Sandau R. Digital Airborne Camera: Introduction and Technology

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2.10 Sensor Orientation 121

2.10.3 Basics of Inertial Navigation

Inertial guidance systems or inertial navigation systems (from the Latin “inertia”)

were originally developed for the navigation of aircraft or ships. Inertial navigation

may be deﬁned as the real-time determination of the trajectory of a moving vehicle

using sensors that react on the basis of Newton’s laws of motion. In general these

laws describe the dynamics of a body in an inertial reference system. If no external

forces are applied, the body will remain at rest or in uniform rectilinear motion.

If external forces are applied, the acceleration is directly proportional to the acting

forces. From the measurement of such accelerations the velocity and position of a

vehicle are ﬁnally obtained in two integration steps. The principle of inertial navi-

gation is the two-fold integration of acceleration with respect to time. Depending on

the degrees of freedom, a set of three accelerometers is normally used to measure the

linear accelerations in three dimensions. This allows for the three-dimensional posi-

tioning of the vehicle. Nevertheless, using three accelerometers on their own on a

vehicle is inadequate for three-dimensional inertial navigation, since the accelerom-

eters are not aligned with the axes of the reference coordinate frame (i.e. inertial or

earth-related frame). Thus the continuous orientation of the vehicle with respect of

the reference frames has to be provided by measurements using gyroscopes. From

their integrated angular rate measurements three directions are obtained to orient

fully the accelerometer triad in three-dimensional space. First attempts to estimate

the travelled distance from measurements of linear accelerations (so called dead

reckoning) were done in the 1920s, mainly in Germany. The ﬁrst operational iner-

tial navigation systems were used in military applications and later civilian use also

became accepted.

In general, an inertial navigation system comprises the inertial measurement

unit itself (i.e. the set of three gyros and three accelerometers), a platform for

the mounting of the sensors and ﬁnally an appropriate computer algorithm trans-

forming the IMU measurements into relevant navigational information. Different

system designs have been developed, depending on the conﬁguration of accelerom-

eters and gyros: stabilized platform systems or strap-down systems. In stabilized

platform systems the sensors are ﬁxed on a space-stabilized platform which de-

couples the sensors from the angular movements of the vehicle. Thus, the sensitive

axes remain with their orientations ﬁxed relative to a space-stabilized or a local

level coordinate frame. Within the local level installation the platform physically

provides the actual local level frame. The vertical sensor axis is in the plumb line

direction and the other two components measure in the horizontal plane. From this,

the changes in horizontal position are directly obtained from the integration of hor-

izontal accelerometer measurements and the change in vertical position, from the

vertical axis accelerometer. This minimizes the amount of processing time. On the

other hand the mechanical attainment of stabilized platform systems is quite com-

plex and therefore expensive. Thus stabilized platform systems are typically used

only for highest accuracy demands. Due to their mechanical complexity the sys-

tems have to be handled with great care and high mechanical wear is possible.

Alternatively, strap-down inertial navigation systems are more favourable for use

122 2 Foundations and Deﬁnitions

in small and ﬂexible environments. In contrast to the stabilized platform approach,

the sensors are rigidly ﬁxed (“strapped down”) to the vehicle’s axes and follow the

full motion of the vehicle. This obviates the use of complex mechanics and therefore

the production of these systems is less demanding and less expensive. Nevertheless,

the computational effort is slightly higher because now the horizontal platform has

to be accomplished analytically. Furthermore, the sensors are subjected to the entire

range of dynamics of the vehicle, which degrades their maximum performance.

With the advent of smaller and less expensive gyros and accelerometers, strap-

down technology is the favoured approach for medium to less accurate inertial

navigation applications. This trend will be much more evident as new develop-

ments emerge, especially in the design of gyros. With the use of micro-electro

mechanical system (MEMS) technologies, considerable potential for miniaturisa-

tion and cost decrease is apparent. Even though these MEMS-based sensors are of

somewhat reduced accuracy, continuous improvement in design and manufacturing

processes may ﬁnally lead to the replacement of mechanically or optically based

inertial sensors, even for navigation purposes.

Another approach to classiﬁcation of inertial navigation systems besides their

individual system design relies on the potential accuracy of their positioning

and attitude measurements. The systems are classiﬁed into three different accu-

racy groups according to the positioning error after 1 h of unaided navigation

(Table 2.10-2): high accuracy or strategic grade systems with positional error well

below 1 nautical mile

(nmi) (up to 0.1–0.2 nmi/h or better) after 1 h of unaided

navigation, medium accuracy or navigation grade systems with positional error in

the range of 1 nmi/h (0.5–2 nmi/h); and low accuracy or tactical grade systems with

positional error substantially more than 2 nmi/h (in many cases up to several tens of

nautical miles). The attitude accuracy in Table 2.10-2 is valid for the performance

Table 2.10-2 Accuracy of INS positions and attitude determination (Schwarz et al., 1994)

System accuracy (RMS)

Error budget High (strategic-grade) Medium (navigation-grade) Low (tactical-grade)

Position

1 h 0.3–0.5 km 1–3 km 200–300 km

1 min 0.3–0.5 m 0.5–3.0 m 30–50 m

1 s 0.01–0.02 m 0.03–0.10 m 0.3–0.5 m

Attitude

1 h 10"–30"

(0.003

◦

–0.008

◦

)



–3



(0.016

◦

–0.050

◦

)

◦

–3

◦

1 min 1"–2"

(0.0003

◦

–0.0006

◦

)

15"–20"

(0.004

◦

–0.006

◦

)

0.2

◦

–0.3

◦

1 s 0.1"–0.2"

(0.00003

◦

–0.00006

◦

)

1"–2"

(0.0003

◦

–0.0006

◦

)

0.01

◦

–0.03

◦

1 nautical mile (nmi) = 1,852 m

2.10 Sensor Orientation 123

of the roll and pitch angles; for the heading angle the performance is worse by a

factor of about 3–5. Since positioning and attitude data is obtained from integration

processes, the absolute system accuracy is time dependent owing to the strong sys-

tematic error propagation. Within the table the corresponding accuracy is given for

the 1 s and 1 min time interval as well. Depending on the individual performance of

sensors, even larger errors are possible. The overall accuracy is inﬂuenced mainly

by the gyro performance, which is characterised by the gyro’s speciﬁc gyro drift

value (given in [deg/h]). On the other hand, the accelerometer offset (given in [μg])

describes the accelerometer performance. The corresponding sensor accuracies for

the three system performance classes are given as follows (guidance values only):

0.0001 deg/h and 1 μg (strategic grade), 0.015 deg/h and 50–100 μg (navigation

grade), 1–10 deg/h and 100–1,000 μg (tactical grade) (El-Sheimy, 2003).

With the use of appropriate external update information the time dependent error

behaviour is reduced signiﬁcantly or almost eliminated. In the ideal case consistent

absolute accuracies in the range of the 1 s interval are possible for high to medium

accuracy inertial navigation systems as long as sufﬁcient update information is avail-

able (Schwarz, 1995). Owing to the lower quality speciﬁcations for sensor bias

stabilisation and sensor noise this 1 s performance cannot be obtained for the tactical

grade systems, even with high quality updates.

As mentioned in the ﬁrst paragraph of this section, Newton’s laws underlie the

basics of inertial navigation. If no external forces are applied, the body will remain at

rest or in uniform rectilinear motion. If external forces are applied, the acceleration

is directly proportional to the acting forces. The acting force f is obtained from the

product of mass m and acceleration a, as given in (2.10-10). The acceleration is

directly portional to the force assuming constant mass of the body.

f = m ·a (2.10-10)

It is interesting to see that acceleration as a unit of length is not obtained from a

scale of length but from the measurement of inertial forces acting on the accelerated

mass.

All this has to be performed in an inertial coordinate frame, which is deﬁned

as the system in which Newton’s laws hold.

Based on this, differential equations describing the interrelationship between

body acceleration a(t), body velocity v(t) and position r(t) are given as follows:

v =

a =

v =

(2.10-11)

The inertial coordinate frame satisﬁes Newton’s laws only within the bounds of the given mea-

surement accuracy of the sensors used. Thus, such a frame is named a quasi-inertial frame or

operational inertial frame.

124 2 Foundations and Deﬁnitions

Assuming a quasi-continuous measurement of accelerations a(t) the instanta-

neous velocity of a body after a certain interval of time t − t

is obtained from

v =



a · dt +v

(2.10-12)

with instantaneous position

r =



v · dt +r



⎛

⎝



a · dt

⎞

⎠

·dt +v

·(t −t

) + r

. (2.10-13)

If the integration process starts with zero velocity v(t

) = 0, Equation (2.10-13)

is reduced to

r =



⎛

⎝



a · dt

⎞

⎠

·dt +r

. (2.10-14)

Typically, navigation is done in the Earth’s gravity ﬁeld, where additional accel-

erations occur and act independently on the mass being accelerated. Thus, the effect

of such gravitational accelerations also has to be considered in the integration pro-

cess. The modiﬁed second order differential equation valid for a moving object in

the Earth’s gravity ﬁeld is given by

= f

, (2.10-15)

where r

is the object position, f

is the sensed linear acceleration (speciﬁc force)

and g

describes the gravity acceleration. All magnitudes are given in the inertial

coordinate frame (i-frame). The second order differential equation is transformed to

a set of ﬁrst order equations:

= v

= f

(2.10-16)

Since all linear accelerations f

are originally sensed in system speciﬁc body b-

frame, which is deﬁned by the axes of the sensors, and not equal to the inertial

coordinate frame, an orthogonal transformation matrix R

is necessary to transform

the measurements from the body b to the inertial frame i:

= R

·f

. (2.10-17)

is obtained from

= R

·

, (2.10-18)

where the skew-symmetric matrix 

is formed from the measurements of the

angular rates ω

, given in the b-frame with respect to the i-frame. Similar to

2.10 Sensor Orientation 125

(2.10-17), the matrix R

is used to transform the gravity vector g

, typically given

with respect to the earth-related e-system, to the inertial i-frame.

= R

·g

(2.10-19)

Now all magnitudes in (2.10-16) are given and the motion equations for an object

in the inertial frame i are described as follows:

⎡

⎣

⎤

⎦

⎡

⎣

·f

·g

·

⎤

⎦

(2.10-20)

From (2.10-20) the navigational information (position, velocity and attitude) is

determined with respect to the inertial i-frame.

In regular applications, however, the inertial frame is a somewhat abstract coor-

dinate frame. Navigation information, therefore, is determined in earth-related

coordinates. Thus the resulting navigation states r

have to be transformed

to such an earth-related coordinate frame. For the navigation of a vehicle on Earth

a local level topocentric navigation frame l is commonly used. Alternatively, nav-

igation in a geocentric earth-ﬁxed coordinate frame e is possible as well. If the

navigation process is conducted in such a frame, the rotation of the Earth with

respect to the previously used inertial frame is also taken into account. Without

going into details here, the navigation equations for a moving body with respect

to the earth-centred, earth-ﬁxed e – frame are obtained as follows (Wei and

Schwarz, 1990):

⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

·f

−2 ·

·v

·(

+

)

⎤

⎥

⎦

(2.10-21)

where 

is the skew-symmetric matrix formed from the components of the Earth

rotation vector ω

, which is transformed via R





from the e –frameto

the sensor-speciﬁc body frame b.Theterm2·

·v

corrects for the coriolis force,

which is dependent on the instantaneous speed of the vehicle on the rotating Earth.

The correction for gravity depends on the instantaneous position and is obtained

from appropriate gravity models. The navigation angles obtained from

= R

·R

(2.10-22)

always relate the body frame to the local topocentric navigation frame n.Thever-

tical axis of the n – frame always coincides with the local plumb line. Changes in

the local plumb line directions have to be considered if one transfers the naviga-

tion angles to another, ellipsoid-related, local topocentric frame l. The matrix R

126 2 Foundations and Deﬁnitions

a function of the instantaneous position given in geographic coordinates (,).

From R

the navigation angles (r roll, p pitch, y yaw) are ﬁnally obtained. In the

case where attitude information from inertial navigation is used in photogrammetric

sensor orientation, the different deﬁnition of photogrammetric angles (i.e. ω,φ,κ)

has to be considered. This results in additional transformation steps. The transfor-

mation of navigation angles, the role of the Geoid and the inﬂuence of deﬂections

of the vertical are addressed in more detail in Section 4.9.2.

In order to solve the navigation equations in (2.10-21), initial values for the posi-

tion r

) , velocity v

) and attitude information R

) have to be provided. The

initial values for position and velocity are typically obtained from a static initiali-

sation on a known coordinated reference point or alternatively from GPS surveys.

In such stationary cases the vehicle has zero velocity with respect to the Earth. The

determination of the initial orientation of the system is more complex. As already

mentioned during discussion of stabilized platform systems versus strap-down sys-

tem design, the initial orientation in strap-down navigation has to be accomplished

strictly analytically. Thus a static alignment is performed where the components of

the gravity vector are sensed in the three accelerometer axes. If the system is tilted

from the horizontal, two levelling angles are obtained analytically from the rotation

to achieve zero acceleration measurements in the now levelled two accelerometers.

The heading alignment is then performed in the second step. Based on the mea-

surement of the Earth angular rate vector, the yaw angle is estimated analytically

from the rotation to obtain zero angular rate measurements within one of the two

already levelled angular rate sensors (so called gyro compassing). This zero angular

rate axis is then pointing towards the direction east, because the Earth angular rate

vector consists only of a north and vertical direction component. The whole static

alignment is typically performed as a two-step procedure – the so-called course

alignment ﬁrst followed by a ﬁne alignment afterwards. In the ﬁrst step the pure

sensor signals are used for the rough alignment only, whereas in the reﬁnement step

sensor errors are also estimated and taken into account besides orientation errors.

It has to be mentioned that the Earth angular rate signal is relatively weak com-

pared to the accuracy and noise of most of the gyros in current use. Thus gyro

compassing is not possible for medium or lower accuracy inertial navigation sys-

tems. The noise of the gyro measurements prevents convergence of yaw angle

determination. In such cases, the static alignment process is replaced by so-called

in-motion alignment techniques. Here external information on vehicle position and

velocity (typically provided by GPS) is used to obtain the initial orientation of the

inertial navigation system. This approach is based on the coupling of INS orienta-

tion errors with INS velocity errors, which are observable from the external GPS

The local topocentric coordinate frame and the navigation coordinate frame are both tangential

frames but differ in their axis directions. From the geodetic point of view, the local topocentric

frame is deﬁned as an east–north–vertical (up) right-handed system, whereas the navigation frame

is a north–east–vertical (down) right-handed system. The relationship between these frames is

given as follows:

= R

(π,0, −

) · R

2.10 Sensor Orientation 127

velocity measurements. Such a kinematic approach to system alignment is quite

often used in integrated GPS/inertial systems (i.e. in airborne environments). Thus

it is also named in-air or in-motion alignment.

As was already shown in Table 2.10-2, the accuracy of INS positioning, velocity

and attitude determination is not constant but dependent on time. This is mainly due

to the individual sensor errors, which affect the integration processes to obtain posi-

tion and attitude from observed linear accelerations and angular rates. Additional

errors in the initial starting values, sub-optimal software used for processing (i.e.

incorrectly modelled gravity anomalies) and technical imperfections of the inertial

sensors themselves are the main sources of the accumulating error budget. The sen-

sor errors are mainly due to offset errors (gyro drift, non-zero bias accelerometers),

scale factor errors, sensor noise, non-orthogonalities of sensor axes and acceleration

dependent effects. All this has to be considered in an appropriate error model for

accelerometers and gyros. The navigation equations (2.10-21) are supplemented by

sensor-dependent magnitudes, as can be seen from (2.10-23). With the availability

of sufﬁcient update information these error terms are estimated within the navigation

process.

⎡

⎢

⎣

δ ˙ω

⎤

⎥

⎦

⎡

⎢

⎣

·(f

+δf

) − 2 · 

·v

·(

+δ

+

)

−I

δω

−I

δf

⎤

⎥

⎦

(2.10-23)

In the extended model given above, the accelerometer bias δf

and gyro drift

terms δω

are modelled as time-variant quantities (ﬁrst-order Gauss-Markov pro-

cess; I

α,β

are diagonal matrices containing the reciprocal process correlation times).

The equation system in (2.10-23) now comprises 15 different states. The six sensor-

related error states are independent of the coordinate frame used for the navigation

process. They are modiﬁed according to the instantaneous error behaviour of the

sensors. Although the error model is now modiﬁed and extended, the true error

behaviour is still not modelled perfectly. This again will induce errors in all naviga-

tion steps. Their inﬂuences are only known approximately and have to be controlled

via additional update information. In traditional inertial navigation this error control

is done in static periods, where the integrated vehicle velocity from inertial mea-

surements is compared to zero (so-called zero-velocity update point, ZUPT). If the

vehicle position is known too, the inertially obtained position is also compared to

this reference value (coordinate update point, CUPT). Due to the coupling between

error states, these updates are sufﬁcient to control the inertial error behaviour.

With the advent of GPS, nearly continuous, high-performance position and velocity

information became available. In almost all cases GPS is integrated with inertial

navigation systems to take advantage of this update data. In addition, GPS and INS

are of almost complementary system behaviour: GPS offers high long-term accuracy

128 2 Foundations and Deﬁnitions

but has shortcomings due to short-term noise and limited frequency, whereas iner-

tial navigation provides very frequent data with high short-term accuracy. Within

integrated GPS/inertial systems the absolute positioning accuracy is mainly depen-

dent on the performance of GPS positioning. On the other hand, the accuracy in

integrated attitude determination is mainly due to the speciﬁc gyro noise. Within

integrated inertial/GPS systems continuous positioning is possible even if the GPS

signals are blocked owing to shadowing.

2.10.4 Concepts of Inertial/GPS Integration

The integration of inertial/GPS systems can be done via software- or hardware-

based approaches. If one focuses on the former, the data from GPS and inertial

measurements are combined via decentralized ﬁltering. Within this concept two or

more ﬁlters, separate but working in parallel, interact at certain times. In the case

of an integrated inertial/GPS system, two separate ﬁlters are implemented. Firstly,

the GPS measurements are ﬁltered, but the second ﬁlter is the so-called master ﬁl-

ter. This ﬁlter does more than process IMU data to obtain the position, velocity

and attitude information. Furthermore, the output from the local GPS ﬁlter is used

as pseudo-observations within the master ﬁlter to update its error states. In princi-

ple, the differences between GPS and inertial positions and velocities are used to

estimate the error budget of inertial data processing. Since the errors in positioning

are continuously coupled with the vehicle’s attitude, GPS positioning and velocity

update are sufﬁcient to control fully the error behaviour of inertial attitude determi-

nation as well. The errors of inertial attitudes and IMU sensor errors are estimated

from GPS position and velocity. This is possible because vehicle position always

relies on the relationship between the inertial measurement frame (as deﬁned by

the IMU sensor axes) and the Earth-related navigation coordinate frame. Since this

orientation information is obtained from the IMU angular rate measurements, the

inﬂuence of IMU attitude on positioning and velocity is obvious.

The optimal estimation of system error states is typically done via Kalman ﬁl-

tering. This ﬁlter is based on the set of differential equations given in (2.10-23).

Kalman ﬁltering has been used for the recursive estimation of time-discrete linear

processes, primarily in navigation applications, since around 1960, mainly for the

following reasons (Brown and Hwang, 1992).

Kalman ﬁltering facilitates real-time processing of measurements. The errors

in navigation are controlled by appropriate update information in real time.

Secondly, the dynamics of the system become a linear model of the errors, i.e.

the linearisation of the system dynamics is done with sufﬁcient accuracy.

Thirdly, in navigation, multi-sensor system conﬁgurations are regularly used,

so different types of input and output data have to be considered. Since all

of this data has its speciﬁc accuracy behaviour, such a ﬁlter constellation is

ideal for two-way updates.

2.10 Sensor Orientation 129

Fourthly, the ﬁnal output is obtained with the best estimated accuracy, based on

an optimal combination of input values in the ﬁlter.

Detailed information on the generic Kalman ﬁlter equations can be obtained from

the literature, for example, Grewal and Andrews (1993), Brown and Hwang (1992)

or Gelb (1974). In navigation the time-discrete formulation of the algorithm is of

relevance.

The update in Kalman ﬁltering is possible in two different ways: In the

feed-forward conﬁguration, the navigation errors and system states are estimated

and used for the correction of the integrated navigation information afterwards.

Alternatively, within so-called feedback or closed-loop error control, the IMU sen-

sors are continuously calibrated ranther than the output navigation information.

After each update cycle the estimated error states are fed back into the navigation

process of inertial data processing. In this case the inertial raw measurements (linear

acceleration and angular rate) themselves are improved.

Filtering always provides the best estimated trajectory information at a certain

point of time t

, which already includes all previous measurements at epochs t

≤ t

. Smoothing is done, on the other hand, the opposite way, i.e. in the negative time

direction. Here all future observations are used to estimate the states at the cur-

rent time epoch t

. Obviously, ﬁlter processes have real-time capability, which is

an indispensable requirement for navigation applications. Smoothing is only per-

formed during post-processing. The combination of ﬁltering and smoothing in a

post-processing approach increases the accuracy of the estimation of system errors.

Such an approach is recommended for high precision trajectory determination.

Highly precise navigation data, for example, is required for the direct georeferencing

of airborne sensors.