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Sensor Aided Inertial Navigation...
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Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor Aided Inertial Navigation Systems
Arvind Ramanandan
Department of Electrical EngineeringUniversity of California, Riverside
Riverside, CA 92507
April, 28th 2011
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Acknowledgements:
1 Prof. Jay A. Farrell
2 Anning Chen
3 Anh Vu
4 Prof. Matthew J. Barth
5 Sharat Suvarna
6 Sheng Zhao
7 Behlul Sutarwala
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Outline
Today, I will discuss:
Inertial Navigation System.
Carrier phase differential GPS (CDGPS) - INS.
CDGPS - Vision - INS.
Stationary updates - INS.
Near Real Time estimation.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Inertial Sensors
Inertial Sensors
Popularity of inertial Sensors are due to:
Immunity to jamming, No external reference required (in theory).Advent of MEMS sensors [1]:
Low cost, Small footprint (∼ 22 × 33 × 11 mm).Well understood and quantifiable error models [5], [8], [12].High frequency updates (> 100 Hz), High Bandwidth (> 330 Hz),High operating ranges (∼ ±400 deg/s, ±10g m/s/s).
Supplies full 6 degrees-of-freedom pose information.Consumer driven demand for applications such as
Navigation: routing, vehicle guidance & control [4], [19] etc.High accuracy mobile mapping [18], [17].Life-critical systems: Vehicle collision avoidances, automotive airbags etc.Hand-held devices: Cellphones, Cameras, Electronics readers etc.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Inertial Sensors
Inertial Sensors
Potentially unbounded error growth in dead reckoning Inertialsystems.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Aiding Sensors
Aiding Sensors
“GPS, Vision, LIDAR, Magnetometer, Stationary updates etc.,”.Features usually complement inertial sensors:
Independent and bounded long term errors.Low update frequency.Do not provide 6-DOF information.
Aided INS problems can be formulated and solved under the Bayesianframework:
Extended Kalman Filters, Particle Filters, Unscented Kalman Filtersetc.,
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Inertial Navigation System
Navigation state:
x⊤ =[
npnb⊤ nvnb
⊤ nbq
⊤]
, x ∈ R6 × S
3.
Inertial Measurements: bu = bu + bb + n, bu ∈ R6.
Bias Gauss-Markov model: bb = −Λbb + nb.
Kinematic equations: x = f (x ,u)npnb = nvnbnvnb = n
bRbf ib − ng − 2 [nωie×] nvnb
nbR = n
bR([
bωib×
]
−[
bωie×
])
INS augmented error state: δx⊤ =[
δx⊤ bδb⊤g
bδb⊤a
]
∈ R15.
Linearized error propagation model: δ ˙x = Aδx + GN .Frames: n=navigation, e=ECEF, i=inertial, b=body, c=camera
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Outline
Today, I will discuss:
Inertial Navigation System.
Carrier phase differential GPS (CDGPS) - INS.
CDGPS - Vision - INS.
Stationary updates - INS.
Near Real Time estimation.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Carrier Phase Differential GPS - INS
Tightly coupled GPS-INS is the de-facto standard for outdoor naviga-tion:
Performance with double differenced, differential carrier-phase pro-cessing with differential corrections: [6, 7, 8],.
1σ positioning accuracy in the order of 0.01 − 0.1 m.1σ attitude accuracy in the order of 1 deg.
Well understood conditions to achieve full state observability [3,8, 9, 10, 11, 16].
Updates with a minimum of 2 satellite measurements (Looselycoupled needs at least 4).
Can do sequential updates (HPH⊤ + R is a scalar).
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Carrier Phase Differential GPS - INS
Generic GPS measurement model:
ρjk = ||epeb − epesj
||+ ν jρ ; ν j
ρ ∼ N (0, 0.52)
φj = ||epeb − epesj||+ λN j + ν
jφ ; ν
jφ ∼ N (0, 0.012)
Dj = ddt ||
epeb − epesj||+ ν
jD ; ν
jφ ∼ N (0, 0.022)
δφj = φ
j − φjk = ||epeb − epesj
|| − ||epeb − epesj||+ ν
jφ
= hjδx + ν
jφ
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Carrier phase residuals
Carrier Phase Differential GPS - INS
Example EKF Phase residuals for Satellite 3, Stationary rover, base-line: 7 km
0 100 200 300 400 500 600 700
−0.1
−0.05
0
0.05
0.1
0.15
δ ψ
(m)
Time(s)
Phase
−0.2 −0.1 0 0.1 0.20
20
40
60
80
100
120
140
µ: −0.010112 , σ: 0.021891Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Carrier phase residuals
Carrier Phase Differential GPS - INS
Example EKF Phase residuals for Satellite 18 driving on I-215:
0 100 200 300 400
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
δ ψ
(m)
Time(s)
Phase
−0.4 −0.2 0 0.2 0.4 0.6 0.8
20
40
60
80
100
120
140
160
180
200
µ: 0.0096183 , σ: 0.075345
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Carrier phase residuals
Outline
Today, I will discuss:
Inertial Navigation System.
Carrier phase differential GPS (CDGPS) - INS.
CDGPS - Vision - INS.
Stationary updates - INS.
Near Real Time estimation.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
CDGPS - Vision - INS
Tightly coupled CDGPS - Vision - INS has several advantages:Under some well defined conditions, can contain drifts in:
Velocity and gyroscope bias.Certain directions in attitude and accelerometer biases.
Provides updates even with a single feature (unlike a loosely cou-pled system).
Not computationally expensive like SLAM.
Can be performed in real-time unlike Bundle adjustment.
Can naturally extend to applications like Mapping, Surveying etc.
“Need to calibrate transformation from Body to Camera frames”.
δx⊤∗ =
[
nδpnb⊤ nδvnb
⊤ nρ⊤ bδb⊤
gbδb⊤
abδp⊤
bcbρ⊤
]
∈ R21.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Perspective projection model
CDGPS - Vision - INS
Feature vector in the Camera frame cp⊤cfj
=[
xj yj zj]
.
Ideal perspective projection model:
cq⊤cfj =
[
uj vj]
=1zj
[
xj yj]
(5.1)
Non-ideal Camera model [2]:
cq⊤cfj =
[
fxx ′j + cx fy y ′
j + cy]
+ nc (5.2)
where
x ′j = uj(1 + k1r2 + k2r4) + 2p1ujvj + 2p2(r2 + 2u2
j )
y ′j = vj(1 + k1r2 + k2r4) + 2p2ujvj + 2p1(r2 + 2v2
j )
r = ||cqcfj ||2
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Perspective projection model
Observability Analysis
Key results [14]:
Proposition 1
Assuming that the Camera is fully calibrated (i.e.(
bpbc ,bcR
)
areknown), then the INS error state δx(0) is fully observable with N0 ≥ 3measurements at 3 time instants such that the set of points{npnf0 , . . . ,
n pnfN0,n pnck
} are not coplanar for all 0 ≤ k ≤ 2.
Proposition 2
If the rover, initially at rest, is accelerates along a straight line andcomes back to rest, aided by both GPS and Vision with N0 ≥ 3features, then the observability gramian has full column rank.Therefore we have full state observability.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Perspective projection model
CDGPS-Vision-INS
Demo
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Perspective projection model
CDGPS - Vision - INS
Data association:
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Perspective projection model
Outline
Today, I will discuss:
Inertial Navigation System.
Carrier phase differential GPS (CDGPS) - INS.
CDGPS - Vision - INS.
Stationary updates - INS.
Near Real Time estimation.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Stationary updates - INS
Reset velocity (bvnb = 0) and rotation (bωnb = 0) to zero when
system is at rest.
δx⊤ =[
nδpnb⊤ nδvnb
⊤ nρ⊤nb
bδb⊤g
bδb⊤a
]
Given stationarity, stationary updates or zero updates arepreferable.Stationary updates corrects errors in:
velocitygyroscope biasessome linear combination of attitude and accelerometer biases.
Helps contain errors in position.position
Detection of stationarity is a challenge.False detection introduces errors in sensor bias estimates directly.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
Measurement model:
by i = s(iTs) + e(iTs) +bb(iTs) + ν(iTs) + n(iTs)
Component Region in the DFT (f Hz)B 0E [8, 85]S (0, 10)
Legend:Symbol : ManeuverBlue asterisks : StationaryRed squares : DeceleratingBlack circles : AcceleratingMagenta triangles : Constant speed
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
For each sensor k , choose appropriate m(k) ∈ S [13].kϕm(k) ∈ C, k
ϕm(k) ∼ N (µ,P).Define f : R2 → [0,+∞) as
f (kϕm(k)) =
kϕ
⊤m(k)P
−1kϕm(k)
Under stationarity, f (kϕm(k)) is an i.i.d. Rayleigh random process
with χ = 1.
Stationarity test
Given a chosen harmonic, m(k) ∈ S, for each sensor k, the rover isstationary if
maxk∈{1,...,6}
f (kϕm(k)) < λ2
for a chosen threshold λ ∈ R+.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
Enables determination of λ using stochastic principles:If λ = 2.4477, the p{ max
k∈{1,...,6}f (k
ϕm(k)) < λ2} = 0.7351.
Conservative choice of λ = 0.05 resulting in pd/s = 0.001 (1detection every 5 s when Fs = 130 Hz).
Upper bounds on probability of false detection [13].
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
Observability analysis [15]:
δx S M − S S − M − S, (ω = 0) S − M − Snδpnb R
3R
3R
3R
3
nδvnb 0 M(t1)ei 0 0nρnb ei ei ei
ngbδbg 0 0 0 0bδba
bnR[ng×]ei
bnR(t1)[ng×]ei
bnR[ng×]ei 0
Fixed point optimal smoother:
δx0 = E{δx0|δy1 . . . δyM}.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Stationary updates - INS
δx S M − S S − M − S, (ω = 0) S − M − Sbδba
bnR[ng×]ei
bnR(t1)[ng×]ei
bnR[ng×]ei 0
Legend:Color : ManeuverBlue : SRed : M − SBlack : S − M − S(ω = 0)Magenta : S − M − S(ω 6= 0)
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Sensor & Vehicle model
Outline
Today, I will discuss:
Inertial Navigation System.
Carrier phase differential GPS (CDGPS) - INS.
CDGPS - Vision - INS.
Stationary updates - INS.
Near Real Time estimation.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Near Real Time estimation
Unlikely residuals: δy⊤p S−1
p δyp > λ.Validation from measurements in future time yn, yn+1, . . .
Updating δxn with δyp violates “white-noise” assumption.
A possible solution:Append state vector: δα⊤ =
[
δx⊤p δx⊤
n
]
.
Append the covariance: Pα =
[
Pp E{δxpx⊤n }
E{xnδx⊤p } Pn
]
.
Update: δyp =[
hp 0]
δα.
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Thanks for listening!
Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
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Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
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Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
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Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
Position Location and Navigation Symposium (PLANS), 2010IEEE/ION. IEEE, pp. 1197–1203.
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Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems
Outline Introduction Inertial Navigation Systems Tightly coupled GPS-INS Tightly coupled GPS-Vision-INS Stationary updates - INS Near Real Time estimation
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Arvind Ramanandan Department of Electrical Engineering University of California, Riverside Riverside, CA 92507
Sensor Aided Inertial Navigation Systems