UAV Circumnavigating an Unknown Target Under a GPS-denied Environment with Range-only Measurements

# UAV Circumnavigating an Unknown Target Under a GPS-denied Environment with Range-only Measurements

## Abstract

One typical application of unmanned aerial vehicles is the intelligence, surveillance, and reconnaissance mission, where the objective is to improve situation awareness through information acquisition. For examples, an efficient way to gather information regarding a target is to deploy UAV in such a way that it orbits around this target at a desired distance. Such a UAV motion is called circumnavigation. The objective of the paper is to design a UAV control algorithm such that this circumnavigation mission is achieved under a GPS-denied environment using range-only measurement. The control algorithm is constructed in two steps. The first step is to design a UAV control algorithm by assuming the availability of both range and range rate measurements, where the associated control input is always bounded. The second step is to further eliminate the use of range rate measurement by using an estimated range rate, obtained via a sliding-mode estimator using range measurement, to replace actual range rate measurement. Such a controller design technique is applicable in the control design of other UAV navigation and control missions under a GPS-denied environment.

{keywords}

UAV, Autonomy, Joint estimation and control, Sliding-mode estimator, GPS-denied environment

## I Introduction

As Unmanned Aerial Vehicles (UAVs) gain more and more favor in both military and civilian applications due to its advantages over manned aircraft, it is now a demanding technology that UAVs can be used to accomplish missions with minimum human supervision. In many cases, complete autonomy is often desired in UAV operations unless human supervision is necessary. Technologies are thus needed to increase autonomy for UAV operations [1].

The need of autonomy for UAV operations comes from two perspectives: performance and cost. From the performance’s point of view, UAV with autonomy capability is more reliable than manned aircraft. Human operators are one of the most common sources of errors in complex systems. In many situations, human operators are more likely to make mistakes. It is also well-known that efficiency of human operators is expected to decrease after a certain period of time. All these drawbacks can be addressed if autonomy becomes available to replace human operators. From the cost’s point of view, UAV with autonomy is less expensive than manned aircraft as the cost of training and/or replacing a pilot is high. Another factor of cost is that manned aircraft are generally larger in size and more complex in capabilities than UAVs due to the existence of human operators onboard.

Due to FAA regulations, UAVs are now mainly used in military and security applications, such as border patrol [2], mapping [3], and surveillance [4]. For instance, UAVs can be used to gather information of a target by orbiting around it at some desired distance. Such a UAV mission is often called circumnavigation. If GPS is available, both target location and UAV location can be obtained and this circumnavigation mission can be accomplished by using existing orbiting algorithms [5]. However, the vulnerability of UAVs to GPS jamming and spoofing poses a significant threat to the safety of UAV operation. Recent tests confirm that GPS can be denied due to jamming [6] or spoofing [7]. In military operations, GPS is more often jammed due to contested environment under which UAVs are operated. Refs. [8, 9, 10, 11, 12] were devoted to achieve circumnavigation under a GPS-denied environment. A localization-and-control framework was first proposed in [8, 9] to solve the circumnavigation problem. The main idea is to first estimate the location of the target based on the location of the UAV under some local coordinate frame and then design controllers based on estimated target location and additional bearing/range measurement. Under a GPS-denied environment, the location of the UAV can be tracked down using inertial sensors at the cost of integration drift. An increasing drift means performance degradation. It is thus desired that the location of the UAV is not used in the controller design. A different control strategy was developed in [10, 11] where both range and range rate measurements are used to design control algorithms. Rigorous analysis was provided in [10, 11] to demonstrate the efficacy of the proposed control law when the UAV does not stay close to the target initially. Due to the nature of local stability, it is necessary to design a controller that guarantees global convergence. To overcome the local stability issue, a “aiming” controller using both range and range rate measurements was developed in [12]. The aiming controller is to control the heading of the UAV such that the UAV moves towards a well-designed circle. Global stability was shown in [12] regardless of the initial state of the UAV. Although range measurement is possible under a GPS-denied environment [13], the need of range rate measurement in the controller design in [12] is impractical as this measurement is typical unavailable or otherwise suffers from large errors. To further remove range rate measurement required in the controller design, the objective of this paper is to develop a new controller using range-only measurement such that circumnavigation is achieved under a GPS-denied environment by expanding on the preliminary work reported in [14].

The objective of the paper is fulfilled via a two-step analysis. First, a control algorithm based on range and range rate measurements is proposed to accomplish the circumnavigation mission. One promising feature of the control algorithm is that the associated control input is always bounded. Second, a sliding-mode estimator using range measurement is designed to accurately estimate range rate in finite time when applying the proposed control algorithm with range rate measurement being replaced by the estimated value. The design of this control algorithm is motivated by the study on sliding-mode observer in [15, 16, 17], where numerous velocity observers were designed based on position information. By combining the two steps, the circumnavigation mission can be accomplished using the proposed control algorithm when actual range rate measurement is replaced by its estimated value obtained from the sliding-mode estimator. To our best knowledge, this is the first paper that solves the circumnavigation problem using range-only measurement.

The rest of the paper is structured as follows. Section II introduces the circumnavigation problem. In Section III, a control algorithm based on range and range rate measurements is proposed to solve the circumnavigation problem, where the associated control input is always bounded. It is shown that UAV can always circumnavigate the target at the desired radius regardless of its initial state. In Section IV, range rate measurement used in the control algorithm in Section III is replaced by its estimated value obtained via a sliding-mode estimator. Such a replacement is valid because the estimated value and the actual value become identical after a finite period of time. In Section V, two illustrative simulation examples are provided as a proof of concept. Finally, Section VI is given to conclude the paper.

## Ii Problem Statement

Circumnavigation concerns with the behavior that a UAV orbits around an unknown target at some desired distance. For instance, in Fig. 1, let denote the unknown target whose location is and the blue triangle denote the UAV. The objective is to have the UAV orbit around at some desired distance . In other words, the desired UAV trajectory is the black solid circle regardless of a counter clockwise or clockwise motion.

Assuming that UAV can maintain its altitude, we consider the following UAV dynamics given by

 ˙x =Vcos(ψ) ˙y =Vsin(ψ) (1) ˙ψ =ω,

where denotes the 2D location of the UAV, denotes the heading of the UAV, is the control input to be designed, and is the (constant) velocity of the UAV. Although this is a simplified model, it serves as a good approximation of practical UAV dynamics. Let the range measurement be denoted by . The objective is to design control input such that as . Such a motion that UAV orbits around a target at a fixed radius is referred to as a stable circular motion, which is defined as follows.

###### Definition II.1

A stable circular motion refers to the behavior that the UAV, with dynamics (II), moves around a target with a constant speed and a constant radius.

Notice that (II) characterizes the relationship between the controller and the state of the UAV . Since the objective is to control by designing an appropriate controller , it is thus desirable that (II) can be rewritten into a new form, in which the relationship between and can be explicitly identified. Clearly, needs to be a state variable in the new form. Another variable used in the new form is the bearing angle, defined as follows.

###### Definition II.2

Denote the reference vector as the vector from the current location of the UAV to . The bearing angle at time is defined as the angle from the reference vector to the current heading of the UAV measured counterclockwise.

As seen in Fig. 1, if the bearing angle is or , the heading of UAV is perpendicular to the vector from the UAV to the target. Physically, this means that the UAV cannot get closer to or farther away from the target. Indeed, if and are chosen as the state variables, the dynamics (II) can be rewritten as

 ˙r =−Vcos(θ) ˙θ =ω+Vsin(θ)r. (2)

The first equation in (II), illustrating the relationship between the bearing angle and the rate of , matches the analyzed physical property. To distinguish (II) and (II), we call (II) “Cartesian dynamics” and (II) “Polar dynamics”. Considering the objective of the paper, i.e., to design based on such that as , (II) is used when referring to UAV dynamics although the two dynamics (Cartesian dynamics and Polar dynamics) are physically equivalent. As can be noticed from (II), the rate of is controlled by the bearing angle , which can then be controlled by . Intuitively, it is possible to design a controller to meet the desired property. However, it remains unclear if such a controller exists when only range measurement is available.

## Iii A Controller Based on Range and Range Rate Measurements

In this section, a control algorithm based on range and range rate measurements is proposed to accomplish the circumnavigation mission. We show that the proposed control algorithm is able to guarantee globally asymptotic convergence and the equilibrium is exponentially stable. Globally asymptotic convergence means that the desired orbit motion is always guaranteed for an arbitrary initial state. System with exponentially convergence property has the benefit of improved robustness against disturbances.

Following the idea behind the control algorithm design in [12], a new control algorithm is proposed as

 ω={k[Vcos(π−sin−1(rar(t)))−˙r(t)],r(t)≥ra,0,otherwise, (3)

where is a constant gain and is a parameter to be determined later. Notice that (3) is a switching control law. To understand how the controller works, let’s refer to Fig. 1 for an illustrative situation. When , i.e., the UAV is on or outside the red dashed circle, the control input is determined by the difference between and . One may consider as a reference for to track. In fact, refers to the change rate of when the UAV moves towards one of the two tangent points on the red dashed circle. It should be emphasized that the two tangent points are not static as the UAV moves with a nonzero velocity. When , i.e., the UAV is inside the red dashed circle, the control input is zero, meaning that the UAV will move forward along its current heading. Because the UAV has a nonzero (constant) velocity, it takes a finite period of time before it exits the red dashed circle. Zero control when is used to drive the UAV outside the undesirable zone while the feedback control when is to drive the UAV in such a way that the desired stable circular motion is reached eventually.

For sake of conciseness, we adopt the following definition even if it is a slight abuse of notation.

###### Definition III.1

The circle centered at with a radius is defined as . The UAV is inside (resp., outside) if (resp., ).

According to Definition III.1, the red dashed circle is in Fig. 1. As mentioned earlier, the radius of , i.e., , is to be designed. Before choosing a proper , let’s first analyze the radius of the stable circular motion under the proposed control algorithm (3) assuming that such a stable circular motion does exist.

###### Lemma III.1

Consider system dynamics (II) subject to control input (3). If a stable circular motion exists, the radius is given by . In addition, the UAV rotates clockwise (resp., counter clockwise) when (resp. ).

Proof: When a stable circular motion exists, let the radius of the stable circular motion be given by . Then the magnitude of the nominal angular velocity is given by . Because the angular velocity of the UAV is equal to the control input (3), the magnitude of the nominal angular velocity is also equal to . For a stable circular motion, or , which implies that

• The UAV cannot be inside because otherwise the UAV moves along a straight line;

• based on (II).

Then becomes for the stable circular motion. It then follows that and should be identical, which happens if and only if .

When and , it can be computed that , indicating that the UAV rotates clockwise. When , one can obtain that , indicating that the UAV rotates counter clockwise.

Lemma III.1 illustrates the relationship between the radius of the stable circular motion, if existing, and the parameter in (3). By choosing

 ra=√r2d−1k2, (4)

one can obtain that according to Lemma III.1. One may observe that is strictly smaller than , meaning that the UAV should aim towards a circle with a smaller radius in order to establish the desired circular motion. This is due to the nonlinear dynamics (II). For a nonnegative , one can obtain that .

The validity of Lemma III.1 is based on the assumption that a stable circular motion does exist. It remains an open question whether this assumption is true. Our effort next is to show that the assumption is indeed true. For the simplicity of presentation, we only consider the case when . A similar analysis can be applied to the case when .

Our focus next is to show that a stable circular motion is guaranteed using the control algorithm (3) with . As shown in Fig. 2, we arrange the proof based on three different phases. Phase 1 is to show that UAV moves inside at most once. The detailed analysis is shown in Lemma III.3 based on Lemma III.2. Phase 2 is to show that always stays in the set after a finite period of time. The detailed analysis is shown in Lemma III.4. Phase 3 is to show that converges to asymptotically. The detailed analysis is shown in Theorem III.2. Following this logic, we next present these three lemmas as well as the main theorem. The proofs of these lemmas and theorem are similar to those in [12] with some variations. To make the paper self-contained, complete proofs of these lemmas and theorem are provided next.

###### Lemma III.2

Consider the UAV dynamics in (II) subject to the control policy in (3). Let there be such that and , then .

Proof: The proof of the lemma can be divided into the following two steps:

Step 1: holds for all . Based on the control algorithm (3), at since when . As a consequence, the UAV will rotate clockwise initially. Note that both and are continuous with respect to . Therefore, always holds before happens. Consequently, the UAV will stop rotating clockwise once . Indeed, if and only if or . Thus, . To prove Step 1, it suffices to show that cannot hold. When and , never holds for all because the UAV moves along a straight line and the straight line is always outside the circle . As a consequence, never holds for all when and . Combining the previous arguments completes the proof of Step 1.

Step 2: for all . When , it follows that . From Step 1, it is known that . This means that at the time when . By recalling the second equation in (II), one can obtain that at the time when . This implies that the UAV cannot get any closer to the target if . At the time when becomes larger than , the bearing angle has to be in the set , which satisfies the condition that . By repeating the analysis in Steps 1 and 2, it is clear that the UAV will never move inside .

###### Lemma III.3

Consider the UAV dynamics in (II) subject to the control policy in (3). The UAV can only move inside at most once.

Proof: When the UAV enters at some time , i.e., , holds true in order to guarantee that the UAV enters . Recall that no control input is imposed on the UAV when it is inside . Then at the time when it exits ,

 θ(tx)={π−θ(te),θ(te)∈[0,π2),3π−θ(te),θ(te)∈(3π2,2π),

because the UAV moves along a straight line due to the fact . As a consequence, . When , it follows that . Therefore, when . By considering the current time be the time in Lemma III.2, it follows from Lemma III.2 that the UAV will never move inside again. Therefore, the UAV can only move inside at most once.

###### Lemma III.4

Consider the UAV dynamics in (II) subject to the control policy in (3). For any , there exists such that for any .

Proof: By Lemma III.3, the UAV can move inside at most once. When the UAV never moves inside , let . When the UAV moves inside once, let be the time when the UAV moves from inside to outside . It is clear that is finite. The lemma is proved if the following two statements are valid:
(1) For any , there exists such that ; and
(2) Once for some , for any .

The first statement is proved by considering the following three cases:

• : From (II), one can obtain that , i.e., will decrease, whenever because while . When is (approximately) zero, is (approximately) . Because both and are continuous with respect to , is always upper bounded by some negative constant. Similarly, when is (approximately) zero, is upper bounded by some negative constant, indicating that is always upper bounded by some negative constant as well. Therefore, in finite time. Noting that , it follows from Step 1 in the proof of Lemma III.2 that cannot get smaller than , which implies that .

• : Under this case, the UAV is heading towards the tangent point such that the bearing is . By computation, , which implies that the UAV will move along a straight line towards the tangent point. As a consequence, the UAV will move towards the tangent point whenever . Given a constant nonzero velocity, it takes a finite period of time before happens. Notice that cannot hold for an arbitrary period of time because otherwise a contradiction happens by noting that (i) based on (3) during that period of time, indicating that the UAV cannot rotate; and (ii) the UAV has to rotate such that holds for that period of time. For , the UAV cannot move inside , as discussed in the first paragraph of the proof. This implies that will increase to be greater than as soon as happens. The bearing angle must be in the interval at the time when increases to be greater than . When , it follows from Case (i) that will be in the set after a finite period of time. When , it is already in the set .

• : If always holds, it takes a finite period of time before happens. Since the UAV never moves inside for , either Case (i) or Case (ii) will happen after a finite period of time. By following the analysis in Cases (i) and (ii), will be in the set after a finite period of time.

To prove the second statement, it is essential to study when or . Because the UAV is outside , it can be computed that

 ω=k[Vcos(π−sin−1(rar(t)))+V]>0,

where the first equation in (II) was used to derive the equality. This indicates that will increase as soon as happens. Similarly, when , one can obtain that

 ω=k[Vcos(π−sin−1(rar(t)))−V]<0,

which indicates that will decrease as soon as happens. Therefore the second statement holds as well.

It can be noted that the proof of Lemma III.3 depends on Lemma III.2 and the proof of Lemma III.4 depends on Lemma III.3. With these three lemmas, we now present the main result in this section.

###### Theorem III.2

Consider the UAV dynamics in (II) subject to the control policy in (3). If and is chosen satisfying (4), then and as .

Proof: According to Lemmas III.3 and III.4, there exists a time instant such that and . Then the control input can be simplified as

 ω=k[Vcos(π−sin−1(rar(t)))+Vcos(θ)],t≥t⋆.

For , consider a Lyapunov function candidate given by

 V=1−sin(θ)+φ,

where Because

 1rd−1z⎧⎪⎨⎪⎩>0,z>rd,=0,z=rd,<0,z

and satisfies such a property as well, combining with the property of integration shows that , indicating that as . When is chosen satisfying (4), one can obtain that by computation. Then can be simplified as . Taking derivative of yields that

 ˙V =−cos(θ)˙θ+˙φ =−cos(θ)[kV(−cossin−1(rar)+cos(θ))+Vsin(θ)r] =Vcos(θ)[−kcos(θ)−sin(θ)r+1r],

where (II) was used to derive the second equality. When , because and . When , because and . Therefore, . Note that is uniformly continuous when and , it follows from Lemma 4.3 in [18] that as . When , implies that . It then follows from (II) that when , is constant. Therefore, a stable circular motion does exist. It then follows from the analysis in the paragraph right after Lemma III.2 that if is chosen satisfying (4). Therefore, and as .

###### Remark III.3

Although it is assumed that the velocity of the UAV, , is constant, Lemmas III.1III.2III.3III.4, and Theorem III.2 are still valid if is varying within a set , where and are two positive constant. In other words, the assumed constant velocity for the UAV is not required as long as the velocity is both lower bounded and upper bounded. Lemma III.1 also shows that the final stable radius remains unchanged even if the velocity of the UAV changes.

In the previous part of this section, the proposed controller (3) was shown to guarantee global asymptotic stability regardless of the initial state. By observation, one can find that the control input associated with (3) is always bounded by because is bounded by and due to (II). Although this controller (3) uses both range and range rate measurements, we will show in the next section that the requirement of range rate measurement can be eliminated thanks to the boundedness of the control input.

Before proceeding to the next section, we show that the closed-loop system of (II) subject to (3) is exponentially stable at its equilibrium. If the equilibrium is exponentially stable, the associated closed-loop system not only converges, but in fact converges at a rate faster or at least as fast as some know rate near the equilibrium. One benefit of such a system is its robustness against disturbances around the equilibrium.

###### Theorem III.4

Consider the closed-loop system given by (II) subject to (3). If and is chosen satisfying (4), then is a locally exponentially stable equilibrium.

Proof: For the UAV dynamics in (II) subject to the control policy in (3), the corresponding closed-loop system can be written as

 ˙r(t)= −Vcos(θ(t)) (5) ˙θ(t)= k[Vcos(π−sin−1(rar(t)))+Vcos(θ(t))] +Vsin(θ(t))r(t), (6)

where (5) was substituted into the first equation in (II) to obtain (6). Let’s define

 f1(r(t),θ(t))\lx@stackrel△=−Vcos(θ(t))

and

 f2(r(t),θ(t))\lx@stackrel△= k[Vcos(π−sin−1(rar(t)))+Vcos(θ(t))] +Vsin(θ(t))r(t).

The linearization of (5) and (6) around the equilibrium is given by

 ˙x(t)=A(t)x(t), (7)

where and with

 A11=∂f1(r(t),θ(t))∂r(t)∣r(t)=rd,θ(t)=π2=0
 A12=∂f1(r(t),θ(t))∂θ(t)∣r(t)=rd,θ(t)=π2=V
 A21= ∂f2(r(t),θ(t))∂r(t)∣r(t)=rd,θ(t)=π2 = −kVr2ar3d√1−r2ar2d−Vr2d

and

 A22=∂f2(r(t),θ(t))∂θ(t)∣r(t)=rd,θ(t)=π2=−kV.

Then the eigenvalues of are given by

 −kV±  ⎷(kV)2−V(kVr2ar3d ⎷1−r2ar2d+Vr2d)2.

Clearly, is Hurwitz. It then follows from [19] that the equilibrium is exponentially stable.

## Iv A Revised Controller Based on Range Measurement and Estimated Range Rate

In Section III, a controller based on range and range rate measurements was presented to guarantee the desired circular motion for the UAV. Direct range rate measurement is typically unavailable or suffers from significant uncertainties. The purpose of this section is to remove range rate measurement needed in the control algorithm (3). In particular, an estimated range rate, obtained via a sliding-mode estimator using range measurement, is used to replace the range rate measurement used in (3).

Here control algorithm (3) is revised as

 ˆω={k[Vcos(π−sin−1(rar(t)))−^x2],r(t)≥ra,0,otherwise, (8)

where is an estimate of . In particular, is obtained via the following sliding-mode estimator as

 ˙^x1 =^x2+k1|r−^x1|12% sgn(r−^x1), ˙^x2 =k2sgn(r−^x1)+k3(r−^x1), (9)

if and

 ˙^x1 =0,^x1(tx)=2rd−^x1(te) ˙^x2 =0,^x2(tx)=−^x2(te), (10)

if , where is the sign function, is the time when UAV moves inside from outside 1 and denotes the first subsequent time when the UAV moves outside , and are positive constants. Because the UAV could move inside and outside multiple times, multiple and may be expected. In fact, the number of and the number of should be exactly the same since the UAV cannot stabilize inside due to the zero control applied when the UAV is inside . Here can be considered an estimate of . Briefly speaking, the main idea behind the estimator is that (i) and satisfy (IV) if the UAV is outside ; (ii) and remain unchanged if the UAV is inside ; and (iii) once the UAV moves outside , is reset as its negate while is reset as plus its negate. As shown in the proof of the following Theorem IV.3, the reset of and at the time when the UAV moves from inside to outside is crucial in establishing a finite-time convergence of to .

###### Remark IV.1

Instead of using actual range rate in controller (3), estimated range rate is used in controller (8). Because the designed sliding mode estimator guarantees that the estimation error converges to zero in finite time, the wellknown âseparation principleâ can be applied in the controller design. That is, the design of range rate estimator and the design of a feedback controller based on estimated range rate can be decoupled into two separated problems.

Before presenting the main result in this section, the following lemma is needed.

###### Lemma IV.1

Consider the differential equation given by

 ˙p =q−k1|p|12sgn(p), ˙q =−k2sgn(p)−k3p+f(t,p,q), (11)

where with . If , , and , approaches in finite time.

Proof: Let and consider the following Lyapunov function candidate given by

 V=ξTPξ, (12)

where

 P=12⎡⎢⎣4k2+k210−k102k30−k102⎤⎥⎦.

Note that is positive-definite and is differentiable almost everywhere except at . When , the derivative of is given by . Notice that the derivative of is given by . By recalling (11), can be rewritten as

 ˙V= −|p|−12ξTQ1ξ−ξTQ2ξ−k1f(t,p,q)|p|12sgn(p) +2qf(t,p,q), (13)

where

 Q1=k12⎡⎢⎣2k2+k210−k102k30−k101⎤⎥⎦

and

 Q2=k2⎡⎢⎣k2+2k21000k40001⎤⎥⎦.

Because under the assumption of the lemma, it can be further obtained that

 ∣∣∣k1f(t,p,q)|p|12sgn(p)∣∣∣ ≤ δ1k1|p|12+k1δ2|q||p|12 = δ1k1|p|−12[|p|12sgn(p)]2+12{δ22q2+k21[|p|12sgn(p)]2}

and

 |2qf(t,p,q)| ≤ 2δ1|q|+2δ2q2 = 2δ1|p|−12∣∣∣|p|12sgn(p)∣∣∣|q|+2δ2q2 ≤ δ1|p|−12{[|p|12sgn(p)]2+q2}+2δ2q2.

Then it follows from (IV) that

 ˙V≤ −|p|−12ξTQ1ξ−ξTQ2ξ+|p|−12ξTQ3ξ+ξTQ4ξ,

where

 Q