Adaptive Trajectory Tracking Control of a High ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
2781
Adaptive Trajectory Tracking Control of a High Altitude Unmanned Airship Yongmei Wu and Ming Zhu School of Aeronautical Science and Engineering, Beihang University, Beijing, China Email: [email protected]
Zongyu Zuo and Zewei Zheng School of Automation Science and Electrical Engineering, Beihang University, Beijing, China Email: [email protected]
Abstract—Nonlinear dynamic model of a high-altitude unmanned airship, expressed by generalized coordinate, was built. A nonlinear compensation was introduced into the control loop to linearize and decouple the nonlinear system globally. In view of the imprecisely known inertia parameters of the airship, an adaptive law was proposed based on the feedback linearization to realize asymptotic tracking of any continuous time-varying desired trajectory from an arbitrary initial condition. The stability of the closed-loop control system was proved via the use of Lyapunov stability theory. Finally, numerical simulation results demonstrate the validity and effectiveness of the proposed adaptive control law. Index Terms—adaptive control, feedback linearization, trajectory tracking, high-altitude unmanned airships
II. KINETIC MODEL OF THE AIRSHIP A. Defination of Coordinate This paper studies an ellipsoid full-actuated high altitude unmanned airship which is symmetrical with respect to the vertical axis, and its tail fin with the cross elevator and rudder, is bisymmetric. The gondola is equipped with a pair of differential propellers under the body. The earth reference frame is denoted by Oe xe ye ze , and the body reference frame Oxyz whose origin is located at the center of volume, as shown in Fig.1.
I. INTRODUCTION High-altitude unmanned airships, which have a wide application prospect in communication, surveillance and investigation, are capable of hovering for a long time. According to the task demands, desired trajectory is designated. Modeling, control method and verification test of high altitude unmanned airships are the focus of the domestic and international studies [1]~[10]. Trajectory tracking, based on adaptive feedback linearization, is designed to solve the control problem on imprecisely known inertia parameters of a high altitude unmanned airship. This paper is organized as follows: nonlinear dynamic model of a conventional airship is built, expressed by generalized coordinate in section II. In section III the feedback linearization control law is designed. Adaptive feedback linearization control law and estimation law of inertia parameters are designed, and stability is proved in section IV. The effectiveness of tracking desired continuous time-varying trajectory is validated via simulation without wind disturbance in section V. Finally, conclusion and future work are summarized in section VI. Manuscript received September 16, 2011; revised September 30, 2011; accepted December 1, 2011. Corresponding author: Yongmei Wu, [email protected]
© 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.11.2781-2787
Oe
xe
ye
ze
Figure 1. Definition of coordinate
B. Dynamics Fundamental Equations of Airship Several basic assumptions are needed: A1. The volume center coincides with the gross center of buoyancy. A2. The airship forms a rigid body such that elastic effects can be ignored. A3. The shape and the whole mass are constant in hovering. In view of the symmetry, the center of mass is located under the center of volume in longitudinal profile, and products of inertia satisfy I xy = I yz = 0 , The dynamic equations of follows[1]~[10] :
the
airship
can
be
formulated
as
2782
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
MV& = N + G + Bu
where V
⎡ − zc mg cos θ sin φ ⎤ ⎢ ⎥ − zc mg sin θ ⎢ ⎥ ⎢ ⎥ 0 ⎥ G = ⎢⎢ ( B − mg ) sin θ ⎥ f ⎢ ⎥ ⎢ − ( B f − mg ) cos θ sin φ ⎥ ⎢ ⎥ ⎢⎣ − ( B f − mg ) cos θ cos φ ⎥⎦
(1)
[ p, q, r , u, v, w]T , [u, v, w]T denotes linear
velocity vector, and [ p, q, r ]T angular velocity vector of the airship. 0 − I xz 0 − mzc 0 ⎡ Ix ⎤ ⎢ 0 I + ρ∇ k ⎥ 0 mz 0 0 3 y c ⎢ ⎥ ⎢ − I xz ⎥ 0 I z + ρ∇ k 3 0 0 0 M =⎢ ⎥ 0 mz 0 m + ∇ k 0 0 ρ 1 c ⎢ ⎥ ⎢ − mzc ⎥ 0 0 0 m + ρ∇ k 2 0 ⎢ ⎥ 0 0 0 0 m + ρ∇k2 ⎥⎦ ⎢⎣ 0
where I x , I y , I z , I xz are inertia parameters, k1 , k2 , k3 are inertial factors of the airship, ∇ is the volume of the airship, ρ is atmospheric density of the flying height, zc is the position coordinates of the center of mass, and m is the whole mass of the airship. N = [ a1 a2 a3 a4 a5 a6 ]
T
where g is gravity acceleration, B f is buoyancy acted on the airship, θ ,ψ , φ are attitudes. cξ 0 0 0 0 ⎡ cξ ⎤ ⎢ sξ 0 0 0 − sξ −2QCM 4 ⎥⎥ ⎢ ⎢ 0 ⎥ 0 1 1 −2QCN 4 0 B=⎢ ⎥ 0 0 ⎢ − z p sξ − z p sξ y p − y p ⎥ ⎢ z p cξ z p cξ − x p − x p 2QCY 4 ⎥ 0 ⎢ ⎥ −2QCZ 4 ⎥⎦ h2 0 0 0 ⎢⎣ h1
where
where a1 = − ( I z − I y ) qr + I xz pq + mzc ( ur − wp ) +
QCL 2 sin β sin ( β ) , a2 = − ( I x − I z ) pr − I xz ( p 2 − r 2 ) − mzc ( wq − vr ) − Q[CM 1 cos (α 2 ) sin ( 2α ) + CM 2 sin ( 2α ) + CM 3 sin α sin ( α )], a3 = − ( I y − I x ) pq − I xz qr + Q[CN 1 cos ( β 2 ) sin ( 2β ) +
CN 2 sin ( 2 β ) + CN 3 sin β sin ( β )], a4 = − ( m + ρ∇k1 )( wq − vr ) − mzc pr − Q ⎡⎣C X 1 cos 2 α cos 2 β + C X 2 sin ( 2α ) sin (α 2 ) ⎤⎦ , a5 = − ( m + ρ∇k2 )( ur − wp ) − mzc qr − Q[CY 1 cos ( β 2 ) sin ( 2β ) + CY 2 sin ( 2 β ) + CY 3 sin β sin ( β )], a6 = − ( m + ρ∇k2 )( vp − uq ) + mzc ( p 2 + q 2 ) − Q[CZ 1 cos (α 2 ) sin ( 2α ) + CZ 2 sin ( 2α ) + CZ 3 sin α sin ( α )].
where α = arctan( w / u ) and β = arctan(v cos α / u ) are the flow angle when wind speed is zero, Q = ρV 2 / 2 is dynamic pressure, V is the flow speed from a distance, CLi , CMi , CNi , C Xi , CYi , CZi , i = 1, 2,3 are the aerodynamic coefficients[7]. © 2012 ACADEMY PUBLISHER
sξ
sin ξ
,
cξ
cos ξ
h1 = x p sξ − y p cξ ,
,
h2 = x p sξ + y p cξ , ( x p , y p , z p ) and ( x p , − y p , z p ) are position coordinates of left and right propellers about the body reference frame, ξ is an angle toward outside of propellers, CM 4 , CN 4 , CY 4 , CZ 4 are the aerodynamic coefficients[7]. u = [ F1cζ 1 F2 cζ 2 F1sζ 1 F2 sζ 2 δ RUD δ ELV ]
T
Six control variables of the airship are thrust F1 and F2 , turning angles ζ 1 , ζ 2 about axis y , rudder angle δ RUD and elevator angle δ ELV , respectively. C. Dynamics Model Expessed by Generalized Coordinate Define generalized coordinate μ
⎡⎣θ ,ψ , φ , xg , y g , z g ⎤⎦
T
where ( xg , yg , z g ) is the coordinate of the center of volume about the earth reference frame. Based on the fundamental kinematics we have ⎡ S11 O3×3 ⎤ V =⎢ ⎥ μ& b ⎣O3×3 Se ⎦
Sμ&
(2)
where b Se is homogenous transformation matrix from the earth reference frame to the body reference frame of the airship. − s μ1 1 ⎤ ⎡ 0 S11 = ⎢⎢ c μ3 c μ1s μ3 0 ⎥⎥ ⎢⎣ − s μ3 c μ1c μ3 0 ⎥⎦
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
b
s μi
2783
&&d + K d ( μ& d − μ& ) + K p ( μd − μ ) r=μ &&d + K d e& + K p e =μ
⎡c μ1c μ 2 c μ1 s μ 2 − s μ1 ⎤ f2 c μ1 s μ3 ⎥⎥ Se = ⎢⎢ f1 ⎢⎣ f3 f4 c μ1c μ3 ⎥⎦
sin μi , c μi
cos μi
where K d and K p are positive definite matrices, then (8) can be rewritten as
i = 1, 2,3, 4,5, 6 ,
e&& + K d e& + K p e = 0
f1
s μ1c μ2 s μ3 − s μ2 cμ3 , f 2
s μ1s μ2 s μ3 + cμ2 c μ3 ,
f3
s μ1c μ2 c μ3 + s μ2 s μ3 , f 4
s μ1s μ2 c μ3 − c μ2 s μ3 .
initial condition ( μ0 , μ& 0 ) , there exists ( μ, μ& ) → ( μd , μ& d ) .
(3)
Substituting (8) into (7)yields the expression of the feedback linearization control law &&d + K d e& + K p e ) + N ( μ, μ& ) + G ( μ ) τ =M ( μ ) ( μ
Multiply both sides of (3) by M , and we have & & + MSμ && MV& = MSμ
(4)
where
,
B ( μ ) = B . Because M and S are invertible, then M ( μ ) is invertible too. Since B ( μ ) ≠ 0 , B ( μ ) is
invertible. According to (5), we can derive
where τ
To this end, the actual control input can be calculated as
(5)
& & − N , G ( μ ) = −G M ( μ ) = MS , N ( μ, μ& ) = MSμ
&& + N ( μ, μ& ) + G ( μ ) = τ M ( μ) μ
(10)
&&d + K d e& + K p e ) + N ( μ, μ& ) + G ( μ ) ⎤⎦ u = B −1 ( μ ) ⎡⎣ M ( μ ) ( μ
Combining (1) with (4) obtains && + N ( μ, μ& ) + G ( μ ) = B ( μ ) u M ( μ) μ
(9)
Thus, ( e, e& ) = ( 0, 0 ) is exponentially stable. For any
Differentiating equation (2) yields & & + Sμ && V& = Sμ
(8)
(6)
IV. ADAPTIVE CONTROL LAW DESIGN A. Adaptive Control Law Denote the imprecisely known inertia parameter vector as η = [ I x , I y , I z , I xz , mzc ]T and the estimated one as ηˆ = [ Iˆx , Iˆy , Iˆz , Iˆxz , mzˆc ]T . The feedback linearization
control law is modified as
B ( μ ) u.
ˆ ( μ)( μ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ ) τ =M
III. FEEDBACK LINEARIZATION CONTROL DESIGN
(11)
The actual control input can be obtained:
A. Control Objective In view of inertia parameter uncertainty, design feedback linearization and adaptive control law[11] to realize asymptotic tracking of any desired trajectory from an arbitrary initial condition. Let μd (t ) denote an arbitrary twice differentiable time-varying trajectory, &&d (t ) are bounded. with μ& d (t ) and μ B. Control Law We choose
ˆ ( μ) ( μ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ )] u = B −1 ( μ ) [ M
ˆ , Nˆ , Gˆ are the estimated matrices of M , N , G of where M ηˆ . Substituting (11) into (6) yields && + N ( μ, μ& ) + G ( μ ) M ( μ) μ ˆ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ ) = M ( μ)( μ
(12)
&&d = e&& + μ && and (12) can be formulated as a Since μ linear function about the dynamics parameter vector:
τ = N ( μ, μ& ) + G ( μ ) + M ( μ ) r
(7)
where r will be designed later. Substituting (7) into (6) yields && + N ( μ, μ& ) + G ( μ ) = N ( μ, μ& ) + G ( μ ) + M ( μ ) r M ( μ) μ
Then we get && = M ( μ ) r M ( μ) μ
which is equivalent to a decoupling linear time-invariant && = r . When μd ( t ) is given, μ& d ( t ) and system μ &&d ( t ) are known. Let error be e = μd − μ , and μ
© 2012 ACADEMY PUBLISHER
ˆ ( μ) ( e&& + K d e& + K p e ) ) M % ( μ, η − ηˆ ) μ && + N% ( μ, μ& , η − ηˆ ) + G% ( μ, η − ηˆ ) =M ) ) &&, η% ) + N ( μ, μ& , η% ) + G% ( μ, η% ) = M ( μ, μ )) )) ) && ) + N ( μ, μ& ) + G ( μ ) ⎤ η% = ⎡⎢ M ( μ, μ ⎣ ⎦⎥ % & && Y ( μ, μ, μ ) η
(13)
N − Nˆ , G% G − Gˆ , η% η − ηˆ . % ≡/ 0 , N% ≡/ 0 , G% ≡/ 0 . Usually we have ηˆ ≠ η, and thus M Since (13) can’t be changed to linear constant system like (9), a real-time estimator to the parameters needs to be designed to realize e → 0, e& → 0 .
where M%
ˆ , N% M −M
2784
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
Assume the estimated parameter ηˆ renders Mˆ ( μ) invertible, so the following the closed-loop system can be obtained ˆ −1 ( μ ) Y ( μ, μ& , μ && ) η% e&& + K d e& + K p e = M
Φη%
T
Hx + DΦη%
(15)
H T R + RH = −Q
Which has an unique positive definite matrix solution R , Choose parameter estimator
B. Stability Proof Since the unmanned airship operates in the vicinity of the cruise altitude 20km, inertial parameters η can be regarded as a constant vector, and the parameter estimation law (16) can be written as
⎡e ⎤ ⎡ x⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ e& ⎥ ⎣ η% ⎦ ⎢ η% ⎥ ⎣ ⎦
Select the candidate Lyapunov function as L(t ) = x T Rx + η% T Γη%
The derivative of trajectory can be calculated along the closed-loop system (15) and (17): L& (t ) = x& T Rx + x T Rx& + η%& T Γη% + η% T Γη%&
= x T ( H T R + RH ) x + 2η% T ⎡⎣Φ T D T Rx + Γη%& ⎤⎦ (18) = − x T Qx ≤ 0
Therefore, the state of closed-loop system [ x T , η% T ]T is bounded. From the definition of the Lyapunov function L ( t ) and
and
© 2012 ACADEMY PUBLISHER
2 2
∫
∞
t0
x
2
≤−
2
∞ 1 1 L& ( t ) dt = ⎡ L ( t0 ) − L ( ∞ ) ⎤⎦ ∫ t λm ( Q ) 0 λm ( Q ) ⎣
1
λm ( Q )
L ( t0 ) < ∞, ∀t ≥ t0 ≥ 0
so x is square integrable. &&d , η are bounded, from Because x, η% , μd , μ& d , μ control law expression (11) μ, μ& , τ are bounded, The && yields the boundedness of boundedness of μ ⎡ e& ⎤ ⎡ μ& − μ& ⎤ x& = ⎢ ⎥ = ⎢ d &&d − μ && ⎥⎦ ⎣e&&⎦ ⎣ μ
Since x is square integrable and x& is bounded, x is uniformly continuous. Barbalat lemma guarantees x → 0 ( t → ∞ ) , i.e. e → 0, e& → 0, μ → μd , μ& → μ& d . V. SIMULATION A. Parameter Values && )6×5 can be obtained The elements of matrix Y ( μ, μ& , μ
from (13) , where (17)
The state vector is defined as
0 ≤ L ( ∞ ) ≤ L ( t ) ≤ L ( t0 ) < ∞, ∀t ≥ t0 ≥ 0
L& ( t ) ≤ −λm ( Q ) x
(16)
The parameter estimator (16) and control law (11) constitute the adaptive feedback linearization trajectory tracking control design of the airship.
(18), one has
x, η% are bounded, and from (18) we get
≤
H is Hurwitz, and there exists a positive definite matrix Q such that
η&% = − Γ −1Φ T D T Rx
2
Thus,
In view of the positive definite matrices K d and K p
η&ˆ = Γ −1Φ T D T Rx （ Γ > 0 ）
2
(14)
Let x = ⎡⎣ e T , e& T ⎤⎦ ，and then (14) can be written as I ⎤ ⎡ 0 ⎡0 ⎤ x& = ⎢ x + ⎢ ⎥ Φη% ⎥ ⎣I ⎦ ⎣− K p − K d ⎦
0 ≤ λm ( R ) x 2 + λm ( Γ ) η% 2 ≤ L ( t ) ≤ L ( t0 ) < ∞
y11 = μ&&3 − μ&&2 s μ1 − μ&1 μ& 2 c μ1 , y12 = ( μ&1c μ3 + μ& 2 c μ1 s μ3 )( μ&1 s μ3 − μ& 2 c μ1c μ3 ) ,
y14 = μ&&1 s μ3 − μ&&2 c μ1c μ3 + μ& 22 s μ1cμ1 s μ3 + 2μ&1 μ& 2 s μ1cμ3 , y15 = gc μ1 s μ3 + μ&&4 ( s μ 2 c μ3 − s μ1c μ 2 s μ3 ) −
μ&&5 ( c μ2 c μ3 + s μ1 s μ2 s μ3 ) − μ&&6 cμ1 s μ3 , y21 = − μ&1 μ& 3 s μ3 + μ& 2 μ& 3 cμ1cμ3 +
μ&1 μ& 2 s μ1 s μ3 − μ& 22 s μ1c μ1cμ3 , y22 = μ& 2 ( μ& 3 c μ1c μ3 − μ&1 s μ1 s μ3 ) + μ&&1c μ3 −
μ&1 μ& 3 s μ3 + μ&&2 c μ1 s μ3 , y23 = μ&1 μ& 3 s μ3 − μ& 2 μ& 3 c μ1c μ3 −
μ&1 μ& 2 s μ1 s μ3 + μ& 22 s μ1 μ1cμ3 , y24 = − μ& 12 + μ& 22 + μ& 32 − ( μ& 2 c μ1 ) + ( μ&1c μ3 ) − 2
2
2μ& 2 μ& 3 s μ1 − ( μ& 2 c μ1c μ3 ) + 2 μ&1 μ& 2 c μ1c μ3 s μ3 , 2
y25 = gs μ1 + μ&&4 c μ1c μ2 + μ&&5 c μ1 s μ2 − μ&&6 s μ1 ,
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
y31 = ( μ& 2 s μ1 − μ& 3 )( μ&1c μ3 + μ& 2 c μ1 s μ3 ) ,
2785
B. Simulation A helix is chosen as the desired trajectory to be followed and expressed as
y32 = ( μ& 3 − μ& 2 s μ1 )( μ&1cμ3 + μ& 2 c μ1 s μ3 ) ,
⎧ xd = 500sin ( 0.01t ) ⎪ ⎨ yd = 500 cos ( 0.01t ) ⎪ z = −20000 − 0.01t ⎩ d
y33 = μ&&2 c μ1c μ3 − μ& 2 ( μ&1 s μ1c μ3 + μ& 3 c μ1 s μ3 ) −
μ&&1 s μ3 − μ&1 μ& 3 cμ3 , y34 = − μ&&3 + μ&&2 s μ1 + μ&1 μ& 2 cμ1 −
Referred to [10], the desired attitude can be calculated through the airship dynamic model and the corresponding Frenet –based kinematic description:
( μ&1cμ3 + μ& 2 cμ1 s μ3 )( μ&1 s μ3 − μ& 2 cμ1cμ3 ) , y35 = 0,
⎧θ d = 0.1974rad ⎪ ⎨ψ d = arctan 2 ( − sin ( 0.01t ) , cos ( 0.01t ) ) rad ⎪ ⎩φd = 0rad
y41 = y42 = y43 = y44 = 0, y45 = − μ& 22 s μ1cμ1c μ3 + 2 μ& 2 μ& 3 c μ1c μ3 +
μ&&1cγ 3 − 2 μ&1 μ& 3 s μ3 + μ&&2 c μ1 s μ3 ,
The initial condition and control parameters in simulation are summarized as follows:
y51 = y52 = y53 = y54 = 0,
y55 = μ&1 μ& 2 c μ1 ( c μ3 ) − μ&12 c μ3 s μ3 − μ&1 μ& 2 c μ1 ( s μ3 ) + 2
T μ0 = [ 0, 0, 0, 0,505, −20150] , Iˆx 0 = 1.5 × 107 kg ⋅ m 2 ,
2
μ& 2 μ&1cμ1 + μ& 22 ( c μ1 ) cμ3 s μ3 + μ&&2 s μ1 − μ&&3 ,
Iˆy 0 = Iˆz 0 = 1× 108 kg ⋅ m 2 , Iˆxz 0 = −4 × 104 kg ⋅ m 2 ,
2
mzˆc 0 = 4.4 × 105 kg ⋅ m, B f = mg , K d = 2 I 6×6 ,
y61 = y62 = y63 = y64 = 0,
K p = I 6×6 , Γ = diag(10−4 ,10−7 ,10−7 ,5 × 10−4 ,10−4 ),
y65 = − ( μ& 3 − μ& 2 s μ1 ) − ( μ&1c μ3 + μ& 2 c μ1 s μ3 ) . 2
2
Q = I12×12 .
In order to validate the feedback linearization adaptive control algorithm, the airship simulation model parameters[7] are shown in table I and table II. The parameter values in table II are dimensionless. TABLE I. PARAMETER VALUES OF THE MODEL Parameter
m
Value
Unit
Parameter
Value
Unit
55749.7
kg
ρ
0.072157
kg/m3
3
D Iy
736311
m
Ix
5x10
kg·m2
2.9x108
kg·m2
Iz
2.9x108
kg·m2
I xz
-6x104
kg·m2
zc
15
m
7
It is noted that the solution matrix R can be sovled by the Lyapunov equation, and control law expressions are obtained by (16) and (11) , and the simulation result can be therefore performed. Without considering wind disturbance the system cruise simulation results are shown in Fig. 2 ~ Fig. 4. The trajectory tracking errors are asymptotically stable, as depicted in Fig. 2. The attitudes track the desired ones quickly and accurately as depicted in Fig. 3. As shown in Fig. 4, all of the inertial parameters estimations are stabile, although they do not converge to the true values, which is consistent with the theoretic analysis. It is found from the values of the longitudinal coordinate in Fig. 4 that the inertia parameters are very large so that the variation of the estimated values is quite subtle.
TABLE II. PARAMETER VALUES OF THE MODEL Param -eter
Value
Parameter
Value
Parameter
Value
k1
0.1054
k2
0.8259
k3
1773.2
CX 1
227.8
CX 2
2307.1
CL1
24059
CL 2
8080
CY 1
2307.1
CY 2
3037.6
CY 3
9932.7
CY 4
657.3
CZ 1
2307.1
CZ 2
3037.6
CZ 3
9730.7
CZ 4
657.3
CM 1
384515.9
CM 2
356916.5
CM 3
373391
CM 4
77238.5
CN1
384515.9
CN 2
356916
CN 3
373391
CN 4
77238.5
© 2012 ACADEMY PUBLISHER
Figure 2. Trajectory tracking errors
2786
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
under a grant from Advanced Aircraft Design Center of Beihang University. REFERENCES
Figure 3. Euler attitudes
Figure 4. Estimated inertia values
VI. CONCLUSION An adaptive feedback linearization control method is proposed, with online inertial parameter compensation, which renders asymptotic tracking of any given continuous time-varying trajectory from any initial conditions. The closed-loop stability is proved, although the estimated values of the inertial parameters don’t converge to their true values. The simulation result is coincident with the theoretical analysis. The method is only suitable for a full-actuated airship with the inertial parameters imprecisely known. The trajectory tracking control method for the under-actuated airship [12] [13] under the control constraint [14], as well as the actuator saturation should be studied further. ACKNOWLEDGMENT The authors would like to thank Professor Zhe WU of Beihang University. This work was supported in part © 2012 ACADEMY PUBLISHER
[1] Gomes, S.B.V., Ramos, J.J.G., “Airship dynamic modeling for autonomous operation,” IEEE International Conference on Robotics and Automation, Leuven, Belgium, May 1998. pp. 3462-3467. [2] Azinheira,J.R., Moutinho, A., de Paiva, E.C., “Airship hover stabilization using a backstepping control approach,” in Journal of Guidance, Control and Dynamic, vol. 29, no. 4, 2006, pp.903-914. [3] Azinheira J.R., Paiva E.C., Ramos J.J.G., Bueno S.S. “Mission path following for an autonomous unmanned airship,” Proceedings of the 2000 IEEE International Conference on Robotics & Automation. San Francisco: IEEE, 2000, pp.1269-1275. [4] De P., E.C., Benjovengo, Fábio, Bueno, Samuel Siqueira, “Sliding mode control for the path following of an unmanned airship,” 6th IFAC Symposium on Intelligent Autonomous Vehicles, IAV2007, Toulouse: France, September 2007, vol. 6, no. 1, pp.221-227. [5] Segio B. Varella Gomes, Josue Jr. G. Ramos, “Airship dynamic modeling for autonomous operation,” Proceedings of IEEE International Conference on Robotics and Automation. Belgium: IEEE, 1998, pp.3462-3467. [6] Schmidt, David K, “Dynamic modeling, control, and station-keeping guidance of a large high-altitude "nearspace" airship,” AIAA Guidance, Navigation, and Control Conference. Keystone, CO, USA: AIAA, August 2006, 2006-6781, pp.5222-5235. [7] Muller J.B. Paluszek M.A and Yiyuan Zhao, “Development of an aerodynamic model and control law design for a high altitude airship,” AIAA 3rd “Unmanned Unlimited” Technical Conference, Workshop and Exhibit. Chicago: AIAA, September 2004, 2004-6479, pp. 415-431. [8] Michael T. Frye, Stephen M. Gammon, Chunjiang Qian, “The 6-DOF dynamic model and simulation of the triturbofan remote-controlled airship,” Proceedings of the American Control Conference. New York, USA: IEEE, July 2007, pp.816-821, dio: 10.1109/ACC.2007.4283087. [9] David K, Schmidt, “Modeling and near-space station keeping control of a large high-altitude airship,” in Journal of Guidance, Control, and Dynamics, Eds. USA: AIAA, March/April 2007, vol. 30, no. 2, pp.540-547, dio: 10.2514/1.24865. [10] Filoktimon Repoulias, Evangelos Papadopoulos, “Dynamically feasible trajectory and open-loop control design for unmanned airships,” Mediterranean Conference on Control and Automation. Athens, Greece: IEEE, July 2007: T34-008, dio: 10.1109/MED.2007.4433820. [11] Craig J, Hsu P, Sasry S, Adaptive control of mechanical manipulators, Int. J. Robotics Research, 1987, pp.16-28. [12] A.Pedro Aguiar, João P.Hespanha, “Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty,” in IEEE Transactions on Automatic Control, vol. 52, no. 8, pp.1362-1379, 2007, dio: 10.1109/TAC.2007.902731. [13] Ofelia Begovich, Edgar N. Sanchez. “Takagi-Sugeno fuzzy scheme for real-time trajectory tracking of an underactuated robot,” in IEEE Transactions on Control Systems Technology, vol. 10, no. 1, pp.14-20, January 2002, dio:10.1109/87.974334. [14] Emeterio Aguiñaga-Ruiz, Arturo Zavala-Río, Víctor santibáñez, “Global trajectory tracking though static feedback for robot manipulators with bounded input,” in
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
IEEE Transactions on Control Systems Technology, vol. 17(4), pp. 934-944, 2009. Yongmei Wu (1974 - ) is currently a Ph.D. candidate in School of Aeronautical Science and Engineering, Beihang University, Beijing, China. Her research interests include spacecraft design,
2787
Zongyu Zuo (1982 - ) is currently a Ph.D. candidate with the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include nonlinear systems, adaptive control and control of autonomous aerial vehicle.
nonlinear systems and adaptive control of aerial vehicle.
Ming Zhu (1976 - ) is currently a Ph.D. Associate Professor in School of Aeronautical Science and Engineering, Beihang University, Beijing, China. His research interests include spacecraft design and avionics of aerial vehicle.
© 2012 ACADEMY PUBLISHER
Zewei Zheng (1985 - ) is currently a Ph.D. candidate in School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include nonlinear systems, adaptive control and avionics of autonomous aerial vehicle.
2781
Adaptive Trajectory Tracking Control of a High Altitude Unmanned Airship Yongmei Wu and Ming Zhu School of Aeronautical Science and Engineering, Beihang University, Beijing, China Email: [email protected]
Zongyu Zuo and Zewei Zheng School of Automation Science and Electrical Engineering, Beihang University, Beijing, China Email: [email protected]
Abstract—Nonlinear dynamic model of a high-altitude unmanned airship, expressed by generalized coordinate, was built. A nonlinear compensation was introduced into the control loop to linearize and decouple the nonlinear system globally. In view of the imprecisely known inertia parameters of the airship, an adaptive law was proposed based on the feedback linearization to realize asymptotic tracking of any continuous time-varying desired trajectory from an arbitrary initial condition. The stability of the closed-loop control system was proved via the use of Lyapunov stability theory. Finally, numerical simulation results demonstrate the validity and effectiveness of the proposed adaptive control law. Index Terms—adaptive control, feedback linearization, trajectory tracking, high-altitude unmanned airships
II. KINETIC MODEL OF THE AIRSHIP A. Defination of Coordinate This paper studies an ellipsoid full-actuated high altitude unmanned airship which is symmetrical with respect to the vertical axis, and its tail fin with the cross elevator and rudder, is bisymmetric. The gondola is equipped with a pair of differential propellers under the body. The earth reference frame is denoted by Oe xe ye ze , and the body reference frame Oxyz whose origin is located at the center of volume, as shown in Fig.1.
I. INTRODUCTION High-altitude unmanned airships, which have a wide application prospect in communication, surveillance and investigation, are capable of hovering for a long time. According to the task demands, desired trajectory is designated. Modeling, control method and verification test of high altitude unmanned airships are the focus of the domestic and international studies [1]~[10]. Trajectory tracking, based on adaptive feedback linearization, is designed to solve the control problem on imprecisely known inertia parameters of a high altitude unmanned airship. This paper is organized as follows: nonlinear dynamic model of a conventional airship is built, expressed by generalized coordinate in section II. In section III the feedback linearization control law is designed. Adaptive feedback linearization control law and estimation law of inertia parameters are designed, and stability is proved in section IV. The effectiveness of tracking desired continuous time-varying trajectory is validated via simulation without wind disturbance in section V. Finally, conclusion and future work are summarized in section VI. Manuscript received September 16, 2011; revised September 30, 2011; accepted December 1, 2011. Corresponding author: Yongmei Wu, [email protected]
© 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.11.2781-2787
Oe
xe
ye
ze
Figure 1. Definition of coordinate
B. Dynamics Fundamental Equations of Airship Several basic assumptions are needed: A1. The volume center coincides with the gross center of buoyancy. A2. The airship forms a rigid body such that elastic effects can be ignored. A3. The shape and the whole mass are constant in hovering. In view of the symmetry, the center of mass is located under the center of volume in longitudinal profile, and products of inertia satisfy I xy = I yz = 0 , The dynamic equations of follows[1]~[10] :
the
airship
can
be
formulated
as
2782
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
MV& = N + G + Bu
where V
⎡ − zc mg cos θ sin φ ⎤ ⎢ ⎥ − zc mg sin θ ⎢ ⎥ ⎢ ⎥ 0 ⎥ G = ⎢⎢ ( B − mg ) sin θ ⎥ f ⎢ ⎥ ⎢ − ( B f − mg ) cos θ sin φ ⎥ ⎢ ⎥ ⎢⎣ − ( B f − mg ) cos θ cos φ ⎥⎦
(1)
[ p, q, r , u, v, w]T , [u, v, w]T denotes linear
velocity vector, and [ p, q, r ]T angular velocity vector of the airship. 0 − I xz 0 − mzc 0 ⎡ Ix ⎤ ⎢ 0 I + ρ∇ k ⎥ 0 mz 0 0 3 y c ⎢ ⎥ ⎢ − I xz ⎥ 0 I z + ρ∇ k 3 0 0 0 M =⎢ ⎥ 0 mz 0 m + ∇ k 0 0 ρ 1 c ⎢ ⎥ ⎢ − mzc ⎥ 0 0 0 m + ρ∇ k 2 0 ⎢ ⎥ 0 0 0 0 m + ρ∇k2 ⎥⎦ ⎢⎣ 0
where I x , I y , I z , I xz are inertia parameters, k1 , k2 , k3 are inertial factors of the airship, ∇ is the volume of the airship, ρ is atmospheric density of the flying height, zc is the position coordinates of the center of mass, and m is the whole mass of the airship. N = [ a1 a2 a3 a4 a5 a6 ]
T
where g is gravity acceleration, B f is buoyancy acted on the airship, θ ,ψ , φ are attitudes. cξ 0 0 0 0 ⎡ cξ ⎤ ⎢ sξ 0 0 0 − sξ −2QCM 4 ⎥⎥ ⎢ ⎢ 0 ⎥ 0 1 1 −2QCN 4 0 B=⎢ ⎥ 0 0 ⎢ − z p sξ − z p sξ y p − y p ⎥ ⎢ z p cξ z p cξ − x p − x p 2QCY 4 ⎥ 0 ⎢ ⎥ −2QCZ 4 ⎥⎦ h2 0 0 0 ⎢⎣ h1
where
where a1 = − ( I z − I y ) qr + I xz pq + mzc ( ur − wp ) +
QCL 2 sin β sin ( β ) , a2 = − ( I x − I z ) pr − I xz ( p 2 − r 2 ) − mzc ( wq − vr ) − Q[CM 1 cos (α 2 ) sin ( 2α ) + CM 2 sin ( 2α ) + CM 3 sin α sin ( α )], a3 = − ( I y − I x ) pq − I xz qr + Q[CN 1 cos ( β 2 ) sin ( 2β ) +
CN 2 sin ( 2 β ) + CN 3 sin β sin ( β )], a4 = − ( m + ρ∇k1 )( wq − vr ) − mzc pr − Q ⎡⎣C X 1 cos 2 α cos 2 β + C X 2 sin ( 2α ) sin (α 2 ) ⎤⎦ , a5 = − ( m + ρ∇k2 )( ur − wp ) − mzc qr − Q[CY 1 cos ( β 2 ) sin ( 2β ) + CY 2 sin ( 2 β ) + CY 3 sin β sin ( β )], a6 = − ( m + ρ∇k2 )( vp − uq ) + mzc ( p 2 + q 2 ) − Q[CZ 1 cos (α 2 ) sin ( 2α ) + CZ 2 sin ( 2α ) + CZ 3 sin α sin ( α )].
where α = arctan( w / u ) and β = arctan(v cos α / u ) are the flow angle when wind speed is zero, Q = ρV 2 / 2 is dynamic pressure, V is the flow speed from a distance, CLi , CMi , CNi , C Xi , CYi , CZi , i = 1, 2,3 are the aerodynamic coefficients[7]. © 2012 ACADEMY PUBLISHER
sξ
sin ξ
,
cξ
cos ξ
h1 = x p sξ − y p cξ ,
,
h2 = x p sξ + y p cξ , ( x p , y p , z p ) and ( x p , − y p , z p ) are position coordinates of left and right propellers about the body reference frame, ξ is an angle toward outside of propellers, CM 4 , CN 4 , CY 4 , CZ 4 are the aerodynamic coefficients[7]. u = [ F1cζ 1 F2 cζ 2 F1sζ 1 F2 sζ 2 δ RUD δ ELV ]
T
Six control variables of the airship are thrust F1 and F2 , turning angles ζ 1 , ζ 2 about axis y , rudder angle δ RUD and elevator angle δ ELV , respectively. C. Dynamics Model Expessed by Generalized Coordinate Define generalized coordinate μ
⎡⎣θ ,ψ , φ , xg , y g , z g ⎤⎦
T
where ( xg , yg , z g ) is the coordinate of the center of volume about the earth reference frame. Based on the fundamental kinematics we have ⎡ S11 O3×3 ⎤ V =⎢ ⎥ μ& b ⎣O3×3 Se ⎦
Sμ&
(2)
where b Se is homogenous transformation matrix from the earth reference frame to the body reference frame of the airship. − s μ1 1 ⎤ ⎡ 0 S11 = ⎢⎢ c μ3 c μ1s μ3 0 ⎥⎥ ⎢⎣ − s μ3 c μ1c μ3 0 ⎥⎦
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
b
s μi
2783
&&d + K d ( μ& d − μ& ) + K p ( μd − μ ) r=μ &&d + K d e& + K p e =μ
⎡c μ1c μ 2 c μ1 s μ 2 − s μ1 ⎤ f2 c μ1 s μ3 ⎥⎥ Se = ⎢⎢ f1 ⎢⎣ f3 f4 c μ1c μ3 ⎥⎦
sin μi , c μi
cos μi
where K d and K p are positive definite matrices, then (8) can be rewritten as
i = 1, 2,3, 4,5, 6 ,
e&& + K d e& + K p e = 0
f1
s μ1c μ2 s μ3 − s μ2 cμ3 , f 2
s μ1s μ2 s μ3 + cμ2 c μ3 ,
f3
s μ1c μ2 c μ3 + s μ2 s μ3 , f 4
s μ1s μ2 c μ3 − c μ2 s μ3 .
initial condition ( μ0 , μ& 0 ) , there exists ( μ, μ& ) → ( μd , μ& d ) .
(3)
Substituting (8) into (7)yields the expression of the feedback linearization control law &&d + K d e& + K p e ) + N ( μ, μ& ) + G ( μ ) τ =M ( μ ) ( μ
Multiply both sides of (3) by M , and we have & & + MSμ && MV& = MSμ
(4)
where
,
B ( μ ) = B . Because M and S are invertible, then M ( μ ) is invertible too. Since B ( μ ) ≠ 0 , B ( μ ) is
invertible. According to (5), we can derive
where τ
To this end, the actual control input can be calculated as
(5)
& & − N , G ( μ ) = −G M ( μ ) = MS , N ( μ, μ& ) = MSμ
&& + N ( μ, μ& ) + G ( μ ) = τ M ( μ) μ
(10)
&&d + K d e& + K p e ) + N ( μ, μ& ) + G ( μ ) ⎤⎦ u = B −1 ( μ ) ⎡⎣ M ( μ ) ( μ
Combining (1) with (4) obtains && + N ( μ, μ& ) + G ( μ ) = B ( μ ) u M ( μ) μ
(9)
Thus, ( e, e& ) = ( 0, 0 ) is exponentially stable. For any
Differentiating equation (2) yields & & + Sμ && V& = Sμ
(8)
(6)
IV. ADAPTIVE CONTROL LAW DESIGN A. Adaptive Control Law Denote the imprecisely known inertia parameter vector as η = [ I x , I y , I z , I xz , mzc ]T and the estimated one as ηˆ = [ Iˆx , Iˆy , Iˆz , Iˆxz , mzˆc ]T . The feedback linearization
control law is modified as
B ( μ ) u.
ˆ ( μ)( μ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ ) τ =M
III. FEEDBACK LINEARIZATION CONTROL DESIGN
(11)
The actual control input can be obtained:
A. Control Objective In view of inertia parameter uncertainty, design feedback linearization and adaptive control law[11] to realize asymptotic tracking of any desired trajectory from an arbitrary initial condition. Let μd (t ) denote an arbitrary twice differentiable time-varying trajectory, &&d (t ) are bounded. with μ& d (t ) and μ B. Control Law We choose
ˆ ( μ) ( μ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ )] u = B −1 ( μ ) [ M
ˆ , Nˆ , Gˆ are the estimated matrices of M , N , G of where M ηˆ . Substituting (11) into (6) yields && + N ( μ, μ& ) + G ( μ ) M ( μ) μ ˆ &&d + K d e& + K p e ) + Nˆ ( μ, μ& ) + Gˆ ( μ ) = M ( μ)( μ
(12)
&&d = e&& + μ && and (12) can be formulated as a Since μ linear function about the dynamics parameter vector:
τ = N ( μ, μ& ) + G ( μ ) + M ( μ ) r
(7)
where r will be designed later. Substituting (7) into (6) yields && + N ( μ, μ& ) + G ( μ ) = N ( μ, μ& ) + G ( μ ) + M ( μ ) r M ( μ) μ
Then we get && = M ( μ ) r M ( μ) μ
which is equivalent to a decoupling linear time-invariant && = r . When μd ( t ) is given, μ& d ( t ) and system μ &&d ( t ) are known. Let error be e = μd − μ , and μ
© 2012 ACADEMY PUBLISHER
ˆ ( μ) ( e&& + K d e& + K p e ) ) M % ( μ, η − ηˆ ) μ && + N% ( μ, μ& , η − ηˆ ) + G% ( μ, η − ηˆ ) =M ) ) &&, η% ) + N ( μ, μ& , η% ) + G% ( μ, η% ) = M ( μ, μ )) )) ) && ) + N ( μ, μ& ) + G ( μ ) ⎤ η% = ⎡⎢ M ( μ, μ ⎣ ⎦⎥ % & && Y ( μ, μ, μ ) η
(13)
N − Nˆ , G% G − Gˆ , η% η − ηˆ . % ≡/ 0 , N% ≡/ 0 , G% ≡/ 0 . Usually we have ηˆ ≠ η, and thus M Since (13) can’t be changed to linear constant system like (9), a real-time estimator to the parameters needs to be designed to realize e → 0, e& → 0 .
where M%
ˆ , N% M −M
2784
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
Assume the estimated parameter ηˆ renders Mˆ ( μ) invertible, so the following the closed-loop system can be obtained ˆ −1 ( μ ) Y ( μ, μ& , μ && ) η% e&& + K d e& + K p e = M
Φη%
T
Hx + DΦη%
(15)
H T R + RH = −Q
Which has an unique positive definite matrix solution R , Choose parameter estimator
B. Stability Proof Since the unmanned airship operates in the vicinity of the cruise altitude 20km, inertial parameters η can be regarded as a constant vector, and the parameter estimation law (16) can be written as
⎡e ⎤ ⎡ x⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ e& ⎥ ⎣ η% ⎦ ⎢ η% ⎥ ⎣ ⎦
Select the candidate Lyapunov function as L(t ) = x T Rx + η% T Γη%
The derivative of trajectory can be calculated along the closed-loop system (15) and (17): L& (t ) = x& T Rx + x T Rx& + η%& T Γη% + η% T Γη%&
= x T ( H T R + RH ) x + 2η% T ⎡⎣Φ T D T Rx + Γη%& ⎤⎦ (18) = − x T Qx ≤ 0
Therefore, the state of closed-loop system [ x T , η% T ]T is bounded. From the definition of the Lyapunov function L ( t ) and
and
© 2012 ACADEMY PUBLISHER
2 2
∫
∞
t0
x
2
≤−
2
∞ 1 1 L& ( t ) dt = ⎡ L ( t0 ) − L ( ∞ ) ⎤⎦ ∫ t λm ( Q ) 0 λm ( Q ) ⎣
1
λm ( Q )
L ( t0 ) < ∞, ∀t ≥ t0 ≥ 0
so x is square integrable. &&d , η are bounded, from Because x, η% , μd , μ& d , μ control law expression (11) μ, μ& , τ are bounded, The && yields the boundedness of boundedness of μ ⎡ e& ⎤ ⎡ μ& − μ& ⎤ x& = ⎢ ⎥ = ⎢ d &&d − μ && ⎥⎦ ⎣e&&⎦ ⎣ μ
Since x is square integrable and x& is bounded, x is uniformly continuous. Barbalat lemma guarantees x → 0 ( t → ∞ ) , i.e. e → 0, e& → 0, μ → μd , μ& → μ& d . V. SIMULATION A. Parameter Values && )6×5 can be obtained The elements of matrix Y ( μ, μ& , μ
from (13) , where (17)
The state vector is defined as
0 ≤ L ( ∞ ) ≤ L ( t ) ≤ L ( t0 ) < ∞, ∀t ≥ t0 ≥ 0
L& ( t ) ≤ −λm ( Q ) x
(16)
The parameter estimator (16) and control law (11) constitute the adaptive feedback linearization trajectory tracking control design of the airship.
(18), one has
x, η% are bounded, and from (18) we get
≤
H is Hurwitz, and there exists a positive definite matrix Q such that
η&% = − Γ −1Φ T D T Rx
2
Thus,
In view of the positive definite matrices K d and K p
η&ˆ = Γ −1Φ T D T Rx （ Γ > 0 ）
2
(14)
Let x = ⎡⎣ e T , e& T ⎤⎦ ，and then (14) can be written as I ⎤ ⎡ 0 ⎡0 ⎤ x& = ⎢ x + ⎢ ⎥ Φη% ⎥ ⎣I ⎦ ⎣− K p − K d ⎦
0 ≤ λm ( R ) x 2 + λm ( Γ ) η% 2 ≤ L ( t ) ≤ L ( t0 ) < ∞
y11 = μ&&3 − μ&&2 s μ1 − μ&1 μ& 2 c μ1 , y12 = ( μ&1c μ3 + μ& 2 c μ1 s μ3 )( μ&1 s μ3 − μ& 2 c μ1c μ3 ) ,
y14 = μ&&1 s μ3 − μ&&2 c μ1c μ3 + μ& 22 s μ1cμ1 s μ3 + 2μ&1 μ& 2 s μ1cμ3 , y15 = gc μ1 s μ3 + μ&&4 ( s μ 2 c μ3 − s μ1c μ 2 s μ3 ) −
μ&&5 ( c μ2 c μ3 + s μ1 s μ2 s μ3 ) − μ&&6 cμ1 s μ3 , y21 = − μ&1 μ& 3 s μ3 + μ& 2 μ& 3 cμ1cμ3 +
μ&1 μ& 2 s μ1 s μ3 − μ& 22 s μ1c μ1cμ3 , y22 = μ& 2 ( μ& 3 c μ1c μ3 − μ&1 s μ1 s μ3 ) + μ&&1c μ3 −
μ&1 μ& 3 s μ3 + μ&&2 c μ1 s μ3 , y23 = μ&1 μ& 3 s μ3 − μ& 2 μ& 3 c μ1c μ3 −
μ&1 μ& 2 s μ1 s μ3 + μ& 22 s μ1 μ1cμ3 , y24 = − μ& 12 + μ& 22 + μ& 32 − ( μ& 2 c μ1 ) + ( μ&1c μ3 ) − 2
2
2μ& 2 μ& 3 s μ1 − ( μ& 2 c μ1c μ3 ) + 2 μ&1 μ& 2 c μ1c μ3 s μ3 , 2
y25 = gs μ1 + μ&&4 c μ1c μ2 + μ&&5 c μ1 s μ2 − μ&&6 s μ1 ,
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
y31 = ( μ& 2 s μ1 − μ& 3 )( μ&1c μ3 + μ& 2 c μ1 s μ3 ) ,
2785
B. Simulation A helix is chosen as the desired trajectory to be followed and expressed as
y32 = ( μ& 3 − μ& 2 s μ1 )( μ&1cμ3 + μ& 2 c μ1 s μ3 ) ,
⎧ xd = 500sin ( 0.01t ) ⎪ ⎨ yd = 500 cos ( 0.01t ) ⎪ z = −20000 − 0.01t ⎩ d
y33 = μ&&2 c μ1c μ3 − μ& 2 ( μ&1 s μ1c μ3 + μ& 3 c μ1 s μ3 ) −
μ&&1 s μ3 − μ&1 μ& 3 cμ3 , y34 = − μ&&3 + μ&&2 s μ1 + μ&1 μ& 2 cμ1 −
Referred to [10], the desired attitude can be calculated through the airship dynamic model and the corresponding Frenet –based kinematic description:
( μ&1cμ3 + μ& 2 cμ1 s μ3 )( μ&1 s μ3 − μ& 2 cμ1cμ3 ) , y35 = 0,
⎧θ d = 0.1974rad ⎪ ⎨ψ d = arctan 2 ( − sin ( 0.01t ) , cos ( 0.01t ) ) rad ⎪ ⎩φd = 0rad
y41 = y42 = y43 = y44 = 0, y45 = − μ& 22 s μ1cμ1c μ3 + 2 μ& 2 μ& 3 c μ1c μ3 +
μ&&1cγ 3 − 2 μ&1 μ& 3 s μ3 + μ&&2 c μ1 s μ3 ,
The initial condition and control parameters in simulation are summarized as follows:
y51 = y52 = y53 = y54 = 0,
y55 = μ&1 μ& 2 c μ1 ( c μ3 ) − μ&12 c μ3 s μ3 − μ&1 μ& 2 c μ1 ( s μ3 ) + 2
T μ0 = [ 0, 0, 0, 0,505, −20150] , Iˆx 0 = 1.5 × 107 kg ⋅ m 2 ,
2
μ& 2 μ&1cμ1 + μ& 22 ( c μ1 ) cμ3 s μ3 + μ&&2 s μ1 − μ&&3 ,
Iˆy 0 = Iˆz 0 = 1× 108 kg ⋅ m 2 , Iˆxz 0 = −4 × 104 kg ⋅ m 2 ,
2
mzˆc 0 = 4.4 × 105 kg ⋅ m, B f = mg , K d = 2 I 6×6 ,
y61 = y62 = y63 = y64 = 0,
K p = I 6×6 , Γ = diag(10−4 ,10−7 ,10−7 ,5 × 10−4 ,10−4 ),
y65 = − ( μ& 3 − μ& 2 s μ1 ) − ( μ&1c μ3 + μ& 2 c μ1 s μ3 ) . 2
2
Q = I12×12 .
In order to validate the feedback linearization adaptive control algorithm, the airship simulation model parameters[7] are shown in table I and table II. The parameter values in table II are dimensionless. TABLE I. PARAMETER VALUES OF THE MODEL Parameter
m
Value
Unit
Parameter
Value
Unit
55749.7
kg
ρ
0.072157
kg/m3
3
D Iy
736311
m
Ix
5x10
kg·m2
2.9x108
kg·m2
Iz
2.9x108
kg·m2
I xz
-6x104
kg·m2
zc
15
m
7
It is noted that the solution matrix R can be sovled by the Lyapunov equation, and control law expressions are obtained by (16) and (11) , and the simulation result can be therefore performed. Without considering wind disturbance the system cruise simulation results are shown in Fig. 2 ~ Fig. 4. The trajectory tracking errors are asymptotically stable, as depicted in Fig. 2. The attitudes track the desired ones quickly and accurately as depicted in Fig. 3. As shown in Fig. 4, all of the inertial parameters estimations are stabile, although they do not converge to the true values, which is consistent with the theoretic analysis. It is found from the values of the longitudinal coordinate in Fig. 4 that the inertia parameters are very large so that the variation of the estimated values is quite subtle.
TABLE II. PARAMETER VALUES OF THE MODEL Param -eter
Value
Parameter
Value
Parameter
Value
k1
0.1054
k2
0.8259
k3
1773.2
CX 1
227.8
CX 2
2307.1
CL1
24059
CL 2
8080
CY 1
2307.1
CY 2
3037.6
CY 3
9932.7
CY 4
657.3
CZ 1
2307.1
CZ 2
3037.6
CZ 3
9730.7
CZ 4
657.3
CM 1
384515.9
CM 2
356916.5
CM 3
373391
CM 4
77238.5
CN1
384515.9
CN 2
356916
CN 3
373391
CN 4
77238.5
© 2012 ACADEMY PUBLISHER
Figure 2. Trajectory tracking errors
2786
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
under a grant from Advanced Aircraft Design Center of Beihang University. REFERENCES
Figure 3. Euler attitudes
Figure 4. Estimated inertia values
VI. CONCLUSION An adaptive feedback linearization control method is proposed, with online inertial parameter compensation, which renders asymptotic tracking of any given continuous time-varying trajectory from any initial conditions. The closed-loop stability is proved, although the estimated values of the inertial parameters don’t converge to their true values. The simulation result is coincident with the theoretical analysis. The method is only suitable for a full-actuated airship with the inertial parameters imprecisely known. The trajectory tracking control method for the under-actuated airship [12] [13] under the control constraint [14], as well as the actuator saturation should be studied further. ACKNOWLEDGMENT The authors would like to thank Professor Zhe WU of Beihang University. This work was supported in part © 2012 ACADEMY PUBLISHER
[1] Gomes, S.B.V., Ramos, J.J.G., “Airship dynamic modeling for autonomous operation,” IEEE International Conference on Robotics and Automation, Leuven, Belgium, May 1998. pp. 3462-3467. [2] Azinheira,J.R., Moutinho, A., de Paiva, E.C., “Airship hover stabilization using a backstepping control approach,” in Journal of Guidance, Control and Dynamic, vol. 29, no. 4, 2006, pp.903-914. [3] Azinheira J.R., Paiva E.C., Ramos J.J.G., Bueno S.S. “Mission path following for an autonomous unmanned airship,” Proceedings of the 2000 IEEE International Conference on Robotics & Automation. San Francisco: IEEE, 2000, pp.1269-1275. [4] De P., E.C., Benjovengo, Fábio, Bueno, Samuel Siqueira, “Sliding mode control for the path following of an unmanned airship,” 6th IFAC Symposium on Intelligent Autonomous Vehicles, IAV2007, Toulouse: France, September 2007, vol. 6, no. 1, pp.221-227. [5] Segio B. Varella Gomes, Josue Jr. G. Ramos, “Airship dynamic modeling for autonomous operation,” Proceedings of IEEE International Conference on Robotics and Automation. Belgium: IEEE, 1998, pp.3462-3467. [6] Schmidt, David K, “Dynamic modeling, control, and station-keeping guidance of a large high-altitude "nearspace" airship,” AIAA Guidance, Navigation, and Control Conference. Keystone, CO, USA: AIAA, August 2006, 2006-6781, pp.5222-5235. [7] Muller J.B. Paluszek M.A and Yiyuan Zhao, “Development of an aerodynamic model and control law design for a high altitude airship,” AIAA 3rd “Unmanned Unlimited” Technical Conference, Workshop and Exhibit. Chicago: AIAA, September 2004, 2004-6479, pp. 415-431. [8] Michael T. Frye, Stephen M. Gammon, Chunjiang Qian, “The 6-DOF dynamic model and simulation of the triturbofan remote-controlled airship,” Proceedings of the American Control Conference. New York, USA: IEEE, July 2007, pp.816-821, dio: 10.1109/ACC.2007.4283087. [9] David K, Schmidt, “Modeling and near-space station keeping control of a large high-altitude airship,” in Journal of Guidance, Control, and Dynamics, Eds. USA: AIAA, March/April 2007, vol. 30, no. 2, pp.540-547, dio: 10.2514/1.24865. [10] Filoktimon Repoulias, Evangelos Papadopoulos, “Dynamically feasible trajectory and open-loop control design for unmanned airships,” Mediterranean Conference on Control and Automation. Athens, Greece: IEEE, July 2007: T34-008, dio: 10.1109/MED.2007.4433820. [11] Craig J, Hsu P, Sasry S, Adaptive control of mechanical manipulators, Int. J. Robotics Research, 1987, pp.16-28. [12] A.Pedro Aguiar, João P.Hespanha, “Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty,” in IEEE Transactions on Automatic Control, vol. 52, no. 8, pp.1362-1379, 2007, dio: 10.1109/TAC.2007.902731. [13] Ofelia Begovich, Edgar N. Sanchez. “Takagi-Sugeno fuzzy scheme for real-time trajectory tracking of an underactuated robot,” in IEEE Transactions on Control Systems Technology, vol. 10, no. 1, pp.14-20, January 2002, dio:10.1109/87.974334. [14] Emeterio Aguiñaga-Ruiz, Arturo Zavala-Río, Víctor santibáñez, “Global trajectory tracking though static feedback for robot manipulators with bounded input,” in
JOURNAL OF COMPUTERS, VOL. 7, NO. 11, NOVEMBER 2012
IEEE Transactions on Control Systems Technology, vol. 17(4), pp. 934-944, 2009. Yongmei Wu (1974 - ) is currently a Ph.D. candidate in School of Aeronautical Science and Engineering, Beihang University, Beijing, China. Her research interests include spacecraft design,
2787
Zongyu Zuo (1982 - ) is currently a Ph.D. candidate with the School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include nonlinear systems, adaptive control and control of autonomous aerial vehicle.
nonlinear systems and adaptive control of aerial vehicle.
Ming Zhu (1976 - ) is currently a Ph.D. Associate Professor in School of Aeronautical Science and Engineering, Beihang University, Beijing, China. His research interests include spacecraft design and avionics of aerial vehicle.
© 2012 ACADEMY PUBLISHER
Zewei Zheng (1985 - ) is currently a Ph.D. candidate in School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. His research interests include nonlinear systems, adaptive control and avionics of autonomous aerial vehicle.
