3.1. Design of a 3-DOF spatial double-Delta mechanism with RCM

As mentioned above, a key step to generate new RCM mechanisms is to find a suitable mechanism that can drive a parallelogram to execute pure translation. The Delta mechanism proposed by Clavel [Reference Clavel35] is the most famous parallel mechanism that can generate purely translational motion. This feature makes the mechanism suitable as the basis to generate spatial RCM mechanisms.

Based on the mechanism constructing method proposed in [Reference Li, Zhang, Xing, Liu and Wang3], and the principle proposed in Section 2, a kind of spatial mechanism based on the Delta mechanism is constructed in this paper, as shown in Fig. 3. From the figure, it can be seen that this spatial mechanism contains two layers of Delta-like linkages. In this design, to make the motions of the two Delta linkages coupled with each other, the parallelograms in the two Delta linkages share the same proximal linkage. Three platforms, i.e., movable platform I, movable platform II, and static platform are used in the mechanism. To fulfill the mobility constraint, the End-link is mounted on the movable platform I through a universal joint and connected with the movable platform II via a universal joint together with a cylindrical joint. The actuators used for driving the mechanism can be set stationary on the static platform, as shown in Fig. 3. The End-link is the end-effector of the proposed double-Delta mechanism with remote center of motion. Different with the common Delta mechanism in the literature, in this design, the three proximal links and the three groups of distal links in each Delta linkage have different lengths, and the dimensions of the three platforms, the static platform and the two movable platforms are identical.

Figure 3. A spatial 3-DOF double-Delta mechanism with RCM.

There are n = 36 links and j = 42 joints (including 39 revolute joints, 2 universal joints and 1 cylindrical joint) in the proposed mechanism. Through the test in the model, it is found that once any joint constraint is removed, the remote center of motion cannot be realized, so there is no redundant constraint. Therefore, according to the Grülber–Kutzbuch formula [Reference Dai, Huang and Lipkin38], the mobility of the mechanism is

(1) \begin{equation} m=\lambda \left(n-j-1\right)+\sum _{i=1}^{j}f_{i}=6\left(36-42-1\right)+39+2+2+2=3 \end{equation}

where m denotes the mobility of the mechanism and λ is the dimension of the space in which the mechanism is presented, λ = 6 for spatial mechanism and λ = 3 for planar and spherical mechanisms.

3.2. Conditions for remote center of motion

To realize remote center of motion for the End-link of the proposed mechanism, there are two conditions need to be satisfied: first, lengths of all the proximal links and lengths of all the distal links in each Delta linkage are same, respectively; and second, the corresponding distal links in the two Delta linkages are parallel with each other. Based on these two conditions, kinematic analysis of the mechanism will be investigated in Section 4 to verify whether the mechanism can execute the RCM motion.

The schematic model of the mechanism is shown in Fig. 4, from the first condition, it is expected that the two movable platforms, simplified as points C₁ and C₂ in the figure (it is noted that the static platform is also simplified as a point A), will have pure translational motion due to the constraints of the parallelograms.

Figure 4. The schematic model of the mechanism under the first condition.

From Fig. 4, it can be seen that line C₁E₁ is the normal line of the plane formed by points B₁₁, B_12, and B₁₃ (denoted as ΠB₁₁B₁₂B₁₃), and line C₂E₂ is the normal line of the plane formed by points B₂₁, B_22, and B₂₃ (denoted as ΠB₂₁B₂₂B₂₃). Then, based on the conditions that $C_{1}E_{1}\bot \unicode{x03C0} B_{11}B_{12}B_{13}$ and $C_{1}B_{11}=C_{1}B_{12}=C_{1}B_{13}$ , it has

(2) \begin{equation} E_{1}B_{11}=E_{1}B_{12}=E_{1}B_{13} \end{equation}

thus point $E_{1}$ is the circumcenter of triangle $\unicode{x03B4} B_{11}B_{12}B_{13}$ .

Similarly, with the conditions that $C_{2}E_{2}\bot \unicode{x03C0} B_{21}B_{22}B_{23}$ and $C_{2}B_{21}=C_{2}B_{22}=C_{2}B_{23}$ , it yields

(3) \begin{equation} E_{2}B_{21}=E_{2}B_{22}=E_{2}B_{23} \end{equation}

hence, point $E_{2}$ is the circumcenter of triangle $\unicode{x03B4} B_{21}B_{22}B_{23}$ .

Similarly, we can prove that line AE ₁ is perpendicular to plane $\unicode{x03C0} B_{11}B_{12}B_{13}$ with point E ₁ and line AE ₂ is orthogonal with plane $\unicode{x03C0} B_{21}B_{22}B_{23}$ with point E ₂. With these results, it can be seen that the two lines, i.e., $AE_{1}$ and $C_{1}E_{1}$ , are both perpendicular to the plane $\unicode{x03C0} B_{11}B_{12}B_{13}$ , which means lines $AE_{1}$ and $C_{1}E_{1}$ are collinear. Similarly, it can be concluded that lines $AE_{2}$ and $C_{2}E_{2}$ are collinear.

Further, considering that the lengths of the corresponding links in the two Delta linkages are the same, there exist the following relations:

(4) \begin{equation} \frac{\overline{AB_{21}}}{\overline{AB_{11}}}=\frac{\overline{AB_{22}}}{\overline{AB_{12}}}\text{ and }\frac{\overline{AB_{21}}}{\overline{AB_{11}}}=\frac{\overline{AB_{23}}}{\overline{AB_{13}}} \end{equation}

which leads to the relations that $B_{11}B_{12}//B_{21}B_{22}$ and $B_{11}B_{13}//B_{21}B_{23}$ , and thus $\unicode{x03C0} B_{11}B_{12}B_{13}//\unicode{x03C0} B_{21}B_{22}B_{23}$ . Moreover, with the relations that $AC_{1}\bot \unicode{x03C0} B_{11}B_{12}B_{13}$ and $AC_{2}\bot \unicode{x03C0} B_{21}B_{22}B_{23}$ , it can be found that lines $AC_{1}$ and $AC_{2}$ are collinear.

Therefore, from the above derivation, it can be found that the three points $A, C_{1}$ , and $C_{2}$ in Fig. 4 are collinear. Further, since points A, C₁, and C₂ in Fig. 4 stand for the corresponding static and movable platforms in Fig. 3, there exist the relations that the three points C₁₁, C₂₁, and A₁ in one branch of the mechanism (see in Fig. 3) are always collinear. Similarly, points C₁₂, C_22, and A₂ and points C₁₃, C_23, and A₃ in the other two branches are also collinear, respectively.

Considering the above relations, in the mechanism, three generalized double parallelograms can be formed denoting as P ₁: $C_{12}A_{2}A_{3}C_{13}\,\& \, C_{22}A_{2}A_{3}C_{23}$ , P ₂: $C_{11}A_{1}A_{2}C_{12}\,\& \,C_{21}A_{1}A_{2}C_{22}$ , and P ₃: $C_{11}A_{1}A_{3}C_{13}\,\& \,C_{21}A_{1}A_{3}C_{23}$ as shown in Fig. 5(a). Then referring to Fig. 5(a), the End-link that passes through points C₁, C₂, and A is parallel with lines A ₁ C ₁₁, A ₂ C ₁₂, and A ₃ C ₁₃ and always has point A (which is the circumcenter of triangle $\Delta A_{1}A_{2}A_{3}$ ) as is remote center of motion. The remote center of motion is visually located in the center of the static platform, but does not need to be supported by physical links or joints. The operation space can be reserved by expanding the size of the static platform, so that human surgeons can clearly enter the incision. Based on the above constraints, the size of platform C1 can be kept unchanged and the size of platform C2 and base a can be expanded in proportion to ensure that the volume is not excessively expanded. It is important to point out that the End-link can also be located outside of the mechanism, as illustrated in Fig. 5(b). In this configuration, two links of equal length, i.e., Link I and Link II, are fixed on the movable platform I and movable platform II, respectively. The joint connect Link I and the End-link is a universal joint and a universal joint and a cylindrical joint are adopted to connect Link II and the End-link. For practical application, if the mechanism is to be used as the end effector of a surgical robot, for the case in Fig. 5(b), the mechanism body can be set away from the patient, making the mechanism better adapted to the surgical environment.

Figure 5. The mechanism can execute RCM motion under the mentioned constraint.

The second condition is that the corresponding distal links in the two Delta linkages are parallel with each other. This condition will secure that the three points A, C_1, and C₂ are collinear. To provide this, we simplify the kinematic model of the mechanism into a schematic model as shown in Fig. 6 and assume that three points A, C_1, and C₂ are not collinear. Based on the condition, it has lines B ₂₁ C ₂ and B ₁₁ C ₁ are parallel with each other, and lines AC ₂ and B ₁₁ C ₁ are in the same plane. Then point $C^{\prime}_{1}$ is used to denote the intersection of the extension of the line $AC_{2}$ and the line $B_{11}C_{1}$ under the above assumption.

Figure 6. The schematic model of the mechanism under the second condition.

The mathematical description of the studied condition is

(5) \begin{equation} \left\{\begin{array}{l@{\quad}l} \dfrac{AB_{21}}{AB_{11}}=\dfrac{AB_{22}}{AB_{12}}=\dfrac{AB_{23}}{AB_{13}}=k & \left(0\lt k\lt 1\right)\\ \\[-6pt] B_{2j}C_{2}//B_{1j}C_{1}& j=1,2,3 \end{array}\right. \end{equation}

Based on the intercept theorem of parallel lines, we have

(6) \begin{equation} \frac{AB_{21}}{AB_{11}}=\frac{AC_{2}}{AC^{\prime}_{1}} \end{equation}

Further, considering that the line $B_{22}C_{2}$ is parallel with the line $B_{12}C_{1}$ , these two lines can form a plane, i.e., the plane $\unicode{x03C0} C_{1}B_{12}B_{22}C_{2}$ . Obviously, point $A$ and point $C_{2}$ locate in that plane. It can be seen that line $B_{12}C_{1}$ and line $AC_{2}$ are the two intersecting lines within that plane. Let us assume that the intersecting point of the two lines is point $C^{\prime\prime}_{1}$ , as shown in figure. Then, based on the same theorem, it yields

(7) \begin{equation} \frac{AB_{22}}{AB_{12}}=\frac{AC_{2}}{AC^{\prime\prime}_{1}} \end{equation}

According to Eq. (5) throughout Eq. (7), it is straightforward to find that

(8) \begin{equation} \frac{AC_{2}}{AC^{\prime}_{1}}=\frac{AC_{2}}{AC^{\prime\prime}_{1}} \end{equation}

Since both of the points $C^{\prime}_{1}$ and $C^{\prime\prime}_{1}$ are on the extension of line $AC_{2}$ , Eq. (8) implies that points $C^{\prime}_{1}$ and $C^{\prime\prime}_{1}$ are coincide. This result indicates that there is a unique intersecting point among three lines $AC_{2}, B_{12}C_{1}$ , and $B_{11}C_{1}$ . Since $C_{1}$ is the intersecting point of lines $B_{12}C_{1}$ and $B_{11}C_{1}$ , the above-mentioned unique intersecting point is point $C_{1}$ . Similarly, it can be found that point $C_{1}$ is also the intersecting point of lines $AC_{2}$ and $B_{13}C_{1}$ . From above analysis, it can be proved that under the second condition, the three points $A, C_{1}$ , and $C_{2}$ are collinear.

Further, it should be pointed out that the RCM feature for the proposed mechanism is independent with the inputs of the mechanism as long as the proposed mechanism satisfies the two conditions discussed above. In addition, RCM mechanisms with similarly topology but different geometric constraint as the one illustrated in Fig. 6 can also be obtained. The famous external positioning mechanism named Dion is proposed by Ricardo et al. for robotic single port surgery [Reference Clavel35] is a typical application example of this kind of RCM mechanism.

4.1. Forward kinematic analysis

To carry out the forward kinematic analysis, the mathematical model of the mechanism is established, as shown in Fig. 7. In this model, the coordinate system $A-xyz$ is the reference coordinate system. The x-axis coincides with the projection of the link $AB_{11}$ on the horizontal plane, z-axis points upward, and the y-axis complies with the right-hand rule. The values of the dimensional parameters are defined as follows: The length of link $AB_{1j} \left(j=1,2,3\right)$ is $l_{i}$ , the length of link $AB_{2i}$ is $kl_{i} \left(0\lt k\leq 1\right)$ , the length of link $B_{2i}C_{2}$ is $l_{2i}$ , the length of link $B_{1i}C_{1}$ is $l_{1i}$ , and the length of the End-link is $l$ .

Figure 7. The kinematic model of the RCM mechanism based on two Delta linkages.

Since the End-link, usually is the surgical tool in a MIS robot, has a fixed value of length, the position of at the end of the End-link, i.e., point $O'$ , can be expressed by the position of points $C_{1}$ and point $C_{2}$ as

(9) \begin{equation} \boldsymbol{{r}}_{O'}=\boldsymbol{{r}}_{C1}-l\left(\frac{\boldsymbol{{r}}_{C1}-\boldsymbol{{r}}_{C2}}{\left\| \boldsymbol{{r}}_{C1}-\boldsymbol{{r}}_{C2}\right\| }\right) \end{equation}

where $\boldsymbol{{r}}_{O'}$ is the position vector of reference point $O', \boldsymbol{{r}}_{C1}$ , and $\boldsymbol{{r}}_{C2}$ denote the position vectors of points $C_{1}$ and $C_{2}$ , respectively.

Figure 8. The mathematic model of the tetrahedron.

Then the kinematic analysis of the mechanism is equivalent to find the final positions of point $C_{1}$ and point $C_{2}$ under a given set of input joint angles, i.e., $\theta _{1}, \theta _{2}$ , and $\theta _{3}$ of the three proximal links, as shown in Fig. 8. To carry out the calculation, two auxiliary lines are employed, they are lines $C_{1}E_{1}$ and $C_{2}E_{2}$ (see Fig. 7), which are perpendicular to plane $\unicode{x03C0} B_{11}B_{12}B_{13}$ and plane $\unicode{x03C0} B_{21}B_{22}B_{23}$ , respectively. Then the position vectors of points $C_{1}$ and $C_{2}$ can be expressed as

(10) \begin{equation} \boldsymbol{{r}}_{C1}=\boldsymbol{{r}}_{E1}+\overline{{{E_{1}C_{1}}}} \end{equation}

(11) \begin{equation} \boldsymbol{{r}}_{C2}=\boldsymbol{{r}}_{E2}+\overline{{{E_{2}C_{2}}}} \end{equation}

where $\boldsymbol{{r}}_{E1}$ and $\boldsymbol{{r}}_{E2}$ are the position vectors of points $E_{1}$ and $E_{2}$ expressed in the reference coordinate system $A-xyz$ . The coordinates of point $B_{ij} \left(i=1,2;\ j=1,2,3\right)$ can be determined by the input joint angles as

(12) \begin{equation} \boldsymbol{{r}}_{B1j}=l_{j}\mathrm{Rot}\left(z\mathit{,}\beta _{\left(j-1\right)}\right)\mathrm{Rot}\left(y\mathit{,}-\theta _{j}\right)\boldsymbol{{e}}_{1} \end{equation}

(13) \begin{equation} \boldsymbol{{r}}_{B2j}=k_{j}l_{j}\mathrm{Rot}\left(z\mathit{,}\beta _{\left(j-1\right)}\right)\mathrm{Rot}\left(y\mathit{,}-\theta _{j}\right)\boldsymbol{{e}}_{1} \end{equation}

where $\beta _{\left(j-1\right)}$ is the dimensional parameter, it is the angle of the projections of link $AB_{ij}$ and link $AB_{i\left(j+1\right)}$ on the horizontal plane, and $\beta _{0}=0, \beta _{1}\neq \beta _{2}$ and $\beta _{0}+\beta _{1}+\beta _{2}=2\pi$ . In addition,

$\mathrm{Rot}\left(z\mathit{,}\beta _{\left(j-1\right)}\right)=\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \beta _{\left(j-1\right)} & -\sin \beta _{\left(j-1\right)} & 0\\ \sin \beta _{\left(j-1\right)} & \cos \beta _{\left(j-1\right)} & 0\\ 0 & 0 & 1 \end{array}\right], \mathrm{Rot}\left(y\mathit{,}-\theta _{j}\right)=\left[\begin{array}{c@{\quad}c@{\quad}c} \cos \left(-\theta _{j}\right) & 0 & \sin \left(-\theta _{j}\right)\\ 0 & 1 & 0\\ -\sin \left(-\theta _{j}\right) & 0 & \cos \left(-\theta _{j}\right) \end{array}\right]$ , and $\boldsymbol{{e}}_{1}=\left[\begin{array}{l} 1\\ 0\\ 0 \end{array}\right]$ .

So far, all the lengths of the links in tetrahedrons $C_{1}-B_{11}B_{12}B_{13}, C_{2}-B_{21}B_{22}B_{23}, A-B_{11}B_{12}B_{13}$ , and $A-B_{21}B_{22}B_{23}$ are known. Then, $\overline{{{E_{1}C_{1}}}}$ and ${\overline{{E_{2}C_{2}}}}$ can be calculated in $C_{1}-B_{11}B_{12}B_{13}$ and $C_{2}-B_{21}B_{22}B_{23}$ , respectively, while ${\overline{{AE_{1}}}}$ and ${\overline{{AE_{2}}}}$ can be calculated in $A-B_{11}B_{12}B_{13}$ and $A-B_{21}B_{22}B_{23}$ , respectively. Let us solve the expression of ${\overline{{E_{1}C_{1}}}}$ and ${\overline{{E_{2}C_{2}}}}$ first. Their directional vectors can be obtained by

(14) \begin{equation} \boldsymbol{{n}}_{{\overline{{E_{1}C_{1}}}}}=\frac{{\overline{{B_{11}B_{12}}}}\times \overline{{{B_{11}B_{13}}}}}{\left\| \overline{{{B_{11}B_{12}}}}\times \overline{{{B_{11}B_{13}}}}\right\| } \end{equation}

(15) \begin{equation} \boldsymbol{{n}}_{{\overline{{E_{2}C_{2}}}}}=\frac{\overline{{{B_{21}B_{22}}}}\times \overline{{{B_{21}B_{23}}}}}{\left\| \overline{{{B_{21}B_{22}}}}\times \overline{{{B_{21}B_{23}}}}\right\| } \end{equation}

where $\overline{{{B_{11}B_{12}}}}=\boldsymbol{{r}}_{B12}-\boldsymbol{{r}}_{B11}, \overline{{{B_{11}B_{13}}}}=\boldsymbol{{r}}_{B13}-\boldsymbol{{r}}_{B11}, \overline{{{B_{21}B_{22}}}}=\boldsymbol{{r}}_{B22}-\boldsymbol{{r}}_{B21}$ , and $\overline{{{B_{21}B_{23}}}}=\boldsymbol{{r}}_{B23}-\boldsymbol{{r}}_{B21}$ .

To solve the length of $\overline{{{E_{i}C_{i}}}}$ which is perpendicular to the plane $\unicode{x03C0} B_{i1}B_{i2}B_{i3}$ , the mathematic model of the tetrahedron is established, as shown in Fig. 8. The thin-dotted lines in the model are the auxiliary lines and both of the two lines $B_{i2}G_{i1}$ and $B_{i2}G_{i2}$ are perpendicular with the line $B_{i1}B_{i2}$ . Line $G_{i1}B_{i2}$ locates in the plane $\unicode{x03C0} C_{i}\hspace{0pt}B_{i1}B_{i2}$ , while the line $B_{i2}G_{i1}$ locates in the plane $\unicode{x03C0} B_{i1}B_{i2}B_{i3}$ . According to cosine theorem, $\gamma _{i1}$ which is the angle formed by line $C_{i}B_{i1}$ and the line $B_{i1}B_{i2}$ can be calculated, it is

(16) \begin{equation} \gamma _{i1}=\arccos \frac{\left\| {\overline{{B_{i1}C_{i}}}}\right\| ^{2}+\left\| {\overline{{B_{i1}B_{i2}}}}\right\| ^{2}-\left\| {\overline{{B_{i2}C_{i}}}}\right\| ^{2}}{2\left\| {\overline{{B_{i1}C_{i}}}}\right\| \left\| {\overline{{B_{i1}B_{i2}}}}\right\| } \end{equation}

Then the length of $\overline{{{B_{i2}G_{i1}}}}$ can be solved by

(17) \begin{equation} \left\| \overline{{{B_{i2}G_{i1}}}}\right\| =\left\| \overline{{{B_{i1}B_{i2}}}}\right\| \tan \gamma _{i1} \end{equation}

By using same method, the length of ${\overline{{B_{i2}G_{i2}}}}$ can be obtained

(18) \begin{equation} \left\| \overline{{{B_{i2}G_{i2}}}}\right\| =\left\| \overline{{{B_{i1}B_{i2}}}}\right\| \tan \gamma _{i2} \end{equation}

where $\gamma _{i2}=\arccos \frac{\left\| \overline{{{B_{i1}B_{i2}}}}\right\| ^{2}+\left\| \overline{{{B_{i1}B_{i3}}}}\right\| ^{2}-\left\| \overline{{{B_{i2}B_{i3}}}}\right\| ^{2}}{2\left\| \overline{{{B_{i1}B_{i2}}}}\right\| \left\| \overline{{{B_{i1}B_{i3}}}}\right\| }$ .

The length of $\left\| \overline{{{G_{i2}G_{i1}}}}\right\|$ can be calculated by

(19) \begin{equation} \left\| \overline{{{G_{i2}G_{i1}}}}\right\| =\sqrt{\left\| \overline{{{B_{i1}G_{i1}}}}\right\| ^{2}+\left\| \overline{{{B_{i1}G_{i2}}}}\right\| ^{2}-2\left\| \overline{{{B_{i1}G_{i1}}}}\right\| \cdot \left\| \overline{{{B_{i1}G_{i2}}}}\right\| \cos \gamma _{i3}} \end{equation}

where $\cos \gamma _{i3}=\frac{\left\| \overline{{{B_{i1}C_{i}}}}\right\| ^{2}+\left\| \overline{{{B_{i1}B_{i3}}}}\right\| ^{2}-\left\| \overline{{{B_{i3}C_{i}}}}\right\| ^{2}}{2\left\| \overline{{{B_{i1}C_{i}}}}\right\| \left\| \overline{{{B_{i1}B_{i3}}}}\right\| }, \left\| \overline{{{B_{i1}G_{i1}}}}\right\| =\frac{\left\| \overline{{{B_{i1}B_{i2}}}}\right\| }{\cos \gamma _{i1}}$ , and $\left\| \overline{{{B_{i1}G_{i2}}}}\right\| =\frac{\left\| \overline{{{B_{i1}B_{i2}}}}\right\| }{\cos \gamma _{i2}}$ .

Then the angle form by plane $C_{i}\hspace{0pt}B_{i1}B_{i2}$ and plane $B_{i1}B_{i2}B_{i3}$ can be calculated by

(20) \begin{equation} \phi _{i}=\arccos \frac{\left\| \overline{{{B_{i2}G_{i1}}}}\right\| ^{2}+\left\| \overline{{{B_{i2}G_{i2}}}}\right\| ^{2}-\left\| \overline{{{G_{i2}G_{i1}}}}\right\| ^{2}}{2\left\| \overline{{{B_{i2}G_{i1}}}}\right\| \left\| \overline{{{B_{i2}G_{i2}}}}\right\| } \end{equation}

Then the length of $\overline{{{E_{i}C_{i}}}}$ can be calculated consequently, it is

(21) \begin{equation} \left\| \overline{{{E_{i}C_{i}}}}\right\| =\left\| \overline{{{C_{i}F_{i}}}}\right\| \sin \phi _{i} \end{equation}

where $C_{i}F_{i}\bot B_{i1}B_{i2}$ and $\left\| \overline{{{C_{i}F_{i}}}}\right\| =\left\| \overline{{{B_{i1}C_{i}}}}\right\| \sin \gamma _{i1}$ . Till now, the directional vector and the length of $\overline{{{E_{i}C_{i}}}}$ are solved.

The position vector of point $E_{i}$ can be calculated in tetrahedrons $A-B_{i1}B_{i2}B_{i3}$

(22) \begin{equation} \boldsymbol{{r}}_{Ei}=\boldsymbol{{r}}_{Bi1}+\overline{{{B_{i1}F_{i}}}}+\overline{{{F_{i}E_{i}}}} \end{equation}

where $\overline{{{B_{i1}F_{i}}}}=\left\| \overline{{{B_{i1}C_{i}}}}\right\| \cos \gamma _{i1}\frac{\overline{{{B_{i1}B_{i2}}}}}{\left\| \overline{{{B_{i1}B_{i2}}}}\right\| }, \overline{{{F_{i}E_{i}}}}=\left(\left\| \overline{{{C_{i}F_{i}}}}\right\| \cos \phi _{i}\right)\frac{\overline{{{E_{i}C_{i}}}}\times \overline{{{B_{i1}B_{i2}}}}}{\left\| \overline{{{E_{i}C_{i}}}}\times \overline{{{B_{i1}B_{i2}}}}\right\| }$ .

Since all the unknowns are expressed by the dimensional parameters and the input joint angles of the mechanism, the positions of points $C_{1}$ and $C_{2}$ can be derived by Eqs. (10), (11), (14), (15), (21), and (22). And consequently, the forward position analysis result, the final position of the reference point $O'$ at the end of the End-link, can be derived by Eq. (9). It can be seen that the unique solution of the forward kinematic problem can be obtained based on the employed geometrical analysis method. One advantage of this process is that it can avoid the choosing procedure in the analytical technique, which may yield multiple solutions. Another advantage is that it can also avoid the time-consuming iteration calculation in the forward kinematic analysis based on numerical techniques.

4.2. Numerical simulation and verification

To verify the validity of the forward kinematic calculation process, a numerical example is employed in this section. The RCM mechanism is generated based on the second condition, which means the corresponding distal links in the two Delta mechanisms are parallel to each other, the forward kinematics of which is more complicated than the first condition. The dimensional parameters in this example are listed in Table I.

Table I. The dimensional parameters of the research object.

Table II. The dimensional parameters of the research object.

The verification process is that a set of reference positions whose values and the corresponding input joint angles were measured from SolidWorks^® software were given first. After that, the forward kinematic analysis was carried out to solve the positions of the reference point based on the joint angles. Then comparison between the given reference positions and the positions obtained through forward kinematic calculation was carried out to check the accuracy of the procedure. The comparison results are listed in Table II. It can be seen that the forward position calculation method is effective. For the third calculation, however, the error between the given position and the calculated position is large. The reason is that the absolute values of two angular positions are small, which makes the relative measured errors of these two angles large, and consequently results in a relative larger calculation error. The configurations of the mechanism with the 3 given angular positions are shown in Fig. 9.

Figure 9. Different configurations of the mechanism with the given positions in Table II.

4.3. Workspace analysis

In the process of motion, the boundary of workspace is determined by the limit position of rotating joint. Number the outer circle joints in sequence as shown in Fig. 10(a), and the diagonal joints of each parallelogram interfere at the same time. When joint 1 reaches the limit position, the joints in the four circles reach the limit position at the same time. Drag the moving platform clockwise in a plane, and every two pairs of joints 1 and 4, 1 and 6, 6 and 3, 3 and 2, 2 and 5, 5 and 4 interfere at the same time. The moving track of the moving platform is a regular hexagon, and the sides of the hexagon are equal in different planes. Figure 10(a) shows the interference state of joints 1 and 4. The moving range of the moving platform in the vertical direction is 119–393 mm, from which the working space can be calculated. As shown in Fig. 10(b), the orange hexagonal prism is the motion range of the top moving platform, and the blue part is the space that can be reached at the end of the instrument. When the insertion depth of the instrument is the maximum, the maximum deflection angle is 62.6°. When the insertion depth of the instrument is the smallest, the maximum deflection angle is 23.7°. Obviously, the rotation range of the rotating joint can be increased by optimizing the connecting rod structure, so as to effectively increase the adjustment angle of the RCM mechanism. Chen et al. proposed a parallel RCM manipulator with a workspace of 60° and explained the reliability of this workspace by analyzing the human body size [Reference Chen, Wang, Wang, Chen, Parenti-Castelli and Angeles39].

Figure 10. Interference configuration and workspace of the proposed mechanism.

4.4. Prototype manufacturing and principle verification

To verify the correctness of the proposed mechanism, a prototype of the mechanism was fabricated, as shown in Fig. 11(a). All the linkages have been manufactured by 3D printing, and all the revolute joints are connected by bearings. The parameters of the prototype are identical with those in Table I. The diameter of the End-link is 4.3 mm. A laser pointer with green laser was utilized to point the remote center of the prototype. Once the End-link cross the remote center, the green spot will be observed on the End-link.

Figure 11. Different configurations of the proposed prototype.

Different configurations have been reached manually, and the positions of End-link relative the remote center have been captured. For each configuration in Fig. 11, the End-link is always with the green spot, verifying the motion of the End-link is restrained by the remote center.