# Neural Network Mapping of Industrial Robots’ Task Times for Real-Time Process Optimization

^{*}

## Abstract

**:**

## 1. Introduction

- the scheduling, the optimal intercept computation, and the trajectory execution time minimization are treated in isolation, without considering the interdependencies between them;
- the scheduling algorithms often do not consider accurately that different tasks are associated with different execution times;
- the cycle or task time maps are developed without a focus on the optimal (as opposed to feasible) times;
- the cycle time maps or tables cover only a portion of the useful workspace of the manipulator and appear tailored to a specific application;
- the methods employed for the generation of the cycle time maps are known to scale poorly to problems featuring a high number of degrees of freedom.

- the development of a systematic architecture for the representation of the task time maps as neural networks;
- the illustration of a computational procedure for the generation of the dataset needed to train the task time models;
- the demonstration of the use of the task time maps to solve the optimal intercept problem in an industrial application case.

## 2. Methods

#### 2.1. Problem Statement and Solution Outline

- the kinematic and dynamic properties of the robot;
- the parameters of the actuators and transmission systems;
- the behavior of the Geometric Planning Module;
- the behavior of the Task Time Optimizer.

- the parametrization of the tasks to be executed by the robot using the GTP;
- the offline generation of a dataset of optimal execution times using the TTO;
- the generation, using deep learning models, of an accurate and efficiently queryable map of the optimal task times over the entire workspace of the robot.

#### 2.2. Geometric Trajectory Planning

#### 2.3. Task Time Optimization

- the actual performances of the actuation drives and of the transmissions;
- the model of the mechanical dynamics of the manipulator;
- the properties of the payload.

#### 2.4. Task Times Map Based on Neural Networks

- sample the position and velocity boundary conditions ${\mathit{p}}_{\mathit{s}}$, ${\mathit{p}}_{\mathit{f}}$, ${\mathit{v}}_{\mathit{s}}$, and ${\mathit{v}}_{\mathit{f}}$;
- for each set of boundary conditions, generate a geometric path using the GTP;
- for each motion path, compute the optimal execution time ${T}_{task}$ using the TTO and store explicitly the association between the inputs $({\mathit{p}}_{\mathit{s}},{\mathit{p}}_{\mathit{f}},{\mathit{v}}_{\mathit{s}},{\mathit{v}}_{\mathit{f}})$ and ${T}_{task}$.

- ability to sample the task space in an unstructured way;
- lower persistent memory usage (which is used not for the ${T}_{task}$ samples directly but rather for the parameters of the neural network);
- more straightforward adaptability to different kinds of manipulators;
- tuneability of the internal architecture for achieving the desired performances;
- application-independent software interface and generation procedure.

## 3. Case Study Description

- length of the proximal links ${l}_{prox}=250$ mm;
- length of the distal links ${l}_{dist}=250$ mm;
- frame length ${l}_{f}=180$ mm;
- mass of the proximal links ${m}_{prox,1}={m}_{prox,2}=2.9$ kg;
- mass of the distal links ${m}_{dist,1}={m}_{dist,2}=2.9$ kg;
- barycentric inertia of the proximal links ${J}_{prox,1}={J}_{prox,2}$ = 5.22 × 10
^{−2}kg m^{2}; - barycentric inertia of the distal links ${J}_{dist,1}={J}_{dist,2}$ = 5.22 × 10
^{−2}kg m^{2}; - mass of the screw-spline and of the end-effector ${m}_{ee}=0.36$ kg;
- screw-spline pitch ${p}_{ss}=2$ mm;
- rotational inertia of the end-effector ${J}_{ee}$ = 6.40 × 10
^{−6}kg m^{2}; - rotational inertia of the transmission system actuating the screw-spline helical joint ${J}_{3}$ = 1.20 × 10
^{−6}kg m^{2}; - rotational inertia of the transmission system actuating the screw-spline prismatic joint ${J}_{4}$ = 1.20 × 10
^{−6}kg m^{2}.

#### Geometric Planning for the 5R Robot

- stationary pick and place tasks, where the initial and final velocities are null;
- on-the-fly pick and place tasks, such as those represented in Figure 1, where the initial and final velocities are non-null and aligned to the x-axis.

- an intercept motion (a), which happens in a slightly elevated plane and which terminates with an x-y position, a rotation and a velocity that match those of the pick and place target; its terminal position can be found, as will be shown, by solving the optimal intercept problem;
- a descent motion (b) through which the gripper approaches the target from above;
- a grasping/release motion (c) during which the gripping device operates to grasp or release the item;
- an ascent motion (d) through which the gripper attains the proper vertical clearance.

- assists the acceleration phase of the descent motion (b);
- opposes the deceleration phase of the descent motion (b);
- opposes the acceleration phase of the ascent motion (d);
- assists the deceleration phase of the ascent motion (d).

## 4. Computational Results and Discussion

#### 4.1. Software and Hardware Setup

- an abstract class representing a generic n-DOF manipulator, from which specific implementations (among which the one relative to the 5R robot) are derived;
- an abstract class representing a generic TTO, from which specific implementations are derived according to the different optimization methods;
- an abstract class representing a generic n-DOF geometric path, with its derived implementations;
- a set of routines implementing the GTP for the 5R robot;
- a utility class dedicated to the multithreaded generation of the dataset.

- a rotation range for the end-effector equal to $\Delta {\varphi}_{ee\in}\left[-\pi \phantom{\rule{0.166667em}{0ex}},\pi \right]$;
- a range for the initial and final velocities equal to $v\in $ [0.05 m s
^{−1}, 0.5 m s^{−1}];

- loss function: MSE (Mean Squared Error)
- training algorithm: SGD (Stochastic Gradient Descent)
- batch size: 2048 data points.

- ${\overline{T}}_{plan}$, the average execution time of the entire motion planning and optimization pipeline for a single trajectory;
- ${\overline{T}}_{eval,cpu}$, the average evaluation time of the task time map on a single CPU core;
- ${\overline{T}}_{eval,gpu}$, the average evaluation time of the task time map on the GPU.

- CPU: Intel
^{®}Core™(Santa Clara, CA, USA) i7-8750H processor $2.2$ $\mathrm{G}$$\mathrm{Hz}$ (9 $\mathrm{M}$ cache, up to $4.1$ $\mathrm{G}$$\mathrm{Hz}$, 6 processors); - GPU: Nvidia
^{®}(Santa Clara, CA, USA) GeForce^{®}(Santa Clara, CA, USA) GTX 1050, 4 $\mathrm{G}$, GDDR5; - RAM: 16 $\mathrm{G}$ DDR4 SO-DIMM, 2400 $\mathrm{M}$$\mathrm{Hz}$.

#### 4.2. Stationary Pick and Place Task Time Map

- ${\overline{T}}_{eval,cpu}=81.38\mathrm{\mu s}$ on a single CPU core, using the Python-3 interpreter;
- ${\overline{T}}_{eval,gpu}=5.96\mathrm{\mu s}$ on the GPU, using the Python-3 interpreter;
- ${\overline{T}}_{plan}=49.4\mathrm{m}\mathrm{s}$ on the CPU, using the C++ implementation of the GTP and TTO.

#### 4.3. On-the-Fly Pick and Place Task Time Map

- ${\overline{T}}_{eval,cpu}=2.31$ ms on a single CPU core;
- ${\overline{T}}_{eval,gpu}=73.42\mathrm{\mu s}$ on the GPU;
- ${\overline{T}}_{plan}=56.9$ ms on the CPU, using the C++ implementation of the GTP and TTO.

^{−1}, in this case, at the end of the intercept task, the target reaches a position different from the robot’s, with a spatial error equal to 10 mm. This might be acceptable or not according to the picked item’s geometry and size. Concerning the computational costs associated with the full planning pipeline, the following considerations can be made: Figure 13, Figure 14, Figure 15 and Figure 16 and Figure 18, Figure 19, Figure 20 and Figure 21 show that even the shortest trajectories require more than 100 ms for their execution; on the other hand, in the histogram depicted in Figure 11, it can be seen that the execution of the complete motion planning pipeline, composed of the GTP and TTO, requires practically always less than 80 ms; this result, moreover, is tuneable by acting on the parameters of the optimization routines (chiefly the number of decision variables and constraints), sacrificing, if needed, accuracy for speed. Even in the most unfortunate cases (with comparatively short trajectory execution times and relatively long planning times), it is possible to plan the next trajectory while the current one is being executed. Since the conveyors move at quasi-constant velocities, it is indeed feasible to predict the motion of the targets with reasonable accuracy in order to plan in advance the next task assigned to the robot. This kind of plan-in-advance strategy is needed in any case for the on-the-fly pick and place tasks since the robot terminates its current trajectory with a non-null velocity and therefore immediately needs a new setpoint to safely prosecute its motion. As a further benefit, this strategy makes it unnecessary to consider, during the optimal intercept computation, the calculation times. The implementation of the overall planning system here briefly delineated should therefore allow the parallel execution of the control algorithms and of the planning operations, and additionally, it should perform short-term forecasts of the pick and place targets’ states.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of a non-deterministic and time-constrained on-the-fly manipulation task.

**Figure 2.**Pictorial representation of the scheduling, geometric motion planning, and task time optimization pipeline.

**Figure 3.**Input–output representation of the scheduling, geometric motion planning, and task time optimization pipeline.

**Figure 4.**Statistical comparison between the trajectory execution times obtained through convex optimization and linear programming.

**Figure 5.**Conceptual diagram of the training process of the task time model; ${\mathit{p}}_{\mathit{s}}$, ${\mathit{p}}_{\mathit{f}}$, ${\mathit{v}}_{\mathit{s}}$, and ${\mathit{v}}_{\mathit{f}}$ represent the position and velocity boundary conditions of the trajectory, while ${T}_{task}$ and ${\widehat{T}}_{task}$ denote each the actual task time of the trajectory and its estimation.

**Figure 6.**5R manipulator at the University of Bergamo and schematic representation of its actuation and transmission systems.

**Figure 10.**Examples of point-to-point paths (

**a**), tracking paths (

**b**), and paths for stationary pick and place tasks (

**c**).

**Figure 13.**Stationary pick and place task time model evaluated at different end-effector rotational displacements $\Delta {\varphi}_{ee}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.0\mathrm{m}$, ${y}_{ee}=0.25\mathrm{m}$.

**Figure 14.**Stationary pick and place task time model evaluated at different end-effector rotational displacements $\Delta {\varphi}_{ee}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=-0.25\mathrm{m}$, ${y}_{ee}=0.15\mathrm{m}$.

**Figure 15.**Stationary pick and place task time model evaluated at different end-effector rotational displacements $\Delta {\varphi}_{ee}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.25\mathrm{m}$, ${y}_{ee}=0.15\mathrm{m}$.

**Figure 16.**Stationary pick and place task time model evaluated at different end-effector rotational displacements $\Delta {\varphi}_{ee}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.0\mathrm{m}$, ${y}_{ee}=0.47\mathrm{m}$.

**Figure 18.**On-the-fly pick and place task time model evaluated for different end-effector initial and final velocities ${v}_{0}$ and ${v}_{f}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.0\mathrm{m}$, ${y}_{ee}=0.25\mathrm{m}$.

**Figure 19.**On-the-fly pick and place task time model evaluated for different end-effector initial and final velocities ${v}_{0}$ and ${v}_{f}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=-0.25\mathrm{m}$, ${y}_{ee}=0.15\mathrm{m}$.

**Figure 20.**On-the-fly pick and place task time model evaluated for different end-effector initial and final velocities ${v}_{0}$ and ${v}_{f}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.25\mathrm{m}$, ${y}_{ee}=0.15\mathrm{m}$.

**Figure 21.**On-the-fly pick and place task time model evaluated for different end-effector initial and final velocities ${v}_{0}$ and ${v}_{f}$; the initial configuration of the robot, shown as a red marker, has been fixed at ${x}_{ee}=0.0\mathrm{m}$, ${y}_{ee}=0.47\mathrm{m}$.

Servoaxis 1 | Servoaxis 2 | Servoaxis 3 | Servoaxis 4 | |
---|---|---|---|---|

${\tau}_{pds,rated}\phantom{\rule{0.166667em}{0ex}},\left[\mathrm{N}\mathrm{m}\right]$ | 0.7 | 0.7 | 0.36 | 0.36 |

${\tau}_{pds,max}\phantom{\rule{0.166667em}{0ex}},\left[\mathrm{N}\mathrm{m}\right]$ | 1.4 | 1.4 | 0.72 | 0.72 |

${\omega}_{pds,max}\phantom{\rule{0.166667em}{0ex}},\left[\mathrm{rad}{\mathrm{s}}^{-1}\right]$ | 500 | 500 | 500 | 500 |

${J}_{m}\phantom{\rule{0.166667em}{0ex}},\left[\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}\right]$ | $1.7\times {10}^{-5}$ | $1.7\times {10}^{-5}$ | $2.4\times {10}^{-6}$ | $2.4\times {10}^{-6}$ |

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

${J}_{t}\phantom{\rule{0.166667em}{0ex}},\left[\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}\right]$ | $2.485\times {10}^{-5}$ | $2.485\times {10}^{-5}$ | $2.05\times {10}^{-5}$ | $2.05\times {10}^{-5}$ |

${i}_{t}$ | 64 | 64 | 10 | 10 |

**Table 2.**Comparison of the evaluation times of the neural network models and of the GTP-TTO algorithms.

Task Type | ${\overline{\mathit{T}}}_{\mathit{eval},\mathit{cpu}},\left[\mathbf{\mu s}\right]$ | ${\overline{\mathit{T}}}_{\mathit{eval},\mathit{gpu}},\left[\mathbf{\mu s}\right]$ | ${\overline{\mathit{T}}}_{\mathit{plan}},\left[\mathbf{\mu s}\right]$ | ${\overline{\mathit{T}}}_{\mathit{plan}}/{\overline{\mathit{T}}}_{\mathit{eval},\mathit{cpu}}$ | ${\overline{\mathit{T}}}_{\mathit{plan}}/{\overline{\mathit{T}}}_{\mathit{eval},\mathit{gpu}}$ |
---|---|---|---|---|---|

Stationary Pick and Place | 81 | 6 | 49,419 | 610.11 | 8236.5 |

On-the-fly Pick and Place | 2310 | 73 | 56,985 | 24.7 | 780.6 |

**Table 3.**Main parameters of the neural network architecture for the stationary pick and place task time map.

Layer 1 | Layer 2 | Layer 3 | |
---|---|---|---|

Type | Fully connected | Fully connected | Fully connected |

Activation | ReLU | ReLU | Linear |

Dropout probability | 0.20 | 0.20 | 0.20 |

${n}_{in}$ | 5 | 2048 | 2048 |

${n}_{out}$ | 2048 | 2048 | 1 |

**Table 4.**Main parameters of the neural network architecture used for the on-the-fly pick and place neural network model.

Layer 1 | Layer 2 | Layer 3 | Layer 4 | |
---|---|---|---|---|

Type | Fully connected | Fully connected | Fully connected | Fully connected |

Activation | ReLU | ReLU | ReLU | Linear |

Dropout probability | 0.05 | 0.05 | 0.05 | 0.05 |

${n}_{in}$ | 5 | 8192 | 8192 | 8129 |

${n}_{out}$ | 8192 | 8192 | 8192 | 1 |

Task Configuration Data | Task 1 | Task 2 | Task 3 | Task 4 |
---|---|---|---|---|

${x}_{rbt,0}$, [mm] | −300 | 175 | −300 | 175 |

${y}_{rbt,0}$, [mm] | 150 | 300 | 150 | 300 |

${\phi}_{rbt,0}$, [${}^{\circ}$] | 0.0 | 0.0 | 0.0 | 0.0 |

${v}_{rbt,0}$, [m s^{−1}] | 0.2 | 0.2 | 0.2 | 0.2 |

${x}_{tgt,0}$, [mm] | −150 | −150 | 150 | 150 |

${y}_{tgt,0}$, [mm] | 400 | 400 | 175 | 175 |

${\phi}_{tgt,0}$, [${}^{\circ}$] | 0.0 | 0.0 | 0.0 | 0.0 |

${v}_{tgt,0}$, [m s^{−1}] | 0.2 | 0.2 | 0.2 | 0.2 |

${T}_{task}$ evaluations | 15 | 15 | 17 | 12 |

${\widehat{T}}_{task}$ evaluations | 14 | 15 | 16 | 11 |

Solution time with GTP-TTO, [ms] | 869 | 869 | 985 | 695 |

Solution time with task time map (CPU), [ms] | 32 | 34 | 37 | 25 |

Solution time with task time map (GPU), [ms] | 1.02 | 1.09 | 1.17 | 0.80 |

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## Share and Cite

**MDPI and ACS Style**

Righettini, P.; Strada, R.; Cortinovis, F.
Neural Network Mapping of Industrial Robots’ Task Times for Real-Time Process Optimization. *Robotics* **2023**, *12*, 143.
https://doi.org/10.3390/robotics12050143

**AMA Style**

Righettini P, Strada R, Cortinovis F.
Neural Network Mapping of Industrial Robots’ Task Times for Real-Time Process Optimization. *Robotics*. 2023; 12(5):143.
https://doi.org/10.3390/robotics12050143

**Chicago/Turabian Style**

Righettini, Paolo, Roberto Strada, and Filippo Cortinovis.
2023. "Neural Network Mapping of Industrial Robots’ Task Times for Real-Time Process Optimization" *Robotics* 12, no. 5: 143.
https://doi.org/10.3390/robotics12050143