An Adaptive B-Spline Neural Network and Its Application in Terminal Sliding Mode Control for a Mobile Satcom Antenna Inertially Stabilized Platform

The mobile satcom antenna (MSA) enables a moving vehicle to communicate with a geostationary Earth orbit satellite. To realize continuous communication, the MSA should be aligned with the satellite in both sight and polarization all the time. Because of coupling effects, unknown disturbances, sensor noises and unmodeled dynamics existing in the system, the control system should have a strong adaptability. The significant features of terminal sliding mode control method are robustness and finite time convergence, but the robustness is related to the large switching control gain which is determined by uncertain issues and can lead to chattering phenomena. Neural networks can reduce the chattering and approximate nonlinear issues. In this work, a novel B-spline curve-based B-spline neural network (BSNN) is developed. The improved BSNN has the capability of shape changing and self-adaption. In addition, the output of the proposed BSNN is applied to approximate the nonlinear function in the system. The results of simulations and experiments are also compared with those of PID method, non-singularity fast terminal sliding mode (NFTSM) control and radial basis function (RBF) neural network-based NFTSM. It is shown that the proposed method has the best performance, with reliable control precision.


Introduction
A mobile satcom antenna (MSA) is a kind of satellite communication antenna which can maintain continuous and reliable communications between satellites and users in movement. MSAs are widely applied in vehicle carriers, flying platforms and vessels. The Jet Propulsion Laboratory (JPL) has developed several mobile vehicular antenna systems for the satellite-based applications [1]. Due to the uncertain working environment, such as bumpy roads, sea waves and airflows, there might be strong disturbances in actual systems [2]. MSAs seeking to achieve high data-rate communications require the inertial antenna to be capable of pointing to within fractions of a degree [3]. Stabilization is used in the antenna pointing control system to maintain the line of sight by isolating the disturbances caused by vehicle motion [4]. Generally speaking, the attitude sensor can be an attitude and heading reference system (AHRS) or an inertial measurement unit (IMU) [5]. The outputs of these inertial sensors are widely used as feedbacks for stabilization in antenna control systems [6]. The accuracy of control is influenced by many factors, mainly including the existence of unmodeled nonlinear uncertainty, random bias of states and sensor noises [7] and friction restriction, which decreases the accuracy most. references, control points are trained by rules. In [33,37], the position of control points can be changed to a known nearby value, but only one control point can be changed once by this method. Additionally, the input and output of neural network are not relevant directly in this group.
In this paper, a novel B-spline neural network with the capability of shape adjustment is proposed and applied to approximate the external disturbance and the unmolded dynamics of the system. The equation of the B-spline curve is reconstructed in numeric form, therefore the new B-spline curve possesses the properties of the radial basis function, making it different from the previous approaches. In this research, the activation function of proposed BSNN is a pp-form spline.
The paper is organized as follows: in Section 2, the dynamic model of MSA and the control block diagram are established respectively. Then a new terminal sliding mode control strategy is proposed and proved. In Section 3, a novel BSNN is proposed and the parameter updating rules are built. In Section 4, experiments and simulations are carried out. The performances of the proposed methods are also compared with other methods. Finally, conclusions and contributions are summarized in Section 5.

Modeling of a Mobile Satcom Antenna
The paper focuses on a double gimbal MSA. The innermost gimbal is an elevation gimbal on which the antenna array is mounted, while the outmost gimbal is an azimuth gimbal which is mounted on the base. According to the coordinates system definition and the conversion relationships between different coordinates, a dynamic model of the mobile satcom antenna is derived.

Gimbal Coordinates Definition
An orthogonal coordinate system is defined: Base coordinate frame (ox b y b z b ), azimuth gimbal frame (ox a y a z a ) and elevation gimbal frame (ox e y e z e ). The rotating transformations are shown in According to the rotation matrix C a b and C e a , the angular rate relations of the two gimbals can be obtained as follows: ω e ie = C e a ω a ia + ω e ae =    p cos θ a + q sin θ a + . θ e −p cos θ e sin θ a + q cos θ a cos θ e + r sin θ e + . θ a sin θ e p sin θ a sin θ e − q cos θ a sin θ e + r cos θ e + . θ a cos θ e    (2) in which, ω b ib is the angular rate of the base carrier; ω a ia is the angular rate of the azimuth gimbal;

Dynamic of the Azimuth Gimbal
The gimbals in this system are all rigid, therefore the basic Newton-Euler rotation equation can be written as follows: in which ∑ is the resultant moment of the force added to the rigid body; is the inertial angular momentum; is the inertial moment; is the absolute angular rate. Since the structures of the two gimbals are similar (the difference is that the azimuth gimbal is effected by coupling of elevation gimbal), only the azimuth gimbal is considered in this paper.
According to the Equation (3), the dynamic model of azimuth gimbal can be obtained as follows: where, is the disturbance torque added to elevation gimbal; is the driving torque of elevation motor; is the coupling torque that effected by elevation gimbal. They are given by Equations (5) and (6) , , are the moments of inertia of the azimuth gimbal related to the , , axes in azimuth coordinates, respectively; , , are the moments of inertia of the elevation gimbal related to the , , axes in the elevation gimbal, respectively; is the azimuth motor torque constant; is the back-EMF coefficient of azimuth motor; is the azimuth gear ratio; is the motor resistance; is the moment of inertial of driver gear; is the viscous friction coefficient of driver gear; is the moment of inertial of passive gear; is the viscous friction coefficient of passive gear; is the input driven voltage of the motor.
Based on the preceding analysis, system (4) can be simplified into a standard second order control system: in which, = , = , is external disturbance and is the coupling effect torque. To further simplify the analysis, we define that | + | ≤ . Then the proposed control block diagram can be obtained as follows:

Dynamic of the Azimuth Gimbal
The gimbals in this system are all rigid, therefore the basic Newton-Euler rotation equation can be written as follows: in which ∑ M is the resultant moment of the force added to the rigid body; H is the inertial angular momentum; J is the inertial moment; ω is the absolute angular rate. Since the structures of the two gimbals are similar (the difference is that the azimuth gimbal is effected by coupling of elevation gimbal), only the azimuth gimbal is considered in this paper.
According to the Equation (3), the dynamic model of azimuth gimbal can be obtained as follows: where, M ad is the disturbance torque added to elevation gimbal; M am is the driving torque of elevation motor; M aez is the coupling torque that effected by elevation gimbal. They are given by Equations (5) and (6): . ω e iez sin θ e + J ex − J ey ω e iex ω e iey sin θ e + J ez . ω e iez cos θ e + J ex − J ey ω e iex ω e iey cos θ e (5) J ax , J ay , J az are the moments of inertia of the azimuth gimbal related to the X, Y, Z axes in azimuth coordinates, respectively; J ex , J ey , J ez are the moments of inertia of the elevation gimbal related to the X, Y, Z axes in the elevation gimbal, respectively; k at is the azimuth motor torque constant; k ab is the back-EMF coefficient of azimuth motor; N a is the azimuth gear ratio; R aa is the motor resistance; J ad is the moment of inertial of driver gear; f ad is the viscous friction coefficient of driver gear; J a f is the moment of inertial of passive gear; f a f is the viscous friction coefficient of passive gear; u aa is the input driven voltage of the motor. Based on the preceding analysis, system (4) can be simplified into a standard second order control system: x a2 (t) = A a x a2 (t) + B a u a (t) + C a d a (t) + D a w a (t) (7) Sensors 2017, 17, 978 θ a , d a (t) is external disturbance and w a (t) is the coupling effect torque. To further simplify the analysis, we define that |C a d a (t) + D a w a (t)| ≤ l d . Then the proposed control block diagram can be obtained as follows: and d a (t) = M ad + (J az + J ez ) . r + J ay + J ey − J ax − J ex ω a iax ω a iay . Then the control block diagram shown in Figure 2 can be obtained.
Then the control block diagram shown in Figure 2 can be obtained. The NFTSM control applied in this paper can guarantee that the system state arrives at the equilibrium point in a finite time with fast response and higher precision. In the following section, a new NFTSM controller is proposed and proved.

Non-Singularity Fast Terminal Sliding Mode
The nonlinear system is defined by Equation (7). In order to improve the control performance of terminal sliding mode, the paper proposed an improved NFTSM as shown in Equation (8) the NFTSM manifold will be reached in a finite time. In addition, the tracking error on the sliding mode surface will also converge to zero in a finite time.
Proof. Consider the Lyapunov candidate function as: The NFTSM control applied in this paper can guarantee that the system state arrives at the equilibrium point in a finite time with fast response and higher precision. In the following section, a new NFTSM controller is proposed and proved.

Non-Singularity Fast Terminal Sliding Mode
The nonlinear system is defined by Equation (7). In order to improve the control performance of terminal sliding mode, the paper proposed an improved NFTSM as shown in Equation (8), which is similar to reference [14].
in which, α, β > 0, 0 < p < 1 and r > p,ê = θ a − θ c a . The first derivative can be expressed as Theorem 1. For System (7) with the adopted NFTSM, if the control law is designed as: the NFTSM manifold will be reached in a finite time. In addition, the tracking error on the sliding mode surface will also converge to zero in a finite time.
Proof. Consider the Lyapunov candidate function as: The derivative of V is given by the following function, It is obvious that Thus, the condition for Lyapunov stability is satisfied. Then the system state will converge to zero within finite time. As shown in Equation (10), the value of 1 − p < 0 is always greater than zero, therefore, the controller is non-singular.
We can also obtain the reaching time t r satisfies (see Appendix A) That completes the proof. Therefore, the system can reach the terminal sliding mode surface within a finite time.
For terminal sliding mode control methods, the upper disturbance bound is generally required, but in practical applications, a large upper bound can be used to eliminate the external disturbance item and guarantee the robustness. However, the chattering phenomena is enhanced correspondingly. The paper attempts to approximate nonlinear disturbance by using BSNN. A novel BSNN is proposed in this paper. As its name implies, the B-spline curve is adopted as activation function.

B-Spline Neural Network
Artificial neural networks have high nonlinear approximation abilities. A spline function is a piecewise polynomial function of degree k. The spline neural network can play a role in local features. In this section, a new kind of B-spline function is introduced herin, then the proposed BSNN is deduced.

B-Spline Basis Function Definition
A B-spline curve differs from a Hermite or Bérzier curve, because a B-spline curve usually consists of more than one curve segment. The B-spline curve is widely used in the field of computer graphics for its excellent performance. The B-spline basis function is the foundation of BSNN.
Definition 1 [39]. Given a knot vector T = {t 0 ≤ t 1 ≤ · · · ≤ t m }, the ith B-spline basis function of degree k can be written as: is a constant, the B-spline curve is called uniform; otherwise, it is called non-uniform. Definition 1 is usually referred to the Cox-de Boor recursion formula. The jth B-spline basis curve with degree k can be written as B i j,k (t) on an arbitrary vector [ t i , t i+1 ) : An order k + 1 B-spline is formed by joining several pieces of polynomials of degree k with at most C k−1 continuity at the breakpoints [40], hence the following formula can be obtained: The derivative of Equation (15) can be written as follows: An arbitrary order derivative formula can be obtained as Equation (19) shows: According to Equations (14), (15), (18) and (19), the coefficients a i j,k , a i j,k−1 , · · · , a i j,1 , a i j,0 in Equation (16) can be obtained. Then the jth B-Spline basis curve B i j,k (t) with degree k on vector [ t i , t i+1 ) can be calculated.
The B-Spline basis function B j,k (t) with degree k can also be represented in matrix form in the The expression of different B-spline basis functions in an interval [ t i , t i+1 ) can be written as follows: Sensors 2017, 17, 978 The coefficient matrixes M e in Equation (20) and M e in Equation (21) can be obtained by Equations (14), (15), (18) and (19).
The B-spline curve is a linear combination of control points P i and a B-spline basis function B i,k (t). The definition is as follows:

B-Spline Curve Definitions
Definition 2 [39]. Given n + 1 control points P i , a knot vector T = {t 0 ≤ t 1 ≤ · · · ≤ t m } and B-spline basis functions B i,k (t), a B-spline curve is given by: Remark 2. Indexes m, n, k must satisfy m = n + k + 1.

Remark 3.
If the first knot and the last knot have multiplicity with value k + 1, the B-spline curve is called closed clamped; otherwise, it is called an open clamped B-spline curve. More specific illustrations can be seen in Figure 3.
The B-spline curve is a linear combination of control points and a B-spline basis function , . The definition is as follows:

Remark 3.
If the first knot and the last knot have multiplicity with value k + 1, the B-spline curve is called closed clamped; otherwise, it is called an open clamped B-spline curve. More specific illustrations can be seen in Figure 3.  From Equation (22) on the ith vector the B-spline function can be written as: If the horizontal value x in of an arbitrary point (x in , y in ) on the B-spline curve is known, the following formula can be obtained by: Therefore, the unknown internal vector vˆt can be calculated. Furthermore, the vertical value y in can be obtained: In this paper, order 3 B-spline curve is adopted. Then Equation (23) can be simplified to the following form: According to the known horizontal coordinate value x in , the corresponding internal knott can be obtained: Furthermore: in which γ = Py 0 Py 1 Py 2 T . According to the Definition 2, the B-spline function is improved in this paper by proposing two deformation factors called translation factor and scaling factor. Definition 3. Given n + 1 control points P i and a knot vector T = {t 0 ≤ t 1 ≤ · · · ≤ t m }, a new B-spline function of degree k can be written as: Remark 4. At the beginning, the value of the two deformation factors λ and κ are zero. thus, the formulas in Definition 2 and 3 are equivalent. λ is called translation factor, and κ is called scaling factor. Assumption 1. The two factors specify how the shape changes, assuming that b = 1 1 · · · 1 1×n , w = 1 1 1 and 2 × ρ + τ = n (τ can be 0 or 1).

The B-Spline Neural Network
Compared with some previous studies, the proposed B-spline adopts an order 3 B-spline function as activation function rather than a B-spline basis function as in some references [29,31]. Generally speaking, neural networks always have more than one hidden layer. The structure of our BSNN is shown in Figure 4.

The B-Spline Neural Network
Compared with some previous studies, the proposed B-spline adopts an order 3 B-spline function as activation function rather than a B-spline basis function as in some references [29,31]. Generally speaking, neural networks always have more than one hidden layer. The structure of our BSNN is shown in Figure 4. As shown in Figure 4, the neural network system has multiple inputs and one output. Each input corresponds to several activation functions. The output of BSNN can be mapped from inputs to an output by using B-splines as activation functions. The output of the hidden layer can be written as follows: In which, , means the th i input corresponding to the th activation function; means the number of inputs; means the number of activation functions. Therefore, the output of BSNN is as follows: The training method is to minimize the error function which is defined as follows: The chain rules of parameter update can be written as follows: As shown in Figure 4, the neural network system has multiple inputs and one output. Each input corresponds to several activation functions. The output of BSNN can be mapped from inputs x i to an output L d by using B-splines as activation functions. The output of the hidden layer can be written as follows: In which, η i,j (x i ) means the ith input corresponding to the jth activation function; ρ means the number of inputs; a means the number of activation functions. Therefore, the output of BSNN is as follows: The training method is to minimize the error function which is defined as follows: The chain rules of parameter update can be written as follows: ∂E ∂s In these equations, η is the learning rate; α is the inertial coefficient.
The B-spline basis function has an important property which is called "Partition of Unity", indicating that the sum of all non-zero degree k basis functions on span [ t i , t i+1 ) is 1, i.e., ∑ n s=0 B s,k (t) = 1. According to Assumption 1, the following Equations (42) and (44) can be obtained: Therefore, an adaptive BSNN is proposed. Experiments and simulations carried out to verify the performance of the BSNN are discussed in the following section.

Results of Experiment and Simulation
Experiments and simulations were carried out to validate the performance of the proposed BSNN-based NFTSM control method in this section. The experimental system is shown in Figure 5. Comparisons are made between the results of the conventional PID method, the TSM in reference [19], the NTSM in reference [15], the NFTSM in this paper, the RBF neural network-based NFTSM. The parameters of the MSA control system are listed in Table 1.
Experiments and simulations were carried out to validate the performance of the proposed BSNN-based NFTSM control method in this section. The experimental system is shown in Figure 5. Comparisons are made between the results of the conventional PID method, the TSM in reference [19], the NTSM in reference [15], the NFTSM in this paper, the RBF neural network-based NFTSM. The parameters of the MSA control system are listed in Table 1. In this system, the BSNN has two inputs and one output. Each of the inputs corresponds to three B-spline activation functions which have + 1 = 7 control points separately. The internal knot vector of B-Spline curve is 1 1 1 2 3 4 5 6 6 6 and the degree of B-spline curve is 2. Moreover, the learning rate = 0.6, the inertial coefficient = 0.05 and = 5. = 7 is adopted for NFTSM control method.

Description of the Coupling Effect
In this section, the amplitude of the elevation gimbal is 0 degrees in the first 47.9 s (Stage A); the gimbal is assigned to perform a sinusoidal motion 0.5 × sin 0.334 × × from 47.9 s to 89.8 s (Stage B); the gimbal is assigned to perform a sinusoidal motion 1.0 × sin 0.1336 × × from 89.8 s to 200 s (Stage C), as shown in Figure 6. In order to describe the coupling effect caused by the elevation and vehicle movement, there is no control and no input applied to the azimuth gimbal. The vehicle is assigned to move with speed of 0.5°/s. Simulation and experiment results are demonstrated in Figure  7.
As shown in Figure 7, the azimuth gimbal has a large angular output in both the simulation and experiment. The azimuth gradually increases due to the motion of the elevation gimbal. Apparently,  In this system, the BSNN has two inputs and one output. Each of the inputs corresponds to three B-spline activation functions which have n + 1 = 7 control points separately. The internal knot vector of B-Spline curve is 1 1 1 2 3 4 5 6 6 6 and the degree of B-spline curve is 2. Moreover, the learning rate η = 0.6, the inertial coefficient α = 0.05 and c = 5. l d = 7 is adopted for NFTSM control method.

Description of the Coupling Effect
In this section, the amplitude of the elevation gimbal is 0 degrees in the first 47.9 s (Stage A); the gimbal is assigned to perform a sinusoidal motion 0.5 × sin(0.334 × π × t) from 47.9 s to 89.8 s (Stage B); the gimbal is assigned to perform a sinusoidal motion 1.0 × sin(0.1336 × π × t) from 89.8 s to 200 s (Stage C), as shown in Figure 6. In order to describe the coupling effect caused by the elevation and vehicle movement, there is no control and no input applied to the azimuth gimbal. The vehicle is assigned to move with speed of 0.5 • /s. Simulation and experiment results are demonstrated in Figure 7.
As shown in Figure 7, the azimuth gimbal has a large angular output in both the simulation and experiment. The azimuth gradually increases due to the motion of the elevation gimbal. Apparently, the results are not what the controller engineer expected. Therefore, a decoupling controller is required.

Description of De-Coupling Effect
In order to validate the decoupling ability of the proposed method, the azimuth gimbal is assigned to perform a sinusoidal motion with amplitude 0 in the first 48.8 s (Stage A); a sinusoidal motion 1.0 × sin 0.164 × × is applied in the following period (Stage B), as shown in Figure 8. The elevation gimbal is still assigned to perform a sinusoidal motion as in Section 4.1.

Simulation Results
The simulation results are shown in Figure 9.

Description of De-Coupling Effect
In order to validate the decoupling ability of the proposed method, the azimuth gimbal is assigned to perform a sinusoidal motion with amplitude 0 in the first 48.8 s (Stage A); a sinusoidal motion 1.0 × sin 0.164 × × is applied in the following period (Stage B), as shown in Figure 8. The elevation gimbal is still assigned to perform a sinusoidal motion as in Section 4.1.

Simulation Results
The simulation results are shown in Figure 9.

Description of De-Coupling Effect
In order to validate the decoupling ability of the proposed method, the azimuth gimbal is assigned to perform a sinusoidal motion with amplitude 0 in the first 48.8 s (Stage A); a sinusoidal motion 1.0 × sin(0.164 × π × t) is applied in the following period (Stage B), as shown in Figure 8. The elevation gimbal is still assigned to perform a sinusoidal motion as in Section 4.1.

Description of De-Coupling Effect
In order to validate the decoupling ability of the proposed method, the azimuth gimbal is assigned to perform a sinusoidal motion with amplitude 0 in the first 48.8 s (Stage A); a sinusoidal motion 1.0 × sin 0.164 × × is applied in the following period (Stage B), as shown in Figure 8. The elevation gimbal is still assigned to perform a sinusoidal motion as in Section 4.1.

Simulation Results
The simulation results are shown in Figure 9.

Simulation Results
The simulation results are shown in Figure 9. From the simulation results, the following conclusions can be obtained: the conventional PID method has the worst performance, the peak to peak (p-p) value of tracking error is 0.131°, the tracking error at time 48.8 s is 0.058°; while the tracking error ∆ of the conventional TSM method is 1.75 × 10 ° and the tracking error at time 48.8 s is −26.1 × 10 ° ; the tracking error ∆ of the NTSM method in reference is 6.9 × 10 ° , the tracking error is −20.0 × 10 ° ; the tracking error ∆ of NFTSM is 6.7 × 10 ° , and the tracking error at time 48.8 s is 18.0 × 10 ° ; for RBFNN based NFTSM the tracking error ∆ is 7.3 × 10 ° and the tracking error at time 48.8 s is 23.0 × 10 ° ; the best performance corresponds to the BSNN-based NFTSM with a tracking error of 1.1 × 10 ° and the tracking error at time 48.8 s of 7.8 × 10 ° . At time 48.8 s, the azimuth gimbal begins to perform the assigned sinusoidal motion from a still state. The simulation results show that the proposed BSNN has a better adaptive capability and the proposed controller shows better decoupling effect than other controllers.
The forms of the B-spline curves are shown in Figure 10. It can be seen from the figure that both the positions and the shapes of the B-spline curves are changed. From the simulation results, the following conclusions can be obtained: the conventional PID method has the worst performance, the peak to peak (p-p) value of tracking error is 0.

Experiment Results
Decoupling effect experiments are carried out in this section. The results are shown in Figure 11. As shown in Figure 11, the proposed BSNN-based NFTSM performs better than the other three methods. The p-p value of ∆ is 0.162° when the PID controller is applied; the p-p value of ∆ is 0.12° when the NFTSM controller is applied; the p-p value of ∆ is 0.077° when the RBFNN-based NFTSM is applied; the p-p value of ∆ is 0.051° when the BSNN-based NFTSM is applied.

Disturbance Rejecting Ability Simulations and Experimental Results
In order to validate the disturbance rejecting ability of the proposed method, pulse disturbances of 50 N·m and 100 N·m are added at time 70 s and 130 s, respectively, as shown in Figure 12.

Experiment Results
Decoupling effect experiments are carried out in this section. The results are shown in Figure 11. As shown in Figure 11, the proposed BSNN-based NFTSM performs better than the other three methods. The p-p value of ∆θ a is 0.162 • when the PID controller is applied; the p-p value of ∆θ a is 0.12 • when the NFTSM controller is applied; the p-p value of ∆θ a is 0.077 • when the RBFNN-based NFTSM is applied; the p-p value of ∆θ a is 0.051 • when the BSNN-based NFTSM is applied.

Experiment Results
Decoupling effect experiments are carried out in this section. The results are shown in Figure 11. As shown in Figure 11, the proposed BSNN-based NFTSM performs better than the other three methods. The p-p value of ∆ is 0.162° when the PID controller is applied; the p-p value of ∆ is 0.12° when the NFTSM controller is applied; the p-p value of ∆ is 0.077° when the RBFNN-based NFTSM is applied; the p-p value of ∆ is 0.051° when the BSNN-based NFTSM is applied.

Disturbance Rejecting Ability Simulations and Experimental Results
In order to validate the disturbance rejecting ability of the proposed method, pulse disturbances of 50 N·m and 100 N·m are added at time 70 s and 130 s, respectively, as shown in Figure 12.

Disturbance Rejecting Ability Simulations and Experimental Results
In order to validate the disturbance rejecting ability of the proposed method, pulse disturbances of 50 N·m and 100 N·m are added at time 70 s and 130 s, respectively, as shown in Figure 12.

Simulation Results
The disturbance rejecting ability simulation results are shown in Figure 13. From the simulation results in Figure 13

Simulation Results
The disturbance rejecting ability simulation results are shown in Figure 13. From the simulation results in Figure 13

Simulation Results
The disturbance rejecting ability simulation results are shown in Figure 13. From the simulation results in Figure 13  However, when the RBFNN-based NFTSM is applied, the p-p values of ∆θ a are 7.7 × 10 −5 ( • ) and 17 × 10 −3 ( • ) at time 70 s and 130 s, respectively; the p-p values of ∆θ a are 4.0 × 10 −5 ( • ) and 6.2 × 10 −5 ( • ) at time 70 s and 130 when the BSNN-based NFTSM is applied, respectively. Therefore, it is apparent that the neural network is good at nonlinear approximation, and the proposed BSNN performs better. The forms of B-spline curves are shown in Figure 14. However, when the RBFNN-based NFTSM is applied, the p-p values of ∆ are 7.7 × 10 ° and 17 × 10 ° at time 70 s and 130 s, respectively; the p-p values of ∆ are 4.0 × 10 ° and 6.2 × 10 ° at time 70 s and 130 when the BSNN-based NFTSM is applied, respectively. Therefore, it is apparent that the neural network is good at nonlinear approximation, and the proposed BSNN performs better. The forms of B-spline curves are shown in Figure 14.

Experimental Results
Four different methods are applied to the control system of MSA. From the experimental results in Figure 15 According to the simulations and experiments, it can be seen that the proposed BSNN based NFTSM is good at nonlinear approximate and has strong self-adaptability.

Experimental Results
Four different methods are applied to the control system of MSA. From the experimental results in Figure 15, the following conclusions can be obtained: the p-p values of ∆θ a are 0. However, when the RBFNN-based NFTSM is applied, the p-p values of ∆ are 7.7 × 10 ° and 17 × 10 ° at time 70 s and 130 s, respectively; the p-p values of ∆ are 4.0 × 10 ° and 6.2 × 10 ° at time 70 s and 130 when the BSNN-based NFTSM is applied, respectively. Therefore, it is apparent that the neural network is good at nonlinear approximation, and the proposed BSNN performs better. The forms of B-spline curves are shown in Figure 14.

Experimental Results
Four different methods are applied to the control system of MSA. From the experimental results in Figure 15 According to the simulations and experiments, it can be seen that the proposed BSNN based NFTSM is good at nonlinear approximate and has strong self-adaptability. According to the simulations and experiments, it can be seen that the proposed BSNN based NFTSM is good at nonlinear approximate and has strong self-adaptability.

Conclusions
The paper focuses on the inertial sensor-based two gimbal mobile satcom antenna. In order to obtain high quality communications, the MSA should point to the specific satellite when the carrier is moving. The dynamic model of MSA is established based on traditional Newton-Euler method and the corresponding control block diagram is built. In this paper, the non-singular fast terminal sliding mode control is adopted and developed to increase the line of sight stabilization accuracy. Meanwhile, the features of existence and convergence in finite time are proved. Then a neural network is employed to approximate the nonlinear item in the system. In addition, a novel BSNN is proposed and used in this paper. A brief study of B-spline basis and B-spline function is also carried out, then the computational function used to obtain the arbitrary point on the curve is derived. The B-spline function is reformed to enhance its adaptive capacity. To validate the effectiveness of the proposed NFTSM and BSNN, simulations and experiments are conducted. Results of different methods, including PID, NFTSM, NFTSM-RBF, NFTSM-BSNN, are compared in this paper. It is shown that the proposed method has better decoupling effects and disturbance rejecting ability than the others. Because the B-spline curve has an excellent ability called "local control", it can be used to approximate arbitrarily shaped curves. According to the analysis in this paper, we can conclude that the proposed BSNN is good at nonlinear approximation owing to its local features. The robustness of the system can also be improved by applying this method.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The Terminal sliding mode surface (8) can be written as follows: The following formula can be obtained: Then Equation (A1) can be further simplified as: It is apparent that: β ê + β α ê r+1  Then the convergence time can be obtained: This ends the proof.