Guidance systems are important to autonomous rendezvous with uncooperative targets such as an active debris removal (ADR) mission. A novel guidance frame is established in rotating line-of-sight (LOS) coordinates, which resolves the coupling effect between pitch and yaw planes in a general 3D scenario. The guidance law is named augmented proportional navigation (APN) by applying nonlinear control along LOS and classical proportional navigation normal to LOS. As saving time is a critical factor in space rescue and on-orbit service, the finite time convergence APN (FTCAPN) is further proposed which proves to possess convergence and high robustness. This paper builds on previous efforts in polynomial chaos expansion (PCE) to develop an efficient analysis technique for guidance algorithms. A large scope of uncertainty sources are considered to make state evaluation trustworthy and provide precise prediction of trajectory bias. The simulation results show that the accuracy of the proposed method is compatible with Monte Carlo simulation which requires extensive computational effort.
A critical challenge for missions such as on-orbit servicing and active debris removal is to perform autonomous rendezvous with uncooperative target which provides limited measurement information and even evades with unexpected motion [
Augmented proportional navigation (APN) is initially proposed in our peer-review paper [
The rotating LOS coordinate system and relative motion are shown in Figure
Reference frame for relative motion.
Coordinate systems
3D rendezvous geometry in inertial coordinates
Figure
APN implements feedback control along LOS, and classical proportional navigation is applied in transverse direction to impel the LOS rate to zero which guides the chaser to the rendezvous trajectory. With feedback control along LOS, the dynamic model is decoupled and linearized to obtain analytical solutions of the LOS rate and then a covariance analysis (CA) technique can be applied for performance assessment of the APN guidance system [
In this paper, nonlinear APN is further developed by introducing finite time convergence sliding mode control along LOS, which is named finite time convergence augmented proportional navigation (FTCAPN) for the first time. Its aim is ensuring that the state variable converges to the sliding surface in finite time; as a result, the chaser finishes the midrange rendezvous mission in finite time. FTCAPN is a highly nonlinear guidance system, and consequently, the linear CA technique is no longer proper for performance assessment. For a nonlinear system, the output is likely to be non-Gaussian even with Gaussian input. So the mean and covariance are not sufficient to represent the probability density function (PDF) of a state variable, which is ignored in current references [
One important part of performance assessment is to determine the effect of input uncertainty on output result, which is also called uncertainty propagation or sensitivity analysis. A proper propagation method could make a precise prediction of trajectory bias, through which it is possible to evaluate performance of the guidance system and provide safety analysis result for trajectory programming and following operations.
Many techniques [ Cost: cost of Monte Carlo (MC) is high with a number of propagations although it is commonly used in practice. An MC approach is based on random sampling. MC shows slow convergence, and a large number of executions are needed. (2) Linearization: the most popular nonsampling method is the covariance analysis method. It is a method for directly determining the statistical properties of solutions of linear systems with random inputs. Covariance analysis describing function technique (CADET) is developed to solve nonlinear systems. A nonlinear system is approximated by linear function using quasilinearization methods. The essence undying quasilinear analysis is the substitution of one or more approximate linear gains (describing functions) for each system nonlinearity. The analytic form of linearization is determined by the nonlinearity, number of state variables, and error criterion. Besides, the validity of CADET is established on the assumption that the state variable is Gaussian. Non-Gaussian nature of state variables after the nonlinearity leads to inaccurate CADET results. (3) Non-Gaussian: covariance realism is necessary while not a sufficient realism in describing uncertainty [
This paper consists of 5 sections. In Section
Different navigation and control strategies should be adopted corresponding to specific measurement information in different rendezvous phases [
The 3D relative dynamic equations are established in a rotating LOS coordinate system which resolves the coupling effect between pitch and yaw planes in a general 3D scenario. Introduction of IRPL (instantaneous rotation plane of LOS) simplifies control of chaser; besides, it improves guidance efficiency with smaller miss distance and less fuel consumption. The applications of a rotating LOS coordinate system are also investigated in 3D missile intercept scenarios in references [
A set of dynamic equations is derived based on equation (
If the target is nonmaneuvering, the relative dynamic equation (
As mentioned above, the sliding mode control is introduced along the direction of LOS. The tracking error is defined as
Rendezvous occurs when the relative distance and relative velocity are equal to zero; thus, a sliding surface function is defined as
Differentiating equation (
So,
The stability of the control system along LOS can be guaranteed. Further, the convergence time can be calculated according to the following lemma.
Suppose that there is a continuously differentiable function
Assuming
To guarantee that the acceleration is within the limitation of maximum thrust, a saturation function is introduced:
In order to avoid chatter at the end of proximity, the sliding surface
From the above derivation, we can see that the system will converge and the upper bound of converge time can be derived from initial conditions and control parameter settings. The midrange rendezvous guidance system, FTCAPN, is a highly nonlinear system. Its effectiveness is demonstrated by simulation results as follows. Its performance will be further analyzed in Section
FTCAPN proposed in our paper can be used for autonomous rendezvous with space debris to avoid risk to humans. As we know, debris remediation in LEO is more urgent compared with GEO and MEO, so we choose a target (NORAD catalog number: 39357) at 700 km altitude, and its orbital elements are listed in Table
State of the target.
Orbital elements | ||||||
---|---|---|---|---|---|---|
Target | 7043.14 | 0 | 98 | 14 | 60 | 30 |
This paper focuses on the midrange proximity. The initial state of the chaser is provided by a relative state to the target as shown in Table
Initial relative state between the target and the chaser.
Relative state | ||||||
---|---|---|---|---|---|---|
Chaser/target | 4386.28 | 308.60 | 3204.43 | -57.7 | 2.1 | -38.5 |
According to Tables
Figure
Three-dimensional rendezvous trajectory.
Rate of line of sight.
When relative distance reaches 50 meters, simulation stops and the relative velocity is 5 m/s at the instant of stop (Figure
Relative velocity.
Relative distance.
Zero-Effort-Miss (ZEM) is defined as the closest distance between the chaser and target after control stops [
The deviation of equation (
So, the uncertainty of ZEM could also be deduced by uncertainty in the LOS rate (
Section
Gaussian distribution is generally a good approximation for initial state uncertainty and measurement uncertainty. Propagating the uncertainty of the state variable through a nonlinear dynamic model transforms the state variable into a non-Gaussian distribution. Non-Gaussian behavior is determined by the following two terms: (1) nonlinearity of function and (2) magnitude of initial distribution uncertainty (standard deviation). Quantification of uncertainty sources is shown in Table
Quantification of uncertainty sources.
Uncertainty sources | Mean | Standard deviation |
---|---|---|
Initial position (m) | [0 0 0] | [1.5 1.5 1.5] |
Initial velocity (m/s) | [0 0 0] | [0.3 0.3 0.3] |
LOS rate (rad/s) | 0 |
Figures
Mean of the LOS rate.
Standard variance of the LOS rate.
Although the skewness value fluctuates around 0 in Figure
Skewness of the LOS rate.
Kurtosis of the LOS rate.
This part will investigate the non-Gaussian nature of FTCAPN by the MC technique with larger magnitude of initial uncertainty. It takes about 1405 seconds to run 1000 times of MC simulation which is implemented single thread in MATLAB environment on a laptop with Intel Core i7 2.4 GHz and 8.0 GB of RAM. Propagations of the LOS rate are shown below.
Larger uncertainty in initial position error (Figures
Skewness of the LOS rate (10 times of initial position error).
Kurtosis of the LOS rate (10 times of initial position error).
Figures Larger uncertainty in initial position and velocity error (Figures
Skewness of the LOS rate (10 times of initial position/velocity error).
Kurtosis of the LOS rate (10 times of initial position/velocity error).
With position and velocity initial uncertainty enlarged by 10 times, the non-Gaussian feature of variable distribution is significant as shown in Figures
Finally, we come to the conclusion that it is not accurate enough to study uncertainty propagation through the covariance analysis method which is based on propagation of the mean and covariance. We need to develop a more precise method.
Robustness of APN is already demonstrated with input error in the initial state and the LOS rate. The efficiency with input of other types of uncertainty is yet to be discovered. In near-distance proximity, uncertainty sources include initial position error, initial velocity error, LOS measurement error, target maneuver acceleration and uncertainty, distance measurement error, commanded acceleration saturation, control delay, and nonlinearity of the control system. Consequently, the APN guidance system is a nonlinear and time-varying system with noisy input and other disturbances.
As mentioned in Introduction, existing performance assessment methods are constrained when dealing with a highly nonlinear system. In this section, PCE is introduced for performance assessment of the proposed guidance system, i.e., FTCAPN. In space community, PCE is already applied in orbit propagation of space debris by Jones et al. [
The term polynomial chaos (PC) is invented by Norbert Wiener and Hermite who used polynomials as an orthogonal basis to study the Gaussian stochastic processes [
An efficient numerical method is the stochastic collocation approach, which is a deterministic sampling method. The full tensor product of one-dimensional nodes is extensively used, and postprocessing is conducted to obtain the statistical feature from the solution ensemble. However, the number of total nodes surges exponentially along with the increasing quantities of uncertainty sources, which is known as curse of dimensionality. Sparse PCE, a subset of full tensor PCE, is gaining increasing interest for computational saving with higher uncertainty dimensions.
The approximation of function,
The basis functions
Estimation of coefficients is the result of expectation operation, which is actually integral operation. There are two methods to calculate the integration, i.e., tensor product and sparse grid.
Once the coefficients
From the above equation, we can see that the mean value of
The covariance matrix is
As mentioned in Section
The extension from univariate to multivariate is easily achieved by a tensor product of the univariate node set. Thus, the total number of evaluations is
Figures
Mean of the LOS rate.
Standard deviation of the LOS rate.
The result curve of PCE matches with MC, which means they can achieve the same precision in performance analysis of the terminal guidance system. However, the computation time of MC and TPCE is 1400 and 182 seconds, respectively. So, the simulation results substantially demonstrate that TPCE outperform the MC method in computation efficiency.
Sparse polynomial chaos is proposed to reduce the number of nodes by the Smolyak algorithm. The reduction is achieved by eliminating high-order interactions. In the polynomials,
So, the SPCE achieves high computation efficiency with fewer nodes and expense of accuracy. To be noted, computational burden of SPCE is larger than TPCE in lower dimensions (generally,
In this case, the SPCE of level 2 is used to calculate the uncertainty propagation of the LOS rate. The model is evaluated for merely 15 times, and the computation time is reduced to 22 seconds. Compared with TPC, SPC achieves a high-precision result with less computation time as shown in Figures
Mean of the LOS rate.
Standard deviation of the LOS rate.
Without mitigation techniques, the PCE method suffers from the curse of dimensionality. Computation time increases exponentially with respect to input dimensions
The size of uncertainty dimensionality is assumed to be 9 which includes uncertainty in initial position
Uncertainty sources.
Uncertainty sources | Mean value | Standard deviation |
---|---|---|
Initial position (m) | [0 0 0] | [15 15 15] |
Initial velocity (m/s) | [0 0 0] | [3 3 3] |
LOS rate measurement (rad/s) | 0 | 10-5 |
Distance measurement (m) | 0 | 5 |
Target maneuver (m/ |
[0 0 0] | [0.1 0.1 0.1] |
Compared with former simulations, two more uncertainty sources are included, i.e., distance measurement error and target maneuver. Consequently, equation (
From Table
Result of adaptive PCE.
Statistics feature | Monte Carlo (1000 times) | Polynomial chaos |
---|---|---|
Mean | 3.8 | 3.2 |
Standard deviation | 3.4 | 3.1 |
Skewness | 1.5 | 0.8 |
Kurtosis | 6.67 | 6.6 |
Computation cost (s) | 1330 | 76.6 |
This paper proposed the finite time convergence augmented proportional navigation (FTCAPN) which can be used in time-critical autonomous rendezvous missions such as active debris removal or space rescue. The time to converge is verified and worked out through the Lyapunov function and formula derivation. Robustness of FTCAPN is proved considering input uncertainty in the initial state and LOS rate measurement.
Further, this paper investigated non-Gaussian nature of the proposed guidance law (FTCAPN) which is not tackled in current literatures. Then, polynomial chaos expansion (PCE) is applied to performance assessment of FTCAPN. At last, adaptive PCE is proposed to reduce the computation burden when there is a large scope of random input sources. Simulation results demonstrate that the accuracy of the proposed method is compatible with Monte Carlo, whereas the latter requires extensive computational effort.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This work was supported by the Major Program of National Natural Science Foundation of China under Grant Numbers 61690210 and 61690213.