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A CATMULL-ROM SPLINE BASED ANALYTICAL C3 CONTINUOUS TOOL PATH SMOOTHING METHOD FOR ROBOTIC MACHINING Cover

A CATMULL-ROM SPLINE BASED ANALYTICAL C3 CONTINUOUS TOOL PATH SMOOTHING METHOD FOR ROBOTIC MACHINING

By: Xu-Lin CAI,  Wen-An YANG and  You-Peng YOU  
Open Access
|Dec 2025

Figures & Tables

Fig. 1.

Transformation of tool frame related to workpiece frame in robotic machining system
Transformation of tool frame related to workpiece frame in robotic machining system

Fig. 2.

Rotation around fixed global coordinate system
Rotation around fixed global coordinate system

Fig. 3.

Flowchart of the proposed tool path smoothing method
Flowchart of the proposed tool path smoothing method

Fig. 4.

Corner smoothing of tool tip position
Corner smoothing of tool tip position

Fig. 5.

Corner smoothing of tool orientation
Corner smoothing of tool orientation

Fig. 6.

Tool orientation error in the WCS
Tool orientation error in the WCS

Fig. 7.

The deviation between spline segment and linear segment
The deviation between spline segment and linear segment

Fig. 8.

Synchronization of the tool tip position and orientation
Synchronization of the tool tip position and orientation

Fig. 9.

The testing tool path utilized in numerical simulation
The testing tool path utilized in numerical simulation

Fig. 10.

The smoothed tool tip position and tool orientation
The smoothed tool tip position and tool orientation

Fig. 11.

Synchronization errors of the tool orientation with respect to the tool tip displacement with εp = 0.8mm and ε0 = 0.01rad
Synchronization errors of the tool orientation with respect to the tool tip displacement with εp = 0.8mm and ε0 = 0.01rad

Fig. 12.

Synchronization errors of the tool orientation with respect to the tool tip displacement with εp = 1.2mm andε0 = 0.005 rad
Synchronization errors of the tool orientation with respect to the tool tip displacement with εp = 1.2mm andε0 = 0.005 rad

Fig. 13.

Kinematic profile after interpolation
Kinematic profile after interpolation

Fig. 14.

Kinematic profiles of each joint of robot
Kinematic profiles of each joint of robot

Parameters of the testing tool path

No. of cornersNo. of tool pathTool tip position (x, y, z)(mm)Tool orientation(α,β,γ) (rad)
11(240, 200, 220)(0.6283, 0.5236, 0.3142)
2(230, 240, 250)(1.0472, 0.7854, 0.3927)
23(270, 230, 252.2)(0.3142, 0.1571, 0.2094)
34(250, 210, 242)(0.6283, 0.7854, 0.5236)
(235, 230, 246.8)(0.3142, 0.3927, 0.5236)

Deviation errors of the tool tip and tool orientation with error tolerances εp = 0_8mm and ε0 = 0_01rad

No. of corners123
Tool tip Position errorεp (mm)0.80.80.8
εp(mm3) (κ,λ)5.4858e-06 (0.48, - 5.4e-4)2.3160e-06 (0.51, -1.4e-4)3.9818e-05 (0.47, 5.3e-4)
Tool orientation errorε0 (rad)0.010.010.01
ε0 (rad3) (κ,λ)2.2647e-11 (0.46, -2.4e-3)1.2184e-10 (0.49, 7.2e-3)2.8408e-11 (0.47, -3.0e-3)

Deviation errors of the tool tip and tool orientation with error tolerances εp = 1_2mm andε0 = 0_005rad

No. of corners123
Tool tip Position errorεp (mm)1.21.21.2
ep (mm3)(κ,λ)1.8557e-05 (0.48, -8.1e-4)7.8179e-06 (0.51, -2.2e-4)1.3406e-04 (0.47, 7.9e-4)
Tool orientation errorε0 (rad)0.0050.0050.005
e0 (rad3) (κ,λ)2.8111e-12 (0.46, -1.2e-3)1.5905e-11 (0.49, 3.6e-3)3.5244e-12 (0.47, -1.5e-3)
DOI: https://doi.org/10.2478/ama-2025-0088 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088
Language: English
Page range: 790 - 803
Submitted on: Aug 13, 2025
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Accepted on: Dec 10, 2025
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Published on: Dec 31, 2025
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2025 Xu-Lin CAI, Wen-An YANG, You-Peng YOU, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.