## Abstract

This paper addresses the time-critical rendezvous problem for a pursuing autonomous unmanned vehicle, e.g., an unmanned aerial vehicle (UAV), guided using the concept of true proportional-navigation guidance, which is a variant of proportional-navigation guidance. In existing vehicle routing and flight time-constrained guidance techniques, specific rendezvous guidance commands are designed based on the specific motion of the target. In contrast to that, we propose a unified guidance command for a UAV that guarantees a time-critical rendezvous with a target that moves arbitrarily. We explore the purview of true proportional-navigation guidance and posit that a guidance law thus designed may be a potential candidate for designing time-critical rendezvous strategies against various target motions, even when the pursuer does not necessarily have a speed advantage over the target. We first derive a closed-form expression for the flight duration until rendezvous, over which we exercise control to make the pursuing vehicle rendezvous with the target at any feasible time prescribed a priori. Next, we ensure that the necessary flight-time-based error variable converges to zero with an optimal convergence pattern with respect to a suitable cost function. We finally validate the efficacy of the proposed unified guidance command via numerical simulations.

## 1 Introduction

Autonomous unmanned vehicles play an essential role in a variety of military and civilian applications, such as exploration, search and rescue, target capture, localization, cargo delivery, and monitoring, to name a few. As such, accurate motion planning for autonomous vehicles is crucial, e.g., see, Refs. [1–10], especially when the given application places a constraint on its trajectory. Meanwhile, the widespread usage of unmanned aerial vehicles also poses several potential threats, including safety, security, privacy, and environmental threats. According to an article published in the *New York Times* [11], drone attacks inflicted damage on installations responsible for processing the majority of Saudi Arabia’s crude oil production, heightening the potential for disruption in global oil supplies. To avert such incidents and safeguard against potential threats, extensive research endeavors have been undertaken to effectively thwart the intrusion of rogue autonomous vehicles. These efforts can be broadly categorized into two primary approaches. The first method focuses on incapacitating the threat through non-lethal techniques like jamming or signal manipulation [12], whereas the second technique entails the physical neutralization or containment of the intruding vehicle [13]. In the context of the second technique, the application of the proportional-navigation (PN) guidance approach, a well-established method commonly employed in target interception, has found extensive use.

However, terminal constraints on the pursuing vehicle’s trajectory are known to greatly enhance its interception capability [14–16]. A particular constraint on an autonomous vehicle’s trajectory is its time of rendezvous at a certain location or with a mobile target/adversary, and this has received significant attention in recent times (e.g., see, Refs. [8,14,17–21] and references therein). Since the time of rendezvous is a terminal constraint on the autonomous vehicle’s trajectory, controlling its flight duration is a challenging problem, particularly because of the lack of a closed-form expression for the vehicle’s flight duration against an adversary that moves arbitrarily.

In recent literature, most of the time-constrained guidance techniques have been presented assuming that the adversary is a stationary point [19,21–25] or moves with a constant velocity [26–31]. Against a stationary target, time-critical rendezvous guidance was first presented in Ref. [22] where a biased pure proportional-navigation guidance (pure proportional-navigation command with flight-time-based error feedback) was used, and the engagement kinematics were linearized. Following the seminal work of Jeon et al. [22], a vast majority of time-constrained guidance strategies to date rely on pure proportional-navigation guidance or its close variants. Note that such strategies have shown satisfactory performance against stationary targets in a time-critical scenario. However, the lack of a closed-form expression for the duration of the flight time, otherwise known as vehicle-target engagement duration or the *time-to-go*, against a mobile target presents formidable challenges in the time-critical motion planning of autonomous vehicles.

To alleviate the challenges associated with the lack of obtaining a closed-form expression for the engagement duration, some works have proposed to resort to a predicted intercept position (PIP) guidance [32–34]. In such a scheme, the mobile target is deemed a virtual stationary target at a predicted rendezvous location in the future, and the pursuing autonomous vehicle issues a guidance command that steers it toward the future rendezvous position. There are several limitations to this method. The prediction of rendezvous position may be computationally intensive if the target is aggressive. Moreover, the predicted location is just an approximation and, as such, may be erroneous if there are errors in the estimation of the engagement duration. Recently, deviated pursuit-based time-critical guidance has been proposed in Ref. [26] that mitigates the limitations of PIP. However, it cannot be used to rendezvous with a stationary target because it demands an unbounded control effort toward interception, which may not be feasible in practice.

In this paper, we approach the time-critical rendezvous guidance from a different perspective, based on the concept of *true proportional-navigation* guidance. While the true PN guidance has been explored for target interception without any constraints [35], we explore the purview of the true PN guidance to enforce a time-critical rendezvous guidance behavior for an autonomous vehicle. At this point, we clarify that the fundamental difference between the pure and the true PN guidance approaches is in the manner the guidance command is applied. In the pure PN principle, the vehicle is steered using acceleration proportional to the vehicle’s line-of-sight angular rate (lateral acceleration) such that the acceleration is applied perpendicular to the velocity vector, whereas speed control (radial acceleration) is typically unavailable. On the other hand, the acceleration is perpendicular to the line-of-sight in the true PN approach. Consequently, both radial and tangential components are available as steering controls. Such an approach can be utilized in agile autonomous unmanned vehicles to capture intruders or adversaries.

The novelty of this paper lies in designing a unified guidance command that allows an autonomous vehicle to rendezvous with a target at a time specified a priori and that the proposed design remains valid for a wider class of autonomous vehicles. In contrast to the standard intercept guidance literature (where different guidance principles are employed to intercept targets that move differently), the main focus of this paper is to present an optimal time-constrained guidance law that is applicable against all kinds of targets (stationary, constant velocity, and arbitrarily maneuvering). We summarize and list the merits of the present paper below.

We exploit true proportional-navigation as a potential candidate in rendezvous guidance scenarios, an area hitherto unexplored when constraints on the terminal time of interception are imposed. This concept paves a new way of designing guidance strategies (in addition to existing methods) for a wider class of autonomous vehicles where the pursuing vehicle’s trajectory has terminal time constraints.

We design a unified guidance command for an autonomous vehicle that is steered by its acceleration components in radial and lateral channels against a target that has only lateral acceleration capability. The pursuing autonomous vehicle may be an unmanned aerial vehicle (e.g., multirotor) or a marine surface vehicle (ships or vessels) for practical purposes, while the mobile target may be a fixed-wing unmanned aerial vehicle (reconnaissance and tactical vehicles) or a marine decoy.

The expression of time-to-go indicates that if the target remains stationary or moves with a constant velocity, then the exact value of the interception time may be known a priori. This observation is unlike time-to-go expressions realized using pure PN guidance (e.g., as in Refs. [8,14] with/without small heading angle assumptions, respectively). However, if the target moves arbitrarily, then the same expression could still be used as a reasonable approximation that is corrected over time. The same expression also underlines the fact that possessing a speed advantage over the target is not a stringent condition for the pursuer due to the availability of radial steering controls. Additionally, the desired time of rendezvous with the target may be set arbitrarily, respecting the physical constraints.

The proposed guidance strategy is meticulously designed in a manner to force the (flight-time-based) error variable to zero with an optimal convergence pattern. Introducing a notion of optimality in the proposed design is aimed at reducing energy expenditure during the engagement.

The proposed unified guidance command does not induce any switching or multi-phase structure in the design, resulting in less abrupt and more predictable changes in the pursuing vehicle’s motion. Furthermore, fine-tuning and precise adjustments of the design parameters in the guidance command become straightforward because of the absence of any discrete modes.

It is important to note that consideration of the aforementioned class of vehicles for pursuing autonomous vehicle and the target represents many practical applications, both military and civilian. It would not be correct to say that the target is weaker because it only has lateral acceleration capabilities or the pursuing vehicle is stronger owing to acceleration capabilities in radial and lateral channels. However, such consideration only represents the manner in which a particular vehicle is steered. Having precise rendezvous guidance under constraints on terminal time could be leveraged in aerial cinematography, search and rescue in complex landscapes, time-critical recharging of aerial vehicles landing on ground vehicles/stations, reducing human intervention in coverage control, etc. Furthermore, the current work is focused on developing theoretical foundations for unified time-constrained guidance to ensure the pursuer rendezvouses with a target capable of executing various target motions. The goal is to develop a single robust and reliable guidance strategy that can adapt to different target motions. Therefore, we emphasize the mathematical models, algorithms, and frameworks that will underpin the guidance system under time-constrained rendezvous. In the future, we plan to validate and test the theoretical foundations through experiments and simulations in the next stage of our research. This will involve implementing the guidance system in a real-world setting or a simulated environment to evaluate its performance, robustness, and effectiveness in achieving successful rendezvous with targets exhibiting various motions.

## 2 Problem Statement

*a*)

*b*)

*c*)

*d*)

*e*)

The goal of this paper is to design a unified guidance strategy for the pursuing vehicle that enables it to rendezvous with the target at a prescribed time appointed a priori regardless of the target’s motion. Our design follows the above nonlinear kinematic model to circumvent any errors arising due to linear approximations. Effectively, the proposed strategy can guarantee a rendezvous even when the engagement happens far from the collision course.

## 3 Main Results

We first obtain a closed-form expression of the time of arrival/rendezvous, followed by the derivation and analysis of the proposed guidance law.

### 3.1 Analytical Computation of the Time of Arrival/Rendezvous.

Owing to the unpredictable motion of the target, obtaining an accurate expression of the engagement duration when the target moves arbitrarily may not be tractable. However, for all intents and purposes, we first derive an analytical expression for the engagement duration, assuming that the target is non-maneuvering, and show that the approximation remains valid even when the target moves arbitrarily.

From (13), one may observe that the time remaining till rendezvous is a function of the relative distance between the vehicles and their relative velocity components. Thus, one may speculate that manipulating these variables through a suitable guidance command may be sufficient to manipulate the engagement duration.

Now it readily follows that the nullification of time-to-go is both necessary and sufficient for rendezvous since $r=0$ implies $tgo=0$ and vice versa when $c$ is chosen large enough to satisfy $c> >(VM+VT)/2$ [36]. It is also worth noting that the above condition (nullification of $tgo$ for rendezvous) holds regardless of the target’s maneuver, $aT$. This essentially points to the fact that ensuring $tgo=0$ is essential to the pursuing vehicle’s guidance system to rendezvous with the target and that it is independent of the latter’s maneuver. However, when the target does not maneuver, that is, when $aT=0$, the expression in (14) is an exact value by design. Otherwise, it is merely an estimate, and rendezvous can still be guaranteed if $tgo=0$.

### 3.2 Proposed Guidance Law for Prescribed Time of Arrival.

Our objective, now, is to design a suitable $aMc$ so that the error, $e(t)$, converges to zero optimally, which is, essentially, to design a corrective command for time-critical rendezvous. While many existing guidance laws are focused on designing finite/fixed-time error convergence at the expense of high control magnitude (e.g., see Refs. [26,25,30,37] and references therein), we propose to drive the necessary error variable to zero in an optimal fashion with respect to a meaningful performance index. Toward this objective, we recall an important result related to optimal error convergence.

#### [38]

The corrective control component, together with the nominal one obtained in (19), yields the expression for the acceleration given in (31).

We now consider a radially unbounded Lyapunov function candidate, $W=(1/2)e2$, whose time derivative is given by $W\u02d9=ee\u02d9$. On substituting the expression for $aM$ given by (31) in $W\u02d9$, one may obtain $W\u02d9=\u2212(K/tgo)e2$. This clearly points to the fact that $W$ and hence $e$ will eventually decay (with a rate in accordance with the desired error dynamics), thus steering the pursuing vehicle on a time-critical rendezvous course with the target.

The first term in (31) is the baseline true proportional-navigation command. The second term in (31) compensates for the target’s motion, whereas the third term is responsible for correcting the time-to-go error. The variable gain in the third term in (31) improves the convergence toward the steady state. Toward rendezvous, $V\theta \u21920$, however, since $c> >(VM+VT)/2$, the numerator in (31) vanish earlier than the denominator, thus preventing any singularity in the proposed guidance command. Furthermore, the tangential/lateral and radial acceleration components, given by $aMt=aMcos\theta M$ and $aMr=aMsin\theta M$, respectively, are used to steer the pursuer toward the target.

## 4 Simulations

We now demonstrate through simulations the efficacy of the proposed time-critical rendezvous guidance in various engagement scenarios. Our aim is to establish a foundational understanding that can be applied in various contexts. We assume that the pursuer, which is located at the origin, is initially separated by the target at 1000 m radially with a $0$ deg line-of-sight angle. The pursuing vehicle has an initial speed of $50m/s$, and an initial heading angle of $10$ deg. We consider three different cases based on the motion of the target vehicle (stationary, constant velocity, or maneuvering). In each case, the controller gain, $K=3$, and the parameter, $c=3(VM+VT)$. In each plot, a black square marker is the initial position of a vehicle, while a circle marker is used to denote the presence of a stationary target.

Figure 2 depicts a typical scenario where the time of rendezvous is chosen as 40 s. Based on the initial engagement geometry, the target’s initial location can be computed as $(rcos\theta ,rsin\theta )$, which is essentially $(1000,0)$. In the first case, the target remains stationary, whereas it moves with a constant velocity of 20 m/s in the second one and maneuvers according to $aT=5sin(\pi t/50)\u22123$ m/s^{2} in the third case with $VT=20$ m/s. In practice, sensors mounted on the pursuing vehicle can determine the target’s velocity. The initial heading angle of the target is chosen as 60 deg. Figure 2(a) shows the trajectories of the adversaries for various engagements, which portrays the different routes taken by the pursuer to rendezvous with the target. It can be seen from Fig. 2(b) that irrespective of the target’s motion, the pursuer is able to rendezvous with the former at exactly the time specified a priori. Figure 2(c) shows the steering control inputs, that is, the tangential and the radial acceleration components. It is seen from Fig. 2(c) that the steering control inputs are smooth, and their magnitudes start to decrease as the engagement proceeds. Eventually, the terminal control magnitudes are near zero, which is an alluring feature in autonomous vehicle guidance. Figure 2(d) depicts the pursuer’s lead angle and the velocity profile during the engagements. One may observe that the lead angle becomes zero in the case of stationary and constant velocity targets. For these cases, the velocity magnitudes are also lesser compared to the case of the maneuvering target, where the lead angle does not necessarily become zero.

We also subject the pursuer to various rendezvous time requirements when the target executes different motions starting from the same location. From Figs. 3–5, we can see that the pursuing vehicle is required to rendevous with the target at various times specified before the engagement starts. In all these cases, the pursuer adjusts its trajectories toward the target in such a way as to arrive at the target at exactly the time specified a priori. For the results shown in Figs. 3–5, the initial heading angle of the pursuer is taken $20$ deg, and the target maneuvers according to $aT=1\u22120.5[cos(\pi t/50)+sin(\pi t/50)]m/s2$. It is also seen from Figs. 3–5 that the pursuer has to detour more and thus increases its lead angle when the rendezvous time is prescribed a large value. However, regardless of that, the pursuer can rendezvous (exactly at the prescribed time) with the target capable of executing various motions. This further attests to the superior performance of the proposed strategy when compared to other guidance laws that are designed against specific motions of the target.

The proposed unified guidance command does not place a stringent requirement on the pursuer to always have a speed advantage over the target. This is unlike most previous guidance strategies, where the pursuer must possess a speed advantage (or have an equal speed in a few cases) to rendezvous with the target. In Figs. 6 and 7, the target moves with a speed of $50$ m/s and maneuvers according to $aT=3+0.5[cos(\pi t/50)\u22123sin(\pi t/100)]m/s2$, whereas the pursuer’s initial velocity is $40m/s$. Figures 6 and 7 point to the fact that the pursuer does not have a speed advantage over the target at all times, yet it is able to rendezvous with a target that moves arbitrarily. In Fig. 6, the time of rendezvous is chosen as a small value ($tf=50s$) to demonstrate the quick response of the proposed guidance strategy. On the other hand, Fig. 7 illustrates that the pursuer is also able to modulate its speed and adjust its trajectory to rendezvous with the target (which is maneuvering arbitrarily) at a larger value of terminal time ($tf=150s$). Inferences from Figs. 6 and 7 lead us to conclude that a true proportional-navigation-inspired optimal guidance strategy may yield the desired performance in engagements that require the pursuer to be agile. As seen from the profiles of the acceleration components in Figs. 6 and 7, one may notice the smoothness and consistent behavior in line with the previous cases.

Since the overall design is based on an optimal error convergence pattern, the time-to-go converges to its desired value optimally in all the cases. This is unlike any finite/fixed-time guidance laws and thus prevents the high control magnitudes at the beginning of the engagement. The very nature of the proposed guidance strategy makes it suitable to be employed in various applications, e.g., target capture, rendezvous guidance, and pursuit-evasion, where the adversary may execute various kinds of motions to satisfy its objective.

## 5 Conclusions

Using the concept of true proportional-navigation guidance, we proposed a unified guidance command to enable a pursuing vehicle to rendezvous with a target capable of exhibiting various motions. We first computed the closed-form expression for the time of rendezvous and then designed a guidance law that steers the pursuing vehicle toward the target in an optimal manner. The time-to-go thus obtained is found to be exact when the target remains stationary or moves with a constant velocity. However, the same expression becomes a reasonable estimate when the target moves arbitrarily. Unlike pure proportional-navigation and deviated pursuit-based laws in the literature, the proposed strategy may apply to time-critical rendezvous problems for various targets. Extending the proposed time-critical rendezvous guidance under incomplete information and investigating actors such as aerodynamics, sensor resolution, noise, communication latency, etc., as well as real-time experimental evaluations may be interesting to explore in the future.

## Acknowledgment

This work was supported in part by the Ministry of Electronics and Information Technology, India, via capacity building for Human Resource Development in Unmanned Aircraft Systems (Drone and Related Technology)—Guidance Navigation and Control (Reference No. L-14011/29/2021-HRD).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.