2. Forward and inverse kinematic analyses of a PPP open-chain manipulator

In this paper, the Denavit–Hartenberg convention [Reference Denavit and Hartenberg22] is adopted to describe the three-link PPP manipulator kinematically. According to the coordinate system definition of D-H notation, the transformation matrix ${}^{i-1}{}_{\!\!i}{T}{}$ from the previous coordinate system $i-1$ to the next coordinate system $i$ can be expressed as

(1) \begin{equation}{}^{i-1}{}_{\!\!i}{T}{}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \cos \theta _{i} & -\sin \theta _{i}\cos \alpha _{i} & \sin \theta _{i}\sin \alpha _{i} & a_{i}\cos \theta _{i}\\[5pt] \sin \theta _{i} & \cos \theta _{i}\cos \alpha _{i} & -\cos \theta _{i}\sin \alpha _{i} & a_{i}\sin \theta _{i}\\[5pt] 0 & \sin \alpha _{i} & \cos \alpha _{i} & d_{i}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

After properly attaching frames $\textrm{X}_{\textrm{i}}\textrm{Y}_{\textrm{i}}\textrm{Z}_{\textrm{i}}$ ( $\textrm{i}=0,1,2,\ \text{and}\ 3$ ) to each link of the PPP open-chain manipulator in Fig. 1, its D-H parameters can be tabulated in Table I. Notice that all joints are prismatic such that the parameter $\theta _{i}$ is constant and specifically denoted by $\theta _{ci}$ for distinction.

Figure 1. PPP open-chain manipulator.

Table I. Link parameters for the PPP open-chain manipulator.

Substitute the D-H parameters of Table I into the transformation matrix ${}^{i-1}{}_{\!\!i}{T}{}$ in Eq. (1) and multiply the transformation matrices of three adjacent links to obtain the matrix ${}^{0\!}{}_{\!\!3}{T}{}$ in Eq. (2). This matrix ${}^{0\!}{}_{\!\!3}{T}{}$ represents the relative position and orientation relationship between the end-effector coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and the fixed coordinate system $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ of the PPP open-chain manipulator in Fig. 1.

(2) \begin{equation}{}^{0\!}{}_{\!\!3}{T}{}={}^{0\!}{}_{\!\!1}{T}{}\;{}^{1\!}{}_{\!\!2}{T}{}\;{}^{2\!}{}_{\!\!3}{T}{}\end{equation}

Matrix ${}^{0\!}{}_{\!\!3}{T}{}$ contains a 3 × 3 rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ and a 3 × 1 translation matrix ${}^{0\!}{}_{\!\!3}{P}{}$ , which can be expressed as

(3) \begin{equation}{}^{0\!}{}_{\!\!3}{T}{}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}|c} &&&\\& {}^{0\!}{}_{\!\!3}{R} & & {}^{0\!}{}_{\!\!3}{P} \\&&&\\\hline0 & 0& 0& 1\end{array}\right]\end{equation}

where the 3 × 3 rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ can be expressed as

(4) \begin{equation}{}^{0\!}{}_{\!\!3}{R}{}=\left[\begin{array}{c@{\quad}c@{\quad}c} a_{11} & a_{12} & a_{13}\\[5pt] a_{21} & a_{22} & a_{23}\\[5pt] a_{31} & a_{32} & a_{33} \end{array}\right]\end{equation}

The elements of the rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ in Eq. (4) can, respectively, be expressed as

(5) \begin{equation}a_{11}=c \theta _{c3}\!\left(c \theta _{c1}c \theta _{c2}-c \alpha _{1}s \theta _{c1}s \theta _{c2}\right)+s \theta _{c3}\!\left(-c \theta _{c2}c \alpha _{1}c \alpha _{2}s \theta _{c1}+s \alpha _{1}s \alpha _{2}s \theta _{c1}-c \theta _{c1}c \alpha _{2}s \theta _{c2}\right)\end{equation}

(6) \begin{align}a_{12} & =-c \alpha _{3}s \theta _{c3}\!\left(c \theta _{c1}c \theta _{c2}-c \alpha _{1}s \theta _{c1}s \theta _{c2}\right)+c \alpha _{3}c \theta _{c3}\!\left(-c \alpha _{1}c \alpha _{2}c \theta _{c2}s \theta _{c1}-c \alpha _{2}c \theta _{c1}s \theta _{c2}+s \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\nonumber\\ &\quad +s \alpha _{3}\!\left(c \alpha _{2}s \theta _{c1}s \alpha _{1}+c \alpha _{1}c \theta _{c2}s \theta _{c1}s \alpha _{2}+c \theta _{c1}s \theta _{c2}s \alpha _{2}\right)\end{align}

(7) \begin{align}a_{13}&=c \alpha _{3}\!\left(c \alpha _{2}s \theta _{c1}s \alpha _{1}+c \alpha _{1}c \theta _{c2}s \theta _{c1}s \alpha _{2}+c \theta _{c1}s \theta _{c2}s \alpha _{2}\right)+s \theta _{c3}s \alpha _{3}\!\left(c \theta _{c1}c \theta _{c2}-c \alpha _{1}s \theta _{c1}s \theta _{c2}\right)\nonumber \\ & \quad -c \theta _{c3}s \alpha _{3}\!\left(-c \alpha _{1}c \alpha _{2}c \theta _{c2}s \theta _{c1}-c \alpha _{2}c \theta _{c1}s \theta _{c2}+s \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\end{align}

(8) \begin{equation}a_{21}=c \theta _{c3}\!\left(c \alpha _{1}c \theta _{c1}s \theta _{c2}+c \theta _{c2}s \theta _{c1}\right)+s \theta _{c3}\!\left(c \alpha _{1}c \alpha _{2}c \theta _{c1}c \theta _{c2}-c \alpha _{2}s \theta _{c1}s \theta _{c1}-c \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\end{equation}

(9) \begin{align}a_{22}&=-c \alpha _{3}s \theta _{c3}\!\left(c \alpha _{1}c \theta _{c1}s \theta _{c2}+c \theta _{c2}s \theta _{c1}\right)+c \alpha _{3}c \theta _{c3}\!\left(c \alpha _{1}c \alpha _{2}c \theta _{c1}c \theta _{c2}-c \alpha _{2}s \theta _{c1}s \theta _{c2}-c \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\nonumber \\&\quad +s \alpha _{3}\!\left(-c \alpha _{2}c \theta _{c1}s \alpha _{1}-c \alpha _{1}c \theta _{c1}c \theta _{c2}s \alpha _{2}+s \theta _{c1}s \theta _{c2}s \alpha _{2}\right)\end{align}

(10) \begin{align}a_{23}&=c \alpha _{3}\!\left(-c \alpha _{2}c \theta _{c1}s \alpha _{1}-c \alpha _{1}c \theta _{c1}c \theta _{c2}s \alpha _{2}+s \theta _{c1}s \theta _{c2}s \alpha _{2}\right)+s \theta _{c3}s \alpha _{3}\!\left(c \alpha _{1}c \theta _{c1}s \theta _{c2}+c \theta _{c2}s \theta _{c1}\right)\nonumber \\ &\quad -c \theta _{c3}s \alpha _{3}\!\left(c \alpha _{1}c \alpha _{2}c \theta _{c1}c \theta _{c2}-c \alpha _{2}s \theta _{c1}s \theta _{c2}-c \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\end{align}

(11) \begin{equation}a_{31}=c \theta _{c3}s \theta _{c2}s \alpha _{1}+s \theta _{c3}(c \alpha _{2}c \theta _{c2}s \alpha _{1}+c \alpha _{1}s \alpha _{2})\end{equation}

(12) \begin{equation}a_{32}=-c \alpha _{3}s \theta _{c2}s \theta _{c3}s \alpha _{1}+c \alpha _{3}c \theta _{c3}\!\left(c \alpha _{2}c \theta _{c2}s \alpha _{1}+c \alpha _{1}s \alpha _{2}\right)+s \alpha _{3}\!\left(c \alpha _{1}c \alpha _{2}-c \theta _{c2}s \alpha _{1}s \alpha _{2}\right)\end{equation}

(13) \begin{equation}a_{33}=c \alpha _{3}\!\left(c \alpha _{1}c \alpha _{2}-c \theta _{c2}s \alpha _{1}s \alpha _{2}\right)-c \theta _{c3}s \alpha _{3}\!\left(c \alpha _{2}c \theta _{c2}s \alpha _{1}+c \alpha _{1}s \alpha _{2}\right)+s \theta _{c2}s \theta _{c3}s \alpha _{1}s \alpha _{3}\end{equation}

Notice that in rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ there are only six angular parameters $\alpha _{1}$ , $\alpha _{2}$ , $\alpha _{3}$ , $\theta _{c1}$ , $\theta _{c2}$ , and $\theta _{c3}$ , which are all constant. In other words, the orientation of the end-effector of the PPP manipulator remains unchanged, while the input parameters of three prismatic joints $d_{1}$ , $d_{2},\;\textrm{and}\; d_{3}$ are actuated.

In addition, the 3 × 1 matrix ${}^{0\!}{}_{\!\!3}{P}{}$ in Eq. (3) can be expressed as

(14) \begin{equation}{}^{0\!}{}_{\!\!3}{P}{}=\left[\begin{array}{c} c_{1}d_{2}+c_{2}d_{3}+c_{3}a_{1}+c_{4}a_{2}+c_{5}a_{3}\\[5pt] c_{6}d_{2}+c_{7}d_{3}+c_{8}a_{1}+c_{9}a_{2}+c_{10}a_{3}\\[5pt] d_{1}+c_{11}d_{2}+c_{12}d_{3}+c_{13}a_{2}+c_{14}a_{3} \end{array}\right]\end{equation}

where the coefficients of the translation matrix ${}^{0\!}{}_{\!\!3}{P}{}$ in Eq. (14) can, respectively, be expressed as

(15) \begin{equation}c_{1}=s \theta _{c1}s \alpha _{1}\end{equation}

(16) \begin{equation}c_{2}=c \alpha _{2}s \theta _{c1}s \alpha _{1}+c \alpha _{1}c \theta _{c2}s \theta _{c1}s \alpha _{2}+c \theta _{c1}s \theta _{c2}s \alpha _{2}\end{equation}

(17) \begin{equation}c_{3}=c \theta _{c1}\end{equation}

(18) \begin{equation}c_{4}=c \theta _{c1}c \theta _{c2}-c \alpha _{1}s \theta _{c1}s \theta _{c2}\end{equation}

(19) \begin{equation}c_{5}=c \theta _{c3}\!\left(c \theta _{c1}c \theta _{c2}-c \alpha _{1}s \theta _{c1}s \theta _{c2}\right)-s \theta _{c3}\!\left(c \alpha _{1}c \alpha _{2}c \theta _{c2}s \theta _{c1}+c \alpha _{2}c \theta _{c1}s \theta _{c2}-s \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\end{equation}

(20) \begin{equation}c_{6}=-c \theta _{c1}s \alpha _{1}\end{equation}

(21) \begin{equation}c_{7}=-c \alpha _{2}c \theta _{c1}s \alpha _{1}-c \alpha _{1}c \theta _{c1}c \theta _{c2}s \alpha _{2}+s \theta _{c1}s \theta _{c2}s \alpha _{2}\end{equation}

(22) \begin{equation}c_{8}=s \theta _{c1}\end{equation}

(23) \begin{equation}c_{9}=c \alpha _{1}c \theta _{c1}s \theta _{c2}+c \theta _{c2}s \theta _{c1}\end{equation}

(24) \begin{equation}c_{10}=c \theta _{c3}\!\left(c \alpha _{1}c \theta _{c1}s \theta _{c2}+c \theta _{c2}s \theta _{c1}\right)+s \theta _{c3}\!\left(c \alpha _{1}c \alpha _{2}c \theta _{c1}c \theta _{c2}-c \alpha _{2}s \theta _{c1}s \theta _{c2}-c \theta _{c1}s \alpha _{1}s \alpha _{2}\right)\end{equation}

(25) \begin{equation}c_{11}=c \alpha _{1}\end{equation}

(26) \begin{equation}c_{12}=c \alpha _{1}c \alpha _{2}-c \theta _{c2}s \alpha _{1}s \alpha _{2}\end{equation}

(27) \begin{equation}c_{13}=s \theta _{c2}s \alpha _{1}\end{equation}

(28) \begin{equation}c_{14}=c \theta _{c3}s \theta _{c2}s \alpha _{1}+s \theta _{c3}\!\left(c \alpha _{2}c \theta _{c2}s \alpha _{1}+c \alpha _{1}s \alpha _{2}\right)\end{equation}

From Eq. (14), it is known that the three elements of the translation matrix ${}^{0\!}{}_{\!\!3}{P}{}$ have a linear relationship with three prismatic joints $d_{1}$ , $d_{2},\;\textrm{and}\;d_{3}$ . Suppose that the ranges of three prismatic joints $d_{1}$ , $d_{2},\textrm{and}\;d_{3}$ are infinite. Thus, the origin of the end-effector coordinate system $\textrm{O}_{3}$ has no boundary workspace. In other words, the end-effector of the PPP manipulator can be moved to any point in the three-dimensional space theoretically.

If the three sliding joint axes are arranged to be perpendicular to each other, as described by the D-H parameters tabulated in Table II. By substituting these D-H parameters into homogeneous transformation matrices, a relatively simple expression for the elements can be obtained as expressed in Eq. (29):

Table II. Link parameters for the three-link Cartesian robot.

(29) \begin{equation}{}^{0\!}{}_{\!\!3}{\textrm{T}}{}=\left[\begin{matrix} 0\;\;\;\;\; & 0\;\;\;\;\; & -1\;\;\;\;\; & -d_{3}\\[5pt] 0\;\;\;\;\; & -1\;\;\;\;\; & 0\;\;\;\;\; & d_{2} \\[5pt] -1\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & d_{1}\\[5pt] 0\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 1 \end{matrix} \right]\end{equation}

Suppose that the endpoint of this manipulator, relative to the end-effector coordinate system and the fixed coordinate system, is expressed as $[\textrm{x},\textrm{y},\textrm{z}]^{\textrm{T}}$ and $[\textrm{X},\textrm{Y},\textrm{Z}]^{\textrm{T}}$ , respectively. Coordinates $[\textrm{x},\textrm{y},\textrm{z}]^{\textrm{T}}$ and $[\textrm{X},\textrm{Y},\textrm{Z}]^{\textrm{T}}$ can be related by the following coordinate transformation:

(30) \begin{equation}\left[\begin{array}{c} \begin{array}{c} \textrm{X}\\[5pt] \textrm{Y}\\[5pt] \textrm{Z} \end{array}\\[5pt] 1 \end{array}\right]={}^{0\!}{}_{\!\!3}{\textrm{T}}{}\left[\begin{array}{c} \begin{array}{c} \textrm{x}\\[5pt] \textrm{y}\\[5pt] \textrm{z} \end{array}\\[5pt] 1 \end{array}\right]\end{equation}

Hence, expanding Eq. (30) yields

(31) \begin{equation}\left[\begin{array}{c} \begin{array}{c} \textrm{X}\\[5pt] \textrm{Y}\\[5pt] \textrm{Z} \end{array}\\[5pt] 1 \end{array}\right]=\left[\begin{array}{c} \begin{array}{c} -d_{3}-\textrm{z}\\[5pt] d_{2}-\textrm{y}\\[5pt] d_{1}-\textrm{x} \end{array}\\[5pt] 1 \end{array}\right]\end{equation}

From Eq. (31), it can be found that the end-point of the manipulator is controlled by three joint variables $d_{1}$ , $d_{2},\textrm{and}\;d_{3}$ . From Eq. (29), we found that this manipulator can be configured into a Cartesian robot, which has linear movements in the three Cartesian coordinates, that is, in x, y, and z axes that are mutually perpendicular.

3. Synthesis of a spatial PPPP mechanism

In Section 2, it has been demonstrated that the three-link PPP manipulator has a fixed orientation and an unlimited workspace. Therefore, its end-effector can be constrained to trace a trajectory identical to that of another open-chain manipulator. If another open-chain manipulator is a single link adjacent to the fixed link by a prismatic joint in Fig. 2(a), the PPP manipulator and the single link can be concatenated to form a spatial closed-loop PPPP mechanism in Fig. 2(b).

Figure 2. Spatial PPPP mechanism by combining two open-chain manipulators.

When such a closed-loop chain containing four prismatic joints is formed, the multiplication of transformation matrices between adjacent links of such a closed-loop chain is equal to an identity matrix, namely,

(32) \begin{equation}{}^{0\!}{}_{\!\!1}{T}{}^{1\!}{}_{\!\!2}{T}{}^{2\!}{}_{\!\!3}{T}{}^{3\!}{}_{\!\!4}{T}{}=I_{4}\end{equation}

To find geometric constraints of twisted angles between the links, a 3 × 3 rotation matrix ${}^{i-1}{}_{\!\!i}{R}{}$ is extracted from Eq. (32) and a closed-loop equation can then be expressed as

(33) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}^{3\!}{}_{\!\!4}{R}{}=I_{3}\end{equation}

To avoid the complexity faced by Pamidi and Soni [Reference Pamidi12, Reference Pamidi and Soni13], this paper proposes an alternative way to simplify the solutions of the closed-loop equation. First, multiply Eq. (33) by an inverse matrix ${}^{3\!}{}_{\!\!4}{R^{-1}}{}$ to obtain the following equation, namely,

(34) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{R}{}={}^{3\!}{}_{\!\!4}{R^{-1}}{}\end{equation}

From the perspective of kinematics, rotation matrix ${}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}$ represents the end-effector orientation of the PPP manipulator relative to the fixed coordinate system. Similarly, rotation matrix ${}^{3\!}{}_{\!\!4}{R^{-1}}{}$ represents the end-effector orientation of the P-joint single-arm manipulator relative to the fixed coordinate system. Therefore, Eq. (34) implies that when the PPPP mechanism satisfies the closed-loop equation, the end-effector orientation of the PPP manipulator and the P-joint single-arm manipulator must be identical. Therefore, this paper attempts to meet the closed-loop equation of the PPPP mechanism by exploring the end-effector orientation of the PPP manipulator. To this end, rotation matrix ${}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}={}^{0\!}{}_{\!\!3}{R}{}$ of the PPP manipulator is set to be equal to Euler angle representation $R_{\textit{Euler}}\!\left(\alpha,\beta,\gamma \right)$ , as shown in Appendix A, namely,

(35) \begin{equation}{}^{0\!}{}_{\!\!3}{R}{}=R_{\textit{Euler}}(\alpha,\beta,\gamma)\end{equation}

Eq. (35) can further be rearranged as

(36) \begin{equation}{}^{0\!}{}_{\!\!3}{R}{}R_{\textit{Euler}}^{-1}(\alpha,\beta,\gamma)=I_{3}\end{equation}

From Eq. (36), multiplying rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}\,$ and $\,R_{\textit{Euler}}^{-1}\!\left(\alpha,\beta,\gamma \right)$ together forms an identity matrix $I_{3}$ . This constraint in the Equation can be regarded as the necessary condition to meet the closed-loop equation of the PPPP mechanism. In addition, once Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ are determined and used to characterize the P-joint single-arm manipulator, the closed-loop equation of the PPPP mechanism can be met satisfactorily. However, the P-joint single-arm manipulator only requires two angular parameters according to the D-H notation. To properly substitute Euler angle parameters into the D-H parameters, this paper proposes decomposing the Euler angles convention such that Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ can be appropriately fit into the D-H notation.

Two types of Euler angles notations, ZXZ and XZX, are used for demonstration.

First, suppose that rotation matrix ${}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}$ of the PPP manipulator is represented by the ZXZ Euler angles, which is

(37) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{z}\!\left(\gamma \right)R_{x} \!\left(\beta \right)R_{z} \!\left(\alpha \right)\end{equation}

Eq. (37) is deliberately rearranged as

(38) \begin{equation}R _{z}^{-1}\! \left(\gamma \right){}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x}\! \left(\beta \right)R_{z}\! \left(\alpha \right)\end{equation}

such that the right side of the equal sign only contains matrix $R_{x}\!\left(\beta \right)$ and $R_{z}\!\left(\alpha \right)$ . In doing so, only two Euler angle parameters $\alpha$ and $\beta$ are required and thus can be fit into the D-H notation for describing the P-joint single-arm manipulator.

Next, matrix ${}^{0\!}{}_{\!\!1}{R}{}$ in Eq. (38) is further decomposed into a multiplication of matrices $R_{z}\!\left(\theta _{c1}\right)$ and $R_{x}\!\left(\alpha _{1}\right)$ , namely,

(39) \begin{equation}R _{z}^{-1} \!\left(\gamma \right)R_{z}\! \left(\theta _{c1}\right)R_{x} \!\left(\alpha _{1}\right){}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x}\!\left(\beta \right)R_{z} \!\left(\alpha \right)\end{equation}

In Eq. (39), matrix $R _{z}^{-1}\!\left(\gamma \right)$ and $R_{z}\!\left(\theta _{c1}\right)$ can be combined as one rotation of $\theta _{c1}-\gamma$ around the z-axis, namely,

(40) \begin{equation}R_{z} \!\left(\theta _{c1}-\gamma \right)R_{x} \!\left(\alpha _{1}\right){}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x} \!\left(\beta \right)R_{z} \!\left(\alpha \right)\end{equation}

Let $R_{z}\!\left(\theta _{c1}-\gamma \right)R_{x}\!\left(\alpha _{1}\right)$ and $R_{x}\!\left(\beta \right)R_{z}\!\left(\alpha \right)$ be denoted by one matrix ${}^{0'\!\!}{}_{\!\!1}{R}{}$ and ${}^{3\!}{}_{\!\!4}{R^{-1}}{}$ , respectively, such that Euler angle parameters can be appropriately fit into the D-H notation and the successively rotational transformations along the closed-loop PPPP mechanism can be expressed as

(41) \begin{equation}{}^{0'\!\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}={}^{3\!}{}_{\!\!4}{R^{-1}}{}\end{equation}

The ZXZ Euler angles embedded in Eq. (41) can be interpreted using Fig. 3 from the perspective of kinematics. Fig. 3(a) illustrates an open-loop PPP manipulator whose end-effector and initial coordinate system are $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ , respectively. If a P-joint single-arm manipulator is connected to the open-loop PPP manipulator to form a closed-loop mechanism, the P-joint single-arm manipulator must share identical end-effector and initial coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ . For this purpose, as shown in Fig. 3(b), angular parameters of the P-joint single-arm manipulator can be substituted with Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ so that end-effector coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ can be transformed to initial coordinate system $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ via the newly added single-arm manipulator. Besides, from Fig. 3(b), it can be interpreted that matrix ${}^{3\!}{}_{\!\!4}{R^{-1}}{}$ represents a coordinate system transformation from $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ to $\textrm{O}_{0'}\textrm{X}_{0'}\textrm{Y}_{0'}\textrm{Z}_{0'}$ , and matrix ${}^{0'\!\!}{}_{\!\!1}{R}{}$ represents a coordinate system transformation from $\textrm{O}_{1}\textrm{X}_{1}\textrm{Y}_{1}\textrm{Z}_{1}$ to $\textrm{O}_{0'}\textrm{X}_{0'}\textrm{Y}_{0'}\textrm{Z}_{0'}$ .

Figure 3. Synthesis of spatial PPPR mechanism using ZXZ Euler angles convention.

Because Euler angle $\gamma$ can be used to orientate $\textrm{X}_{0\textrm{'}}$ toward $\textrm{X}_{0}$ , $\theta _{c1}$ in Eq. (40) can be assumed as a dummy parameter equal to zero. At last, D-H parameters in Eq. (41) can be tabulated in Table III.

Table III. Link parameters involving ZXZ Euler angles convention for the closed-loop PPPP mechanism.

Now, suppose that rotation matrix ${}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}$ of the PPP manipulator is represented by the XZX Euler angles, which is

(42) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)R_{x}\!\left(\alpha \right)\end{equation}

Eq. (42) is deliberately rearranged as

(43) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}R _{x}^{-1}\!\left(\alpha \right)=R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)\end{equation}

such that the right side of the equal sign only contains matrix $R_{x}\!\left(\gamma \right)$ and $R_{z}\!\left(\beta \right)$ . In doing so, only two Euler angle parameters $\gamma$ and $\beta$ are required and thus can be fit into D-H notation for describing the P-joint single-arm manipulator.

Next, matrix ${}^{2\!}{}_{\!\!3}{R}{}$ in Eq. (43) is further decomposed into a multiplication of matrices $R_{z}\!\left(\theta _{c3}\right)$ and $R_{x}\!\left(\alpha _{3}\right)$ , namely,

(44) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}R_{z}\!\left(\theta _{c3}\right)R_{x}\!\left(\alpha _{3}\right)R _{x}^{-1}\!\left(\alpha \right)=R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)\end{equation}

In Eq. (44), matrix $R_{x}\!\left(\alpha _{3}\right)$ and $R _{x}^{-1}\!\left(\alpha \right)$ can be combined as one rotation of $\alpha _{3}-\alpha$ around the z-axis, namely,

(45) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}R_{z}\!\left(\theta _{3}\right)R_{x}\!\left(\alpha _{3}-\alpha \right)=R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)\end{equation}

Let $R_{z}\!\left(\theta _{3}\right)R_{x}\!\left(\alpha _{3}-\alpha \right)$ and $R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)$ be denoted by one matrix ${}^{2\!}{}_{\!\!3'}{R}{}$ and ${}^{3'\!\!}{}_{\!\!4}{R^{-1}}{}$ , respectively, such that Euler angle parameters can be appropriately fit into the D-H notation and the successively rotational transformations along the closed-loop PPPP mechanism can be expressed as

(46) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3'}{R}{}={}^{3'\!\!}{}_{\!\!4}{R^{-1}}{}\end{equation}

The XZX Euler angles embedded in Eq. (46) can be interpreted using Fig. 4 from the perspective of kinematics. Fig. 4(a) illustrates an open-loop PPP manipulator whose end-effector and initial coordinate system are $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ , respectively. If a P-joint single-arm manipulator is connected to the open-loop PPP manipulator to form a closed-loop mechanism, the P-joint single-arm manipulator must share identical end-effector and initial coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ . For this purpose, as shown in Fig. 4(b), angular parameters of the P-joint single-arm manipulator can be substituted with Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ so that end-effector coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ can be transformed to initial coordinate system $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ via the newly added single-arm manipulator. Besides, from Fig. 4(b), it can be interpreted that matrix ${}^{3'\!\!}{}_{\!\!4}{R^{-1}}{}$ represents a coordinate system transformation from $\textrm{O}_{3'}\textrm{X}_{3'}\textrm{Y}_{3'}\textrm{Z}_{3'}$ to $\textrm{O}_{0'}\textrm{X}_{0'}\textrm{Y}_{0'}\textrm{Z}_{0'}$ and matrix ${}^{2}{}_{\!\!3'}{R}{}$ represents a coordinate system transformation from $\textrm{O}_{3'}\textrm{X}_{3'}\textrm{Y}_{3'}\textrm{Z}_{3'}$ to $\textrm{O}_{2}\textrm{X}_{2}\textrm{Y}_{2}\textrm{Z}_{2}$ .

Figure 4. Synthesis of spatial PPPR mechanism using XZX Euler Angles convention.

Because Euler angle $\alpha$ can be used to orientate $\textrm{Z}_{3}$ toward $\textrm{Z}_{3'}$ , $\alpha _{3}$ in Eq. (45) can be assumed as a dummy parameter equal to zero. At last, D-H parameters in Equation can be tabulated in Table IV.

Table IV. Link parameters involving XZX Euler angles convention for the closed-loop PPPP mechanism.

Because the motion of the PPPP mechanism can be characterized by picturing its end-effector being constrained to trace a linear trajectory, the kinematics of the PPPP mechanism can be analyzed based on the kinematics of the PPP manipulator. In reference to Fig. 5, the end-effector and initial coordinate system of the PPP manipulator are denoted as $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ , respectively. When the PPPP mechanism is moving, the origin $\textrm{O}_{3}$ of the end-effector is constrained to move along axis $\textrm{Z}_{3}$ , a straight line whose equation can be expressed as

(47) \begin{equation}\textrm{L}\!\left(d_{4}\right)=\left[\begin{array}{c} -a_{4}\\[5pt] -d_{4}\;\mathit{\sin }\; \alpha _{4}\\[5pt] -d_{4}\;\mathit{\cos }\; \alpha _{4} \end{array}\right]\end{equation}

Figure 5. Kinematic analysis of a spatial PPPP mechanism.

By setting the elements of the 3 × 1 matrix ${}^{0\!}{}_{\!\!3}{P}{}$ , as expressed in Eq. (14), equal to these of $\textrm{L}\!\left(d_{4}\right)$ in Eq. (47), joints variables $d_{1}$ , $d_{2}$ , and $d_{3}$ can be expressed in terms of $d_{4}$ , namely,

(48) \begin{align}d_{1}&=-\left(a_{2}+a_{1}c \theta _{c2}+a_{3}c \theta _{c3}+a_{4}c \theta _{c1}c \theta _{c2}+d_{4}c \alpha _{4}s \alpha _{1}s \theta _{c2}+d_{4}s \alpha _{4}c \theta _{c2}s \theta _{c1}-a_{4}c \alpha _{1}s \theta _{c1}s \theta _{c2}\right. \nonumber \\ & \left. \quad +d_{4}c \alpha _{1}s \alpha _{4}c \theta _{c1}s \theta _{c2}\right)/s\alpha _{1}s\theta _{c2}\end{align}

(49) \begin{align}d_{2} & =\left(c \alpha _{2}s \alpha _{1}c \theta _{c1}-s \alpha _{2}s \theta _{c1}s \theta _{c2}+c \alpha _{1}s \alpha _{2}c \theta _{c1}c \theta _{c2}\right)\left(a_{4}+a_{1}c \theta _{c1}+a_{2}c \theta _{c1}c \theta _{c2}+a_{3}c \theta _{c1}c \theta _{c2}c \theta _{c3} \right. \nonumber \\ & \left. \quad -\, a_{2}c \alpha _{1}s \theta _{c1}s \theta _{c2}-a_{3}c \alpha _{1}c \theta _{c3}s \theta _{c1}s \theta _{c2}-a_{3}c \alpha _{2}c \theta _{c1}s \theta _{c2}s \theta _{c3}+a_{3}s \alpha _{1}s \alpha _{2}s \theta _{c1}s \theta _{c3}\right. \nonumber \\ & \left. \quad-\, a_{3}c \alpha _{1}c \alpha _{2}c \theta _{c2}s \theta _{c1}s \theta _{c3}\right)/\left(s \alpha _{1}s \alpha _{2}s \theta _{c2}+\left(c \alpha _{2}s \alpha _{1}s \theta _{c1}+s \alpha _{2}c \theta _{c1}s \theta _{c2}+c \alpha _{1}s \alpha _{2}c \theta _{c2}s \theta _{c1}\right)\left(d_{4}s \alpha _{4} \right.\right. \nonumber \\ & \left.\left.\quad +\, a_{1s} \theta _{c1}+a_{3}c \theta _{c3}\!\left(c \theta _{c2}s \theta _{c1}+c \alpha _{1}c \theta _{c1}s \theta _{c2}\right)\hbox{--}a_{3}s \theta _{c3}\!\left(s \alpha _{1}s \alpha _{2}c \theta _{c1}+c \alpha _{2}s \theta _{c1}s \theta _{c2}\hbox{--}c \alpha _{1}c \alpha _{2}c \theta _{c1}c \theta _{c2}\right) \right.\right. \nonumber \\ & \left.\left.\quad +\, a_{2}c \theta _{c2}s \theta _{c1}+a_{2}c \alpha _{1}c \theta _{c1}s \theta _{c2}\right)\right)/s\alpha _{1}s\alpha _{2}s\theta _{c2}\end{align}

(50) \begin{equation}d_{3}=-\left(a_{1}+a_{2}c \theta _{c2}+a_{4}c \theta _{c1}+a_{3}c \theta _{c2}c \theta _{c3}+d_{4}s \alpha _{4}s \theta _{c1}-a_{3}c \alpha _{2}s \theta _{c2}s \theta _{c3}\right)/s\alpha _{2}s\theta _{2}\end{equation}

A numerical example is provided to illustrate using the ZXZ Euler angles convention to synthesize an over-constrained PPPP mechanism. First, five arbitrary angular parameters $\left\{\theta _{c2}, \theta _{c3}, \alpha _{1}, \alpha _{2}, \alpha _{3}\right\}=\left\{-70^{\circ},-60^{\circ},45^{\circ},55^{\circ},70^{\circ}\right\}$ and four-link parameters $\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}=\left\{130,140,160,160\right\}$ are assumed and substituted into Table III, which is tabulated in Table V.

Table V. ZXZ Euler angles convention for synthesized PPPP mechanism.

Next, substituting five arbitrary angular parameters $\left\{\theta _{c2}, \theta _{c3}, \alpha _{1}, \alpha _{2}, \alpha _{3}\right\}=\left\{-70^{\circ},-60^{\circ},45^{\circ},55^{\circ},70^{\circ}\right\}$ and $\theta _{c1}=0$ into rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ in Eq. (4) yields,

(51) \begin{equation}{}^{0\!}{}_{\!\!3}{R}{}\!\left(\theta _{c1}=0,\theta _{c2},\theta _{c3},\alpha _{1},\alpha _{2},\alpha _{3}\right)=\left[\begin{array}{c@{\quad}c@{\quad}c} a_{11} & a_{12} & a_{13}\\[5pt] a_{21} & a_{22} & a_{23}\\[5pt] a_{31} & a_{32} & a_{33} \end{array}\right]\end{equation}

Then, use to calculate Euler angle $\beta$ , which is

(52) \begin{equation}\beta =\pm \cos ^{-1} a_{33}=\pm 87.273^{\circ}\end{equation}

Here, the negative value of Euler angle $\beta$ is adopted to solve for Euler angle $\alpha$ and $\gamma$ using Eqs. (A4) and (A5), which are

(53) \begin{equation}\alpha =\text{atan}2(\!-a_{31},-a_{32})=97.2240^{\circ}\end{equation}

(54) \begin{equation}\gamma =\text{atan}2(\!-a_{13},a_{23})=55.7477^{\circ}\end{equation}

The aforementioned results, which are tabulated in Table VI, can be used to structure a closed-loop PPPP mechanism, as shown in Fig. 6.

Table VI. D-H table for synthesized PPPP mechanism.

Figure 6. Synthesized PPPP mechanism using presented methodology.

4. Synthesis of a spatial over-constrained five-bar mechanisms and their isomers PPPRR mechanism

4.1. PPPRR mechanism

Because the end-effector of the PPP manipulator can be constrained to move along an arbitrary space curve, in Section 3, its trajectory is purposely assigned to move along a straight line and connected to a P-joint single-arm manipulator to form a closed-loop PPPP mechanism. This synthesis concept is further extended by assigning the end-effector trajectory of the PPP manipulator to be circular, as shown in Fig. 7(a). Hence, the end-effector now can be connected to a two-arm RR manipulator, as shown in Fig. 7(b).

Figure 7. Two open-chain manipulators used to form A spatial PPPRR mechanism.

Recall that the orientation of the PPP manipulator is fixed. To form a closed-loop mechanism by combining a PPP manipulator with a RR manipulator, the end-effector of the RR manipulator must be constrained to move along a circle without changing its orientation. In reference to Fig. 7(b), when joint axes of two revolute joints are parallel and the second joint variable $\theta _{2}$ is a summation of adding the first joint variable $\theta _{1}$ with an arbitrary constant $\phi$ , endpoint $\textrm{O}_{2}$ of the RR manipulator traces a circular trajectory and the orientation of the end-effector coordinate system $\textrm{O}_{2}\textrm{X}_{2}\textrm{Y}_{2}\textrm{Z}_{2}$ is always identical to that of $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ . Hence, the RR manipulator can be combined with the PPP manipulator by sharing the same end-effector trajectory and orientation.

For a better illustration, coordinate systems of the PPP and RR manipulator defined according to the D-H notation are labeled in Fig. 8(a). $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ and $\textrm{O}_{0}\textrm{X}_{0}\textrm{Y}_{0}\textrm{Z}_{0}$ are the end-effector and initial coordinate systems, respectively, for the PPP manipulator. $\textrm{O}_{4}\textrm{X}_{4}\textrm{Y}_{4}\textrm{Z}_{4}$ and $\textrm{O}_{6}\textrm{X}_{6}\textrm{Y}_{6}\textrm{Z}_{6}$ are the end-effector and initial coordinate systems, respectively, for the RR manipulator. Notice that revolute joint $\textrm{Z}_{4}$ is parallel to revolute joint $\textrm{Z}_{5}$ . Now, let origin $\textrm{O}_{3}$ be coincident with origin $\textrm{O}_{4}$ and the orientation of coordinate system $\textrm{O}_{3}\textrm{X}_{3}\textrm{Y}_{3}\textrm{Z}_{3}$ remain unchanged relative to that of $\textrm{O}_{4}\textrm{X}_{4}\textrm{Y}_{4}\textrm{Z}_{4}$ . In doing so, a closed-loop five-bar PPPRR mechanism is formed, as shown in Fig. 8(b).

Figure 8. Spatial PPPRR mechanism by combining two open-chain manipulators.

The DH parameters for the PPPRR mechanism are presented in Table VII. Notice that all joint variables $\theta _{i}$ for prismatic pairs are specifically denoted by a constant $\theta _{ci}$ for distinction. Likewise, all joint offsets $d_{i}$ for revolute pairs are specifically denoted by a constant $d_{ci}$ for distinction. In addition, to form a closed-loop mechanism by combining a PPP and RR manipulator, two revolute joint axes of the RR manipulator must be parallel and the difference of their joint variables must be constant. Thus, two more constraints that must be met can be expressed as

Table VII. Link parameters for the closed-loop PPPRR mechanism.

(55) \begin{equation}\alpha _{4}=0\end{equation}

(56) \begin{equation}\theta _{5}=\phi -\theta _{4}\end{equation}

Next, by substituting the D-H parameters in Table VII into the transformation matrix of the PPPRR mechanism, a multiplication of five 3 × 3 rotation matrices can be extracted and expressed as

(57) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}^{3\!}{}_{\!\!4}{R}{}^{4\!}{}_{\!\!5}{R}{}=I_{3}\end{equation}

Table VIII. Link parameters involving ZXZ Euler angles convention for the closed-loop PPPRR mechanism.

Multiply Eq. (57) by inverse matrices ${}^{4\!}{}_{\!\!5}{R^{-1}}\,{}^{3\!}{}_{\!\!4}{R^{-1}}{}$ to obtain the following equation, namely,

(58) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}={}^{4\!}{}_{\!\!5}{R^{-1}}\,{}^{3\!}{}_{\!\!4}{R^{-1}}{}\end{equation}

Substituting Eqs. (55) and (56) into matrices ${}^{4\!}{}_{\!\!5}{R^{-1}}\,{}^{3\!}{}_{\!\!4}{R^{-1}}{}$ yields the following equation:

(59) \begin{equation}{}^{4\!}{}_{\!\!5}{R^{-1}}\,{}^{3\!}{}_{\!\!4}{R^{-1}}{}=\left[\begin{array}{c@{\quad}c@{\quad}c} \textrm{c} \phi & \textrm{s} \phi & 0\\[5pt] -\textrm{s} \phi \textrm{c} \alpha _{5} & \textrm{c} \phi \textrm{c} \alpha _{5} & \textrm{s} \alpha _{5}\\[5pt] \textrm{s} \phi \textrm{s} \alpha _{5} & -\textrm{c} \phi \textrm{s} \alpha _{5} & \textrm{c} \alpha _{5} \end{array}\right]\end{equation}

It can be found that ${}^{4\!}{}_{\!\!5}{R^{-1}}\,{}^{3\!}{}_{\!\!4}{R^{-1}}{}$ in Eq. (59) can be further decomposed into a multiplication of matrices $R_{x}\!\left(-\alpha _{5}\right)$ and $R_{z}\!\left(-\phi \right)$ , namely,

(60) \begin{equation}R_{x}\!\left(-\alpha _{5}\right)R_{z}\!\left(-\phi \right)=\left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0\\[5pt] 0 & \textrm{c} \alpha _{5} & \textrm{s} \alpha _{5}\\[5pt] 0 & -\textrm{s} \alpha _{5} & \textrm{c} \alpha _{5} \end{array}\right]\left[\begin{array}{c@{\quad}c@{\quad}c} \textrm{c} \phi & \textrm{s} \phi & 0\\[5pt] -\textrm{s} \phi & \textrm{c} \phi & 0\\[5pt] 0 & 0 & 1 \end{array}\right]\end{equation}

Therefore, from Eqs. (58), (59), and (60), we can find that

(61) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x}\!\left(-\alpha _{5}\right)R_{z}\!\left(-\phi \right)\end{equation}

Now, a similar manipulation of rotation matrices, as mentioned in the previous section, can be applied to properly fit Euler angle parameters into D-H parameters. Two types of Euler angles notations, ZXZ and XZX, are used for demonstration. First, we discuss the case using the ZXZ Euler angles. In reference to Eq. (40), this equation is deliberately rearranged as

(62) \begin{equation}{}^{0'\!\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3}{R}{}=R_{x}\!\left(\beta \right)R_{z}\!\left(\alpha \right)\end{equation}

where,

(63) \begin{equation}{}^{0'\!\!}{}_{\!\!1}{R}{}=R_{z}\!\left(\theta _{c1}-\gamma \right)R_{x}\!\left(\alpha _{1}\right)\end{equation}

By comparing Eq. (61) with (62), it can be found that the ZXZ Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ can be appropriately fit into the D-H parameters in Table VIII describing the closed-loop equation of the PPPRR mechanism.

Next, we discuss the case using the XZX Euler angles. In reference to Eq. (45), this equation is deliberately rearranged as

(64) \begin{equation}{}^{0\!}{}_{\!\!1}{R}{}^{1\!}{}_{\!\!2}{R}{}^{2\!}{}_{\!\!3'}{R}{}=R_{x}\!\left(\gamma \right)R_{z}\!\left(\beta \right)\end{equation}

where,

(65) \begin{equation}{}^{2\!}{}_{\!\!3'}{R}{}=R_{z}\!\left(\theta _{3}\right)R_{x}\!\left(\alpha _{3}-\alpha \right)\end{equation}

By comparing Eq. (61) with (64), it can be found that the XZX Euler angle parameters $\left(\alpha,\beta,\gamma \right)$ can be appropriately fit into the D-H parameters in Table IX describing the closed-loop equation of the PPPRR mechanism.

Table IX. Link parameters involving XZX Euler angles convention for the closed-loop PPPRR mechanism.

A numerical example is provided to illustrate using the ZXZ Euler angles convention to synthesize an over-constrained PPPRR mechanism. First, five arbitrary angular parameters $\left\{\theta _{c2}, \theta _{c3}, \alpha _{1}, \alpha _{2}, \alpha _{3}\right\}=\left\{-60^{\circ},-70^{\circ},50^{\circ},40^{\circ},55^{\circ}\right\}$ and seven link parameters $\left\{d_{c4},d_{c5},a_{1}, a_{2}, a_{3}, a_{4},a_{5}\right\}=\left\{85,55,180,140,130,100,120\right\}$ are assumed and substituted into Table III, which is tabulated in Table X.

Table X. ZXZ Euler angles convention for synthesized PPPRR mechanism.

Next, substituting five arbitrary angular parameters $\left\{\theta _{c2}, \theta _{c3}, \alpha _{1}, \alpha _{2}, \alpha _{3}\right\}=\left\{-60^{\circ},-70^{\circ},50^{\circ},40^{\circ},55^{\circ}\right\}$ and $\theta _{c1}=0$ into rotation matrix ${}^{0\!}{}_{\!\!3}{R}{}$ in Eq. (4) yields,

(66) \begin{equation}{}^{0\!}{}_{\!\!3}{R}{}\!\left(\theta _{c1}=0,\theta _{c2},\theta _{c3},\alpha _{1},\alpha _{2},\alpha _{3}\right)=\left[\begin{array}{c@{\quad}c@{\quad}c} a_{11} & a_{12} & a_{13}\\[5pt] a_{21} & a_{22} & a_{23}\\[5pt] a_{31} & a_{32} & a_{33} \end{array}\right]\end{equation}

Then, use Eq. (A3) to calculate Euler angle $\beta$ , which is

(67) \begin{equation}\beta =\pm \cos ^{-1} a_{33}=\pm 87.273^{\circ}\end{equation}

Here, the negative value of Euler angle $\beta$ is adopted to solve for Euler angle $\gamma$ using Eq. (A5), which are

(68) \begin{equation}\gamma =\text{atan}2 (\!-a_{13},a_{23})=63.0047^{\circ}\end{equation}

The aforementioned results, which are tabulated in Table XI, can be used to structure a closed-loop PPPRR mechanism, as shown in Fig. 9.

Table XI. D-H table for synthesized PPPRR mechanism.

Figure 9. Synthesized PPPRR mechanism using presented methodology.

4.2. Isomers of PPPRR mechanism

The cylindrical joint is a joint that provides one translational and one rotational of freedom along the same joint axis. Therefore, when two adjacent R joints and P joints are shown in Fig. 10(a) where the R joint axis $\textrm{Z}_{\textrm{R}}$ and P joint axis $\textrm{Z}_{\textrm{P}}$ are coaxial, the R joint and P joint will be configured to a cylindrical joint, as shown in Fig. 10(b).

Figure 10. Cylindrical joint formed by a revolute and a prismatic joint along the same axis.

Because of a cylindrical joint that can be formed by a revolute and a prismatic joint along the same axis, the synthesized PPPRR mechanism can be transformed into another over-constrained mechanism with a cylindrical joint. For example, set $a_{3}=0$ and $\alpha _{3}=0$ in Table VIII, a PPPRR mechanism based on the ZXZ Euler angles convention can be synthesized in Fig. 11(a), where P joint axis $\textrm{Z}_{2}$ and R joint axis $\textrm{Z}_{3}$ and are coaxial. Thus, a PPCR mechanism can be formed, in Fig. 11(b), by combining the P joint and R joint into a cylindrical joint. In addition, the D-H parameters describing the closed-loop equation of the PPCR mechanism can be listed in Table XII.

Table XII. Link parameters involving ZXZ Euler angles convention for the closed-loop PPCR mechanism.

Figure 11. Synthesized PPPRR mechanism and its isomer.

5. Mobile assemblies derived from PPPRR and PPCR mechanisms

In this section, synthesized PPPRR and PPCR mechanisms can be further used as source modules to form other mobile assemblies. These assemblies may provide a much large archive allowing the engineers to choose desired motions and trajectories. To synthesize new mechanisms configurationally, link parameters of PPPRR and PPCR mechanisms are correspondingly adjusted such that two source modules can be assembled as another over-constrained mechanism.

5.1. RRPRRP mechanism

Figure 12 shows an assembly configuration of the two PPPRR mechanisms and their schematics. In Fig. 12(a), $\textrm{Z}_{\textrm{i}}$ is the direction of axis i, and $[a_{i}, \alpha _{i}]$ represent the common normals and the skew angles between joint axes $\textrm{Z}_{\textrm{i}}$ and $\textrm{Z}_{\textrm{i}+1}$ . The joint variable of a revolute joint of axis $\textrm{Z}_{\textrm{i}}$ is denoted by $\theta _{i}$ , while the joint variable of a prismatic joint is denoted by $\theta _{ci}$ , a constant value. When the PPPRR mechanism is configured based on the presented method using the ZXZ Euler angles convention, angular parameters $\theta _{c2}$ , $\theta _{c3}$ , $\alpha _{1}$ , $\alpha _{2}$ , and $\alpha _{3}$ can be freely assigned. Hence, $\left[a_{1}, \alpha _{1}\right]$ of the first PPPRR mechanism can be set equal to $[a_{2}, \alpha _{2}]$ of the second PPPRR mechanism. In addition, $\theta _{c2}$ of the first PPPRR mechanism can be set equal to $\theta _{c2'}$ of the second PPPRR mechanism. In doing so, $\textrm{Z}_{1}$ , $\textrm{Z}_{2}$ , and $\textrm{Z}_{3}$ are, respectively, parallel to $\textrm{Z}_{1'}$ , $\textrm{Z}_{2'}$ , and $\textrm{Z}_{3'}$ , as schemed in Fig. 12(b).

Figure 12. Assembly of the two PPPRR mechanisms and their schematics.

Next, combining these PPPRR mechanisms by aligning $[a_{1}, \alpha _{1}]$ with $[a_{1'}, \alpha _{1'}]$ and $[a_{2}, \alpha _{2}]$ with $[a_{2'}, \alpha _{2'}]$ leads to coincidences of $\textrm{Z}_{1}$ and $\textrm{Z}_{1'}$ , $\textrm{Z}_{2}$ and $\textrm{Z}_{2'}$ , and $\textrm{Z}_{3}$ and $\textrm{Z}_{3'}$ , as shown in Fig. 12(c).

Now, joint variable $d_{2}$ of the first PPPRR mechanism is constrained to be joint variable $d_{2'}$ of the second PPPRR mechanism. Therefore, two PPPRR mechanisms with independent motion are synchronized. Afterward, collinearly overlapped links $[a_{1}, \alpha _{1}]$ and $[a_{1'}, \alpha _{1'}]$ as well as $[a_{2}, \alpha _{2}]$ with $\left[a_{2'}, \alpha _{2'}\right]$ now can be removed to form a single closed-loop six-bar RRPRRP mechanism in Fig. 13(a). Since the prismatic joints along axis $\textrm{Z}_{3}$ and are coaxial, they can be equivalently replaced by one prismatic joint adjacent to link $[a_{3}, \alpha _{3}]$ and $[a_{3'}, \alpha _{3'}]$ . Besides, the twisted angle of the substitute prismatic joint is $\theta _{c3'}-\theta _{c3}$ . Likewise, the prismatic joints along axis $\textrm{Z}_{1}$ and $\textrm{Z}_{1'}$ can be equivalently replaced by one prismatic joint adjacent to link $[a_{1}, \alpha _{1}]$ and $[a_{1'}, \alpha _{1'}]$ . Besides, the twisted angle of the substitute prismatic joint is $\theta _{c1}-\theta _{c1'}$ . The rest of $[a_{i}, \alpha _{i}]$ for the synthesized RRPRRP mechanism are all identical to these of original PPPRR mechanisms. In addition, due to the constraint $d_{2}=d_{2'}$ , the movement of the synthesized RRPRRP mechanism is the same as the two PPPRR mechanisms run synchronously with equal input variables, as simulated in Fig. 13(b).

Figure 13. Synthesized RRPRRP mechanism and its motion animation.

5.2. RRPRC mechanism

Figure 14(a) shows an assembly configuration of the PPPRR and PPCR mechanism. As mentioned, five angular parameters $\theta _{c2}$ , $\theta _{c3}$ , $\alpha _{1}$ , $\alpha _{2}$ , and $\alpha _{3}$ of the PPPRR mechanism can be freely assigned. However, for the PPCR mechanism, only three angular parameters $\theta _{c2}$ , $\alpha _{1}$ , and $\alpha _{2}$ can be freely assigned. Hence, $\left[a_{1}, \alpha _{1}\right]$ and $[a_{2}, \alpha _{2}]$ of the PPPRR mechanism can be set equal to $\left[a_{1\textrm{'}}, \alpha _{1\textrm{'}}\right]$ and $[a_{2\textrm{'}}, \alpha _{2\textrm{'}}]$ of the PPCR mechanism. In addition, $\theta _{c2}$ of the PPPRR mechanism can be set equal to $\theta _{c2'}$ of the PPCR mechanism. In doing so, $\textrm{Z}_{1}$ , $\textrm{Z}_{2}$ , and $\textrm{Z}_{3}$ are, respectively, parallel to $\textrm{Z}_{1'}$ , $\textrm{Z}_{2'}$ , and $\textrm{Z}_{3'}$ , as schemed in Fig. 14(b).

Figure 14. Assembly of the PPPRR and PPCR mechanisms and their schematics.

Next, aligning $[a_{1}, \alpha _{1}]$ with $[a_{1'}, \alpha _{1'}]$ and $[a_{2}, \alpha _{2}]$ with $[a_{2'}, \alpha _{2'}]$ yield coincidences of $\textrm{Z}_{1}$ and $\textrm{Z}_{1'}$ , $\textrm{Z}_{2}$ and $\textrm{Z}_{2'}$ , and $\textrm{Z}_{3}$ and $\textrm{Z}_{3'}$ , as shown in Fig. 14(c). Now, joint variable $d_{2}$ of the PPPRR mechanism is constrained to be joint variable $d_{2'}$ of the PPCR mechanism. Therefore, two mechanisms with independent motion are synchronized. Afterward, removing collinearly overlapped links $[a_{1}, \alpha _{1}]$ and $[a_{1'}, \alpha _{1'}]$ as well as $[a_{2}, \alpha _{2}]$ with $\left[a_{2'}, \alpha _{2'}\right]$ yields a single closed-loop five bar RRPRC loop in Fig. 15(a). In the combined loop, the prismatic joint along axis $\textrm{Z}_{3}$ and the cylindrical joint along $\textrm{Z}_{3'}$ are coaxial, they can be equivalently replaced by one cylindrical joint adjacent to link $[a_{3}, \alpha _{3}]$ and $[a_{3'}, \alpha _{3'}]$ . Besides, the twisted angle of the substitute prismatic joint is $\theta _{c3'}-\theta _{c3}$ . Likewise, the prismatic joints along axis, $\textrm{Z}_{1}$ and $\textrm{Z}_{1'}$ can be equivalently replaced by one prismatic joint adjacent to link $[a_{1}, \alpha _{1}]$ and $[a_{1'}, \alpha _{1'}]$ . Besides, the twisted angle of the substitute prismatic joint is $\theta _{c1}-\theta _{c1'}$ . The rest of $[a_{i}, \alpha _{i}]$ for the synthesized RRPRC mechanism are all identical to these of original PPPRR and PPCR mechanisms. The synthesized RRPRC mechanism should be immobile since its mobility is zero based on the prediction of Chebychev–Grübler–Kutzbach criterion. However, its mobility inherit from the synchronized motion of input variables $d_{2}=d_{2'}$ driving the combined PPPRR and PPCR mechanism, as simulated in Fig. 15(b).

Figure 15. Synthesized RRPRC mechanism and its motion animation.

5.3. RCRC mechanism

By the same token, two PPCR mechanisms are assembled as shown in Fig. 16(a) and schematized in Fig. 16(b). Three angular parameters $\theta _{c2}, \alpha _{1}$ , and $\alpha _{2}$ for these two PPCR mechanisms can be correspondingly assigned such that $\left[a_{1}, \alpha _{1}\right]$ and $[a_{2}, \alpha _{2}]$ of the first PPCR mechanism are equal to $\left[a_{1\textrm{'}}, \alpha _{1\textrm{'}}\right]$ and $[a_{2\textrm{'}}, \alpha _{2\textrm{'}}]$ of the second PPCR mechanism. In addition, $\theta _{c2}$ of the first PPCR mechanism can be set equal to $-\theta _{c2'}$ of the second PPCR mechanism such that $\textrm{Z}_{1}$ , $\textrm{Z}_{2}$ , and $\textrm{Z}_{3}$ are, respectively, parallel to $\textrm{Z}_{3'}$ , $\textrm{Z}_{2'}$ , and $\textrm{Z}_{1'}$ , as schemed in Fig. 16(b).

Figure 16. Assembly of two PPCR mechanisms and their schematics.

Next, two PPCR mechanisms can be combined by aligning $[a_{1}, \alpha _{1}]$ with $[a_{1\textrm{'}}, \alpha _{1\textrm{'}}]$ and $[a_{2}, \alpha _{2}]$ with $[a_{2\textrm{'}}, \alpha _{2\textrm{'}}]$ so as to achieve coincidences of $\textrm{Z}_{3}$ and $\textrm{Z}_{1\textrm{'}}$ , $\textrm{Z}_{2}$ and $\textrm{Z}_{2\textrm{'}}$ , and $\textrm{Z}_{1}$ and $\textrm{Z}_{3\textrm{'}}$ , as shown in Fig. 16(c). Now, two PPCR mechanisms move synchronously with equal input variables due to the constraint $d_{2}=d_{2'}$ . Afterward, a single closed-loop four-bar RCRC loop in Fig. 17(a) can be formed by removing collinearly overlapped links $[a_{1}, \alpha _{1}]$ and $[a_{2'}, \alpha _{2'}]$ as well as $[a_{2}, \alpha _{2}]$ with $\left[a_{1'}, \alpha _{1'}\right]$ . In the combined loop, the prismatic joint along axis $\textrm{Z}_{1}$ and the cylindrical joint along $\textrm{Z}_{3'}$ can be equivalently substituted by a cylindrical joint adjacent to link $[a_{1}, \alpha _{1}]$ and $[a_{3'}, \alpha _{3'}]$ . Besides, the twisted angle of the substitute cylindrical joint is $\theta _{c1}+\theta _{3'}$ . Likewise, the prismatic joints along axis $\textrm{Z}_{3}$ and $\textrm{Z}_{1'}$ can be equivalently substituted by a cylindrical joint adjacent to link $[a_{3}, \alpha _{3}]$ and $[a_{1'}, \alpha _{1'}]$ . Besides, the twisted angle of the substitute prismatic joint is $\theta _{3}+\theta _{c1'}$ . The rest of $[a_{i}, \alpha _{i}]$ for the synthesized RCRC mechanism are all identical to these of original PPCR mechanisms. The synthesized RCRC loop is also an over-constrained mechanism, as simulated in Fig. 17(b). An intriguing research that relates the RCRC linkage and its ressemble linkage, the RCCC linkage, is the passive coupling method proposed by Dimentberg [Reference Dimentberg23]. He suggests synthesizing the RCRC linkage by locking the degree of freedom of one revolute joint of the RCCC linkage. This approach may provide an extended investigation of the ongoing discussion but is not within the scope of this paper.

Figure 17. Synthesized RCRC mechanism and its motion animation.

6. Closing remarks

This paper presents a synthesis method of spatial over-constrained mechanisms. First, the kinematics of a three-link PPP manipulator is investigated based on the Denavit–Hartenberg notation. Next, the end-effector of the PPP manipulator is constrained to trace a straight trajectory such that its end-effector can be combined with that of a P-joint single-arm manipulator. By doing so, these two open-chain manipulators can be assembled to form a spatial closed-loop PPPP mechanism. By the same token, the end-effector of the PPP manipulator can be constrained to trace a circular trajectory such that the end-effectors of the PPP and RR manipulators can be combined to form a PPPRR mechanism. In addition, the synthesized PPPRR mechanism can be transformed into an over-constrained PPCR mechanism by aligning the revolute and prismatic joints of the original PPPRR mechanism. The DH parameters of those synthesized mechanisms are defined by incorporating them with the Euler angles notations. Solving the Euler angle parameters is advantageous to avoid excessive algebraic assumptions. At last, PPPRR and PPCR mechanisms are further adopted as source modules to concatenate other mobile assemblies. These assemblies can be correspondingly configured into over-constrained RRPRRP, RRPRC, and RCRC mechanisms by the common link-pair (CLP) method by Wohlhart [Reference Wohlhart24]. Compared with deriving geometrical constraints from closed-loop equation analytically, the modular method presented in this research is more intuitive to find the over-constrained mechanisms.

In addition, our work aims to provide a synthesis method of spatial over-constrained mechanisms based on the kinematics of a specific serial manipulator. Some concepts are similar to the previous studies by Li and Angeles’s work [Reference Li and Angeles25, Reference Li and Angeles26]. They are devoted to promote a 3-CCC parallel manipulator by refining its kinematic, singularity, workspace, and dexterity analyses. To the goal for both of our research, kinematic analyses that describe the input/output function among mating parts are required. One noticeable difference between our methods is the mathematical foundations adopted. We apply D-H notations to relate link dimensions and joint variables for the serial manipulator and derive the position and orientation of the manipulator end-effector based on the homogeneous transformation matrix representation. Li and Angeles’s work establishes the constraint-screw systems for every CCC limb [Reference Li and Angeles25] and solves a quartic resolvent polynomial to yield the orientation of the moving platform. Although the mathematical formulations are distinct, both of our works attempt to simplify the analyses by decoupling the rotational and translational motion of the end-effector. We verify that the serial PPP manipulator has an unchanged orientation and unlimited workspace by separately examining the elements governing rotation and translation. Hence, we can concentrate on aligning the orientations of the end-effector coordinate system of two manipulators without paying attention to coinciding the origin of the end-effector coordinate system. Li and Angeles reveal that the orientation of 3-CCC PKM can be individually solved without considering the linear displacement of the cylindrical joints. Besides, they found that the singularity of the 3-CCC PKM only results from the orientation of the moving platform. Both of our works indicate that the complexity of a kinematic problem may be reduced by considering the rotation and translation of the links individually. This feature, as suggested by Li and Angeles [Reference Li and Angeles25, Reference Li and Angeles26], may provide an intriguing enlightenment on relevant research in the future.