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It is a comparatively convenient technique to investigate the motion of a particle with the help of the differential geometry
theory, rather than directly decomposing the motion in the Cartesian coordinates. The new model of three-dimensional (3D)
guidance problem for interceptors is presented in this paper, based on the classical differential geo...
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Citations
... Classical guidance laws are represented by the tracking method [1], the parallel approach method [2], proportional navigation guidance method, etc. [3][4][5]. Both the tracking method and parallel approach method can be classified into PNG methods [6]. As a classical guidance laws, PNG method has a wide application in engineering practice [7], owing to its simple structure, easy implementation, less information required, straight trajectory, high guidance accuracy, applicability for maneuvering targets, and other desirable qualities [8]. ...
This paper proposed a united proportional navigation guidance (UPNG) method to alleviate the guidance command saltation with an impact angle constraint under the condition of no real-time distance between the vehicle and the target (line-of-sight (LOS) distance). Firstly, based on the biased proportional navigation guidance (BPNG), a smooth-biased proportional navigation guidance (SBPNG) method was proposed, whose bias term was designed as a trigonometric function. In SBPNG method, due to the continuous smooth change of the bias term, the guidance command would not saltus anymore, and the impact angle was controlled by the bias integral component. Secondly, biased on SBPNG method, the united proportional navigation guidance (UPNG) method combining SBPNG and variable coefficient proportional navigation guidance (VCPNG) was established. In UPNG method, because there was no LOS distance, the guidance coefficient was designed as a function of the difference between the expected impact angle and the estimated impact angle, so the closed-loop control of impact angle was realized. Finally, a lot of simulation experiments on different guidance laws were carried out without real-time LOS distance. The results verify that the UPNG method proposed in this paper solves the problem of guidance command saltation effectively and has better robustness in impact angle control.
... Additionally, formation reconfiguration strategy is designed considering how some typical unexpected situations influence cooperative formations. Combining the guidance law of the parallel-approach method [8,9] and the calculation method of virtual dynamic tracking points, a method is proposed for cooperative path-point following in formation control [10][11][12] to realize multi-machine formation flight of fixed-wing UAVs. It has the advantages of fast tracking response in the low-altitude formation flight, and rapid formation reconstruction in unexpected situations. ...
... In the ground coordinate system, the line-of-sight angle between the wingman In Figure 3, v m is the velocity vector of the UAV tracking the target point whose azimuth angle is η m . The velocity vector V d of the UAV has components v i that are equal to v m , i.e., v i = v m (8) In the formation of the lead-aircraft-wingman control mode selected herein, the lead aircraft sends real-time status information through an airborne data link, and the wingmen in the formation obtain v m and η m of the lead aircraft through calculation. ...
This paper reports on the formation and transformation of multiple fixed-wing unmanned
aerial vehicles (UAVs) in three-dimensional space. A cooperative guidance law based on the classic
missile-type parallel-approach method is designed for the multi-UAV formation control problem.
Additionally, formation transformation strategies for multi-UAV autonomous assembly, disbandment,
and special circumstances are formed, effective for managing and controlling the formation. When
formulating the management strategy for formation establishment, its process is divided into three
steps: (i) selecting and allocating target points, (ii) forming loose formations, and (iii) forming short�range formations. The management of disbanding the formation is formulated through reverse
thinking: the assembly process is split and recombined in reverse, and a formation disbanding
strategy that can achieve a smooth transition from close to lose formation is proposed. Additionally,
a strategy is given for adjusting the formation transformation in special cases, and the formation
adjustment is completed using the adjacency matrix. Finally, a hardware-in-the-loop simulation and
measured flight verification using a simulator show the practicality of the guidance law in meeting
the control requirements of UAV formation flight for specific flight tas
... The study of absolute and relative motion of vehicles based on differential geometric curve principle is a new and popular research method in the field of missile interception. Anderson and Yang et al. [23][24][25][26][27][28][29][30][31][32] obtained a more simplified relative motion equation through studying the rotation law of LOS by curvature and torsion. Taking the orbit of chaser and target as a curve in three-dimensional space, the theory of differential geometry curve can be extended to the control problem of approaching and forced fly-around against the tumbling target in space. ...
... The Fig. 7 indicates that the control laws in (25) and (27) drive LOS and IRPL rotation rate to the desired values in finite time. In this case, the desired values of LOS and IRPL rotation rate are both precession angular rate of tumbling target, namely ̇= 2.25 • ∕s. ...
Forced fly-around of spacecraft is the basis for tracking and capturing space tumbling target in active debris removal mission. In this paper, the relative kinematic equation between chaser and target is first obtained with the help of classical differential geometry principle. Then, taking the attitude dynamic model of tumbling target into consideration, the finite time convergence sliding model control (FTCSMC) is designed to enable the forced fly-around in the line of sight rotation coordinate system. Finally, the effectiveness and robustness of the control strategy are verified in two typical cases by numerical simulation.
... However, with enhancement of target maneuverability, PN suffers from a significant degradation of intercept performance because of limited capability to suppress rotation of the line of sight (LOS) between a missile and a target induced by target maneuver. Then, the augmented PN was proposed to compensate target maneuver, but the price paid is the information of target acceleration which cannot be measured directly and is difficult to be accurately estimated [4,5]. To meet the challenge of precisely intercepting agile targets, some advanced control algorithms have been used to develop robust guidance laws, such as sliding mode control [6][7][8][9][10][11], nonlinear H ∞ control [12], dynamic surface control [13], and finite time control [14][15][16][17][18]. ...
... One is the commonly used spherical LOS coordinate system, as shown in [9]. The other is the rotating LOS coordinate system proposed in [5]. In this paper, we adopt the rotating LOS coordinate system where a decoupled relative-motion equation set is obtained. ...
... Let e θ = e ω × e r , and then the set ðe r , e θ , e ω Þ is the unit vectors along the axes of the rotating LOS coordinate system. According to [5], we have the following relations: ...
For the terminal guidance problem of a missile intercepting a maneuvering target, a profile-tracking-based adaptive guidance law is proposed with inherent continuity in this paper. To flexibly and quantitatively control the convergence rate of the line-of-sight rate, a standard tracking profile is designed where the convergence rate is analytically given. Then, a nonsingular fast terminal sliding-mode control approach is used to track the profile. By estimating the square of the upper bound of target maneuver, an adaptive term is constructed to compensate the maneuver. Therefore, no information of target acceleration is required in the derived law. Stability analysis shows that the tracking error can converge to a small neighborhood of zero in finite time. Furthermore, a guidance-command-conversion scheme is presented to convert the law into the one appropriate for endoatmospheric interceptions. Simulation results indicate that the proposed law is effective and outperforms existing guidance laws.
... [47][48][49] However, in this way the description of the relative motion was quite complex due to the cross-coupling and too many variables were involved. Using the kinematic equations established in the rotating coordinate system of LOS can simplify the description of the 3D relative motion so that it can be divided into two decoupled sub-motions: [50][51][52][53][54] (1) the relative motion in the engagement plane spanned by the relative position and velocity vectors and (2) the rotation of this plane. In this section, we will realize the NASMG in 3D space by directly constructing a planar NASMG in the relative engagement plane. ...
A novel adaptive sliding mode guidance law is proposed in this article. The target is assumed to have an arbitrarily but upper bounded maneuvering acceleration which is considered as the system disturbances and uncertainties. The guidance law is consisted of three terms. The first one is a proportional navigation–type term. The second one is a term used for compensating the target maneuvering acceleration. And the last one is a term for controlling the convergence time of the line-of-sight angular rate. In this guidance law, the upper bound of the target acceleration is estimated by an adaptive estimator with a tunable updating law. Hence, the prior knowledge of the upper bound of the target acceleration is not essential for this guidance law. The novel adaptive sliding mode guidance law can guarantee the asymptotical convergence of the line-of-sight rate to zero or its neighborhood, or even the finite time convergence of the line-of-sight rate conditionally. Finally, the new theoretical findings are demonstrated by numerical simulations.
... Advantages of utilizing this type of coordinate system were also demonstrated. For example, Li et al. [25] and Li et al. [26]~ [30] used this kinematic equation set to study 3D differential geometric guidance Law (DGGL). Liu et al. [31] used this kinematic equation set to investigate the performance of an augmented proportional navigation guidance law for the relative motion control between a service spacecraft and its target. ...
... Proof: Firstly, (26) is proven by contradiction. If (26) does not hold, from the continuities of r(t) and rt, it is trivial that there must exist a constant t 1 ∈(0, +∞) such that ...
... As shown in (60), the first two equations can be decoupled from the third one. For more details, the reader is referred to [23], [26]~ [31]. ...
Using the Lyapunov-like approach, the capturability of a recently-proposed sliding mode guidance law which is used for exoatmospheric interception and can guide the line-of-sight (LOS) angular rate to converge to zero or its small neighborhood in a finite time is thoroughly analyzed. The target is assumed to have an arbitrary but upper-bounded maneuvering acceleration. A more realistic definition of capture is considered. The upper-bound of the commanded acceleration is obtained, and so is the capture region. The new theoretical findings are extended to the three-dimensional space with the help of the rotating line-of-sight coordinate system.
... [20,21]. In fact, the DG approach provides the unique fundamental foundation, and it is often combined with other guidance laws, such as PNG and its variants [22]. Drawing lessons from the PNG laws design, a series of three-dimensional DG guidance laws has emerged and was qualitatively analyzed in Ref. [23][24][25][26][27]. Recently, a planar spacecraft rendezvous DG guidance law was developed in Ref. [28], which was the first attempt to apply the DG approach to a two-dimensional problem. ...
... According to Eq. (22), ε will be kept as zero when the missile flight distance is larger than μD f , and the design parameter μ can be used to adjust the convergence rate of the flight direction error. ...
This paper presents a new differential geometric guidance design methodology applied to an impact-time control (ITC) problem of constant-speed missiles subject to lateral control acceleration. For such an ITC problem, a simple circular arc guidance equation of desired heading angle is derived, its approximate closed-form solutions are obtained, and a circular predictive guidance (CPG) law is developed. Furthermore, a new dual-virtual-target (DVT) concept is incorporated into the CPG law to guide a missile along a straight-line flight path (with zero lateral acceleration) during its terminal gliding phase toward a stationary target. The integrated CPG-DVT strategy does not involve any form of time-to-go estimation or numerical iterations, and it can be easily employed to accommodate a variety of practical missile engagement requirements.
... For further information about (16) and (17), the reader is referred to Refs. [11]~ [14] . Equation (8) can be rewritten as (18) The geometric relationship between iM and iR is shown in Fig. 2, and so is the geometric relationship between iT and iR. ...
The capturability of two-dimensional (2D) pure proportional navigation (PPN) guidance law against lower-speed arbitrarily maneuvering target for homing phase had been thoroughly analyzed by using the nonlinear output regulation (NOR) method before. However, due to the complexity of the three-dimensional (3D) relative kinematics, the NOR method has not been applied to the capturability analysis of 3D PPN, which leads to the capturability discrepancy of 2D PPN and its 3D extension. Thanks to the 3D relative kinematic equation between the missile and target established in the rotating line of sight (LOS) coordinate system, the capturability of 3D PPN against the lower-speed arbitrarily maneuvering target for the homing phase is restudied by extending the NOR method of 2D PPN to the 3D space. The necessary and sufficient condition for the missile guided by 3D PPN to intercept this type of target is obtained. It is proven that the capturability of 3D PPN is identical with that of 2D PPN.
... Introduction of IRPL (instantaneous rotation plane of LOS) simplifies control of chaser; besides, it improves guidance efficiency with smaller miss distance and less fuel consumption. The applications of a rotating LOS coordinate system are also investigated in 3D missile intercept scenarios in references [15,16]. A set of dynamic equations is derived based on equation (1): ...
Guidance systems are important to autonomous rendezvous with uncooperative targets such as an active debris removal (ADR) mission. A novel guidance frame is established in rotating line-of-sight (LOS) coordinates, which resolves the coupling effect between pitch and yaw planes in a general 3D scenario. The guidance law is named augmented proportional navigation (APN) by applying nonlinear control along LOS and classical proportional navigation normal to LOS. As saving time is a critical factor in space rescue and on-orbit service, the finite time convergence APN (FTCAPN) is further proposed which proves to possess convergence and high robustness. This paper builds on previous efforts in polynomial chaos expansion (PCE) to develop an efficient analysis technique for guidance algorithms. A large scope of uncertainty sources are considered to make state evaluation trustworthy and provide precise prediction of trajectory bias. The simulation results show that the accuracy of the proposed method is compatible with Monte Carlo simulation which requires extensive computational effort.
... Drawing lessons from the PN guidance, a simple and explicit three-dimensional DG guidance algorithm was designed and its guidance performance was qualitatively analyzed by a Lyapunov-like approach in Refs. [17][18][19][20]. Apart from the above researches, some distinct guidance algorithms also have been proposed for missile interception using the differential geometry concepts in Refs. ...
... If the gravitational acceleration is assumed to be an explicit function of only time, the analytical optimal solution can be found [24,25]. The rendezvous problem can then be regarded as an optimization problem of determining the acceleration time history a(t) of the chaser subject to Eqs. (17) and (18) and a given set of initial and final conditions. ...
In this paper, the performance of two distinct classes of feedback guidance algorithms is evaluated for a spacecraft rendezvous problem utilizing a continuous low-thrust propulsion system. They are the DG (Differential Geometric) and ZEM/ZEV (Zero-Effort-Miss/Zero- Effort-Velocity) feedback guidance algorithms. Even though these two guidance algorithms do not attempt to minimize the onboard fuel consumption or ΔV directly, the ΔV requirement is used as a measure of their orbital rendezvous performance for various initial conditions and a wide range of the rendezvous time (within less than one orbital period of the target vehicle). For the DG guidance, the effects of its guidance parameter and terminal time on the closed-loop performance are evaluated by numerical simulations. For the ZEM/ZEV guidance, its nearfuel- optimality is further demonstrated for a rapid, short-range orbital rendezvous, in comparison with the corresponding open-loop optimal solutions. Furthermore, the poor ΔV performance of the ZEM/ZEV guidance for a slow, long-range orbital rendezvous is remedied by simply adding an initial drift phase. The ZEM/ZEV feedback guidance algorithm and its appropriate variants are then shown to be a simple practical solution to a non-impulsive rendezvous problem, in comparison with the DG guidance as well as the open-loop optimal guidance.
