Smooth trajectory generation for five-axis machine tools Alexander Yuen a , Ke Zhang b , Yusuf Altintas a,n a Manufacturing Automation Laboratory, The University of British Columbia, Vancouver, BC, Canada V6T1Z4 b Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100190, China article info Article history: Received 18 February 2013 Accepted 16 April 2013 Available online 23 April 2013 Keywords: Five-axis interpolation Splined tool path C 3 continuity abstract This paper presents a smooth spline interpolation technique for five-axis machining of sculptured surfaces. The tool tip and orientation locations generated by the CAM system are first fitted to quintic splines independently to achieve geometric jerk continuity while decoupling the relative changes in position and orientation of the cutter along the curved path. The non-linear relationship between spline parameters and displacements along the path is approximated with ninth order and seventh order feed correction splines for position and orientation, respectively. The high order feed correction splines allow minimum deviation from the reference axis commands while preserving continuous jerk on three translational and two rotary drives. The proposed method has been experimentally demonstrated to show improvements in reducing the excitation of inertial vibrations while improving tracking accuracy in five-axis machining of curved paths found in dies, molds and aerospace parts. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Five-axis machining provides a highly productive method for producing freeform parts, often found in the aerospace, biomedi- cal, and die and mold industry. In order to machine freeform parts, curved tool paths must be followed at desired feedrates, which can only be achieved if the interpolator can generate desired displace- ments at fixed control intervals along the curved tool path. If the curved path is approximated by a series of linear segments, discontinuities are created in the displacement, velocity, accelera- tion and jerk commands during trajectory generation. The curved paths are usually approximated by spline functions to minimize the discontinuities. However, the curved tool path is not a linear function of discrete path lengths, creating problems when it is necessary for the tool path to be followed at desired feedrates. Several interpolation solutions to this problem exist for three-axis splines [1–3], while research in five-axis spline interpolation is still studied due to added complexity of interpolating orientations. Fleisig and Spence [4] generated quintic arc-length parameter- ized splines to interpolate position and orientation, while main- taining coordination between position and orientation of the tool path with a reparameterization spline. Liu et al. [5] improved on [4] by generating the reparameterization spline with respect to arc-length instead of chord length as originally proposed. Later, Langeron et al. [6] took a different approach by interpolating tool position and orientation with a B-spline located at the tool tip and end of the tool axis, respectively. The same method has been extended to Non-Uniform Rational B-Splines (NURBS) in [7] and quaternions in [8]. In this paper, an overall approach is presented to interpolate five-axis tool paths with C 3 continuity. A quintic C 3 B-spline is fitted by assigning control points and knots to force the spline to pass through the tool tip position data [9]. To ensure the correct spline parameter is scheduled to yield the desired arc displace- ment, and subsequently the desired feed profile accurately, the non-linear relationship between the spline parameter and dis- placement [1] is approximated with an adaptive ninth order C 3 feed correction polynomial. As demonstrated in Section 3, orienta- tion is interpolated with a quintic C 3 B-spline that is fit to spherical coordinates of the tool orientation data, then mapped back to Cartesian coordinates to avoid infeasible orientations. Like the interpolation of position, the non-linear relationship between the orientation spline parameter and angular change is approximated with a seventh order Bézier spline, which is optimized to mini- mize jerk while preserving C 3 continuity. In Section 4, the proposed method is compared to dual spline and C 2 decoupled methods, in simulations and experiments in terms of fitting, experimental frequency content, and experimental tracking accuracy. 2. C 3 tool tip position spline generation The splines are generated to interpolate tool tip position and orientation as functions of tool tip displacement l as outlined in Fig. 1. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijmactool International Journal of Machine Tools & Manufacture 0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.04.002 n Corresponding author. Tel.: +1 604 822 5622; fax: +1 604 822 2403. E-mail addresses: altintas@mail.ubc.ca, altintas@mech.ubc.ca (Y. Altintas). International Journal of Machine Tools & Manufacture 71 (2013) 11–19