SENSOR FUSION FOR LINEAR MOTORS, AN APPROACH FOR LOW-COST MEASUREMENTS
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- Derick Phelps
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1 Proc. of Mechatronics 22, University of Twente, June 22 SENSOR FUSION FOR LINEAR MOTORS, AN APPROACH FOR LOW-COST MEASUREMENTS Bas J. de Kruif, Bastiaan van Wermeskerken, Theo J. A. de Vries and Maarten J. Korsten University of Twente, Drebbel Institute for Mechatronics, P.O. Box 217, 75 AE Enschede, The Netherlands for information please Abstract A new measurement setup for a linear motor is investigated. It is custom to use an optical ruler as a measurement device for a linear motor. The high position measurement accuracy that is obtained with this ruler, is required to get an accurate estimate of the velocity to be used in the velocity feedback loop. The accuracy is not required for the position feedback loop. An optical ruler is an expensive measurement device and because the accuracy is higher than needed for the position measurement, it could be advantageous to use a cheaper sensory system to measure the position and to use auxiliary sensors to get a good estimate of the velocity. The permanent magnets that are inherently present in a synchronous PM motor can be used as a coding such that no external ruler is necessary. Hall sensors are used to measure the magnetic field of these magnets and determine the position based on this magnetic field. The velocity estimate derived from this measurement does not meet the specifications and therefore an acceleration sensor is added. The acceleration sensor in combination with the position sensors estimate the velocity with an accuracy that meets the specifications for a considerably smaller price than the optical ruler. 1 Introduction Linear motors are typically used in applications that require high speed, high force and high precision actuation. The high precision, i.e. position accuracy, can only be achieved if good measurements are present for the feedback. For fluent control a good estimation of the position is not enough and a good velocity estimate is also needed. It is common to use an optical ruler to measure the position (Gieras & Piech 2). The velocity is not actually measured but calculated from the position measurement. This velocity estimation uses differentiation and is therefore noise sensitive. High quality position measurements are needed to get satisfactory velocity estimates. Typically, the deciding factor in selecting a position sensor for a linear motor is the resolution that is
2 Proc. of Mechatronics 22, University of Twente, June 22 required in order to obtain a sufficiently good velocity estimation, and not the accuracy that is required for the position feedback loop. An optical ruler is an expensive measurement device and it would be economically attractive to replace it with a less costly substitute. The positional accuracy of the substitute may degenerate for the control of the motor, however, the quality of the velocity estimation may not decrease. The required accuracy, the distance between the measured quantity and the true quantity, is given to be 5 µm for the position estimate and should be accurate within 1.7 mm/s for the velocity estimate. In this paper, we propose such a measurement system. This paper is organized as follows. In section 2 the background of the method is treated. The concepts by which we try to substitute the optical ruler are given here including some theory. In section 3 the requirements placed on the hardware are discussed. With the new measurement setup a set of experiments is performed. These experiments are treated in section 4. Finally a conclusion is drawn in section 5. 2 Background Optical rulers can be used to obtain a good position measure and based on this measurement a velocity estimate can be deduced. An optical ruler consists of two parts: 1. a strip with some form of coding that can be read with optical means 2. a reader that measures position by reading the coding of the strip High accuracies are achievable with such systems. However, for medium accuracies, a ruler is relatively expensive, due to the fact that the cost price for an optical reader is high and because a specific strip with accurate coding always has to be added to the motion system. It is constructional unattractive to include a strip in the motion system. In order to substantially lower the costs, we should look for a measurement system based on much cheaper hardware. This may be realized by considering solutions that meet the following principles: 1. functional interaction: measure motion characteristics on basis of inherently present actuator properties instead of using a separately added measurement device. 2. sensor fusion: measure motion characteristics with multiple low cost devices instead of one costly device Function interaction: We wish to combine the inherently present actuator properties for the estimation of the position. The most obvious choice is to use the magnets that are present for the generation of a force as a coding. The magnetic field can be related to the position of the translator. Sensor fusion: We want to combine several magnetic field measurements to one position estimate with a higher accuracy than a single sensor can achieve. Secondly, we want to fuse this position estimate with an acceleration measurement to get a velocity estimate. All these combinations are done at a signal level and it is therefore called signal-level sensor fusion (Lou & Kay 1992). This two-stage fusion scheme is depicted in figure 1. In this figure the block SF measures the position from several magnetic field measurements and the block K estimates the states from the position and acceleration measurement.
3 Proc. of Mechatronics 22, University of Twente, June 22 X1 X2 X3 Xn SF xˆ a K xˆˆ ∠vˆˆ Figure 1: Sensor fusion scheme for estimation of the states 2.1 Position measurement The position is measured by a combination of functional interaction and sensor fusion. If only one Hall sensor is used to measure the magnetic field of the permanent magnets, the spatial sensitivity might be too low in certain regions, and therefore several magnetic field sensors can be used. The combination of these measurements is the sensor fusion part. The spatial sensitivity of one Hall sensor that can measure the complete range variation of the magnetic field might be too low to obtain the required positional accuracy as stated before. The spatial sensitivity is given as the derivative of the output of the Hall sensor relative to the position. The output of one sensor as function of the position is given in figure 2. The location of two magnets is included in this figure. The spatial sensitivity is zero at the top and the bottom of the sine because the tangent is horizontal. If an AD-card is used to read the output, the position range that corresponds to this digital value would be large compared with other positions. Should the Hall sensor be moved closer to the magnets, the output will saturate. Both unsaturated and saturated sensor outputs are given in figure 2. It can be seen that the spatial sensitivity is larger for the saturated sensor if it is not saturated. If we use multiple sensitive Hall sensors that saturate most of the time but that are located in such a way that there is always one out of saturation, we are able to increase the spatial sensitivity as much as desirable. This is illustrated in figure 3. In this figure it can be seen that if several sensors are used, the spatial sensitivity is nearly equal in the complete domain. The numbers in this figure denote the sensor number. The number of required Hall sensors for the given accuracy will be calculated in section 3. To obtain an estimate of a position based on the strength of the magnetic field, the magnetic field should be known as function of the position. If all the magnets are the same and their position relative to each other is equal, the position can be calculated by the strength of the magnetic field and the number of magnets that are passed. However, there is some tolerance on the magnet strength and their placement which is unknown. This tolerances don t need to introduce new inaccuracies. The output of the sensor can be stored in a table as function of the position during an identification phase. By comparing an output with the stored outputs and by including the number of passed magnets, the position can be determined. The tolerances on the spacing and the strength are incorporated in the table and shall only give small errors due to interpolation between measured values. 2.2 State estimate The set of Hall sensors gives a measurement of the position. This measurement can be combined with an acceleration measurement to get an estimate of the velocity. This
4 Proc. of Mechatronics 22, University of Twente, June 22 Hall output output ma Hall output output ma output min output min N S position Figure 2: The sensor output as a function of the position at different distances to the permanent magnets N S position Figure 3: The output of several sensors close to the permanent magnets and displaced relative to each other combination is depicted with the block K in figure 1. The combination of measurements to obtain non-measured, but dependent, variables can be done using a Kalman filter (Ljung 1999). To obtain an estimate of nonmeasured variables, the Kalman filter includes an internal model of the system. The measurements of the variables are compared with the states of this model and the states of the internal model are updated depending on the variances on the measurements and the predicted variance of the model states. The discrete internal model of the Kalman filter is given in state space by: x(k + 1) = A(k)x(k) + B(k)u(k) + w(k) z(k) = C(k)x(k) + v(k) In this equation the x is the state vector, the u the control vector, the z is the output vector, w is the system noise and v is the measurement noise. The noise sources are assumed to be white. If this is not true the noise can be coloured using augmented states. The covariance matrix of the system noise equals E{ww T } = Q while the covariance matrix of the measurement noise is given as E{vv T } = R. The crosscovariance matrix between these is assumed to be zero. The error covariance matrix is denoted by P ee. This is the variance of the difference between the state and the measurements. With this internal model a prediction can be made for the next state and the next error covariance. This one-step ahead prediction is given as: x(k k 1) = A(k 1)x(k 1 k 1) + B(k 1)u(k 1) P ee (k k 1) = A(k 1)P ee (k 1 k 1)A T (k 1) + Q(k 1) The estimated state vector is a combination of the predicted state vector based on the internal model and the measured state vector. The Kalman gain matrix M gives a measure for the difference in uncertainties in predicted and measured state vectors, that is based on the innovation matrix S. S is a measure for the total uncertainty: S(k) = C(k)P ee (k k 1)C T (k) + R(k) M(k) = P ee (k k 1)C T (k)s 1 (k)
5 Proc. of Mechatronics 22, University of Twente, June 22 If a new measurement, z, becomes available, the states and the error matrix are updated as follows x(k k) = x(k k 1) + M(k) ( z(k) C(k)x(k k 1)) P ee (k k) = P ee (k k 1) M(k)S(k)M T (k) The internal model that is used in the Kalman filter is a kinematic model. The knowledge that is inserted is that the derivative of the position is the velocity and the derivative of the velocity is the acceleration. A physical model is not used to increase the portability. The measurement should give results according to its specifications for each mass independent of the setup that is used. The inclusion of model knowledge will improve the accuracy. The discrete internal model is given as: A = 1 T s 1 2 T 2 s 1 T s 1, B =, C = [ 1 1 Note the matrix B which consists of only zeros. This is done because the control signal is assumed to be unknown. If the desired acceleration is present from the reference profile, this can be used in the input matrix B. But in our case we assume not to posses this knowledge. This results that for this model all alterations in accelerations are due to the system noise. The covariance matrix Q should get such a value that the filter can react fast enough on changes. The value of this matrix is determined in the next section by simulation. The covariance of the final estimate depends on the variance of the measurements and thus on the chosen sensors. The accuracy of the position has to be 5 µm while the velocity should be estimated within the 1.7 mm/s of the true value. These specifications state that the maximum absolute error is bound. However, the noise sources and the covariance matrices are given as variances. With a given variance, a bound on the absolute error can never be given with absolute certainty. Therefore we interpret the given specifications such that the standard deviation of the accuracies should be below the given values. To estimate the variance allowed on the sensors that will give this accuracy, a set of simulations is performed. 3 Hardware requirements In the previous section it was shown how the different sensor signals were combined to get an estimate of the position and the velocity. In this section the requirements on the hardware are determined. First the allowable variances of the measurement signals that serve as input of the Kalman filter are determined. The position that is used by the Kalman filter is also the combination of several Hall sensors. In the second part of this section the number of required Hall sensors is calculated. 3.1 Variances Unfortunately all sensor outputs are corrupted with noise. In this section we shall determine how large the variance of the noise on the output of the position and acceleration measurement are allowed to be such that the estimated position and velocity is still accurate enough. The variance of the estimated variables is calculated for a set of different variances on the measured variables. The variance of the estimation is determined by calculation ]
6 Proc. of Mechatronics 22, University of Twente, June 22 the error covariance matrix P ee in steady state which can be done using the Riccati equations. The variances of the estimated variables are the diagonal elements of this matrix. The standard deviation of the position estimation is given in figure 4 and the standard deviation of the velocity estimation is given in figure 5. In these figures the standard deviation is plotted as function of the standard deviation of the acceleration measurement. This is done for several values of the standard deviation of the position measurement. The horizontal line denotes the required accuracy. The exchangeability between variance on the position and acceleration measurement can be observed. The calculations are made in the steady state situation. The variance on the sensors is chosen well below the required value. The choice of the sensor variances is given by the diamond in the figures. This is done to play it save and because no control signal is included in the model so the sensors should have to give good measurements to follow fast changes. The cost of the sensors is kept in mind while making this choice. The standard deviation on the measurement is given by: σˆx = [m] σ a = [m/s 2 ] This will result in a steady state deviation of: σˆx σˆv = [m] = [m/s] With these variances the matrix R is given as: [ ] R = 1 3. Next to the calculation of the matrix R the matrix Q should be given a value. This has nothing to do with the hardware selection, but it can be determined during the same simulations for the testing of the measurement setup. This matrix gives the variances on the system noise. Only the acceleration is influenced by the system noise and therefore only the (3,3)-element of this matrix has a value unequal to zero. The change in acceleration is only due to the system noise because the control signal was not available for the internal model. This means that the system noise should be chosen such that this noise is capable of driving the model to the same accelerations as the physical system. If the system noise is chosen too small, the filter will not be capable to estimate the position fast enough and a phase lag will occur. On the other hand, if this value is too high, the noise of the acceleration sensor will not be filtered and is assumed to be the true value of the acceleration. A compromise has to be found. Through simulations the variance of the system noise is determined. The motion that was performed consisted of several sines such that the maximal acceleration was 1 m/s 2 while the maximum velocity was 2 m/s. A reasonable compromise between phase lag and noise was to set the variance on 1. With the given matrices a set of simulations was performed to test if the determined co-variance matrices gave the correct behaviour. The first simulation was to estimate the velocity and the position while standing still. Both estimations were located for most of the time within their predicted standard deviation, as expected. The second simulation was a simulation were the acceleration and the velocity of the motion were at their maximum value of 2 m/s and 1 m/s 2. In this case the maximal error in the
7 Proc. of Mechatronics 22, University of Twente, June σˆx [m] 1 6 σˆx = 1 3 σˆx = 1 4 σˆx = 1 5 σˆx = σˆx = σˆv [m/s] 1 4 σˆx = 1 3 σˆx = σ a [m/s 2 ] Figure 4: The standard deviation of the position estimate for different set of sensors σ a [m/s 2 ] Figure 5: The standard deviation of the position estimate for different set of sensors velocity estimate was (1 2 m/s). The estimation lagged behind and this also resulted in large position errors (36 µm). This error occurred at the highest jerk. If the jerk was limited to remain within 25 m/s 3 the specifications were met. It should be noted that the velocity estimation of the present controller gave worse results. 3.2 Number of Hall sensors The position measurement ˆx is the result of combining the outputs of the Hall sensors. Several Hall sensors are used for this measurement. If the sensors are placed close to the magnets, they are saturated most of the time, but if they are not saturated, the position sensitivity is high. This was already illustrated in figure 3. The closer the sensors are placed by the magnets, the larger the position sensitivity and the shorter the time they are out of saturation. In this paragraph the number of sensors is calculated to achieve the required resolution of 5 µm. The number of sensors depends on the noise of the sensors and the number of bits used in the AD-card. If no noise is present on the sensor output, any resolution could be achieved if enough bits were used for the conversion. However noise is present and it should not introduce errors that are larger than 5 µm for each position of the motor. The output of one Hall sensor can be written as: ( ) 2πx y = 2.5 sin + ɛ [V] p in which p is the pitch of the magnets and ɛ is the noise that is measured to have a standard deviation of 6 µv. The x is the displacement and the amplitude of the output was 2.5 V. The worst spatial sensitivity occurs right above a magnet because the tangent is horizontal there (fig. 2). To estimate the position within 5µm the deviation to the right or the left is only allowed to be 2.5µm. By moving 2.5 µm to the right we find a change of the output of ( 2π y = sin π ) = 2.1 [µv] 2 The standard deviation of the sensor is 6 µv while a change of 2.1 µv gives already rise to a deviation of 2.5 µm. It is clear that the output noise is too large and a larger spatial sensitivity is required.
8 Proc. of Mechatronics 22, University of Twente, June 22 If two sensors are used that are out of saturation successively, the output of these can be given by: 2.5 ( ) 2πx 2 sin p [V] if y y 1 = ( ( )) πx 2.5sign sin p [V] otherwise 2.5 ( ) 2πx 2 sin p y 2 = + 1π 2 [V] if y ( ( )) πx 2.5sign sin [V] otherwise p + 1π 2 The worst sensitivity occurs if one of the sensors is just out of saturation or is just about to enter. A deviation of 2.5 µm would result in a change in the output of: y 1 = π sin( π ) = 3.2 [mv] 4 So in the worst position of the motor, a change in position of 2.5 µm correspond to a change in the output signal of 3.2 mv. This is about 5 times the standard deviation of the sensor noise, and it is therefore unlikely that the sensor noise will introduce measurement errors that are larger than 5 µm. The AD-card should be able to detect the changes of 2.5 µm and should therefore have a maximal quantization step of 3.2 mv and have a reach from to 5 V. This requires a minimum of 11 bits. A set of two Hall sensors read by AD card with 11 bits should be enough to measure the position within 5µm. However, to play it save and to increase the accuracy, four sensors are used with 12-bits AD-converters. 4 Experiments Several experiments were performed on a linear motor that is present in our laboratory. This motor was extended with four Hall sensors that were displaced relative to each other for the measurement of the magnetic field such that one was always out of saturation. Furthermore, two acceleration sensors were mounted on the motor. One was mounted on the moving part while the other was mounted on the stationary part. All these signals were read by a computer through a 12-bit AD-card at a sample frequency of 4 KHz. Three experiments were performed. The first was the estimation of the position and the velocity while standing still. The second was the estimation of the position and the velocity at a constant velocity of 1 m/s and the third was the estimation while accelerating at 5 m/s 2. These experiments were performed in closed loop and the used measurement device for the feedback was still the optical ruler that was already present in the setup. The optical ruler is also used as the true value of the position for comparison reasons although it has a time delay. The position and the velocity estimation based on the acceleration measurement and the Hall outputs was done by the gathered data off-line. The estimation of the variables was done as if the data came in per sample and no techniques were used that could not be used in a real-time setting like anti-causal filtering. First a set of experiments were performed to estimate the time delays in the system and the sensor biases. These estimates were used to compensate for these effects while estimating the position and velocity.
9 Proc. of Mechatronics 22, University of Twente, June Position error [m] Velocity error [m/s] Figure 6: Steady state position error Figure 7: Steady state estimate error Position error [m] 1-1 Velocity error [m/s] Figure 8: Position estimate error while Figure 9: Velocity estimate error while moving with constant velocity moving with constant velocity The error in the position estimate and the velocity estimate in the steady state case are given in figure 6 and 7. The bias in the position estimate is 3.3 µm while the standard deviation is only.6 µm. The bias in the velocity estimate is.7 mm/s while the standard deviation is.13 mm/s. These biases are introduced by a bias in the acceleration measurement. The steady state estimation stays within the specifications. The error in the position and velocity estimate while moving with a constant velocity are given in the figures 8 and 9. It can be observed that a periodic error is present, with a period corresponding to the magnet spacing. This effect is assumed to originate for differences in time delays. Next to the periodic error, the bias of the acceleration sensor can still be observed. The specifications are almost met. The error in the position and velocity estimate while moving with a constant acceleration is given in the figures 1 and 11. The same periodic error as with the constant velocity can be observed. The amplitude of this error grows as the motor is moving faster which can supports the hypothesis that the periodic error is introduced by due to time delays, but it can also be contributed to the magnetic field generated by the motor coils. The specifications are not met, although it is close.
10 Proc. of Mechatronics 22, University of Twente, June Position error [m] Velocity error [m/s] Figure 1: Position estimate error while moving with constant acceleration 5 Conclusions Figure 11: Velocity estimate error while moving with constant acceleration In this paper a method is presented that uses functional interaction and sensor fusion to estimate the position and the velocity of a linear motor. This measurement setup makes it possible to omit the expensive optical ruler and use low cost acceleration sensors and Hall sensors. By using the magnets that are already present in the linear motor to generate the driving force no ruler has to be placed next to the motor which makes this setup constructual attractive. The specifications were met in the steady state case, slightly missed while moving with a constant velocity of 1 m/s and not met while accelerating with constant acceleration. However it is a most promising technique for the replacement of the optical ruler. It is expected that the performance can be easily increased such that it will meet the specifications if the time delays are compensated for as for the motor currents. References Gieras, J. & Piech, Z. (2), Linear Synchronous Motors, Transportation and Automation Systems, CRC Press, Boca Raton, Florida. Ljung, L. (1999), System Identification, Theory for the user, PTR Prentice-Hall information and system sciences series, 2nd edn, Prentice-Hall PTR, Upper Saddle River, New Jersey. Lou, R. & Kay, M. (1992), Data Fusion in Robotics and Machine Intelligence, Academic Press, Inc., chapter Data Fusion and Sensor Integration: State-of-the-art 199 s, pp
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