## Abstract

Motion camouflage is a stealth strategy observed in nature. We formulate the problem as a feedback system for particles moving at constant speed, and define what it means for the system to be in a state of motion camouflage. (Here, we focus on the planar setting, although the results can be generalized to three-dimensional motion.) We propose a biologically plausible feedback law, and use a high-gain limit to prove the accessibility of a motion-camouflage state in finite time. We discuss connections to work in missile guidance. We also present simulation results to explore the performance of the motion-camouflage feedback law for a variety of settings.

## 1. Introduction

Motion camouflage is a stealth strategy employed by various visual insects and animals to achieve prey capture, mating or territorial combat. In one type of motion camouflage, the predator camouflages itself against a fixed background object so that the prey observes no relative motion between the predator and the fixed object. In the other type of motion camouflage, the predator approaches the prey such that from the point of view of the prey, the predator always appears to be at the same bearing. (In this case, we say that the object against which the predator is camouflaged is a point at infinity.) The motion-camouflage strategy serves to minimize *motion parallax* cues that moving prey would be able to extract from the apparent relative motion of objects at various distances (Srinivasan & Davey 1995). For example, insects with compound eyes are quite sensitive to optical flow (which arises from the transverse component of the relative velocity between the predator and the prey), but are far less sensitive to looming (which arises from the component of the relative velocity between the predator and the prey along the line joining them). More broadly, such interactions may also apply to settings of mating activity or territorial manoeuvre as well. In Srinivasan & Davey (1995), it was suggested that the data on visually mediated interactions between two hoverflies *Syritta pipiens*, obtained earlier by Collett & Land (1975), support a motion-camouflage hypothesis. Later, Mizutani *et al*. (2003), observing the territorial aerial manoeuvres of dragonflies *Hemianax papuensis*, concluded that the flight pattern is motivated by motion camouflage (see fig. 1 in Mizutani *et al*. 2003). See also Srinivasan & Zhang (2004) for a review of related themes in insect vision and flight control.

Motion camouflage can be used by a predator to stealthily pursue the prey, but a motion-camouflage strategy can also be used by the prey to evade a predator. The only difference between the strategy of the predator and the strategy of the evader is that the predator seeks to approach the prey while maintaining motion camouflage, whereas the evader seeks to move away from the predator while maintaining motion camouflage. Besides explaining certain biological pursuit strategies, motion camouflage may also be quite useful in certain military scenarios (although the ‘predator’ and ‘prey’ labels may not be descriptive). In some settings, as is the case in Collett & Land (1975), Srinivasan & Davey (1995) and Mizutani *et al*. (2003), it is more appropriate to substitute the labels ‘shadower’ and ‘shadowee’ for the predator–prey terminology.

The essential geometry of motion camouflage (see §2*a*) is not limited to encounters between visual insects. In a recent work on the neuroethology of insect-capture behaviour in echolocating bats, a strategy geometrically indistinguishable from motion camouflage is observed (Ghose *et al*. 2006).

In this work, we take a structured approach to derive feedback laws for motion camouflage, which incorporates biologically plausible (vision) sensor measurements. We model the predator and prey as point particles moving at constant (but different) speeds, and subject to steering (curvature) control. For an appropriate choice of feedback control law for one of the particles (as the other follows a prescribed trajectory), a state of motion camouflage is then approached as the system evolves. (In the situation where the predator follows a motion-camouflage law, and the speed of the predator exceeds the speed of the prey, the predator is able to pass ‘close’ to the prey in finite time. In practice, once the predator is sufficiently close to the prey, it would change its strategy from a pursuit strategy to an intercept strategy.)

What distinguishes this work from an earlier study on motion-camouflage trajectories in Glendinning (2004) is that we present *biologically plausible feedback laws* leading to motion camouflage. Furthermore, unlike the neural-network approach used in Anderson & McOwan (2003*a*) to achieve motion camouflage using biologically plausible sensor data, our approach gives an explicit form for the feedback law which has a straightforward physical interpretation.

The study of motion-camouflage problems also naturally extends earlier work on interacting systems of particles, using the language of curves and moving frames (Justh & Krishnaprasad 2002, 2003, 2005; Zhang *et al*. 2004).

## 2. Planar pursuit–evasion model

For concreteness, we consider the problem of motion camouflage in which the predator (which we refer to as the ‘pursuer’) attempts to intercept the prey (which we refer to as the ‘evader’) while it appears to the prey as if it is always at the same bearing (i.e. motion camouflaged against a point at infinity). In this model, we consider that the pursuer moves at unit speed in the plane, while the evader moves at a constant speed *ν*<1. The dynamics of the pursuer are given by(2.1)where is the position of the pursuer, is the unit tangent vector to the trajectory of the pursuer, is the corresponding unit normal vector (which completes a right-handed orthonormal basis with ), and the plane curvature is the steering control for the pursuer. Similarly, the dynamics of the evader are(2.2)where is the position of the evader, is the unit tangent vector to the trajectory of the evader, is the corresponding unit normal vector and is the steering control for the evader. Figure 1 illustrates equations (2.1) and (2.2). Note that and are planar natural Frenet frames for the trajectories of the pursuer and evader, respectively.

We model the pursuer and evader as point particles (confined to the plane), and use natural frames and curvature controls to describe their motion, because this is a simple model for which we can derive both physical intuition and concrete control laws. (Furthermore, this approach generalizes nicely for three-dimensional motion, as in Justh & Krishnaprasad (2005) and Reddy *et al*. (2006).) Flying insects and animals (also unmanned aerial vehicles) have limited manoeuvrability and must maintain sufficient airspeed to stay aloft, so that treating their motion as constant speed with steering control is physically reasonable, at least for some range of flight conditions. (Note that the steering control directly drives the angular velocity of the particle, and hence is actually an acceleration input. However, this acceleration is constrained to be perpendicular to the instantaneous direction of motion, and therefore, the speed remains unchanged.)

We refer to (2.1) and (2.2) as the ‘pursuit–evader system’. In what follows, we assume that the pursuer follows a *feedback* strategy to drive the system towards a state of motion camouflage, and close in on the evader. The evader, on the other hand, follows an open-loop strategy. The analysis we present for the pursuer feedback strategy also suggests (with a sign change in the control law) how the evader could use feedback and a motion-camouflage strategy to conceal its flight from the pursuer. Ultimately, it would be interesting to investigate the game-theoretic problem in which both the pursuer and the evader follow feedback strategies, so that the system would truly be a pursuit–evader system. (What we address in this work would be more properly described as a pursuer–pursuee system. However, we keep the pursuer–evader terminology because it sets the stage for analysing the true pursuer–evader system, which we aim to discuss in a future paper.)

Equations (2.1) and (2.2) represent curve evolution for two particles in the Lie group of rigid motions in the plane. In the setting of cooperative interacting particles, generalization to the three-dimensional setting is developed in Justh & Krishnaprasad (2005).

### (a) Motion camouflage with respect to a point at infinity

Motion camouflage with respect to a point at infinity is given by(2.3)where is a fixed unit vector and *λ* is a time-dependent scalar (see also §5 in Glendinning (2004)). In the rest of this paper, it should be understood that by ‘motion camouflage’ we mean *with respect to a point at infinity* (see, however, §6).

Let(2.4)be the vector from the evader to the pursuer. We refer to as the ‘baseline vector’, and as the ‘baseline length’. We restrict our attention to non-collision states, i.e. . In that case, the component of the pursuer velocity transverse to the base line isand similarly, that of the evader isThe *relative* transverse component is(2.5)

### Lemma 2.1 **Infinitesimal characterization of motion camouflage.**

*The pursuit–evasion system* *(2.1)*–*(2.2)* *is in a state of motion camouflage without collision on an interval iff* *on that interval*.

Suppose motion camouflage holds. Thus,(2.6)Differentiating, . Hence, noting that under non-collision,(2.7) Suppose on . Thus,(2.8)so that(2.9)where and . ▪

It follows from lemma 2.1 that the set of all motion-camouflage states constitutes a five-dimensional smooth manifold with two connected components (one corresponding to approach and the other corresponding to retreat), each diffeomorphic to in the six-dimensional state space of the problem. In practice, we are interested in knowing how far is the pursuit–evasion system from a state of motion camouflage. In what follows, we offer a measure of this.

### (b) Cost function

Consider the ratio(2.10)which compares the rate of change of the baseline length to the absolute rate of change of the baseline vector. If the baseline experiences pure lengthening, then the ratio assumes its maximum value, . If the baseline experiences pure shortening, then the ratio assumes its minimum value, . If the baseline experiences pure rotation, but remains of the same length, then . Noting that(2.11)we see that may alternatively be written as(2.12)Thus, is the dot product of two unit vectors: one in the direction of , and the other in the direction of . Note that is well defined except at , since(2.13)For convenience, we define the notation to represent the vector rotated counter-clockwise in the plane by an angle . Thus, for example, . The transverse component of relative velocity, expression (2.5), then becomes(2.14)For convenience, we define to be the (signed) magnitude of , i.e.(2.15)and refer also to as the transverse component of the relative velocity. From the orthogonal decomposition(2.16)it follows that(2.17)Thus, is a measure of departure from motion camouflage.

### (c) Feedback law derivation

Differentiating along trajectories of (2.1) and (2.2) gives(2.18)

From (2.4) we obtain(2.19)and(2.20)Also,(2.21)Then, from (2.18), we obtain(2.22)Noting that(2.23)and(2.24)we obtain(2.25)Suppose that we take(2.26)where *μ*>0, so that the steering control for the pursuer consists of two terms: one involving the motion of the evader, and the other involving the transverse component of the relative velocity. Then,(2.27)and for any choice of *μ*>0, there exists such that(2.28)for all such that . Thus, for control law (2.26),(2.29)

## 3. The high-gain limit

Control law (2.26) has the nice property that for any value of the gain *μ*>0, there is a disc of radius (depending on *μ*) such that outside the disc. However, the problem with (2.26) is that the pursuer needs to know (i.e. sense and estimate) the evader's steering programme . Here we show that by taking *μ* as sufficiently large, motion camouflage can be achieved (in a sense we will make precise) using a control law depending only on the transverse relative velocity,(3.1)in place of (2.26), provided is bounded. Comparing (3.1) to (2.15), we see that, indeed, is proportional to the signed length of the relative transverse velocity vector. We will designate this as the *motion camouflage proportional guidance* (MCPG) law for future reference (see §5).

As is further discussed in §5, (3.1) requires range information as well as pure optical flow sensing. However, the range information can be coarse since range errors (within appropriate bounds) have the same effect in (3.1) as gain variations. We say that (3.1) is *biologically plausible* because the only critical sensor measurement required is optical flow sensing. Optical flow sensing does not yield the relative transverse velocity directly, but rather the angular speed of the image of the evader across the pursuer's eye. In fact, it is the sign of the optical flow that is most critical to measure correctly, since errors in the magnitude of the optical flow, like range errors, only serve to modulate the gain in (3.1).

For biological systems, the capabilities of the sensors *vis-à-vis* the sensing requirements for implementing (3.1) constrain the range of conditions for which (3.1) represents a feasible control strategy. In the high-gain limit we focus on below, the sensor noise (which is amplified by the high gain) would be expected to have significant impact. However, to illustrate the essential behaviour, here we neglect both sensor limitations and noise.

### (a) Bounds and estimates

Let us consider control law (3.1), and the resulting behaviour of as a function of time. From (2.25), we obtain the inequality(3.2)where we have used (2.13). For convenience, we define the constant as(3.3)

For any *μ*>0, we can define and such that(3.4)(and it is clear that many such choices of and exist). Note that (3.4) implies(3.5)Thus, for , (3.2) becomes(3.6)Suppose that given , we take . Then, for ,(3.7)where(3.8)

There are two possibilities for(3.9)The state we seek to drive the system toward has ; however, (3.9) can also be satisfied for . (Recall that .) There is always a set of initial conditions such that (3.9) is satisfied with . We can address this issue as follows: let denote how close to −1 we wish to drive , and let denote the initial value of . Take(3.10)so that (3.7) with (3.8) applies from time .

From (3.7), we can write(3.11)which, on integrating both sides, leads to(3.12)where . Noting that(3.13)we see that for , (3.7) implies(3.14)where we have used the fact that is a monotone increasing function.

Now we consider estimating how long , which in turn determines how large can become in inequality (3.14), and hence how close to −1 will be driven. From (2.12), we have(3.15)which from (2.13) and , , implies(3.16)From (3.16), we conclude that(3.17)and, more to the point,(3.18)For (3.18) to be meaningful for the problem at hand, we assume that . Then, defining(3.19)to be the minimum interval of time over which we can guarantee that , we conclude that(3.20)

From (3.20), we see that by choosing to be sufficiently large (which can be accomplished by choosing to be sufficiently large), we can force . Noting that(3.21)for , we see that(3.22)Thus, if is taken to be sufficiently large such that(3.23)then we are guaranteed (under the conditions mentioned in the above calculations) to achieve at some finite time .

### (b) Statement of result

Given the system (2.1)–(2.2) with defined by (2.12), we say that ‘motion camouflage is accessible in finite time’ if for any *ϵ*>0 there exists a time such that .

*Consider the system* *(2.1)–(2.2)* *with* *defined by* *(2.12)**, and control law* *(3.1)**, with the following hypotheses:*

*(and ν is constant),**is continuous and**is bounded,**, and*.

*Motion camouflage is accessible in finite time using high-gain feedback (i.e. by choosing μ>0 sufficiently large)*.

Choose such that . Choose sufficiently large so as to satisfy (3.23), and choose accordingly to ensure that (3.7) holds for . Then, defining *μ* according to (3.4) ensures that , where is defined by (3.19). ▪

Assumption (A 1) can be generalized to . (The case *ν*=0 corresponds to a stationary ‘evader’, so that the natural Frenet frame (2.2) and steering control for the evader are not defined.)

## 4. Simulation results

The following simulation results illustrate the behaviour of the pursuit–evasion system (2.1)–(2.2), under the control law (3.1) for the pursuer and various open-loop controls for the evader. The simulations also confirm the analytical results presented earlier. Figure 2 shows the behaviour of the system for the simplest evader behaviour, , which corresponds to straight-line motion. Because control law (3.1) is the same as (2.26) when , tends monotonically towards −1 (for the initial conditions and choice of gain *μ* used in the simulation shown). In figure 2, as in the subsequent figures showing pursuer and evader trajectories, the solid light lines connect the pursuer and evader positions at evenly spaced time instants. For a pursuit–evasion system in a state of motion camouflage, these lines would all be parallel to one another. Also, each simulation is run for a finite time, at the end of which the pursuer and evader are in close proximity. (The ratio of speeds is in all the simulations shown.)

Figure 3 illustrates the behaviour of the pursuer for a sinsusoidally varying steering control of the evader, and the corresponding behaviour of . Increase in the value of the feedback gain *μ* by a factor of three is observed to decrease the peak difference between and −1 by a factor of about . This is consistent with the calculations in the proof of proposition 3.3. Figure 4 illustrates the behaviour of the pursuer for a randomly varying steering control of the evader, as well as the corresponding behaviour of . Similar to figure 3, figure 4 shows that increase in the feedback gain *μ* by a factor of three decreases the peak difference between and −1 by a factor of about . Figure 4 also shows the value of initial transient in for as small. As would be expected, increase in the gain *μ* increases the convergence rate. Figure 5 suggests a graceful degradation in performance in the presence of sensor noise—a key requirement for biological plausibility.

## 5. Connections to missile guidance

There is a vast literature on the subject of missile guidance in which the problem of pursuit of a (evasively) manoeuvring target by a tactical missile is of central interest. A particular class of feedback laws, known as *pure proportional navigation guidance* (PPNG) occupies a prominent place (Shneydor 1998). For planar missile–target engagements, the PPNG law determining the steering control for the missile/pursuer is(5.1)where denotes the rate of rotation (in the plane) of the line-of-sight vector from the pursuer to the evader (not to be confused with the *λ* in §2*a*). Here the gain is a dimensionless positive constant known as the navigation constant. Note that our motion camouflage proportional guidance (MCPG) law given by (3.1) has a gain *μ*, which has the dimensions of . Also, it is easy to see that(5.2)So, to make a proper comparison we let as in §3 be a length-scale for the problem and define the dimensionless gain(5.3)Thus, our MCPG law takes the form(5.4)It follows that motion camouflage uses range information to support a high gain in the initial phase of the engagement, ramping down to a lower value in the terminal phase . In nature, this extra freedom of gain control is particularly relevant for echolocating bats (see Ghose *et al*. 2006), which have remarkable ranging ability.

Analysis of the performance of the PPNG law is carried out in Ha *et al*. (1990) and Oh & Ha (1999), using arguments similar to ours (although our sufficient conditions appear to be weaker). While motion camouflage as a strategy is discussed in Shneydor (1998), under ‘parallel navigation’, to the best of our knowledge, the current work is the first to present and analyse a feedback law for motion camouflage.

## 6. Directions for further work

In work under preparation, we have generalized the analysis to the case of planar motion camouflage with respect to a fixed background object (finite point). This requires a feedback law more complicated than (3.1), since the pursuer needs to keep track of the background object in addition to the evader. We have also generalized the analysis of motion camouflage (with respect to a point at infinity) to the three-dimensional setting (Reddy *et al*. 2006). The three-dimensional analysis is made possible by the use of natural Frenet frames, analogously to the three-dimensional unit-speed particle interaction laws described in Justh & Krishnaprasad (2005).

Because we are able to treat the motion-camouflage problem within the same framework as our earlier formation control and obstacle-avoidance work (Justh & Krishnaprasad 2002, 2003, 2005; Zhang *et al*. 2004), we would like to understand how teams of vehicles can make use of motion camouflage, and whether we can determine the convergence behaviour of such systems. Various biologically inspired scenarios for motion camouflage with teams have been described in Anderson & McOwan (2003*b*). Considering additional military applications without biological analogues, there are thus a variety of team motion-camouflage problems to study.

## Acknowledgements

The authors would like to thank M. V. Srinivasan of the Research School of Biological Sciences at the Australian National University for valuable discussions and helpful comments on an earlier draft of this paper. This research was supported in part by the Naval Research Laboratory under grant nos. N00173-02-1G002, N00173-03-1G001, N00173-03-1G019 and N00173-04-1G014; by the Air Force Office of Scientific Research under AFOSR grant nos. F49620-01-0415 and FA95500410130; by the Army Research Office under ODDR&E MURI01 Program grant no. DAAD19-01-1-0465 to the Center for Communicating Networked Control Systems (through Boston University); and by NIH-NIBIB grant 1 R01 EB004750-01, as part of the NSF/NIH Collaborative Research in Computational Neuroscience Program.

## Footnotes

- Received January 12, 2006.
- Accepted May 11, 2006.

- © 2006 The Royal Society