2.1. Reconfigurable base

A GSLPM-RCM is presented based on the spherical surface geometric model introduced in previous work [Reference Li, Zhang and Tian13], as depicted in Figure 1(a). This mechanism consists of a fixed rigid base, two R[Pa]^IIR limbs, and a specific connecting structure (SCS), wherein a double parallelogram linkage constitutes a [Pa]^II joint. To accommodate diverse task requirements, the mechanism features a reconfigurable base designed to adjust its parameters, thereby adjusting the mechanism’s characteristics to fit different needs. The occupied space of the mechanism can be changed through its reconfigurable base, while also allowing for adjustments in key performance aspects, including workspace shape and operating posture range. These adjustments enable the mechanism to adapt to various operational scenarios and to execute a range of tasks effectively. A key challenge in the design of a reconfigurable base lies in ensuring it meets the necessary constraints to facilitate coordinated movement between the limbs. Specifically, the rigid base is depicted as the circular arc A₁A₂ in Figure 1(b), characterized by a radius of r and a central angle of 2α. The base determines three key parameters: the radius r, the central angle 2α, and the position of the RCM point N (x _N, y _N, z _N). Since the endpoint B _i of the limb A _i B _i always moves on a spherical surface S, the reconfigurable base needs to ensure that the first axis a _i of limb-i always passes through the RCM point N, while point A _i remains on a virtual circle with the same fixed radius r. A concept diagram of the reconfigurable base is established as depicted in Figure 1(c), where the central angle 2α and the position N (x _N, y _N, z _N) serve as variable parameters.

Figure 1. The concept of the mechanism with a reconfigurable base.

Figure 2. The evolution process of the reconfigurable base linkage.

Then, the configuration evaluation process of a reconfigurable base is described as follows:

Step 1: Construct a 1-DOF linkage with a link and a R joint as shown in Figure 2(a). The link NA rotates around point N, while its endpoint A is constrained to move along circle C with a constant radius r. There is a physical R joint at the center point N to constrain the link NA.

Step 2: Construct a parallelogram linkage ONAD, as shown in Figure 2(b). As a result, the actuated R joint at point N, shown in Figure 2(a), can be moved to point O. To obtain a virtual center point N of the circle C, the link ON and NA need to be removed.

Step 3: Extend the link NA to point F as shown in Figure 2(c). Then construct a virtual parallelogram linkage OAFH, where the length of OA is variable. The parallelogram linkage ONAD constrains the motion of the virtual parallelogram linkage OAFH.

Step 4: Keep the length of link OA constant and then construct a parallelogram linkage OAFH, as shown in Figure 2(d). It is essential that the parallelogram linkage OAFH is constrained by the virtual parallelogram linkage ONAD. This constraint ensures that point A consistently resides on circle C with a fixed radius r. The length of ON is variable, while the length of NA is fixed and equal to r. The center point N can move along line L_v. Although the links ON and NA are removed, a virtual constraint still exists.

Step 5: Extend link FH to point G, ensuring that point G is positioned on the line L_v, as shown in Figure 2(e). By applying the judgment theorem of similar triangles, it is established that △NAO is always similar to △OHG. Utilizing the properties of these similar triangles, maintain the length of GH, a segment of link FG, as constant while ensuring that point G remains on the line L_v. As a result, the length of NA is consistently fixed at r, and the center point N is always located on the line L_v. To accomplish this, a prismatic (P) joint is introduced between points O and G, then a crank-slider linkage GOH is constructed to constrain the motion of the parallelogram linkage OAFH. This arrangement eliminates the virtual parallelogram linkage ONAD, as shown in Figure 2(d).

Step 6: Construct the symmetric linkage OGHFA along the line L_v to obtain a reconfigurable base, as shown in Figure 2(f). The maximum width denoted as d _f, of the reconfigurable base, serves as an indicator of the footprint of the mechanism. The position of the center point N can be characterized by the distance between points O and N denoted as d _N.

The CAD model of the reconfigurable base is built as depicted in Figure 3, which also displays various configurations with distinct central angles of 2α.

Figure 3. The configurations of the reconfigurable base with different central angle 2 α.

Note that the RCM point N can translate along the X-axis; however, this translation is a parasitic motion. Given that the reconstruction of the mechanism is finalized before the surgical intervention, the RCM point remains stationary during the surgery itself. Consequently, the parasitic motion associated with the RCM point does not interfere with the surgical process. As the central angle 2α increases, the footprint index d _f correspondingly increases, while the distance d _N of the center point N decreases.

2.2. A GSLPM-RCM with a reconfigurable base

A GSLPM-RCM with a reconfigurable base is constructed, as depicted in Figure 4(a). This 4-DOF mechanism is capable of rotating and translating around point N, classifying it as a 3R1T RCM mechanism. A based frame, denoted as { O }: O-XYZ, is established on the fixed base of the reconfigurable base. To describe the RCM motion, an RCM frame { G }: N-xyz is set at the point N, where the unit directional vectors for the axes are defined as u = [1, 0, 0]^T for the x-axis, v = [0, 1, 0]^T for the y-axis, and w = [0, 0, 1]^T for the z-axis. Additionally, a limb frame { L _i }: N-x _i y _i z _i is built to analyze limb-i, as shown in Figure 4(b), where the x _i -axis follows a _i , and the y _i -axis is perpendicular to the plane P _li . The unit vectors t _i and e _i are along the y _i and z _i -axis, respectively. Notably, ι is the angle between a _i and XY-plane, and α _i is the angle between the projection NA _i ’ of NA _i onto the XY-plane and X-axis. Furthermore, the position vector of endpoint B _i of the limb-i in{ G } is denoted as b _i . Define b and k as the unit directional vectors along line B₁B₂ and the output link FP, respectively. P = [θ, β, γ, l _NP]^T represents the position and orientation of the output link FP, where θ, β, and γ correspond to rotations around v , t , and k , respectively, and l _NP is the distance between the points P and N, where t · v = 0. Set P ₀ = [0, 0, 0, 0]^T as the initial configuration of the GSLPM-RCM, where, t = [1, 0, 0]^T and b = [0, 1, 0]^T. Following the design principle of the GSLPM-RCM, the position of the point B _i determines the position and orientation of the output link. The rotation matrix from { L _i } to{ G }is formulated as follows:

Figure 4. Kinematic diagram of the GSLPM-RCM.

(1) \begin{align} {}^{G}{\boldsymbol{{R}}}{_{\boldsymbol{{L}}i}^{}}=\boldsymbol{{R}}_{\boldsymbol{{w}}}\!\left(\alpha _{i}\right)R_{{y_{i}}}\left(\iota \right)=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\alpha _{i} & -\mathrm{s}\alpha _{i} & 0\\[3pt] \mathrm{s}\alpha _{i} & \mathrm{c}\alpha _{i} & 0\\[3pt] 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\iota & 0 & \mathrm{s}\iota \\[3pt] 0 & 1 & 0\\[3pt] -\mathrm{s}\iota & 0 & \mathrm{c}\iota \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\alpha _{i}\mathrm{c}\iota & -\mathrm{s}\alpha _{i} & \mathrm{c}\alpha _{i}\mathrm{s}\iota \\[3pt] \mathrm{s}\alpha _{i}\mathrm{c}\iota & \mathrm{c}\alpha _{i} & \mathrm{s}\alpha _{i}\mathrm{s}\iota \\[3pt] -\mathrm{s}\iota & 0 & \mathrm{c}\iota \end{array}\right] \end{align}

where symbols s and c are sine and cosine functions, respectively.

3.1. Inverse kinematics

Based on the modular analysis method, the inverse kinematics analysis of the GSLPM-RCM is divided into three parts: the reconfigurable base, limb-i, and SCS.

(a) SCS

According to the length geometric relation, the distance between the points N and G is derived as

(2) \begin{align} l_{\mathrm{GN}}=l_{\mathrm{NP}}+l_{\mathrm{FP}}-l_{\mathrm{GE}}-2l_{1}l_{\mathrm{GE}}/l_{2} \end{align}

where l ₁ and l ₂ represent the length of the link I _i E and link B _i E, respectively; The distance between the points N and P (G and E, F and P) is l _NP (l _GE, l _FP). Considering the z-coordinate of point P is negative when the reference point P is below the xNy-plane, in this instance, l _NP is defined as negative, and vice versa. For a right triangle, yields:

(3) \begin{align} r^{2}={l_{\mathrm{GN}}}^{2}+\left(\frac{l_{{\mathrm{B}_{1}}{\mathrm{B}_{2}}}}{2}\right)^{2} \end{align}

Combining Eqs. (2) and (3), we get:

(4) \begin{align} \left\{\begin{array}{l} l_{\mathrm{GE}}=\frac{-\mathrm{E}_{2}\pm \sqrt{\mathrm{E}_{2}^{2}-4\mathrm{E}_{1}\mathrm{E}_{3}}}{2\mathrm{E}_{1}}\\[3pt] l_{{\mathrm{B}_{1}}{\mathrm{B}_{2}}}=2\sqrt{l_{2}^{2}-{l_{\mathrm{GE}}}^{2}} \end{array}\right. \end{align}

where

\begin{align*} \left\{\begin{array}{l} \mathrm{E}_{1}=4l_{1}^{2}+4l_{1}l_{2},\\[3pt] \mathrm{E}_{2}=-4l_{1}l_{2}l_{\mathrm{FP}}-4l_{1}l_{2}l_{\mathrm{NP}}-2l_{2}^{2}l_{\mathrm{FP}}-2l_{2}^{2}l_{\mathrm{NP}},\\[3pt] \mathrm{E}_{3}=-r^{2}l_{2}^{2}+l_{2}^{4}+l_{2}^{2}l_{\mathrm{FP}}^{2}+l_{2}^{2}l_{\mathrm{NP}}^{2}+2l_{2}^{2}l_{\mathrm{FP}}l_{\mathrm{NP}}\mathit{.} \end{array}\right. \end{align*}

In the frame{ G }, through rotation transformation, k and b can be expressed as

(5) \begin{align} \left\{\begin{array}{l} \boldsymbol{{k}}\boldsymbol{=}\boldsymbol{{R}}_{\boldsymbol{{v}}}\left(\theta \right)\boldsymbol{{R}}_{\boldsymbol{{t}}}\left(\beta \right)\cdot \boldsymbol{{w}}\boldsymbol{=}\left[\mathrm{c}\beta \mathrm{s}\theta, -\mathrm{s}\beta, \mathrm{c}\beta \mathrm{c}\theta \right]^{\mathrm{T}}\\[3pt] \boldsymbol{{b}}\boldsymbol{=}\boldsymbol{{R}}_{\boldsymbol{{v}}}\left(\theta \right)\boldsymbol{{R}}_{\boldsymbol{{t}}}\left(\beta \right)\cdot \boldsymbol{{R}}_{\boldsymbol{{k}}}\left(\gamma \right)\cdot \boldsymbol{{v}}\boldsymbol{=}\left[\mathrm{s}\theta \mathrm{s}\beta \mathrm{c}\gamma -\mathrm{c}\theta \mathrm{s}\gamma, \mathrm{c}\beta \mathrm{s}\gamma, \mathrm{c}\theta \mathrm{s}\beta \mathrm{c}\gamma +\mathrm{s}\theta \mathrm{s}\gamma \right]^{\mathrm{T}} \end{array}\right. \end{align}

where in

$\boldsymbol{{R}}_{\boldsymbol{{v}}}(\theta )= \left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\theta & 0 & \mathrm{s}\theta \\[3pt] 0 & 1 & 0\\[3pt] -\mathrm{s}\theta & 0 & \mathrm{c}\theta \end{array} \right],\ \boldsymbol{{R}}_{\boldsymbol{{t}}}(\beta )\boldsymbol{=} \left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0\\[3pt] 0 & \mathrm{c}\beta & -\mathrm{s}\beta \\[3pt] 0 & \mathrm{s}\beta & \mathrm{c}\beta \end{array} \right],\ \boldsymbol{{R}}_{\boldsymbol{{k}}}(\gamma ) = \left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\gamma & -\mathrm{s}\gamma & 0\\[3pt] \mathrm{s}\gamma & \mathrm{c}\gamma & 0\\[3pt] 0 & 0 & 1 \end{array} \right]$ .

Based on the vector closed-loop equation, the input of the SCS can be written as

(6) \begin{align} \boldsymbol{{b}}_{i}=l_{\mathrm{GN}}\boldsymbol{{k}}+\left({-}1\right)^{i}\frac{l_{{\mathrm{B}_{1}}{\mathrm{B}_{2}}}}{2}\boldsymbol{{b}} = \left[b_{ix},\ b_{iy},\ b_{iz}\right]^{\mathrm{T}} \end{align}

(b) Limb- $i$

The position vector b _i of the point B _i at { G } is formulated as

(7) \begin{align} \boldsymbol{{b}}_{i}={}^{G}{\boldsymbol{{R}}}{_{\boldsymbol{{L}}i}^{}}\cdot \boldsymbol{{R}}_{{\boldsymbol{{a}}_{i}}}\!\left(\theta _{i}\right)\boldsymbol{{R}}_{{\boldsymbol{{t}}_{i}}}\!\left(\beta _{i}\right)\cdot r\boldsymbol{{w}} \end{align}

where

\begin{align*} \boldsymbol{{R}}_{{\boldsymbol{{a}}_{i}}}\left(\theta _{i}\right)=\left[\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0\\[3pt] 0 & \mathrm{c}\theta _{i} & -\mathrm{s}\theta _{i}\\[3pt] 0 & \mathrm{s}\theta _{i} & \mathrm{c}\theta _{i} \end{array}\right],\ \boldsymbol{{R}}_{{\boldsymbol{{t}}_{i}}}\left(\beta _{i}\right)\boldsymbol{=}\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{c}\beta _{i} & 0 & \mathrm{s}\beta _{i}\\[3pt] 0 & 1 & 0\\[3pt] -\mathrm{s}\beta _{i} & 0 & \mathrm{c}\beta _{i} \end{array}\right] \end{align*}

Both sides of Eq. (7) left multiplied by the inverse of ^G R _{L i} yields:

(8) \begin{align} {}^{G}{\boldsymbol{{R}}}{_{\boldsymbol{{L}}i}^{}}-^{1}\cdot \boldsymbol{{b}}_{i}=\boldsymbol{{R}}_{{\boldsymbol{{a}}_{i}}}\!\left(\theta _{i}\right)\boldsymbol{{R}}_{{\boldsymbol{{t}}_{i}}}\!\left(\beta _{i}\right)\cdot r\boldsymbol{{w}}\boldsymbol{=}\left[r\mathrm{s}\beta _{i},\ -r\mathrm{s}\theta _{i}\mathrm{c}\beta _{i},\ r\mathrm{c}\theta _{i}\mathrm{c}\beta _{i}\right]^{\mathrm{T}} \end{align}

Let

(9) \begin{align} \boldsymbol{{d}}_{i}={}^{G}{\boldsymbol{{R}}}{_{\boldsymbol{{L}}i}^{}}^{-1}\cdot \boldsymbol{{b}}_{i}=\left[d_{ix},\ d_{iy},\ d_{iz}\right]^{\mathrm{T}} \end{align}

The input angles of the limb-i are solved as

(10) \begin{align} \left\{\begin{array}{l} \theta _{i}=\mathrm{t}^{-1}\left(-d_{iy}/d_{iz}\right)\\[3pt] \beta _{i}=\mathrm{s}^{-1}\left(d_{ix}/r\right) \end{array}\right. \end{align}

where t⁻¹ is arctangent function, and s⁻¹ is arcsine function. Then, two motor angles of the limb-i are obtained as

(11) \begin{align} \left\{\begin{array}{l} \varphi _{i1}=\theta _{i}/k\\[3pt] \varphi _{i2}=\beta _{i} \end{array}\right. \end{align}

where k is the transmission ratio of the synchronous belt transmission.

(c) The reconfigurable base

The reconfigurable base is actuated by a P joint allowing for a simplified kinematics diagram as depicted in Figure 2(e). l _GO represents the distance between points G and O, l _NA represents the distance between points A and N, l _GH, and l _HF denote GH and HF segments of the link FH, respectively, and l _OH corresponds to the length of the link OH. Applying the law of cosines, the distance l _GO can be derived as

(12) \begin{align} l_{\mathrm{GO}}=l_{\mathrm{OH}}\mathrm{c}\alpha \pm \sqrt{{l_{\mathrm{OH}}}^{2}\mathrm{c}^{2}\alpha -{l_{\mathrm{OH}}}^{2}+{l_{\mathrm{GH}}}^{2}} \end{align}

Utilizing the properties of similar triangles, the position vector of RCM point N in { O } is derived as

(13) \begin{align} \boldsymbol{{n}}=\left[l_{\mathrm{NA}}l_{\mathrm{GO}}/l_{\mathrm{OH}},\ 0,\ 0\right]^{\mathrm{T}} \end{align}

3.2. Workspace

The workspace is a significant feature of PMs, determining the type of operation task and serving as a foundation for planning robot motion trajectories. To determine the reachable workspace, the reachable extent of limbs and the SCS need to be assessed. Furthermore, the conditions for rising interference between the limb and the SCS also need to be determined. According to the structure of the GSLPM-RCM, the reachable extent of the limbs corresponds to the output range for the limb, denoted as θ _i ∈(θ _i ^min, θ _i ^max) and β _i ∈(β _i ^min, β _i ^max), where θ _i ^min, θ _i ^max, β _i ^min, and β _i ^max represent the limiting output values of limb-i, occurring when collisions are detected between the links. Similarly, the reachable extent of the SCS is determined by l _B1B2, expressed as l _B1B2∈(l _B1B2 ^min, l _B1B2 ^max), where l _B1B2 ^min and l _B1B2 ^max correspond to the distances between points B₁ and B₂ when the links of the SCS collide. Whether interference between the limb and the SCS occurs can be determined by examining the angle between the two planes where limb-i and the SCS are respectively situated, i.e.,

(14) \begin{align} \delta _{{\mathrm{P}_{i}}}=\cos ^{-1}\left(\boldsymbol{{n}}_{{\mathrm{L}_{i}}}\cdot \boldsymbol{{n}}_{\mathrm{SCS}}\right)\lt {\delta _{{\mathrm{P}_{i}}}}^{\max } \end{align}

where $\boldsymbol{{n}}_{Li} = (\boldsymbol{{b}}_{i} \times \boldsymbol{{a}}_{i}) / || \boldsymbol{{b}}_{i} \times \boldsymbol{{a}}_{i} || $ and $\boldsymbol{{n}}_{\rm scs} = (\boldsymbol{{k}} \times \boldsymbol{{b}})/||\boldsymbol{{k}} \times \boldsymbol{{b}} ||$ are the unit normal vectors of the planes where the limb-i and the SCS are situated, respectively. δ _Pi ^max is the limit angle when limb-i and the SCS collide.

With the different central angle 2α, the workspace of the GSLPM-RCM is analyzed based on the inverse kinematics. Through analysis, it was found that there is not much difference between the orientation workspace with different l _NP, which can be verified by the results of ref. [Reference Li, Zhang and Tian13]. In this work, the orientation workspaces with l _NP = −260 mm are drawn as shown in Figure 5. It is observed that the orientation workspace demonstrates symmetry about the plane-XZ, as indicated by the boundary of angle γ in Figure 5. To provide a quantitative description of the workspace dimensions, a maximum inscribed circle within the boundary of angle γ is introduced, as depicted in Figure 6. The boundary value γ _B of the angle γ and the radius r _B of the maximum inscribed circle are utilized as indices to characterize the workspace. As the central angle 2α decreases, it is necessary to reduce γ _B to obtain a larger r _B.

Figure 5. The orientation workspace with l_NP = −260 mm.

Figure 6. The description of the size of the workspace.

4.1. Transmission wrenches of the GSLPM-RCM

In the GSLPM-RCM, the output of limb-i is represented by the vector b _i . The output link of the SCS exhibits three rotational degrees of freedom and one translational degree of freedom centered around point N. To facilitate analysis, a simplified equivalent model of the GSLPM-RCM is constructed, reflecting its inherent characteristics as depicted in Figure 7. This simplified mechanism comprises two limbs with a R ^{a i} R ^{t i} R ^{b i} R ^g R ^g -structure, where the three axes a _i , t _i, and b _i intersect at point N, and i = 1, 2. Additionally, a virtual middle limb with an R ^v R ^t R ^k P ^k -structure is introduced to constrain the output link. Each limb in the GSLPM-RCM features two actuated joints, which are arranged in series and mounted near the base.

Figure 7. A simplified equivalent model of the GSLPM-RCM.

Within the frame { G }, the twist system for limb-i is written as

(15) \begin{align} \left\{\boldsymbol{{\$ }}_{i}\right\}=\left\{\begin{array}{l} \boldsymbol{{\$}}_{\mathrm{I},i1}=\left(\boldsymbol{{a}}_{i};\ \mathbf{0}\right)\\[3pt] \boldsymbol{{\$}}_{\mathrm{I},i2}=\left(\boldsymbol{{t}}_{i};\ \mathbf{0}\right)\\[3pt] \boldsymbol{{\$}}_{i3}=\left(\boldsymbol{{b}}_{i}/r;\ \mathbf{0}\right)\\[3pt] \boldsymbol{{\$}}_{i4}=\left(\boldsymbol{{g}};\ \boldsymbol{{b}}_{i}\times \boldsymbol{{g}}\right)\\[3pt] \boldsymbol{{\$}}_{i5}=\left(\boldsymbol{{g}};\ l_{\mathrm{EN}}\boldsymbol{{k}}\times \boldsymbol{{g}}\right) \end{array}\right. \end{align}

where $\boldsymbol{{g}} = \boldsymbol{{b}} \times \boldsymbol{{k}} $ . The twist system { $ _i } is a 5-system. The twists corresponding to each joint in limb-i are denoted by $ _I,i1, $ _I,i2, $ _i3, $ _i4, and $ _i5, and $ _I,i1 and $ _I,i2 are input twists of limb-i. By applying the principle of reciprocity, the corresponding constraint wrenches for Eq. (15) is calculated as

(16) \begin{align} \left\{\boldsymbol{{\$}}_{i}^{r}\right\}=\boldsymbol{{\$}}_{i}^{r}=\left(\boldsymbol{{g}};\ 0\right) \end{align}

$ _i ^r is a pure force along g passing through point N. Because limb-i contains two actuated joints, and the i1^th and i2^th actuated joints are connected in series and move independently, the transmission wrenches associated with two actuated joints in limb-i need to be derived, respectively. For the transmission wrench associated with the i1^th actuated joint, it is reciprocal to all the twists of joints in limb-i except for $ _I,i1, i.e.,

(17) \begin{align} \boldsymbol{{\$}}_{\mathrm{T},i1}\circ \boldsymbol{{\$}}_{ij}=0\left(\boldsymbol{{\$}}_{ij}\in \left\{\boldsymbol{{\$}}_{\mathrm{I},i2},\boldsymbol{{\$}}_{i3},\boldsymbol{{\$}}_{i4},\boldsymbol{{\$}}_{i5}\right\}\right) \end{align}

To find the transmission wrench $ _T,i1 accurately, two planes are constructed using limb-1 as an example as shown in Figure 8(a). In Figure 8(a), the plane P _i1 is formed by the twists $ _I,i2 and $ _i3 and determined by the unit vectors t _i and b _i . The plane P _i3 is formed by the twists $ _i4 and $ _i5 and determined by the unit vectors g and m _i . The intersecting line L _i1 of the planes P _i1 and P _i3 corresponds to the desired transmission wrench $ _T,i1 associated with the i1^th actuated joint. The transmission wrench $ _T,i1 is a pure force, and it can be derived as

Figure 8. The wrenches associated with the actuated joints in limb-1.

(18) \begin{align} \boldsymbol{{\$}}_{\mathrm{T},i1}=\left(\boldsymbol{{f}}_{i};\ \boldsymbol{{b}}_{i}\times \boldsymbol{{f}}_{i}\right) \end{align}

where the unit vector f _i is the directional vector of the intersecting line L _i1, and $\boldsymbol{{f}}_{i} = (( \boldsymbol{{g}} \times \boldsymbol{{m}}_{i}) \times (\boldsymbol{{b}}_{i}/ r \times \boldsymbol{{t}}_{i}))/ || (\boldsymbol{{g}} \times \boldsymbol{{m}}_{i}) \times (\boldsymbol{{b}}_{i}/ r \times \boldsymbol{{t}}_{i})||$ , the unit vector m _i is the directional vector of the link B _i E, and $\boldsymbol{{m}}_{i} = (l_{\rm EN} \boldsymbol{{k}} - \boldsymbol{{b}}_{i} )/l_{2} $ .

Similarly, for the transmission wrench associated with the i2^th actuated joint, the following reciprocal relationship exists, i.e.,

(19) \begin{align} \boldsymbol{{\$}}_{\mathrm{T},i2}\circ \boldsymbol{{\$}}_{ij}=0\left(\boldsymbol{{\$}}_{ij}\in \left\{\boldsymbol{{\$}}_{\mathrm{I},i1},\boldsymbol{{\$}}_{i3},\boldsymbol{{\$}}_{i4},\boldsymbol{{\$}}_{i5}\right\}\right) \end{align}

The transmission wrench $ _T,i2 can be found according to Figure 8(b). In Figure 8(b), the plane P _i2 is formed by the twists $ _I,i1 and $ _i3 and determined by the unit vectors a _i and b _i . The transmission wrench $ _T,i2 associated with the i2^th actuated joint along the intersecting line L _i2 of the planes P _i2 and P _i3, and it is a pure force and can be derived as

(20) \begin{align} \boldsymbol{{\$}}_{\mathrm{T},i2}=\left(\boldsymbol{{h}}_{i};\ \boldsymbol{{b}}_{i}\times \boldsymbol{{h}}_{i}\right) \end{align}

where h _i represents the directional vector of the intersecting line L _i2, and $\boldsymbol{{h}}_{i} = ((\boldsymbol{{g}} \times \boldsymbol{{m}}_{i}) \times (\boldsymbol{{a}}_{i} \times \boldsymbol{{b}}_{i}/r))/|| (\boldsymbol{{g}} \times \boldsymbol{{m}}_{i}) \times (\boldsymbol{{a}}_{i} \times \boldsymbol{{b}}_{i}/r)|| $ . The span of the two transmission wrenches $ _T,i1 and $ _T,i2 forms a 2-system that covers all pure forces in the plane P _i3 which pass through the point B _i .

Since the motions of the output link of the GSLPM-RCM are three rotations around the RCM point N and translation along k , passing through the RCM point N. The constraint wrenches of the GSLPM-RCM can be directly obtained as

(21) \begin{align} \left\{\boldsymbol{{\$}}_{\mathrm{C}}\right\}=\left\{\begin{array}{l} \boldsymbol{{\$}}_{\mathrm{C},1}=\left(\boldsymbol{{b}};\ \mathbf{0}\right)\\[3pt] \boldsymbol{{\$}}_{\mathrm{C},2}=\left(\boldsymbol{{g}};\ \mathbf{0}\right) \end{array}\right. \end{align}

where the unit vectors b , g , and k are orthogonal to each other. $ _C,₁ and $ _C,₂ are two constraint forces passing through the RCM point N. Consequently, the output link of the GSLPM-RCM is constrained by 6 wrenches ( $ _C,₁, $ _C,₂, $ _T,11, $ _T,12, $ _T,21, $ _T,22).

Locking all other actuated joints except for the i1^th actuated joint, the GSLPM-RCM is constrained by a 5-order system:

(22) \begin{align} \boldsymbol{{C}}_{i1}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} \boldsymbol{{\$}}_{\mathrm{C},1} & \boldsymbol{{\$}}_{\mathrm{C},2} & \boldsymbol{{\$}}_{\mathrm{T},i2} & \boldsymbol{{\$}}_{\mathrm{T},\neg i1} & \boldsymbol{{\$}}_{\mathrm{T},\neg i2} \end{array}\right] \end{align}

The GSLPM-RCM becomes a 1-DOF mechanism. The output twist $ _O, _i1 associated with the i1^th actuated joint, which is reciprocal to C _i1, is derived as

(23) \begin{align} \boldsymbol{{\$}}_{\mathrm{O},i1}=\left(\boldsymbol{{s}}_{i1};\ \chi _{i1}\boldsymbol{{k}}\right) \end{align}

where

\begin{align*}{s_i}_1 & = {\chi _i}_1{\left[ {{K_i}_{1,1},{K_i}_{1,2},{K_i}_{1,3}} \right]^T}; \\[3pt]{A_i}_{1,1} & = \boldsymbol{{u}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{h}}_i}} \right),{B_i}_{1,1} = \boldsymbol{{v}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{h}}_i}} \right),{C_i}_{1,1} = \boldsymbol{{w}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{h}}_i}} \right),{D_i}_{1,1} = - \boldsymbol{{k}}\cdot{\boldsymbol{{h}}_i};\\[3pt]{A_i}_{1,2} & = \boldsymbol{{u}} \cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{B_i}_{1,2} = \boldsymbol{{v}} \cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{C_i}_{1,2} = \boldsymbol{{w}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{D_i}_{1,2} = - \boldsymbol{{k}}\cdot{\boldsymbol{{f}}_\neg }_i;\\[3pt]{A_i}_{1,3} & = \boldsymbol{{u}} \cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{B_i}_{1,3} = \boldsymbol{{v}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{C_i}_{1,3} = \boldsymbol{{w}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{D_i}_{1,3} = - \boldsymbol{{k}}\cdot{\boldsymbol{{h}}_\neg }_i;\\[3pt]K_{i1,1} & = \frac{B_{i1,3}C_{i1,2}D_{i1,1}-B_{i1,2}C_{i1,3}D_{i1,1}-B_{i1,3}C_{i1,1}D_{i1,2}+B_{i1,1}C_{i1,3}D_{i1,2}+B_{i1,2}C_{i1,1}D_{i1,3}-B_{i1,1}C_{i1,2}D_{i1,3}}{A_{i1,3}B_{i1,2}C_{i1,1}-A_{i1,2}B_{i1,3}C_{i1,1}-A_{i1,3}B_{i1,1}C_{i1,2}+A_{i1,1}B_{i1,3}C_{i1,2}+A_{i1,2}B_{i1,1}C_{i1,3}-A_{i1,1}B_{i1,2}C_{i1,3}},\\[3pt]K_{i1,2} & = \frac{-A_{i1,3}C_{i1,2}D_{i1,1}+A_{i1,2}C_{i1,3}D_{i1,1}+A_{i1,3}C_{i1,1}D_{i1,2}-A_{i1,1}C_{i1,3}D_{i1,2}-A_{i1,2}C_{i1,1}D_{i1,3}+A_{i1,1}C_{i1,2}D_{i1,3}}{A_{i1,3}B_{i1,2}C_{i1,1}-A_{i1,2}B_{i1,3}C_{i1,1}-A_{i1,3}B_{i1,1}C_{i1,2}+A_{i1,1}B_{i1,3}C_{i1,2}+A_{i1,2}B_{i1,1}C_{i1,3}-A_{i1,1}B_{i1,2}C_{i1,3}},\\[3pt]\end{align*}

\begin{align*}K_{i1,3}& =\frac{A_{i1,3}B_{i1,2}D_{i1,1}-A_{i1,2}B_{i1,3}D_{i1,1}-A_{i1,3}B_{i1,1}D_{i1,2}+A_{i1,1}B_{i1,3}D_{i1,2}+A_{i1,2}B_{i1,1}D_{i1,3}-A_{i1,1}B_{i1,2}D_{i1,3}}{A_{i1,3}B_{i1,2}C_{i1,1}-A_{i1,2}B_{i1,3}C_{i1,1}-A_{i1,3}B_{i1,1}C_{i1,2}+A_{i1,1}B_{i1,3}C_{i1,2}+A_{i1,2}B_{i1,1}C_{i1,3}-A_{i1,1}B_{i1,2}C_{i1,3}};\\[2pt]\chi _{i1} & =\pm 1/\sqrt{{K_{i1,1}}^{2}+{K_{i1,2}}^{2}+{K_{i1,3}}^{2}}.\end{align*}

The symbol “ $\neg i$ ” represents that it is not i. Because i = 1 or 2, if i is 1, then “ $\neg i$ ” is 2; if i is 2, then “ $\neg i$ ” is 1. Similarly, the output twist $ _O, _i2 associated with the i2^th actuated joint is derived as

(24) \begin{align} \boldsymbol{{\$}}_{\mathrm{O},i2}=\left(\boldsymbol{{s}}_{i2};\ \chi _{i2}\boldsymbol{{k}}\right) \end{align}

where

\begin{align*} {s_i}_2 & = {\chi _i}_2{\left[ {{K_i}_{2,1},{K_i}_{2,2},{K_i}_{2,3}} \right]^T};\\[2pt]{A_i}_{2,1} & = \boldsymbol{{u}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{f}}_i}} \right),{B_i}_{2,1} = \boldsymbol{{v}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{f}}_i}} \right),{C_i}_{2,1} = \boldsymbol{{w}} \cdot\left( {{\boldsymbol{{b}}_i} \times {\boldsymbol{{f}}_i}} \right),{D_i}_{2,1} = -\boldsymbol{{k}}\cdot{\boldsymbol{{f}}_i};\\[2pt]{A_i}_{2,2} & = \boldsymbol{{u}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{B_i}_{2,2} = \boldsymbol{{v}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{C_i}_{2,2} = \boldsymbol{{w}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{f}}_\neg }_i),{D_i}_{2,2} = - \boldsymbol{{k}}\cdot{\boldsymbol{{f}}_\neg }_i;\\[2pt]{A_i}_{2,3} & = \boldsymbol{{u}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{B_i}_{2,3} = \boldsymbol{{v}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{C_i}_{2,3} = \boldsymbol{{w}}\cdot({\boldsymbol{{b}}_\neg }_i \times {\boldsymbol{{h}}_\neg }_i),{D_i}_{2,3} = - \boldsymbol{{k}}\cdot{\boldsymbol{{h}}_\neg }_i;\\[2pt]K_{i2,1}& =\frac{B_{i2,3}C_{i2,2}D_{i2,1}-B_{i2,2}C_{i2,3}D_{i2,1}-B_{i2,3}C_{i2,1}D_{i2,2}+B_{i2,1}C_{i2,3}D_{i2,2}+B_{i2,2}C_{i2,1}D_{i2,3}-B_{i2,1}C_{i2,2}D_{i2,3}}{A_{i2,3}B_{i2,2}C_{i2,1}-A_{i2,2}B_{i2,3}C_{i2,1}-A_{i2,3}B_{i2,1}C_{i2,2}+A_{i2,1}B_{i2,3}C_{i2,2}+A_{i2,2}B_{i2,1}C_{i2,3}-A_{i2,1}B_{i2,2}C_{i2,3}},\\[2pt]K_{i2,2}& =\frac{-A_{i2,3}C_{i2,2}D_{i2,1}+A_{i2,2}C_{i2,3}D_{i2,1}+A_{i2,3}C_{i2,1}D_{i2,2}-A_{i2,1}C_{i2,3}D_{i2,2}-A_{i2,2}C_{i2,1}D_{i2,3}+A_{i2,1}C_{i2,2}D_{i2,3}}{A_{i2,3}B_{i2,2}C_{i2,1}-A_{i2,2}B_{i2,3}C_{i2,1}-A_{i2,3}B_{i2,1}C_{i2,2}+A_{i2,1}B_{i2,3}C_{i2,2}+A_{i2,2}B_{i2,1}C_{i2,3}-A_{i2,1}B_{i2,2}C_{i2,3}},\\[2pt]K_{i2,3}& =\frac{A_{i2,3}B_{i2,2}D_{i2,1}-A_{i2,2}B_{i2,3}D_{i2,1}-A_{i2,3}B_{i2,1}D_{i2,2}+A_{i2,1}B_{i2,3}D_{i2,2}+A_{i2,2}B_{i2,1}D_{i2,3}-A_{i2,1}B_{i2,2}D_{i2,3}}{A_{i2,3}B_{i2,2}C_{i2,1}-A_{i2,2}B_{i2,3}C_{i2,1}-A_{i2,3}B_{i2,1}C_{i2,2}+A_{i2,1}B_{i2,3}C_{i2,2}+A_{i2,2}B_{i2,1}C_{i2,3}-A_{i2,1}B_{i2,2}C_{i2,3}};\\[2pt]\chi _{i2} & =\pm 1/\sqrt{{K_{i2,1}}^{2}+{K_{i2,2}}^{2}+{K_{i2,3}}^{2}}. \end{align*}

4.2. Transmission index of the GSLPM-RCM

In ref. [Reference Liu, Wu and Wang43], Liu. et al. introduced the input, output, and local transmission index (ITI, OTI, and LTI). For the GSLPM-RCM, firstly, the transmission index of each limb is characterized by taking the smaller of the transmission indices associated with its two actuated joints. That is,

(25) \begin{align} \eta _{\mathrm{I},i}=\min _{k=1,2}\left(\frac{\left| \boldsymbol{{\$}}_{\mathrm{I},ik}\circ \boldsymbol{{\$}}_{\mathrm{T},ik}\right| }{\left| \boldsymbol{{\$}}_{\mathrm{I},ik}\circ \boldsymbol{{\$}}_{\mathrm{T},ik}\right| _{\max }}\right) \\[-10pt] \nonumber \end{align}

(26) \begin{align} \eta _{\mathrm{O},i}=\min _{k=1,2}\left(\frac{\left| \boldsymbol{{\$}}_{\mathrm{T},ik}\circ \boldsymbol{{\$}}_{\mathrm{O},ik}\right| }{\left| \boldsymbol{{\$}}_{\mathrm{T},ik}\circ \boldsymbol{{\$}}_{\mathrm{O},ik}\right| _{\max }}\right) \\[6pt] \nonumber \end{align}

Then, the local and global transmission indices for the GSLPM-RCM are written as

(27) \begin{align} \eta _{\mathrm{LTI}}=\min _{i=1,2}\left\{\eta _{\mathrm{I},i},\eta _{\mathrm{O},i}\right\} \\[-10pt] \nonumber \end{align}

(28) \begin{align} \tau _{\mathrm{GTI}}=\frac{\int _{\mathrm{W}}\eta _{\mathrm{LTI}}\mathrm{dW}}{\int _{\mathrm{W}}\mathrm{dW}} \\[6pt] \nonumber \end{align}