Fuzzy Proportional Navigation Guidance for a Complete 3-DOF Model

1. Introduction

As the maneuverability of aerial threats increases, guidance and control systems must provide greater agility and robustness. Atmospheric disturbances, variations in seeker and missile parameters, sensor noise, nonlinear missile dynamics, and actuator saturation can all affect guidance performance. Consequently, there is a need for guidance methods that remain effective under uncertain and nonlinear operating conditions.

Proportional navigation (PN) has been widely used as a practical homing-guidance rule because of its relative simplicity and closed-loop structure. However, the conventional PN law is generally derived from simplified and partially linearized engagement dynamics. These assumptions can reduce performance when nonlinearities, uncertainties, and high target maneuvers become significant.

Fuzzy-logic guidance has been investigated as an alternative because fuzzy controllers can operate effectively for nonlinear and complex systems without requiring a highly precise mathematical model. Fuzzy logic has previously been applied to missile-guidance-law design [1], fuzzy homing-guidance evaluation [2], proportional-navigation implementation [3], and fuzzy predictive pursuit guidance [4].

Related studies have also considered trajectory-correction and guidance problems in a variety of computational and engineering settings, including trajectory correction for semi-Lagrangian schemes [5], trajectory correction in free-breathing radial cine MRI [6], trajectory-correction projectiles [7], ballistic and weaponeering analysis [8], one-dimensional trajectory-correction strategies [9], and six-degree-of-freedom digital simulations for missile guidance and control [10]. Although several of these studies address different application domains, they collectively illustrate the importance of trajectory modelling and correction under dynamic conditions.

The broader homing-guidance literature provides a well-established foundation for proportional navigation, seeker filtering, flight-control dynamics, and high-fidelity simulation. Standard treatments discuss proportional-navigation geometry, miss-distance analysis, homing-loop dynamics, filtering, autopilot effects, and tactical engagement simulation [11,12,13,14,15,16]. Practical missile guidance and control design has also been studied through integrated guidance-and-control architectures and numerical tuning of guidance and control algorithms [17,18].

A substantial body of research has extended conventional guidance laws to address nonlinear engagement geometry, highly maneuvering targets, terminal constraints, and uncertainty. Examples include guidance laws for highly maneuvering targets [19], optimal virtual-target guidance [20], nonlinear modified bias proportional navigation [21], field-of-view-constrained interception [22], and hit-to-kill interception under demanding maneuver requirements [23]. Sliding-mode and robust-control formulations have further addressed finite-time convergence, autopilot dynamics, acceleration saturation, and precision interception [24,25,26,27].

Fuzzy and intelligent guidance methods have been developed specifically to improve adaptability when the engagement dynamics or measurements are uncertain. In addition to fuzzy proportional-navigation guidance [28], fuzzy logic has been combined with inverse-kinematics guidance [29], integrated proportional-navigation strategies for highly maneuvering targets [30], and adaptive second-order sliding-mode guidance [31]. Neural-network-assisted proportional navigation has also been investigated for ballistic-target interception [32], while full missile models have been used to compare alternative fuzzy guidance structures [33]. These developments support evaluating fuzzy guidance in a model that retains seeker, autopilot, actuator, noise, and velocity effects rather than relying only on idealized engagement equations.

More broadly, intelligent computational methods have been applied across a wide range of nonlinear modelling, recognition, prediction, and control problems. Representative applications include face detection and recognition [34,35,36,37], palmprint and signature recognition [38,39], deformable texture matching and computer vision [40,41], and fuzzy-inference-based load forecasting [42]. Data-driven and machine-learning approaches have also been used for healthcare prediction and screening [43,44,45,46,47], retinal-image analysis [48], and general multiclass classification software [49]. Related optimization and intelligent-system research spans wireless computing and communications [50], antenna design [51], mobile-robot control [52], scalable load balancing in distributed computing [53], and studies of human cognitive factors such as thinking styles and self-efficacy [54]. Although these applications differ from missile guidance, they demonstrate the broad use of computational intelligence, fuzzy reasoning, learning, and optimization for complex systems and motivate continued investigation of intelligent guidance and control methods.

The main contribution of the present study is to compare fuzzy proportional-navigation guidance with conventional PN using a missile model that includes seeker/tracker dynamics, autopilot and airframe dynamics, actuator effects, sensor noise, and missile-velocity variation.

The performance criterion used in the simulations is the integral of the absolute lateral acceleration command, which is treated as an indicator of control effort and fuel consumption. Four scenarios are considered: a fixed target, a target maneuvering at 3g, targets maneuvering at 5g and 6g, and a fixed target with seeker noise.

2. Proportional Navigation Guidance Rule

Proportional navigation commands lateral acceleration in proportion to the closing velocity and the line-of-sight angular rate. The guidance law used in the source manuscript is

$$a_c=N{V}_c\dot{\lambda },$$

(1)

where $a_c$ is the commanded lateral acceleration, N is the navigation constant, $V_c$ is the missile–target closing velocity, and $\dot{\lambda }$ is the line-of-sight angular rate.

Figure 1. Missile–target engagement geometry used in the proportional-navigation formulation.

Figure 1. Missile–target engagement geometry used in the proportional-navigation formulation.

3. Fuzzy Guidance System

A fuzzy system performs inference using fuzzy sets and fuzzy logic. The controller considered in this work consists of a database, fuzzification stage, decision-making unit, and defuzzification stage. Input signals are first converted into fuzzy linguistic variables. The rule base is then used to determine the fuzzy output, which is finally converted into a crisp acceleration command [55].

3.1. Inputs and Output

The two fuzzy-controller inputs are the closing velocity $V_c$ and the line-of-sight angular rate $\dot{\lambda }$. The controller output is the commanded acceleration.

Figure 2. MATLAB fuzzy-inference-system editor showing closing velocity and line-of-sight angular rate as inputs and acceleration as the output.

Figure 2. MATLAB fuzzy-inference-system editor showing closing velocity and line-of-sight angular rate as inputs and acceleration as the output.

3.2. Membership Functions

Seven linguistic labels are used:

• NB: negative big,
• NM: negative medium,
• NS: negative small,
• ZE: zero,
• PS: positive small,
• PM: positive medium,
• PB: positive big.

The membership functions used for the controller inputs and output are shown in Figure 3.

3.3. Fuzzy Rule Base

The source paper uses a $7\times 7$ rule base. Table 1 reproduces the supplied fuzzy guidance table in LaTeX form. Rows correspond to $\dot{\lambda }$ and columns correspond to $V_c$.

The fuzzy rules can be expressed in linguistic form. For example, a rule has the structure

$$\mathrm{IF}{V}_c\mathrm{is}A\mathrm{AND}\dot{\lambda }\mathrm{is}B,\mathrm{THEN}{a}_c\mathrm{is}C.$$

The rule-editor implementation in MATLAB is shown in Figure 4.

For each rule, membership values are evaluated for the conditional terms. The fuzzy output is then passed to a defuzzification stage. The source manuscript uses a centroid-type defuzzification approach.

Figure 5. Example relationship between line-of-sight angular rate and the fuzzy-controller acceleration output.

Figure 5. Example relationship between line-of-sight angular rate and the fuzzy-controller acceleration output.

Figure 6. Relationship between the two fuzzy inputs, closing velocity and line-of-sight angular rate.

Figure 6. Relationship between the two fuzzy inputs, closing velocity and line-of-sight angular rate.

4. Simulation Model

The missile model was implemented in MATLAB/Simulink. The model contains five main blocks, as shown in Figure 7.

The main blocks are:

1.

Target model: provides the target trajectory, with or without maneuvering.

2.

Seeker/tracker: includes seeker dynamics, associated gyroscopes, line-of-sight calculations, and sensor-control dynamics.

3.

Guidance: contains the guidance law and the required flight sequencing.

4.

Airframe and autopilot: includes the autopilot, navigation sensors, control-surface actuators, aerodynamics, flight equations, and atmosphere model.

5.

3-DOF animation: provides a graphical representation of missile motion.

In the comparative simulations, the guidance block is alternatively configured with conventional PN or the fuzzy guidance controller. The principal performance index is the integral of the absolute lateral acceleration, which is treated as a measure of control effort and fuel consumption.

5. Simulation Results

5.1. Scenario 1: Fixed Target

For the first scenario, the target is fixed and the initial missile angle is zero. Figure 8 shows the missile and target trajectories.

Figure 9 compares the accumulated absolute lateral acceleration for PN and fuzzy guidance.

The source manuscript reports a final value of approximately 304 energy units for PN guidance and 316 energy units for fuzzy guidance. Therefore, PN performs slightly better for this fixed-target case.

5.2. Scenario 2: Target Maneuvering at 3g

The second scenario introduces a target maneuver of 3g. Figure 10 shows the missile and target trajectories.

The accumulated absolute lateral acceleration is shown in Figure 11.

The source paper reports approximately 456 energy units for PN and 455 energy units for fuzzy guidance. Thus, at 3g the two methods are nearly equivalent, with fuzzy guidance providing a small reduction.

5.3. Scenario 3: Higher Target Maneuvers

The third scenario considers 5g and 6g target maneuvers. Figure 12 compares the accumulated absolute lateral acceleration.

For the 5g case, the source manuscript reports approximately 573 energy units for PN guidance and 565 energy units for fuzzy guidance. For the 6g case, the corresponding values are approximately 643 and 606 energy units. These results indicate that the relative advantage of fuzzy guidance increases as target maneuver intensity increases.

5.4. Scenario 4: Fixed Target with Seeker Noise

The fourth scenario considers a fixed target with added seeker noise. Figure 13 compares the accumulated absolute lateral acceleration.

The source manuscript reports approximately 363 energy units for PN guidance and 335 energy units for fuzzy guidance. Under this noisy-sensor condition, fuzzy guidance requires less control effort according to the selected performance index.

6. Discussion

The simulations indicate that conventional PN performs slightly better for a fixed and non-maneuvering target, whereas fuzzy guidance becomes more competitive as the target maneuver intensity increases. For the 3g maneuver, the two methods are nearly equivalent. For 5g and 6g maneuvers, the fuzzy guidance controller achieves lower accumulated absolute lateral acceleration.

The simulations also show that fuzzy guidance is less sensitive to seeker noise under the conditions studied. This is consistent with the general motivation for fuzzy control in nonlinear systems with uncertain or incomplete model information.

The results are specific to the supplied simulation model, controller settings, membership functions, and rule base. The source paper treats the integral of absolute lateral acceleration as a proxy for fuel consumption; therefore, the comparisons reported here should be interpreted in terms of that selected control-effort measure.

7. Conclusions

This paper compares fuzzy proportional-navigation guidance with conventional proportional navigation using a comparatively complete 3-DOF missile model. The simulations include seeker/tracker dynamics, airframe and autopilot dynamics, actuator effects, sensor noise, and other nonlinearities represented in the supplied model. For a fixed target, conventional PN produces a slightly lower control-effort index. As target maneuver intensity increases, fuzzy guidance becomes more effective and produces lower accumulated absolute lateral acceleration. The fuzzy controller also performs better in the tested seeker-noise scenario. A possible extension suggested in the source manuscript is a hybrid guidance strategy in which PN is used for low-maneuvering targets and fuzzy guidance is used for highly maneuvering targets, with the transition between the two strategies determined by a fuzzy supervisory mechanism.