Determination of uncertain parameters of a two-axis gimbal and motion tracking via Fuzzy logic control approach

Abstract

This paper proposes a new method to improve fuzzy control performance accuracy in the stabilization of the two-axis gimbal system. To this end, due to the fact that the knowledge of the accurate behavior of the system plays a substantial role in fuzzy control performance, all the uncertain parameters of the dynamic model such as friction, mass imbalance and moments of inertia are estimated prior to the controller design and without imposing any computational burden on the control scheme. To estimate the uncertainties and disturbances which exist in the dynamic equations, an identification process formulated as an inverse problem is utilized, and the Gauss– Newton method is adopted for the optimization process. Regarding the severe sensitivity of inverse problems to measurement errors, this undesirable effect is reduced by using a proper smoothing technique. In order to increase the accuracy of the final results, a novel procedure for calculation of the sensitivity coefficients of the inverse problem is proposed. This procedure is based on the direct differentiation of the governing equations with respect to the unknown parameters. At the end, simulation results are presented to confirm the effectiveness of the proposed scheme.

Keywords

Fuzzy control parameter estimation two-axis gimbal

1 Introduction

1.1 Literature review

Two-axis gimbals are major components of the inertially stabilized platforms (ISP) that carry out the task of isolating the effects of internal or environmental disturbances on the desired object. According to the widespread use of the gimbals in the space industry, filmmaking and arms industry, numerous studies have been conducted on stabilization of the line of sight (LOS) of gimbals’ payload relative to an inertial frame [1 –3].

In gimbal control process, a primary concern in the wide range of control strategies is the presence of various disturbances which adversely affect the overall performance of the system [4, 5]. Also, precise values of some quantities are not always available for many parameters of a dynamic system, although they play an important role in the control process [6]. As a consequence, use of inaccurate values of the system parameters can lead to failure of the control process or degradation of the performance of the closed loop system’s response. To address this crucial challenge, various control approaches have been developed over recent decades.

One of the frequent robust control approaches is sliding mode control (SMC) due to its insensitivity to uncertainty, bounded disturbances and noise rejection [7 –9]. The switching control law enables this approach to drive the system states from any initial state to a desired surface, and maintain the intended state for the following time [10]. However, switching action between the different sliding mode surfaces may lead to a serious chattering problem. Hence, dealing with uncertainties and unknown disturbances makes the design of the sliding mode control more complex. Ding et al. proposed the idea of combining an adaptive radial basis function neural network with a backstepping sliding mode controller to approximate uncertain disturbances in the two-axis ISP [11]. Mao et al. developed a higher order sliding mode observer to cope with the uncertainties caused by cross-couplings, mass imbalance, and other unmodeled dynamics of the two-axis ISP [12]. They also investigated a continuous finite-time SMC to eliminate the undesirable influences caused by the multiple disturbances in a 2-DOF ISP system [13]. In their research, multiple disturbances were treated as a lumped uncertainty which was estimated and compensated by using a Finite-time disturbance observer. An adaptive fractional-order sliding mode method was proposed by Naderolasli and Tabatabaei to stabilize a two-axis gimbal system in the presence of disturbance effects, mass unbalance and cross-coupling terms which demonstrated better transient response compared with the integer-order one [14].

Due to the capability of handling nonlinearities, parameter uncertainties, and disturbances, active disturbance rejection control (ADRC) is one of the nonlinear control methods which has been extensively used by researchers for the control of gimbals. Bai and Zhang presented a least mean squares based ADRC for an ISP composed three-axis gimbal [15]. Ahi and Nobakhti derived an inclusive dynamic model of a one-axis gimbal mechanism under the influence of various sources of disturbances, including base angular motion, mass imbalance, and friction torque. They applied ADRC for the control of the gimbal [16]. Wang et al. adopted the same method to improve the stabilization accuracy of a two-axis gimbal which was a component of the ISP for airborne star tracker application [17]. The proposed simplified noise reduction disturbance observer effectively improved anti-disturbance performance of the proposed control scheme. A critical requirement of the ADRC controller is to determine a large number of parameters which are very difficult to be identified at the same time. Zhou et al. proposed a genetic algorithm based method to tune these parameters which significantly improved the stabilization accuracy of the ISP composed three-axis gimbal [18].

Another control strategy which has exhibited acceptable performance in controlling gimbal systems is disturbance observer control (DOBC). This approach has the capability of compensating uncertainty and suppressing disturbance, however, it should be noted that large uncertainties and sensor noise adversely affect the robustness of the control scheme. Ren et al. proposed a method in which uncertainties, friction, and coupling shortcoming of the gimbal system is summed up as a disturbance suppression problem [19]. They achieved disturbance compensation through feedforward channel of DOBC. To guarantee robust stability of DOBC when the controlled object changes, a H_∞ based filter design was adopted.

In addition to the above-mentioned researches, other types of advanced control techniques like adaptive dynamic surface [20] and adaptive robust output feedback tracking [21] which are capable of managing uncertainties have been utilized to control the gimbal systems. Guo designed a H_∞ robust controller to damp the negative influence of the complex structure of a novel ISP with a magnetically suspended gimbal [22]. Pan used a robust PID control for a two-axis ISP driven by ultra-sonic motor to meet high performance requirements [23]. The controller was designed based on the derived dynamic model in which the mechanical resonance, the coupling of inertia moment and torque, and nonlinearity of ultrasonic motor were considered.

In some works, neural network has been combined with the main control approach to improve stabilization accuracy by estimating disturbances and uncertainties [10 , 25]. According to the conducted investigations which use the fuzzy approach for the control of gimbals, for instance [26 –30] can be seen, this method is often applied to improve the performances of the control schemes that are designed based on other kinds of control methods; however, generally fuzzy logic control attains good control performance. It demonstrates remarkable adaptability when faced with change of control parameters, good robustness and strong anti-interference ability against nonlinearity or complexity of the control object. Besides, fuzzy control brings system fast response and also properly manages the system to maintain a small overshoot. This is due largely to the fact that the determination of fuzzy rule sets and membership functions depend much on experience. Moreover, the control performances of these approaches rely heavily on the accuracy of the system model which is used to predict the system behavior [29, 31].

1.2 Research motives

Regarding the aforementioned issues, accurate identification of system’s fundamental dynamic parameters from its behavior can significantly reduce the difficulties of the controller design. Most often, the mentioned control schemes need online calculations posing a heavy computational burden on the control design. Other schemes require a lot of information about the system before operating, which is difficult to provide according to the working conditions of the gimbal system.

In the present work, according to the crucial role of the system information in determining structural parameter of fuzzy control, an accurate mathematical model of the system is provided to shape a clear view of the system at the primary stage of control design. Relying on the simplicity of the fuzzy control, this approach attempts to improve the fuzzy control immunity and stability by a remedy in which no online calculation is required. Hence, the proposed method is a common fuzzy control which is not combined with other methods. To this end, first, all the uncertain parameters of the two-axis gimbal dynamic equation are identified. The identification problem is formulated as an inverse problem. Subsequently, a technique for estimation of the unknowns, i.e. the mass imbalance, friction, and moments of inertia, is proposed. In this trend, the angular positions of the azimuth and the elevation are recorded in response to a sinusoidal stimulation. These data are used as reference input data for the inverse problem. Due to the severe sensitivity of the inverse problems to the measurement errors, a proper smoothing technique is applied to minimize this adverse effect. Then, the damped Gauss-Newton method is implemented for the solution of the inverse problem. Also, a novel method for calculating sensitivity coefficients of the inverse problem is presented. The proposed method is based on direct differentiation of the dynamic equations with respect to the unknown parameters, and is more accurate than the conventional techniques. At the end of the identification process, all the unknown parameters of system’s dynamic are obtained and will be used in control design. Finally, the effectiveness of the proposed control approach is validated via some numerical examples.

The remainder of this paper is organized as follows. Section 2 describes the dynamic model parameters. Section 3 develops the fuzzy logical control based on the estimated parameters. To assess the effectiveness of the proposed method, the detailed simulation procedure of the parameters identification and the fuzzy logical controller is presented in Section 4. Finally, Section 5 concludes the paper.

2 Parameter estimation

Generally, a two-axis gimbal configuration consists of an inner and an outer gimbal frame. The payloads which are to be stabilized are placed on the inner gimbal frame; also, two encoders are mounted on the two frames to measure angular displacements of the both frames. The angular displacement, velocity and acceleration of the azimuth axis (the outer frame) are represented by q₁, ${\dot{q}}_{1}$ and ${\ddot{q}}_{1}$ respectively. Similarly,q₂, ${\dot{q}}_{2}$ and ${\ddot{q}}_{3}$ represent the angular displacement, velocity, and acceleration of the elevation axis (the inner frame). Considering the disturbances that arise from the mass imbalance and friction in joints on the system performance, the dynamic model of the two-axis gimbal can be described by Equations (1–4). These equations are derived by using the Denavit-Hartenberg convention and by assuming that each gimbal frame is rigid [7, 32]. $D (q) \ddot{q} + C (q, \dot{q}) \dot{q} + F_{s} (q, \dot{q}) + G (q) = τ$ (1) $D (q) = [\begin{matrix} I_{1_{33}} + I_{2_{11}} {sin}^{2} q_{2} + I_{2_{33}} {cos}^{2} q_{2} & 0 \\ 0 & I_{2_{22}} \end{matrix}]$ (2)

$\begin{matrix} C (q, \dot{q}) \\ = [\begin{matrix} \frac{1}{2} {\dot{q}}_{2} (I_{2_{11}} - I_{2_{33}}) sin (2 q_{2}) & \frac{1}{2} {\dot{q}}_{1} (I_{2_{11}} - I_{2_{33}}) sin (2 q_{2}) \\ - \frac{1}{2} {\dot{q}}_{1} (I_{2_{11}} - I_{2_{33}}) sin (2 q_{2}) & 0 \end{matrix}] \end{matrix}$ (3)

$F_{s} = {\begin{matrix} F_{s} sgn ({\dot{q}}_{1}) \\ F_{s} sgn ({\dot{q}}_{2}) \end{matrix}} G (q) = [\begin{matrix} 0 \\ k sin (q_{2}) \end{matrix}]$ (4)

Where D represents the inertia matrix, C represents the coriolis matrix, F_s represents the frictional torque vector, and G represents the vector of the torque resulting from mass imbalance. I_1₃₃ represents the azimuth gimbal moment of inertia and I_2₁₁, I_2₂₂, and I_2₃₃ represent the elevation gimbal moments of inertia. F_s represents the friction coefficient, and k represents the product of multiplication of the payload’s weight by the distance from the center of mass to the axis of rotation. Equation (1) can be represented in the form of the system of Equations (5) and (6) which show the dynamic equations governing the azimuth and elevation axes, respectively: $\begin{matrix} ((I_{1_{33}} + I_{2_{33}}) + (I_{2_{11}} - I_{2_{33}}) {sin}^{2} (q_{2})) {\ddot{q}}_{1} \\ + (I_{2_{11}} - I_{2_{33}}) sin (2 q_{2}) {\dot{q}}_{1} {\dot{q}}_{2} + F_{s} sgn ({\dot{q}}_{1}) = τ_{1} \end{matrix}$ (5) $\begin{matrix} I_{2_{22}} {\ddot{q}}_{2} - \frac{1}{2} (I_{2_{11}} - I_{2_{33}}) sin (2 q_{2}) {\dot{q}}_{1}^{2} \\ + k sin (q_{2}) + F_{s} sgn ({\dot{q}}_{2}) = τ_{2} \end{matrix}$ (6)

According to Equations (5) and (6), the response of the system does not depend independently on values of I_2₁₁, I_2₃₃ and I_1₃₃. A careful review of the equations reveals that the following independent parameters determine the response of the system: $\begin{matrix} A_{1} = I_{1_{33}} + I_{2_{33}}, A_{2} = I_{2_{11}} - I_{2_{33}}, \\ A_{3} = I_{2_{22}}, A_{4} = k, A_{5} = F_{s} \end{matrix}$ (7)

In this paper, it is assumed that the values of these five parameters are not known accurately, and should be determined prior to the controller design. By substituting identities of Equation (7) into Equations (5) and (6), the new form of the system’s dynamic equations can be expressed as follows:

$\begin{matrix} (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{1} + A_{2} sin (2 q_{2}) {\dot{q}}_{1} {\dot{q}}_{2} \\ + F_{s} sgn ({\dot{q}}_{1}) = τ_{1} \end{matrix}$ (8)

$A_{3} {\ddot{q}}_{2} - \frac{1}{2} A_{2} sin (2 q_{2}) {\dot{q}}_{1}^{2} + A_{4} sin (q_{2}) + F_{s} sgn ({\dot{q}}_{2}) = τ_{2}$ (9)

2.1 The Gauss-Newton’s method

For solving an inverse problem, first, an objective function must be defined, such that its minimization would lead to identification of unknown parameters of the problem. In this paper, to estimate values of the five unknown parameters present in the two-axis gimbal dynamic equations, the cost function is expressed in terms of squares of the differences between the measured and the computed values of the angular positions of the azimuth and elevation axes. The cost function is then minimized by applying the damped Gauss–Newton method [33]. The cost function is defined as: $f (X) = [\bar{Y} (X) - Y (X)]^{T} [\bar{Y} (X) - Y (X)]$ (10)

The vector X contains the unknown parameters of the system’s equations. Vectors Y and $\bar{Y}$ contain the values of the calculated and measured positions of the azimuth and the elevation axes: $\bar{Y} = {[{\bar{Y}}_{1} {\bar{Y}}_{2} \dots {\bar{Y}}_{N}]}^{T}$ (11)

In this expression ${\bar{Y}}_{N}$ is the last datum which is measured during the data collection procedure. At each stage of the analysis of the inverse problem, the values of these quantities, which are calculated in terms of an approximation of the vector X, are placed in the vector Y (X).

Based on the damped Gauss-Newton method, to minimize the objective function, the derivative of the objective function with respect to unknowns’ should be set to zero. After performing a series of matrix calculations, the vector of unknowns is found using the following iterative Equation (12). $X^{k + 1} = X^{k} + γ^{k} W^{k}$ (12)

Where the superscripts (k) and (k + 1) represent the step number, γ^(k) is the step length, and γ^(k) is the search direction. In the damped Gauss-Newton method, with γ^(k) updated in each step, the search direction can be obtained using the following equation [34]:

$\begin{matrix} W^{(k + 1)} = & {[{(E^{(k + 1)})}^{T} (E^{(k + 1)})]}^{- 1} \\ [{(E^{(k + 1)})}^{T} (\bar{Y} - Y^{(k + 1)})] \end{matrix}$ (13) where E is the sensitivity matrix and is expressed as follows: $E = [\begin{matrix} E_{11} & E_{12} & \dots & E_{15} \\ E_{21} & E_{22} & \dots & E_{25} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ E_{N 1} & E_{N 2} & \dots & E_{N 5} \end{matrix}]$ (14)

The elements of this matrix represent the derivative of the measured quantities with respect to the unknown parameters of the problem; for example, E_nr represents the derivative of the measured quantity at the n’th time step with respect to the r’th unknown parameter. In the present study, the angular positions of the azimuth and the elevation are considered as the measured quantity and are used to identify the unknown parameters. The number of columns in this matrix equals the number of unknowns of the two-axis gimbal’s dynamic equations. To evaluate the sensitivity of the dynamic equations response to the unknown parameters, the derivative of the system angular position should be taken with respect to each of the unknown parameters at each time step. To simplify the representation of the derivative equations, new variables q_ij are defined which represents the derivative of the i’th degree of freedom with respect to the unknown parameter j, i.e.: $\begin{matrix} q_{ij} = \frac{\partial q_{i}}{\partial A_{j}} {\dot{q}}_{ij} = \frac{\partial {\dot{q}}_{i}}{\partial A_{j}} {\ddot{q}}_{ij} = \frac{\partial {\ddot{q}}_{i}}{\partial A_{j}} \\ i = 1, 2; j = 1, 2, 3, 4, 5; \end{matrix}$ (15)

Differentiating both sides of the dynamic equations with respect to the first unknown parameter, A₁ leads to the following system of equations:

$\begin{matrix} (1 + A_{2} q_{21} sin (2 q_{2})) {\ddot{q}}_{1} + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{11} \\ + A_{2} ({\dot{q}}_{11} {\dot{q}}_{2} + {\dot{q}}_{1} {\dot{q}}_{21}) sin (2 q_{2}) + 2 A_{2} {\dot{q}}_{1} {\dot{q}}_{2} q_{21} cos (2 q_{2}) = 0 \end{matrix}$ (16)

$\begin{matrix} A_{3} {\ddot{q}}_{21} - A_{2} {\dot{q}}_{11} {\dot{q}}_{1} sin (2 q_{2}) - {\dot{q}}_{1}^{2} A_{2} cos (2 q_{2}) \\ + A_{4} q_{21} cos (q_{2}) = 0 \end{matrix}$ (17)

It should be noted that the unknown quantity in these equations is the new variable q_ij, while q_i are known quantities which are obtained from solving the original equation of motion. The solution of the system of Equations (16) and (17) gives the derivative of angular position with respect to the parameter A₁. These derivatives are the components of the first column of the sensitivity matrix, E.

Similarly, the derivative of the gimbal’s angular position with respect to the other unknown parameters can be obtained from the solution of the system of equations that are presented in following. The differentiation of the dynamic equations with respect to the parameter A₂ is expressed by Equations (18) and (19). The solution of this system of equations gives the second column of sensitivity matrix.

$\begin{matrix} ({sin}^{2} (q_{2}) + A_{2} q_{22} sin (2 q_{2})) {\ddot{q}}_{1} + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{12} \\ + A_{2} ({\dot{q}}_{12} {\dot{q}}_{2} + {\dot{q}}_{1} {\dot{q}}_{22}) sin (2 q_{2}) + {\dot{q}}_{12} {\dot{q}}_{2} sin (2 q_{2}) \\ + 2 A_{2} {\dot{q}}_{1} {\dot{q}}_{2} q_{22} cos (2 q_{2}) = 0 \end{matrix}$ (18)

$\begin{matrix} A_{3} {\ddot{q}}_{22} - A_{2} {\dot{q}}_{12} {\dot{q}}_{1} sin (2 q_{2}) - \frac{1}{2} {\dot{q}}_{1}^{2} A_{2} sin (2 q_{2}) \\ - q_{22} {\dot{q}}_{1}^{2} A_{2} cos (2 q_{2}) + A_{4} q_{22} cos (q_{2}) = 0 \end{matrix}$ (19)

In the same way, the third, the forth, and the last column of the sensitivity matrix are obtained by solving systems of equations 20 and 21, 22 and 23, and 24 and 25, respectively.

$\begin{matrix} A_{2} q_{23} sin (2 q_{2}) {\ddot{q}}_{1} + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{13} \\ + A_{2} ({\dot{q}}_{13} {\dot{q}}_{2} + {\dot{q}}_{1} {\dot{q}}_{23}) sin (2 q_{2}) + 2 A_{2} {\dot{q}}_{1} {\dot{q}}_{2} q_{23} cos (2 q_{2}) = 0 \end{matrix}$ (20)

$\begin{matrix} {\ddot{q}}_{2} + A_{3} {\ddot{q}}_{23} - A_{2} {\dot{q}}_{13} {\dot{q}}_{1} sin (2 q_{2}) - {\dot{q}}_{1}^{2} A_{2} sin (2 q_{2}) \\ - q_{23} {\dot{q}}_{1}^{2} A_{2} cos (2 q_{2}) + A_{4} q_{22} cos (q_{2}) = 0 \end{matrix}$ (21)

$\begin{matrix} A_{2} q_{24} sin (2 q_{2}) {\ddot{q}}_{1} + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{14} \\ + A_{2} ({\dot{q}}_{14} {\dot{q}}_{2} + {\dot{q}}_{1} {\dot{q}}_{24}) sin (2 q_{2}) + 2 A_{2} {\dot{q}}_{1} {\dot{q}}_{2} q_{24} cos (2 q_{2}) = 0 \end{matrix}$ (22)

$\begin{matrix} A_{3} {\ddot{q}}_{24} - A_{2} {\dot{q}}_{14} {\dot{q}}_{1} sin (2 q_{2}) - q_{24} {\dot{q}}_{1}^{2} A_{2} cos (2 q_{2}) \\ + sin (q_{2}) + A_{4} q_{24} cos (q_{2}) = 0 \end{matrix}$ (23) $\begin{matrix} A_{2} q_{25} sin (2 q_{2}) {\ddot{q}}_{1} + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{15} \\ + A_{2} ({\dot{q}}_{15} {\dot{q}}_{2} + {\dot{q}}_{1} {\dot{q}}_{25}) sin (2 q_{2}) \\ + 2 A_{2} {\dot{q}}_{1} {\dot{q}}_{2} q_{25} cos (2 q_{2}) + sgn ({\dot{q}}_{1}) = 0 \end{matrix}$ (24) $\begin{matrix} A_{3} {\ddot{q}}_{25} - A_{2} {\dot{q}}_{15} {\dot{q}}_{1} sin (2 q_{2}) - q_{25} {\dot{q}}_{1}^{2} A_{2} cos (2 q_{2}) \\ - A_{4} q_{25} cos (q_{2}) - sgn ({\dot{q}}_{2}) = 0 \end{matrix}$ (25)

It is worth mentioning that in the usual method of calculating the sensitivity matrix, the derivatives of the measured quantities are obtained using a finite difference scheme in which the selection of the finite difference step is very important. Value of the difference step highly influences the results accuracy. This issue is more pronounced in nonlinear systems, such as the two-axis gimbal studied in this research. Therefore, by using the method presented in this study, not only the sensitivity coefficients are obtained accurately, but also the reliability of the inverse problem solution and the design of the controller are improved.

The identification process begins with taking initial guess for the vector of unknowns, X. After calculating and forming the sensitivity matrix, the direction of the search vector is obtained according to Equation (13), and by using Equation (12), the next guess is determined for the unknown parameters. If the value of the objective function which is obtained based on the new values of the unknowns is less than the value of the previous step’s objective function, the process continues, otherwise the length of the step is halved, and again the objective function will be calculated and the inequality f (X^(k+1)) < f (X^(k)) will be checked; this process will be continued until the inequality is satisfied.

In the identification process two error indicators are considered to check the convergence of the algorithm. One is the value of the objective function, and the other one is the difference of the unknowns of each iteration to the previous ones. At each step of the identification process, the algorithm continues until these two quantities are less than predetermined tolerances e₁ and e₂. Hence, the convergence conditions of the identification algorithm can be described as: $∥ X^{(k + 1)} {- X}^{(k)} ∥ < e_{1}, f (X^{(k + 1)}) < e_{2}$ (26)

2.2 Smoothing technique

Since the measured data is most often noisy, it is reasonable to filter the noise prior to the estimation procedure. Consider that it is desired to reduce the oscillations of the elements of a vector P and obtain a smoothed vector _P′. To this end, the following functional is defined and minimized subsequently [34]: $R = \sum_{i = 1}^{M} {({P^{'}}_{i} {- P}_{i})}^{2} + β \sum_{i = 2}^{M - 1} {({P^{'}}_{i + 1} {- 2 P^{'}}_{i} {+ P^{'}}_{i - 1})}^{2}$ (27)

In Equation (27), M is the number of elements of the vector P and β is the smoothing parameter. If the value of this parameter is too large, the amount of smoothing would be very high, but the values of the two vectors will be far apart. In contrast, small values of β would not smooth the vector P effectively. The first term in the expression of R guarantees a vector P^′ with elements as close as possible to the elements of the vector P. The second term reduces the oscillation of vector P′ to produce a nearly piecewise linear distribution. Minimization of R is equivalent to minimization of the expression ∥BP′ - L∥; where L and B are defined as: $B = {\begin{matrix} I \\ β H \end{matrix}} L = {\begin{matrix} P \\ 0 \end{matrix}} H = [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 1 & - 2 & 1 & 0 & 0 \\ 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & 1 & - 2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$ (28)

In general, the matrix H is of size M × M, herein, H is written for M=5. By minimizing the function R, vector P′ will be obtained as follows: $P^{'} = {(B^{T} B)}^{- 1} B^{T} L$ (29)

3 Fuzzy controller design

According to the previous section, the dynamic equations of a two-axis gimbal are available at the preliminary controller design stage. Knowing mathematical model provides the ability of predicting and analyzing the designed closed-loop fuzzy control system theoretically. As a consequence, the structure and parameters of the fuzzy controller are designed in such a way that the desired tracking performance is achieved. The control objective is to develop proper control laws for line of sight stabilization of a two-axis gimbal. To this end, two fuzzy controllers are designed for the azimuth and the elevation of the gimbal. Each controller has two inputs, the tracking error and the tracking error velocity, and also one output, control torque. A typical fuzzy control structure is composed of four principal components: a fuzzifier, a fuzzy rule base, an inference engine and a defuzzifier.

3.1 Fuzzifier

Fuzzifier is an input interface of the fuzzy controller, which determines the input position deviation and the rate of change of the position deviation and transforms them into fuzzy quantities. The Gaussian fuzzifier which is used in this study has the task of converting real crisp values $x_{i}^{*} \in U_{i} \subset ℝ$ into a fuzzy set A′ in U = U₁ × U₂ × . . . × U_n according to the following Gaussian membership function Equation (30). $\underset{A^{'}}{μ} (X) = e^{{(\frac{x_{1} - x_{1}^{*}}{a_{1}})}^{2}} * \dots * e^{{(\frac{x_{n} - x_{n}^{*}}{a_{n}})}^{2}}$ (30) where X = [x₁, x₂, . . . , x_n]. In Equation (30), a_is are positive parameters and the t-norm * is usually chosen as algebraic product or minimum.

3.2 Fuzzy rule base

A fuzzy rule base consists of a set of fuzzy IF-THEN rules. It is the heart of the fuzzy system in the sense that all other components are used to implement these rules in a reasonable and efficient manner. Choosing an appropriate number of rules is important in designing the fuzzy systems, because too many rules result in a complex fuzzy system that may be unnecessary for the problem, whereas too few rules produce a less powerful fuzzy system that may be insufficient to achieve the objective [35].

The fuzzy rules set applied in this study (Table 1) are determined based on torque’s variations effects on system behavior.

Table 1
Fuzzy rule base

q $\dot{q}$ τ

P P NN

P N Z

P Z N

Z P N

Z Z Z

Z N P

N P Z

N Z P

N N PP

q	$\dot{q}$	τ
P	P	NN
P	N	Z
P	Z	N
Z	P	N
Z	Z	Z
Z	N	P
N	P	Z
N	Z	P
N	N	PP

3.3 Fuzzy inference engine

The inference engine is the fuzzy logic control core, and has the ability of simulating human-decision-making process by performing approximate reasoning in order to achieve a desired control strategy. In this research, the minimum inference engine is chosen. in this inference engine, individual-rule based inference with union combination, minimum for all the t-norm operators, maximum for all the s-norm operators and Mamdani’s minimum implication are used. Consequently, when the input of the minimum inference engine is the fuzzy set A′ in U, gives the fuzzy set B′ in V according to Equation (31).

$μ_{B^{'}} (y) = {max}_{l = 1}^{M} [sup_{x \in U} (min (\begin{matrix} μ_{A^{'}} (x), μ_{A_{1}^{l}} (x_{1}), \\ \dots, μ_{A_{n}^{l}} (x_{n}), μ_{B^{l}} (y) \end{matrix}))]$ (31)

where l is the rule number.

3.4 Defuzzifier

Inference engine output which should be used for the control signal is a fuzzy set. Therefore, a defuzzifier is needed for converting the fuzzy set to an actual value in order to be utilized as the control torque. In the other words, defuzzifier is a mapping from fuzzy set B′ in $V \subset ℝ$ (which is the output of the fuzzy inference engine) to a crisp value y^* ⊂ V. In this study, the centroid or center of gravity defuzzifier is applied due to its consistency, section invariance, monotonicity, linearity, and scale invariance [36]. Centroid defuzzification method calculates the y* as the center of the area covered by the membership function of B′ (Equation (32)). $y^{*} = \frac{\int_{V} y μ_{B^{'}} (y) dy}{\int_{V} μ_{B^{'}} (y) dy}$ (32)

3.5 Membership functions

The tracking error and tracking error rate are the inputs of the purposed fuzzy controllers. Three fuzzy sets named negative (N), zero (Z) and positive (P) are defined to represent inputs. Figure 1 shows the corresponding membership functions for the tracking error of the azimuth and the elevation axes. Also, tracking error rate membership functions of these two axes are defined in a similar way as Fig. 1.

Fig. 1

Membership functions of the tracking error.

Positive positive (PP), positive (P), zero (Z), negative (N) and negative negative (NN) sets are also defined to describe the applied torques. The membership functions which are considered for output torques of the azimuth and the elevation axes are shown in Figs. 2 and 3, respectively.

Fig. 2

Membership functions of the azimuth control torque.

Fig. 3

Membership functions of the elevation control torque.

4 Simulation results

In this section, results of the proposed algorithms for identification of the unknown parameters of the dynamic equations of the gimbal and also the results of the control algorithm for an example problem are presented.

4.1 Identification results

To demonstrate the accuracy of the proposed method in identification of the unknown parameters of the dynamic equation, i.e. the inertia components, the mass imbalance and friction, a set of examples are presented.

As outlined earlier, the identification process begins with measuring the system response to a known stimulation. In the present work, a sinusoidal stimulation is used for collecting the measurements. Instead of performing real experiments, the response of the system is obtained by solving the equations of the gimbal. However, in order to simulate the experimental conditions, random errors with Gaussian distribution are added to the system response to resemble the inherent measurement errors. In this way, the effect of the level of measurement error on the obtained results can be investigated effectively.

After providing essential experimental data for the identification process, these data are smoothed in order to reduce the ill-posedness of the identification problem. Following these steps, a set of initial guesses for the unknowns are selected and using the proposed algorithm, the unknowns are determined. It is worth mentioning that in a well-developed algorithm, the identified unknowns should not depend on the initial guesses. To investigate the effect of initial guesses on the identified unknowns, the identification results are obtained by considering three sets of initial guesses. The Initial guesses as well as the identified unknowns are reported in Table 2. The actual values of these parameters are extracted from a real two-axis gimbal [7] and [32]. The results of Table 2 are obtained in presence of 5% measurement error.

Table 2
Initial guesses and corresponding estimated values with 5% measurement error

A ₁ A ₂ A ₃ A ₄ A ₅ f(x) e₂

Actual value 0.0950 –0.0070 0.0250 0.0112 0.0050

Initial guess (a) 0.2500 –0.0100 0.0450 0.0300 0.0025

Estimated value 0.0963 –0.0157 0.0240 0.0112 0.0050 0.0023 0.1014

Initial guess (b) 0.0400 –0.1000 0.0100 0.0040 0.0075

Estimated value 0.0959 –0.0089 0.0242 0.0113 0.0050 0.0022 0.0016

Initial guess (c) 0.0100 0.0100 0.0600 0.0600 0.001

Estimated value 0.0968 –0.0217 0.0240 0.0112 0.0050 0.0020 0.0212

	A ₁	A ₂	A ₃	A ₄	A ₅	f(x)	e₂
Actual value	0.0950	–0.0070	0.0250	0.0112	0.0050
Initial guess (a)	0.2500	–0.0100	0.0450	0.0300	0.0025
Estimated value	0.0963	–0.0157	0.0240	0.0112	0.0050	0.0023	0.1014
Initial guess (b)	0.0400	–0.1000	0.0100	0.0040	0.0075
Estimated value	0.0959	–0.0089	0.0242	0.0113	0.0050	0.0022	0.0016
Initial guess (c)	0.0100	0.0100	0.0600	0.0600	0.001
Estimated value	0.0968	–0.0217	0.0240	0.0112	0.0050	0.0020	0.0212

According to the Table 2, although the initial guesses are selected in a wide range, all of them have resulted in small values of the cost function and therefore a good accuracy for the identified unknowns has been obtained. Figures 4 and 5 show the response of the system obtained with the estimated unknowns. The estimated responses are quite similar to the measured response which assesses that the identified values have enough accuracy to estimate the system’s behavior. As Figs. 4 and 5 clearly illustrate, all sets of estimated values can precisely model system’s response. Although in some cases the values for the initial guesses are several times more or less than the actual ones, the identification results are the same and accurate.

Fig. 4

Estimated and measured angular positions of the azimuth axis with 5% measurement error.

Fig. 5

Estimated and measured angular positions of the elevation axis with 5% measurement error.

Figures 6 and 7 present the estimated response in comparison to the measured one, when the level of measurement error is 10%. These Figures demonstrate the superb performance of the proposed identification algorithm even in presence of 10% measurement error. Both the azimuth and elevation angles are estimated with a high accuracy. Table 3 presents the estimated values of the unknown parameters of the dynamic equation of the gimbal. The main parameters of the identification algorithm are given in Table 4.

Fig. 6

Estimated and measured angular positions of the azimuth axis with 10% measurement error.

Fig. 7

Estimated and measured angular positions of the elevation axis with 10% measurement error.

Table 3

Initial guess and corresponding estimated value with 10% measurement error

Unknown parameter	Actual value	Estimated value
A ₁	0.0950	0.0962
A ₂	–0.0070	–0.0199
A ₃	0.0240	0.0238
A ₄	0.0112	0.0111
A5	0.0050	0.0050

Table 4

Main parameters of identification algorithm with 10% measurement error

Main parameters	Value
Time step (sec)	0.1
The azimuth axis smoothing coefficient (β)	3
The elevation axis smoothing coefficient (β)	5
The final answer’s distance with the previous guess	0.0222
The value of the objective function	0.0046

4.2 Control results

To validate the performance of the proposed method, the comparison between the control performances of the sliding mode control (SMC) and the proposed fuzzy logic control (FLC) is presented in this section. Some of the basic points considered in the sliding mode output torque design are briefly described as follows.

To stabilize the LOS of the two-axis gimbal by the SMC, a Lyapunov function candidate V = S²/2 is utilized, where S is the sliding surface and defined as:

$\begin{matrix} S = \dot{e} + λ e \\ e = q - q_{d}; \dot{e} = \dot{q} - {\dot{q}}_{d} \end{matrix}$ (33)

Both reference signals for angular position and angular velocity of the gimbal are determined a sinusoidal signal via Equation (34) $\begin{matrix} q_{d} = 0.1 sin (t) \\ {\dot{q}}_{d} = 0.1 cos (t) \end{matrix}$ (34)

Equation (35) and Equation (36) represent the final results of the SMC design for azimuth and elevation control torques, respectively.

$\begin{matrix} τ_{1} = \\ [A_{2} sin (2 q_{2}) {\dot{q}}_{1} {\dot{q}}_{2} + F_{s} sgn ({\dot{q}}_{1})] + (A_{1} + A_{2} {sin}^{2} (q_{2})) {\ddot{q}}_{1 d} \\ - λ_{1} (A_{1} + A_{2} {sin}^{2} (q_{2})) ({\dot{q}}_{1} - {\dot{q}}_{1 d}) \\ - K_{1} (A_{1} + A_{2} {sin}^{2} (q_{2})) Sat (S_{1} / - φ) \end{matrix}$ (35)

$\begin{matrix} τ_{2} = \\ [\frac{1}{2} A_{2} sin (2 q_{2}) {\dot{q}}_{1}^{2} + A_{4} sin (q_{2}) + F_{s} sgn ({\dot{q}}_{2}) + A_{3} {\ddot{q}}_{2 d}] \\ - A_{3} λ_{2} ({\dot{q}}_{2} - {\dot{q}}_{2 d}) - K_{2} A_{3} Sat (S_{2} / - φ) \end{matrix}$ (36)

where parameters A_i, i = 1,2,3,4,5 are estimated parameters presented in Table 3 and the control parameters are chosen as k₁ = k₂ = 5 and λ ₁ =λ ₂ = 5. It must be mentioned that in order to prevent chattering phenomenon, saturation function is used instead of sign function. The main difference between SMC and FLC methods is that the dynamic equations is used to offer the necessary information for making decision and fuzzy rule base construction in the fuzzy controller design process while the sliding mode control utilizes the dynamic equations directly in its control commands. Therefore, the comparison between their results reveals the effect of the proposed approach in accuracy of the control performance. To this end, system initial angular positions are set to q₁ = 0.09 rad and q₂ = -0.06 rad; and initial angular velocities are set to ${\dot{q}}_{1} = 0.02$ rad/sec and ${\dot{q}}_{2} = 0.15$ rad/sec.

Figures 8 and 9 illustrate the dynamic responses of FLC and SMC methods to the sinusoidal reference signal. The tracking performance of the system responses are reported in Table 5.

Fig. 8

Angular position of the azimuth axis.

Fig. 9

Angular position of the elevation axis.

Table 5

Control performance of FLC and SMC

Axis	Average tracking error (rad)		Settling time (sec)
	FLC	SMC	FLC	SMC
Azimuth	0.0099	0.0091	2.8	1.15
Elevation	0.0061	0.0064	2.4	1.15

According to Table 5, the settling time in the azimuth axis is about 2.8 sec via FLC and 1.15 sec via SMC.

In the elevation axis, settling time is about 2.4 sec via FLC and 1.15 sec via SMC. The obtained values for average tracking errors of the proposed approach and SMC are close to each other in both axes. The average tracking error in the azimuth axis is 0.0099 rad via FLC and 0.0091 rad via SMC. In the elevation axis, this quantity is 0.0064 rad and 0.0061 rad for FLC and SMC design, respectively. From these results, it can be concluded that although FLC has a slower convergence, it has a similar performance in comparison with the SMC in the steady state response.

The gimbal’s response to sinusoidal reference signal for the azimuth and the elevation angular velocities are demonstrated in Figs. 10 and 11 which confirm gimbal fine performance in tracking velocity command of 0.1cos(t). According to the transient part of the velocity responses shown in Figs. 10 and 11, the overshoot is not observed in the response of FLC method and its performance are smoother in comparison with SMC from the aspect of the transient response.

Fig. 10

Angular velocity of the azimuth axis.

Fig. 11

Angular velocity of the elevation axis.

According to Figs. 12 and 13, the SMC and FLC output torques indicate sharp changes at the beginning of the transient state. In the case of FLC design, the maximum required torques are 0.0673 N.m and 0.0533 N.m for azimuth and elevation axes, respectively. These values are 0.0846 N.m and 0.0283 N.m in the case of SMC design. Due to the considering determinate values for mass imbalance and friction disturbance torques along with fundamental dynamic parameters, it can be seen that the FLC and SMC results obtain the steady-state oscillation without sacrificing the disturbance rejection performance. As a consequence, the similarity between the responses of the FLC and SMC shows that the proposed fuzzy controller can achieve high stabilization performance.

Fig. 12

Obtained control torque for the azimuth axis.

Fig. 13

Obtained control torque for the elevation axis.

5 Conclusions

In this paper, a Gauss-Newton gradient-based method for identification of the most important uncertain parameters of the mathematical model of a two-axis gimbal was presented. For the first time, the sensitivity analysis of the gimbal’s coupled system of equations was performed by direct differentiation of the governing equations, instead of the conventional finite difference approach. A suitable smoothing technique was adopted to reduce the adverse effect of the measurement errors in the identification process. Afterwards, based on the dynamic equations in which all the unknown parts were identified, the fuzzy controller was designed for the system. The proposed controller performance was compared with sliding mode control performance which utilizes same estimated values for the uncertain parameters in its control commands.

The obtained results demonstrated the following major conclusions:

By taking the proposed identification method, the uncertainties and unknown parts of the two-axis gimbal dynamics were properly estimated without posing any online or complex computational burden on the controller design.

Increasing the measurement error from 5% to 10% did not affect the accuracy of the identification results. It was also found that adopting different initial guesses does not have a great impact on the precision of the estimated values.

In tracking sinusoidal reference signal, the FLC has a slower convergence and smoother behavior in transient state in comparison with the SMC response, while both of these methods have similar performances in steady-state response.

By using precise information of system dynamics in fuzzy controller design, the proposed fuzzy control approach attained significant efficiency in sinusoidal tracking performance.

In a further research, the precision of the dynamic model of the system can be investigated by considering more complex relations for the friction force and additional terms for the other disturbance sources. Also, all terms of the dynamic model of the system can be regarded as unknown and identified through the identification process. Additionally, it would be beneficial to consider actuator saturation in the design of the controller in further researches as it possibly results in degradation of the control performance.

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Determination of uncertain parameters of a two-axis gimbal and motion tracking via Fuzzy logic control approach

Abstract

Keywords

1 Introduction

1.1 Literature review

1.2 Research motives

2 Parameter estimation

3.1 Fuzzifier

Table 1 Fuzzy rule base q q ̇ τ P P NN P N Z P Z N Z P N Z Z Z Z N P N P Z N Z P N N PP

4.1 Identification results

References

Table 1
Fuzzy rule base

q $\dot{q}$ τ

P P NN

P N Z

P Z N

Z P N

Z Z Z

Z N P

N P Z

N Z P

N N PP