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An introduction by Christoph Arndt (ZESS University of Siegen)
Processing multidimensional Data
There are, in general, two possibilities to extract information out of noisy multi-dimensional measurements, in order to obtain multi-dimensional state estimates. The first, called Data-Integration, uses every set of data gathered by a single sensor, to obtain an optimal estimate of some desired state. As there are more sensors, and therefore more sets of data, one could use these to increase the knowledge of the state and to provide an improved estimate. Nevertheless, Data-Integration ignores this possibility and processes every set of sensor data on its own. The following figure shows the flow of information, which is typical for Data-Integration.
Summarising the main points of Data-Integration:
This concept of data processing, though applied in a lot of practical cases, does not provide optimal estimates, because of the waste of the extra information in the additional measurements. Thus these applications only permit suboptimal performance and the need for optimal estimation finally leads to a more sophisticated concept:
The second method of multidimensional data processing is known as Data-Fusion. This approach uses any information that might be gathered from additional measurements. This information may be partly redundant but certainly not superfluous. The main strategy of Data-Fusion is to process the actual multi-dimensional measurement data, given in the measurement vector, at once, and find the optimal combination to produce the best possible state estimate. Therefore, one has to combine the measurements and the state variables due to the physical laws.
This often requires a sophisticated modelling of the state dynamics and observation mapping, because every error in modelling leads to a loss of information. These losses of information lead to suboptimal and inefficient estimates. An optimal estimator, however, extracts as much information as possible out of the available measurements to generate the best estimate. So a lot of care has to be taken about the exact modelling of the process.
Figure 2 shows the flow of information in Data-Fusion, which indeed is much greater than in Data-Integration. Therefore it seems to be obvious that Data-Fusion clearly outperforms Data-Integration. The mathematical proof of this superior performance is given in the paper 'Information gained by Data-Fusion'.
Summarising the main points of Data-Fusion:
To demonstrate these statements in pactice, we are now going to examine a simple linear example, where Data-Integration as well as Data-Fusion are applied to obtain the actual position of an Automatically Guided Vehicle.
Example: Automatically Guided Vehicles
In Automatically Guided Vehicles (AGV) the movement along a coordinate axis is described by the distance between the AGV and some obstacle, by the AGV's actual velocity and its acceleration. The knowledge of these variables in all three coordinates (x,y,z) determines the dynamic state of the vehicle.
Restricting the description to one coordinate, the movement can be represented by the 'Singer-Model', which combines distance, velocity and acceleration
in a state space model.
In the simplest case the connection between observation space (where we take the measurements) and state space (which contains the actual state of the system) is given by an identity mapping and additional measurement noise. The desciption of the dynamical relations between the actual and the following state vector is given in vector-matrix-notation:
The state vector vector is mapped onto the observation vector by:
Applying Data-Integration to obtain the actual motional state of the AGV essentially says that any measurement is processed on its own, e.g. distance measurements are only utilized to determine the distance, velocity measurements are only exploited to get velocity information, and the acceleration measurements only serve to estimate the AGV's actual acceleration. Any 'cross'-information is 'thrown' away. For example we simply ignore the implicit information about the distance which is contained in the velocity and acceleration measurements or we simply do not incorporate the velocity information which becomes available in the difference of two successive distance measurements. Data-Fusion on the other hand uses all the information that is contained in the available measurements and this leads to a considerably increased accuracy in the estimation of the vehicle's motional state.
It is easy to verify that Data-Fusion accumulates more information than Data-Integration, which results in
The latter benefit is especially interesting when we are dealing with dynamical systems, where the actual state varies with time. Here the increased flow of information created by Data-Fusion leads to a better tracking of the real state and therefore minimizes the risk of divergence, which inherently occurs in nonlinear estimation.