Analytical modeling of the CNC machine axis motion in high-speed milling with local smoothing

  • Mohamed EssidEmail author
  • Bassem Gassara
  • Maher Baili
  • Moncef Hbaieb
  • Gilles Dessein
  • Wassila Bouzid Saï


In high-speed milling (HSM) of free-form surfaces, the tool path is mainly characterized by a set of short discontinuous segments in tangency. These discontinuities bring an intense kinematic parameter fluctuation of the computer numerical control (CNC) machine axes. The smoothing of the programmed tool path and the optimal choice of the manufacturing process parameters ensure a high productivity with the required quality of the machined surfaces. Precise estimation of the machining time allows an accurate evaluation of the machined product cost. Thereby, this is achieved by studying the effect of the tool path smoothing and the CNC controller parameters on the axis kinematics and the following errors. This paper focuses on the geometrical modeling of the local smoothing block adopted by Sinumerik CNC. Further, a kinematic model is proposed to simulate the axis motion in linear interpolation mode with local smoothing. The identification of the interpolator reduction effect on the programmed tolerance leads to the identification of the smoothing model adopted by the CNC unit. Then, the axis kinematic behavior is modeled while taking into account the drive parameter axes defined by the manufacturer of the CNC unit. The experimental results showed a smooth variation of the axis feed rate along the smoothing blocks and a good correlation with the proposed models.


Tool path setpoint Interpolator Following error Hermit polynomial Feed rate 



axis position along the smoothing block at point Pi

\( {S}_i^{\hbox{'}}(u) \)

first parametric derivative of Si(u)

\( {S}_i^{{\prime\prime} }(u) \)

second parametric derivative of Si(u)

\( {T}_{{\mathrm{L}}_i} \)

transition length of Si(u)


maximum contour tolerance of Si(u)


X-axis tolerance


Y-axis tolerance


programmed contour tolerance


j-axis following error


weight of two tangential vectors (Ti and Ti + 1) to Si(u)


reduction rate of the programmed tolerance bound to Si(u)


deviation angle of two successive linear blocks at point Pi


bisector of the angle formed by two successive segments of the programmed tool path at point Pi


curvature of the smoothing block Si(u)

j = 1 : 3

for the X-, Y-, and Z-axes of the machine tool


unit tangential vector to Si(u)


unit normal vector to Si(u)

\( {V}_{{\mathrm{M}}_j} \)

maximum velocity of axis j

\( {A}_{{\mathrm{M}}_j} \)

maximum acceleration of axis j

\( {J}_{{\mathrm{M}}_j} \)

maximum jerk of axis j in linear interpolation mode

\( Jt{b}_{{\mathrm{M}}_j} \)

maximum jerk of axis j specific to the transition block


rate of acceleration allowed in radial acceleration


rate of jerk allowed in radial jerk

\( {A}_{{\mathrm{RM}}_j} \)

maximum radial acceleration of axis j

\( {A}_{{\mathrm{TM}}_j} \)

maximum tangential acceleration of axis j

\( Jt{b}_{{\mathrm{RM}}_j} \)

maximum radial jerk of axis j

\( Jt{b}_{{\mathrm{TM}}_j} \)

maximum tangential jerk of axis j


cycle time


sampling factor


sampling time


parametric position of the axes along Si(u)

\( {\dot{u}}_i(t) \)

parametric feed rate along Si(u)

\( {\left.{\dot{u}}_{i_{\mathrm{L}}}\right|}_{u=0} \)

parametric feed rate limit at the input of Si(u)

\( {\left.{\dot{u}}_i\right|}_{u=0} \)

parametric feed rate at the input of Si(u)

\( {\ddot{u}}_i(t) \)

parametric acceleration along Si(u)

\( {\left.\ddot{u}\right|}_{u=0} \)

parametric acceleration at the input ofSi(u)

\( {\overset{\dddot{}}{u}}_i(u) \)

parametric jerk along Si(u)


programmed feed rate


feed rate of axis j along the tool path


axis feed rate along Si(u)


feed rate along the tool path

\( Vi{n}_{i_{\mathrm{L}}} \)

limit feed rate at the input of Si(u)


feed rate at the input of Si(u)


feed rate at the output of Si(u)


axis acceleration along Si(u)


axis jerk along Si(u)



This work is carried out thanks to the support and funding allocated to the Unit of Mechanical and Materials Production Engineering (UGPMM/UR17ES43) by the Tunisian Ministry of higher Education and Scientific Research.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Unité de Génie de Production Mécanique et MatériauxENISSfaxTunisia
  2. 2.Institut Supérieur des Etudes Technologiques de SfaxEl Bustan SfaxTunisia
  3. 3.Laboratoire de Génie de Production (LGP)ENITTarbesFrance

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