2.1. The calculation of trajectory curvature

The NC codes obtained directly from CAM software are a series of discrete points, but the curvature is calculated based on curve. Hence, a appropriate curve is selected to fit the discrete points into a curve. In the present study, the B-spline curve is used to construct the toolpath, because it can provide a unified and accurate mathematical expression. The p-order B-spline curve is defined as:

(1) \begin{equation} C(\tau)=\sum _{m=0}^{M}E_{m}D_{m,p}(\tau)\begin{array}{ll} & 0\leq \tau \leq 1 \end{array} \end{equation}

where $\tau$ is the parameter of B-spline curve, $E_{m}$ is the mth control point of the curve, M + 1 is the total number of control points, and $D_{m,p}(\tau)$ is the p-order B-spline basis function of the mth control point and satisfies the Cox-deBoor recurrence relation:

(2) \begin{equation} \left\{\begin{array}{l} D_{m,0}(\tau)=\left\{\begin{array}{l} 1\begin{array}{ll} & \tau _{m}\leq \tau \leq \tau _{m+1} \end{array}\\ \\[-8pt]0\begin{array}{ll} \begin{array}{ll} \begin{array}{ll} & \textit{others} \end{array} & \end{array} & \end{array} \end{array}\right.\\ \\[-8pt]D_{m,p}(\tau)=\dfrac{\tau -\tau _{m}}{\tau _{m+p}-\tau _{m}}D_{m,p-1}(\tau)+\dfrac{\tau _{m+p+1}-\tau }{\tau _{m+p+1}-\tau _{m+1}}D_{m+1,p-1}(\tau) \end{array}\right. \end{equation}

By using the cumulative chord length method, the parameter corresponding to any points can be obtained as:

(3) \begin{equation} \left\{\begin{array}{l} \tau _{0}=0\\ \tau _{n+1}=\tau _{n}+\dfrac{\left| P_{n+1}-P_{n}\right| }{\sum\limits_{n=0}^{N-1}\left| P_{n+1}-P_{n}\right| }\\ \tau _{N}=1 \end{array}\right.\left(n=0\sim N\right) \end{equation}

where $P_{n}$ and $P_{n+1}$ are the nth point and the (n + 1)th point on the trajectory, respectively, N + 1 is the total number of points on the trajectory, and N = m-p-1.

To ensure the accuracy of toolpath fitting, the toolpath is piecewise fitted based on the fitting error. The toolpath fitting error $f$ can be expressed as:

(4) \begin{equation} f=\sum _{n=0}^{N}\left| P_{n}-C(\tau _{n})\right| ^{2} \end{equation}

where $f$ is a scalar valued function of M-1 variables of $E_{1}$ , … $\,E_{M-1}$ .

In order to minimize the fitting error, the partial derivative of $f$ related to the trajectory control point should be equal to zero. Therefore, we can get a linear system of equations composed of M-1 equations to express the control points:

(5) \begin{align} \boldsymbol{{E}}&=\left(\boldsymbol{{G}}_{1}^{\textrm{T}}\boldsymbol{{G}}_{1}\right)^{-1}\boldsymbol{{G}}_{2}, \boldsymbol{{G}}_{1}=\left[\begin{array}{c@{\quad}c@{\quad}c} D_{1,p}\left(\tau _{1}\right) & \cdot \cdot \cdot & D_{M-1,p}\left(\tau _{1}\right)\\ \colon & \cdot \cdot \cdot & \colon \\ D_{1,p}\left(\tau _{N-2}\right) & \cdot \cdot \cdot & D_{1,p}\left(\tau _{N-2}\right) \end{array}\right], \nonumber\\ \boldsymbol{{G}}_{2}&=\left[\begin{array}{c} D_{1,p}\left(\tau _{1}\right)N_{3,1}+\cdot \cdot \cdot +D_{1,p}\left(\tau _{N-2}\right)N_{3,N-2}\\ \colon \\ D_{M-1,p}\left(\tau _{1}\right)N_{3,1}+\cdot \cdot \cdot +D_{M-1,p}\left(\tau _{N-2}\right)N_{3,N-2} \end{array}\right] \end{align}

where

(6) \begin{equation} N_{3,\zeta }=P_{\zeta }-D_{0,p}(\tau _{\zeta })-D_{M,\zeta }(\tau _{\zeta })P_{\zeta }\ \ \zeta=1\sim N-2 \end{equation}

The specific strategies are as shown in Fig. 3. First, the points to be processed are taken as the fitting point of the B-spline curve. Second, the fitting error is calculated to judge whether the fitting toolpath is appropriate. If the fitting error is less than the error threshold, the current points are taken as a segment of the toolpath. On the contrary, if the fitting error is larger than the error threshold, the number of points is gradually reduced until the fitting trajectory error is less than the error threshold.

Figure 3. Process of fitting the points into B-spline curve.

The k-order derivative of any points in the toolpath can be obtained by using the de Boor algorithm.

(7) \begin{align} \left[C(\tau)\right]^{(k)}&=\frac{\textrm{d}^{(k)}\!\sum\limits_{\overline{m}=0}^{M}\boldsymbol{{E}}_{\overline{m}}D_{\overline{m},p}(\tau)}{\textrm{d}\mu ^{(k)}}=\sum _{\overline{m}=m-p}^{m-k}\boldsymbol{{E}}_{\overline{m}}^{(k)}D_{\overline{m},p-k}(\tau) \nonumber \\&\quad \tau \in \left[\tau _{m},\tau _{m+1}\right]\subset \left[\tau _{p},\tau _{M+1}\right] \end{align}

where

(8) \begin{equation} \boldsymbol{{E}}_{\overline{m}}^{\left(r\right)}=\left\{\begin{array}{ll} \boldsymbol{{E}}_{\overline{m}} & r=0\\ \\[-8pt]\left(p-r+1\right)\!\dfrac{\boldsymbol{{E}}_{\overline{m}+1}^{\left(r-1\right)}-\boldsymbol{{E}}_{\overline{m}}^{\left(r-1\right)}}{\mu _{\overline{m}+p+1}-\mu _{\overline{m}+r}} & \overline{m}=m-p,m-p+1,\cdots,m-r;\ \ r=1,2\cdots k \end{array}\right. \end{equation}

The curvature $\kappa (\mu)$ of the toolpath can be expressed as:

(9) \begin{equation} \kappa (\mu)=\frac{\left\| \left[C(\mu)\right]^{\left(1\right)}\times \left[C(\mu)\right]^{\left(2\right)}\right\| }{\left\| \left[C(\mu)\right]^{\left(1\right)}\right\| ^{3}} \end{equation}

2.2. Determination of limited velocity

The constrained velocity points are defined in this paper as points on the spline where the allowable velocity is lower than the rated velocity value. The allowable velocity is obtained by applying chord error, centripetal acceleration, and contouring error constraints. The rated velocity value is given by user. The velocity control can effectively decrease the frequent acceleration variation and limit the machining error, which are favorable to high-quality machining.

As shown in Fig. 4, under the constraint of the chord error and the centripetal acceleration, the limited velocity [Reference Yeh and Hsu34, Reference Feng, Li, Wang and Chen35] can be obtained as follows:

(10) \begin{equation} v_{lim,1}=\frac{2\cdot \sqrt{\frac{1}{{\kappa ^{2}}_{n}}-\!\left(\frac{1}{\kappa _{n}}-\delta _{\max }\right)}}{T}, v_{lim,2}=\sqrt{\frac{a_{\max }}{\kappa _{n}}} \end{equation}

where $\kappa _{n}$ is the curvature of toolpath $\boldsymbol{{P}}_{n}$ , which can be obtained from Eq. (9), $\delta _{\max }$ is the set maximum permissible chord error of $\boldsymbol{{P}}_{n}\boldsymbol{{P}}_{n+1}$ , $a_{\max }$ is the set maximum permissible centripetal acceleration of $\boldsymbol{{P}}_{n}\boldsymbol{{P}}_{n+1}$ , which is determined by the performance of the motor.

Figure 4. The constraints of chord error and centripetal acceleration.

Limited to the system bandwidth, the output of the servo drive cannot trace the reference input timely and accurately, which leads to the contouring error. As shown in Fig. 5, the included angle of two adjacent small segments is

(11) \begin{equation} \gamma _{n}=\arcsin \!\left(\frac{\boldsymbol{{P}}_{n-1}\boldsymbol{{P}}_{n}\cdot \boldsymbol{{P}}_{n}\boldsymbol{{P}}_{n+1}}{\left| \boldsymbol{{P}}_{n-1}\boldsymbol{{P}}_{n}\right| \cdot \left| \boldsymbol{{P}}_{n}\boldsymbol{{P}}_{n+1}\right| }\right) \end{equation}

Figure 5. The constraint of contour error.

If $\boldsymbol{{v}}_{n}=\boldsymbol{{v}}_{n+1}$ , the contouring error $\sigma$ can be obtained, and the contouring error does not exceed the set maximum permissible contouring error $\sigma _{\max }$ :

(12) \begin{equation} \sigma =\frac{1}{4}\cdot v_{n}\cdot T\cdot \sin \!\frac{\gamma _{n}}{2}\leq \sigma _{\max } \end{equation}

So under the constraint of the contouring error, the limited velocity can be obtained by:

(13) \begin{equation} v_{\textrm{lim,3}}=v_{n}=\frac{4\sigma _{\max }}{T\sin \!\frac{\gamma _{n}}{2}} \end{equation}

Clearly, if the rated velocity value is set higher, the chord error, centripetal acceleration, and contouring error will be greater for reducing the machining accuracy. So the velocity of any original tool tip can be obtained as shown in Eq. (14):

(14) \begin{equation} v_{\lim }=\min \!\left\{v_{\textrm{lim,1}},v_{\textrm{lim,2}},v_{\textrm{lim,3}},F\right\} \end{equation}

where F is the rated velocity value. If $v_{\lim }$ < F, the current tool tip is defined as the constrained velocity point, and the corresponding velocity of the tool tip is defined as the limited velocity.

Obviously, the tool tip needs to be accelerated and decelerated between different velocities. Without proper acceleration and deceleration control, impact and vibration will be caused, which will affect the machining accuracy. For a constrained velocity point, the corresponding acceleration and deceleration control must be implemented to ensure the constancy of acceleration and deceleration. Thus, the corresponding acceleration and deceleration control is used to further improve the machining accuracy.

2.3. The acceleration and deceleration control

The five-phase S-shape curve is adopted to realize the flexible acceleration and deceleration control in the present study due to its merits of continuous acceleration and small flexible impact. According to the distance formula of S-shape velocity curve [Reference Du, Jie and Zhu36, Reference Yuxiang, Xiaoguang and Zhu37], the distance $S_{\textrm{acc}}$ from the previous limited velocity acceleration to the desired feedrate can be obtained by:

(15) \begin{equation} S_{\textrm{acc}}=\left(v_{\lim }^{\textrm{pre}\textrm{vious}}+F\right)\cdot \sqrt{\frac{\left(F-v_{\lim }^{\textrm{previous}}\right)}{J_{\textrm{set}}}} \end{equation}

where $v_{\lim }^{\textrm{previous}}$ is the velocity of the previous constrained velocity point, $J_{\textrm{set}}$ is the set jerk, and $T^{\textrm{previous}}$ is the interpolation period of the previous constrained velocity point.

The distance $S_{\textrm{dec}}$ from the desired feedrate deceleration to the next limited velocity can obtained by:

(16) \begin{equation} S_{\textrm{dec}}=\left(v_{\lim }^{\textrm{next}}+F\right)\cdot \sqrt{\frac{\left(F-v_{\lim }^{\textrm{next}}\right)}{J_{\textrm{set}}}} \end{equation}

where $v_{\lim }^{\textrm{next}}$ is the velocity of the next constrained velocity point and $T^{\textrm{next}}$ is the interpolation period of the next constrained velocity point.

The Simpson method is used to approximate the distance $l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)$ between the adjacent constrained velocity points.

(17) \begin{align} l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)&=\frac{\left(\mu _{\textrm{next}}-\mu _{\textrm{previous}}\right)}{6}\cdot \left(\left\| \left[C_{\textrm{tip}}\!\left(\mu _{\textrm{previous}}\right)\right]^{\left(1\right)}\right\| +4\cdot \left\| \left[C_{\textrm{tip}}\!\left(\left(\mu _{\textrm{previous}}+\mu _{\textrm{next}}\right)/\textrm{2}\right)\right]^{\left(1\right)}\right\| \right.\nonumber\\[4pt] & +\left.\left\| \left[C_{\textrm{tip}}\!\left(\mu _{\textrm{next}}\right)\right]^{\left(1\right)}\right\| \right) \end{align}

where $\mu _{\textrm{previous}}$ is the parameter value corresponding to the previous constrained velocity point and $\mu _{\textrm{next}}$ is the parameter value corresponding to the next constrained velocity point.

According the relationship between $S_{\textrm{acc}}+S_{\textrm{dec}}$ and $l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)$ , one may find that there are two types of velocity curve of tool tip between two adjacent constrained velocity points with usage of five-phase S-shape curve as shown in Fig. 6.

Figure 6. Two typess of velocity curve of tool tip between two adjacent constrained velocity points.

As shown in Fig. 6(a), the sum of the distances $S_{\textrm{acc}}$ and $S_{\textrm{dec}}$ satisfies: $S_{\textrm{acc}}+S_{\textrm{dec}}\lt l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)$ . This indicates that the tool tip has enough distance to achieve a smooth acceleration–deceleration curve. Therefore, five-phase S-shape curve can be adopted to realize the acceleration and deceleration control of tool tip under this circumstance.

As shown in Fig. 6(b), the sum of the distances $S_{\textrm{acc}}$ and $S_{\textrm{dec}}$ satisfies: $S_{\textrm{acc}}+S_{\textrm{dec}}\geq l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)$ . This indicates that the acceleration of tool tip will change suddenly. The tool tip of acceleration changed suddenly is defined as catastrophe point. Under this circumstance, the five-phase S-shape curve will be not suitable for acceleration and deceleration control of tool tip.

To address this problem, a modified curve combing five-phase S-shape curve and trigonometric function curve is considered here as shown in Fig. 7. The modified curve is used to replace the original S-shape curve. It is obvious that the catastrophe point disappeared.

Figure 7. A modified velocity curve of tool tip.

Taking the velocity of two adjacent constrained velocity points as the boundary condition, the expressions of the velocity and acceleration of the interference section is derived as follows:

(18) \begin{equation} V_{\textrm{tri}}(u)=\frac{1}{2}\!\left[1-\cos\!\left(\pi u\right)\right]\left(v_{\lim }^{\textrm{next}}-v_{\lim }^{\textrm{previous}}\right)+v_{\lim }^{\textrm{previous}} \end{equation}

(19) \begin{equation} A_{\textrm{tri}}(u)=\frac{\pi }{2}\!\left(v_{\lim }^{\textrm{next}}-v_{\lim }^{\textrm{previous}}\right)\cdot \sin\!\left(\pi u\right) \end{equation}

The time of the acceleration and deceleration control by using the trigonometric function curve is

(20) \begin{equation} T_{\textrm{tri}}=\frac{l\!\left(\mu _{\textrm{previous}},\mu _{\textrm{next}}\right)}{v_{\lim }^{\textrm{previous}}+v_{\lim }^{\textrm{next}}} \end{equation}

By adopting the above velocity control strategy, a smooth velocity curve without a sudden change in acceleration will be obtained to improve the machining quality.

3.1. Interpolation algorithm

The tool can be regarded as a rigid body moving between the curve of tool tip and the curve of tool-axis point. The original tool tips are obtained from CAM software. In order to distinguish the original tool tip from the interpolated tool tip, the position vector of the original tool tip is set as $\boldsymbol{{P}}_{m}^{0}=\left[x_{P,m}^{0},y_{P,m}^{0},z_{P,m}^{0}\right]^{\textrm{T}}$ with respect to workpiece coordinate system. The original tool tip to be processed is taken as the fitting point of the B-spline curve. The original tool tip is fitted into a B-spline curve, and it is set as the tool tip curve $\boldsymbol{{C}}_{\textrm{tip}}(\mu)$ .

According to Eq. (1), the velocity along the B-spline curve can be expressed as:

(21) \begin{equation} v=\left\| \frac{\textrm{d}\boldsymbol{{C}}_{\textrm{tip}}(\mu)}{\textrm{d}t}\right\| =\left\| \frac{\textrm{d}\boldsymbol{{C}}_{\textrm{tip}}(\mu)}{\textrm{d}\mu }\right\| \cdot \frac{\textrm{d}\mu }{\textrm{d}t} \end{equation}

where $\left\| \boldsymbol{{M}}\right\|$ represents the two-norm of M .

Due to the complexity of B-spline curve, the nodal parameters have no exact solution. Therefore, the second-order Taylor expansion method is used to approximate the nodal parameters:

(22) \begin{equation} \mu _{i+1}=\mu _{i}+T\cdot \frac{\textrm{d}\mu }{\textrm{d}t}+\frac{T^{2}}{2}\cdot \frac{\textrm{d}\!\left(\frac{\textrm{d}\mu }{\textrm{d}t}\right)}{\textrm{d}t} \end{equation}

where T is the interpolation period and $\frac{\textrm{d}\mu }{\textrm{d}t}$ is the first derivative of the parameter $\mu$ with respect to time t, which can be solved by Eq. (21).

By substituting $\mu _{i}$ into $\boldsymbol{{C}}_{\textrm{tip}}(\mu)$ , one may obtain the interpolated tool tip $\boldsymbol{{P}}_{i}$ .

3.2. Parameter synchronization

To synchronize the interpolated tool tip and the interpolated tool-axis point, the parameter synchronization is conducted. The unit tool-axis vector corresponding to the tool tip is set as $\boldsymbol{{R}}_{m}^{0}=\left[x_{R,m}^{0},y_{R,m}^{0},z_{R,m}^{0}\right]^{\textrm{T}}$ with respect to workpiece coordinate system. Like the tool tip, the original unit tool-axis vectors are obtained from CAM software. So the position vector of the original tool-axis point $\boldsymbol{{Q}}_{m}^{0}$ can be expressed as:

(23) \begin{equation} \boldsymbol{{Q}}_{m}^{0}=\boldsymbol{{P}}_{m}^{0}+H\cdot \boldsymbol{{R}}_{m}^{0} \end{equation}

where H is the length of tool.

The original tool-axis points to be processed are taken as the fitting points of the B-spline curve. The original tool-axis points are fitted into a B-spline curve, and it is set as the tool-axis point curve $\boldsymbol{{C}}_{\textrm{tool}}(\omega )$ .

Although the dual B-spline toolpath has been generated, it cannot be used directly for NC machining, because the parameter $\mu$ of tool tip curve and the parameter $\omega$ of tool-axis point curve are not synchronized in the toolpath and machine motion, especially where the direction of the toolpath changes greatly.

As can be seen from Eq. (23), the distance between $\boldsymbol{{C}}_{\textrm{tip}}(\mu)$ and $\boldsymbol{{C}}_{\textrm{tool}}(\omega )$ always keeps the same height value H along the toolpath. Therefore, the dual B-spline curves are parallel structure during the process of tool interpolation. As shown in Fig. 9, if the parameter $\mu$ locates in [ $\mu _{\textrm{m}}, \mu _{\textrm{m}+1}$ ], there must be a corresponding parameter $\omega$ belonging to [ $\omega _{\textrm{m}}, \omega _{\textrm{m}+1}$ ]:

(24) \begin{equation} \frac{\omega -\omega _{m}}{\omega _{m+1}-\omega _{m}}=\frac{\mu -\mu _{m}}{\mu _{m+1}-\mu _{m}} \end{equation}

when $\mu _{m+1}-\mu _{m}=0, \omega =\omega _{m}$ .

Figure 9. Diagram of toolpath.

By substituting $\omega _{i}$ into $\boldsymbol{{C}}_{\textrm{tool}}(\omega )$ , the interpolated tool-axis point $\boldsymbol{{Q}}_{i}$ can be obtained. And the interpolated tool-axis vector $\boldsymbol{{R}}_{\textrm{tool}}$ corresponding to the interpolated tool tip and the interpolated tool-axis point can be expressed as:

(25) \begin{equation} \boldsymbol{{R}}_{\textrm{tool}}=\left[\begin{array}{l} x_{\textrm{tool}}\\ \\[-9pt]y_{\textrm{tool}}\\ \\[-9pt]z_{\textrm{tool}} \end{array}\right]=\frac{\boldsymbol{{Q}}_{i}-\boldsymbol{{P}}_{i}}{\left\| \boldsymbol{{Q}}_{i}-\boldsymbol{{P}}_{i}\right\| } \end{equation}

Combined with Eqs. (1, 22, 24, 25), the interpolated tool tip and the interpolated tool-axis vectors can be obtained.

4.1. Description of a hybrid machine tool

Figure 11 depicts a laboratory prototype and a schematic diagram of the hybrid machine tool.

Figure 11. Laboratory prototype and schematic diagram of the hybrid machine tool.

As can be seen from Fig 11(a), the proposed hybrid machining tool consists of a 2UPR&2RPS parallel module and an X-Y sliding gantry module. Herein, "U", "P", "R", and "S" represent a universal joint, an actuated prismatic joint, a revolute joint, and a spherical joint, respectively. The parallel module is composed of a fixed base, a moving platform, and four limbs. Specially, limb 1 and limb 2 are two symmetrically arranged UPR limbs, while limb 3 and limb 4 are two symmetrically arranged RPS limbs. The moving platform is connected to the fixed base by the four limbs. According to our previous study, the moving platform possesses two rotational and one translational motion capabilities. Therefore, the spindle installed on the moving platform can fulfill 5-axis motions with respect to the workpiece fixed on the X-Y sliding gantries.

As shown in Fig 11(b), A _j and B _j (j = 1 ∼ 4) are the geometric center points of the joints connecting to the fixed base and the moving platforms, respectively. The X sliding gantry and the Y sliding gantry are along A ₃ A ₄ and A ₁ A ₂, respectively. O and o are the central points of the quadrate of □A ₁ A ₂ A ₃ A ₄ and □B ₁ B ₂ B ₃ B ₄, while o $^{\prime}$ denotes the lower left corner of the upper surface of the workpiece. p is the tool tip. To facilitate kinematic analysis, some coordinate systems are defined. A reference coordinate system O-XYZ is established at O, with X axis pointing to A ₄, Y axis pointing to A ₁, and Z axis satisfying the right-hand rule. A moving coordinate system o-uvw is established at o, with u axis pointing to B ₄, v axis pointing to B ₁ and w axis satisfying the right-hand rule. A workpiece coordinate system $o'-x'y'z'$ is established at $o'$ , with $x'$ axis, $y'$ axis, and $z'$ axis parallel to those of the reference frame of O-XYZ.

4.2. Inverse position solution model

Measured in $o'-x'y'z'$ , the position vector of the tool tip and the tool-axis vectors are defined as $\boldsymbol{{o}}\boldsymbol{'}\boldsymbol{{P}}=\left[\textrm{x}_{P},y_{P},z_{P}\right]^{\textrm{T}}$ and $\boldsymbol{{R}}_{\text{tool}}=\left[x_{\textrm{tool}},y_{\textrm{tool}},z_{\textrm{tool}}\right]^{\textrm{T}}$ , respectively. Measured in O-XYZ, the position vector of $o'$ and o are defined as $\boldsymbol{{Oo}}\boldsymbol{'}=\left[\textrm{X}_{\textrm{o}'},\textrm{Y}_{o'},\textrm{Z}_{o'}\right]^{\textrm{T}}$ and $\boldsymbol{{Oo}}=\left[\textrm{X}_{\textrm{o}},\textrm{Y}_{o},\textrm{Z}_{o}\right]^{\textrm{T}}$ , respectively. The transformation matrix $\boldsymbol{{R}}_{\boldsymbol{{o}}}^{\boldsymbol{{O}}}$ of the frame of o-uvw with respect to the reference frame of O-XYZ can be written as:

(26) \begin{equation} \boldsymbol{{R}}^{O}_{o}=\left[\begin{array}{ccc}\textrm{c}\varphi \textrm{c}\theta& \textrm{c}\varphi \textrm{s}\theta \textrm{s}\psi- \textrm{s}\varphi \textrm{c}\psi & \textrm{c}\varphi \textrm{s}\theta \textrm{c}\psi- \textrm{s}\varphi \textrm{s}\psi \\ \\[-8pt] \textrm{s}\varphi \textrm{c}\theta& \textrm{s}\varphi \textrm{s}\theta \textrm{s}\psi- \textrm{c}\varphi \textrm{c}\psi & \textrm{s}\varphi \textrm{s}\theta \textrm{c}\psi- \textrm{c}\varphi \textrm{s}\psi \\ \\[-8pt]-\textrm{s}\theta &\textrm{s}\psi \textrm{c}\theta& \textrm{c}\theta \textrm{c}\psi \end{array}\right]\end{equation}

where $\psi, \theta$ , and $\varphi$ are the precession angle, the nutation angle, and the rotation angle; "s" and "c" mean the sine function and the cosine function, respectively.

Measured in O-XYZ, the position vector of the tool tip can be obtained as:

(27) \begin{equation} \boldsymbol{{OP}}\boldsymbol{=}\boldsymbol{{Oo}}\boldsymbol{+}\boldsymbol{{R}}_{o}^{O}\cdot \left[0,0,L\right]^{\textrm{T}}=\boldsymbol{{Oo}}\boldsymbol{'}\boldsymbol{+}\boldsymbol{{R}}_{o'}^{O}\cdot \boldsymbol{{o}}\boldsymbol{'}\boldsymbol{{P}} \end{equation}

where L is the distance from point p to point o $^{\prime}$ ; the transformation matrix $\boldsymbol{{R}}_{o'}^{O}$ is a third-order unit diagonal matrix.

Taking $\psi, \theta$ and $Z_{o}$ as the independent motion coordinates of the moving platform, the parasitic motions of the moving platform can be obtained as:

(28) \begin{equation} \varphi =0, Y_{o}=0, X_{o}=Z_{o}\cdot \tan \theta \end{equation}

Since the movement of o $^{\prime}$ is realized by the X sliding gantry and the Y sliding gantry, the position of the two sliding gantries can be obtained by combining Eqs. (27) and (28):

(29) \begin{equation} \left\{\begin{array}{l} X_{o'}=Z_{o}\cdot \tan \theta +L\cdot \textrm{s}\theta \textrm{c}\psi -x_{P}\\ \\[-9pt]Y_{o'}=-L\cdot s\psi -y_{P}\\ \\[-9pt]Z_{o}=Z_{o'}-z_{P}-L\cdot c\theta c\psi \end{array}\right. \end{equation}

where $Z_{o'}$ denotes the distance between plane XOY and plane $x'o'y'$ .

Measured in O-XYZ, the tool-axis vector can be expressed as:

(30) \begin{equation} \boldsymbol{{R}}_{\textrm{tool}}=\boldsymbol{{R}}_{o'}^{O}\cdot \left[x_{\textrm{tool}},y_{\textrm{tool}},z_{\textrm{tool}}\right]^{\textrm{T}}=\boldsymbol{{R}}_{o}^{O}\cdot \left[0,0,1\right]^{\textrm{T}}=\left[\textrm{s}\theta \textrm{c}\psi,-\textrm{s}\psi,\textrm{c}\theta \textrm{c}\psi \right]^{\textrm{T}} \end{equation}

By solving Eq. (30), one may obtain the following equation:

(31) \begin{equation} \left\{\begin{array}{l} \psi =\arcsin \!\left(\!-y_{\textrm{tool}}\right)\\ \\[-9pt] \theta =\arctan \!\left(x_{\textrm{tool}}/z_{\textrm{tool}}\right) \end{array}\right. \end{equation}

The length of $\left| \boldsymbol{{B}}_{j}\boldsymbol{{A}}_{j}\right|$ ( $q_{j}$ ) and its unit direction vector ( $\boldsymbol{{v}}_{{q_{j}}}$ ) can be calculated by Eqs. (32) and (33), respectively:

(32) \begin{equation} q_{j}=\left| \boldsymbol{{Oo}}+\boldsymbol{{R}}_{o}^{O}\cdot \boldsymbol{{b}}_{j}-\boldsymbol{{a}}_{j}\right|\,(j=1\sim4) \end{equation}

(33) \begin{equation} \boldsymbol{{v}}_{{q_{j}}}=\left(\boldsymbol{{Oo}}+\boldsymbol{{R}}_{o}^{O}\cdot \boldsymbol{{b}}_{j}-\boldsymbol{{a}}_{j}\right)/q_{j} \ \ (j=1\sim4)\end{equation}

where, $\boldsymbol{{a}}_{j}$ is the position vector of $\boldsymbol{{O}}\boldsymbol{{A}}_{j}$ (j = 1 ∼ 4) with respect to O-XYZ and $\boldsymbol{{b}}_{j}$ is the position vector of $\boldsymbol{{o}}\boldsymbol{{B}}_{j}$ (j = 1 ∼ 4) with respect to o-uvw.

4.3. Velocity mapping model

Measured in O-XYZ, letting $\boldsymbol{{V}}_{\boldsymbol{{o}}}=\left[V_{oX},V_{oY},V_{oZ}\right]^{\textrm{T}}$ denotes the linear velocity of the geometric center point of the moving platform, $\boldsymbol{{W}}_{\boldsymbol{{o}}}=\left[W_{oX},W_{oY},W_{oZ}\right]^{\textrm{T}}$ is the angular velocity of the geometric center point of the moving platform, and $\boldsymbol{{V}}_{P}=\left[V_{PX},V_{PY},V_{PZ}\right]^{\textrm{T}}$ represents the linear velocity of tool tip.

The velocity V _j (j = 1 ∼ 4) of the jth limb along $\boldsymbol{{A}}_{\boldsymbol{{j}}}\boldsymbol{{B}}_{\boldsymbol{{j}}}$ (j = 1 ∼ 4) can be expressed as:

(34) \begin{equation} V_{j}=\left[\boldsymbol{{V}}_{o}+\boldsymbol{{W}}_{o}\times \left(\boldsymbol{{R}}_{o}^{O}\cdot \boldsymbol{{b}}_{j}\right)\right]\cdot \boldsymbol{{v}}_{{q_{\boldsymbol{{j}}}}} \ \ (j=1\sim4)\end{equation}

The relationship between $\boldsymbol{{W}}_{\boldsymbol{{o}}}$ and the Euler angular velocity can be expressed as:

(35) \begin{equation} \boldsymbol{{W}}_{\boldsymbol{{o}}}=\boldsymbol{{J}}_{0}\cdot \left[\begin{array}{c} \dfrac{\textrm{d}\psi }{\textrm{d}t}\\ \\[-9pt]\dfrac{\textrm{d}\theta }{\textrm{d}t}\\ \\[-9pt]\dfrac{\textrm{d}\varphi }{\textrm{d}t} \end{array}\right], \boldsymbol{{J}}_{0}=\left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & -\sin\!\theta \\ \\[-9pt]0 & \cos\!\psi & \sin\!\psi \cos\!\theta \\ \\[-9pt]0 & -\sin\!\psi & \cos\!\psi \cos\!\theta \end{array}\right] \end{equation}

The derivative of $X_{o}, Y_{o}, Z_{o}, \psi, \theta$ and $\varphi$ with respect to time t can be obtained as:

(36) \begin{equation} \left[\frac{\textrm{d}X_{o}}{\textrm{d}t},\frac{\textrm{d}Y_{o}}{\textrm{d}t},\frac{\textrm{d}Z_{o}}{\textrm{d}t},\frac{\textrm{d}\psi }{\textrm{d}t},\frac{\textrm{d}\theta }{\textrm{d}t},\frac{\textrm{d}\varphi }{\textrm{d}t}\right]^{\textrm{T}}=T_{P}\cdot \left[\frac{\textrm{d}Z_{o}}{\textrm{d}t},\frac{\textrm{d}\psi }{\textrm{d}t},\frac{\textrm{d}\theta }{\textrm{d}t}\right]^{\textrm{T}} \end{equation}

where

(37) \begin{equation} \boldsymbol{{T}}_{P}=\left[\begin{array}{c@{\quad}c@{\quad}c} \tan \theta & 0 & \frac{\textrm{d}Z_{o}}{\textrm{d}t}\!\left(1+\left(\tan \theta \right)^{2}\right)\\ \\[-9pt]0 & 0 & 0\\ \\[-9pt]1 & 0 & 0\\ \\[-9pt]0 & 1 & 0\\ \\[-9pt]0 & 0 & 1\\ \\[-9pt]0 & 0 & 0 \end{array}\right] \end{equation}

Combining Eqs. (34) and (36), the velocity mapping relationship between the actuating joints of the parallel module and the center point of the moving platform is obtained as:

(38) \begin{equation} \left[\begin{array}{l} V_{1}\\ \\[-8pt] V_{2}\\ \\[-8pt] V_{3}\\ \\[-8pt] V_{4} \end{array}\right]=\boldsymbol{{J}}_{a}\cdot \boldsymbol{{T}}_{P}\cdot \left[\begin{array}{c} \dfrac{\textrm{d}Z_{o}}{\textrm{d}t}\\ \\[-7pt] \dfrac{\textrm{d}\psi }{\textrm{d}t}\\ \\[-7pt] \dfrac{\textrm{d}\theta }{\textrm{d}t} \end{array}\right], \boldsymbol{{J}}_{a}=\left[\begin{array}{c@{\quad}c} \boldsymbol{{v}}_{{q_{1}}} & \boldsymbol{{r}}_{1}\times \boldsymbol{{v}}_{{q_{1}}}\cdot \boldsymbol{{J}}_{0}\\ \\[-8pt] \boldsymbol{{v}}_{{q_{2}}} & \boldsymbol{{r}}_{2}\times \boldsymbol{{v}}_{{q_{2}}}\cdot \boldsymbol{{J}}_{0}\\ \\[-8pt] \boldsymbol{{v}}_{{q_{3}}} & \boldsymbol{{r}}_{3}\times \boldsymbol{{v}}_{{q_{3}}}\cdot \boldsymbol{{J}}_{0}\\ \\[-8pt] \boldsymbol{{v}}_{{q_{4}}} & \boldsymbol{{r}}_{4}\times \boldsymbol{{v}}_{q4}\cdot \boldsymbol{{J}}_{0} \end{array}\right] \end{equation}

where $\boldsymbol{{r}}_{j}=\boldsymbol{{R}}_{o}^{O}\cdot \boldsymbol{{b}}_{j}$ is the geometric center points of the jth joints connecting to the moving platforms.

It can be seen from Fig. 11(b) that the tool-axis vector can be expressed as:

(39) \begin{equation} \boldsymbol{{oP}}=\boldsymbol{{R}}_{\boldsymbol{{o}}}^{\boldsymbol{{O}}}\cdot \left[0,0,L\right]^{\textrm{T}} \end{equation}

The velocity of the tool tip can be expressed as:

(40) \begin{equation} \boldsymbol{{V}}_{P}=\boldsymbol{{V}}_{o}+\boldsymbol{{W}}_{o}\cdot \boldsymbol{{oP}}=\boldsymbol{{J}}_{2}\cdot \left[\begin{array}{l} \boldsymbol{{V}}_{o}\\ \\[-8pt] \boldsymbol{{W}}_{o} \end{array}\right], \boldsymbol{{J}}_{2}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & {L}\text{c}\theta \text{c}\psi & L\text{s}\psi \\ \\[-8pt] 0 & 1 & 0 & -{L}\text{c}\theta \text{c}\psi & 0 & {L}\text{s}\theta \text{c}\psi \\ \\[-8pt] 0 & 0 & 1 & -L\text{s}\psi & -{L}\text{s}\theta \text{c}\psi & 0 \end{array}\right] \end{equation}

By substituting Eqs. (34) and (39) into Eq. (40), one may obtain the following equation:

(41) \begin{equation} \boldsymbol{{V}}_{P}=\boldsymbol{{J}}_{3}\cdot \left[\begin{array}{c} \dfrac{\textrm{d}Z_{o}}{\textrm{d}t}\\ \\[-7pt] \dfrac{\textrm{d}\psi }{\textrm{d}t}\\ \\[-7pt] \dfrac{\textrm{d}\theta }{\textrm{d}t} \end{array}\right], \boldsymbol{{J}}_{3}=\boldsymbol{{J}}_{2}\cdot \left[\begin{array}{c@{\quad}c} \boldsymbol{{E}}_{3\times 3} & \textbf{0}_{3\times 3}\\ \\[-8pt] \textbf{0}_{3\times 3} & \boldsymbol{{J}}_{0} \end{array}\right]\cdot \boldsymbol{{T}}_{P} \end{equation}

where $\textbf{0}_{3\times 3}$ is a three-order null matrix.

By combining Eqs. (38) and (41), the mapping relationship between the velocity of the tool tip and the velocity of the actuating joints can be expressed as:

(42) \begin{equation} \left[\begin{array}{l} V_{1}\\ \\[-8pt] V_{2}\\ \\[-8pt] V_{3}\\ \\[-8pt] V_{4} \end{array}\right]=\boldsymbol{{J}}_{1}\cdot \left(\boldsymbol{{J}}_{3}\right)^{-1}\cdot \boldsymbol{{V}}_{P} \end{equation}

Since the X sliding gantry and the Y sliding gantry are connected in series, the velocity of the serial module is the velocity of the actuating joints. So the velocity of the actuating joints can be obtained as:

(43) \begin{equation} \left\{\begin{array}{l} V_{5}=V_{Cx}=\dfrac{X_{oe}-X_{os}}{T}\\ \\[-7pt]V_{6}=V_{Cy}=\dfrac{Y_{oe}-Y_{os}}{T}\\ \\[-7pt]V_{Cz}=0 \end{array}\right. \end{equation}

where $V_{\boldsymbol{{Cx}}}, V_{\boldsymbol{{Cy}}}$ , and $V_{\boldsymbol{{Cz}}}$ are the velocity components of the serial module measured in $o'-x'y'z'$ , respectively. $X_{os}, Y_{os}, X_{oe}$ , and $Y_{oe}$ are the displacement of the X sliding gantry and the Y sliding gantry corresponding to adjacent tool tip measured in $o'-x'y'z'$ .

4.4. Velocity allocation

Since the joint space and the operation space are nonlinear, the velocity of joint space must be allocated in order to determine the velocities of parallel mechanism and serial mechanism. Then combined with the velocity mapping model of hybrid machine tool, the velocity corresponding to the actuating joints is obtained.

When the velocity curve of tool tip is known, the interpolated tool tip and the interpolation period can be obtained by trajectory control strategy as proposed in Section 3. After trajectory control, the position vectors of adjacent interpolation tool tip are expressed as $\boldsymbol{{P}}_{\textrm{s}}=\left[x_{Ps},y_{Ps},z_{Ps}\right]^{\textrm{T}}$ and $\boldsymbol{{P}}_{\boldsymbol{{e}}}=\left[x_{Pe},y_{Pe},z_{Pe}\right]^{\textrm{T}}$ , respectively. So the tangential velocity of tool tip along the toolpath can be expressed as:

(44) \begin{equation} \left\{\begin{array}{l} V_{Fx}=\dfrac{x_{Pe}-x_{Ps}}{T}\\ \\[-7pt]V_{Fy}=\dfrac{y_{Pe}-y_{Ps}}{T}\\ \\[-7pt]V_{Fz}=\dfrac{z_{Pe}-z_{Ps}}{T} \end{array}\right. \end{equation}

where $\boldsymbol{{V}}_{\boldsymbol{{F}}}=\left[V_{Fx},V_{Fy},V_{Fz}\right]^{\textrm{T}}$ is the tangential velocity of the tool tip along the toolpath measured in $o'-x'y'z'$ .

Because the tangential velocity of tool tip along the toolpath is the synthetic velocity of parallel mechanism and serial mechanism, the velocity of parallel mechanism with respect to O-XYZ can be expressed as:

(45) \begin{equation} \left\{\begin{array}{l} V_{PX}=V_{Fx}-V_{Cx}\\ \\[-7pt]V_{PY}=V_{Fy}-V_{Cy}\\ \\[-7pt]V_{PZ}=V_{Fz}-V_{Cz} \end{array}\right. \end{equation}

By substituting Eq. (45) into Eq. (42), the velocity of each actuating joint can be obtained.

5.1. Developments of the proposed method

To perform the machining tests, an open-architecture NC system is constructed as shown in Fig. 12.

Figure 12. NC system of the developed 5-axis hybrid machine tool.

As shown in Fig. 12, the NC system is composed of a host computer (personal computer), a slave computer, six servo modules, and other electrical components. The slave computer is SMC606 multi-axis motion controller. The model of six servo modules is HG-KR43BJ, whose pulse frequencies are all set as 10 kHz. By considering the lead of ball screw (5 mm), the resolution of each prismatic joint can be calculated as 5 × 10^-4mm.

Meanwhile, based on the proposed smooth toolpath interpolation method, a post-processor and a Human–Machine Interface (HMI) system are developed and demonstrated in Figs. 13 and 14.

Figure 13. The Graphical User Interface of post-processor.

Figure 14. The Graphical User Interface of HMI system

As shown in Fig. 13, the post-processor is to generate a smooth toolpath and export a set of optimized NC codes by importing the original NC codes, which integrates the functions of velocity control and trajectory control. Herein, the velocity control is applied to calculate a smooth velocity curve of tool tip for avoiding sudden changes in acceleration and reducing velocity fluctuation. The trajectory control is applied to generate a smooth toolpath and calculate the corresponding interpolated tool-axis vectors and the corresponding interpolated tool tips.

As shown in Fig. 14, the HMI system is used to process user instructions and realize the motion control of a hybrid machine tool. It integrates the functions of kinematics transformation. To be specific, the inverse position solution model is to transform the tool tips and the tool-axis vectors into the displacement of actuated joints. The velocity mapping model is to transform the velocity of tool tip into the velocity of actuated joints.

5.2. Machining test

Figure 15 depicts an overview of 5-axis machining operation by using the proposed toolpath generation method.

Figure 15. Overview of 5-axis machining operation.

As can be seen from Fig. 15, the proposed methodology includes three stages, that is, pre-processing, post-processing, and machining tests.

(1) During the pre-processing stage, a 3D CAD model of the workpiece is developed by using SolidWorks software. This CAD model is then transformed into a CAM model to generate the original NC codes, which include the tool tips and the tool-axis vectors data.

(2) During the post-processing stage, the NC codes are imported into a self-developed post-processor to generate a smoother toolpath and export a set of optimized NC codes.

(3) During the machining tests stage, the optimized NC codes are input into the HMI system of the hybrid machine tool. Based on the inverse position solution and the velocity mapping model, the imported NC codes are transformed into the displacements and the velocities of actuated prismatic joints. Driven by the six servo motors simultaneously, the proposed hybrid machine tool can implement machining tasks.

In order to verify the effectiveness of the proposed smooth toolpath interpolation method, we carried out two sets of machining tests on S-shape workpieces [Reference Wang, Wu and Wang19, Reference Geng, Yu, Zheng, Zhang and Wang38] to verify the effectiveness of the proposed smooth toolpath interpolation method. For clarity, the experimental conditions between the two sets of machining tests are given and compared in Table II. Where the PVT (P: position; V: velocity; T: time) is a interpolation motion pattern based on Hermite cubic spline curve. By using this motion pattern, the users can control the position and the velocity of each actuated joint at any time.

As can be seen from Table II, the two sets of machining tests are set to the same experimental conditions except for the NC codes. In the first machining test, the NC codes are obtained directly from the CAM software. In the second machining test, the NC codes are optimized by using the proposed smooth toolpath interpolation method.

The velocity feedback values of the tool tip are recorded in real time to reflect the velocity fluctuation in the two set of machining tests. The results are depicted in Fig. 16.

Figure 16. The velocity fluctuation in two set of machining test.

As can be seen from Fig. 16(a), the velocity vary in the intervals of [0.24, 29.84] mm/s in the first machining test. The average value of velocity fluctuation can be calculated by Eq. (46) as 0.59 mm/s. As can be seen from Fig. 16(b), the velocity vary in the range of [2.07, 14.26] mm/s in the second machining test. The average value of velocity fluctuation can be obtained as 0.4 mm/s. It is obvious that the average value of velocity fluctuation is decreased by 32.2% after using the proposed velocity control strategy. This indicates that the proposed velocity control strategy helps to reduce the velocity fluctuation:

(46) \begin{equation} \overline{v}_{f}=\frac{\sum\limits_{i=0}^{N_{P}}\left(v_{i}-F\right)}{N_{P}} \end{equation}

where $\overline{v}_{f}$ is the average value of velocity fluctuation, $v_{i}$ is the velocity of ith acquisition point, and $N_{P}$ is the total number of acquisition points.

Figure 17 depicts the two machined S-shape workpiece.

Figure 17. Two set of S-shape workpiece after machining.

As shown in Fig. 17, the total machining time of the first machining test is about 648s, and the total machining time of the second machining test is about 689s. Obviously, the time increase after using the smooth toolpath interpolation method is only 6.33%, which has little impact on the overall machining efficiency.

As can be observed from Fig. 17(a), there are some tool marks on the surface of the first machining test. As can be observed from Fig. 17(b), most of tool marks are disappeared from the surface of the second machining test. This indicates that the proposed smooth toolpath interpolation method can effectively improve the surface quality of machining workpiece without reducing machining efficiency.

As shown in Fig. 18, to further compare the surface quality of the two set of S-shape workpiece, geometric errors of surface contour are tested with MarSurf PS 10 roughness instrument. The position of the S-shape workpiece is positioned by the gauge block to ensure that the starting point and the ending point of the evaluation length are in the same position.

Figure 18. S-shape workpiece test platform of geometric error of surface contour.

Figure 19 depicts the results of geometric error of surface contour.

Figure 19. Geometric error in two set of machining test.

As shown in Fig. 19, the maximum height Rz of the contour of the first machining test is 5.612 µm, which of the second machining test is 4.398 µm. It is obvious that Rz is decreased by 21.6%. The calculated average deviation Ra of the contour of the first machining test is 1.016 µm, which of the second machining test is 0.660 µm. It shows that Ra is decreased by 35.0%. This further indicates that the proposed smooth toolpath interpolation method can effectively improve the surface quality of machining workpiece.

From the above comparison, it can be concluded that the first machining test is smoother than the second machining test. At the same time, the velocity fluctuation can be significantly reduced by using the proposed velocity control strategy.