# A Comprehensive Multibody Model of a Collaborative Robot to Support Model-Based Health Management

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the UR5

^{®}i7-8750H processor at 3.91 GHz equipped with 16 GB of RAM DDR4 takes about five minutes.

#### 2.1. Control Logic and Power Electronics

#### 2.2. Electric Motor

#### 2.3. Gearbox

- Group 1: comprised of the first four equations describing, respectively, the dynamic equilibrium of the wave generator, the flexspline radial and tangential contributions, and the one of the circular spline;
- Group 2: comprised of the last five equations, representing the interactions between the input shaft and the elliptical cam, through the WG bearing, between the FS and the CS teeth, and between the FS and the joint case and the CS with the output shaft.

#### 2.4. Friction

#### 2.5. Sensors

- Optical encoder: an encoder integral to the motor/input shaft used to derive the motor angular velocity (${\dot{q}}_{M}$) used to provide the feedback signal (${\dot{q}}_{M,FB}$) to the velocity control loop;
- Magnetic encoder: an encoder integral to the joint/output shaft used to measure the joint actual angular position ($q$) used to close the position control loop through the feedback signal (${q}_{FB}$);
- Current sensor: used to measure the motor current ($i$) necessary to close the current control loop through the feedback signal (${i}_{FB}$).

#### 2.6. Six-DoF Forward Kinematics and Dynamics

^{i}

^{−1}

**A**

_{i}as:

## 3. Experimental Testing

#### 3.1. Dynamic Parameters Excitation Trajectory

^{2}, 0 rad/s, 0.08 ms, 0.03, and 500, respectively.

^{0}

**A**

_{6}), in fact, are described by the combination of rigid body transformation matrices described in Equation (8) for each joint as:

- Implementation of simplified DC models of the joint motors instead of using three-phase ones, whose parameters, such as resistance and inductance, are not fully known. Moreover, the motor back-EMF constant (${k}_{e}$) was assumed to be constant for the entire trajectory, while it slightly changes with the applied load.
- A possible incorrect description of the joint friction. As already mentioned, the identification algorithms provided by [40,52] only take into account viscous and Coulomb friction, while they are not able to describe more complex trends, such as the ones affecting the real robot. To do so, friction was modeled through Equation (7), whose coefficients were tuned according to several experimental campaigns aiming to reach a good overall fit between simulated and measured motor currents.
- Lack of both the kinematic error and the gear torsional hysteresis in the translation-equivalent models of the strain wave gears.
- The joint control logic schematized in Figure 2 is not the same as the one implemented in the real robot, which is unknown. As an example, studies such as [69,70] highlight how manipulators from Universal Robots are equipped with vibration suppression algorithms, which were not implemented in the current model. Having two different control logics deeply affects the torque trends. Since, according to the scheme reported in Figure 2, the current control loop is the most internal one, the reference value of the motor current is affected by both the position and the velocity control loops. As a direct consequence, since, as depicted in Figure 8, the joint position error is different between the two manipulators, the velocity reference signal will also be different. Hence, based on the adopted PI gains, the motor angular velocity error differs from the one of the real robot as well.
- While the control parameters reported in Table 1 are constant, it should not be excluded that Universal Robots A/S might adopt variable settings according to the specific operating conditions of the manipulator. such as very fast or very slow movements.

#### 3.2. Pick-and-Place Trajectory

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Name | Symbol | Name |

$\mathit{q}$ | Joint position | ${\alpha}_{t}$ | Teeth pressure angle |

${\mathit{q}}_{set}$ | Joint set position | ${K}_{WG}$ | WG torsional stiffness |

${\mathit{q}}_{FB}$ | Joint feedback position | ${c}_{WG}$ | WG damping coefficient |

${e}_{pos}$ | Joint position error | ${K}_{b}$ | Bearing radial stiffness |

${\dot{q}}_{M}$ | Motor velocity | ${c}_{b}$ | Bearing radial damping coefficient |

${\dot{q}}_{M,ref}$ | Motor reference velocity | ${K}_{FS}$ | FS torsional stiffness |

${\dot{q}}_{M,FB}$ | Motor feedback velocity | ${c}_{FS}$ | FS damping coefficient |

${e}_{vel}$ | Motor velocity error | ${K}_{m}$ | Meshing stiffness between FS and CS |

$i$ | Motor current | ${c}_{m}$ | Meshing damping coefficient |

${i}_{ref}$ | Motor reference current | ${K}_{CS}$ | CS torsional stiffness |

${i}_{FB}$ | Motor feedback current | ${c}_{CS}$ | CS damping coefficient |

${e}_{cur}$ | Motor current error | $\tau $ | Gear ratio |

${V}_{ref}$ | Reference voltage | ${J}_{WG}$ | WG inertia |

${V}_{sup}$ | Supply voltage | ${J}_{FS}$ | FS inertia |

${V}_{net}$ | Network voltage | ${J}_{CS}$ | CS inertia |

$L$ | DC motor equivalent inductance | ${q}_{WG}$ | WG angular position |

$R$ | DC motor equivalent resistance | ${q}_{CS}$ | CS angular position |

${k}_{t}$ | Motor torque constant | ${x}_{WG}$ | Radial displacement of the bearing outer rings |

${k}_{e}$ | Motor voltage constant | ${x}_{FS}$ | Radial displacement of the FS |

${T}_{M}$ | Motor electromechanical torque | ${y}_{FS}$ | Tangential displacement of the FS free edge |

${J}_{M}$ | Motor rotor moment of inertia | ${y}_{CS}$ | Tangential displacement of the CS |

${T}_{SWG,in}$ | Strain wave gear input torque | ${T}_{\mathrm{f}}$ | Joint friction torque |

${T}_{SWG,out}$ | Strain wave gear output torque | ${T}_{f,s}$ | Static friction torque |

${r}_{g}$ | Strain wave gear primitive equivalent radius | ${T}_{f,C}$ | Coulomb friction torque |

${Z}_{FS}$ | FS teeth number | ${f}_{v}$ | Viscous friction coefficient |

${Z}_{CS}$ | CS teeth number | ${T}_{u}$ | Joint useful torque |

## Appendix A

- Current loop: switch A not influent, switch B in position 2, switches C and D open;
- Velocity loop: switches A and B both in position 1, switch C closed, switch D open;
- Position loop: switch A in position 2 and switch B in position 1, switches C and D closed.

**Figure A2.**(

**a**) Base inertia as a function of shoulder and elbow angles; (

**b**) shoulder inertia as a function of elbow and wrist 1 angles; (

**c**) elbow inertia as a function of wrist 1 and 2 angles; (

**d**) wrist 1 inertia as a function of wrist 2 and 3 angles.

[kgm^{2}] | Base | Shoulder | Elbow | Wrist 1 | Wrist 2 | Wrist 3 |
---|---|---|---|---|---|---|

Maximum | 5.628 | 5.719 | 2.649 | 0.228 | 0.215 | 0.212 |

Mean | 3.194 | 4.144 | 2.526 | 0.227 | 0.215 | 0.212 |

Minimum | 2.061 | 2.752 | 2.405 | 0.226 | 0.215 | 0.212 |

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**Figure 3.**(

**a**) Schematic representation of the strain wave gear mounted; (

**b**) equivalent translational model of the gearbox.

**Figure 6.**(

**a**) Joints’ angular positions of the excitation trajectory; (

**b**) joints’ angular positions of the pick and place trajectory.

**Figure 7.**Comparison between the joints’ angular positions measured from the UR5 and estimated through the proposed model (${\mathit{q}}_{FB}$) compared with the target/set (${\mathit{q}}_{set}$) values for the excitation trajectory.

**Figure 8.**Mismatch between the target/set (${\mathit{q}}_{set}$) and the feedback/measured joints’ angular positions (${\mathit{q}}_{FB}$) of the UR5 and the proposed model for the excitation trajectory for the excitation trajectory.

**Figure 9.**Comparison between the joints’ torques derived from the UR5 motor currents, that estimated through the proposed model (${\mathit{i}}_{FB}$) for the excitation trajectory and that calculated by exploiting the Corke model.

**Figure 10.**Comparison between the joints’ angular positions measured from the UR5 and estimated through the proposed model (${\mathit{q}}_{FB}$) compared with the target/set (${\mathit{q}}_{set}$) values for the pick-and-place trajectory.

**Figure 11.**Mismatch between the target/set (${\mathit{q}}_{set}$) and the feedback/measured joints’ angular positions (${\mathit{q}}_{FB}$) of the UR5 and the proposed model for the excitation trajectory for the pick-and-place trajectory.

**Figure 12.**Comparison between the joints torques derived from the UR5 motor currents, the one estimated through the proposed model (${\mathit{i}}_{FB}$) for the pick-and-place trajectory and the one calculated by exploiting the Corke model.

Joint | Position | Velocity | Current | ||
---|---|---|---|---|---|

P [s^{−1}] | P [As/rad] | I [A/rad] | P [V/A] | I [V/As] | |

Base | 7400 | 1 | 0.1 | 10 | 10,000 |

Shoulder | 7500 | 1 | 0.2 | 10 | 10,000 |

Elbow | 7500 | 0.5 | 0.1 | 30 | 10,000 |

Wrist 1 | 2900 | 0.8 | 0.2 | 30 | 10,000 |

Wrist 2 | 3000 | 1 | 0.2 | 30 | 10,000 |

Wrist 3 | 3250 | 1 | 0.2 | 30 | 10,000 |

Joint | ${\mathit{J}}_{\mathit{M}}\left[{\mathbf{kgm}}^{2}\right]$ | $\mathit{R}[\mathbf{\Omega}]$ | $\mathit{L}\left[\mathsf{H}\right]$ | ${\mathit{k}}_{\mathit{t}}[\mathbf{Nm}/\mathbf{A}]$ |
---|---|---|---|---|

Base, shoulder, elbow | 1.877 × 10^{−4} | 0.30 | 0.83 × 10^{−3} | [0.1350, 0.1361, 0.1355] |

Wrist 1, Wrist 2, Wrist 3 | 2.076 × 10^{−5} | 1.65 | 2.50 × 10^{−3} | [0.0957, 0.0865, 0.0893] |

Parameter | Units | Size A | Size B |
---|---|---|---|

${r}_{g}$ | mm | 30.65 | 16.90 |

${\alpha}_{t}$ | deg | 30 | 30 |

${K}_{WG}$ | Nm/rad | 1.92 × 10^{5} | 1.77 × 10^{4} |

${K}_{b}$ | N/m | 1.9 × 10^{8} | 1.2 × 10^{8} |

${K}_{FS}$ | Nm/rad | 3.99 × 10^{5} | 7.26 × 10^{4} |

${K}_{m}$ | N/m | 5.70 × 10^{8} | 3.20 × 10^{8} |

${K}_{CS}$ | Nm/rad | 1.98 × 10^{7} | 6.23 × 10^{6} |

${J}_{WG}$ | kgm^{2} | 2.60 × 10^{−5} | 2.21 × 10^{−6} |

${J}_{FS}$ | kgm^{2} | 2.65 × 10^{−4} | 3.75 × 10^{−5} |

${J}_{CS}$ | kgm^{2} | 2.43 × 10^{−4} | 1.85 × 10^{−5} |

Link | $\mathit{a}$ [mm] | $\mathit{\alpha}$ [°] | $\mathit{\beta}$ [°] | $\mathit{d}$ [mm] | $\mathit{\theta}$ [°] |
---|---|---|---|---|---|

Link 1 | 0.110 | 89.905 | 0 | 89.084 | 0.003 |

Link 2 | −425.156 | 0.004 | 0.013 | 0 | −0.019 |

Link 3 | −392.066 | −0.232 | 0.071 | 0 | −0.013 |

Link 4 | 0.025 | 89.966 | 0 | 110.212 | −0.008 |

Link 5 | −0.069 | −90.013 | 0 | 94.879 | 0.009 |

Link 6 | 0 | 0 | 0 | 82.494 | −0.006 |

Link | Mass [kg] | Center of Mass [x, y, z] [m] | Inertia Tensor [I _{xx}, I_{yy}, I_{zz}, I_{xy} = I_{yx}, I_{xz} = I_{zx}, I_{yz} = Iz_{zy}] [kgm^{2}] |
---|---|---|---|

Link 1 | 3.7 | [0, −0.02561, 0.00193] | [67, 64, 67, 0, 0, 0] |

Link 2 | 8.393 | [0.2125, 0, 0.11336] | [149, 3564, 3553, 0, 0, 0] × 10^{−4} |

Link 3 | 2.33 | [0.15, 0, 0.0265] | [25, 551, 546, 0, 34, 0] × 10^{−4} |

Link 4 | 1.219 | [0, −0.0018, 0.01634] | [12, 12, 9, 0, 0, 0] × 10^{−4} |

Link 5 | 1.219 | [0, 0.0018, 0.01634] | [12, 12, 9, 0, 0, 0] × 10^{−4} |

Link 6 | 0.1879 | [0, 0, −0.001159] | [1, 1, 1, 0, 0, 0] × 10^{−4} |

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## Share and Cite

**MDPI and ACS Style**

Raviola, A.; Guida, R.; Bertolino, A.C.; De Martin, A.; Mauro, S.; Sorli, M.
A Comprehensive Multibody Model of a Collaborative Robot to Support Model-Based Health Management. *Robotics* **2023**, *12*, 71.
https://doi.org/10.3390/robotics12030071

**AMA Style**

Raviola A, Guida R, Bertolino AC, De Martin A, Mauro S, Sorli M.
A Comprehensive Multibody Model of a Collaborative Robot to Support Model-Based Health Management. *Robotics*. 2023; 12(3):71.
https://doi.org/10.3390/robotics12030071

**Chicago/Turabian Style**

Raviola, Andrea, Roberto Guida, Antonio Carlo Bertolino, Andrea De Martin, Stefano Mauro, and Massimo Sorli.
2023. "A Comprehensive Multibody Model of a Collaborative Robot to Support Model-Based Health Management" *Robotics* 12, no. 3: 71.
https://doi.org/10.3390/robotics12030071