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Terminal Time and Angle-Constrained Nonlinear Intercept Guidance
Shivam Bajpai, Abhinav Sinha
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 7/9/2026, 12:44:22 AM
Summary
This paper addresses the underactuated problem of simultaneously controlling an interceptor's impact time and impact angle using only lateral acceleration. It proposes a hierarchical sliding mode-based guidance law featuring a two-layer sliding manifold structure. The first layer defines sub-sliding surfaces for time and angle error dynamics, while the second layer combines them into a composite manifold. A variable-gain adaptive guidance law is designed to achieve precise terminal constraints for both stationary and constant-velocity targets, with simulations validating its effectiveness and flexibility.
Entities (6)
Relation Signals (6)
Lateral Acceleration → controls → Impact Time
confidence 95% · simultaneously controlling an interceptor's impact time and impact angle using its lateral acceleration as the sole control input
Lateral Acceleration → controls → Impact Angle
confidence 95% · simultaneously controlling an interceptor's impact time and impact angle using its lateral acceleration as the sole control input
Variable-Gain Adaptive Guidance Law → ensures → Time and Angle-Constrained Interception
confidence 92% · a variable-gain adaptive guidance law is designed to ensure time and angle-constrained interception against a stationary target
Sub-Sliding Surface → combinedinto → Composite Sliding Manifold
confidence 90% · the second layer introduces a composite sliding manifold that combines the two individual sub-surfaces
Hierarchical Sliding Manifold → comprises → Sub-Sliding Surface
confidence 90% · The proposed architecture consists of a two-layer sliding manifold. The first layer comprises two sub-sliding surfaces
Underactuated Kinematics → complicates → Guidance Law Synthesis
confidence 88% · With a single control input, the nonlinear engagement kinematics is inherently underactuated, which complicates guidance law synthesis
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Abstract
Abstract:This paper considers the problem of simultaneously controlling an interceptor's impact time and impact angle using its lateral acceleration as the sole control input. With a single control input, the nonlinear engagement kinematics is inherently underactuated, which complicates guidance law synthesis. To overcome this challenge, a hierarchical sliding mode-based guidance law is developed to concurrently regulate the two terminal constraints. The proposed architecture consists of a two-layer sliding manifold. The first layer comprises two sub-sliding surfaces corresponding to the impact time and impact angle error dynamics, respectively, while the second layer introduces a composite sliding manifold that combines the two individual sub-surfaces. Then, a variable-gain adaptive guidance law is designed to ensure time and angle-constrained interception against a stationary target, which is further extended to intercept a constant velocity target. Simulations are conducted for various engagement scenarios to attest to the efficacy of the proposed approach.
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- Source: https://arxiv.org/abs/2606.02872v1
- Canonical: https://arxiv.org/abs/2606.02872v1
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Terminal Time and Angle-Constrained Nonlinear Intercept Guidance Shivam Bajpai111Ph.D. Research Scholar, email: bajpaism@mail.uc.edu and Abhinav Sinha 222Assistant Professor, email: abhinav.sinha@uc.edu (Senior Member, AIAA). Abstract This paper considers the problem of simultaneously controlling an interceptor’s impact time and impact angle using its lateral acceleration as the sole control input. With a single control input, the nonlinear engagement kinematics is inherently underactuated, which complicates guidance law synthesis. To overcome this challenge, a hierarchical sliding mode-based guidance law is developed to concurrently regulate the two terminal constraints. The proposed architecture consists of a two-layer sliding manifold. The first layer comprises two sub-sliding surfaces corresponding to the impact time and impact angle error dynamics, respectively, while the second layer introduces a composite sliding manifold that combines the two individual sub-surfaces. Then, a variable-gain adaptive guidance law is designed to ensure time and angle-constrained interception against a stationary target, which is further extended to intercept a constant velocity target. Simulations are conducted for various engagement scenarios to attest to the efficacy of the proposed approach. 1 Introduction In last few years, rapid advancements in autonomous systems have necessitated the demand for terminal performance by requiring control over impact time and impact angle simultaneously to enhance mission effectiveness in adversarial scenarios. Control over the impact angle enables the interceptor to realize a prescribed terminal approach geometry, which is often essential for exploiting directional vulnerabilities or satisfying mission-dependent engagement requirements [1]. At the same time, control over the impact time is central to coordinated multi-interceptor operations, especially in salvo settings where temporal synchronization can significantly improve the likelihood of mission success against defended targets [2]. Despite substantial progress on impact-time guidance and impact-angle guidance as largely separate problems, the development of a unified guidance framework that achieves interception under both terminal constraints using only the interceptor’s lateral acceleration remains a challenging and still unresolved problem. A substantial body of literature has addressed the design of guidance laws for enforcing a prescribed impact angle at interception [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Much of this development has been rooted in proportional-navigation (PN)-type constructions [1, 3, 4, 5, 6, 7]. The earliest studies established the feasibility of shaping terminal approach geometry through modified PN laws, first for reentry guidance [1] and subsequently in the presence of first-order autopilot dynamics [3]. Later works introduced biased PN formulations to explicitly drive the impact-angle error to zero [4, 5], while other variants combined PN with pursuit-based strategies or switching logic to improve terminal angle regulation against stationary or nonmaneuvering targets [6, 7]. Beyond the PN framework, optimal-control-based formulations have also been developed to achieve impact-angle regulation from a more systematic synthesis perspective [8, 9]. These include linearized optimal guidance laws for moving-target interception [8] and linear time-varying designs derived through inverse optimality arguments to jointly reduce terminal miss and angle error [9]. In parallel, sliding-mode-based strategies have been proposed to enhance robustness with respect to target maneuvers, model uncertainty, and bounded disturbances [10, 11, 12]. Representative developments include dual-surface sliding-mode guidance with model-free target acceleration estimation for maneuvering targets [10], three-dimensional partial integrated guidance laws with angle constraints [11], and finite-time sliding-mode designs that guarantee convergence to the desired terminal geometry [12]. Against this backdrop, significant attention has also been devoted to guidance strategies that enforce impact time constraints, motivated by the need for simultaneous target interception and enhanced mission effectiveness [2, 13, 14, 15, 16, 17, 18, 19, 20, 21]. The earliest formulation of this problem appears in [2], where a guidance strategy was developed by augmenting the PN guidance with the feedback of impact time error. Subsequent efforts extended this line of research in several directions. A modified pure PN guidance law was proposed in [13] to achieve interception of stationary targets at a prescribed impact time under nonlinear engagement kinematics, while the work in [14] introduced a varying-gain PN guidance strategy capable of precise impact-time regulation without resorting to linearized engagement models. Beyond shaping the navigation constant, alternative design paradigms have also been explored. In [15], a data-driven impact-time guidance law was developed to reduce timing error, with particular relevance to hypersonic engagements involving substantial velocity variation. Sliding-mode-based formulations have likewise been employed, including the construction in [16], where a sliding surface combining relative range and time-to-go was used to guarantee interception at a prescribed time, and the three-dimensional nonlinear design in [17], which addressed impact-time-constrained interception within an SMC framework. In addition, the work in [22] developed a salvo guidance strategy against a stationary target. The authors in [19] derived a guidance strategy by solving a minimum-effort optimal control problem with fixed impact time and a quadratic approximation of the kinematic equations. Based on the relative virtual framework and the classical differential geometry curve theory, a polynomial guidance method was proposed in [20] to intercept maneuvering targets at the desired impact time. In [21], an output trajectory shaping guidance law with a Bezier curve was proposed for impact time control against a stationary target. Unlike the formulations that regulate only one terminal objective, the simultaneous treatment of both impact time and angle substantially increases the difficulty of the guidance design. Recently, the problem of simultaneously controlling impact time and impact angle has attracted significant attention. Early developments along this direction include radial-tangential guidance formulations in [23, 24]. The work in [23] developed multi-interceptor guidance laws based on finite-time sliding-mode control and the super-twisting algorithm, whereas the authors in [24] proposed a cooperative strategy founded on fixed-time control and a leader-follower architecture to achieve simultaneous arrival under prescribed impact-angle constraints. In parallel, several studies pursued time-to-go-estimation-independent designs [25, 26, 27, 28]. These include the shaping-function-based analytical construction in [25], the optimal-control formulation in [26], the time-varying lead-angle tracking law in [27], and the geometric guidance approach reported in [28]. Sliding-mode-based solutions have also continued to play a prominent role. The work in [29] addressed the simultaneous enforcement of impact time and impact angle through a dedicated sliding-mode structure, while [30] incorporated explicit time-to-go estimation within the sliding mode framework. Related developments include the polynomial time-to-go based design in [31] and the two-stage strategy in [32], which combines sliding mode and PN guidance through a virtual-target switching mechanism. In our previous work [33], this line of research was further extended to account for first-order autopilot dynamics, and the interception of a non-maneuvering target via the concept of predicted interception point [34]. Although the aforementioned studies have significantly advanced the state of the art, several limitations remain. Note that a class of previous methods is based on radial-tangential guidance formulations for regulating the desired impact time and impact angle. While such formulations are analytically convenient, their practical realizability is limited in many interceptor settings. The reason is that direct manipulation of the radial velocity component is generally not available as an independent control input. Instead, the interceptor is typically equipped to generate only lateral acceleration normal to its velocity vector, while its longitudinal speed is either fixed, weakly actuated, or constrained by propulsion and airframe limitations. Consequently, guidance laws that presuppose direct regulation of radial motion may not be consistent with the true control authority of the vehicle and may therefore be difficult to implement in realistic engagements. A different class of methods achieves simultaneous terminal constraint satisfaction by imposing a particular sliding-surface structure or by explicitly embedding a time-to-go estimate into the guidance design. Although such constructions can be effective under nominal conditions, they often inherit reduced flexibility and robustness. Hence, the guidance performance gets largely dictated by the chosen surface parameterization or by the fidelity of the chosen time-to-go estimate. Moreover, optimal-control-based and polynomial-parameterization-based schemes often incur a nontrivial computational burden, especially when extended to cooperative or multi-interceptor scenarios. In such settings, the simultaneous enforcement of impact time and angle constraints across multiple agents can lead to increased online computational complexity, which may limit scalability and real-time deployability. Accordingly, there remains a clear need for guidance designs that enforce both terminal constraints through the interceptor’s available lateral-acceleration input alone, without being tied to a particular sliding-mode structure or to explicit time-to-go estimation. We summarize the main contribution of this work below. A general hierarchical sliding-manifold structure is introduced to regulate the terminal objectives through submanifolds. This construction provides a systematic mechanism for decomposing the simultaneous impact-time and impact-angle regulation problem into interconnected objectives while preserving an integrated design architecture. Owing to this general construction, the proposed method is flexible, not tied to any particular time-to-go expression, and can accommodate arbitrary time-to-go estimates and varying target motions. A variable-gain adaptive guidance law is proposed for the simultaneous control of impact time and impact angle using the interceptor’s lateral acceleration as the sole control input. This is particularly important from a practical standpoint, since lateral acceleration constitutes the physically available guidance input in many interceptor systems, whereas direct regulation of radial motion is generally not realizable. By constructing the guidance law directly through the available control channel, the proposed design remains consistent with realistic interceptor control authority and is therefore more suitable for practical implementation. The resulting framework is modular and flexible, making it amenable to the incorporation of additional terminal or in-flight constraints without fundamentally altering the baseline guidance architecture. Any new constraint can be accommodated by introducing an associated submanifold, after which an augmented composite manifold may be constructed to integrate the new objective within the existing framework. The proposed guidance framework is formulated directly under nonlinear engagement kinematics, which avoids reliance on linearized approximations and broadens its applicability across diverse operating conditions. The design is first established for stationary targets and is then extended to non-maneuvering targets through the predicted interception point concept, thus enlarging the class of target scenarios that can be handled within a unified framework. The proposed guidance law is systematically compared with existing (simultaneous) impact time and impact angle guidance strategies to highlight its structural and practical advantages. In addition, the efficacy and generality of the proposed framework are further demonstrated by redesigning the guidance law using a different time-to-go expression, thereby showing that the methodology is not tied to a particular time-to-go model and can retain its effectiveness under alternative time-to-go constructions. 2 Problem Formulation As shown in Figure˜1, we consider a planar two-agent engagement scenario consisting of an interceptor (PP) and a target (TT). Figure 1: Interceptor-target engagement geometry in a plane. The interceptor is assumed to move with constant speed vPv_P and is tasked with intercepting the target subject to prescribed terminal constraints on both the impact time and the impact angle. The planar engagement assumption is appropriate for scenarios in which the interaction occurs at approximately constant altitude and captures the essential geometric features of many air-combat settings. In Figure˜1, r and θ denote the relative range and the line-of-sight (LOS) angle, respectively, while γP _P and σP _P represent the interceptor heading angle and lead angle. Under these definitions, the kinematics of relative motion between the agents are given by r˙= r= −vPcosσP, -v_P _P, (1a) rθ˙= r θ= −vPsinσP, -v_P _P, (1b) where σP=γP−θ _P= _P-θ, and γ˙P= γ_P= aPvP, ~ a_Pv_P, (2) where aPa_P is the interceptor’s sole control authority. Lemma 1. The dynamics of the lead angle of the interceptor has a relative degree of one with respect to its lateral acceleration, aPa_P. Proof. Differentiating the lead angle σP _P with respect to time yields σ˙P=γ˙P−θ˙, σ_P= γ_P- θ, (3) which can be further simplified using (1b) and (2) to σ˙P=aPvP+vPsinσPr. σ_P= a_Pv_P+ v_P _Pr. (4) It can be observed that from (4) that the interceptor’s lateral acceleration appears in the first derivative of the lead angle, indicating that it influences σPσ_P. This completes the proof. ∎ Lemma˜1 indicates that the lead angle responds directly to the interceptor’s lateral acceleration. Therefore, by modulating the lateral acceleration, one shapes the evolution of the lead angle, which in turn modifies the interceptor trajectory, the engagement geometry, and ultimately the terminal conditions of the engagement. Definition 1. Impact time, tft_f, is the time instant when the interceptor intercepts the target. At any instant of time, the remaining time till interception (also known as the time-to-go), tgot_go, can be defined as the difference between tft_f and the current time, t, that is tgo=tf−t_go=t_f-t. Definition 2. Impact angle, θimp _imp, is the angle between the velocity vectors of the target and the interceptor at the interception time. Mathematically, it is equivalent to θimp=γPf−γTf, _imp= _P_f- _T_f, (5) where γTf _T_f and γPf _P_f are the headings of the target and the interceptor at the interception time, respectively. Without loss of generality, one can consider the target’s heading, γTf _T_f, to be zero in the case of a stationary target, which results in θimp=γPf _imp= _P_f. When the interceptor is on a collision course, i.e. rθ˙=0r θ=0, it follows from (1b) θ˙=−vPsin(γPf−θd)r=0⟹−vPsin(θimp−θd)r=0. θ=- v_P ( _P_f- _d)r=0 - v_P ( _imp- _d)r=0. (6) where θd _d is the desired impact angle. Solving (6) over (−π,π](-π,π] yields two solutions as θd=θimp,π+θimp _d= _imp,~π+ _imp. Accordingly, the control objective is to design the interceptor lateral acceleration command such that the interceptor achieves interception at a prescribed impact time and a prescribed impact angle simultaneously. Problem. For given desired terminal values tdt_d and θd _d, design a nonlinear guidance law (the interceptor’s lateral acceleration command) that leads to impact time- and angle-constrained interception of a stationary target, that is, r(td)=0r(t_d)=0 and θimp(td)=θd _imp(t_d)= _d. 3 Impact Time- and Angle-Constrained Guidance Strategy From Definition˜1, it follows that the desired time-to-go tgod=td−t_go^d=t_d-t, and hence, regulating the impact time is essentially regulating the engagement duration time-to-go. As a representative formulation of the engagement duration, consider the time-to-go estimate against the stationary target, which accounts for larger heading angle errors [35], given by tgo=rvP(1+sin2σPK),t_go= rv_P (1+ ^2 _PK ), (7) where K=4N−2K=4N-2, with N≥3N≥ 3 denotes the navigation constant in PN guidance. It follows from (7) that the interception occurs when tgo=0t_go=0, which corresponds to r=0r=0 (and vice versa). Thus tgo=0⇔r=0t_go=0 r=0. Lemma 2. The dynamics of the time-to-go of the interceptor has a relative degree of one with respect to its lateral acceleration, aPa_P. Proof. Differentiating (7) with respect to time, one may obtain t˙go t_go =r˙vP+1KvP(r˙sin2σP+2rsinσPcosσPσ˙P). = rv_P+ 1Kv_P ( r ^2 _P+2r _P _P σ_P ). (8) Further simplification using the result from Lemma˜1 and substituting (1a) into (8) yields t˙go t_go =−cosσP+1KvP(2vPsin2σPcosσP−vPsin2σPcosσP+rsin2σPvPaP), =- _P+ 1Kv_P (2v_P ^2 _P _P-v_P ^2 _P _P+ r 2 _Pv_Pa_P ), =−cosσP(1−sin2σPK)+(rsin2σPKvP2)aP. =- _P (1- ^2 _PK )+ ( r 2 _PKv^2_P )a_P. (9) It is evident from (3) that the time-to-go dynamics possess a relative degree of one with respect to the lateral acceleration of the interceptor. ∎ Lemma˜2 shows that the interceptor’s lateral acceleration directly influences the time-to-go, which is critical for the design of time-constrained guidance law. Lemma 3. The dynamics of the LOS angle has a relative degree of two with respect to the interceptor’s lateral acceleration. Proof. Differentiating (1b) with respect to time yields θ¨ θ =−(vPcosσPσ˙Pr−vPsinσPr˙r2). =- ( v_P _P σ_Pr- v_P _P rr^2 ). (10) Using the results from Lemma˜1 and substituting (1a) into (10) leads to θ¨ θ =−(vPcosσPr(aPvP+vPsinσPr)+vP2sinσPcosσPr2) =- ( v_P _Pr ( a_Pv_P+ v_P _Pr )+ v^2_P _P _Pr^2 ) =−vP2sin2σPr2−cosσPraP. =- v_P^2 2 _Pr^2- _Pra_P. (11) From (3), it can be observed that the dynamics of the LOS angle is also influenced by the interceptor’s lateral acceleration. ∎ Consider the sub-sliding surface representing the error between the estimated time-to-go and the desired time-to-go, given as st=tgo−tgod=tgo−(td−t),s_t=t_go-t_go^d=t_go-(t_d-t), (12) where t˙god=−1 t_go^d=-1. Lemma 4. Consider the sub-sliding surface (12). The equivalent control associated with the manifold st=0s_t=0 is given by ut=−KvP2tanσP22rcosσP−vP2sinσP2r. u_t=- Kv_P^2 _P22r _P- v_P^2 _P2r. (13) Under this control, the manifold st=0s_t=0 is rendered invariant, that is, st=0⟹s˙t=0s_t=0 s_t=0. Proof. On differentiating (12) with respect to time, we obtain s˙t=t˙go+1. s_t= t_go+1. (14) Using the results from Lemma˜2, we obtain the dynamics of the sub-sliding surface sts_t as s˙t=1−cosσP(1−sin2σPK)+rsin2σPKvP2aP. s_t=1- _P (1- ^2 _PK )+ r 2 _PKv_P^2a_P. (15) By definition, the equivalent control is the continuous control input required to maintain motion on the sliding manifold once the trajectory has reached it. Hence, on the manifold st=0s_t=0, one imposes the invariance condition s˙t=0 s_t=0. Using the control-affine dynamics of sts_t in (15), this yields 0=1−cosσP(1−sin2σPK)+rsin2σPKvP2aP, 0=1- _P (1- ^2 _PK )+ r 2 _PKv_P^2a_P, (16) which leads to the expression in (13) after simplifying using cosσP=1−2sin2σP2 _P=1-2 ^2 _P2. ∎ Remark 1. The quantity (13) represents the nominal lateral acceleration command required to exactly preserve the time-channel manifold sts_t once it is reached. This essentially means that it characterizes the ideal control action associated with the impact time objective alone. In the present problem, however, the same input aPa_P must also regulate the impact angle channel. To enforce the impact angle constraints, we first define the impact angle error as eθ=θ−θd.e_θ=θ- _d. (17) Differentiating eθe_θ with respect to time yields e˙θ=θ˙=−vPsinσPr. e_θ= θ=- v_P _Pr. (18) The second time derivative of eθe_θ, together with the results from Lemma˜1, yields e¨θ=−vP2sin2σPr2−cosσPrap. e_θ=- v_P^2 2 _Pr^2- _Pra_p. (19) Now, consider the second sub-sliding surface corresponding to the impact angle error, given as sθ=eθ+me˙θc1/c2,s_θ=e_θ+m e^c_1/c_2_θ, (20) where m>0m>0 is a design parameter governing the rate of convergence of eθe_θ, whereas c1c_1 and c2c_2 are odd positive integers such that c1>c2c_1>c_2 and 1<c1/c2<21<c_1/c_2<2. Such a sub-sliding surface facilitates a finite-time error convergence once sliding mode is enforced. Lemma 5. Consider the sub-sliding surface (20). The equivalent control associated with the manifold sθ=0s_θ=0 is given by uθ=rc2mc1cosσP[(−vPsinσPr)(2−c1/c2)−mc1vP2sin2σPr2c2]. u_θ= rc_2mc_1 _P [ (- v_P _Pr )^(2-c_1/c_2)- mc_1v^2_P 2 _Pr^2c_2 ]. (21) Under this control, the manifold sθ=0s_θ=0 is rendered invariant, that is, sθ=0⟹s˙θ=0s_θ=0 s_θ=0. Proof. On differentiating (20) with respect to time, one may obtain s˙θ=e˙θ+(mc1c2)e˙θ(c1/c2−1)e¨θ. s_θ= e_θ+ ( mc_1c_2 ) e_θ (c_1/c_2-1 ) e_θ. (22) Substituting (18) and (19) in (22) yields s˙θ= s_θ= −vPsinσPr+mc1c2(−vPsinσPr)(c1/c2−1)(−vP2sin2σPr2−cosσPrap) - v_P _Pr+ mc_1c_2 (- v_P _Pr )^(c_1/c_2-1) (- v_P^2 2 _Pr^2- _Pra_p ) = = −(vPsinσPr+mc1vP2sin2σPr2c2(−vPsinσPr)(c1/c2−1))−mc1cosσPrc2(−vPsinσPr)(c1/c2−1)aP. - ( v_P _Pr+ mc_1v^2_P 2 _Pr^2c_2 (- v_P _Pr )^(c_1/c_2-1) )- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)a_P. (23) On the manifold sθ=0s_θ=0, one imposes the invariance condition s˙θ=0 s_θ=0. Using the control-affine dynamics of sθs_θ in (3), this yields 0=−(vPsinσPr+mc1vP2sin2σPr2c2(−vPsinσPr)(c1/c2−1))−mc1cosσPrc2(−vPsinσPr)(c1/c2−1)aP, 0=- ( v_P _Pr+ mc_1v^2_P 2 _Pr^2c_2 (- v_P _Pr )^(c_1/c_2-1) )- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)a_P, (24) which leads to the expression in (21) after simplifying using sin2σP=4sinσP2cosσP2cosσP 2 _P=4 _P2 _P2 _P. ∎ Remark 2. From (15) and (3), it is evident that the dynamics of both sub-sliding surfaces depend on the interceptor’s lead angle. Moreover, since the interceptor has access only to a single control input, namely the lateral acceleration aPa_P, the resulting guidance problem, as seen from (15) and (3), is underactuated in nature. Remark 3. The validity of the equivalent-control expressions in Lemmas˜4 and 5 requires that the corresponding input gain remain nonzero along the engagement. This essentially means that the equivalent-control construction is well defined only when σP _P avoids the singular configurations associated with σP∈0,π/2,π _P∈\0,π/2,π\. The treatment of such isolated cases and the conditions under which they can be excluded or systematically handled will be addressed later to demonstrate that the proposed design is nonsingular throughout. The underactuated nature of the problem constitutes difficulty in the guidance design since improving performance with respect to one objective may directly influence the evolution of the other. In other words, the same lateral acceleration designed to stabilize sts_t alone does not guarantee stabilization of sθs_θ and vice versa. The challenge is to design a single unified lateral acceleration that can drive both sts_t and sθs_θ to 0 simultaneously. To satisfy the terminal constraints on both time and angle, we consider a composite sliding surface as s=λst+sθ,s=λ s_t+s_θ, (25) where λ is a time-varying design parameter. Lemma 6. The dynamics of the composite sliding surface possesses a relative degree of one with respect to the interceptor’s lateral acceleration. Proof. Differentiating (25) with respect to time yields s˙=λ˙st+λs˙t+s˙θ. s= λs_t+λ s_t+ s_θ. (26) On substituting (15) and (3) in (26), one may obtain s˙ s =λ˙st+λ(1−cosσP(1−sin2σPK)+rsin2σPKvP2aP)−(vPsinσPr+mc1vP2sin2σPr2c2(−vPsinσPr)(c1/c2−1)) = λs_t+λ (1- _P (1- ^2 _PK )+ r 2 _PKv_P^2a_P )- ( v_P _Pr+ mc_1v^2_P 2 _Pr^2c_2 (- v_P _Pr )^(c_1/c_2-1) ) −mc1cosσPrc2(−vPsinσPr)(c1/c2−1)aP, - mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)a_P, (27) which indicates that the dynamics of the composite sliding surface is related to the lateral acceleration command with a relative degree of one. ∎ We now present the proposed lateral acceleration command aPa_P in the next theorem. Theorem 1. Consider the interceptor-target engagement kinematics whose relative motion is governed by (1), the time-to-go formulation in (7), and the impact angle error (17). The proposed interceptor’s lateral acceleration command, aP= a_P= rc2mc1cosσP[(−vPsinσPr)(2−c1/c2)−mc1vP2sin2σPr2c2]−KvP2tanσP22rcosσP−vP2sinσP2r−ψ¯1sign(s)−ψ¯2s, rc_2mc_1 _P [ (- v_P _Pr )^(2-c_1/c_2)- mc_1v^2_P 2 _Pr^2c_2 ]- Kv_P^2 _P22r _P- v_P^2 _P2r- ψ_1~sign(s)- ψ_2s, (28) where ψ¯1= ψ_1= ψ1χ2+(−mc1cosσPrc2)(−vPsinσPr)(c1/c2−1)+λ(χ1+(rsin2σPKvP2)), _1 _2+ (- mc_1 _Prc_2 ) (- v_P _Pr )^(c_1/c_2-1)+λ ( _1+ ( r 2 _PKv_P^2 ) ), (29) ψ¯2= ψ_2= ψ2χ2+(−mc1cosσPrc2)(−vPsinσPr)(c1/c2−1)+λ(χ1+(rsin2σPKvP2)). _2 _2+ (- mc_1 _Prc_2 ) (- v_P _Pr )^(c_1/c_2-1)+λ ( _1+ ( r 2 _PKv_P^2 ) ). (30) are adaptive gains such that the design parameters satisfy ψ1,ψ2>0 _1, _2>0, χ1,χ2≥0 _1, _2≥ 0, ensures that the target is intercepted at the prescribed impact time and impact angle. Proof. We design the lateral acceleration as a sum of three components: aP=ut+uθ+uc,a_P=u_t+u_θ+u_c, (31) where utu_t and uθu_θ are equivalent control terms corresponding to (15) and (3), respectively (see Lemmas˜4 and 5), and ucu_c is to be designed as a switching term that ensures sliding mode convergence. To derive the switching control law uctu_ct, consider a Lyapunov function candidate as V=s22. V= s^22. (32) Differentiating V with respect to time yields V˙ V =ss˙, =s s, =s(λs˙t+λ˙st+s˙θ). =s (λ s_t+ λs_t+ s_θ ). (33) Using the results in Lemma˜6, one may write V˙ V =s[λ˙st+λ(1−cosσP(1−sin2σPK)+rsin2σPKvP2aP)−(vPsinσPr+mc1vP2sin2σPr2c2(−vPsinσPr)(c1/c2−1)) =s [ λs_t+λ (1- _P (1- ^2 _PK )+ r 2 _PKv_P^2a_P )- ( v_P _Pr+ mc_1v^2_P 2 _Pr^2c_2 (- v_P _Pr )^(c_1/c_2-1) ) . −mc1cosσPrc2(−vPsinσPr)(c1/c2−1)aP]. .- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)a_P ]. (34) Using (31) and the results from Lemma˜4 and Lemma˜5, one may write V˙= V= s[λrsin2σPKvP2(uθ+uct)+λ˙st−mc1cosσPrc2(−vPsinσPr)(c1/c2−1)(ut+uc)], s [λ r 2 _PKv^2_P (u_θ+u_ct )+ λs_t- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1) (u_t+u_c ) ], = = s[λrsin2σPKvP2uθ−mc1cosσPrc2(−vPsinσPr)(c1/c2−1)ut s [λ r 2 _PKv^2_Pu_θ- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)u_t . +λ˙st+(λrsin2σPKvP2−mc1cosσPrc2(−vPsinσPr)(c1/c2−1))uc]. .+ λs_t+ (λ r 2 _PKv^2_P- mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1) )u_c ]. (35) To ensure the stability of the composite sliding surface, s, let the design parameter, λ, evolve according to the adaptation law [36] λ˙=−λ(χ1+rsin2σPKvP2)uθst‖st‖2+Ω−χ2+(−mc1cosσPrc2)(−vPsinσPr)(c1/c2−1)utst‖st‖2+Ω λ=-λ ( _1+ r 2 _PKv_P^2 )u_θs_t||s_t||^2+ - _2+ (- mc_1 _Prc_2 ) (- v_P _Pr )^(c_1/c_2-1)u_ts_t||s_t||^2+ (36) where Ω≥0 ≥ 0 is a design parameter which keeps λ˙ λ non-singular. It is apparent from (36) that λ converges to a constant value as st→0s_t→ 0. Then one has, λ˙st λs_t =mc1cosσPrc2(−vPsinσPr)(c1/c2−1)ut−λrsin2σPKvP2uθ = mc_1 _Prc_2 (- v_P _Pr )^(c_1/c_2-1)u_t-λ r 2 _PKv^2_Pu_θ (37) and the switching component is chosen as uc u_c =−ψ¯1sign(s)−ψ¯2s, =- ψ_1~sign(s)- ψ_2s, (38) where ψ¯1 ψ_1 and ψ¯2 ψ_2 are given in (29)–(30). On substituting the expressions from (37) and (38) into (3), one may obtain V˙ V =−ψ1ssign(s)−ψ2s2, =- _1~s~sign(s)- _2s^2, =−ψ1|s|−ψ2s2, =- _1|s|- _2s^2, (39) which is negative definite for all s≠0s≠ 0, thus ensuring the stability of the composite sliding mode dynamics. Moreover, integrating (3) gives V(t)−V(0)=∫0t(−ψ1|s|−ψ2s2)t, V(t)-V(0)= _0^t (- _1|s|- _2s^2 )dt, (40) which implies V(0)= V(0)= V(t)+∫0t(ψ1|s|+ψ2s2)t≥∫0t(ψ1|s|+ψ2s2)t. V(t)+ _0^t ( _1|s|+ _2s^2 )dt≥ _0^t ( _1|s|+ _2s^2 )dt. (41) From (41), it follows that limt→∞∫0t(ψ1|s|+ψ2s2)t<∞. _t→∞ _0^t ( _1|s|+ _2s^2 )dt<∞. (42) From (42), we infer that limt→∞∫0tψ1|s|t<∞ _t→∞ _0^t _1|s|dt<∞ and limt→∞∫0tψ2s2t<∞ _t→∞ _0^t _2s^2dt<∞. This implies s∈ℒ1s _1 and s∈ℒ2s _2. From (32) and (41), we obtain s22=V(t)=V(0)−∫0t(ψ1|s|+ψ2s2)t<∞;∀t≥0, s^22=V(t)=V(0)- _0^t ( _1|s|+ _2s^2 )dt<∞; ∀~t≥ 0, (43) which implies that s∈ℒ∞s _∞, thus showing that the composite sliding surface s is bounded. From (3), we have dVdt=sdsdt=−ψ1|s|−ψ2s2<∞∀t≥0, dVdt=s dsdt=- _1|s|- _2s^2<∞ ∀~t≥ 0, (44) so that s˙∈ℒ∞ s _∞. Therefore, by Barbalat’s lemma, the composite sliding surface, s, is asymptotically stable. Once the sliding mode is enforced, (25) becomes equal to zero. To guarantee target interception at the prescribed impact time and angle, it is essential to ensure that both sts_t and sθs_θ become zero thereafter. To this end, we now demonstrate that the sub-sliding surfaces, sts_t, and sθs_θ are also asymptotically stable after sliding mode is enforced on s. The time horizon can be divided into two phases by the time instant tαt_α. In the interval [0,tα][0,t_α], the system state trajectories move towards the composite sliding surface, s. In the interval (tα,∞)(t_α,∞), the trajectories stay on the composite sliding surface and converge to the origin. To examine the stability of the sub-sliding surfaces within the interval [0,tα][0,t_α], define a truncation operator as htα(t)=1,t≤tα0,t>tαh_t_α(t)= \ aligned &1, t≤ t_α\\ &0, t>t_α aligned . (45) and define the truncated sub-sliding surface variables as sitα=sihtα(t),s˙itα=s˙ihtα(t), s_i_t_α=s_ih_t_α(t), s_i_t_α= s_ih_t_α(t), (46) where i∈t,θi∈\t,θ\. With this construction333Here s˙i,tα s_i,t_α is defined as above rather than obtained by differentiation. It is to be considered only as an analysis device., the analysis is restricted to the interval [0,tα][0,t_α], over which si,tαs_i,t_α and s˙i,tα s_i,t_α coincide with sis_i and s˙i s_i, respectively. For t>tαt>t_α, both truncated variables vanish identically. Consequently, si,tαs_i,t_α and s˙i,tα s_i,t_α remain bounded on [0,tα][0,t_α], and once sliding mode is enforced at tαt_α, the truncated representation is consistent with convergence to the corresponding sliding manifold. In the subsequent interval (tα,∞)(t_α,∞), the state trajectory has reached the composite sliding manifold s and the reduced-order dynamics evolve autonomously on this manifold. To characterize the limiting behavior, define the compact positively invariant set Sw≔s∈ℝ2|V˙(s)≤0, S_w \s ^2 | V(s)≤ 0 \, (47) and the set Su≔s∈Sw|V˙(s)=0. S_u \s∈ S_w | V(s)=0 \. (48) Since V˙≤0 V≤ 0 in SwS_w, LaSalle’s invariance principle implies that every trajectory starting in SwS_w approaches, as t→∞t→∞, the largest invariant set contained in SuS_u. It therefore remains to identify this invariant set. On the composite manifold, s=λst+sθ. s=λ s_t+s_θ. (49) Hence, any invariant motion in SuS_u must satisfy λst+sθ=0,dt(λst+sθ)=0. λ s_t+s_θ=0,~~ ddt (λ s_t+s_θ )=0. (50) The only invariant solution consistent with V˙=0 V=0 is the coordinate origin of the sub-sliding-surface space, namely st=0,sθ=0 s_t=0,~~s_θ=0 (51) Therefore, the largest invariant set contained in SuS_u reduces to the origin. By LaSalle’s invariance principle, it follows that the sub-sliding surfaces sts_t and sθs_θ also converge asymptotically to zero. Hence, the sub-sliding surfaces are asymptotically stable. After sliding mode is enforced on sts_t, the interceptor aligns on the requisite trajectory that would lead to a time-constrained interception since st=0s_t=0 implies that tgo=tgodt_go=t_go^d from (12). Furthermore, after sθ=0s_θ=0, the impact angle error (17) also converges to zero. ∎ Remark 4. If (51) does not hold, then the composite sliding surface would converge to a point other than the coordinate origin defined by the axes sts_t and sθs_θ. In the phase plane spanned by sts_t and sθs_θ, limt→∞s=β;β∈ℝ∖0. _t→∞s=β;~β \0\. (52) This indicates that the composite sliding surface is not asymptotically stable, which contradicts the asymptotic stability of s. Remark 5. Following Lemma˜4 and (31), one may write s˙t=1−cosσP(1−sin2σPK)+rsin2σPKvP2(ut+uθ+uc), s_t=1- _P (1- ^2 _PK )+ r 2 _PKv_P^2 (u_t+u_θ+u_c ), (53) which, after the composite surface is reached, is equivalent to s˙t=rsin2σPKvP2uθ, s_t= r 2 _PKv_P^2u_θ, (54) since uc=0u_c=0 in the ideal sliding-mode sense. If we let W=12st2W= 12s_t^2, then it follows that sts˙t=strsin2σPKvP2uθ. s_t s_t=s_t r 2 _PKv_P^2u_θ. (55) To guarantee that st→0s_t→ 0, one may impose a sufficient condition on the reduced-order dynamics such that sts˙t≤−ϱtst2=−2ϱtW;∀ϱt>0, s_t s_t≤- _ts_t^2=-2 _tW;~~∀~ _t>0, (56) so one has W(t)≤W(tr)exp−2ϱt(t−tr). W(t)≤ W(t_r) \-2 _t (t-t_r ) \. (57) The above result implies that limt→∞st→0 _t→∞s_t→ 0. On the composite manifold, from (50), sθ=−λsts_θ=-λ s_t. Assume that ∃λ¯>0∃~ λ>0 such that supt≥tr|λ(t)|≤λ¯ _t≥ t_r|λ(t)|≤ λ. Then, |sθ(t)|≤|λ(t)||st(t)|≤λ¯|st(t)|. |s_θ(t)|≤|λ(t)|\,|s_t(t)|≤ λ|s_t(t)|. (58) Since st→0s_t→ 0, it follows immediately that sθ→0s_θ→ 0. Thus, the proposed strategy allows for time correction first, and thereafter angle correction is effected once sθs_θ (and therefore eθe_θ) converges to zero. Recall Remark˜3 that the equivalent-control expressions remain well defined only when σP∉0,π/2,π _P∉\0,π/2,π\. This motivates the following analysis of equilibrium points of the lead angle dynamics and nonsingularity conditions, through which the admissible operating region of the proposed guidance command in Theorem˜1 is characterized. Theorem 2. Consider the interceptor-target engagement kinematics whose relative motion is governed by (1), the time-to-go formulation in (7), and the impact angle error (17). Under the proposed interceptor’s lateral acceleration command (28), the lead angle dynamics admits no equilibrium at σP=±π2 _P=± π2. Moreover, the configuration σP=π _P=π corresponds to inverse collision course, whereas σP=0 _P=0 can occur only at the final impact time if σP(0)≠0 _P(0)≠ 0. Proof. Recall the lead angle dynamics from Lemma˜1 and note that σP=π _P=π corresponds to an inverse-collision geometry and is incompatible with the interception scenario of interest. Hence, this configuration is excluded from the admissible operating region. Next, consider σP(0)=0 _P(0)=0. If the prescribed terminal conditions satisfy tf=r(0)/vPt_f=r(0)/v_P and θd=0 _d=0, then the interceptor is already on the required collision course and no maneuver is needed, so that σP(t)≡0 _P(t)≡ 0 and aP(t)≡0a_P(t)≡ 0. This is a degenerate but nonsingular case. Excluding this trivial situation, if tf≠r(0)/vPt_f≠ r(0)/v_P and σP(0)≠0 _P(0)≠ 0, then σP=0 _P=0 cannot occur for any t<tft<t_f. Otherwise, the interceptor would already be on the final collision course before satisfying the full terminal requirements. The same argument holds even if σP(0)=0 _P(0)=0. In this case, since tf≠r(0)/vPt_f≠ r(0)/v_P, the lead angle will immediately deviate from this zero initial value to place the interceptor on the requisite trajectory. Consequently, we focus our attention on the steady-state, i.e., after the respective errors have converged. Hence, for nontrivial engagements, σP=0 _P=0 is attained only at interception. This may be confirmed by obtaining a relation between the range-to-go and the lead angle, assuming a small lead angle. Substituting (28) into (4) leads to σ˙P= σ_P= rc2mvPc1cosσP[(−vPsinσPr)(2−c1/c2)−mc1vP2sin2σPr2c2]−KvPtanσP22rcosσP−vPsinσP2r+vPsinσPr. rc_2mv_Pc_1 _P [ (- v_P _Pr )^(2-c_1/c_2)- mc_1v^2_P 2 _Pr^2c_2 ]- Kv_P _P22r _P- v_P _P2r+ v_P _Pr. (59) Using (1a) and (59), one may write drdσP drd _P =−vPcosσP(−vPsinσPr)(2−c1/c2)(rc2mvPc1cosσP)−2vPsinσPr−KvPtanσP22rcosσP−vPsinσP2r+vPsinσPr = -v_P _P (- v_P _Pr )^(2-c_1/c_2) ( rc_2mv_Pc_1 _P )- 2v_P _Pr- Kv_P _P22r _P- v_P _P2r+ v_P _Pr =−vPcosσP(−vPsinσPr)(2−c1/c2)(rc2mvPc1cosσP)−KvPtanσP22rcosσP−3vPsinσP2r. = -v_P _P (- v_P _Pr )^(2-c_1/c_2) ( rc_2mv_Pc_1 _P )- Kv_P _P22r _P- 3v_P _P2r. (60) In the terminal phase (near interception), the interceptor’s lead angle is typically small because its velocity is nearly aligned with the LOS angle. Then, (3) can be simplified further as drdσP= drd _P= −vP(−vPσPr)(2−c1/c2)(rc2mvPc1)−KvPσP4r−3vPσP2r ~ -v_P (- v_P _Pr )^(2-c_1/c_2) ( rc_2mv_Pc_1 )- Kv_P _P4r- 3v_P _P2r = = −vP(vPc2σP2mrc1)(−rvPσP)c1/c2−(vPσPr)(K4+32) ~ -v_P ( v_Pc_2 _P^2mrc_1 ) ( -rv_P _P )^c_1/c_2- ( v_P _Pr ) ( K4+ 32 ) (61) under the assumption of a small lead angle in the terminal phase. Upon taking the reciprocal of (3) and after further simplifications, one has dσPdr d _Pdr =σPr(K4+32)−c2σP2mrc1(−rvPσP)c1/c2, =~ _Pr ( K4+ 32 )- c_2 _P^2mrc_1 ( -rv_P _P )^c_1/c_2, (62) which, upon integrating from r(0)r(0) to r (with corresponding σP(0) _P(0) to σP _P) yields σP=r(K4+32)(σP(0)r(0)(K4+32)−c1−c2mc1vpc1/c2r(0)1−α1−α+c1−c2mc1vPc1/c2r1−α1−α), _P=r ( K4+ 32 ) ( _P(0)r(0) ( K4+ 32 )- c_1-c_2mc_1v_p^c_1/c_2 r(0)^1-α1-α+ c_1-c_2mc_1v_P^c_1/c_2 r^1-α1-α ), (63) where α=(c1−c2)(K−2)4c2α= (c_1-c_2)(K-2)4c_2. Since r(0)≠0r(0)≠ 0, vP≠0v_P≠ 0, α≠1α≠ 1, and the design parameters m,c1,c2m,c_1,c_2 are also nonzero, it is now evident from (63) that the lead angle vanishes only at the time of interception (i.e., when r→0r→ 0), and hence aPa_P remains nonsingular in the steady-state. It remains to exclude σP=±π/2 _P=±π/2. For this purpose, suppose, for contradiction, that σP=π2 _P= π2 is an equilibrium point, which means once the lead angle reaches π2 π2, it stays there, thereby σ˙P=0 σ_P=0. Then it follows from (59) that KvPtanσP2=mc1vP2sinσPcosσP+2r2c2[(−vPsinσPr)(2−c1/c2)−mc1vP2sin2σPr2c2]mc1vP. Kv_P _P2= mc_1v^2_P _P _P+2r^2c_2 [ (- v_P _Pr )^(2-c_1/c_2)- mc_1v^2_P 2 _Pr^2c_2 ]mc_1v_P. (64) If σP=π2 _P= π2, then (64) reduces to KvP Kv_P =2r2c2mc1(−vPr)2−c1/c2. = 2r^2c_2mc_1 (- v_Pr )^2-c_1/c_2. Solving (3) for r yields r=[Kmc12c2(−vP)c1/c2]c2/c1. r= [ Kmc_12c_2 (-v_P )^c_1/c_2 ]^c_2/c_1. (66) In (66), the LHS varies with time, while the RHS remains constant, and thus the equality cannot be satisfied. Hence, σP=π2 _P= π2 is not an equilibrium point. The case for σP=−π2 _P=- π2 can be argued similarly. ∎ It can be inferred from Theorem˜2 that the proposed guidance command, (28), is nonsingular by design everywhere, except in the situation when the trajectories cross isolated points σP=±π2 _P=± π2 in the transient phase. Since the interceptor’s lateral acceleration remains bounded in practice, such isolated points, even if encountered, do not pose any implementation issues. Figure˜2 shows a typical r−σPr- _P plane where one may observe that the σP→0 _P→ 0 as r→0r→ 0 for any nontrivial case. Figure 2: Range versus lead angle for different initial and terminal conditions showing that the lead angle vanishes at the impact time only. To demonstrate the generality of the proposed framework, we redesign the guidance law using a different time-to-go expression and show that the proposed approach retains its effectiveness under this alternative formulation. Let the time-to-go be chosen as [30] tgo=rvP(1+σP2+σPf215−σPσPf30), t_go= rv_P (1+ _P^2+σ^2_P_f15- _P _P_f30 ), (67) where σPf=γPf−θ, _P_f= _P_f-θ, (68) is the lead angle at the impact, and γPf _P_f is the interceptor’s heading at that time. Lemma 7. The dynamics of the interceptor’s time-to-go in (67) has a relative degree of one with respect to its lateral acceleration. Proof. Differentiating the time-to-go estimate, (67), with respect to time, yields t˙go=r˙vP(1+σP2+σPf215−σPσPf30)+rvP(215(σPσ˙P+σPfσ˙Pf)−130(σ˙PσPf+σPσ˙Pf)). t_go= rv_P (1+ _P^2+ _P_f^215- _P _P_f30 )+ rv_P ( 215 ( _P σ_P+ _P_f σ_P_f )- 130 ( σ_P _P_f+ _P σ_P_f ) ). (69) Using the results from Lemma˜1 and substituting for the term, σ˙Pf σ_P_f by differentiating (68) with respect to time, (69) can be simplified as t˙go=r˙vP(1+σP2+σPf215−σPσPf30)+rvP(2σP15vPaP−2σP15θ˙−2σPf15θ˙+σP30θ˙−σPf30vPaP+σPf30θ˙). t_go= rv_P (1+ _P^2+σ^2_P_f15- _P _P_f30 )+ rv_P ( 2 _P15v_Pa_P- 2 _P15 θ- 2 _P_f15 θ+ _P30 θ- _P_f30v_Pa_P+ _P_f30 θ ). (70) On rearranging the terms above yields t˙go=r˙vP(1+σP2+σPf215−σPσPf30)−rθ˙10vP(σP+σPf)+rvP2(2σP15−σPf30)aP. t_go= rv_P (1+ _P^2+σ^2_P_f15- _P _P_f30 )- r θ10v_P ( _P+ _P_f )+ rv_P^2 ( 2 _P15- _P_f30 )a_P. (71) Substituting the range rate and LOS rate from (1a) and (1b), (71) can be further simplified to t˙go=−cosσP(1+σP2+σPf215−σPσPf30)+sinσP(σP+σPf)10+rvP2(2σP15−σPf30)aP. t_go=- _P (1+ _P^2+σ^2_P_f15- _P _P_f30 )+ _P ( _P+ _P_f )10+ rv_P^2 ( 2 _P15- _P_f30 )a_P. (72) It can be observed from (72) that this time-to-go dynamics also possesses a relative degree of one with respect to the interceptor’s lateral acceleration. ∎ Lemma˜7 reveals that the lateral acceleration design corresponding to the time-to-go expression in (67) follows directly along the same lines as the previous development, because (7) and (67) share the same relative degree. Therefore, we present the following results without proof. Lemma 8. Consider the sub-sliding surface defined in (12). Suppose the time-to-go dynamics induced by the chosen time-to-go expression is given by (72). Then, the equivalent control associated with the manifold st=0s_t=0 is given by ut u_t =−(1−cosσP(1+σP2+σPf215−σPσPf30)+sinσP(σP+σPf)10)30vP2r(4σP−σPf). =- (1- _P (1+ _P^2+σ^2_P_f15- _P _P_f30 )+ _P ( _P+ _P_f )10 )30v_P^2r (4 _P- _P_f ). (73) Under this control, the manifold st=0s_t=0 remains invariant, which implies s˙t=0 s_t=0. Lemma˜5 still holds since the impact angle error and its definition remain unchanged under the adoption of a different time-to-go dynamics. Hence, we present the modified lateral acceleration directly in the next theorem. Theorem 3. Consider the interceptor-target engagement kinematics whose relative motion is governed by (1), the time-to-go formulation in (67), and the impact angle error (17). The proposed interceptor’s lateral acceleration command, aP a_P =−(1−cosσP(1+σP2+σPf215−σPσPf30)+sinσP(σP+σPf)10)30vP2r(4σP−σPf)−ψ¯1sign(s)−ψ¯2s =- (1- _P (1+ _P^2+σ^2_P_f15- _P _P_f30 )+ _P ( _P+ _P_f )10 )30v_P^2r (4 _P- _P_f )- ψ_1sign(s)- ψ_2s −rc2mc1cosσP(−vPsinσPr)(c1/c2−1)[vPsinσPr+mc1vP2sin2σPr2c2(−vPsinσPr)(c1/c2−1)], - rc_2mc_1 _P (- v_P _Pr )^(c_1/c_2-1) [ v_P _Pr+ mc_1v^2_P 2 _Pr^2c_2 (- v_P _Pr )^(c_1/c_2-1) ], (74) where ψ¯1=ψ1χ2+(−mc1cosσPrc2)(−vPsinσPr)(c1/c2−1)+λ(χ1+λr(4σP−σPf)30vP2), ψ_1= _1 _2+ (- mc_1 _Prc_2 ) (- v_P _Pr )^(c_1/c_2-1)+λ ( _1+ λ r (4 _P- _P_f )30v_P^2 ), (75) ψ¯2=ψ2χ2+(−mc1cosσPrc2)(−vPsinσPr)(c1/c2−1)+λ(χ1+λr(4σP−σPf)30vP2), ψ_2= _2 _2+ (- mc_1 _Prc_2 ) (- v_P _Pr )^(c_1/c_2-1)+λ ( _1+ λ r (4 _P- _P_f )30v_P^2 ), (76) are adaptive gains such that the design parameters satisfy ψ1,ψ2>0 _1, _2>0, χ1,χ2≥0 _1, _2≥ 0, ensures that the target is intercepted at the prescribed impact time and impact angle. Proof. Similar to the proof of Theorem˜1, one has aP=ut+uθ+uc,a_P=u_t+u_θ+u_c, (77) where utu_t is obtained from Lemma˜8, and uθu_θ is the same as given in Lemma˜5. The corrective term ucu_c is also designed to have the same structure as in the proof of Theorem˜1. Thereafter, one may choose the same Lyapunov function candidate (32) and follow the procedures outlined in the proof of Theorem˜1 to obtain the modified command (74) under the time-to-go (67). ∎ The proposed strategy may be extended for the impact time- and angle-constrained interception of a moving non-maneuvering target by leveraging the concept of predicted interception point (PIP) [34]. This point is a virtual or predicted point at which the target is expected to be intercepted. The interceptor steers its heading towards the PIP, where it perceives the moving target as stationary. If the actual position of the target coincides with the PIP at the desired impact time, then the target is guaranteed to be captured. In Figure˜3, the actual relative range between the target and the interceptor is represented by r. However, the interceptor aims to capture the moving target at P, whose relative range is given by rPr_P, and the corresponding LOS angle in this direction is predicted LOS. The position of the target at PIP for the desired impact time tdt_d can be written as xT(td)=xT(t)+vTcosγT(tgo), x_T(t_d)=x_T(t)+v_T _T(t_go), (78a) yT(td)=yT(t)+vTsinγT(tgo), y_T(t_d)=y_T(t)+v_T _T(t_go), (78b) where (xT(t),yT(t)) (x_T(t),y_T(t) ) and (xT(td),yT(td)) (x_T(t_d),y_T(t_d) ) represent the target coordinates at time instants t and tdt_d, and σT=γT−θ _T= _T-θ is the lead angle of the target. rrvTtgov_Tt_gorpr_pXLX_LPPTTPIPγT _TPredicted LOS Figure 3: Illustration of the PIP for interception of a moving target. 4 Simulation Results We now demonstrate the performance of the proposed approach via simulations. Without loss of generality, we assume that the heading of the target is 0∘0 . The interceptor’s speed remains fixed at 150150 m/s, and the lateral acceleration bound on the interceptor is |20||20| g, where g is the acceleration due to gravity. In all trajectory plots, the initial positions of the target and the interceptor are depicted by red and black diamond markers (⋄)( ), respectively. The navigation constant N is selected as 33, thus K=10K=10. The effectiveness of the proposed framework against a stationary target is shown in Figure˜4 for various different initial and terminal conditions. The trajectories of the interceptor are depicted in Figure˜4(a). It can be observed that by adopting the guidance command (28), the interceptor captures the target successfully in all cases. Figure˜4(b) illustrates that the interceptor follows the required trajectories to intercept the target at the desired impact time. The time-to-go profiles exhibit initial deviation, which arises from their trajectories initially deviating and subsequently converging toward the desired path (as shown in Figure˜4(a)). This signifies that the interceptor takes a detour necessary to attain the requisite time-constrained geometry. The interceptor’s lateral acceleration profiles are demonstrated in Figure˜4(c). It is apparent from Figure˜4(c) that during the transient phase, the interceptor requires high lateral acceleration to achieve course correction, followed by a smooth convergence to zero in the terminal phase. As shown in Figure˜4(d), the LOS angles converge to their desired values in the terminal phase, whereas sliding mode is enforced on the composite sliding surfaces within around 2020 s in all cases. The lead angles in Figure˜4(e) show initial variations due to initial detour, but then decrease to zero monotonically after sliding mode is enforced. This is consistent with the results established in Theorem˜2. The range profiles are demonstrated in Figure˜4(f). It is apparent that the individual ranges converge to zero at the desired impact time, thereby satisfying the impact-time constraint. Note that both range and the lead angle are zero at the impact time. (a) Interceptor’s trajectories. (b) Time-to-go profiles. (c) Lateral acceleration profiles. (d) LOS angles and composite sliding manifold profiles. (e) Lead angle profiles. (f) Range profiles. Figure 4: Performance of the proposed strategy against a stationary target. Next, simulations are performed for the interception of a non-maneuvering target. The target’s heading, impact time, and impact angle are fixed at 0∘0 , 100100 s, and 135∘135 , respectively. The target is moving with a speed of 6565 m/s. The initial range and the LOS angle between the interceptor and the target are 77 km and 150∘150 , respectively, while the initial heading of the interceptor is 45∘45 . The design parameters are c1=11,c2=9,m=185,η=75,κ=1.8,χ1=χ2=0.2c_1=11,c_2=9,m=185,η=75,κ=1.8, _1= _2=0.2, and Ω=5 =5. The simulation results are demonstrated in Figure˜5. From the agents’ trajectories plot in Figure˜5(a), it is evident that the moving target is intercepted successfully, indicating the effectiveness of the proposed approach. The time-to-go profile in Figure˜5(b) confirms that the interceptor achieves the target interception at the desired impact time. The control input of the interceptor is portrayed in Figure˜5(c), indicating a higher control effort requirement in the transient phase for course correction, followed by a smooth convergence to zero in the steady state. The LOS angle and the composite sliding manifold profiles are shown in Figure˜5(d). The LOS angle converges to its desired value monotonically to help achieve the desired impact angle. The composite sliding surface increases from a negative value and ultimately converges to zero smoothly. As shown in Figure˜5(e), the lead angle initially increases as the interceptor detours, followed by a smooth convergence to zero during the terminal phase. The range profile depicted in Figure˜5(f) decreases to zero as the lead angle converges to zero. (a) Agents’ trajectories. (b) Time-to-go profile. (c) Lateral acceleration (steering control). (d) LOS angle and composite sliding manifold profiles. (e) Lead angle profile. (f) Range profile. Figure 5: Performance of the proposed strategy against a moving target. (a) Trajectories. (b) Lateral accelerations. (c) LOS angles. (d) Range profiles. (e) Lead angles’ profiles. (f) Sliding manifold profiles. Figure 6: Performance comparison of the proposed guidance strategy with the existing guidance strategies. The performance of the proposed strategy is also compared with the ones in [29] and [30], and the corresponding results are shown in Figure˜6. The initial range and the LOS angle between the interceptor and the target are 77 km and 150∘150 , respectively, while the initial heading of the interceptor is 45∘45 . The desired impact time and impact angle are set to 6060 s and 110∘110 . All simulations are performed under the same conditions. It is evident from Figure˜6(a) that, under all guidance strategies, the target is intercepted successfully, though the interceptor’s trajectories are different. The lateral acceleration profiles are depicted in Figure˜6(b). It can be observed that the guidance strategy proposed in [30] results in a sharp drop in lateral acceleration at the time instant when the sliding mode is enforced, which is due to the time-varying weight assigned in the beginning to allow a correction to the impact time. Such correction is unnecessary in the proposed design. One may also notice that both the proposed strategy and that in [29] have smooth lateral acceleration profiles. Furthermore, the acceleration demand is compared based on the control effort calculation using ∫t0tfaP2t _t_0^t_fa_P^2\,dt. Based on this integral-of-square effort, the lateral acceleration demand using the strategy in [29] is 157%157\% higher than the proposed strategy, whereas the control demand using [30] is 13%13\% higher. The LOS angle profiles are depicted in Figure˜6(c). It can be observed that the LOS angle profile converges to its desired value faster using the strategy in [29], while for the proposed strategy and the strategy presented in [30], the LOS profiles converge to their desired value in the terminal phase to meet the impact angle constraint. Forcing angle correction early may require larger lateral acceleration demand, and could be too stringent. The range profiles of the interceptor are depicted in Figure˜6(d), indicating that range profiles converge to zero at the prescribed impact time. The proposed approach thus demonstrates effectiveness even under large initial heading errors, successfully caters to non-maneuvering targets, and operates independently of any specific sliding surface or time-to-go formulation. The lead angle profiles are shown in Figure˜6(e). It is apparent from the profiles that the lead angle profile converges to zero faster using [29], as the LOS angle converges to zero, while for the proposed strategy and the strategy presented in [30], the lead angles converge to zero in the terminal phase. The sliding manifold profiles are depicted in Figure˜6(f). The sliding manifold converges to zero within around 55 s using the strategy in [30], while for the proposed strategy and the strategy proposed in [29], the sliding manifold profiles converge to zero in around 4040 s. The performance of the proposed guidance strategy is further compared with the ones in [29] and [30] in the presence of an autopilot modeled as a first-order lag with a time-constant of 0.10.1 s. The simulation results for this case are presented in Figure˜7. The initial conditions are kept the same as in the previous case of Figure˜6. The trajectories of the interceptor are depicted in Figure˜7(a), where the interceptor successfully intercepts the target under all guidance strategies under the presence of autopilot lag. The lateral acceleration profiles are shown in Figure˜7(b), indicating a similar trend as in Figure˜6(b). From the control effort calculation, the lateral acceleration demand for the strategy in [29] is now 132%132\% higher than the proposed strategy, while the control demand for the strategy in [30] is 32%32\% high. The LOS angle profiles are represented in Figure˜7(c), showing a similar behavior as in the absence of autopilot lag (Figure˜6(c)). The range profiles are demonstrated in Figure˜7(d), where it converges to zero at the desired impact time under all guidance strategies. The lead angle profiles are shown in Figure˜7(e). Similar to the previous case (Figure˜6(e)), the lead angle profile converges to zero faster using the strategy in [29], while for the proposed strategy and the strategy presented in [30], the lead angles converge to zero in the terminal phase. The sliding manifold profiles are demonstrated in Figure˜7(f), showing a similar trend as in the absence of autopilot lag (Figure˜6(f)). The performance comparison of control efforts in different guidance laws is presented in Table˜1. Reference Guidance Law Control Effort without Autopilot (m2/s) Control Effort with Autopilot (m2/s) [29] aP=−vP2sinσPr−vPksignc2/c1(s)+4vP2r(cosσP−1)eθr2eθ2+4(vPtd−vPt−r)2+4vP2(cosσPeθ+sinσP)(vPtd−vPt−r)r2eθ2+4(vPtd−vPt−r)2 aligned a_P=&- v_P^2 _Pr-v_Pksign^c_2/c_1(s)\\ &+ 4v^2_Pr ( _P-1 )e_θr^2e_θ^2+4 (v_Pt_d-v_Pt-r )^2\\ & +4v^2_P ( _Pe_θ+ _P ) (v_Pt_d-v_Pt-r )r^2e_θ^2+4 (v_Pt_d-v_Pt-r )^2 aligned 8008.9 6112.8 [30] aP=−kft+(e˙θ−mc1c2e˙θc1/c2−12r˙θ˙r)tgod−td(k−(eθ+me˙θc1/c2)(tgod−td)2)gt+ht−(eθ+me˙θc1/c2)(tgod−td)2(ft−1)(k−(eθ+me˙θc1/c2)(tgod−td)2)gt+ht aligned a_P=&- kf_t+ ( e_θ- mc_1c_2 e_θ^c_1/c_2-1 2 r θr )t_go^d-t_d (k- (e_θ+m e_θ^c_1/c_2 ) (t_go^d-t_d )^2 )g_t+h_t\\ &- - (e_θ+m e_θ^c_1/c_2 ) (t_go^d-t_d )^2 (f_t-1 ) (k- (e_θ+m e_θ^c_1/c_2 ) (t_go^d-t_d )^2 )g_t+h_t aligned where ft=1+r˙vP(1+σP2+σPf215−σPσPf30)+rθ˙vP(−2(σP+σPf)15+σP+σPf30), aligned where f_t&=1+ rv_P (1+ _P^2+ _P_f^215- _P _P_f30 )\\ &+ r θv_P (- 2 ( _P+ _P_f )15+ _P+ _P_f30 ), aligned gt=(rvP2(2σP15−σPf30)) aligned g_t&= ( rv_P^2 ( 2 _P15- _P_f30 ) ) aligned, ht=−mc1e˙θc1/c2−1cosσPrc2(tgod−td) aligned h_t=- mc_1 e_θ^c_1/c_2-1 _Prc_2 (t_go^d-t_d ) aligned 3513.3 3479.2 Proposed Given in (28) 3108.1 2634.8 Table 1: Comparison of performance of various guidance laws. (a) Trajectories. (b) Lateral accelerations. (c) LOS angles. (d) Range profiles. (e) Lead angle profiles. (f) Sliding manifold profiles. Figure 7: Performance comparison of proposed strategy with existing guidance strategies under first-order autopilot lag. The robustness of the proposed approach with respect to another time-to-go estimate given in (67), is validated through the simulation results next. The initial range and LOS angle between the interceptor and the target are 66 km and 170∘170 , respectively, while the initial heading of the interceptor is 70∘70 . The desired time and angle are set to 4545 s and 110∘110 , respectively. The design parameters are c1=9,c2=7,m=250,η=0.1,κ=5,χ1=χ2=0c_1=9,c_2=7,m=250,η=0.1,κ=5, _1= _2=0, and Ω=0.75 =0.75. The simulation results for this case are shown in Figure˜8. The interceptor’s trajectory is illustrated in Figure˜8(b), from which one may notice that the interceptor successfully intercepts the target even if the time-to-go is different. The time-to-go profile is depicted in Figure˜8(b), in which it is apparent that the target is intercepted at the desired impact time. Additionally, even though the initial time-to-go estimate is greater than the desired impact time, the interceptor effectively adjusts its trajectory using the strategy following the command in (74) to ensure interception at the desired impact time. The interceptor’s lateral acceleration profile is depicted in Figure˜8(c). It can be observed that the interceptor requires higher control authority in the transient phase to achieve its course correction, followed by a smooth convergence close to zero. Furthermore, at around 4040 s, there appears a smooth increase in the lateral acceleration demand to achieve the impact angle constraint. The LOS angle and the composite sliding manifold are demonstrated in Figure˜8(d). The LOS angle approaches its desired value in the terminal phase. The composite sliding surface rapidly converges to zero and remains on it. The lead angle in Figure˜8(e) converges to zero at the interception instant. As depicted in Figure˜8(f), the range eventually converges to zero as the lead angle decreases, which is consistent with the previous cases. (a) Interceptor’s trajectory. (b) Time-to-go profile. (c) Lateral acceleration (steering control). (d) LOS angle and composite sliding manifold profiles. (e) Lead angle profile. (f) Range. Figure 8: Performance of the proposed strategy for the time-to-go (67) for tf=45t_f=45 s and θd=110∘ _d=110 . The variations of the adaptive gain (λ) for different initial and terminal conditions are depicted in Figure˜9. It is apparent from the plot that the gain varies during the transient phase as the interceptor adjusts its course to drive the impact time error to zero, followed by convergence to a constant value in the steady state. Additionally, it is evident that the gain does not cross zero, indicating that the sub-sliding surface associated with the impact angle error, sθs_θ, converges to zero when the sub-sliding surface associated with the impact time error, sts_t, approaches zero, which is consistent with Remark˜5. Figure 9: Variation of adaptive gain (λ). 5 Conclusions In this work, we designed an adaptive guidance strategy to control both impact time and impact angle simultaneously using the interceptor’s lateral acceleration as the sole control input. The proposed approach was built upon a hierarchical sliding mode framework comprising two layers– the first layer consists of two sub-sliding surfaces associated with impact time and impact angle error variables, while the second layer integrates these sub-sliding surfaces into a composite sliding surface. An adaptive gain was assigned to the sub-sliding surface associated with impact time. 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