The efficient calculation of the Cartesian geometry of non-cartesian structures J.M. Freeman and D.G. Ford Engineering Control and Metrology Research Group, The School of Engineering, University of Huddersfield. Email: J.M.Freeman@hud.ac.uk Abstract There is currently much interest in non-cartesian machines in the machine tool industry. For example, an adaptation of the Hexapod configuration with six supporting struts attached to a machine head has successfully been used in the Variax machine by Giddings and Lewis. Such machines have the advantage of stiffness and lightness compared with conventional machines, where bending moments of the structure provide the machining forces. In the hexapod the machining forces are along the struts and each member is uniformly stressed. The disadvantage of the non-cartesian machine is the complex mathematics required to convert a predetermined path in Cartesian space into changes in lengths of the struts. This paper analyses the general non-cartesian machine, enabling the calculation of the Cartesian coordinates of the sensor on a co-ordinate measuring machine, or the machine head on a cutting machine, from the strut lengths. Since these distances can be determined accurately by laser interferometry, it is possible to derive the co-ordinates of the head to very high precision. The methods are computationally efficient and lead to relatively simple application. Introduction Non-Cartesian machines such as the hexapod have the advantage of stiffness and lightness compared with conventional machines, where bending moments of the structure provide the machining forces. In the hexapod the machining forces are along the struts and each member is uniformly stressed. A useful introduction to non-cartesian machines is available on Internet. ' The disadvantage of the non-cartesian machine is the complex mathematics required to convert a predetermined path in Cartesian space into changes in lengths of the struts. This is referred to as the serial/parallel manipulator duality and is discussed by Waldron.^ The present paper will present a general
318 Laser Metrology and Machine Performance mathematical treatment of non-cartesian machines. In the analysis it is necessary to concentrate on the mathematics of the calculation of the Cartesian co-ordinates of the machining head from a knowledge of the distances of the head from fixed reference points. Since these distances can be determined accurately by laser interferometry it is possible to derive co-ordinates to very high precision. The reference points are usually fixed points on the structural frame of the machine. The methods are computationally efficient and lead to relatively simple application. THEORETICAL CONSIDERATION OF THE GENERAL NON-CARTESIAN MACHINE The machines under consideration include scanners, co-ordinate measuring machines and machine tools. In a scanner an analogue probe traces a continuous contour of an artefact and the co-ordinates of the trace are calculated from the known x, y, z positions of the scanner and the small deflections measured by the analogue sensor. These deflections are in the same direction as the normal x, y z co-ordinates for a Cartesian machine, but will be in a different reference frame for a non-cartesian machine. The probe requires a controller capable of keeping the sensor on the surface of the object while performing a trace at a reasonable speed. In a co-ordinate measuring machine a touch probe takes samples of the co-ordinates of the surface of an artefact, very accurate measurements being possible. A machine tool requires accurate positioning of a cutter in three dimensional space, any tilt of the cutter being critical. The essential difference between the measurement machines and the machine tool is that any errors due to the tilt of a sensor can be compensated for in software, whereas the tilt of a cutting tool requires to be within tolerance, no software corrections being possible. NON-CARTESIAN MACHINE ARCHITECTURE In principle the location of a sensor or a cutter head exhibits six degrees of freedom. These may be specified in various ways, the usual constituting the x, y and z Cartesian co-ordinates of the head and the angular position of the head rotated about the Cartesian axes, say xrot, yrot and zrot. Cartesian machines use the x, y and z co-ordinates directly by utilising drives in these directions and various schemes allow rotation of the head about up to three axes. Calculations of head position are simple in principle, although geometrical and thermal error corrections require linear error mapping techniques. Non-Cartesian machines use drives which are not along the Cartesian axes and therefore present a formidable mathematical challenge. Various designs of six degree of freedom positional devices are discussed by Merkle. * The
Laser Metrology and Machine Performance 319 obvious design is to use six struts anchored to mechanical earth at one end and to the head assembly at the other. Variations in the lengths of the six struts would then be sufficient to control the head in all six degrees of freedom. This is the principle used in the Hexapod machine, although various schemes exist for positioning the struts. For example, they may be anchored in pairs at the head end as in the Sandia Hexapod. Another scheme adopted in the Variax machine by Giddings and Lewis utilises an adaptation of the Stewart platform, * used in flight simulators. If we include in our consideration machines with less than six struts, then it is necessary to compensate for the lack of control in one or more degrees of freedom by in some way restricting motion of the head in that mode. This can be done by mechanically constraining the movement of the head in that direction. For example, if only three struts are used then it is necessary to control the remaining three degrees of freedom. This can be done by controlling or constraining the head to remain vertical at all times and for it to have a pre-determined rotation about its axis. Ideally this rotation should be constant, but it should at least be calculable from the location of the head. MATHEMATICAL ANALYSIS Consider a machine with n struts (n<=6) of length p^ connected to mechanical earth at the known points A; ( 1 <= i <=n ), where A; is a vector with component A^, A,y and A^, representing the Cartesian co-ordinates of the anchor at mechanical earth. The struts are located at unknown points Bj on a sensor or machining head as shown in Fig 1. A similar notation is used for B and it is assumed that the struts can pivot about the points Bj. Although the absolute co-ordinates B in the global reference frame are unknown, the relative positions of the Bj to each other are known from a knowledge of the local geometry of the head and its anchor points.
320 Laser Metrology and Machine Performance A Mechanical ^ KEarth Position of Sensor Head Q^ _ Fig. 1: The General Non-Cartesian Structure The head will require a mechanical mechanism or some other form of control to remove the remaining 6-n degrees of freedom. Only then will it be fully locatable in space and useful as a scanning or machining head. The problem of the calculation of PJ from x, y and z presents no real difficulty. The B, can be found from x, y and z providing the orientation and geometry of the head are known. This will require a knowledge of how the mechanism to remove the 6-n degrees of freedom operates. Then calculation of p, is a simple application of Pythagoras Theorem. However, this calculation will be specific to each mechanism. The more difficult task is to calculate the x, y and z co-ordinates of a point on the head, usually the tip of the sensor or cutter, from the values of the strut lengths PJ. If the head were infinitesimally small the problem would reduce to finding the co-ordinates of a point given its distances from n fixed points. However, in practice it will be necessary to include corrections for the finite size of the head. It is convenient to introduce a vector notation, p; representing a vector with components p^, pjy, p^ for the strut pj resolved in x, y and z directions. Similarly, r, represents the resolved distances of point Bj from the sensor or cutter. Call the latter point X, its position being represented by the vector \. In effect r- defines the local geometry of the sensor or cutter head. It is now possible to write down n vector equations involving vector addition to relate the points A; to the point X ie. X + r, +p- = A, (!</<«) For the system to be solvable it is necessary to know r^ This will require m=6-n further relationships which define the way in which the head is constrained by the controlling mechanism. This can conveniently be represented by a transformation matrix T relating the ^ to a reference frame for the head defined by s^. The S; contain the known co-ordinates of the n points Bi when the head is in a fixed reference position. The matrix T will
Laser Metrology and Machine Performance 32 1 often be formed from rotation matrices for each of the m controls. The composite matrix T will be a matrix product of the individual rotation matrices represented below as Tj(0j). The 9^ are the rotational angles caused by the mechanism and must be known or be calculable from x. Hence and Note that there is also a non-linear relationship between each of the n vectors p; and the co-ordinates x due to Pythagoras of the form p- = (4,-Brf + (4,-B^ + (4,-Brf We therefore have 4n equations and 4n unknowns and their solution is theoretically possible. However, the non-linear nature of n of these equations makes their solution intractable. It is desirable to remove the non-linearities. The following analysis is for the special case of a machine with three struts attached at one end to the sensor head and at the other to fixed reference points in a horizontal plane. The more general case of reference points located at any position is considered in our next paper, * If the head is of negligible size then the problem is equivalent to the triangulation of a point from three reference points in a horizontal plane. By triangulation is meant the determination of the three dimensional Cartesian co-ordinates of a point from its known distances to three reference points. The mathematical derivation follows.
322 Laser Metrology and Machine Performance TRIANGULATION OF AN UNKNOWN POINT TO THREE POINTS IN A HORIZONTAL REFERENCE PLANE If we consider two struts to be connected to earth at points A and B and each other at the point X, then X will trace a circular locus in a plane at right angles to the reference plane containing the three reference points A, B and C. These planes are shown in Fig 2, but the circles are omitted for clarity. In principle it is possible to find equations defining the three circles and then to find the desired point X by finding the point of intersection of any two circles. However, the equations are non-linear and difficult to solve. Another approach is to find the equations defining the planes containing the circles, which will be linear, and then to find where these planes intersect. Of course, they will intersect in a line perpendicular to the reference plane as shown in Fig. 2. Since all three planes intersect in the same line it is necessary to consider only two of the three planes. When the point of intersection of the line with the reference plane has been determined, it remains to find the distance along the line from that point of intersection to X ie. the z value for point X, since the reference plane is horizontal Fig 2: Triangulation to a Horizontal Plane The planes in Fig. 2 are those containing the circular loci as described above. Hence, we obtain their representation from the equations for those circles, which are actually described by the intersection of any two spheres of radii equal to the strut lengths p, q and r and centred on A, B and C respectively. The equations of the two spheres on A and B are:
Laser Metrology and Machine Performance 323 where xp, yp, zp is any point on the circle. These two equations taken together represent the circle, but the plane containing that circle can be found by eliminating the squared terms involving the co-ordinates of the unknown point X. This is accomplished by expanding the brackets and then subtracting the second equation from thefirstgiving This is a linear equation since all squared terms are constants. It is the equation of the plane containing the circular locus formed using the struts anchored on A and B. Two more planes can be found if we take each pair of points (ie A-B, A-C, B-C) in turn, but only one more is required. The equation for the plane formed between the struts on A and C is +2* - x J + 2y, (y, - y J + 2z, (z, - z, ) These planes taken together define a line of intersection which is normal to the reference plane. However, note that the coefficient of z in these equations is zero, since the z values of the reference points are all equal. Thus the equations reduce to two simultaneous equations in x, and yp, which define the x and y co-ordinates of the point of intersection of the normal with the plane. ie. in matrix notation '(*6-*J (yb-ya) (x - x ) (v - y ) \ C Q. J \./ c * Cl J -*.'- >.'-._ These two simultaneous equations can easily be solved for x_ and y matrix inversion or by direct substitution. The z value of X can now be obtained by back substitution of x, and the original equation. by into ie. P~ = This simple method can still be applied even if the reference points are not exactly horizontal. If the reference points A, B and C are not exactly in a
324 Laser Metrology and Machine Performance horizontal plane, but their exact positions are known, it is possible to apply a correction as follows: -6.2,,.2,_2 _2 _2 _2,^,, _^,,,.^_ derived in a similar way to above, but using all three reference points. Since the value of Zp is unknown, it is necessary to use an estimate of Zp and an iterative scheme to converge on a consistent solution for Zp. CORRECTION FOR FINITE HEAD SIZE If the head has a finite size then the struts are not located at the point X but at some displacement from X which is determined by the design of the head. The displacements are best represented as rotational transformations of the head from a reference position of the head in which the geometry is known. The angles through which to rotate the head must be known or must be calculable from x, y and z for the sensor. The displacements are best thought of as equivalent displacements of the reference points A, B and C. This simplifies the mathematics as follows: In the equation the term 1 L j^ j^i represents the displacements of the supports on 7=1 the head. These can be thought of as corrections to A; ( Hence, % + p- A V This can now be solved in the way described for the infinitesimally small head, the only difference being the adjustments made to A;. APPLICATION TO A NON-CARTESIAN SCANNING MACHINE Using the techniques described above, software has been written in Pascal and C to perform the transformation from Cartesian Co-ordinates to strut lengths and also to perform the reverse transformation. The Pascal software
Laser Metrology and Machine Performance 325 provides an animation of the machine in three projections. The C software is designed to run in real-time and to provide an interface from a Cartesian controller to the non-cartesian machine. To achieve accurate transformations double precision arithmetic is required. To give sub-micron accuracy it is necessary to perform several iterations of the algorithm, although less iterations are required if a scan is being performed and a good estimate is available as a starting point from the last sample, the head having moved little between samples. The speed of the algorithms depend on the number of iterations performed but is a few milliseconds on a Pentium 100. CONCLUSION It has been possible to construct software based on the techniques described that has a high re-usability and wide application/ The speed of the software is sufficient to allow real-time operation of a scanning machine but not to perform high speed cutting. Mapping techniques may provide a faster solution in the future. This will enable transformation maps generated off-line to be used for real-time operation by utilising three-dimensional interpolation techniques. REFERENCES 1. Merlet J.P. parallel Manipulators: State of the Art and Perspectives, on the web at http://www.inria.fr/prisme/personnel/merlet/etat/etat_de_lart.html. 2. Waldron K.J. and Hunt K.H., Series-Parallel Dualities in Actively Coordinated Mechanisms. The Int. J. of Robotics Research, 10(2):473-480, April 1991. 3. Merkle R.C., A New Family of Six Degree of Freedom Positional Devices, To be published in Nanotechnology, IOPP, draft available on the web at http://nano.xerox.com/nanotech/6dof.html. 4. Stewart D., A Platform with Six Degrees of Freedom, The Institute of Mechanical Engineers, Proceedings, 1965-66, 180 Part 1, No 15, pp 371-386. 5. Freeman J.M. and Ford D.G., The Analysis of Geometrical and Thermal Errors of Non-Cartesian Structures. This publication.