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IMPROVING THE ACCURACY OF COMPUTER CONTROLLED INDUSTRIAL ROBOT L. M. Bolotin and Vu. V. Stolin /11'/;/11/1'

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Computer control of an industrial robot allows its Abstract. analyt i cal programming. In this case . its movemen ts are generated from the data provided by sensors and those stored in a data base characterizing the robot's model and its environment. Th e discepancy which exists between the relations describing relative coordinate Systems of various items of the environment. industrial robot and sensors. as well as geometrical. rigidity and other parameters of the theoretical model of the industria l robot and its effective structure induces considerable systematic error of the industrial robot mo vements, The paper discusses methods of experimental attesting of the rigidity and geometrical parameters of the industrial robopt moving system. On the basis of attested parameters and proposed accuracy models of industrial robopts. algorithms of programmed compensation of robot control programs allowing consigerable reduction of systematic positioning errors in case of analytical programming of industrial robopt have been developed,

Keywords. Industrial robot. computer control. error correction. modal control. matrix algebra. accuracy. experimental attesting.

INTRODUCTION The accuracy of computer controlled and analytically programmed industrial robot (IR) is achieved presently by the way of "calibration" of their working area (Gremaylo. 1981: Tarvin. 1980). This procedure implies the use of a table establishing the relationship between the carthesian coordinates of the hand and generalized coordinates (joint displacements) of the arm. In order to compose such a table. a bi- or tridimensional network of nodular points is constructed in the IR working area. at which points the hand is positioned in turns and corrsponding generalized coordinates are recorded. This aproach. similar to the teaching method. implies compensation of syste matic positioning errors due to various factors without revealing the nature of the latter. Hence the " calibraton " method requires a great amount of measurment all over the IR working area.

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Its disadvantages. which limit considerably its application. are: a large volume of memory required on the control computer and a great amount of labour. The analysis shows Tarvin (1980). Wu (1983) that the major part of the systematic positioning error of such IRs is due to two types of positioning errors (which are usually not taken into account in IR theoretical models): - "static" due to backlashes in mechanical transmissions between the sensors and the arm skeleton links. as well as to elastic properties of these transmissions: and "geometrical" due to IR manufacture and assembly defects. as well as to sensor zero setting errors in original configurations of the arm skeleton links. The accuracy in analytical programming can be improved by gradually solving the following problems: 1. creating hardand software for attesting geometrical

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and rigidity parameters of the IR manipulators having N degrees of mobility; 2. constructing a specific IR manipulator model used for control which takes into account its effective (attested) geometrical and rigidity paramrters and. from this. developing algorithms ensuring highly accurate movements of IR. These tasks are discussed below.

THE MODEL In the model under examination the mechanical transmissions structure is described by an A matrix composed of elements determining the partial ratios of kinematic drive chains between sensor output shafts Yl coordinates i=1. Nand IR arm skeleton links ~i generalized coordinates i=1.N. what gives. the transmissions being considered as perfect:

(1) Besides that. the model includes the IR arm skeleton (Fig. 2.a) composed of perfectly rigid elem~nts linked with each other by means of 1 degree of mobility kinematic couples. Relative positions of successive arm links are determined by a group of constant parameters dt (Fig. 2.b). which characterize the effective links geometry. and generalized coordinates. for example. relative rota... tion angles ~, i=1.N. The effective geometrical parameters of an i link are related to corresponding nominal values by the relations:
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where 7:.; is the 4x4 matrix characterizing the position and orientation of the i carthesian coordinate system related to the i link in j carthesian coordinate system . The backlashes and elasticity in mechanical transmissions alter the relation (1). This alteration makes necessary their at-

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ATTESTING OF IR Attesting of IR rigidity parameters. The experimental method of attesting rigidity parameters - determination of generalized elasticity values. backlashes in transmissions for each mobility degree is based on the use of generalized coordinates in which the unitary elasticity matrix of the IR drive transmission is diagonal. The same coordinates are used for constructing a backlash diagonal mat rix. For the most of IR manipulators this requirement is satisfied with the coordinates of motor shaft rotation angles. A model of IR mo ving system taking into account elasticity and backlashes between the elements o f mechanical transmission linking the arm links with the drive mot o rs and sensors is shown in Fig . l.a .

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For measurment. the IR gripper is fixed with respect to the base . The motor shafts are loaded with variable referenced torques and the readings of displacement sensors related to the motor shafts are recorded. Then elasticity curves are constru c ted ( Fig. l.b ) for each degree o f mobility of the IR . which are used for determining mechanical tran smission rigidity and backlashes. Different "behav iour" of the elasticity and backlashes in response to the loading allows their easy differen c iation . Measurment accura c y can be improved by removing the impact of the backlashes in the arm skeleton joints. For that the arm links should be fixed with respect to the base or to each other . Atte s ting of IR geometri c al parameters. There is no other but experimental - direct or indirect - of attesting IR manufa c ture and assembly geometrical errors ad;. d al. a d. and those of posi t ion sen sor zero setting A~i. i=1.N. The direct method implying measurement of each IR arm link geometri c al parameters with subsequent surning up of the dimensions does not ensure achie v ing sufficiently accurate results. The known (Hayati. 1983: Klepov. 1984) indirect methods of attesting use a n o n-linear model of the

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impact of geometrical parameters ~" ai. \I{.' . i=l.N on the end effector position in a carthesian coordinate system. The proposed method differs from them by the fact that it is based on a linear model of the impact of the errors Alii. .la.,. Ild,. IlY,oIi.. i=l. N. accounting static drifts

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The experimental method of IR 4N geometrical parameters attesting is based on the end effector effective position measurment with respect to a reference geometrical object placed into the IR working area. The method consists in following. A reference object with attested geometrical parameters is placed into the IR working area (Fig. 2.a). Then the IR arm is put. under manual control. into contact with characteristic points or plans of the object. In this moment. the readings of the arm link position sensors~. i=1.N are recorded. From these results ( ~. i=1.N) and the nominal IR arm link geometrical parameters di. ai. di. i=1.N. the "altered" geometrical parameters of the reference object are computed using formulas of direct problem on the arm position. They may be distances between the normal to a plan or coordinates of the object characteristic points in its own coordinates system. After that effective geometrical parameters of the IR are computed from their discrepancy with attested geometrical parameters. For that it is necessary to compose and solve the following simultaneously equations:


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The coefficients of the <:P matrix are determined from the parameters c(j. a •.

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(3) due to backlashes and elasticity. on the end effector positioning errors. Besides that. in all the above mentioned papers and in (Hayati. 1983: Wu. 1983) the problem of attesting has not been solved for the entire system (4N geometrical parameters) but only for some types of errors. usually for arm link length errors and pOSition sensor zero setting. It becomes obvious from the analysis that. generally. the attesting of two other types of errors is not less important.

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where matrices 1/ ,i / tA:il . k=1.4 are determined by "analytical" differenciation of ~ - ~i matrix elements by corresponding parameters tA:. k=1.4. Vector elements are the difference between the attested and corresponding "altered" geometric values of the reference object It is supposed that IR nominal geometrical parameters are but slightly different from the corresponding effective ones. And this equation. in the first approximation. can be considered as linear with respect to the unknown errors <10(,. 4a,. 4di. t.~". i=1.N. Hence the simulteneous equations are linearly independent when the configurations differ from each other by displacement. at least in one arm joint. equal to not less than 5~

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In order to improve the accuracy of IR geometrical parameters attesting. extra position sensors should be placed on the arm links for the period of experimentation. When the attesting is performed using the "own" feedback sensors of the IR. the method allows taking into account the static drift (3) of the end effector due to the backlashes in mechanical transmissions and their elasticity under external static loading. Solving the problem of attesting the conditions for obtaining from the total number of 4N errors their "undivisable" sets which are determined only by the structural arm skeleton properties and produce a degenerated system of linear equations. Analytical and numerical methods of characterizing such structural blocks of geometrical errors and their replacement with one or several equivalent "divisible" errors are proposed. Besides that. conditions for choosing the end effector geometrical parameters necessary for attesting the four geometrical errors of the last link of the arm A method of scaling linear values which improve the conditionality of the system is proposed. as well as iterative method of levelling the effect of linearization

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of the direct problem of IR manipulator position which improve the accuracy of the solution. On the basis of this general approach hard - and software have been developed which allow attesting geometrical parameters of IR with any number of arm links including translational and rotational joints. Numerical relatious for necessary number of check IR arm configurations depending on the number of mobility degrees and method of attesting being used are defined. For instance. complete attesting of an IR having six mobility degrees reguires 12 arm configuratious. Statistical reliability eva luations of the solutions are defined. From the general solution a method of IR reattesting is developed which serves for only zero setting of position sensors in c ase they are replaced. determining the relationship between the coordinate systems of various items of the environ ment and manip~lators and sensors in order to ensure the functioning of the IR in adaptive mode.

COMPENSATION OF IR END EFFECTOR POSITION ERRORS On the basis o f specified rigidity ( fi. Si. i=1. N) and geometrical a7. dt. i=1.N) parameters of the IR model used for control. algorithms of computing compensations for IR control program are developed. They compensate static and geometrical position errors of the IR end effector.

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The essence of the progammed compensation algorithms is the following. From the known rigidity parameters of mechanical transmissions a matrix of IR unitary elastici ties Fy is composed in mo tor shaft coordinate system. For the backlashes a backlash matrix 4Y is constructed. Both are diagonal in these coordinates. Then static external load H9 (gravity forces) reduced to the same generalized coordinates is determined. Static drifts of the system reduced to the programmed drive coordinates and corresponding compensation values to be introduced into the IR control program are determined by suming up the drifts due to elasticity and backlashes calculated respectively by multiplying the unitary elasticity matrix by the generalized forces vector and the backlash

matrix by the vector Ht ~ 11 sisn rig, /1 the elements of which are signs of these forces (+1.-1): (6)

Calculating static position errors it is taken into account that due to the forces provided by balancing devices. sectors of mechanical transmissions between the drives and the points where these forces are applied are unloaded as well as the drives. In order to formalize the algorithm of static position errors calculation taking into account balancing forces the model is devided into two successive N- dimensional sub-systems (Fig. 1.a). Then the matrix of elasticity. backlashes and static drifts are determined for each of the sub-systems. Suming up the latters the static drifts of the whole system are found out. The essence of the algorithm of geometrical position errors compensation is the following . First. a 6-dimensional vector .d = /1 AX, £I Y, ~ Z, 8.) tJr, Bz//~f IR end effector positioning erro rs ( ./lX. AY. AZ) and its orientation ( B). Br. (lz. ) in the carthesian coo rdina te s yst em are de termined fr om the difference between two solution of the direct problem of the IR arm posi tion Ton and ro~ respectively for nominal and effective arm link parameters and sensor reading:

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Then by sol vi ng the direct linearized position problem (6 simultaneous linear equatious with N unknown values) (8 )

the pseudo-reverse matrix and pseudo-solution procedure for the system an N-dimensional vector of programmed compensations is determined:

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where J: 1/ a~jll is J acoby matrix determining the relation between generalized coordinate increments ~ and the elements A of the error vector . The elements of the J matrix are calculated for IR arm link effective parameters ~:. .. cI-'t 'f/ i=1.N. Cl,.

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Improving the accuracy REFERENCES The definite N-dimensional vector of compensation values for IR control program which compensates geometrical position errors is determined as the sum of two vectors: the first (~1r) defines the pseudo~solution of the simultaneous linear equations and compensates only the errors in arm link parameters. the other (d~) is equal with opposite sign to the vector of position sensor zero setting errors and compensates the effect of the latters on the accuracy of the IR: (10)

CONCLUSION So. all the elements of the compensation values (11) to be introduced into IR control program in order to compensate static and geometrical IR end effector positioning error calculated for nominal values of arm link parameters are determined. A flow-diagram of the IR control device performing the above-cited algorithms of compensation of static and geometrical positininig errors is given in Fig. 3. The application of the above-described methods of improving the accuracy of IR has allowed to reduce positioning errors in analytically programmed robots TUR-10 by 5-10 and limit them to within 1mm.

Grimaylo S.I. et.al (1981) . The use of the robot "An eye - 2 arms" for manipulating non-oriented parts. Preprint IPM. Moscow (in Russian) Hayati S.A. (1983) Robot Arm Geometric Link Parameter Estimation. Proc. 22-nd IEEE Conf. Decis. and Contr. San Antonio. Tex. Vol . 3. N.V. pp. 1II77-11!83. Klepov A. E. et.al (1981!) Attesting of geometrical parameters of industrial robot. Proc. 3-ed Conf. Ind. Robots. pp. 81-82. Voronezh (in Russian). Paul R (1972) Modelling. Trajectory CalcUlation and Servoing of a Computer controlled Arm. Stanford Artificial Intelligence Project. Memo AIM-177. Comp. Science Dep . . Rep. CS-311. Tarvin R.L. (1980) Considerations for Off-line Programming a Heavy Duty Industrial Robots. Conf. Ind. Robot Technolo.. Milan. pp. 109-11 7. Wu C. -H. (1983) The Kinematic Error Mode for the Design of Robot Manipulator. Proc. Amer. Contr. Conf. San Francisko. Calif. V.2. pp. 1!97-502 .

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