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Abstract

It is difficult to obtain a precise mathematical model of free-floating space robot for the uncertain factors, such as current measurement technology and external disturbance. Hence, a suitable solution would be an adaptive robust control method based on neural network is proposed for free-floating space robot. The dynamic model of free-floating space robot is established; a computed torque controller based on exact model is designed, and the controller can guarantee the stability of the system. However, in practice, the mathematical model of the system cannot be accurately obtained. Therefore, a neural network controller is proposed to approximate the unknown model in the system, so that the controller avoids dependence on mathematical models. The adaptive learning laws of weights are designed to realize online real-time adjustment. The adaptive robust controller is designed to suppress the external disturbance and compensate the approximation error and improve the robustness and control precision of the system. The stability of closed-loop system is proved based on Lyapunov theory. Simulations tests verify the effectiveness of the proposed control method and are of great significance to free-floating space robot.

Keywords

Uncertainty space robot neural network robust control stability

Introduction

With the development of space technology, space robots are playing an increasingly important role. Unlike the robot with fixed base on the ground, the space robot base is floating freely and belongs to a rootless multi-body system. It has dynamic coupling between the floating base and each member of the manipulator, and its space liquid fuel consumption also causes some disturbance to the system, therefore, it has stronger coupling and nonlinearity than ground robot. Many control methods applied to ground robot are difficult to achieve satisfactory control results in space robot.

Space fuel directly determines the operating life of space robots, so it is very precious. If the position and attitude of the base are not controlled, it can save fuel and prolong the service life of the spacecraft. Therefore, it is of great significance to study free-floating space robot (FFSR).^1
–3

The FFSR arm movement must cause the coupling motion of the base, and the system is more nonlinear.^4
–6 In particular, if the influences of uncertainty factors are considered, such as measurement errors, machining accuracy, external interferences, and gear clearances, the precise mathematical model is difficult to get. These problems increase the difficulty of high precision control.^7,8

Pan and Yu⁹ provided the first result of parameter convergence without the PE condition for adaptive control of a general class of robotic systems. Yu and Chen¹⁰ proposed a terminal sliding mode control scheme for FFSR system, which is not controlled by the position and attitude of the base. According to the problem of stability control of space robot in task space, Sun et al.¹¹ proposed an adaptive tracking controller using neuro-fuzzy dynamic inversion for a robotic manipulator, with its dynamics approximated by a dynamic Takagi–Sugeno fuzzy model. Chen¹² proposes a novel dynamic structure neural-fuzzy network (DSNFN) via a robust adaptive sliding-mode approach to address trajectory tracking control of an n-link robot manipulator. Wai¹³ proposed a neural network control method based on inversion. Wu and Wang¹⁴ propose an adaptive neural network stability control method for robot system in task space. For the nonlinear space robot with friction dead zone, a compensation control strategy based on neural network is proposed by He et al.,¹⁵ and neural network controller is used to identify nonlinear models.

For the unknown uncertainty of the space manipulator model, Xie et al.¹⁶ proposed an adaptive control method based on neural network, the method used the neural network to approach the uncertain part, and the approximation error is eliminated by variable structure controller. To solve the trajectory tracking control problem of the robot in joint space, Yang¹⁷ proposed the model control strategy based on the determination of learning theory. Zhang et al.¹⁸ studied the redundant control method of neural variable structure under the condition of terminal mass mutation. In Cheng and Chen’s study,¹⁹ the system stability problem of the space dual-arm robot after capturing the spacecraft is presented, and they propose an adaptive neural network control strategy.

In the above literature, the neural network adaptive controller is used for the overall approximation of the unknown nonlinear model, the approximation errors or disturbances are usually eliminated by robust controller or variable structure controller, and the lack of adaptive characteristics will lead to chattering. In this article, a neural network control method for different nonlinear parts is proposed. At the same time, the approximation error and disturbance are suppressed by the adaptive robust control method, which avoids the chattering.

For these reasons, an adaptive robust control method based on neural networks is proposed. The dynamics model of floating base space robot is established, a computed-torque controller based on precise model is designed. For the unknown nonlinear part in the model, neural network controller is designed to approximate the unknown models, adaptive learning laws of network weights are designed to realize online real-time adjustment. An adaptive robust controller is designed to suppress external disturbance and compensate approximation errors and improve the robustness and control precision of the system. Then, the stability of the closed-loop system can be proven by Lyapunov’s method for stability. The simulation test verifies the effectiveness of the proposed control method.

Dynamics equation of FFSR

The dynamic parameters of space robot are defined as shown in figure 1.^16
–18

Figure 1.

Parameters of space robot.

The base of the FFSR is not fixed, thus the system added six degrees of freedom, and then n joints of space manipulators have n + 6 degrees of freedom. Because of the space microgravity, the weak gravity gradient is ignored. Therefore, the Lagragian function can be reduced to sole a kinetic component. The dynamical equation of n joint space robot based on the second Lagrange equation can be deduced, the dynamic equation has been widely used in space robot research.^18

–21

M^{*} (q) \ddot{q} + C^{*} (q, \dot{q}) \dot{q} + d^{*} = τ^{*}

where $M^{*} (q) \in R^{(n + 6) (n + 6)}$ is the inertial matrix in the dynamics equation of space robot, $C^{*} (q, \dot{q}) \in R^{(n + 6) (n + 6)}$ is centrifugal force and Coriolis force. $q = {[\begin{matrix} q_{b}^{T} & q_{m}^{T} \end{matrix}]}^{T} \in R^{n + 6}$ , $q_{b} \in R^{6}$ is the carrier position and attitude vector.

$q_{m} = {[\begin{matrix} q_{1} & q_{2} & \dots & q_{n} \end{matrix}]}^{T} \in R^{n}$ is the generalized displacement vector of the n-joint manipulators, $d^{*} \in R^{n + 6}$ is the external interference, $τ^{*} = {[\begin{matrix} β_{6 \times 1}^{T} & τ_{n \times 1}^{T} \end{matrix}]}^{T} \in R^{n + 6}$ , $β_{6 \times 1}^{}$ is the base control moment, $τ_{n \times 1}^{}$ is the joints torque of space manipulators.

When position and posture of the base are not controlled, that is $β_{6 \times 1}^{} = 0$ , $τ^{*} = {\begin{matrix} [\underset{1 \times 6}{\underset{︸}{\begin{matrix} 0 & \dots & 0 \end{matrix}}} & τ_{n \times 1}^{T} \end{matrix}]}^{T} \in R^{n + 6}$ belongs to the underactuated mode. This model can save valuable space energy, and space robot of the movement is called FFSR. Free-floating space manipulators system is not only conservation of momentum but also the conservation of momentum moment. Then, the FFSR dynamic equation of fully actuated form is^21

–24:

M (q_{b}, q_{m}) {\ddot{q}}_{m} + C (q_{b}, q_{m}, {\dot{q}}_{m}) {\dot{q}}_{m} + d = τ

where $d \in R^{n}$ , $C (q_{b}, q_{m}, {\dot{q}}_{m}) \in R^{n \times n}$ , $M (q_{b}, q_{m}) \in R^{n \times n}$ , $τ \in R^{n}$ .

The dynamics equation (2) of FFSR has the following properties^19

–22:

Property 1: the inertial matrix $M (q_{b}, q_{m})$ is a symmetric positive definite matrix.

Property 2: proper selection of $C (q_{b}, q_{m}, {\dot{q}}_{m})$ . $\dot{M} (q_{b}, q_{m}) - 2 C (q_{b}, q_{m}, {\dot{q}}_{m})$ is the skew symmetric matrix.

Design of computed-torque controller for precise model

From property (positive definite), the dynamics can be written as

{\ddot{q}}_{m} = M^{- 1} [τ - (C {\dot{q}}_{m} + d)]

{\ddot{q}}_{m} = M^{- 1} τ - M^{- 1} (C {\dot{q}}_{m} + d)

where $\ddot{y} = {\ddot{q}}_{m}$ , $u = τ$ , $G (x) = M^{- 1}$ , $F (x) = - M^{- 1} (C {\dot{q}}_{m} + d)$ .

Then, the kinetic equation can be transformed into

\ddot{y} = F (x) + G (x) u

The desired trajectory is defined as

y_{d} (t) = {[y_{d 1} (t), \dots, y_{d p} (t)]}^{T}

Hypothesis 1: $G (x)$ is the positive definite matrix, and there is real number $σ_{0} > 0$ , which satisfies $G (x) \geq σ_{0} I_{p}$ .

Hypothesis 2: Desired trajectory $y_{d i} (t)$ is bounded, there is $r_{i}$ derivative, and all derivatives are bounded, $i = 1, \dots, p$ .

The trajectory tracking errors are defined as

\begin{array}{l} e_{1} (t) = y_{d 1} (t) - y_{1} (t) \\ ⋮ \\ e_{p} (t) = y_{d p} (t) - y_{p} (t) \end{array}

The filter tracking error function is defined as

\begin{array}{l} s_{1} (t) = {(\frac{d}{d t} + λ_{1})}^{r_{1} - 1} e_{1} (t), \begin{matrix} λ_{1} > 0 \end{matrix} \\ ⋮ \\ s_{p} (t) = {(\frac{d}{d t} + λ_{p})}^{r p - 1} e_{p} (t), \begin{matrix} λ_{p} > 0 \end{matrix} \end{array}

From equation (7), it can be seen that if $s_{i} \to 0$ , then $e_{i} \to 0 (i = 1, 2, \dots, p)$ .

According to Newton binomial theorem:

{(a + b)}^{n} = \sum_{i = 0}^{n} c_{n}^{i} a^{i} b^{n - i}

where $c_{n}^{i} = \frac{n!}{(n - i)! i!}$ are the quadratic expansion coefficients.

From equation (8), it follows that equation (7) can be written as

\begin{array}{l} s_{i} (t) = {(\frac{d}{d t} + λ_{i})}^{r i - 1} e_{i} (t) \\ = \sum_{j = 0}^{r_{i} - 1} \frac{(r_{i} - 1)!}{(r_{i} - 1 - j)! j!} {(\frac{d}{d t})}^{j} e_{i} (t) λ_{i}^{r_{1} - 1 - i} \\ = \sum_{j = 1}^{r_{i}} \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} {(\frac{d}{d t})}^{j - 1} e_{i} (t) λ_{i}^{r_{1} - j} \end{array}

Then

\begin{array}{l} {\dot{s}}_{i} (t) = \sum_{j = 1}^{r_{i}} \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} e_{i}^{(j)} (t) λ_{i}^{r_{i} - j} \\ = e_{i}^{(r i)} + \sum_{j = 1}^{r_{i} - 1} \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} e_{i}^{(j)} (t) λ_{i}^{r_{i} - j} \\ = y_{d_{i}}^{(r_{i})} - y_{i}^{(r_{i})} + \sum_{j = 1}^{r_{i} - 1} \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} e_{i}^{(j)} (t) λ_{i}^{r_{i} - j} \\ = y_{d_{i}}^{(r_{i})} - f_{i} (x) - \sum_{j = 1}^{i} g_{i j} (x) u_{i} + \sum_{j = 1}^{r_{i} - 1} \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} e_{i}^{(j)} (t) λ_{i}^{r_{i} - j} \end{array}

namely

\begin{array}{l} {\dot{s}}_{1} = v_{1} - f_{1} (x) - \sum_{j = 1}^{p} g_{1 j} (x) u_{j} \\ ⋮ \\ {\dot{s}}_{p} = v_{p} - f_{p} (x) - \sum_{j = 1}^{p} g_{p j} (x) u_{j} \end{array}

where

\begin{array}{l} v_{1} = y_{d 1}^{(r_{1})} + β_{1, r_{_{1}} - 1} e_{1}^{(r_{1} - 1)} + \dots + β_{1, 1} {\dot{e}}_{1} \\ ⋮ \\ v_{p} = y_{d p}^{(r_{p})} + β_{p, r_{p} - 1} e_{p}^{(r_{p} - 1)} + \dots + β_{p, p} {\dot{e}}_{p} \end{array}

β_{i, j} = \frac{(r_{i} - 1)!}{(r_{i} - j)! (j - 1)!} λ_{i}^{r_{i} - j}, i = 1, \dots, p; j = 1, \dots, r_{i} - 1 .

Define

s (t) = {[s_{1} (t) \dots s_{p}]}^{T}, v (t) = {[v_{1} (t) \dots v_{p} (t)]}^{T}

Then, equation (10) can be written as

\dot{s} = v - F (x) - G (x) u

For equation (2), if M, C, and d can be precisely obtained, namely, the nonlinear function $F (x)$ and $G (x)$ is known.

The computed torque control law is designed as follows

u = G^{- 1} (v + K_{0} s - F)

where, $K_{0} = diag [k_{01}, \dots, k_{0 p}]$ , $k_{0 i} > 0$ , $i = 1, \dots, p$ .

Theorem 1

For the space robot system (5), the control law (14) is adopted. Then, the trajectory tracking error and its derivatives are bound to converge to zero.

Proof

Substituting equation (14) into equation (13)

\dot{s} (t) = - K_{0} s (t)

where ${\dot{s}}_{i} (t) = - K_{0 i} s_{i} (t)$

To solve the differential equation

s_{i} (t) = s_{i} (o) e^{- {k_{0 i}^{}}^{t}}

Obviously, when $t \to \infty$ , $s_{i} (t) \to o$ , $e_{i} (t)$ , and its $r_{i} - 1$ derivatives converge uniformly to zero.

However, in practice, due to the measurement level limitation and external interference factors, the nonlinear function $f_{i} (x)$ and $g_{i j} (x)$ cannot be precisely obtained, considering the superior nonlinear approximation ability of neural network, the neural network controller can be designed to approximate the nonlinear function $f_{i} (x)$ and $g_{i j} (x)$ .

Design of adaptive robust controller based on neural network for unknown model

Because RBF neural network (RBFNN) is a local generalization network with fast-learning speed and can avoid local minimum problems.^19

–22 Therefore, this network is used to approach the unknown nonlinear $f_{i} (x)$ and $g_{i j} (x)$ .

RBFNN belongs to the three-layer forward network, which is the input layer, the hidden layer and the output layer, as shown in figure 2. The input vector is mapped by the hidden layer basis function, if the base function center and width are determined, the mapping relationship is determined. There is a linear relationship between the hidden layer and the output layer.^23

–26

Figure 2.

Radial basis neural network structure.

Define $f_{i}^{*}$ as the optimal output of neural network for $f_{i}$ , $ε_{f i}$ is its approximation error. $g_{i j}^{*}$ is the optimal output of the neural network for $g_{i j}$ , and $ε_{g i j}$ is its approximation error.

Namely

ε_{f i} = f_{i} - f_{i}^{*}

ε_{g i j} = g_{i j} - g_{i j}^{*}

Its respective radial basis neural network optimal output is

f_{i}^{*} = θ_{f i}^{* T} φ_{f i} (x)

g_{i j}^{*} = {θ_{g i j}^{*}}^{T} φ_{g i j} (x)

where x is input of the neural network, $θ_{f i}^{*}$ and $θ_{_{g i j}}^{*}$ are the optimal weights. $φ_{f i} (x)$ and $φ_{g i j} (x)$ are the Gaussian function of the network hidden layer.

According to the approximation ability of RBFNN, hypothesis^16

–22

Hypothesis 3: For neural network optimal output $f_{i}^{*}$ and $g_{i j}^{*}$ , its minimum approximation error are $ε_{f i}$ and $ε_{g i j}$ , and there is a very small positive, which satisfy $| | ε_{f i} | | \leq {\bar{ε}}_{f i}$ and $| | ε_{g i j} | | \leq {\bar{ε}}_{g i j}$ .

Hypothesis 4: The optimal weight parameter $θ_{f i}^{*}$ and $θ_{g i j}^{*}$ are bounded, that is, there is a positive, which satisfy $| | θ_{f i}^{*} | | \leq {\bar{θ}}_{f i j}$ and $| | θ_{g i}^{*} | | \leq {\bar{θ}}_{g i j}$ .

In practice, it is difficult to obtain the optimal approximation for neural networks. Therefore, ${\hat{f}}_{i}$ is defined as the actual approximation of the neural network to the nonlinear function $f_{i}$ . ${\hat{g}}_{i j}$ is the actual approximation of the neural network to the nonlinear function $g_{i j}$ , ${\hat{θ}}_{f i}$ , and ${\hat{θ}}_{g i j}$ are the corresponding network weight parameters.

Namely

{\hat{f}}_{i} = {\hat{θ}}_{f i}^{T} φ_{f i} (x)

{\hat{g}}_{i j} = {\hat{θ}}_{g i j}^{T} φ_{g i j} (x)

For the actual weights ${\hat{θ}}_{f i}$ and ${\hat{θ}}_{g i j}$ , the optimal weights $θ_{_{f i}}^{*}$ and $θ_{_{g i j}}^{*}$ , the following definitions: ${\tilde{θ}}_{f i} = θ_{f i}^{*} - {\hat{θ}}_{f i}$ , ${\tilde{θ}}_{g i j} = θ_{g i j}^{*} - {\hat{θ}}_{g i j}$

Meanwhile

\hat{F} (x) = {[{\hat{f}}_{1} (x), \dots, {\hat{f}}_{p} (x)]}^{T}

\hat{G} (x) = [\begin{matrix} {\hat{g}}_{11} & \dots & {\hat{g}}_{1 p} \\ ⋮ & ⋱ & ⋮ \\ {\hat{g}}_{p_{1}} & \dots & {\hat{g}}_{p_{2}} \end{matrix}]

ε_{g} (x) = [\begin{matrix} ε_{g 11} (x) & \dots & ε_{g 1 p} (x) \\ ⋮ & ⋱ & ⋮ \\ ε_{g p 1} (x) & \dots & ε_{g p p} (x) \end{matrix}]

{\bar{ε}}_{1} (x) = {[{\bar{ε}}_{11} (x), \dots, {\bar{ε}}_{f p} (x)]}^{T}

{\bar{ε}}_{g} (x) = [\begin{matrix} {\bar{ε}}_{g 11} (x) & \dots & {\bar{ε}}_{g 1 p} (x) \\ ⋮ & ⋱ & ⋮ \\ {\bar{ε}}_{g p 1} (x) & \dots & {\bar{ε}}_{g p p} (x) \end{matrix}]

Then

\begin{array}{l} F (x) - \hat{F} (x, {\hat{θ}}_{f}) = F^{*} (x, θ_{f}^{*}) + ε_{f} (x) - \hat{F} (x, {\hat{θ}}_{f}) \\ G (x) - \hat{G} (x, {\hat{θ}}_{g}) = \hat{G} (x, {\hat{θ}}_{g}) + ε_{g} (x) - G^{*} (x, θ_{_{g}}^{*}) \end{array}

If $\hat{F} (x, {\hat{θ}}_{f})$ and $\hat{G} (x, {\hat{θ}}_{g})$ are used instead of $F (x)$ and $G (x)$ in the linear PID controller (14). The control law is modified as follows

u_{c} = {\hat{G}}^{- 1} (- \hat{F} + v + k_{0} s)

As $\hat{G}$ is obtained by online approximation, it is difficult to guarantee the nonsingularity of ${\hat{G}}^{- 1}$ , so the generalized inverse $\hat{G} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1}$ is used instead of ${\hat{G}}^{- 1}$ , and the control law is modified as

u_{c} = {\hat{G}}^{T} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} (v + K_{0} s - \hat{F})

where ε₀ is any small positive integer, I_P is unit matrix.

To eliminate the approximation error of neural network and external interference, improve the control precision. Robust controller is designed as follows

u_{r} = \frac{s | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{K_{s} {‖ s ‖}^{2} + δ}

u_{0} = ε_{0} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} (K_{0} s - \hat{F} + v)

where K_s is a positive, δ is a variable.

An adaptive neural network robust controller is designed as follows

u = u_{c} + u_{r}

Adaptive control law is designed

{\dot{θ}}_{f i} = - η_{f i} φ_{f i} s_{i}

{\dot{θ}}_{g i j} = - η_{g i j} φ_{g i j} s_{i} u_{c j}

\dot{δ} = - η_{0} \frac{| s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ}

where $η_{f i} > 0$ , $η_{g i j} > 0$ , $η_{0} > 0$ , $δ (0) > 0$ .

Theorem 2

If the hypothesis is satisfied, the adopt control law equations (25) to (28) and adaptive law equations (29) to (31), All signals of the system’s closed-loop system can be guaranteed to be bounded. Moreover, the trajectory tracking error and its derivatives converge uniformly to zero.

Proof

Substituting equation (28) into equation (14)

\dot{s} = v - F - (G - \hat{G}) u_{c} - \hat{G} u_{c} - \hat{G} u_{r}

Because

\begin{array}{l} \hat{G} {\hat{G}}^{T} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} = \frac{\hat{G} {\hat{G}}^{T} + ε_{0} I_{P} - ε_{0} I_{P}}{ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}} \\ = I_{p} - ε_{0} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} \end{array}

In combination with equations (25) and (33), the following formula can be obtained

\begin{array}{l} v - \hat{G} u_{c} = v - \hat{G} {\hat{G}}^{T} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} (v + K_{0} s - \hat{F}) \\ = v - \hat{G} {\hat{G}}^{T} {[ε_{0} I_{P} + \hat{G} {\hat{G}}^{T}]}^{- 1} (v + K_{0} s - \hat{F}) \\ = \hat{F} - K_{0} s + u_{0} \end{array}

Substituting equation (34) into equation (32)

\begin{array}{l} \dot{s} = - K_{0} s - (F - \hat{F}) - (G - {\hat{G}}^{T}) u_{c} + u_{0} - G u_{r} \\ = - K_{0} s - (F^{*} - \hat{F}) - (G^{*} - \hat{G}) u_{c} - G u_{r} + u_{0} - ε_{f} - ε_{g} u_{c} \end{array}

Equation (35), multiply both sides by $s^{T}$ , the following formula can be obtained

\begin{array}{l} s^{T} \dot{s} = - s^{T} K_{0} s - \sum_{i = 1}^{p} φ_{f_{i}}^{T} {\tilde{θ}}_{f_{i}} s_{i} - \sum_{i = 1}^{p} \sum_{j = 1}^{p} φ_{g_{i j}}^{T} {\tilde{θ}}_{g_{i j}} s_{i} u_{c j} \\ - s^{T} G u_{r} + s^{T} u_{0} - s^{T} ε_{1} - s^{T} ε_{g} u_{c} \end{array}

The Lyapunov function is defined as follows

V = \frac{1}{2} s^{T} s + \frac{1}{2} \sum_{i = 1}^{p} \frac{1}{η_{f_{i}}} {\tilde{θ}}_{f_{i}}^{T} {\tilde{θ}}_{f_{i}} + \frac{1}{2} \sum_{i = 1}^{p} \sum_{j = 1}^{p} \frac{1}{η_{g_{i j}}} {\tilde{θ}}_{g_{i j}}^{T} {\tilde{θ}}_{g_{i j}} + \frac{1}{2 η_{0}} δ^{2}

Then

\dot{V} = s^{T} \dot{s} - \sum_{i = 1}^{p} \frac{1}{η_{f_{i}}} {\tilde{θ}}_{f_{i}}^{T} {\dot{θ}}_{g_{i j}} - \sum_{i = 1}^{p} \sum_{j = 1}^{p} \frac{1}{η_{g_{i j}}} {\tilde{θ}}_{g_{i j}}^{T} {\dot{θ}}_{g_{i j}} + \frac{1}{η_{0}} δ \dot{δ}

Substituting equation (36) into equation (38)

\dot{V} = {\dot{V}}_{1} + {\dot{V}}_{2} - s^{T} K_{0} s

where

\begin{array}{l} {\dot{V}}_{1} = - \sum_{i = 1}^{p} {\tilde{θ}}_{_{f_{i}}}^{T} (φ_{f_{i}} s_{i} + \frac{1}{η_{f_{i}}} {\dot{θ}}_{f_{i}}) \\ - \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{θ}}_{g_{i j}}^{T}_{} (φ_{g_{i j}} s_{i} u_{c j} + \frac{1}{η_{g_{i j}}} {\dot{θ}}_{g_{i j}}) \end{array}

{\dot{V}}_{2} = s^{T} u_{0} - s^{T} G u_{r} - s^{T} ε_{f} - s^{T} ε_{g} u_{c} + \frac{1}{η_{0}} δ \dot{δ}

Substituting equations (29) and (30) into equation (40)

{\dot{V}}_{1} = 0

$s^{T} G (x) s > σ_{0} {‖ s ‖}^{2}$ can be derived from hypothesis 1, according to formula (26), the following is obtained

\begin{array}{l} s^{T} G u_{r} \\ = s^{T} G \frac{s | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ} \geq \frac{σ_{0} {‖ s ‖}^{2} | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ} \\ = \frac{(σ_{0} {‖ s ‖}^{2} + δ - δ) | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ} \\ = | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |) - \frac{δ | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ} \end{array}

Then, equation (41) can be written as

{\dot{V}}_{2} \leq | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |) - s^{T} G u_{r} + \frac{1}{η_{0}} δ \dot{δ}

Substituting equation (42) into equation (43)

{\dot{V}}_{2} \leq \frac{δ | s^{T} | ({\bar{ε}}_{f} + {\bar{ε}}_{g} | u_{c} | + | u_{0} |)}{σ_{0} {‖ s ‖}^{2} + δ} + \frac{1}{η_{0}} δ \dot{δ}

Substituting equation (31) into the above equation: ${\dot{V}}_{2} < 0$

Then

\begin{array}{l} \dot{V} \leq - s^{T} K_{0} s \\ = - \sum_{i = 1}^{p} k_{0 i} s_{i}^{2} \\ \leq 0 \end{array}

That is, $\dot{V}$ is negative semidefinite, $V \subset L_{\infty}$ , then $V (t)$ is non-increasing function and bounded, the signals $s_{i} (t)$ , ${\tilde{θ}}_{f_{i}} (t)$ and ${\tilde{θ}}_{g_{i j}} (t)$ are bounded, that is, $θ_{f i}$ , $θ_{g_{i j}}$ ,x,u and ${\dot{s}}_{i} (t)$ are bounded.

Integral the above formula (45)

\int_{0}^{\infty} \sum_{i = 1}^{p} k_{0 i} s_{i}^{2} (t) d t \leq V (0) - V (t) < \infty

That is $s_{i} (t) \in L_{2} \cap L_{\infty}$ , because of ${\dot{s}}_{i} (t) \in L_{\infty}$ . Adaptive system stability Barbalat’s lemma^25,26: if $g \in L^{2} \cap L^{\infty}$ and $\dot{g}$ is bounded, then $lim_{t \to \infty} g (x) = 0$ . This is known, when t→∞, $s_{i} (t) \to 0$ . That is, all signals in the closed loop system are bounded, and when t→0, $e_{i}^{(j)} (t) \to 0$ , $i = 1, L, p$ , $j = 0, 1, L, r_{i} - 1$ .

Control system validation and analysis

The validity of the control method is validated for double joints free-floating space manipulators, The simulation parameters of space manipulators are shown in Table 1. m₀, m₁, m₂, and I₀, I₁, I₂ are the mass and inertia of the base, rod 1, and rod 2 respectively. b₀, b₁, b₂, respectively, represent the center of mass of the matrix, rod 1 and rod 2 to the distance of the joint, a₁ and a₂ are joints to each center of mass. The physical meaning of specific parameters is as shown in the Figure 3:

The increased disturbance is mainly used to verify the effectiveness of the controller, the disturbance is supposed:

d = {[q_{1} {\dot{q}}_{1} 0.2 sin t, q_{2} {\dot{q}}_{2} 0.15 sin t]}^{T},

Desired trajectories:

q_{1 d} = 0.1 sin t - 0.2 cos t

q_{2 d} = 0.1 sin 2 t + 0.1 cos t

Controller gain and parameters:

K_{0} = diag (10, 10), λ_{1} = 25, λ_{2} = 35, γ = 50, K_{s} = diag (5, 5), η_{f i} = 0.8, η_{g i j} = 0.8, η_{0} = 0.005, {\bar{ε}}_{g} = diag (0.5, 0.5), {\bar{ε}}_{f} = diag (0.8, 0.8), δ (0) = 0 .

Joint initial values:

q_{1} (0) = q_{2} (0) = 0 .

Table 1.

Parameters of space manipulator.

m ₀	420 kg	a ₁	0.6 m
m ₁	10 kg	a ₂	0.5 m
m ₂	5 kg	b ₀	0.6 m
I ₀	50 kg m²	b ₁	0.7 m
I ₁	1.5 kg m²	b ₂	0.5 m
I ₂	0.4 kg m²

Figure 3.

Schematic diagram of two-rod space manipulator parameters.

The initial weights of the neural network are all set to 0, and the width and center of each base function are randomly selected in the input and output domain, and the hidden layer node number is 40.

Three control methods are used to verify the advantages of the proposed control methods. Computed-torque control method is used by method 1; sliding mode variable structure control method is used by method 2; adaptive robust controller base on neural network is used by method 3.

Method 1: Computed-torque control

Method 1 is computed-torque controller (equation (14)). The simulation results are shown in Figures 4 and 5.

Figure 4.

Trajectory tracking of method 1 (computed torque controller): (a) trajectory tracking of joint 1 and (b) trajectory tracking of joint 2.

Figure 5.

Control torque of method 1 (computed torque controller): (a) control torque diagram of joint 1 and (b) control torque of joint 2.

It can be seen from Figures 4 and 5, method 1 can guarantee the stability of the control system, and the control torque is small, but there is a large steady state error. It is shown that the simple computed-torque controller cannot achieve the control precise for the coupled nonlinear system.

Method 2: Sliding mode variable structure control

Method 2 is Sliding mode variable structure controller. The simulation results are shown in Figures 6 and 7.

Figure 6.

Trajectory tracking of method 2 (sliding mode variable structure controller): (a) trajectory tracking of joint 1 and (b) trajectory tracking of joint 2.

Figure 7.

Control torque of method 2 (sliding mode variable structure controller): (a) control torque of joint 1 and (b) control torque of joint 2.

This article further compares with the sliding mode variable structure control to prove the advantages of the control method in this article.

Simulation shows that the sliding mode variable structure control can also achieve better control accuracy from Figures 6 and 7, but its control torque output is large, reaching 80 Nm, which is a waste of precious space energy for space robots. In addition, the torque output is seriously “shaken,” which may cause damage to the equipment in practical engineering, so it is not the best control method.

Method 3: Adaptive robust control base on neural network

Adaptive robust control base on neural network (equations (28) to (31)) is proposed by the article. The simulation results are shown in Figures 8 to 11. Trajectory tracking diagrams are shown in Figure 8; $‖ F ‖$ , $‖ G ‖$ , and their neural network approximate are shown in Figure 9; $‖ F ‖$ and $‖ G ‖$ are their respective norm; control torque are shown in Figure 10. Neural weights evolution curves have be given in Figure 11, because the neural network has more nerve nodes and more weight. Here, norm form of neural weights is used to show the evolution process of the whole weight.

Figure 8.

Trajectory tracking of method 3 (neural network adaptive robust controller): (a) trajectory tracking of joint 1 and (b) trajectory tracking of joint 2.

Figure 9.

$‖ F ‖$ , $‖ G ‖$ and their neural network approximate. (a) $‖ F ‖$ and its neural network approximate comparison and (b) $‖ G ‖$ and its neural network approximate comparison

Figure 10.

Control torque of method 3 (neural network adaptive robust controller): (a) control torque of joint 1 and (b) control torque of joint 2.

Figure 11.

Neural weights evolution curves.

As can be seen from Figure 8 in method 3, the expected trajectory is well tracked in 1.5 s, and the control accuracy is high (Figure 8). As can be seen from Figure 9, the neural network controller is used to approximate $‖ F ‖$ and $‖ G ‖$ , and the acceptable approximation result is achieved within about 1 s. Due to the nonlinearity of $‖ F ‖$ and $‖ G ‖$ themselves, the approximation values of neural networks fluctuate up and down mainly around their expected value. This also indicates that the designed neural network controller is effective.

It can be seen that although there is a certain approximation error in Figure 9, the trajectory tracking can still obtain higher control accuracy. It is found that, since the total control law is composed of two parts, namely the neural network controller and the robust controller u_r, although the neural network controller has the approximation error, the robust controller can play the role of compensating control, thus eliminating the approximation error and achieving higher control accuracy. This also indicates that the designed robust controller is effective.

It can be seen from Figure 11 that the weight of the neural network starts to evolve from the initial value of 0. Under the action of adaptive regulation, the weight parameters quickly evolved to a more stable range. At the moment, the control precision is satisfied, which indicates that the weight adaptive control law is effective.

The control method proposed in this article has only 5 N·m output torque (Figure 10), which is not only small but also relatively smooth. At the same time, better control accuracy is obtained, so it is a better control method.

Further study, it is also found that the approximation property of neural network is not only related to weight but also related to network parameters. This article adopts the radial basis neural network, and center and width of Gaussian function have great influence on the approximation property. Because this article adopts the form of fixed parameters for both center parameters and width parameters, only weight parameters can be adjusted adaptively, so the approximation accuracy is affected. In the future, to further improve the approximation accuracy, how to design the adaptive regulation law of the center parameter and the width parameter is a subject that needs further research.

Conclusions

An adaptive robust control method base on neural network adaptive robust control is proposed for the unknown FFSR with interferences.

Firstly, the dynamic model of FFSR is established. A computed-torque controller based on accurate model is designed and the stability of the system is proved.

A neural network controller is designed to approximate the unknown model, the adaptive learning laws are designed to realize the online real-time adjustment of network weights. The designed controller avoids the dependence on the precise mathematical model of the space robot.

Adaptive robust controller is designed to eliminate external disturbance and approximation error and improve the robustness of the system.

The stability of closed-loop system is proved based on Lyapunov theory. The simulation test verified the effectiveness of the proposed control method. The proposed control method is of great value for the study of similar nonlinear systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (61772247), Lishui Science and Technology Plan Project (2015KCPT03) and (2015RC04).

ORCID iD

Zhang Wenhui

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Adaptive robust control for free-floating space robot with unknown uncertainty based on neural network

Abstract

Keywords

Introduction

Dynamics equation of FFSR

Design of computed-torque controller for precise model

Theorem 1

Proof

Design of adaptive robust controller based on neural network for unknown model

Theorem 2

Proof

Control system validation and analysis

Method 1: Computed-torque control

Method 2: Sliding mode variable structure control

Method 3: Adaptive robust control base on neural network

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References