Heuristic Reduction of Gyro Drift in Gyro-based Vehicle Tracking

Transcription

1 Presented at the SPIE Defense, Securty + Sensng, Conference 7305: Sensors, and Command, Control, Communcatons, and Intellgent (C3I) Technologes for Homeland Securty and Homeland Defense VIII, Orlando, FL, Aprl Reference: Proc. SPIE Vol Heurstc Reducton of Gyro Drft n Gyro-based Vehcle Trackng Johann Borensten and Lauro Ojeda The Unversty of Mchgan, 2260 Hayward Street, Ann Arbor MI 4809, USA johannb@umch.edu, lojeda@umch.edu ABSTRACT Ths paper pertans to the reducton of measurement errors due to drft n rate gyros used for trackng the poston of movng vehcles. In these applcatons, gyros and odometry are often used to augment GPS when GPS recepton s unavalable. Drft n gyros causes the unbounded growth of errors n the estmaton of headng, renderng low-cost gyros almost entrely useless n applcatons that requre good accuracy for more than just a few seconds or utes. Our proposed method, called Heurstc Drft Reducton (HDR), apples a unque closed-loop control system approach to estmate drft n real-tme and remove the estmated drft nstantaneously from the gyro readng. The paper presents results of experments, n whch a gyro-equpped car was drven hundreds of mles on hghways, rural roads, and cty streets. HDR reduced the average headng error over all of these drves by one order of magntude. Keywords: IMU, gyro, drft, vehcle trackng, car, fleet, heurstc, poston, estmaton INTRODUCTION For almost all outdoor land navgaton tasks, GPS s the obvous sensor of choce. However, t s well known that dense tree coverage or so-called urban canyons can occlude satellte sgnals. In mltary applcatons ntentonal jamg of GPS rado sgnals s also a concern. For these reasons, addtonal sensor modaltes, such as Inertal Measurement Unts (IMUs) should be ntegrated wth GPS to offer poston estmates when GPS s unavalable. Indoors or underground, GPS s unavalable altogether, makng good nertal sensng even more crtcal for vehcle trackng applcatons. In ths paper we propose an nnovatve method for mprovng the accuracy of gyro-based headng estmaton. For smplcty, we assumed that the vehcle s movng on near-horzontal terran for most of the tme and therefore we used a sngle one-axs gyro to measure the yaw-rate, from whch relatve changes n headng can be computed. Of course, the proposed method can be appled to 6-axs IMUs, just as well. The output of a gyro s rate-of-turn, ω. In vehcle trackng applcatons, one s usually nterested n headng, whch can be computed from ω by ntegratng the output sgnal numercally. The numerc ntegraton has a tendency to cause errors due to bas nstablty, more commonly referred to as drft. Drft s produced when small, slow-changng devatons from the correct sgnal are ntegrated wth respect to tme. The hghly undesrable result of drft s that the error of the computed output relatve headng ncreases contnuously and wthout bound. Gyros are also senstve to changes n temperature, whch ncur slow-changng devatons, just lke drft does. Our proposed drft reducton method counteracts all slow-changng errors regardless of whether they were caused by the physcal phenomena of drft or temperature senstvty. For that reason, throughout ths paper, we lump all slowchangng error sources together and call them collectvely drft. Yet another source of errors n computng headng s hgh-frequency nose n the gyro s output sgnal. The accumulated error due to the ntegraton of ths sgnal s called Angle Random Walk (ARW). However, n the computaton of headng from the gyro s rate of turn measurements, errors due to ARW are relatvely small, snce the ARW s average s about zero. A more comprehensve verson of ths paper was accepted for publcaton n the Internatonal Journal of Vehcle Informaton and Communcaton Systems 9.

2 Before explanng our proposed drft reducton method n detal n Secton 2, we dscuss here brefly several others approaches. The most common method for reducng the effects of gyro drft s to ntegrate data from an Inertal Measurement Unt (IMU) and GPS 2,3,4. Another method ntegrates data from a magnetc compass as well 5. The man drawback of these approaches s that they requre GPS and/or magnetometer data, whch are not avalable all the tme. Basnayake et al. proposed a method that makes use of avalable maps and uses map matchng technques for further enhancement. Some other methods have been proposed that try to mprove accuracy by fndng a mathematcal model for bas errors 6,7. However, these technques have lmted applcablty and can only estmate the deterstc part, f any, of the bas drft. Our proposed method, called Heurstc Drft Reducton (HDR), explots the fact that streets, roads, and hghways are man-made and straght over sgnfcant portons. Ths s partcularly true n downtown areas Urban Canyons are a major source of GPS outages. At any moment, the HDR method estmates the lkelhood that the vehcle s drvng along a straght lne. If that lkelhood s hgh, HDR apples a correcton to the gyro output that would result n a reducton of drft f ndeed the vehcle was drvng along a straght lne. If HDR assesses that the vehcle s not travelng straght, then t does nothng. Ths way, and on an abstract level, HDR uses landmarks (.e., man-made straght-lne features), but there s no requrement that the locaton of these landmarks be known n advance. The remander of ths paper s organzed as follows. Secton 2 explans the basc HDR method n the context of deal condtons. Secton 4 dscusses how real condtons dffer from deal condtons, and provdes several enhancements to the HDR algorthm to help cope wth real condtons. Secton 5 shows expermental results obtaned wth HDR appled to a gyro n a vehcle trackng system. Conclusons are presented n Secton 5. 2 HEURISTIC DRIFT REDUCTION Suppose a vehcle s drvng straght forward. Movng straght forward, the output of the gyro should be exactly zero throughout the trp. However, due to drft the actual output s off by some small value, ε. Suppose further that we dvded the total travel dstance nto one-second ntervals. Due to the drft error ε, n each nterval the rate of rotaton computed based on the gyro measurements s thus ω raw, = ω true, + ε 0 + ε d, () ω raw, Rate of rotaton measurement. Ths s the drect output of the gyro. ω true, True rate of rotaton. In realty ω true s not known, but n our dealzed straght-lne example ω true = 0. ε 0 Statc bas drft, measured mmedately pror to a drve. ε d, Drft (.e., bas nstablty). Immedately pror to each drve and wth the gyro held completely motonless, the statc bas drft ε 0 s measured by averagng T bas = 30 seconds worth of gyro output. The value for T bas depends on the qualty of the gyro and can be estmated by usng the Allan Varance analyss 8. T bas s also called bas tme. Durng the drve, ε 0 s subtracted from every readng of ω raw, : ω raw, = ω raw, - ε 0 = ω true, + ε d, (2) Then, the new headng ψ s computed ψ = ψ - + ω raw, T (3) ψ New computed headng [ ]. Duraton of tme nterval [sec]. Throughout ths paper, T = sec. T 2

3 3 THE BASIC HEURISTIC ASSUMPTION Durng straght-lne travel, f ε d, s postve, then the perceved change of headng n ths nterval s counterclockwse (we wll call ths a left turn, for smplcty) and f ε d, s negatve, then the perceved change of headng s clockwse or a rght turn. It s unpredctable whether ε d wll be postve or negatve and ε d may change sgns durng a drve. However, f ε d changed sgns very often durng a drve, then drft errors wll partally cancel each other out and the overall error s less severe. Our greater concern s thus for cases ε d keeps the same sgn for prolonged perods of tme and thereby accrues headng errors n the same drecton. For the sake of smplcty, we assume n the remander of ths secton that ε d keeps the same sgn throughout the drve. In practce, however, ths s not a necessary requrement for the HDR algorthm and ε d may change sgns. If ε d keeps the same sgn throughout the drve, then n each nterval T the headng error wll have the same drecton regardless of the unpredctable and ever-changng value of ε d. For a straght-lne drve of,000 seconds and assug ε d s postve, there are,000 ntervals, n whch the vehcle trackng system erroneously perceved that t had turned left (due to the postve drft assumed n ths example), and zero ntervals, n whch t erroneously perceved t had turned rght. Nose n the gyro output can blur ths sharp rato somewhat, but as long as the average of the nose s zero there wll stll be a sgnfcantly larger count of perceved left turns than rght turns. Moreover, once the absolute drft value exceeds the magntude of the nose, nose wll no longer have any nfluence on the left turn/rght turn count. More mportantly, even a sharp turn wll affect ths countng scheme only brefly, namely for the duraton of the turn. Once the turn s over, there wll agan be many more perceved left turns, due to the postve drft. Next we wll ntroduce a practcal method that can effectvely estmate the magntude of drft based on the proceedng observatons. 3. The basc HDR algorthm In order to explan the basc HDR algorthm, we contnue to make the smplfyng assumpton that when a vehcle drves, t moves along straght lnes at least some of the tme. In a later secton we wll ntroduce enhancements to the basc HDR method that allow us to drop the smplfyng assumpton of ths secton and deal effectvely wth all realstc condtons. The basc HDR algorthm functons essentally lke a closedloop control system. Ths s dfferent from most other measurng systems, sgnals pass from the sensor to the nstrument s output n open-loop fashon. Fgure shows a block dagram of the closed-loop control system mplemented n the HDR algorthm. Our above-stated smplfyng assumpton means that ω true = 0 s correct at least some of the tme. When ω true = 0, then the only output from the gyro (after subtractng the statc bas drft ε 0 ) s ε d. For our closed-loop control system, ε d s a dsturbance. Because of the I-controller, and provded the control parameters are properly chosen, the control sgnal I wll track (but wth an opposte sgn) slow changes of ε d wth no offset. That s, I -ε d under deal condtons. ω set = 0 + Bnary I-controller Gyro (adds ε 0 +ε d ) ω true +ε 0 +ε d + The deal condton ω true = 0 s, of course, rarely met. Indeed, ω true can brefly be orders of magntude larger that ε d, for example, when the vehcle takes a turn. In that case a conventonal I-controller does not work well, snce t would respond strongly to large values of ω true, thereby overwhelg the ntegrator n the I-controller. To avod ths ptfall, the I-controller should be nsenstve to the magntude of the error sgnal E. Ths can be acheved by treatng the error sgnal E as a bnary sgnal that can have only one of two values: postve or negatve. Ths way, the ntegrator I reflects the dfference between the numbers of perceved left and rght turns, as explaned n the proceedng secton. For the mplementaton of the algorthm we should note that snce the setpont ω set s permanently set to zero, the followng s true: - E ω - Z - ε 0 - ω true I + + ω true +ε d ω Fgure : The basc HDR algorthm vewed as a closed loop control system. The bnary I-controller s explaned n the text. 3

4 When ω - > 0 (a perceved left turn), E s negatve When ω - < 0 (a perceved left turn), E s postve We can now formulate the bnary I-controller I c for ω > 0 (a perceved left turn) I = (4a) I + c for ω < 0 (a perceved rght turn) and I = I - for ω - = 0 c Fxed ncrement [ /sec] An alternatve way of wrtng Eq. (4) s I = I ) (4b) SIGN( ω c SIGN() s a programg functon that deteres the sgn of a number. SIGN returns f the number s postve, 0 f the number equals zero, and - f the number s negatve. The next element n the control loop adds the controller output to the raw measurement ω = ω true + ε d + I (5) ω Corrected rate of rotaton [ /sec]. If I -ε d, as we assume for now to be the case under deal condtons, n steady state, and because of the closed loop control system, then by substtutng I -ε d n Eq. (5) ω ω true, (6) Ths result s desrable, snce the unknown drft s removed. 4 REALISTIC CONDITIONS In order to explan the basc HDR algorthm we assumed that much of the drvng was along perfectly straght lnes. Durng deal straght-lne moton, any perceved rght or left turn s ndeed the result of drft. In realty, however, vehcles don t drve exactly straght. In Secton 4. we dscuss typcal detractons from deal straght-lne drvng, such as swayng, curvng, and turnng. Then, n Secton 4.2, we present enhancements to the basc HDR algorthm, amed at counteractng these detractons. 4. Detractons from deal straght-lne moton 4.. Swayng We call moton that s ntended to be straght but s not entrely straght swayng. Swayng s the result of ute steerng correctons that a human drver performs ntutvely n order to stay near the center of a traffc lane. An example for swayng s shown n Fgure 2. Errors due to swayng can be reduced sgnfcantly by low-pass flterng, as wll be explaned n Secton Curvng Curvng s the moton along an extended arc. Ths moton s potentally the least favorable one for the HDR method. That s because extended drvng along an arc wth a very large radus wll be perceved contnuously as a turn n the same drecton and counteracted just lke drft. In the worst case, f the arc curves, say, to the left, whle the actual drft s 4

5 to the rght, then the HDR algorthm wll try to counteract the perceved left turns and ncrease I, thereby actually ncreasng the error caused by drft alone. Ths s a partcular concern wth or roads that follow natural terran and consst manly of curvng segments. In those cases, the basc HDR wll not work well. Beng aware of ths lmtaton, we developed a method for detectng curvng moton. Ths method, descrbed n detal n Secton 4.2.3, gradually reduce the effect of HDR as long as the curvng moton perssts, so as to avod the ntroducton of greater errors. The effectveness of ths measure wll become apparent n the Suburbs experment lsted n Secton 5. In that experment we ntentonally drove along a route that comprsed mostly of the curvng streets found n many typcal suburban subdvsons n the Unted States. Even then HDR reduced headng errors by a factor of 3.4 (see Secton 5). Ths mprovement was possble because even curvng roads have short straght-lne sectons, and HDR tracks drft whenever the vehcle s on a straght lne secton Turnng Turnng s a sharp but short change of drecton. Examples are street corners or ntersectons of rural roads. Turns are easly dentfed because the gyro measures much larger rates of rotaton durng turnng than what can be expected as a result from drft. Thus, a smple test can be performed n software: f ω s larger than some threshold, then the vehcle s turnng and Eq. 4 can be skpped altogether. 4.2 Refnements to the basc HDR algorthm In ths secton we dscuss several enhancements to the basc HDR algorthm. Together, these enhancements overcome the two man challenges for the HDR method under real condtons: swayng and curvng Low-pass flter Passng raw gyro readngs through a low-pass flter s normally not necessary for estmatng headng, snce the act of ntegratng the measured rate-of-turn to estmate headng acts by tself as a low-pass flter. However, n the HDR system a low-pass flter for smoothng the nosy gyro sgnal has a dramatc effect, snce HDR acts on each ndvdual gyro readng pror to the numerc ntegraton. The low-pass flter s mplemented n software ω' raw, T + τω' ω' = (7) T + τ T Samplng tme ω Low-pass fltered rate of turn τ Low-pass flter tme constant Swayng Curvng Turnng Fgure 2: Three types of non-straght moton: swayng, curvng, and turnng Turn swtch Another enhancement to the basc HDR system, albet a trval one, s what we call the Turn Swtch. When the vehcle takes a sharp turn, then the absolute value of the gyro-measured rate of turn, ω, spkes up to values that can be one or two orders of magntude larger than any change n ω due to drft. To prevent large rates of turn from affectng the Integrator I, a smple bnary threshold can be used to dstngush between large ω due to turnng and small ω that mght be due to drft. If a readng of ω exceeds that threshold, then the algorthm reduces the value of c to zero, for as long as ω remans above the threshold. Ths measure effectvely suspends the HDR algorthm, snce HDR keeps ncrementng I by zero untl the condton s resolved. By dong so, HDR prevents tself from modfyng I n response to sgnals that are caused by true turnng, not by drft. 5

6 0 for ω- < Θw W = (8) for ω- Θw W Turn swtch Turn threshold Θ w Repetton attenuator Whle sharp turns are easy to dentfy and handle by the Turn Swtch, a challenge for the HDR algorthm s the dffculty of dstngushng between the effect of curvng (e.g., drvng along a slghtly curved road) and drft. To address ths problem, we make use of the observaton that straght-lne moton, -I properly tracks drft, and curvng, whch msleads the HDR algorthm, can be dstngushed by lookng at the number of tme ntervals durng whch ω retans ts sgn. Ths s because at steady state, durng straght lne travel, ω tends to oscllate about zero as -I oscllates about drft. In contrast, durng curvng, ω retans ts sgn, because -I chases ω. Wth that dstngushng characterstc n d, we can now mplement the thrd enhancement for the HDR algorthm: the Repetton Attenuator, R. The task of R s to gradually reduce the value of the ncrement c, for every nterval n whch the sgn of ω remaned unchanged. For that, we frst defne a repetton counter, r. r s ncremented by for every teraton, n whch the sgn of ω - remans unchanged from that of ω -2 (note that snce r s used to compute ω we have to go back two teratons, to ω -2, to see f the sgn has changed). r + for SIGN( ω ) = SIGN( ω 2 ) r = (9) for SIGN( ω ) SIGN( ω 2 ) Then, the orgnal ncrement c s reduced n some nverse proporton to r. In our system we created ths nverse proporton by ths functon: + c R = (0) + c c2 r c Repetton Attenuator constant c 2 Repetton Attenuator power Equaton (0) allows the shapng of an attenuaton curve by adjustng the parameters c and c 2. A more detaled dscusson of the repetton attenuator enhancement, as well as a detaled descrpton of the combned mplementaton of all three enhancements s gven n our journal paper 9. 5 EXPERIMENTAL RESULTS In order to evaluate the effectveness of HDR for vehcle trackng, we mounted a sngle-axs rate gyro on the floor of a small sports utlty vehcle (SUV), a Subaru Forester. The gyro s the CRS03-04 made by Slcon Sensng 0, shown n Fgure 3. Table I lsts some of the key specfcaton for the CRS In all experments we sampled the gyro data at a rate of 0 Hz, whle drvng. Smultaneously we logged GPS data to serve as ground truth. Snce the GPS data was sampled at Hz, we down-sampled the Gyro data to Hz by averagng every 0 samples 0 Hz 0Hz ω = ω0 + j for =, 2. n () 0 j = Consequently, all expermental results n ths paper were obtaned from computatons performed on the bass of a samplng tme of T = sec. Table I: Key specfcatons for the Slcon Sensng CRS03-04 rate gyro. Sze Bandwdth Dynamc range mm 0 Hz ±200 deg/sec <. deg/ Bas drft (66 deg/hr) Approxmate cost 350 USD Fgure 3: The Slcon Sensng CRS03-04 sngle-axs rate gyro. 6

7 5. A typcal drve experment The noal route of a typcal drve experment wth HDR s depcted n Fgure 4. Ths drve, called the Mxed Drve, comprsed of 8 km of rural roads (from Start to A ),.5 km of hghway (from A to B ), and 3.75 km of cty streets (from B to Stop ). The route was purposefully desgned to nclude stretches of hghway that are curvng, not just straght segments. The total route was 23.6 km long and took ~20 utes to drve. Fgure 5 shows a plot of headng versus tme for the Mxed Drve experment. The thck blue curve represents ground truth headng computed from GPS data. Because of the varance of the GPS updates, headng estmaton at slow speeds was often very nosy. Whenever that was the case, we smoothed the nosy GPS data manually. The dotted, red curve shows headng as estmated from uncorrected gyro data, whle the thn green curve shows headng estmated wth HDR-corrected data. Note how closely the HDR-corrected curve follows the ground truth data. Quanttatve results for ths drve and others are gven n the followng secton. 5.2 Multple drve experments It s qute easy to adjust the tunable parameters of the HDR algorthm so as to provde excellent performance for a sngle, specfc drve experment, n postprocessng. In practce, however, parameters tuned for optmal performance n one experment wll almost certanly not produce optmal performance n another one. The only practcal way for tunng parameters s to collect data from a large number of experments and observe how a sngle set of parameters performs when appled to all avalable data sets. Parameters tuned ths way wll not provde optmal performance for any ndvdual expermental drve, but they are far more lkely to produce acceptable results wth any future drve n a producton vehcle trackng system, re-tunng the parameters s out of the queston. One condton for ths to be true s that the so-called tranng data sets be based on a wde range of dfferent drve condtons. Wth these consderatons n d, we performed nne dfferent drve experments under a wde range of drvng condtons. Specfcally, the drves ncluded hghways, rural roads, and cty streets, and they vared between 3 and 52 utes n duraton and between and 9 km n dstance. Each experment started by measurng the statc bas drft, ε 0, whle the vehcle was standng stll for about 20 seconds. In order to express the results of these experments quanttatvely, we use two metrcs: the Average Headng Error, E ψ, and the Normalzed Average Headng Error, NE ψ, descrbed next. Headng [deg] Absolute headng from GPS Uncorrected headng HDR-corrected headng Fgure 4: The noal route for the Mxed Drve experment Tme [] Fgure 5: A plot of headng data from the Mxed Drve experment of Secton 5.. The thck blue curve shows ground truth, derved from GPS data. The dotted red curve shows headng estmated from gyro data, wthout HDR. The dashed green curve shows headng estmated after applyng the HDR algorthm. 7

8 5.2. The average headng error (E ψ ) Ths metrcs s useful for judgng the accuracy acheved n a sngle drve. In order to compute E ψ we frst computed the momentary headng error by comparng the gyro-derved estmated headng wth the GPS-derved ground truth headng n each samplng nterval (.e., once every second). Then, the averagng of all momentary headng errors yelds the Average Headng Error: E Ψ = n n = ψ ψ, Gyro, GPS (2) E ψ Average Headng Error n degrees. Index for the nterval number wthn each drve. n Total number of samples n the drve (the effectve samplng frequency was Hz). ψ,gyro Gyro-derved headng n nterval, n degrees. ψ,gps GPS-derved headng n nterval, n degrees The normalzed average headng error (NE ψ ) Snce headng errors due to drft can ncrease wthout bound, one can expect to see only small momentary headng errors shortly after startng a drve, and ncreasngly larger momentary headng errors later nto the drve. Consequently, one can expect the Average Headng Error to be smaller for short-duraton drves and larger for long-duraton drves. In order to be able to compare the performance between short and long drves, we defne a second metrcs, the Normalzed Average Headng Error per ute of travel tme, NE ψ, whch s useful for comparng the accuracy acheved n multple drves of dfferent duratons. NE ψ s measured n degrees per ute. E NE = (3) Ψ Ψ T d T d duraton of the drve n utes. Table II lsts relevant detals and results for all nne drves. A graphcal representaton of these results s shown n Fgure 6. For each drve we specfy both E ψ and NE ψ. Average Headng Error [degrees] No correcton HDR-corrected 84 o Cty Suburbs Mxed Rural Rural 2 Hghway Hghway 2 Hghway mx Cty 2 Averages: Drve Name (a) We should emphasze that a sngle set of parameters was used for all drves, ψ ; (b) Normalzed Average Headng Errors, NE ψ. Fgure 6: Graphcal representaton of the results of nne drve experments wth and wthout HDR correcton. (a) Average Headng Errors E and all results of Table II and Fgure 6 were obtaned wth that fxed set of parameters. These parameters and ther values are lsted n Table III. Dscusson Several ssues related to the cted results deserve further dscusson:. It s qute apparent that some of the uncorrected headng errors are substantally larger n some drves than n others. In partcular, the Hghway drve has an uncorrected error of 84, much more than what on mght expect due to drft alone. To understand these large dfferences, one should consder the followng addtonal error sources, each of whch can cause errors many tmes larger than bas drft alone. Normalzed Average Headng Error [degrees per ute of travel tme] No correcton HDR-corrected Cty Suburbs Mxed Rural Rural 2 Hghway Hghway 2 Hghway mx Cty 2 Averages: Drve Name (b) 8

9 Table II: Descrpton and results for the nne drve experments. Cty Suburbs Mxed* Rural Rural 2 Wthout HDR Wth HDR Relatve HDR Improvement E ψ : 7 E ψ : 3. NE ψ : 0.87 / NE ψ : 0.6 / 5.4 E ψ : 35 E ψ : 0.4 NE ψ :.4 / NE ψ : 0.43 / 3.4 E ψ : 44 E ψ : 4.7 NE ψ : 2.2 / NE ψ : 0.24 / 9.4 Hghway Hghway 2 Hghway Mx Cty 2 Total for all nne drves: Experment Duraton Dstance Descrpton Mostly straght streets, few 0.8 km turns Mostly curvng streets n 7.0 km the suburbs Mx of cty streets, rural 23.6 km roads, and hghway Mostly straght streets, few 6. km turns Mostly straght rural roads, 5.5 km some curvng suburban streets Average Headng Errors E ψ : 42 NE ψ : 3.3 / E ψ : 56 NE ψ : 3.0 / 90% hghway, mostly E 9.4 km ψ : 84 straght NE ψ : 3.5 / 90% hghway, some curvng NE ψ :.3 / E 80.6 km ψ : 63 70% hghway wth some E 8.2 km curvng, 30% rural, mostly ψ : 32 NE straght ψ : 0.66 /.9 km 4.45 hrs 348 km 00% downtown streets, over degree turns Average of all nne drves: E ψ : 74 NE ψ : 3.4 / E ψ : 6 E ψ : 5.5 NE ψ : 0.42 / E ψ : 5.5 NE ψ : 0.29 / E ψ : 6.4 NE ψ : 0.2 / E ψ : 9.3 NE ψ 0.9 / E ψ : 7. NE ψ : 0.5 / E ψ :.3 NE ψ : 0.52 / a. Some of the drves were performed at near-freezng outdoor temperatures. The statc bas drft was measured shortly after the test vehcle was drven out of a garage, the nteror temperature was on the order of 5-0 C (4-50 F). Then, durng the drve, the nteror temperature of the vehcle was ncreased to room temperature, for drver comfort. Ths large change n temperature has a profound effect on drft, whch may be much larger than the manufacturer-specfed noal bas drft rate. b. Even small changes n the way the statc bas drft s measured before a drve can have profound effects on headng errors, especally on long drves. For example, n the Hghway Mx drve, we measured a statc bas drft of ε 0, =.432 /sec and the average headng error, accordng to Table II, was E ψ = 32. Had we measured a slghtly dfferent statc bas drft, say, ε 0,2 =.446 /sec (whch dffers from ε 0, by just %), then that would have yelded E ψ = n the same drve. Varatons on the order of % or larger n measurng ε 0 are ndeed possble. For example: when the statc bas drft test s done for less than the recommended amount of tme; due to the gyro not beng perfectly level durng the statc bas test (MEMS gyros are senstve to acceleraton, e.g., gravtaton); or due to changes n temperature durng the statc bas test. 2. The Hghway Drve has an exceptonally strong Relatve HDR Improvement score of 29-fold. That s almost three tmes hgher than the next-best score of 0.2-fold for the Rural 2 Drve. Ths possble excepton nflates the Average Relatve HDR Improvement to 9.-fold (bottom rght cell n Table II). If we exclude the Hghway Drve from the overall average, then that revsed Average Relatve HDR Improvement would be reduced from 9.2-fold wth a standard devaton of ) σ = to 6.6-fold wth a standard devaton of ) σ = However, we do not see any compellng reason for actually excludng ths drve. The uncorrected headng error, whle large, s not much greater than that of other drves when normalzed (.e., dvded by the duraton of the drve), as s apparent from Fgure 6b. N E ψ : 2.2 / E ψ : 7.0 N E ψ :0.28 / *Note: Ths s the Mxed Drve experment that was descrbed n more detal n Secton ( ) σ = ) 9

10 5.2.3 Test set The tunable parameters of the HDR algorthm were panstakngly hand-tuned to mze the bottom lne n Table II, and specfcally the Average of all Normalzed Average Headng Errors ( N Eψ = 0.28 ). Ths s partcularly apparent n Table III, whch shows that some of the parameters were tuned wth the precson of three sgnfcant dgts. Ths rases the queston of how well the algorthm would perform wth addtonal drves, for whch the parameters were not specfcally tuned. To answer ths queston, we performed another set of fve drves and appled the HDR algorthm to those drves wth the exact same parameter values as those of Table III. In the Test Set drves HDR provdes a 6.9-fold mprovement ( σ ) = ) over uncorrected headng estmates Anmaton Vdeo For llustraton purposes only, we nclude a vdeo anmaton that shows the trajectory of a 50-mle hghway drve as computed wth HDR. Ths partcular experment was not part of the set of experments descrbed up to ths pont. In ths experment we used a dfferent gyro, namely, the Z-axs gyro of a low-cost sx-axs IMU made by Memsense. Ths gyro has smlar performance characterstcs as the CRS03-04 sngle-axs rate gyro that was used n all other experments of Secton 5. The HDR parameters used n ths experment were dfferent from those of Table III and they were tuned for best performance n ths experment. The noal path s shown n cyan color n the fgure labeled Vdeo, and the HDR-computed trajectory s shown n the vdeo clp as a fant red lne. 6 CONCLUSIONS In ths paper we proposed the HDR method for reducng errors due to gyro drft n vehcle trackng applcatons. The basc heurstc s that much drvng s done along reasonably straght lnes. Whenever that s the case, the closed-loop control approach n our system leads the controller output, -I, to track the drft. Subtractng I from the gyro-measured rate of turn data then effectvely removes the estmated drft. Although n smulatons HDR vrtually elates the effects of drft (n real runs one cannot measure drft, thus makng t mpossble to prove or dsprove ths clam for real runs) some errors reman, due to two reasons:. The basc HDR algorthm cannot dstngush well between drft and actual curvng moton. To reduce ths undesrable behavor, we ntroduced enhancements: the Lowpass Flter, the Turn Swtch, and the Repetton Attenuator. Wth these enhancements the HDR algorthm effectvely suspends ts operaton for as long as the actual curvng Table III: A sngle set of parameters was used for all drves. Parameter Symbol Value Unts Noal ncrement c /sec Low-pass flter tme const. τ 0.75 sec Turn threshold Θ w /sec Rep. attenuator constant c 0.6 none Rep. attenuator power c 2.80 none Vdeo : Anmated vdeo llustraton of a 50-mle drve on hghways. Thck cyan lne: Noal path. Thn red lne: Trajectory from Gyro wth HDR. To vew vdeo, clck ths lnk: moton contnues. Then, when straght lne moton resumes, HDR correctons resume and -I resumes ts trackng of drft. Headng errors due to drft whle HDR was suspended cannot be recovered and they are the man contrbutor to errors n the HDR system. 2. Even durng straght-lne moton, there s always a resdual offset between -I and drft because drft never reaches steady state. Ths resdual offset ncurs headng errors that cannot be recovered. 0

11 Despte the errors that HDR ntroduces, ts correctve effect outweghs ts errors by a wde margn, as s apparent from the expermental results. For the gyro used n our vehcle trackng system, HDR reduced headng errors by a factor of 9.2 for the Tranng Set and a factor of 6.9 n the Test Set. Other advantages of the HDR method are The HDR method can be mplemented n software only, n as lttle as 20 or so lnes of code, and, besdes the gyro, no addtonal sensors or data are needed. Another, potentally major advantage that we dd not address n ths paper s the fact that HDR can elate the need for measurng statc bas drft for seconds before each drve. In nformal testng we found that just one second of measured statc bas drft, e.g., the tme between turnng the gnton key and startng to drve, s suffcent for an HDR system to produce very usable results wth errors just slghtly larger than those reported n ths paper. Acknowledgements Ths work was supported by the U.S. Department of Energy under Award No. DE FG NA REFERENCES [] Basnayake, C., Mezentsev, O., Lachapelle, G., and Cannon, M.E., An HSGPS, nertal and map-matchng ntegrated portable vehcular navgaton system for unnterrupted real-tme vehcular navgaton. Internatonal Journal of Vehcle Informaton and Communcaton Systems (), 3 5, (2005). [2] Cavallo, F., Sabatn, A.M., and Genovese, V., A step toward GPS/INS personal navgaton systems: real-tme assessment of gat by foot nertal sensng. Proc. IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems, 87-9, (2005). [3] Grewal, M.S., Well, L., and Andrews, A., Global Postonng Systems, Inertal Navgaton, and Integraton. John Wley & Sons, (2007). [4] Grejner-Brzeznska, D.A, Toth, C., Moafpoor, S., Jwa, Y., and Kwon, J., Mult-sensor personal navgator supported by human moton dynamcs model. Proc. 3rd IAG / 2th FIG Symposum, Baden, Austra, May, (2006). [5] Cho, S.Y., Lee, K.W., Park, C.G., and Lee, J.G., A Personal Navgaton System Usng Low-Cost MEMS/GPS/Fluxgate. Proc. 59th Insttute of Navgaton (ION) Annual Meetng, (2003). [6] Pant, S.M., and Zhang, W., Modelng Random Gyro Drft Rate by Data Dependent Systems. IEEE Transactons on Aerospace and Electronc Systems 22(4), , (986). [7] Chen, X., Modelng Temperature Drft of FOG by Improved BP Algorthm and by Gauss-Newton Algorthm. Lecture Notes n Computer Scence, Sprnger Berln/Hedelberg, , (2004). [8] IEEE Standards, IEEE Standard Specfcaton Format Gude and Test Procedure for Sngle-Axs Interferometrc Fber Optc Gyros, (996). [9] Borensten, J. and Ojeda, L., Heurstc Reducton of Gyro Drft n Vehcle Trackng Applcatons. Accepted for publcaton n the Internatonal Journal of Vehcle Informaton and Communcaton Systems, (2009). [0] Slcon Sensng, (2009). [] MemSense, (2009).