A novel kinematic model for five-axis machine tools and its CNC applications

Abstract

In conventional five-axis CNC machining, the machine structure is treated as a single kinematic chain just like a robotic manipulator while the cutter is treated as an end effector. In this paper, besides the machine kinematic chain, a cutter kinematic chain is introduced, and the two subkinematic chains are combined to form one machine–cutter kinematic chain. Forward, inverse kinematics and constrained inverse kinematics for the proposed machine–cutter kinematic chain are further put forward. Two applications are presented to demonstrate the advantages and effectiveness of the proposed kinematic model. The proposed kinematic model unifies the structure of the machine and cutter; therefore, the flexibility of the five-axis machine tool can be fully explored.

Appendix

According to D–H convention, the transformation matrices of prismatic pair and revolute pairs are defined as follows:

$$ \text{Trans}\left( {a,b,c} \right)=\left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \\ 0 & 0 & 0 & 1 \\ \end{array}} \right] $$

(29)

$$ \text{Rot}\left( {\prime x\prime ,\theta } \right)=\left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 & 0 \\ 0 & {\cos \theta } & {-\sin \theta } & 0 \\ 0 & {\sin \theta } & {\cos \theta } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right] $$

(30)

$$ \text{Rot}\left( {\prime y\prime ,\theta } \right)=\left[ {\begin{array}{*{20}{c}} {\cos \theta } & 0 & {\sin \theta } & 0 \\ 0 & 1 & 0 & 0 \\ {-\sin \theta } & 0 & {\cos \theta } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right] $$

(31)

$$ \text{Rot}\left( {\prime z\prime ,\theta } \right)=\left[ {\begin{array}{*{20}{c}} {\cos \theta } & {-\sin \theta } & 0 & 0 \\ {\sin \theta } & {\cos \theta } & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right] $$

(32)

where [a, b c] is the translation vector of a prismatic pair, and θ is the rotation angle of a revolute pair.

For a kinematic chain composed of a series of kinematic pairs, the transformation can be described by the product of the transformation matrices in the order of the corresponding kinematic pairs.