# A New Energy-Efficient Approach to Planning Pick-and-Place Operations

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## Abstract

**:**

## 1. Introduction

- Minimum cycle time duration;
- Minimum energy consumption, minimum total torque or minimum driving force;
- Smoothness of the higher-order derivatives of trajectories.

- Optimisation based on hardware solutions;
- Optimisation based on software solutions.

- Minimising the specific energy consumption of the transport cycle;
- Minimisation of transport cycle time;
- Minimising the economic cost of realising the transport cycle.

- A proposal of a new, energy-efficient and implementable approach to path and trajectory design by optimising the trajectory of TCP and solving the 6DoF robot placement problem.
- The industrially acceptable solution for setting an optimal position of the robot integrated with an autonomous mobile platform.
- The proposal of cost-cutting for pick-and-place operation via a multi-criteria optimisation, where both energy consumption and process efficiency are addressed.
- Experimental proof of the proposed approach.

## 2. Characterisation of the Pick-and-Place Tasks

## 3. Minimisation of Specific Energy Consumption

- The determination of an analytical model of the electric energy consumption of each robot drive as a function of the geometric, kinematic, and dynamic parameters;
- The design of a function to determine the cost and constraints of a trajectory optimisation task;
- Performing optimisation calculations.

#### 3.1. The Analytical Model

- Forward and inverse kinematics;
- Forward and inverse dynamics;
- Determining the energy consumption.

#### 3.1.1. Forward and Inverse Kinematics

- ${a}_{i-1}$—distance between the ${z}_{i-1}$ and ${z}_{i}$ axes measured along the ${x}_{i-1}$ axis;
- ${\alpha}_{i-1}$—the angle between the ${z}_{i-1}$ and ${z}_{i}$ axes;
- ${d}_{i}$—distance between the ${x}_{i-1}$ and ${x}_{i}$ axes measured along the ${z}_{i}$ axis;
- ${\theta}_{i}$—angle between the axes ${x}_{i-1}$ and ${x}_{i}$.

#### 3.1.2. Forward and Inverse Dynamics

- The Newton–Euler method;
- The Lagrange equation of the second kind.

- Uncertainties in the size and mass of the mechanical components, resulting in an inaccurate determination of the centres of gravity of the links;
- Uncertainties in the estimation of the parameters of the drive chain components;
- Nonlinear properties of the control system;
- Nonlinearities of harmonic transmissions;
- Neglecting power losses in inverter, cables and connectors.

## 4. Definition of the Optimisation Problem

- Minimises the duration of the transport task (cycle);
- Minimises the specific energy consumption required for a single transport task;
- Minimises the economic cost of the transport task taking into account both the energy cost and the depreciation cost of the robotics workspace.

#### 4.1. Optimisation of the Duration of the Transport Task

#### 4.2. Optimisation of the Specific Energy Consumption

#### 4.3. Optimisation of Economic Cost

## 5. Case Study

#### 5.1. ES5 Robot

#### 5.2. Research Methodology

- A quite typical variant of the transportation task was chosen. It imitates a typical pick-and-place task encountered in industrial practice;
- This variant relies on the movement of the workpiece from the pick-up point to the drop-off point, with a change in tool orientation between both positions. A graphical illustration of this task is shown in Figure 7.

- The effectiveness of the developed approach in terms of searching for the optimal position of the robot in the $xy$-plane due to the minimisation of the specific cycle time;
- The effectiveness of the developed approach in terms of searching for the optimal position of the robot in the $xy$-plane of the task due to minimum energy consumption;
- The effectiveness of the developed approach in terms of searching for the optimal position of the robot in the $xy$-plane of the task, due to a mixed criterion imposing a penalty on the cycle execution time and rewarding the savings of the energy expenditures necessary for its execution. In this case, the coefficients $\alpha $ and $\beta $ (37) are calculated based on the parameters given in Table 5.

#### 5.3. Test Bench

## 6. Research Experiment

#### 6.1. Verification of the Analytical Model of Specific Energy Consumption

#### 6.2. Minimisation of Transport Cycle Time

- The minimum cycle time;
- The minimum energy consumed;
- The minimum economic cost.

#### 6.3. Minimisation of Specific Energy Consumption

#### 6.4. Minimisation of the Economic Cost

## 7. Discussion

#### 7.1. Verification of the Analytical Model of Specific Energy Consumption

#### 7.2. Minimisation of Transport Cycle Time

- The robot base location has a significant impact on both the cycle time and the energy expenditure required to complete the cycle;
- The criterion that optimises cycle time is inefficient from an energy point of view in the sense that even a small increase in this time allows for significant savings in energy consumption;
- In the absence of an upper limit on cycle time, a mixed optimisation criterion should be used.

#### 7.3. Minimisation of the Specific Energy Consumption

- The choice of the location of the robot base relative to the task space has a significant impact on the value of specific energy consumption;
- The optimising specific energy consumption is recommended to be used before the searching for a minimum cycle time. Significant energy savings can be achieved at the expense of a small increase in cycle time.

#### 7.4. Minimisation of the Economic Cost

#### 7.5. Comparison of the Achieved Results

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The annual number of newly installed robots in 2020, in thousands of pieces, broken down by categories of tasks. There are three main task categories: pick-and-place, processing and unclassified. The drawing was elaborated based on [1].

**Figure 3.**Three-dimensional drawing of the ES5 robot with assigned local coordinate systems according to the Denavit–Hartenberg representation.

**Figure 6.**Illustration of the result of optimising the location of the robot base position relative to the task space.

**Figure 7.**Graphical illustration of a transport task. The task relies on moving a piece, combined with the change of tool orientation. Symbols: A—pick-up point, B—put-down point.

**Figure 10.**Results of the experimental verification of the specific energy consumption. The charts (

**a**–

**f**) show the instantaneous power consumption of all six robot’s motors. The green plot indicates instantaneous power of the model output; red plot indicates experimentally measured instantaneous power. Blue plot indicates the relative modelling error.

**Figure 13.**Specific transport cost [USD] in terms of the displacement vector of the robot base [dx,dy].

Joint Number | n | ${\mathit{k}}_{\mathit{T}}$ [Nm/A] | ${\mathit{k}}_{\mathit{e}}$ [V/rad/s] | ${\mathit{R}}_{\mathit{t}}$ [$\mathsf{\Omega}$] | L [mH] |
---|---|---|---|---|---|

1 | 101:1 | 0.1418 | 0.12 | 0.7 | 0.9 |

2 | 121:1 | 0.1418 | 0.12 | 0.7 | 0.9 |

3 | 101:1 | 0.1418 | 0.12 | 0.7 | 0.9 |

4 | 101:1 | 0.1636 | 0.08 | 3.5 | 3.4 |

5 | 101:1 | 0.1636 | 0.08 | 3.5 | 3.4 |

6 | 101:1 | 0.1636 | 0.08 | 3.5 | 3.4 |

_{T}—motor mechanical constant; k

_{e}—motor electrical constant; R

_{t}—motor resistance; L—inductance of motor windings.

Joint Number | ${\mathit{a}}_{\mathit{i}-1}$ [m] | ${\mathit{\alpha}}_{\mathit{i}-1}$ [°] | ${\mathit{d}}_{\mathit{i}}$ [m] | ${\mathit{\theta}}_{\mathit{i}}$ [°] |
---|---|---|---|---|

1 | 0.0 | 0 | 0.0 | ${\theta}_{1}$ |

2 | 0.0 | 90 | 0.0 | ${\theta}_{2}$ |

3 | 0.425 | 0 | 0.0 | ${\theta}_{3}$ |

4 | 0.395 | 0 | 0.1105 | ${\theta}_{4}$ |

5 | 0.0 | 90 | 0.101 | ${\theta}_{5}$ |

6 | 0.0 | 90 | 0.0765 | ${\theta}_{6}$ |

Link Number | ${\mathit{m}}_{\mathit{i}}$ [kg] | ${\mathit{P}}_{\mathit{C}\mathit{i}}^{\mathit{i}}$ |
---|---|---|

1 | 3.931 | [0.0, −0.008, −0.031] |

2 | 10.442 | [0.207, 0.0, 0.124] |

3 | 2.846 | [0.228, 0.0, 0.018] |

4 | 1.37 | [0.0, −0.010, −0.005] |

5 | 1.3 | [0.0, −0.010, −0.005] |

6 | 0.365 | [0.0, 0.0, −0.012] |

Parameter | Unit | Circumstances |
---|---|---|

Orientation of the approach vector | [°] | Parallel to the $xy$-plane of the base coordinate system when picking up, and perpendicular when putting down the piece. |

Default pick-up point | [m] | [0.7, 0.3, 0.4] |

Tool orientation at the pick-up point | $[\xb0]$ | [90.0, 0.0, 90.0] |

Default put-down point | [m] | [0.7, −0.4, 0.0] |

Tool orientation at the put-down point | $[\xb0]$ | [90.0, 0.0, 180.0] |

Maximum TCP speed | [m/s] | 0.4 |

Maximum TCP acceleration | [m/s${}^{2}$] | 2.0 |

Lower joint constraint | [°] | [0, 0, −160, −70, 0, −360] |

Upper joint constraint | [°] | [360, 180, 160, 250, 360, 360] |

Maximum joint speed | [/s] | [32, 32, 32, 36, 36, 36] |

Maximum joint acceleration | [/s${}^{2}$] | [240, 240, 240, 360, 360, 360] |

Parameter | Unit | Value |
---|---|---|

Depreciation period | [year] | 5 |

Unit energy cost per kWh | USD | 0.20 |

Total cost of workstation | USD | 20,000 |

Joint Number | Energy-Analytic Model [J] | Energy-Experiment [J] |
---|---|---|

1 | 26.13 | 42.97 |

2 | 105.19 | 101.24 |

3 | 20.47 | 14.93 |

4 | 4.48 | 3.09 |

5 | 0.83 | 0.95 |

6 | 0.47 | 0.66 |

all axes | 157.56 | 163.85 |

Case Index | Energy-Analytic Model [J] | Energy-Experiment [J] | Model - Experiment Difference [%] |
---|---|---|---|

1 | 76.64 | 81.69 | 6.1 |

2 | 73.23 | 78.97 | 7.3 |

3 | 74.73 | 81.43 | 8.2 |

4 | 118.90 | 116.97 | 1.6 |

5 | 176.22 | 160.66 | 9.6 |

6 | 138.61 | 126.13 | 9.8 |

7 | 106.69 | 97.62 | 9.2 |

8 | 80.41 | 85.40 | 5.8 |

Criterion | Cycle Time [s] | Energy [J] | Displacement [mm] |
---|---|---|---|

Minimum of cycle time | 2.86 | 231.01 | [−65, 229] |

Minimum of specific energy | 3.90 | 92.53 | [−297, 90] |

Minimum economic cost | 2.94 | 159.14 | [−41, 214] |

Item | Unit | Minimal Energy | Average Energy |
---|---|---|---|

Energy—model output | [J] | 90.18 | 134.27 |

Energy—measurements | [J] | 92.53 | 126.87 |

Cycle time | [s] | 3.90 | 3.65 |

Displacement vector | [mm] | [−297, 90] | - - - |

Criterion | Cost [USD] | Energy [J] | Displacement [mm] |
---|---|---|---|

Minimum of cycle time | 0.371 | 231.01 | [−65, 229] |

Minimum of specific energy | 0.389 | 92.53 | [−297, 90] |

Minimum economic costs | 0.358 | 159.14 | [−41, 214] |

Without any criterion (random robot location) | 0.383 | 126.87 | [−168, 58] |

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**MDPI and ACS Style**

Gruszka, Ł.; Bartyś, M.
A New Energy-Efficient Approach to Planning Pick-and-Place Operations. *Energies* **2022**, *15*, 8795.
https://doi.org/10.3390/en15238795

**AMA Style**

Gruszka Ł, Bartyś M.
A New Energy-Efficient Approach to Planning Pick-and-Place Operations. *Energies*. 2022; 15(23):8795.
https://doi.org/10.3390/en15238795

**Chicago/Turabian Style**

Gruszka, Łukasz, and Michał Bartyś.
2022. "A New Energy-Efficient Approach to Planning Pick-and-Place Operations" *Energies* 15, no. 23: 8795.
https://doi.org/10.3390/en15238795