MARVIN A. GRlFFIN RICHARD S. SfMPSON Project Directors I H. PAUL HASSELL, JR. Research Astociote h GPO CFST 1 Jma, 1% 1 ff 853 July 65 I! I! TECMNtCU REPORT NUMBER 13 II SYSTEMS ENGINEERING GROUP 1 BUREAU OF ENGINEERING RESEARCH UNIVERSITY OF ALABAMA UNIVERSITY, ALABAMA n I- STATISTICAL QUALITY CONTROL t! APPLIED TO A TELEMETRY SYSTEM ACCEPTANCE PROCEDURE Marvin A. Griffin Richard S. Simpson Project Directors H. Paul Hassell, Jr. Research Associate William S. Spivey 1 Graduate Assistant I I June, 1966 I TECHNICAL REPORT NUMBER 13 I Prepared for I National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama Under CONTRACT NUMBER NAS8-20172 Systems Engineering Group Bureau of Engineering Research University of Alabama ABSTRACT This is the thirteenth of a series of technical reports concerned with the Telemetry Systems on the Saturn vehicle. The purpose of this report is to develop a methodology for implementing statistical control charts as a basis for telemetry package acceptance procedures. The methodology is developed for both the control chart for mean values and for the control chart for standard deviations. A total expected cost model which relates alpha and beta errors as well as the sample size is developed. This model is used to establish optimum upper and lower control limits for the chart for mean values. Control limits for the chart for standard deviations are then established based on this model. An experiment that was designed and conducted for the purpose of testing the feasibility of the optimum control limits is reported. Results of this experiment confirm the reasonableness of the assumptions made in the cost model. Subcarrier oscillators in the experimental telemetry package that were intentionally maladjusted are detected by the control charts. Estimates of the accuracy and precision of the telemetry package are obtained and ninety-nine per cent confidence limits are established for these limits. Standards for future control chart analysis are established for both charts. These standards may be used for future package checkout procedures. I I- 8 TABLE OF CONTENTS Page I ABSTRACT ............................ if 1 LISTOFTABLES e. 0 0. 0 . V LISTOF ILLUSTRATIONS. .................... vi Chapter I I. INTRODUCTION ...................... 1 Statement of the Problem t The Proposed Methodology 11, THE GENERAL THEORY OF CONTROL CHARTS ......... 8 I 111. THE DEVELOPMENT OF THE METHODOLOGY .......... 14 The X-Chart t The u-Chart Estimation of uc Effects of Non-Normality 1 Swmtary of the Methodology IV. THE DEVELOPMENT OF THE PROBABILITY FUNCTIONS 1 FOR THE TYPE I AND TYPE I1 ERRORS .......... 37 Operating- Characteristic Function for the X-Chart t Operating Characteristic Function for the a-Chart Probability Function for a for the %Chart I Probability Function for a for the u-Chart V. THE DETERMINATION OF THE PROPER LIMIT CONSTANT .... 47 I VI. APPLICATION OF THE METHowLoGy ............ 55 Description of the Experimental Output 1 Determination of the Optimum K and n Analysis by Control Charts 8 I iii 1 I Chapter Page S-rY Sources of Possible Error Reconmendations for Future Research and Application APPENDICES . 0 85 A. Miscellaneous Proofs, Theorems, and Sample Calculations ................ 86 B. Glossary of Symbols ................. 93 LISTOFREFERENCES 0 . 0 . 97 BIBLIOGRAPHY ......................... 100 LIST OF TABLES Table Page - 1. Values of O--, u, and u for 0%Input Level .... 56 E, 0 2. ?kans and Standard Deviations for 0%Input Level......... .............. 57 3. Optimum Values of K ................. 58 4. Values of Total Cost (TC = c 8 + c a + c n) ..... 59 1 2 3 5. Means and Standard Deviations for 25X, SOX, 75%, and 100% Input Levels ............ 60 6. Values of 01, q, and Ujz and Channels Requiring Investigation for 5 IAput Levels ......... 64 V LIST OF ILLUSTRATIONS Figure Page 1. Distribution of Chance Variations in a Sample MeasureofQuality . 9 2, Illustration of the Theoretical Basis for a Controlchart . - . 10 3. Control Chart for 51 for Shoulder Depth of Fragmentation Bds . 12 4. Control Chart for u for Shoulder Depth of FragmentationBombs . 12 5. Block Diagram of the FM/FM Experimental Telemetry System . 16 6. Graph of f(F ) Depicting Areas Under the Curve . 22 ij 7. Control Chart for man Values of SCO's . 24 8. Control Chart for Standard Deviation of SCO's . 29 9. Possible States of the Process . 38 10. Graphic Reprecentation of a Shift in the Process wan from to Xc6................. 41 11. Control Charts for u, 0% Input . 63 12. Control Charts for X, OX Input . 65 13. Control Charts for a, 25% Input . 66 14. Control Charts for x, 25% Input . 67 15. Control Charts for u, 50% Input . 68 16. Control Charts for 51, 50% Input . 69 17. Control Charts for u, 75% Input . 70 18. Control Charts for 51, 75% Input . 71 vi Figure Page 19. Control Charts for u, 100%Input ........... 72 20. Control Charts for Z, 100% Input ........... 73 21. OC Curve for --Chart Based on Standard Values, 0% Input ..................... 78 22. OC Curve for u-Chart Based on Standard Values. .... 78 vii CHAPTER1 INTRODUCTION While the attempt to control the quality of a manufactured product is as old as industry itself, the concept of statistical quality control is relatively new. The greatest development in statistics has occurred in the last sixty years, and it was not until the 1920's that statistical theory began to be applied effectively to quality control (4).1 In recent years great progreats has been Bade in applying statistical methods to problems of research and development. At the same time the application of statiatical quality control to the manufacturing process has become recognized as a major factor in the reduction of the costs asso- ciated with improved quality and in the improvement of product quality. The integration of statistical quality control into the area of research and development has also been a tremendous aid in attaining both process and product control. Statement of the Problem A particular quality control problem has recently become 'Numbere is parentheses throughout the thesis indicate the references as listed in the LIST OF REFEFUDICES. 1 2 cvident in the field of aerospace telemetry. Aerospace telemetry la the "science of transmission of information from air and space vehicles to accessible locations" (15). The advent of the missile age has brought about a pheno- me~lincreaae in the usage of telemetry equipment. With the evolution of new techniques and equipment for telemeterlng in- flight apace vehicle data, the need for increased accuracy and precision of the transmitting equipment is obvious. An airborne teleaetry package2 ie placed on the spacecraft for the purpose of transritting the most critical measurements to a telemetry ground station. At the ground station personnel continuously monitor and analyze these measurement data to determine the effect of flight conditions at the vehicle. Therefore it is of prime necessity for the telewtry package to be of sufficient quality to assure that the transmitted data is actually that measured at the vehicle. During the the required for each telemetry package to be sent from the manufacturer to the space vehicle, there are several places where the control of the quality of the package needs to be utabltehed. The first of these is at the manufacturing plant hmedlately before hipping the package to the telemetry personnel. Another is at the test laboratory immediately after the telemetry LA telemetry package is an electrical system coneleting of a set of subcarrier oscillators for converting measured voltage into frequency, a dxer amplifier, a transmitter, and a paver amplifier used for transmitting signals from space vehicle to a ground receiving station. 8 1 - 3 pernounel have received the equipment. A third place for a quality 8 control progrm is at the test laboratory lmediately before sending I the package to the vehicle. A fourth area for controlling the quality of the equipment Sa at the vehicle prior to launch tire, A final I area at which the control of the telemetry package performance is a necessity is in the spacecraft during flight. The mitoring of I actual flight calibration data and subsequent analysis by statis- tical methods would indicate whether or not the package ua~ 1 performing adequately during this critical phase. I It la believed that the ethodology presented in this thesis could be applied to any of these areas, However, the research vi11 1 be conducted at, and applied to, the third of these areas; the MA telemetry test laboratory imediately prior to sending the package 8 to the vehicle. 0 The present program for dettnnfning if a telemetry package ie operating in a satisfactory manner and is ready to be sent to 8 the spacecraft consists of a series of rigorous electrical tests. After these tests have been conducted and adjustments made on the I components, a five point calibration sequence3 is fed through the 8 package aud trhtted over a cable to the ground station. Eere it is received and sent througra bank of discr~ors4and then five point calibration sequence consists of supplying B voltage in five distinct steps to a telemetry package. The steps represent 0, 25, 50, 7S, and 100%of 5 volts. 8 4A discrlminator is sn instrwnent for separating a mixed frequency signal into various frequency bands corresponding to 1 thoae produced by a related subcarrier oscillator. 1 8 4 recorded by an oscillograph. This record made by the oscillograph is at present the only meam for dyzing the quality of the assembled package. It is thought that this method furnishes inconclusive answers to the questions "what is the accuracy and precisian of the package?" and "are there auy assignable causes of varirtiaap within the package?" Accuracy is a measure of systematic errors while precision is a measure of rand- (chance) errors (5,6) .