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Singularity-free backstepping controller for model helicopters
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Singularity-free backstepping controller for model helicopters Yao Zou a,b, Wei Huo a,b,n a b
The Seventh Research Division, Beihang University, Beijing 100191, China School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2015 Received in revised form 5 June 2016 Accepted 20 June 2016 This paper was recommended for publication by Dr. Y. Chen.
This paper develops a backstepping controller for model helicopters to achieve trajectory tracking without singularity, which occurs in the attitude representation when the roll or pitch reaches 7 π2. Based on a simplified model with unmodeled dynamics, backstepping technique is introduced to exploit the controller and hyperbolic tangent functions are utilized to compensate the unmodeled dynamics. Firstly, a position loop controller is designed for the position tracking, where an auxiliary dynamic system with suitable parameters is introduced to warrant the singularity-free requirement for the extracted command attitude. Then, a novel attitude loop controller is proposed to obviate singularity. It is demonstrated that, based on the established criteria for selecting controller parameters and desired trajectories, the proposed controller realizes the singularity-free trajectory tracking of the model helicopter. Simulations confirm the theoretical results. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model helicopter Trajectory tracking Singularity-free controller Backstepping technique Auxiliary dynamic system
1. Introduction In recent decades, model helicopters have attracted much attention due to their capacities of vertically taking-off and landing, hovering and low-speed flight. They have been widely utilized in reconnaissance, surveillance and other stationary flight missions [1,2]. Since model helicopter dynamics are featured with under-actuation, aerodynamic complexity and strong coupling, the autonomous control of a model helicopter is still a challenge for researchers. Since the model helicopter is an under-actuated system, whose 3-dof (degree of freedom) translational motions and 3-dof rotational motions are driven by four controls, command attitude extraction is a necessary step for controller designs. Generally, an inner-outer loop control structure is used to design controllers for the model helicopter [3]: firstly, a position loop controller is synthesized for the position tracking; then, a main rotor thrust (as one control) and a command attitude are extracted from the position loop controller; and finally, an attitude loop controller (including three controls) is exploited for the attitude tracking to the command attitude. Nevertheless, under this control structure, when the attitude is represented with Euler angles, the inner attitude loop controller cannot offset the error from the outer position loop due to the nonlinear command attitude extraction. In this case, the n
Corresponding author. E-mail addresses: [email protected] (Y. Zou), [email protected] (W. Huo).
time-scale separation assumption is available [3], which requires a faster convergence of the attitude loop than the position loop. Under the time-scale separation assumption, various control approaches have been used for the position and attitude loop controller developments, such as nonlinear H 1 with model predictive control [4], feedback linearization [5–7], sliding mode control [8], PID with model inversion blocks [9]. However, with the time-scale separation assumption, the stable position tracking is on the premise of the stable attitude tracking. Backstepping technique is available for the controller development without the time-scale separation assumption [10,11]. Based on the backstepping strategy, He et al. [12] developed a controller to achieve the bounded trajectory tracking of a model helicopter. Lee et al. [13,14] combined the integral backstepping and dynamic extension to improve the tracking performance. The controllers in [12–14] are valid under the condition that the roll and pitch of the helicopter lie in π2 ; π2 , but they fail to ensure the condition. Further, to obviate the nonlinear command attitude extraction, Raptis et al. [15] proposed a new attitude representation instead of Euler angles. Nevertheless, transformation between the Euler angles and the new attitude representation appears singularity when the roll or pitch of the helicopter reaches 7 π2 . Although the backstepping controller in [15] achieves the nonsingular exponentially stable trajectory tracking of the model helicopter, it will be invalid once the unmolded dynamics of the helicopter are taken into account. Using the attitude representation proposed in [15], the backstepping controllers in [16–18] are
http://dx.doi.org/10.1016/j.isatra.2016.06.010 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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based on the hypothesis that no singularity occurs, but they cannot obviate it. Moreover, the unmodeled dynamics of the model helicopter may affect the tracking performance, thus, they should be compensated. Neural networks [7,19], nonlinear damping [17] and disturbance observers [12,20] have been applied to compensate the unmodeled dynamics. However, the neural network may increase computational complexity [21], the nonlinear damping may introduce a large undesired control input [22], and the disturbance observer may bring in additional intricate dynamics [23]. With the prior knowledge of the upper bounds of the unmodeled dynamics, the most effective way is to directly compensate them with sign functions and their upper bounds [24]. However, when the sign function appears in the position loop controller, chattering phenomenon may occur in the extracted command attitude, which is unbeneficial for the attitude tracking. Although the sliding-mode observers proposed in [25–27] can relieve chattering, they do not establish a parameterized relation between the chattering attenuation and the tracking performance. With the attitude representation proposed in [15], this paper develops a singularity-free trajectory-tracking controller for the model helicopter. The helicopter system is firstly simplified into a cascaded structure with unmodeled dynamics. Based on the simplified model, the backstepping technique is used to exploit the controller and the hyperbolic tangent function is applied to compensate the unmodeled dynamics. During the backstepping procedure, to avoid the aforementioned singularity in the attitude representation, it is necessary to impose the same non-singular requirement on the extracted command attitude. An auxiliary dynamic system with appropriate parameters is introduced to guarantee such non-singular requirement. It is proven that, with suitable controller parameters and desired trajectories, the proposed controller achieves the non-singular trajectory tracking of the model helicopter. Main contributions are listed as follows: (i) during the position loop controller design, an auxiliary dynamic system is introduced, and criteria for choosing parameters are established to ensure the stability of it and the non-singular requirement of the extracted command attitude; (ii) a singularity-free attitude controller with appropriate parameters and desired trajectory constraint is put forward; (iii) instead of the sign function, the hyperbolic tangent function is introduced to compensate the unmodeled dynamics, and the parameterized relation between the chattering attenuation and the tracking performance is built. The following sections are organized as follows: some mathematical preliminaries are presented in Section 2; control problems are stated in Section 3, controller design and stability analysis are provided in Section 4; Simulations are carried out in Section 5; and conclusions are drawn in Section 6.
2. Preliminaries In this paper, j j denotes the absolute value of a scalar, J J denotes the Euclidean norm of a vector, Sx and Cx with x A R are short for the trigonometric functions sin ðxÞ and cos ðxÞ, λ ðÞ and λ ðÞ denote the maximum and minimum eigenvalues of a square matrix, e1 ¼ ½1; 0; 0T , e2 ¼ ½0; 1; 0T and e3 ¼ ½0; 0; 1T denote three unit vectors. For x ¼ ½x1 ; x2 ; x3 T A R3 , the superscript denotes the transformation from x to a skew-symmetric matrix, namely, 2 3 0 x3 x2 6 x 0 x1 7 x ¼4 3 5: x2 x1 0
x
e For x A R, define the hyperbolic tangent function tanhðxÞ ¼ eex þ e x, tanhðxÞ which satisfies j tanhðxÞj o1 and 0 o x o 1. For x ¼ ½x1 ; …; xn T A Rn , define the hyperbolic tangent function vector tanhðxÞ ¼ ½tanhðx1 Þ; …; tanhðxn ÞT . x
Lemma 1 ([28]). Suppose that hðxÞ : D-Rn is a smooth function defined on D Rn , and the Jacobian matrix of h is nonsingular at a point x ¼ x0 , then on a suitable open subset D0 of D, containing x0 , h defines a local diffeomorphism. Lemma 2 ([18,29]). Given ϵ 4 0, the following inequality holds for x A R: x ð1Þ 0 r j xj x tanh r kq ϵ;
ϵ
where kq satisfies kq ¼ e ðkq þ 1Þ , i.e., kq ¼0.2785. Lemma 3 ([30,31]). The x A fx A R∣j xj o kg: 2
k
inequality
x2
holds
for
: k k x2 Lemma 4. Consider the system
ð2Þ
ξ€ þ α tanhðkξ þ lξ_ Þ þ β tanhðlξ_ Þ dðtÞ ¼ 0;
ð3Þ
ln
2
x2
r
following
2
where ξ A Rn and ξ_ A Rn are states, dðtÞ : R þ -Rn is a disturbance, k; l; α and β are positive constants and satisfy sffiffiffiffiffiffiffi k 1 k β α r β þ : ð4Þ r 2 2 2 l l If there exist d 4 0 and t 4 0 such that J dðtÞ J od o
kβ 2ðl β þ kÞ 2
;
8t Zt:
ð5Þ
then ξ and ξ_ ultimately converge to the attractive set T T T T T Z ¼ f½ξ ; ξ_ T ∣½kξ þ lξ_ ; lξ_ T o μ g;
where μ satisfies
ð6Þ
2 l2 β þ k d o tanhμ ðμ Þ o 12. kβ
Proof. See Appendix A.
3. Problem statements 3.1. Helicopter model In this paper, the helicopter is considered as a 6-dof rigid body. Let I ¼ fOxyzg denote an inertial frame whose origin O is located at a fixed point on the earth, and B ¼ fOb xb yb zb g denote a body frame whose origin Ob is located at the helicopter c.g. (center of gravity) (see Fig. 1). During the modeling, p ¼ ½px ; py ; pz T and v ¼ ½vx ; vy ; vz T respectively denote the position and velocity of the helicopter c.g. in I , ω ¼ ½ωx ; ωy ; ωz T denotes the angular velocity of the helicopter in B, γ ¼ ½ϕ; θ; ψ T denotes the Euler angle vector (roll, pitch, yaw), and the corresponding rotation matrix R from B to I , which is parametrized by γ , is expressed as 2 3 R11 R12 R13 6R 7 R ¼ 4 21 R22 R23 5 R31 R32 R33 2 3 C θ C ψ Sϕ Sθ C ψ C ϕ Sψ C ϕ Sθ C ψ þ Sϕ Sψ 6 7 ¼ 4 C θ S ψ S ϕ S θ S ψ þ C ϕ C ψ C ϕ S θ S ψ S ϕ C ψ 5: ð7Þ Sθ Sϕ C θ CϕCθ Referring to [32], the kinematic equations of the helicopter are described as p_ ¼ v;
ð8Þ
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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respect to the Ob xb zb plane, f and τ denote the applied force and torque in B, which are mainly generated by the main and tail rotors during the hovering or low-speed maneuver, and according to [18], they are expressed as 2 3 2 3 T m Sa T m hm Sb þ Lb þ T t ht þ τm Sa 6 T S þT 7 6 T l þ T h S þ Ma þ τ τ S 7 τ ¼4 m m m m a f ¼4 m b t 5; t m b 5; T m CbCa
T m lm S b T t lt þ τ m C a C b
where subscripts m and t denote the main and tail rotors, T i and τi (i¼m or t) denote the thrusts and anti-torques generated by the rotors, ½hi ; 0; li T (i¼m or t) denotes the relative position between the corresponding rotor and the helicopter c.g., L and M denote the stiffness coefficients of the main rotor, a and b denote the longitudinal and lateral flapping angles. Further, in terms of [18], τi (i¼m or t) is determined with τi ¼ C i j T i j 1:5 þ Di , where Ci and Di are the aerodynamic constants. Fig. 1. Model helicopter configuration.
R_ ¼ Rω :
3.2. Simplified model ð9Þ
Due to strong coupling between the applied force f and torque
τ , some simplifications should be undertaken to facilitate the
The attitude representation of the helicopter requires three independent components [15,17], whereas the attitude kinematic equation (9) is described in the form of a matrix differential equation, from which it is intricate to design a controller. In order to design a backstepping controller, an alternative attitude representation proposed in [15] is adopted. Let R3 ¼ Re3 , and from (9), it follows that
controller design. According to [17,32], due to physical constraint, the flapping angles of the helicopter are fairly small, so that the small angle approximation is feasible, namely, cos ðÞ ¼ 1 and sin ðÞ ¼ ðÞ. Further, the Ob xb -axis and Ob yb -axis force components as well as the tail rotor anti-torque make tiny contributions [15]. To this end, f and τ are rewritten as
_ 3 ¼ Rω e3 ¼ Re ω: R_ 3 ¼ Re 3
f ¼ T m e3 þ f Δ ;
ð15Þ
τ ¼ Aτ ρ þ τ B þ Δτ ¼ τ γ þ Δτ
ð16Þ
ð10Þ
In view of J R3 J ¼ 1, R3 incorporates just two independent components. Extracting first two constitutes the reduced-dimension attitude R 3 ¼ ½R31 ; R32 T . From (10), we have R_ 3 ¼ R^ ω ;
h
i
ð11Þ
is an invertible matrix and ω ¼ ½ωx ; ωy . where R^ ¼ Further, from (9), we can derive the following yaw kinematic equation: R12 R11 R22 R21
T
Sϕ C ωy þ ϕ ωz : ð12Þ Cθ Cθ Due to det ∂∂γγR ¼ C 2ϕ C θ , in terms of Lemma 1, the map from γ to
ψ_ ¼
T
γ R ¼ ½R 3 ; ψ T defines a local diffeomorphism on the set
j ϕ j o π2 ; j θ j o π2 , and singularity occurs at j ϕ j ¼ π2 or j θ j ¼ π2. Due π π to J R3 J ¼ 1 and continuity of R3 , under ϕð0Þ; θð0Þ A 2; 2 , we know that ϕðtÞ; θðtÞ A π2 ; π2 is equivalent to R33 ðtÞ ¼ C ϕðtÞ C θðtÞ 4 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
or J R 3 ðtÞ J ¼ 1 R33 ðtÞ2 o 1, for t Z0. Thus, given ϕð0Þ; θð0Þ A π2; π2 , as long as J R 3 ðtÞ J o 1 (t Z 0), singularity would not occur in the attitude representation γ R In addition, according to [17,32], with the Newton–Euler formula, the dynamic equations of the helicopter are derived as 1 v_ ¼ ge3 þ Rf ; m
ð13Þ
_ ¼ ω J ω þ τ ; Jω
ð14Þ
where m denotes the mass of the helicopter, g denotes the gravitational acceleration, J denotes the inertial matrix with respect to B, which is defined as 2 3 Jx 0 J xz 6 Jy 0 7 J¼4 0 5; Jz J xz 0 with zero elements resulting from the symmetric structure with
where 2
3 2 3 τm T m hm þ L ht Tt 6 0 7 7 τ m 5; ρ ¼ 6 T m hm þM Aτ ¼ 4 4 a 5; lt 0 T m lm b 2 3 2 3 T m Sa 0 6T l 7 6 T S þT 7 τ B ¼ 4 m m 5; f Δ ¼ 4 m b t 5; T m ðC a C b 1Þ τm 2
3 τm ðSa aÞ þT m hm ðSb bÞ 6 τ τ ðS bÞ þT h ðS aÞ 7 Δτ ¼ 4 t m b m m a 5: τm ðC a C b 1Þ T m lm ðSb bÞ Substituting (15) into (13) and (16) into (14) yield v_ ¼ ge3 þ
Tm R 3 þ Δf ; m
_ ¼ ω J ω þ τ γ þ Δτ ; Jω
ð17Þ ð18Þ
1 Rf Δ . During the controller design, Δf and Δτ are where Δf ¼ m treated as unmodeled dynamics, and their upper bounds will be compensated.
3.3. Control objective Given desired smooth trajectory pr ¼ ½prx ; pry ; prz T and yaw ψr, the objective is to develop control inputs Tm, Tt, a and b for the model helicopter, so that it can track P r and ψr with small errors. Further, under ϕð0Þ; θð0Þ A ð π2 ; π2 Þ, in order for the singularity-free tracking, the developed control inputs are required to warrant J R 3 ðtÞ J o 1 (t Z 0).
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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Assumption 1. The second-order derivative of the desired trajectory pr satisfies j p€ rz ðtÞj r δ o g ðt Z 0Þ. Assumption 2. The unmodeled dynamics Δf ¼ ½Δf x ; Δf y ; Δf z and Δτ ¼ ½Δτx ; Δτy ; Δτz T are bounded and satisfy j Δf i j r σ i ; j Δτi j r ςi ;
i ¼ x; y; z;
T
ð19Þ
where σ ¼ ½σ x ; σ y ; σ z with σ z o g δ and ς ¼ ½ςx ; ςy ; ςz are two known upper bound vectors. T
T
Remark 1. Assumption 2 seems strong, due to the dependence of Δf and Δτ on the system inputs. However, during the low-speed maneuver, affected by physical restrictions arising from mechanical constraint of flapping angles and stiffness of rotor blades, Δf and Δτ of typical helicopters are usually extremely small [32]. Here, we assume that they are bounded by known constants, which can be acquired by experiment results or practical experience.
where cp 4 0. Substituting (22) into (21) yields L_ 1 ¼ cp J p e J 2 þ p Te v e ; where v e ¼ ½v ex ; v ey ; v ez ¼ v αp η_ ¼ ve η_ . 4.1.2. Velocity error dynamics From (17), v e satisfies Tm v_ e ¼ ge3 þ R3 þ Δf α_ p η€ : m
From Assumption 2 and Lemma 2, it follows that X X v v Te Δf r j v ei j σ i r v ei tanh ei þ kq ϵ σ i i ¼ x;y;z
¼ v Te Tanh
T
ð24Þ
Choose a Lyapunov function L2 ¼ L1 þ 12v Te v e , and its derivative along (20) and (24) satisfies Tm L_ 2 ¼ cp J p e J 2 þ v Te ge3 þ R3 þ Δf α_ p η€ þ p e : m
4. Controller design The controller is designed with the backstepping technique and its block diagram is illustrated in Fig. 2. The controller design procedure is streamlined based on the backstepping technique in [22]. First, in the position loop, a controller u ¼ Tmm R3c is designed for the position to track pr ; second, the main rotor thrust Tm and the reduced-dimension command attitude R 3c are extracted from u; third, in the attitude loop, a controller τ γ is developed for the
ð23Þ T
ve
ϵ
i ¼ x;y;z
ϵ
σ þ kq ϵd1 ; v
Tanhðvϵe Þ ¼ diag½tanhðvϵex Þ; tanhð ϵey Þ; tanhðvϵez Þ and d1 ¼ 2 € _ i ¼ x;y;z σ i . Further, it can be derived that α p ¼ cp p e cp v e þ p r . _ Thus, L 2 satisfies Tm ve σ η€ L_ 2 r cp J p e J 2 þ v Te ½ ge3 p€ r þ R3 þ Tanh m ϵ
where P
ðc2p 1Þp e þ cp v e þ kq ϵd1 :
ð25Þ
attitude to track γ c ¼ ½R 3c ; ψ c T ; and finally, the control inputs Tt, a and b are derived from τ γ .
Develop a second-order auxiliary dynamic system about η:
4.1. Position loop controller design
η€ ¼ αη tanhðkη η þ lη η_ Þ βη tanhðlη η_ Þ þ Δη ;
4.1.1. Position error dynamics Define the position tracking error pe ¼ ½pex ; pey ; pez T ¼ p pr and the virtual position tracking error p e ¼ ½p ex ; p ey ; p ez T ¼ pe η, where η is an auxiliary variable. From (8), p e satisfies
where Δη ¼ ðc2p 1Þp e þðcp þ cv Þv e with cv 4 0, and positive parameters αη , β η , kη and lη satisfy vffiffiffiffiffiffiffiffiffiffiffi ! u ukη kη 1 t β η r αη r β þ : ð27Þ 2 2 η l2 l
p_ e ¼ v p_ r η_ :
ð20Þ
η
ð26Þ
η
Assign a Lyapunov function L1 ¼ 12p Te p e , and its derivative along (20) satisfies
The initial η and η_ can be arbitrary values; however, to facilitate analysis, let ηð0Þ ¼ η_ ð0Þ ¼ ½0; 0; 0T . Then, design a position loop controller:
L_ 1 ¼ p Te ðv p_ r η_ Þ:
u¼
ð21Þ
Design a virtual control:
αp ¼ cp p e þ p_ r ;
ð22Þ
Tm R3c ¼ αη tanhðkη η þ lη η_ Þ β η tanhðlη η_ Þ m ve þ ge3 þ p€ r Tanh σ;
ϵ
ð28Þ
Fig. 2. Controller block diagram.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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where R3c is a command attitude. And the main rotor thrust Tm is extracted as Tm ¼ mJuJ:
ð29Þ
Substituting (26) and (28) into (25) yield Tm L_ 2 r cp J p e J 2 cv J v e J 2 þ v Te R3e þ kq ϵd1 m
ð30Þ
where R3e ¼ R3 R3c . 4.2. Non-singular command attitude extraction From (28) and (29), the command attitude R3c is derived as R3c ¼
m u: Tm
ð31Þ
Extracting first two components of R3c constitutes R 3c ¼ ½R31c ; R32c T , which, together with the desired yaw ψr, serves as the tracking objective for the attitude loop controller design. To ensure J R 3 J o 1 for non-singularity, the same requirement should be imposed on its tracking objective, namely, J R 3c J o 1. If the parameters αη and β η in (26) and (28) satisfy
αη þ β η o g δ σ z ;
ð32Þ
from Assumptions 1 and 2, the third row of (28) satisfies
5
4.3.2. Yaw error dynamics Define the yaw error ψ e ¼ ψ ψ r . From (12), it satisfies
ψ_ e ¼
Sϕ C ωy þ ϕ ωz ψ_ r : Cθ Cθ
ð37Þ
Choose a Lyapunov function L4 ¼ ψ 2e , and its derivative along (37) satisfies Sϕ C ð38Þ ωy þ ϕ ωz ψ_ r : L_ 4 ¼ ψ e Cθ Cθ Design a virtual control: S C αψ ¼ θ cψ ψ e þ ψ_ r ϕ ωy ; Cϕ Cθ
ð39Þ
where cψ 4 0. Substituting (39) into (38) yields Cϕ L_ 4 ¼ cψ ψ 2e þ ψ e ωez ; Cθ
ð40Þ
where ωez ¼ ωz αψ . 4.3.3. Angular velocity error dynamics Define αγ ¼ ½αTR ; αψ T and ωe ¼ ½ω Te ; ωze T , ωe ¼ ω αγ . From (18), ωe satisfies
which
satisfy
_ e ¼ ω J ω þ τ γ þ Δτ J α_ γ : Jω
Tm R33c ¼ αη tanhðkη ηz þ lη η_ z Þ βη tanhðlη η_ z Þ m v ez σz þ g þ p€ rz tanh
ð41Þ
Choose a Lyapunov function L5 ¼ ω ωe , and its derivative along (20), (24), (33), (37) and (41) satisfies L3 þL4 þ 12
ϵ
Z αη β η þ g δ σ z 4 0
L5 r cp J p e J 2 cv J v e J 2 R 3e R^ ω e
cR J R 3e J 2 2 bR ðkR J R 3e J 2 Þ
T eJ
cψ ψ 2e þ
Cϕ ψ ωez Cθ e
T
which means R33c 4 0. Further, from J R3c J ¼ 1, R33c 4 0 implies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J R 3c J ¼ 1 R233c o1. 4.3. Attitude loop controller design 4.3.1. Reduced-dimension attitude error dynamics Define the reduced-dimension attitude error R 3e ¼ R 3 R 3c and a time-varying parameter kR ¼ 1 J R 3c J 4 0. If J R 3e J o kR , then J R 3 J r J R 3e J þ J R 3c J o1, so that non-singularity can be guaranteed. From (11), R 3e satisfies _ _ R 3e ¼ R^ ω R 3c :
ð33Þ
Assign a Lyapunov function L3 ¼ L2 þ 2b1R ln
2
kR 2 kR J R 3e J 2
with bR 4 0, and
its derivative along (20), (24) and (33) satisfies Tm L_3 r cp J p e J 2 cv J v e J 2 þ v Te R3e m T k_ R R 3e ðR^ ω R_ 3c R 3e Þ þ kq ϵd1 : þ 2 kR bR ðk J R 3e J 2 Þ
ð35Þ
T
derived that Tmm v Te R3e Tmm R 3e v^ e ¼ 0 (See Appendix B). Substituting (35) into (34) yields
T R 3e R^ e þ kq þ 2 bR ðkR J R 3e J 2 Þ
ω
where ω e ¼ ω αR .
cR J R 3e J
Similarly, from Assumption 2 and Lemma 2, it follows that ω ωTe Δτ r ωTe Tanh e ς þ kq ϵd2 ;
ϵ
ω
Tanhðωϵe Þ ¼ diag½tanhðωϵex Þ; tanhð ϵey Þ; tanhðωϵez Þ _ ς i ¼ x;y;z i . Thus, L 5 satisfies
where P
L5 r cp J p e J 2 cv J v e J 2
cR J R 3e J 2 2
bR ðkR J R 3e J 2 Þ
2 bR ðkR
ϵ
T C R 3e R^ ; ϕ J R 3e J 2 Þ C θ
ψe
and
d2 ¼
cψ ψ 2e
h i ωe þ ωTe ω J ω þ τ γ þ Tanhð Þς J α_ γ þ ϱ þkq ϵd;
ð42Þ
T
and d ¼ d1 þ d2 . Design an attitude
τ γ ¼ cω ωe þ ω J ω þ J α_ γ ϱ Tanh
ð36Þ
e
ϵ
ς;
ρ ¼ Aτ 1 ðτ γ τ B Þ:
ð43Þ
ð44Þ
Substituting (43) into (42) yields L5 r cp J p e J 2 cv J v e J 2 cω J ωe J 2 þ kq ϵd;
2
ω
where cω 40. From (16), the control input ρ ¼ ½T t ; a; bT is derived as
2
bR ðkR J R 3e J 2 Þ
ϵd1 ;
þ ωTe ð ω J ω þ τ γ þ Δτ J α_ γ Þ þ kq ϵd1
loop controller: ð34Þ
þ R13c þ R23c where cR 4 0 and v^ e ¼ ½v ex RR13 v ez ; v ey RR23 v ez T . It can be 33 þ R33c 33 þ R33c
L_3 r cp J p e J 2 cv J v e J 2
2 bR ðkR J R 3e J 2 Þ
where ϱ ¼
R
In order to warrant J R 3e J o kR , design a virtual control: " ! 2 1 k_ R bR T m ðkR J R 3e J 2 Þ ^ αR ¼ R cR þ j j R 3e þ R_ 3c v^ e ; m kR
þ
cR J R 3e J 2 2
bR ðkR J R 3e J 2 Þ
cψ ψ 2e ð45Þ
Remark 2. From (35) and (43), R_ 3c and α_ γ are required to determine αR and τ γ , respectively; however, it is intricate to analytically calculate them. Instead, they can be approximated with the
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command filter introduced in [33]. Using appropriate parameters, this approximation is of high accuracy.
inferred from (51) that, if J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J o kR ðtÞ ðt Z 0Þ. □
Remark 3. Since kR is not differentiable at R 3c ¼ 0, a complementary definition of k_ R at R 3c ¼ 0 is required. With Filippov Theory [24], the derivative of kR can be determined by
Remark 4. The bounds of the ultimate convergent sets Z p , Z v , Z R , Z ψ and Z ω can be made arbitrarily small by increasing the parameter c.
k_ R ¼
8 < :
R 3c _ R ; J R 3c J 3c T
R 3c a0;
ð46Þ
R 3c ¼ 0:
0;
Lemma 6. Consider the auxiliary dynamic system (26) with the parameters kη, lη, αη and β η satisfying (27) and (32), if p e and v e are bounded and ultimately converge to the sets Z p and Z v , and the controller parameter ϵ in (28) and (43) satisfies 2
ϵo
4.4. Stability analysis and singularity discussion
cΔ η
; 2c 2 kq d
ð52Þ
where c ¼ maxðj c2p 1j ; cp þ cv Þ and Δ η A 0; Lemma 5. Consider the closed-loop system composed by (20), (24), (33), (37) and (44), if the initial reduced-dimension attitude error R 3e ð0Þ satisfies J R 3e ð0Þ J okR ð0Þ, then the developed control inputs (29) and (44) can guarantee that (i) the tracking errors p e , v e , R 3e , ψe and ωe are bounded and ultimately converge to sets around the origin; (ii) J R 3e ðtÞ J o kR ðtÞ ðt Z0Þ. Proof. (i) In terms of Lemma 3, it follows that ln
2 kR 2
kR J R 3e J 2
r
J R 3e J 2
2
kR J R 3e J 2
:
Consider the definition of L5, then (45) becomes ð47Þ L_ 5 r 2cL5 þ kq ϵd; where c ¼ min cp ; cv ; cR ; cψ ; cω . From Comparison Principle [21], λ ðJÞ it follows that kq ϵd 2ct kq ϵd e ; þ L5 ðtÞ r L5 ð0Þ 2c 2c
8 t Z 0:
ð48Þ
r L5 ð0Þe 2ct þ
kq ϵd ; 2c
8 t Z 0;
which implies that p e , v e ,
ð49Þ
ψe and ωe are bounded and
qffiffiffiffiffiffiffi kq ϵd , ultimately converge to the sets Z p ¼ p e A R3 ∣J p e J r c
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi kq ϵd k q ϵd Z v ¼ v e A R3 ∣ J v e J r , Z ψ ¼ ψ e A R∣j ψ e j r and c c
rffiffiffiffiffiffiffiffi k ϵd Z ω ¼ ωe A R3 ∣ J ωe J r cλq ðJÞ , respectively. Moreover, in view of the definition of L5, we also have kq ϵd 1 kR ; ln r L5 ð0Þe 2ct þ 2c 2bR k2 J R 3e J 2 2
8 t Z 0:
ð50Þ
R
Taking exponential on (50) yields J R 3e ðtÞ J r kR ðtÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2ct þ kq ϵd 2c o kR ðtÞ; 1 e 2bR L5 ð0Þe
8 t Z 0;
ð51Þ
2ðlη βη þ kη Þ
, then the
Proof. In view of the definition of Δη , we obtain J Δη J 2 r 2ðc2p 1Þ2 J p e J 2 þ2ðcp þ cv Þ2 J v e J 2 r 2c 2 ð J p e J 2 þ J v e J 2 Þ: In terms of (49), then k q ϵd ; J Δη ðtÞ J 2 r 4c 2 L5 ð0Þe 2ct þ 2c If ϵ is chosen satisfying (52), then exists a tη satisfying 8 > > > > 0; > > < 2 0 2 13 tη ¼ > Δ k ϵ d 1 1 > η q > ln4 @ A5 > > > 2c Lð0Þ 4c 2 : 2c
8 t Z0:
ð53Þ
2c kq ϵd o c
Δ η . In this case, there
2
2
2
if Lð0Þ r
Δη
4c 2
kq ϵd ; 2c
kq ϵd ; 2c
2
if Lð0Þ 4
Δη
4c 2
ð54Þ
such that J Δη ðtÞ J r Δ η (t Z t η ). From Lemma 4, it follows that η and η_ ultimately converge to the attractive set
where μ η satisfies
1 1 ð J p e ðtÞ J 2 þ J v e ðtÞ J 2 þ ψ e ðtÞ2 þ J ω ðtÞ J 2 Þ 2 λ ðJÞ e
auxiliary variable η and its derivative η_ ultimately converge to sets around the origin.
Z η ¼ f½ηT ; η_ T T ∣ J ½kη ηT þ lη η_ T ; lη η_ T T J r μ η g;
Based on the definition of L5, we know
kη βη
2
lη βη þ kη kη βη 2
Δη o
tanh ðμ η Þ 2
μη
ð55Þ
o 12. Further, it can be derived
that η and η_ are bounded and ultimately converge to the sets Z η1 n o n o μ μ ¼ η A R3 ∣ J η J r kηη and Z η2 ¼ η_ A R3 ∣ J η_ J r lηη , respectively.□ Remark 5. From Lemma 5, a smaller ϵ results in smaller ultimately convergent bounds of the tracking errors p e , v e , R 3e , ψe and ωe , however, too small ϵ may lead to the chattering phenomenon in the position loop controller u, which is unfavorable for the subsequent attitude tracking. Thus, when choosing ϵ satisfying (52), we should make a tradeoff between the good tracking performance and the chattering attenuation. Theorem 1. Consider the model helicopter system (8), (11)–(14) with Assumptions 1 and 2, given the initial states pð0Þ A R3 , vð0Þ A R3 , ωð0Þ A R3 , ϕð0Þ; θð0Þ A ð π2; π2Þ (i.e., J R 3 ð0Þ J o 1) and ψ ð0Þ A ð π ; π , if the parameters kη, lη, αη and βη satisfy (27) and (32), ϵ satisfies (52), and the desired trajectory pr as well as the upper bound vector σ make the initial reduced-dimension command attitude R 3c ð0Þ satisfy J R 3 ð0Þ R 3c ð0Þ J þ J R 3c ð0Þ J o1;
ð56Þ
where which implies that R 3e is bounded and ultimately converges to the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bR kq ϵd . (ii) It can be compact set Z R ¼ R 3e A R2 ∣J R 3e J r kR 1 e c
R 3c ð0Þ ¼
½e1 ; e2 T uð0Þ ; J uð0Þ J
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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3 v ð0Þ þ cp ðpx ð0Þ prx ð0ÞÞ p_ rx ð0Þ p€ rx ð0Þ tanh x σx ϵ 6 7 6 7 6 p€ ry ð0Þ tanh vy ð0Þ þ cp ðpy ð0Þ pry ð0ÞÞ p_ ry ð0Þ σ y 7 ϵ 6 7; uð0Þ ¼ 6 7 6 7 vz ð0Þ þ cp ðpz ð0Þ prz ð0ÞÞ p_ rz ð0Þ 4 g þ p€ rz ð0Þ tanh σz 5 ϵ 2
the designed control inputs (29) and (44) can guarantee that (i) the tracking errors pe and ve are bounded and converge to sets around the origin; (ii) no singularity occurs during the tracking progress. Proof. (i) From Lemmas 5 and 6, if the parameters kη, lη, αη , βη
and ϵ satisfy (27), (32) and (52), the designed control inputs (29) and (44) can ensure that the tracking errors p e , v e and the auxiliary variables η, η_ are bounded and ultimately converge to the sets Z p , Z v , Z η1 and Z η2 , respectively. From the definitions of
7
pe and ve , we have J pe J r J p e J þ J η J and J ve J r J v e J þ J η_ J . Thus, pe and ve are bounded and ultimately converge to the
qffiffiffiffiffiffiffi kq ϵd μ η and Z v ¼ ve A R3 j J ve J r sets Z p ¼ pe A R3 j J pe J r c þ kη qffiffiffiffiffiffiffi kq ϵd μ η þ , respectively. c lη (ii) If R 3c ð0Þ satisfies (56), then J R 3e ð0Þ J o kR ð0Þ. We have proven in Lemma 5 that, if J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J o kR ðtÞ ðt Z 0Þ. In this case, with the triangle inequality, we obtain J R 3 ðtÞ J ¼ J R 3c ðtÞ þ R 3e ðtÞ J r J R 3c ðtÞ J þ J R 3e ðtÞ J o J R 3c ðtÞ J þ kR ðtÞ ¼ 1;
8 t Z 0;
ð57Þ
which means ϕðtÞ; θðtÞ A ð π2; π2 Þ ðt Z 0Þ. Thus, no singularity occurs during the tracking progress.
□
Remark 6. Generally, the roll ϕ and pitch θ of the model helicopter is initialized at the origin, which means R 3 ð0Þ ¼ ½0; 0T . In
Fig. 5. Velocity tracking error.
Fig. 3. 3D trajectory tracking.
Fig. 4. Position and yaw tracking errors.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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this case, the relation (56) can be simplified as pffiffiffi 3ð J p€ r ð0Þ J þ J σ J Þ og þ p€ rz ð0Þ σ z ;
ð58Þ
where p€ r ¼ ½p€ rx ; p€ ry T and σ ¼ ½σ x ; σ y T . Actually, from (56), R 3 ð0Þ ¼ ½0; 0T implies J R 3c ð0Þ J o 12; if the desired trajectory pr and the upper bound vector σ satisfy (58), then p€ r ð0Þ J þ J σ J ffi J R 3c ð0Þ J r J qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð J p€ r ð0Þ J þ J σ J Þ2 þ ðg þ p€ rz ð0Þ σ z Þ2 p€ r ð0Þ J þ J σ J 1 ¼ : oJ 2ð J p€ r ð0Þ J þ J σ J Þ 2
5. Simulations As far as the helicopter model composed by (8) and (11)–(14) is concerned, simulations with MATLAB/Simulink are carried out to verify the effectiveness of the proposed controller. The physical and aerodynamic parameters [16] are as follows: m ¼ 7:4 kg, J x ¼ 0:16 kg m2 , J y ¼ 0:30 kg m2 , J z ¼ 0:32 kg m2 , J xz ¼ 0:05 kg m2 , lm ¼ 0:01 m, hm ¼ 0:25 m, lt ¼ 0:9 m, ht ¼ 0:12 m, M ¼ L ¼ 25:23, C m ¼ 0:00452, Dm ¼ 0:08488, C t ¼ 0:005066 and Dt ¼ 0:008488. Choose σ ¼ ½1; 1; 1T and ς ¼ ½0:1; 0:1; 0:1T , which mean d ¼3.3. 5.1. Performance verification
tracking. Fig. 7 depicts the bounded angular velocity tracking error. Further, denote μη ¼ ½4ηT þ η_ T ; η_ T . Fig. 8 shows that η and η_ are bounded and ultimately converge to the attractive set Z η with μ η ¼ 0:47, which verifies the result proposed in Lemma 6. 5.2. Performance comparison In order to highlight the proposed controller, it will be compared with [17], which also adopted the attitude representation introduced in [15]. The desired trajectory is described as 8 > ½0:1t 2 ; 0; 5T m; if t r 10 s; > > 2 3 > < 10 cos ð0:2ðt 10ÞÞ pr ¼ 6 7 > 4 10 sin ð0:2ðt 10ÞÞ 5 m; otherwise; > > > : 5 with ψ r ¼ 0 rad. The controller parameters and the initial conditions are the same with the above maneuver. The simulation results are illustrated in Figs. 9 and 10. It can be observed from Fig. 9 that, the proposed controller achieves the position and yaw tracking of the model helicopter with small errors, however, the controller in [17] fails and its position and yaw diverge to infinity. From Fig. 10, the proposed controller can guarantee the roll ϕ satisfying j ϕ j o π2, however, the roll ϕ with the controller in [17] exceeds π2 and diverges to infinity, which leads to singularity and the divergence of the position and yaw. By comparison, we conclude that the proposed controller is
The desired trajectory in this maneuver is a circular line: pr ¼ ½5 cos ð0:2tÞ; 5 sin ð0:2tÞ; 5T m with ψ r ¼ 0 rad. The procedure of the controller parameter choice is as follows: first, choose cp ¼ cv ¼ 0:5, cR ¼4, bR ¼0.2, cψ ¼ 1 and cω ¼ 15, which mean c ¼ 1; then, based on (27) and (32), choose kη ¼ αη ¼ βη ¼ 4 and lη ¼ 1;
finally, set Δ η ¼ 0:8, and based on (52), choose ϵ ¼ 0:1. With the selected controller parameters, it is derived that the convergent tanh ðμ Þ 2
bound μ η satisfies 0:4 o μ η o0:5. It is calculated by MATLAB/ Simulink that μ η A ð0:46; 0:64Þ. The model helicopter is initially still at pð0Þ ¼ ½0; 5; 0T m with ψ ð0Þ ¼ 0 rad. Since (58) holds, set R 3 ð0Þ ¼ ½0; 0T . The simulation results are illustrated in Figs. 3–8. Fig. 3 depicts the 3D tracking trajectory with respect to the desired one, which shows the accomplishment of the claimed tracking objective. Fig. 4 illustrates the bounded position and yaw tracking errors. Fig. 5 depicts the bounded velocity tracking error. Fig. 6 indicates that the reduced-dimension attitude tracking error is bounded and that once J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J okR ðtÞ always holds, which, from Theorem 1, means ϕðtÞ; θðtÞ A ð π2 ; π2 Þ during the tracking progress and the singularity-free attitude
Fig. 7. Angular velocity tracking errors.
Fig. 6. Lateral-longitudinal attitude tracking error.
Fig. 8. Auxiliary variable tracking errors.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
Y. Zou, W. Huo / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
Fig. 9. Position and yaw (straight line: desired ones; dash line: ones with the proposed controller; dot dash line: ones with the controller in [17]).
Acknowledgement This work is supported by National Natural Science Foundation of China under Grant No. 61074010.
Appendix A. Proof of Lemma 4 Lemma 7 ([34]). Consider the system (3) and the convergent set Z in (6). If there exist d 4 0 and t 4 0, such that " # 2 2 tanh ðμ Þχ 1 tanh ðμ Þχ 1 ; αþβ ; 8t Zt; ; ðA:1Þ J dðtÞ J od omin α lμ βl þ kl μ
Fig. 10. Roll ϕ (dash line: one with the proposed controller; dot dash line: one with the controller in [17]).
able to achieve the non-singular trajectory tracking of the model helicopter.
6. Conclusions A singularity-free controller is proposed for model helicopters to accomplish trajectory tracking missions. The backstepping technique is employed to design the controller, and the hyperbolic tangent function is applied to compensate the unmodeled dynamics. An auxiliary dynamic system is introduced in the position loop to ensure the nonsingular requirement of the extracted command attitude. A novel singularity-free attitude control strategy is also presented. Stability analysis has proven that, based on the established selection criteria for controller parameters and desired trajectories, the proposed controller achieves the trajectory tracking of the model helicopter.
where χ 1 ¼ minðα2 l; klβÞ, then ξ and ξ_ ultimately converge to the attractive set Z. qffiffiffiffiffiffi k 1 k Proof of Lemma 4. In view of 2 β r 2ðβ þ 2 Þ, there exists an α l l satisfying (4). From the right inequality of (4), αl r 12ðβl þ klÞ o βl þ kl, which implies tanh ðμ Þχ 1 tanh μχ 1 o : α lμ ðβl þ klÞμ 2
The
2
left
inequality
of
(4)
yields
χ 1 ¼ min½α2 l; klβ ¼ klβ. Further, we have kβ
tanh ðμ Þ
kβ
2
μ
l βþk 2
o
l β þk 2
Thus, it follows that 2
α2 l Z klβ, which implies
o β o α þ β:
3
6tanh2 ðμ Þχ 1 tanh2 ðμ Þχ 1 7 tanh2 ðuÞkβ ; α þ β7 ; min6 4 5¼ 2 k αlμ ðl β þ kÞμ βl þ μ l Since tanhμ ðμ Þ is continuous about μ A ð0; 1Þ and its maximum value 2
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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10
lies in the interval (0.5,0.6), there exists a μ such that do
kβ
tanh ðμ Þ
l βþk 2
2
μ
o
kβ : 2 2ðl β þ kÞ
ðA:2Þ
Therefore, from Lemma 7, ξ and ξ_ ultimately converge to the attractive set Z in (6), where, in view of (A.2), μ satisfies 2 l2 β þ k d o tanhμ ðμ Þ o 12. kβ
□ T
Appendix B. Derivation of R 3e v^ e ¼ RT3e v e From J R3 J 2 ¼ R231 þ R232 þR233 ¼ 1 and J R3c J 2 ¼ R231c þ R232c þ R233c ¼ 1, it follows that R13 þ R13c R23 þ R23c T R 3e v^ e ¼ R13e v ex v ez þR23e v ey v ey v ez R33 þ R33c R33 þ R33c R13 þ R13c R23 þ R23c R13e þ R23e v ez ¼ R13e v ex þ R23e v ey R33 þ R33c R33 þ R33c ¼ R13e v ex þ R23e v ey
R213 R213c þ R223 R223c v ez R33 þ R33c
R233 R233c v ez R33 þ R33c ¼ R13e v ex þ R23e v ey þ ðR33 R33c Þv ez ¼ R13e v ex þ R23e v ey þ R33e v ez ¼ R13e v ex þ R23e v ey þ
¼ RT3e v e :
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Singularity-free backstepping controller for model helicopters Yao Zou a,b, Wei Huo a,b,n a b
The Seventh Research Division, Beihang University, Beijing 100191, China School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 December 2015 Received in revised form 5 June 2016 Accepted 20 June 2016 This paper was recommended for publication by Dr. Y. Chen.
This paper develops a backstepping controller for model helicopters to achieve trajectory tracking without singularity, which occurs in the attitude representation when the roll or pitch reaches 7 π2. Based on a simplified model with unmodeled dynamics, backstepping technique is introduced to exploit the controller and hyperbolic tangent functions are utilized to compensate the unmodeled dynamics. Firstly, a position loop controller is designed for the position tracking, where an auxiliary dynamic system with suitable parameters is introduced to warrant the singularity-free requirement for the extracted command attitude. Then, a novel attitude loop controller is proposed to obviate singularity. It is demonstrated that, based on the established criteria for selecting controller parameters and desired trajectories, the proposed controller realizes the singularity-free trajectory tracking of the model helicopter. Simulations confirm the theoretical results. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model helicopter Trajectory tracking Singularity-free controller Backstepping technique Auxiliary dynamic system
1. Introduction In recent decades, model helicopters have attracted much attention due to their capacities of vertically taking-off and landing, hovering and low-speed flight. They have been widely utilized in reconnaissance, surveillance and other stationary flight missions [1,2]. Since model helicopter dynamics are featured with under-actuation, aerodynamic complexity and strong coupling, the autonomous control of a model helicopter is still a challenge for researchers. Since the model helicopter is an under-actuated system, whose 3-dof (degree of freedom) translational motions and 3-dof rotational motions are driven by four controls, command attitude extraction is a necessary step for controller designs. Generally, an inner-outer loop control structure is used to design controllers for the model helicopter [3]: firstly, a position loop controller is synthesized for the position tracking; then, a main rotor thrust (as one control) and a command attitude are extracted from the position loop controller; and finally, an attitude loop controller (including three controls) is exploited for the attitude tracking to the command attitude. Nevertheless, under this control structure, when the attitude is represented with Euler angles, the inner attitude loop controller cannot offset the error from the outer position loop due to the nonlinear command attitude extraction. In this case, the n
Corresponding author. E-mail addresses: [email protected] (Y. Zou), [email protected] (W. Huo).
time-scale separation assumption is available [3], which requires a faster convergence of the attitude loop than the position loop. Under the time-scale separation assumption, various control approaches have been used for the position and attitude loop controller developments, such as nonlinear H 1 with model predictive control [4], feedback linearization [5–7], sliding mode control [8], PID with model inversion blocks [9]. However, with the time-scale separation assumption, the stable position tracking is on the premise of the stable attitude tracking. Backstepping technique is available for the controller development without the time-scale separation assumption [10,11]. Based on the backstepping strategy, He et al. [12] developed a controller to achieve the bounded trajectory tracking of a model helicopter. Lee et al. [13,14] combined the integral backstepping and dynamic extension to improve the tracking performance. The controllers in [12–14] are valid under the condition that the roll and pitch of the helicopter lie in π2 ; π2 , but they fail to ensure the condition. Further, to obviate the nonlinear command attitude extraction, Raptis et al. [15] proposed a new attitude representation instead of Euler angles. Nevertheless, transformation between the Euler angles and the new attitude representation appears singularity when the roll or pitch of the helicopter reaches 7 π2 . Although the backstepping controller in [15] achieves the nonsingular exponentially stable trajectory tracking of the model helicopter, it will be invalid once the unmolded dynamics of the helicopter are taken into account. Using the attitude representation proposed in [15], the backstepping controllers in [16–18] are
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Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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based on the hypothesis that no singularity occurs, but they cannot obviate it. Moreover, the unmodeled dynamics of the model helicopter may affect the tracking performance, thus, they should be compensated. Neural networks [7,19], nonlinear damping [17] and disturbance observers [12,20] have been applied to compensate the unmodeled dynamics. However, the neural network may increase computational complexity [21], the nonlinear damping may introduce a large undesired control input [22], and the disturbance observer may bring in additional intricate dynamics [23]. With the prior knowledge of the upper bounds of the unmodeled dynamics, the most effective way is to directly compensate them with sign functions and their upper bounds [24]. However, when the sign function appears in the position loop controller, chattering phenomenon may occur in the extracted command attitude, which is unbeneficial for the attitude tracking. Although the sliding-mode observers proposed in [25–27] can relieve chattering, they do not establish a parameterized relation between the chattering attenuation and the tracking performance. With the attitude representation proposed in [15], this paper develops a singularity-free trajectory-tracking controller for the model helicopter. The helicopter system is firstly simplified into a cascaded structure with unmodeled dynamics. Based on the simplified model, the backstepping technique is used to exploit the controller and the hyperbolic tangent function is applied to compensate the unmodeled dynamics. During the backstepping procedure, to avoid the aforementioned singularity in the attitude representation, it is necessary to impose the same non-singular requirement on the extracted command attitude. An auxiliary dynamic system with appropriate parameters is introduced to guarantee such non-singular requirement. It is proven that, with suitable controller parameters and desired trajectories, the proposed controller achieves the non-singular trajectory tracking of the model helicopter. Main contributions are listed as follows: (i) during the position loop controller design, an auxiliary dynamic system is introduced, and criteria for choosing parameters are established to ensure the stability of it and the non-singular requirement of the extracted command attitude; (ii) a singularity-free attitude controller with appropriate parameters and desired trajectory constraint is put forward; (iii) instead of the sign function, the hyperbolic tangent function is introduced to compensate the unmodeled dynamics, and the parameterized relation between the chattering attenuation and the tracking performance is built. The following sections are organized as follows: some mathematical preliminaries are presented in Section 2; control problems are stated in Section 3, controller design and stability analysis are provided in Section 4; Simulations are carried out in Section 5; and conclusions are drawn in Section 6.
2. Preliminaries In this paper, j j denotes the absolute value of a scalar, J J denotes the Euclidean norm of a vector, Sx and Cx with x A R are short for the trigonometric functions sin ðxÞ and cos ðxÞ, λ ðÞ and λ ðÞ denote the maximum and minimum eigenvalues of a square matrix, e1 ¼ ½1; 0; 0T , e2 ¼ ½0; 1; 0T and e3 ¼ ½0; 0; 1T denote three unit vectors. For x ¼ ½x1 ; x2 ; x3 T A R3 , the superscript denotes the transformation from x to a skew-symmetric matrix, namely, 2 3 0 x3 x2 6 x 0 x1 7 x ¼4 3 5: x2 x1 0
x
e For x A R, define the hyperbolic tangent function tanhðxÞ ¼ eex þ e x, tanhðxÞ which satisfies j tanhðxÞj o1 and 0 o x o 1. For x ¼ ½x1 ; …; xn T A Rn , define the hyperbolic tangent function vector tanhðxÞ ¼ ½tanhðx1 Þ; …; tanhðxn ÞT . x
Lemma 1 ([28]). Suppose that hðxÞ : D-Rn is a smooth function defined on D Rn , and the Jacobian matrix of h is nonsingular at a point x ¼ x0 , then on a suitable open subset D0 of D, containing x0 , h defines a local diffeomorphism. Lemma 2 ([18,29]). Given ϵ 4 0, the following inequality holds for x A R: x ð1Þ 0 r j xj x tanh r kq ϵ;
ϵ
where kq satisfies kq ¼ e ðkq þ 1Þ , i.e., kq ¼0.2785. Lemma 3 ([30,31]). The x A fx A R∣j xj o kg: 2
k
inequality
x2
holds
for
: k k x2 Lemma 4. Consider the system
ð2Þ
ξ€ þ α tanhðkξ þ lξ_ Þ þ β tanhðlξ_ Þ dðtÞ ¼ 0;
ð3Þ
ln
2
x2
r
following
2
where ξ A Rn and ξ_ A Rn are states, dðtÞ : R þ -Rn is a disturbance, k; l; α and β are positive constants and satisfy sffiffiffiffiffiffiffi k 1 k β α r β þ : ð4Þ r 2 2 2 l l If there exist d 4 0 and t 4 0 such that J dðtÞ J od o
kβ 2ðl β þ kÞ 2
;
8t Zt:
ð5Þ
then ξ and ξ_ ultimately converge to the attractive set T T T T T Z ¼ f½ξ ; ξ_ T ∣½kξ þ lξ_ ; lξ_ T o μ g;
where μ satisfies
ð6Þ
2 l2 β þ k d o tanhμ ðμ Þ o 12. kβ
Proof. See Appendix A.
3. Problem statements 3.1. Helicopter model In this paper, the helicopter is considered as a 6-dof rigid body. Let I ¼ fOxyzg denote an inertial frame whose origin O is located at a fixed point on the earth, and B ¼ fOb xb yb zb g denote a body frame whose origin Ob is located at the helicopter c.g. (center of gravity) (see Fig. 1). During the modeling, p ¼ ½px ; py ; pz T and v ¼ ½vx ; vy ; vz T respectively denote the position and velocity of the helicopter c.g. in I , ω ¼ ½ωx ; ωy ; ωz T denotes the angular velocity of the helicopter in B, γ ¼ ½ϕ; θ; ψ T denotes the Euler angle vector (roll, pitch, yaw), and the corresponding rotation matrix R from B to I , which is parametrized by γ , is expressed as 2 3 R11 R12 R13 6R 7 R ¼ 4 21 R22 R23 5 R31 R32 R33 2 3 C θ C ψ Sϕ Sθ C ψ C ϕ Sψ C ϕ Sθ C ψ þ Sϕ Sψ 6 7 ¼ 4 C θ S ψ S ϕ S θ S ψ þ C ϕ C ψ C ϕ S θ S ψ S ϕ C ψ 5: ð7Þ Sθ Sϕ C θ CϕCθ Referring to [32], the kinematic equations of the helicopter are described as p_ ¼ v;
ð8Þ
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respect to the Ob xb zb plane, f and τ denote the applied force and torque in B, which are mainly generated by the main and tail rotors during the hovering or low-speed maneuver, and according to [18], they are expressed as 2 3 2 3 T m Sa T m hm Sb þ Lb þ T t ht þ τm Sa 6 T S þT 7 6 T l þ T h S þ Ma þ τ τ S 7 τ ¼4 m m m m a f ¼4 m b t 5; t m b 5; T m CbCa
T m lm S b T t lt þ τ m C a C b
where subscripts m and t denote the main and tail rotors, T i and τi (i¼m or t) denote the thrusts and anti-torques generated by the rotors, ½hi ; 0; li T (i¼m or t) denotes the relative position between the corresponding rotor and the helicopter c.g., L and M denote the stiffness coefficients of the main rotor, a and b denote the longitudinal and lateral flapping angles. Further, in terms of [18], τi (i¼m or t) is determined with τi ¼ C i j T i j 1:5 þ Di , where Ci and Di are the aerodynamic constants. Fig. 1. Model helicopter configuration.
R_ ¼ Rω :
3.2. Simplified model ð9Þ
Due to strong coupling between the applied force f and torque
τ , some simplifications should be undertaken to facilitate the
The attitude representation of the helicopter requires three independent components [15,17], whereas the attitude kinematic equation (9) is described in the form of a matrix differential equation, from which it is intricate to design a controller. In order to design a backstepping controller, an alternative attitude representation proposed in [15] is adopted. Let R3 ¼ Re3 , and from (9), it follows that
controller design. According to [17,32], due to physical constraint, the flapping angles of the helicopter are fairly small, so that the small angle approximation is feasible, namely, cos ðÞ ¼ 1 and sin ðÞ ¼ ðÞ. Further, the Ob xb -axis and Ob yb -axis force components as well as the tail rotor anti-torque make tiny contributions [15]. To this end, f and τ are rewritten as
_ 3 ¼ Rω e3 ¼ Re ω: R_ 3 ¼ Re 3
f ¼ T m e3 þ f Δ ;
ð15Þ
τ ¼ Aτ ρ þ τ B þ Δτ ¼ τ γ þ Δτ
ð16Þ
ð10Þ
In view of J R3 J ¼ 1, R3 incorporates just two independent components. Extracting first two constitutes the reduced-dimension attitude R 3 ¼ ½R31 ; R32 T . From (10), we have R_ 3 ¼ R^ ω ;
h
i
ð11Þ
is an invertible matrix and ω ¼ ½ωx ; ωy . where R^ ¼ Further, from (9), we can derive the following yaw kinematic equation: R12 R11 R22 R21
T
Sϕ C ωy þ ϕ ωz : ð12Þ Cθ Cθ Due to det ∂∂γγR ¼ C 2ϕ C θ , in terms of Lemma 1, the map from γ to
ψ_ ¼
T
γ R ¼ ½R 3 ; ψ T defines a local diffeomorphism on the set
j ϕ j o π2 ; j θ j o π2 , and singularity occurs at j ϕ j ¼ π2 or j θ j ¼ π2. Due π π to J R3 J ¼ 1 and continuity of R3 , under ϕð0Þ; θð0Þ A 2; 2 , we know that ϕðtÞ; θðtÞ A π2 ; π2 is equivalent to R33 ðtÞ ¼ C ϕðtÞ C θðtÞ 4 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
or J R 3 ðtÞ J ¼ 1 R33 ðtÞ2 o 1, for t Z0. Thus, given ϕð0Þ; θð0Þ A π2; π2 , as long as J R 3 ðtÞ J o 1 (t Z 0), singularity would not occur in the attitude representation γ R In addition, according to [17,32], with the Newton–Euler formula, the dynamic equations of the helicopter are derived as 1 v_ ¼ ge3 þ Rf ; m
ð13Þ
_ ¼ ω J ω þ τ ; Jω
ð14Þ
where m denotes the mass of the helicopter, g denotes the gravitational acceleration, J denotes the inertial matrix with respect to B, which is defined as 2 3 Jx 0 J xz 6 Jy 0 7 J¼4 0 5; Jz J xz 0 with zero elements resulting from the symmetric structure with
where 2
3 2 3 τm T m hm þ L ht Tt 6 0 7 7 τ m 5; ρ ¼ 6 T m hm þM Aτ ¼ 4 4 a 5; lt 0 T m lm b 2 3 2 3 T m Sa 0 6T l 7 6 T S þT 7 τ B ¼ 4 m m 5; f Δ ¼ 4 m b t 5; T m ðC a C b 1Þ τm 2
3 τm ðSa aÞ þT m hm ðSb bÞ 6 τ τ ðS bÞ þT h ðS aÞ 7 Δτ ¼ 4 t m b m m a 5: τm ðC a C b 1Þ T m lm ðSb bÞ Substituting (15) into (13) and (16) into (14) yield v_ ¼ ge3 þ
Tm R 3 þ Δf ; m
_ ¼ ω J ω þ τ γ þ Δτ ; Jω
ð17Þ ð18Þ
1 Rf Δ . During the controller design, Δf and Δτ are where Δf ¼ m treated as unmodeled dynamics, and their upper bounds will be compensated.
3.3. Control objective Given desired smooth trajectory pr ¼ ½prx ; pry ; prz T and yaw ψr, the objective is to develop control inputs Tm, Tt, a and b for the model helicopter, so that it can track P r and ψr with small errors. Further, under ϕð0Þ; θð0Þ A ð π2 ; π2 Þ, in order for the singularity-free tracking, the developed control inputs are required to warrant J R 3 ðtÞ J o 1 (t Z 0).
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Assumption 1. The second-order derivative of the desired trajectory pr satisfies j p€ rz ðtÞj r δ o g ðt Z 0Þ. Assumption 2. The unmodeled dynamics Δf ¼ ½Δf x ; Δf y ; Δf z and Δτ ¼ ½Δτx ; Δτy ; Δτz T are bounded and satisfy j Δf i j r σ i ; j Δτi j r ςi ;
i ¼ x; y; z;
T
ð19Þ
where σ ¼ ½σ x ; σ y ; σ z with σ z o g δ and ς ¼ ½ςx ; ςy ; ςz are two known upper bound vectors. T
T
Remark 1. Assumption 2 seems strong, due to the dependence of Δf and Δτ on the system inputs. However, during the low-speed maneuver, affected by physical restrictions arising from mechanical constraint of flapping angles and stiffness of rotor blades, Δf and Δτ of typical helicopters are usually extremely small [32]. Here, we assume that they are bounded by known constants, which can be acquired by experiment results or practical experience.
where cp 4 0. Substituting (22) into (21) yields L_ 1 ¼ cp J p e J 2 þ p Te v e ; where v e ¼ ½v ex ; v ey ; v ez ¼ v αp η_ ¼ ve η_ . 4.1.2. Velocity error dynamics From (17), v e satisfies Tm v_ e ¼ ge3 þ R3 þ Δf α_ p η€ : m
From Assumption 2 and Lemma 2, it follows that X X v v Te Δf r j v ei j σ i r v ei tanh ei þ kq ϵ σ i i ¼ x;y;z
¼ v Te Tanh
T
ð24Þ
Choose a Lyapunov function L2 ¼ L1 þ 12v Te v e , and its derivative along (20) and (24) satisfies Tm L_ 2 ¼ cp J p e J 2 þ v Te ge3 þ R3 þ Δf α_ p η€ þ p e : m
4. Controller design The controller is designed with the backstepping technique and its block diagram is illustrated in Fig. 2. The controller design procedure is streamlined based on the backstepping technique in [22]. First, in the position loop, a controller u ¼ Tmm R3c is designed for the position to track pr ; second, the main rotor thrust Tm and the reduced-dimension command attitude R 3c are extracted from u; third, in the attitude loop, a controller τ γ is developed for the
ð23Þ T
ve
ϵ
i ¼ x;y;z
ϵ
σ þ kq ϵd1 ; v
Tanhðvϵe Þ ¼ diag½tanhðvϵex Þ; tanhð ϵey Þ; tanhðvϵez Þ and d1 ¼ 2 € _ i ¼ x;y;z σ i . Further, it can be derived that α p ¼ cp p e cp v e þ p r . _ Thus, L 2 satisfies Tm ve σ η€ L_ 2 r cp J p e J 2 þ v Te ½ ge3 p€ r þ R3 þ Tanh m ϵ
where P
ðc2p 1Þp e þ cp v e þ kq ϵd1 :
ð25Þ
attitude to track γ c ¼ ½R 3c ; ψ c T ; and finally, the control inputs Tt, a and b are derived from τ γ .
Develop a second-order auxiliary dynamic system about η:
4.1. Position loop controller design
η€ ¼ αη tanhðkη η þ lη η_ Þ βη tanhðlη η_ Þ þ Δη ;
4.1.1. Position error dynamics Define the position tracking error pe ¼ ½pex ; pey ; pez T ¼ p pr and the virtual position tracking error p e ¼ ½p ex ; p ey ; p ez T ¼ pe η, where η is an auxiliary variable. From (8), p e satisfies
where Δη ¼ ðc2p 1Þp e þðcp þ cv Þv e with cv 4 0, and positive parameters αη , β η , kη and lη satisfy vffiffiffiffiffiffiffiffiffiffiffi ! u ukη kη 1 t β η r αη r β þ : ð27Þ 2 2 η l2 l
p_ e ¼ v p_ r η_ :
ð20Þ
η
ð26Þ
η
Assign a Lyapunov function L1 ¼ 12p Te p e , and its derivative along (20) satisfies
The initial η and η_ can be arbitrary values; however, to facilitate analysis, let ηð0Þ ¼ η_ ð0Þ ¼ ½0; 0; 0T . Then, design a position loop controller:
L_ 1 ¼ p Te ðv p_ r η_ Þ:
u¼
ð21Þ
Design a virtual control:
αp ¼ cp p e þ p_ r ;
ð22Þ
Tm R3c ¼ αη tanhðkη η þ lη η_ Þ β η tanhðlη η_ Þ m ve þ ge3 þ p€ r Tanh σ;
ϵ
ð28Þ
Fig. 2. Controller block diagram.
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where R3c is a command attitude. And the main rotor thrust Tm is extracted as Tm ¼ mJuJ:
ð29Þ
Substituting (26) and (28) into (25) yield Tm L_ 2 r cp J p e J 2 cv J v e J 2 þ v Te R3e þ kq ϵd1 m
ð30Þ
where R3e ¼ R3 R3c . 4.2. Non-singular command attitude extraction From (28) and (29), the command attitude R3c is derived as R3c ¼
m u: Tm
ð31Þ
Extracting first two components of R3c constitutes R 3c ¼ ½R31c ; R32c T , which, together with the desired yaw ψr, serves as the tracking objective for the attitude loop controller design. To ensure J R 3 J o 1 for non-singularity, the same requirement should be imposed on its tracking objective, namely, J R 3c J o 1. If the parameters αη and β η in (26) and (28) satisfy
αη þ β η o g δ σ z ;
ð32Þ
from Assumptions 1 and 2, the third row of (28) satisfies
5
4.3.2. Yaw error dynamics Define the yaw error ψ e ¼ ψ ψ r . From (12), it satisfies
ψ_ e ¼
Sϕ C ωy þ ϕ ωz ψ_ r : Cθ Cθ
ð37Þ
Choose a Lyapunov function L4 ¼ ψ 2e , and its derivative along (37) satisfies Sϕ C ð38Þ ωy þ ϕ ωz ψ_ r : L_ 4 ¼ ψ e Cθ Cθ Design a virtual control: S C αψ ¼ θ cψ ψ e þ ψ_ r ϕ ωy ; Cϕ Cθ
ð39Þ
where cψ 4 0. Substituting (39) into (38) yields Cϕ L_ 4 ¼ cψ ψ 2e þ ψ e ωez ; Cθ
ð40Þ
where ωez ¼ ωz αψ . 4.3.3. Angular velocity error dynamics Define αγ ¼ ½αTR ; αψ T and ωe ¼ ½ω Te ; ωze T , ωe ¼ ω αγ . From (18), ωe satisfies
which
satisfy
_ e ¼ ω J ω þ τ γ þ Δτ J α_ γ : Jω
Tm R33c ¼ αη tanhðkη ηz þ lη η_ z Þ βη tanhðlη η_ z Þ m v ez σz þ g þ p€ rz tanh
ð41Þ
Choose a Lyapunov function L5 ¼ ω ωe , and its derivative along (20), (24), (33), (37) and (41) satisfies L3 þL4 þ 12
ϵ
Z αη β η þ g δ σ z 4 0
L5 r cp J p e J 2 cv J v e J 2 R 3e R^ ω e
cR J R 3e J 2 2 bR ðkR J R 3e J 2 Þ
T eJ
cψ ψ 2e þ
Cϕ ψ ωez Cθ e
T
which means R33c 4 0. Further, from J R3c J ¼ 1, R33c 4 0 implies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J R 3c J ¼ 1 R233c o1. 4.3. Attitude loop controller design 4.3.1. Reduced-dimension attitude error dynamics Define the reduced-dimension attitude error R 3e ¼ R 3 R 3c and a time-varying parameter kR ¼ 1 J R 3c J 4 0. If J R 3e J o kR , then J R 3 J r J R 3e J þ J R 3c J o1, so that non-singularity can be guaranteed. From (11), R 3e satisfies _ _ R 3e ¼ R^ ω R 3c :
ð33Þ
Assign a Lyapunov function L3 ¼ L2 þ 2b1R ln
2
kR 2 kR J R 3e J 2
with bR 4 0, and
its derivative along (20), (24) and (33) satisfies Tm L_3 r cp J p e J 2 cv J v e J 2 þ v Te R3e m T k_ R R 3e ðR^ ω R_ 3c R 3e Þ þ kq ϵd1 : þ 2 kR bR ðk J R 3e J 2 Þ
ð35Þ
T
derived that Tmm v Te R3e Tmm R 3e v^ e ¼ 0 (See Appendix B). Substituting (35) into (34) yields
T R 3e R^ e þ kq þ 2 bR ðkR J R 3e J 2 Þ
ω
where ω e ¼ ω αR .
cR J R 3e J
Similarly, from Assumption 2 and Lemma 2, it follows that ω ωTe Δτ r ωTe Tanh e ς þ kq ϵd2 ;
ϵ
ω
Tanhðωϵe Þ ¼ diag½tanhðωϵex Þ; tanhð ϵey Þ; tanhðωϵez Þ _ ς i ¼ x;y;z i . Thus, L 5 satisfies
where P
L5 r cp J p e J 2 cv J v e J 2
cR J R 3e J 2 2
bR ðkR J R 3e J 2 Þ
2 bR ðkR
ϵ
T C R 3e R^ ; ϕ J R 3e J 2 Þ C θ
ψe
and
d2 ¼
cψ ψ 2e
h i ωe þ ωTe ω J ω þ τ γ þ Tanhð Þς J α_ γ þ ϱ þkq ϵd;
ð42Þ
T
and d ¼ d1 þ d2 . Design an attitude
τ γ ¼ cω ωe þ ω J ω þ J α_ γ ϱ Tanh
ð36Þ
e
ϵ
ς;
ρ ¼ Aτ 1 ðτ γ τ B Þ:
ð43Þ
ð44Þ
Substituting (43) into (42) yields L5 r cp J p e J 2 cv J v e J 2 cω J ωe J 2 þ kq ϵd;
2
ω
where cω 40. From (16), the control input ρ ¼ ½T t ; a; bT is derived as
2
bR ðkR J R 3e J 2 Þ
ϵd1 ;
þ ωTe ð ω J ω þ τ γ þ Δτ J α_ γ Þ þ kq ϵd1
loop controller: ð34Þ
þ R13c þ R23c where cR 4 0 and v^ e ¼ ½v ex RR13 v ez ; v ey RR23 v ez T . It can be 33 þ R33c 33 þ R33c
L_3 r cp J p e J 2 cv J v e J 2
2 bR ðkR J R 3e J 2 Þ
where ϱ ¼
R
In order to warrant J R 3e J o kR , design a virtual control: " ! 2 1 k_ R bR T m ðkR J R 3e J 2 Þ ^ αR ¼ R cR þ j j R 3e þ R_ 3c v^ e ; m kR
þ
cR J R 3e J 2 2
bR ðkR J R 3e J 2 Þ
cψ ψ 2e ð45Þ
Remark 2. From (35) and (43), R_ 3c and α_ γ are required to determine αR and τ γ , respectively; however, it is intricate to analytically calculate them. Instead, they can be approximated with the
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command filter introduced in [33]. Using appropriate parameters, this approximation is of high accuracy.
inferred from (51) that, if J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J o kR ðtÞ ðt Z 0Þ. □
Remark 3. Since kR is not differentiable at R 3c ¼ 0, a complementary definition of k_ R at R 3c ¼ 0 is required. With Filippov Theory [24], the derivative of kR can be determined by
Remark 4. The bounds of the ultimate convergent sets Z p , Z v , Z R , Z ψ and Z ω can be made arbitrarily small by increasing the parameter c.
k_ R ¼
8 < :
R 3c _ R ; J R 3c J 3c T
R 3c a0;
ð46Þ
R 3c ¼ 0:
0;
Lemma 6. Consider the auxiliary dynamic system (26) with the parameters kη, lη, αη and β η satisfying (27) and (32), if p e and v e are bounded and ultimately converge to the sets Z p and Z v , and the controller parameter ϵ in (28) and (43) satisfies 2
ϵo
4.4. Stability analysis and singularity discussion
cΔ η
; 2c 2 kq d
ð52Þ
where c ¼ maxðj c2p 1j ; cp þ cv Þ and Δ η A 0; Lemma 5. Consider the closed-loop system composed by (20), (24), (33), (37) and (44), if the initial reduced-dimension attitude error R 3e ð0Þ satisfies J R 3e ð0Þ J okR ð0Þ, then the developed control inputs (29) and (44) can guarantee that (i) the tracking errors p e , v e , R 3e , ψe and ωe are bounded and ultimately converge to sets around the origin; (ii) J R 3e ðtÞ J o kR ðtÞ ðt Z0Þ. Proof. (i) In terms of Lemma 3, it follows that ln
2 kR 2
kR J R 3e J 2
r
J R 3e J 2
2
kR J R 3e J 2
:
Consider the definition of L5, then (45) becomes ð47Þ L_ 5 r 2cL5 þ kq ϵd; where c ¼ min cp ; cv ; cR ; cψ ; cω . From Comparison Principle [21], λ ðJÞ it follows that kq ϵd 2ct kq ϵd e ; þ L5 ðtÞ r L5 ð0Þ 2c 2c
8 t Z 0:
ð48Þ
r L5 ð0Þe 2ct þ
kq ϵd ; 2c
8 t Z 0;
which implies that p e , v e ,
ð49Þ
ψe and ωe are bounded and
qffiffiffiffiffiffiffi kq ϵd , ultimately converge to the sets Z p ¼ p e A R3 ∣J p e J r c
qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi kq ϵd k q ϵd Z v ¼ v e A R3 ∣ J v e J r , Z ψ ¼ ψ e A R∣j ψ e j r and c c
rffiffiffiffiffiffiffiffi k ϵd Z ω ¼ ωe A R3 ∣ J ωe J r cλq ðJÞ , respectively. Moreover, in view of the definition of L5, we also have kq ϵd 1 kR ; ln r L5 ð0Þe 2ct þ 2c 2bR k2 J R 3e J 2 2
8 t Z 0:
ð50Þ
R
Taking exponential on (50) yields J R 3e ðtÞ J r kR ðtÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2ct þ kq ϵd 2c o kR ðtÞ; 1 e 2bR L5 ð0Þe
8 t Z 0;
ð51Þ
2ðlη βη þ kη Þ
, then the
Proof. In view of the definition of Δη , we obtain J Δη J 2 r 2ðc2p 1Þ2 J p e J 2 þ2ðcp þ cv Þ2 J v e J 2 r 2c 2 ð J p e J 2 þ J v e J 2 Þ: In terms of (49), then k q ϵd ; J Δη ðtÞ J 2 r 4c 2 L5 ð0Þe 2ct þ 2c If ϵ is chosen satisfying (52), then exists a tη satisfying 8 > > > > 0; > > < 2 0 2 13 tη ¼ > Δ k ϵ d 1 1 > η q > ln4 @ A5 > > > 2c Lð0Þ 4c 2 : 2c
8 t Z0:
ð53Þ
2c kq ϵd o c
Δ η . In this case, there
2
2
2
if Lð0Þ r
Δη
4c 2
kq ϵd ; 2c
kq ϵd ; 2c
2
if Lð0Þ 4
Δη
4c 2
ð54Þ
such that J Δη ðtÞ J r Δ η (t Z t η ). From Lemma 4, it follows that η and η_ ultimately converge to the attractive set
where μ η satisfies
1 1 ð J p e ðtÞ J 2 þ J v e ðtÞ J 2 þ ψ e ðtÞ2 þ J ω ðtÞ J 2 Þ 2 λ ðJÞ e
auxiliary variable η and its derivative η_ ultimately converge to sets around the origin.
Z η ¼ f½ηT ; η_ T T ∣ J ½kη ηT þ lη η_ T ; lη η_ T T J r μ η g;
Based on the definition of L5, we know
kη βη
2
lη βη þ kη kη βη 2
Δη o
tanh ðμ η Þ 2
μη
ð55Þ
o 12. Further, it can be derived
that η and η_ are bounded and ultimately converge to the sets Z η1 n o n o μ μ ¼ η A R3 ∣ J η J r kηη and Z η2 ¼ η_ A R3 ∣ J η_ J r lηη , respectively.□ Remark 5. From Lemma 5, a smaller ϵ results in smaller ultimately convergent bounds of the tracking errors p e , v e , R 3e , ψe and ωe , however, too small ϵ may lead to the chattering phenomenon in the position loop controller u, which is unfavorable for the subsequent attitude tracking. Thus, when choosing ϵ satisfying (52), we should make a tradeoff between the good tracking performance and the chattering attenuation. Theorem 1. Consider the model helicopter system (8), (11)–(14) with Assumptions 1 and 2, given the initial states pð0Þ A R3 , vð0Þ A R3 , ωð0Þ A R3 , ϕð0Þ; θð0Þ A ð π2; π2Þ (i.e., J R 3 ð0Þ J o 1) and ψ ð0Þ A ð π ; π , if the parameters kη, lη, αη and βη satisfy (27) and (32), ϵ satisfies (52), and the desired trajectory pr as well as the upper bound vector σ make the initial reduced-dimension command attitude R 3c ð0Þ satisfy J R 3 ð0Þ R 3c ð0Þ J þ J R 3c ð0Þ J o1;
ð56Þ
where which implies that R 3e is bounded and ultimately converges to the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bR kq ϵd . (ii) It can be compact set Z R ¼ R 3e A R2 ∣J R 3e J r kR 1 e c
R 3c ð0Þ ¼
½e1 ; e2 T uð0Þ ; J uð0Þ J
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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3 v ð0Þ þ cp ðpx ð0Þ prx ð0ÞÞ p_ rx ð0Þ p€ rx ð0Þ tanh x σx ϵ 6 7 6 7 6 p€ ry ð0Þ tanh vy ð0Þ þ cp ðpy ð0Þ pry ð0ÞÞ p_ ry ð0Þ σ y 7 ϵ 6 7; uð0Þ ¼ 6 7 6 7 vz ð0Þ þ cp ðpz ð0Þ prz ð0ÞÞ p_ rz ð0Þ 4 g þ p€ rz ð0Þ tanh σz 5 ϵ 2
the designed control inputs (29) and (44) can guarantee that (i) the tracking errors pe and ve are bounded and converge to sets around the origin; (ii) no singularity occurs during the tracking progress. Proof. (i) From Lemmas 5 and 6, if the parameters kη, lη, αη , βη
and ϵ satisfy (27), (32) and (52), the designed control inputs (29) and (44) can ensure that the tracking errors p e , v e and the auxiliary variables η, η_ are bounded and ultimately converge to the sets Z p , Z v , Z η1 and Z η2 , respectively. From the definitions of
7
pe and ve , we have J pe J r J p e J þ J η J and J ve J r J v e J þ J η_ J . Thus, pe and ve are bounded and ultimately converge to the
qffiffiffiffiffiffiffi kq ϵd μ η and Z v ¼ ve A R3 j J ve J r sets Z p ¼ pe A R3 j J pe J r c þ kη qffiffiffiffiffiffiffi kq ϵd μ η þ , respectively. c lη (ii) If R 3c ð0Þ satisfies (56), then J R 3e ð0Þ J o kR ð0Þ. We have proven in Lemma 5 that, if J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J o kR ðtÞ ðt Z 0Þ. In this case, with the triangle inequality, we obtain J R 3 ðtÞ J ¼ J R 3c ðtÞ þ R 3e ðtÞ J r J R 3c ðtÞ J þ J R 3e ðtÞ J o J R 3c ðtÞ J þ kR ðtÞ ¼ 1;
8 t Z 0;
ð57Þ
which means ϕðtÞ; θðtÞ A ð π2; π2 Þ ðt Z 0Þ. Thus, no singularity occurs during the tracking progress.
□
Remark 6. Generally, the roll ϕ and pitch θ of the model helicopter is initialized at the origin, which means R 3 ð0Þ ¼ ½0; 0T . In
Fig. 5. Velocity tracking error.
Fig. 3. 3D trajectory tracking.
Fig. 4. Position and yaw tracking errors.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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this case, the relation (56) can be simplified as pffiffiffi 3ð J p€ r ð0Þ J þ J σ J Þ og þ p€ rz ð0Þ σ z ;
ð58Þ
where p€ r ¼ ½p€ rx ; p€ ry T and σ ¼ ½σ x ; σ y T . Actually, from (56), R 3 ð0Þ ¼ ½0; 0T implies J R 3c ð0Þ J o 12; if the desired trajectory pr and the upper bound vector σ satisfy (58), then p€ r ð0Þ J þ J σ J ffi J R 3c ð0Þ J r J qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð J p€ r ð0Þ J þ J σ J Þ2 þ ðg þ p€ rz ð0Þ σ z Þ2 p€ r ð0Þ J þ J σ J 1 ¼ : oJ 2ð J p€ r ð0Þ J þ J σ J Þ 2
5. Simulations As far as the helicopter model composed by (8) and (11)–(14) is concerned, simulations with MATLAB/Simulink are carried out to verify the effectiveness of the proposed controller. The physical and aerodynamic parameters [16] are as follows: m ¼ 7:4 kg, J x ¼ 0:16 kg m2 , J y ¼ 0:30 kg m2 , J z ¼ 0:32 kg m2 , J xz ¼ 0:05 kg m2 , lm ¼ 0:01 m, hm ¼ 0:25 m, lt ¼ 0:9 m, ht ¼ 0:12 m, M ¼ L ¼ 25:23, C m ¼ 0:00452, Dm ¼ 0:08488, C t ¼ 0:005066 and Dt ¼ 0:008488. Choose σ ¼ ½1; 1; 1T and ς ¼ ½0:1; 0:1; 0:1T , which mean d ¼3.3. 5.1. Performance verification
tracking. Fig. 7 depicts the bounded angular velocity tracking error. Further, denote μη ¼ ½4ηT þ η_ T ; η_ T . Fig. 8 shows that η and η_ are bounded and ultimately converge to the attractive set Z η with μ η ¼ 0:47, which verifies the result proposed in Lemma 6. 5.2. Performance comparison In order to highlight the proposed controller, it will be compared with [17], which also adopted the attitude representation introduced in [15]. The desired trajectory is described as 8 > ½0:1t 2 ; 0; 5T m; if t r 10 s; > > 2 3 > < 10 cos ð0:2ðt 10ÞÞ pr ¼ 6 7 > 4 10 sin ð0:2ðt 10ÞÞ 5 m; otherwise; > > > : 5 with ψ r ¼ 0 rad. The controller parameters and the initial conditions are the same with the above maneuver. The simulation results are illustrated in Figs. 9 and 10. It can be observed from Fig. 9 that, the proposed controller achieves the position and yaw tracking of the model helicopter with small errors, however, the controller in [17] fails and its position and yaw diverge to infinity. From Fig. 10, the proposed controller can guarantee the roll ϕ satisfying j ϕ j o π2, however, the roll ϕ with the controller in [17] exceeds π2 and diverges to infinity, which leads to singularity and the divergence of the position and yaw. By comparison, we conclude that the proposed controller is
The desired trajectory in this maneuver is a circular line: pr ¼ ½5 cos ð0:2tÞ; 5 sin ð0:2tÞ; 5T m with ψ r ¼ 0 rad. The procedure of the controller parameter choice is as follows: first, choose cp ¼ cv ¼ 0:5, cR ¼4, bR ¼0.2, cψ ¼ 1 and cω ¼ 15, which mean c ¼ 1; then, based on (27) and (32), choose kη ¼ αη ¼ βη ¼ 4 and lη ¼ 1;
finally, set Δ η ¼ 0:8, and based on (52), choose ϵ ¼ 0:1. With the selected controller parameters, it is derived that the convergent tanh ðμ Þ 2
bound μ η satisfies 0:4 o μ η o0:5. It is calculated by MATLAB/ Simulink that μ η A ð0:46; 0:64Þ. The model helicopter is initially still at pð0Þ ¼ ½0; 5; 0T m with ψ ð0Þ ¼ 0 rad. Since (58) holds, set R 3 ð0Þ ¼ ½0; 0T . The simulation results are illustrated in Figs. 3–8. Fig. 3 depicts the 3D tracking trajectory with respect to the desired one, which shows the accomplishment of the claimed tracking objective. Fig. 4 illustrates the bounded position and yaw tracking errors. Fig. 5 depicts the bounded velocity tracking error. Fig. 6 indicates that the reduced-dimension attitude tracking error is bounded and that once J R 3e ð0Þ J o kR ð0Þ, then J R 3e ðtÞ J okR ðtÞ always holds, which, from Theorem 1, means ϕðtÞ; θðtÞ A ð π2 ; π2 Þ during the tracking progress and the singularity-free attitude
Fig. 7. Angular velocity tracking errors.
Fig. 6. Lateral-longitudinal attitude tracking error.
Fig. 8. Auxiliary variable tracking errors.
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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9
Fig. 9. Position and yaw (straight line: desired ones; dash line: ones with the proposed controller; dot dash line: ones with the controller in [17]).
Acknowledgement This work is supported by National Natural Science Foundation of China under Grant No. 61074010.
Appendix A. Proof of Lemma 4 Lemma 7 ([34]). Consider the system (3) and the convergent set Z in (6). If there exist d 4 0 and t 4 0, such that " # 2 2 tanh ðμ Þχ 1 tanh ðμ Þχ 1 ; αþβ ; 8t Zt; ; ðA:1Þ J dðtÞ J od omin α lμ βl þ kl μ
Fig. 10. Roll ϕ (dash line: one with the proposed controller; dot dash line: one with the controller in [17]).
able to achieve the non-singular trajectory tracking of the model helicopter.
6. Conclusions A singularity-free controller is proposed for model helicopters to accomplish trajectory tracking missions. The backstepping technique is employed to design the controller, and the hyperbolic tangent function is applied to compensate the unmodeled dynamics. An auxiliary dynamic system is introduced in the position loop to ensure the nonsingular requirement of the extracted command attitude. A novel singularity-free attitude control strategy is also presented. Stability analysis has proven that, based on the established selection criteria for controller parameters and desired trajectories, the proposed controller achieves the trajectory tracking of the model helicopter.
where χ 1 ¼ minðα2 l; klβÞ, then ξ and ξ_ ultimately converge to the attractive set Z. qffiffiffiffiffiffi k 1 k Proof of Lemma 4. In view of 2 β r 2ðβ þ 2 Þ, there exists an α l l satisfying (4). From the right inequality of (4), αl r 12ðβl þ klÞ o βl þ kl, which implies tanh ðμ Þχ 1 tanh μχ 1 o : α lμ ðβl þ klÞμ 2
The
2
left
inequality
of
(4)
yields
χ 1 ¼ min½α2 l; klβ ¼ klβ. Further, we have kβ
tanh ðμ Þ
kβ
2
μ
l βþk 2
o
l β þk 2
Thus, it follows that 2
α2 l Z klβ, which implies
o β o α þ β:
3
6tanh2 ðμ Þχ 1 tanh2 ðμ Þχ 1 7 tanh2 ðuÞkβ ; α þ β7 ; min6 4 5¼ 2 k αlμ ðl β þ kÞμ βl þ μ l Since tanhμ ðμ Þ is continuous about μ A ð0; 1Þ and its maximum value 2
Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i
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10
lies in the interval (0.5,0.6), there exists a μ such that do
kβ
tanh ðμ Þ
l βþk 2
2
μ
o
kβ : 2 2ðl β þ kÞ
ðA:2Þ
Therefore, from Lemma 7, ξ and ξ_ ultimately converge to the attractive set Z in (6), where, in view of (A.2), μ satisfies 2 l2 β þ k d o tanhμ ðμ Þ o 12. kβ
□ T
Appendix B. Derivation of R 3e v^ e ¼ RT3e v e From J R3 J 2 ¼ R231 þ R232 þR233 ¼ 1 and J R3c J 2 ¼ R231c þ R232c þ R233c ¼ 1, it follows that R13 þ R13c R23 þ R23c T R 3e v^ e ¼ R13e v ex v ez þR23e v ey v ey v ez R33 þ R33c R33 þ R33c R13 þ R13c R23 þ R23c R13e þ R23e v ez ¼ R13e v ex þ R23e v ey R33 þ R33c R33 þ R33c ¼ R13e v ex þ R23e v ey
R213 R213c þ R223 R223c v ez R33 þ R33c
R233 R233c v ez R33 þ R33c ¼ R13e v ex þ R23e v ey þ ðR33 R33c Þv ez ¼ R13e v ex þ R23e v ey þ R33e v ez ¼ R13e v ex þ R23e v ey þ
¼ RT3e v e :
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Please cite this article as: Zou Y, Huo W. Singularity-free backstepping controller for model helicopters. ISA Transactions (2016), http: //dx.doi.org/10.1016/j.isatra.2016.06.010i