Kinematic Modelling and Workspace Prediction of a Hybrid Kinematic Machining Unit Integrating Redundantly Actuated Parallel Manipulator

Abstract

To meet the increasing requirements for high-performance machine tools in manufacturing industry, a 5-axis hybrid kinematic machining unit (HKMU) is proposed with a topology of 2UPR&1RPS&1RPU-XY. Firstly, the inverse/forward position of the HKMU is analyzed by using the closed-loop vector method. Secondly, based on the position analysis, a 3-axis and a 5-axis reachable workspace are defined to demonstrate the motion capabilities of the proposed HKMU. And a ‘hierarchical’ searching algorithm is proposed to calculate the above two types of reachable workspace. Finally, the relationship between dimensional parameters and reachable workspace is graphically revealed by using an illustrative example of the proposed HKMU. 

1 Introduction

Due to the inherent merits of large stiffness/volume ratio, high flexibility, rapid dynamic response, serial-parallel hybrid kinematic machining units (HKMUs) have been regarded as promising alternative solution for efficient 5-axis machining of complex structural parts [1, 2]. This proposition has been successfully exemplified by the commercial application of the Exechon machining unit [3] in aircraft industry. Other propositions and applications of 5-axis HKMUs can be traced in recent publications [2, 4].

Compared with their counterparts of non-redundantly actuated forms, redundantly actuated parallel manipulators (RAPMs) often claim comparative advantages of better dexterity, enhanced stiffness and better dynamic performances [5,6,7,8,9]. For these merits, the authors recently proposed an RAPM with a topology of 2UPR&1RPS&1RPU (R, revolute joint; U, universal joint; S, spherical joint; P, prismatic actuated joint). This newly presented RAPM is designed as a spindle head and integrated with an X–Y sliding gantries to construct a novel hybrid kinematic machining unit (HKMU). Before the newly proposed 2UPR&1RPS&1RPU-XY HKMU can be applied in 5-axis machining, two critical issues need to be considered. One is to find out the kinematic motion of the HKMU; the other is to evaluate the workspace of the HKMU. These two issues are the fundamentals for guiding basic structure design and spatial layout of functional modules.

As to the first critical issue of kinematic analysis, numerous efforts have been conducted in the past decades. For example, Wang et al. [6] designed a 3PRS&1UPS RAPM and analyzed its inverse kinematics, singularity and dexterity characteristics. Li et al. [10] proposed a variety of RAPMs and analyzed their inverse kinematics and motion/force transfer characteristics. Schreiber et al. [11] established a kinematic model for two types of planar RAPMs, based on which their inverse position solutions and workspace analysis were completed. Dong et al. [12] designed a 4UPS&1UP RAPM and analyzed its motion transmission performance. Compared with abundant investigations on the kinematics of RAPMs, efforts on the kinematic analysis of HKMUs are few [13,14,15,16,17]. By reviewing previous studies on HKMUs, the authors select the close-vector method to establish the kinematic model for the proposed 2UPR&1RPS&1RPU-XY HKMU.

As to the second critical issue of workspace prediction, it mainly follows two different tracks: one is to obtain the workspace by using analytical methods; the other is to predict the workspace with numerical methods. Following the first track, Bihari et al. [18] proposed a geometric approach to predict the workspace of two symmetric planar parallel manipulators. Using the same method, Fu et al. [19] obtained the position and orientation workspace of a 6-DOFs parallel manipulator with decoupling three branches. Following the second track, Tang et al. [20] introduced a ‘sliced’ search algorithm to evaluate the workspace of a parallel manipulator by considering its geometry constraints. Geoffrey et al. [21] investigated the reachable workspace and dexterous workspace of three parallel manipulators based on homogeneous Jacobian matrix condition number. Inspired by the aforementioned literatures, the authors propose a hierarchical search algorithm to evaluate the workspace of proposed 2UPR&1RPS&1RPU-XY HKMU.

Bearing with the above two considerations, the authors establish a closed-loop vector based kinematic model to reveal the motion characteristics of the proposed 2UPR&1RPS&1RPU-XY HKMU. By analyzing the inverse/forward position property of the HKMU, the authors further evaluate the 3-axis and the 5-axis reachable workspace of the HKMU. The rest of this paper is organized as follows. Section 2 briefly describes the structure of the HKMU followed by some necessary kinematic definitions. Section 3 sequentially carry out the inverse/forward position analysis. In Sect. 4, the 3-axis and 5-axis reachable workspace of the HKMU are defined and evaluated. Finally, some conclusions are drawn to close the paper.

2 Structure Description

As shown in Fig. 1, the proposed HKMU integrates a 1T2R RAPM with an X–Y sliding gantry module. The topological architecture behind the 1T2R RAPM is a 2UPR-1RPS-1RPU parallel manipulator. It composes of a moving platform, a base and four limbs (denoted as limb 1, limb 2, limb 3 and limb 4). To be specific, limb 1 and limb 2 are two identical UPR limb, which are distributed symmetrically with respect to limb 3 and limb 4. Limb 3 and limb 4 are a RPS limb and a RPU limb, respectively. X–Y module is composed of two orthogonal sliding gantries. In addition, a spindle is installed on the moving platform while the workpiece is fixed to the X–Y sliding gantry module.

Fig. 1

Structure of the proposed HKMU

For clarity, a schematic diagram of the proposed HKMU is demonstrated in Fig. 2.

Fig. 2

A kinematic diagram of the proposed HKMU

As shown in Fig. 2, $B_{i}$ and $A_{i}$ $(i = 1,2,3,4)$ denote the geometric centers of the prismatic joints connected to the base and the moving platform, respectively. Without loss of generality, $B_{i}$ and $A_{i}$ $(i = 1,2,3,4)$ are set to be two squares $\square B_{1} B_{2} B_{3} B_{4}$ and $\square A_{1} A_{2} A_{3} A_{4}$. O and O′ denote the geometric centers of $\square B_{1} B_{2} B_{3} B_{4}$ and $\square A_{1} A_{2} A_{3} A_{4}$, respectively. $O_{W}$ represents the workpiece origin. For derivation facility, some kinematic definitions and coordinate systems are set as follows. A reference frame $\{ O - xyz\}$ is attached at O, with its x axis pointing to B₃, its y axis pointing to B₂, and its z axis satisfying the right-hand rule. A moving frame $\{ O^{\prime} - uvw\}$ is built at O′, with its x′ axis pointing to A₃, its y′ axis pointing to A₂, and its z′ axis satisfying the right-hand rule. A workpiece frame is set at O_W, whose axes are always parallel to those of $\{ O - xyz\}$. Specially, X sliding gantry is parallel to x axis, while Y sliding gantry is parallel to y axis. Assuming that $d_{{\text{X}}}$ and $d_{{\text{Y}}}$ are the sliding displacement of the X–Y sliding gantry module.

The basic dimensional parameters of the HKMU are listed in Table 1. Herein, r_A and r_B are the exterior radius of the moving platform and the base, respectively. $d_{{{\text{min}}}}$ and $d_{{{\text{max}}}}$ are the minimum and maximum displacement of the prismatic joint, respectively; $\theta_{{{\text{U11}}}}$ and $\theta_{{{\text{U12}}}}$ are the rotation angles of the two rotating shafts of the universal joint in the first and the second branch limb; $\theta_{{{\text{U21}}}}$ and $\theta_{{{\text{U22}}}}$ are the rotation angles of the two rotating shafts of the universal joint in the fourth limb. $\theta_{{\text{R}}}$ is the rotation angle of the rotating joint. $\theta_{{{\text{S1}}}}$, $\theta_{{{\text{S2}}}}$ and $\theta_{{{\text{S3}}}}$ are the rotation angles of the three rotating shafts of the spherical joint; $d_{{{\text{Xmin}}}}$ and $d_{{{\text{Xmax}}}}$ are the minimum and maximum displacements of X sliding gantry. $d_{{{\text{Ymin}}}}$ and $d_{{{\text{Ymax}}}}$ are the minimum and maximum displacements of Y sliding gantry. $d_{{\text{t}}}$ is the distance from point $P$ to point $O^{\prime}$.

Table 1 Dimension parameters of the HKMU

3 Kinematic Modelling

3.1 Inverse Position Analysis

In this section an inverse position analysis is carried out to provide fundamentals for reachable workspace prediction.

As shown in Fig. 2, the position vector of point P and point $O^{\prime}$ measured in $O - xyz$ is

$${\varvec{r}}_{P}^{O} = {\varvec{R}}_{{O_{W} }}^{O} {\varvec{r}}_{P}^{{O_{W} }} + {\varvec{r}}_{{O_{W} }}^{O}$$

(1)

$${\varvec{r}}_{{O^{\prime}}}^{O} = {\varvec{r}}_{P}^{O} - d_{t} ({\varvec{R}}_{{O_{W} }}^{O} {\varvec{e}}_{{{\text{t0}}}} ) = [x_{{O^{\prime}}} ,y_{{O^{\prime}}} ,z_{{O^{\prime}}} ]^{{\text{T}}}$$

(2)

where ${\varvec{R}}_{{O_{W} }}^{O}$ is a transformation matrix between $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$ and $\{ O - xyz\}$. Accordingly, ${\varvec{r}}_{P}^{{O_{W} }}$ and ${\varvec{r}}_{{O_{W} }}^{O}$ represent the vectors of the points P and $O_{W}$ measured in the frame $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$ and the frame $\{ O - xyz\}$, respectively; ${\varvec{e}}_{{{\text{t0}}}}$ is the unit vector of the tool-axis measured in the frame $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$.

Since the proposed RAPM possess two rotational DOFs and one translational DOF, the Euler angles ψ, θ and the coordinate $z_{{O^{\prime}}}$ can be taken as independent orientation and position parameters to describe the motion of the moving platform. The transformation matrix between $O^{\prime} - uvw$ and $O - xyz$ can be formulated as

$${\varvec{R}} = \left[ {\begin{array}{*{20}c} {{\text{c}} \varphi {\text{c}} \theta } & {{\text{c}} \varphi {\text{s}} \theta {\text{s}} \psi - {\text{s}} \varphi {\text{c}} \psi } & {{\text{c}} \varphi {\text{s}} \theta {\text{c}} \psi + {\text{s}} \varphi {\text{s}} \psi } \\ {{\text{s}} \varphi {\text{c}} \theta } & {{\text{s}} \varphi {\text{s}} \theta {\text{s}} \psi + {\text{c}} \varphi {\text{c}} \psi } & {{\text{s}} \varphi {\text{s}} \theta {\text{c}} \psi - {\text{c}} \varphi {\text{s}} \psi } \\ { - {\text{s}} \theta } & {{\text{s}} \psi {\text{c}} \theta } & {{\text{c}} \theta {\text{c}} \psi } \\ \end{array} } \right]$$

(3)

where ψ, θ and $\varphi$ are Euler angles defined as precession, nutation and rotation respectively; and “c” and “s” represent the “cosine’’ and “sine” functions, respectively.

According to the constraints of joints of RAPMs as described in Fig. 2, we obtain

$${\varvec{s}}_{1,1} \cdot {\varvec{s}}_{1,2} = 0, \, {\varvec{s}}_{1,4} \cdot {\varvec{r}}_{{O^{\prime}}}^{O} = 0, \, {\varvec{s}}_{3,1} \cdot {\varvec{r}}_{{O^{\prime}}}^{O} = 0$$

(4)

where $\varvec{s}_{1,1}$ and ${\varvec{s}}_{1,2}$ represent the unit vectors of the universal joint in limb 1. Accordingly, ${\varvec{s}}_{1,4}$ and ${\varvec{s}}_{3,1}$ denote the unit vectors of the revolute joint in limb 1 and limb 3, respectively.

By solving Eq. (4), one can obtain the parasitic motions of the proposed RAPM

$$\varphi = 0,\quad x_{{O^{\prime}}} = z_{{O^{\prime}}} \tan \theta ,\quad y_{{O^{\prime}}} = 0$$

(5)

Since the unit vector of the tool-axis in frame $\{ O^{\prime} - uvw\}$ can be expressed as $[0,0,1]$$^{{\text{T}}}$, ${\varvec{e}}_{{\mathbf{t}}}$ in the frame $\{ O - xyz\}$ can be expressed by

$${\varvec{e}}_{{\text{t}}} = {\varvec{Re}}_{{{\text{t0}}}} = {\varvec{R}}[0,0,1]^{{\text{T}}}$$

(6)

From Eq. (6) we can derive that

$$\psi = - {\text{a}}\sin(\varvec{e}_{{\text{t}}} (2)),\theta = {\text{atan}}(\varvec{e}_{{\text{t}}} (1)/\varvec{e}_{{\text{t}}} (3))$$

(7)

where ${\varvec{e}}_{{\text{t}}} (i)$ represents the ith element of the vector ${\mathbf{e}}_{{\text{t}}}$.

To obtain the actuator’s displacement $d_{i}$ and its unit direction vector ${\mathbf{v}}_{i}$, we construct the following closed-loop equation

$${\varvec{r}}_{{O^{\prime}}}^{O} { = }{\varvec{b}}_{i} + d_{i} {\varvec{v}}_{i} - {\varvec{Ra}}_{i} \quad (i = {1},{2},{3},{4)}$$

(8)

By substituting Eqs. (2), (5) and (7) into the closed-loop Eq. (8), we can obtain $d_{i}$ and ${\varvec{v}}_{i}$ as

$$\left\{ \begin{array}{*{20}l} d_{i} = \left| {{\varvec{r}}_{{\varvec{O}^{\prime}}} + {\varvec{Ra}}_{i} - {\varvec{b}}_{i} } \right| \hfill \\ {\varvec{v}}_{i} { = }({\varvec{r}}_{{\varvec{O}^{\prime}}} + {\varvec{Ra}}_{i} - {\varvec{b}}_{i} )/d_{i} \hfill \\ \end{array} \right.$$

(9)

where ${\varvec{a}}_{i}$ is the position vector of the point $A_{i}$ in $O^{\prime} - uvw$. ${\varvec{b}}_{i}$ is the position vector of the point $B_{i}$ in the coordinate system $\{ O - xyz\}$. They can be expressed as follows:

$$\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\varvec{b}_{i1} = r_{B} \cdot [\cos \upalpha_{1} ,\sin \upalpha_{1} ,0]^{\text{T}} } \\ {\varvec{b}_{i2} = r_{B} \cdot [\cos \upalpha_{2} ,\sin \upalpha_{2} ,0]^{\text{T}} } \\ \end{array} } \right.} \\ {\left\{ {\begin{array}{*{20}l} {\varvec{a}_{i1} = r_{A} \cdot [\cos \upalpha_{1} ,\sin \upalpha_{1} ,0]^{\text{T}} } \\ {\varvec{a}_{i2} = r_{A} \cdot [\cos \upalpha_{2} ,\sin \upalpha_{2} ,0]^{\text{T}} } \\ \end{array} } \right.} \\ \end{array} \quad (i_{1} = 1,2;\,i_{2} = 3,4)$$

(10)

Herein, $\alpha_{1} = (2 - i_{1} ) \cdot\uppi ;\alpha_{2} = (3 - i_{2} ) \cdot\uppi +\uppi /2$.

Subsequently, by substitute Eqs. (2) and (5) into Eq. (1), one can obtain

$${\varvec{r}}_{{O_{W} }}^{O} = {\varvec{r}}_{P}^{O} - {\varvec{R}}_{{O_{W} }}^{O} {\varvec{r}}_{P}^{{O_{W} }}$$

(11)

where ${\varvec{r}}_{P}^{{O_{W} }}$ represents the vector of the point $P$ measured in $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$. Meanwhile $d_{\text{X}}$ and $d_{\text{Y}}$ measured in $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$ can be express as

$$d_{{\text{X}}} = x_{{O_{W} }} ,\quad d_{{\text{Y}}} = y_{{O_{W} }}$$

(12)

3.2 Forward Position Analysis

In this subsection, the forward position of the HKMU is carried out to determine the pose of the tool tip relative to the frame $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$.

Expanding Eq. (10) and taking the square of $d_{i}$, one can obtain

$$\left\{ {\begin{array}{*{20}l} {d_{2}^{2} - d_{1}^{2} = \displaystyle\frac{{4r_{A} z_{{O^{\prime}}} \sin \psi }}{\cos \theta }} \\ {d_{4}^{2} - d_{3}^{2} = 4r_{B} z_{{O^{\prime}}} \tan \theta } \\ {d_{4}^{2} + d_{3}^{2} = 2\left( {\left( {\frac{{d_{4}^{2} - d_{3}^{2} }}{{4r_{B} }}} \right)^{2} + r_{A}^{2} + r_{B}^{2} + z_{{O^{\prime}}}^{2} } \right) - 4r_{A} r_{B} \cos \theta } \\ \end{array} } \right.$$

(13)

By simplifying the constant term of Eq. (13), we obtain

$$\left\{ {\begin{array}{*{20}l} {k_{1} = d_{2}^{2} - d_{1}^{2} \, } \\ {k_{2} = 4r_{{\text{A}}} } \\ {k_{3} = d_{4}^{2} - d_{3}^{2} } \\ {k_{4} = 4r_{{\text{B}}} } \\ {k_{5} = d_{4}^{2} + d_{3}^{2} } \\ {k_{6} = - 4r_{{\text{A}}} r_{{\text{B}}} } \\ {k_{7} = 2\left( {\left( {\frac{{d_{4}^{2} - d_{3}^{2} }}{{4r_{{\text{B}}} }}} \right)^{2} + r_{{\text{A}}}^{2} + {\mathbf{r}}_{B}^{2} } \right)} \\ \end{array} } \right.$$

(14)

From this equation, one can obtain

$$k_{1} = \frac{{k_{2} z_{{O^{\prime}}} \sin \psi }}{\cos \theta }$$

(15)

$$k_{3} = k_{4} z_{{O^{\prime}}} \tan \theta$$

(16)

$$k_{5} = k_{7} + 2z_{{O^{\prime}}}^{2} + k_{6} \cos \theta$$

(17)

Assume $g_{0} = \cos \theta$, $k_{8} = k_{5} - k_{7}$, $k_{9} = 2 \cdot (k_{3} /k_{4} )^{2}$. By substituting Eq. (16) into Eq. (15), we can obtain

$$g_{0}^{3} - \frac{{(k_{8} + k_{9} )}}{{k_{6} }}g_{0}^{2} - g_{0} + \frac{{k_{8} }}{{k_{6} }} = 0$$

(18)

By solving the above equation, one can obtain

$$\theta = {\text{sign}}(d_{4}^{{}} - d_{3}^{{}} )a\cos (g_{0} )$$

(19)

where ${\text{sign}}( \cdot )$ is the symbolic function.

By substituting Eq. (18) into Eq. (16) and Eq. (14), one can obtain

$$z_{{O^{\prime}}} = \sqrt {\frac{{k_{5} - k_{6} \cos \theta - k_{7} }}{2}} ,\quad \psi = {\text{a}}\sin \left( {\frac{{k_{1} \cos \theta }}{{k_{2} z_{{O^{\prime}}} }}} \right)$$

(20)

By substituting Eqs. (5),  (18) and (20) into Eq. (2), the position vector ${\varvec{r}}_{p}$ of point $P$ in the reference frame $\{ O - xyz\}$ is

$${\varvec{r}}_{P}^{O} = {\varvec{r}}_{{O^{\prime}}}^{O} + d_{t} ({\varvec{R}}_{{O_{W} }}^{O} {\varvec{e}}_{{{\text{t0}}}} )$$

(21)

By substituting Eqs. (12) and (21) into Eq. (1), we obtain

$${\varvec{r}}_{P}^{{O_{W} }} = ({\varvec{R}}_{{O_{W} }}^{O} )^{ - 1} ({\varvec{r}}_{P}^{O} - {\varvec{r}}_{{O_{W} }}^{O} )$$

(22)

According to Eq. (6), the unit vector ${\varvec{e}}_{{{\text{t0}}}}$ of the tool-axis relative to $\{ O_{W} - x^{\prime}y^{\prime}z^{\prime}\}$ can be expressed as

$${\varvec{e}}_{{{\text{t0}}}} = ({\varvec{R}}_{{O_{W} }}^{O} )^{ - 1} {\varvec{R}} \cdot [0,0,1]^{{\text{T}}}$$

(23)

4 Reachable Workspace Calculation

4.1 Workspace Definition

As depicted in Fig. 3, the 3-axis workspace is the collection of spatial position points that can be reached when e_t0 = [0, 0, 1]^T. If the tool-axis can arbitrarily change when its posture within the range of cone angle $\beta_{0}$, the collection of spatial position points that can be reached is defined as the 5-axis workspace. Following this definition, one may take the coordinate value of tool tip x'_p, y'_p,z'_p as independent variables and formulate the 3-axis and 5-axis workspaces as

$$\begin{aligned} W_{3} & = \left\{ [x^{\prime}_{p} ,y^{\prime}_{p} ,z^{\prime}_{p} ]\left| {\gamma_{t} } \right. \in U_{{\gamma_{t} }} (t = 1,2, \ldots ,n),\right.\\ & \left.\quad d_{i} \in U_{{d_{i} }} (i = 1,2,3,4,X,Y),\,\psi = 0,\theta = 0 \right\} \end{aligned}$$

(24)

$$\begin{aligned}W_{5}& = \left\{ [x^{\prime}_{p} ,y^{\prime}_{p} ,z^{\prime}_{p} ]\left| {\gamma_{t} } \right. \in U_{{\gamma_{t} }} (t = 1,2, \ldots ,n),\right.\\ & \left. \quad d_{i} \in U_{{d_{i} }} (i = 1,2,3,4,X,Y),\,U_{\varphi \theta } \supseteq U_{{\beta_{0} }} \right\} \end{aligned}$$

(25)

Fig. 3

The posture of tool axis within $\beta_{0}$ conical space

where $\gamma_{t}$ denotes the instantaneous rotation angle of the tth passive joint; $U_{{\gamma_{t} }}$ is the reachable rotation space of the tth passive joint; $U_{{d_{i} }}$ is the maximum stroke of the ith prismatic joint; $U_{{\beta_{0} }}$ is the collection of all rotation space of tool-axis within the cone angle $\beta_{0}$ range; $U_{\psi \theta }$ are the collection of reachable rotation space of tool-axis.

According to the above definition, the 5-axis reachable workspace can be expressed as

$$W_{5} = \bigcap\limits_{\beta = 0}^{{\beta_{0} }} {W_{3}^{\beta } }$$

(26)

where $W_{3}^{\beta }$ is the 3-axis workspace which the position of tool axis is within angle $\beta$; $\bigcap\nolimits_{\beta = 0}^{{\beta_{0} }} {W_{3}^{\beta } }$ is all intersections within the cone angle $\beta_{0}$ range.

4.2 Workspace Searching Algorithm

Without loss of generality, the singularity should be considered when predict the workspace of the HKMU. Due to the series structure, there is no motion singularity in the X–Y sliding gentries. Therefore, the singularity of the HKMU is only determined by the RAPM. According to reference [22], when the Jacobian matrix is singular, the 2UPR&1RPS&1RPU parallel mechanism will demonstrate singularity, which can be formulated as

$${\text{rank}}({\varvec{J}}_{{\text{a}}} ) < f||{\text{rank}}({\varvec{J}}_{{\text{c}}} ) < {6 - }f||{\text{rank}}({\varvec{J}}) < 6$$

(27)

where ${\varvec{J}}_{{\text{a}}}$ is the Jacobian of actuations; ${\varvec{J}}_{{\text{c}}}$ is the Jacobian of constraints.; ${\varvec{J}}$ is the overall Jacobian matrix; $f$ is the number of degrees of freedom of the parallel mechanism.

To evaluate the reachable workspace of the HKMU, a hierarchical searching algorithm is proposed in this paper. For clarity, the basic idea and corresponding flowchart are illustrated in Figs. 4 and 5, respectively.

Fig. 4

Hierarchical searching for reachable workspace

Fig. 5

Flowchart of reachable workspace prediction

As shown in Fig. 4, the hierarchical searching algorithm can be roughly described as the following seven steps.

Step 1: discrete the cone angle $\beta$ into a set of cone angles with an increment of $\Delta \beta$;

Step 2: discrete $\Delta \beta$ space into a set of cone angles with an increment of $\Delta \alpha$;

Step 3: discrete the potential workspace into a series of workplane along z₁ axis with an increment of $\Delta z_{1}$;

Step 4: discrete the workplane $x_{1} y_{1}$ into grids with numerous nodes along $x_{1}$ and $y_{1}$ axes with increments of $\Delta x_{1}$ and $\Delta y_{1}$, respectively;

Step 5: initialize $\beta$, $\alpha$, $z_{1}$, $y_{1}$ and $x_{1}$, and judge whether the node [x'_p, y'_p, z'_p] locates in the workspace or not;

Step 6: judge whether the HKMU is at a singular pose according to Eq. (27);

Step 7: export the 3-axis and the 5-axis reachable workspace of HKMU graphically.

4.3 Illustrative Example

With the above searching algorithm, the reachable workspace of the proposed HKMU can be predicted in an accurate yet efficient manner. To demonstrate the proposed algorithm, the 3-axis and 5-axis workspace of an exemplary HKMU is evaluated numerically. The basic dimensional parameters of exemplary HKMU are listed in Table 1. For simplicity, the coordinate z_ow is set to be z_ow = 350 mm.

The predicted reachable workspace is demonstrated in Fig. 6.

Fig. 6

Reachable workspace of the HKMU

As illustrated in Fig. 6a, the proposed HKMU possesses a regular 3-axis reachable workplace with a volume of 5.054 × 10⁶ mm³. The contour of the 3-axis reachable workspace is similar to a symmetric polyhedron with respect to the planes of x_p = 0, y_p = 0. The motion range along the directions of $x_{1}$, $y_{1}$ and $z_{1}$ are [− 70, 70] mm, [− 95, 95] mm and [− 78, 112] mm, respectively.

As shown in Fig. 6b, the 5-axis reachable workspace of the HKMU is similar to a triangular pyramid with a volume of 2.604 × 10⁵ mm³. The 5-axis reachable workspace is symmetrical about x₁ axis and $y_{1}$ axis. The motion ranges along the directions of x₁, y₁ and z₁ are [− 17.5, 17.5] mm, [− 67.5, 67.5] mm and [− 55.25, 44.75] mm. It can be easily observed from Fig. 6b that $x^{\prime}_{p}$ and $y^{\prime}_{p}$ decreases rapidly with the increase of $z^{\prime}_{p}$.

5 Conclusions

This paper investigates the kinematics of a newly patented HKMU with a topology of 2UPR&1RPS&1RPU-XY. The inverse/forward position analysis and reachable workspace of the proposed HKMU are conducted for further investigations about kinematics enhancement, trajectory planning and motion control of 5-axis machining task. Based on the conducted research, the following conclusions can be drawn:

• (1) The inverse and forward kinematic models are established for the proposed 5-axis HKMU through the close-vector method with a concise form.

• (2) The 3-axis and 5-axis reachable workspace are defined to graphically evaluate the position-orientation capabilities of the end platform of the HKMU.

• (3) A ‘hierarchical’ searching algorithm is proposed to predict the reachable workspace of the proposed HKMU in an accurate yet efficient manner.