (m/s) 2 Vn Ve pochs rom 1 to 8 pochs rom 8 to 36 Circle Line Rectangle S-turn Time (sec) Fig. 3 Coariance simulation results or the heading error state poch rom 1 to 8 pochs rom 1 to 36 Circle Line Rectangle S-turn S-turn asting (Km) Time (sec) Time (sec) Fig. 4 Coariance simulation results or the horiontal accelerometer bias (-ais) Fig. 2 Four dierent trajectories or the simulations Figs. 3 to 5 show the noise ariances o some crucial and less obserable states (i.e. heading error and horiontal accelerometer biases). In these igures the top graphs show the results rom epoch 1 to epoch 8 whereas the bottom graphs illustrate those rom epoch 81 to epoch 36. It is possible to draw the ollowing conclusions rom these results: (a) the ilter perormance can be improed by steady-turning ( Circle Rectangle S-turn cases) when compared with the Line (constant-elocity) (b) the quickest ilter initialisation o the three states is achieed with the S-turn ; c) the perormance o heading error estimation o the Circle Rectangle and S-turn becomes similar; and d) the Circle trajectory proides the best estimation perormance in the horiontal accelerometer biases pochs rom 1 to 8 Circle Line Rectangle S-turn pochs rom 8 to Time (sec) Fig. 5 Coariance simulation results or the horiontal accelerometer bias (y-ais) 3.2 aigation error behaior during a GPS blockage Further simulation analyses were carried out to study the eect o ehicle dynamics and trajectory on naigation

5 Wang et al.: Inluence o ynamics and Trajectory on Integrated GPS/IS 113 error estimation in the Kaman ilter during GPS outages. The scenarios or the two tests are as ollows (Fig. 6): - The ehicle remained in stationary mode or 6 seconds beore moing in circles within the same trajectory or 68 seconds ( Circle ); - The ehicle stayed in static mode or 6 seconds beore moing in circles within the same trajectory or 4 seconds then in a straight line at a constant speed or 28 seconds ( Line ). The GPS blockage was simulated or the last 28 seconds. It is also important to note that all the naigation and sensor error states in the ilter reach the steady-state condition beore GPS signals are blocked. Fig. 7 shows the coariance simulation results which indicate error behaiours o naigation parameters (e.g. position elocity and attitude). ote that Circular rror Probability (CFP) and Probable rror (P) in the igure represent horiontal and ertical positioning errors respectiely. These results indicate that all the naigation errors rapidly deteriorate according to the length o time that GPS is unaailable. Among these results the most interesting is the position and heading error behaiour since the dierence in the error increase between the two trajectories are relatiely large (e.g. around 15 meters in position and 1 arc-minutes in heading). Comparing these results those obtained rom the Line are superior to the Circle. This is opposite to the result obtained in the preceding section. Thereore it can be noted that the ehicle s lateral acceleration changes enhance the integration ilter s perormance when the ilter can be continuously updated by eternal measurements whereas it degrades the integrated GPS/IS naigation perormance during the GPS blockage. orth[m] S Line case ast[m] orth[m] S : Starting Point : nding Point GPS Blockage Circle case ast[m] S Fig. 7 rror behaiours o naigation parameters during the GPS blockage 4 Real ata Analysis 4.1 ata acquisition and processing To analyse the inluence o ehicle dynamics and trajectory on the integrated GPS/IS system perormance kinematic eperiments were carried out in Cloelly Bay Car park Sydney on the 24 th and 25 th o March 23. The IS used in this research was the Boeing C-MIGITS II system which is considered to be a tactical-leel accuracy unit (5 deg/h 5µg) two Lieca 5 GPS receiers were used at both the base and roer (ehicle) stations. uring the data acquisition raw IS and GPS measurements were recorded at 1H and 1H respectiely and there were 6 isible satellites (aboe the cut-o angle o 15 ). The C-MIGITS unctions as an integrated GPS (MicroTracker single board)/imu naigation system calculating a Kalman-iltered naigation solution in realtime. In this study the raw IS and GPS data rom the Leica 5 receier were processed using an in-house sotware package - the modiied ersion o the AIMS TM naigation processing sotware (Grejner-Breinska et al 1998a & b; Lee et al 22). Fig. 6 Trajectories used in the GPS blockage simulation

6 114 Journal o Global Positioning Systems 4.2 Vehicle ynamics Inluences in the Filter estimation horiontal acceleration and 2.5 arc-minutes in heading error estimation at the last epoch (see Figures 9 and 1). In order to study the inluence o ehicle dynamics and trajectory on the estimation o the error states that hae poor obserability (e.g. horiontal accelerometer biases and heading error) our eperiments were carried out with controlled-trajectories. Fig. 8 depicts the ehicle trajectories and dynamics during manoeuring. For conenience each o the trajectories is named as Circle Line Rectangle and S-turn. ote that the ertical dynamics (Vd) are ery low compared with those o the horiontal components (Vn and Ve). Fig. 9 RMS errors or horiontal accelerometer bias and heading error estimation Figure 8 Vechicle trajectories and dynamics during the tests Fig. 9 and 1 show the Root-Mean-Square (RMS) errors in horiontal accelerometer biases and heading error estimation indicating the dierent ehicle dynamic contributions to the Kalman ilter estimation procedure. The alues were obtained rom the diagonal components o the coariance matri. It can be seen rom these results that the ilter estimation precision is improed by steadyturn manoeures (i.e. Circle Rectangle and S-turn ) when compared with the constant-elocity manoeure (e.g. Line ). This improement can also be eriied rom the Rectangle results showing that the ilter precision is considerably increased when the ehicle makes its irst right angle turn. In addition the results in Figs. 9 and 1 show that the S-turn proides the best ilter estimation perormance among the our trajectories considered in these tests. This is due to reersing o the ehicle s lateral acceleration that occurs in the S-turn manoeure (Port and Bar-Ithack 1981). ote that the precision dierence between the S-TUR and LI cases is about 2 µ g in Fig. 1 Magniied results or heading RMS error in Fig aigation perormance during GPS blockage Two tests were conducted to inestigate the eect o ehicle dynamics on naigation error estimation in the Kalman ilter during a GPS outage. The scenarios or the two tests are as ollows (Fig. 11): - The ehicle remained in static mode or a duration o 6 seconds (initialisation) beore moing in circles within the same trajectory or 42 seconds; - The ehicle stayed in static mode or a duration o 6 seconds (initialisation) beore moing in circles within the same trajectory or 34 seconds then in a straight line at a constant speed or 8 seconds.

7 Wang et al.: Inluence o ynamics and Trajectory on Integrated GPS/IS 115 The GPS blockage was simulated or the last 8 seconds. Fig. 12 depicts test results showing error growth o naigation parameters during the GPS blockage. There is no abrupt error growth or the initial ew tens o seconds immediately ater the outage. This seems to be due to the naigation and sensor errors being well calibrated during the irst cycle with precise double-dierenced carrier phase measurements. Otherwise the results would be dierent rom those presented. As seen rom Fig. 12 the naigation parameter errors in the case o Circle increase more rapidly than those in the Line case during the simulated GPS blockage (i.e. stand-alone IS). Hence highlighting that the integration ilter perormance is strengthened when continuously updated by eternal GPS measurements. This may be due to the act that the equilibrant relationship among error parameters quickly deteriorates the leel o which is dependent on the magnitude o the ehicle dynamic change during the GPS blockage. 5. Concluding Remarks The impact o the ehicle trajectories and dynamics on the perormance o the integrated GPS/IS system has been inestigated in this paper through both coariance simulation and real data analyses. When the GPS measurements were used or all the data processing (no signal blockages) the results hae showed that (a) ehicle dynamics aect the Kalman ilter initialisation time and estimation perormance especially or the heading component; (b) the higher the dynamic changes in the lateral direction the shorter the initialisation time; (c) the S-turn shaped trajectory proided the quickest ilter initialisation o the our trajectories considered in these tests. Furthermore throughout these simulations the perormance o heading error estimation o the Circle Rectangle and S-turn becomes similar and the Circle trajectory proides the best perormance in the horiontal accelerometer biases. On the other hand when GPS signal blockage was simulated relatiely high dynamic changes degrade the system perormance; thus resulting in the rapid growth o the naigation errors. Acknowledgements The second author (HKL) is supported in his Ph research by a Scholarship unded by the Kwanjeong ducational Foundation o Korea. Reerences Fig. 11 Vehicle trajectories and elocities during the GPS blockage Fig. 12 rror growth o naigation parameter during the GPS blockage Bar-Ithac IY and Porat B (198) Aimuth obserability enhancement during Inertial aigation System in-light alignment AIAA Journal o Guidance Control & ynamics ol. 3 no Bar-Ithack IY and Berman (1988) Control theoretic approach to inertial naigation system. AIAA Journal o Guidance Control & ynamics ol a R Inestigation o a Low-Cost and High-Accuracy GPS/IMU System Proceedings o IO ational Technical Meeting Santa Monica Caliornia January Farrell RA and Barth M (1988) The Global Positioning System & Inertial aigation McGraw-Hill companies Yew York 34pp. Greenspan RL (1996) GPS and Inertial Integration Parkinson B and Spilker JJ (eds) Global Positioning System: Theory and Applications ol. 2 chapter 7 American Institute o Aeronautics and Astronautics Inc. Washington. Grejner-Breinska a R and Toth C (1998) GPS rror Modeling and OTF Ambiguity Resolution or High- Accuracy GPS/IS Integrated System Journal o Geodesy ol Grejner-Breinska a R & Toth C (1998) Positioning accuracy o the Airborne Integrated Mapping System

8 116 Journal o Global Positioning Systems Proceedings o IO ational Technical Meeting Long Beach Caliornia January Ham FM and Brown RG (1983) Obserability igenalues and Kalman iltering I Transactions on Aerospace and lectronic Systems ol. 19 no Hong S Lee MH Rios J and Speyer JL (2) Obserability Analysis o GPS Aided IS Proceeding o IO GPS September 2 Salt Lake City UT Kwon JH and Jekeli C (21) A ew Approach or Airborne Vector Graimetry Using GPS/IS Journal o Geodesy ol Lee HK Wang J and Rios C (22) Kinematic Positioning with an Integrated GPS/Pseudolite/IS Proceedings o 2nd Symp. on Geodesy or Geotechnical & Structural Applications Berlin Germany May Maybeck PS (1979) Stochastic Models stimation and Control : Volume 1 Academic Press Inc. 423pp. Porat B and Bar-Ithac IY (1981) ect o acceleration switching during IS in-light alignment AIAA Journal o Guidance Control & ynamics ol. 4 no Saage PG (2) Strapdown Analytics : Part II Strapdown Associates Inc. 65pp. Wang J. Lee H.K. and Rios C (23) GPS/IS integration: A perormance sensitiity analysis Wuhan Uniersity Journal o ature Sciences Vol. 8 pp