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A new homing navigation guidance law is presented for a pursuer to intercept a moving target with a prescribed impact angle: the impact angle control problem. The proposed guidance law can solve the impact angle control problem for virtually every initial geometry of pursuer and target on a plane and any desired impact attitude angle. The impact an...
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... is at our disposal which will be called as the free launching condition. Under this condition (12) can be safely ignored and remaining conditions uniquely determine a circle C * ref in Fig.3 as a reference curve. ...
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... be precise the reference curve is only a part (arc) of the circle C * ref but this difference will not cause confusions as the directional line in Fig.3 indicates the direction of velocity vector ( ˙ x, ˙ y) at impact. ...
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... be precise the reference curve is only a part (arc) of the circle C * ref but this difference will not cause confusions as the directional line in Fig.3 indicates the direction of velocity vector ( ˙ x, ˙ y) at impact. This justifies a convention of calling the circle in Fig.3 as simply a reference circle from now on. ...
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... the pursuit dynamics (7)- (8) and (16). Also the equality φ − φ de = 2(π + σ − φ de ) holds while (x, y) moves on the reference circle C * ref ; see Fig. 3. It should be also noticed that R ref (·) can be ...
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... that the reference circle is not well defined if an initial point (x(0), y(0)) lies on a straight line overlapping the directional line of Fig. 3. That is, the following ...
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... order to clarify the implications of this requirement, let us briefly consider situations where (18) does not hold. Firstly, suppose 0 < φ de < π as in Fig. 3 and σ(0) = φ de . Then the reference circle C * ref becomes a straight line overlapping the line . As such, if y(0) < 0, then a pursuer is well aligned initially and a perfect impact angle follows with no controls. On the contrary, if y(0) > 0, then a capturing fails. These two special situations can be shown to occur in other cases ...
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Citations
... One used the current missile velocity direction and the relative position of missile to determine the circular flight path and the other used the desired terminal missile velocity direction. From the concept of the reference circle on a moving frame fixed to target, Yoon developed a navigation guidance law called a relative circular navigation guidance (RCNG) for the impact angle control problem under the planar engagement in [22]. And RCNG was later extended to the three-dimensional scenario [23]. ...
... In [33], a circular trajectory could be updated by controlling LOS angle equal to the approach-angle error. Yoon [34] further proposed a relative circle guidance method for tracking moving target. ...
A novel trajectory-shaping guidance law with impact angle constraint is proposed for attacking stationary targets in a three-dimensional environment. The guidance concept, derived from the inverse dynamic method, is to design an analytical curve trajectory satisfying the impact angle in advance and obtain reasonable control commands. The vehicle motion is decomposed in the horizontal and vertical planes of the inertial coordinate system. Firstly, the trajectory in the vertical plane is designed as ellipse, which can be shaped by adjusting its axis direction. An improved sliding mode control (SMC) method, which adds position-dependent correction to the weight in sliding mode surface, is adopted to track the nominal trajectory. Therefore, the vehicle approaches the ellipse quickly and smoothly in the early stage and hits the stationary target accurately. Secondly, a third-order Bézier curve with adjustable parameters is employed as the prior nominal trajectory in the lateral plane. When the vehicle deviates from the original trajectory due to perturbation or self-limitation, it will turn to the updated curve in real time according to its own condition. Moreover, coupling of acceleration commands in the two planes is resolved through acceleration decomposition, which qualifies independent trajectory design in two planes and paves a new way to more curve combinations. Nominal testing and Monte Carlo simulations on the proposed method are carried out. Simulation results demonstrate that the proposed guidance law is highly designable and strongly robust.
... Another possible approach is to shape the trajectory to be a circular one. Circular trajectories have been used for impact angle guidance [17][18][19][20][21]. Manchester and Savkin [17] proposed a geometric guidance concept named circular navigation guidance. ...
... It makes the interceptor follow a circular arc to the target and ensures interception with a desired impact angle under certain conditions. Yoon [18] proposed another guidance concept by introducing a circular reference curve on a moving frame fixed to the target and using the Frenet-Serret formulation to impose an impact angle constraint. In Ref. [19], a three-point guidance law, named inscribed angle guidance, was introduced by the authors. ...
A guidance concept to impose a desired impact time is presented and investigated. The guidance concept is based on the geometrical principle that constrains the interceptor to follow a circular trajectory to the target. The concept is extended for moving targets, varying interceptor speed, and a constrained field of view of the interceptor. Implementation is demonstrated for a salvo-attack scenario where, by scheduling the launches of the interceptors, not only the desired impact time but also the desired impact angles are enforced. Application of the guidance concept for standoff tracking of stationary and moving targets is also demonstrated. Performance in various scenarios is investigated via nonlinear simulations.
... For instance, anti-ship missile systems are additionally demanded to meet the terminal impact time constraint [4][5]. Satisfaction of a desired terminal impact angle is an additional important requirement for the anti-tank missile systems to increase the lethality [6][7][8]. If the interception condition is only consideration for the missile system, additional guidance goals can be typically represented as achieving desired patterns of zero-effort-miss (ZEM) or guidance commands: for the surface-to-air missile systems controlled by the aerodynamic force only, the decreasing pattern of the guidance command is desirable according to the change of the dynamic pressure. ...
This paper suggests a new approach in designing homing guidance laws to enable direct shaping of the pattern of zero-effort-miss (ZEM) as desired. The proposed approach uses the concept of weighted ZEM and its specific desired error dynamics: the former is to provide an additional degree of freedom in shaping actual ZEM and the latter is to guarantee a finite-time convergence. Utilization of these two concepts allows simple determination of the guidance law that can achieve the desired pattern of ZEM. The resultant guidance law is shown a type of proportional navigation guidance (PNG) law with the specific form of time-varying gain not revealed in previous studies. It provides unique information on how the time-varying gain should be shaped to obtain the desired pattern of ZEM. Accordingly, the resultant guidance laws can cope with various operational objectives in a more direct way compared with the previously existing approaches. This paper also performs theoretical analysis to investigate the properties of designed guidance laws including the closed-loop solutions of ZEM and acceleration command. Also, we determine the feasible set of desired ZEM patterns that can be achieved in the proposed framework. Two illustrative examples are considered to show how to design guidance laws using the proposed approach. Moreover, the characteristics of the guidance laws designed are validated and demonstrated via numerical simulations.
... A biased proportional navigation guidance law was proposed in [11] by adding a time-varying term to control the flight path angle. In [12], the guidance law with LOS information sought convergence to a circular path by using the PPN guidance law with a specific navigation gain. A two-phased proportional navigation guidance law was introduced in [13] for achieving all impact angles against stationary targets. ...
... In (12), the expected bias amount is ...
Traditional guidance laws with range-to-go information or time-to-go estimation may not be implemented in passive homing missiles since passive seekers cannot measure relative range directly. A time-varying biased proportional guidance law, which only uses line-of-sight (LOS) rate and look angle information, is proposed to satisfy both impact angle constraint and seeker’s field-of-view (FOV) limit. In the proposed guidance law, two time-varying bias terms are applied to divide the trajectory into initial phase and terminal phase. The initial bias is designed as a function of LOS rate and look angle to maintain the seeker’s lock-on while the final bias eliminates the deviation between the integral value of angle control bias and the expected bias amount. A switching logic is adopted to change the biases continuously so that there is no abrupt acceleration change during the engagement. Extensive simulations considering both kinematic and realistic missile models are performed to illustrate the efficiency of the proposed method.
... To drive the relation between impact angle θimp and LOS angle (q), we assume that the missile and the target lie at the collision course. At the collision course, the LOS angle rate (q) is 0. According to (2) and (4), we can derive the corresponding LOS angle, denoted as q f , as follows: ...
This paper presents a novel guidance law to intercept non-maneuvering targets with impact angle and lateral acceleration command constraints. Firstly, we formulate the impact angle control to track the desired line-of-sight (LOS) angle, which is achieved by selecting the missile’s lateral acceleration to enforce the sliding mode on a sliding surface at impact time. Secondly, we use the Lyapunov stability theory to prove the stability and finite time convergence of the proposed nonlinear sliding surface. Thirdly, we introduce the wavelet neural network (WNN) to adaptively update the additional control command and reduce the high-frequency chattering of sliding mode control (SMC). The proposed guidance law, denoted WNNSMC guidance law with impact angle constraint, combines the SMC methodology with WNN to improve the robustness and reduce the chattering of the system. Finally, numerical simulations are performed to demonstrate the validity and effectiveness of the WNNSMC guidance law. © 2015 Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
... To enhance the penetration probability of a cruise missile, controlling both the impact angle and the impact time is of special significance in practice. In the past three decades, several guidance laws with terminal impact angle specifications were proposed within different frameworks [1][2][3][4][5][6][7][8][9][10][11]. It should be noted that the impact time is a crucial factor in the salvo attack against a highly valued surface target equipped with advanced air defense installations such as close-in weapon systems (CIWS). ...
... Using (11), (8) can be reformulated in terms of as ...
The trajectory optimization problem subject to terminal impact time and angle specifications can be reformulated as a nonlinear programming problem using the Gauss pseudospectral method. The cost function of the trajectory optimization problem is modified to reduce the terminal control energy. A receding horizon optimization strategy is implemented to reject the errors caused by the motion of a surface target. Several simulations were performed to validate the proposed method via the C programming language. The simulation results demonstrate the effectiveness of the proposed algorithm and that the real-time requirement can be easily achieved if the C programming language is used to realize it.
... for the purpose of steering pendulum cart to a given destination r f :=(x f , y f ), assuming that a local controller designed for that particular equilibrium point is stabilizing pendulum. Our idea is motivated from the classical pure proportional navigation (PPN) law in missile guidance field, e.g., see [11]. ...
This paper studies a stabilization problem of an inverted spherical pendulum on a wheeled cart assuming that pure rotation
of cart cannot move support of pendulum. It is shown that in spite of nonholonomic constraints of wheels and the above geometric
condition on pendulum support the pendulum can be locally stabilizable at a certain non-static equilibrium point by applying
a standard linearization approach. Furthermore it is shown that equilibrium points can be chosen to make cart move to a given
location. Numerical simulations are performed to confirm theoretical findings and illustrate interesting dynamics of closed
loop pendulum system.
KeywordsLinearization–nonholonomic constraints–spherical inverted pendulum
