# Design, Analysis, and Comparison of Control Strategies for an Industrial Robotic Arm Driven by a Multi-Level Inverter

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Two-Level Inverter

- (2)
- Inverter coupled by Diodes and their Derivations

- Neutral-Point-Clamped (NPC), which delivers three levels of voltage from different DC voltage sources, or from a single one that is divided with capacitors in series.
- Diode/Capacitor Clamped (DCC), which is a modification of the NPC category as it includes an additional capacitor to reduce the voltage peaks in the switches during commutation.
- New Diode Clamped (NDC), which is a modification of the NPC category as it includes additional diodes that block a same voltage in series.

- (3)
- Flying-Capacitor Inverter

- (4)
- Cascaded Full-Bridge Inverter

- Inverter with cascaded symmetrical monophase bridges that uses a monophasic H-bridge to deliver 3 levels of output voltage. However, a cascaded connection of 2 H-bridges enables 5 output voltage levels. By increasing the number of cascaded H-bridges, the number of output voltage levels can be increased with a small quantity of components and without the need of clamping diodes or flying capacitors.
- Cascaded hybrid asymmetrical inverter that has the same topology described above but with only a fraction of the voltage level that feeds each H-bridge, allowing for a 7-level increase in output voltage.
- Inverters coupled by converters in which only one DC voltage source for each H bridge is required, reducing the possibilities of short circuits during switch commutation. Additionally, the turns ratio of the converters allows for generating different voltage levels. Although this configuration has the advantage of employing only one voltage source, it exhibits the economic disadvantage of adding converters.
- Cascaded multi-level inverters used to work with high voltage and power levels. With this purpose, the cascaded H-bridge stages are replaced by flying capacitors or clamping diodes, thereby reducing the number of voltage sources isolated in the inverter input. For example, to generate 9 voltage levels, 2 converters with cascaded flying capacitors can be used so only two independent voltage sources are necessary, whereas the cascaded H-bridge configuration requires 4 independent voltage sources.

- (1)
- More power for the same size.
- (2)
- Wider speed range, as they do not have mechanical limitations.
- (3)
- More efficiency as they lose less heat.
- (4)
- Better performance.
- (5)
- Longer shelf life.
- (6)
- Better speed/motor torque ratio.
- (7)
- More heat dissipation.
- (8)
- Lower weight.
- (9)
- Less maintenance due to absence of wear.
- (10)
- Less electronic noise.

## 2. Materials and Methods

- ${T}_{1}$: Torque applied to the first link (N · m).
- ${F}_{2}$: Force applied to the second link (N).
- ${\mathsf{\theta}}_{1}$: Angular position of the first link (°).
- ${\dot{\mathsf{\theta}}}_{1}$: Angular speed of the first link (°/s).
- ${\ddot{\mathsf{\theta}}}_{1}$: Angular acceleration of the first link (°/s
^{2}). - ${d}_{2}$: Longitudinal position of the second link (m).
- ${m}_{1}$: First link mass (Kg).
- ${m}_{2}$: Second link mass (Kg).
- ${l}_{1}$: First link length (m).
- ${l}_{c1}$: Length from the first link origin to its center of mass (m).
- ${I}_{1}$: Inertia moment of the first link (Kg · m
^{2}). - ${I}_{2}$: Inertia moment of the second link (Kg · m
^{2}). - $g$: Constant of the gravity exerted by the gravitational system in which the robot is (m/s
^{2}).

- ${T}_{f}$: Friction torque in the arm rotation axis (N · m).
- ${p}_{1}$: Slope of viscous friction for positive torque (N · m · s/rad).
- ${p}_{2}$: Slope of viscous friction for negative torque (N · m · s/rad).
- ${n}_{1}$: Coulomb’s friction constant for positive torque (N · m).
- ${n}_{2}$: Coulomb’s friction constant for negative torque (N · m).

- ${T}_{e}$: Electromagnetic torque of the motor (N · m).
- $J$: Inertia moment of the motor (Kg · m).
- ${w}_{r}$: Angular speed of the motor (rad/s).
- $B$: Viscous friction of the motor (N · m ·s/rad).
- $r$: Gear ratio (times).

## 3. Implemented Control Strategies

#### 3.1. Gain Scheduling per Trenches

_{p}, K

_{i}, and K

_{d}, respectively) depends on the angular position of the robotic arm (θ

_{1}= 15°; θ

_{2}= 35°; θ

_{3}= 60°, and θ

_{4}= 75°).

#### 3.2. Gain Scheduling by Interpolation

#### 3.3. Adaptive Control

- $\phi $: Parameter to be adjusted.
- $\mathsf{\Gamma}$: Positive constant that represents the adjustment mechanism.
- $e$: Difference between desired and real positions (°).
- ${\mathsf{\theta}}_{\mathrm{real}}$: Real position (°).

#### 3.4. Fuzzy Logic

^{®}as a base controller taking error and the error derivative as input values. The controller is configured to work with the two input variables above and the output variable, which is the action sent to the multi-level inverter.

## 4. Computer Simulations and Synthesis of Results

#### 4.1. Analysis of Control Strategies Applied to the First Link of the Robotic Arm

#### 4.2. Analysis of Control Strategies Applied in the Two Links of the Robotic Arm

#### 4.3. Analysis of the Control Strategies Applied in the Pattern Tracking of the Robotic Arm

## 5. Conclusions

## 6. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- (1)
- List of components used in the implementation of each H-bridge board:

- IRF741 MOSFET.
- 74HC04N Hex inverter.
- 2N7000 MOSFET.
- MCT6 Optocoupler.
- 15 V Zener diode.
- 1N4007 Diode.

- (2)
- List of components used in the implementation of the control board:

- ATMEGA 1281 Microcomputer.
- LM1086 (5 V) Linear voltage regulator.
- LM317 (12 V) Adjustable linear voltage regulator.
- 2N7002 MOSFET.
- LM741OPAM.
- MAX3238 Voltage adapter for serial port.
- ADR550 Voltage reference.

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**Figure 1.**Electric scheme of three-phase multi-level inverter with 27 levels of voltage per phase connected to an ACBPMM.

**Figure 5.**Interpolation scheme for the calculation of the P proportional gain of a PID controller using gain scheduling by interpolation of a linear polynomial.

**Figure 10.**Analysis of IA, RMS, and RSD errors of the control strategies applied in the first link of the robotic arm.

**Figure 15.**Analysis of the IA, RMS, and RSD errors of the control strategies applied in the two links of the robotic arms.

**Figure 17.**Movement of each link of the robotic arm when following the reference pattern in the Cartesian plane.

**Figure 18.**Movement of each link of the robotic arm when this follows the reference pattern, and mechanical disturbances.

**Figure 19.**Analysis of IA, RMS, and RSD errors of the control strategies when the robotic arm follows the reference pattern.

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

Power | 750 (W) |

Number of pole pairs | 4 |

Synchronous inductance | 6.4 (mH) |

Synchronous resistance | 2.88 (Ω) |

Moment of inertia | 2.42 · 10^{−4} (Kg · m ^{2}) |

Viscous friction | 0.002 (N · m · s) |

Flux density generated by the rotor magnet | 0.17351 (Wb) |

Range | K_{p} | K_{i} | K_{d} |
---|---|---|---|

0°–θ_{1} | K_{p1} = 0.093 | K_{i1} = 0.008 | K_{d1} = 0.5 |

θ_{1}–θ_{2} | K_{p2} = 0.178 | K_{i2} = 0.010 | K_{d2} = 1.0 |

θ_{2}–θ_{3} | K_{p3} = 0.212 | K_{i3} = 0.012 | K_{d3} = 1.1 |

θ_{3}–θ_{4} | K_{p4} = 0.249 | K_{i4} = 0.139 | K_{d4} = 1.3 |

θ_{4}–90° | K_{p5} = 0.296 | K_{i5} = 0.224 | K_{d5} = 2.1 |

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**MDPI and ACS Style**

Urrea, C.; Jara, D.
Design, Analysis, and Comparison of Control Strategies for an Industrial Robotic Arm Driven by a Multi-Level Inverter. *Symmetry* **2021**, *13*, 86.
https://doi.org/10.3390/sym13010086

**AMA Style**

Urrea C, Jara D.
Design, Analysis, and Comparison of Control Strategies for an Industrial Robotic Arm Driven by a Multi-Level Inverter. *Symmetry*. 2021; 13(1):86.
https://doi.org/10.3390/sym13010086

**Chicago/Turabian Style**

Urrea, Claudio, and Daniel Jara.
2021. "Design, Analysis, and Comparison of Control Strategies for an Industrial Robotic Arm Driven by a Multi-Level Inverter" *Symmetry* 13, no. 1: 86.
https://doi.org/10.3390/sym13010086