Differential GPS upported navigation for a mobile robot Jakob Raible, Michael Blaich, Oliver Bittel HTWG Kontanz - Univerity of Applied Science, Department of Computer Science, Brauneggertraße 55, D-78462 Kontanz, Germany {jraible,mblaich,bittel}@htwg-kontanz.de Abtract: The objective of thi work i the development of a low cot differential GPS ytem uited for mobile robotic application which enhance poitioning accuracy compared to a ingle receiver ytem. In order to keep cot minimal we ued ingle frequency (L1) receiver, namely U-Blox AEK-4T. We adapted the GPS Toolkit (GPSTk) to work with ingle frequency (L1) obervation in real-time. Thi allowed u to apply an already exiting algorithm, originally intended for Precie Point Poitioning (PPP) application uing a double frequency receiver. The core of thi algorithm i a Kalman filter that procee code and carrier phae ingle difference. Carrier phae ambiguitie are treated a real (float) value, we do not try to fix them to their correct integer value. In a tatic tet with a baeline length of 11m, obervation were collected for five minute. The developed ytem achieved a horizontal RMS of 6.9cm. Furthermore we carried out a dynamic tet where the rover drove around in a circle. Seven circle were driven in about five minute. The ytem determined the circle radiu with an RMS error of 13.2cm. Keyword: Differential GPS, DGPS, carrier phae, ingle difference, ingle frequency, Kalman filter, mobile robot 1. INTRODUCTION Many outdoor robotic application require poition that are more accurate than thoe obtained by a ingle GPS receiver. Exiting RTK-GPS ytem already provide accuracie in the ub-centimeter level. However, the geodetic grade double frequency receiver that are uually required for thee ytem are expenive. Other important factor a ize, weight or power conumption limit the poible form of application and the acceptance in the field of mobile robotic. Nowaday, inexpenive, mall, light and power aving device exit, but uually they do not provide raw obervation data, which i abolutely needed for any differential GPS (DGPS) olution. Some new generation module promie to bridge thi gap by combining the above mentioned factor while remaining inexpenive (U- Blox AEK-4T: AC295) and providing raw peudorange, carrier phae and Doppler obervation at an update rate of up to ten Hertz. Thi allow the ue of thee receiver in the context of a cientific application. Epecially the carrier phae obervation promie to enhance poitioning accuracy to a level which hould be ufficient for mot mobile robotic application. Compared to exiting RTK-GPS olution that uually ue double frequency receiver, the AEK- 4T repreent a ingle frequency receiver. Thi make the olution of the carrier phae ambiguity problem, involved in carrier phae baed DGPS application, more complicated. Already accomplihed reearch how that fixing the ambiguitie within reaonable time i a problem when inexpenive ingle frequency receiver and cheap antenna are ued (Liu et al. (2003), Pinchin et al. (2008)). Inexpenive receiver and antenna feature high noie level in the peudorange obervation, which i a problem even for very hort baeline. Odijk et al. (2007) examined the popular LAMBDA ambiguity reolving algorithm with a ingle frequency receiver. They conclude that intantaneou ambiguity reolution baed on ingle-frequency data i only ucceful with many (> 10) atellite. Takau and Yauda (2008) obtained a mean time to firt fix with ambiguity reolution of almot eleven minute for the U-Blox AEK-4T receiver with ANN-MS antenna. Note that loe of lock, which are likely to occur with inexpenive ingle frequency receiver under dynamic condition, require the reinitialization of the ambiguitie. Baed on thee reult, we decided to implement a float approach baed on Salazar et al. (2008). However, we work with peudorange and carrier phae ingle difference at the L1 frequency intead of linear combination of double frequency obervation a preented by Salazar, who worked with double frequency receiver. The olution preented in thi paper require an initialization phae of about four to five minute. During thi period, the poitioning accuracy increae. However, the ytem doe not need to be reinitialized after loe of lock or when new atellite are introduced in contrat to ingle frequency approache that fix the ambiguitie to integer number. In our cae, poitioning accuracy decreae temporary, but quickly reache an acceptable level when the receiving condition improve again. The tet preented in thi paper reult in a ub-decimeter horizontal RMS error in a tatic tet and a horizontal RMS error of about 13cm in a dynamic tet after an initialization phae of five minute in each cae.
2. DIFFERENTIAL GPS BASED ON CARRIER PHASE OBSERVATIONS The goal of a DGPS ytem i to enhance poitioning accuracy by uing two GPS receiver. Uually, one i tationary and it poition i exactly known. It i called reference- or bae tation. The econd receiver whoe poition i to be determined i called mobile receiver or rover. A DGPS ytem enhance the accuracy becaue the common-mode error (error common to both receiver) can be determined and eliminated when the two receiver operate in a limited geographic region. Thi i uually done by calculating the baeline vector (vector between mobile receiver and reference tation), where the common-mode error are canceled out. In order to determine the baeline vector, a claical DGPS ytem ue the coare/acquiition (C/A) code. By utilizing the carrier ignal, on which the C/A code i modulated, the poitioning accuracy can be enhanced again. 2.1 Carrier phae obervation Becaue the dicued method i baed on phae obervation of the L1 carrier ignal (1575.42MHz), a hort explanation of the baic of thi meaurement i provided here according to Odijk et al. (2007). A tandard GPS receiver compute it poition baed on range meaurement to the GPS atellite by applying the trilateration technique. Thee range meaurement are uually obtained by tracking the coare/acquiition (C/A) code. A the name ugget, thi code i rather coare becaue of it hort code length compared to a long chip rate ( 300km/ 300m). The ignal propagation delay i obtained by cro correlation between the received C/A code and a replica code generated by the receiver. The coare nature of the C/A code lead to range meaurement that are affected by high noie level. Epecially inexpenive receiver and antenna which are ued in thi work are affected by thi problem. But, in order to track the C/A code, the receiver need to track the carrier ignal a well. Thi i uually done via phae-locked loop (PLL) filter, which enable the receiver to compute the o called phae range. Becaue of the hort wavelength of the L1 ignal ( 19cm), thee range are very precie and characterized by low noie level. The problem i that the phae range are offet to the C/A code range by an ambiguou number of whole phae cycle. One can imagine thi problem when trying to read from a meaurement tape, but only a mall area i viible around the meauring point. For example, you would read 47,3cm but you do not know if it i 47,3cm or 147,3cm or even 1047,3cm. The phae range Φ [m] can be decribed mathematically a follow: where Φ(t) = ρ(t) + c (τ r (t) + τ (t)) + λ 1 N (1) ρ [m] true geometric range between atellite and receiver c [m/] peed of light τ r, [] receiver clock error τ, [] atellite clock error λ 1, [m] L1 wavelength ( 19cm) N R, [cycle] Carrier phae float ambiguity In fact, the carrier phae float ambiguity term N conit of three component: N = Φ (t 0 ) Φ r (t 0 ) + N (2) Where Φ (t 0 ) i the atellite offet in cycle at initial time t 0 and Φ r (t 0 ) the receiver offet, repectively. N N i the unknown initial number of whole carrier phae cycle between atellite and receiver. Becaue we do not try to fix N to an integer number, we tay with the term N a preented in Gao (2006). Once the ambiguity problem i completely olved, which mean that the exact integer number of whole phae cycle between atellite and receiver i known, accuracie in the ub-centimeter level can be reached. Such a ytem i mot commonly referred to a Real Time Kinematic GPS (RTK- GPS) or Carrier Phae Enhancement GPS (CPGPS). Uually, thee ytem ue geodetic grade double frequency receiver. Thi i needed in order to be able to fix the ambiguitie to the correct integer value within hort time. A tated in ection 1, the ambiguity reolution i not poible within hort time in our cae. Therefore, we float the ambiguitie in thi approach. Thi mean that the accuracy will not reach the ub-centimeter level but hould tay in the decimeter or centimeter level. A the double difference that are uually applied in RTK-GPS ytem achieve their full potential only when fixing the ambiguitie to integer value, which we are not doing here, we decided to work with ingle difference to implify matter. Thu, we do not have to elect a mater atellite which would introduce the hand-over problem and additional meaurement noie when obtaining the double difference. 2.2 Kalman filter deign Our approach i baed on the well-known Kalman filter. A good introduction to the filter i given in Welch and Bihop (1995). The deign of the filter that we ue for our approach i preented in the following ection. Sytem model The Kalman filter i trying to etimate the tate vector x t, which decribe the ytem. The tate vector of our ytem ha the following form: b τ m,r x t = N 1 m,r (3). N n m,r The ymbol b repreent the baeline vector change compared to the lat epoch. The baeline vector i given in a local north (N) eat (E) up (U) ytem. Thu, it conit of the three component N, E and U. Becaue we work with ingle difference, the combined receiver clock error τ m,r of the mobile (roving) receiver m and the reference (bae) tation receiver r doe not cancel out. Therefore, we have to etimate thi error. N i m,r are the float phae ambiguity etimate of the i-th atellite, which are received by both the mobile receiver and the reference tation.
The ytem tate for the current epoch t i predicted a follow: ˆx t = Aˆx t 1 (4) Where A i the ytem tate tranition matrix, ˆx t the etimate of the ytem tate x and ˆx t the predicted ytem tate, which i not yet updated by the meaurement. While we aume a white noie model for the change of the baeline vector b and for the combined receiver clock error τ m,r, the combined phae ambiguitie N are treated a contant. Beide the white noie model, we alo invetigated a random walk model for the change of the baeline vector. The initial phae ambiguitie are determined by ubtracting the code obervation from the carrier phae obervation. Thi i alo the cae when the ambiguitie need to be reinitialized due to cycle lip or complete loe of lock. Cycle lip detection i a problem for ingle frequency receiver. We compare the bia between code and phae obervation with a computed mean bia in order to decide whether a cycle lip occurred or not. Additionally, the lo of lock indicator (LLI), which i et by the receiver i ued to guide thi deciion. When a cycle lip i detected, the phae ambiguity N i not projected into the next epoch but it i initialized again, a mentioned above. Becaue the phae obervation are characterized by coniderably le noie compared to the code obervation, a weight factor i implemented to benefit from thi advantage. We weight the phae obervation 100 time higher, baed on the fact that we aume σ = 1m for code obervation and σ = 1cm for phae obervation. Matrix A depend on the model that i applied to decribe the poition component of the proce noie covariance matrix Q. If we apply the white noie model, the lat poition tate ˆx t 1 i not projected into the etimated new a priori tate ˆx t. Thu, the correponding element of A are 0: 0 0... 0 0 1... 0 A =...... (5) 0 0... 1 In contrat, if the random walk model i applied, the poition component of A are 1: 1 0... 0 0 1... 0 A =...... (6) 0 0... 1 Thi mean that the lat tate etimation i ued to predict the new a priori tate, which correpond to a walking rover that cannot jump in contrat to the white noie model. Meaurement model The meaurement model of our Kalman filter i decribed by the following equation: P i m,r = P i m P i r (7) = be i + cτ m,r (8) Φ i m,r = Φ i m Φ i r (9) = be i + cτ m,r + λ 1 N m,r (10) Where Pm,r i i the ingle differenced peudorange prefit reidual to atellite i, which i obtained by ubtracting the peudorange pre-fit reidual of the reference tation Pm i from the peudorange pre-fit reidual of the mobile receiver Pr. i Accordingly, Φ i m,r repreent the ingle differenced carrier phae pre-fit reidual. In our cae, the pre-fit reidual are obtained by ubtracting the computed range to the atellite from the according peudorange obervation. We calculate the range to the atellite baed on the computed poition of the mobile receiver in the previou epoch and ubtract it from the peudorange obervation of the current epoch. Thi equate to the movement b of the mobile receiver in the direction of the the line-of-ight (LOS) unit vector e i ince the lat epoch. The LOS vector point from the mobile receiver to the atellite. By forming ingle difference (between mobile receiver and reference tation), the common-mode error uch a the atellite clock error and ionopheric and tropopheric delay are canceled out. However, the combined receiver clock error τ m,r i till exitent in the reidual, a decribed by equation (8) and (10). The ingle differenced carrier phae reidual Φ i m,r additionally contain the differenced (between mobile receiver and reference tation) number N m,r of carrier phae cycle. The meaurement vector z t look a follow: P 1 m,r P 2 m,r. z t = P n m,r Φ 1 m,r Φ 2 m,r. Φ n m,r (11) The vector length depend on the number n of atellite imultaneouly received by the mobile receiver and the reference tation. The meaurement z t can be obtained from the ytem tate x t a follow: z t = H t x t (12) Where the filter meaurement matrix H t correpond to the GPS geometry matrix, which relate the ytem tate x t to the meaurement z t and look a follow: e 1 c 0 0... 0 e 2 c 0 0... 0........ e H t = n c 0 0... 0 e 1 (13) c λ 1 0... 0 e 2 c 0 λ 1... 0........ e n c 0 0... λ 1 Again, n denote the number of atellite received by both receiver. A in the meaurement vector z, the upper half i related to the ingle differenced peudorange pre-fit reidual and the lower half to the ingle differenced phae range pre-fit reidual. The one on the main diagonal in the lower right part apply the phae ambiguitie N.
Implementation We implemented our approach in C++ uing the GPS toolkit (GPSTk), an open ource library and uite of application for atellite navigation purpoe (Tolman et al. (2004)). Becaue the GPSTk i intended for pot-proceing application uing double frequency receiver, we had to implement an online converter, that tranlate the raw obervation from the proprietary U- Blox protocol to the RINEX baed clae contained in the GPSTk prior to the development of the application itelf. After that, we were able to modify an exiting cla which wa originally intended for the ue in Precie Point Poitioning (PPP) application which ue a ingle geodetic grade receiver, a preented in Salazar et al. (2008). Our development and teting etup conited of two Au Eee PC 901 running Ubuntu Eee, which i an Ubuntu 8.04 Linux derivative adapted to the Eee PC hardware. The communication link wa etablihed through a WLAN Ad-Hoc network. The rover PC wa running the application, acceing the bae tation receiver through ocat, a linux command line multi purpoe relay tool. We were alo able to etablih a link uing a mobile phone which enable our ytem to be ued for application that involve far range between bae tation and rover. The bae tation PC wa connected to the Internet via conventional broadband acce. The rover PC etablihed a dial-up GPRS connection via Bluetooth and acceed the bae tation receiver through ocat, a explained before. 3. EXPERIMENTAL EVALUATION We carried out a tatic and a dynamic tet to determine the poitioning accuracy of our DGPS application. 3.1 Static tet Firt, we intalled a tatic tet etup to determine the poitioning accuracy of the dicued ytem. For thi purpoe, a parking lot in the indutrial area of Contance, Germany wa choen. While only urrounded by humble building and mall tree, it offer good receiving condition. The line of the parking lot and, for the orientation Google Earth, helped u to determine the reference poition of the mobile receiver with a meauring tape: Eating: 11.0m Northing: -0.05m It hould be tated here, that an error of approximately five centimeter i poible when uing thi method for determining the reference poition. We ued the active patch antenna U-Blox ANN-MS that hip with the AEK-4T evaluation kit. Becaue thee antenna work better when intalled on a metal plate, we put the antenna on a 25cm x 25cm ordinary teel plate in both tet. The raw obervation update rate wa et to 10Hz, which i the maximum of the AEK-4T receiver. We collected obervation for ten minute, thi correpond to 6000 epoch. During thi tet, ix atellite were received by both the reference tation and the mobile receiver. The GDOP (geometric dilution of preciion) value ranged between 3.12 at the beginning and 2.95 at the end. Becaue we are not intereted in the calculated height information, we do not preent thi information here. The tatitical reult only contain horizontal information, RMSE i therefore the horizontal RMS error. Becaue our approach focue on mobile robotic application, we modeled the proce noie Q t of the Kalman filter a random walk with different model for horizontal and vertical movement. We choe a proce pectral denity σ 2 of 5 m2 for horizontal movement and 1 m2 for vertical movement. The maller value for vertical movement wa choen baed on the fact that a mobile robot uually drive on flat ground, meaning mall vertical movement. We alo invetigated a white noie model with σ = 100m. Thi correpond to a fully kinematic ytem, alo uitable for airborne application. The reult achieved with thi model are almot equal to the random walk approach, which performed lightly better. The poitioning error over time uing the random walk model i preented in figure 1 eating error (blue), northing error (red) [m] 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 1000 2000 3000 4000 5000 6000 epoch (update rate 10Hz) Fig. 1. Static tet: Eating (blue) / Northing (red) error in meter over ten minute of tatic obervation at an update rate of ten Hertz. Proce error Q t a random walk with horizontal σ2 = 5 m2 and vertical σ 2. The tatitical reult are preented in table 1. Table 1. Statitical reult of tatic tet Eating Northing Without initialization phae (Epoch 0-6000) RMSE [m] 0.110 0.298 STD [m] 0.102 0.282 Minimum [m] 10.719-0.421 Maximum [m] 11.240 0.873 With initialization phae (Epoch 3001-6000) RMSE [m] 0.037 0.044 STD [m] 0.037 0.044 Minimum [m] 10.949-0.141 Maximum [m] 11.063 0.002 3.2 Dynamic tet For the dynamic tet we attached the rover, a radio controlled model car, to a fix pile uing a leah. The rover wa able to move in a circle with a known radiu. The bae tation antenna wa mounted on the pile at the center of the circle. A radiu of 5m wa meaured uing a meaurement tape. Thu, the calculated baeline length hould alway match thi radiu. Becaue we were only intereted in the two-dimenional poitioning accuracy, we calculated the euclidean ditance d a follow: d = E 2 + N 2 (14)
Where E i the eating component and N the northing component of the baeline vector obtained by the preented DGPS ytem. During thi tet, 5740 epoch were collected at an update rate of 10Hz. The rover wa tanding till for four minute at the beginning. Thi i regarded a initialization phae. Thu, the reult of the firt 2400 epoch are not incorporated in the tatitical reult. During the remaining 3340 epoch, the rover wa moving in circle at a roughly contant velocity. Seven complete circle were accomplihed during thi time which yield to an average of 48 econd per circle. In thi tet, we modeled the proce noie Q t a white noie with σ = 100m. Again, we obtained almot the ame reult uing the random walk model with horizontal σ2 = 5 m2 σ and vertical 2. Figure 2 how the computed track. receiving condition, too. For thi purpoe, the HTWG Kontanz campu wa choen a teting ite. It i urrounded by high building and many tree are cloe to the choen track which i depicted in figure 3. The ame radio controlled model car that wa ued in the dynamic tet wa employed here. Fig. 3. HTWG Kontanz campu with the etimated reference track in red (ource: Microoft Bing Map TM ) Fig. 2. Dynamic tet: Poition plot of the dynamic tet (all 7 circle) The tatitical reult are preented in table 2. Table 2. Statitical reult of dynamic tet Becaue we did not have a reference ytem to obtain the rover true poition, we can only preent an etimation of the driven track. Firt, we applied the generic white noie model with σ = 100m a proce error. Figure 4 how the poition jump that occur in the eatern part of the campu, where the nearby building obcure the view to mot of the received atellite. Many cycle lip and complete loe of lock lead to reinitialization of the ambiguitie N with the peudorange obervation (ee ection 2.2.1). The peudorange obervation are biaed by heavy multipath effect in thi cae, reulting in evere jump in the calculated poition. All 7 circle Lat circle Mean [m] 5.009 5.011 RMSE [m] 0.132 0.084 STD [m] 0.131 0.084 Minimum [m] 4.712 4.876 Maximum [m] 5.271 5.134 Additionally, we determined the center of the circle through circle fitting in a leat-quare ene according to Gander et al. (1994). The reult are preented in table 3. Table 3. Circle center obtained through circle fitting All 7 circle Lat circle Eating [m] 0.097 0.114 Northing [m] 0.131 0.037 3.3 Challenging condition Becaue perfect receiving condition cannot be aumed for every outdoor robotic application, the performance of the preented ytem wa invetigated under difficult Fig. 4. Google Earth TM poition plot of HTWG campu tet. Q t a white noie with σ = 100m. Red: etimated reference track; green: computed poition
In order to reduce the jump in the poition output, we applied a different model for the proce error Q t. Becaue the radio controlled model car nearly doe not move vertically, it make ene to aign a mall σ value to vertical movement. Furthermore, horizontal movement are limited to the maximum velocity of the RC car. The reult wa a random walk model with proce pectral denity σ2 for the horizontal component and σ 2 = 0.05 m2 for the vertical component. A expected, thi limit the poition jump becaue the Kalman filter now etimate lower movement and weight the erroneou obervation le than before. Thi can be een in figure 5. Fig. 5. Google Earth TM poition plot of HTWG campu tet. Q t a random walk with horizontal σ2 and vertical σ2 = 0.05 m2. Red: etimated reference track; green: computed poition The downide of thi modification i, however, that the random walk model need to be tuned carefully to the kinematic characteritic of the rover. If the σ2 value are too mall, the poition tend to drift away and do not converge to the true poition anymore. On the other hand, too big value do not limit the poition jump ufficiently. A enor fuion with the robot odometer enor could olve thi problem in a better way. The odometer information could be ued a control input to the Kalman filter. 4. CONCLUSION In thi paper we preented the development of a differential GPS ytem for the ue in mobile robotic environment. In contrat to already exiting RTK olution which ue expenive geodetic grade double frequency receiver, our goal wa to keep cot minimal. By utilizing the carrier phae obervation and benefiting from hort baeline, the developed approach provide a poitioning accuracy that hould be ufficient for many deirable application. Problem occurred under bad receiving condition, for example when high building caued cycle lip or complete loe of lock. We were able to reduce the reulting jump in the poition output partially by modeling the proce noie a random walk intead of white noie. Thereby, we applied different model for horizontal and vertical movement and adjuted them to the kinematic characteritic of our robot. By fuing the robot odometer enor with the Kalman filter preented in thi paper, we expect an effective reduction of thee jump. Finally, only more high quality enor information allow to enhance accuracy. We will invetigate thi approach in future work. ACKNOWLEDGEMENTS We want to thank Dagoberto Salazar for guiding u in the right direction and helping u with quetion concerning the GPSTk. REFERENCES Gander, W., Golub, G.H., and Strebel, R. (1994). Leatquare fitting of circle and ellipe. BIT, 43. Gao, Y. (2006). What i precie point poitioning (ppp), and what are it requirement, advantage and challenge? InideGNSS, 1(8), 16 18. Liu, J., Cannon, M.E., Alve, P., Petovello, M.G., Lachapelle, G., MacGougan, G., and degroot, L. (2003). A performance comparion of ingle and dual frequency GPS ambiguity reolution trategie. GPS Solution, 7(2), 87 100. Odijk, D., Traugott, J., Sach, G., Montenbruck, O., and Tiberiu, C. (2007). Two approache to precie kinematic gp poitioning with miniaturized l1 re ceiver. In Proceeding of ION GNSS 20th International Technical Meeting of the Satellite Diviion, 827 838. The Intitute of Navigation, Fort Worth, TX. Pinchin, J., Hide, C., Park, D., and XiaoQi, C. (2008). Precie kinematic poitioning uing ingle frequency GPS receiver and an integer ambiguity contraint. In Poition, Location and Navigation Sympoium, 2008 IEEE/ION, 600 605. Salazar, D., Hernandez-Pajare, M., Juan, J., and Sanz, J. (2008). High accuracy poitioning uing carrier-phae with the open ource GPSTk oftware. In Proceeding of the 4th. ESA Workhop on Satellite Navigation Uer Equipment Technologie. NAVITEC 2008, Noordwijk, The Netherland. Takau, T. and Yauda, A. (2008). Evaluation of RTK- GPS Performance with Low-cot Single-frequency GPS Receiver. In Proceeding of International Sympoium on GPS/GNSS 2008. Tokyo, Japan. Tolman, B., Harri, R.B., Gauiran, T., Munton, D., Little, J., Mach, R., Nelen, S., and Renfro, B. (2004). The GPS Toolkit: Open Source GPS Software. In Proceeding of the 16th International Technical Meeting of the Satellite Diviion of the Intitute of Navigation. Long Beach, California. Welch, G. and Bihop, G. (1995). An Introduction to the Kalman Filter. Technical report, Chapel Hill, NC, USA.