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In the automotive industry, electromagnetic variable reluctance (VR) sensors have been extensively used to measure engine position and speed through a toothed wheel mounted on the crankshaft. In this work, an application that already uses the VR sensing unit for engine and/or transmission has been chosen to infer, this time, the indirect position of the electric machine in a parallel Hybrid Electric Vehicle (HEV) system. A VR sensor has been chosen to correct the position of the electric machine, mainly because it may still become critical in the operation of HEVs to avoid possible vehicle failures during the start-up and on-the-road, especially when the machine is used with an internal combustion engine. The proposed method uses Chi-square test and is adaptive in a sense that it derives the compensation factors during the shaft operation and updates them in a timely fashion.

Motion control is a sub-field of automation, in which the position and/or velocity of machines are controlled using some type of device such as a hydraulic pump, linear actuator, or an electric motor, generally a servo. Motion control system provides precise control of the movement of various actuating elements of a device or system. Motion control is an important part of robotics and CNC machine tools, however it is more complex than in the use of specialized machines, where the kinematics are usually simpler. Motion control is widely used in the packaging, printing, textile, automotive and assembly industries. There are a number of extensive works that can be found in literature related to the motion control applications as some described in [

Magnetic sensors are largely applied to detect position in automotive systems. The conversion of speed or position to a magnetic signal enables non-contact magnetic detection that is unsusceptible to contamination and wear. For instance, magnetic detectors can be used to measure wheel and transmission speed, crank and cam shaft position for engine timing, throttle valve position for air intake, steering wheel position, pedal position, fluid level, chassis height, and in electronic door locks [

A wheel has teeth disposed radially along the periphery of the wheel at a predetermined angular spacing and is mounted on the revolving shaft that transmits motion. The wheel has a first tooth and a second tooth positioned a meaningful distance apart from the first tooth. Ideally, the position of the second tooth, relative to the first tooth is indicated by an angular displacement. In this work, the wheel has a tooth every 20 degrees, with one tooth missing. Since one revolution is comprised of 360 degrees of shaft rotation, there will be 17 teeth excluding one missing for each shaft revolution. The missing tooth on the wheel indicates a fixed absolute position of the wheel. This missing tooth marker is used to synchronize the control of the machine dependent on this fixed position. Using the signal difference caused by the missing tooth, the sensor can provide the signals for the control unit to obtain shaft position in degrees and velocity in rpm. If the displacement causes an error, the feedback sensor will produce an erroneous result. This can never be tolerated for systems that rely on an accurate speed profile [

In order to determine the shaft velocity, only Δ_{i}_{i}^{*} is defined as:

This paper is based on a method using the chi-square test that compensates for wheel profile irregularities in measured shaft positions resulting from errors in the physical placement of target teeth or position markers. These errors are known as position errors and a significant source of such irregularity in determining the rotational arcs during each measuring interval. During manufacture of a wheel, errors occur between the desired and actual positions for position markers on the wheel. Any deviation of the actual angle

The proposed method is an adaptive one in a sense that it compensates for the wheel profile irregularities that derives the compensation factors during actual machine operation. The compensation factors are updated in a timely fashion.

The remaining of the paper is arranged as follows. Section 2 describes the proposed sensor, their operation and properties. Section 3 briefly explains the basics of the chi-square test, the calculation of the test statistic and interpretation of the test results. Section 4 describes the chi-square based iterative error compensation method, and studies an example of an induction machine by applying the method under the effect of viscous friction and presents the simulation results. Section 5 presents the experimental results of the actual system. Finally, conclusions are drawn in Section 6.

Advantages of the VR sensor can be summarized as follows: low cost, robust proven speed and position sensing technology, self-generating electrical signal which requires no external power supply, fewer wiring connections which contribute to excellent reliability, and finally meeting a wide range of output, resistance, and inductance requirements so that the sensor can be tailored to meet specific customer requirements.

As the teeth pass through the sensor’s magnetic field, the amount of magnetic flux passing through the permanent magnet and consequently the coil varies. When the tooth gear is close to the sensor, the flux is at a maximum. When the tooth is further away, the flux drops off. The moving target results in a time-varying flux that induces a voltage in the coil, producing an electrical analog wave. The frequency and voltage of the analog signal is proportional to velocity of the rotating toothed wheel. Each passing discontinuity in the target causes the VR sensor to generate a pulse. The cyclical pulse train or a digital waveform created can be interpreted by a signal processing unit equipped with the required instrumentation.

One significant advantage that VR sensors offer is their low cost; coils of wire and magnets are relatively inexpensive. However, the low cost of the sensor is partially offset by the cost of the additional signal processing unit needed to recover a useful signal. Since they are currently used in wide range of automotive applications including engine and transmission, in this work, an application that already uses the VR sensing circuitry for engine and/or transmission has been chosen to infer, this time, the indirect position of the electric motor in the considered parallel HEV application. An alternative but more expensive technology is Hall-effect sensor.

VR sensors can be made to operate at temperatures in excess of 300 °C. Such high temperature applications include sensing the turbine speed of a jet engine, and engine cam shaft and crankshaft position control in an automobile. VR sensor is well-suited for a variety of other industrial applications, such as conveyer belts, truck, construction equipment, railroad and marine transmissions, automatic transmission in vehicles, All Terrain Vehicles (ATV) tachometer sensors, and ABS brake systems for wheel slip and traction control.

The chi-square test is one of the most commonly used methods for comparing frequencies, distributions, or proportions. The chi-square test is a statistical method used to determine if observed data deviate from those expected under a particular hypothesis. The chi-square test is also referred to as a test of a measure of fit or “goodness of fit” between data. Typically, the hypothesis tested is whether or not two samples are different enough in a particular characteristic to be considered members of different populations. The distribution of the test statistic under the null hypothesis fits the theoretical chi-square distribution. This means that once we know the chi-square test statistic, we can calculate the probability of getting that value of the chi-square statistic [

The chi-square analysis is used to test the null hypothesis (_{0}_{0}_{0}

The test statistic is calculated by taking an observed number (

The larger the difference between observed and expected, the larger the deviation from the null hypothesis, or the larger the test statistic, becomes. Squaring the differences makes them all positive. Each difference is divided by the expected number, and these standardized differences are summed. The test statistic is conventionally called a “chi-square” statistic [

The general form for a test of a hypothesis concerning multinomial probabilities is given as _{0}_{1}_{1,0}_{2}_{2,0}_{k}_{k,0}_{1,0}_{2,0}_{k,0}_{0}_{i}_{i,0}_{0}

A multinomial experiment is conducted. This is generally satisfied by taking a random sample from the population of interest.

The sample size _{i}

The sample size of our measuring/test examples is 18, which is plentiful. Therefore the observed accuracy of the considered hypothesis over the measuring examples will be assumed at least not a poor estimator of its accuracy over future examples. Furthermore, variance in the estimate will not be greater because of the size of the set of test examples [

The shape of the chi-square distribution depends on the number of degrees of freedom,

Assume a critical value and a number called degrees of freedom for the chi-square test. Critical values for the chi-square are determined from a statistical table based on the significance level at which the test is being performed. If the calculated chi-square value is equal to or greater than this critical value, it can be concluded that the probability of the null hypothesis being correct is some very small probability or less. If a calculated value is greater than the critical value, then the null hypothesis is rejected, and it is concluded that there is a significant difference between the observed and expected distributions.

As the velocity changes nonlinearly with the distance in coasting mode, exhibition of this nonlinear behavior has been effectuated by considering, at least, a simple model of the friction with a viscous term that is proportional to the velocity. Hence a rotational system that is an induction motor attached to a revolving shaft under the effect of viscous friction will be considered.

The system equation of motion is:
_{s}^{2}/rad^{2}; ^{st} order ordinary differential equation (ODE) with constant coefficients. The complete response is the sum of the homogeneous term and the forced term:

The homogeneous term is due to the initial conditions in the system. In this case, a non-zero initial condition _{h}^{−t/τ}

Note that the unknown coefficient

The forced term is due to the excitation or input of the system, as the name implies. The forced response can be found by firstly applying a step input to the system, and then by allowing the system to elapse a sufficiently long time (

The complete response is, therefore:

All that is left is to find the unknown coefficient _{0}

Replacing

It is important to note that if _{0} < _{0} >

The compensation factors are made for systematic irregularities arising from the position errors. Speed and load change will also change the compensations dynamically [

Assume that an actual velocity for each delta time interval is equal to an average velocity per revolution as follows:
_{f}_{f}

In the method, there are always two chi-square (^{2}_{α}^{2}^{2}^{2}^{2}

The chi-square test based compensation method does not require a continuous check of the manufacturer tolerances against the compensation factors during the computations of

The following is the flow chart of the algorithm used for the error compensation (see next page). In the algorithm, the centre of the set comprising one complete shaft revolution is also known as the (current) measured time and will be used as the observed number in the chi-square equation in _{0}^{2}, _{0}

In this simulation the time interval was 0 ≤

A simplified block diagram of the Parallel HEV system used in the experiment is shown in

The controller directly outputs the signal for the power circuit and accepts a digital waveform of the position information (VR sensor signals) from a signal processing unit equipped with the required instrumentation. Signal processing unit produces a pulse train consisting of pulses synchronized with the passage of the leading edges of the toothed wheel past the sensor. VR sensor block includes the sensor pickup and the toothed wheel as shown in

An AC induction motor has been chosen as an actual system for experimenting purposes. AC induction motors are the most common motors used in industrial motion control systems, as well as in main powered home appliances. Recently, they have also been increasingly one of the most demanding machines in HEV applications. Because the operating point of the traction engine does not stay in the most efficient operating range in vehicle applications. Series and parallel HEVs are known to have a 10 to 30 percent improvement in fuel economy by choosing the operating point within a higher efficiency range. Therefore, many automotive manufacturers have developed their own design using the induction machine as motor and/or generator, targeting the HEV system applications.

In an automotive industry, electromagnetic VR sensors have been extensively used to measure engine position and speed through a toothed wheel mounted on the crankshaft. However, in this work, the VR sensor has been applied to correct the position of the electric motor, mainly because it may still become critical in the operation of HEVs to avoid possible vehicle failures during the start-up and on-the-road, especially when it is used with an internal combustion engine. Excess acceleration, excess deceleration, and unintended vehicle acceleration as hazards need to be detected and mitigated within the required response time in the case of HEVs.

Single point of failure in velocity or accelerator may lead to overestimated driver request for acceleration, or underestimated wheel torque acceleration, or to detection and estimation which occurs too late and exceeds required response time, or too early or too sensitive. False detection of single point of velocity failure due to the engine speed, motor or generator speed may have the potential effect of false detection of the vehicle speed especially at low speeds developing in the form of creep torque for engine, and motor in generator mode. Potential effect of failing to detect single point of velocity failure may be to potentially disable detection of excess acceleration and excess deceleration, and detection of wheel torque sign/direction while in drive, low, and reverse.

Complete experimental results for 415 V, two-pole, 22-kW induction machine are presented. The machine needs to be operated in an unpowered, coasting mode as mentioned previously. Coast-down (a.k.a. run-down or spin-down) is defined to be the behavior of the rotating system that goes unpowered until the system comes to rest. When the power to the rotating system is cut off, the energy dissipation in the system due to the frictional effects of bearings, windage,

An experiment was carried out in which the motor was firstly brought up to the vicinity of synchronous speed of 3000 rpm and then brought down or launched from a measured speed of 2500 rpm at 50-Hz supply (a.k.a. cut-off speed) to 0 rpm in 20 seconds as can be seen in

In the following part, the induction motor of the actual system was used, and the algorithm for the wheel profile irregularities was implemented to obtain the compensation factors that were based on the chi-square test. These factors were iteratively determined during in-use machine operation and were updated on-line.

Now, the experimental results for the actual system of the 22-kW induction motor as based on the coast-down data above are presented.

The most noticeable distinction between

It is observed from the experiment that the proposed algorithm compensates for the toothed wheel profile irregularities thus position errors accurately and not only allows for changing velocity and not only handles geometric constraints, but also is capable of compensating for the irregularities quickly and efficiently. This is very important performance criterion in the position error compensation problem. The method also presents many advantages of which some are the reduced risk of the control system’s failure or malfunction, and the computationally simple implementation without extensive memory or processing requirements. The compensation factors are obtained iteratively without the need for a look-up table or stored compensation factors from previous operations or experience.

A VR sensor has been used to correct the position of the electric machine of a parallel HEV system, mainly because it may still become critical in the operation of HEVs to avoid possible vehicle failures during the start-up and on-the-road, especially when the electric machine is used with an internal combustion engine. Both simulated and actual results have been presented to demonstrate the accuracy of the method. Specifically, the actual results show the effectiveness of the presented method and that the method is capable of compensating for the irregularities and the compensation was rather accurate and the algorithm was rather quick and efficient. In the future, the plan is to extend this study to investigate the performance of the proposed compensation method under the transient operating conditions, which is usually considered for the HEVs that could only use the motor to assist or start the engine.

Variable Reluctance (VR) sensor that senses movement of the toothed wheel past point of sensor.

Simulated coast-down curve.

Angular delta shaft position vector before compensation.

Angular delta shaft position vector corresponding with unique set of compensation factors.

Unique set of compensation factors through the search of

Angular delta shaft position error vector before- and after- compensation.

Measured time Δ

Structure of the Parallel system considered in the experiment. The letters indicate: CE: Internal combustion engine, EC: Energy conversion unit (Motor/Generator), ES: Energy storage unit (Battery), TR: Transmission, VR: Variable Reluctance sensor and its signal processing unit.

Main components of the proposed experimental system.

Experimental coast-down curve.

Angular delta shaft position vector corresponding with unique set of compensation factors.

Angular delta shaft position error vector for both simulated and actual systems after compensation.