Real time detection through Generalized Likelihood Ratio Test of position and speed readings inconsistencies in automated moving objects

Transcription

1 Real time detection through Generalized Likelihood Ratio Test of position and speed readings inconsistencies in automated moving objects Sternheim Misuraca, M. R. Degree in Electronics Engineering, University of Buenos Aires, Argentina Project Engineer, Sistemas Industriales, ABB Argentina Abstract In the present work we describe a system to detect inconsistencies of position and speed readings in automated moving objects to prevent collisions. The system is implemented through a Generalized Likelihood Ratio Test (GLRT) strategy. Position and speed measurements are used as inputs of an hypothesis testing system for consistency checking. Type I error (false alarm) probability can be specified and set, while minimizing type II error (mis-detection) probability. Also, we show the results of the successful implementation of this strategy in the transelevator control system we installed in April 2013, in the Distribution Center of Molinos Rio de La Plata in Barracas, Buenos Aires, Argentina. 1 Introduction and problem set up There are several industrial applications where automated moving objects relay on position and speed readings to control its movement, such as cranes and transelevators. Independent position and speed measurements are easily implemented through laser positioning, bar codes or encoders. However they are usually noisy, leaving consistency checking systems open to errors such as stating that the readings are not consistent when they are (type I or false alarm error), or stating that the readings are consistent when they are not (type II or mis-detection error). We examine the Distribution Center of Molinos Río de la Plata S.A, located in Barracas, Buenos Aires, Argentina, where we implemented the system in the trans-elevator control program we developed and installed in April The Distribution Center consists on a vertical warehouse that storages pallets carrying comestible loads of up until one ton. The warehouse has three corridors, each of 100m long x 12m high, with capacity for 2200 pallets. Through each corridor, a fully automated trans-elevator of two ton moves at 10km/h - see figure 1. 1

2 1 Introduction and problem set up 2 In the event of a mechanical or electronic issue in the position or speed readings used for positioning control, it is of the essence that the control system stop the device in order to protect the people and the facilities involved. Fig. 1: Empty trans-elevator moving through the warehouse In this work we present a strategy of hypothesis testing using Generalized Likelihood Ratio Test (GLRT). The system is implemented in real time with negligible computational burden, and allows to set false alarm probability while minimizing mis-detection error probability. The aforementioned trans-elevators are automated by means of a AC800M controller, model PM851. Movement is controlled individually for each axis, by means of an ACS800 drive attached controlling the corresponding motor. The drive receives motor speed measurements through an incremental encoder plugged to the motor axis, and position measurements of the trans-elevator jail through a different incremental encoder plugged to an independent mechanical system - (elevation is taken from a wire, and horizontal movement from a timing chain). We added the consistency check layer to the existing control system without adding computational burden to the controller. In the next section we detail the

3 2 Mathematical model 3 implemented algorithm and we compare it with more conventional integration and differentiation methods to address this issue. 2 Mathematical model We start modeling available measurements. Time is discretized according to the execution time of the controller s task running the program, which we will call t. Time is referenced by a subindex (e.g. x i ). For each axis, at instant i, independent position - x i - and speed - v i measurements are available. They are, however, noisy, with mean equal to the actual position and speed, respectively. Noise statistics are approximated by a gaussian distribution (due to it being mathematically easy to use). We thus have x i = x i + η i (1) v i = v i + ν i (2) With η i R and ν i R gaussian, zero mean and variances σx 2 and σv 2 respectively. We assume they are both ergodic processes (measurement noises are uncorrelated if measurements are taken at different times, though the statistical parameters are equal). Next we derive the most common approaches to this problem, along with the issues they present - differentiation and integration, and finally we describe the method we chose to implement, the generalized likelihood ratio test. 2.1 Differentiation strategy The most simple strategy to check for consistency is to approximate the derivative of the position via finite differences, and then compare it to the speed measurement. We start by defining Λ dx dt (i) x i x i 1 t Λ = x i x i 1 t (3) v i (4) Now we define ɛ > 0 as the bound for the difference between measured speed and approximated speed through derivatives. Thus, when Λ > ɛ, inconsistency warning is set. This strategy presents several issues. First, position measurement noise is amplified, since 3 is a high pass filter, which effect is to filter out the mean and to amplify the random component. Since successive position measurements are taken very fast compared to the trans-elevator speed, the means are similar, rendering it vulnerable to the filter. Moreover, since the subtraction is divided by a very small factor ( t), the noise is even more amplified. Secondly, speed measurement v i is unfiltered, so its noise is not mitigated, which is not desirable. Finally, there is no control over the error probabilities.

4 2 Mathematical model Integration strategy Instead of derivating position to compare it with speed, we will try integrating speed to compare it with the position. Speed measurement, at each moment, is multiplied by the time interval t (and by a factor ρ if speed needs scaling); and then it is added to approximate the integral of the speed. We then calculate the discriminant as Λ = (x N x 1 ) ρ t Again, the decision strategy boils down to comparing the absolute value of the discriminant with the bound ɛ - if it is greater, we decide there is an inconsistency in the measurements. This strategy is better than derivating since speed measurement noise is mitigated. Proof lies on averaging zero mean noise components. Due to eq 2, we have v i = v i + Since adding independent, identically distributed (i.i.d.) samples and dividing by the number of samples is an unbiased estimator of the mean (remember measurement noise is ergodic by hypothesis) we then write η i v i = v i + N ˆµ η However µ η = 0 by hypothesis, thus noise effect is mitigated. Another improvement is that position measurement noise is not as highly amplified, since means of position separated by the data window (N) are bound to be less similar (unless the trans-elevator is not moving at all). Besides, we are no longer dividing it by the sampling factor t. We must underline the fact that, as in the previous case, we don t have means to quantify the errors, much less set them according to specifications. 2.3 Likelihood ratio strategy Constructing the sample set To overcome the problems inherent to integration, we will use hypothesis testing. We start defining the random variable z as follows z N = (x N x 1 ) ρ t v i v i (5) This random variable is gaussian, due to being a linear combination of gaussian, independent variables, with mean equal to the linear combination of their means, and variance equal to the sum of individual variances, weighted by the squared coefficients. Since the linear combination of the means is equal to zero (in normal

5 2 Mathematical model 5 conditions, the mean value of the integral of the speed is equal to the position difference) we have z N (0, 2σ 2 x + ρ 2 t 2 Nσ 2 v) = N (0, σ 2 z) (6) Using 5 we build the sample set {z 1... z N } of (i.i.d.) variables. Samples will be i.i.d. if data windows are independent. However, this will delay sample production, so data overlapping will be permitted if needed out of the application sample speed requirement, assuming i.i.d. samples to simplify the mathematical model Generalized Likelihood ratio test - GLRT Next we define two hypothesis. Null hypothesis is that the mean of z is zero, which corresponds to speed and position measurements consistency. Alternative hypothesis is defined as having non-zero mean, which signifies that there is a deterministic difference between speed and position measurements dynamics, and thus an inconsistency in the data which might be caused by electrical or mechanical issues. Formalizing H 0 : µ z = 0 H A : µ z 0 To contrast such hypothesis, we use Likelihood Ratio Test, which consists on obtaining the discriminant comparing the probability of obtaining the sample set conditioned to the null hypothesis or the alternative hypothesis. At this point, we must redefine the alternative hypothesis (to avoid comparing the zero mean probability to every mean probability) - alternative mean is now equal to the maximum likelihood estimator of the mean computed from the sample set, which is equal to the data set average. We don t lose generality in the process, in the sense that even if the alternative hypothesis will always be more likely, the key is to accept the null hypothesis when it is not likely enough. Later we will see that the error probability, which can now be computed, will help us set the bound to choose the alternative or null hypothesis. See [1] and [2] for details of this detection strategy. Next we formally redefine the hypothesis H 0 : µ z = 0 H A : µ z = µ MLE z = 1 N z i Λ 0 = p(z 1... z N H 0 ) p(z 1... z N H A ) = p(z 1... z N µ z = 0) p(z 1... z N µ z = µ MLE z ) Since samples are independent and gaussian, we write (7)

6 2 Mathematical model 6 Λ 0 = N N 1 e σ z 2π 1 σ z 2π e z 2 i 2σ 2 z (z i µ MLE z ) 2 2σ 2 z Taking logarithms and simplifying constants, we have Λ 1 = z 2 i 2σ 2 z + (z i µ MLE z ) 2 2σ 2 z dividing by 2σz 2 and expanding (z i µ MLE z ) 2 we obtain Λ 2 = Nµ MLE 2 z ( N ) 2 = z i Finally, dividing by the constant and taking squared root (note that Λ 1 is always negative or zero since Λ 0 1, since the denominator of 7 is always greater than the numerator by definition of maximum likelihood estimator) Λ = z i (8) Discriminant Λ is then compared with ɛ. Should it be smaller, null hypothesis is accepted and the system is branded consistent. Else, null hypothesis is rejected, branding the system inconsistent Si Λ ɛ null hypothesis is accepted, system is consistent Si Λ > ɛ null hypothesis is rejected, system is inconsistent The most important difference with the previously described methods is that we are now allowed to compute the probability of false alarm (type I error) α, that is, the probability of Λ being greater than the discriminant conditioned to z having zero mean (inconsistency warning is set, though system is consistent) α = p(λ > ɛ µ z = 0) Statistics of Λ can be derived from the statistics of z. According to (8), Λ is the absolute value of a zero mean gaussian variable, with variance (N)σ z (see 6), namely Folded Normal Distribution. The variance is a function of the ratio ρ (linear-angular speed ratio), sampling time t and σ x y σ v standard deviations. Applying the same principle we derive a mis-detection error bound β Det, which means Λ being smaller that the discriminant given non-zero mean (readings are inconsistent but warning is not set) β Det = p(λ ɛ µ z > µ βdet )

7 3 Implementation 7 According to Neyman-Pearson theorem ([1]), likelihood ratio test minimizes mis-detection error once false alarm error is set, which renders the method optimal when such error is set by the system s specifications. High values of mis-detection errors are more acceptable than high values of false alarm errors. The reason is that monitoring is done in real time (a check for each time instant), thus the probability of a standing type II error through k instants is βdet k. This is the reason why a low type I (false alarm) error should be set to a small value, and then type II error be minimized. Details on how to choose the bound for the discriminant are given in the next section. To minimize the global error, Bayesian estimation methods should be used, which requires a priori knowledge of the class probability density functions (i.e. the a priori probability for the system of being consistent/inconsistent), see [2] for details. 3 Implementation 3.1 Algorithm description and computational burden The algorithm consists on building position and speed measurements vectors, and the sample vector, to choose for system consistency/inconsistency 1. Initialize position and speed measurements vectors, and sample vector z of size N. Initialize the cumulative sum of speed measurements and the cumulative sum of samples z. 2. Refresh measurement vectors, eliminating the oldest measurement and adding the newest one. By the same token, refresh cumulative speed measurement. 3. Compute a new sample z, according to 5. Refresh samples vector cumulative sum of samples vector z, as seen in the previous step. 4. Take absolute value of z to build discriminant Λ, according to 8, and compare it to the decision boundary ɛ. Set inconsistency warning if the discriminant surpasses the decision boundary. 5. Increment time and go back to step 2. The algorithm is O(1) regarding vector size, since the number of operations is independent of such size. The system is therefore implementable in real time without burdening the process controller. 3.2 Practical considerations Next we detail the implementation of the system in the elevation axis of the transelevators in the Distribution Center of Molinos Rio de La Plata S.A. located in Barracas, Buenos Aires, Argentina. Procedures for horizontal movement is analogous. Linear-angular speed ration ρ is identified setting the trans-elevator to move at constant speed to have constant movement difference and constant speed integral. We selected t = 60ms, and vector size N=20. We approximated standard deviation of speed measurement as σ v = 0.1mHz, and position measurement as

8 3 Implementation 8 σ x = 1mm. 3.3 Type I and II error selection Using data according to the previous section, we obtained curves that allows to set false alarm and mis-detection errors. Next we show false alarm error probability function, setting the decision boundary to ɛ = 0.15, which determines type I error α < 1%. For this type of error, increasing the decision boundary improves performance. Fig. 2: Estimation of P (Λ x H 0 ) Finally, we show detection error probability function, setting the same decision boundaryɛ = 0.15, which determines a mis-detection error of β 1seg 15cm = 0.1%. For this type of error, lowering the decision boundary improves performance. Fig. 3: Estimation of P (Λ < x H A ) It is not possible to lower both error types probabilities simultaneously. However, due to Neyman-Pearson theorem, fixing one type of error minimizes the other.

9 4 Conclusions 9 In this case, selecting ɛ = 0.15 renders a false alarm error of α < 1% and a mis-detection lower error bound (for a displacement of 15cm) of β 15cm = 50%. Recall that the probability of a detection error of a displacement over 15cm to last over 600ms is β15cm 10 = 0.1%. The value 15cm is an acceptable distance for the trans-elevator to stop in the event of an inconsistency in the speed and position readings. Test conducted on site supported the simulated results. In the event of inducted failures regarding encoders, trans-elevators stopped safely and reported inconsistency warnings. 4 Conclusions We have implemented, in real time, a GLRT strategy to detect inconsistencies in the position and speed readings of an automated moving object, with minimum computational and hardware requirements. Implemented system constitutes a security layer to avoid collisions, and its performance can quantified and false alarm error adjusted to have it meet the application requirements, whereas mis-detection error is optimally minimized. Finally, we describe the implemented solution in the trans-elevator system of the Distribution Center of Molinos Rio de La Plate S.A. located in Barracas, Buenos Aires, Argentina. Acknowledgments To Alejandro Carrasco, Engineering Manager of Industrial Systems - ABB Argentina, for his assistance in the writing of the present paper. To Juan Pablo Giuttari, Service Engineer - ABB Argentina, for his assistance in the testing process. 5 Bibliography [1] S. Kay, Fundamental of Statistical Signal Processing Vol II Detection Theory. New Jersey: Prentice Hall, [2] O. Duda, P. Hart, and D. Stork, Pattern Classification 2da. Ed. Nueva York: John Wiley & Sons, Inc., [3] S. Kay, Fundamentals of statistical signal processing, volume I: Estimation theory. Prentice Hall, [4] Kailath, A. H. Sayed, and Hassibi, Linear Estimation. New Jersey: Prentice Hall, 2000.