Sensors 2005, 5, 613-632
`
`sensors
`ISSN 1424-8220
`© 2005 by MDPI
`http://www.mdpi.org/sensors
`
`Improving the Response of a Rollover Sensor Placed in a Car
`under Performance Tests by Using a RLS Lattice Algorithm
`
`Wilmar Hernandez
`
`Department of Circuits and Systems in the EUIT de Telecomunicación at the Universidad Politécnica
`
`de Madrid (UPM), Campus Sur UPM, Ctra. Valencia km 7, Madrid 28031, Spain
`
`Phone: +34913367830. Fax: +34913367829. E-mail: whernan@ics.upm.es
`
`Received: 5 December 2005 / Accepted: 21 December 2005 / Published: 21December 2005
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`Abstract: In this paper, a sensor to measure the rollover angle of a car under performance tests
`
`is presented. Basically, the sensor consists of a dual-axis accelerometer, analog-electronic
`
`instrumentation stages, a data acquisition system and an adaptive filter based on a recursive
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`least-squares (RLS) lattice algorithm. In short, the adaptive filter is used to improve the
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`performance of the rollover sensor by carrying out an optimal prediction of the relevant signal
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`coming from the sensor, which is buried in a broad-band noise background where we have little
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`knowledge of the noise characteristics. The experimental results are satisfactory and show a
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`significant improvement in the signal-to-noise ratio at the system output.
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`Keywords: dual-axis accelerometer; rollover angle; RLS lattice algorithm; adaptive noise
`
`canceller; uncertainty
`
`1. Introduction
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`Unfortunately, the highest fatality rates per year in car accident statistics are attributed to rollover
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`crashes [1]. In fact, this type of crash accounts for about 40% of passenger vehicle fatalities. This is
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`why today’s cars are designed with the highest priority placed on passenger safety.
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`On the question of what the causes for this kind of crash are. It has been shown that tall, narrow
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`vehicles with high center of gravity are bound to roll over if they are involved in single-vehicle crashes.
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`In addition, excessive speed, alcohol consumption, poor roads and environmental factors, among
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`others, are all contributory factors to rollover crashes.
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`Therefore, with this scenario in mind, no one would dispute the need to improve the design of the
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`sensors used for industrial measurements [2-22]. This is why in the last decades researchers from all
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`around the world have been working hard to invent intelligent devices consisting of not only sensors,
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`but also advanced materials [23] and microprocessors, among other devices, that incorporate a certain
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`amount of intelligence to the sensors themselves, transforming them into better prepared measuring
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`systems [24-47].
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`Nevertheless, the process of improving the performance of sensors is not an easy task. Actually, we
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`have to deal with disturbances and/or interferences whose characteristics we have little or no prior
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`knowledge of, and we have to use efficient methods of estimation, prediction and control in order to
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`get clear information about the physical magnitude or process that we want to measure or control [38-
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`47].
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`In short, this paper shows the improvement of the real-time response of a sensor to measure the
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`rollover angle of a car under performance tests. Furthermore, this system can be easily integrated into
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`technologies that aim to improve the comfort and safety of the passengers in most of today’s cars.
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`Electronic stability control, variable ride-height suspension and rollover airbag systems are examples
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`of the above mentioned technologies.
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`1. Accelerometers
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`2.1. Principles
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`In the industrial world, the most common design of sensors to measure acceleration is the
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`accelerometer design based on a combination of Newton’s law of mass acceleration and Hooke’s law
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`of spring action.
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`Figure 1 shows the basic spring-mass system accelerometer. Such a mass is connected to the base
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`by a spring that is in its unextended state and exerts no force on the mass. In addition, the mass is free
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`to slide on the base [20].
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`Figure 1. Basic spring-mass system accelerometer.
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`According to Johnson [20], if the seismic mass m, is undergoing an acceleration a, then there must
`(cid:32)
`(cid:152)
`be a force F acting on the mass and given by
`. In addition, the spring of spring constant k is
`amF1
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`2
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`extended (or stretched) from its equilibrium position for a distance x(cid:39) , with a force F2 (opposite to F1)
`(cid:39)(cid:152)(cid:32)
`. This condition is described by equating Newton’s and
`acting on the spring and given by
`k
`x
`F2
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`Hooke’s laws. Thus, the measurement of acceleration is reduced to a measurement of spring extension
`
`(linear displacement):
`
` (1)
`
`(cid:39)
`x
`
`mk
`
`(cid:32)
`
`a
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`The spring-mass principle applies to many common accelerometer designs, but most designs differ
`
`from each other in how they carry out the measurement of the displacement of the spring.
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`In analyzing the time domain performance of the spring-mass system and carrying out the impulse
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`response analysis, it can be seen that such a system exhibits oscillations at a natural frequency with
`
`damping:
`
`f
`
`osc
`
`(cid:32)
`
`f
`
`N
`
`(cid:93)(cid:16)
`
`1
`
`2
`
` (2)
`
`1
`(cid:83)
`2
` is the natural frequency and (cid:93) is the dimensionless
`
`
`
` (3)
`
`
`
`
`
`mk
`
`(cid:32)
`
`f N
`
`where oscf
`
` is the frequency of oscillation, Nf
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`damping ratio.
`
`At this point, it is important to point out that the greater the friction associated with the seismic
`
`mass and the base, the sooner the mass settles to equilibrium. That is to say, as the friction increases,
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`the damping becomes greater and the response will reach zero sooner. In addition, if the spring-mass
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`system is underdamped, the mass will swing back and forth with decreasing amplitude. In this case,
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`the frequency of oscillation is given by Eq. (2). However, if the friction tends to zero (lossless case),
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`the seismic mass will oscillate at some characteristic natural frequency (see Eq. (3)).
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`2.1 Types of accelerometers and the selection of the sensor
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`There is a wide variety of accelerometers that could be used in various applications depending on
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`the requirements of range, natural frequency, damping, temperature, size, weight, hysteresis, low noise,
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`and so on. Piezoelectric accelerometers, piezoresistive accelerometers, variable capacitance
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`accelerometers, linear variable differential transformers (LVDT), variable reluctance accelerometers,
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`potentiometric accelerometers, gyroscopes used for sensing acceleration, strain gauges accelerometers,
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`among others, are some of the numerous accelerometers.
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`In this paper, we are interested in measuring steady-state accelerations. That is to say, we are
`
`interested in a measure of acceleration that may vary in time but that is nonperiodic [20]. In this
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`application, the frequency at which the acceleration changes is low (< 50 Hz).
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`Generally speaking, the accelerometers should have the following characteristics:
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`1) Adequate range to cover expected acceleration magnitudes.
`2) A natural frequency higher than twice the frequency at which the measured acceleration
`changes.
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`Therefore, taking into consideration the above statements, in this paper the Analog Devices dual-
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`axis accelerometer ADXL202 was used. Such a sensor is a low cost, low power, complete 2-axis
`accelerometer with a measurement range of (cid:114) 2g and sensitivity 312 mV/g, where g is the gravitational
`acceleration (9.81m/s2). Also, this sensor can measure both dynamic acceleration (e.g., vibration) and
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`static acceleration (e.g., gravity). Furthermore, the bandwidth of such an accelerometer may be set
`500 (cid:80)
`g
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`from 0.01 Hz to 5 kHz, and the typical noise floor is
`
`Hz
`
` allowing signals below 5 mg to be
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`resolved for bandwidths below 60 Hz.
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`2. Rollover sensor system and mathematical modeling
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`In order to obtain an approximation of the model of the system, it is usually assumed that the
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`movement of the car’s center of gravity can be discarded [48, 49]. That is to say, it is assumed that the
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`car has a stiff suspension. Thus, the movement of the car’s center of gravity owing to the flexibility of
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`the suspension is disregarded.
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`Figure 2 shows the mechanical model of the system. In such a figure the car is seen from the front.
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`Furthermore, experience tells us that the influence that the movement of the car’s center of gravity
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`exerts on the measure of both the acceleration of the car in the y direction and the acceleration of the
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`car in the z direction (see Fig. 2) can be discarded.
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`Figure 2. Mechanical model of the system.
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`In Fig. 2 a dual-axis accelerometer is situated at the car’s center of gravity with one of its axes
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`sensing the acceleration in the y direction and the other in the z direction. Also, T represents the track
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`width, r is the radius of the curvature of the path of motion, Fc is the centrifugal force, h is the height of
`the car’s center of gravity, and Fyl and Fyr are friction forces that prevent the car from sliding off of the
`road. In addition, W represents the weight of the car, (cid:69) is the degree of inclination of the curve in the
`road, Nl is the normal force between the left front tire and the ground, and Nr is the normal force
`between the right front tire and the ground. Furthermore, the total forces in the y and z axis are
`
`(cid:32)
`
`F
`y
`
`ma
`
`y
`
`(cid:32)
`
`F
`c
`
`cos
`
`(cid:16)(cid:69)
`
`sinW
`
`(cid:69)
`
`
`
`(cid:32)
`
`F
`z
`
`ma
`
`z
`
`(cid:32)
`
`cosW
`
`(cid:14)(cid:69)
`
`F
`c
`
`sin
`
`(cid:69)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
` (4)
`
` (5)
`
`
`
` (6)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
` (7)
`
`2
`
`rv
`
`c (cid:32)
`mF
`
`mgW (cid:32)
`
`
`
`where Fy is the total force in the y axis, Fz is the total force in the z axis, v is the longitudinal speed of
`the car, m is the mass of the car, g is the gravitational acceleration, ay is the acceleration in the y
`direction and az is the acceleration in the z direction.
`Therefore, analyzing the mechanical model shown in Fig. 2, it can be said that the car is bound to
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`roll over if the vector indicating the force Ft strikes the ground at any point lying to the right of the
`center of the front right wheel. What is more, it can also be said that Eq. (4) and Eq. (5) are the
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`orthogonal projections of Ft on the y and z axis, respectively; and that the car will roll over if
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
` (8)
`
`(cid:32)
`
`T
`
`h2
`
`
`
`zy
`FF
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`Also, as it can be seen from Eq. (4) and Eq. (5) that
`
`zy
`aa
`
`(cid:32)
`
`zy
`FF
`
`(cid:32)
`
`T
`
`h2
`
`zy
`aa
`
`
`
`
`
`
`
`
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`Eq. (8) can also be given by
`
`The rollover angle is
`
`
`
`
`
`
`
`
`
` (9)
`
`(cid:184)(cid:185)(cid:183)
`
`T
`
`h2
`
`(cid:168)(cid:169)(cid:167)
`
`(cid:32)(cid:74)
`
`(cid:16)
`tan 1
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`Equations (8) and (9) can be obtained by using standard techniques of classical mechanics and
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`vector analysis [50, 51].
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`3. General considerations of noise characterization
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`Basically, it can be said that the sources of disturbances or interferences are many, but the most
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`important are those caused by the mechanical and electrical systems that today’s cars have, and by
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`poor roads and environmental factors. As a matter of fact, the designer has to deal with vibrations of
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`the framework, chassis, front axle, rear axle, and engine. In addition, the noise generated by the
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`vertical movement, yaw, pitch, roll, and forces and moments on each wheel has to be taken into
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`consideration as noise affecting the signal of interest.
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`According to Aparicio et al. [49], the mechanical vibrations have the following eigenfrequencies:
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`1) 1 Hz < eigenfrequencies < 3 Hz. Vibrations of the framework and the car sprung masses,
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`vertical movement, yaw, roll and pitch.
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`2) 4 Hz < eigenfrequencies < 8 Hz. Vibrations of the wheels at low speed.
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`3) 10 Hz < eigenfrequencies < 20 Hz. Vibrations of the car sprung and unsprung masses at
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`medium and high speed. Also, vibrations of the engine, framework, chassis, front and rear axle,
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`etc.
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`4) eigenfrequencies > 20 Hz. Vibrations caused by direct actions, vibrations of the tires, etc.
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`The above vibrations are considered to be mechanical disturbances affecting the measurements. In
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`addition, in order to reflect real vehicle driving conditions, the variable road characteristics are treated
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`as a random process [52-54]. Moreover, there are other noise sources that are generated by the car’s
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`electrical systems. Such noises and/or interferences are a mixture of random signals that fluctuate
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`rapidly in an unpredictable manner and interfere with limited band spectrum.
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`4. Design of the adaptive filter
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`5.1.
`
`Introduction
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`The idea of canceling or diminishing the unwanted information without causing damage to the
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`relevant signal is very complex, and the more the noise and the relevant signal share the same
`
`frequency spectrum, the less the designer can remove the unwanted information by using classical
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`filters. In fact, when the unwanted information and the relevant signal share the same (or a very similar)
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`frequency spectrum, the use of classical filters should be discarded [55-60]. When this happens, it is
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`often said that the signal is buried in a broad-band noise background.
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`According to Anderson and Moore [60], the classical approach to filtering postulates that the useful
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`signals lie in one frequency band and unwanted signals, normally termed noise, lie in another, though
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`on occasions there can be overlap. And when this happens, it is very difficult to eliminate small-
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`magnitude disturbance or interference, and the background noise causes serious difficulties. In
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`addition, the use of fixed filters with very sharp cut-off regions can introduce problems such as:
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`additional delays, overshoots, undesirable amplification at some frequency range, and so on.
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`Butterworth filters, Chebyshev filters and elliptical filters are some of the landmarks in the body of
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`classical filter design theory.
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`In addition, the earliest statistical ideas of Wiener and Kolmogorov [61, 62] relate to processes with
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`statistical properties which do not change with time, i.e., to stationary processes. For such processes, it
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`proved possible to relate the statistical properties of the useful signal and unwanted noise to their
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`frequency domain properties. The assumption that the underlying signal and noise processes are
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`stationary is crucial to the Wiener and Kolmogorov theory. Nevertheless, this is a drawback of such a
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`theory.
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`The Wiener-Kolmogorov theory is inadequate to deal with problems in which the relevant signal
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`and/or the noise are not stationary processes. This is the case under study. Therefore, a new theory, the
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`Kalman filter theory [55-60], was developed in the late 1950s and early 1960s to deal with situations
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`in which nonstationarity of the signal and/or noise is intrinsic to the problem. The optimal adaptive
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`filter proposed in this paper is based on the Kalman filtering theory.
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`Therefore, taking into consideration the above statments, it is recommended that designers use
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`optimal signal processing techniques to cancel noise and interference that are usually very hard to
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`cancel using the classical approach to filtering. These reasons, among others, justify the need to apply
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`optimal filtering [60].
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`What is more, due to the characteristics of the noise corrupting the important information, it is
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`recommended that such an optimal filter be implemented as an adaptive filter [63-67]. In short, an
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`adaptive filter is a filter with a mechanism for adjusting its own parameters automatically by using a
`
`recursive algorithm at the same time that it is in active interaction with the environment. In addition,
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`all this happens in such a way that the performance of the adaptive filter is continuously improved
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`according to a specified performance criterion (or cost function) which has been previously established
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`by the designer.
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`Taking into consideration the above statements and the need to use an optimal adaptive filter with
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`low computational burden, good numerical behavior, robustness, ease of implementation, a
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`satisfactory rate of convergence and good round-off error rejection characteristics, a recursive least-
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`squares (RLS) lattice algorithm was chosen to carry out the process of optimal estimation of the
`
`relevant signal [65-69]. In the present paper, the application of such an adaptive filter is an interference
`
`or noise canceller [64, 66, 70], and a block diagram representation of it is shown in Fig. 3. Furthermore,
`
`a summary of it follows.
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`5.2. Summary of the RLS lattice algorithm (from Haykin [66])
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`5.2.1 Introduction
`
`According to Haykin [66], this algorithm is based on a priori estimation errors, and the reflection
`
`and joint-process estimation coefficients are all derived directly. The algorithm is called the RLS
`
`lattice algorithm using a priori estimation errors with error feedback. To derive this algorithm, we may
`
`proceed in one of two ways (among others):
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`1) Apply the squaring procedure to an extended form of square-root adaptive filtering algorithms
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`for exponentially weighted RLS estimation (QR-RLS). The QR-RLS algorithm, or, more
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`precisely, the QR decomposition-based RLS algorithm (QRD-RLS), derives its name from the
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`fact that the computation of the least-squares weight vector in a finite-duration impulse
`
`response (FIR) filter implementation of the adaptive filtering algorithm is accomplished by
`
`working directly with the incoming data matrix via QR decomposition [71], rather than
`
`working with the (time-average) correlation matrix of the input data as in the standard RLS
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`algorithm. Accordingly, the QRD-RLS algorithm is numerically more stable than the standard
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`RLS algorithm.
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`2) Apply Kalman filter theory [55-60, 66, 67] in conjunction with the covariance Kalman
`
`filtering algorithm for the special unforced dynamical model of the state-space model of the
`
`RLS filter for an exponential weighting factor greater than zero and lower than or equal to one
`
`[66].
`
`In this paper, it is followed procedure 2) because this is insightful and fairly straightforward. In
`
`addition, this algorithm, naturally enough, has both advantages and disadvantages.
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`Figure 3. Block diagram representation of the adaptive filter.
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`On the one hand, as the updating of the reflection and regression coefficients is performed directly,
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`the numerical behavior of the RLS lattice algorithm using a priori estimation errors with error
`
`feedback is very good. The good numerical behavior of this algorithm is one of its strength in the case
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`under study. This issue is very important for us because of the effects of quantization, or round-off,
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`errors that arise when adaptive filters are implemented digitally. In short, the quantization process has
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`the effect of causing the performance of a digital implementation of the algorithm to deviate from the
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`ideal (i.e., infinite-precision) form of the adaptive filtering algorithm. In addition, the nature of this
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`deviation is influenced by a combination of the following factors:
`
`a) The design details of the adaptive filtering algorithm
`
`b) The degree of ill conditioning (i.e., the eigenvalue spread) in the underlying correlation matrix
`
`that characterizes the input data
`
`c) The form of numerical computation (fixed point or floating point) employed
`
`Understanding the numerical properties of adaptive filtering algorithms helps the designers to meet
`
`design specifications.
`
`Another advantage of the RLS lattice algorithm using a priori estimation errors with error feedback
`
`is the high quality of the estimation of unknown parameters of the stochastic model of the physical
`
`process under study. This is another very important strength of this algorithm.
`
`On the other hand, the input data and internal calculations are all quantized to a finite precision that
`
`is determined by design and cost considerations. The cost of a digital implementation of an algorithm
`
`is influenced by the number of bits (i.e., the precision) available for performing the numerical
`
`computations associated with the algorithm. In general, the cost of implementation increases with the
`
`number of bits employed.
`
`Another disadvantage of this algorithm is that, generally speaking, algorithms based on Kalman
`
`filtering algorithms have no good long-term stability. However, this problem can be solved by
`
`reinitializing the algorithm at any appropriate interval of time, whether fixed or variable.
`
`On balance, although there are disadvantages, experience tells us that the RLS lattice algorithm
`
`using a priori estimation errors with error feedback is one of the best that the designers can use when
`
`estimating unknown parameters of the stochastic models of many physical process.
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`5.2.2 The RLS lattice algorithm using a priori estimation errors with error feedback
`
`5.2.2.1.
`
`Initialization
`
`To initialize the algorithm, at time n = 0, set
`
`(cid:11) (cid:12) (cid:71)(cid:32)
`(cid:41) (cid:16) 01r
`
`
`
` (cid:11) (cid:12) (cid:71)(cid:32)(cid:16)(cid:52) (cid:16)
`1
`
`1r
`
`(cid:74)
`
`(cid:41)
`
`r,
`
`(cid:11) (cid:12)
`0
`
`(cid:83)(cid:32)
`
`(cid:52)
`
`r,
`
`(cid:11) (cid:12) 0
`(cid:32)
`0
`
`(cid:78)
`
`(cid:11) (cid:12) 1
`(cid:32)
`00
`
`where (cid:71) is a small positive constant, (cid:41) is the forward prediction-error energy, (cid:52) is the backward
`prediction-error energy, r is the order of the least-squares predictor and r = 1, 2, …, R, where R is the
`final order of the least-squares predictor. In addition, (cid:74) is the forward reflection coefficient, (cid:83) is the
`backward reflection coefficient, and (cid:78) is the conversion factor.
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`For each instant n (cid:149) 1, generate the zeroth-order variables:
`(cid:11) (cid:12)
`(cid:11) (cid:12)
`(cid:11) (cid:12)nx
`n
`n
`
`(cid:75)
`
`0
`
`(cid:69)(cid:32)
`
`0
`
`(cid:32)
`
`(cid:41)
`
`0
`
`(cid:11) (cid:12)
`n
`
`(cid:52)(cid:32)
`
`0
`
`(cid:11) (cid:12)
`n
`
`(cid:41)(cid:79)(cid:32)
`
`0
`
`(cid:11)
`(cid:12)
`(cid:14)(cid:16)
`1n
`
`(cid:11) (cid:12) 2
`nx
`
`(cid:78)
`
`(cid:12) 1
`(cid:11)
`(cid:32)(cid:16)
`1n0
`
`(cid:100)(cid:79)(cid:31)
`where the constant (cid:79) ,
`, is the forgetting factor and its typical values used are the real
`0
`1
`numbers in the range from 0.99 to 1, (cid:75) is the forward a priori prediction error,(cid:69) is the backward a
`priori prediction error, and x is the reference input.
`
`For joint-process estimation, at time n = 0, set
`
`(cid:11) (cid:12) 0
`(cid:32)
`(cid:86) (cid:16)
`01r
`
`At each instant n (cid:149) 1, generate the zeroth-order variable
`(cid:11) (cid:12)
`(cid:11) (cid:12)ny
`n0
`
`(cid:32)
`
`(cid:72)
`
`where (cid:86) is the tap-weight vector of the transversal filter. It contains R + 1 taps. Also, y is the primary
`input and (cid:72) is the system output.
`
`5.2.2.2. Predictions
`
`For n = 1, 2, 3,…, compute the various order updates in the sequence r = 1, 2, …, R.
`
`(cid:11) (cid:12)
`n
`
`(cid:16)
`1r
`
`(cid:41)(cid:79)(cid:32)
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:78)(cid:14)(cid:16)
`1n
`
`(cid:11)
`(cid:12)
`(cid:75)(cid:16)
`1n
`
`(cid:11) (cid:12) 2
`n
`
`(cid:16)
`1r
`
`(cid:16)
`1r
`
`(cid:52)
`
`(cid:16)
`1r
`
`(cid:12)
`(cid:52)(cid:79)(cid:32)(cid:16)
`1n
`
`(cid:11)
`(cid:16)
`2n
`
`(cid:12)
`
`(cid:78)(cid:14)
`
`(cid:11)
`(cid:12)
`(cid:69)(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12) 2
`(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:16)
`1r
`
`(cid:41) (cid:11)
`
`(cid:75)
`
`(cid:75)(cid:32)
`
`
`
`(cid:11) (cid:12)1n
`(cid:16)
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:69)(cid:16)
`1n
`
`r,
`
`(cid:13)(cid:41)
`
`(cid:74)(cid:14)
`
`(cid:11) (cid:12)
`n
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)
`n
`
`r
`
`(cid:11) (cid:12)n
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:75)(cid:16)
`1n
`
`r,
`
`(cid:13)(cid:52)
`
`(cid:11)
`(cid:12)
`(cid:83)(cid:14)(cid:16)
`1n
`
`(cid:69)(cid:32)
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)
`n
`
`(cid:69)
`
`r
`
`(cid:74)
`
`(cid:41)
`
`r,
`
`(cid:11) (cid:12)
`n
`
`(cid:74)(cid:32)
`
`(cid:41)
`
`r,
`
`(cid:11)
`(cid:12)
`(cid:16)(cid:16)
`1n
`
`(cid:78)
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:11)
`(cid:12)
`(cid:12)
`(cid:69)(cid:16)
`(cid:16)
`1n
`1n
`(cid:16)
`1r
`(cid:11)
`(cid:12)
`(cid:52)
`(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)n
`
`(cid:75)
`
`(cid:13)
`r
`
`(cid:83)
`
`(cid:52)
`
`r,
`
`(cid:11) (cid:12)
`n
`
`(cid:83)(cid:32)
`
`(cid:52)
`
`r,
`
`(cid:11)
`(cid:12)
`(cid:16)(cid:16)
`1n
`
`(cid:78)
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:75)(cid:16)
`1n
`(cid:16)
`1r
`(cid:11)
`(cid:12)
`(cid:52)
`(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)
`n
`
`(cid:11) (cid:12)n
`
`(cid:69)
`
`(cid:13)
`r
`
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`
`(cid:11)
`(cid:12)
`(cid:78)(cid:32)(cid:16)
`1n
`
`(cid:78)
`
`r
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:16)(cid:16)
`1n
`
`(cid:78)
`
`2
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:11)
`(cid:12)
`(cid:69)(cid:16)
`(cid:16)
`1n
`1n
`(cid:16)
`1r
`(cid:11) (cid:12)1n
`
`(cid:16)
`(cid:52)
`
`(cid:16)
`1r
`
`2
`
`5.2.2.3. Filtering
`
`For n = 1, 2, 3,…, compute the various order updates in the sequence r = 1, 2, …, R + 1:
`
`(cid:11) (cid:12)
`n
`
`r
`
`(cid:72)(cid:32)
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)
`n
`
`(cid:86)(cid:16)
`
`(cid:13)
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:69)(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:11) (cid:12)n
`
`(cid:12)
`
`(cid:86)(cid:32)
`
`(cid:16)
`1r
`
`(cid:11)
`(cid:12)
`(cid:14)(cid:16)
`1n
`
`(cid:78)
`
`(cid:11) (cid:12)
`(cid:11) (cid:12)
`(cid:69)
`n
`n
`(cid:16)
`1r
`(cid:11)
`(cid:12)
`(cid:16)
`1n
`
`(cid:16)
`1r
`
`(cid:16)
`1r
`(cid:52)
`
`(cid:11) (cid:12)n
`
`(cid:72)
`
`(cid:13)
`r
`
`(cid:72) (cid:11)
`
`(cid:86)
`
`(cid:16)
`1r
`
`n
`
`where the asterisk denotes complex conjugation.
`
`Before moving on to the calibration of the system, it is important to point out that, as stated in the
`1R (cid:14)(cid:72)
`initialization step, the output of the filtering process (
`) is the system output (see Fig. 3). Moreover,
`R(cid:75) and R(cid:69) ) are the variables used in this paper to obtain the cost
`
`the outputs of the lattice predictor (
`
`function (see Section 7).
`
`5. Calibration of the system
`
`Before moving on to the results of the experiment, it is important to carry out the calibration of the
`
`sensor presented in this paper, so that the costumer can have an estimation of the uncertainty of
`
`measurement of the system [72-76].
`
`In this paper, a computer-controlled platform was used to generate the reference signals for the
`
`calibration process of the intelligent sensor. In addition, an inclinometer was used as the working
`
`standard in this research.
`
`Specifically, a TESA ClinoBEVEL1 inclinometer was used, which, in order to guarantee the
`
`traceability of the measurements carried out in the experiment, was calibrated by the National
`
`Accreditation Body of Spain. Also, the result of such a calibration was an uncertainty of measurement
`of ±0.02o, where the superscript o represents degrees.
`Furthermore, the previously mentioned uncertainty estimate represents an expanded uncertainty
`
`expressed at approximately the 95% level of confidence using a coverage factor of k = 2. The
`
`calibration steps carried out in the laboratory experiments are described below.
`
`First, the zero of the computer-controlled platform is adjusted. In short, the angle of inclination of
`the computer-controlled platform is set at 0o and the inclinometer (or working standard in this paper) is
`situated on the surface of the platform. Then angle of inclination of the computer-controlled platform
`is adjusted until the inclinometer reads 0o. After that, the system is allowed to stabilize for
`approximately 15 minutes, and the zero is readjusted if necessary.
`Second, the inclination of the platform is increased step by step from 0o to 45o, which is the
`working range, and the working standard is used to guarantee that the platform inclination is at the
`
`desired calibration point. Consequently, a measurement is carried out by the system under calibration
`
`at the calibration point and the result of this measurement is written in a table specifically prepared to
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`satisfactorily record this information. Here, the calibration points are the following: 0o, 5o, 10o, 15o, 20o,
`25o, 30o, 35o, 40o and 45o.
`Third, once the platform has reached the 45o of inclination, this is decreased step by step to 0o, and
`also one measurement is carried out at each of the i’th calibration points. Note that this step is
`
`performed similarly to the second step except that the movement of the platform is in the opposite
`
`direction.
`Fourth, when the platform reaches the 0o of inclination, this is again increased and the processes
`explained in the second and third steps are repeated for four more times.
`
`In the end, a record of 100 measurements is obtained. What is more, such measurements are well
`
`organized in groups of 10 measurements carried out at each one of the 10 calibration points.
`
`Fifth, the expanded uncertainty of the system at the i’th calibration point is
`
`
`
` (10)
`
`
`
`
`
`2 i
`
`c
`
`(cid:185)(cid:183)
`p1
`
`(cid:14)(cid:184)(cid:184)
`s
`
`2 i
`
`(cid:14)
`
`1
`
`(cid:169)(cid:167)
`
`(cid:168)(cid:168)
`
`(cid:14)
`
`2 w
`
`s
`
`(cid:32)
`
`uk
`
`U
`
`i
`
`with a coverage factor of k = 2. Furthermore, uws is the uncertainty of measurement of the working
`standard using a coverage factor of k = 1, si is the experimental standard deviation of the p
`measurements (p = 10) taken by the measuring instrument of the computer-controlled platform at the
`
`i’th calibration point, and ci is the difference between the mean of such measurements and the
`conventional true value indicated by the working standard at the i’th calibration point.
`
`Sixth, the uncertainty of the system is the maximum among the ten Ui values obtained in step five.
`Finally, the calibration of the system was carried out following the above mentioned steps and its
`expanded uncertainty was ±0.10o using a coverage factor of k = 2, the laboratory temperature was
`20oC (cid:114) 1oC. Furthermore, the repeatability was 0.04o, the non-linearity was 0.03o and the sensitivity
`was 1000.2 mo/o. Moreover, the response characteristic of this system is completely linear.
`
`6. Results of the experiment
`
`Generally speaking, in order to get the best information about the complex dynamics of cars,
`
`several accelerometers, speedometers, inclinometers, dynamometers, position sensors, etc., should be
`
`used and placed in the most important sections of the car under performance tests. However, the
`
`instrumentation of a car is sometimes expensive. For this reason, in this paper, the use of the optimal
`
`adapative filtering algorithm presented in previous sections was proposed to carry out the real-time
`
`measure of the rollover angle of a car under performace tests. The knowledge of this information by
`
`car drivers can save millions of lives every year [1]. Therefore, this section is devoted to showing the
`
`results of the laboratory experiments.
`
`The optimal estimation of the filter parameters is obtained from experimental curves, which have
`
`been obtained by ensemble-averaging the mean-squared error (MSE) of the filter over 360 independent
`
`trials of the experiment.
`
`In addition, after studying the bandwidth of the relevant signal, a sampling frequency of 500 Hz
`
`was chosen. Furthermore, the signal treatment was carried out by using a laptop computer and the
`
`National Instruments Data Acquisition Card DAQCard-700, both placed in the car under performance
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`tests. Moreover, a computer-controlled platform was used to generate the reference signals or desired
`
`responses necessary to test the sensor.
`
`Fig. 4 shows a typical signal coming from the sensor (i.e., the primary input). In addition, the
`
`reference input is a signal that is uncorrelated with the relevant signal but correlated in some way with
`
`its noise. The reference signal consists of additive noise coming from another accelerometer placed
`
`close to the dual-axis accelerometer that is carrying out the measurements.
`
`Figure 4. A typical signal coming from the sensor versus n (1V = 4.50).
`
`In addition, and Table 1 shows the normalized ensemble-averaged mean-squared error of the filter
`
`(i.e., the cost function) over 360 independent trials of the experiment. Such independent trials were
`
`