Algebraic modelling of 5 axis tool path motions

Lead Research Organisation: University of Birmingham
Department Name: Mechanical Engineering

Abstract

The use of computers to aid the manufacturing process has been long established. In particular, the use of computer numerically controlled machines to progressively remove material from a solid block to produce a finished component is now an integral part of the component industry. The hardware, software and methodologies for material removal have all matured into state of the art software packages and multi-axis machining centres. Despite this, there remains issues with machining high precision components that should require little or no finishing. There are many parameters involved in the machining process and even when optimised the physical part may exhibit unexpected errors. These may be due to a number of effects (ignoring such issues as tool wear and vibration) including: simplifications made in the computer aided manufacturing software regarding the model of the cutting tool; the need to discretise the tool to send to machine tool controller; the need for the controller to re-interpolate the required tool path and the need to control the tool to follow the (newly re-interpolated) path.

To offset these effects, time consuming and expensive physical cutting trials are required in order to produce a high quality surface finish. An alternative would be to have the ability to accurately simulate the cutting process that predicts the true cutting conditions and reproduces the machined surface finish. It would be over ambitious to attempt to construct a simulation that is capable of modelling every aspect of the entire machining process in a single project. However, if a framework can be established and demonstrated for a manageable set of parameters, this could then be developed by others to incorporate other aspects including vibration, tool wear, etc. This project aims at providing such a framework for the realistic simulation of material removal using multi-axis machining tools and a robust, integrated framework for the modelling of tool path motions.

Previous work at the University of Birmingham has considered the problem of determining what the actual machined part is going to be. This work was based on the principle of generating surface normals to a CAD model of the part. Machining is then simulated by using an exact model of the cutting tool and using this to truncate the normals. This has been used successfully to predict the small cusps that can remain during manufacture. It is proposed to extend this approach to deal with more complicated surface forms so that defects can be predicted and hence means for attenuating them investigated.

Interest at the University of Bath has been in the use of geometric algebra to describe motions. Geometric algebra provides a framework in which rigid-body motions, both translation and rotations, can be handled in a common form. This allows techniques for free-form curves to be extended in a natural way to deal with free-form motions. It is proposed to investigate the use of the approach for describing the motion of the cutting tool and hence to improve means for interpolating (and re-interpolating) tool-path information.

Planned Impact

Industrial Impact
The primary pathway to impact is the collaboration with Delcam. The tool path motion algorithms and the material removal simulation methodology will be developed in line with current systems within the existing Delcam CADCAM packages. In this way the underlying mathematical modelling will be made available to end users in a well established user-friendly environment specifically tuned for the manufacturing community. The close collaboration with Delcam involving key individuals will simplify this knowledge transfer and provide practical two way dialogue to enhance the research and to maximise the success of the project. This will also present the ideal platform for simulation experimentation by providing a template for other researchers in the field looking at other aspects of CNC machining, by enabling tool path motion to be better controlled.

This impact will be achieved in a number of ways. Firstly there will be quarterly formal meetings involving academic and industrial collaborators. These will review progress and plan work for the subsequent quarter. These will normally take place in-company. In addition there will be informal contact as and when required.
The collaborators will also be involved in the manufacturing of the planned test pieces. The most significant industrial impact will be the transfer of research computer code and algorithms to the companies for commercial testing and development with a view to being made available to the wider manufacturing community through
our collaborators existing product base. Furthermore, the work will be presented to our collaborators via a series of formal seminars.

Academic
Creating a rigorous approach to modelling true tool path motion using geometric algebra will provide a solid platform for the development of greater understanding of the interaction between CNC control, machine tool path movement and the process of material removal under the machining process. The algebraic model will also contribute to the greater task of fully understanding the complex process of material removal by enabling other factors, both dynamic and static, to be studied
in a more controlled way. Finally, the successful tool path motion model will be of great interest to a wider field of academic disciplines including robotics and motion studies in general. There may also be some impact in the field of biomedical engineering especially in the area of prosthetic design and robotic surgery.

The results of the research will be widely disseminated through published papers in international journals and high quality established conferences.

Economic
The economic impact of the successful research is directly related to more efficient manufacturing processes. The ability to accurately simulate the quality of a component without the time, money and resource required to carry out real physical trials will ultimately lead to cheaper development cost which in turn will
enable products to be developed and manufactured at lower costs whilst maintaining profitable margins for successful enterprises. The primary beneficiaries will be the CNC tool manufacturers, control motor manufacturers and software companies who provide milling software solutions. Users of the enhanced manufacturing systems will benefit due to the efficiency savings. Finally, consumers should benefit from the increased efficiency of development of products at cheaper costs.

Societal
Society will benefit directly from either cheaper products or enhanced functionality. There may also be a benefit to society due to the fact that with greater understanding of the fundamental issues involved in the manufacturing process comes a greater confidence and a wider uptake in its use and hence a greater participation in the sector.

Publications

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Cripps R (2015) Using geometric algebra to represent and interpolate tool poses in International Journal of Computer Integrated Manufacturing

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Cripps R (2017) Singularities in five-axis machining: Cause, effect and avoidance in International Journal of Machine Tools and Manufacture

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Cripps RJ (2014) Design of freeform motions

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Cripps, R. J. (2015) Singularities and 5-axis machining

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Cross B. (2014) G4 linear motion

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Cross B. (2014) Pole avoidance

 
Description Key Findings

One of the interests of this research project has been the use of geometric algebra as a means for representing free-form motions such as those of cutter paths in machine tools. While the ideas of geometric algebra go back to the 1800s, it is only in the last twenty or so years that interest in their use has re-emerged. This is partly because of their inherent speed and robustness as compared with the more traditional use of matrix transforms.

A free-form motion is a (smoothly) varying rigid-body transform which is applied to body to be moved. A number of variations have emerged in the research community for representing such transforms including: dual quaternions [1], the homogeneous model [2, 3], conformal geometric algebra (CGA) [4], and the geometric algebra G4 used by the researchers on this project [5].

What is required is an environment in which the geometry of the moving body can be defined and manipulated, and in which transforms cam be applied in a straightforward manner. All the above variations satisfy these requirements to some extent: in particular, geometric algebra can represent the free-form motions of machine tool cutters [6] [objective OB1]. A comparison between the variations [7] also established that:

• dual quaternions provide only a limited environment to handle geometry: in
particular the concepts of \line" and \plane" are not supported;
• the homogeneous model provides means for dealing with geometry and transforms
but does so with vectors representing planes rather than points which
seem unnatural and can proving confusing;
• the CGA can represent geometrical objects including points, lines, planes,
circles and spheres and as well as supporting rigid-body transforms offer other
such as inversion; however, to do this it requires points to be held as null vectors which can be non-intuitive and means that some combinations of transforms are not applicable;
• the algebra G4 supports the geometry of points, lines and planes in a natural
way (but not circles and spheres) and deals well with rigid-body transforms;
both additive and multiplicative combinations of transforms generate other
transforms; the need to deal with the ? symbol is a drawback.

In particular, it was found that the CGA approach can be adapted to remove the need to use null vectors [8], and that in G4 there is a natural relation between a rigid-body transform and the line of its instantaneous Chasles axis [9].

A software environment was successfully created to allow simulation of machine tool motions using the geometric algebra approach. This also enabled simulation of machine removal and so obtain an indication of surface finish. The environment was used to investigate issues of singularities in the path planning and control of multi-axis machine tools [10, 11]. [This was objective OB2 with, following the advice of the collaborating company, the environment being based upon geometric algebra rather than surface normals.]

The algebra G4 allows transform to be combined either additively or multiplicatively. The latter is complicated by the need to use logarithms and exponentiation and leads to extensions of the slerp (spherical linear interpolation) construction [12]. The additive approach seems to have received little attention in the literature and has the advantage of being simpler to deal with. Either approach enables free-form motions to be created using the Bezier and B-spline techniques familiar from curve and surface work. In particular, the additive approach enables motions to be generated passing exactly through prescribed precision poses [13, 14] and to incorporate constraints on speed [15]. The corresponding problem for the multiplicative approach is significantly more challenging, but a method has been found in the case when certain restrictions are placed on the form of the motion [16]. It has also been established than further freedom in motion design can be achieved by composing two (or more) motions, and by combining the exponents of control poses written in exponential form [17]. One of the advantages of the approaches investigated is that they deal with motion of a body as a whole, rather than considering separately the translation motion of a reference point in the body, and the rotation relative to that point (e.g. [18]). The latter approach has the disadvantages of being dependent on the choice of reference point and the need to handle two separate motion with possibly distinct parameterisations. Related to this, it was also found that the method used by the collaborating company for assessing cutter paths generated by its software did rely upon a choice of reference point in the cutting tool, and a different choice could lead to a different assessment being made. [This covers the representational aspects of objectives OB3 and OB4.]

A number of possible metrics for assessing the quality of a motion and its accuracy compared to design constraints were proposed and investigated. [This was to cover the error assessment aspects of objectives OB3, OB4 and OB5.] However, none was found to be fully satisfactory. The need for a good error measure is partly made unnecessary by the ability to fit motions through prescribed precision poses. The investigations did lead to some findings, including the conclusion that any metric is likely to be dependent upon the shape of the body in motion, and greater familiarity with the properties of the derivatives of a motion.

Discussion with the collaborating company revealed some of the problems associated with specifying tool paths in terms of NC instructions for a machine tool. Specifically, the tool path needs to be discretised and it is not known how the machine tool controller will reassemble the pieces. A consequence is that the specification of the path uses a finer discretisation than is needed (to restrict the actions of the controller) resulting in more data and a slower response. It has been proposed that additional NC instructions based upon elemental spiral motions could help to resolve the problem and these can be specified using parameters based upon geometric algebra representations [19].

References

[1] Leclercq, G., Lefèvre, P., Blohm, G., 2013. 3D kinematics using dual quaternions: theory and applications in neuroscience. Frontiers in Behavioral Neuroscience. 7, 7:1-25.
[2] Selig, J. M., 2000. Clifford algebra of points, lines and planes. Robotica. 18(5), 545-556.
[3] Gunn, C., 2011. On the homogeneous model of Euclidean geometry. In: Dorst, L., Lasenby, J. (eds). Guide to Geometric Algebra in Practice. Springer, London, pp. 297-327.
[4] Cibura, C., Dorst, L., 2011. Determining conformal transformations in Rn from minimal correspondence data. Mathematical Methods in the Applied Sciences. 34(16), 2031-2046.
[5] Mullineux, G., Simpson, L. C., 2011. Rigid-body transforms using symbolic infinitesimals. In: Dorst, L., Lasenby, J. (eds). Guide to Geometric Algebra in Practice. Springer, London, pp. 353-369.
[6] Cripps, R. J., Mullineux, G., 2016, "Using geometric algebra to represent and interpolate tool poses", International Journal of Computer Integrated Manufacturing, 29(4), 406-423.
[7] Cross, B., Cripps, R. J., Mullineux, G., "Representations of geometry and transforms: a comparison of approaches", in preparation, 2018.
[8] Hunt, M., Mullineux, G., Cripps, R. J., Cross, B., 2017, "Free-form additive motions using conformal geometric algebra", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, on-line version available.
[9] Mullineux, G., Cripps, R. J., Cross, B., "Lines and axes in geometric algebra", in preparation, 2018.
[10] Cross, B., Cripps, R. J., Hunt, M., Mullineux, G., 2015, "Singularities and 5-axis machining", in: ICMR2015: 13th International Conference on Manufacturing Research 2015, Newnes, L. B., Nassehi, A., Dhokia, V., eds., University of Bath, 45-50.
[11] Cripps, R. J., Cross, B., Hunt, M., Mullineux, G., 2017, "Singularities in five-axis machining: cause, effect and avoidance", International Journal of Machine Tools & Manufacture, 116, 40-51.
[12] Shoemake, K., 1985. Animating rotation with quaternion curves. ACM SIGGRAPH. 19(3), 245-254.
[13] Hunt, M., Mullineux, G., Cripps, R. J., Cross, B., 2015. Representing cutter tool paths using geometric algebra. In: Newnes, L. B., Dhokia, V., Nassehi, A. (eds). Proceedings of the 13th International Conference on Manufacturing Research (ICMR2015). University of Bath, pp. 1-6.
[14] Hunt, M., Mullineux, G., Cripps, R. J., Cross, B., 2016, "Smooth tool motions through precision poses", in: Proc. Tools and Methods for Competitive Engineering (TMCE) 2016, Horváth, I., Pernot, J.-P., Rusák, Z., Delft University of Technology, 551-562.
[15] Mullineux, G., Cripps, R. J., Cross, B., "Bézier motions with end-constraints on speed", submitted to Computer Aided Geometric Design.
[16] Hunt, M., Mullineux, G., Cripps, R. J., Cross, B., "Fitting a planar quadratic slerp motion", submitted to Computer Aided Geometric Design.
[17] Cross, B., Cripps, R. J., Mullineux, G., 2018, "Types of free-form motion", in: Proc. Tools and Methods for Competitive Engineering (TMCE) 2018, Horváth, I., Suárez, J. P., Delft University of Technology, accepted.
[18] Jaklic, G., Jüttler, B., Krajnc, M., Vitrih, V., Žagar, E., 2013. Hermite interpolation by rational Gk motions of low degree. Journal of Computational and Applied Mathematics. 240, 20-30.
[19] Cross, B., Cripps, R. J., Matthews, J., Mullineux, G., 2018, "G-codes and free-form motions", in: Proc. Tools and Methods for Competitive Engineering (TMCE) 2018, Horváth, I., Suárez, J. P., Delft University of Technology, accepted
Exploitation Route Please see above
Sectors Aerospace, Defence and Marine,Education,Manufacturing, including Industrial Biotechology,Transport,Other

 
Description Narrative impact It is expected that the main impact of the research will be academic. As reported under "key findings", the project has shown that a particular form of geometric algebra can handle both three-dimensional geometry and the rigid-body transforms that act upon it, and that free-form motions (of objects such as cutting tools) can be constructed from control poses using additive and multiplicative combinations and by composing elementary motions. The economic and societal impact has been less than originally anticipated. It had been planned to embed appropriate free-form constructions within the software of the collaborating company. At the start of the project, the company was Delcam International. During the course of the project the company merged with Autodesk with the result that, while the company's involvement in the project in terms of collaboration and advice was undiminished, the opportunity to embed code was no longer available. There are however three areas where the results of discussions have been beneficial to the company. The first of these relates to tests for singularities in proposed tool paths. One test is to consider the path of a point along the length of the cutting tool. It was shown that this need did always make the appropriate predictions and that it was better to consider the motion of the tool as a whole (that is as a rigid body). Secondly, the company required a suitable measure of the \distance" between poses along a potential path for a robot end-effector so that the \length" of the path could be optimised. As the motion involves both translation and rotation, the terms \distance" and "length" are ill-defined. Suggestions for suitable measures were made based on the work on metrics for motions described by geometric algebra. The third area is perhaps the most significant. As noted in the "key findings", the task of communicating a required tool path to the controller of a machine tool is complicated by the need to discretise the path before transmission and then the need to reassemble it within the controller. Lack of confidence in the actions of a general controller can lead to the path being over-specified. It has been proposed that the introduction of additional NC commands based on well-defined elementary \spiral motion" could help to alleviate this difficulty. It is hoped to continue work in this area to prove the validity of the proposal. As yet, it has not been possible to gain the support of a commercial producer of machine tool controllers.
First Year Of Impact 2017
Sector Manufacturing, including Industrial Biotechology
Impact Types Economic

 
Description Collaboration with Delcam International 
Organisation Delcam International
Country United Kingdom 
Sector Private 
PI Contribution Working jointly on the research project
Collaborator Contribution Working jointly on the research project
Impact Joint research into ideas of geometric algebra, free-form motions and applications, especially in manufacturing.
 
Description Collaboration with Delcam International 
Organisation University of Bath
Department Department of Mechanical Engineering
Country United Kingdom 
Sector Academic/University 
PI Contribution Working jointly on the research project
Collaborator Contribution Working jointly on the research project
Impact Joint research into ideas of geometric algebra, free-form motions and applications, especially in manufacturing.
 
Description Joint research collaboration with Bath 
Organisation University of Bath
Department Department of Biology and Biochemistry
Country United Kingdom 
Sector Academic/University 
PI Contribution University of Bath: joint research outcomes
Collaborator Contribution See research outcomes: Algebraic modelling of 5 axis tool path motions
Impact See research outcomes: Algebraic modelling of 5 axis tool path motions
Start Year 2013
 
Description ICMR Conference 2016 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Conference attendance and presentation
Year(s) Of Engagement Activity 2016
URL https://www.designsociety.org/news/event/225
 
Description ICMR Conference Bath, 2015 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Conference attendance and presentation
Year(s) Of Engagement Activity 2015
URL http://epc.ac.uk/events/13th-international-conference-on-manufacturing-research-icmr2015
 
Description MMCS Conference 2016 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Conference attendance and presentation
Year(s) Of Engagement Activity 2016
URL http://www.mn.uio.no/math/english/research/groups/cm/events/conferences/ninth-curves-and-surfaces/in...
 
Description TMCE Conference 2014 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Conference presentation
Year(s) Of Engagement Activity 2014
URL http://tmce.io.tudelft.nl/?year=2014
 
Description TMCE Conference 2016 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Conference attendance and presentation
Year(s) Of Engagement Activity 2016
URL http://tmce.io.tudelft.nl/?year=2016