# A Sliding Mode Control-Based Guidance Law for a Two-Dimensional Orbit Transfer with Bounded Disturbances

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Mathematical Model

## 3. State-Feedback Control Design

#### Disturbance Modeling

## 4. Case of an Unperturbed System

#### 4.1. The $\tau $-Variation of Tracking Errors and Controls

#### 4.2. Control Parameter Selection

## 5. Numerical Simulations and Mission Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

a | propulsive acceleration magnitude $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

${a}_{r}$ | radial component of the propulsive acceleration $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

${a}_{t}$ | transverse component of the propulsive acceleration $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

c | speed of approach to ${x}_{3}=0$ when ${Z}_{t}=0$ |

${d}_{r}$ | radial component of the disturbance acceleration $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

${D}_{r}$ | maximum of $|{d}_{r}|$, $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

${d}_{t}$ | transverse disturbance acceleration, $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

${D}_{t}$ | maximum of $|{d}_{t}|$ $\phantom{\rule{0.166667em}{0ex}}(\mathrm{mm}/{\mathrm{s}}^{2})$ |

K | speed of approach to $s=0$ when ${Z}_{r}=0$ |

${K}_{H}$ | value of K corresponding to ${\tau}_{f}={\tau}_{H}$ |

${K}_{v}$ | value of K that minimizes the total velocity change |

n | dimensionless positive parameter; see Equation (59) |

P | primary body center of mass |

r | orbital radius $\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{au}\right)$ |

s | linear combination of $\{{x}_{1},\phantom{\rule{0.166667em}{0ex}}{x}_{2}\}$; see Equation (22) |

S | spacecraft center of mass |

$\mathcal{S}$ | sigmoid-like function; see Equation (35) |

t | time $\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{days}\right)$ |

u | magnitude of command signal |

${u}_{r}$ | dimensionless value of ${a}_{r}$ |

${u}_{t}$ | dimensionless value of ${a}_{t}$ |

${v}_{r}$ | radial velocity component $\phantom{\rule{0.166667em}{0ex}}(\mathrm{km}/\mathrm{s})$ |

${v}_{t}$ | transverse velocity component $\phantom{\rule{0.166667em}{0ex}}(\mathrm{km}/\mathrm{s})$ |

X | normally distributed random number |

$\{{x}_{1},{x}_{2},{x}_{3}\}$ | dimensionless tracking errors along $\{r,{v}_{r},{v}_{t}\}$ |

${z}_{r}$ | dimensionless radial component of the disturbance acceleration |

${Z}_{r}$ | maximum magnitude of ${z}_{r}$ |

${z}_{t}$ | dimensionless transverse component of the disturbance acceleration |

${Z}_{t}$ | maximum magnitude of ${z}_{t}$ |

${\alpha}_{t}$ | thrust angle $\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{rad}\right)$ |

$\beta $ | ratio of ${\tau}_{{x}_{3}}$ to ${\tau}_{s}$ |

$\delta $ | auxiliary parameter; see Equation (27) |

$\Delta v$ | dimensionless velocity change |

$\gamma $ | auxiliary parameter; see Equation (30) |

${\u03f5}_{r}$ | percentage error in orbital radius |

$\theta $ | polar angle $\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{rad}\right)$ |

$\lambda $ | convergence rate of ${x}_{1}$ and ${x}_{2}$ |

${\lambda}_{H}^{\u2605}$ | value of ${\lambda}^{\u2605}$ corresponding to ${\tau}_{f}={\tau}_{H}$ |

$\mu $ | primary body gravitational parameter $\phantom{\rule{0.166667em}{0ex}}({\mathrm{km}}^{3}/{\mathrm{s}}^{2})$ |

$\rho $ | ratio of ${r}_{f}$ to ${r}_{0}$ |

$\sigma $ | specific standard deviation |

$\tau $ | dimensionless time |

${\tau}_{H}$ | dimensionless Hohmann transfer time |

${\tau}_{s}$ | time to reach the condition $s=0$ |

${\tau}_{{x}_{3}}$ | time to reach the condition ${x}_{3}=0$ |

Subscripts | |

0 | initial |

f | final |

Superscripts | |

· | derivative with respect to t |

${}^{\prime}$ | derivative with respect to $\tau $ |

★ | design value |

∼ | measured |

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**Figure 4.**Variation in ${\tau}_{f}$ with $\{K,\phantom{\rule{0.166667em}{0ex}}\rho \}$ when $\lambda ={\lambda}^{\u2605}$ and $n=4$.

**Figure 5.**Variation in ${\beta}^{\u2605}$ with K when $\lambda ={\lambda}^{\u2605}$ and $n=4$. (

**a**) $\rho =0.723$; (

**b**) $\rho =1.524$.

**Figure 6.**Variation in ${\Delta v|}_{\beta ={\beta}^{\u2605}}$ with K when $\lambda ={\lambda}^{\u2605}$ and $n=4$. (

**a**) $\rho =0.723$; (

**b**) $\rho =1.524$.

**Figure 7.**Transfer trajectories when $K={K}_{v}$ in the two interplanetary mission scenarios. (

**a**) Earth–Venus case; (

**b**) Earth–Mars case.

**Figure 8.**Time-variations of ${a}_{r}$ and ${a}_{t}$ when $K={K}_{v}$ in two typical interplanetary mission scenarios. (

**a**) Radial component, Earth–Venus case; (

**b**) transverse component, Earth–Venus case; (

**c**) radial component, Earth–Mars case; (

**d**) transverse component, Earth–Mars case.

**Figure 9.**Transfer trajectories when ${\tau}_{f}={\tau}_{H}$ in two typical interplanetary mission scenarios. (

**a**) Earth–Venus case; (

**b**) Earth–Mars case.

**Figure 10.**Time-variations of ${a}_{r}$ and ${a}_{t}$ when ${\tau}_{f}={\tau}_{H}$ in two typical interplanetary mission scenarios. (

**a**) Radial component, Earth–Venus case; (

**b**) transverse component, Earth–Venus case; (

**c**) radial component, Earth–Mars case; (

**d**) transverse component, Earth–Mars case.

**Figure 11.**Time-variations of a in the four test mission scenarios. (

**a**) Earth–Venus case, $K={K}_{v}$; (

**b**) Earth–Mars case, $K={K}_{v}$; (

**c**) Earth–Venus case, ${\tau}_{f}={\tau}_{H}$; (

**d**) Earth–Mars case, ${\tau}_{f}={\tau}_{H}$.

**Figure 12.**Trade-off solution between flight time and total velocity change in two typical interplanetary mission scenarios. (

**a**) Earth–Venus case; (

**b**) Earth–Mars case.

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**MDPI and ACS Style**

Bassetto, M.; Mengali, G.; Abu Salem, K.; Palaia, G.; Quarta, A.A.
A Sliding Mode Control-Based Guidance Law for a Two-Dimensional Orbit Transfer with Bounded Disturbances. *Actuators* **2023**, *12*, 444.
https://doi.org/10.3390/act12120444

**AMA Style**

Bassetto M, Mengali G, Abu Salem K, Palaia G, Quarta AA.
A Sliding Mode Control-Based Guidance Law for a Two-Dimensional Orbit Transfer with Bounded Disturbances. *Actuators*. 2023; 12(12):444.
https://doi.org/10.3390/act12120444

**Chicago/Turabian Style**

Bassetto, Marco, Giovanni Mengali, Karim Abu Salem, Giuseppe Palaia, and Alessandro A. Quarta.
2023. "A Sliding Mode Control-Based Guidance Law for a Two-Dimensional Orbit Transfer with Bounded Disturbances" *Actuators* 12, no. 12: 444.
https://doi.org/10.3390/act12120444