Kinematics Analysis of a Parallel Coordinate Measuring Machine

Abstract

A 4-UPS-UPU 5-Degrees of Freedom PCMM(Parallel coordinate measuring machine), which can achieve three translation degrees of freedom and two rotational degrees of freedom, is presented in this paper. The inverse position model of the 4-UPS-UPU PCMM is established. The Jacobian matrix, and the expressions of velocity and acceleration are deduced. Furthermore, a numerical analysis of the inverse solutions of the position, velocity and acceleration are carried out. The results are confirmed by a virtual simulation of the 4-UPS-UPU PCMM prototype. This research has laid a solid foundation for the motion control and movement performance analysis of the 4-UPS-UPU PCMM.

1. Introduction

The PCMM, as a type of new-structure measuring machine tool, has been utilized for many practical applications[1], such as machinery manufacturing, automation, aerospace, assembling robots, etc. Compared with the traditional CMM, the PCMM has the advantages of good kinematic/dynamic performance, high stiffness, tight construction and no accumulated precision errors.

In recent years, most of the PCMMs have been developed based on a 3-DOF parallel mechanism and a 6-DOF Stewart platform [2–6]. However, the 3-DOF PCMMs cannot measure points on complex surfaces. Since the 6-DOF PCMMs must have six motors, the manufacturing cost of 6-DOF PCMMs is high. Due to the nature of its design, the general 5-DOF PCMMs with five motors can meet the measuring requirement. Therefore, the PCMM with 5-DOF has received more and more attention from various industries and institutions. The kinematics analysis of PCMM provides an important foundation for the following research, including knowledge on motion control, movement performance analysis, working space, control algorithms, accuracy analysis, and so forth.

This paper focuses on the kinematics analysis of 4-UPS-UPU PCMM. The inverse position model of 4-UPS-UPU PCMM is constructed, and the Jacobian matrix, and the expressions of velocity and acceleration are deduced. Based on the kinematics model mentioned above, a numerical analysis of 4-UPS-UPU PCMM is carried out and verified by an ADAMS simulation.

2. Description of 4-UPS-UPU PCMM

The 4-UPS-UPU PCMM is shown in Figure 1; it consists of a stationary platform, a moving platform, four UPS (Universal joint-Prismatic pair- Spherical joint) actuating limbs, a UPU (universal joints-prismatic pairs- universal joints) actuating limb and a working head. The moving platform and the stationary platform are connected by four UPS actuating limbs and a UPU actuating limb. The probe is attached to the centre of the working head. When varying the length of the five actuating limbs, the probe was able to perform five DOF movements. Thus, the aim of measuring a complicated surface can be achieved. The 4-UPS-UPU PCMM has some characteristics such as an easy inverse and forward solution, small movement inertia, nimble position and pose of its moving platform, et al.

Figure 1.

Mechanism diagram of PCMM.

3. Kinematics analysis of 4-UPS-UPU PCMM

3.1. Inverse position analysis

As shown in Fig. 1, we establish the reference frame {A} on the stationary platform and the moving frame {B} on the moving platform.

Given the position and orientation parameters $({}^{\mathrm{A}}\mathrm{X}{}_{BO},{}^{\mathrm{A}}\mathrm{Y}{}_{BO},{}^{\mathrm{A}}\mathrm{Z}{}_{BO},\alpha ,\beta )$ of the moving platform, the inverse solution can be obtained. Let R denote the orientation cosine matrix from the moving frame {B} relative to the stationary frame {A} and ${}^{\mathrm{A}}P{}_{BO}={\left[\begin{array}{ccc}{}^{\mathrm{A}}\mathrm{X}{}_{BO}& {}^{\mathrm{A}}\mathrm{Y}{}_{BO}& {}^{\mathrm{A}}\mathrm{Z}{}_{BO}\end{array}\right]}^{\mathrm{T}}$ denote the vector of the origin of {B} expressed in {A}.

The five universal joints of the stationary platform are arranged as shown in Fig. 1. One universal joint U₁ is along the reverse X-axis of {A} and its coordinate value is W. The rest of the four universal joints are arranged uniformly, the radius is denoted as ${\mathrm{r}}_{\mathrm{A}}$ and the angle between the two near universal joints of the rest of the four universal joints is θ ( $=\frac{\pi }{2}$ ). The coordinate of ${\mathrm{U}}_{\mathrm{i}}(\mathrm{i}=1,\hspace{0.17em}2,\cdots ,5)$ expressed in {A} is given by

${\hspace{0.17em}}^{\mathrm{A}}{P}_{\text{Ui}}=\left[{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\text{Ui}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Y}}_{\text{Ui}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Z}}_{\text{Ui}}\right]$

(1)

The four spherical joints and one universal joint of the moving platform are arranged uniformly. The universal joint is along the reverse X-axis of {B}. The radius is denoted as ${\mathrm{r}}_{\mathrm{B}}$ and the angle between the two near joints is φ ( $=\frac{2\pi }{5}$ ). The coordinate of the five joints $(\mathrm{i}=1,\hspace{0.17em}2,\cdots ,5)$ expressed in {B} is given by

${\hspace{0.17em}}^{\mathrm{B}}{P}_{\mathrm{i}}={\left[{\hspace{0.17em}}^{\mathrm{B}}{\mathrm{X}}_{\mathrm{i}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{B}}{\mathrm{Y}}_{\mathrm{i}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{B}}{\mathrm{Z}}_{\mathrm{i}}\right]}^{\mathrm{T}}$

(2)

The coordinate of the five joints $(\mathrm{i}=1,\hspace{0.17em}2,\cdots ,5)$ expressed in {A} is given by

${\hspace{0.17em}}^{\mathrm{A}}{P}_{\mathrm{i}}=\left[{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\mathrm{i}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Y}}_{\mathrm{i}}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Z}}_{\mathrm{i}}\right]$

(3)

Then the length vectors of the five actuated limbs with respect to {A} is given by

$L_{\mathrm{i}}={\hspace{0.17em}}^{\mathrm{A}}{P}_{\mathrm{i}}-{\hspace{0.17em}}^{\mathrm{A}}{P}_{\text{Ui}}=\hspace{0.17em}{\mathit{R}}^{\mathrm{B}}{P}_{\mathrm{i}}+{\hspace{0.17em}}^{\mathrm{A}}{P}_{\text{BO}}-{\hspace{0.17em}}^{\mathrm{A}}{P}_{\text{Ui}}\hspace{0.17em}({\mathrm{B}}_5)$

(4)

where

$\mathrm{R}=\left[\begin{array}{l}\hspace{0.17em}{\mathrm{r}}_{11}\hspace{0.17em}{\mathrm{r}}_{12}\hspace{0.17em}{\mathrm{r}}_{13}\\ \hspace{0.17em}{\mathrm{r}}_{21}\hspace{0.17em}{\mathrm{r}}_{22}\hspace{0.17em}{\mathrm{r}}_{23}\\ \hspace{0.17em}{\mathrm{r}}_{31}\hspace{0.17em}{\mathrm{r}}_{32}\hspace{0.17em}{\mathrm{r}}_{33}\end{array}\right]$ is the orientation cosine matrix.

When $R$ , which is the transformation matrix of the moving frame {B} relative to the stationary frame {A}, is denoted by $\mathrm{Z}-\mathrm{Y}-\mathrm{X}$ Euler angles $\left(\alpha ,\beta ,\gamma \right)$ . $R$ can be simplified as below

$R={}_{\mathrm{B}}{}^{\mathrm{A}}R{}_{\mathrm{Z}-\mathrm{Y}-\mathrm{X}}\text{(}\alpha ,\beta ,\gamma =0°\text{)}=\left[\begin{array}{ccc}\mathrm{c}\beta & 0& \mathrm{s}\beta \\ \mathrm{s}\alpha \mathrm{s}\beta & \mathrm{c}\alpha & -\mathrm{s}\alpha \mathrm{c}\beta \\ -\mathrm{c}\alpha \mathrm{s}\beta & \mathrm{s}\alpha & \mathrm{c}\alpha \mathrm{c}\beta \end{array}\right]$

(5)

Thus the length of the five actuating limbs is given by

${\mathrm{l}}_{\mathrm{i}}=|L_{\mathrm{i}}|=|{\hspace{0.17em}}^{\mathrm{A}}{P}_{\mathrm{i}}-{\hspace{0.17em}}^{\mathrm{A}}{P}_{\text{Ui}}|={\left[\begin{array}{l}{({\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\mathrm{i}}-{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\text{Ui}})}^2+\hfill \\ +{({\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Y}}_{\mathrm{i}}-{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\text{Ui}})}^2+{({\hspace{0.17em}}^{\mathrm{A}}{\mathrm{Z}}_{\mathrm{i}}-{\hspace{0.17em}}^{\mathrm{A}}{\mathrm{X}}_{\text{Ui}})}^2\hfill \end{array}\right]}^{1/2}$

(6)

Where

$\mathrm{i}=1,2,\cdots ,5$

Based on the inverse solution equation (6), the position and orientation of the working head on the moving platform planned in the working space of the PCMM can be transformed to the length of the actuated limb in the joint space. In this way, the control of the position and orientation of the working head when measuring is carried out.

3.2. Inverse velocity solution analysis

As shown in Fig. 2, ${\mathrm{v}}_{BO}$ is the linear velocity of the centre of the moving platform ${\mathrm{O}}_{\mathrm{B}}$ , ${\omega }_{\mathrm{B}}$ is the angular velocity of the moving platform, ${\mathrm{r}}_{\mathrm{i}}$ is the vector radius of the joints from ${\mathrm{O}}_{\mathrm{B}}$ , ${\mathrm{V}}_{si}$ is the linear velocity of joint ${\mathrm{S}}_{\mathrm{i}}$ , ${\mathrm{n}}_{\mathrm{i}}$ is the unit vector of ${\mathrm{L}}_{\mathrm{i}}$ , ${\stackrel{.}{l}}_{\mathrm{i}}$ is the length rate change of ${\mathrm{L}}_{\mathrm{i}}$ , where $\mathrm{i}=1,2,\cdots ,5$ .

Figure 2.

Velocity diagram of the mechanism.

Then the equation is obtained as follows

${\mathrm{v}}_{Si}={\mathrm{v}}_{BO}+{\omega }_{\mathrm{B}}\times {\mathrm{r}}_{\mathrm{i}}$

(7)

$\stackrel{•}{{\mathrm{L}}_{\mathrm{i}}}={\mathrm{v}}_{Si}\cdot {\mathrm{n}}_{\mathrm{i}}=\left[\begin{array}{cc}{\mathrm{n}}_{\mathrm{i}}^{\mathrm{T}}& {({\mathrm{r}}_{\mathrm{i}}\times {\mathrm{n}}_{\mathrm{i}})}^{\mathrm{T}}\end{array}\right]\left[\begin{array}{c}{\mathrm{v}}_{BO}\\ {\omega }_{\mathrm{B}}\end{array}\right]$

(8)

For all the five actuated limbs, the relationship between the velocity of the moving platform and the velocity of the actuated limbs is given by

$\stackrel{•}{{\mathrm{L}}_{\mathrm{i}}}={\left[\begin{array}{ccccc}\stackrel{•}{{\mathrm{L}}_1}& \stackrel{•}{{\mathrm{L}}_2}& \stackrel{•}{{\mathrm{L}}_3}& \stackrel{•}{{\mathrm{L}}_4}& \stackrel{•}{{\mathrm{L}}_5}\end{array}\right]}^{\mathrm{T}}=\left[{\mathrm{J}}_1\right]\left[\begin{array}{c}{\mathrm{v}}_{BO}\\ {\omega }_{\mathrm{B}}\end{array}\right]$

(9)

Where

$\begin{array}{l}\left[{\mathrm{J}}_1\right]=\left[\begin{array}{cc}{\mathrm{n}}_1^{\mathrm{T}}& {({\mathrm{r}}_1\times {\mathrm{n}}_1)}^{\mathrm{T}}\\ {\mathrm{n}}_2^{\mathrm{T}}& {({\mathrm{r}}_2\times {\mathrm{n}}_2)}^{\mathrm{T}}\\ {\mathrm{n}}_3^{\mathrm{T}}& {({\mathrm{r}}_3\times {\mathrm{n}}_3)}^{\mathrm{T}}\\ {\mathrm{n}}_4^{\mathrm{T}}& {({\mathrm{r}}_4\times {\mathrm{n}}_4)}^{\mathrm{T}}\\ {\mathrm{n}}_5^{\mathrm{T}}& {({\mathrm{r}}_5\times {\mathrm{n}}_5)}^{\mathrm{T}}\end{array}\right]=\\ {\left[\begin{array}{ccccc}{\mathrm{n}}_1& {\mathrm{n}}_2& {\mathrm{n}}_3& {\mathrm{n}}_4& {\mathrm{n}}_5\\ {\mathrm{r}}_1\times {\mathrm{n}}_1& {\mathrm{r}}_2\times {\mathrm{n}}_2& {\mathrm{r}}_3\times {\mathrm{n}}_3& {\mathrm{r}}_4\times {\mathrm{n}}_4& {\mathrm{r}}_5\times {\mathrm{n}}_5\end{array}\right]}^{\mathrm{T}}\end{array}$

(10)

The Jacobian matrix $\left[{\mathrm{J}}_1\right]\in {\mathrm{R}}^{5\times 6}$ , called the velocity transfer matrix of the parallel mechanism, is the Jacobian matrix between the six-dimension velocity of the moving platform and the driving velocity.

The angular velocity of the moving platform ${}^{\mathrm{A}}\omega {}_{\mathrm{B}}$ is expressed by the differential coefficient with respect to the time of the orientation parameter denoted by $\mathrm{X}-\mathrm{Y}-\mathrm{Z}$ Euler angles $\left(\alpha ,\beta ,\gamma \right)$ and is given by

$\begin{array}{c}{\hspace{0.17em}}^A\omega \mathrm{B}=\mathrm{R}(\mathrm{X},\alpha )\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\stackrel{•}{\alpha }+\mathrm{R}(\mathrm{X},\alpha )\mathrm{R}(\mathrm{Y},\beta )\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\stackrel{•}{\beta }+\mathrm{R}(\mathrm{X},\alpha )\mathrm{R}(\mathrm{Y},\beta )\mathrm{R}(\mathrm{Z},\gamma )\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\stackrel{•}{\gamma }\\ ={\mathrm{T}}_{\mathrm{W}}\left[\begin{array}{c}\stackrel{•}{\alpha }\\ \stackrel{•}{\beta }\\ \stackrel{•}{\gamma }\end{array}\right]\end{array}$

(11)

Where

$\mathrm{R}(\mathrm{X},\alpha )$ denotes the direction cosine matrix for a rotation around the X axis in frame {A} by angle α.

Because the rotation of the moving platform around its center axis is constrained by the construction of the mechanism, $\gamma (=0^{\circ })$ is a constant, $\stackrel{.}{\gamma }=0$ , equation (11) can be written as

${\hspace{0.17em}}^A\omega {}_{\mathrm{B}}=\mathrm{A}\left[\begin{array}{c}\stackrel{•}{\alpha }\\ \stackrel{•}{\beta }\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& \cos \alpha \\ 0& \text{sin}\alpha \end{array}\right]\left[\begin{array}{c}\stackrel{•}{\alpha }\\ \stackrel{•}{\beta }\end{array}\right]$

(12)

Put equation (12) into equation (8), So the Jacobian matrix $\left[\mathrm{J}\right]$ is given by

$\left[\mathrm{J}\right]=\left[{\mathrm{J}}_1\right]{\mathrm{T}}_{\mathrm{w}}$

(13)

Where

$T_{\mathrm{W}}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& \cos \alpha \\ 0& 0& 0& 0& \sin \alpha \end{array}\right]$

The Jacobian matrix $\left[\mathrm{J}\right]\in {\mathrm{R}}^{5\times 5}$ , is the velocity transfer matrix of the novel 4-UPS-UPU PCMM, which describes the homogenous transformation between five actuated limbs and the velocity of the moving platform.

3.3. Inverse acceleration solution analysis

Assuming two vectors are $\eta \hspace{0.17em}$ and $\delta \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ , we can get

$\eta \text{=}\left[\begin{array}{l}{\eta }_{\mathrm{x}}\\ {\eta }_{\mathrm{y}}\\ {\eta }_{\mathrm{z}}\end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\delta =\left[\begin{array}{l}{\delta }_{\mathrm{x}}\\ {\delta }_{\mathrm{y}}\\ {\delta }_{\mathrm{z}}\end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{\wedge }{\eta }=\left[\begin{array}{l}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\eta }_{\mathrm{z}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\eta }_{\mathrm{y}}\\ {\eta }_{\mathrm{z}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\eta }_{\mathrm{x}}\\ -{\eta }_{\mathrm{y}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\eta }_{\mathrm{x}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\end{array}\right]$

$\eta \times \delta =\stackrel{\wedge }{\eta }\delta$

(14)

${\stackrel{\wedge }{\eta }}^{\mathrm{T}}=-\stackrel{\wedge }{\eta }$

(15)

When presuming a is the linear acceleration of the moving platform, ε is the angular acceleration of the moving platform, and ${\mathrm{a}}_{Li}$ is the acceleration of $L_{\mathrm{i}}$ , a and ε can be expressed as

$\mathrm{a}={\left({\mathrm{a}}_{\mathrm{x}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{a}}_{\mathrm{y}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{a}}_{\mathrm{z}}\right)}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\epsilon ={\left({\epsilon }_{\mathrm{x}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_{\mathrm{y}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_{\mathrm{z}}\right)}^{\mathrm{T}}$

From the equation $\left[\begin{array}{l}{}^{\mathrm{A}}\mathrm{V}{}_{bo}\\ {}^{\mathrm{A}}\omega {}_{\mathrm{b}}\end{array}\right]={\mathrm{J}}_{2\mathrm{A}}\left[\begin{array}{l}{}^{\mathrm{A}}\mathrm{V}{}_{xbo}\\ {}^{\mathrm{A}}\mathrm{V}{}_{ybo}\\ {}^{\mathrm{A}}\mathrm{V}{}_{zbo}\\ \hspace{0.17em}\hspace{0.17em}\stackrel{.}{\alpha }\\ \hspace{0.17em}\hspace{0.17em}\stackrel{.}{\beta }\end{array}\right]$ derivative, we can obtain

$\mathrm{A}={\mathrm{J}}_{2\mathrm{A}}{\mathrm{a}}_{\mathrm{s}}+{\mathrm{v}}_{\mathrm{s}}{}^{\mathrm{T}}\mathrm{H}\hspace{0.17em}{\mathrm{v}}_{\mathrm{s}}$

(16)

Where

$\begin{array}{l}\mathrm{A}={\left(\mathrm{a}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\epsilon \right)}^{\mathrm{T}}\\ {\mathrm{a}}_{\mathrm{s}}={\left[\stackrel{·}{{}^{\mathrm{A}}\mathrm{V}{}_{xbo}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{·}{{}^{\mathrm{A}}\mathrm{V}{}_{ybo}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{·}{{}^{\mathrm{A}}\mathrm{V}{}_{zbo}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{··}{\alpha }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\stackrel{··}{\beta }\right]}^{\mathrm{T}}\\ \mathrm{H}={\left[{\mathrm{H}}_1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{H}}_2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{H}}_3\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{H}}_4\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{H}}_5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{H}}_6\right]}^{\mathrm{T}}\end{array}$

The Hessian matrix ${\mathrm{H}}_{\mathrm{i}}\in {\mathrm{R}}^{5\times 5}$ , where $\mathrm{i}=1,2,\cdots ,5$ , can be given by

$\begin{array}{l}{\mathrm{H}}_1=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{\mathrm{H}}_2=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\\ {\mathrm{H}}_3=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{\mathrm{H}}_4=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\\ {\mathrm{H}}_5=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \text{-}\sin \alpha \\ 0& 0& 0& 0& 0\end{array}\right]{\mathrm{H}}_6=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \cos \alpha \\ 0& 0& 0& 0& 0\end{array}\right]\end{array}$

From the equation

${\stackrel{·}{\mathrm{L}}}_{\mathrm{i}}=\left[{}^{\mathrm{A}}\mathrm{n}{{}_{\mathrm{i}}}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({}^{\mathrm{A}}\mathrm{r}{}_{si}\times {}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}\right)}^{\mathrm{T}}\right]\left[\begin{array}{l}{}^{\mathrm{A}}\mathrm{V}{}_{bo}\\ {}^{\mathrm{A}}\omega {}_{\mathrm{b}}\end{array}\right]$

derivative, we can obtain

$\begin{array}{c}{\mathrm{a}}_{Li}=\\ \left({\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}\times {\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}\right)}^{\mathrm{T}}\right)\left(\begin{array}{l}\mathrm{a}\\ \epsilon \end{array}\right)+\left({\hspace{0.17em}}^{\mathrm{A}}\mathrm{V}{{}_{bo}}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\hspace{0.17em}}^{\mathrm{A}}\omega {{}_{\mathrm{b}}}^{\mathrm{T}}\right){\mathrm{h}}_{\mathrm{i}}\left(\begin{array}{l}{\hspace{0.17em}}^{\mathrm{A}}\mathrm{V}{}_{bo}\\ {\hspace{0.17em}}^{\mathrm{A}}\omega {}_{\mathrm{b}}\end{array}\right)\end{array}$

(17)

Where

${\mathrm{h}}_{\mathrm{i}}=\frac{1}{{\mathrm{L}}_{\mathrm{i}}}{\left(\begin{array}{l}\hspace{0.17em}-{\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}}}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}}}^2\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}}\\ -\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}}{\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}}}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{L}}_{\mathrm{i}}\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}}\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}}+\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}}{\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{n}{}_{\mathrm{i}}}}^2\stackrel{\wedge }{{\hspace{0.17em}}^{\mathrm{A}}\mathrm{r}{}_{si}}\end{array}\right)}_{6\times 6}$

According to equation (17), If ${}^{\mathrm{A}}\mathrm{V}{}_{xbo},{}^{\mathrm{A}}\mathrm{V}{}_{ybo},\stackrel{·}{\alpha },\stackrel{·}{\beta },\stackrel{·}{\gamma },\stackrel{·}{{}^{\mathrm{A}}\mathrm{V}{}_{xbo}},\stackrel{·}{{}^{\mathrm{A}}\mathrm{V}{}_{ybo}},\stackrel{··}{\alpha },\stackrel{··}{\beta },\stackrel{··}{\gamma }$ is known, the acceleration of the driving limbs can be computed.

4. Kinematics verification of 4-UPS-UPU PCMM

The structural parameters of the 4-UPS-UPU are illustrated as follows. The distance between the first universal joint and the centre point on the stationary platform is 780mm, the distribution radius of the other universal joints on the stationary platform is 720mm. The distribution radius of the spherical joints on the moving platform is 200mm.

The motion of the centre of the moving platform of the 4-UPS-UPU PCMM is described as

$\{\begin{array}{l}\mathrm{x}=150\cos (2\mathrm{t})\begin{array}{cc}\begin{array}{cc}\hfill & \hfill \end{array}\hfill & \hfill \end{array}\hfill \\ \mathrm{y}=-150\sin (2\mathrm{t})\begin{array}{cc}\begin{array}{cc}\hfill & \hfill \end{array}\hfill & \hfill \end{array}\hfill \\ \mathrm{z}=980\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}\hfill & \hfill \end{array}\hfill & \hfill \end{array}\hfill & \hfill & \hfill \end{array}\hfill \\ \alpha =-pi/36+(pi/72)\mathrm{t}\hfill \\ \beta =-pi/36+(pi/72)\mathrm{t}\hfill \end{array}\mathrm{t}\in \left[1,4\right]$

(18)

The numerical analysis of the length, velocity and acceleration of the actuating limbs for 4-UPS-UPU PCMM by using inverse kinematics model, are shown in Figure 3, Figure 5 and Figure 7. According to Figure 3, Figure 5 and Figure 7, the length, velocity and acceleration of the actuating limbs vary greatly with the different poses of the moving platform and the length value of the actuating limbs is between 950mm and 1200mm, the velocity value of the actuating limbs is from −150mm/s mm to 150mm/s, the acceleration value of actuating limbs is between −250mm/s² mm and 350mm/s². The inverse solutions of the position, velocity and acceleration obtained by the ADAMS simulation are shown in Figure 4, Figure 6 and Figure 8. As shown in Figure 3 to Figure 8, the inverse solutions obtained by numerical calculation and the ADAMS simulation for 4-UPS-UPU PCMM are in basic agreement.

Figure 3.

Inverse position solution by numerical calculation

Figure 4.

Inverse position solution by ADAMS

Figure 5.

Inverse velocity solution by numerical calculation

Figure 6.

Inverse velocity solution by ADAMS

Figure 7.

Inverse acceleration solution by numerical calculation

Figure 8.

Inverse acceleration solution by ADAMS

5. Conclusion

The equation for the inverse solution of position, and the expressions of velocity and acceleration for 4-UPS-UPU PCMM are deduced. The kinematics analysis of position, velocity and acceleration for 4-UPS-UPU PCMM are realized by numerical calculation. The calculation results are verified by virtual simulation on the ADAMS software. The research can provide an important theoretical base for the controlling and optimization design of these parallel mechanisms.