Fixed-Wing UAV Guidance Law for Surface-Target
`Tracking and Overflight
`
`Niki Regina
`DEIS
`University of Bologna
`Via Fontanelle 40 Forli’ 47100 Forli’, Italy
`+39-0543-786931
`niki.regina2@unibo.it
`
`Matteo Zanzi
`ARCES
`University of Bologna
`Via Fontanelle 40 Forli’ 47100 Forli’, Italy
`+39-0543-786933
`matteo.zanzi@unibo.it
`
`Abstract—This study presents a new guidance algorithm for a
`fixed-wing Unmanned Aerial Vehicle (UAV) used for surveil-
`lance tracking purposes. In particular, the algorithm ensures
`continual overflying of the target whether it is fixed or in motion.
`Attention was paid to the definition of the guidance specifica-
`tions in order to derive a suitable guidance algorithm capable
`of fulfilling the requirements under tight constraints including
`the constancy of airspeed and bounded lateral accelerations.
`An assessment of the performance of the presented algorithm
`is given by means of hardware-in-the-loop tests.
`
`TABLE OF CONTENTS
`INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1
`1
`2 MODEL OF THE TARGET-TRACKING PROBLEM .
`2
`3 PROPOSED GUIDANCE LAW . . . . . . . . . . . . . . . . . . . . .
`3
`4 TARGET VELOCITY ESTIMATION . . . . . . . . . . . . . . .
`4
`5 SIMULATION RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . .
`5
`6 HIL ARDUPILOT SIMULATION . . . . . . . . . . . . . . . . . .
`8
`APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
`REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
`BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
`
`1. INTRODUCTION
`During the past decade, a growth in UAV technologies has
`made possible different tasks. One of the most important is
`the capability of the air vehicle to track a moving ground
`target through the use of a gimbaled camera. Usually a
`gimbals operator on the ground selects a target of interest
`using a joystick that moves the gimbaled camera. Once
`the target is selected the UAV and camera automatically
`track the target. This kind of system also performs a real
`time estimation of the target’s velocity using UAV-gimbal
`telemetry data and the extracted target position on the image
`plane. Information about this scientific theme is available in
`Ref. [1], [2] and in Ref. [3] where similar approaches are
`presented.
`
`In this paper the target is assumed to be friendly and its
`position known by the pursuer guidance system. The target
`position is assumed to be determined by the target itself and
`transmitted via data-link to the flying pursuer. It is an aim of
`the pursuer UAV to autonomously follow the target during its
`maneuvers, by avoiding to go too far away from it, as well as
`if the UAV is kept in a virtual leash by the target.
`
`1978-1-4577-0557-1/12/$26.00 c(cid:13)2012 IEEE.
`2 IEEEAC Paper #1126, Version 3, Updated 02/01/2012.
`
`The aim of this paper is to develop a guidance law able
`to ensure a continuous loitering over the target when it is
`keeping still and to create a bounded trajectory around the
`position of the target when it moves.
`
`In order to create a guidance algorithm two main different
`scenarios are described in literature:
`
`1. The UAV flies autonomously along a predefined trajectory
`created on the fixed position of the target;
`2. The UAV creates by itself a trajectory around the moving
`target.
`
`For the first scenario a representative guidance law design
`technique recently presented in literature is the Lyapunov
`vector field (see Ref. [4], [5]). An inner feedback control
`loop is assumed to ensure the vehicle tracks the vector field
`by actuating the aircraft control surfaces in order to produce
`aerodynamics moments to achieve the desired vehicle attitude
`and altitude hold mode. The direction and the magnitude
`of the computed vector field velocity are transformed by
`the guidance outer loop into heading commands, suitable to
`assure the maneuvers of the UAV to be consistent to its flight
`performances in terms of minimum required air-speed and
`maximum turn-rate. In Ref. [6] a vector field for continuous
`live sensing is created through the use of a particular figure
`known as Lemniscate. Within the first scenario also the
`design techniques based on a imaginary point moving along
`the predefined trajectories must be included. This point is
`called pseudo-target or ghost-target. This approach is used in
`Ref. [7], [8] and [10] (here a 3D tracking is considered) where
`a guidance method for tracking straight line and curved path
`is presented. This guidance law is similar to a pure-pursuit
`guidance described in Ref. [9].
`
`An hybrid solution is presented in Ref. [11], [12] and [13]. In
`these papers an oscillatory ghost-target is created around the
`moving or fixed position of the target. Consequently the UAV
`is forced to track the factious target by different guidance
`laws. This approach is definitely an improvement respect
`to the approaches cited until now although neither of them
`consider the problem of out-of-frame target images.
`
`With reference to the second scenario, the guidance laws
`presented in Ref. [14] are able to track a moving target with
`a fixed horizontal range chosen by the operator. Besides, it
`is able to track the target even if it is out-of-frame. One of
`the limit of this guidance law is when the horizontal range is
`equal to zero due to disturbances, for instance.
`
`The guidance law presented in Ref. [15] solves this problem.
`It can be seen in the simulations that, even if the distance
`is chosen equal to zero, the UAV stabilizes itself on a circle
`
`1
`
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`

`
`whose radius meets the UAV dynamics (see Ref. [16])
`
`In this paper, using an approach similar to the last two cited
`articles, a new non-linear two dimension (2D) guidance law
`is presented.
`It generates petal-like trajectories along an
`horizontal plane at an established flight altitude, centered
`around a ground target, that ensures a continuous loitering
`over the target whether it keeps still or moves. The main
`features of the proposed law are twofold: its simplicity and
`adaptability to various tracking scenarios. In fact, according
`to this law, the UAV does not follow a path with a predefined
`shape created around the target; rather, it creates a trajectory
`whose points are computed in real time by the UAV itself
`depending on target available position and velocity data.
`
`Similarly as in most flight applications, a separate inner
`and outer feedback-loop control approach is assumed in this
`work. This because of its simplicity and the availability of
`good autopilots for attitude stabilization and altitude hold.
`The outer guidance loop transforms the lateral acceleration
`computed by the guidance law into heading commands. The
`guidance law is suitable to guarantee the maneuvers of the
`UAV to be consistent to its flight performances in terms of
`minimum required air-speed and maximum turn-rate. The
`computation of the guidance law requires the knowledge
`of the UAV/Target Line-Of-Sight (LOS) and the LOS rate.
`Moreover, a few parameters have to be chosen and tuned
`according to the UAV flight performances and mission re-
`quirements (related to target over-flying repetition rate).
`
`The significance of the proposed guidance algorithm respect
`to the existing ones consists in the fact that no fuel, time or
`distance-based cost function are minimized by the control
`law; instead, it allows a UAV for the continuous overflight
`of a target without the need of the design of a predefined
`trajectory.
`
`Results are shown by providing simulations and tests. Nu-
`merical simulations of different tracking scenarios are pro-
`posed in order to show the behavior of the law, also taking
`into account wind effect. Besides, tests with a well known X-
`Plane 6DoF simulator are accomplished in order to evaluate
`the realistic behavior of the implemented guidance law on an
`existing aerial vehicle.
`
`The paper is organized as follows. Section 2 describes the
`mathematical model of the target tracking problem. Section
`3 presents the proposed guidance law and its main features.
`Section 4 is dedicated to the target velocity estimation.
`In
`section 5 some simulation results in order to evaluate the
`effectiveness of the law. Section 6 implements the proposed
`guidance law on X-Plane 6DoF simulator.
`
`2. MODEL OF THE TARGET-TRACKING
`PROBLEM
`In order to develop a guidance law for a fixed-wing UAV that
`tracks a ground target, some assumptions have been stated
`in this work. Firstly, the motion of the UAV is considered
`at constant altitude, herein the problem can be considered
`a two dimensional (2D) tracking problem. Moreover, the
`airspeed of the UAV is assumed to be constant in order
`to provide the necessary lift to hold the altitude as much
`as possible. Besides, the control input of the airplane is
`the lateral acceleration caused by aileron deflections and the
`lateral acceleration in its turn gives raise a heading change.
`Herein the UAV has been modeled as a mass-point moving on
`
`a horizontal plane according to the following mathematical
`model:
`
`˙x = V cos(ψ) + Wx
`˙y = V sin(ψ) + Wy
`˙ψ = an
`V
`
`
`
`
`(1)
`
`where [x, y, ψ]T is the state vector of the UAV model
`and the state variables represent the two Cartesian position
`coordinates along a North-East reference frame and the head-
`ing angle, respectively. Wx and Wy are the wind velocity
`components. The angle ψ is positive in an anti-clockwise
`sense and it represents the angle between the x (North)
`axis and the longitudinal UAV axis; this, in turn, coincides
`with the direction of the UAV airspeed vector V because
`no sideslip angle is considered. V is the norm of V. an
`is the single input signal of the model and represents the
`value of lateral acceleration, i. e. the acceleration of the air
`vehicle perpendicular to its longitudinal axis. No longitudinal
`acceleration respect to the air flow is considered because
`˙V = 0. As told before, the effect of the lateral acceleration is
`to cause a change in the rate of turn while leaving the airspeed
`unchanged.
`
`In case of no wind the heading angle coincides with the
`the ground-track velocity-vector angle
`course angle χ, i.e.
`respect to the North direction.
`
`In a target tracking problem is interesting to analyze the
`evolution of the geometry of relative target-pursuer motion
`rather than the position of the single points.
`In particular,
`according to [9] , the behavior of the projection of the relative
`target-pursuer distance on the horizontal plane, R, together
`with the Line-Of-Sight (LOS) angle σ, is described by the
`following dynamic model:
`
`(cid:26) ˙R = VT cos(σ − χT ) − VG cos(σ − χ)
`
`R ˙σ = −VT sin(σ − χT ) + VG sin(σ − χ)
`
`(2)
`
`where VT and χT are the target speed and course angle,
`respectively, while VG is the UAV ground speed, R is the
`target-UAV distance.
`
`Of course there is a relationship between model (1) and (2). In
`
`fact, VG = p ˙x2 + ˙y2 and R = p(x − xT )2 + (y − yT )2,
`
`with [xT , yT ]T the position coordinates of the target; more-
`over, tan χ = ˙y/ ˙x.
`
`A representation of the involved variables is given in figure 1.
`
`As a consequence of
`VG(ψ, Wx, Wy) where
`
`the above discussion is VG =
`
`VG(ψ, Wx, Wy) ≡
`
`p(V cos(ψ) + Wx)2 + (V sin(ψ) + Wy)2
`
`(3)
`
`and, by defining the track angle
`
`g(ψ, Wx, Wy) ≡
`arctan 2 (V sin(ψ) + Wy, V cos(ψ) + Wx)
`
`(4)
`
`2
`
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`

`
`N
`
`ΧT
`
`VT
`
`TARGET
`
`V
`
`R
`
`Χ
`
`σ
`
`UAV
`
`Figure 1. Target-Pursuer Relative Geometry
`
`how position and velocity estimations are accomplished for
`the particular mission considered in this work). Then, the
`state of system (8) is observable. Moreover, the wind velocity
`is assumed to be known. According to this hypothesis, in eq.
`(8) VT , χT , Wx and Wy can be considered as measurable
`parameters.
`
`The requirements of the problem of guidance here considered
`can be summarized as follows.
`
`The UAV has to track the target continuously. In order to do
`this, its airspeed is not lower than the target ground speed;
`rather, its airspeed is almost always greater then that of the
`target. As a consequence of this, the UAV usually reaches
`and overflies the target: hence a maneuver for turning back
`on the target after an overtaking is necessary.
`
`E
`
`The turning back has to be accomplished through turn rates
`compliant with the mechanical characteristics of the aircraft,
`and lateral accelerations required for these maneuvers have to
`be bounded. As previously recalled, during all the tracking
`maneuvers the airspeed of the UAV has to be maintained
`unchanged.
`
`where z = arctan 2 (b, a) is the usual four-quadrant arc
`a , it is
`tangent function such that tan (arctan 2(b, a)) = b
`
`In order to derive a guidance law fulfilling the stated re-
`quirements, it can be designed according to the following
`specifications.
`
`χ = g(ψ, Wx, Wy)
`
`(5)
`
`At first, the guidance law has to act on the lateral acceleration
`the only one controllable input of the system. The
`an,
`guidance law has to provide bounded signals.
`
`For the discussion in the next section it is also worth noting
`that
`
`and
`
`g (ψ, 0, 0) = ψ
`
`VG(ψ, 0, 0) = V
`
`(6)
`
`(7)
`
`Finally, a mathematical model suitable for the UAV-Target
`tracking problem studied in this work is obtained by using
`eq. (1), eq. (2), eq. (3) and eq. (5) that yield to:
`
`˙R = VT cos(σ − χT )−
`−VG(ψ, Wx, Wy) cos(σ − g(ψ, Wx, Wy))
`
`R ˙σ = −VT sin(σ − χT )+
`+VG(ψ, Wx, Wy) sin(σ − g(ψ, Wx, Wy))
`
`(8)
`
`˙ψ = an
`V
`
`
`
`
`Model (8) is a third order dynamic model where [R, σ, ψ]T
`is the state vector, an is the input, VT and χT are time varying
`parameters, V , Wx and Wy are constant parameters.
`
`3. PROPOSED GUIDANCE LAW
`For the development of the guidance law the position and
`velocity of the target are assumed to be known by the UAV
`guidance system. (In [17] and [18] a description of typical
`sensors useful for this aim are given. Next section explains
`
`3
`
`Moreover, the guidance law has to steer the UAV towards the
`target when the UAV is approaching it. In this, the guidance
`law has to behave as a typical proportional navigation guid-
`ance. In particular, the guidance law has to be a function of
`the angle among the LOS and UAV ground velocity vector,
`i.e. σ − χ.
`
`Nevertheless, during overflying and after, the guidance law
`needs to give the UAV time sufficient to move away the target
`enough in order to gain the necessary range to be able to:
`
`• turn back with a maneuver not so strong as to require
`an excessive lateral acceleration, that is to let admissible
`curvature radii;
`• come back by heading toward the target avoiding to remain
`trapped on a loitering circle with a fixed radius around the
`target itself. This specification is necessary to avoid behaviors
`that imply steady limit cycles as in [15].
`
`The proposed guidance law has therefore the following form:
`
`an = K1(R, ˙R) arctan(K2(σ − χ))
`
`(9)
`
`where
`
`• K1() : R+ × R → R+;
`• K2 is a positive constant owning to the interval ]0,1]
`
`Function K1 acts as a state-dependent gain that modulates the
`strength of the lateral acceleration provided by the guidance
`law. It acts according to the following criterion: when the
`UAV is going away after overflying the target but is yet near
`it within a circular area specified by a predefined radius R0,
`the gain must be at its lowest value; otherwise it reaches its
`maximum value.
`
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`

`
`As it has been discussed in the previous sections, the aim
`of this guidance law is to ensure with a fixed position of
`the target a continuos over-loiter on it. For this reason the
`equilibrium points which would drive the UAV on a circle
`trajectory must be avoided. To this aim R < R0 must hold.
`Hence, from eq. (12) and the considerations presented in the
`remark it holds:
`
`(16)
`
`tan(
`
`πRmin
`2R0
`
`)
`
`2 π
`
`K2 >
`
`While it is possible to find out a smooth version of gain K1,
`a discrete behavior has been selected in this work in order
`to evaluate the overall performances of the guidance law.
`Hence, the gain function is specified in table 1.
`
`Table 1. Gain Selection for the Guidance Mode
`
`˙R < 0
`˙R ≥ 0
`
`R < R0 R ≥ R0
`C
`C
`0
`C
`
`with C ∈ R+. Value of C and R0 have to be selected by a
`tuning phase, even if an analytical approach is possible.
`
`As a consequence the new condition Rmin < R0 can be
`derived in order to satisfy eq. (16).
`
`4. TARGET VELOCITY ESTIMATION
`In order to compute the guidance law,
`the position and
`velocity vectors of both the UAV and the ground target are
`necessary. Position and velocity of the UAV are available
`from on-board sensors (like a GPS receiver); in this context
`the position of the target is assumed to be transmitted from
`ground, while the target velocity is assumed to be unknown.
`This is mainly due to limit the data-link bandwidth.
`
`The availability of position and velocity information allows
`for the determination of the parameters used in the com-
`putation of the guidance law according to the following
`relationships:
`
`~R = [xT − x, yT − y]T , R = (cid:13)(cid:13)(cid:13)
`
`tan(σ) = xT −x
`yT −y
`
`~R(cid:13)(cid:13)(cid:13)
`
`~VG = [ ˙x, ˙y]T , ~VT = [ ˙xT , ˙yT ]T , ~VR = ~VT − ~VG
`
`where ~VR indicates the target-UAV relative velocity; more-
`over it can be easily verified that the following expression
`holds
`
`˙R =
`
`~R·~VR
`k ~Rk
`
`In fact, it must be highlighted that the proposed guidance law
`(9) is bounded and its maximum value is
`
`(10)
`
`(11)
`
`π 2
`
`|an|max = C
`
`and it can be easily derived a formula for the gain C:
`
`|an|max
`
`2 π
`
`C =
`
`It is also possible the equilibrium point of relative kinematic
`system and to derive a condition for K2. In fact:
`
`(12)
`
`tan(
`
`V 2
`CR0
`
`)
`
`2 π
`
`K2 <
`
`Remark
`
`In order to have a reference value for the maximum lateral
`acceleration, it is worth noting that: by considering a plane
`flying along constant rate turn manouveour, it experiences a
`lateral acceleration given by V 2
`R where R is the turn radious.
`Hence, the maximum value for the lateral acceleration can be
`chosen from this expression by selecting a suitable minimum
`value for Rmin. That is:
`
`|an|max =
`
`V 2
`Rmin
`
`(13)
`
`Hence, a filter capable to provide an estimate of the target ve-
`locity, together with a smoothed target position information,
`is needed on-board.
`
`Moreover, by considering the well known flight mechanical
`equation for the coordinate turn [19] is
`
`g tan(φ) =
`
`V 2
`R
`
`(14)
`
`where φ is the roll angle. By selecting a maximum roll angle
`value φmax = |φ|max for a chosen UAV, it is possible to
`compute an expression for the minimun value for turn radious
`Rmin.
`
`Many filters proposed for this task exist in literature based
`on variations of Kalman filters (EKF, Unscented, Particle
`filters, etc.): unfortunately the robustness of the estimation
`process remains usually a critical problem. Here a simple
`filter structure is given: its derivation is partially based on the
`idea of the fast estimator described in [14].
`
`The filter has to estimate the North and East components of
`target position and velocity: due to the fact that North and
`East coordinates are de-coupled, only one channel of the filter
`is described here, the other one being similar.
`
`The mathematical model of the process to be observed is:
`
`Rmin =
`
`V 2
`g tan(φmax)
`
`(15)
`
`4
`
`(cid:26) ˙xT = u
`
`y = xT
`
`(17)
`
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`

`
`where u denotes the target unknown North velocity, xT is
`the target North position, y is the measurement received on-
`board. The proposed filter is illustrated in figure 2 where the
`hat over the variables indicate the corresponding estimated
`values. R(s) is a transfer function to be selected in order to
`make the filter estimation process convergent.
`
`with
`
`M = 2
`c Be
`
`eu0 (0)
`B −1
`
`B = k ˙eu0(0) + c
`2 eu0(0)k
`
`Figure 2. Target velocity estimation scheme
`
`The updating equations of the filter, expressed in the s-
`domain, are:
`
`ˆxT (s) =
`
`R(s) 1
`s
`1 + R(s) 1
`s
`
`y(s)
`
`ˆu(s) =
`
`R(s)
`1 + R(s) 1
`s
`
`y(s)
`
`(18)
`
`(19)
`
`and the velocity estimation error can be expressed by
`
`ˆu(s) − u(s) = G(s)u(s)
`
`(20)
`
`where
`
`Figure 3. Distance with no wind effect
`
`G(s) =
`
`R(s)
`s + R(s)
`
`− 1
`
`(21)
`
`Figure 4. Distance UAV-Target with constant wind effect
`
`By choosing a first order stable transfer function of the type
`
`R(s) =
`
`k
`s + c
`
`(22)
`
`with k and c positive constants, it results that the filter is stable
`and the velocity estimation error is bounded.
`
`In particular, by selecting k = c2
`4 , the filter is a second order
`system with two coinciding poles. An error estimation bound
`is given by (see the appendix for the derivation of the results
`and for the definition of L1 and L∞ norms):
`
`kˆut − utkL∞
`
`≤ keu0tkL∞
`
`+ kG(s)kL1 kutkL∞
`
`(23)
`
`with eu0(t) being the free response of the system G(s) to the
`initial estimation errors.
`
`Because of the limited speed of the target, it is
`
`kutkL∞
`
`≤ umax
`
`hence, from (23), by substuting the results in appendix, it
`holds
`
`kˆut − utkL∞
`
`≤ M + 2umax
`
`(24)
`
`5. SIMULATION RESULTS
`In this section several simulations results are presented in
`order to test the effectiveness of the guidance law presented
`in section 3.
`In these simulations the velocity of the UAV
`is considered constant at 10 m/s while, when the wind effect
`is considered, it has a constant east-direction and a constant
`speed equal to 3 m/s. The gains of the guidance law are:
`C = 5, R0 = 40 and K2 = 1. Three different cases
`are simulated, both with the target fixed and moving with
`different trajectories.
`
`Through these simulations the effectiveness of the guidance
`law can be shown for all the possible motions of the target
`(fixed, moving with constant heading, moving with variable
`heading).
`
`A. Fixed Target
`In this first subsection the target is considered fixed in the
`origin and two different simulations are presented: the first
`one with no wind effect while the second with a constant
`wind. The initial position of the UAV is (100,100) with
`the initial heading pointing in north-east direction. The
`simulation lasts 100 s.
`
`In figure 5 and 6 the continuous line represents the trajectories
`of the UAV while the star in the origin is the fixed target po-
`sition. In figure 5 no wind effect are considered and the main
`
`5
`
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`

`
`Figure 5. Trajectories of UAV with No Wind Effect
`
`Figure 7. Distance UAV-Target with No Wind Effect
`
`Figure 8. Distance UAV-Target with constant Wind Effect
`
`B. Straight-line Target
`In this case the target is supposed with a fixed north heading
`direction in order to create a straight-line trajectory. However,
`the speed of the target is variable always slower than the UAV
`cruise airspeed. The target’s profile velocity is shown in fig. 9
`where, again, from 70 to 250 s the target is considered fixed.
`It can be seen that the maximum velocity supposed is 7 m/s.
`As in the previous case, the simulations are presented with
`and without wind effect.
`
`Figure 6. Trajectories of UAV with constant Wind Effect
`
`charateristic of the guidance law designed are highlighted:
`the UAV is able to always pass over the target. In fig. 6 the
`trajectory of the UAV in presence of wind effect are shown.
`In this case the UAV is also able to pass over the target but
`with a different trajectory.
`
`These considerations are also documented in fig. 7 and 8
`where in both cases the distance between the UAV and the
`target goes periodically to zero. The only slight difference
`that can be seen between the two figures is the period of
`interception. Considering wind effects the period is longer.
`This could be an advantage when the trajectory will be
`implemented on the 6DoF flight simulator because the lateral
`acceleration required is smaller.
`
`Figure 9. Target Velocity Profile
`
`The trajectories of the UAV and the target are represented
`by, respectively, the blue line and the red line in fig. 10. In
`both cases, through the use of the guidance law designed, the
`UAV is able to track the target. Moreover, when the target
`is fixed the UAV flies over the pursuer continuously. The
`
`6
`
`Yuneec Exhibit 1020 Page 6
`
`

`
`different trajectories created by the guidance law are due to
`the presence of the wind.
`
`Without the wind effect the trajectory in similar to a constant
`oscillation around the position of the target until it stops (see
`fig. 10). Observing fig. 11 it can be seen that the UAV, due
`to the presence and the direction of the wind that reduce the
`speed of the air-vehicle, points its heading behind the target.
`However, also in this case, the UAV starts to loiter around the
`final point reached by the target.
`
`shows how the UAV intercept the target just before 50th s,
`avoiding the oscillation motion. Once the UAV reaches the
`target it starts to loiter around the target as it is shown in the
`previous case.
`
`Figure 12. Distance with no wind effect
`
`Figure 13. Distance UAV-Target with constant wind effect
`
`C. Circular Target
`In order to complete the simulations of the proposed guidance
`law, a circular trajectory is supposed for the target. The aim
`of this simulations is to test the guidance law when the target
`is completing a bend trajectory. In this case the velocity of
`the target is supposed to be constant at 5 m/s, with a constant
`lateral acceleration equal to 0.05 m/s2 and, as in the previous
`cases wind and no-wind simulations are shown.
`
`The initial position of the UAV is (100,0) with north-east ini-
`tial heading angle. The target is initially placed in the origin
`(0,0). The simulation lasts 100 s. In fig. 14 the continuous
`line represents the UAV trajectory while the dashed line is the
`target trajectory. Both in fig. 14 (without wind) and in fig. 15
`(constant wind) it is shown that the UAV is able to intercept
`the target and starts to loiter around.
`
`Figure 16 shows that the distance between the UAV and
`the target goes to zero. The periodic behavior cited in the
`previous case is confirmed here where the maximum distance
`between the target and the UAV is around 60 m. The target is
`intercepted with a constant period 25 s.
`
`Figure 17 shows that the UAV is able to reach the target but,
`due to the presence of the wind, the trajectory created by the
`guidance law is not periodic anymore.
`
`7
`
`Figure 10. Trajectories of UAV with no wind effect
`
`Figure 11. Trajectories of UAV with constant wind effect
`
`Figure 12 shows the distance between the target and the UAV
`during the simulation.
`In fig. 12 the oscillation motion is
`shown in the first 50 s where the distance between the target
`and the UAV never goes to zero. Once the target stops itself
`the distance goes, again, periodically to zero. Figure 13
`
`Yuneec Exhibit 1020 Page 7
`
`

`
`Figure 14. Trajectories of UAV with no wind effect
`
`Figure 17. Distance UAV-Target with constant wind effect
`
`Comments
`In all the cases analyzed in this section (A, B, C) the guidance
`law designed has the expected behavior.
`In particular, the
`aim to create a guidance law able to fly over the target
`continuously is reached. Moreover, the fig. 10 and the fig.
`16 shows that the guidance law ensures that the distance goes
`to zero also when the target is moving.
`
`6. HIL ARDUPILOT SIMULATION
`The proposed guidance law described in sec. 3, has been
`implemented on the board ardupilot-mega. One of the main
`features of this microcontroller is the open source firmware
`which can be easily modified by any user. In particular, the
`APM 2.12 source code has been changed to ensure a target
`tracking hardware in the loop (HIL) simulation. The setup is
`explained in fig.
`
`Figure 15. Trajectories of UAV with constant wind effect
`
`Figure 18. Setup of HIL simulation
`
`The 6DoF flight simulator chosen for HIL simulation is X-
`Plane and, in order to have a more realistic simulation, the
`PT-60 RC airplane has been chosen as RC model aircraft. The
`charateristics of the plane are described in tab. 2.
`
`Together with the APM source code, ardupilot-mega offers
`the chance to use the Mission Planner (known also as ground
`station) which serves as a bridge between the board and X-
`Plane. The ground station reads the data coming from the
`fligth simulator and sends it, through the serial port, to the
`APM. At the same time it sends the servo outputs calculated
`
`8
`
`Figure 16. Distance with no wind effect
`
`Yuneec Exhibit 1020 Page 8
`
`

`
`As can be seen the UAV follows the target and its attitude
`shows a bounded roll angle and a stable pitch which can be
`also seen by the altitude profile. The distance plot shows
`sometimes values over 500 [m] but this is due to the speed
`reached by the ground target. It is obvious that if the target
`goes faster than the UAV the distance increases but it drops
`quickly around 150 [m] when the car decreases its speed.
`
`Figure 19. GP-PT 60 RC model
`
`GP-PT 60
`
`5.4 [f t]
`0.95 [f t]
`NACA 2412
`0.0 [deg]
`±20[deg]
`
`1.85 [f t]
`0.52 [f t]
`NACA 0006
`8.5 [deg]
`±20[deg]
`
`UAV Model
`Wing
`Span
`Mean Aerodynamic Chord
`Airfoil
`Incidence
`Aileron Deflection
`Horizontal Stabilizer
`Span
`Mean Aerodynamic Chord
`Airfoil
`Incidence
`Elevator Deflection
`Vertical Stabilizers
`Span
`1.4 [f t]
`Mean Aerodynamic Chord
`0.5 [f t]
`Airfoil
`NACA 0009
`Rudder Deflection
`±20[deg]
`Table 2. GP-PT Geometric Characteristics
`
`Figure 20. 3D HIL trajectories
`
`Figure 21. Roll and Pitch angles
`
`Figure 22. Height and Distance
`
`Fixed Target
`In this simulation a fixed target has been supposedly placed
`near Forl Airport. As can be seen a continuous over-loitering
`of the fixed point is performed by the UAV. As in the previous
`subsection the roll angle remains bounded as the pitch angle
`and the altitude. It is important to see how the distance goes
`close to zero cyclically.
`
`9
`
`by the autopilot to X-Plane.
`
`Two different simulations results are presented. In the first
`one a moving target has been simulated while in the second
`simulation a fixed target position has been chosen in order
`to highlight the main charateristic of the proposed guidance
`law to ensure a continuos over-loitering on it. Both the
`simulations are made with these parameters:
`
`• Aircraft Cruise Speed V = 15 [m/s];
`• Wind Speed 3 [m/s];
`• Wind Direction 30;
`• Desired radious 100 [m]
`• C = 15, K2 = 0.3.
`
`Moving Target
`In order to simulate a target tracking, a trajectory has been
`completed by a car and its data (position, speed and heading)
`sampled by a GPS receiver at 0.3 Hz. All these data has been
`implemented on ardupilot-mega and with different changes
`made to the APM 2.12 it has been possible to simulate the
`UAV that follows a moving target.
`
`Figure 20 shows the 3D trajectory of the UAV. The yellow
`poiters represent the data sampled by the GPS’s car while
`the red line is the trajectory completed by the UAV. It is also
`important to show the attitude (fig.21), the altitude and the
`distance between the UAV and the moving target (fig. 22).
`
`Yuneec Exhibit 1020 Page 9
`
`

`
`where y0(t) is the free response to initial conditions and
`g(t) is impulse response od the system, obtained as the anti-
`transform of G(s).
`
`Hence, it holds
`
`kytkL∞
`
`≤ ky0tkL∞
`
`Z t
`+(cid:13)(cid:13)(cid:13)(cid:13)
`
`0
`
`g(τ )u(t − τ )dτ(cid:13)(cid:13)(cid:13)(cid:13)L∞
`
`(27)
`
`and, from (27),
`
`Figure 23. 3D HIL trajectories
`
`kytkL∞
`
`≤ ky0tkL∞
`
`+ kG(s)kL1 kutkL∞
`
`(28)
`
`Figure 24. Roll and Pitch angles
`
`If
`
`G(s) =
`
`−s(s + c)
`s(s + c) + k
`
`by selecting k = c2
`4 , c > 0, yields to
`
`G(s) = −s
`
`(s + c)
`(s + c
`2 )2
`
`whose impulse response g(t) is
`
`g(t) = δ(t) −
`
`c2
`4
`
`te− c
`2 t
`
`(29)
`
`(30)
`
`(31)
`
`Figure 25. Height and Distance
`
`where δ(t) is the Dirac impulse.
`
`From easy passages the following expression can be obtained
`
`APPENDICES
`The definition of L1 and L∞ norms is:
`
`kutkL∞
`
`= sup
`0≤τ ≤t
`
`|u(τ )|
`
`kG(s)kL1 = Z ∞
`
`0
`
`|g(τ )| dτ
`
`kG(s)kL1 = Z ∞
`
`0
`
`|g(τ )| dτ = 2
`
`Moreover, in this case the free response can be computed after
`some algebra and gives:
`
`y0(t) = (y0(0) + Bt)e− c
`2 t
`
`(32)
`
`If the Input-Output model of a system in the s-domain is
`
`with
`
`y0(0)
`
`c 2
`
`B = ˙y0(0) +
`
`y(s) = G(s)u(s) =
`
`−s(s + c)
`s(s + c) + k
`
`(25)
`
`|y0(t)| admits a maximum for t > 0 whose value is M =
`y0 (0)
`B −1. Hence
`|B|e
`
`2 c
`
`then the time response y(t) is
`
`y(t) = y0(t) +Z t
`
`0
`
`g(τ )u(t − τ )dτ
`
`(26)
`
`10
`
`ky0tkL∞
`
`= M
`
`(33)
`
`Yuneec Exhibit 1020 Page 10
`
`

`
`Proceedings of Aerospace Conference. Big Sky, MT,
`USA: AIAA/IEEE, 2009.
`[16] V. Dobrokhodov, I. Kaminer, K. Jones, and R. Ghabch-
`eloo, “Vision-based tracking and motion estimation for
`moving targets using unmanned air vehicles,” Journal of
`Guidance, Control and Dynamics, vol. 31, no. 4, 2008.
`[17] K. ZuWhan and R.Sengupta, “Target detection and
`position likelihood using an aerial image sensor,” in
`Proceedings of International Conference on Robotics
`and Automation.
`Pasadena, CA, USA: ICRA/IEEE,
`2008.
`[18] L. Bertuccelli and J. How, “Search for dynamic targets
`with uncertain probability maps,” in Proceedings of
`American Control Conference. Minneapolis, MI, USA:
`IEEE, 2006.
`[19] D. Raymer, Aircraft Design: a conceptual approach.
`AIAA Education Series, 1999.
`
`BIOGRAPHY[
`
`Niki Regina is a Ph.D researcher with
`the Department of Electronics, Com-
`puter Systems and Telecommunications
`(DEIS) of the university of Bologna. He
`was graduated in 2007 in Aerospace En-
`gineering