# Algebraic modelling of 5 axis tool path motions

Lead Research Organisation:
University of Bath

Department Name: Mechanical Engineering

### Abstract

Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.

## People |
## ORCID iD |

G Mullineux (Principal Investigator) |

### Publications

Cripps R
(2017)

*Singularities in five-axis machining: Cause, effect and avoidance*in International Journal of Machine Tools and Manufacture
Cripps R
(2015)

*Using geometric algebra to represent and interpolate tool poses*in International Journal of Computer Integrated Manufacturing
Cripps R
(2020)

*Self-Reverse Elements and Lines in an Algebra for 3D Space*in Advances in Applied Clifford Algebras
Cripps R J
(2015)

*Singularities and 5-axis machining*
Cripps R J
(2016)

*Machinability of surfaces via motion analysis*
Cripps, R. J.
(2014)

*Design of free-form motions*
Cross B
(2018)

*Types of free-form motion*
Cross B
(2014)

*G4 linear motion*
Cross B
(2016)

*C1 motion example*
Cross B
(2016)

*Pose distance 1*Description | One of the interests of this research project as 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 requirement 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 epsilon 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., Lefevre, 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 R^n 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, Horvath, I., Pernot, J.-P., Rusak, Z., Delft University of Technology, 551-562. [15] Mullineux, G., Cripps, R. J., Cross, B., Bezier 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, 2018. [17] Cross, B., Cripps, R. J., Mullineux, G., 2018, Types of free-form motion, in: Proc. Tools and Methods for Competitive Engineering (TMCE) 2018, Horvath, I., Suarez, J. P., Delft University of Technology, accepted. [18] Jaklic, G., Juttler, B., Krajnc, M., Vitrih, V., Zagar, 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, Horvath, I., Suarez, 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 | 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 benficial 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 did not always make the optimum 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 dificulty. 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. Research in the area of free-form motions has continued (although unfunded) since the end of the project. This has resulted in a better understanding of the ideas of additive and multiplicative motions and of the ways of designing motion segments which join together smoothly. Added March 2022: Dr Cripps at University of Birmingham and I have now both retired. We are continuing some research (of a general nature) in the area of geometric algebra and its application to free-form motion. As noted above, Delcam was merged with or taken over by Autodesk during the course of the last project - something which was foreseen at the start of the project. Steve Hobbs, who was an invaluable supporter of many university-industry research projects and other senior colleagues have now retired from (or simply left) the company and any links that Dr Cripps and I had have now disappeared. So it is unclear how any of the research undertaken during the project is been used by what is now Autodesk. |

Sector | Manufacturing, including Industrial Biotechology |

Impact Types | Economic |

Description | collaboration with Delcam International plc |

Organisation | Delcam International |

Country | United Kingdom |

Sector | Private |

PI Contribution | Discussions with Delcam on issues relating to accurate and well-formed manufacture |

Collaborator Contribution | Advice, accesss to staff and software relating to issues of accurate manufacture |

Impact | Too early to give specific outcomes |

Start Year | 2013 |

Description | collaboration with university of birmingham |

Organisation | University of Birmingham |

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 | 9th International Conference on the Mathematics of Curves and Surfaces |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Attendance and presentation at international conference |

Year(s) Of Engagement Activity | 2016 |

Description | International Conference on Manufacturing Research 2015 |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Attendance and presentation at international Conference |

Year(s) Of Engagement Activity | 2015 |

Description | Poster presentation at international conference |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | Poster presentation ("Geometry and motion using geometric algebra") at "Curves and Surfaces 2018 Conference" in Arcachon, June 2018. Presentation of ideas to other researchers and discussion. |

Year(s) Of Engagement Activity | 2018 |

Description | TMCE conference |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | Attendance, presentation, session chair, member of international program committe, member of international paper review panel for TMCE 2014 (Tools and Methods of Competitive Engineering) held in Budapest in May 2014. Conintinued invovlement with that community |

Year(s) Of Engagement Activity | 2014 |

Description | Tools and Methods for Competitive Engineering (TMCE) 2016 |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Attendance and presentation at international conference |

Year(s) Of Engagement Activity | 2016 |