It is known that the inverse kinematics in general 3R manipulators can be solved through a fourth-degree polynomial P in the variable of the last active joint. The Groebner Basis Elimination is used to eliminate the first two active joint variables

$\left({\mathsf{\theta}}_{1},{\mathsf{\theta}}_{2}\right)$ [

22], and to produce a solution that stands for the orthogonal 3R metamorphic manipulator. Following the method presented in [

22] the polynomial is derived in the following form:

with

$t=\mathrm{tan}\frac{{\theta}_{3}}{2},\alpha ={w}_{0}-{w}_{2}$,

$b=2\left({w}_{1}-{w}_{3}\right)$,

$c=2\left(2{w}_{4}+{w}_{0}\right)$,

$d=2\left({w}_{1}+{w}_{3}\right)$,

$e={w}_{2}+{w}_{0}$,

${w}_{0}={(R+K)}^{2}+4{{d}_{3}}^{2}{{d}_{4}}^{2}+4{{d}_{2}}^{2}({{r}_{2}}^{2}-{\rho}^{2})$,

${w}_{1}=4{d}_{4}{r}_{2}(L-R)$,

${w}_{2}=-4{d}_{3}{d}_{4}(K+R)$,

${w}_{3}=8{d}_{3}{{d}_{4}}^{2}{r}_{2}$,

${w}_{4}=4{{d}_{4}}^{2}({{d}_{2}}^{2}-{{d}_{3}}^{2}+{{r}_{2}}^{2})$,

${\rho}^{2}={x}^{2}+{y}^{2}$, and

$R={\rho}^{2}+{z}^{2}$,

$S={d}_{3}{}^{2}+{d}_{4}{}^{2}+{r}_{2}{}^{2}+{r}_{3}{}^{2}$,

$K={d}_{2}{}^{2}-S$,

$L={d}_{2}{}^{2}+S$. The coefficients of the polynomial P depend on the DH-parameters, including the pseudo-joint variables, and the TCP coordinates

$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)$.

#### 3.1. Necessary and Sufficient Conditions to Investigate Cuspidality

The necessary and sufficient conditions to recognize a cuspidal 3R manipulator has been introduced in [

28]. If and only if there is at least one singular point in its workspace such that the inverse kinematics admits a real triple root, then the manipulator is considered as cuspidal. Therefore, it is equivalent to prove that

$t$ in Equation (3) admits at least one triple root. Based on the method introduced in [

22], a fourth-degree polynomial P has at least one or more triple roots if and only if the polynomial system

$\mathrm{P},$ $\frac{\mathrm{dP}}{\mathrm{dt}}$,

$\frac{{\mathrm{d}}^{2}\mathrm{P}}{{\mathrm{dt}}^{2}}$ admits real radicals. In this way, an algebraic parametric polynomial system S derived to identify and investigate the cuspidal anatomies in orthogonal 3R metamorphic manipulators:

The parametric polynomial system is considered a zero-dimensional system of three equations with three unknowns $\left\{t,z,\rho \right\}$. Without loss of generality, it is assumed that $y=0$ because a complete rotation around the z-axis of the first active joint lets the system invariant. In addition, ${\mathrm{d}}_{2}>0,{\mathrm{d}}_{3}>0,{\mathrm{d}}_{4}>0,{\mathrm{r}}_{2}>0$ and ${r}_{3}=0$ are the constraints for the solution of S. Since ${d}_{2}\left({\theta}_{{\pi}_{1}}\right)=A\ast sin\left({\theta}_{{\pi}_{1}}\right)$ is a continuous and differentiable function in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, then, ${d}_{2}\left({\theta}_{{\pi}_{1}}\right)$ and ${\mathrm{d}}_{3}\left({\theta}_{{\pi}_{2}}\right)$ are monotonically increasing function in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ $\left({\mathrm{always}\mathrm{d}}_{2}>0\right)$. The same stands for ${\mathrm{d}}_{3}\left({\theta}_{{\pi}_{2}}\right)$ since A and B are positive quantities. ${r}_{2}\left({\theta}_{{\pi}_{2}}\right)>0$ is always positive i.e., $d>B\ast \mathrm{cos}{\theta}_{{\pi}_{2}}$ in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

The design parameter space $\left\{{\mathrm{d}}_{2},{\mathrm{d}}_{3},{\mathrm{d}}_{4},{\mathrm{r}}_{2}\right\}\in {\mathbb{R}}^{4}$ must be divided in subspaces such that the sign of the polynomials in Equation (4) is constant. The set of variables $\left\{t,z,\rho \right\}$ should be eliminated to derive polynomials that depend only on DH-parameters.

For this reason, it has been developed general and efficient algorithms to solve parametric algebraic polynomial systems [

29,

30]. In this way, the parametric polynomial system shown in Equation (4) is solved and the discriminant variety is obtained. After removing the imaginary polynomials, the desired algebraic polynomials are derived that depend only on the following four DH-kinematic parameters

$\left\{{\mathrm{d}}_{2},{\mathrm{d}}_{3},{\mathrm{d}}_{4},{\mathrm{r}}_{2}\right\}$. The following system of polynomials indicates the bifurcating equations:

However, only three real polynomials out of five

$\left\{{h}_{2},{h}_{3},{h}_{5}\right\}$ in Equation (5) can be used to the classification according to the number of real roots of Equation (4) i.e., cusp points [

31].

Last but not least, it is worth mentioning that Equation (5) has the most general form, as well as the separating equations, are valid for any 3R orthogonal metamorphic manipulator with the selected four kinematic parameters $\left\{{\mathrm{d}}_{2},{\mathrm{d}}_{3},{\mathrm{d}}_{4},{\mathrm{r}}_{2}\right\}$.

In the following sections the investigation of the set of the Equation (5) to classify the metamorphic manipulator according to the number of cusps and the number of nodes.

#### 3.2. Separating Algebraic Equations through Investigation of det$\left(J\right)=0$

The algebraic set of Equation (5) is used to the classification of open chain 3R orthogonal metamorphic manipulators according to the number of cusp points. The analysis presented in this section is based on the method introduced in [

31] without taking into account the assumption that

${\mathrm{d}}_{2}=1$, since in the considered metamorphic structure this parameter depends on the first pseudo-joint angle

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$. The bifurcating surfaces separate the metamorphic design parameters space is subspaces according to the number of cusps as it is shown in

Figure 3. The discrete transition of the metamorphic parameters is shown only for the positive angles of the two pseudo-joints

$\left({\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}\right)$. The classification is the same for all possible combinations of pseudo-joints exploiting the symmetry for all the distinct kinematic configurations of the mechanism i.e., (169 kinematic postures). Using

Figure 3 the anatomy is derived by selecting the metamorphic parameters based on the number of cusp points.

The bifurcating equation

${h}_{5}$ in Equation (5) is a biquadratic polynomial in

${d}_{4}$ providing the following two roots:

Equations (6) and (7) apply to manipulators with a singular point in the workspace where two cusp points coincide with a node such that Equation (4) has a quadruple root [

32]. Equation (6) defines the transition between binary and quaternary manipulators. The surfaces

${\mathrm{C}}_{0\mathsf{\alpha}}$ and

${\mathrm{C}}_{0\mathrm{b}}$ does not appear in

Figure 3 since they are valid for negative values of the metamorphic parameters

$\left({\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}\right)$.

The rest bifurcating equations are derived from the investigation of the determinant of Jacobian det

$\left(\mathrm{J}\right)=0$:

Since for the last joint offset, it is assumed that

${r}_{3}=0$, Equation (

**8**) could be written as two product factors providing the following equations:

Taking into account that

${\mathrm{s}}_{3}=\mathsf{\epsilon}\sqrt{1-{\left(\frac{{\mathrm{d}}_{3}}{{\mathrm{d}}_{4}}\right)}^{2}}$, where

$\mathsf{\epsilon}=\pm 1$ for

${\mathrm{d}}_{3}\le {\mathrm{d}}_{4}$ and substituting in Equation (9) the following equation is obtained:

Assuming that $0\le {c}_{2}\le 1$ with $\epsilon =-1$ then, ${d}_{4}\le \left(\frac{{d}_{3}}{1+{d}_{3}}\right)\sqrt{{r}_{2}{}^{2}+{\left({d}_{2}+{d}_{3}\right)}^{2}}$.

As it shown in

Figure 4a the transition from subspace 1 to subspace 2 is characterized by a manipulator for which the singular branch (line)

${\mathrm{E}}_{1}$ defined by

${\mathsf{\theta}}_{3}=-{\mathrm{cos}}^{-1}\left(-\frac{{\mathrm{d}}_{3}}{{\mathrm{d}}_{4}}\right)$ in the joint space is tangent to the singular curve

${\mathrm{S}}_{1}$. So, the bifurcating surface

${\mathrm{C}}_{1}$ separates the metamorphic anatomies with four and two cusps such that,

Since

${\mathrm{d}}_{2}$ and

${\mathrm{d}}_{3}$ depend on

${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}$ and

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$ respectively then the bifurcating surface

${\mathrm{C}}_{1}$, separates the subspace 1 from subspace 2 with four and two cups respectively, as it is shown in

Figure 3. Assuming that

$0\ge {\mathrm{c}}_{2}\ge -1$ with

$\mathsf{\epsilon}=-1$, then

${\mathrm{d}}_{4}\ge \frac{{\mathrm{d}}_{3}}{\left|{\mathrm{d}}_{2}-{\mathrm{d}}_{3}\right|}\sqrt{{\mathrm{r}}_{2}{}^{2}+{\left({\mathrm{d}}_{3}-{\mathrm{d}}_{2}\right)}^{2}},{\mathrm{d}}_{3}>{\mathrm{d}}_{2}$ or

$\left|{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}\right|>\left|{\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}\right|$.

As it shown in

Figure 4b the transition from subspace 2 with two cusps to subspace 3 with four cusps is characterized by a metamorphic anatomy for which the singular branch (line)

${\mathrm{E}}_{1}$ defined by

${\mathsf{\theta}}_{3}=-{\mathrm{cos}}^{-1}\left(-\frac{{\mathrm{d}}_{3}}{{\mathrm{d}}_{4}}\right)$ in joint space is tangent to the singular curve

${\mathrm{S}}_{2}$. So, the bifurcating surface

${\mathrm{C}}_{2}$ separates the manipulators with two and four cusps such that,

The final bifurcating surface is ${\mathrm{d}}_{4}=\frac{{\mathrm{d}}_{3}}{\left|{\mathrm{d}}_{2}-{\mathrm{d}}_{3}\right|}\sqrt{{\mathrm{r}}_{2}{}^{2}+{\left({\mathrm{d}}_{3}-{\mathrm{d}}_{2}\right)}^{2}}$ with ${\mathrm{d}}_{2}>{\mathrm{d}}_{3}$ or $\left|{\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}\right|>\left|{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}\right|$.

As it shown in

Figure 4c the transition from subspace 2 to subspace 4 is characterized by a manipulator for which the singular branch (line)

${\mathrm{E}}_{2}$ defined by

${\mathsf{\theta}}_{3}{=+\mathrm{cos}}^{-1}\left(-\frac{{\mathrm{d}}_{3}}{{\mathrm{d}}_{4}}\right)$ in joint space is tangent to the singular curve

${\mathrm{S}}_{1}$. So, the bifurcating surface

${\mathrm{C}}_{3}$ separates the manipulators with two and no cusp points (regular workspace topology) given by,

The above surfaces

${\mathrm{C}}_{\mathrm{i}},\mathrm{i}=0,1,2,3$ can be verified through Equation (5). The separating equation

${\mathrm{h}}_{2}$ in Equation (5) is a second-degree polynomial in

${\mathrm{d}}_{4}$ such that:

Similarly, the bifurcating algebraic equation

${\mathrm{h}}_{3}$ can be simplified as:

#### 3.3. Classification According to the Number of Nodes

Another important topological feature is the node which is a singular point in the workspace where two singular curves (internal or external) intersect and the polynomial P admits two double roots. In the present section, the distinct kinematic anatomies are classified according to the number of nodes in order to show the deformation of the workspace and hence the non-isomorphism of the kinematic topology.

The method to classify 3R orthogonal fixed manipulators according to the number of nodes is introduced in [

23], where analytical algebraic expressions of the surfaces in the parameter space were derived. Analytical algebraic expressions of the surfaces of the parameter space were produced in [

23] and are used in this paper to classify the anatomies derived from the considered orthogonal metamorphic structure. The bifurcating surfaces subdivide the metamorphic design space

$\left\{{\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}{,\mathrm{d}}_{4}\right\}$ into eight distinct non-isomorphic subspaces with a constant number of cusps and nodes shown in

Figure 5. The number of cusp and node points are indicated in parentheses of separating subspaces, respectively. The production of these subspaces is based on the following analysis.

Subspace 1 in

Figure 3 represents metamorphic anatomies with four cusp points and is divided into three distinct subspaces with different numbers of nodes. The transition between subspace 1.1

$\left(4,2\right)$ to subspace 1.2

$\left(4,0\right)$ is given by the following boundary surface:

Figure 6a shows the singularity curves of a representative metamorphic anatomy from subspace 1.1

$\left(4,2\right)$ that includes generic anatomies with four cusps, two nodes, a void, two subregions with four and one with two inverse kinematic solutions (IKS), respectively shown in

Figure 5. Subspace 1.2

$\left(4,0\right)$ in

Figure 6b includes metamorphic anatomies with 4 cusps, no nodes, one subregion with four IKS and another one with two IKS, respectively shown in

Figure 5. The surface that divides the subspace 1.2

$\left(4,0\right)$ and subspace 1.3

$\left(4,2\right)$ is the following:

Subspace 1.3

$\left(4,2\right)$ in

Figure 6c contains metamorphic anatomies with four cusps, two nodes, one region with two and three regions with four IKS respectively and five c-sheets shown in

Figure 5. Moreover, subspace 2 in

Figure 3 includes metamorphic anatomies with two cusps and it can be subdivided into two neighboring subspaces. The bifurcating surface is formulated as follows,

The transition between subspace 1.3

$\left(4,2\right)$ to subspace 2.1

$\left(2,1\right)$ is expressed by Equation (12) in

Figure 4a subspace 2.1

$\left(2,1\right)$ exhibits non-generic metamorphic anatomies with two cusps, one node, two subregions with four and one subregion with 2 IKS respectively and 5 aspects. On the other hand, the transition from subspace 2.1

$\left(2,1\right)$ to subspace 2.2

$\left(2,3\right)$ is defined through the boundary strict surface in Equation (17). In

Figure 7a subspace 2.2

$\left(2,3\right)$ includes metamorphic anatomies with two cusps, three nodes, five c-sheets, two subregions with four and two IKS, too as well as the internal intersect with the external boundaries.

Furthermore, subspace 3

$\left(4,4\right)$ in

Figure 4b is a region with four cusps, four nodes, six c-sheets, three subregions with four and two subregions with two IKS, respectively shown in

Figure 5.

Finally, the regular subspace 4 in

Figure 3 is classified into two spaces through the surface in Equation (17). Subspace 4.1

$\left(0,0\right)$ in

Figure 7b includes regular metamorphic anatomies with no nodes, 4 c-sheets, one region with four and two IKS, respectively. Finally, the subspace 4.2

$\left(0,2\right)$ in

Figure 4c includes regular non-generic metamorphic anatomies with two nodes, two subregions with 2 and one subregion with four IKS, respectively and four aspects shown in

Figure 5.

Moreover, three more arm anatomies are exhibited with at least one zero DH-parameter in

Figure 7c,d from subspaces 1.3, 4.1 and 1.3, respectively. The anatomy appeared in

Figure 7c is regular with one region of 4 IKS and 2 aspects. Similarly, the manipulator anatomy in

Figure 7d has one region of 4 IKS but 4 aspects.

Figure 8a illustrates the transformation of metamorphic workspace with a variation of

${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}$ and

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm \frac{\mathsf{\pi}}{2}{,\mathrm{d}}_{4}=0.13$ m and the corresponding topologies are discerned with different colors. However, the number of cusp points remains constant and equal to four. Moreover, the internal singular segments tend to deform in both axes

$\left(\mathsf{\rho},\mathrm{z}\right)$ as well as the location of cusp points is changed too. Moreover, the ratios between the internal subregion i.e., 4 IKS and external one i.e., 2 IKS is changed too.

The two DH-parameters

${\mathrm{d}}_{3}{,\mathrm{r}}_{2}$ depend on the variation of the second passive joint angle

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$ as it is shown in

Table 1. The variation of

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$ in

$\left[-\frac{\mathsf{\pi}}{2},\frac{\mathsf{\pi}}{2}\right]$. causes increasing variation for

${\mathrm{d}}_{3}$, decreasing and increasing for

${\mathrm{r}}_{2}$ in

$\left[0,\frac{\mathsf{\pi}}{2}\right]$,

$\left[-\frac{\mathsf{\pi}}{2},0\right]$ respectively. Consequently, the continuous change of the second pseudo-joint with

${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm \frac{\mathsf{\pi}}{2}{,\mathrm{d}}_{4}=0.13$ m changes in a continuous manner the topology of the workspace as it is plotted in

Figure 8b. The ratios of internal and external regions are varied, the number of cusp and node points is changing as well as the maximum reach of the end-effector of the mechanism is increasing. Moreover, it is also feasible to switch from generic to non-generic manipulators. Finally, the perpendicular distance from the first joint axes is decreased and as a result, the total workspace is placed closer to the local coordinate system of the base (see

Figure 8b in horizontal axis

$\mathsf{\rho}$). Besides, the topological transition from cuspidal to regular anatomy is feasible only with the activation of angular rotation steps of

${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$.

In conclusion, it is worth mentioning that the metamorphosis provides various anatomies from a single structure. Kinematic singularities in the workspace of cuspidal manipulators especially the internal boundaries cause serious drawbacks in planning smooth and continuous trajectories and control. However, metamorphic manipulators overcome this fact since it provides a wide spectrum of arm anatomies and hence a variety of regular or cuspidal topological workspaces with varied shape or volume are created. Therefore, engineers can easily select the anatomy required by the given task, based on the classification and analysis introduced in this work. Then, the position of the trajectory or the points for moving objects based on [

14,

16] can be optimized. In the next section examples of these trajectories are presented.