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INTEGRATED MOTION MEASUREMENT OF MULTIBODY SYSTEMS AND FLEXIBLE VEHICLE STRUCTURES
INTEGRATED MOTION MEASUREMENT OF MULTIBODY SYSTEMS AND FLEXIBLE VEHICLE STRUCTURES Jörg F. Wagner, Thorsten Örtel Universität Stuttgart Institute of Statics and Dynamics of Aerospace Structures 70550 Stuttgart, Germany
Abstract: Integrated navigation systems based on inertial sensors and GPS are wellestablished devices for vehicle guidance. The system design is traditionally based on the assumption that the vehicle is a rigid body. However, generalizing such integrated systems to rigid multibody or even more to flexible structures is possible. It is based on distributed sensors and provides an interesting basis for motion control and system identification. The kernel of the integrated system consists of an observer that estimates the motion state of the mechanical structure. The paper presents the multibody and the flexible structure approach as well as first motion estimation results. Copyright © 2006 IFAC Keywords: Navigation systems, observers, data fusion, inertial sensors, robotics, large structures, kinematics.
1. INTRODUCTION Integrated navigation systems based on gyros, accelerometers, and satellite navigation receivers are wellestablished devices for vehicle guidance. The central system design requires modelling the kinematics of the vehicle, which is traditionally assumed to be simply a single rigid body (Farrell and Barth, 1999). However, this premise is no longer reasonable if the system layout includes sensors being distributed spaciously over the vehicle and if in parallel the vehicle changes its shape due to structural flexibilities (e.g. large aircraft) or due to a multibody structure (e.g. mobile robots). In this case, generalizing the theory of integrated navigation systems becomes necessary. This approach is outlined in the following and provides not only an interesting basis for an extensive motion control but also for system identification. Firstly, Section 2 presents the principle of integrated navigation systems and illustrates that this is more precisely a matter of integrated motion measurement
of a given mechanical structure. Including simulation and experimental results, Section 3 contains a synopsis of the extended system theory for multibody structures, and Section 4 for flexible vehicles. Section 5 summarises the outcome and specifies future work.
2. INTEGRATED MOTION MEASUREMENT The idea of integrated navigation systems consists of combining complementary motion measuring principles and of utilising their specific advantages: Inertial sensors like classical or like modern micro-electromechanical gyros and accelerometers are used to obtain reliable signals being usable for a short period of time and allowing a high resolution with time. On the other hand, less dependable sensors (often with relevant signal delays) like GPS receivers and radar units are used due to their good long-term accuracy. The kernel of integrated navigation systems is an observer (typically realised by an extended Kalman filter (Gelb, 1989)) blending the sensor signals and estimating the
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relevant vehicle motion (Farrell and Barth, 1999). Besides the sensor combination employed, the theoretical basis for the filter requires a kinematical model of the vehicle motion considered, which has to be set up individually (nevertheless, established models exist (Wagner and Wieneke, 2003)). This model describes the standardised dynamics of the vehicle, mostly by means of specific forces, i.e. accelerations, and of angular rates. Hence, there are no dynamometers or mass properties needed in this approach. There are different system integration variants of such navigation systems (Wagner and Wieneke, 2003). However as mentioned, they all have the observer principle in common with the signal flow depicted in Figure 1: Reliable sensors of high availability permit a good resolution with time (like accelerometers and gyros) and provide the input signal vector u generating the vehicle motion considered (state vector x). Based on x and u, so-called aiding sensors (being mostly attached to the vehicle) like a GPS receiver or a laser altimeter provide measurement signals (vector y) with good long-term accuracy. Furthermore, there is a parallelism between the performance of the real moving structure and its aiding equipment on one side and a motion and aiding simulation on the other side. The simulation takes place within the actual observer, it is based on two kinematical models, and it leads to estimates xˆ and yˆ of x and y. The first model describes the motion considered and is a set of ordinary nonlinear differential equation (being solved numerically); the second one emulates the aiding: xˆɺ (t ) = f (xˆ (t ), u(t )) , (1) yˆ (t ) = h (xˆ (t ), u(t )) .
(2)
Due to sensor, modelling and initialisation errors, the estimates show inaccuracies, which increase usually with time t and which require therefore a correction: The feedback of the difference between y and yˆ serves as input of a compensation device adjusting xˆ by K (y−yˆ ) . The correction matrix K(t) is typically part of the algorithm of an extended Kalman filter, however sometimes alternatives like particle filters are used as well (Yi and Grejner-Brzezinska, 2006). Independently from the observer type, the system stability requires that x is completely observable. Reflecting the type and geometrical array of the sensors used (see below), the content of f and h determines this property. The observability is ensured if the matrix
observer
input u
moving structure
motion state x u
aiding aiding signals equipment y
+ motion model estimate xˆ aiding model estimate u yˆ = h ( xˆ , u ) – yˆ xˆɺ = f ( xˆ , u ) compensation ∆xˆ = K (y −yˆ )
Fig. 1. Observer principle used for signal fusion.
n−1 (3) Ξ = H T FT ⋅ H T … (FT ) ⋅ H T employing the Jacobians F(x(t),u(t)) and H(x(t),u(t)) of f and h has full rank n (Gelb, 1989) with n being the number of state variables in x. As F and H vary with time, it is possible that phases of complete observability alternate with phases of reduced observability. This applies especially for the classical combination of inertial sensors with a single antenna GPS receiver during periods of steady vehicle motion (Hong et al., 2000). To solve this problem effectively, several aiding antennas are required that have to be distributed spaciously over the vehicle (Wagner, 2003).
Assuming consequently distributed sensors, the idea of a single rigid body being the moving structure becomes doubtful. On the other hand, the theory of motion modelling for integrated navigation can be extended to rigid multibody systems and also to flexible structures. This is outlined in the next two sections.
3. RIGID MULTIBODY SYSTEMS The kinematics (i.e. time variable geometry) of rigid multibody systems can be conveniently and completely described with a minimal set of generalised coordinates q(t) and minimal set of (pseudo) velocities πɺ (t ) (Bremer and Pfeifer, 1992). In case, the system is holonomic, qɺ and πɺ are identical. Otherwise, usual mechanical structures show a linear relation between qɺ and πɺ , which reduces the number of degrees of freedom from the level of position/attitude to the level of velocities. Employing the Jacobian Jq and the term q′, this equation reads in general:
qɺ = J q ( q, t ) πɺ + q′( q, t ) .
(4)
Assuming a rigid multibody system equipped with µ accelerometers j (i.e. j = 1, …, µ) and ν gyros k (i.e. k = µ+1, …, µ+ν), each being strapped down to one of the rigid components, the following general relations i describe firstly the position irj , the velocity i rɺ j , and i the acceleration i rɺɺj of an accelerometer attachment point. In these relations, the subscript on the lower left side indicates the coordinate system for representing the respective vector, whereas the upper left subscript describes that the vector has been differentiated with respect to the indicated coordinate system. In both cases, an inertial frame i is employed here: i
r j = i r j(q, t ) ,
i i
rɺ j = J r (q, t ) qɺ +
i i
ɺɺr j = J r (q, t ) q ɺɺ +
j
j
(5a)
∂ i rj ∂t d Jr
dt
with J r = j
j
qɺ +
∂ i rj ∂q
d ∂ i r j . d t ∂t
, (5b) (5c)
The next equations describe the attitude iγk (e.g. three Euler angles), the angular rate iωk , and the angular acceleration ii ωɺ k at a gyro attachment point. As large rotations iγk have not the character of a Cartesian vector like irj or iωk , only similar relations between the atti-
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tude and the angular velocity can be established. Fortunately, this has no influence to the following theory. i
γ k = i γ k(q, t ) ,
i
ω k = J γ (q, t ) qɺ + ω ′k(q, t ) k
ζ
i
A
η
i
(6a)
ρ 1 = y1 l1
q1 q2
g
u3
(6b)
d γk ∂γ ∂γ ⇒ J γ ≠ k , ω ′k ≠ k , k dt ∂q ∂t d Jγ d ω k′ i k ɺ ɺɺ qɺ + . (6c) i ω k = J γ k (q, t ) q + dt dt
e2
with i ω k ≠
An inertial sensor detects the projection of the local acceleration and local angular rate vector respectively on the relevant sensor input axis. This can be expressed with the unit vector ej and accordingly ek having the direction of this axis, which is a function of γ at the respective attachment point. Furthermore, a compensation of the negative gravity vector (– g) is necessary if accelerometers are used. Then, the acceleration signal aj and the angular rate signal Ωk at the sensor points can be described as follows.
ɺɺ ) + i g) , a j = i e j(q)⋅( ii rɺɺj(q, qɺ , q
(7a)
Ωk = i e k(q )⋅ i ω k(q, qɺ ).
(7b)
A combination of all left sides leads to
a = [a1 … aµ ]T ,
(8a) T
Ω = [Ωµ+1 … Ωµ+ν ] .
(8b)
Furthermore, equations (4) to (6) have to be introduced in the according arrangement of the right sides of equations (7). This leads to vector functions being ɺɺ or π ɺɺ for accelerometer siglinear with respect to q nals and qɺ or πɺ for gyro signals. If now the state vector x is appropriately composed of q and qɺ or q and πɺ , equation (1) results from rearranging these linear vector functions with respect to xɺ (Wagner, 2004). Then a and Ω form together u. Assuming moreover aiding signals that consist of range and angle measurements and form together the vector y, the function h can easily be derived from equations (5a) and (6a). Example. Without referring completely to the consideration above, which has partially rather the character of a proof, it is possible to illustrate this theory by the simple example of a double pendulum. For this, Figure 2 shows the mechanical structure, the definition of all generalised coordinates, of all components of the inertial input vector u, and of all components of the aiding vector y: Two stiff rods (length l1 , l2) are pivoted to each other. Freely swivelling, the end A of the upper one slides additionally on a rail and is joined to a wall with a spring. The general coordinates are the transverse distance q1 and the angular positions q2 , q3 of the rods. Two accelerometers are attached to the lower joint (u1 , u2) and two gyros are fixed elsewhere to one rod each (u3 , u4); the measurement axes are indicated by four arrows. The aiding equipment shall consist of three radar units measuring the distances ρ1 , ρ2 , ρ3 to a nearby wall.
u1
u2
e1 ρ 2 = y2
l2 q3 u4 ρ 3 = y3
d
Fig. 2. Example of a rigid multi body system. The state vector x consists of all generalised coordinates and the time derivative of q1 :
x = [ q1
q2
q3
T
qɺ1 ] := [ x1
x2
x3
T x4 ] . (9)
The reason for omitting qɺ 2 and qɺ 3 in x is that these values are directly measurable by the gyros and form therefore a part of u instead of x. This is in conformity with solving equation (7b) for qɺ . Detecting qɺ1 is only feasible indirectly through qɺɺ1 provided by the accelerometers. This corresponds to solving equation (7a) for q ɺɺ . Appropriately, the function f reads: x4 u3 . (10) xɺ = f ( x,u ) = u 4 2 (u1 − l1uɺ3 ) cos x2 − (u2 − l1u3 ) sin x2
The last component of f shows two special features. Firstly, it was possible to remove the influence of the gravity. Secondly, effects of angular acceleration and of centripetal forces appear. The latter characteristic is a consequence of distributing accelerometers over the structure. It is new compared to classical inertial navigation systems. Thus, angular acceleration measurements have to augment u. Such signals can be generated by numerically differentiating some Ωk or by new inertial sensors (STM, 2004). Accordingly, the input vector for the example considered is: u = [u1
u2
u3
u4
T uɺ3 ] .
(11)
The aiding vector y consists of three distance measurements. The geometrical relations for modelling these signals can be directly inferred from Figure 2: y = [ρ1
ρ2
T ρ3 ] ,
(12)
d − x1 . (13) h ( x, u ) = d − x1 − l1 sin x2 d − x − l sin x − l sin x 1 1 2 2 3 The Kalman filter requires completing the modelling with stochastic sensor errors, which is achieved using white noise with a specific variance for each measurement device (Wagner, 2004). Furthermore, the
539
functions f and h are nonlinear, f, x, and u are quasicontinuous with time (the sampling rate of u is typically high), h and y are discrete with time (the sampling rate of y is often low and unsteady). Thus, the so-called continuous-discrete extended Kalman filter has to be employed (Gelb, 1989). System test. Prepared by simulated measurements, the system as described above was experimentally realised to test its functionality and performance. For this, the usual compensation of the inertial sensor biases had additionally to enter the system design (Farrell and Barth, 1999). As an example of the results, Figure 3 shows a comparison of the aiding signal y1 and the estimated transverse distance q1 for the pendulum being at rest. All of the sensors used were products designed for mass markets with an accuracy of 0.05 m/s2 (accelerometers), 0.2°/s (gyros), 0.02 m (radar noise), and 2 m (radar jamming). Nevertheless, the remaining noise of the estimate is remarkably low. Wagner (2004) gives further results and additional details about the general modelling procedure, the sensors used, and the interference resistance being a special feature of integrated measurement systems.
body frame b
bζ
rB
ℓ0j
rj
η sk
ζ sk
η bη
iη
inertial frame i iζ
Fig. 4. Aircraft fuselage cross section with distorted wing half and peripheral sensors j and k. any peripheral sensor. Again, the subscript on the upper left side indicates the coordinate system, in which the differentiation takes place. The angular rate ωbi describes additionally the angular velocity between the body and the inertial frames b and i. Assuming furthermore small distortions of the structure (a presumption being largely adequate for flexible vehicles), the actual lever arm ℓ j and its body oriented time derivatives can be approximated by a finite series reflecting e.g. the main vibration modes: Γa
Figure 4 represents the exemplary initial point of the corresponding theory for flexible structures. It shows an aircraft fuselage with an exaggeratedly distorted wing half attached, the original wing shape being indicated by the dashed line. A body-fixed coordinate system b with origin B serves for describing the timevariant structural geometry. An inertial measurement unit (IMU; 3 gyros, 3 accelerometers) is located at B and µ additional accelerometers j and ν further gyros k are placed on the wing measuring aj and Ωk in the sensor frames. The accelerometer attachment points are subject to the acceleration i i b ɺɺ b ɺ ɺɺ ɺɺ b r j = b rB + b ℓ j + 2 ( b ω bi × b ℓ j ) (14) + bi ωɺ bi × b ℓ j + b ω bi ×( b ω bi × b ℓ j ) .
In this equation, r is an inertial position vector, and ℓ denotes the actual lever arm between the IMU and 1.85
qˆ1
qˆ1 , ρ1 [m]
1.80 t [s]
sj
∆ℓ j
b
2000
sensor frames j, k
ζ
ℓj
B
4. FLEXIBLE STRUCTURES
ρ1
sj
3000
Fig. 3. Comparison of the estimate qˆ1 and of ρ1 for the pendulum of Figure 2 being at rest.
ℓ j(t )= b ℓ 0 j + b ∆ℓ j(t ) ≈ b ℓ 0 j + ∑ bχ(t )⋅ b s χ( b ℓ 0 j ) , (15) χ=1
b b
i
Γa
ℓ j(t ) ≈ ∑ bɺχ(t )⋅ bs χ( b ℓ 0 j ) ,
(16)
χ=1
b b
ii
Γa
ℓ j(t ) ≈ ∑ bɺɺχ(t )⋅ bs χ( b ℓ 0 j ).
(17)
χ=1
All time dependent “amplitudes” bχ(t) represent extra degrees of freedom, all s χ (ℓ 0 ) are the vectorial descriptions of Γa deformation modes selected for describing the structural distortions. Besides, every of these functions of the structural position ℓ 0 is defined to have at least at one point the displacement value 1. Thus, bχ(t) is the resultant position shift of this point if only the respective mode sχ is excited. Introducing equations (15) - (17) into (14) for all µ accelerometers and considering again the unit vectors ej = ej(b1, ..., bµ) for the measurement axes (part of the sensor frames), a set of linear function for all bɺɺχ results on the right. If µ = Γa holds and the sensors are placed appropriately, solving this set of equations for each single bɺɺχ is possible (if µ > Γa holds, the employment of a Least Squares procedure is viable). Composing now the state vector x of all bχ and their first time derivatives, the composition of f(x,u) (equation (1)) follows directly. However, a completion of f by the classical equations of an inertial navigation system is also necessary (Wagner, 2003). Then, the input vector u contains all signals of the IMU and of the peripheral accelerometers. Instead of the accelerometers, the use of gyros as input sensors is also feasible. A gyro k is subject to the angular velocity
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1 bd (18) (curl b∆ℓ k (t )) . 2 dt For modelling the structural deformations ΓΩ other functions sχ , χ = Γa+1, ..., Γa+ΓΩ shall be used: b
ω ki =bω bi +
Γ α +Γ Ω d (curl b∆ℓ k (t )) ≈ ∑ bɺχ(t )⋅ curl bs χ ( b ℓ 0 k ) . (19) dt χ=Γ α +1
b
Introducing equations (19) into (18) for all ν gyros and employing also the unit vectors ek for the measurement axes, a set of linear function for the new bɺχ results on the right. For ν ≥ ΓΩ and appropriately positioned gyros, solving this set of equations for each single bɺχ is now possible. Compiling the state vector x by all new bχ, the components of f(x,u) follow directly (the input u contains now all gyro signals including those of the IMU). Employing a mixture of gyros and accelerometers is also possible. However, this case requires some decoupling conditions (Wagner, 2003) to avoid that some bχ are detected equally both from gyros and from accelerometers (this would lead to the problem that these bχ would appear twice in the state vector): (20a) curl s χ |ℓ = 0 , χ = 1, ..., Γ a , 0k
s χ |ℓ = 0 , χ = Γ a + 1, ..., Γ a + Γ Ω . 0j
(20b)
Besides these equations, the numbers Γa and ΓΩ as well as the determination of the deformation modes sχ depend on the vehicle structure, the relevant motion excitation and on the accuracy to be achieved. Assuming furthermore aiding sensors like GPS antennas, radar units, or strain gauges, the components of h follow from applying equation (15) for modelling distances, structural strains, etc. Example. This theory is illustrated as well by an example representing now a simple model for a wing half (Figure 5). The moving structure consists of a flexible beam of length l, constant thickness h, and flexural stiffness EI. Again, one end slides on a rail
and is joined to a wall with a spring. Representing the IMU, two accelerometers and one gyro are fixed to the bar at the swivel joint and generate the signals u1 , u2 (linear accelerations), u6 (angular rate), and u7 (angular acceleration). Furthermore, three peripheral accelerometers (u3 , u4 , u5) supply input signals, and three deformation modes sχ , χ = 1, 2, 3 are considered. The aiding consists now of four radar units measuring the distances ρ and velocities ρɺ (using Doppler shifts). Additionally, strain gauges provide three aiding strain signals ε. The state vector x contains now nine components:
x = d dɺ ψ b1 bɺ1 b2 = [ x1 ... x9 ].
bɺ2
b3
T bɺ3
(21)
The reason for omitting ψɺ in x is again that this term can be measured directly. Hence, it is part of u (i.e. ψɺ = u6 ). Employing moreover especially equations (14) – (17), the kinematical model (1) describing the motion of the beam reads: x2 u1 cos x3 + u2 sin x3 u6 x5 f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...) { } 1 3 7 2 4 7 3 5 7 χ=1 . f = x7 { f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...)} 2 4 7 3 5 7 1 3 7 χ= 2 x9 { f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...)} 1 3 7 2 4 7 3 5 7 χ=3 (22) The expressions of bɺɺ1 = xɺ5 , bɺɺ2 = xɺ 7 , bɺɺ3 = xɺ9 are not explicitly shown due to an extensive amount of space to be needed (Wagner, 2003). Nevertheless, it is intended to publish them in due time in a PhD thesis. The aiding vector y consists now of eleven signals as shown in Figure 5: T
y = [ ρ0 ρɺ 0 ρ1 ρɺ1 ρ2 ρɺ 2 ρ3 ρɺ 3 ε1 ε2 ε3 ] . (23)
u2
u6 , u7
ρ0 , ρɺ 0 u1
g
ψ bη
ε1 ru , rρ 3
3
rρ
2
...
iζ iη
u3
ε2
ρ1 , ρɺ1
u4
ρ 2 , ρɺ 2 ε3 u5
The last part is the aiding model: To reproduce the strain measured, the theory of bending elastic beams with thickness h and flexural stiffness EI has to be used (Roark and Young, 1986). In this case, the deformation modes are no longer vector functions but scalar functions sχ(bη) representing the cross displacements. Furthermore, the introduction of several auxiliary variables is helpful. For this, the sensor distances r as indicated in Figure 5 have to be considered, as well as derivatives of the bending line with respect to the structural coordinate bη: γ j = rρ ⋅ sin ψ + ν j ⋅ cos ψ , j
ρ3 , ρɺ 3
( ) ( ) ( ) = bɺ ⋅ s (r ) + bɺ ⋅ s (r ) + bɺ ⋅ s (r ) ,
ν j = b1 ⋅ s1 rρ + b2 ⋅ s2 rρ + b3 ⋅ s3 rρ , j
d
Fig. 5. Example of a flexible structure.
νɺ j
1
1
ρj
j
2
2
ρj
j
3
3
ρj
541
j
0.20
d 2 sχ
,
d bη 2 b
η = rε
0.15
j
x1 x2 x1 + γ1 x2 + u6 γ1 + νɺ1 cos ψ x1 + γ 2 x2 + u6 γ 2 + νɺ 2 cos ψ x1 + γ 3 h = x2 + u6 γ 3 + νɺ 3 cos ψ h b1 ⋅ s1′′ rε + b2 ⋅ s2′′ rε + b3 ⋅ s3′′ rε 1 1 1 2 EI h 2 EI b1 ⋅ s1′′ rε2 + b2 ⋅ s2′′ rε2 + b3 ⋅ s3′′ rε2 h b ⋅ s ′′ r + b2 ⋅ s2′′ rε + b3 ⋅ s3′′ rε 3 3 2 EI 1 1 ε3
( ( ) ( ( ) ( ( )
( ) ( ) ( )
. (24)
( )) ( )) ( ))
The continuous-discrete extended Kalman filter used for data fusion requires again completing the modelling with stochastic sensor errors being modelled by white noise with a specific variance for each measurement device. System test. To get an assessment about the modelling quality and system performance for the example given above and to prepare an analogous experimental setup, simulated measurement and reference data were generated by a finite element program (ABAQUS) using two dimensional beam elements and a nonlinear dynamics analysis. A further analysis calculated the linear Eigenmodes of the system, which served as deformation modes sχ . A compensation of the inertial sensor biases was again introduced into f and h. Once more, the low-cost sensors mentioned above were the basis for determining the sensor noise characteristics. The spring at the upper end of the beam and the flexural stiffness EI got such values that the first vibration modes had an order of magnitude of 1 Hz. The length l of the beam was 4 m. As an example of first simulation results for integrated system, Figure 6 shows the estimation error of the elastic deformation at the lower end of the beam. For this, the beam was exited by a sinusoidal force at its upper end. The outcome shows a stochastic inaccuracy of about 5 cm. Nevertheless, the filter performance can be optimised by filter tuning, by sensor positioning, and by adding also gyros as peripheral sensors. Instead of vibration modes, other methods from the theory of model reduction can also be considered to find suitable deformation functions sχ . 5. CONCLUSIONS The paper above is an outline of the theory of generalizing integrated navigation systems to spaciously distributed sensors. This requires especially an adequate kinematical description of the moving struc-
estimation error [m]
( )
sχ′′ rε =
0.10 0.05 0 -0.05 -0.10 -0.15 900
920
940
t [s]
960
980
1000
Fig. 6. Estimation error of the elastic displacement bˆ1s1 (l ) + bˆ2 s2 (l ) + bˆ3 s3 (l ) at the end of the beam. ture. As demonstrated with two examples, such an approach is feasible and enables an extensive motion measurement of mechatronic structures. Compared to classical inertial navigation, now the measurement of angular acceleration is necessary. Furthermore, elastic structures require in addition a model reduction. Future work will focus on the experimental verification of the flexible structure approach and on improving the ongoing simulations by filter tuning, optimisation concerning sensor type and placement, and employing other types of model reduction.
REFERENCES Bremer, H. and F. Pfeiffer (1992). Elastische Mehrkörpersysteme. Teubner, Stuttgart. Farrell, J.A. and M. Barth (1999). The Global Positioning System and inertial navigation. McGrawHill, New York. Gelb, A. (Ed.) (1989). Applied optimal estimation. 11th Printing. The M.I.T. Press, Cambridge/MA. Hong, S., J.-H. Lee, J. Rios, J.L. Speyer (2000). Observability analysis of GPS aided INS. In: Proceedings of ION GPS 2000, pp. 2618–2624. The Institute of Navigation, Alexandria/VA. Roark, R.J. and W.C. Young (1986). Formulas for stress and strain. 5th ed. McGraw-Hill, Auckland. STM (2004). LIS1Y08AS4 Inertial sensor. Product Preview. Rev. 1.1. STMicroelectronics, Geneva. Wagner, J.F. (2003). Zur Verallgemeinerung integrierter Navigationssysteme auf räumlich verteilte Sensoren und flexible Fahrzeugstrukturen. VDIVerlag, Düsseldorf. Wagner, J.F. (2004). Adapting the principle of integrated navigation systems to measuring the motion of rigid multibody systems. Multibody Systems Dynamics, 11, 87-110. Wagner, J. and T. Wieneke (2003). Satellite and inertial navigation – conventional and new fusion approaches. Control Engin. Practice, 11, 543-550. Yi, Y. and D.A. Grejner-Brzezinska (2006). Performance comparison of the nonlinear Bayesian filters supporting GPS/INS integration. In: Proceedings of the ION NTM 2006, pp. 977-983. The Institute of Navigation, Fairfax/VA.
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Abstract: Integrated navigation systems based on inertial sensors and GPS are wellestablished devices for vehicle guidance. The system design is traditionally based on the assumption that the vehicle is a rigid body. However, generalizing such integrated systems to rigid multibody or even more to flexible structures is possible. It is based on distributed sensors and provides an interesting basis for motion control and system identification. The kernel of the integrated system consists of an observer that estimates the motion state of the mechanical structure. The paper presents the multibody and the flexible structure approach as well as first motion estimation results. Copyright © 2006 IFAC Keywords: Navigation systems, observers, data fusion, inertial sensors, robotics, large structures, kinematics.
1. INTRODUCTION Integrated navigation systems based on gyros, accelerometers, and satellite navigation receivers are wellestablished devices for vehicle guidance. The central system design requires modelling the kinematics of the vehicle, which is traditionally assumed to be simply a single rigid body (Farrell and Barth, 1999). However, this premise is no longer reasonable if the system layout includes sensors being distributed spaciously over the vehicle and if in parallel the vehicle changes its shape due to structural flexibilities (e.g. large aircraft) or due to a multibody structure (e.g. mobile robots). In this case, generalizing the theory of integrated navigation systems becomes necessary. This approach is outlined in the following and provides not only an interesting basis for an extensive motion control but also for system identification. Firstly, Section 2 presents the principle of integrated navigation systems and illustrates that this is more precisely a matter of integrated motion measurement
of a given mechanical structure. Including simulation and experimental results, Section 3 contains a synopsis of the extended system theory for multibody structures, and Section 4 for flexible vehicles. Section 5 summarises the outcome and specifies future work.
2. INTEGRATED MOTION MEASUREMENT The idea of integrated navigation systems consists of combining complementary motion measuring principles and of utilising their specific advantages: Inertial sensors like classical or like modern micro-electromechanical gyros and accelerometers are used to obtain reliable signals being usable for a short period of time and allowing a high resolution with time. On the other hand, less dependable sensors (often with relevant signal delays) like GPS receivers and radar units are used due to their good long-term accuracy. The kernel of integrated navigation systems is an observer (typically realised by an extended Kalman filter (Gelb, 1989)) blending the sensor signals and estimating the
537
relevant vehicle motion (Farrell and Barth, 1999). Besides the sensor combination employed, the theoretical basis for the filter requires a kinematical model of the vehicle motion considered, which has to be set up individually (nevertheless, established models exist (Wagner and Wieneke, 2003)). This model describes the standardised dynamics of the vehicle, mostly by means of specific forces, i.e. accelerations, and of angular rates. Hence, there are no dynamometers or mass properties needed in this approach. There are different system integration variants of such navigation systems (Wagner and Wieneke, 2003). However as mentioned, they all have the observer principle in common with the signal flow depicted in Figure 1: Reliable sensors of high availability permit a good resolution with time (like accelerometers and gyros) and provide the input signal vector u generating the vehicle motion considered (state vector x). Based on x and u, so-called aiding sensors (being mostly attached to the vehicle) like a GPS receiver or a laser altimeter provide measurement signals (vector y) with good long-term accuracy. Furthermore, there is a parallelism between the performance of the real moving structure and its aiding equipment on one side and a motion and aiding simulation on the other side. The simulation takes place within the actual observer, it is based on two kinematical models, and it leads to estimates xˆ and yˆ of x and y. The first model describes the motion considered and is a set of ordinary nonlinear differential equation (being solved numerically); the second one emulates the aiding: xˆɺ (t ) = f (xˆ (t ), u(t )) , (1) yˆ (t ) = h (xˆ (t ), u(t )) .
(2)
Due to sensor, modelling and initialisation errors, the estimates show inaccuracies, which increase usually with time t and which require therefore a correction: The feedback of the difference between y and yˆ serves as input of a compensation device adjusting xˆ by K (y−yˆ ) . The correction matrix K(t) is typically part of the algorithm of an extended Kalman filter, however sometimes alternatives like particle filters are used as well (Yi and Grejner-Brzezinska, 2006). Independently from the observer type, the system stability requires that x is completely observable. Reflecting the type and geometrical array of the sensors used (see below), the content of f and h determines this property. The observability is ensured if the matrix
observer
input u
moving structure
motion state x u
aiding aiding signals equipment y
+ motion model estimate xˆ aiding model estimate u yˆ = h ( xˆ , u ) – yˆ xˆɺ = f ( xˆ , u ) compensation ∆xˆ = K (y −yˆ )
Fig. 1. Observer principle used for signal fusion.
n−1 (3) Ξ = H T FT ⋅ H T … (FT ) ⋅ H T employing the Jacobians F(x(t),u(t)) and H(x(t),u(t)) of f and h has full rank n (Gelb, 1989) with n being the number of state variables in x. As F and H vary with time, it is possible that phases of complete observability alternate with phases of reduced observability. This applies especially for the classical combination of inertial sensors with a single antenna GPS receiver during periods of steady vehicle motion (Hong et al., 2000). To solve this problem effectively, several aiding antennas are required that have to be distributed spaciously over the vehicle (Wagner, 2003).
Assuming consequently distributed sensors, the idea of a single rigid body being the moving structure becomes doubtful. On the other hand, the theory of motion modelling for integrated navigation can be extended to rigid multibody systems and also to flexible structures. This is outlined in the next two sections.
3. RIGID MULTIBODY SYSTEMS The kinematics (i.e. time variable geometry) of rigid multibody systems can be conveniently and completely described with a minimal set of generalised coordinates q(t) and minimal set of (pseudo) velocities πɺ (t ) (Bremer and Pfeifer, 1992). In case, the system is holonomic, qɺ and πɺ are identical. Otherwise, usual mechanical structures show a linear relation between qɺ and πɺ , which reduces the number of degrees of freedom from the level of position/attitude to the level of velocities. Employing the Jacobian Jq and the term q′, this equation reads in general:
qɺ = J q ( q, t ) πɺ + q′( q, t ) .
(4)
Assuming a rigid multibody system equipped with µ accelerometers j (i.e. j = 1, …, µ) and ν gyros k (i.e. k = µ+1, …, µ+ν), each being strapped down to one of the rigid components, the following general relations i describe firstly the position irj , the velocity i rɺ j , and i the acceleration i rɺɺj of an accelerometer attachment point. In these relations, the subscript on the lower left side indicates the coordinate system for representing the respective vector, whereas the upper left subscript describes that the vector has been differentiated with respect to the indicated coordinate system. In both cases, an inertial frame i is employed here: i
r j = i r j(q, t ) ,
i i
rɺ j = J r (q, t ) qɺ +
i i
ɺɺr j = J r (q, t ) q ɺɺ +
j
j
(5a)
∂ i rj ∂t d Jr
dt
with J r = j
j
qɺ +
∂ i rj ∂q
d ∂ i r j . d t ∂t
, (5b) (5c)
The next equations describe the attitude iγk (e.g. three Euler angles), the angular rate iωk , and the angular acceleration ii ωɺ k at a gyro attachment point. As large rotations iγk have not the character of a Cartesian vector like irj or iωk , only similar relations between the atti-
538
tude and the angular velocity can be established. Fortunately, this has no influence to the following theory. i
γ k = i γ k(q, t ) ,
i
ω k = J γ (q, t ) qɺ + ω ′k(q, t ) k
ζ
i
A
η
i
(6a)
ρ 1 = y1 l1
q1 q2
g
u3
(6b)
d γk ∂γ ∂γ ⇒ J γ ≠ k , ω ′k ≠ k , k dt ∂q ∂t d Jγ d ω k′ i k ɺ ɺɺ qɺ + . (6c) i ω k = J γ k (q, t ) q + dt dt
e2
with i ω k ≠
An inertial sensor detects the projection of the local acceleration and local angular rate vector respectively on the relevant sensor input axis. This can be expressed with the unit vector ej and accordingly ek having the direction of this axis, which is a function of γ at the respective attachment point. Furthermore, a compensation of the negative gravity vector (– g) is necessary if accelerometers are used. Then, the acceleration signal aj and the angular rate signal Ωk at the sensor points can be described as follows.
ɺɺ ) + i g) , a j = i e j(q)⋅( ii rɺɺj(q, qɺ , q
(7a)
Ωk = i e k(q )⋅ i ω k(q, qɺ ).
(7b)
A combination of all left sides leads to
a = [a1 … aµ ]T ,
(8a) T
Ω = [Ωµ+1 … Ωµ+ν ] .
(8b)
Furthermore, equations (4) to (6) have to be introduced in the according arrangement of the right sides of equations (7). This leads to vector functions being ɺɺ or π ɺɺ for accelerometer siglinear with respect to q nals and qɺ or πɺ for gyro signals. If now the state vector x is appropriately composed of q and qɺ or q and πɺ , equation (1) results from rearranging these linear vector functions with respect to xɺ (Wagner, 2004). Then a and Ω form together u. Assuming moreover aiding signals that consist of range and angle measurements and form together the vector y, the function h can easily be derived from equations (5a) and (6a). Example. Without referring completely to the consideration above, which has partially rather the character of a proof, it is possible to illustrate this theory by the simple example of a double pendulum. For this, Figure 2 shows the mechanical structure, the definition of all generalised coordinates, of all components of the inertial input vector u, and of all components of the aiding vector y: Two stiff rods (length l1 , l2) are pivoted to each other. Freely swivelling, the end A of the upper one slides additionally on a rail and is joined to a wall with a spring. The general coordinates are the transverse distance q1 and the angular positions q2 , q3 of the rods. Two accelerometers are attached to the lower joint (u1 , u2) and two gyros are fixed elsewhere to one rod each (u3 , u4); the measurement axes are indicated by four arrows. The aiding equipment shall consist of three radar units measuring the distances ρ1 , ρ2 , ρ3 to a nearby wall.
u1
u2
e1 ρ 2 = y2
l2 q3 u4 ρ 3 = y3
d
Fig. 2. Example of a rigid multi body system. The state vector x consists of all generalised coordinates and the time derivative of q1 :
x = [ q1
q2
q3
T
qɺ1 ] := [ x1
x2
x3
T x4 ] . (9)
The reason for omitting qɺ 2 and qɺ 3 in x is that these values are directly measurable by the gyros and form therefore a part of u instead of x. This is in conformity with solving equation (7b) for qɺ . Detecting qɺ1 is only feasible indirectly through qɺɺ1 provided by the accelerometers. This corresponds to solving equation (7a) for q ɺɺ . Appropriately, the function f reads: x4 u3 . (10) xɺ = f ( x,u ) = u 4 2 (u1 − l1uɺ3 ) cos x2 − (u2 − l1u3 ) sin x2
The last component of f shows two special features. Firstly, it was possible to remove the influence of the gravity. Secondly, effects of angular acceleration and of centripetal forces appear. The latter characteristic is a consequence of distributing accelerometers over the structure. It is new compared to classical inertial navigation systems. Thus, angular acceleration measurements have to augment u. Such signals can be generated by numerically differentiating some Ωk or by new inertial sensors (STM, 2004). Accordingly, the input vector for the example considered is: u = [u1
u2
u3
u4
T uɺ3 ] .
(11)
The aiding vector y consists of three distance measurements. The geometrical relations for modelling these signals can be directly inferred from Figure 2: y = [ρ1
ρ2
T ρ3 ] ,
(12)
d − x1 . (13) h ( x, u ) = d − x1 − l1 sin x2 d − x − l sin x − l sin x 1 1 2 2 3 The Kalman filter requires completing the modelling with stochastic sensor errors, which is achieved using white noise with a specific variance for each measurement device (Wagner, 2004). Furthermore, the
539
functions f and h are nonlinear, f, x, and u are quasicontinuous with time (the sampling rate of u is typically high), h and y are discrete with time (the sampling rate of y is often low and unsteady). Thus, the so-called continuous-discrete extended Kalman filter has to be employed (Gelb, 1989). System test. Prepared by simulated measurements, the system as described above was experimentally realised to test its functionality and performance. For this, the usual compensation of the inertial sensor biases had additionally to enter the system design (Farrell and Barth, 1999). As an example of the results, Figure 3 shows a comparison of the aiding signal y1 and the estimated transverse distance q1 for the pendulum being at rest. All of the sensors used were products designed for mass markets with an accuracy of 0.05 m/s2 (accelerometers), 0.2°/s (gyros), 0.02 m (radar noise), and 2 m (radar jamming). Nevertheless, the remaining noise of the estimate is remarkably low. Wagner (2004) gives further results and additional details about the general modelling procedure, the sensors used, and the interference resistance being a special feature of integrated measurement systems.
body frame b
bζ
rB
ℓ0j
rj
η sk
ζ sk
η bη
iη
inertial frame i iζ
Fig. 4. Aircraft fuselage cross section with distorted wing half and peripheral sensors j and k. any peripheral sensor. Again, the subscript on the upper left side indicates the coordinate system, in which the differentiation takes place. The angular rate ωbi describes additionally the angular velocity between the body and the inertial frames b and i. Assuming furthermore small distortions of the structure (a presumption being largely adequate for flexible vehicles), the actual lever arm ℓ j and its body oriented time derivatives can be approximated by a finite series reflecting e.g. the main vibration modes: Γa
Figure 4 represents the exemplary initial point of the corresponding theory for flexible structures. It shows an aircraft fuselage with an exaggeratedly distorted wing half attached, the original wing shape being indicated by the dashed line. A body-fixed coordinate system b with origin B serves for describing the timevariant structural geometry. An inertial measurement unit (IMU; 3 gyros, 3 accelerometers) is located at B and µ additional accelerometers j and ν further gyros k are placed on the wing measuring aj and Ωk in the sensor frames. The accelerometer attachment points are subject to the acceleration i i b ɺɺ b ɺ ɺɺ ɺɺ b r j = b rB + b ℓ j + 2 ( b ω bi × b ℓ j ) (14) + bi ωɺ bi × b ℓ j + b ω bi ×( b ω bi × b ℓ j ) .
In this equation, r is an inertial position vector, and ℓ denotes the actual lever arm between the IMU and 1.85
qˆ1
qˆ1 , ρ1 [m]
1.80 t [s]
sj
∆ℓ j
b
2000
sensor frames j, k
ζ
ℓj
B
4. FLEXIBLE STRUCTURES
ρ1
sj
3000
Fig. 3. Comparison of the estimate qˆ1 and of ρ1 for the pendulum of Figure 2 being at rest.
ℓ j(t )= b ℓ 0 j + b ∆ℓ j(t ) ≈ b ℓ 0 j + ∑ bχ(t )⋅ b s χ( b ℓ 0 j ) , (15) χ=1
b b
i
Γa
ℓ j(t ) ≈ ∑ bɺχ(t )⋅ bs χ( b ℓ 0 j ) ,
(16)
χ=1
b b
ii
Γa
ℓ j(t ) ≈ ∑ bɺɺχ(t )⋅ bs χ( b ℓ 0 j ).
(17)
χ=1
All time dependent “amplitudes” bχ(t) represent extra degrees of freedom, all s χ (ℓ 0 ) are the vectorial descriptions of Γa deformation modes selected for describing the structural distortions. Besides, every of these functions of the structural position ℓ 0 is defined to have at least at one point the displacement value 1. Thus, bχ(t) is the resultant position shift of this point if only the respective mode sχ is excited. Introducing equations (15) - (17) into (14) for all µ accelerometers and considering again the unit vectors ej = ej(b1, ..., bµ) for the measurement axes (part of the sensor frames), a set of linear function for all bɺɺχ results on the right. If µ = Γa holds and the sensors are placed appropriately, solving this set of equations for each single bɺɺχ is possible (if µ > Γa holds, the employment of a Least Squares procedure is viable). Composing now the state vector x of all bχ and their first time derivatives, the composition of f(x,u) (equation (1)) follows directly. However, a completion of f by the classical equations of an inertial navigation system is also necessary (Wagner, 2003). Then, the input vector u contains all signals of the IMU and of the peripheral accelerometers. Instead of the accelerometers, the use of gyros as input sensors is also feasible. A gyro k is subject to the angular velocity
540
1 bd (18) (curl b∆ℓ k (t )) . 2 dt For modelling the structural deformations ΓΩ other functions sχ , χ = Γa+1, ..., Γa+ΓΩ shall be used: b
ω ki =bω bi +
Γ α +Γ Ω d (curl b∆ℓ k (t )) ≈ ∑ bɺχ(t )⋅ curl bs χ ( b ℓ 0 k ) . (19) dt χ=Γ α +1
b
Introducing equations (19) into (18) for all ν gyros and employing also the unit vectors ek for the measurement axes, a set of linear function for the new bɺχ results on the right. For ν ≥ ΓΩ and appropriately positioned gyros, solving this set of equations for each single bɺχ is now possible. Compiling the state vector x by all new bχ, the components of f(x,u) follow directly (the input u contains now all gyro signals including those of the IMU). Employing a mixture of gyros and accelerometers is also possible. However, this case requires some decoupling conditions (Wagner, 2003) to avoid that some bχ are detected equally both from gyros and from accelerometers (this would lead to the problem that these bχ would appear twice in the state vector): (20a) curl s χ |ℓ = 0 , χ = 1, ..., Γ a , 0k
s χ |ℓ = 0 , χ = Γ a + 1, ..., Γ a + Γ Ω . 0j
(20b)
Besides these equations, the numbers Γa and ΓΩ as well as the determination of the deformation modes sχ depend on the vehicle structure, the relevant motion excitation and on the accuracy to be achieved. Assuming furthermore aiding sensors like GPS antennas, radar units, or strain gauges, the components of h follow from applying equation (15) for modelling distances, structural strains, etc. Example. This theory is illustrated as well by an example representing now a simple model for a wing half (Figure 5). The moving structure consists of a flexible beam of length l, constant thickness h, and flexural stiffness EI. Again, one end slides on a rail
and is joined to a wall with a spring. Representing the IMU, two accelerometers and one gyro are fixed to the bar at the swivel joint and generate the signals u1 , u2 (linear accelerations), u6 (angular rate), and u7 (angular acceleration). Furthermore, three peripheral accelerometers (u3 , u4 , u5) supply input signals, and three deformation modes sχ , χ = 1, 2, 3 are considered. The aiding consists now of four radar units measuring the distances ρ and velocities ρɺ (using Doppler shifts). Additionally, strain gauges provide three aiding strain signals ε. The state vector x contains now nine components:
x = d dɺ ψ b1 bɺ1 b2 = [ x1 ... x9 ].
bɺ2
b3
T bɺ3
(21)
The reason for omitting ψɺ in x is again that this term can be measured directly. Hence, it is part of u (i.e. ψɺ = u6 ). Employing moreover especially equations (14) – (17), the kinematical model (1) describing the motion of the beam reads: x2 u1 cos x3 + u2 sin x3 u6 x5 f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...) { } 1 3 7 2 4 7 3 5 7 χ=1 . f = x7 { f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...)} 2 4 7 3 5 7 1 3 7 χ= 2 x9 { f *(u , u ,...) + f *(u , u ,...) + f *(u , u ,...)} 1 3 7 2 4 7 3 5 7 χ=3 (22) The expressions of bɺɺ1 = xɺ5 , bɺɺ2 = xɺ 7 , bɺɺ3 = xɺ9 are not explicitly shown due to an extensive amount of space to be needed (Wagner, 2003). Nevertheless, it is intended to publish them in due time in a PhD thesis. The aiding vector y consists now of eleven signals as shown in Figure 5: T
y = [ ρ0 ρɺ 0 ρ1 ρɺ1 ρ2 ρɺ 2 ρ3 ρɺ 3 ε1 ε2 ε3 ] . (23)
u2
u6 , u7
ρ0 , ρɺ 0 u1
g
ψ bη
ε1 ru , rρ 3
3
rρ
2
...
iζ iη
u3
ε2
ρ1 , ρɺ1
u4
ρ 2 , ρɺ 2 ε3 u5
The last part is the aiding model: To reproduce the strain measured, the theory of bending elastic beams with thickness h and flexural stiffness EI has to be used (Roark and Young, 1986). In this case, the deformation modes are no longer vector functions but scalar functions sχ(bη) representing the cross displacements. Furthermore, the introduction of several auxiliary variables is helpful. For this, the sensor distances r as indicated in Figure 5 have to be considered, as well as derivatives of the bending line with respect to the structural coordinate bη: γ j = rρ ⋅ sin ψ + ν j ⋅ cos ψ , j
ρ3 , ρɺ 3
( ) ( ) ( ) = bɺ ⋅ s (r ) + bɺ ⋅ s (r ) + bɺ ⋅ s (r ) ,
ν j = b1 ⋅ s1 rρ + b2 ⋅ s2 rρ + b3 ⋅ s3 rρ , j
d
Fig. 5. Example of a flexible structure.
νɺ j
1
1
ρj
j
2
2
ρj
j
3
3
ρj
541
j
0.20
d 2 sχ
,
d bη 2 b
η = rε
0.15
j
x1 x2 x1 + γ1 x2 + u6 γ1 + νɺ1 cos ψ x1 + γ 2 x2 + u6 γ 2 + νɺ 2 cos ψ x1 + γ 3 h = x2 + u6 γ 3 + νɺ 3 cos ψ h b1 ⋅ s1′′ rε + b2 ⋅ s2′′ rε + b3 ⋅ s3′′ rε 1 1 1 2 EI h 2 EI b1 ⋅ s1′′ rε2 + b2 ⋅ s2′′ rε2 + b3 ⋅ s3′′ rε2 h b ⋅ s ′′ r + b2 ⋅ s2′′ rε + b3 ⋅ s3′′ rε 3 3 2 EI 1 1 ε3
( ( ) ( ( ) ( ( )
( ) ( ) ( )
. (24)
( )) ( )) ( ))
The continuous-discrete extended Kalman filter used for data fusion requires again completing the modelling with stochastic sensor errors being modelled by white noise with a specific variance for each measurement device. System test. To get an assessment about the modelling quality and system performance for the example given above and to prepare an analogous experimental setup, simulated measurement and reference data were generated by a finite element program (ABAQUS) using two dimensional beam elements and a nonlinear dynamics analysis. A further analysis calculated the linear Eigenmodes of the system, which served as deformation modes sχ . A compensation of the inertial sensor biases was again introduced into f and h. Once more, the low-cost sensors mentioned above were the basis for determining the sensor noise characteristics. The spring at the upper end of the beam and the flexural stiffness EI got such values that the first vibration modes had an order of magnitude of 1 Hz. The length l of the beam was 4 m. As an example of first simulation results for integrated system, Figure 6 shows the estimation error of the elastic deformation at the lower end of the beam. For this, the beam was exited by a sinusoidal force at its upper end. The outcome shows a stochastic inaccuracy of about 5 cm. Nevertheless, the filter performance can be optimised by filter tuning, by sensor positioning, and by adding also gyros as peripheral sensors. Instead of vibration modes, other methods from the theory of model reduction can also be considered to find suitable deformation functions sχ . 5. CONCLUSIONS The paper above is an outline of the theory of generalizing integrated navigation systems to spaciously distributed sensors. This requires especially an adequate kinematical description of the moving struc-
estimation error [m]
( )
sχ′′ rε =
0.10 0.05 0 -0.05 -0.10 -0.15 900
920
940
t [s]
960
980
1000
Fig. 6. Estimation error of the elastic displacement bˆ1s1 (l ) + bˆ2 s2 (l ) + bˆ3 s3 (l ) at the end of the beam. ture. As demonstrated with two examples, such an approach is feasible and enables an extensive motion measurement of mechatronic structures. Compared to classical inertial navigation, now the measurement of angular acceleration is necessary. Furthermore, elastic structures require in addition a model reduction. Future work will focus on the experimental verification of the flexible structure approach and on improving the ongoing simulations by filter tuning, optimisation concerning sensor type and placement, and employing other types of model reduction.
REFERENCES Bremer, H. and F. Pfeiffer (1992). Elastische Mehrkörpersysteme. Teubner, Stuttgart. Farrell, J.A. and M. Barth (1999). The Global Positioning System and inertial navigation. McGrawHill, New York. Gelb, A. (Ed.) (1989). Applied optimal estimation. 11th Printing. The M.I.T. Press, Cambridge/MA. Hong, S., J.-H. Lee, J. Rios, J.L. Speyer (2000). Observability analysis of GPS aided INS. In: Proceedings of ION GPS 2000, pp. 2618–2624. The Institute of Navigation, Alexandria/VA. Roark, R.J. and W.C. Young (1986). Formulas for stress and strain. 5th ed. McGraw-Hill, Auckland. STM (2004). LIS1Y08AS4 Inertial sensor. Product Preview. Rev. 1.1. STMicroelectronics, Geneva. Wagner, J.F. (2003). Zur Verallgemeinerung integrierter Navigationssysteme auf räumlich verteilte Sensoren und flexible Fahrzeugstrukturen. VDIVerlag, Düsseldorf. Wagner, J.F. (2004). Adapting the principle of integrated navigation systems to measuring the motion of rigid multibody systems. Multibody Systems Dynamics, 11, 87-110. Wagner, J. and T. Wieneke (2003). Satellite and inertial navigation – conventional and new fusion approaches. Control Engin. Practice, 11, 543-550. Yi, Y. and D.A. Grejner-Brzezinska (2006). Performance comparison of the nonlinear Bayesian filters supporting GPS/INS integration. In: Proceedings of the ION NTM 2006, pp. 977-983. The Institute of Navigation, Fairfax/VA.
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