Nomenclature

$U$

upper compartment

$L$

lower compartment

$rel$

relative

$V$

separation velocity

$a$

acceleration

$\alpha$

angular acceleration

$J$

moment of inertia

$\omega$

angular velocity

$\textbf{M}$

torque

$F$

force

$s$

displacement

$r_i$

actual value of the deviation parameter

$\mu_i$

mean deviation

$\sigma_i$

standard deviation

$f$

maximum force deviation of springs

$\Delta F_i$

working force deviation relative to reference spring

$\theta_i$

force line angle deviation

$f_{i,final}$

spring residual force deviation

Abbreviations

Adaboost

adaptive boosting

ADAMS

Automated Dynamic Analysis of Mechanical Systems

ADRC

active disturbance rejection control

RSM

response surface method

SIASE

Shanghai Institute of Aerospace Systems Engineering

2.0 Problem formulation

The evaluation indexes of the separation motion of the upper and lower modules mainly include the relative separation speed, module separation overload, module angular velocity and relative distance. In order to express the above indexes, it is necessary to define the co-ordinate system related to each rigid body of the separation system, in which is the centroid co-ordinate system of modules, and the base co-ordinate system is the centroid co-ordinate system of the lower module. The spacecraft attitude is described by the relationship between the three axes of the spacecraft body co-ordinate system and the three axes of the orbit co-ordinate system, both of them are right-handed orthogonal co-ordinate systems, and co-ordinate systems are given in Fig. 2. X-axis direction as the positive direction of separation.

Figure 2.

Coordinate systems of separation motion.

In Fig. 3, red circles represent seven sets of forces, from 1 to 4 are separation springs, 5 and 6 are separation disconnectors, and the separation plug is 7. Because of the deviation between separation springs and the asymmetrical layout of forces, the dissociative motion state of modules might be disturbed. In fact, the mission will be decided failure if the separate velocity can not reach the stated objective that was set for entering orbit, or collision happened between upper and lower module.

Figure 3.

Forces distributed on contact surface of modules.

In deriving the kinetic equations, the upper and lower compartment modules are considered as homogeneous rigid bodies. The nonlinear model of separation system of reusable spacecraft contains input and output relationship of multiple parameters. According to the dynamics of the spatial motion multi-rigid body system, where the descriptive equations for the main parameters are as follows Newton-Euler equations [Reference Aslanov, Kruglov and Yudintsev19]:

1. 1) Relative separation speed between modules

(1) \begin{align}{V_{rel}} = {V_U} - {V_L} \\[-24pt] \notag \end{align}

${V_U}$ , ${V_L}$ are the centre-of-mass velocities of the upper and lower module, the upper module velocity ${V_U}$ for example:

(2) \begin{align}{V_U} = \int\limits_0^t {{a_U}dt} \end{align}

${a_U}$ is the centre-of-mass acceleration of the upper module:

(3) \begin{align}{a_U} = {\dot V_U} = \frac{{{F_U}}}{{{m_U}}} \\[-30pt] \notag \end{align}

(4) \begin{align}{F_U} = \sum\limits_{i = 1}^7 {{f_i}} \end{align}

1. 2) Angular acceleration of separation of the upper module

(5) \begin{align}{J_U} \cdot {\dot \omega _U} + {\omega _U} \times \left( {{J_U} \cdot {\omega _U}} \right) = {{\bf{M}}_U} \\[-25pt] \notag \end{align}

(6) \begin{align}{\alpha _U} = {\dot \omega _U} = \frac{{{{\bf{M}}_U}{\rm{ - }}{\omega _U} \times \left( {{J_U} \cdot {\omega _U}} \right)}}{{{J_U}}} \\[-30pt] \notag \end{align}

(7) \begin{align}{{\bf{M}}_U} = \sum\limits_{i = 1}^7 {{{\bf{M}}_{U,i}}} \\[-30pt] \notag \end{align}

(8) \begin{align}{{\bf{M}}_{U,i}}={r_{U,i}} \times {f_i} \\[-25pt] \notag \end{align}

(9) \begin{align}{r_{U,i}}={D_{U,i}}{\rm{ - }}{C_U} \\[-25pt] \notag \end{align}

(10) \begin{align}{\omega _U} = \int\limits_0^t {{\alpha _U}dt} \\[-25pt] \notag \end{align}

1. 3) Overload of separation of lower module

(11) \begin{align}{a_L} = {\dot V_L} = \frac{{{F_L}}}{{{m_L}}} \\[-25pt] \notag \end{align}

(12) \begin{align}{{\rm{g}}_L} = \frac{{{a_L}}}{G} \\[-25pt] \notag \end{align}

1. 4) Relative distance between upper and lower module

(13) \begin{align}\Delta s = {s_U} - {s_L} \\[-25pt] \notag \end{align}

(14) \begin{align}{s_U} = \int_0^t {{V_U}d} t \\[-25pt] \notag \end{align}

(15) \begin{align}{s_L} = \int_0^t {{V_L}d} t \end{align}

Using the dynamics simulation technology of Automated Dynamic Analysis of Mechanical Systems (ADAMS) software, we import the simplified combined model of the separated system into it and directly solve the attitude parameters after separation. We mainly focus on the overall separated attitude of the upper and lower modules, and perform Boolean operations on the upper and lower modules separately after importing into ADAMS to reduce the number of parts in the overall model. In ADAMS, in addition to the local coordinate system of the part, there is a global coordinate system that coincides with the above coordinate system.

3.0 Dynamic simulation of spacecraft separation event

The upper and lower modules are separated by a series of components, including springs, a separation plug and disconnectors. The seven sets of external forces, as well as the contact surfaces that may generate contact between the upper and lower modules, were set up using the ADAMS built-in dynamics model and the mathematical and displacement function statements in the runtime function.

The separating direction of the upper module is positive, i.e. in the direction of the X-axis, and the force between the upper and lower modules mainly consists of the spring force generated by the four sets of springs, which is a positive separating thrust for the upper module and a reaction force of opposite magnitude and direction for the lower module.

There are two sets of disconnectors between the upper and lower modules. The disconnectors provide a thrust to the upper module at the beginning of the separation section, 0–0.01 m, and a resistance to the upper module at the end of the separation section, 0.01–0.0442 m, and the forces acting on the lower module are equal in magnitude and opposite in direction.

In addition to the above seven sets of external forces, there is a separating plug between the upper and lower modules, which produces a separating resistance on the upper module when the relative displacement distance between the upper and lower modules reaches 0.01, with a value of 80 N, which rapidly decreases to 0 after the displacement increment reaches 0.002 m.

It is the work of this section to transform the force under the above conditions into a time-varying function and to introduce the numerical model of the separation system for calculation. In addition, it should be noted that the disconnectors and seperation plugs do not have a deviation from the line of force.

According to Figs. 4 and 5, it can be analysed that the initial force of the separation spring acting on the upper module is 2102 N, i.e., the preload is 2102 N. The force decreases linearly from 0 to 0.1073 s, and the value is 533.5 N at 0.1073 s. The spring force no longer decreases after that, and it can be concluded that its time of action is 0.1073 s, and it is 0.1073 s for the ideal working condition.

Figure 4.

Upper module separation attitude.

Figure 5.

Lower module separation attitude.

It can be seen that the separation spring separation thrust has a large influence on the separation attitude of the upper and lower modules, and the asymmetric spring layout has a cross-strong external moment on its Y-axis, which generates a certain angular acceleration and angular velocity, and then generates angular displacement. And the angular velocity, relative velocity, and separation overload between the upper and lower modules can be used as the evaluation indexes of the separation performance to analyse the reliability and parameter sensitivity of the spacecraft separation dynamics performance.

4.0 Numerical analysis simulation results

After the verification of model accuracy and reasonableness, this section brings a new method to assess the reliability and sensitivity analysis of spacecraft whole star separation performance based on Monte Carlo method.

Model with deviation parameter

(16) \begin{align} f{({r_i})_{({r_{i\min }} \le x \le {r_{i\max }})}} = \int\limits_{{r_{i\min }}}^{{r_{i\max }}} {\frac{1}{{\sqrt {2\pi } \cdot {\sigma _i}}}{e^{ - \frac{{{{({r_i} - {\mu _i})}^2}}}{{2{{({\sigma _i})}^2}}}}}d{r_i}} \end{align}

Where ${r_i}$ is the actual value of the deviation parameter, ${\mu _i} $ and ${\sigma _i} $ are the mean and standard deviation, respectively, assuming that the upper and lower limits of the inputs are ${r_{i\min }} $ and ${r_{i\max }} $ , respectively. then its mean value can be derived, which is:

(17) \begin{align}{\mu _i} = \frac{1}{2}({r_{i\min }} + {r_{i\max }}) \end{align}

According to the principle of $3\sigma $ , the standard deviation can be found to be:

(18) \begin{align}{\sigma _i} = \frac{1}{6}({r_{i\min }} - {r_{i\max}}) \end{align}

The deviation parameters are specified in Tables 1 and 2, starting with the internal deviations of the separation system.

Table 1.

Internal deviations of the separation system

*The variable distribution probabilities all follow a normal distribution

Measurement objects are established for the performance indicators that are to be observed and collected, and they can be observed during the simulation process, with the specific names corresponding to those shown in the Table 3.

Establish the simulation target that needs to collect data, the name of the simulation target, and the type of data to be collected during the simulation as shown in the Table 4.

Table 2.

Deviation of quality characteristics

*The variable distribution probabilities all follow a normal distribution

Table 3.

Measurement objects

Table 4.

Simulation target

The last step in setting up the simulation test is to create a simulation script. Due to the large number of model components, simulation model variables and simulation objectives, the calculation process is complex, so after several rounds of comparison, the optimal script termination time and step size were finally determined, where the termination time is 0.2 s and the step size is 0.01 s.

A key step in the design of the experiment within ADAMS insight is the setting of the sampling parameters, where the variables set in the previous section are added to the experimental influences and a random sample is set.

A normality test on the sample set generated by ADAMS found that the random sample data set generated by the set variables followed a normal distribution.

Separation parameters meet the performance index calculation

After completing the experimental design, the simulation test is run. This simulation test is a total of 3000 rounds of cycle, time 4 hours and 47 minutes. After each simulation, the programme will automatically collect the simulation target parameters are specified as follows:

Table 5.

Overview of angular velocity

Table 6.

Overview of relative velocity

Figure 6.

Distribution of relative separation velocity data between upper and lower modules.

Figure 7.

Normality test for relative separation velocity between upper and lower modules.

Relative separation speed between modules

The performance index of the relative separation velocity between the upper and lower modules is ≮0.4 m/s. According to the simulation model, it can be inferred that the main source of angular velocity is the spring group, disconnector and separating plug fixed between the upper and lower modules, and the thrust resistance generated between the two compartments at the stage of separation, as shown in the scatter diagram of angular velocity amplitude generated after randomly hitting the target 3,000 times, Table 5 is a overview of angular velocity, Figs. 6 and 7 describe the data distribution:

According to the results measured by the simulation target of the relative separation velocity between upper and lower modules, it can be seen that the relative separation velocity between upper and lower modules is mainly distributed in the x-direction, and the data show a spindle-shaped distribution, with less on both sides and more in the middle, mainly concentrated near 0.667 m/s.

The data are further tested and analysed for normality.

It can be seen that the relative separation speed data between upper and lower modules basically conforms to a normal distribution. The specific data are shown in the Table 6.

Random target shooting calculations show that the relative separation amplitude between the upper and lower modules is in the range of [0.612,0.707], and its performance specification is ≮0.4 m/s, so it satisfies the performance specification.

Angular acceleration of separation of the upper module

The upper module separation angular velocity performance index is required to be ≯1.2 deg/s. According to the simulation model, it can be inferred that the main source of its angular velocity is the y-direction tilting force brought by the asymmetric design of the spring, and the angular velocity dataset generated after 3,000 times of random target hitting is shown in Fig. 8.

Figure 8.

Distribution of separation angular velocity data for the upper module.

Figure 9.

Upper module separation angular velocity normality test.

According to the results of the separation angular velocity of the upper module, it can be seen that the separation angular velocity of the upper module is mainly distributed in the y-direction, and the data show a spindle-shaped distribution, with less on either side and more in the middle, mainly concentrated around 0.357 deg/s.

The data are further checked and analysed for normality in Fig. 9.

It can be seen that the separation angular velocity data of the upper module basically conforms to the normal distribution. The specific data are shown in the Table 5.

Random target shooting calculations show that the upper module separation angular velocity amplitude range is [0.282,0.43], while its performance specification is ≯1.2 deg/s, so it meets the performance specification. The overview of Overhold is shown in Table 7.

Overload of separation of lower module

The separation overload performance index of the lower modules is required to be ≯0.7 g, and according to the simulation model, it can be assumed that the overload mainly comes from the spring group, disconnector and separation plug fixed between the upper and lower modules, and the thrust resistance to the two compartments in the separation stage, as shown in Fig. 10. The separation overload data set of the lower modules generated after 3,000 times of random target hits.

According to the results of the lower module separation overload simulation target, it can be seen that the lower module separation overload is mainly distributed in the x-direction, and the data show a spindle-shaped distribution, with fewer on both sides and more in the middle, which is mainly concentrated around 5.228 m/s².

The data are further tested and analysed for normality in Fig. 11.

Random target calculations show that the lower module separation overload amplitude is in the range [4.51,5.997], while its performance specification is ≯0.7 g (6.86 m/s²), so it meets the performance specification.

Collision analysis

In this paper, the safety and success of the separation process is assessed by measuring whether or not the contact force is generated between the main components of the upper and lower modules.

From the above Fig. 12 it can be concluded that the contact force between the upper and lower modules was always 0 during the simulation of the 3000 rounds. Therefore, it was judged that all simulated rounds were safely and successfully separated.

It can be seen in Fig. 13 that the on-orbit separation performance of the separation system fully meets the design requirements and its reliability reaches 100%.

Figure 10.

Lower module separation overload data distribution.

Figure 11.

Lower module separation overload normality test.

Figure 12.

Upper and lower modules collision detection.

Figure 13.

Separation performance distribution.

Figure 14.

Adaboost regression algorithm steps.

Dynamic change of separation gap

The core of reliability assessment for separation based on the separation gap between upper and lower cabin modules lies in analysing whether the gap between modules during the separation process meets the requirements of collision-free and smooth separation, which necessitates the integration of multi-dimensional data to quantify the correlation between the gap and reliability.

5.0 Parameter importance analysis based on adaboost regression

The Adaboost regression algorithm gives a high weight to learners with a low error rate and a low weight to learners with a high error rate, combining the weak learners and the corresponding weights to generate strong learners. The difference between the regression and classification algorithms is that the error rate is calculated in a different way. Classification problems generally use a 0/1 loss function, while regression problems generally use a squared loss function or a linear loss function.

Table 7.

Overview of overload

The AdaBoost algorithm can be used not only for classification tasks but also for regression tasks. In this section, the parameter set of separation dynamics formed by ADAMS simulation is used to analyse the degree of influence of error inputs obeying normal distribution on the performance of the spacecraft separation system based on Adaboost regression.

Suppose there is a dataset that consists of m samples as follows:

(19) \begin{align} D = \big\{ {\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right), \ldots ,\left( {{x_m},{y_m}} \right)} \big\} \end{align}

Among them, ${x_i} \in {{\bf{R}}^d} $ , means each sample data has d features, $ {y_i} $ is the target value. The specific steps of the AdaBoost regression algorithm are as follows in Fig. 14.

Take relative velocity as an example. The model parameters are set as Table 8.

Table 8.

Regression model parameterisation

The existing separation performance data is sliced and reorganised for training and testing through three steps: data slicing, data shuffling and cross-validation.

In the following Fig. 15, the prediction evaluation metrics of the cross-validation set, the training set and the test set are shown to measure the prediction effect of Adaboost through quantitative metrics. Among them, the evaluation metrics of the cross-validation set can continuously adjust the hyperparameters to get a reliable and stable model. Tables 9–11 shows the module assessment results of seperation parameter.

From the analysis results in Fig. 16, it can be seen that the maximum preload of the separation spring has a significant effect on the separation overload of the lower module and the relative separation speed of the two modules, which are 60.50% and 46.70%, respectively, followed by the mass of the lower module, which is 37.60% and 22.70%, respectively; the variables that have significant effects on the angular speed of the upper module separation are the Z-axis coordinates of the centre of mass of the upper module and the rotational moment of inertia of the upper module, which are 22.20% and 22.20%, respectively. 22.20% and 29.40%, respectively.

Figure 15.

Adaboost regression prediction.

7.0 On-orbit validation

In December 2024, the reusable spacecraft studied in this article, developed by the Shanghai Institute of Aerospace Systems Engineering (SIASE), was released on orbit for separation and successfully operated, and the actual measured values of the angular velocity of the spacecraft and the overload of the lower module during the separation process fell within the prediction intervals of the Monte Carlo stochastic target. They also met the design specifications for separation performance. During the actual on-orbit separation mission, the assembled body has a trace initial angular velocity in the pitch, roll and yaw directions, which is input into the model built in this paper, and the output results still have stability and reliability.