2.1. Main architecture assumptions

The proposed geometrical description of the qSPM, schematized in Figure 3a, derives from the following architecture assumptions:

1. M1) We define ${{\boldsymbol{r}}_{\boldsymbol{nK}}}|_{n=(1,\cdots, 5), K=(A,B,C)}$ the unitary vector referenced on the origin of the absolute RF $({{\boldsymbol{O}},\hat{\boldsymbol{x}},\hat{\boldsymbol{y}},\hat{\boldsymbol{z}}})$ identifying the axis of revolution of a generic rotational joint $K_n|_{n=(1,\cdots, 5), K=(A,B,C)}$ . We impose ${\boldsymbol{r}}_{\boldsymbol{1A}} \equiv {\hat{\boldsymbol{z}}}, {\boldsymbol{r}}_{\boldsymbol{1B}} \equiv {\hat{\boldsymbol{x}}}, {{\boldsymbol{r}}_{\boldsymbol{1C}}} \equiv {\hat{\boldsymbol{y}}}$ ;

2. M2) For the superposition principle, universal joints of the URU leg can be divided in two rotative joints lying on perpendicular axes, namely $A_1 \equiv A_2$ and $A_4 \equiv A_5$ . Thus: ${\boldsymbol{r}}_{\boldsymbol{1A}} \perp {\boldsymbol{r}}_{\boldsymbol{2A}}$ and ${\boldsymbol{r}}_{\boldsymbol{4A}} \perp {\boldsymbol{r}}_{\boldsymbol{5A}}$ ;

3. M3) With such formulation, we define the constant geometrical angles $(\alpha, \beta, \gamma )$ , respectively, identifying the proximal links’ and distal links’ angular span (i.e., respectively, angles $K_1\hat{O}K_2|_{K=(B,C)}$ and $K_2\hat{O}K_3|_{K=(B,C)}$ ), and the angle $E\hat{O}X|_{X=(A_4 \equiv A_5,B_3,C_3)}$ , as shown in Figure 3b;

4. M4) Due to the architecture’s spherical geometry, and thus with minimal exceptions in the URU leg, links’ fundamental shapes only depend on geometrical angles of Assumption (M3). The overall description is thus scalable, having a quasi-spherical description;

5. M5) For the sake of brevity, we implicitly assume, unless stated otherwise, the formulation of vector ${{\boldsymbol{r}}_{\boldsymbol{nK}}}|_{n=(1,\cdots, 5), K=(A,B,C)} = R_{nK} \cdot {\hat{\boldsymbol{z}}}$ , being $R_{nK}$ a suitable $(3 \times 3)$ rotational matrix;

6. M6) In order to be consistent with previous articles’ work [Reference Chaker, Mlika, Laribi, Romdhane and Zeghloul17–Reference Saafi, Laribi, Zeghloul and Arsicault19], the EE orientation ${{\boldsymbol{r}}_{\boldsymbol{E}}}$ is expressed in Euler $zxz$ angles, as in (1);

   (1) \begin{equation} {\boldsymbol{r}}_{\boldsymbol{E}}(\psi, \theta, \phi ) = R_E(\psi, \theta, \phi ) {\hat{\boldsymbol{z}}} = R_z(\psi ) \cdot R_x(\theta ) \cdot R_z(\phi ) \end{equation}
   In which: $R_E$ and the triplet ( $\psi$ , $\theta$ , $\phi$ ) the rotational matrix and the angles describing the $zxz$ Euler description; inscriptions $R_z (\!\cdot\!)$ and $R_x (\!\cdot\!)$ rotation matrices around axes ${\hat{\boldsymbol{z}}}$ and ${\hat{\boldsymbol{x}}}$ .

7. M7) As described in Section 1, three motors are mounted on the proximal links and thus act on the active angles $\theta _{1A}{\boldsymbol{r}}_{\boldsymbol{1A}}$ , $\theta _{1B}{\boldsymbol{r}}_{\boldsymbol{1B}}$ , and $\theta _{1C}{\boldsymbol{r}}_{\boldsymbol{1C}}$ .

2.2. Inverse kinematics model and working modes

Adopting the geometrical description as in Section 2.1, we can derive the inverse kinematics model (IKM) resolution as a trigonometric function of the active angles (2).

\begin{align*} \;\;\;\qquad\qquad\qquad\qquad\qquad\qquad \left \{\begin{array}{c@{\quad}c@{\quad}c} A_1 \cdot \cos (\theta _{1A}) + A_2 \cdot \sin (\theta _{1A}) + A_3 &= 0 \qquad \qquad \qquad\qquad \qquad (2\textrm{a})\\[5pt] B_1 \cdot \cos (\theta _{1B}) + B_2 \cdot \sin (\theta _{1B}) + B_3 &= 0 \qquad \qquad \qquad \qquad\qquad (2\textrm{b})\\[5pt] C_1 \cdot \cos (\theta _{1C}) + C_2 \cdot \sin (\theta _{1C}) + C_3 &= 0 \qquad \qquad \qquad \qquad\qquad (2\textrm{c}) \end{array}\right .\nonumber \end{align*}

In which: $K_i|_{(K=A,B,C),i=(1,2,3)}$ are functions of constant and imposed angles $(\alpha, \beta, \gamma, \psi, \theta, \phi )$ and are reported in (34)(35)(36) in Appendix; $A_3=0$ due to the different structure of leg A.

It can be demonstrated that the IKM admits eight solutions, or working modes $m_i|_{i=(1,\dots, 8)}$ , directly depending on the configuration of legs A, B, and C, as in Figure 4.

(3) \begin{equation} \begin{cases} \begin{alignedat}{3} \theta _{1A} =& \, atan2(\!-A_1,A_2)& \qquad (A_3 = 0), \, \, & m_i|_{i=(1,\dots, 8)} \\[5pt] \theta _{1B} =& \, atan2(y_{B1},x_{B1}) & \qquad & m_{1,5},m_{2,6} \\[5pt] & \, atan2(y_{B2},x_{B2}) & \quad & m_{3,7},m_{4,8} \\[5pt] \theta _{1C} =& \, atan2(y_{C1},x_{C1}) & \qquad & m_{1,5},m_{3,7} \\[5pt] & \, atan2(y_{C2},x_{C2}) & \qquad & m_{2,6},m_{4,8} \end{alignedat} \end{cases} \end{equation}

In which expressions of $x_{Ki} |_{K =(B,C), i = (1,2)}$ and $y_{Ki} |_{K =(B,C), i = (1,2)}$ are reported in (37) in Appendix.

2.3. Jacobian computation

The Jacobian matrix can be reconstructed by time deriving the geometric description of the three legs [Reference Saafi, Laribi, Zeghloul and Arsicault19]. The system can be reduced as in (4).

(4) \begin{equation} \boldsymbol{\omega } = J_p^{-1} J_s \dot{\boldsymbol{\Theta }} = J \dot{\boldsymbol{\Theta }} \end{equation}

In which: $\boldsymbol{\Theta } = [\theta _{1A},\theta _{1B},\theta _{1C}]^t$ is the joint vector; $\dot{\boldsymbol{\Theta }}$ and $\boldsymbol{\omega }$ are the angular speeds of the motors and the platform; $J_p$ (5a) and $J_s$ (5b) are the parallel and serial parts of the Jacobian.

\begin{align*} \,\qquad\qquad\qquad\qquad\qquad \left \{ \begin{array}{l} J_p = \begin{bmatrix} ({\boldsymbol{r}}_{\boldsymbol{4A}} \times {\boldsymbol{r}}_{\boldsymbol{5A}})^t \\[5pt] ({\boldsymbol{r}}_{\boldsymbol{2B}} \times {\boldsymbol{r}}_{\boldsymbol{3B}})^t \\[5pt] ({\boldsymbol{r}}_{\boldsymbol{2C}} \times {\boldsymbol{r}}_{\boldsymbol{3C}})^t \\[5pt] \end{bmatrix}\qquad \qquad \qquad\qquad\qquad \, \qquad \qquad\qquad \qquad \qquad (5\textrm{a})\\[5pt] J_s = \begin{bmatrix} {\boldsymbol{r}}_{\boldsymbol{1A}} \cdot ({\boldsymbol{r}}_{\boldsymbol{4A}} \times {\boldsymbol{r}}_{\boldsymbol{5A}}) & 0 & 0\\[5pt] 0 & {\boldsymbol{r}}_{\boldsymbol{1B}} \cdot ({\boldsymbol{r}}_{\boldsymbol{2B}} \times {\boldsymbol{r}}_{\boldsymbol{3B}}) & 0\\[5pt] 0 & 0 & {\boldsymbol{r}}_{\boldsymbol{1C}} \cdot ({\boldsymbol{r}}_{\boldsymbol{2C}} \times {\boldsymbol{r}}_{\boldsymbol{3C}}) \end{bmatrix}\qquad \qquad \qquad (5\textrm{b}) \end{array}\right . \end{align*}

Note that $J$ implicitly depends on working modes $m_i|_{i=(1,\dots, 8)}$ due to the IKM structure.

Figure 4. Admitted working modes $m_i|_{i=(1,\dots, 8)}$ of the qSPM system.

2.4. Singularity definition

Singular positions can thus be achieved in three different ways:

• • Serial singularity: happening if $det(J_s)=0$ , that is, when factors of at least one triple product in expression (5b) are linearly dependent. This happens when the vectors lie on the same plane or at least two of them are parallel, that is, the legs are either fully retracted or fully extended, as shown in Figure 5a, and two solutions $m_i|_{i=(1,\dots, 8)}$ coincide. Serial singularities tend to be outside of the operative workspace, defined in Section 3.1;

• • Parallel singularity: happening if $det(J_p)=0$ , that is, when the matrix in (5a) is composed by linearly dependent vector entries, that is, when at least two of the planes containing $(r_{3K},r_{2K})|_{K=(B,C)}$ or $(r_{4A},r_{5A})$ are parallel, as shown in Figure 5b;

• • Structural singularity: happening when both determinants $det(J_s)$ and $det(J_p)$ are null.

Figure 5. Serial singularity example, obtained in $(\psi, \theta, \phi )=(179,54.7,0) (^{\circ })$ , in which leg B is fully extended and $m_{1}$ and $m_{3}$ coincide (a); parallel singularity example, obtained with $m_{3}$ in $(\psi, \theta, \phi )=(109.5,41.7,-40) (^{\circ })$ , in which planes described by vectors ${({\boldsymbol{r}}_{\boldsymbol{4A}},{\boldsymbol{r}}_{\boldsymbol{5A}})}$ (plane A, red) and ${({\boldsymbol{r}}_{\boldsymbol{2B}},{\boldsymbol{r}}_{\boldsymbol{3B}})}$ (plane B, green) are parallel (b).

2.5. Forward kinematics model

The forward kinematics model (FKM) of the architecture depends on the analysis of spherical four-bar loops presented in ref. [Reference Bai, Hansen and Angeles29]. In a similar way of what was done in ref. [Reference Saafi, Laribi and Zeghloul20], we can identify the $B_2B_3C_3C_2$ four-bar loop, expressed as in (6) (7).

(6) \begin{equation} {\boldsymbol{z}}^t \cdot R_x(\delta ) \cdot R_z(\pi - \xi ) \cdot R_x(\beta ) \cdot R_z(\pi - \sigma ) \cdot R_x(\gamma ') \cdot {\boldsymbol{z}} - {\boldsymbol{z}}^t \cdot \left (R_x(\beta )\right )^t \cdot {\boldsymbol{z}} = 0 \end{equation}

(7) \begin{equation} R_E = R_{2C} \cdot R_z(\theta _{2C}) \cdot R_x(\!-\beta ) \cdot R_z\left(\frac{5}{6} \cdot \pi +\sigma \right) \cdot R_x(\!-\gamma ) \cdot R_z(\!-\phi _E) \end{equation}

In which, as shown in Figure 6: $\xi$ and $\sigma$ are the input and output angles of the four-bar equations [Reference Bai, Hansen and Angeles29]; $\delta$ is the angle between ${\boldsymbol{r}}_{\boldsymbol{2B}}$ and ${\boldsymbol{r}}_{\boldsymbol{2C}}$ , depending on the robot’s pose; $\gamma '$ is the constant angle between ${\boldsymbol{r}}_{\boldsymbol{3B}}$ and ${\boldsymbol{r}}_{\boldsymbol{3C}}$ ; $R_{2C}$ derives from (M5); $\phi _E$ is a constant adjustment angle, as in Table II in Appendix.

Assuming as knowns the active angles $\theta _{1K}|_{K=(A,B,C)}$ , $\delta$ can be computed online. On the other hand, $\xi$ can be computed, with great computational gains, by adding an additional encoder measuring $\theta _{2C}$ . The Euler angles triplet ( $\psi, \theta, \phi$ ) can then be computed from (7) with reference to the Euler $zxz$ construction formulas.

Figure 6. Spherical four-bar loop used for the resolution of the FKM . Nomenclature from Figure 3a.

2.6. Haptic feedback

An haptic feedback system can then be built through the FKM, as in Section 2.5, providing the online computation of the Jacobian $J$ , and (8).

(8) \begin{equation} \boldsymbol{\tau } \stackrel{\text{def}}{=} \begin{bmatrix} \tau _{1A} \\[5pt] \tau _{1B} \\[5pt] \tau _{1C} \\[5pt] \end{bmatrix} = J^t \cdot {\boldsymbol{T}} + \boldsymbol\tau _{\boldsymbol{ctrl}} + \boldsymbol\tau _{\boldsymbol{comp}} \end{equation}

In which, $\tau _{1K}|_{K=(A,B,C)}$ are the active torques acting on the active angles $\theta _{1K}|_{K=(A,B,C)}$ ; ${\boldsymbol{T}}$ are the operational torques acting on the slave robot’s tool and received by the master’s control algorithm; $\boldsymbol{\tau}_{\boldsymbol{ctrl}}$ are additional control torques that can be applied directly on the master device; and $\boldsymbol{\tau}_{\boldsymbol{comp}}$ are static compensation torques due to gravity acting on the qSPM mechanical elements.

Figure 7. Robot workspace $W_{op}$ , defined in (9) on plane $\psi _{r}\theta _{r}$ , defined in (11). The red circle represents the workspace center, defined by (10).

3.1. Operative workspace definition

As mentioned in Section 1, the slave robot must operate on an RCM conic workspace built through a parametric cone demi-angle $\delta _C$ , as in Figure 2b. The operative workspace can be expressed in function of the Euler angles triplet ( $\psi, \theta, \phi$ ) as in (9):

\begin{align*} \;\;\;\,\quad\qquad W_{op}= \left \{\begin{array}{l@{\quad}c} &\sin (\psi ) - \cos (\psi ) \cdot \sin (\theta ) + \cos (\theta ) \ge \sqrt{3} \cdot \cos (\delta /2)\\[-4pt] (\psi, \theta, \phi )\Bigg| \qquad & -50 ^{\circ } \leq \phi \leq 50 ^{\circ }\\[-4pt] & \delta = 50 ^{\circ } \end{array}\right \} \begin{array}{l@{\quad}c} &\qquad \qquad (9\textrm{a})\\[5pt] &\qquad \qquad (9\textrm{b})\\[5pt] &\qquad \qquad (9\textrm{c}) \end{array} \end{align*}

In which, $\delta _C$ is the cone demi-angle as in Figure 7; (9a) does not depend on the self-rotation angle $\phi$ due to conic symmetry of the workspace, and thus (9b) on $\phi$ is imposed; we assume, due to the geometric description presented in the previous paragraphs, a central workspace position $w_c$ as in (10).

(10) \begin{equation} \begin{aligned} {\boldsymbol{r}}_{\boldsymbol{wc}} =\frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\[5pt] 1 \\[5pt] 1 \\[5pt] \end{bmatrix} & \quad & \Rightarrow &\quad & \begin{cases} \psi _{wc} &\thickapprox 135^{\circ }\\[5pt] \theta _{wc} &\thickapprox 54.7^{\circ }\\[5pt] \phi _{wc} &= 0^{\circ } \end{cases} \\[5pt] \end{aligned} \end{equation}

For the sake of brevity, we introduce the relative Euler angles triplet ( $\psi _{r}, \theta _{r}, \phi _{r}$ ) in (11). Workspace $W_{op}$ can be represented on the offset plane $\psi _r\theta _r$ independently from angle $\phi$ as in Figure 7.

(11) \begin{equation} \begin{cases} \psi _{r}(\psi ) = \psi - \psi _{wc}\\[5pt] \theta _{r}(\theta ) = \theta - \theta _{wc}\\[5pt] \phi _{r}(\phi ) = \phi - \phi _{wc} \end{cases} \end{equation}

We define $\delta _{E}$ as the angle between workspace center vector ${{\boldsymbol{r}}_{\boldsymbol{wc}}}$ (10), and the EE orientation vector ${\boldsymbol{r}}_{\boldsymbol{E}}(\psi _r,\theta _r,\phi _r)$ (1) (11), as in (12). Inside the workspace, $\delta _E \leq \delta _C$ due to (9).

(12) \begin{equation} \delta _E(\psi _r,\theta _r,\phi _r) = atan2({||{\boldsymbol{r}}_{\boldsymbol{wc}} \boldsymbol\times {\boldsymbol{r}}_{\boldsymbol{E}}||, {\boldsymbol{r}}_{\boldsymbol{wc}} \boldsymbol\cdot {\boldsymbol{r}}_{\boldsymbol{E}}}) \end{equation}

3.2. Dexterity maps

To enhance the analysis of singularities within the operative workspace, described in Section 3.1, for each working mode $m_i|_{i=(1,\dots, 8)}$ (3), we introduce the dexterity parameter $\eta (J)$ , defined as in (13).

(13) \begin{equation} \begin{aligned} \eta (J)=\frac{1}{||J|| \cdot ||J^{-1}||}, & \qquad & 0\lt \eta (J)\lt 1 \end{aligned} \end{equation}

By imposing geometrical angles values as in Table II in Appendix, and a minimum dexterity threshold $\eta _{thr}=0.02$ , singularity areas $S_J$ can be defined by (14).

(14) \begin{equation} S_J = \{ (\psi _{r}, \theta _{r}, \phi _{r}) \mid \eta (J(\psi, \theta, \phi )) \lt \eta _{thr} = 0.02 \} \end{equation}

Figure 8. Dexterity map in the Euler angles space for $m_{1,5}$ . Planes $\psi _{r}\theta _{r}$ range with a discrete step of $\phi _{r,step} =12.5^{\circ }$ . Relative angles defined in (11). Red areas correspond to singularity areas $S_J$ (14) inside the operative workspace $W_{op}$ (9).

Figure 9. Performance index $\delta _{E,min,S_J}(\phi _r)$ of working modes $m_{i,i+4}|_{i=(1,\dots, 4)}$ (red, green, blue, and cyan). Black dashed line is $\delta _C$ (9).

Map results, depending on working modes $m_i|_{i=(1,\dots, 8)}$ , are shown inside the Euler $zxz$ space $(\psi _r, \theta _r,\phi _r)$ by varying discretely the Euler angle $\phi _r$ and analyzing planes $\psi _r\theta _r$ .

It can be demonstrated that, due to Assumptions (M1) and (M2) on the overall description of the architecture and (3), leg A’s configuration does not influence in any way the dexterity of the manipulator inside $W_{op}$ . Therefore, pairs of working modes $m_{i,i+4}|_{i=1,\dots, 4}$ produce the same dexterity maps. Figure 8 shows discrete results for $m_{1,5}$ and Figure 22 in Appendix for the other solutions.

3.3. Dexterity performance index and reachable workspace

As a mean to assess and compare working modes’ performances on dexterity, we introduce the performance index $\delta _{E,min,S_J}(\phi _r)$ . Said index corresponds to the partial minimization on planes $\psi _r\theta _r$ of all possible $\delta _E$ (12) corresponding to critical points $(\psi _r,\theta _r,\phi _r) \in S_J$ (14). Therefore, the index corresponds to the angular distance from workspace center to the nearest singularity point in the cartesian space. The index is expressed in (15) and is portrayed in Figure 9 for all working modes.

(15) \begin{equation} \delta _{E,min,S_J} (\phi _r) = \min _{\psi _r,\theta _r}{\delta _{E}(\psi _r,\theta _r,\phi _r)} \quad (\psi _r,\theta _r,\phi _r) \in S_J \end{equation}

Figure 10. Aspects $A_i|_{i=(1,\dots, 8)}$ (red, green, blue, and cyan) defined in (17) inside the configuration space $(\theta _{1A}, \theta _{1B}, \theta _{1C})$ . Point-symmetry virtual center, to be discussed in Section 5.2, is a black dot highlighted by an arrow (a); Intersections (red) and tangential zones (blue) among aspects (b).

Figure 11. Parameters involved in the definition of a general link, with a generic point ${\boldsymbol{P}}$ , as in (20): generic view (a); View on plane $(r_{i,K},r_{i+1,k})$ (b).

Focusing, for example, on $m_{1,5}$ and thus Figure 8, note that, for $\phi _r = 50 ^{\circ }$ , there are parts of the workspace that are unreachable due to singularity crossings. The defined reachable workspace thus restricts according to the expression (16).

(16) \begin{equation} W_{reach} = \{ P=(\psi, \theta, \phi ) \in W_{op} \mid \exists s(w_c, P) \in C^0 \;:\; s(w_c, P) \cap S_J = \emptyset \} \end{equation}

In which, $s(w_c, P)$ is a random continuous path between point $P$ and the operative workspace center $w_c$ (10); $W_{reach}$ implicitly depends on to working modes $m_i|_{i=(1,\dots, 8)}$ through $S_J$ due to the Jacobian defined in Section 2.3.

3.4. Configuration space and aspects

Inside the configuration space, we define the Aspects $A_i|_{i=(1,\dots, 8)}$ as the images of the IKM function $f_{IKM}$ in the workspace $W_{reach}$ according to working modes $m_i|_{i=(1,\dots, 8)}$ as in (17) and Figure 10a.

(17) \begin{equation} A_i|_{i=(1,\dots, 8)} = \{ \Theta = (\theta _{1A}, \theta _{1B}, \theta _{1C}) \mid \exists (\psi, \theta, \phi ) \in W_{reach} \;:\; f_{IKM}(\psi, \theta, \phi ) = \Theta \} \end{equation}

As a direct product of the dexterity maps portrayed in Section 3.2, also Aspects $A_{i,i+4}|_{i=1,\dots, 4}$ are paired, providing the same results inside the configuration space.

4.1. Self-collision architecture assumptions

The self-collision phenomenon can involve either the legs, either, and possibly, the platform. To efficiently analyze the said phenomenon between legs, the following assumptions are needed:

1. A8) Every link in legs B and C can be defined, as in Figure 11, by the following parameters $(K=(B,C), \, i=(1,2), \, j=(p,d)$ stands for “proximal” and “distal”):

   • • $\epsilon = (\alpha, \beta )$ is the geometrical angle of the link (M3);

   • • $\tilde{\epsilon }$ is the angular span which involves the non-lumped rotational joint;

   • • ${{\boldsymbol{r}}_{\boldsymbol{i,K}}}$ and ${\boldsymbol{r}}_{{\boldsymbol{i}}\textbf{+1},{\boldsymbol{K}}}$ are the axes of its rotational joints;

   • • $r_{j,K}$ is the radial distance between the link’s geometrical center and the fixed RF origin;

   • • $l_{j,K}, h_{j,K}$ are, respectively, the tangential and radial dimensions of the link.

   • • $V_{d,K}|_{K=(A,B)}$ is the portion of space occupied by the link, function of the aforementioned parameters and the pose of the manipulator, that is, the active angles $\Theta$ and $m_i|_{i=(1,\dots, 8)}$ ;

   • • $c_{d,K}|_{K=(A,B)}$ is the set of points defining the contour of the link lying in the spherical sector identified by $r_{j,d}$ .

2. A9) Due to the involved rotational joint, the proximal and distal links of the same leg $K$ do not lie within the same spherical sectors. Nevertheless, for simplicity, symmetry, system robustness, and manufacturing, it is assumed that $r_{p,B} = r_{p,C}$ and $r_{d,C} = r_{d,C}$ ;

3. A10) It is assumed that, in the qSPM, self-collision between legs happens only between distal links. Proximal links can collide only with working modes $m_{2,6}$ and values of $\alpha +\tilde{\alpha } \geq 45 ^{\circ }$ . Having imposed, as in Table II in Appendix, $\alpha +\tilde{\alpha } \approx 42 ^{\circ }$ , from now on we will focus only on distal links.

In addition with the legs’ self-collision phenomenon, in prospect of a future axial displacement actuation along ${\boldsymbol{r}}_{\boldsymbol{E}}$ , this article also analyzes the self-collision phenomenon between the legs and a cylindric volume having as axis ${\boldsymbol{r}}_{\boldsymbol{E}}$ , as in Figure 12a. Additional assumptions are then required:

1. A11) Said cylinder can be defined with the following parameters, as in Figure 12b:

   • • $r_{cyl}$ and $h_{cyl}$ are the radius and the height of the cylinder, reported in Table II in Appendix;

   • • $V_{cyl}$ is the portion of space occupied by the cylinder;

   With such formulation, the cylinder develops from point ${\boldsymbol{O}}$ (origin of the absolute RF) to point $h_{cyl}{\boldsymbol{r}}_{\boldsymbol{E}}$ ;

2. A12) It can be demonstrated that the cylindric volume can collide only with the distal links inside the reachable workspace enriched with the collision phenomenon, to be presented in (25). Therefore, for the sake of brevity, collision between said volume and the proximal links is not analyzed.

Due to the superposition principle, the self-collision problem can then be divided in three subproblems, which can be analyzed separately: the collisions between distal links $V_{d,B}$ and $V_{d,C}$ ; between distal link $V_{d,B}$ and cylinder $V_{cyl}$ ; between distal link $V_{d,C}$ and cylinder $V_{cyl}$ .

Figure 12. Parameters involved in the definition of a general cylinder, with a generic point ${\boldsymbol{P}}$ , as in (20): contextual view (in $m_3$ , $(\psi _r,\theta _r,\phi _r)=(0,0,0)$ ) (a); detailed view (b).

4.2. Self-collision description

4.2.1. Self-collision between distal links

With the premises made in the first part of Section 4.1, we can note that collision between distal links, labeled for brevity as $C1$ , happens with the expression (18):

(18) \begin{equation} C1 \iff \exists {\boldsymbol{P}} \;:\; {\boldsymbol{P}} \in V_{d,B} \cap V_{d,C} \end{equation}

From the perspective of logical computation, this condition can be reduced, with great computational gains, in the expression (19):

(19) \begin{equation} C1 \iff \exists {\boldsymbol{P}}_{\textbf{1}} \;:\; ({\boldsymbol{P}}_{\textbf{1}} \in c_{d,B} \cap V_{d,C}) \lor \exists {\boldsymbol{P}}_{\textbf{2}} \;:\; ({\boldsymbol{P}}_{\textbf{2}} \in c_{d,C} \cap V_{d,B}) \end{equation}

The problem then translates in identifying the logical conditions for which a general point ${\boldsymbol{P}}$ is inside the dominion $V_{d,K}$ (20) and apply said conditions to a discrete number of points in the contour $c_{d,K}$ of the other link.

(20) \begin{equation} {\boldsymbol{P}} \in V_{d,K} \iff \begin{cases} \begin{aligned} |\beta _{Pt}| &\leq \frac{\beta }{2} + \tilde{\beta }\\[5pt] r_{d,K} - \frac{h_{d,K}}{2} &\leq |{\boldsymbol{P}}_{\boldsymbol{t}}| \leq r_{d,K} + \frac{h_{d,K}}{2} \\[5pt] |{\boldsymbol{P}}_{\boldsymbol{n}}| &\leq \frac{l_{d,K}}{2} \\[5pt] \end{aligned} \end{cases} \end{equation}

In which: ${\boldsymbol{P}} = {\boldsymbol{P}}_{\boldsymbol{t}} + {\boldsymbol{P}}_{\boldsymbol{n}}$ , being ${\boldsymbol{P}}_{\boldsymbol{t}}$ and ${\boldsymbol{P}}_{\boldsymbol{n}}$ the tangential and normal projection vectors of ${\boldsymbol{P}}$ on the plane identified by vectors ${{\boldsymbol{r}}_{\boldsymbol{2K}},{\boldsymbol{r}}_{\boldsymbol{3K}}}$ ; $\beta _{Pt}$ is the angle between $r_{d,K}{\hat{\boldsymbol{r}}}$ and ${\boldsymbol{P}}_{\boldsymbol{t}}$ , as portrayed in Figure 11.

4.2.2. Self-collision between distal links and cylinder

With the premises made in the second part of Section 4.1, we can note that collisions between the cylindric volume and the distal links, labeled for brevity as $C2$ and $C3$ , happens only when (21) is satisfied.

(21) \begin{equation} C2 \iff \exists {\boldsymbol{P}} \;:\; {\boldsymbol{P}} \in V_{d,B} \cap V_{cyl}, \quad C3 \iff \exists {\boldsymbol{P}} \;:\; {\boldsymbol{P}} \in V_{d,C} \cap V_{cyl} \end{equation}

From the computational point of view, as previously done with $C1$ , the problem can be reduced by identifying the logical conditions for which a general point ${\boldsymbol{P}}$ is inside the dominion $V_{cyl}$ (22) and apply them to a discrete number of points of the contours $c_{d,B}$ and $c_{d,C}$ , respectively, identifying $C2$ and $C3$ .

(22) \begin{equation} {\boldsymbol{P}} \in V_{d,K} \iff \begin{cases} \begin{aligned} |{\boldsymbol{P}}_{\boldsymbol{r}}| \leq r_{cyl} \\[5pt] |{\boldsymbol{P}}_{\boldsymbol{a}}| \leq h_{cyl} \\[5pt] \end{aligned} \end{cases} \overset{h_{cyl} \equiv r_p}{\implies } |{\boldsymbol{P}}_{\boldsymbol{r}}| \leq r_{cyl} \end{equation}

In which: ${\boldsymbol{P}} = {\boldsymbol{P}}_{\boldsymbol{r}} + {\boldsymbol{P}}_{\boldsymbol{a}}$ , being ${\boldsymbol{P}}_{\boldsymbol{r}}$ and ${\boldsymbol{P}}_{\boldsymbol{a}}$ the radial and axial projection vectors of ${\boldsymbol{P}}$ on axis ${\boldsymbol{r}}_{\boldsymbol{E}}$ , as in Figure 12b; $r_P$ is the actual distance between the EE center and the origin, reported in Table II in Appendix. Note that, imposing $h_{cyl} \equiv r_P$ , the condition on ${\boldsymbol{P}}_{\boldsymbol{a}}$ is always satisfied for every point inside $V_{d,K}|_{K=(B,C)}$ , and therefore can be simplified.

Experimental evaluation of this kind of collision is presented in Section 5.5.

4.3. Reachable workspace, dexterity maps, and performance index with collision

We can then enrich the reachable workspace introduced in (16) and the dexterity maps of Section 3.2 with the self-collision phenomenon between the two distal links and the cylindric volume. The resulting collision dominion $C_V$ , the reduced collision dominion $\tilde{C}_V$ , considering only $C1$ , and the reachable workspace $W_{reach,c}$ are described by (23) and (25), and discretely shown in Figure 13.

\begin{equation*} \;\;\qquad\qquad\qquad\qquad\qquad\qquad C_V = \left \{\begin{array}{l@{\quad}l} & C1: \, V_{d,B} \cap V_{d,C} \neq \emptyset \\[-4pt] (\psi _{r}, \theta _{r}, \phi _{r})\Bigg| & C2: \, V_{d,B} \cap V_{cyl} \neq \emptyset \\[-4pt] & C3: \, V_{d,C} \cap V_{cyl} \neq \emptyset \end{array} \right \} \begin{array}{l@{\quad}c} &\qquad\qquad\qquad \qquad (23\textrm{a})\\[5pt] &\qquad\qquad \qquad\qquad (23\textrm{b})\\[5pt] &\qquad \qquad\qquad\qquad (23\textrm{c}) \end{array} \end{equation*}

(24) \begin{equation} \tilde{C}_V = \{ (\psi _{r}, \theta _{r}, \phi _{r}) \mid C1: \, V_{d,B} \cap V_{d,C} \neq \emptyset \} \end{equation}

(25) \begin{equation} W_{reach,c} = \{ P \in W_{op} \mid \exists s(w_c, P) \in C^0 \;:\; s(w_c, P) \cap (S_J \cup C_V) = \emptyset \} \end{equation}

Figure 13. Dexterity map in the Euler angles space for $m_{1,5}$ . Planes $\psi _{r}\theta _{r}$ with a discrete-step of $\phi _{r,step} =12.5^{\circ }$ . Relative angles defined in (11). Colored areas correspond, respectively, to singularity areas $S_J$ (red) (14) and collision areas $C_V$ (blue: $C1$ , cyan: $C2$ , magenta: $C3$ as in Section 4.2) (23) inside the operative workspace $W_{op}$ (9).

In a similar manner of Section 3.3, we can define two other performance indices, considering $C_V$ (23) and $\tilde{C}_V$ (24), as in (26) and (27). Figure 14 shows the two indices for all possible working modes $m_i|_{i=(1,\dots, 8)}$ .

(26) \begin{equation} \delta _{E,min,C_V} (\phi _r) = \min _{\psi _r,\theta _r}{\delta _{E}(\psi _r,\theta _r,\phi _r)} \quad (\psi _r,\theta _r,\phi _r) \in C_V \end{equation}

(27) \begin{equation} \delta _{E,min,\tilde{C}_V} (\phi _r) = \min _{\psi _r,\theta _r}{\delta _{E}(\psi _r,\theta _r,\phi _r)} \quad (\psi _r,\theta _r,\phi _r) \in \tilde{C}_V \end{equation}

Figure 14. Performance indices $\delta _{E,min,C_V}(\phi _r)$ (continuous lines) and $\delta _{E,min,\tilde{C}_V}(\phi _r)$ (dashed lines) of working modes $m_{i,i+4}|_{i=(1,\dots, 4)}$ (red, green, blue, and cyan) with collision. Black dashed line is $\delta _C$ (9).

4.4. Aspects enriched with collision

The knowledge of the workspace $W_{reach,c}$ (25), enriched with the collision phenomenon, allows for the updated definition of the Aspects $A_i|_{i=(1,\dots, 8),c}$ according to working modes $m_i|_{i=(1,\dots, 8)}$ inside the configuration space (28).

(28) \begin{equation} \begin{aligned} A_i|_{i=(1,\dots, 8),c} = \{\Theta = (\theta _{1A}, \theta _{1B}, \theta _{1C}) \mid \exists (\psi, \theta, \phi ) \in W_{reach,c} \;:\; f_{IKM}(\psi, \theta, \phi ) = \Theta \} \end{aligned} \end{equation}