Adaptive coordinated motion control with variable forgetting factor for a dual-arm space robot in post-capture of a noncooperative target

Abstract

To overcome the problem of dynamics coupling between a space robot and a target satellite, this study introduces a new coordinated motion control approach with an adaptive filtering algorithm for a dual-arm free-floating space robot. Based on the reaction null space control scheme, one arm is used to complete the capture task and the other to counteract disturbance to the space base. However, when space robot captures a noncooperative target, the system may experience abrupt changes in dynamic parameters and output measurement noise, which can cause traditional control methods to achieve poor results in practical applications. Thus, an adaptive filtering algorithm with a variable forgetting factor is proposed to improve the tracking capabilities and robustness of the system. The convergence analysis is performed based on a Lyapunov function. The simulation results demonstrate the effectiveness of the proposed algorithm.

Introduction

With the development of astronautic technology, space robots have been playing an important role in space exploration. Their main missions include capturing and repairing noncooperative space objects or debris and supporting astronauts in replacing or assembling components on space stations. Therefore, many countries have paid significant attention to the development of space robotic technologies. The SUMO/FREND project and the Phoenix Program ¹ exemplify typical orbital applications of space robots. The main characteristics of the two projects are that the space robots have more than one manipulator, and the inertial parameters of the target spacecraft are much larger than those of the robot.

A number of investigations on the capture of satellites and space debris have been conducted. To describe the coupling relationship between a satellite base and its mounted manipulators, researchers in this field generally tend to separate the on-orbit capture missions into four phases. ² The first is the observing and approaching phase where a space manipulator is controlled and moved toward the grasping location by gradually following the motion of the target. The second phase is the capture (physical contact) phase in which the end-effector of the space manipulator physically captures the target. In the third phase, the space manipulator firmly captures the target satellite and applies the control strategy to deal with the tumbling motion and dynamic uncertainties. The fourth is the compound stabilization phase in which the space robot dampens the motion of the target. In this study, we address the problems that arise in the post-capture and compound stabilization phase. The main topic presented in this study is the minimization of the disturbance to the base after capturing a large noncooperative target satellite. This task is necessary since the antennas of the servicing base must be pointed toward the Earth ³ ; therefore, the base attitude must be maintained.

To resolve the dynamic interaction problems of free-floating robots, a well-known concept of reaction null space (RNS) control law has been widely employed. The RNS control law was originally proposed by Nenchev et al. ⁴ to achieve the attitude control of a free-floating space robot. Yoshida et al. ⁵ applied RNS control to stabilize the base attitude in the ETS-VII project, which proved useful. In Dimitrov and Yoshida, ² a distributed momentum control strategy was proposed for capturing a tumbling satellite. The RNS motion control was employed to control the joint motion and spacecraft attitude. Recently, based on RNS control, Huang et al. ⁶ planned zero-disturbance end-effector paths for a dual-arm space robot using a dynamic balance control algorithm. However, most of the approaches mentioned above have relied on the accurate dynamic parameters of the target, such as mass and moment of inertia.

In the presence of parameter uncertainties, a wide range of adaptive controllers have been developed for space robots. After capturing an unknown target, adaptive techniques were proposed in the literature ^7,8 to avoid the effect of parameter uncertainties on the base attitude and achieve trajectory tracking of the end-effector. Nguyen-Huynh and Sharf ^9,10 presented an adaptive reaction null space (ARNS) control algorithm to satisfy the objective of maintaining a minimum disturbance to the base without knowledge of the target dynamics. In the proposed adaptive approach, the recursive least squares (RLS) algorithm was employed to update the reactionless joint rates for parameter adaptation in an online manner. An adaptive filter was used to update the estimated parameters at each time sample. In the classical RLS algorithm, the forgetting factor is constant with values between 0 and 1. However, it is unsuitable for tracking time-varying parameters since the algorithm gain converges to zero, which leads to an exponential growth of the filter gain matrix ¹¹ To resolve the conflicts, numerous variable forgetting factor RLS (VFF-RLS) algorithms have been developed. ^12–14 In this study, we improve a variable forgetting factor weighted recursive least square (VFF-WRLS) algorithm for system identification and apply it to the coordinated motion control for a dual-arm space robotic system. This algorithm avoids the covariance explosion problem arising in the RLS algorithm with a constant forgetting factor. In this case, past data are gradually discarded on the assumption that more recent data are more informative.

The main contribution of the proposed algorithm can be stated as follows:

The conventional RNS motion control scheme is implemented in a dual-arm space robot in which both arms execute ARNS motion. This adaptive control scheme is developed to stabilize a noncooperative target with an unknown momentum without the use of attitude control system (ACS) devices such as thrusters or reaction wheels.

In the presence of parameter uncertainties, an improved adaptive filtering algorithm with a VFF that improves the tracking capabilities and robustness of the system is implemented.

The VFF is defined based on the prediction errors, and a basic convergence analysis is performed to make this approach more practical to the robotic system.

This article is organized as follows. In the second section, the kinematic model of a dual-arm space robot is built and the coordinated motion equation is obtained. Then, the ARNS algorithm for the dual-arm space robot is developed with a VFF-WRLS algorithm, and a convergence analysis of the algorithm is conducted. In the fifth section, a set of simulations verify the proposed methods. The conclusions are summarized in the last section.

Dual-arm space robot system

Basic assumptions

During the operation, the space robot is in a free-floating mode. We assume that:

The system is composed of rigid bodies only, and the origin of the inertial frame ${\Sigma }_I$ is located at the center of mass of the entire system to reduce the computational complexity.

After the target is grasped by the space robot, it is fixed to the end-effector. Thus, there is no relative motion between the end-effector and the target.

There is no initial angular momentum on the space robot. In the absence of any external forces or torques on the system, the total momentum is conserved. We do not consider any momentum exchange devices in this study such as reaction wheels or thrusters.

Kinematic modeling of a dual-arm space robot

As shown in Figures 1 and 2, ¹⁵ a dual-arm space robotic system for a capture task typically consists of three major parts: ¹⁶ a space base or servicing satellite, two arms mounted on the space base, and the satellite or debris to be captured. Figure 1 shows the space base and the two arms that comprise the servicing system. In this scenario, we use one arm to complete the capture mission and the other to counteract the disturbance to the base.

Figure 1.

Dual-arm space robot system.

Figure 2.

Part of a target captured by the space robot.

The principal difference between a space robot and a ground-fixed robot is that the base of a space manipulator is allowed to be uncontrolled (free-floating mode operation) in the orbital environment. Special attention must be paid to the dynamic coupling between the space base and its manipulators. The kinematic and dynamic model of a dual-arm space robot has been described in Jiao et al. ^17,18 For clarification purposes, we recall the process of building the kinematic model here. The body 0 in Figure 1 represents the space base of the space robot, which is connected to two manipulators, each with three links. Manipulator joints are revolute and have a single degree of freedom. The inertial position of the link $r_i^k$ and end-effector $p_e^k$ in arm-k (k = a, b) can be written, respectively, as

$\begin{array}{l}{r}_i^k=r_0+b_0^k+\sum _{j=1}^{i-1}\left(a_j^k+b_j^k\right)+a_i^k\hfill \end{array}$

1

$\begin{array}{l}{p}_e^k=r_0+b_0^k+{\displaystyle \sum _{j=1}^{n_k}}\left(a_j^k+b_j^k\right)\hfill \end{array}$

2

where $r_0$ is the position vector of the center of mass of the base satellite and $a_i^k$ and $b_i^k$ are the position vectors of the links. Then, the linear and angular velocity equations can be obtained according to equations (1) and (2)

$\begin{array}{l}{v}_i^k=v_0+{\omega }_0\times \left(r_i^k-r_0\right)+{\displaystyle \sum _{j=1}^i}\left[z_i^k\times \left(r_i^k-p_j^k\right)\right]{\dot{\theta }}_j^k\hfill \end{array}$

3

$\begin{array}{l}{v}_e^k=v_0+{\omega }_0\times \left(p_e^k-r_0\right)+{\displaystyle \sum _{j=1}^i}\left[z_i^k\times \left(p_e^k-p_j^k\right)\right]{\dot{\theta }}_j^k\hfill \end{array}$

4

$\begin{array}{l}{\omega }_i^k={\omega }_0+{\displaystyle \sum _{j=1}^i}\left(z_j^k{\dot{\theta }}_j^k\right)\hfill \end{array}$

5

$\begin{array}{l}{\omega }_e^k={\omega }_0+{\displaystyle \sum _{j=1}^{n_k}}\left(z_j^k{\dot{\theta }}_j^k\right)\hfill \end{array}$

6

where $v_0$ and ${\omega }_0$ are the linear and angular velocity of the spacecraft, respectively; $z_i^k$ is the unit vector for the rotation direction of joint i; and ${\dot{\theta }}_j^k$ is the joint rates of joint j in arm-k.

To decrease the fuel consumption, we eliminated the use of ACS devices such as thrusters or reaction wheels. Under Assumption (3), there are no external forces or torques acting on the space robot system; therefore, the linear and angular momentum of the system are conserved, which means $\left(P,L\right)$ remains constant. With Assumption (3), the initial momentum is zero, and we can obtain the following equation, the elements of the matrix are explicitly described in the study by Jiao et al. ¹⁸

$\begin{array}{l}P=m_0{v}_0+{\displaystyle \sum _{i=1}^{n_a}}{m}_i^a{\dot{r}}_i^a+{\displaystyle \sum _{i=1}^{n_b}}{m}_i^b{\dot{r}}_i^b=0\hfill \end{array}$

7

$\begin{array}{ll}L=\hfill & I_0{\omega }_0+r_0\times m_0{\dot{r}}_0+{\displaystyle \sum _{i=1}^{n_a}}\left(I_i^a{\omega }_i^a+r_i^a\times m_i^a{\dot{r}}_i^a\right)\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{n_b}}\left(I_i^b{\omega }_i^b+r_i^b\times m_i^b{\dot{r}}_i^b\right)=0\hfill \end{array}$

8

where ${\omega }_i^k$ is the angular velocity of the links and $I_0,I_i^k$ are the inertia matrices of the spacecraft and link $B_i^k$ , respectively. Defining $L_0$ as the angular momentum of the system in the frame of the mass center of the base, we then can obtain

$\begin{array}{l}L=L_0+r_0\times P\hfill \end{array}$

9

According to equations (7) to (9), we have

$\begin{array}{l}{L}_0=0\hfill \end{array}$

10

Reformulating equations (7) and (10) in a matrix form, we obtain

$\left[\begin{array}{c}P\\ L_0\end{array}\right]=\left(\begin{array}{cc}ME& M{\stackrel{\sim }{r}}_{0g}^{\text{T}}\\ 0& H_{\omega }\end{array}\right)\left[\begin{array}{c}{v}_0\\ {\omega }_0\end{array}\right]+\left[\begin{array}{c}{J}_{T\omega }^a\\ I_{\varphi }^a\end{array}\right]{\dot{\Theta }}^a+\left[\begin{array}{c}{J}_{T\omega }^b\\ I_{\varphi }^b\end{array}\right]{\dot{\Theta }}^b=0$

11

where ${\stackrel{\sim }{r}}_{0g}^{\text{T}}$ is the position of the mass center of the entire system and $J_{T\omega }^k$ is the Jacobian matrix of arm-k. Then, equation (11) can be represented as

$\begin{array}{l}\left[\begin{array}{c}P\\ L_0\end{array}\right]=H_b{\dot{x}}_b+H_{bm}^a{\dot{\Theta }}^a+H_{bm}^b{\dot{\Theta }}^b=0\hfill \end{array}$

12

where $H_b$ is the global inertia matrix of the spacecraft and $H_{bm}^k$ represents the coupled inertia matrix of the manipulator

$H_b=\left(\begin{array}{cc}ME& M{\stackrel{\sim }{r}}_{0g}\\ 0& H_{\omega }\end{array}\right)\in R^{3\times 3}$

$H_{bm}^k={\left[\begin{array}{cc}{J}_{T\omega }^k& I_{\omega \varphi }^k\end{array}\right]}^{\text{T}}\in R^{3\times n_k}$

Then, the angular momentum conservation equation is achieved by removing the linear velocity of the space base in equation (12)

$\begin{array}{l}{H}_{\omega }{\omega }_0+H_{\Theta }^a{\dot{\Theta }}^a+H_{\Theta }^b{\dot{\Theta }}^b=0\hfill \end{array}$

13

Assuming that there is no attitude disturbance to the base $\left({\omega }_0=0\right)$

$\begin{array}{l}{H}_{\Theta }^a{\dot{\Theta }}^a+H_{\Theta }^b{\dot{\Theta }}^b=0\hfill \end{array}$

14

In this task, arm-a is used to accomplish the capture task and arm-b is designed as the balance arm mainly used to compensate for the attitude disturbance owing to the motion of arm-a; thus, the mapping relationship between the two arms can be formulated as

$\begin{array}{l}{\dot{\Theta }}_d^b=-{\left(H_{\Theta }^b\right)}^{+}{H}_{\Theta }^a{\dot{\Theta }}_d^a\hfill \end{array}$

15

where ${\left(H_{\Theta }^k\right)}^{+}={\left(H_{\Theta }^k\right)}^{\text{T}}{\left(H_{\Theta }^k{\left(H_{\Theta }^k\right)}^{\text{T}}\right)}^{-1}$ denotes the Moore–Penrose inverse of the inertia coupling matrix $H_{\Theta }^k$ . Arm-a is used as the mission arm and was planned based on the requirement of the mission. Then from equations (14) and (15), the RNS joint rates of the balanced arm can be obtained as

$\begin{array}{l}{\dot{\Theta }}_{\text{RNS}}^b=-{\left(H_{\Theta }^b\right)}^{+}{H}_{\Theta }^a{\dot{\Theta }}_{\text{RNS}}^a\hfill \end{array}$

16

The precondition of utilizing equations (13) to (16) is that the initial angular momentum is zero. For the general case when the precondition is not satisfied and the tumbling target carries an initial angular momentum $L_t$ , equation (14) can be rewritten as

$\begin{array}{l}{H}_{\Theta }^a{\dot{\Theta }}_{\text{RNS}}^a+H_{\Theta }^b{\dot{\Theta }}_{\text{RNS}}^b=L_m+L_t=L\hfill \end{array}$

17

where $L_t$ is the initial angular momentum of the target before capture. After the target is physically attached to the end-effector, the matrix $H_{\Theta }^a$ is suddenly changed and starts to include the inertia term of the target. In this context, the desired RNS motion of the two arms in equation (16) can be rewritten as

${\dot{\Theta }}_{\text{RNS}}^b={\left(H_{\Theta }^b\right)}^{+}\left(L-H_{\Theta }^a{\dot{\Theta }}_{\text{RNS}}^a\right)$

18

After capture, the angular momentum of the entire system becomes $L=L_m+L_t$ , and ${\dot{\Theta }}_{\text{RNS}}^k$ denotes the desired reactionless joint velocities of arm-k. However, because of the unknown properties of the target, we do not have accurate knowledge of matrix $H_{\Theta }^a$ . Thus, the joint motion will be governed by equation (17) with a nonzero ${\omega }_0$ ; the general solution for the joint rates can be written as

$\begin{array}{l}{\dot{\Theta }}^b={\left(H_{\Theta }^b\right)}^{+}\left(L-H_{\omega }{\omega }_0-H_{\Theta }^a{\dot{\Theta }}^a\right)\hfill \end{array}$

19

Substituting for ${\dot{\Theta }}_{\text{RNS}}^b$ from equation (18) into equation (19) yields

$H_{\Theta }^a\left({\dot{\Theta }}_{\text{RNS}}^a-{\dot{\Theta }}^a\right)+H_{\Theta }^b\left({\dot{\Theta }}_{\text{RNS}}^b-{\dot{\Theta }}^b\right)-H_{\omega }{\omega }_0=0$

20

It can also be described as

$\begin{array}{l}\left[\begin{array}{cc}{H}_{\Theta }^a& H_{\Theta }^b\end{array}\right]\left(\left[\begin{array}{c}{\dot{\Theta }}_{\text{RNS}}^a\\ {\dot{\Theta }}_{\text{RNS}}^b\end{array}\right]-\left[\begin{array}{c}{\dot{\Theta }}^a\\ {\dot{\Theta }}^b\end{array}\right]\right)=H_{\omega }{\omega }_0\hfill \end{array}$

21

Apparently, the expressions of ${\dot{\Theta }}_{\text{RNS}}^a$ and ${\dot{\Theta }}_{\text{RNS}}^b$ share the same form; thus, they can be rewritten in a more compact form as

${\dot{\Theta }}_{\text{RNS}}-\dot{\Theta }=H_{\Theta }^{+}{H}_{\omega }{\omega }_0$

22

where

${\dot{\Theta }}_{\text{RNS}}=\left[\begin{array}{c}{\dot{\Theta }}_{\text{RNS}}^a\\ {\dot{\Theta }}_{\text{RNS}}^b\end{array}\right],\dot{\Theta }=\left[\begin{array}{c}{\dot{\Theta }}^a\\ {\dot{\Theta }}^b\end{array}\right],H_{\Theta }=\left[\begin{array}{c}{H}_{\Theta }^a\\ H_{\Theta }^b\end{array}\right]$

or, more succinctly,

${\dot{\Theta }}_{\text{RNS}}-\dot{\Theta }=K{\omega }_0$

23

where $K=H_{\Theta }^{+}{H}_{\omega }\in R^{6\times 1}$ .

Thus, the regression form in equation (23) is the foundation of the ARNS control scheme for a dual-arm space robot system. From this regression form of equation (23), it can be viewed that if the joint rates $\dot{\Theta }$ closely follow the desired RNS joint rates ${\dot{\Theta }}_{\text{RNS}}$ , ${\omega }_0$ may converge to zero, which means that zero attitude disturbance to the base is produced. In the ARNS control scheme, equation (23) is coupled with the well-known recursive VFF-WRLS algorithm for parameter adaption.

Weighted recursive adaptation algorithm with VFFs

If perfect knowledge of the system properties is available, equation (23) can be considered as an alternative to compute the RNS motion. However, to capture a noncooperative target, the mission will involve an unpredictable change in the inertia properties as well as the total momentum of the space robotic system. When the system has parameter uncertainties, one way to cope with such an issue is to develop an adaptive control algorithm to reduce the uncertainties. In the system identification context, the RLS algorithm with exponential data weighting was employed to adaptively update the joint velocities online.

The time-varying system commonly can be represented by a linear regression equation, that is, equation (23), which is the fundamental scheme of the ARNS motion for a dual-arm space robot system. In our conception of ARNS algorithm, we assumed that it was not necessary for the manipulators to follow any specific trajectory because the main objective was to minimize the disturbance to the base immediately after the capture of the tumbling target. Thus, the desired joint velocity ${\dot{\Theta }}_{\text{RNS}}$ in equation (23) can be redefined as

$\begin{array}{l}{\widehat{\dot{\Theta }}}_{\text{RNS}}-\dot{\Theta }=\widehat{K}{\omega }_0\hfill \end{array}$

24

where $\dot{\Theta }$ and ${\omega }_0$ are the exact measured joint velocities and base angular velocity from the sensors, respectively, and $\left\{\widehat{{}^{•}}\right\}$ denotes the current estimates of the unknown variables from the noncooperative target.

According to the relation equations (23) and (24), one can obtain the data generating mechanism

$\begin{array}{l}y\left(n\right)=K\left(n-1\right){\omega }_0\left(n\right)\hfill \end{array}$

25

where the time index n is introduced to describe the discrete nature of the process in a practical control system, assuming a sampling rate of $\Delta t$ and $t=n\Delta t$ . $y\left(n\right)$ is the reference output error vector, which is a sequence of independent random variables with zero mean values (white noise).

Then, it is natural to define the a priori estimation error as

$\begin{array}{l}e\left(n\right)=y\left(n\right)-\widehat{K}\left(n-1\right){\omega }_0\left(n\right)\hfill \end{array}$

26

$e\left(n\right)$ is the sequence of independent Gaussian variables with $E\left[e\left(n\right)\right]=0$ . $\widehat{K}\left(n-1\right)$ is the estimation of the regression coefficient $K\left(n-1\right)$ in equation (25). The prediction of the reference output vector $y\left(n\right)$ , based on the input measurement vector ${\omega }_0\left(n\right)$ and the previous estimation of the filter parameter $\widehat{K}\left(n-1\right)$ , is given by

$\begin{array}{l}\widehat{y}\left(n\right)=\widehat{K}\left(n-1\right){\omega }_0\left(n\right)\hfill \end{array}$

27

Weighted RLS algorithm with exponential forgetting factor

It is more common in the adaptive filtering scenario expressed as a system parameter identification problem in Figure 3 to employ a weighted regularized least-squares cost function. Here $\omega \left(n\right)$ is the input signal that passes through the system $K\left(n\right)$ and resulting in the sensor output signal $y\left(n\right)$ . The noise $\delta \left(n\right)$ is also picked up by the sensors. $d\left(n\right)$ is the desired filter response and $e\left(n\right)$ is the prediction error or residual. The uncertain system parameter $K\left(n\right)$ is estimated at each time step by minimizing the criterion function J. More specifically, greater weighting was selectively applied to those data points deemed to be more informative based on some criteria. Here we consider the case where the most recent data were assumed to be more informative than past data, and hence we exponentially discarded old data, leading to the exponentially weighted cost function

$\text{min}J\left(n\right)=\frac{1}{n}{\displaystyle \sum _{i=0}^n}\beta \left(n,i\right)e^2\left(i\right)$

28

where $\beta \left(n,i\right)$ is the profile of the forgetting variable with typical increase in i for a given n. The prediction error $e\left(n\right)$ is defined by equation (26).

Figure 3.

System identification problem.

The corresponding mean of the criterion function J is readily seen to be a mathematical expectation, that is, the criterion function equation (26) reduces to

$J\left(n\right)=E\left[\beta \left(n,i\right)e^2\left(i\right)\right]$

29

A recursive form for the weight factor $\beta \left(n,i\right)$ was proposed in the study by Kovacevic et al. ¹⁹ as

$\beta \left(n,i\right)=\lambda \left(n\right)\beta \left(n-1,i\right)$

30

According to equation (30), one further concludes that

$\begin{array}{l}\beta \left(n,i\right)=\left[{\displaystyle \prod _{j=i+1}^n}\lambda \left(j\right)\right]\alpha \left(i\right),\alpha \left(i\right)=\beta \left(i,i\right)\hfill \end{array}$

31

The typical value of $\lambda \left(j\right)$ is bounded by $0<\lambda \left(j\right)\le 1$ , and in particular, if $\lambda \left(j\right)$ is constant, that is, $\lambda \left(j\right)=\lambda$ , one obtains

$\begin{array}{l}\beta \left(n,i\right)={\lambda }^{n-i}\alpha \left(i\right)\hfill \end{array}$

32

Basically, the cost function in equation (28) represents the mean of squares of the prediction errors $e\left(n\right)$ . The purpose of this algorithm is to emphasize artificially the effect of current data by exponentially weighting past data values, because while a space robot is grasping a target, a huge system impact may occur that will cause large deviations in motion and measurement errors in the sensors. Since the sequence $\lambda \left(j\right)$ has the effect of attaching a smaller weight to the terms in which $y\left(n\right)$ is expected to have larger errors, the algorithm is frequently used in time-varying systems. Another use of the sequence $\lambda \left(j\right)$ is to discard initial data in nonlinear estimation problems. The erroneous initial data may deteriorate the performance of the algorithm and must be discarded once the algorithm begins.

According to equations (26) and (27), $J\left(n\right)$ is a quadratic function of the argument $\widehat{K}\left(n\right)$ . Note that according to the prediction error of equation (26), the parameters are slowly changing, that is, $\widehat{K}\left(n\right)\approx \widehat{K}\left(n-1\right)$ . Its minimum is obtained by solving the equation

$\begin{array}{ll}\frac{\partial J\left(n\right)}{\partial \widehat{K}\text{(}n\text{)}}\hfill & \text{=}2n^{-1}{\displaystyle \sum _{i=1}^n}\beta \left(n,i\right)\left[y\left(i\right)-{\omega }_0{(i)}^{\text{T}}\widehat{K}\left(i\right)\right]\frac{\partial {\omega }_0{(i)}^{\text{T}}\widehat{K}\left(i\right)}{\partial \widehat{K}\left(i\right)}\hfill \\ \hfill & \text{=}0\hfill \end{array}$

33

Solving the obtained algebraic equation over $\widehat{K}\left(n\right)$ , one obtains

$\widehat{K}(n)=\frac{{\displaystyle {\sum }_{i=1}^n\beta \left(n,i\right){\omega }_0{\left(i\right)}^{\text{T}}y\left(i\right)}}{{\displaystyle {\sum }_{i=1}^n\beta \left(n,i\right){\omega }_0{\left(i\right)}^{\text{T}}{\omega }_0\left(i\right)}}$

34

Equation (34) is the non-recursive form of the WRLS algorithm. According to the relation equation (30), a possible recursive version can be obtained as

$\begin{array}{l}R\left(n\right)={\displaystyle \sum _{i=1}^n}\beta \left(n,i\right){\omega }_0{\left(i\right)}^{\text{T}}{\omega }_0\left(i\right)\\ ={\displaystyle \sum _{i=1}^{n-1}}\beta \left(n-1,i\right){\omega }_0{\left(i\right)}^{\text{T}}{\omega }_0\left(i\right)+\beta \left(n,n\right){\omega }_0{\left(n\right)}^{\text{T}}{\omega }_0\left(n\right)\\ =\lambda \left(n\right)R\left(n-1\right)+\alpha \left(n\right){\omega }_0{\left(n\right)}^{\text{T}}{\omega }_0\left(n\right)\end{array}$

35

The matrix $R\left(n\right)$ represents the correlation matrix of the input signal ${\omega }_0\left(n\right)$ . $R\left(n\right)$ also represents the estimation of the Hessian ${\nabla }^{2}J\left(n\right)$ at the moment n.

After applying the lemma on matrix inversion ²⁰ in equation (35), one obtains

$\begin{array}{l}P\left(n\right)=R^{-1}\left(n\right)\\ =\frac{1}{\lambda \left(n\right)}\left[P\left(n-1\right)-\frac{P\left(n-1\right){\omega }_0\left(n\right){\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right)}{\lambda \left(n\right)/\alpha \left(n\right)+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)}\right]\end{array}$

36

$P\left(n\right)$ , initialized by $P\left(-1\right)={\sigma }^{2}E$ for ${\sigma }^2\gg 1$ , is the inverse of the correlation matrix $R\left(n\right)$ . In this manner, the solution $\widehat{K}\left(n\right)$ of the exponentially weighted RLS problem (equation (28)) can be computed recursively as

$\begin{array}{l}\widehat{K}\left(n\right)=\widehat{K}\left(n-1\right)+G\left(n\right)e\left(n\right)\hfill \end{array}$

37

$G\left(n\right)=\frac{P\left(n-1\right){\omega }_0\left(n\right)}{\lambda \left(n\right)/\alpha \left(n\right)+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)}$

38

where $G\left(n\right)$ is the Kalman filtering gain vector. The matrix $P\left(n\right)$ has the meaning of an error covariance matrix of the estimated parameters ^.19 The variable $\alpha \left(n\right)$ is the convergence factor that controls the convergence speed of the algorithm. Thus, the role of matrix P is to accelerate the convergence of the adaptive algorithm, and the price to pay is an increase of computational complexity. If the value $P\left(n-1\right)$ in equation (38) is replaced by a unit matrix E, one can obtain the well-known least mean square algorithm, which has worse convergence properties but a lower complexity. ¹⁹

Choice of forgetting factor on prediction error

The forgetting factor $\lambda \left(n\right)$ suggests that it represents the measure by considering the previous measurements in the estimation process. In other words, the choice of the values for the forgetting factor determines how quickly one neglects the influence of the previous measurements, which is one form of data windowing, whereby the effective length of the window is $\approx 1/\left(1-\lambda \right)$ samples. Typical values for λ are between 0.9 and 0.99, which corresponds to the effective length of a window between 10 and 100 samples, respectively.

In the articles, ^9,10 a constant forgetting factor λ was employed, but problems can occur in an adaptive control situation. Since the matrix $\widehat{K}\left(n\right)$ changes with time, matrix $P\left(n\right)$ may become excessively large or approach zero. $P\left(n\right)$ needs to be reset to its initial value whenever it surpasses the preset thresholds; otherwise, the parameter estimator will become unstable. When the algorithm gain $P\left(n\right)$ approaches zero, the corresponding data point will not be included in the parameter update and the estimator switches off. Then, the sequential least-squares algorithm might become unsuitable for time-varying problems.

To overcome this problem, the fixed exponential forgetting factor $\beta \left(n,i\right)={\lambda }^{n-i}$ should be replaced by some weight function (forgetting factor variable in time) that represents an increasing function of the argument i $\left(1\le i\le n\right)$ for a given moment of time t.

The idea of exponential data weighting with VFF was employed. It has been shown in the study by Goodwin and Sin ²¹ that a good choice for $\lambda \left(n\right)$ in such cases is

$\begin{array}{l}\lambda \left(n\right)=1-{\lambda }_0\frac{e{\left(n\right)}^2}{{\overline{e}}^2}\hfill \end{array}$

39

where $e{\left(n\right)}^2$ is the current prediction error, ${\overline{e}}^2$ is the mean value of $e{\left(n\right)}^2$ over a certain period, and ${\lambda }_0$ is a small constant to satisfy the desired estimation quality in the stationary operation mode. Based on the knowledge of Kalman filtering theory, ²¹ it can be shown that ${\overline{e}}^2$ is proportional to $1+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)$ , and hence equation (39) can be rewritten as

$\begin{array}{l}\lambda \left(n\right)=1-{\lambda }_0\frac{e{\left(n\right)}^2}{1+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)}\hfill \end{array}$

40

Unlike traditional VFF schemes, the improved VFF-WRLS algorithm is built on prediction errors. Equation (40) represents a normalized error because the term $1+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)$ represents an estimation of the error variance $e\left(n\right)$ . The effect of the algorithm (equations (39) and (40)) can be explained as follows. When the space robot captures the target satellite, a sudden impact in the control system occurs and $e{\left(n\right)}^2$ increases; this reduces $\lambda \left(n\right)$ temporarily but increases $P\left(n\right)$ quickly, and rapid adaptation occurs. After adaptation, $e{\left(n\right)}^2$ decreases and $\lambda \left(n\right)$ returns to a value near 1 and the cycle repeats.

In the typical RLS algorithm with a fixed exponential forgetting factor, $\alpha \left(n\right)$ is a constant scalar variable, that is, $\alpha \left(n\right)\text{=}1$ . Since $\alpha \left(n\right)$ controls the convergence speed of the gradient method in such a case, we also redefine the series of $\alpha \left(n\right)$ as

$\alpha \left(n\right)=1-\lambda \left(n\right)={\lambda }_0\frac{e{\left(n\right)}^2}{1+{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)}$

41

Hence, the new VFF-WRLS algorithm can be obtained by equations (24) to (40). The new VFF-WRLS has a higher numerical stability and faster convergence than the conventional RLS algorithm in the study by Nguyen-Huynh and Sharf. ^9,10

Once $\widehat{K}\left(n\right)$ is calculated, the desired ARNS motion of the arms can be obtained as

$\begin{array}{l}{\widehat{\dot{\Theta }}}_{\text{RNS}}\left(n\right)=\dot{\Theta }\left(n\right)+\widehat{K}\left(n\right){\omega }_0\left(n\right)\hfill \end{array}$

42

The proposed ARNS scheme with VFF requires measurements of the base angular velocity ${\omega }_0\left(n\right)$ and the current joint rates $\dot{\Theta }\left(n\right)$ . To compute the VFF in equation (40), the current prediction error $e\left(n\right)$ is also required. In this way, the parameter matrix $\widehat{K}\left(n\right)$ and the ARNS joint rates ${\dot{\Theta }}_{\text{RNS}}\left(n\right)$ are updated.

It is emphasized that the principal objective of the proposed control algorithm is to maintain the attitude of the space base, that is, ${\omega }_0\left(n\right)\approx 0$ after capturing a noncooperative target. Since the total prediction error is minimized using the VFF-WRLS approach and the VFF plays a significant role during the process of converging, this control objective can be satisfied even though there is a mismatch between the estimated matrix $\widehat{K}\left(n\right)$ and its true value $K\left(n\right)$ .

The process of the complete algorithm is shown in Figure 4, and the following primary steps are obtained below, assuming a sampling rate of $\Delta t$ .

Step 1. Initialize the system based on pre-capture parameters, ${\dot{\Theta }}_{\text{RNS}}\left(0\right)=H_{\Theta }^{+}{L}_m,n=0$ .

Step 2. Measure $\Theta \left(n\right)$ , $\dot{\Theta }\left(n\right)$ , and ${\omega }_0\left(n\right)$ .

Step 3. Compute $\lambda \left(n\right)$ from equation (40) and $\widehat{K}\left(n\right)$ from equations (36) to (38).

Step 4. Using $\widehat{K}\left(n\right)$ , update the desired motion from equation (42)

Step 5. $n=n+1$ , return to step 2 until finished.

Figure 4.

ARNS control scheme with VFF-WRLS algorithm. ARNS: adaptive reaction null space; VFF-WRLS: variable forgetting factor weighted recursive least square.

As illustrated in Figure 4, the control architecture lies on two loops: the inner loop with the proportional–derivative (PD) joint velocity-based controller and the outer loop with the improved VFF-WRLS algorithm. The inner loop controller can drive the joint velocity error $y={\widehat{\dot{\Theta }}}_{\text{RNS}}-\dot{\Theta }$ to zero where the joint torques are computed as per the PD control law with constant gains; the outer loop can update the ARNS motion for the two arms with the online parameter adaptation algorithm (VFF-WRLS).

Convergence analysis

Following the proposed approach to deal with a time-varying system, the convergence properties are discussed in this section. A nonnegative Lyapunov function $V\left(n\right)$ is defined as

$\begin{array}{l}V\left(n\right)=\frac{\stackrel{\sim }{K}{\left(n\right)}^{\text{T}}P{\left(n\right)}^{-1}\stackrel{\sim }{K}\left(n\right)}{{\displaystyle {\prod }_{i=1}^n\lambda \left(i\right)}}\hfill \end{array}$

43

Using equations (25), (26), (37), and (38), we have

$\begin{array}{ll}\stackrel{\sim }{K}\left(n\right)\hfill & =\widehat{K}\left(n\right)-K\left(n\right)\hfill \\ \hfill & =\left(E-G\left(n\right){\omega }_0{\left(n\right)}^{\text{T}}\right)\stackrel{\sim }{K}\left(n-1\right)\hfill \end{array}$

44

From equations (36) and (38), we obtain

$\begin{array}{l}P{\left(n\right)}^{-1}\stackrel{\sim }{K}\left(n\right)=\lambda \left(n\right)P{\left(n-1\right)}^{-1}\stackrel{\sim }{K}\left(n-1\right)\hfill \end{array}$

45

The difference of $V\left(n\right)$ is obtained by

$\begin{array}{l}\Delta V=V\left(n\right)-V\left(n-1\right)\\ =\frac{\stackrel{\sim }{K}{\left(n\right)}^{\text{T}}P{\left(n\right)}^{-1}\stackrel{\sim }{K}\left(n\right)}{{\displaystyle {\prod }_{i=1}^n\lambda \left(i\right)}}\\ -\frac{\stackrel{\sim }{K}{\left(n-1\right)}^{\text{T}}P{\left(n-1\right)}^{-1}\stackrel{\sim }{K}\left(n-1\right)}{{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}}\end{array}$

46

Inserting equations (44) and (45) into equation (46) gives

$\begin{array}{l}\Delta V=V\left(n\right)-V\left(n-1\right)\\ =\frac{\stackrel{\sim }{K}{\left(n\right)}^{\text{T}}P{\left(n\right)}^{-1}\stackrel{\sim }{K}\left(n\right)}{{\displaystyle {\prod }_{i=1}^T\lambda \left(i\right)}}\\ -\frac{\stackrel{\sim }{K}{\left(n-1\right)}^{\text{T}}P{\left(n-1\right)}^{-1}\stackrel{\sim }{K}\left(n-1\right)}{{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}}\\ =\frac{{\left[\stackrel{\sim }{K}\left(n\right)-\left(\stackrel{\sim }{K}\left(n-1\right)\right)\right]}^{\text{T}}P{\left(n-1\right)}^{-1}\stackrel{\sim }{K}\left(n-1\right)}{{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}}\\ =-\frac{1}{{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}}\left[\frac{\stackrel{\sim }{K}{\left(n-1\right)}^{\text{T}}{\omega }_0\left(n\right){\omega }_0{\left(n\right)}^{\text{T}}\stackrel{\sim }{K}\left(n-1\right)}{\lambda \left(n\right)+{\omega }_0{\left(n\right)}^{T}P\left(n-1\right){\omega }_0\left(n\right)}\right]\\ =-\frac{e{\left(n\right)}^2}{{\displaystyle {\prod }_{i=1}^n\lambda \left(i\right)}+{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}{\omega }_0{\left(n\right)}^{\text{T}}P\left(n-1\right){\omega }_0\left(n\right)}\\ \le -\frac{e{\left(n\right)}^2}{{\displaystyle {\prod }_{i=1}^n\lambda \left(i\right)}+{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}{\sigma }_{\text{max}}\left(P\left(n\right)\right){\omega }_0{\left(n\right)}^{\text{T}}{\omega }_0\left(n\right)}\\ \le 0\end{array}$

47

Recall that $0<\lambda \left(n\right)\le 1$ and $P\left(-1\right)={\sigma }^{2}E$ for ${\sigma }^2\gg 1$ is a positive definite matrix and ${\sigma }_{\text{max}}\left(P\left(n\right)\right)$ is the maximum eigenvalue of the matrix $P\left(n\right)$ which is bounded, that is

$0<{\sigma }_{\text{max}}\left(P\left(n\right)\right)\le B$

48

It is clear that $V\left(n\right)$ is a nonnegative, nonincreasing function, and hence it converges. Thus, we have

$\underset{t\to \infty }{{\displaystyle \text{lim}}}\frac{e{\left(n\right)}^2}{{\displaystyle {\prod }_{i=1}^n\lambda \left(i\right)}+{\displaystyle {\prod }_{i=1}^{n-1}\lambda \left(i\right)}{\sigma }_{\text{max}}\left(P\left(n\right)\right){\omega }_0{\left(n\right)}^{\text{T}}{\omega }_0\left(n\right)}=0$

49

It is clear from the equations above that $\stackrel{\sim }{K}\left(n\right)$ will converge to zero. This is a desirable property and tends to improve the robustness of the algorithm.

Simulation study

In this section, the proposed control algorithms are evaluated and compared using a planar dual-arm space robot. The free-floating space robot includes three components, two of which are three-link manipulators, while the third is the space base. The target is assumed to be firmly held by the end-effector of the mission arm with no relative motion as illustrated in Figure 5.

Figure 5.

Dual-arm space robot with captured target.

The dynamic models of the system were created primarily in MATLAB/SimMechanics with S-functions. The geometric and dynamic parameters of the space robot and the target are presented in Tables 1 and 2. The desired motion generated by ARNS was produced by driving the joints with torques computed using the PD control law with constant gains for the entire motion. The parameters of the PD controller are presented in Table 3.

Table 1.

Parameters for dual-arm space robot.

Symbol	Mass (kg)	Length (m)	Inertia
		$a_i^k,\hspace{0.17em}\hspace{0.17em}{b}_i^k$	$I_i^k$
$B_0$	44	0, 0.3	44
$B_1^a$	2.5	0.2, 0.2	0.21
$B_2^a$	2.5	0.2, 0.2	0.21
$B_3^a$	2.5	0.2, 0.2	0.21
$B_1^b$	2.5	0.2, 0.2	0.21
$B_2^b$	2.5	0.2, 0.2	0.21
$B_3^b$	2.5	0.2, 0.2	0.21

Table 2.

Initial joint state for dual-arm space robot.

Joint	$J_1^a$	$J_2^a$	$J_3^a$	$J_1^b$	$J_2^b$	$J_3^b$
${\Theta }_0$	45	−10	−20	−20	45	−60

Table 3.

PD controller parameters for dual-arm space robot.

Gains	$J_1^a$	$J_2^a$	$J_3^a$	$J_1^b$	$J_2^b$	$J_3^b$
$K_p$	300	200	150	300	200	150
$K_d$	20	15	10	20	15	10

PD: proportional–derivative.

To demonstrate the capability of the proposed algorithm, the noncooperative target was much larger than the space robot. The mass and inertia were assumed to be almost three times as large as that of the servicer (refer to Table 4).

Table 4.

Parameters for noncooperative target.

Mass (kg)	Length (m)	Inertia (kg·m2)
$m_t$	$b_t$	$I_t$
120	0.3	120

The space robot initially has no momentum before capture, and the target is tumbling with an initial angular velocity. The initial adaptation gain matrix was chosen as $P\left(-1\right)=500$ . The initial forgetting factor for the RLS algorithm was defined as $\lambda \left(0\right)=0.99$ and was reset when $\lambda \left(n\right)\le 0.1$ or $\lambda \left(n\right)\ge 0.9999$ . The step size was $\Delta t=0.005\hspace{0.17em}\text{s}$ and the actual capture occurred at $t=0\hspace{0.17em}\text{s}$ .

The proposed algorithm was tested for two basic cases: $L_t=0$ and $L_t\ne 0$ . The first case verified the accuracy of the simulation platform by examining the momentum conservation of the system. This case also demonstrated the adaptation of ARNS algorithm in producing the RNS motion when the inertia parameters of the space manipulator were modified as a result of target capture. For the case $L_t\ne 0$ , the simulation of the post-capture of an unknown tumbling target aimed to verify the capability of the proposed scheme to minimize the disturbance to the base.

Case A: Test case with $L_t=0$

Considering the “noise” from inaccurate modeling and computer round-off errors, before the implementation of the algorithm, we first checked the accuracy of the simulation platform by examining the momentum conservation of the system during the ARNS motion. In this case, the initial momentum of the target was $L_t=0$ ; the space robot was initially at rest and the manipulators were commanded with joint motions as per equation (42). The convergence speed factor α was typically set as 1. Figure 6 shows that the angular momentum distribution as the total angular momentum was zero. It is apparent that the total momentum of the system was conserved. Figure 7 shows the attitude change of the base. The attitude disturbance remained on the scale of ${10}^{-3}$ degrees after 100 s. A base-10 log scale was used for the x-axis (t = 100 s) to show the rate of change in some figures. In Figure 8, the minimum base disturbance was achieved by the ARNS motion, while one arm held the tumbling target. When the end-effector contacted the target, a significant rate disturbance (as high as 0.017° s⁻¹) was produced by the impact between the manipulator and the target. However, the perturbation was quickly damped out by the ARNS motion. The results of the ARNS joint rates are shown in Figure 9. Figure 10 displays the error profiles of the PD velocity-based controller, from which one can observe that the joint rate error profiles were consistent with the base angular velocity profile in Figure 9. The result in Figure 11 shows that the prediction errors or residual of matrix $\widehat{K}\left(n\right)$ converges to 0. This is a desired property and tends to improve the robustness of the control system. The obtained results showed the accuracy of the simulation platform and the effectiveness of the proposed ARNS motion algorithm.

Figure 6.

Angular momentum of system.

Figure 7.

Base attitude $L_t=0$ .

Figure 8.

Base angular velocity $L_t=0$ .

Figure 9.

Joint rates $L_t=0$ .

Figure 10.

Joint rate errors $L_t=0$ .

Figure 11.

Prediction errors $e\left(n\right)$ , $L_t=0$ .

Case B: Test case with $L_t\ne 0$

In this subsection, the improved VFF-WRLS algorithm tested by simulating the capture of a tumbling target is described. As in the previous simulations, we assumed that the target was attached rigidly to the end-effector after capture and its tumbling motion was emulated with an external impulse torque of 240 N·m applied to the target for a duration of 0.005 s. The initial forgetting factor for the RLS algorithm was $\lambda \left(0\right)=0.99$ . In this simulation, we utilized the factor $\alpha \left(n\right)=1-\lambda \left(n\right)$ to accelerate the convergence of the system. The actual capture occurred at $t=0\hspace{0.17em}\text{s}$ . The corresponding results for the VFF-WRLS algorithm are shown in Figures 12 to 17.

Figure 12.

Joint rates $L_t\ne 0$ .

Figure 13.

Base attitude $L_t\ne 0$ .

Figure 14.

Base angular velocity, $L_t\ne 0$ .

Figure 15.

Joint rate errors $L_t\ne 0$ .

Figure 16.

Prediction error $e\left(n\right)$ , $L_t\ne 0$ .

Figure 17.

Variable forgetting factor $L_t\ne 0$ .

Figure 12 shows the joint rates, from which one can observe that the arms were initialized at the instant of capture with rates ${\dot{\Theta }}_{\text{RNS}}$ computed from equation (42), and the ARNS adaptively updated the reactionless motion. The base attitude and base angular velocity are shown in Figures 13 and 14, respectively. In Figure 13, the base attitude remained on the scale of $O\left({10}^{-3}\right)$ . As shown in Figure 14, the capture created an initial angular disturbance on the base. This disturbance, however, was successfully reduced with the ARNS motion to the level of $O\left({10}^{-5}\right)$ within 2 s (400 iterations) after capture. The joint rate error profiles of the velocity-based PD controller shown in Figure 15 match the base angular velocity profile in Figure 13. Figure 16 shows the prediction errors or residual of matrix $\widehat{K}\left(n\right)$ . Further details about the convergence properties are presented in the next subsection. Figure 17 shows the forgetting factor curves. When the space robot captures the target, the algorithm obtains a small forgetting factor to discard the estimation because of the large error. These initial data may deteriorate the performance of the algorithm unless they are discarded once the algorithm operation begins. Later, it increases the forgetting factor to attach more recent data to the parameter adaptation problems. Finally, the forgetting factor converges to 1, and the algorithm deteriorates to the well-known standard least-squares algorithm. In this way, the adaptive forgetting factor speeds up the convergence process substantially.

Case C: Convergence analysis

To evaluate the performance of the proposed VFF-WRLS algorithm, we compared the classical RLS algorithm with different constant values of forgetting factor with the improved VFF-WRLS algorithm. The forgetting factors were set with typical values of $\lambda =0.9$ and $\lambda =0.99$ , respectively. Other initial parameters were the same as those of the previous simulation in case B. The results are presented in Figures 18(a) and 19(b), and the relevant results of the improved VFF-WRLS are shown in Figures 13 and 16. According to these results, all the algorithms were able to achieve the ARNS motion regardless the value of the forgetting factor. However, problems can occur with a constant value of λ. In this situation, this can result in a “burst” phenomenon in parameter estimates that occur at around t = 50 s in Figure 18(a) and t = 80 s in Figure 18(b). The “blips” occurring in these figures reflect the non-robustness of the algorithm. This can be explained as follows.

Figure 18.

Prediction errors $e\left(n\right)$ with RLS, constant forgetting factor: (a) $\lambda =0.9$ and (b) $\lambda =0.99$ . RLS: recursive least squares.

Initially, with poor parameter estimates (within the first 0.1 s), the resulting feedback leads to bad regulation, and hence the data are rich in information. Then, as the estimates converge, the system under feedback tends to settle down, but simultaneously the estimation covariance matrix $P\left(n\right)$ begins to grow because of the loss of persistent excitation. After some time, the parameter estimator can become unstable because $P\left(n\right)$ appears as a gain in the algorithm. This can give rise to poor estimates, and the resulting feedback controller will begin to perform poorly. From Figures 13, 19(a) and (b), in terms of the disturbance to the base, the proposed VFF-WRLS outperformed the other algorithms. From Figure 19(a) and (b), one can observe that the ARNS motion with a constant forgetting factor produced a significant rate disturbance (as high as 0.021° s⁻¹) on the base, and the time to converge was much longer.

Figure 19.

Disturbance to base with RLS, constant forgetting factor: (a). $\lambda \hspace{0.17em}=\hspace{0.17em}0.9$ and (b) $\lambda \hspace{0.17em}=\hspace{0.17em}0.99$ . RLS: recursive least squares.

The obtained experimental results based on the simulations pointed out that the use of a VFF leads to a better adaptability of the filtering parameter estimation in comparison to the conventional algorithm with a fixed forgetting factor. With the correct choice of forgetting factor, the improved VFF-WRLS algorithm achieved a faster convergence and a greater robustness.

Conclusion

This study presented a practical implementation of a weighted RLS algorithm with a VFF for a dual-arm space robot capturing task. In the course of on-orbit servicing, the tumbling target was assumed to be much larger than the space robot, which meant that the uncertainties of the inertia properties of the target would degrade the control performance and the compound stabilization. To address this problem, the ARNS algorithm was extended to a dual-arm space robot and was enhanced by incorporating the VFF-WRLS technique. The novelty of the proposed VFF-WRLS algorithm lies in the time-varying function of determining the forgetting factor as well as relating the forgetting factor to the prediction error in the estimated parameters. The convergence properties of this algorithm were analyzed. Simulation results revealed the good performance of the proposed algorithm for both maintaining a minimum disturbance to the base and guaranteeing the convergence of tracking errors. We conclude that the proposed methods are applicable to a dual-arm space robot supplying on-orbit services.

Based on the proposed methods, we provide here a brief account of some implementation considerations in adaptive control design to raise some issues relevant to practical applications.

Adaptive control is a useful way of approaching control problems, but in practice, one must keep in mind the practical realities of the problem under study and include as much physical insight as possible. One of the first decisions that must be made in digital control is the sampling speed. This varies greatly depending on the application and hardware (from milliseconds to minutes). Roughly, the sampling period should depend on the computation speed, system time delay, choice of control law, and other factors.

To apply the algorithm into practice, we must investigate the performance of our prototype estimation scheme in the presence of bounded noise, such as measurement noise, inaccurate modeling, and computer round-off errors. The ARNS algorithm was analyzed under somewhat idealized assumptions. However, in practice, one must be aware of the degree of robustness of the algorithms to conditions that do not strictly comply with the assumptions.

The computation cost should be analyzed for the adaptive filtering algorithm because the high number of estimated states leads to a larger number of calculations that increase the time delay of the system and possibly deteriorate the overall performances. In particular, for a space robotic system, the performance of a microcomputer is worse than that of a normal computer on the ground.

To ensure security of the operation, self-collision detection algorithm should be considered for multi-arm robotics. Space robot is a kind of complex multi-body system, which may cause the collision between the links, the spacecraft module and external objections while in the on-orbit tasks. A challenge of real-time self collision detection is its computation cost, since computation resource for a space robot is limited. Servo loop, sensor-based motion generator, and collision checker need to share the resource.