# An Experimental Investigation of the Dynamic Performances of a High Speed 4-DOF 5R Parallel Robot Using Inverse Dynamics Control

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## Abstract

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## 1. Introduction

- The literature does not show research conducted on an industrial 5R robot for the inverse dynamics control evaluation. Some authors use simple mechanical systems, such as in [12] or [13], where experimental tests are conducted on 2-DOF prototypes that move slowly and hence with negligible dynamic contributions to the joint torques;
- In almost all articles, except for [7], the trajectories used to evaluate the performance of control systems are not meaningful to the pick-and-place application. Specifically, the trajectories with low dynamics, or not representative of a typical pick-and-place motion, are used;
- None of the reviewed papers quantifies the contributions of the various components of validated control systems;
- The inverse dynamics controller is almost always applied only in the joint space.

- The experimental activities are conducted on a robot designed for high-speed pick-and-place operations in industrial applications; furthermore, the performances of the system are investigated using a pick-and-place trajectory [14] that fully exploits the manipulator’s characteristics;
- The pick-and-place trajectory used for the experimental activities does not lie just on a plane, but it also has out-of-plane sections; moreover, it features both highly dynamic and quasi-static portions;
- The ID controller is applied in the task space and not, as is commonly done, in the joint space;
- The different contributions that constitute the global controller signal are evaluated and discussed.

## 2. Methods

#### 2.1. Dynamic Model of the Manipulator

**τ**are the torques applied at the output shaft of the four gearboxes. It should be noted that the planar linkage operates in the horizontal plane, and hence its potential energy is constant; also it may be remarked that the BSS is a linear time-invariant system and, therefore, not subject to Coriolis and centrifugal actions. Finally, the following matrix:

#### 2.2. Control Schemes

- Both the mass matrix ${\mathit{M}}_{\mathit{p}}$ of the system and the feedback linearization term ${\mathit{n}}_{\mathit{p}}$ are computed as a function of the four task space coordinates of the system according to the dynamic model expressed in Equation (15);
- Each torque setpoint is calculated according to the errors on more than one coordinate, since the mass matrix is not purely diagonal.

#### 2.3. Workcycle Description

- (a)–(b): Pick intercept motion that brings manipulator’s end-effector directly above and aligned with its target item; at point (b) the velocity of the robot matches that of the pick conveyor;
- (b)–(c): Descent motion that brings the end-effector in contact with the item to be picked; during this phase the velocity of the conveyor belt is tracked in the longitudinal direction;
- (c)–(d): Grasping motion, during which the conveyor velocity is tracked as the gripping tool on the end-effector operates to collect the item;
- (d)–(e): Ascent motion, in which the end-effector moves away from the conveyor while still tracking its longitudinal velocity to avoid any collision with other items on the belt;
- (e)–(f): Intercept motion needed to reach the moving place position (e.g., an empty box carried by the place conveyor);
- (f)–(g): Descent motion, in which the item held by the robot is lowered to the level of the place conveyor;
- (g)–(h): Deposit motion, in which the item is released on the conveyor;
- (h)–(i): Ascent motion;
- (i)–(a): Auxiliary deceleration motion that brings the robot to a resting state.

^{−2}. Concerning the workcycle geometry, it can be seen in Figure 6 that it occupies a large portion of the useful workspace. The more dynamic portions of the workcycle are constituted by the target intercept motions, during which the end-effector should quickly reach its destination; within these phases the setpoint tracking accuracy is not paramount, while the attainment of high speed is more important given the need to reduce the cycle time. As will be shown in the following section, the required torques are quite significant during these phases, and briefly reach the saturation levels, which were set to the maximal torque compatible with the power drive systems. Torque saturations were purposefully allowed in order to ensure the full exertion of the system’s dynamic capabilities. On the contrary, the conveyor tracking motions are performed at a lower and constant end-effector velocity, compatible with typical industrial applications. In these portions of the workcycle, the manipulator should accurately track the conveyor and guarantee high precision, due to the finer nature of the grasping and release operations. Therefore, the proposed test cycle covers a wide range of situations, and allows the investigation both of highly dynamic working conditions, characterized by high accelerations and velocities, and of the quasi-stationary motions experienced during the conveyor tracking phases.

#### 2.4. Experimental Setup

- The state machine, which implements the main operating logic;
- The safety logic subsystem protecting the experimental setup from user or programming errors;
- The EtherCAT communication Master, natively included in Simulink Real-Time, which deals with communication from and to the field devices;
- The implementation of the previously described controllers.

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the 4-DOF manipulator with its main functional subelements. All the notable points and angles are also indicated.

**Figure 8.**Experimental setup showing the main components of the robot, the SLRT target PC, and the development laptop.

**Figure 9.**Control system architecture, including the main software functions and the EtherCAT fieldbus.

**Figure 11.**Error norms in the xy plane, projected along the path normal, and magnified by a factor of 20.

**Figure 12.**Norm of the in-plane errors; absolute values of the vertical translation and rotation errors.

Description | Symbol | Value |
---|---|---|

Length of the proximal links | ${l}_{p}$ | $250\mathrm{m}\mathrm{m}$ |

Length of the distal links | ${l}_{d}$ | $250\mathrm{m}\mathrm{m}$ |

Frame length | ${l}_{f}$ | $180\mathrm{m}\mathrm{m}$ |

Mass of the proximal links | ${m}_{p,1},{m}_{p,2}$ | $2.9\mathrm{k}\mathrm{g}$ |

Mass of the distal links | ${m}_{d,1},{m}_{d,2}$ | $2.9\mathrm{k}\mathrm{g}$ |

Barycentric inertia of the proximal links | ${J}_{p,1},{J}_{p,2}$ | 5.22 × 10^{−2} kg m^{2} |

Barycentric inertia of the distal links | ${J}_{d,1},{J}_{d,2}$ | 5.22 × 10^{−2} kg m^{2} |

Mass of the BSS and of the end-effector | ${m}_{ee}$ | $0.36\mathrm{k}\mathrm{g}$ |

BSS pitch | ${p}_{bss}$ | $20\mathrm{m}\mathrm{m}$ |

Rotational inertia of the end effector | ${J}_{ee}$ |
6.40 × 10^{−6} kg m^{2} |

Ball screw nut’s moment of inertia | ${J}_{3}$ |
1.20 × 10^{−6} kg m^{2} |

Spline nut’s moment of inertia | ${J}_{4}$ |
1.20 × 10^{−6} kg m^{2} |

Description | Symbol | Axis 1 | Axis 2 | Axis 3 | Axis 4 |
---|---|---|---|---|---|

Rated torque | ${\mathit{\tau}}_{pds,rated},\left[\mathrm{N}\mathrm{m}\right]$ | 0.7 | 0.7 | 0.36 | 0.36 |

Peak torque | ${\mathit{\tau}}_{pds,max},\left[\mathrm{N}\mathrm{m}\right]$ | 1.4 | 1.4 | 0.72 | 0.72 |

Peak velocity | ${\omega}_{pds,max},\left[\mathrm{rad}/\mathrm{s}\right]$ | 500 | 500 | 500 | 500 |

Motor inertia | ${J}_{m},\left[\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{m}}^{2}\right]$ | 17 | 17 | 2.4 | 2.4 |

Transmission efficiency | ${\eta}_{t}$ | $\sim 1$ | $\sim 1$ | $\sim 1$ | $\sim 1$ |

Transmission inertia | ${J}_{t},\left[\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{m}}^{2}\right]$ | 24.85 | 24.85 | 20.5 | 20.5 |

Reduction ratio | ${i}_{t}$ | 64 | 64 | 40 | 40 |

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**MDPI and ACS Style**

Righettini, P.; Strada, R.; Cortinovis, F.; Tabaldi, F.; Santinelli, J.; Ginammi, A.
An Experimental Investigation of the Dynamic Performances of a High Speed 4-DOF 5R Parallel Robot Using Inverse Dynamics Control. *Robotics* **2024**, *13*, 54.
https://doi.org/10.3390/robotics13030054

**AMA Style**

Righettini P, Strada R, Cortinovis F, Tabaldi F, Santinelli J, Ginammi A.
An Experimental Investigation of the Dynamic Performances of a High Speed 4-DOF 5R Parallel Robot Using Inverse Dynamics Control. *Robotics*. 2024; 13(3):54.
https://doi.org/10.3390/robotics13030054

**Chicago/Turabian Style**

Righettini, Paolo, Roberto Strada, Filippo Cortinovis, Federico Tabaldi, Jasmine Santinelli, and Andrea Ginammi.
2024. "An Experimental Investigation of the Dynamic Performances of a High Speed 4-DOF 5R Parallel Robot Using Inverse Dynamics Control" *Robotics* 13, no. 3: 54.
https://doi.org/10.3390/robotics13030054