Nonlinear Trajectory Tracking for Fixed Wing UAVs

via Backstepping and Parameter Adaptation

Wei Ren∗

Department of Electrical and Computer Engineering, Utah State University, Logan, UT, 84322, USA

Ella Atkins†

Space Systems Laboratory, University of Maryland, College Park, MD 20742, USA

Equipping a fixed wing unmanned air vehicle (UAV) with low-level autopilots, we derivehigh-level velocity and roll angle control laws for the UAV. Backstepping techniques areapplied to design the velocity and roll angle control laws from known velocity and headingangle control laws that explicitly account for velocity and heading rate constraints of theUAV. Regarding unknown autopilot constants, a parameter adaptation technique is usedto estimate autopilot parameters. Simulation results on a fixed wing UAV are presentedto show the effectiveness of our approach.

Nomenclature

x, y inertial position, mψ heading angle, radφ roll angle, radv Airspeed, m/sh Altitude, mα∗ Autopilot Parametersg gravitational constant, m/s2

Subscriptr referenceSuperscriptc command

I. Introduction

Advanced control technologies for unmanned air vehicles (UAVs) have received significant attentionin recent years. Potential applications of autonomous UAVs in both civilian and military sectors in-

clude environment monitoring, search and rescue, communication relays, border patrol, situation awareness,surveillance, and battle damage assessment.

Fully-automating UAVs poses both theoretical and practical challenges.1 One important research is UAVpath planning.2,3 Another important aspect focuses on trajectory optimization for UAVs.4,5 In addition,cooperative control of multiple UAVs is also studied extensively in the literature.6–8

Effective trajectory tracking algorithms guarantee that a UAV can accurately follow its pre-specifieddesired trajectory. In addition, the study of nonlinear tracking control techniques for UAVs is essential forthe success of cooperative timing (e.g. Refs. 9,10) and formation keeping missions (e.g. Ref. 11). In Ref. 12,velocity and heading control laws that explicitly account for stall conditions, thrust limitations, and saturated

∗Assistant Professor, Department of Electrical and Computer Engineering, Utah State University, Email: weiren@ieee.org†Assistant Professor, Department of Aerospace Engineering, University of Maryland, Email: ella@ssl.umd.edu, Senior Mem-

ber.

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heading rate constraints are designed for small fixed wing UAVs equipped with velocity hold, heading hold,and altitude hold autopilots. In Ref. 12, it is assumed that the autopilot response to heading commands isfirst order in nature. However, the headings of the UAVs are controlled by their roll motions, which implythat the kinematic model for UAVs should explicitly take into account the roll angle command. In addition,the autopilot constants are generally unknown and depend on specific designs. Using inaccurate autopilotconstants in a control law design may result in degraded performance for the UAV. In this paper, we usea more accurate kinematic model that takes into account the roll motion for heading control. We applybackstepping techniques to derive velocity and roll angle control laws. In addition, a parameter adaptationtechnique is developed for the control law design regarding unknown autopilot constants. This paper is thecontinuation of the previous work presented in Ref. 12.

II. Problem Statement

Let (x, y), ψ, v, φ, and h denote the inertial position, heading angle, velocity, roll angle, and altitudeof the UAV respectively. We assume that the UAV is equipped with standard autopilots as described inRef. 13. The kinematic equations of motion are given by

x = v cos(ψ)y = v sin(ψ)

ψ = gtan(φ)

v(1)

v =1αv

(vc − v),

φ =1

αφ(φc − φ),

h = − 1αh

h +1αh

(hc − h),

where φc, vc, and hc are the commanded roll angle, velocity, and altitude to the autopilots, g is the gravi-tational constant, and α∗ are positive constants.13,14

In the following, we assume that an effective altitude controller exists and focus on the design of velocityand roll angle control laws.

Due to the stall conditions, thrust limitations, and roll angle and pitch rate constraints of fixed wingaircraft, the following input constraints are imposed on the UAV:

0 < vmin ≤ v ≤ vmax

−φmax ≤ φ ≤ φmax (2)

where φmax > 0.We assume that the desired reference trajectory (xr, yr, ψr, vr, ωr) generated by a trajectory generator5

satisfies

xr = vr cos(ψr)yr = vr sin(ψr) (3)

ψr = ωr

where vr and ωr are continuous and satisfy that vr and ωr are bounded, inft≥0 vr(t) > vmin, supt≥0 vr(t) <vmax, and supt≥0 |ωr(t)| < ωmax with ωmax > 0 denoting the heading rate constraint of the UAV.

Transforming the tracking errors expressed in the inertial frame to the UAV frame, the error coordinates15

become

xe

ye

ψe

=

cos(ψ) sin(ψ) 0− sin(ψ) cos(ψ) 0

0 0 1

xr − x

yr − y

ψr − ψ

. (4)

Note that the motivation for this transformation is only to simplify the mathematics so that a constrainedLyapunov function can be easily derived.

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Accordingly, the tracking error model can be represented as

xe = gtan(φ)

vye − v + vr cos(ψe)

ye = −gtan(φ)

vxe + vr sin(ψe) (5)

ψe = ωr − gtan(φ)

v.

III. Trajectory Tracking Using Backstepping and Parameter AdaptationTechniques

In Ref. 12, the authors use the following simplified kinematic model for the UAV:

x = vc cos(ψ)y = vc sin(ψ) (6)

ψ =1

αψ(ψc − ψ),

where vc and ψc are the commanded velocity and heading to the autopilots and αψ > 0 is an autopilotconstant.

However, the autopilot response to the heading command is not truly first order in reality. As a result,Eq. (1) represents a more accurate model of the UAV equipped with standard autopilots.

Let

v0 =

vmin, ηvxe < v

vr cos(ψe) + ηvxe, v ≤ ηvxe ≤ v

vmax, ηvxe > v

, (7)

ω0 =

ωmax, −ηωσω < ω

ωr + ηωσω, ω ≤ −ηωσω ≤ ω

−ωmax, −ηωσω > ω

, (8)

where v4= vmin − vr cos(ψe), v

4= vmax − vr cos(ψe), ω

4= ωr − ωmax, ω

4= ωr + ωmax, σω

4= λψe + ye√

x2e+y2

e+1,

and ηv and ηω are sufficiently large positive constants expressed precisely in Ref. 12. In Ref. 12, it is shownthat vc = v0 and ψc = αψω0 + ψ guarantees that |x − xr| + |y − yr| + |ψ − ψr| → 0 asymptotically. Inaddition, v0 and ω0 satisfy the following input constraints

0 < vmin ≤ v0 ≤ vmax

−ωmax ≤ ω0 ≤ ωmax,

which represents the case that the UAV has limited velocity and heading rate constraints.However, the velocity and heading rate commands (7) and (8) are no longer valid for Eq. (1), where the

control commands to the autopilot are the velocity and roll angle commands. In the following, we applybackstepping techniques to derive velocity and roll angle commands.

Note that Eqs. (1) and (5) can be rewritten as

χ = f(t, χ) + g(χ)ξ (9)

ζ = ν, (10)

where χ = [xe, ye, ψe]T ,

ξ =

[v

g tan(φ)v

]

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ζ =

[v

φ

]

f(t, χ) =

vr cos(ψe)vr sin(ψe)

ωr

g(χ) =

−1 ye

0 −xe

0 −1

ν =

[νv

νφ

]=

[1

αv(vc − v)

1αφ

(φc − φ)

].

Assume that ξ is the control input to Eq. (9) and suppose that we have found a control law ξd =[vd, g tan(φd)

vd ]T that stabilizes χ. Furthermore, assume that there exists a Lyapunov function V1 satisfyingV1 ≤ −W (χ) with ξd as the control input, where W (·) is a positive definite function.

However, ξ is not a true control input in reality, so we will try to design vc and φc such that ξ →[vd, g tan(φd)

vd ]T in the next step.

Let ω = g tan(φ)v and ωd = g tan(φd)

vd . Consider a Lyapunov function candidate

V2 = V1 +12zT z, (11)

where

z = ξ − ξd =

[v − vd

ω − ωd

]. (12)

Differentiating Eq. (11), we get that

V2 =∂V1

∂χf(t, χ) +

∂V1

∂χg(χ)ξ + zT z

=∂V1

∂χf(t, χ) +

∂V1

∂χg(χ)[ξd + (ξ − ξd)]

+ zT

[v − vd

g sec2(φ)vφ−tan(φ)vv2 − wd

]

= −W (χ)

+ zT

([v − vd

g sec(φ)2vφ−tan(φ)vv2 − wd

]+ (

∂V1

∂χg(χ))T

),

where vd and ωd represent the time derivative of vd and ωd in the case that vd and ωd are differentiable. Inthe case that vd and ωd are not differentiable, we let ˙vd and ˙ωd denote the generalized time derivative of vd

and ωd respectively and let vd and ωd represent the minimum norm element of ˙vd and ˙ωd respectively (seeRefs. 16,17).

Define

P =

[1 0

−g tan(φ)v2 g sec2(φ)

v

].

Letting [νv

νφ

]= P−1

(−Kz − (

∂V1

∂χg(χ))T +

[vd

ωd

]),

we have

vc = αvνv + v

φc = αφνφ + φ. (13)

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Given the control command (13), we can see that V2 = −W (χ) − zT Kz, which is negative definite. As aresult, it is straightforward to see that the control command (13) guarantees that |xe| + |ye| + |ψe| + |v −vd|+ |g tan(φ)

v − g tan(φd)vd | → 0, which in turn implies that |x− xr|+ |y − yr|+ |ψ − ψr| → 0.

Now we need to find vd and ωd and a Lyapunov function V such that

V =∂V

∂χf(t, χ) +

∂V

∂χg(χ)

[vd

ωd

]

is negative definite.In Ref. 12 we have shown that

V0(χ) =

√√√√(

λψe +ye√

x2e + y2

e + 1

)2

+ 1 (14)

+ k√

x2e + y2

e + 1− (1 + k) (15)

is a constrained Lyapunov function for system χ = f(t, χ) + g(χ)[v, ω]T with virtual control inputs v and ωgiven by Eqs. (7) and (8) such that V0(χ) ≤ −W (χ), where W (χ) is a continuous positive-definite function,k > 1

2 , λ > κ, where κ is a positive constant expressed precisely in Ref. 12.Therefore, we have the following lemma.

Lemma III.1 Let vd = v0 and ωd = ω0, where v0 and ω0 are given by Eqs. (7) and (8). The controlcommand (13) guarantees that |x− xr|+ |y − yr|+ |ψ − ψr|+ |v − vr|+ |g tan(φ)

v − ωr| → 0 asymptotically.

Proof: Letting V1 = V0(χ) in Eq. (11), we can see that V2 = −W (χ) − zT Kz, which implies that χ → 0and z → 0 asymptotically, that is, |xe| + |ye| + |ψe| → 0 and |v − vd| + |g tan(φ)

v − ωd| → 0. The factthat χ → 0 also implies that |vd − vr| + |ωd − ωr| → 0. Combining the above arguments, we know that|x− xr|+ |y − yr|+ |ψ − ψr|+ |v − vr|+ |g tan(φ)

v − ωr| → 0 asymptotically.In reality, the parameter vector θ = [αv, αφ]T is unknown and depends on the autopilot design. The

commanded velocity vc and roll angle φc are given by vc = αvνv +v and φc = αφνφ +φ, where θ = [αv, αφ]T

is an estimate of θ.Let the parameter estimate vector be updated as

˙θ = −Γ

[νv 00 νφ

]PT z, (16)

where Γ is a diagonal positive definite matrix.Consider a Lyapunov function candidate

V3 = V2 +12θT

[1

αv0

0 1αφ

]Γ−1θ, (17)

where θ = θ − θ.Differentiating Eq. (17), we get that

V3 = −W (χ)− zT Kz − zT P

[νv

αv0

0 νφ

αφ

][αv

αφ

]

+ θT

[1

αv0

0 1αφ

]Γ−1 ˙

θ

= −W (χ)− zT Kz,

where the last equality comes from the fact that

˙θ = Γ

[νv 00 νφ

]PT z.

As a result, we know that W (χ) → 0 and z → 0 from Theorem 5.27 in Ref. 18. Note that here θ maynot approach zero.

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IV. Simulation Results

In this section, we simulate a small Zagi airframe19 based UAV that tracks a trajectory generated bya trajectory generator described in Ref. 5. The simulation results in this section are based on a full sixdegree-of-freedom twelve-state model equipped with low-level autopilots described in Ref. 13. The UAV isequipped with a standard autopilot and the autopilot constants α∗ are generally unknown. However, wehave tuned the autopilot design to find out that αv ≈ 1

2 and αφ ≈ 10.55 for comparison purposes. Table 1

shows the specifications of the UAV and the control law parameters.

Table 1. Specifications of the UAV and the control law parameters.

Parameter Value

vmin 7.5 (m/s)vmax 13.5 (m/s)ωmax 0.671 (rad/s)φmax

π4 (rad)

vr ∈ [9.5, 11.5] (m/s)ωr ∈ [−0.471, 0.471] (rad/s)λ 1ηv 10ηω 10K diag{10, 10}Γ I

In the first case, we let vc = αvνv + v and φc = αφνφ + φ, where αv = 12 and αφ = 1

0.55 . Fig. 1 shows theactual and desired trajectories of the UAV without parameter adaptation. Here we use circles to representthe starting position of the UAV and squares to represent the ending position of the UAV. Also, we usediamonds to represent the position of the UAV at t = 10, 20, 30, 40 (secs). It can be seen that the UAV cantrack its desired trajectory accurately with known αv and αφ. Fig. 2 shows the tracking error for positionand heading. Fig. 3 shows the velocity and roll angle of the UAV. Note that the velocity and roll angle ofthe UAV satisfy their constraints. The switching phenomena of the roll angle are due to the fact that thereference velocity and heading rate vr and ωr are only piecewise continuous.

−600 −500 −400 −300 −200 −100 0−100

−50

0

50

x (m)

y (m

)

Reference and Actual Trajectories

ReferenceSaturation

Figure 1. Actual and desired trajectories of the UAV with accurate αv and αφ but without parameter adap-tation.

In the second case, we assume that no parameter adaptation law is applied. Here we let vc = αvνv + vand φc = αφνφ + φ, where αv and αφ are arbitrarily chosen as 0.192 and 0.55. Fig. 4 shows the actual and

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0 5 10 15 20 25 30 35 40 45 50−10

0

10

20

30

40

x r−x

(m)

0 5 10 15 20 25 30 35 40 45 50−30

−20

−10

0

10

y r−y

(m)

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

t (s)

ψr−

ψ (

rad)

Figure 2. Tracking errors of the UAV with accurate αv and αφ but without parameter adaptation.

0 5 10 15 20 25 30 35 40 45 507

8

9

10

11

12

13

14

t (s)

v (m

/s)

0 5 10 15 20 25 30 35 40 45 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t (s)

phi (

rad)

Figure 3. Velocity and roll angle of the UAV with accurate αv and αφ but without parameter adaptation.

desired trajectories of the UAV with neither accurate αv and αφ nor parameter adaptation. It can be seenthat the UAV cannot track its desired trajectory accurately. Fig. 5 shows the tracking error for position andheading. Fig. 6 shows the velocity and roll angle of the UAV.

In the third case, Eq. (16) is applied to update the estimated parameters αv and αφ, where we assumethat αv(0) = 0.192 and αφ(0) = 0.55. Fig. 7 shows the actual and desired trajectories of the UAV withinaccurate αv and αφ but with parameter adaptation. It can be seen that the UAV can track its desiredtrajectory more accurately in this case than in the second case. Fig. 8 shows the tracking error for positionand heading with adaptation. Fig. 9 shows the velocity and roll angle of the UAV with adaptation. Fig. 10shows the actual and estimated parameters of the UAV. Note that although the estimated parameters donot match the actual parameters here, trajectory tracking errors are still guaranteed to converge to zero dueto the inherent properties of adaptive control.20

V. Conclusion

With a UAV equipped with low-level autopilots, the twelve-state model of the UAV is reduced to a seven-state model with velocity, roll angle, and altitude command inputs. We have used backstepping techniquesto design velocity and roll angle control laws from known velocity and heading angle control laws for smallUAVs equipped with effective autopilots. The velocity and roll angle control laws have extended the velocityand heading controllers in Ref. 12, where the velocity and heading controllers explicitly account for limited

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−600 −500 −400 −300 −200 −100 0−100

−50

0

50

x (m)

y (m

)

Reference and Actual Trajectories

ReferenceSaturation

Figure 4. Actual and desired trajectories of the UAV with neither accurate αv and αφ nor parameter adaptation.

0 5 10 15 20 25 30 35 40 45 50−10

0

10

20

30

40

x r−x

(m)

0 5 10 15 20 25 30 35 40 45 50−30

−20

−10

0

10

y r−y

(m)

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

t (s)

ψr−

ψ (

rad)

Figure 5. Tracking errors of the UAV with neither accurate αv and αφ nor parameter adaptation.

velocity and heading rate constraints of fixed wing UAVs. A parameter adaptation technique has beenapplied to estimate unknown autopilot parameters. Simulation results on a small fixed wing UAV haveshown the effectiveness of our approach.

Acknowledgments

The authors would like to gratefully acknowledge Prof. Randy Beard and Derek Kingston from BrighamYoung University for discussing the ideas of this paper.

References

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t (s)

v (m

/s)

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−0.5

0

0.5

1

t (s)

phi (

rad)

Figure 6. Velocity and roll angle of the UAV with neither accurate αv and αφ nor parameter adaptation.

−600 −500 −400 −300 −200 −100 0−100

−50

0

50

x (m)

y (m

)

Reference and Actual Trajectories

ReferenceSaturation

Figure 7. Actual and desired trajectories of the UAV with inaccurate αv and αφ but with parameter adaptation.

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6McLain, T. W. and Beard, R. W., “Cooperative Rendezvous of Multiple Unmanned Air Vehicles,” Proceedings of theAIAA Guidance, Navigation and Control Conference, Denver, CO, August 2000, Paper no. AIAA-2000-4369.

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9McLain, T. W. and Beard, R. W., “Coordination Variables, Coordination Functions, and Cooperative Timing Missions,”AIAA Journal of Guidance, Control, and Dynamics, 2004, (to appear).

10Beard, R. W., McLain, T. W., Goodrich, M., and Anderson, E. P., “Coordinated Target Assignment and Intercept forUnmanned Air Vehicles,” IEEE Transactions on Robotics and Automation, Vol. 18, No. 6, 2002, pp. 911–922.

11Beard, R. W., Lawton, J. R., and Hadaegh, F. Y., “A Coordination Architecture for Formation Control,” IEEE Trans-actions on Control Systems Technology, Vol. 9, No. 6, November 2001, pp. 777–790.

12Ren, W. and Beard, R. W., “Trajectory Tracking for Unmanned Air Vehicles with Velocity and Heading Rate Con-straints,” IEEE Transactions on Control Systems Technology, Vol. 12, No. 5, September 2004, pp. 706–716.

13Kingston, D., Beard, R., McLain, T., Larsen, M., and Ren, W., “Autonomous Vehicle Technologies for Small Fixed WingUAVs,” AIAA 2nd Unmanned Unlimited Systems, Technologies, and Operations–Aerospace, Land, and Sea Conference andWorkshop & Exhibit , San Diego, CA, September 2003, Paper no. AIAA-2003-6559.

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0 5 10 15 20 25 30 35 40 45 50−10

0

10

20

30

40

x r−x

(m)

0 5 10 15 20 25 30 35 40 45 50−30

−20

−10

0

10

y r−y

(m)

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

t (s)

ψr−

ψ (

rad)

Figure 8. Tracking errors of the UAV with inaccurate αv and αφ but with parameter adaptation.

0 5 10 15 20 25 30 35 40 45 507

8

9

10

11

12

13

14

t (s)

v (m

/s)

0 5 10 15 20 25 30 35 40 45 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t (s)

phi (

rad)

Figure 9. Velocity and roll angle of the UAV with inaccurate αv and αφ but with parameter adaptation.

16Shevitz, D. and Paden, B., “Lyapunov Stability Theory of Nonsmooth Systems,” IEEE Transactions on AutomaticControl , Vol. 39, No. 9, 1994, pp. 1910–1914.

17Tanner, H. G. and Kyriakopoulos, K. J., “Backstepping for Nonsmooth Systems,” Automatica, Vol. 39, 2003, pp. 1259–1265.

18Sastry, S., Nonlinear Systems Analysis, Stability, and Control , Springer-Verlag, New York, 1999.19http://zagi.com.20Slotine, J.-J. E. and Li, W., Applied Nonlinear Control , Prentice Hall, Englewood Cliffs, New Jersey, 1991.

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0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

t (s)

α v

estimateactual

0 5 10 15 20 25 30 35 40 45 500.5

1

1.5

2

2.5

3

t (s)

α φ

estimateactual

Figure 10. Parameter estimates of the UAV with inaccurate αv and αφ but with parameter adaptation.

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