Nonlinear Trajectory Tracking for Fixed Wing UAVs via Backstepping and Parameter Adaptation. Wei Ren

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1 AIAA Guidance, Naigation, and Control Conference and Exhibit 5-8 August 25, San Francisco, California AIAA Nonlinear Trajectory Tracking for Fixed Wing UAVs ia Backstepping and Parameter Adaptation Wei Ren Department of Electrical and Computer Engineering, Utah State Uniersity, Logan, UT, 84322, USA Ella Atkins Space Systems Laboratory, Uniersity of Maryland, College Park, MD 2742, USA Equipping a fixed wing unmanned air ehicle (UAV) with low-leel autopilots, we derie high-leel elocity and roll angle control laws for the UAV. Backstepping techniques are applied to design the elocity and roll angle control laws from known elocity and heading angle control laws that explicitly account for elocity and heading rate constraints of the UAV. Regarding unknown autopilot constants, a parameter adaptation technique is used to estimate autopilot parameters. Simulation results on a fixed wing UAV are presented to show the effectieness of our approach. x, y inertial position, m ψ heading angle, rad φ roll angle, rad Airspeed, m/s h Altitude, m α Autopilot Parameters g graitational constant, m/s 2 Subscript r reference Superscript c command Nomenclature I. Introduction Adanced control technologies for unmanned air ehicles (UAVs) hae receied significant attention in recent years. Potential applications of autonomous UAVs in both ciilian and military sectors include enironment monitoring, search and rescue, communication relays, border patrol, situation awareness, sureillance, and battle damage assessment. Fully-automating UAVs poses both theoretical and practical challenges. One important research is UAV path planning. 2, 3 Another important aspect focuses on trajectory optimization for UAVs. 4, 5 In addition, cooperatie control of multiple UAVs is also studied extensiely in the literature. 6 8 Effectie trajectory tracking algorithms guarantee that a UAV can accurately follow its pre-specified desired trajectory. In addition, the study of nonlinear tracking control techniques for UAVs is essential for the success of cooperatie timing (e.g. Refs. 9, ) and formation keeping missions (e.g. Ref. ). In Ref. 2, elocity and heading control laws that explicitly account for stall conditions, thrust limitations, and saturated Assistant Professor, Department of Electrical and Computer Engineering, Utah State Uniersity, weiren@ieee.org Assistant Professor, Department of Aerospace Engineering, Uniersity of Maryland, ella@ssl.umd.edu, Senior Member. of Copyright 25 by the American Institute of Aeronautics and American Astronautics, Institute Inc. All of rights Aeronautics resered. and Astronautics

2 heading rate constraints are designed for small fixed wing UAVs equipped with elocity hold, heading hold, and altitude hold autopilots. In Ref. 2, it is assumed that the autopilot response to heading commands is first order in nature. Howeer, the headings of the UAVs are controlled by their roll motions, which imply that 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 autopilot constants in a control law design may result in degraded performance for the UAV. In this paper, we use a more accurate kinematic model that takes into account the roll motion for heading control. We apply backstepping techniques to derie elocity and roll angle control laws. In addition, a parameter adaptation technique is deeloped for the control law design regarding unknown autopilot constants. This paper is the continuation of the preious work presented in Ref. 2. II. Problem Statement Let (x, y), ψ,, φ, and h denote the inertial position, heading angle, elocity, roll angle, and altitude of the UAV respectiely. We assume that the UAV is equipped with standard autopilots as described in Ref. 3. The kinematic equations of motion are gien by ẋ = cos(ψ) ẏ = sin(ψ) ψ = g tan(φ) = α ( c ), φ = (φ c φ), ḧ = ḣ + (h c h), αḣ α h where φ c, c, and h c are the commanded roll angle, elocity, and altitude to the autopilots, g is the graitational constant, and α are positie constants. 3, 4 In the following, we assume that an effectie altitude controller exists and focus on the design of elocity and roll angle control laws. Due to the stall conditions, thrust limitations, and roll angle and pitch rate constraints of fixed wing aircraft, the following input constraints are imposed on the UAV: < min max φ max φ φ max (2) where φ max >. We assume that the desired reference trajectory (x r, y r, ψ r, r, ω r ) generated by a trajectory generator 5 satisfies ẋ r = r cos(ψ r ) ẏ r = r sin(ψ r ) (3) ψ r = ω r where r and ω r are continuous and satisfy that r and ω r are bounded, inf t r (t) > min, sup t r (t) < max, and sup t ω r (t) < ω max with ω max > denoting the heading rate constraint of the UAV. Transforming the tracking errors expressed in the inertial frame to the UAV frame, the error coordinates 5 become x e y e ψ e = cos(ψ) sin(ψ) sin(ψ) cos(ψ) x r x y r y ψ r ψ (). (4) Note that the motiation for this transformation is only to simplify the mathematics so that a constrained Lyapuno function can be easily deried. 2 of

3 Accordingly, the tracking error model can be represented as ẋ e = g tan(φ) y e + r cos(ψ e ) ẏ e = g tan(φ) x e + r sin(ψ e ) (5) ψ e = ω r g tan(φ). III. Trajectory Tracking Using Backstepping and Parameter Adaptation Techniques In Ref. 2, the authors use the following simplified kinematic model for the UAV: ẋ = c cos(ψ) ẏ = c sin(ψ) (6) ψ = α ψ (ψ c ψ), where c and ψ c are the commanded elocity and heading to the autopilots and α ψ > is an autopilot constant. Howeer, the autopilot response to the heading command is not truly first order in reality. As a result, Eq. () represents a more accurate model of the UAV equipped with standard autopilots. Let min, η x e < = r cos(ψ e ) + η x e, η x e, (7) max, η x e > ω max, η ω σ ω < ω ω = ω r + η ω σ ω, ω η ω σ ω ω, (8) ω max, η ω σ ω > ω where = min r cos(ψ e ), = max r cos(ψ e ), ω = ω r ω max, ω = ω r + ω max, σ ω = λψe + y e x 2 e +ye 2+, and η and η ω are sufficiently large positie constants expressed precisely in Ref. 2. In Ref. 2, it is shown that c = and ψ c = α ψ ω + ψ guarantees that x x r + y y r + ψ ψ r asymptotically. In addition, and ω satisfy the following input constraints < min max ω max ω ω max, which represents the case that the UAV has limited elocity and heading rate constraints. Howeer, the elocity and heading rate commands (7) and (8) are no longer alid for Eq. (), where the control commands to the autopilot are the elocity and roll angle commands. In the following, we apply backstepping techniques to derie elocity and roll angle commands. Note that Eqs. () and (5) can be rewritten as χ = f(t, χ) + g(χ)ξ (9) ζ = ν, () where χ = x e, y e, ψ e T, ξ = g tan(φ) 3 of

4 f(t, χ) = ζ = φ r cos(ψ e ) r sin(ψ e ) ω r y e g(χ) = x e ν ν = α = ( c ) ν φ (φ c. φ) Assume that ξ is the control input to Eq. (9) and suppose that we hae found a control law ξ d = d, g tan(φd ) T that stabilizes χ. Furthermore, assume that there exists a Lyapuno function V d satisfying V W (χ) with ξ d as the control input, where W ( ) is a positie definite function. Howeer, ξ is not a true control input in reality, so we will try to design c and φ c such that ξ d, g tan(φd ) T in the next step. d Let ω = g tan(φ) and ω d = g tan(φd ). Consider a Lyapuno function candidate d V 2 = V + 2 zt z, () where Differentiating Eq. (), we get that z = ξ ξ d = d ω ω d. (2) V 2 = V χ f(t, χ) + V χ g(χ)ξ + zt ż = V χ f(t, χ) + V χ g(χ)ξd + (ξ ξ d ) + z T d g sec2 (φ) φ tan(φ) ẇ d 2 = W (χ) ( + z T d g sec(φ)2 φ tan(φ) ẇ d 2 + ( V χ g(χ))t where d and ω d represent the time deriatie of d and ω d in the case that d and ω d are differentiable. In the case that d and ω d are not differentiable, we let ṽ d and ω d denote the generalized time deriatie of d and ω d respectiely and let d and ω d represent the minimum norm element of ṽ d and ω d respectiely (see Refs. 6, 7). Define Letting we hae ν ν φ P = = P ( g tan(φ) 2 g sec2 (φ) Kz ( V χ g(χ))t + c = α ν +. d ω d ) φ c = ν φ + φ. (3), ), 4 of

5 Gien the control command (3), we can see that V 2 = W (χ) z T Kz, which is negatie definite. As a result, it is straightforward to see that the control command (3) guarantees that x e + y e + ψ e + d + g tan(φ) g tan(φd ), which in turn implies that x x d r + y y r + ψ ψ r. Now we need to find d and ω d and a Lyapuno function V such that V = V V f(t, χ) + χ χ g(χ) is negatie definite. In Ref. 2 we hae shown that ( 2 V (χ) = y e λψ e + + (4) x 2 e + ye 2 + ) d ω d + k x 2 e + y 2 e + ( + k) (5) is a constrained Lyapuno function for system χ = f(t, χ) + g(χ), ω T with irtual control inputs and ω gien by Eqs. (7) and (8) such that V (χ) W (χ), where W (χ) is a continuous positie-definite function, k > 2, λ > κ, where κ is a positie constant expressed precisely in Ref. 2. Therefore, we hae the following lemma. Lemma III. Let d = and ω d = ω, where and ω are gien by Eqs. (7) and (8). The control command (3) guarantees that x x r + y y r + ψ ψ r + r + g tan(φ) ω r asymptotically. Proof: Letting V = V (χ) in Eq. (), we can see that V 2 = W (χ) z T Kz, which implies that χ and z asymptotically, that is, x e + y e + ψ e and d + g tan(φ) ω d. The fact that χ also implies that d r + ω d ω r. Combining the aboe arguments, we know that x x r + y y r + ψ ψ r + r + g tan(φ) ω r asymptotically. In reality, the parameter ector θ = α, T is unknown and depends on the autopilot design. The commanded elocity c and roll angle φ c are gien by c = ˆα ν + and φ c = ˆ ν φ + φ, where ˆθ = ˆα, ˆ T is an estimate of θ. Let the parameter estimate ector be updated as ν ˆθ = Γ P T z, (6) ν φ where Γ is a diagonal positie definite matrix. Consider a Lyapuno function candidate where θ = θ ˆθ. Differentiating Eq. (7), we get that V 3 = V θ T α V 3 = W (χ) z T Kz z T P + θ T α = W (χ) z T Kz, where the last equality comes from the fact that θ = Γ Γ θ ν α ν φ ν P T z. ν φ Γ θ, (7) As a result, we know that W (χ) and z from Theorem 5.27 in Ref. 8. Note that here θ may not approach zero. α 5 of

6 IV. Simulation Results In this section, we simulate a small Zagi airframe 9 based UAV that tracks a trajectory generated by a trajectory generator described in Ref. 5. The simulation results in this section are based on a full six degree-of-freedom twele-state model equipped with low-leel autopilots described in Ref. 3. The UAV is equipped with a standard autopilot and the autopilot constants α are generally unknown. Howeer, we hae tuned the autopilot design to find out that α 2 and.55 for comparison purposes. Table shows the specifications of the UAV and the control law parameters. Table. Specifications of the UAV and the control law parameters. Parameter Value min 7.5 (m/s) max 3.5 (m/s) ω max.67 (rad/s) π φ max 4 (rad) r 9.5,.5 (m/s) ω r.47,.47 (rad/s) λ η η ω K diag{, } Γ I In the first case, we let c = ˆα ν + and φ c = ˆ ν φ + φ, where ˆα = 2 and ˆ =.55. Fig. shows the actual and desired trajectories of the UAV without parameter adaptation. Here we use circles to represent the starting position of the UAV and squares to represent the ending position of the UAV. Also, we use diamonds to represent the position of the UAV at t =, 2, 3, 4 (secs). It can be seen that the UAV can track its desired trajectory accurately with known α and. Fig. 2 shows the tracking error for position and heading. Fig. 3 shows the elocity and roll angle of the UAV. Note that the elocity and roll angle of the UAV satisfy their constraints. The switching phenomena of the roll angle are due to the fact that the reference elocity and heading rate r and ω r are only piecewise continuous. 5 Reference and Actual Trajectories Reference Saturation y (m) x (m) Actual and desired trajectories of the UAV with accurate α and but without parameter adap- Figure. tation. In the second case, we assume that no parameter adaptation law is applied. Here we let c = ˆα ν + and φ c = ˆ ν φ + φ, where ˆα and ˆ are arbitrarily chosen as.92 and.55. Fig. 4 shows the actual and 6 of

7 x r x (m) y r y (m) ψ r ψ (rad) Figure 2. Tracking errors of the UAV with accurate α and but without parameter adaptation (m/s) phi (rad) Figure 3. Velocity and roll angle of the UAV with accurate α and but without parameter adaptation. desired trajectories of the UAV with neither accurate α and nor parameter adaptation. It can be seen that the UAV cannot track its desired trajectory accurately. Fig. 5 shows the tracking error for position and heading. Fig. 6 shows the elocity and roll angle of the UAV. In the third case, Eq. (6) is applied to update the estimated parameters ˆα and ˆ, where we assume that ˆα () =.92 and ˆ () =.55. Fig. 7 shows the actual and desired trajectories of the UAV with inaccurate α and but with parameter adaptation. It can be seen that the UAV can track its desired trajectory more accurately in this case than in the second case. Fig. 8 shows the tracking error for position and heading with adaptation. Fig. 9 shows the elocity and roll angle of the UAV with adaptation. Fig. shows the actual and estimated parameters of the UAV. Note that although the estimated parameters do not match the actual parameters here, trajectory tracking errors are still guaranteed to conerge to zero due to the inherent properties of adaptie control. 2 V. Conclusion With a UAV equipped with low-leel autopilots, the twele-state model of the UAV is reduced to a seenstate model with elocity, roll angle, and altitude command inputs. We hae used backstepping techniques to design elocity and roll angle control laws from known elocity and heading angle control laws for small UAVs equipped with effectie autopilots. The elocity and roll angle control laws hae extended the elocity and heading controllers in Ref. 2, where the elocity and heading controllers explicitly account for limited 7 of

8 5 Reference and Actual Trajectories Reference Saturation y (m) x (m) Figure 4. Actual and desired trajectories of the UAV with neither accurate α and nor parameter adaptation. x r x (m) y r y (m) ψ r ψ (rad) Figure 5. Tracking errors of the UAV with neither accurate α and nor parameter adaptation. elocity and heading rate constraints of fixed wing UAVs. A parameter adaptation technique has been applied to estimate unknown autopilot parameters. Simulation results on a small fixed wing UAV hae shown the effectieness of our approach. Acknowledgments The authors would like to gratefully acknowledge Prof. Randy Beard and Derek Kingston from Brigham Young Uniersity for discussing the ideas of this paper. References Clough, B. T., Unmanned Air Vehicles: Autonomous Control Challenges, A Researcher s Perspectie, Cooperatie Control and Optimization, edited by R. Murphey and P. M. Pardalos, Vol. 66, Kluwer Academic Publishers Series: Applied Optimization, 22, pp Bortoff, S. A., Path Planning for UAVs, Proceedings of the American Control Conference, 2. 3 Chandler, P., Rasumussen, S., and Pachter, M., UAV Cooperatie Path Planning, Proceedings of the AIAA Guidance, Naigation, and Control Conference, Dener, CO, August 2, AIAA Paper No. AIAA Yakimenko, O. A., Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories, AIAA Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, September-October 2, pp Anderson, E. P. and Beard, R. W., An Algorithmic Implementation of Constrained Extremal Control for UAVs, 8 of

9 4 3 2 (m/s) phi (rad) Figure 6. Velocity and roll angle of the UAV with neither accurate α and nor parameter adaptation. 5 Reference and Actual Trajectories Reference Saturation y (m) x (m) Figure 7. Actual and desired trajectories of the UAV with inaccurate α and but with parameter adaptation. Proceedings of the AIAA Guidance, Naigation, and Control Conference, Monterey, CA, August 22, Paper no. AIAA McLain, T. W. and Beard, R. W., Cooperatie Rendezous of Multiple Unmanned Air Vehicles, Proceedings of the AIAA Guidance, Naigation and Control Conference, Dener, CO, August 2, Paper no. AIAA Kang, W. and Sparks, A., Task Assignment in the Cooperatie Control of Multiple UAVs, Proceedings of the AIAA Guidance, Naigation, and Control Conference, Austin, TX, August 23, Paper no. AIAA Chandler, P. R., Cooperatie Control of a Team of UAVs for Tactical Missions, AIAA st Intelligent Systems Technical Conference, Chicago, IL, September 24, Paper no. AIAA McLain, T. W. and Beard, R. W., Coordination Variables, Coordination Functions, and Cooperatie Timing Missions, AIAA Journal of Guidance, Control, and Dynamics, 24, (to appear). Beard, R. W., McLain, T. W., Goodrich, M., and Anderson, E. P., Coordinated Target Assignment and Intercept for Unmanned Air Vehicles, IEEE Transactions on Robotics and Automation, Vol. 8, No. 6, 22, pp Beard, R. W., Lawton, J. R., and Hadaegh, F. Y., A Coordination Architecture for Formation Control, IEEE Transactions on Control Systems Technology, Vol. 9, No. 6, Noember 2, pp Ren, W. and Beard, R. W., Trajectory Tracking for Unmanned Air Vehicles with Velocity and Heading Rate Constraints, IEEE Transactions on Control Systems Technology, Vol. 2, No. 5, September 24, pp Kingston, D., Beard, R., McLain, T., Larsen, M., and Ren, W., Autonomous Vehicle Technologies for Small Fixed Wing UAVs, AIAA 2nd Unmanned Unlimited Systems, Technologies, and Operations Aerospace, Land, and Sea Conference and Workshop & Exhibit, San Diego, CA, September 23, Paper no. AIAA Proud, A. W., Pachter, M., and D Azzo, J. J., Close Formation Flight Control, Proceedings of the AIAA Guidance, Naigation, and Control Conference, Portland, OR, August 999, pp , Paper No. AIAA Kanayama, Y. J., Kimura, Y., Miyazaki, F., and Noguchi, T., A stable tracking control method for an autonomous mobile robot, Proceedings of the IEEE International Conference on Robotics and Automation, 99, pp of

10 x r x (m) y r y (m) ψ r ψ (rad) Figure 8. Tracking errors of the UAV with inaccurate α and but with parameter adaptation (m/s) phi (rad) Figure 9. Velocity and roll angle of the UAV with inaccurate α and but with parameter adaptation. 6 Sheitz, D. and Paden, B., Lyapuno Stability Theory of Nonsmooth Systems, IEEE Transactions on Automatic Control, Vol. 39, No. 9, 994, pp Tanner, H. G. and Kyriakopoulos, K. J., Backstepping for Nonsmooth Systems, Automatica, Vol. 39, 23, pp Sastry, S., Nonlinear Systems Analysis, Stability, and Control, Springer-Verlag, New York, Slotine, J.-J. E. and Li, W., Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey, 99. of

11 5 4 estimate actual α estimate actual Figure. Parameter estimates of the UAV with inaccurate α and but with parameter adaptation. of