PAMM · Proc. Appl. Math. Mech. 7, 4130001–4130002 (2007) / DOI 10.1002/pamm.200700072
Integrated measurement system using accelerometers and gyros as peripheral sensors to estimate the motional state of an elastic beam ∗1 ¨ Thorsten Ortel and J¨org F. Wagner∗∗1 1
Institute of Statics and Dynamics of Aerospace Structures, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany.
Integrated navigation devices for vehicle guidance are the most common example of an integrated motion measurement system combining the signals from an inertial measurement unit (IMU consisting of three accelerometers and three gyros) and a GPS receiver with a single antenna. For this, the vehicle is traditionally assumed to be a single rigid body with six motional degrees of freedom to be determined. During periods of low vehicle dynamics the common integrated navigation systems show, however, stability problems. Nevertheless, the stability of the system can be guaranteed by distributing sensors over the vehicle structure. In this case the rigid body assumption has to be expanded to take the distributed sensors and the flexibility of the structure into account. Integrated systems in general are fusing different measuring signals by combining their benefits and blinding out their disadvantages. For instance, gyros and accelerometers are used to obtain reliable signals with a good time resolution. On the other hand, aiding sensors like radar units and strain gauges are known to be long-term accurate. Furthermore, the kernel of such integrated systems consists of an extended Kalman filter that estimates the motion state of the structure. Besides the sensor signals, the basis for the filter is an additional kinematical model of the structure which has to be developed individually. The example of the motion of an elastic beam being considered here is meant to be an approach to obtain motional measurements of a wing of a large airplane during flight. By means of a modal approach, a kinematical model of the beam was developed. This paper will compare integrated systems utilising accelerometers as peripheral sensors with systems using gyros and systems with a combination of both peripheral sensor types. Based on simulation the paper shows this approach, different sensor configurations, and estimated motion results of an elastic beam. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Basics of Integrated Measurement Systems Integrated measurement systems are typically equipped with sensors that supply highly available signals and contrariwise sensors with a good long-time accuracy. The signal fusion is usually accomplished by a Kalman filter that employs a kinematical model to interpret the signals of these quite different sensors . For this, the Kalman filter (in particular the extended Kalman filter for nonlinear models) estimates the unknown motion state x utilising the integration basis of an observer. Inertial sensors like accelerometers and gyros are providing highly available signals commonly forming an input vector u. On the other hand GPS, laser, radar units, and strain gauges are providing long-term accurate signals (so-called aiding signals) forming the vector y. The kinematical model describing the structure motion and aiding signals is a set of ordinary differential equations f and a ˆ and y ˆ being the estimate of the motion state x and the aiding signal y respectively. set of algebraic equations h, with x ˆ˙ = f (ˆ x x, u) ,
ˆ = h(ˆ y x, u) .
State of the art integrated measurement systems are idealising the structure to be a rigid body with one to six degrees of freedom. Thus, the measurements are referenced to a single point of the vehicle. This approach is valid for small structures with negligible deformations during their use. However, considering large structures, the kinematical model has to take the flexibility of the structures into account. Integrated motion measurement on flexible structures can be used e.g. to estimate the motion state of a wing of a big airplane. The additional data may be used for flight or structural control of the plane.
2 Kinematical Modelling of Flexible Structures The general structure of an airplane with an exaggerated deformed wing half is depicted in Figure 1. The IMU is located at point C. An additional accelerometer ja and a gyro jΩ are placed on the wing. The sensors are subject to the following acceleration i ¨rja and angular rate ωjΩ i (between the sensor system jΩ and the inertial system i) respectively. The superscript on the left side indicates the coordinate system, in which the expression has been differentiated, e.g. b being the body system. i
¨rja = i ¨rC + b ¨ja + 2(ωbi × b ˙ ja ) + i ω˙ bi × ja + ω bi × (ω bi × ja ) ,
ω jΩ i = ω bi + ∗ ∗∗
1 d (curl ∆jΩ ) . 2 dt
¨ Thorsten Ortel: e-mail: [email protected]
, Phone: +49 711 685 69524, Fax: +49 711 685 63706 J¨org F. Wagner: e-mail: [email protected]
, Phone: +49 711 685 67046, Fax: +49 711 685 63706.
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Fig. 1 Fuselage cross section with a distorted wing being equipped with a peripheral accelerometer ja and a gyro jΩ .
The lever arm ja and its variation ∆jΩ respectively, including their time derivatives, can be expressed as the following : b˙ b¨ ¨bχ (t) sχ (¯rj ) . bχ (t) sχ (¯rj ) , (4) j = ¯rj + ∆j ≈ ¯rj + j ≈ b˙ χ (t) sχ (¯rj ) , j ≈ χ
The time dependent amplitudes bχ (t) are additional degrees of freedom and sχ (¯rj ) are unit deformations, that can e.g. be interpreted as eigenmodes of the structure. Putting equation (2), (4) and (3), (4) respectively together and solving for ¨bχ for peripheral accelerometers and b˙ χ for peripheral gyros yields expressions that are added to the kinematical model (1) . The investigated beam structure representing a simplification of a wing of a large airplane shows the Figure 2. Three different sensor sets were tested considering three unit deformations χ = 1, 2, 3. The first model includes three peripheral accelerometers aj and the IMU (2D: two accelerometers, one gyro) as input u. The aiding y consists of four radar units (ρk , ρ˙ k ) and three strain gauges εl . The second model uses three peripheral gyros Ωj and the IMU as input, and again four radar units ρk and three strain gauges εl as aiding. The third model investigated uses the same input and aiding signals as the first model, but uses additionally three gyros as aiding y. All three models are completely observable which is a necessary condition to guarantee a stable behaviour of the filter performance. Reference, input and aiding data were simulated with a non-linear dynamic analysis within a finite element approach and compared to the Kalman filter estimations. The errors of the flexible deformation at the lower end of the beam of the three kinematical models are shown in Figure 3. The best estimation results yield the second model including three peripheral gyros. The error of the elastic deformation of the first model including three peripheral gyros is more than twice as much. At each hundred seconds an increase of the error can be observed, which is due to intensified impulse-type excitation of the system at that time. The second model is less susceptible to this impulse-type excitation of the beam. The better performance of the second model can be explained with the one-time integration of b˙ χ , whereas the models including accelerometers have to integrate ¨bχ twice to obtain the unknown sate bχ . Finally, the estimation results of the third model lies in between the other two models.
Fig. 2 Model of the beam.
Fig. 3 Error of elastic deformation including each three unit deformations.
References  A. Gelb (ed.). Applied Optimal Estimation, 11th printing (The M.I.T. Press, Cambridge, MA, 1989).  J.F. Wagner. Zur Verallgemeinerung integrierter Navigationssysteme auf r¨aumlich verteilte Sensoren und flexible Fahrzeugstrukturen, (VDI-Verlag, D¨usseldorf, 2003). ¨  T. Ortel and J.F. Wagner. Integrated Motion Measurement for Flexible Structures, Tech. Mech. 27, 93-113 (2007).
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim