foot direction uplift Time liftoff time landing time rest time gravity direction 8在-l=0 ax(t+I)>0 a,-1>0 g作-)<0.g0=0 g,)<0 a,0<0 a,<0 trace of foot a,-<0.a,)=0 a. Phase(3) Phasc(4) Phase (5) (a)Walking process Phase (1) Phase (2) Phase (3) Phase (4) Phase (5) 201网 2 10 12 14 16 18 20 uplift time liftoff time landing time rest time 500 -500 6 8 10 12 14 16 18 20 Number of Samples (b)Sensor data of the walking process Fig.8.The process of a complete foot step. To calibrate the acceleration from toe-in and toe-out prob- lem,Eq.(4)gives the way to get the acceleration along the △sr moving direction. am(t)=af(t)cos(p)+ar(t)sin(p),tE[T,Ta] (4) Having the acceleration along the moving direction,we can Fig.10.Estimate the angle. finally get the displacement of the current step by Eq.(5). data,we got an estimated step length of 1.11m.Referring to Ta rt the ground-truth 1.23m,our calibration reduces the error from S= am(t)dtdt' 0.63mto0.12m (5) 40 (af(t)cos(p)+az(t)sin(p))dtdt' "ground-truth estimated 20 m calibrated =Sx+Sy+S: Here S,S,S2 is the real acceleration's projection on the accelerometer's three-axis,which are Eq.(6). -20 Sx= (az(t)sin())dtdt' 0.4 60.10.20.30.4 Time(s) JT JT (a)Acceleration calibration.(b)Displacement calibration. S.= (a(t)cos(0(t))cos())dtdt' (6) JT Fig.11.Case study of the step length estimator. Double feet based calibration.To further reduce the error S.= (a=(t)sin(0(t))cos())dtdt' accumulation,FootStep-Tracker embeds two sensors in both feet and respectively estimates step length.Having the intu- As depicted in Fig.10,to get the angle we let the user to itions that the distance between the two feet can not be too walking ahead for a constant distance s.Then we do double large at any time,if the difference of displacement for each integral on the af to get the displacement along the foot foot is more than one meter.we chose the mean of them as direction sf.As the angle between the Asf and As is the displacement,and restart the estimation process. we can estimate the angle byarccos().Besides. sf is equal to the sum of Asf,and s is equal to the sum of D.Moving Direction Estimation As.Therefore,we can estimate the angle by=arccos().Motivation and Challenge.To depict the user's moving trace, Fig.11 (a)shows the raw data ay and calibrated data am.we also need to figure out the user's moving direction.As Figure (b)shows the corresponding displacement.We do we embed the sensor in the shoes,we should estimate the integral on raw data and on calibrated data.For raw data, relatively variety angle of foot when the user make turns we got an estimated step length of 1.86m.For the calibrated according to the inertial sensor readings.And furthermore,duePhase (1) Phase (2) Phase (3) Phase (4) Phase (5) az ay=0 az ay © az ay=0 az ay z © foot direction gravity direction © © ay az © © ay az © © Ground ay az © © trace of foot dispalcement of foot liftoff time ay(t+1)>0 ay(t)<0 uplift Time gx(t-1)=0 gx(t)<0 landing time ay(t-1)>0 ay(t)<0 rest time gx(t-1)<0, gx(t)=0 ay(t-1)<0, ay(t)=0 gx gx gx gx gx (a) Walking process 0 2 4 6 8 10 12 14 16 18 20 −20 0 20 a y(m/s 2 ) Phase (1) Phase (2) Phase (3) Phase (4) Phase (5) uplift time liftoff time landing time rest time 0 2 4 6 8 10 12 14 16 18 20 −500 0 500 Number of Samples g x ( °/s) (b) Sensor data of the walking process Fig. 8. The process of a complete foot step. To calibrate the acceleration from toe-in and toe-out prob￾lem, Eq. (4) gives the way to get the acceleration along the moving direction. am(t) = af (t)cos(ϕ) + ax(t)sin(ϕ), t ∈ [Tl , Td ] (4) Having the acceleration along the moving direction, we can finally get the displacement of the current step by Eq.(5). S = Z Td Tl Z t 0 Tl am(t) dtdt0 = Z Td Tl Z t 0 Tl (af (t)cos(ϕ) + ax(t)sin(ϕ)) dtdt0 = Sx + Sy + Sz (5) Here Sx, Sy, Sz is the real acceleration’s projection on the accelerometer’s three-axis, which are Eq.(6). Sx = Z Td Tl Z t 0 Tl (ax(t)sin(ϕ)) dtdt0 Sy = Z Td Tl Z t 0 Tl (ay(t)cos(θ(t))cos(ϕ)) dtdt0 Sz = Z Td Tl Z t 0 Tl (az(t)sin(θ(t))cos(ϕ)) dtdt0 (6) As depicted in Fig.10, to get the angle ϕ, we let the user to walking ahead for a constant distance s. Then we do double integral on the af to get the displacement along the foot direction sf . As the angle between the ∆sf and ∆s is ϕ, we can estimate the angle ϕ by ϕ = arccos( ∆s ∆sf ). Besides, sf is equal to the sum of ∆sf , and s is equal to the sum of ∆s. Therefore, we can estimate the angle by ϕ = arccos( s sf ). Fig.11 (a) shows the raw data ay and calibrated data am. Figure (b) shows the corresponding displacement. We do integral on raw data and on calibrated data. For raw data, we got an estimated step length of 1.86m. For the calibrated s φ φ φ φ φ sf Ƹs Ƹs Ƹs Ƹs Ƹs Ƹsf Ƹsf Ƹsf Ƹsf Ƹsf Fig. 10. Estimate the angle ϕ. data, we got an estimated step length of 1.11m. Referring to the ground-truth 1.23m, our calibration reduces the error from 0.63m to 0.12m. 0 0.2 0.4 −40 −20 0 20 40 Time(s) a y(m/s 2 ) a y a m (a) Acceleration calibration. 0 0.1 0.2 0.3 0.4 0 1 2 3 Time(s) S(m) ground−truth estimated calibrated (b) Displacement calibration. Fig. 11. Case study of the step length estimator. Double feet based calibration. To further reduce the error accumulation, FootStep-Tracker embeds two sensors in both feet and respectively estimates step length. Having the intu￾itions that the distance between the two feet can not be too large at any time, if the difference of displacement for each foot is more than one meter, we chose the mean of them as the displacement, and restart the estimation process. D. Moving Direction Estimation Motivation and Challenge. To depict the user’s moving trace, we also need to figure out the user’s moving direction. As we embed the sensor in the shoes, we should estimate the relatively variety angle of foot when the user make turns according to the inertial sensor readings. And furthermore, due