# Guidance and Control for Safe Contactless Plume Impingement Operations to Detumble an Uncooperative Spacecraft

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Plume Impingement Model

## 3. Plume Impingement Control Algorithm

## 4. Chaser Guidance and Control Strategies

#### 4.1. Chaser Translational Guidance

#### 4.2. Chaser Translational Control

#### 4.3. Attitude Guidance and Control

## 5. Simulation Results

Chaser Spacecraft | ||
---|---|---|

Mass | ${m}_{\mathrm{ch}}$ | 600 kg |

Inertia | ${\mathrm{J}}_{\mathrm{ch}}$ | diag([200, 100, 200]) $\mathrm{kg}{\mathrm{m}}^{2}$ |

Initial angular rate | ${\omega}_{0,\mathrm{ch}}$ | [0, 0, 0] deg/s |

Target spacecraft | ||

Mass | ${m}_{\mathrm{tgt}}$ | 150 kg |

Inertia | ${\mathrm{J}}_{\mathrm{tgt}}$ | diag([45, 25, 50]) $\mathrm{kg}{\mathrm{m}}^{2}$ |

Initial angular rate | ${\omega}_{0,\mathrm{tgt}}$ | See Table 8 |

Shape (see Figure 2) | - | Box $1\times 1\times 1$ m Panels $1\times 1$ m |

**Table 6.**Impingement control parameters and guidance and control functions parameters for the simulated test case.

Impingement Control | ||
---|---|---|

Impingement offset | ${D}_{\mathrm{imp}}$ | 1.25 m or 0.3 m |

Angle tolerance | ${\epsilon}_{\theta}$ | 45 deg |

Magnitude tolerance | ${\epsilon}_{\mathrm{m}}$ | ${10}^{-4}$ Nm |

Gain of guidance torque | ${K}_{\mathrm{imp}}$ | ${10}^{-3}$ |

Detumbling limit | ${\omega}_{\mathrm{lim}}$ | 0.05 deg/s |

Limit pointing chaser | ${\theta}_{\mathrm{p},\mathrm{lim}}$ | 5.7 deg |

Distance limit firing | ${r}_{\mathrm{lim}}$ | 16 m |

Translational guidance | ||

Phase angle limit | $\xi $ | 15 deg |

Synch switch limit | ${\omega}_{\mathrm{sync},\mathrm{lim}}$ | 0.5 deg/s |

KOZ radius | ${R}_{\mathrm{KOZ}}$ | 8 m |

V-bar initial condition | $a\delta {\alpha}_{0}$ | [0, 14, 0, 0, 0, 0]^{T} m |

PSO initial condition | $a\delta {\alpha}_{0}$ | [0, 0, 0, 12, 0, 12]^{T} m |

Translational control | ||

ADRC ESO parameter 1 | ${\beta}_{01}$ | 5 |

ADRC ESO parameter 1 | ${\beta}_{02}$ | 200 |

ADRC ESO parameter 1 | ${\beta}_{03}$ | 0.1 |

ADRC controller gain 1 | ${K}_{1}$ | 0.01 |

ADRC controller gain 2 | ${K}_{2}$ | 1 |

ADRC controller parameter | ${b}_{0}$ | 1 |

PD controller gain 1 | ${K}_{\mathrm{p}}$ | 0.12 |

PD controller gain 2 | ${K}_{\mathrm{d}}$ | 1.2 |

Rotational control | ||

Attitude controller gain 1 | ${K}_{\mathrm{q}}$ | 1 |

Attitude controller gain 2 | ${K}_{\omega}$ | 10 |

Maximum torque RWs | ${T}_{\mathrm{ch},\mathrm{max}}$ | 0.4 Nm |

Numerical propagator | Fixed step Runge–Kutta 4th order |

Simulation step | 0.2 s |

Max simulation time | 30 periods |

Absolute attitude errors | 2 deg (1$\sigma $) |

Angular velocity errors | 0.05 deg/s (1$\sigma $) |

Magnitude error impingement torque | 50% (1$\sigma $) |

Pointing error impingement torque | 20 deg (1$\sigma $) |

$\mathit{\omega}$_{0,tgt} = [8, 1, 1] [deg/s] | |||
---|---|---|---|

$\overline{\delta v}$ [m/s] | $std\left(\delta v\right)$ [m/s] | ${N}_{F}$ | |

V-bar hovering | 17.04 | 1.82 | 0 |

ROE no sync | 13.17 | 1.35 | 32 |

ROE sync | 13.48 | 1.31 | 0 |

$\mathit{\omega}$_{0,tgt} = [1, 8, 1] [deg/s] | |||

$\overline{\delta v}$ [m/s] | $std\left(\delta v\right)$ [m/s] | ${N}_{F}$ | |

V-bar hovering | 39.07 | 0.89 | 0 |

ROE no sync | 32.75 | 0.76 | 6 |

ROE sync | 32.59 | 0.76 | 0 |

$\mathit{\omega}$_{0,tgt} = [1, 1, 8] [deg/s] | |||

$\overline{\delta v}$ [m/s] | $std\left(\delta v\right)$ [m/s] | ${N}_{F}$ | |

V-bar hovering | 9.94 | 1.16 | 0 |

ROE no sync | 9.76 | 1.07 | 29 |

ROE sync | 9.67 | 0.94 | 0 |

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Density field of the plume obtained with the Simons model with the parameters of hydrazine shown in Table 1. The reference frame adopted features ${x}_{LOS}$ along the centre line of the thruster.

**Figure 2.**Density field on an example target represented by meshed surfaces. In magenta the thruster centre line pointing is displayed. A 5 N hydrazine thruster is considered.

**Figure 3.**Plume impingement problem geometry exploited in the pointing algorithm definition. All vector quantities are expressed in the RTN frame, a color-code (orange) is used to highlight quantities belonging to the plane $\mathcal{P}$.

**Figure 4.**Plume impingement problem guidance torques, achieved torque with the plume analytic model and threshold included in the algorithm.

**Figure 6.**Representation of the covariance 1$\sigma $ ellipsoids evolution at three different fractions of the orbital period T after the error propagation in the V-bar hovering case, with the parameters detailed in Table 2.

**Figure 7.**Representation of the covariance ellipsoid evolution at three different fractions of the orbital period T after the error propagation in the PSO case, with the parameters detailed in Table 2. (

**a**) 3D view in RTN, (

**b**) view in the RN plane.

**Figure 8.**Example of target rotation angular momentum vector direction motion in RTN for one orbital period.

**Figure 10.**Simulations results of the V-bar hovering case for impingement operations considering the initial angular rate of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s and 50 different initial attitude states. On the

**left**, the time history of the target’s angular rate magnitude is shown for all runs. The

**right**side shows the distribution of the total delta-v spent for the impingement operations in the simulations displayed.

**Figure 11.**Simulation results of the PSO guidance case for impingement operations considering the initial angular rate of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s and 50 different initial attitude. On the

**left**, the time history of the target’s angular rate magnitude is shown for all runs. The

**right**side shows the distribution of the total delta-v spent for the impingement operations in the simulations displayed.

**Figure 12.**Results of the adaptive synchronisation PSO guidance solution considering the initial angular rate of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s and 50 different initial attitude. On the

**left**, the time history of the target’s angular rate magnitude is shown for all runs. The

**right**side shows the distribution of the total delta-v spent for the impingement operations in the simulations displayed.

**Figure 13.**Example trajectories of impingement operations during the adaptive synchronisation PSO guidance cases.

**Figure 14.**Evolution of the target angular velocity components in the body frame during the impingement operations with adaptive synchronisation PSO guidance and initial conditions of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s.

**Figure 15.**(

**Left**) Representation of the relative eccentricity and relative inclination vector during transfers due to impulsive manoeuvres. (

**Right**) Projection in the RN plane of the failure trajectories during transfer, showing an effective E/I separation to ensure passive safety [35]. The red dashed line represents the KOZ limit considered.

**Figure 16.**Evolution of ROEs during the impingement operations with adaptive synchronisation PSO guidance and initial conditions of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s. Simulations obtained with the ADRC and PD controller are shown.

**Figure 17.**Evolution of delta-v spent over time during the impingement operations (

**top**) and percentage of saving of ADRC vs PD (

**bottom**) with adaptive synchronisation PSO guidance and initial conditions of ${\mathit{\omega}}_{0,\mathrm{tgt}}=[1,1,8]$ deg/s. Simulations obtained with the ADRC and PD controller are shown.

**Figure 18.**Evolution of the integral of the total disturbance $G\left(t\right)$ vector components along RTN frame used in the ADRC controller to track the guidance trajectory.

Parameter | Symbol | Value |
---|---|---|

Thrust | F | 5 N |

Specific Impulse | ${I}_{\mathrm{sp}}$ | 220 s |

Expansion ratio (area) | ${E}_{\mathrm{r}}$ | 80 |

Chamber temperature | ${T}_{\mathrm{c}}$ | 1000 K |

Chamber pressure | ${P}_{\mathrm{c}}$ | 1.1 MPa |

Specific heat ratio | $\gamma $ | 1.37 |

Specific gas constant | ${R}_{\mathrm{plume}}$ | 791.85 $\mathrm{J}/\left(\mathrm{kgK}\right)$ |

Boundary layer ratio | $\frac{\delta}{{R}_{\mathrm{E}}}$ | 0.01 |

Nozzle aperture angle | ${\alpha}_{\mathrm{e}}$ | 15 deg |

Normal accommodation coeff. | ${c}_{\mathrm{n}}$ | 1 |

Tangential accommodation coeff. | ${c}_{\mathrm{t}}$ | 0.97 |

Limiting wall temperature | ${T}_{\mathrm{w}}$ | 300 K |

**Table 2.**Conditions and parameters for the error propagation analysis of a failure scenario during impingement operations.

Initial condition V-bar | $a\delta \mathit{\alpha}={[0,14,0,0,0,0]}^{\mathit{T}}$ m |

Initial condition PSO | $a\delta \mathit{\alpha}={[0,0,14,0,14,0]}^{T}$ m |

KOZ radius | $R=8$ m |

Impulsive disturbance | $\mathsf{\Delta}{\mathit{v}}_{\mathrm{d}}$ = 3 mm/s |

Magnitude error | ${\sigma}_{\mathrm{m}}$ = 20% ($1\sigma $) of $\mathsf{\Delta}{v}_{d}$ |

Direction error | ${\sigma}_{\mathrm{p}}$ = 20 deg ($1\sigma $) |

Initial position error | ${\sigma}_{\mathrm{r}}$ = 50 cm ($1\sigma $) |

Initial velocity error | ${\sigma}_{\mathrm{v}}$ = 1 mm/s ($1\sigma $) |

${\mathit{\beta}}_{01}$ | diag([5, 5, 5]) |

${\mathit{\beta}}_{02}$ | diag([200, 200, 200]) |

${\mathit{\beta}}_{03}$ | diag([0.1, 0.1, 0.1]) |

${\mathrm{K}}_{1}$ | diag([0.01, 0.01, 0.01]) |

${\mathrm{K}}_{2}$ | diag([1, 1, 1]) |

Semi-major axis | a = 7578 km |

Eccentricity | e = 0 |

Inclination | i = 87.9 deg |

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## Share and Cite

**MDPI and ACS Style**

Borelli, G.; Gaias, G.; Colombo, C.
Guidance and Control for Safe Contactless Plume Impingement Operations to Detumble an Uncooperative Spacecraft. *Aerospace* **2024**, *11*, 224.
https://doi.org/10.3390/aerospace11030224

**AMA Style**

Borelli G, Gaias G, Colombo C.
Guidance and Control for Safe Contactless Plume Impingement Operations to Detumble an Uncooperative Spacecraft. *Aerospace*. 2024; 11(3):224.
https://doi.org/10.3390/aerospace11030224

**Chicago/Turabian Style**

Borelli, Giacomo, Gabriella Gaias, and Camilla Colombo.
2024. "Guidance and Control for Safe Contactless Plume Impingement Operations to Detumble an Uncooperative Spacecraft" *Aerospace* 11, no. 3: 224.
https://doi.org/10.3390/aerospace11030224