# Investigation of Cyclicity of Kinematic Resolution Methods for Serial and Parallel Planar Manipulators

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## Abstract

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## 1. Introduction

- Null space methods: A particular solution $\dot{\mathbf{q}}$ of (2) is determined by adding a vector in the null-space of the Jacobian. The latter is usually the gradient of a scalar objective function that is to be maximized (or minimized).
- Task augmentation methods: Redundancy is eliminated by adding r auxiliary tasks, in order to make the overall system non-redundant.

## 2. Redundancy Resolution Methods

- PGMThe PGM exploits the fact that a general solution of the differential kinematics can be substituted to Equation (4) when a desired joint rate vector $\mathbf{v}$ is projected into the null space of $\mathbf{J}$:$$\dot{\mathbf{q}}={\mathbf{J}}^{\#}({\dot{\mathbf{p}}}_{d}+\mathbf{G}\mathbf{e})-\mathbf{P}\mathbf{v},$$
- AJMA different approach is followed in the AJM. An additional constraint task is imposed to the original task of the EE. Following [7,12], the objective function $h\left(\mathbf{q}\right)$ is projected onto the null space of $\mathbf{J}$ and imposed to be zero. Formally we can write:$$\mathbf{g}\left(\mathbf{q}\right)={\mathbf{Z}}^{T}\mathbf{v}=\mathbf{0}.$$$\mathbf{Z}\in {\mathbb{R}}^{n\times r}$ is an orthonormal basis for the null space of $\mathbf{J}$ and $\mathbf{v}$ is the gradient of $h\left(\mathbf{q}\right)$ with respect to the joint variables as in the previous method. Therefore, Equation (6) yields r independent constraints keeping $h\left(\mathbf{q}\right)$ at the extreme at each time of the trajectory starting from the initial configuration ${\mathbf{q}}_{0}$.The added Jacobian ${\mathbf{J}}_{a}\in {\mathbb{R}}^{r\times n}$ can simply obtained as$${\mathbf{J}}_{a}=\frac{\partial \mathbf{g}\left(\mathbf{q}\right)}{\partial \mathbf{q}},$$$$\dot{\mathbf{q}}={\mathbf{J}}_{aug}^{-1}(\left[\begin{array}{c}{\dot{\mathbf{p}}}_{d}\\ \mathbf{0}\end{array}\right]+\left[\begin{array}{cc}\mathbf{G}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathbf{e}\\ \mathbf{0}\end{array}\right]),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{J}}_{aug}=\left[\begin{array}{c}\mathbf{J}\\ {\mathbf{J}}_{a}\end{array}\right].$$

## 3. Numerical Simulations

#### 3.1. Serial 4R

#### 3.2. Parallel 2RRP

#### 3.3. Serial 6R

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**$h\left(\mathbf{q}\right)$ in 4R with $\u03f5=0.3$. PGM: − (red line), AJM: $--$ (blue line).

**Figure 5.**AJM: $h\left(\mathbf{q}\right)$ in 4R. $\u03f5=0$: $--$ (black line), $\u03f5=0.25$: $-.-$ (red line), $\u03f5=0.5$: − (blue line).

**Figure 6.**PGM: $h\left(\mathbf{q}\right)$ in 4R. $\u03f5=0$: $--$ (black line), $\u03f5=0.5$: $-.-$ (red line), $\u03f5=1.0$: − (blue line), $\u03f5=1.5$: ⋯ (green line).

**Figure 8.**$h\left(\mathbf{q}\right)$ in 2RRP with $\u03f5=0.15$. PGM: − (red line), AJM: $--$ (blue line).

**Figure 9.**AJM: $h\left(\mathbf{q}\right)$ in 2RRP. $\u03f5=0$: $--$ (black line), $\u03f5=0.1$: $-.-$ (red line), $\u03f5=0.15$: − (blue line).

**Figure 10.**PGM: $h\left(\mathbf{q}\right)$ in 2RRP. $\u03f5=0$: $--$ (black line), $\u03f5=0.5$: $-.-$ (red line), $\u03f5=1.0$: − (blue line), $\u03f5=1.5$: ⋯ (green line).

**Figure 12.**$h\left(\mathbf{q}\right)$ in 6R with $\u03f5=0.5$. PGM: − (red line), AJM: $--$ (blue line).

**Figure 13.**AJM: $h\left(\mathbf{q}\right)$ in 6R. $\u03f5=0$: $--$ (black line), $\u03f5=0.25$: $-.-$ (red line), $\u03f5=0.5$: − (blue line).

**Figure 14.**PGM: $h\left(\mathbf{q}\right)$ in 6R. $\u03f5=0$: $--$ (black line), $\u03f5=0.5$: $-.-$ (red line), $\u03f5=1.0$: − (blue line), $\u03f5=1.5$: ⋯ (green line).

**Figure 15.**PGM with v and ∞-trajectory: $h\left(\mathbf{q}\right)$ in 2RRP. $\u03f5=0$: − (black line), $\u03f5=0.75$: $--$ (red line), $\u03f5=1.5$: $-.$ (blue line).

**Table 1.**4R. a: links length; (${x}_{0},{y}_{0}$), R: coordinates of the centre and radius of the EE trajectory.

a = 0.5 m | ${x}_{0}$ = 1.0 m | ${y}_{0}$ = 1.0 m | R = 0.5 m |

**Table 2.**2RRP. a: links length; H: base width, (${x}_{0},{y}_{0}$), R: coordinates of the centre and radius of the EE trajectory.

a = 0.2 m | H = 1 m | ${x}_{0}$ = 0.5 m | ${y}_{0}$ = 0.5 m | R = 0.1 m |

**Table 3.**6R. (${x}_{0},{y}_{0},{z}_{0}$), R: coordinates of the centre and radius of the EE trajectory. EE trajectory on the plane $y=-0.256$ m. ${d}_{1},{a}_{2},{a}_{3},{d}_{4},{d}_{5},{d}_{6}$: Denavit-Hartenberg geometrical parameters.

${x}_{0}$ = 0.572 m | ${y}_{0}=-0.256$ m | ${z}_{0}$ = 0.496 m | R = 0.153 m | ${d}_{1}$ = 0.127 m |

${a}_{2}$ = 0.612 m | ${a}_{3}$ = 0.572 m | ${d}_{4}$ = 0.164 m | ${d}_{5}$ = 0.116 m | ${d}_{6}$ = 0.092 m |

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Ruggiu, M.; Müller, A.
Investigation of Cyclicity of Kinematic Resolution Methods for Serial and Parallel Planar Manipulators. *Robotics* **2021**, *10*, 9.
https://doi.org/10.3390/robotics10010009

**AMA Style**

Ruggiu M, Müller A.
Investigation of Cyclicity of Kinematic Resolution Methods for Serial and Parallel Planar Manipulators. *Robotics*. 2021; 10(1):9.
https://doi.org/10.3390/robotics10010009

**Chicago/Turabian Style**

Ruggiu, Maurizio, and Andreas Müller.
2021. "Investigation of Cyclicity of Kinematic Resolution Methods for Serial and Parallel Planar Manipulators" *Robotics* 10, no. 1: 9.
https://doi.org/10.3390/robotics10010009