Probbilistic Stnd Still Detection using Foot Mounted IMU Jons Cllmer, Dvid Törnqvist nd Fredrik Gustfsson Division of Automtic Control Linköping University Linköping, Sweden {cllmer, tornqvist, fredrik}@isy.liu.se Abstrct We consider stnd still detection for indoor locliztion bsed on observtions from footmounted inertil mesurement unit (IMU). The min contribution is sttisticl frmework for stnd-still detection, which is fundmentl step in zero velocity updte (ZUPT) to reduce the drift from cubic to liner in time. First, the observtions re trnsformed to test sttistic hving non-centrl chi-squre distribution during zero velocity. Second, hidden Mrkov model is used to describe the mode switching between stnd still, wlking, running, crwling nd other possible movements. The resulting lgorithm computes the probbility of being in ech mode, nd it is esily extendble to dynmic nvigtion frmework where mp informtion cn be included. Results of first mode probbility estimtion, second mp mtching without ZUPT nd third step length estimtion with ZUPT re provided. Keywords: Indoor locliztion, stnd still detection, HMM, ZUPT Introduction The problem of indoor locliztion hs received n incresing mount of ttention in the lst couple of yers [, 2, 3, 5, 4, 8]. The desire to ccurtely trck the position of first responders or militry personel, or to provide positioning id for civilins in shopping mlls nd irports, hs led to trnsition from robot sensor pltforms to humn ones. To trck person, vriety of sensors cn be used. A foot or body mounted IMU with ccelerometers, gyros nd mgnetometers is simple nd chep nd is therefore very common sensor. It is usully supported by rnge mesuring rdio device such s WiFi or UWB [8] or is fused with preexisting mps for enhnced trcking precision [8]. The IMUs used for indoor locliztion re smll nd chep nd consequently perform quite poorly. There is commonly drift in the gyros cusing the orienttion est west 3 4 5 6 7 8 t = 25 9 5 6 7 8 9 2 3 4 south north Figure : Locliztion experiment without ZUPT. Red dots re the position hypotheses nd the blue dot nd line is the men position nd the heding mesurement, respectively. The lck of ZUPT mens tht we hve lrge step length uncertinty, which cuses the hypotheses to spred ll long the corridor. estimte to be incorrect. Since the orienttion is wrong, the direction of down is wrong nd prt of the grvittionl ccelertion will insted be believed to originte from the user moving the sensor. This error is double integrted to estimte the sensor position, resulting in position error tht grows cubiclly in time. This rpidly cuses very lrge positioning errors. The gyro drift in foot mounted IMU cn be corrected if we cn detect tht the foot is on the ground. Then the foot is sttionry nd the gyro should be showing zero ngulr velocity. Insted it is showing the drift which cn be estimted nd then compensted for. This is known s ZUPT nd reduces the positioning error to being liner in time [3]. Previously, stnd still detection hs been performed d hoc, usully by compring the signl to threshold. In this work we put the stnd still detection in probbilistic frmework using test sttistics with known dis-
tributions nd Hidden Mrkov Model (HMM). The result is probbility of stnd still t every time instnt which cn be used for ZUPT in filtering frmework. Figure shows n illustrtive exmple of mp ided locliztion experiment without ZUPT. No stnd still detections results in very uncertin step length estimtes cusing the position hypotheses to spred ll long the corridor. 2 Relted Work Most solutions to the stnd still detection problem use n verged ccelerometer or gyro vlue nd compre it to threshold [, 2, 3, 5]. The threshold is chosen d hoc nd is normlly quite low to minimize flse positives. Another pproch is the moving vrince used in [4] where the vrince computed over sliding window is compred to threshold. Probbilistic zero velocity detection hs previously been proposed in [6] who used hypothesis test to determine if the foot ws sttionry or moving. The hypothesis test ws performed using test sttistic bsed on Generlized Likelihood Rtio Test (GLRT). The pdf of the ccelertion nd/or the ngulr velocity during the swing phse of the step, ws pproximted with n unnormlized uniform distribution. The pdf during stnd still ws bsed on the exponentil of the norm of the ccelertion nd/or the ngulr velocity, which hs n unknown distribution. The resulting test sttistic ws moving verge of the norm of the ccelertion mesurements nd/or the ngulr velocity mesurements. This ws compred to threshold to determine if the foot ws to be rendered sttionry. Since the test sttistic hs n unknown distribution the threshold ws chosen d hoc, mking the frmework similr to the ones in [, 2, 3, 5]. The test sttistics used in [6] re similr to the ones used in this work since we both evlute three different ones where one is ccelertion bsed, one is ngulr velocity bsed nd one is bsed on combintion of ccelertion nd ngulr velocity. The ccelertion bsed test sttistic differs though in tht we hve chosen one which hs known distribution. This is lso the cse in the combined test sttistic. Our frmework to determine the mode probbilities lso differs. 3 Stnd Still Detection The sensor is n Xsens MT motion sensor smpling t Hz. The signls used re the ccelerometers nd the gyros. An exmple of wlking sequence is shown in Figure 2. The foot is sttionry in the time instnts round 55, 66, 77, 87 nd 98. During these phses the norm of the ccelerometer signls is the grvittion constnt with noise. Simultneously, the norm of the ngulr velocity signl is zero with some dditive noise. 3 2 Accelerometer dt 2 5 5 Gyro dt Figure 2: Exmple of ccelerometer dt (where x, y nd z is blue, green nd red) nd gyro dt (ω x, ω y nd ω z is blue, green nd red, ω i is ngulr rottion rte round xis i) during wlking sequence. The foot is sttionry round time instnts 55, 66, 77, 87 nd 98. 3. Sensor Models The signl model is y t = [ ] [ ] y t (θ) v y ω + t t (θ) v ω t () where y t nd y ω t denote ccelertion vector nd ngulr velocity vector, respectively. Further, θ denotes the model dependence of the phse of the humn step sequence. Nturlly, the model differs significntly between when the foot is t stnd still nd when it is swinging. The mesurements hve dditive Gussin noise distributions v N(, σ 2 ) nd v ω N(, σ 2 ω). The noise covrinces re independent, resulting in σ 2 ω = σωi 2 nd σ 2 = σi, 2 where I is the 3x3 identity mtrix. During stnd still the sensor model cn be described s [ ] [ ] [ ] y t gu v y ω = + t t v ω, (2) t where u is grvittionl direction vector nd g is the grvittionl constnt 9.8. When the foot is moving the sensor model chnges to [ ] y t = y ω t [ ] [ ] gu + t v + t ω t v ω t (3) where t nd ω t hve unknown distributions. The problem is to sfely distinguish between these two modes, stnd still nd swing. It is most importnt to minimize the stnd still flse positives, i.e. clling stnd still when the foot is in midir.
3.2 Test Sttistic To be ble to differentite between the two modes, test sttistics with known distributions re computed. Three different ones re evluted, one using only the ccelerometer dt, T, one using only the ngulr velocity dt, T ω, nd one using combintion of both ccelerometer nd ngulr velocity dt, T c. 3.2. Accelertion Mgnitude Detector The ccelerometer mgnitude detector test sttistic is computed s Tt = y t 2 σ 2 (4) where T χ 2 (3, λ) during stnd still. It hs noncentrl chi-squre distribution since y t hs nonzero men when the foot is sttionry. Its noncentrlity prmeter λ = g 2 /σ 2 nd 3 is the number of degrees of freedom. 3.2.2 Angulr Rte Mgnitude Detector The ngulr velocity test sttistic is T ω t = yω t 2 σ 2 ω (5) where T ω χ 2 (3) during stnd still since y ω hs zero men when the foot is sttionry. 3.2.3 Combined Accelertion nd Angulr Rte Detector The lst test sttistic combines ccelertion nd ngulr velocity to incorporte more informtion. It is clculted s T c t = y t 2 σ 2 + yω t 2 σ 2 ω (6) where T c χ 2 (6, λ) during stnd still. λ is the sme s in (4) but the number of degrees of freedom hs doubled to 6. 3.3 Test Sttistic Appernce during Wlking Sequence A plot of the test sttistics of the wlking sequence in Figure 2 cn be seen in Figure 3. The stnd still events occuring round time instnts 55, 66, 77, 87 nd 98 re clerly visble. Figure 4 shows zoom in of the test sttistic with the men of the stnd still distribution mrked with dotted line. This revels some of the problems with using only ccelertion for stnd still detection. The test sttistic T hs movement distribution tht hs significnt overlp of the stnd still distribution, cusing the test sttistic to cross the men of the stnd still distribution during the stride. This is shown round time instnts 53, 65, 63, 7, 75, 825 nd 935. Simply clling stnd still when T is close to the men of the stnd still distribution will therefore cuse lot of flse positives. 5 5 5 5 5 5 T T w T c Figure 3: Test sttistic from top to bottom; Tt, Tt ω nd Tt c. The foot is sttionry round time instnts 55, 66, 77, 87 nd 98. 5 5 5 T 5 5 T w 5 T c Figure 4: Zoom in of the test sttistics with the men of the stnd still distribution mrked with dotted line. The foot is sttionry round time instnts 55, 66, 77, 87 nd 98. T ω hs distribution during movement tht does not hve significnt overlp of the stnd still distribution, mking T ω sfer test sttistic thn T to use for stnd still detection. Still, there re two occsions during the stride where the foot is quite sttionry considering the ngulr velocity; one just fter the foot hs been lifted, in Figure 4 shown round time instnts 6, 7, 85 nd 92, nd one just before set down shown t time instnts 525, 635, 745 nd 96. These cn result in flse positives. The third test sttistic T c combines the strengths of T nd T ω. The bottom plot in Figure 4 shows tht the foot does not pper sttionry during the stride when you look t ccelertion nd ngulr velocity simultneously. This results in robust stnd still detection.
4 Test Sttistic Distribution Vlidtion The test sttistics must be vlidted to ensure tht the distribution of the test sttistic under experimentl stnd still is close to the theoreticl stnd still distribution. We lso estimte the distribution of the test sttistic under experimentl movements to illustrte the empiricl probbility density functions of stnd still nd movement tht need to be seprted. 4. Accelertion Mgnitude Detector The distributions of the ccelertion mgnitude test sttistic T is shown in Figure 5. The theoreticl nd the empiricl stnd still distributions hve similr men but slightly different covrinces. One of the resons why the empiricl density hs smller covrince thn the theoreticl one, could be tht we hve been bit too meticulous selecting the stnd still dt. Note lso the significnt overlp of the probbility distributions of stnd still nd movement. Tht mkes it difficult to sfely identify stnd stills by only looking t T. p(t ) Probbility Density Function, T 2 3 p(t stnd still) estimted p(t stnd still) estimted p(t moving) p(t ω ) Probbility Density Function, T ω 2 3 4 2 2 3 T ω p(t ω stnd still) estimted p(t ω stnd still) estimted p(t ω moving) Figure 6: Theoreticl stnd still distribution of T ω, empiricl estimte of stnd still distribution of T ω from experimentl dt nd empiricl estimte of movement distribution of T ω from experimentl dt. 4.3 Combined Accelertion nd Angulr Rte Detector The combined test sttistic nturlly hs distributions tht look like combintions of the distributions of T nd T ω. The empiricl stnd still distribution hs similr men but slighly smller covrince compred to the theoreticl distribution. The empiricl movement distribution does not overlp the stnd still distributions s much s for T, enbling sfer stnd still detection. 4 2 3 4 T Probbility Density Function, T c p(t c stnd still) estimted p(t c stnd still) estimted p(t c moving) 2 Figure 5: Theoreticl stnd still distribution of T, empiricl estimte of stnd still distribution of T from experimentl dt nd empiricl estimte of movement distribution of T from experimentl dt. p(t c ) 3 4 4.2 Angulr Rte Mgnitude Detector The distributions of T ω is shown in Figure 6. Clerly, the theoreticl stnd still distribution is very similr to the empiricl one, estimted by experimentl dt. Also note the lrge seprtion in mgnitude of the empiricl stnd still nd moving distributions. This enbles more robust stnd still detection thn the distributions of T. 2 3 4 Figure 7: Theoreticl stnd still distribution of T c, empiricl estimte of stnd still distribution of T c from experimentl dt nd empiricl estimte of movement distribution of T c from experimentl dt. T c
5 Hidden Mrkov Model To determine the probbility of stnd still, Hidden Mrkov Model (HMM) is used. It determines the probbility of ech mode using the test sttistic, the probbility density functions of the modes nd the mode trnsition probbility mtrix. There re two modes; mode when the foot is t stnd still nd mode 2 when the foot is moving. The mode trnsition probbility mtrix sttes the probbility of mode switch which induces some dynmics into the probbility estimtion. A lower mode trnsition probbility requires mesurement with higher likelihood for the other mode to induce switch. The mode trnsition probbility mtrix is [ ].95.5 Π = (7).5.95 which sttes tht the probility of going from stnd still to moving or vice vers, is 5%. During norml wlking your right foot tkes bout one step per second which results in roughly 2 mode trnsitions every mesurements. The trnsition probbilities were chosen slightly higher to incorporte lso fster movements. The mode probbilities t time t re clculted using the recursion Hence we hve µ i t = µ i t = P (r t = i y t ) p(y t r t = i)p (r t = i y t ) N r = p(t t r t = i) Π ji µ j t. (8) j= p(t t r t = i) N r j= Π jiµ j t Nr l= p(t t r t = l) N r j= Π jlµ j. (9) t The probbility density function of movement used in the HMM is n pproximtion tht is set to resemble the empiricl movement density functions in Figures 5, 6 nd 7. The HMM frmework thus gives the probbility of movement nd stnd still for ech time instnt. This frmework cn be extended to other modes like running nd crwling, simply by extending the mode trnsition probbility mtrix by incorporting these new modes nd estimting the probbility densities for these movements too. 6 Experimentl Results The mode probbilities provided by the HMM of the dt sequence in Figure 2 is shown in Figure 8. All three test sttistics hve been used to illustrte the difference in stnd still detection performnce. The ccelertion bsed test sttistic T suffers from flse positives round some of the troublesome time instnts mentioned in Section 3.3; 65, 7, 75, 825 nd.99.99 Mode probbility using T Mode probbility using T ω.99 Mode probbility using T c Figure 8: Mode probbilities for the different test sttistics, evluted on the dt set in Figure 2. The foot is sttionry round time instnts 55, 66, 77, 87 nd 98. 935. The frmework does not cll it stnd still fter the first troublesome mesurement, but fter couple of mesurements close to the stnd still men the HMM ssumes the foot is t rest. The ngulr velocity bsed test sttistic T ω gives distict detection of every foot stnce. The sttionry moment is rther short but is often followed by shorter second sttionry moment. Figure 4 shows tht this is becuse there is commonly slight ngulr movement hlfwy through the deemed sttionry prt. This second sttionry moment provides no new informtion nd only the first detection is necessry to perform ZUPT. No flse positives occur during the stride phse of the step. The combined test sttistic T c provides very sfe stnd still detections. A long intervl when the foot is t rest is deemed sttionry nd there re no flse positives. A second dt set is constituted of running phse followed by wlking phse, see Figure 9. The subject is running up until round time instnt. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3, the lst two re during wlking. The mode probbilities provided by the HMM of this sequence is shown in Figure. The sme movement distribution ws used during this whole experiment. T does not provide ny relible stnd still detections. The foot stnces re detected, but lot of flse positives re lso present. This is not surprising considering the ccelerometer dt in Figure 9. The gyro bsed T ω does not result in ny stnd still detections t ll during the running phse. This is bit surprising since the gyro dt looks pretty comprehendble nd is probbly becuse the IMU ws fstened on the side of the foot where only very short periods of low ngulr
2 2 3 Accelerometer dt 65 7 75 8 85 9 95 5 5 5 5 Gyro dt 65 7 75 8 85 9 95 5 5.99.99 Mode probbility using T 65 7 75 8 85 9 95 5 5 Mode probbility using T ω.99 65 7 75 8 85 9 95 5 5 Mode probbility using T c 65 7 75 8 85 9 95 5 5 Figure 9: Exmple of ccelerometer nd gyro dt during sequence contining running, 65, followed by wlking, 5. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3. The dt hs the sme color encoding s in Figure 2. velocity re experienced. The combined test sttistic T c still provides quite sfe stnd still detections. It picks up ll the stnd still sequences reveled by ccelertion but mnges to disregrd the flse ones using the ngulr velocity mesurements. Here, the combined test sttistic hs shown to provide the most robust stnd still detection. Further wlking experiments revel the stnd still detection performnce shown in Tble. All 74 true sttionry phses were detected, but lso some flse positives. The ccelertion bsed test sttistic hs flse positive between pretty much every step. Most flse positives of the T c sttistic occur during sequences when smll movements re performed like when door is opened. During wlking, T ω gives very few flse positives nd is the sfest stnd still detector. T T ω T c Stnd stills detected 74 74 74 Flse positives 69 2 5 Tble : True detected sttionry phses nd flse detected sttionry phses. 74 steps were tken. 6. Step Length Estimtion The foot mounted IMU hs coordinte system following the moving foot. In order to estimte the step length, the orienttion of the foot in world coordintes is described by the unit quternion q. This reltes the mesured ccelertions y t nd ngulr velocities y ω t to movements nd heding chnges in the world coordintes. A filter with the sttes p nd v for position nd velocity in world coordintes nd q cn now be used Figure : Mode probbilities for the different test sttistics during the running followed by wlking sequence. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3. to estimte the length of ech step. For thorough description of the dynmicl model, see [7]. A short experiment of 6 steps covering 5. meters ws performed to evlute the step length estimtion performnce. Stnd still ws detected using the gyro bsed test sttistic nd ZUPT ws performed. Figure shows the estimted movement in world coordintes. Totl step length ws estimted s 4.7 meters rendering step length estimtion error of 6%. position, [m].5.5.5 2 2.5 3 3.5 Movement Estimtion 4 2 4 6 8 2 4 Figure : Movement estimtion in world coordintes. 6 short steps were tken with totl length of 5. meters. 7 Conclusions nd Future Work Three test sttistics with known distributions hve been evluted for stnd still detection. The one bsed on ccelerometer dt only, hs been shown to provide x y z
plenty of flse detections. This is nturl since there is significnt overlp between the test sttistic pdf during stnd still nd the pdf during movements. The gyro bsed hs been shown to provide excellent stnd still detection cpbilities during wlking while the one combining ccelerometer nd gyro dt hs been shown to provide good stnd still detection during both wlking nd running. In conjunction with Hidden Mrkov Model, the mode probbilities re redily clculted nd cn be used for zero velocity updtes. Future work includes extending the HMM frmework to incorporte more modes nd to merge the stnd still detection with our locliztion frmework. We will lso look into whether the sttionry phses detected using T c re unnecessrily long for ZUPT. Wht we wnt to detect is gyro drift when the gyro is sttionry nd wht we detect is when the combintion of gyro nd ccelerometer is sttionry. It is not necessrily the sme thing. Further reserch is needed to decide when to perform the zero velocity updte bsed on T c. References [] S. Beuregrd. Omnidirectionl pedestrin nvigtion for first responders. In Proc. of the 4th Workshop on Positioning, Nvigtion nd Communiction, WPNC7, Hnnover, Germny, 27. [2] R. Feliz, E. Zlm, nd J. G. Grci-Bermejo. Pedestrin trcking using inertil sensors. Journl of Physicl Agents, 3():35 43, 29. [3] E. Foxlin. Pedestrin trcking with shoe-mounted inertil sensors. IEEE Computer Grphics nd Applictions, 25(6):38 46, 25. [4] S. Godh, G. Lchpelle, nd M. E. Cnnon. Integrted GPS/INS system for pedestrin nvigtion in signl degrded environment. In Proc. of ION GNSS, 26. [5] L. Ojed nd J. Borenstein. Non-GPS nvigtion for security personnel nd first responders. Journl of Nvigtion, 6(3):39 47, 27. [6] I. Skog. Low-cost Nvigtion Systems. PhD thesis, KTH, Stockholm, Sweden, 29. [7] D. Törnqvist. Estimtion nd Detection with Applictions to Nvigtion. PhD thesis, Linköping University, Linköping, Sweden, 28. [8] O. Woodmn nd R. Hrle. Rf-bsed initilistion for inertil pedestrin trcking. In Proceedings of the 7th Interntionl Conference on Pervsive Computing, 29.