# A Reconfiguration Algorithm for the Single-Driven Hexapod-Type Parallel Mechanism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanism Architecture

## 3. Kinematic Relations

_{1}′y

_{1}′ is the axis of the rotated coordinate system after the platform rotation by angle φ around axis O

_{1}z

_{1}; O

_{1}″x

_{1}″ is the axis of the rotated coordinate system after the platform rotation by angle θ around axis O

_{1}′y

_{1}′. Elementary rotation matrices included in (2) are calculated in the following way:

**X**:

**X**, i.e., to find vector function

**f**:

**f**(q,

**X**) =

**0**.

**X**with coordinates of other links, which depend on angle q. Thus, (16) presents system (7) that we looked for. In the following section, we will show how to handle the obtained relations to choose the initial orientations of the cranks for different output link trajectories.

## 4. Reconfiguration Algorithm

**X**. Only one of the platform coordinates can have an explicit form. Let us specify the prescribed coordinate as ${X}_{g}$, which represents any component of vector

**X**. We can group the five remaining unknown coordinates as vector ${\mathbf{X}}_{v}$. System (17) in this case can be written as follows:

_{g}(t), variables ${\mathbf{X}}_{v}$ and q, determined in the calculation results, will also be time dependences of ${\mathbf{X}}_{v}$(t) and q(t), which we can represent similarly in a discrete form:

**s**of dimension 6 + 6N:

**f**:

**f**is highly sparse. We can estimate the matrix sparsity, sp, as the ratio of its zero elements to the total number of the elements. In the discussed case, there are 6N × (6 + 6N) elements in total, and the number of nonzero elements is 42N. As a result, we obtain the following equation:

## 5. Examples

^{−8}as a stopping criterion. To find a solution, “fsolve” applies a least-squares method for system (24). At this step, we also provided Jacobian matrix (26), the nonzero elements of which were found in an explicit form by differentiating expression (17) with respect to the corresponding variables (see Appendix A). We set N equal to 125, which corresponded to sp = 0.99 according to expression (28). All variables had zero initial guesses, except output link coordinate ${z}_{P}$ with the initial guess of 195 mm taken for each value of j.

**X**(t), where φ(t) corresponds to expression (29). It follows from Figure 3 that four of the six platform coordinates remain unchanged and equal to zero, while coordinate ${z}_{P}$ changes in a small range.

**X**(t), where ${z}_{P}(t)$ corresponds to expression (31). It follows from Figure 4 that the changes in Cartesian coordinates ${x}_{P}$ and ${y}_{P}$ of the platform center and its orientation angles θ and ψ are negligible, and angle φ varies in a small range. The coordinates have the same values at t = 0 s and t = 5 s.

**X**(t), where ${x}_{P}(t)$ corresponds to expression (33). In this case, all six coordinates of the output link vary and have the same values at t = 0 s and t = 5 s.

## 6. Discussion

- Set desired motion X
_{g}(t) for one output link coordinate and calculate crank initial configuration ${\mathsf{\beta}}^{0}$ together with drive control law q(t) using the proposed reconfiguration algorithm; - On the mechanism, manually disconnect the belts and reorient the cranks according to calculated values ${\mathsf{\beta}}^{0}$. Reconnect the belts;
- Rotate the drive from its initial (zero) position to q(0). At this step, one can use any control law for the drive, e.g., an ordinary rotation with a constant speed;
- Control the drive according to calculated time law q(t).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Components of Jacobian Matrix

**X**. In the expressions below, we omit discretization index j for clarity:

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**Figure 1.**Virtual prototype of the hexapod-type reconfigurable parallel mechanism with single actuation.

**Figure 6.**Angle q calculated for various given output link coordinates ${X}_{g}(t)$: blue line for ${X}_{g}(t)=\mathsf{\phi}(t)$ according to (29); red line for ${X}_{g}{(t)=z}_{P}(t)$ according to (31); green line for ${X}_{g}{(t)=x}_{P}(t)$ according to (33).

**Figure 7.**Output link trajectory with ${X}_{g}(t)=\mathsf{\phi}(t)$ according to (29) and an updated initial guess (35) for ${\mathsf{\beta}}^{0}$.

**Figure 8.**Output link trajectory with ${X}_{g}{(t)=z}_{P}(t)$ according to (31) and an updated initial guess (35) for

**β**

^{0}.

**Figure 9.**Output link trajectory with ${X}_{g}{(t)=x}_{P}(t)$ according to (33) and an updated initial guess (35) for ${\mathsf{\beta}}^{0}$.

**Figure 10.**Angle q calculated for various given output link coordinates ${X}_{g}(t)$ and an updated initial guess (35) for ${\mathsf{\beta}}^{0}$: blue line for ${X}_{g}(t)=\mathsf{\phi}(t)$ according to (29); red line for ${X}_{g}{(t)=z}_{P}(t)$ according to (31); green line for ${X}_{g}{(t)=x}_{P}(t)$ according to (33).

Parameter | i | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

${R}_{1}$, mm | 246.05 | |||||

${R}_{2}$, mm | 62.50 | |||||

${R}_{3i}$, mm | 22.50 | |||||

${R}_{4i}$, mm | 15.00 | |||||

${R}_{5i}$, mm | 30.00 | |||||

${d}_{i}$, mm | 76.00 | |||||

${l}_{i}$, mm | 38.85 | |||||

${\mathsf{\alpha}}_{i}$, deg | 30.0 | 90.0 | 150.0 | 210.0 | 270.0 | 330.0 |

${\mathbf{r}}_{Ei}$, mm | ${\left[\begin{array}{ccc}{R}_{11}{\mathrm{cos}\mathsf{\chi}}_{i}& {R}_{11}{\mathrm{sin}\mathsf{\chi}}_{i}& 0\end{array}\right]}^{\mathrm{T}}$ | |||||

${R}_{11}$, mm | 193.00 | |||||

${\mathsf{\chi}}_{i}$, deg | 10.0 | 110.0 | 130.0 | 230.0 | 250.0 | 350.0 |

${L}_{i}$, mm | 220.05 |

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**MDPI and ACS Style**

Fomin, A.; Antonov, A.; Glazunov, V. A Reconfiguration Algorithm for the Single-Driven Hexapod-Type Parallel Mechanism. *Robotics* **2022**, *11*, 8.
https://doi.org/10.3390/robotics11010008

**AMA Style**

Fomin A, Antonov A, Glazunov V. A Reconfiguration Algorithm for the Single-Driven Hexapod-Type Parallel Mechanism. *Robotics*. 2022; 11(1):8.
https://doi.org/10.3390/robotics11010008

**Chicago/Turabian Style**

Fomin, Alexey, Anton Antonov, and Victor Glazunov. 2022. "A Reconfiguration Algorithm for the Single-Driven Hexapod-Type Parallel Mechanism" *Robotics* 11, no. 1: 8.
https://doi.org/10.3390/robotics11010008