2. Design approach

In this section, the KJM that can represent linkage mechanism is introduced in detail. The basic module of the KJM can be represented as

(1) \begin{equation} \left[\begin{array}{l@{\quad}l@{\quad}l} m_{11} & m_{12} & m_{13}\\[4pt] m_{21} & m_{22} & m_{23}\\[4pt] m_{31} & m_{32} & m_{33} \end{array}\right] \end{equation}

where $m_{ij}$ (i, j = 1,2,3) in the ith row and jth column element of this square matrix denotes the kinematic joint type within the parallel mechanism.

This matrix can be divided into three categories. The diagonal elements m _ii of Eq. (1) can display one kind of kinematic joint. Three elements above or below the main diagonal can indicate additional two kinds of kinematic pairs, respectively. In such a way, this matrix has the ability to imply linkage mechanisms under some conditions. The parallel mechanism with three kinds of joints can be expressed by the KJM, as demonstrated below

(2)

where j ₁, j ₂ and j ₃ indicate three sorts of kinematic joints, respectively. The vertical dash line is employed to distinguish adjacent square matrices.

In the KJM, the written sequence is from left to right. In each group of kinematic pair, the corresponding joint position will be filled once this position in the left square matrix is already filled, to achieve a compact format.

The widely used prismatic (P) joint, revolute (R) joint and parallelogram (Pa) joint are taken as examples. The global coordinate system attached to the fixed platform is utilized as a reference. These joints are further constrained in this case. Each prismatic joint axis can be parallel to X, Y or Z direction, denoted as P _X , P _Y and P _Z , respectively. The rotational joint axis can parallel to X, Y or Z direction separately, represented, respectively, as R _X , R _Y and R _Z . Every Pa joint can be placed in planes that are parallel to XOY, XOZ or YOZ plane, and, respectively, shown as P _XY , P _XZ and P _YZ . Thereafter, each category of kinematic joint is further divided into three cases, which reveals that each component position is unique. According to definition of Eq. (1), P _X , P _Y , P _Z , R _X , R _Y , R _Z , P _XY , P _XZ , P _YZ can be placed in the positions of m ₁₁, m ₂₂, m ₃₃, m ₃₂, m ₃₁, m ₂₁, m ₁₂, m ₁₃ and m _23, respectively. Each kinematic joint has its own position in matrix, which helps to distinguish two kinematic joint matrices. The KJM under this circumstance can be formulated as

(3) \begin{equation} \left[\begin{array}{l@{\quad}l@{\quad}l} \mathrm{P}_{X} & \mathrm{P}_{XY} & \mathrm{P}_{XZ}\\[4pt] \mathrm{R}_{Z} & \mathrm{P}_{Y} & \mathrm{P}_{YZ}\\[4pt] \mathrm{R}_{Y} & \mathrm{R}_{X} & \mathrm{P}_{Z} \end{array}\right] \end{equation}

In this kind of parallel architectures, each kinematic chain has the identical joint type and number. Take the consideration of the coupling movement of the parallelogram pair, only one Pa joint is permitted in one kinematic limb. The analysis is further limited to the fully parallel mechanism, where the number of chains is same as the DOF of the whole manipulator. In addition, the number of joints of each chain is equal to the DOF of the parallel mechanism, and only the joint connected to the fixed platform is equipped with an actuator. For a m ( $2\leq m\leq 6$ ) DOFs parallel mechanism that qualifies the abovementioned requirements, the maximum size of the corresponding matrix is 3-by-3m.

Some feasible parallel structure examples denoted by the proposed matrices are introduced accordingly. The original matrix for a 2-DOF parallel mechanism is given as

(4)

Since Eq. (4) has many null elements and no revolute joint, it can be further simplified as a 2-by-4 matrix after deleting the third-row and third-column of each 3-by-3 matrix module. The simplification format is shown below,

(5)

Both Eqs. (4) and (5) can represent the 2-PPa parallel mechanism, as illustrated in Fig. 1. This symmetrical mechanism belongs to planar translational parallel architecture. The active prismatic joints of two branches are in the same direction, and the parallelogram joints of both branches move in the same plane.

Figure 1. 2-DOF parallel mechanism.

The matrix for a 3-DOF parallel structure is constructed as

(6)

Eq. (6) can describe a spatial 3-PPaP parallel manipulator with three identical chains, as depicted in Fig. 2. In every chain, two prismatic joints are perpendicular to each other, and the active sliding joint is in the plane containing of the Pa kinematic pair.

Figure 2. 3-DOF parallel mechanism.

Figure 3. 4-DOF parallel mechanism.

The matrix for a 4-DOF parallel mechanism is seen in

(7)

Eq. (7) can define a 4-PPaPR parallel manipulator as demonstrated in Fig. 3. This mechanism has four chains, which can be divided into two sets with same joints axes configurations, limb one and limb two, limb three and limb four. The two prismatic joints of each chain are perpendicular to each other. In the limb one or limb two, the parallelogram joint plane is perpendicular to the axis of the passive sliding joint, while in the limb three or limb four, the parallelogram module plane is perpendicular to the axis of the active prismatic joint. All these revolute joints axes are in the same direction.

The matrix for a 5-DOF parallel structure is generated as

(8)

One possible parallel mechanism corresponding to the Eq. (8) is seen in Fig. 4. It is a 5-PPaPRR parallel manipulator. In each kinematic chain, the axes of two sliding joints are perpendicular and the axes of two revolute joints are orthogonal. In each branch, the driving prismatic joint axis moves in the plane containing of the parallelogram unit. Except the kinematic limb five, all the other limbs are identical.

Figure 4. 5-DOF parallel mechanism.

The matrix for a kind of 6-DOF parallel mechanism is listed below

(9)

Eq. (9) may indicate a 6-PPaPRRR parallel manipulator, as illustrated in Fig. 5. Each limb contains of two orthogonal sliding joints, three orthogonal revolute joints and one parallelogram module. The active sliding direction of each kinematic chains is different. The axes of the driving joints in limb one/four, limb two/five/six and limb three are parallel to X, Z and Y directions, respectively.

Figure 5. 6-DOF parallel mechanism.

To better illustrate the function of the KJM, several parallel structures with the corresponding kinematic joint matrices are provided in Appendix.

3. Features of kinematic joint matrix

The size of the KJM is small. For the m degree-of-freedom fully parallel mechanism cases mentioned in Section 2, the largest original matrix size is 3-by-3m, although the number of all linkages is m ²+m + 2. The 2T mechanism matrix has a simplified version due to no rotational joints. The size is 2-by-4.

The KJM can reveal the joints categories, directions and numbers utilized in the proposed parallel architecture. Since many parallel mechanisms are composed of several identical limbs and there are no connection rods among different limbs, the possible parallel structure(s) can be obtained according to the corresponding KJM. The relation from the KJM to the parallel mechanisms is one-to-many mapping, i.e. the KJM shown in the following form

(10)

Eq. (10) can denote both P _X P _XY /P _Y P _XY structure (seen in Fig. 6(a)) and P _XY P _XY /P _X P _Y structure (seen in Fig. 6(b)).

Figure 6. Schematic diagrams of 2-DOF parallel mechanisms. (a) P_XP_XY/P_YP_XY type and (b) P_XYP_XY/P_XP_Y type.

Furthermore, the first branch of Fig. 6(a) can be P _X P _XY or P _XY P _X . This scenario also demonstrates that the specific joint sequence in each kinematic chain is not obtained.

The abovementioned case can be avoided if the parallel mechanism chains are properly predefined, e.g. only one Pa joint is permitted in one kinematic chain and the prismatic joint is connected to the fixed platform. However, the one-to-many mapping relationship might happen when the matrix is complicated. The matrix form shown in the following expression:

(11)

Eq. (11) can represent a 2P _X P _XY P _Z R _X /P _Y P _YZ P _X R _X /P _Z P _XZ P _Y R _X mechanism (shown in Fig. 7(a)) or P _X P _XY P _Z R _X /P _Y P _XY P _Z R _X /P _Z P _YZ P _X R _X /P _X P _XZ P _Y R _X mechanism (shown in Fig. 7(b)).

Figure 7. Prototypes of 4-DOF parallel structures. (a) The first type and (b) the second type.

4. Discrimination of kinematic joint matrix

There are two reasons indicating the KJM is insufficient to directly distinguish any two parallel structures. The first factor is the one-to-many mapping relations with parallel mechanisms. The second factor is that various kinematic joint matrices can be obtained if the parallel manipulator is placed in different coordinate systems (e.g. the P _X joint in one coordinate system might be P _Y or P _Z joint in another reference system).

However, it is meaningful to propose an approach for discriminating different kinds of matrices, since any two parallel manipulators expressed by different categories of kinematic joint matrices are distinct. It also means the second factor can be resolved. It is evident that two kinematic joint matrices are different if the total numbers for prismatic joints or revolute joints or Pa joints are not the same. This question is further constrained to distinguish the matrices with identical numbers for prismatic joints, rotary joints and Pa joints, respectively. For a given KJM representing parallel mechanisms, the total numbers of P _X , P _Y , P _Z , R _X , R _Y , R _Z , P _XY , P _XZ and P _YZ joints are expressed separately as N ₁₁, N ₂₂, N ₃₃, N ₃₂, N ₃₁, N ₂₁, N ₁₂, N ₁₃ and N ₂₃. The i, j, k are integers ranging from 1 to 3. The following numbers are predefined

(12) \begin{equation} \left\{\begin{array}{l@{\quad}l} a1=\min \!\left(i,j\right) & a2=\max\!\left(i,j\right)\\[4pt] a3=\min\!\left(i,k\right) & a4=\max\!\left(i,k\right)\\[4pt] a5=\min\!\left(j,k\right) & a6=\max\!\left(j,k\right) \end{array}\right. \end{equation}

A unique block pattern is introduced, as shown in Fig. 8(a). In this block pattern, the first column is for N _ii . The second column is filled by N ₃₂, N ₃₁ and N ₂₁. The last column is for N ₁₂, N ₁₃ and N ₂₃. An example is illustrated in Fig. 8(b). Take the N ₁₁, N ₃₂, N ₁₂ and N ₁₃ as examples, the physical relevance (inner feature) among them is the common subscript X in P _X , R _X , P _XY and P _XZ . The external feature is based on the subscripts of these filled elements N _ij in this pattern. If the first column is selected, each element of the second column can be decided by the elements of the first column and the other rows. For example, the subscripts a5 and a6 of N _a6a5 are, respectively, the minimal and maximum values between j and k (subscripts of N _jj , N _kk ). Each element of the third column can be concluded by the directly connected two columns (the two elements of the first column), i.e. N _a1a2 is directly connected by N _a6a5 and N _a4a3. Then the subscripts a1 and a2 are solved, respectively, as the minimal and maximum elements of the subscripts of N _ii , N _jj .

Figure 8. Block patterns for kinematic joint matrix. (a) Block pattern and (b) block pattern of a sample. The red block is filled with the sum for one kind of joint. The double arrow line implies two blocks are pertinent.

Before filling elements into the block pattern, comparing N _ii , N _jj , $N_{kk}$ (i, j, k = 1,2,3.) and the following expression can be obtained

(13) \begin{equation} N_{ii}\geq N_{jj}\geq N_{kk} \end{equation}

The following scenarios can be found

(14) \begin{align}\left(\mathrm{I}\right) & \ N_{ii}\gt N_{jj}\gt N_{kk}\nonumber\\[4pt] \left(\mathrm{II}\right) & \ N_{ii}\gt N_{jj}=N_{kk}\ \& \ N_{a1a2}\gt N_{a3a4}\nonumber\\[4pt] \left(\mathrm{III}\right) & \ N_{ii}\gt N_{jj}=N_{kk}\ \& \ N_{a1a2}=N_{a3a4}\nonumber\\[4pt] \left(\mathrm{IV}\right) & \ N_{ii}\gt N_{jj}=N_{kk}\ \& \ N_{a1a2}\lt N_{a3a4}\\[4pt] \left(\mathrm{V}\right) & \ N_{ii}=N_{jj}\gt N_{kk}\ \& \ N_{a3a4}\gt N_{a5a6}\nonumber\\[4pt] \left(\mathrm{VI}\right) & \ N_{ii}=N_{jj}\gt N_{kk}\ \& \ N_{a3a4}=N_{a5a6}\nonumber\\[4pt] \left(\mathrm{VII}\right) & \ N_{ii}=N_{jj}\gt N_{kk}\ \& \ N_{a3a4}\lt N_{a5a6}\nonumber \end{align}

In cases (I), (II), (V), the first column of the block pattern is N _ii , N _jj and N _kk . The final pattern is the same as in Fig. 8(a). For case (III), the first column of the pattern can be N _ii , N _jj and N _kk or N _ii , N _kk and N _jj . The whole patterns are the same. In case (IV), the first column of the pattern is listed as N _ii , N _kk and N _jj . In case (VI), the first column of the block pattern can be N _ii , N _jj and N _kk or N _jj , N _ii and N _kk . The contents are identical in any selection. In case (VII), the first column of the block pattern is listed as N _jj , N _ii and N _kk .

The other situations happen when N ₁₁ = N ₂₂ = N ₃₃. Before classifying the remaining cases, comparing N _a1a2, N _a3a4 and N _a5a6 yields

(15) \begin{equation} N_{b1b2}\geq N_{b3b4}\geq N_{b5b6} \end{equation}

where N _b1b2 and N _b5b6 are, respectively, the largest and smallest among N _a1a2, N _a3a4 and N _a5a6. N _b3b4 is the rest of them.

The remaining scenarios are concluded as

(16) \begin{align} \left(\text{VIII}\right) & \ N_{ii}=N_{jj}=N_{kk}\ \&\ N_{b1b2}\gt N_{b3b4}\gt N_{b5b6}\nonumber\\[4pt] \left(\mathrm{IX}\right) & \ N_{ii}=N_{jj}=N_{kk}\ \&\ N_{b1b2}\gt N_{b3b4}=N_{b5b6}\nonumber\\[4pt] \left(\mathrm{X}\right) & \ N_{ii}=N_{jj}=N_{kk}\ \&\ N_{b1b2}=N_{b3b4}\gt N_{b5b6}\\[4pt] \left(\mathrm{XI}\right) & \ N_{ii}=N_{jj}=N_{kk}\ \&\ N_{b1b2}=N_{b3b4}=N_{b5b6}\nonumber \end{align}

In case (VIII), the first column of the block pattern is N _ii , N _jj and N _kk . In case (IX), the first column of the pattern can be either N _ii , N _jj and N _kk or N _jj , N _ii and N _kk . In case (X), the first column of the pattern can be either N _ii , N _jj and N _kk or N _ii , N _kk and N _jj . In case (XI), the first column of the pattern can be listed in any sequence to finally derive a unique pattern.

All the feasible situations for the block pattern are summarized in Eqs. (14) and (16). As long as two kinematic joint matrices can create identical block patterns, these two matrices belong to the same kind of matrix and can express the same parallel mechanism(s). One sample is provided below. Two kinematic joint matrices are listed as

(17)

(18)

The parameters for these two aforementioned matrices are computed separately as

(19) \begin{equation} N_{11}=6,N_{12}=0,N_{13}=2,N_{21}=6,N_{22}=4,N_{23}=4,N_{31}=6,N_{32}=6,N_{33}=2 \end{equation}

(20) \begin{equation} N_{11}=4,N_{12}=0,N_{13}=4,N_{21}=6,N_{22}=6,N_{23}=2,N_{31}=6,N_{32}=6,N_{33}=2 \end{equation}

Following the above method, these two scenarios can be classified into the case (I). Their final block patterns can both be written as seen in Fig. 9, which indicates they belong to the same kind of KJM. One feasible 6-DOF parallel structure is illustrated in Fig. 10. According to Fig. 10, Eq. (17) is based on the X1-Y1-Z1 coordinate system, while Eq. (18) is generated in accordance with the X2-Y2-Z2 coordinate system. These two coordinate systems can be transformed by rotation operations.

Figure 9. The block pattern for two similar matrices.

Figure 10. Prototype of a 6-DOF parallel structure.

Therefore, this block pattern is helpful for the enumerations of every possible KJM with predefined conditions.

5. Comparisons between KJM and topology matrix

5.1. Illustration of examples

The characteristics of the KJM can be further revealed when compared with the well-known topology matrix proposed by Yan and Kuo [Reference Kuo and Yan29, Reference Kuo, Dai and Yan30]. The topology matrix M_T of a mechanism is a N -by-N square matrix (N denotes the total number of linkages). The ith diagonal elements show the types of the ith linkage. If the ith rod and jth rod (i<j) are connected by a kinematic joint, the ith row and jth column element will be filled by the type of the kinematic pair, and jth row and ith column element will be provided the letter that illustrates this adjacent point. Zero will be assigned to the remaining elements.

Generally, different kinds of topology matrices denote the corresponding mechanisms own different topological structures, which is convenient to distinguish linkage mechanisms. The KJM could not discriminate straightly two distinct linkage mechanisms, as mentioned in Section 4. To further compare the characteristics of the KJM and topology matrix on parallel structures, two similar mechanisms are provided in this section. Figs. 11(a) and (b) illustrate the architectures with detailed numbers of the first and second parallel structures. The linkages 1, 2, 8 and 11 are separately denoted as K _F , K _P1, K _P2, and K _P3. The ith (i = 3−7) linkage is represented as K _L(i-2). The jth (j = 9−11) linkage is represented as K _L(j-3). The nth (n = 12−14) linkage is represented as K _L(n-4). A _ij , B _ij and C _ij stand for the kinematic pairs between adjacent rods. A ₁₆, B ₁₆ and C ₁₇ are the virtual midpoints of A ₁₂ A ₁₄, B ₁₂ B ₁₄ and C ₁₂ C _14, respectively.

Figure 11. 3D models of two 2-DOF parallel manipulators. (a) First structure and (b) second structure.

The KJM of Fig. 11(a) is provided as

(21)

The corresponding KJM of the mechanism in Fig. 11(b) is expressed as

(22)

The topology matrices of the Figs. 11(a) and (b) mechanisms are the same and are generated as

(23) \begin{equation} \mathbf{M}_{\mathrm{T}1}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} K_{F} & \mathrm{J}_{\mathrm{P}} & 0 & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{P}} & 0 & 0 & \mathrm{J}_{\mathrm{P}} & 0 & 0 & 0\\[4pt] A_{15} & K_{P1} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{R}} & 0 & 0\\[4pt] 0 & A_{11} & K_{L1} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & A_{12} & K_{L2} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{R}} & 0 & 0\\[4pt] 0 & 0 & 0 & A_{17} & K_{L3} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & C_{17} & K_{L4} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{R}} & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & C_{12} & K_{L5} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] C_{15} & 0 & 0 & 0 & 0 & 0 & C_{11} & K_{P2} & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{R}} & 0\\[4pt] 0 & 0 & 0 & 0 & B_{17} & 0 & 0 & 0 & K_{L6} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0 & \mathrm{J}_{\mathrm{R}}\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & B_{12} & K_{L7} & \mathrm{J}_{\mathrm{R}} & 0 & 0 & 0\\[4pt] B_{15} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & B_{11} & K_{P3} & 0 & 0 & \mathrm{J}_{\mathrm{R}}\\[4pt] 0 & A_{13} & 0 & A_{14} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{L8} & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & C_{14} & 0 & C_{13} & 0 & 0 & 0 & 0 & K_{L9} & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & B_{14} & 0 & B_{13} & 0 & 0 & K_{L10} \end{array}\right] \end{equation}

where K _F , K _Pi and K _Li denote the fixed platform, ith sliding linkage and ith kinematic link, respectively. J_P and J_R mean the prismatic joint and rotational joint separately.

The distinction methodology in Section 4 is employed. The corresponding parameters for Eqs. (21) and (22) are separately described as

(24) \begin{equation} N_{11}=N_{12}=N_{21}=3,N_{13}=N_{22}=N_{23}=N_{31}=N_{32}=N_{33}=0 \end{equation}

(25) \begin{equation} N_{11}=2,N_{12}=N_{21}=3,N_{22}=1,N_{13}=N_{23}=N_{31}=N_{32}=N_{33}=0 \end{equation}

In accordance with Eqs. (14), (24) and (25), Eq. (21) belongs to case (II) while Eq. (22) is classified into case (I). The relevant block patterns are depicted in Fig. 12. The patterns in Figs. 12(a) and (b) are not the same. Thereby, Eqs. (21) and (22) belong to distinct kinematic joint matrices, and the mechanisms in Figs. 11(a) and (b) are different.

Figure 12. The diagrams for two block patterns. (a) The first pattern and (b) the second pattern.

However, the topology matrix shown in Eq. (23) is capable to demonstrate the connection status between any two linkages, but it cannot distinguish these two parallel mechanisms. There are too many zero elements since the connection feature of parallel mechanisms is simple. The joint directions that are important in parallel structures cannot be revealed by the topology matrix. The matrix is more complicated, and the matrix size is larger. For this class of parallel mechanisms, the size of the topology matrix size is N-by-N (N= m ²+ m + 2). The topology matrix is insufficient to denote some parallel structures with special linkage dimensions or joint configurations (such as the axes of revolute joints are parallel, or perpendicular or intersecting lines), i.e. parallelogram joint, Sarrus mechanism, Bennett mechanism, Myard mechanism, Goldberg mechanism, Bricard mechanism, spherical 5R mechanism, Agile eye wrist-spherical 3-RRR parallel robot.

5.2. Improvements for topology matrix

Inspired by the KJM, some supplementary regulations are suggested for the topology matrix to be employed for parallel mechanisms.

First rule : The commonly used mechanisms (e.g. (R-R)₂, (U-U)₂, (U-U)₃, (S-S)₂, (S-S)₃) in parallel structures can be treated as one module to generate the corresponding topology matrix. Fig. 11(a) is taken as an example. The parallelogram unit of each kinematic chain is regarded as one generalized linkage (linkages 2, 4, 5 in Fig. 13). The modified parallel mechanism is shown in Fig. 13, and the corresponding topology matrix is expressed as

Figure 13. The parallel mechanism with generalized linkages.

(26) \begin{equation} \mathbf{M}_{\mathrm{T}2}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} K_{F} & \mathrm{J}_{\mathrm{P}} & 0 & \mathrm{J}_{\mathrm{P}} & \mathrm{J}_{\mathrm{P}}\\[4pt] A_{15} & K_{P1} & \mathrm{J}_{R} & 0 & 0\\[4pt] 0 & A_{17} & K_{L1} & \mathrm{J}_{R} & \mathrm{J}_{R}\\[4pt] C_{15} & 0 & C_{17} & K_{P2} & 0\\[4pt] B_{15} & 0 & B_{17} & 0 & K_{P3} \end{array}\right] \end{equation}

Compared to Fig. 11(a) and Eq. (23), this mechanism is simplified, and the matrix size is greatly reduced. Furthermore, since the parallelogram module is predefined, the feature of the whole mechanism is evidently expressed and simple to be understood.

Second rule : The kinematic joint with special configurations may be represented in topology matrix with detailed information. Various types of kinematic prismatic and rotational joints with parallel, orthogonal or intersecting axes [Reference Fang and Tsai31–Reference Kong and Gosselin35] have been defined and classified to name different parallel architectures, but these concepts have not yet been employed into the topology matrix to demonstrate more detailed construction information. Take the rotational joint as an example, the classification concept in the cited references [Reference Fang and Tsai31–Reference Kong and Gosselin35] will be utilized with a more specific definition, and the axes directions will be indicated in some cases. Four kinds of revolute joints will be defined as J_RX , J_RY , J_RZ and J_RO1, to denote the axes are parallel to X axis, Y axis, Z axis or intersect at the same point O ₁. Two similar parallel mechanisms equipped with only revolute joints are illustrated in Figs. 14(a) and (b). The corresponding topology matrices for Figs. 14(a) and (b) are separately expressed as

Figure 14. 3-RRR parallel structures. (a) Planar parallel mechanism and (b) spherical parallel mechanism.

(27) \begin{equation} \mathbf{M}_{\mathrm{T}3}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} K_{F} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & \mathrm{J}_{\mathrm{RZ}} & 0 & \mathrm{J}_{\mathrm{RZ}}\\[4pt] A_{21} & K_{L1} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & A_{22} & K_{L2} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & A_{23} & K_{L3} & \mathrm{J}_{\mathrm{RZ}} & 0 & \mathrm{J}_{\mathrm{RZ}} & 0\\[4pt] 0 & 0 & 0 & C_{23} & K_{L4} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0\\[4pt] C_{21} & 0 & 0 & 0 & C_{22} & K_{L5} & 0 & 0\\[4pt] 0 & 0 & 0 & B_{23} & 0 & 0 & K_{L6} & \mathrm{J}_{\mathrm{RZ}}\\[4pt] B_{21} & 0 & 0 & 0 & 0 & 0 & B_{22} & K_{L7} \end{array}\right] \end{equation}

(28) \begin{equation} \mathbf{M}_{\mathrm{T}4}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} K_{F} & \mathrm{J}_{\mathrm{RO}1} & 0 & 0 & 0 & \mathrm{J}_{\mathrm{RO}1} & 0 & \mathrm{J}_{\mathrm{RO}1}\\[4pt] A_{31} & K_{L1} & \mathrm{J}_{\mathrm{RO}1} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & A_{32} & K_{L2} & \mathrm{J}_{\mathrm{RO}1} & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & A_{33} & K_{L3} & \mathrm{J}_{\mathrm{RO}1} & 0 & \mathrm{J}_{\mathrm{RO}1} & 0\\[4pt] 0 & 0 & 0 & C_{33} & K_{L4} & \mathrm{J}_{\mathrm{RO}1} & 0 & 0\\[4pt] C_{31} & 0 & 0 & 0 & C_{32} & K_{L5} & 0 & 0\\[4pt] 0 & 0 & 0 & B_{33} & 0 & 0 & K_{L6} & \mathrm{J}_{\mathrm{RO}1}\\[4pt] B_{31} & 0 & 0 & 0 & 0 & 0 & B_{32} & K_{L7} \end{array}\right] \end{equation}

With the above definitions of revolute joints, the planar and spherical 3-RRR parallel architectures can be indicated and distinguished by different topology matrices. Their particular features will be demonstrated too.

Third rule : A unified naming sequence is essential for topology matrix to state and distinguish parallel mechanisms. A general naming convention is proposed as below.

1. Step one: Define the fixed platform as the first rod.

2. Step two: Identify the shortest loop (minimum number of linkages) containing the mobile platform and the fixed platform. The starting linkage connected to the fixed platform will be chosen from the shortest chain. Then naming the following linkages until the last rod of this closed loop. The first rule will be employed to simplify the procedure.

3. Step three: In the remaining kinematic limbs containing of the moving platform, beginning with the shortest kinematic branch and naming from linkage attached to the moving platform.

4. Step four: Repeat step three until at least one linkage of each kinematic chain has been counted.

5. Step five: Naming the remaining subchains to deal with the coupling chains in parallel mechanisms. Starting from the chain that has the smallest number in its corresponding loop. The initial linkage of this chain will be the rod that is connected to the linkage with the previously defined smallest number.

The priority for these steps is to begin with any chain if there are multiple identical chains. If there are many shortest loops/chains, starting from the loop/chain with the least DOF. One planar parallel structure using this naming sequence is employed as an example, as seen in Fig. 15. Its corresponding topology matrix is listed as

Figure 15. Schematic diagram of a planar mechanism.

(29) \begin{equation} \mathbf{M}_{\mathrm{T}5}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} K_{F} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & 0 & \mathrm{J}_{\mathrm{RZ}} & 0 & 0\\[4pt] A_{41} & K_{L1} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & A_{41} & K_{L2} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & \mathrm{J}_{\mathrm{RZ}} & 0\\[4pt] 0 & 0 & A_{43} & K_{L3} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & B_{44} & K_{L4} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & B_{43} & K_{L5} & \mathrm{J}_{\mathrm{RZ}} & 0 & 0\\[4pt] B_{41} & 0 & 0 & 0 & 0 & B_{42} & K_{L6} & 0 & \mathrm{J}_{\mathrm{RZ}}\\[4pt] 0 & 0 & A_{43} & 0 & 0 & 0 & 0 & K_{L7} & \mathrm{J}_{\mathrm{RZ}}\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & B_{42} & C_{41} & K_{L8} \end{array}\right] \end{equation}

In analyzing the topology matrices of parallel mechanisms, it is okay to use other kinds of naming conventions as long as the same sequence is utilized to maintain consistency. These concepts can also be modified and expanded to be employed in other motion transmission systems (e.g. linkage mechanism and gear transmission system).