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Journal of the Korean Society for Precision Engineering  Vol. 38, No. 2, pp.115122  
Abbreviation: J. Korean Soc. Precis. Eng.  
ISSN: 12259071 (Print) 22878769 (Online)  
Print publication date 01 Feb 2021  
Received 08 Aug 2020 Revised 10 Nov 2020 Accepted 14 Nov 2020  
DOI: https://doi.org/10.7736/JKSPE.020.077  
A Study on Gimbal Motion Control System Design based on SuperTwisting Control Method  
Thinh Huynh^{1} ; YoungBok Kim^{2}^{, #}
 
1Department of Smart Robot Convergence & Application Engineering, Graduate School, Pukyong National University  
2Department of Mechanical System Engineering, Pukyong National University  
SuperTwisting 제어기법 기반 짐벌운동제어시스템 설계에 관한 연구  
틴 휭^{1} ; 김영복^{2}^{, #}
 
1부경대학교 대학원 스마트로봇융합응용공학과  
2부경대학교 기계시스템공학과  
Correspondence to : ^{#}Email: kpjiwoo@pknu.ac.kr, TEL: +82516296197  
Copyright © The Korean Society for Precision Engineering This is an OpenAccess article distributed under the terms of the Creative Commons Attribution NonCommercial License (http://creativecommons.org/licenses/bync/3.0) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.  
Controlling an optical sensor’s line of sight (LOS) with an inertial stabilization system carried out on a dynamic platform is a challenging engineering task. The LOS needs to track a target object accurately despite intentional maneuvers, inadvertent motions, and additional disturbances. In this study, a supertwisting sliding mode controller (STSMC) is implemented to overcome this problem. The controller is designed based on the analysis of system dynamics. The stability is then proved to be satisfactory by the Lyapunov theory. Then, the control law is validated through experimental studies. In addition, a comparison to the performance of a linear controller is derived so that the effectiveness of the proposed controller is validated.
Keywords: Gimbal, Line of sight, Inertial stabilization, Supertwisting algorithm, Sliding mode control 키워드: 짐벌, LOS, 관성안정화, 슈퍼트위스팅 알고리즘, 슬라이딩모드제어 
The main goal of an inertial stabilization is to hold or control the LOS of one object relative to another object or inertial space (Hilkert^{1}). It is used in a large range of applications, such a s surveillance, target tracking, communications, handheld cameras, etc. The optical sensor is carried by a gimbaled structure with at least two orthogonal axes. In direct stabilization system, the sensor is considered as the payload mounted in the inner gimbal channel where its absolute motion is sensed by a gyroscope and the gimbal motions are actuated by two independent servo systems. When disturbances affect the LOS are attenuated, the system can achieve accurate tracking performance with a welldesigned controller. In the case of the gimbal carried on a mobile vehicle, disturbances arise from both intentional maneuvers and inadvertent motions of the vehicle. In addition, external loads, such as wind, friction, and mass unbalance, add unpredicted perturbations into the system.
Modern control techniques based on sliding mode control are well known as robust and effective methods to be applied in these situations. However, chattering is the main drawback of the technique. In applications of inertial stabilization, many approaches based on sliding mode control have been proposed. For example, in the works of Kürkçü^{2} and Li,^{3} the use of an integral sliding mode controller combined with a disturbance/uncertainty estimator^{2} or a state observer^{3} were suggested. In Mao^{4} and Suoliang,^{5} controllers were designed based on terminal sliding mode algorithm. A combination of backstepping control and sliding mode control was proposed in Dong^{6} and Ding.^{7}
Besides that, supertwisting algorithm not only provides finitetime convergence but also reduces chattering effects (Chalanga^{8}). Tran^{9} showed that supertwisting algorithm could be applied for interconnected systems and performed well. Chanlanga^{8} studied an output feedback stabilization of perturbed doubleintegrator systems using supertwisting control. Two methodologies were proposed, one was supertwisting controller based on supertwisting observer, and the other was highorder sliding mode observer based supertwisting controller. While the first approach was proved mathematically that it was impossible to achieve second order sliding on the control sliding variable, the second approach could achieve continuous control with supertwisting algorithm. Applying these results, Reis, et al., designed a sliding mode control strategy for both stabilization and target tracking for 3DOF inertial stabilization (Reis^{10}). Both state and output feedback cases were considered. The full state feedback values were represented in form of quaternions, while a highorder sliding mode observer was proposed to estimate velocities from the outputs. In each case, two supertwisting controllers were employed in cascade topology, providing robust and effective performance evaluated by simulation results. On the other hand, the main difficulty with supertwisting technique is to establish the proof of stability for perturbed systems. Studies of Levant,^{11} Moreno^{12,13} and other authors have suggested several rules for tuning controller gains such that system becomes stable. Especially, a recent study from Seeber^{14} provided a strict Lyapunov function which extends the provably stable parameter range, even for the system with unmatched disturbances.
However, abovementioned studies related to supertwisting sliding mode control are either theoretical or simulation ones. In order to tackle effectively the problem of tracking in twoaxis gimbal system, a practical approach for design and implement of the supertwisting sliding mode controller is needed. In this work, the problem is solved and presented as follows. Section 2 derives system’s mathematical model from the analysis of its kinematics and dynamics. Section 3 presents the procedure of supertwisting controller design from the system’s model. Section 4 provides stability proof encouraged from the work of Seeber and Horn. The system with proper choice of parameters is proved to be finitetime stable in the sense of Lyapunov stability theory despite of the present of disturbances. Section 5 suggests some modifications improving the robustness and effectiveness of the implemented controller. Experimental studies validating the proposed controller are discussed in Section 6. Experimental results are analyzed and compared in this section. Finally, conclusions will be drawn.
Consider a twoaxis gimbal mounted on a dynamics platform illustrated in Fig. 1. The gimbal consists of an outer channel actuating pan motion and an inner channel operating tilt motion. The first step of controller design process for this system is to build up the mathematical model which characterizes the kinematics and dynamics of it. By using the Euler matrix for rotations, the angular rates of the pan and tilt channels can be respectively written as Eq. (1).
(1) 
Where S and C denote for sine and cosine respectively, ω_{b} = [ω_{bx} ω_{by} ω_{bz}]^{T} is the angular rate of the platform,
(2) 
Where T is the applied torque, J is the matrix of inertia, and ω is the angular velocity.
To simplify the analysis, assume that the inner gimbal rotation axes are aligned with the principal axes of inertia so that its inertia matrix is diagonal J_{t} = diag{J_{tx}, J_{ty}, J_{tz}} and assume J_{tx} = J_{tz}. Applying Eqs. (1) and (2), the inner gimbal dynamics about Yaxis and the outer gimbal dynamics about Zaxis are expressed as Eqs. (3) and (4).
(3) 
(4) 
Where ty and pz are index terms denoting elements corresponding to tilt motion about Yaxis and pan motion about Zaxis respectively. φ stands for angular position of LOS, J is inertia matrix element, K and Q are viscous friction coefficient and cable restrain coefficient respectively. T is control torque and T_{d} indicates additional disturbance torques. Disturbances in each channel consist of mass unbalance torque T_{U}, nonlinear parts of friction T_{K} and cable restrain T_{Q} torques, platform maneuver torques transmitted to the channel T_{b}, and mutual interference between two channels (Kennedy,^{15} Ekstrand^{16} and Mokbel^{17}).
(5) 
For the sake of simplification, let us consider the case that the relative angle between the inner and outer gimbal does not change when the outer gimbal is moving. Hence, Eq. (4) becomes Eq. (6).
(6) 
Typical control configurations for inertial stabilization use two control loops: inner stabilization loop and outer tracking loop (Masten^{18}). With the use of a simple P controller for stabilization, the control torques applied to actuators are generated as Eq. (7).
(7) 
Where P are the gains of inner loop controller,
(8) 
Therefore, the procedure of tracking controllers design considers Eq. (8) as the plant, which contains a closedloop system consisted of the gimbal plant and stabilization controller.
The idea behind sliding mode control algorithm is to design a feedback control law such that control error remains on well behaved sliding manifold despite the presence of the model imprecision and of disturbances. In order to design the tracking controller for the gimbal, let us consider the general description of each channel rewritten as follow Eq. (9).
(9) 
Where i = 1, 2 are the index of tilt channel and pan channel respectively; a, b, and c are nominal parameters of the system model corresponding to system dynamics, u is the control signal of proposed tracking controller and T_{d} is the total disturbance.
Firstly, a sliding manifold for each channel is assigned as follows Eq. (10).
(10) 
Where e_{i} = φ_{di}  φ_{i} is the tracking error  the difference between desired position φ_{di} and actual position φ_{i}. With m_{i} > 0, s_{i} is Hurwitz.
Take the time derivative of s_{i} and substitute from Eq. (9).
(11) 
Then, a feedback control law is selected based on SuperTwisting algorithm in order to satisfy sliding condition,
(12) 
where
(13) 
with λ_{1i} and λ_{2i} are positive constants so that the system is stable and well performance.
Substituting the control law into equation of sliding manifold surface yields that Eq. (14).
(14) 
T_{di} = T_{mi} + T_{ui} denotes that in general, unknown disturbances consist of matched disturbances T_{mi} and unmatched disturbances T_{ui}.
Assume that the unmatched disturbances can be expressed as ηs_{i}^{1/2}, and disturbances are bounded by nonnegative constants η ≤ N and
(15) 
where α_{i} is a positive parameter. This function is continuous, positive definite, piecewise differentiable and locally Lipschitz continuous everywhere except in the origin (Seeber^{14}).
In detail, in the first case, in the subdomain of semipositive values of s_{i}, the function
In the first case,
(16) 
(17) 
The restrictions s_{i} ≥ 0 and
(18) 
It can be seen that with λ_{1}_{i} ≥ N, d^{2}g/dx_{i}^{2} ≥ 0, then g(x_{i}) is convex.
Therefore:
(19) 
where
(20) 
Thus, by homogeneity one concludes that:
(21) 
The system stability is preserved if
(22) 
and
(23) 
If s_{i} < 0, the same conclusion is obtained by a similar procedure.
Furthermore, in the case of
(24) 
(25) 
From Eqs. (23) and (25), the system is finite time stable if the conditions λ_{2}_{i} > L and
As a result, with the proper choice of gain values, the stability of closed  loop system is preserved in the sense of Lyapunov stability theory despite of the present of disturbances.
Chattering is well known as one of the main disadvantages of sliding mode controller. It has been shown that this effect is mainly caused by unmodelled cascade dynamics which increase the system’s relative degree and perturb the ideal sliding mode existing in the system (Fridman^{19}).
By including an integrator, the SuperTwisting algorithm is able to attenuate chattering in relative degree one system. In order to reduce the influence of higher relative degree, the proposed control law is calculated from a continuous signumlike function instead of the sign function.
(26) 
Where sgn(s_{i}, δ_{i}) is defined as
Experimental studies are performed on a twoaxis gimbal used in ocean surveillance applications. Nominal parameters of the system defined in Eq. (9) are experimentally obtained using system identification method. Their values are presented as follows.
System configuration is shown in Fig. 3. The payload is mounted at the center of tilt gimbal. Its absolute angular position and rate are measured by an Attitude Heading Reference System (AHRS) attached directly.
Two channels of the gimbal are actuated independently by two servo system through driving belts. There is already a P controller for speed control in each servo as mentioned in Section 3.
All actuators and sensor communicate through RS 232 interface. The controller is implemented in Matlab/Simulink. Sampling time is chosen as 0.05s. The hardware system specifications are listed in Table 1, and the controller gains are represented in Table 2.
Parameter  Value  

Actuators: CoolMuscle CM1C23S30 
Integrated controller  PPI 
Rated power [W]  45  
Max. speed [rpm]  3000  
Rated continuous torque [Nm] 
0.294  
Supply voltage  VDC24±10%  
Supply current [A]  3.9  
Gear ratio  5 : 1  
Sensor: NTRexLAB M WAHRSv1 
Measurement algorithm: Fusion of acceleration and gyro sensor with Kalman filter 

Angle’s resolution [°]  0.001  
Static error [°]  0.1  
Dynamic error [°]  2  
Response time [ms]  1 
Gimbal channel  Tilt  Pan 

STSMC  m_{1} = 2 λ_{11} = 120 λ_{21} = 100 δ = 5 
m_{1} = 1.5 λ_{11} = 200 λ_{21} = 100 δ = 5 
PID  P_{1} = 2.613 I_{1} = 0.068 D_{1} = 0.236 N_{1} = 10 
P_{2} = 2.785 I_{2} = 0.073 D_{2} = 0.170 N_{2} = 10 
At first, two channels are experimented to track target trajectories in forms of step signal value. The response of tilt gimbal and its control signal are shown in Figs. 4(a) and 4(b). Experimental results collected from pan motion are shown in Figs. 4(c) and 4(d). Note that the control signals of the tracking controller are the reference values for the inner stabilizer controllers, so their units are deg/s. One can see that the proposed method achieves desired tracking position in a short time with almost no overshoot. The zoomed figures show in detail the transient responses and steady states of each channel. Evidently, the system is robust and well performed.
(27) 
Then, a comparison of system performance controlled by STSMC and PID controller is presented to evaluate the efficiency of the proposed control law more clearly. The PID controller has the general form as follows Eq. (27).
where tuned controller gains are also represented in Table 2.
Both channels of the gimbal are controlled at the same time so that the LOS tracked a target object following a counterclockwisecircular trajectory in a vertical plane. Tracking performances are shown in Fig. 5. Furthermore, the rootmeansquare errors (RMS) of the experiment data are calculated and shown in Table 3.
RMS  Tilt  Pan  Radial 

STSMC  0.0017  0.0071  0.0043 
PID  0.0298  0.0295  0.0025 
In Fig. 5(a), the circular trajectory in the vertical plane and corresponding system outputs are presented. At the first glance on the results, the PID control seems to achieve better tracking performance. However, it is worth to notice that the results are shown geometrically only in Fig. 5(a), which may lead to the difficulty in evaluating the system responses with respect to time. The zoomed figures, where the reference point and corresponding system outputs at a specific time are highlighted, show that the tracking error by STSMC is smaller than the error generated by PID controller. It is easy to see that the system response controlled by STSMC is very closed to the reference, while the one by the PID controller is always far behind. The tracking errors in Fig. 5(b) also prove that the STSMC archives significantly better performance.
In detail, Fig. 5(c) shows the angle response of each channel controlled by PID controller is always behind and the one of STSMC is almost match to the reference. In addition, Fig. 5(d) shows the control signals from STSMC and PID controller. The element u_{iST} in the STSMC’s control signal compensates unmodelled dynamics and disturbances perturbed the system, which causes the bounce as seen in this Fig. 5.
For the RMS tracking errors shown in Table 3, the proposed STSMC provides much smaller tracking error in both channels. However, the radial residual of PID controller is accidently smaller, due to periodic characteristic of circular trajectory.
In this paper, a supertwisting sliding mode controller has been designed for the gimbal motion control. A system model has been obtained from the analysis of system kinematics and dynamics, and consideration of the presence of stabilization controller. By experimental studies, the effectiveness of the proposed controller has been shown and validated. Comparing with tradition PID controller, the designed supertwisting controller has achieved much better performance. In detail, for the tilt and pan channels, the RMS error of the proposed control method is respectively equal to 5.7 and 24% of those from PID control. The system performance robustness with respect to limited disturbances is proved by Lyapunov theory. Consider complex kinematics coupling and outer disturbance influencing the system, future studies are necessary to enhance system performance and robustness.
ω :  Angular Velocity 
φ :  Angular Position 
T :  Torque 
K :  Viscous Friction Coefficient 
Q :  Cable Restrain Coefficient 
J :  Inertia Moment 
u :  Control Signal 
s_{i} :  Sliding Manifold 
V :  Lyapunov Function 
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Graduate School Student in the Department of Smart Robot Convergence and Application Engineering, the Graduate School, Pukyong National University. His research interests include control engineering, robotics and automotive engineering.
Email: huynhthinh@hcmute.edu.vn
Professor in the Department of Mechanical System Engineering, Pukyong National University. His research interests include control theory and application with dynamic ship positioning and autonomous control system design, etc.
Email: kpjiwoo@pknu.ac.kr
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