#
Position and Singularity Analysis of a Class of Planar Parallel Manipulators with a Reconfigurable End-Effector^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Bilateration

- if points ${P}_{i}$, ${P}_{j}$ and ${P}_{k}$ are ordered in counterclockwise sense (see Figure 1a), the area is positive;
- if points ${P}_{i}$, ${P}_{j}$ and ${P}_{k}$ are ordered in clockwise sense (see Figure 1b), the area is negative.

## 3. Kinematic Analysis

#### 3.1. Inverse Displacement Analysis

#### 3.2. Direct Displacement Analysis

- (a)
- unknown variable: ${s}_{2,6}$
- (b)
- bilateration sequence: $[({P}_{1},{P}_{2})\to {P}_{6},({P}_{2},{P}_{6})\to {P}_{7},({P}_{3},{P}_{7})\to {P}_{8},({P}_{4},{P}_{8})\to {P}_{9},({P}_{5},{P}_{9})$$\to {P}_{10}]$
- (c)
- closure condition: ${s}_{6,10}={\parallel {\mathbf{p}}_{6}-{\mathbf{p}}_{10}\parallel}^{2}={l}_{5}^{2}$

- The method lends itself easily to generalization for more complex architectures: however, as one may expect, the computational complexity of solving the DKP grows with the number of RRR chains. This is mostly due to the algebraic manipulations required to remove all radicals in the closure equation; these operations are well beyond human feasibility even for $n=4$ and have thus required the use of a symbolic analysis package.
- The time and resources needed to tackle the DKP are dependent on the choice of bilateration sequence and can be greatly reduced through a careful choice. Considering again the example in Figure 3, it was found that using the sequence $[({P}_{1},{P}_{2})\to {P}_{6},({P}_{2},{P}_{6})\to {P}_{7},({P}_{3},{P}_{7})\to {P}_{8},({P}_{5},{P}_{6})\to {P}_{10},({P}_{4},{P}_{10})\to {P}_{9}]$ (see Figure 3b) instead of the one previously indicated in point (b) the time required to obtain a solution is much shorter. The script was run with MATLAB R2019a and an Intel Core processor i7-8700 CPU at 3.20: with this setup, the DKP was solved in two days with the first choice for the bilateration sequence and in ten minutes with the second one. Our empirical observation is that the bilateration sequence is optimized by taking approximately the same number of steps in clockwise sense (such as $({P}_{3},{P}_{7})\to {P}_{8}$ in the last sequence, see Figure 3b) and in counterclockwise sense (such as $({P}_{5},{P}_{6})\to {P}_{10}$): under this guideline, there are fewer nested radicals in the final closure conditions, which then becomes easier to simplify.
- We conjecture that, for an n-RRR robot, the characteristic univariate polynomial has degree ${2}^{n+1}-4$, namely 12, 28, 60 and 124 for, respectively, $n=3$, 4, 5 and 6.

`ga`with a population of 200 individuals and a MATLAB interface to Bertini [54]: for each n, the algorithm iteratively searches for the architecture parameters that lead to the maximum number of real and distinct solutions. For each case, the algorithm converges within 18 generations (or fewer) to an architecture that has as many real and distinct solutions as the characteristic univariate polynomial degree.

## 4. Singularity Analysis

#### 4.1. General Classification

- (1)
- A type-1 kinematic singularity occurs when $det{\mathbf{J}}_{\theta}=0$. In this case, the null space of ${\mathbf{J}}_{\theta}$ is not empty, thus there exists some nonzero $\dot{\theta}$ that yields $\dot{\pi}=\mathbf{0}$ in Equation (9). Therefore, infinitesimal motions of the EE along certain directions cannot be accomplished with finite joint rates and the manipulator loses one or more DoFs. Type-1 kinematic singularities usually occur at the workspace boundary or where different branches of the IKP converge [42].
- (2)
- A type-2 kinematic singularity occurs when $det{\mathbf{J}}_{\pi}=0$. In this case, there exist some nonzero $\dot{\pi}$ that yields $\dot{\theta}=\mathbf{0}$. The EE can have infinitesimal motions while all actuators are locked and the EE gains one or more uncontrolled DoFs. Type-2 kinematic singularities usually occur where different branches of the DKP meet.
- (3)
- A type-3 singularity occurs when both ${\mathbf{J}}_{\pi}$ and ${\mathbf{J}}_{\theta}$ are singular. Generally, this type of singularity can only occur for manipulators with special architectures. At these configurations, Equation (8) degenerates to the identity $\mathbf{0}=\mathbf{0}$. The EE can have infinitesimal motions while the actuators are locked and it can also remain stationary while the actuators undergo infinitesimal motions.

#### 4.2. Definition of the Jacobian Matrices

#### 4.3. Jacobian Matrices of a 5-RRR Pcrp

- (a)
- A type-2 singularity occurs when all links $\overline{{P}_{i}{P}_{i+5}}$ are parallel (Figure 5a). In this case, one can see that the first two columns of ${\mathbf{J}}_{\pi}$ from Equation (16) are linearly dependent and thus the matrix is singular. The EE gains a uncontrolled DoF, namely, the rigid translation in the direction orthogonal to the parallel links.
- (b)
- If links $\overline{{P}_{1}{P}_{6}}$, $\overline{{P}_{6}{P}_{7}}$, and $\overline{{P}_{7}{P}_{2}}$ are aligned (Figure 5b), then $\mathbf{k}\xb7{\mathbf{l}}_{1}\times {\mathbf{d}}_{2}=0$ and ${\mathbf{d}}_{1}$ is a scalar multiple of ${\mathbf{d}}_{2}$, so that the first two rows of ${\mathbf{J}}_{\pi}$ are linearly dependent. In this configuration, points ${P}_{6}$ and ${P}_{7}$ move perpendicularly to ${\mathbf{d}}_{1}$, while the EE undergoes small deformations. By symmetry, this singular configuration extends to the cases where the links $\left(\right)$, $\left(\right)$, $\left(\right)$ or $\left(\right)$ are aligned.
- (c)
- In Figure 5c we show another type of singularity, first noted by Crapo in [45] for a similar type of structure, that can also be applied in our case. For a given link $\overline{{P}_{i}{P}_{i+1}}$ on the EE, we define point ${N}_{i}$, at the intersection of the lines through the links connecting $\overline{{P}_{i}{P}_{i+1}}$ to the base, and point ${Q}_{i}$, at the intersection of the lines through the links on the EE connected to $\overline{{P}_{i}{P}_{i+1}}$; also, let T be at the intersection of the lines $\overleftrightarrow{{Q}_{i}{N}_{i}}$ and $\overleftrightarrow{{Q}_{j}{N}_{j}}$. If the line $\overleftrightarrow{{P}_{k}{P}_{n+k}}$ (through the distal link on the remaining RR chain connecting the EE to the base) also passes through point T, we have a type-2 singular configuration. Note that this includes the special case where all lines $\overleftrightarrow{{P}_{i}{P}_{n+i}}$ pass through the same point T.

## 5. Examples

- (i)
- the EE translates and rotates while keeping a fixed configuration, like a conventional (redundantly-actuated) rigid-EE manipulator;
- (ii)
- the robot switches between two different solutions of the IKP for a given EE pose $\pi $;
- (iii)
- the robot passes through a type-1 singularity configuration (see Section 4);
- (iv)
- finally, the EE configures itself in order to grip a ball at a first position and moves it to a different position, to present a potential application of having a configurable platform. A schematic of this motion is reported for clarity in Figure 7.

## 6. Conclusions

- to prove our conjectures regarding the degree of the characteristic polynomial of the direct kinematics, namely, verifying that it is the polynomial of lowest degree for all n. Furthermore, we aim to find a general example architectures having the maximum possible number of real and distinct solutions (at least for the case $n=6$);
- to obtain more general results for planar PRCPs with n kinematic chains, for instance including chains that have prismatic joints;
- to perform a statistical analysis of the time required to solve the direct kinematics, both with bilateration and through conventional analytical methods, thus showing the difference in the required computational effort;
- to find the full set of singular configurations for any number of kinematic chains, extending the work in Section 4;
- to further develop the prototype in order to apply it to practical manipulation tasks, for instance by using the flexible EE as a gripper, as shown in Figure 7 and in the multimedia attachment.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PRCP | Parallel robots with configurable platform |

EE | End-Effector |

DoF | Degree of Freedom |

DKP | Direct Kinematic Problem |

IKP | Inverse Kinematic Problem |

## Appendix A. Example Architectures

**Figure A1.**Schematics of: (a) a general 3-RRR robot; (b) a general 4-RRR robot. They have, respectively, 3 and 4 DoFs.

## Appendix B. Singularity Equations

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**Figure 1.**Two possible solutions for bilateration with three points P

_{i}, P

_{j}and P

_{k}on a plane ((

**a**): solution with points in counterclockwise order, (

**b**): solution in clockwise order).

**Figure 2.**(

**a**) schematic of the n-RRR robot; (

**b**) the corresponding n-RR structure, obtained by fixing the actuated angles θ

_{i}. Angles ψ

_{i}and φ

_{i}are represented only for the first two RRR chains, for simplicity.

**Figure 3.**(

**a**) a bilateration approach for solving the Direct Kinematic Problem (DKP) of a 5-RR structure; (

**b**) a second possible approach. Each bilateration step is denoted in red.

**Figure 4.**The two configurations of a type-1 singularity for a generic n-RR structure: (

**a**) stretched-out; (

**b**) folded-back.

**Figure 5.**Some examples of type-2 kinematic singularities of a 5-RR structure (derived from a 5-RRR mechanism): (

**a**) a singularity occurring when all links $\overline{{P}_{i}{P}_{n+i}}$ are parallel; (

**b**) a singularity occurring when three consecutive links are aligned; (

**c**) a singularity configuration derived from [45]; (

**d**) the auxiliary figure of Figure 5c. For each structure, we also show (in dashed lines) a configuration which is close to the singular one; this approximates the infinitesimal motion that the structure can have around its singularity (for Figure 5c, we added the auxiliary Figure 5d)

**Figure 7.**A schematic of a grasping motion with the prototype (see multimedia attachment): an object (in gray) is grasped by reconfiguring the EE and moved to a new pose in the plane, through three successive poses (

**a**–

**c**).

**Table 1.**Coordinates of fixed points ${P}_{i}$ ($i\in \{1,2,3\}$) and link lengths for a 3-RR structure whose DKP has 12 real and distinct solutions. Without loss of generality, points ${P}_{1}$ and ${P}_{2}$ are taken respectively at the coordinate-system origin and on the x axis; the other values in the table are found by optimization.

i | ${\mathit{x}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{d}}_{\mathit{i}}$ [mm] | ${\mathit{l}}_{\mathit{i}}$ [mm] |
---|---|---|---|---|

1 | 0 | 0 | 2.031 | 1.087 |

2 | 1 | 0 | 1.890 | 1.449 |

3 | −0.775 | 0.889 | 1.385 | 2.204 |

**Table 2.**Architecture parameters for a 4-RR structure having 28 real and distinct assembly configurations (compare with Table 1).

i | ${\mathit{x}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{d}}_{\mathit{i}}$ [mm] | ${\mathit{l}}_{\mathit{i}}$ [mm] |
---|---|---|---|---|

1 | 0 | 0 | 2.196 | 1.373 |

2 | 1 | 0 | 1.952 | 1.606 |

3 | 1.017 | −0.998 | 1.703 | 1.881 |

4 | 2.071 | −1.056 | 2.186 | 2.200 |

**Table 3.**Architecture parameters for a 5-RR structure having 60 real and distinct assembly configurations (compare with Table 1).

i | ${\mathit{x}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{\mathit{i}}}$ [mm] | ${\mathit{d}}_{\mathit{i}}$ [mm] | ${\mathit{l}}_{\mathit{i}}$ [mm] |
---|---|---|---|---|

1 | 0 | 0 | 1.888 | 1.714 |

2 | 1 | 0 | 2.221 | 2.211 |

3 | −1.101 | −0.0284 | 2.131 | 2.049 |

4 | −1.399 | −2.088 | 2.099 | 1.857 |

5 | −2.201 | −0.442 | 1.946 | 2.186 |

**Table 4.**All 12 possible solutions of the DKP for the 3-RR architecture in Table 1. Each solution is defined by the value of the unknown ${s}_{2,4}$ in the characteristic polynomial; from this value, the coordinates of points ${P}_{i}$ ($i\in \{4,5,6\}$) are found through bilateration and the pose $\pi ={\left(\right)}^{{x}_{{P}_{4}}}T$ is derived.

${\mathit{s}}_{2,4}$ | ${\mathit{x}}_{{\mathit{P}}_{4}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{4}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{s}}_{2,4}$ | ${\mathit{x}}_{{\mathit{P}}_{4}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{4}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] |
---|---|---|---|---|---|---|---|

1.419 | 1.852 | 0.832 | 112.254 | 6.803 | −0.840 | −1.849 | −175.797 |

2.649 | 1.237 | −1.610 | −145.848 | 7.686 | −1.281 | 1.575 | −124.472 |

3.991 | 0.566 | 1.950 | 107.386 | 7.925 | −1.401 | −1.470 | −94.413 |

4.921 | 0.101 | 2.028 | 84.062 | 8.175 | −1.526 | 1.340 | −6.086 |

5.717 | −0.297 | −2.009 | 6.196 | 8.366 | −1.621 | 1.222 | −49.419 |

6.327 | −0.602 | 1.939 | −6.847 | 8.728 | −1.803 | 0.935 | −9.103 |

**Table 5.**All 28 possible solutions of the DKP for the 4-RR architecture in Table 2.

${\mathit{s}}_{2,5}$ | ${\mathit{x}}_{{\mathit{P}}_{5}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{5}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{\varphi}}_{2}$ [°] | ${\mathit{s}}_{2,5}$ | ${\mathit{x}}_{{\mathit{P}}_{5}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{5}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{\varphi}}_{2}$ [°] |
---|---|---|---|---|---|---|---|---|---|

1.439 | 2.191 | 0.141 | 88.308 | −82.185 | 2.877 | 1.472 | −1.629 | −175.676 | −34.982 |

1.496 | 2.162 | −0.380 | −100.786 | 66.259 | 3.364 | 1.228 | −1.820 | 23.770 | 103.060 |

1.538 | 2.141 | 0.486 | −60.445 | 129.994 | 3.709 | 1.056 | 1.925 | −161.920 | −29.821 |

1.567 | 2.127 | −0.546 | −109.921 | −156.138 | 4.343 | 0.739 | −2.068 | 17.808 | −139.497 |

1.575 | 2.123 | −0.560 | 57.699 | 176.872 | 4.640 | 0.590 | 2.115 | −16.369 | −139.009 |

1.796 | 2.012 | 0.879 | 128.995 | −64.617 | 5.647 | 0.0867 | 2.194 | −12.185 | −69.531 |

1.811 | 2.005 | −0.895 | 46.566 | −100.651 | 6.318 | −0.249 | −2.181 | 9.747 | 174.167 |

1.812 | 2.004 | −0.897 | −130.048 | 17.285 | 6.432 | −0.306 | −2.174 | 9.346 | 12.964 |

1.885 | 1.968 | −0.974 | −134.621 | 168.071 | 6.551 | −0.365 | −2.165 | 106.607 | −64.919 |

1.898 | 1.961 | −0.987 | 43.857 | 155.226 | 8.334 | −1.257 | 1.800 | −74.637 | −82.443 |

1.910 | 1.955 | −0.999 | −136.083 | 7.983 | 8.774 | −1.477 | −1.625 | 65.920 | −72.265 |

1.924 | 1.948 | −1.013 | 43.135 | −102.777 | 10.076 | −2.128 | −0.542 | −10.938 | 53.300 |

2.119 | 1.851 | 1.181 | −38.567 | −98.047 | 10.166 | −2.173 | −0.316 | −14.058 | −68.241 |

2.291 | 1.765 | −1.306 | −154.742 | 138.322 | 10.172 | −2.176 | −0.295 | 24.982 | 13.511 |

**Table 6.**All 60 possible solutions of the DKP for the 5-RR architecture in Table 3. Notice that some solutions are very close in terms of ${s}_{2,6}$ (and are thus indistinguishable from each other without providing more significant digits that those which can be displayed in the available space), yet lead to clearly different poses.

${\mathit{s}}_{2,6}$ | ${\mathit{x}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{\varphi}}_{2}$ [°] | ${\mathit{\varphi}}_{3}$ [°] | ${\mathit{s}}_{2,6}$ | ${\mathit{x}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{\varphi}}_{2}$ [°] | ${\mathit{\varphi}}_{3}$ [°] |
---|---|---|---|---|---|---|---|---|---|---|---|

0.788 | 1.888 | −0.0132 | 65.8 | −134.9 | −99.3 | 6.241 | −0.839 | −1.691 | −17.7 | 178.1 | −116.0 |

0.788 | 1.887 | 0.0191 | −65.4 | 135.2 | −99.3 | 6.244 | −0.840 | 1.690 | 17.7 | −83.8 | −99.3 |

0.990 | 1.786 | −0.610 | 35.1 | 178.4 | −99.6 | 6.409 | −0.923 | −1.647 | 100.0 | 23.4 | −99.4 |

1.086 | 1.739 | −0.735 | −120.1 | 178.2 | 16.3 | 6.444 | −0.940 | −1.637 | −19.1 | 177.9 | 122.3 |

1.098 | 1.733 | −0.749 | −121.2 | 178.1 | −17.6 | 6.457 | −0.947 | −1.633 | 99.2 | −92.2 | 31.2 |

1.192 | 1.686 | −0.850 | −128.7 | 177.8 | 106.1 | 6.474 | −0.956 | −1.628 | 98.9 | −95.1 | −119.3 |

1.246 | 1.659 | −0.901 | −132.5 | 177.6 | 92.0 | 6.493 | −0.965 | −1.622 | 98.5 | −98.1 | 122.1 |

1.273 | 1.645 | 0.926 | −24.1 | 177.4 | −99.5 | 6.502 | −0.970 | −1.620 | 98.4 | −99.4 | 108.6 |

1.396 | 1.583 | 1.028 | 141.9 | −175.7 | −94.8 | 6.534 | −0.986 | −1.610 | −19.8 | 177.8 | 108.6 |

1.467 | 1.548 | 1.080 | 145.8 | −175.4 | −95.1 | 6.635 | −1.036 | −1.578 | 96.1 | −116.5 | −98.1 |

1.876 | 1.343 | −1.326 | −164.0 | 175.6 | 53.0 | 6.676 | −1.056 | 1.564 | −95.3 | 103.4 | −93.3 |

2.061 | 1.251 | −1.413 | 10.8 | 149.5 | −178.5 | 6.711 | −1.074 | 1.552 | −94.7 | 108.4 | −94.5 |

2.061 | 1.251 | −1.413 | −170.7 | 48.3 | 179.5 | 6.859 | −1.148 | 1.499 | 22.3 | −176.0 | −93.6 |

2.070 | 1.246 | −1.418 | 10.7 | −179.0 | 147.2 | 6.962 | −1.200 | 1.457 | −90.2 | 133.7 | −96.6 |

2.226 | 1.169 | −1.482 | 9.1 | −179.2 | −98.8 | 7.084 | −1.260 | 1.405 | 24.2 | −175.8 | −94.4 |

2.606 | 0.978 | −1.614 | 5.6 | 138.4 | −99.3 | 7.465 | −1.451 | 1.207 | −80.5 | 7.5 | −99.2 |

2.606 | 0.978 | −1.614 | 172.9 | 37.5 | −99.3 | 8.097 | −1.767 | −0.663 | −37.8 | 49.2 | 179.6 |

3.501 | 0.531 | −1.811 | 151.9 | 169.4 | −97.6 | 8.097 | −1.767 | −0.663 | 64.7 | −21.0 | −176.7 |

3.747 | 0.408 | 1.843 | 2.5 | −178.2 | −107.9 | 8.123 | −1.780 | −0.628 | 63.9 | −167.4 | −12.1 |

4.420 | 0.0717 | 1.886 | 6.5 | −109.7 | −179.3 | 8.134 | −1.786 | −0.612 | −38.7 | 175.0 | 47.0 |

4.420 | 0.0716 | 1.886 | −134.1 | −13.8 | −175.8 | 8.163 | −1.800 | −0.569 | −39.5 | 174.9 | −97.9 |

4.605 | −0.0210 | 1.887 | −130.8 | −11.7 | −99.4 | 8.295 | −1.866 | −0.285 | 56.0 | −169.9 | −97.1 |

4.819 | −0.128 | 1.883 | −127.0 | −158.3 | 8.5 | 8.323 | −1.880 | −0.171 | 53.6 | −170.6 | −97.0 |

5.678 | −0.558 | −1.803 | 112.4 | 126.2 | −96.2 | 8.331 | −1.884 | −0.119 | −47.8 | 173.2 | −97.8 |

5.830 | −0.633 | −1.778 | 109.8 | 111.9 | −95.0 | 8.333 | −1.885 | 0.0953 | −52.1 | 172.1 | 27.8 |

6.047 | −0.742 | 1.736 | −106.1 | −104.6 | 86.6 | 8.334 | −1.885 | 0.0904 | 48.4 | −171.8 | −31.1 |

6.053 | −0.745 | −1.734 | −16.4 | 178.2 | 28.8 | 8.336 | −1.886 | 0.0671 | 48.8 | −36.1 | −177.8 |

6.080 | −0.758 | 1.728 | −105.6 | −100.2 | 103.5 | 8.336 | −1.886 | 0.0671 | −51.5 | 33.2 | 178.9 |

6.097 | −0.767 | 1.725 | −105.3 | −97.7 | 127.1 | 8.338 | −1.887 | −0.0241 | −49.7 | 35.1 | −99.3 |

6.126 | −0.782 | 1.718 | −104.8 | −93.2 | 4.2 | 8.338 | −1.887 | −0.0240 | 50.6 | −34.2 | −99.3 |

**Table 7.**Coordinates of the ground joints (points ${A}_{i}$) and the corresponding input angles ${\theta}_{i}$; the links’ lengths are ${c}_{i}=160$, ${d}_{i}=120$ and ${l}_{i}=80$ (for $i=1,\cdots ,5$).

i | ${\mathit{x}}_{{\mathit{A}}_{\mathit{i}}}$ [mm] | ${\mathit{y}}_{{\mathit{A}}_{\mathit{i}}}$ [mm] | ${\mathit{\theta}}_{\mathit{i}}$ [°] |
---|---|---|---|

1 | 0 | 0 | 64.8 |

2 | 330 | 0 | 115.2 |

3 | 432 | 314 | 201.67 |

4 | 165 | 508 | 237.6 |

5 | −102 | 314 | 320.4 |

**Table 8.**The three real solutions of the DKP for the 5-RR prototype defined in Table 7 and the corresponding poses $\pi $.

${\mathit{s}}_{2,6}$ | ${\mathit{x}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{y}}_{{\mathit{P}}_{6}}$ [mm] | ${\mathit{\varphi}}_{1}$ [°] | ${\mathit{\varphi}}_{2}$ [°] | ${\mathit{\varphi}}_{3}$ [°] |
---|---|---|---|---|---|

6022 | 186.620 | 125.830 | 113.294 | 82.320 | −161.487 |

21,190 | 147.477 | 234.790 | −93.704 | 73.103 | 78.503 |

32,029 | 119.506 | 253.215 | −4.348 | 98.934 | −131.219 |

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**MDPI and ACS Style**

Marchi, T.; Mottola, G.; Porta, J.M.; Thomas, F.; Carricato, M.
Position and Singularity Analysis of a Class of Planar Parallel Manipulators with a Reconfigurable End-Effector. *Machines* **2021**, *9*, 7.
https://doi.org/10.3390/machines9010007

**AMA Style**

Marchi T, Mottola G, Porta JM, Thomas F, Carricato M.
Position and Singularity Analysis of a Class of Planar Parallel Manipulators with a Reconfigurable End-Effector. *Machines*. 2021; 9(1):7.
https://doi.org/10.3390/machines9010007

**Chicago/Turabian Style**

Marchi, Tommaso, Giovanni Mottola, Josep M. Porta, Federico Thomas, and Marco Carricato.
2021. "Position and Singularity Analysis of a Class of Planar Parallel Manipulators with a Reconfigurable End-Effector" *Machines* 9, no. 1: 7.
https://doi.org/10.3390/machines9010007