#### 4.3. Simulation Scenarios

Two different scenarios are analyzed. Firstly, the algorithms’ success rate (

SR) is computed considering different positions along a target-chaser relative trajectory designed for safe debris monitoring [

36] using the simulator described in

Section 3.3. This trajectory is a safety ellipse [

37] in which the relative distance ranges from about 25 to 53 m. The evaluated

SR is the percentage of successful TM pose solutions over 241 positions equally separated in time (50 s) along two relative orbits (see the black dots in

Figure 5). The time step is selected so that two consecutive poses are significantly different in terms of both relative attitude and position.

**Figure 5.**
Time variation of the target-chaser relative distance along two consecutive orbits (in blue). The black dots indicate the time instants and the corresponding relative distances at which the TM algorithm is tested.

**Figure 5.**
Time variation of the target-chaser relative distance along two consecutive orbits (in blue). The black dots indicate the time instants and the corresponding relative distances at which the TM algorithm is tested.

The second scenario is introduced to deepen the results from the first analysis, and in particular to separate the effect of the target-chaser relative distance and orientation on algorithms’ performance. In this case, the SR is the percentage of successful pose solutions over a certain number of randomly generated sets of Euler angles, at different values of the target-chaser relative range, i.e., at 20, 30, 40, and 50 m. Specifically, at each range, 500 sets of Euler angles are generated extracting yaw, pitch, and roll values from three uniform distributions.

#### 4.4. Simulation Results

The simulations are performed in MATLAB™ environment and run on a commercial desktop equipped with an Intel™ i7 CPU at 3.4 GHz. As regards the analysis over the relative trajectory, five different sampling steps (

i.e.,

Δ = 60°,

Δ = 40°,

Δ = 30°,

Δ = 20°, and

Δ = 10°) are considered. The effect of

Δ on the on-line TM success rate (

SR_{TM}), expressed in the range (0, 1), as well as on the number of templates to be generated (and consequently on the computational cost) is shown in

Figure 6, considering both the NN and the NS approaches for the matching step.

**Figure 6.**
(**Top**) Effect of Δ on SR_{TM} comparing the NN and NS approaches. (**Center**) Effect of Δ on the number of templates to be generated; (**Bottom**) Effect of Δ on the average computational cost.

**Figure 6.**
(**Top**) Effect of Δ on SR_{TM} comparing the NN and NS approaches. (**Center**) Effect of Δ on the number of templates to be generated; (**Bottom**) Effect of Δ on the average computational cost.

The first fundamental result is that the NN approach appears to be more effective than the NS one for the ICP matching step, at least immediately after the pose acquisition. This can be explained by remarking that highly rough initializations of the pose parameters (as the ones provided by the TM algorithm) can cause sensor-model point associations characterized by larger distances (if compared to the NN approach) when projecting a sensor point on the closest model surface, as requested by the NS logic. So, in the following, only the success rates achieved by the NN variant will be shown. As expected, a reduction of

Δ produces an increase in

SR_{TM}, since it allows restraining the angular gap that the tracking algorithm has to compensate. Specifically, when

Δ is 10°,

SR_{TM} reaches its maximum value of about 76%, but the number of templates is so large (~26,000) that the computational time also becomes unacceptably high (145 s) for close-proximity flight. On the other hand, if

Δ is 60°, the number of templates drops down to 196 and so does the computational time (1 s), but at the same time the

SR_{TM} reduces to 59%. However, it is interesting to notice that the selection of intermediate values of

Δ (

i.e., 20° or 30°) keeps the algorithm’s computational time low enough (20 s and 7 s, respectively) to enable real time operations, while simultaneously providing values of

SR_{TM} slightly higher than 70%. This can be explained observing that increasing

Δ from 10° to 20° the number of templates reduces of one order of magnitude, while the average estimation error in the Euler angles (evaluated considering only the successful pose estimates) is approximately the same, as it is shown in

Table 2.

**Table 2.**
On-line TM average estimation error in the relative Euler angles averaged over the successful pose estimates within the two-orbit sequence of poses.

**Table 2.**
On-line TM average estimation error in the relative Euler angles averaged over the successful pose estimates within the two-orbit sequence of poses.
Average Attitude Estimation Error |
---|

**Δ (°)** | 10 | 20 | 30 | 40 | 60 |

**Successful pose estimates** | 184 | 177 | 173 | 163 | 142 |

**Euler angles** | **α (°)** | 9.73 | 14.04 | 13.83 | 39.83 | 56.88 |

**β (°)** | 6.17 | 6.88 | 8.15 | 14.50 | 18.97 |

**γ (°)** | 21.40 | 22.70 | 37.45 | 47.62 | 78.45 |

By looking at the results in

Table 2, it is possible to notice that the error in the yaw angle is always the largest. This can be explained considering that the simplified model used for ENVISAT has a preferential direction or symmetry axis. This makes the estimation of the rotation around this axis more challenging due to possible pose ambiguity issues. This situation is of course determined also by the fact that the model does not include many details present on the satellite external surface. Another important property of the on-line TM can be noticed by comparing the values of the estimation error in the relative position vector when the algorithm is successful to the same ones corresponding to algorithm failures. These results are collected in

Table 3 where

T_{X},

T_{Y} and

T_{Z} are the components of

T in the SRF. When the LIDAR point cloud centroid is too far from the TRF origin (

i.e., the geometric center of ENVISAT main body) due to particular conditions in terms of target relative attitude, the relative position estimation error becomes too large, thus compromising the algorithm’s capability to find the set of sampled Euler angles adequately close to the real triplet.

**Table 3.**
On-line TM average estimation error in the relative position vector components for the considered two-orbit sequence of poses.

**Table 3.**
On-line TM average estimation error in the relative position vector components for the considered two-orbit sequence of poses.
Relative Position Vector Components | Average Position Estimation Error |
---|

Δ = 10° | Δ = 20° | Δ = 30° | Δ = 40° | Δ = 60° |
---|

**On-line TM success** | **T**_{X} (m) | 2.809 | 2.697 | 2.689 | 2.772 | 2.853 |

**T**_{Y} (m) | 1.324 | 1.268 | 1.251 | 1.343 | 1.369 |

**T**_{Z} (m) | 1.924 | 1.936 | 1.868 | 1.808 | 1.868 |

**On-line TM failure** | **T**_{X} (m) | 4.021 | 4.198 | 4.130 | 3.773 | 3.443 |

**T**_{Y} (m) | 1.588 | 1.714 | 1.732 | 1.478 | 1.412 |

**T**_{Z} (m) | 4.698 | 4.360 | 4.390 | 4.192 | 3.601 |

In

Figure 7 the on-line TM algorithm performance is compared to its fast variant, defined in

Section 2.1, in terms of both success rate (

SR_{fast-TM}) and computational cost.

**Figure 7.**
On-line TM vs. on-line fast-TM. (**a**) Success rate; (**b**) Loss of success rate of the on-line fast-TM; (**c**) Percentage of templates excluded from the correlation function evaluation; (**d**) Computational time reduction.

**Figure 7.**
On-line TM vs. on-line fast-TM. (**a**) Success rate; (**b**) Loss of success rate of the on-line fast-TM; (**c**) Percentage of templates excluded from the correlation function evaluation; (**d**) Computational time reduction.

Since a value of τ equal to 0.1 has been considered in Equation (3), the on-line fast-TM excludes from the evaluation of the correlation function, C, about 66% of the generated templates for any value of Δ, thus getting a reduction of about 15% in the computational time as compared to the basic approach. This reduction is limited by the fact that this technique has no impact on the time required for templates generation, which represents the main contribution to the overall computational burden. In terms of SR, although the decreasing behavior as a function of Δ is confirmed, the fast-TM strategy causes a loss of performance as compared to the basic approach due to the fact that, in some cases, it excludes also good candidate templates. However, if Δ is low enough (i.e., 10° or 20°), the loss of success rate (SR_{TM} - SR_{fast-TM}) is extremely limited (about 1%), while it increases up to 22% when Δ grows to 60°. It is worth noting that when Δ is 40° the loss of success rate (10%) is lower than in the case of Δ equal to 30° (14%). This is related only to the particular sequence of poses analyzed over the relative trajectory.

The results about the success rate reduction derive from the fact that the condition introduced by Equation (3) is a reliable measure of the similarity between the templates and the LIDAR point cloud only if they contain enough information to perform the discrimination process. For instance, this happens when the number of templates is large (

i.e., when

Δ is low), and the sensor point cloud is dense enough. To better explain this effect, the simulation results relevant to 500 randomly generated sets of Euler angles at different target-chaser relative ranges (ρ) are analyzed.

Figure 8 shows the effect of

Δ on

SR_{TM}, on the left side, and on

SR_{TM} -

SR_{fast-TM}, on the right side, while ρ varies from 20 to 50 m.

**Figure 8.**
(**Left**) SR_{TM} as a function of Δ at different ranges; (**Right**) Variation of SR_{fast.TM}-TM with respect to SR_{TM} as a function of Δ at different ranges.

**Figure 8.**
(**Left**) SR_{TM} as a function of Δ at different ranges; (**Right**) Variation of SR_{fast.TM}-TM with respect to SR_{TM} as a function of Δ at different ranges.

Again, a reduction of Δ produces an SR increase. More precisely, at ρ equal to 20 m, by changing the sampling step from 60° to 10° the value of SR_{TM} varies from 65% up to 97%. On the other hand, at ρ equal to 40 m, by changing Δ from 60° to 10° the value of SR_{TM} rises from 61% to only 70%. The effect of Δ on the algorithm’s performance weakens as the target-chaser relative distance increases since, being δ_{LOS} fixed to 1°, the point clouds become so sparse (i.e., their size reduces from about 490 points on average at 20 m to about 120 points on average at 50 m) that templates corresponding to pose parameters different from the actual ones can give rise to ambiguous matches (i.e., they can produce similar values of C), independently of how well the attitude parameters space is sampled. Moreover, as the target moves far away from the chaser, the LIDAR SNR goes down, thus increasing the probability of point misdetection. This completely explains the worsening of the effectiveness of the proposed approach for pose acquisition at larger range. For instance, if Δ is equal to 30°, the value of SR_{TM} goes from 88% at 20 m down to 68% at 50 m.

By focusing the attention on the right side of

Figure 8, it is possible to notice that the increase of the range-to-target has a negative effect also on the

SR performance of the on-line fast-TM. For instance, at 20 m range, the value of

SR_{TM} -

SR_{fast-TM} remains below 1% for Δ lower than 60° (which provides a loss of success rate of 9.9%), and, above all, by adopting the angular sampling step of 20°, it becomes −0.4% thus meaning that the adoption of the fast variant of the proposed TM algorithm is able not only to reduce the computational load but also to slightly improve the performance. On the other hand, at 30 m range,

SR_{TM} -

SR_{fast-TM} is always positive (never any

SR improvement introduced by the fast variant), and gets worse for increasing

∆, ranging from 3.9% to 23% for ∆ varying from 10° to 60°. Basically, this analysis over a wide range of pose parameters confirms that it is convenient to apply the fast-TM strategy only when enough information to solve the pose ambiguity problem is available,

i.e., at close range and/or selecting very low values of ∆.

As regards the computational load, also the time saving provided by the on-line fast-TM with respect to the basic TM algorithm is influenced by the variation of the target range. Specifically, as the target-chaser range enlarges, the size of the LIDAR point cloud tends to reduce. Hence, the contribution of the correlation determination task to the overall TM algorithm computational burden becomes less important, thus limiting the acceleration provided by its fast variant.

#### 4.5. TM Failure Detection Approach

In order to cope with the cases of unsuccessful pose acquisition, a strategy is now introduced to autonomously detect the TM algorithm failure. To this aim, the analysis performed with

Δ equal to 30° over the relative trajectory of

Section 4.3 is considered. Moreover, the attention is focused only on the on-line TM algorithm. In this case

SR_{TM} is 72% and the computational time is 7 s. The strategy for autonomous failure detection is based on the value of the ICP cost function, as defined in [

15], at the convergence (

f_{CONV}) of the tracking algorithm, that is at the first time instant subsequent to the pose acquisition.

Figure 9 clearly shows that when the TM is successful the value of

f_{CONV} is of the order of few square millimeters, thus being significantly lower with respect to the case of TM failure where

f_{CONV} goes from more than 10 square centimeters up to few square meters.

**Figure 9.**
(**Up**) Value of the ICP cost function at convergence for the considered two-orbit sequence of poses; (**Down**) Enlargement of the same graph. The red dots correspond to successes for the TM algorithm, while the blue circles indicate the failures. The maximum value of f_{CONV} which corresponds to algorithm failure is highlighted.

**Figure 9.**
(**Up**) Value of the ICP cost function at convergence for the considered two-orbit sequence of poses; (**Down**) Enlargement of the same graph. The red dots correspond to successes for the TM algorithm, while the blue circles indicate the failures. The maximum value of f_{CONV} which corresponds to algorithm failure is highlighted.

This result can be generalized by analyzing all the simulation runs performed over the relative trajectory, as summarized in

Table 4.

**Table 4.**
Comparison among the values of f_{CONV} in the cases of failure and success of the TM algorithm.

**Table 4.**
Comparison among the values of f_{CONV} in the cases of failure and success of the TM algorithm.
| | f_{CONV} (m^{2}) |
---|

| Δ(°) | TM Failure: Minimum Value | TM Failure: Mean Value | TM Success: Maximum Value | TM Success: Mean Value |
---|

**On-line TM** | **10** | 0.1882 | 1.4620 | 0.0023 | 0.0012 |

**20** | 0.1882 | 1.4738 | 0.0023 | 0.0012 |

**30** | 0.1852 | 1.5309 | 0.0023 | 0.0012 |

**40** | 0.1539 | 1.3478 | 0.0026 | 0.0012 |

**60** | 0.1254 | 1.6681 | 0.0029 | 0.0013 |

**On-line fast-TM** | **10** | 0.3086 | 1.5513 | 0.0023 | 0.0012 |

**20** | 0.3513 | 1.6901 | 0.1702 | 0.0022 |

**30** | 0.1254 | 1.4034 | 0.0023 | 0.0012 |

**40** | 0.1539 | 1.4273 | 0.0027 | 0.0013 |

**60** | 0.1254 | 1.7980 | 0.0026 | 0.0012 |

A difference of three orders of magnitude exists between the average values of

f_{CONV} in the two cases. So, on this basis, it can be defined a maximum threshold value for the ICP cost function (

f_{MAX}) that allows the TM algorithm to autonomously detect its own failure. In case a value of

f_{CONV} larger than

f_{MAX} is obtained, it is advisable to wait for a certain amount of time and then to repeat the pose acquisition step until the condition for

f_{CONV} is satisfied, and the pose solution provided by the TM can be considered reliable for the tracking phase. The logic for the implementation of the proposed algorithms for pose determination during a close range relative navigation maneuver is shown in detail in

Figure 10.

**Figure 10.**
Logic for autonomous verification of successful pose estimation for the proposed TM algorithms.

**Figure 10.**
Logic for autonomous verification of successful pose estimation for the proposed TM algorithms.