# Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notation

## 3. Preliminaries

#### 3.1. Modeling of Swing-Up

#### 3.2. Operator Theory

**Definition**

**1.**

**Definition**

**2.**

- (i)
**Right factorization:**Let the input space be denoted by U and output space by Y. In general, these spaces are different extended linear spaces. Let the plant operator $P:U\to Y$ be such that $y\left(t\right)=P\left(u\left(t\right)\right)$ where $u\left(t\right)\in U$ and $y\left(t\right)\in Y$. In addition, let W be an auxiliary linear space and let the operator $N\to Y$ be stable such that $N\left(w\left(t\right)\right)=y\left(t\right)$, $w\left(t\right)\in W$, and let $D:W\to U$ be stable and invertible such that $D\left(w\left(t\right)\right)=u\left(t\right)$. It follows that the plant P has a right factorization determined by N and ${D}^{-1}$$$\begin{array}{c}\hfill P=N{D}^{-1}\end{array}$$- (ii)
**Right coprime factorization:**Suppose there is a right factorization operator N, D in plant P. The Bezout equation is obtained as$$\begin{array}{c}\hfill AN+BD=M,\exists M\in \mathcal{U}\left(W,\phantom{\rule{4pt}{0ex}}U\right)\end{array}$$- (iii)
**Robust right coprime factorization:**In general, there are uncertainties that are difficult to express in a mathematical model in an actual nonlinear control system. Thus, the nonlinear control system with uncertainties may be unstable. Using robust right coprime factorization to factorize a plant can guarantee the robust stability of a nonlinear feedback system with uncertainties. The nonlinear feedback system with uncertainties is shown in Figure 3. Plant P without uncertainties is the nominal plant, and the actual plant with uncertainties $\Delta P$ is $\tilde{P}=P+\Delta P$$$\begin{array}{c}\hfill \tilde{P}=P+\Delta P=\left(N+\Delta N\right){D}^{-1}\end{array}$$$$\begin{array}{c}\hfill A\left(N+\Delta N\right)+BD=\tilde{M}\end{array}$$$$\begin{array}{c}\hfill A\left(N+\Delta N\right)+BD=AN+BD=M\end{array}$$$$\begin{array}{c}\hfill \parallel \left(A\left(N+\Delta N\right)-AN\right){M}^{-1}{\parallel}_{Lip}<1\end{array}$$

## 4. Nonlinear Control System Design

#### 4.1. Tracking Controller Design of Swing-Up

#### 4.1.1. Determining Target Angle

#### 4.1.2. Tracking Controller Design

#### 4.2. Control System Design Based on Operator Theory and Isomorphism Scheme

#### 4.2.1. Right Factorization of the Swing-Up

#### 4.2.2. Right Coprime Factorization of Underactuated Robot

#### 4.2.3. Robust Stability Condition

#### 4.2.4. Control System Design Based on Operator Theory and Isomorphism Scheme

## 5. Simulation

#### 5.1. Control System Simulation Based on Operator Theory

#### 5.2. Control System Simulation Based on Operator Theory and Isomorphism Scheme

#### 5.3. Robust Stability of a Control System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Nonlinear feedback system without uncertainties based on operator theory and an isomorphism scheme.

**Figure 9.**First link angle ${q}_{1}$ of nonlinear feedback system with uncertainties based on operator theory.

**Figure 10.**Second link angle ${q}_{1}$ of nonlinear feedback system with uncertainties based on operator theory.

**Figure 12.**Robust stability assessment of nonlinear feedback system with uncertainties based on operator theory.

**Figure 13.**First link angle ${q}_{1}$ of nonlinear feedback system with uncertainties based on operator theory and isomorphism scheme.

**Figure 14.**Second link angle ${q}_{2}$ of nonlinear control feedback system with uncertainties based on operator theory and an isomorphism scheme.

**Figure 15.**Robust stability assessment of nonlinear feedback system with uncertainties based on operator theory and isomorphism scheme.

$\mathcal{S}(U,Y)$ | the set of stable operators from U to Y |

$\mathcal{U}(U,Y)$ | the set of unimodular operators |

${\parallel \xb7\parallel}_{{U}_{s}},{\parallel \xb7\parallel}_{{Y}_{S}}$ | norm |

$\parallel A\parallel $ | Lipschitz semi-norm |

$A:{U}_{S}\to {Y}_{S}$ | an operator mapping from ${U}_{s}$ to ${Y}_{S}$ |

$\mathcal{D}\left(A\right)$ | the domain and range of A |

$\mathcal{R}\left(A\right)$ | the range of A |

$\mathcal{N}\left({U}_{s};{Y}_{s}\right)$ | the family of all nonlinear operators mapping |

from $\mathcal{D}\left(A\right)\subseteq {U}_{s}$ into ${Y}_{s}$ | |

${U}_{s},{Y}_{s}$ | normed linear space over the field of complex |

numbers endowed with norms ${\parallel \xb7\parallel}_{{U}_{s}},{\parallel \xb7\parallel}_{{Y}_{S}}$ | |

$Lip\left({D}_{S},{Y}_{s}\right)$ | a Lipschitz operator mapping from ${D}_{s}$ to ${Y}_{s}$ |

P | plant |

$\Delta P$ | uncertainties |

$\tilde{P}$ | the actual plant with uncertainties |

$A,N,B,D,\tilde{A},\tilde{N},\tilde{B},\tilde{D}$ | operators of the system |

$M,\tilde{M}$ | unimodular operators |

L | Lagrangian | [J] |

K | Kinetic energy of acrobot | [J] |

V | Potential energy of acrobot | [J] |

${q}_{i}$ | Target angle | [$\mathrm{rad}$] |

$\tau $ | Torque | [N·m] |

${q}_{1}$ | Angle of Link1 | [rad] |

${q}_{2}$ | Angle of Link2 | [rad] |

${m}_{1}$ | Mass of Link1 | 0.175 kg |

${m}_{2}$ | Mass of Link2 | 0.285 kg |

${l}_{1}$ | Length of Link1 | 0.3 m |

${l}_{2}$ | Length of Link2 | 0.5 m |

${l}_{c1}$ | Lengh from First joint to the | |

center of gravity of Link1 | 0.177 m | |

${l}_{c2}$ | Lengh from Second joint to the | |

center of gravity of Link2 | 0.25 m | |

${I}_{1}$ | Moment of inertia of Link1 | 0.0013 kg$\xb7{\mathrm{m}}^{2}$ |

${I}_{2}$ | Moment of inertia of Link2 | 0.0059 kg$\xb7{\mathrm{m}}^{2}$ |

g | Acceleration of gravity | [m/${\mathrm{s}}^{2}$] |

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**MDPI and ACS Style**

Deng, M.; Kubota, S.
Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme. *Axioms* **2021**, *10*, 62.
https://doi.org/10.3390/axioms10020062

**AMA Style**

Deng M, Kubota S.
Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme. *Axioms*. 2021; 10(2):62.
https://doi.org/10.3390/axioms10020062

**Chicago/Turabian Style**

Deng, Mingcong, and Shotaro Kubota.
2021. "Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme" *Axioms* 10, no. 2: 62.
https://doi.org/10.3390/axioms10020062