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An error identification and compensation method for Cartesian 3D printer based on specially designed test artifact

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Abstract

The printing accuracy is one of the most important metrics to evaluate the additive manufacturing (AM) machine. In this paper, an error identification and compensation method for Cartesian 3D printer is presented based on a specially designed test artifact to improve printing accuracy. The relationship between the geometric errors of the printed object and the kinematic errors of the printer axes is established based on the theory of the multi-body system. A series of formulas are derived to separate the kinematic errors of each axis from the geometric errors. To extract the geometric errors required for the mathematical calculations, an artifact with the special features is proposed and printed. The geometric errors of the characteristic points on the artifact are measured by a coordinate measuring machine (CMM). From the measured geometric errors, kinematic errors of the printer can be identified, and can be further compensated by adjusting the CAD model of the object. Two compensated algorithms are established; one uses the fitted curves of the kinematic errors, and the other uses the average kinematic error values. Printing tests and case studies are performed to verify the effectiveness of the proposed method. The results show that the proposed method can improve printing accuracy of the Cartesian 3D printer.

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Abbreviations

\({}_{b}{}^{a}{T}_{p}\) :

The initial position matrix

\({}_{b}{}^{a}{T}_{pe}\) :

The error matrix of the initial position

\({}_{b}{}^{a}{T}_{k}\) :

The dynamic position matrix

\({}_{b}{}^{a}{T}_{ke}\) :

The error matrix of dynamic position

\({}_{b}{}^{a}{T}_{i}\) :

The ideal position matrix

\({}_{b}{}^{a}{T}_{a}\) :

The actual position matrix

\({E}_{p}\) :

The geometric error of point P

\({E}_{px}\) :

The geometric error of point P in the x-direction

\({E}_{py}\) :

The geometric error of point P in the y-direction

\({E}_{pz}\) :

The geometric error of point P in the z-direction

\({\alpha }_{ab}\) :

The angle around the X-axis between body a and b

\({\beta }_{ab}\) :

The angle around the Y-axis between body a and b

\({\gamma }_{ab}\) :

The angle around the Z-axis between body a and b

\({i}_{ab}\) :

The linear offset between body a and b at the X-direction

\({j}_{ab}\) :

The linear offset between body a and b at the Y-direction

\({k}_{ab}\) :

The linear offset between body a and b at the Z-direction

\({\theta }_{x}\) :

The rotational angle around the X-axis

\({\theta }_{y}\) :

The rotational angle around the Y-axis

\({\theta }_{z}\) :

The rotational angle around the Z-axis

\({x}_{ab}\) :

The linear displacement in the X-axis between body a and b

\({y}_{ab}\) :

The linear displacement in the Y-axis between body a and b

\({z}_{ab}\) :

The linear displacement in the Z-axis between body a and b

\({\delta }_{X\left(x\right)}\) :

The servo error of the X-axis

\({\delta }_{Y\left(x\right)}\) :

The horizontal straightness error of the X-axis

\({\delta }_{Z\left(x\right)}\) :

The vertical straightness error of the X-axis

\({\varepsilon }_{\alpha \left(x\right)}\) :

The roll error of the X-axis

\({\varepsilon }_{\beta \left(x\right)}\) :

The pitch error of the X-axis

\({\varepsilon }_{\gamma \left(x\right)}\) :

The yaw error of the X-axis

\({\delta }_{X\left(y\right)}\) :

The horizontal straightness error of the Y-axis

\({\delta }_{Y\left(y\right)}\) :

The servo error of the Y-axis

\({\delta }_{Z\left(y\right)}\) :

The vertical straightness error of the Y-axis

\({\varepsilon }_{\alpha \left(y\right)}\) :

The roll error of the Y-axis

\({\varepsilon }_{\beta \left(y\right)}\) :

The pitch error of the Y-axis

\({\varepsilon }_{\gamma \left(y\right)}\) :

The yaw error of the Y-axis

\({\delta }_{X\left(z\right)}\) :

The horizontal straightness error of the Z-axis

\({\delta }_{Y\left(z\right)}\) :

The vertical straightness error of the Z-axis

\({\delta }_{Z\left(z\right)}\) :

The servo error of the Z-axis

\({\varepsilon }_{\alpha \left(z\right)}\) :

The roll error of the Z-axis

\({\varepsilon }_{\beta \left(z\right)}\) :

The pitch error of the Z-axis

\({\varepsilon }_{\gamma \left(z\right)}\) :

The yaw error of the Z-axis

\(\Delta {\gamma }_{xy}\) :

The squareness error between the X-axis and Y-axis

\(\Delta {\beta }_{xz}\) :

The squareness error between the X-axis and Z-axis

\(\Delta {\alpha }_{yz}\) :

The squareness error between the Y-axis and the Z-axis

ex1:

The geometric error in the X-direction of the original model

ey1:

The geometric error in the Y-direction of the original model

ez1:

The geometric error in the Z-direction of the original model

ex2:

The geometric error at the X-direction of the compensated model using fitted curves

ey2:

The geometric error at the Y-direction of the compensated model using fitted curves

ez2:

The geometric error at the Z-direction of the compensated model using fitted curves

ex3:

The geometric error at the X-direction of the compensated model using the average value of kinematic error

ey3:

The geometric error at the Y-direction of the compensated model using the average value of kinematic error

ez3:

The geometric error at the Z-direction of the compensated model using the average value of kinematic error

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Tianjian Li, Jungang Li & Xiaohong Ding

  2. Department of Mechanical and Aerospace Engineering, Brunel University London, Uxbridge, UB8 3PH, UK

    Xizhi Sun

  3. R&D Centre, Wuhan Second Ship Design & Research Institute, Wuhan, 430205, China

    Tao Wu

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Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Tianjian Li and Jungang Li. The first draft of the manuscript was written by Jungang Li and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Tianjian Li.

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Li, T., Li, J., Ding, X. et al. An error identification and compensation method for Cartesian 3D printer based on specially designed test artifact. Int J Adv Manuf Technol 125, 4185–4199 (2023). https://doi.org/10.1007/s00170-023-10858-8

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