Dynamic modeling,control and experimental validation of gimballed sensor system for precision pointing applications

Abstract

Gimballed sensor system is a precision electromechanical assembly designed primarily to isolate the optical system from disturbances induced by the operating environment. This paper gives an insight to the design and development of gimballed sensor system for Line of Sight (LOS) stabilization of an electro-optical tracking and pointing system. Initially kinematic equations are formulated to establish a relationship between LOS angle and applied torque. This relationship is used to obtain nested loop transfer function model. First, the parameters of the proposed assembly are determined through experimentation & rigorous analysis process, and then conventional control design methodology is adopted for controller configuration design for current and rate loop. The frequency response analysis of the designed LOS stabilization model with conventional controller is done in simulation and the obtained results are verified experimentally against angular disturbances in real time scenario. Further, Based on prior qualitative information about system dynamics and linguistic performance criteria, a fuzzy logic controller of mamdani type with simplified rule set is developed with an objective to improve the disturbance attenuation and command response performance of designed system irrespective of angular disturbances due to platform vibrations, model uncertainties and mass imbalance in gimbal assembly. Both the Fuzzy logic simulation model and conventional model are tested based on critical performance characteristics such as stability of the loop, responsiveness of the loop and insensitivity to disturbances. Finally, the comparative analysis suggests that, although both the control configuration satisfies the required accuracy, Fuzzy logic control certainly improvised the performance of the gimballed sensor system and hence can be very effective for more precise pointing, tracking and stabilization application.

Keywords

Proportional integral (PI) compensation fuzzy logic control jitter attenuation gimballed electro-optical system tracking and pointing

1 Introduction

Optical equipments such as CCD or IR camera, high resolutions televisions, laser range finders, radars have found numerous applications in the field of surveillance, target tracking system, navigation system, communication system and modern guided weaponry system. The main functional requirement of every such system is most accurate pointing of optical sensor’s line of sight to a fixed or moving target in-spite of various internal, external disturbances and plant modeling related critical issues and assumptions [1]. Inertially Stabilization platforms (ISP) are the best way available so far to stabilize and point the sensor in desired direction, when either the host vehicle or the enemy target or both are stationary or dynamic [2]. ISP electro-mechanical configurations design depends on the concerned application. They are usually made up of an assembly of structure, gimbal with associated motor and bearings and servo sensors. Electro-optical (EO) systems mounted on mobile platform always requires some form of control configuration to stabilize the sensor pointing vector along the target Line of Sight (LOS) [3]. Acquisition, tracking and pointing (ATP) are the basic functions inherent in all systems that control the line-of sight or beam to the target [4].

Over the last few years, gimbal assembly structure has been widely used in various dynamic platforms especially unmanned aerial vehicle and automatic weaponry systems. But it is always a challenging task to obtain precise response characteristics against dynamic platform structure, irregularities in mathematical modeling and angular disturbance in LOS due to internal and external disturbances. A combination of two or more gimbals can be used for fine degree of precision and independency from host vehicle, depending upon application [5]. The ultimate objective of every ISP assembly is to point, hold and control the LOS of optical sensor onto specified target location. The performance of such systems depends on its ability to isolate the gimbaled sensor system from platform vibrations and other operating environment disturbances.

Masten and Hilkert [2, 3] have described the following fundamental requirements of stabilization systems;

Pointing the optical sensor

Stabilizing the sensor to achieve low “jitter”

Measuring the orientation of the sensor’s line-of-sight.

In [6], A. K. Rue derived mathematical techniques for describing the interrelationships of pointing vectors (direction) and angular rates in a variety of technical elements (reflective mirrors and refractive lenses), which are main parameters of interest for modelling stabilization system. Achieving high accuracy and performance in stabilization and tracking loops depends heavily on obtaining sufficient stability and bandwidth [3]. S. B. Kim et al. [7] discussed the feedback method to reduce the effect of electrical as well as mechanical resonances. These resonances limit the maximum achievable bandwidth in practical control system and therefore hamper the ability to achieve precise stabilization [8]. Vishay Netzer [9] discusses techniques for improving image stabilization in which subsystems located within the overall system are used rather than relying entirely upon a stabilized gimbal platform. In this approach, the gimballed platform is used to point the sensor at the target and to achieve relatively coarse stabilization. Similar gimballed platform based systems are modelled and simulated with precise control technique for tracking and pointing applications in [10, 11]. The application of techniques to stabilization systems are described in [12]. Further in recent years, many work has also been published in the field of mirror stabilized system or indirect LOS stabilization approach [13 –15].

All such systems are conventionally designed using classical control technique. However they are not effective to design control laws for complex and nonlinear multivariable systems. A retrospect of ATP system shows fuzzy logic [16, 17], Neural network [18], hybrid control algorithm [19], Adaptive control mechanism [5, 20], evolutionary computation [21], disturbance observer based control [22], and other related control algorithms in [23 –25] which remarkably improve pointing accuracy and disturbance rejection ability using gimballed sensor system. Their ability to deal with highly nonlinear, MIMO, and complex dynamical system help in surpassing every conventional approach. These methods do not rely too much on accurate modelling of the plant. However, knowledge base of designer is an important factor to FLC design. FLC provide a flexible alternative by converting human expertise into appropriate fuzzy rules and membership functions.

In this work, dynamic equations of motion of a dual axis gimbaled platform assembly are formulated. These gimbals are precision electromechanical assemblies designed primarily to isolate the optical system from the disturbances induced by platform vibrations. Gimbal assembly is interfaced with a control system operating through electronics. Nested loop topology is adopted to design basic control loop configuration. In this approach, a very first step is to design a fast current loop, which is being closed by a servo power amplifier itself, followed by the design of fast stabilization loop or rate loop with bore-sight gyros as a rate feedback sensor. The stabilization loop is then nested with in a track or outermost loop which generally employed a video tracker for fine tracking. The main objective of stabilization loop is to attenuate jitter and stabilize the LOS against angular disturbances due to platform vibrations, model uncertainties and mass imbalance in gimbal assembly. The conventional control design methodology based on frequency domain analysis is adopted for controller configuration design and the obtained simulated results are verified based on real time experimentation. Next, a Fuzzy logic control configuration is developed for the designed application with an objective to reduce the residual jitter to the minimum level and to obtain precision in command following. Finally, a comprehensive comparative analysis of the two proposed methodologies is presented.

2 Dynamic modeling of a dual axis gimballed sensor system

Modeling of Plant dynamics is an important milestone in the control loop design of stabilized gimbal platform assembly [2]. In the following section, a simple formulation based on the first principle of modeling the azimuth torsional mode is presented. Figure 1 shows the two axis gimbal flexible model. Appropriate transfer functions representing these modes are derived based on the some simplifying assumptions. The degrees of freedom are restricted to three in orders to keep the formulation tractable and insightful. The model consists of two lumped inertias connected by two linear springs.

Fig. 1

Two-Axis Gimbal Flexible Model.

Where, T ₁

Azimuth motor torque

I ₁

Moment of inertia of azimuth base structure

I ₂

Moment of inertia of elevation Gimbal assembly + lumped inertia of azimuth arms and elevation motor about azimuth axis

M ₂

Mass of elevation Gimbal assembly + lumped mass of azimuth arms and motor

K _a K _b

Equivalent spring stiffnesses at the left and right azimuth arms respectively

δ _{z
_a} δ _{z
_b}

Corresponding bearing point deflections (assuming +ve in the positive Z₁ direction)

δ _{z
_cm}

Deflection of centre of mass of elevation Gimbal assembly w.r.t O along Z₁.

θ ₁

Azimuth servo angle

δ _θ
₁

Incremental angular deflection of elevation Gimbal assembly w.r.t. azimuth base structure about azimuth (X₁) axis

r _a r _b

Distance of left and right bearing points r spectively from the centre

In the modeling of the azimuth structural modes, the following assumptions are made:

Fixed elevation angle.

Cross moments of inertia are taken as zero.

r_a = r_b = r i.e., the bearings are placed symmetrically with respect to O.

δθ₁ ⪡ θ₁

The following geometric relationship can be derived from Fig. 2 as; $δ z_{cm} = \frac{δ z_{a} + δ z_{b}}{2}$ (1) $δ z_{a} = δ z_{cm} + r . δ θ_{1}$ (2) $δ θ_{1} = \frac{δ z_{a} - δ z_{b}}{2 . r}$ (3) $δ z_{b} = δ z_{cm} + r . δ θ_{1}$ (4)

Hence, there are effectively three degrees of freedom in the model, viz θ₁, δθ and δ_{z
_cm}. The rotational equation of motion of the azimuth base structure about the X₁ axis is: $T_{1} + (k_{a} . δ z_{a} - k_{b} . δ z_{b}) . r = I_{1} . {\ddot{θ}}_{1}$ (5)

The rotational equation of motion of the elevation Gimbal assembly is: $- (k_{a} . δ z_{a} - k_{b} . δ z_{b}) . r = I_{2} . ({\ddot{θ}}_{1} + δ {\ddot{θ}}_{1})$ (6)

The equation of translational motion for mass M₂ (elevation Gimbal assembly) along the Z₁ direction is: $- k_{a} δ z_{a} - k_{b} δ z_{b} = M_{2} . (δ {\ddot{z}}_{cm} - {\dot{θ}}_{1}^{2} . δ z_{cm})$ (7) where, the last nonlinear term $(- {\dot{θ}}_{1}^{2} . δ z_{cm})$ represents the centripetal acceleration.

If the elevation Gimbal assembly is allowed to swing along the Y₁ axis then additional nonlinear terms for coriolis and tangential acceleration will appear in the equation.

From Equations (1) to (4), we eliminate δ_{z
_a} and δ_{z
_b} to get the following relations: $T_{1} + (k_{a} - k_{b}) . r . δ z_{cm} + (k_{a} + k_{b}) . r^{2} . δ θ_{1} = I_{1} . {\ddot{θ}}_{1}$ (8) $- (k_{a} - k_{b}) . r . δ z_{cm} - (k_{a} + k_{b}) . r^{2} . δ θ_{1} = I_{2} . ({\ddot{θ}}_{1} + δ {\ddot{θ}}_{1})$ (9) $- (k_{a} + k_{b}) . δ z_{cm} - (k_{a} - k_{b}) . r . δ θ_{1} = M_{2} . (δ {\ddot{z}}_{cm} + {\dot{θ}}_{1}^{2} . δ z_{cm})$ (10)

If K_a = K_b = K and if we drop the nonlinear term, then the above equations reduce to: $T_{1} + 2 . k . r^{2} . δ θ_{1} = I_{1} . {\ddot{θ}}_{1}$ (11) $- 2 . k . r^{2} . δ θ_{1} = I_{2} . ({\ddot{θ}}_{1} + δ {\ddot{θ}}_{1})$ (12) $- 2 . k . δ z_{cm} = M_{2} . δ {\ddot{z}}_{cm}$ (13)

In the case when the translational Equation (13) is decoupled from the rotational Equation (11) & (12); Equation (13) will represent harmonic oscillations with frequency $\sqrt{\frac{2 K}{M_{2}}}$ rad/s and correspond to the mode where both the azimuth arms bend in phase.

Equations (11) & (12) correspond to the torsional azimuth modes. The transfer functions from input to output are obtained as: $\frac{θ_{1} (s)}{T_{1} (s)} = (\frac{1}{I_{1} . s^{2}}) . (\frac{s^{2} + (2 . k . r^{2} / I_{2})}{s^{2} + 2 . k . r^{2} . (1 / I_{1} + 1 / I_{2})})$ (14)

$\begin{matrix} \frac{(θ_{1} + δ θ_{1}) (s)}{T_{1} (s)} = (\frac{1}{(I_{1} + I_{2}) . s^{2}}) . \\ (\frac{2 . k . r^{2} . (1 / I_{1} + 1 / I_{2})}{s^{2} + 2 . k . r^{2} . (1 / I_{1} + 1 / I_{2})}) \end{matrix}$ (15)

Consider, I_azimuth = I₁ + I₂; C_torosional = 2Kr² and $\bar{I} = (I_{1} . I_{2}) / (I_{1} + I_{2})$ ; $ω_{s 1} = \sqrt{c_{torosional} / \bar{I}}$

Then Equation (15) can be re-written as: $\frac{(θ_{1} + δ θ_{1}) (s)}{T_{1} (s)} = (\frac{1}{I_{azimuth} . s^{2}}) . (\frac{ω_{s 1}^{2}}{s^{2} + ω_{s 1}^{2}})$ (16)

Further it must also be noted that there are the peaks of finite magnitude at tuned frequencies and phase is not changing abruptly. This effect can be taken care by implementing suitable notch filter.

Considering, B as the total friction in the assembly. T₁(s) can be replaced by T₁ (s) - B (s) in the Equation (8) and (10).

The part of the transfer function representing the rigid body mode in (16) becomes; $(\frac{1}{s (I_{azimuth} S + B)})$ (17)

And after taking the effect of torsional structural mode in flexible model, the transfer function takes the form as;

$\begin{matrix} \frac{(θ_{1} + {δ θ}_{1}) (s)}{T_{1} (s)} = (\frac{1}{s (I_{azimuth} . S + B)}) . \\ (\frac{s^{2} + 2 ξ_{1} ω_{s 1} + ω_{s 2}^{2}}{s^{2} + 2 ξ_{2} ω_{s 2} + ω_{s 2}^{2}}) \end{matrix}$ (18)

The relationship of Equation 18 can be represented in block diagram form as follows:

Fig. 2

Block Diagram Representation of Generalized Form of Gimbal Mechanical Assembly.

3 Design consideration

Gimbaled sensor system is driven by permanent magnet DC torque drive. Servo power amplifier is used to amplify controller output before given to DC torque. Further, DTG is used to measure inertial angular rate of gimbal. The numerical values of plants mechanical parameters are obtained from Finite Element Model method using ANSYS Software. Moment of Inertia (I_azimuth) and Viscous friction coefficient (B_viscous) are 390 kg-m² and 1225 N-m-sec/rad respectively. The torsional structural modes are assumed to have natural frequencies of aro und 30 and 85 Hz. The first torsional mode at 30 Hz is assumed to have very small effect on the azimuth pointing so that mode is neglected here during the modeling in flexible mode. Structural damping of 7% is included. The ETEL TML0360-150 motor [26] is modeled as a simple DC machine, with torque constant = 28.9 N-m/Amp, L = 14.5 mH, R = 1.6 Ohm, back emf constant k_b = 16.7 V-sec/rad. For a brushless motor model, the above constants get scaled by appropriate constant factors representing conversion from line-line quantities to line-phase quantities. This does not change the time constant of the motor, it only requires scaling of the gain in the power amplifier (drive electronics) by appropriate factor. This will not affect the performance of the control system.

Further, the peak torque and peak current of the motor are 2120 N-m and 115 Amp respectively. Assuming that 10 V DAC will be used for converting the digital control signal into analog signal which will be fed to Servo Power Amplifier, the scale factor for the current feedback can be selected as 10 V = 115 Amp. Desirable control system specifications are as follows:

Closed loop bandwidth: >15 Hz

Gain margin: >6 dB

Phase margin: >30 deg

Damping coefficient: 0.4–0.7

Line of sight (LOS) jitter accuracy:≤50μrad.

Settling time:≤0.2 sec.

Steady state error: Zero or Minimal

Disturbance Spectrum: 2 mrad at 1 Hz and less at higher frequencies (20 dB/decade)

A complete nested loop topology based LOS stabilization and pointing architecture is shown in Fig. 3.

Fig. 3

LOS Stabilization and Pointing Architecture.

4 Conventional controller design and implementation

Considering all the developed dynamics of DC motor [27], Dual axis gimbal assembly and DTG rate sensor [28], an uncompensated transfer function based simplified first step model can be directly designed as shown in Fig. 4.

Fig. 4

Transfer function model of uncompensated Azimuth plant with actuator.

An effective control system design is required to be implemented around Electro mechanical gimbal assembly to accurately point the target, attenuate jitter on the LOS and meet all the required control system specification.

Initially conventional controller configuration has been developed for LOS stabilization as that would not only be the most precise way to tune the controller structure as per desired time and frequency domain characteristics but also provides easier interfacing with real time hardware structure and in depth analysis of performance of each block in simulation.

Complete azimuth gimbal assembly simulink model design consists of two stages:

Current loop Control system Design

Stabilization/Rate loop Control system Design

1) Current loop Control system Design –Consider the designed uncompensated simulink model in Fig. 4, draw the open loop bode plot with reference to actuator input voltage as input and equivalent voltage corresponding to the armature current as output.

The required current loop bandwidth is more than 1 KHz, so the gain cross over frequency of the compensated open loop system must be 1 KHz. It can be observed from Fig. 5 that open loop gain at 1 KHz is –60 dB. This indicates that more than 60 dB gain is required to achieve adequate current loop bandwidth. But closing the loop directly with this gain may introduce some steady state error; so a PI controller or a PI compensator is the best choice to reduce steady state error. However using a PI controller does not satisfy the phase margin specification because of the introduction of phase delay from integral part of PI controller, hence Proportional –Integral compensation is used as a current loop controller.

Fig. 5

Uncompensated open loop Bode plots, (Gain at 1 KHz = –60 dB).

Assume a PI compensator of the form $(K [\frac{1 + S / a}{S}])$ . As the required gain cross over frequency is 1 KHz which is quite high, a zero is needed to be placed very close to origin such that gain roll off (–20 db/decade) does not continue for higher frequencies. To begin with, a zero is placed at 0.25 rad/s. with this value of ‘a’, $[\frac{1 + S / a}{S}]$ part of PI compensator provides a gain of appox. 20*log10 (1/a) = 20*log10 (1/.25) = 12 dB at 1 KHz. Hence for gain cross-over frequency at 1 KHz, K should be selected in such a way that it gives 60–12 = 48 dB gain or equivalent to 250 at 1 KHz [29].

Hence final PI compensator transfer function is $PI : 260 * (4 s + 1) / s$

Figure 6 shows the compensated open loop and closed loop bode plot of the simulated azimuth current loop.

Fig. 6

a) Compensated open loop Bode plots, (Gain Margin = 14.2 dB, Phase Margin = 79.8 deg., Gain cut-off frequency = 999 Hz) b) Azimuth closed current loop bode plot, (Bandwidth = 1.27 KHz).

It is observed from the Fig. 6 that both the gain margin and phase margin are positive and hence the designed PI controller stabilizes the current loop.

2) Stabilization/Rate loop Control system Design

For analysis and design of stabilization loop using bode plot, voltage to the actuator (V) taken as input, while the output is the angular rate. Figure 7 shows the frequency response plot of the uncompensated open loop azimuth plant along with actuator model and inner current loop.

Fig. 7

Uncompensated open loop Bode plots, (Gain at 1 Hz = –3.3 dB & Gain at 15 Hz = –26 dB).

From Fig. 7, it can be observed that the open loop gain at 1 Hz and 15 Hz is –3.3 dB and –26 dB respectively. This indicates that more than 26 dB gain, which translates to around 20, will be required to achieve adequate stabilization loop bandwidth. More importantly, the gain to overcome the worst case disturbances should be roughly equal to the ratio of the disturbance to the allowable residual jitter on the line-of-sight. The disturbance specification is 2 mrad at 1 Hz, while the allowable jitter is 50 μrad. So, 2 mrad / 50 micro-rad = 40, which is equivalent to 32 dB. Since the open loop uncompensated gain is –3.3 dB at 1 Hz, the stabilization loop must provide atleast 32 + 3.3 = 35.3 dB of gain to achieve the stabilization accuracy.

As the required gain cutoff frequency is at 15 Hz, a zero of the PI compensator can be placed at 1/5^th of the gain cutoff frequency i.e. at 3 Hz, then the value of ‘a’ becomes 2*pi*3 = 18.85. Now with this value of a, $[\frac{1 + S / a}{S}]$ part of PI compensator provides a gain of –15.5 dB at 1 Hz & –25.3 dB at 15 Hz. Hence for 32 dB open loop gain at 1 Hz, K should be set in such a way that it gives 35.3 + 15.5 = 50.8 ≈ 51 dB of gain at 1 Hz & 26 + 25.3 = 51.3 dB of gain at 15 Hz. Now if we take 51.3 dB of gain, it is equivalent to 360. Hence, K = 360. $PI : 360 (0 . 05305 s + 1) / s$

Figure 8 shows the PI compensated open loop bode plot.

Fig. 8

a) Compensated open loop Bode plots, (Gain Margin = 12.2 dB, Phase Margin = 71.2 deg., Gain at 1 Hz = 32.4 dB) b) Azimuth closed stabilization loop bode plot, (Bandwidth = 20.5 Hz).

Figure 8(b) shows the closed stabilization loop Bode plot. The stabilization loop bandwidth achieved in this case is around 20.5 Hz. Since the gyro bandwidth (100 Hz) is much higher than the target closed loop bandwidth, it does not significantly affect the response at frequencies of interest for disturbance attenuation and tracking. On the other hand, it does act as a low pass filter for high frequency noise (>100 Hz).

Figure 9 shows the complete block diagram of the stabilization loop along with inner current loop for the azimuth plant.

Fig. 9

Complete Structural model of gimballed LOS Sensor System.

4.1 Analysis of the designed controller with azimuth plant assembly

a) Step Response Analysis

Consider an input command value of 500 mV is given to the designed azimuth assembly complete simulink model, the output of the system is observed to be as follows;

As the gyro scale factor is taken to be 5.73 V/rad/s, which is equivalent to100 mV/deg/s, and hence 500 mV Voltage corresponds to 5 deg/ s angular speed in the output as observed in Fig. 10.

Fig. 10

Step input Command Response.

b) Disturbance Attenuation Analysis

The simulink model is designed by considering the disturbance specification as 2 mrad up to a frequency of 1 Hz and less at higher frequencies (20 dB/decade roll-off is assumed). If θ_d = 2mrad is the amplitude of disturbance and ^′f′ = 1Hz is the frequency than the disturbance can be introduced in the model as a harmonic disturbance torque input of amplitude, I . θ_d . (2πf) ²], where I is the moment of inertia. Alternatively, the disturbance may be introduced in the stabilization loop as an output rate disturbance of amplitude [θ_d . (2πf)], without making much difference in the output characteristics, since the disturbance frequencies of the interest are much lower than the structural mode frequencies. Consider the designed azimuth assembly simulation system and apply rate disturbance, i.e. 0.01256 with no command input.

A harmonic disturbance of 2 mrad at 1 Hz is attenuated to within 50 microradians. The disturbance attenuation is better at lower frequencies due to high open loop gain. Attenuation gets worse as the frequency increases beyond 1 Hz because the open loop gain decreases at a faster rate than the disturbance.

4.2 Experimental validation of the designed simulink model

Figure 13 shows the experimental platform to analyze and measure the frequency response characteristics and disturbance attenuation of the stabilized gimbal assembly in azimuth direction.

Fig. 11

Applied Input Disturbance.

Fig. 12

LOS jitter (μrad) for Applied Disturbance signal.

Fig. 13

Experimental Setup for the frequency response analyses and disturbances attenuation measurement.

The designed PI controller for current loop of azimuth assembly has been implemented in Servo Power Amplifier (SPA) itself. SPA has its digital loops which takes input in form of K_p & T_i values of PI controller and internally convert into digital domain.

Further designed PI controller for the velocity loop of azimuth assembly is implemented in ADSP - 2181 Hardware by considering following steps:

Conversion of S-domain parameter into Z-domain discrete controller using bi-linear transformation.

Compute Z-domain coefficient.

Develop the discrete time difference equation, which need to be directly implemented on DSP Hardware.

Designed PI controller for the azimuth assembly is, $360 * (\frac{0.05305 s + 1}{s})$ , the sampling frequency for the velocity loop is taken as 4 KHz, hence sampling time is 1/4000 = 250μs. Using Bi-linear transformation, PI controller can be represented in form of Z –domain as: $H (z) = \frac{Y (z)}{X (z)} = \frac{360 (0.053175 z - 0.052925)}{z - 1}$ (19)

After taking the inverse Z transform of H(z), the difference equation can be written as: $y (n) = 360 [0.053175 x (n) - 0.052925 x (n + 1)] + y (n - 1)$ (20)

Finally the above equation is directly implemented on ADSP-2181 H/w using assembly language coding.

4.3 Testing of the control system

Figure 13 shows the experimental setup for the frequency response analyses and disturbances attenuation measurement.

In order to test the performance of the control system, Dynamic signal analyzer (DSA) with two input is used, where channel 1 is considered for the input parameter while channel 2 is used for the output parameter. Measurement of frequency response is performed by giving Sine wave an input from the DSA for energizing the system and output signal is taken from the gyro rate sensor, which is further connected to channel 2 for the measurement.

The response obtained from the DSA for Azimuth assembly is shown in Fig. 14.

Fig. 14

a) Open loop Bode plot from DSA (Azimuth) b) Closed loop bode plot from DSA (Azimuth).

A vehicle mounted laser based electro-optical system is the experimental plant under consideration. Two axis gimbal configuration is used to steer the line of sight in desired direction and minimize the effect of various internal and external disturbances during firing, such that LOS can be held steady onto the target with minimal jitter and hence maximum damage effect.

As, GFRP (Glass fibre reinforced polymer) sheet mounted on a mobile platform is used as a target.

Initially the target was locked in track mode and a high power laser beam was transmitted onto the target. During Laser firing, lot of platform disturbances are induced, which in turn act as a source of angular disturbances on Los of the proposed gimbaled sensor system. The gimballed sensor system reduces these disturbances and maintains the LOS steady onto the target. The target tracking and pointing process in defined field of view includes the following stages as shown in Fig. 15.

Fig. 15

Target tracking and Pointing Stages.

Further for the LOS jitter, target to boresight error (in pixels) has been computed through the processing of the captured CCD images. Finally LOS jitter has been computed via target to camera boresight error (in pixels) and corresponding field of view of camera and Zoom lens. The output pattern of the jitter measurement is shown in the Fig. 16.

Fig. 16

Jitter Characteristics with 2mrad platform disturbances.

A frequency domain characteristics comparison of simulated and experimental modeled gimbaled sensor system is drawn in Table 1.

Table 1

Frequency domain characteristics comparison of simulated and experimental model

S.No.	Specification	Simulation value	Actual value
1.	Open stabilization loop gain at 1 Hz	–3.3 dB	–4.5 dB
2.	Open stabilization loop gain at 15 Hz	–26 dB	–28 dB
3.	Gain Margin	12.2 dB	10 dB
4.	Phase Margin	71.2 deg.	36 deg.
5.	Bandwidth	20.5 Hz	18.8 Hz
6.	LOS Jitter	49 μrad	35μrad

5 Fuzzy logic approximation

The basic fuzzy inference system shown in Fig. 17 consists of four principal components: fuzzification inference, knowledge base, rule base and defuzzification inference. The input variables are error (e) and its derivative $(\frac{de}{dt})$ . The output variable is the incremental control force (u). The input for basic fuzzy inference mechanism can be either fuzzy or crisp but output is always produced in the form of fuzzy sets. But in case of fuzzy inference controller, it is required to have crisp output. So, at output stage crisp values are extracted from fuzzy set, and this process is called as defuzzification.

Fig. 17

Basic Structure of fuzzy controller.

Fuzzy sets are completely characterized by its membership function. Membership function can be represented as a mathematical formula.

In this work, Gaussian Membership functions are used, and are given as; $f (x, σ, c) = \exp$ (21)

5.1 Membership functions diagram

The selection of parameters such as mean (c) and standard deviation (σ) for each fuzzy sets of a membership function is the most challenging task, as they directly affect the system performance. The exact determination of these values heavily depends on the efficacy, experience and knowledge base of the designer [30]. Initially the membership function parameter and overlapping of fuzzy sets are selected based on knowledge base of designer and then fine tuned to obtain the desired result. Table 2 indicates the exact numeric values of membership functions parameters such as mean (c) and centroid (σ) for both the input variables ( $eand \dot{e}$ ) and the output variable (u).

Table 2
Fuzzy Membership function parameter values

Variables Error (e) Derivative of error ( $\dot{e}$ ) Output (u)

c σ c σ c σ

NH –1.0 0.27 –1.0 0.20 –1.0 0.22

NM –0.5 0.22 –0.59 0.15 –0.55 0.20

NL –0.2 0.05 –0.22 0.1 –0.20 0.08

Z 0 0.027 0 0.04 0 0.01

PL 0.2 0.05 0.22 0.10 0.20 0.08

PM 0.5 0.22 0.59 0.15 0.55 0.20

PH 1.0 0.27 1.0 0.20 1.0 0.22

Variables	Error (e)	Derivative of error ( $\dot{e}$ )	Output (u)
NH	–1.0	0.27	–1.0	0.20	–1.0	0.22
NM	–0.5	0.22	–0.59	0.15	–0.55	0.20
NL	–0.2	0.05	–0.22	0.1	–0.20	0.08
Z	0	0.027	0	0.04	0	0.01
PL	0.2	0.05	0.22	0.10	0.20	0.08
PM	0.5	0.22	0.59	0.15	0.55	0.20
PH	1.0	0.27	1.0	0.20	1.0	0.22

The final Gaussian membership functions for both inputs i.e. error and derivative of error and output variable (u) are obtained based on above inference mechanism, as shown in Fig. 18.

Fig. 18

Membership Functions Diagram: (a) Error (e) (b) Rate of change of error $\dot{e}$ c) Output (u).

The Mamdani model makes use of set of linguistic rules which describes a mapping from controller inputs to output.

Generalize form of rule R_i is:

If x is A then y is B, where linguistic variables A and B are called as antecedent and conclusion respectively. In this work, seven membership functions are defined for each input and hence fourteen possible combinations are formed. The rule base used is show in Table 3, where NH is negative high, NM is negative medium, NL is negative low, Z is zero, PL is positive low, PM is positive medium and PH is positive high. Finally, centroid defuzzification technique is used to convert fuzzy set output into crisp output. Figure 19 shows the surface view of the fuzzy inference system.

Fig. 19

Surface View representation.

Table 3

Fuzzy Rule Base

e\de	NH	NM	NL	Z	PL	PM	PH
NH	NH	NH	NH	NM	NL	NL	Z
NM	NH	NM	NM	NM	NL	Z	PL
NL	NH	NM	NL	NL	Z	PL	PM
Z	NH	NM	NL	Z	PL	PM	PH
PL	NM	NL	Z	PL	PL	PM	PH
PM	NL	Z	PL	PM	PM	PM	PH
PH	Z	PL	PL	PM	PH	PH	PH

5.2 Fuzzy controller implementation

The fuzzy logic controller structure shown in Fig. 20 consists of two input i.e. error (e), derivative of error $\dot{e}$ and output is the defuzzified value obtained from the aggregated fuzzified value corresponding to rules. Then this output is used to drive LOS stabilization plant.

Fig. 20

Proposed Fuzzy logic Controller configuration.

The overall transfer function model of the system can be obtained by replacing controller configuration block of stabilization loop in Fig. 9 with proposed fuzzy logic controller configuration. The dynamic response and disturbance rejection capabilities of the controller are tested and analyzed in the simulation. The system dynamic response plots corresponding to system states ‘ ${\dot{θ}}_{LOS}$ ’ represented by gyro output (V) and ‘θ_LOS’ (LOS position) in reference to applied disturbance are obtained and compared with the performance characteristics of conventional controller based sensor system. Performance comparison of Step input command response and disturbance attenuation characteristics of the gimbaled sensor system is shown in Figs. 21 and 22 respectively.

Fig. 21

Step input Command Response.

Fig. 22

LOS jitter (μrad) for Applied Disturbance signal.

The time response characteristics comparison of LOS stabilization system model with both the proposed controllers is given in Table 4.

Table 4

Performance Comparison of Conventional and Fuzzy Logic Controller

Characteristics	Conventional controller	Fuzzy controller
	Simulation values	Experimental values	Simulation values
Rise Time	25 ms	14 ms	10 ms
Settling Time	168 ms	10 ms	60 ms
% Overshoot	12	17	14
Steady State Error (%)	0.01	0.01	0.01
LOS jitter	49μrad	35μrad	29μrad

It is clearly evident from the tabular comparison of performance characteristics of gimbaled sensor system for a maximum disturbance spectrum value of 2mrad at 1 Hz and less at higher frequency, that although both the controller configuration were able to stabilize the gimbaled system with required accuracy, the performance of fuzzy logic controller is slightly better than the conventional approach in terms of settling time and disturbance attenuation.

6 Conclusion

This paper starts with the dynamic modeling of the gimbaled stabilization and pointing platform. A nested loop topology is adopted for implementation and design of fast current and stabilization/rate loop. Control configuration in these loop are initially implemented with conventionally tuned PI compensators. The overall dynamics and control configuration design has been implemented in MATLAB/Simulink. The first level validation of the control system design along with plant dynamics is done with several runs in the simulation. Then, this control configuration has been implemented on high end DSP hardware and interfaced with gimbaled assembly. The control parameters were tuned experimentally and finally the efficacy of the overall system was validated during field testing. It is observed that both the conventionally designed simulink model and experimental system are perfectly satisfying the desired criteria in terms of input command response, LOS jitter, bandwidth offered and other stringent time and frequency response characteristics. Furthermore, Fuzzy logic control (FLC) configuration is designed in simulink for the proposed stabilization model and tested based on critical parameters with number of simulation runs. The comparative analysis clearly indicates that FLC based stabilization system provides the improved response both in terms of jitter accuracy and faster response, which proves its utility in more precise pointing and stabilization applications.

References

Kennedy

P.J.

and Kennedy

R.L.

, Direct versus indirect line of sight (LOS) stabilization, IEEE Trans Control Syst Technol 11 (2003), 3–15. doi:10.1109/TCST.2002.806443

Masten

M.K.

, Inertially Stabilized Platforms for Optical Imaging Systems - Tracing Dynamic Target with Mobile Sensors, IEEE Control Syst Mag 28 (2008), 47–64. doi:10.1109/MCS.2007.910201

Hilkert

J.M.

, Platform Technology - Concepts and Principles, IEEE Control Syst Magzine (2008), 26–46. doi:10.1109/MCS.2007.910256

Kaymak

, Rojas-Cessa

, Feng

J.H.

, Ansari

, Zhou

M.C.

and Zhang

, A Survey on Acquisition, Tracking, and Pointing Mechanisms for Mobile Free-Space Optical Communications, IEEE Commun Surv Tutorials (2018). doi:10.1109/COMST.2018.2804323

Deng

, Cong

, Member

, Kong

and Shen

, Discrete-Time Direct Model Reference Adaptive Control Application in a High-Precision Inertially Stabilized Platform, IEEE Trans Ind Electron 66 (2019), 358–367. doi:10.1109/TIE.2018.2831181

Rue

A.K.

, Stabilization of Precision Electrooptical Pointing and Tracking Systems, IEEE Trans Aerosp Electron Syst AES-5 (1969), 805–819. doi:10.1109/TAES.1969.309879

Kim

S.H.

, Kim

S.B.

and Kwak

Y.K.

, Robust control for a two-axis gimbaled sensor system with multivariable feedback systems, IET Control Theory Appl 4 (2010), 539–551. doi:10.1049/iet-cta.2008.0195

Morrison

, Imboden

and Bishop

B.J.

, Tuning the resonance frequecies and mode shapes in a large range multi degree of freedom micromirror, Opt Express 25 (2017), 7895–7906. doi:10.1364/OE.25.007895

Netzer

, Line of sight steering and stabilization. V, (2013)

10.

, Zheng

and Ning

, Precise Control for Gimbal System of Double Gimbal Control Moment Gyro Based on Cascade Extended State Observer, IEEE Trans Ind Electron 64 (2017), 4653–4661. doi:10.1109/TIE.2017.2674585

11.

Arefi

M.M.

and Khayatian

, Adaptive dynamic surface control of a two-axis gimbal system, IET Sci Meas Technol 10 (2016), 607–613. doi:10.1049/iet-smt.2016.0005

12.

F.R.R.

, Ortega

M.G.

, Gordillo

and Vargas

, Application of position and inertial-rate control to a 2-DOF gyroscopic platform, Robot Comput Integr Manuf 26 (2010), 344–353. doi:10.1016/j.rcim.2009.11.012

13.

Zhou

, Design and Analysis of a Fast Steering Mirror for Precision Laser Beams Steering. Sensors Transducers J. (2014)

14.

Hilkert

J.M.

, Kanga

and Kinnear

, Line-of-Sight Kinematics and Corrections for Fast-Steering Mirrors Used in Precision Pointing and Tracking Systems, Spie 9076 (2014), 1–15. doi:10.1117/12.2049857

15.

Tian

, Yang

, Peng

and Tang

, Inertial sensor-based multiloop control of fast steering mirror for line of sight stabilization, SPIE Opt Eng 55 (2019), 1–6. doi:10.1117/1.OE.55.11.111602

16.

Mao

, Indirect fuzzy contour tracking for X –Y PMSM actuated motion system applications, IET Electr Power Appl 12 (2017), 12–24. doi:10.1049/iet-epa.2016.0881

17.

Liu

, Che

and Sun

, Research on stabilizing and tracking control system of tracking and sighting pod, J Control Theory Appl 10 (2012), 107–112. doi:10.1007/s11768-012-9152-8

18.

Rubio-solis

, Melin

, Martinez-hernandez

and Panoutsos

, General Type-2 Radial Basis Function Neural Network: A Data-Driven Fuzzy Model, IEEE Trans Fuzzy Syst 27 (2018), 333–347. doi:10.1109/TFUZZ.2018.2858740

19.

Qadir

and Semke

, Vision Based Neuro-Fuzzy Controller for a Two Axes Gimbal System with Small UAV, J Intell Robot Syst (2013). doi:10.1007/s10846-013-9865-z

20.

Ouda

A.N.

, A robust adaptive control approach to missile autopilot design, Int J Dyn Control 6 (2018), 1239–1271. doi:10.1007/s40435-017-0352-4

21.

Wang

, Shi

, Yang

, Wu

and Chen

, Fuzzy generalised predictive control for a class of fractional-order non-linear systems, 1 (2017), 87–96. doi:10.1049/iet-cta.2017.0239

22.

Chen

W.H.

, Yang

, Guo

and Li

, Disturbance observer based control and related methods—An overview, IEEE Trans Ind Electron 63, 1083–1095. doi:10.1109/TIE.2015.2478397

23.

Hsu

C.-F.

, Self-organizing adaptive fuzzy neural control for a class of nonlinear systems, IEEE Trans Neural Networks 18 (2007), 1232–1241 doi:10.1109/TNN.2007.899178

24.

Chen

, Tian

, Peng

L.I.

, Qingdong

L.I.

, and Z.R., Sliding-Mode-Control Based Robust Guidance Algorithm Using Only Line-of-Sight Rate Measurement, J Syst Sci Complex 29 (2016), 1485–1504. doi:10.1007/s11424-016-5013-8

25.

Pau

, Collotta

, Maniscalco

and Choo

K.R.

, A fuzzy-PSO system for indoor localization based on visible light communications, Soft Comput (2018). doi:10.1007/s00500-018-3212-z

26.

Technology

E.M.

, Torque Motors., Switzerland (2013)

27.

Chaudhary

, Khatoon

and Singh

, ANFIS Based Speed Control of DC Motor. In: Innovative Applications of Computational Intelligence on Power, Energy and Controls with their Impact on Humanity (CIPECH). (2016), pp. 63–67. IEEE.

28.

May

, Modeling the dynamically tuned gyroscope in support of high-bandwidth capture loop design, SPIE Proc Acquis Track Pointing XIII 3692 (1999), 101–111. doi:10.1117/12.352852

29.

Ogata

, Modern Control Engineering Book.

30.

Mahmoud

, Reza

and Reza

, Stabilization loop of a two axes gimbal system using self-tuning PID type fuzzy controller, ISA Trans 53 (2014), 591–602. doi:10.1016/j.isatra.2013.12.008