# Discovering Stick-Slip-Resistant Servo Control Algorithm Using Genetic Programming

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Structure of an Experimental Simulation System

#### 2.2. Friction Model

#### 2.3. Learning Trajectory and Fitness Function

#### 2.4. Fitness of the PID Algorithm

**Figure 4.**Learning trajectory and PID control: (

**a**) reference position and stick-slip effect; (

**b**) detailed view of stick-slip effect; (

**c**) fitness function sensitivity on settling time; (

**d**) control signal.

#### 2.5. Implementation and Configuration of the Genetic Programming Process

## 3. Results

#### 3.1. Unconstrained Bang-Bang Control Algorithms

#### 3.2. Control Algorithms for Servo with Ideal Measurement

#### 3.3. Control Algorithms for Servo with Real Position Sensor

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GP | Genetic Programming |

ECJ | Evolutionary Computation [research system written in] Java |

## References

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**Figure 1.**Servomechanizm with friction: (

**a**) general control structure; (

**b**) cascade control structure; (

**c**) single loop structure with PID; (

**d**) stick-slip-resistant structure with parse-tree-based controller; (

**e**) stick-slip effect.

**Figure 6.**Tracking quality for bang-bang control: (

**a**) $J=2.7\times {10}^{-9}$; (

**b**) $J=1.3\times {10}^{-12}$.

**Figure 10.**Tracking quality for best individual: (

**a**) full range of signal values; (

**b**) signal values in steady state.

**Figure 11.**Servo control sensitivity: (

**a**) $x\in [0.1{x}_{\mathrm{r}},10{x}_{\mathrm{r}}]$; (

**b**) $x\in [0.5{x}_{\mathrm{r}},2{x}_{\mathrm{r}}]$.

**Figure 12.**Tracking quality of composite trajectories: (

**a**) piecewise constant and sinusoidal; (

**b**) parabolic.

Symbol | Arity | Operation | Explanation |
---|---|---|---|

+ | 2 | arithmetic addition | |

− | 2 | arithmetic subtraction | |

* | 2 | arithmetic multiplication | |

÷ | 2 | safe arithmetic division | $\xf7(a,b)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}b=0\phantom{\rule{4pt}{0ex}}\mathrm{then}\phantom{\rule{4pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\mathrm{else}\phantom{\rule{4pt}{0ex}}a/b\phantom{\rule{4pt}{0ex}}\mathrm{end}$ |

I | 3 | if-else expression | $\mathrm{I}(a,b,c)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}a>0\phantom{\rule{4pt}{0ex}}\mathrm{then}\phantom{\rule{4pt}{0ex}}b\phantom{\rule{4pt}{0ex}}\mathrm{else}\phantom{\rule{4pt}{0ex}}c\phantom{\rule{4pt}{0ex}}\mathrm{end}$ |

S | 1 | sin function | $\mathrm{S}\left(a\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}sina$ |

U | 1 | unit delay | |

$\{1,\dots ,100\}$ | 0 | integer constant | randomly initialized and mutated |

R | 0 | reference position | from trajectory generator |

C | 0 | control signal | from previous control cycle |

P | 0 | actual position | from position sensor |

V | 0 | actual velocity | from velocity sensor (or estimator) |

A | 0 | actual acceleration | from acceleration sensor (or estimator) |

Parameter | Setting |
---|---|

Initial population | ramped half-and-half (2:6) algorithm |

Breeding pipelines | reproduction, crossover, mutation |

Reproduction probability | 0.1 |

Crossover probability | 0.9 |

Mutation probability | 0.1, 0.2, 0.3 |

Selection method | tournament without elitism |

Tournament size | 7 |

Size of population | 1000 |

Number of generations | 500 |

Mutation Probability | Full Input | No Acc | No Vel | No Acc, No Vel | No Acc, No Vel, No Ctr |
---|---|---|---|---|---|

0.1 | F01 | NA01 | NV01 | NAV01 | NAVC01 |

0.2 | F02 | NA02 | NV02 | NAV02 | NAVC02 |

0.3 | F03 | NA03 | NV03 | NAV03 | NAVC03 |

No. | Parameter | F01 | F02 | F03 | NA01 | NA02 | NA03 | NV01 | NV02 | NV03 | NAV01 | NAV02 | NAV03 | NAVC01 | NAVC02 | NAVC03 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-2},{10}^{-1}]$ | 44 | 30 | 30 | 28 | 19 | 13 | 16 | 31 | 22 | 5 | 9 | 5 | |||

2 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-3},{10}^{-2})$ | 27 | 29 | 30 | 27 | 24 | 31 | 24 | 16 | 20 | 14 | 11 | 11 | 4 | 2 | |

3 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-4},{10}^{-3})$ | 13 | 18 | 15 | 19 | 14 | 15 | 32 | 28 | 29 | 66 | 68 | 63 | 93 | 97 | 94 |

4 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-5},{10}^{-4})$ | 2 | 5 | 1 | 2 | 5 | 8 | 12 | 7 | 9 | 6 | 4 | 4 | 2 | ||

5 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-6},{10}^{-5})$ | 2 | 2 | 3 | 1 | 7 | 6 | 5 | 9 | 6 | 5 | 5 | 11 | 1 | 1 | |

6 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-7},{10}^{-6})$ | 2 | 3 | 6 | 5 | 7 | 3 | 9 | 5 | 6 | 4 | 1 | 2 | 2 | 3 | 1 |

7 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-8},{10}^{-7})$ | 6 | 6 | 8 | 7 | 9 | 12 | 1 | 1 | 2 | 2 | |||||

8 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-9},{10}^{-8})$ | 1 | 1 | 1 | 3 | 8 | 7 | 1 | 2 | 2 | ||||||

9 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-10},{10}^{-9})$ | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | |||||||

10 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-11},{10}^{-10})$ | 2 | 2 | 1 | 1 | 1 | 1 | |||||||||

11 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-12},{10}^{-11})$ | 1 | 1 | 1 | 1 | |||||||||||

12 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-13},{10}^{-12})$ | 1 | ||||||||||||||

13 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-14},{10}^{-13})$ | 2 | 1 | |||||||||||||

14 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-15},{10}^{-14})$ | 1 | 2 | |||||||||||||

15 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-16},{10}^{-15})$ | 1 | 1 | 1 | ||||||||||||

16 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-17},{10}^{-16})$ | 1 | 1 | |||||||||||||

17 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-18},{10}^{-17})$ | 1 | 1 | 1 | ||||||||||||

18 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-19},{10}^{-18})$ | 1 | 1 | 2 | ||||||||||||

19 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-21},{10}^{-20})$ | 1 | ||||||||||||||

20 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-22},{10}^{-21})$ | 1 | ||||||||||||||

21 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right){J}_{\mathrm{minPID}}$ | 14 | 18 | 24 | 24 | 38 | 33 | 16 | 18 | 20 | 9 | 8 | 17 | 3 | 3 | 2 |

22 | ${\mathrm{mean}}_{s\in \mathbb{S}}J\left(s\right)\phantom{\rule{2.em}{0ex}}[\times {10}^{-4}]$ | $85.5\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $65.3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $61.3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $61.1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $39.2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $34.0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $38.0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $57.1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $45.4\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $16.0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $23.2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $14.2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $4.72\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $3.43\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $3.78\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

23 | ${min}_{s\in \mathbb{S}}J\left(s\right)$ | $8.22\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-17}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.03\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-21}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $4.18\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-19}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.08\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-18}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $3.94\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-22}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.30\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-19}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $6.03\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $3.60\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-14}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.68\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-17}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.87\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-7}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.05\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.12\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-10}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.35\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-7}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $4.60\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-7}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.03\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-7}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

No. | Parameter | F01a | F02a | F03a | NA01a | NA02a | NA03a | F01b | F02b | F03b | NA01b | NA02b | NA03b |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-2},{10}^{-1}]$ | 48 | 32 | 43 | 24 | 14 | 9 | 24 | 17 | 5 | 5 | 11 | 8 |

2 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-3},{10}^{-2})$ | 15 | 21 | 14 | 22 | 30 | 20 | 41 | 41 | 56 | 33 | 38 | 32 |

3 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-4}{10}^{-3},)$ | 15 | 24 | 24 | 23 | 16 | 17 | 21 | 25 | 15 | 31 | 25 | 21 |

4 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-5},{10}^{-4})$ | 8 | 7 | 4 | 10 | 7 | 8 | 5 | 5 | 10 | 8 | 5 | 11 |

5 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-6},{10}^{-5})$ | 6 | 4 | 1 | 4 | 5 | 7 | 2 | 3 | 3 | 3 | 4 | 7 |

6 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-7},{10}^{-6})$ | 7 | 5 | 2 | 7 | 17 | 12 | 4 | 5 | 5 | 7 | 5 | 2 |

7 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-8},{10}^{-7})$ | 1 | 4 | 10 | 9 | 9 | 23 | 2 | 3 | 6 | 11 | 10 | 17 |

8 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-9},{10}^{-8})$ | 3 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 1 | 1 | ||

9 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right)\in [{10}^{-10},{10}^{-9})$ | 1 | 1 | ||||||||||

10 | $\#\left(\right)open="\{"\; close="\}">s\in \mathbb{S}:J\left(s\right){J}_{\mathrm{minPID}}$ | 14 | 16 | 15 | 21 | 33 | 46 | 9 | 12 | 14 | 23 | 21 | 28 |

11 | ${\mathrm{mean}}_{s\in \mathbb{S}}J\left(s\right)\phantom{\rule{2.em}{0ex}}[\times {10}^{-4}]$ | $96.0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $65.5\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $80.2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $53.5\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $36.2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $27.7\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $62.1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $52.3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $42.5\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $25.4\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $37.7\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $31.1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

12 | ${min}_{s\in \mathbb{S}}J\left(s\right)$ | $2.04\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-8}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.42\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.17\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $4.56\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $9.49\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $2.48\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $6.90\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $6.09\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.60\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-8}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $1.07\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-9}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $3.75\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-10}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $9.78\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-10}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

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**MDPI and ACS Style**

Bożek, A.
Discovering Stick-Slip-Resistant Servo Control Algorithm Using Genetic Programming. *Sensors* **2022**, *22*, 383.
https://doi.org/10.3390/s22010383

**AMA Style**

Bożek A.
Discovering Stick-Slip-Resistant Servo Control Algorithm Using Genetic Programming. *Sensors*. 2022; 22(1):383.
https://doi.org/10.3390/s22010383

**Chicago/Turabian Style**

Bożek, Andrzej.
2022. "Discovering Stick-Slip-Resistant Servo Control Algorithm Using Genetic Programming" *Sensors* 22, no. 1: 383.
https://doi.org/10.3390/s22010383