Abstract: In this paper, combined attitude control of an underactuated helicopter experimental system is considered. The controlled helicopter experimental ...
COMBINED ATTITUDE CONTROL OF AN UNDERACTUATED HELICOPTER EXPERIMENTAL SYSTEM Mingcong Deng ∗ , Akira Inoue ∗ , and Tatsunori Shimizu ∗
Dept. Systems Eng., Okayama University 3-1-1 Tsushima-Naka, Okayama, Japan 700-8530 {deng, inoue}@suri.sys.okayama-u.ac.jp ∗
Abstract: In this paper, combined attitude control of an underactuated helicopter experimental system is considered. The controlled helicopter experimental system has two inputs and three outputs, namely, this system is underactuated. The combined attitude controller includes a nonlinear MIMO controller based on adaptive sliding mode control and non-adaptive nonlinear controllers. Control system stability is guaranteed by Lyapunov function based proof. Comparing simulation between the existed design method and the proposed design method shows the effectiveness of the proposed method. Keywords: Helicopter, underactuated system, modeling, control
1. INTRODUCTION Controlling an underactuated helicopter has received a lot of attention, due to the fact that the helicopter system is an underactuated nonlinear system. That is, the number of control inputs is less than the number of the nonlinear system outputs. This paper develops a combined attitude control for an underactuated helicopter experimental system, where the controlled system is a three-degree of freedom helicopter system that has two inputs and three outputs. So far, concerning with this topic of underactuated system control, many works have been developed (for example, [1] - [4], etc.) for different controlled systems. Especially, PD-based control system [3] and backstepping-based control system [4] were used to deal with an underactuated helicopter system. However, controller design of an underactuated helicopter system with uncertainties is an open problem. 1
Partially supported by the JSPS Grant.
In this paper, by extending a combined adaptive and non-adaptive attitude control method of 2-input 2-output helicopter experimental system [5,7], a combined attitude control for a threedegree of freedom helicopter experimental system that has 2-input and 3-output [3] is designed. The combined design method is facilitated by adaptive sliding mode control method and some nonlinear control methods to the uncertain underactuated system. Stability of the control system is guaranteed by Lyapunov function based proof. Comparing with PD-based control system [3], the proposed controller shows a desired control result by simulations.
2. MODEL AND PROBLEM STATEMENT We introduce the model of three-degree of freedom helicopter experimental system with 2-input and 3-output. The photo of the experimental system is shown in Fig. 1.
A : Coef f icient based on shape of the rotor
Based on the relationship, we have that Ir r¨ + Dr r˙ = La Ak 2 (u2 − u1 ) Ip p¨ + Dp p˙ + mLg g sin p = Lm Ak 2 (u1 + u2 ) cos r Iy y¨ + Dy y˙ = Lm Ak 2 (u1 + u2 ) sin r
(9)
That is, 1 a2 r˙ + (u2 − u1 ) a1 a1 b3 1 b2 sin p + (u1 + u2 ) cos r(10) p¨ = − p˙ − b1 b1 b1 1 c2 y¨ = − y˙ + (u1 + u2 ) sin r c1 c1 r¨ = −
Fig. 1. Helicopter experimental system
f
f f
1
L
l
a
f
r
a m 2
a
r
m 2
Ir Ip , b1 = La Ak 2 Lm Ak 2 Dp Dr , b2 = a2 = La Ak 2 Lm Ak 2 Iy mLg g c1 = , b3 = 2 Lm Ak Lm Ak 2 Dy c2 = Lm Ak 2 a1 =
2
a
-g
Fig. 2. Relationships of forces on role angle The experimental system is 2-input 3-output system which attaches the two motors for turning the two rotors, and detects pitch angle, roll angle and yaw angle by rotary encoders. The model on roll angle direction is as follows (Fig. 2). f˜1 = fr + fl f˜2 = La (fl − fr )
(1)
fr = Ak 2 u2r = Ak 2 u1
(3)
fl = Ak 2 u2l = Ak 2 u2
(4)
ωr = kur
(5)
ωl = kul
(6)
fr = Aωr2 fl = Aωl2
(7)
where, f˜1 is a resultant force of fr and fl , f˜2 is a moment of the roll direction, and ωr : Angular speed of right rotor ωl : Angular speed of lef t rotor k : Coef f icient between volt. and angular speed fr : Lif t f orce of right rotor fl : Lif t f orce of lef t rotor
The detailed dynamic equations of the angles are given in Appendix. For the simplification, we rearrange motion equations as follows. r¨ = −d2 r˙ + d1 (u2 − u1 ) p¨ = −e2 p˙ − e3 sin p + e1 (u1 + u2 ) cos r (12) y¨ = −f2 y˙ + f1 (u1 + u2 ) sin r 1 1 , e1 = a1 b1 b2 a2 d2 = , e2 = a1 b1 b3 1 f1 = , e3 = c1 b1 c2 f2 = c1
d1 =
(2)
(8)
(11)
(13)
where the helicopter system is controlled by u1 and u2 . Further, by using the result in [5,7], we obtain the following state equation of the helicopter system. ˙ x(t) = f x(t), t + g x(t), t u(t) (14) where
x4 x5 x6 f x(t), t = −d2 x4 −e2 x5 − e3 sin x2 −f2 x6
T x(t) = [x1 ,· · · , x6 ]T = [r, p, y, r, ˙ p, ˙ y] ˙ 0 0 0 0 0 0 0 0 0 g x(t), t = −d1 0 0 0 e1 cos x1 0 0 0 f1 sin x1 z1 u(t) = z2 (15) z3
where u(t) is the equivalent control input. In this case, the system has three inputs z1 , z2 and z3 . Concerning with the real experimental system, there exists modeling error [5, 7]. Forthis reason, uncertainties are considered. Define f x(t), t and g x(t), t in (14) as f x(t), t = f¯ x(t), t + ∆f x(t), t ¯ x(t), t + ∆g x(t), t g x(t), t = g
3. COMBINED ATTITUDE CONTROLLER DESIGN In this section, a combined attitude controller is designed and the control system stability proof is also given. The proposed combined controller includes a linear feedback controller, a relay controller, a nonlinear controller for the uncertainty of a definite part, a nonlinear controller for the uncertainty of an indefinite part, and a controller for a set point variation. Namely, u(t) = ueq (t) + ulf (t) + unl (t) +uad (t) + uη (t) + ur (t)
(21)
The detailed structure of these controllers is given as follows.
(16)
The equivalent controller The equivalent controller is expressed by the following equation.
˙ x(t) = f¯ x(t), t + ∆f x(t), t + ∆g x(t), t u(t) + g¯ x(t), t u(t) (17)
−1 ueq (t) = − S¯ g x(t), t S f¯ x(t), t (22)
Substituting these variables into (14), then
As a result, we represent the state equation as follows. For brevity, we omitted the detailed calculation (see [5,7]). ˙ ¯ x(t), t h x(t), x(t) = f¯ x(t), t + g u(t), t (18) ¯ x(t), t ∆h x(t), t + g¯ x(t), t u(t) + g where the structure of uncertainty h x(t), u(t), t is known, but parameters are unknown, i.e. uncer- tainty of an indefinite part. Moreover, ∆h x(t), t is called uncertainty of an indefinite part, and is an unknown model. However, it is referred to as:
∆h x(t), t ≤ η x(t), t (19)
and the upper-bound value function η x(t), t is considered and as bounded known. In order for ∆f x(t), t + ∆g x(t), t u(t) to fulfill matching conditions for sliding mode control, it is necessary to calculate h x(t), u(t), t , defined by: ¯ x(t), t h x(t), u(t), t g = ∆f x(t), t + ∆g x(t), t u(t)
(20)
The objective of this paper is to design a combined attitude controller (u1 and u2 ) so that the controller can control the three system outputs, namely, pitch angle p, roll angle r and yaw angle y. The deign method has two steps, that is, to z1 design u(t) = z2 , and then, to realize the z3 control u(t) by u1 and u2 .
where, a switching surface is designed as σ=
σ1 σ2
= S x(t) − y r (t)
ι1 0 1 0 S= 0 ι2 0 1
(23) (24)
and y r (t) is taken as the first order filter output of a step response. Linear feedback controller The linear feedback controller is expressed by the following equation. −1 g x(t), t Kσ ulf (t) = − S¯
(25)
where, constant matrix K is a positive definite. Relay controller The relay controller is expressed by the following equation. −1 unl (t) = − S¯ g x(t), t κ Λ(σ) sgn (σ1 ) Λ(σ) = sgn (σ2 ) sgn (σ3 ) 1, for x > 0 0, for x = 0 sgn (x) = −1, for x < 0
where, κ > 0.
Controller for definite part uncertainty Based on the result in [5], we have
(26) (27)
(28)
h x(t), u(t), t = Φ x(t), u(t), t θ where Φ x(t), u(t), t is a 3 × 7 matrix, and
θ = θ1 θ2 θ3 θ4 θ5 θ6 θ7
T
ˆ = θ(t)
θˆ1 θˆ2 θˆ3 θˆ4 θˆ5 θˆ6 θˆ7
T
As a result, the control input to compensate the right hand side 2nd term of (18), which is the term of a presumed error, is designed as follows. ˆ x(t), u(t), t uad (t) = −h ˆ = −Φ x(t), u(t), t θ(t)
ˆ where θ(t) is obtained by T ˆ˙ θ(t) = Γ S g¯ x(t), t Φ x(t), u(t), t σ
(29)
where
sgn (τ1 ) χ x(t), t = sgn (τ2 ) sgn (τ3 ) τ1 T τ2 = S g¯ x(t), t σ τ3 T = σ T S¯ g x(t), t
ur1 u r2 −1 = S g¯ x(t), t S y˙ r (t)
(31)
y¨ = Fy
(38)
Fp∗ = z2
(39)
= z3
(40)
and z1 + e2 x5 + e3 sin x2 e1 cos x1
(42)
z1 + d2 x4 d1
(43)
u2 − u1 =
where, but the reference input for role angle is x∗1 . In the following, a brief proof of control system stability is shown. Let V be Lyapunov function, T 1 T 1 ˆ ˆ −θ θ(t) − θ Γ−1 θ(t) σ σ+ 2 2 σ1 1 σ1 σ2 σ3 σ2 = 2 σ3 1ˆ ˆ + θ1 − θ1 θ2 − θ2 · · · θˆ7 − θ7 Γ−1 2 ˆ θ1 − θ1 θˆ2 − θ2 ... θˆ7 − θ7
V = (32)
(33)
(34)
where y˙ r (t) is bounded.
In the following, the explanation on the possibility for control of the underactuated system is shown. That is, to realize the control u(t) by u1 and u2 is explained. From (12) and (15), define that Fr = −d2 x4 + d1 (u2 − u1 ) Fp = −e2 x5 − e3 sin x2 + e1 (u1 + u2 ) cos x1(35) Fy = −f2 x6 + f1 (u1 + u2 ) sin x1 Then, we have that
(37)
where, z2 and z3 are given in (15) and (21). From (35), (39) and (40), we obtain the desired role angle as
z3 + f2 x6 ∗ −1 e1 (41) x1 = tan f1 z2 + e2 x5 + e3 sin x2
(30)
Controller for set-point variation ur (t) is considered for a set point varying as follows. ur (t) =
p¨ = Fp
Fy∗
u1 + u2 =
Controller for indefinite part uncertainty The control input to the unknown model is designed as uη (t) = −η x(t), t χ x(t), t
(36)
Now, we design the desired Fp and Fy as follows.
Because θ is unknown, it is calculated using online identification, where, the estimate of θ is set to:
r¨ = Fr
where, Γ is positive define, and γ1 · · · 0 γ2 Γ= ··· 0 γ7
(44)
Then, we have that V =
7 1 2 1 ˆ 1 2 σ1 + σ22 + σ32 + θn − θn (45) 2 2 n=1 an
It is obvious that V (x) > 0 From (23),
(46)
˙ σ˙ = S x(t) − y˙ r (t) = S f¯ x(t), t + g¯ x(t), t h x(t), u(t), t ¯ x(t), t u(t) − y˙ r (t) +¯ g x(t), t ∆h x(t), t + g = S f¯ x(t), t + S¯ g x(t), t h x(t), u(t), t +S¯ g x(t), t ∆h x(t), t +S¯ g x(t), t u(t) − S y˙ r (t) (47)
Further, from (21) we have
σ˙ = S f¯ x(t), t + S¯ g x(t), t h x(t), u(t), t +S¯ g x(t), t ∆h x(t), t + S¯ g x(t), t ueq (t) + ulf (t) + unl (t) + uad (t) +uη (t) + ur (t) − S y˙ r (t) = S f¯ x(t), t + S¯ g x(t), t h x(t), u(t), t +S¯ g x(t), t ∆h x(t), t + S¯ g x(t), t −1 − S¯ g x(t), t S f¯ x(t), t −1 − S¯ g x(t), t Kσ −1 ˆ x(t), u(t), t − S¯ g x(t), t κΛ(σ) − h −1 −η x(t), t χ x(t), t + S¯ g x(t), t S y˙ r (t) − S y˙ r (t) = −Kσ − κΛ(σ) − S¯ g x(t), t ˆ x(t), u(t), t − h x(t), u(t), t h −S¯ g x(t), t η x(t), t χ x(t), t +S¯ g x(t), t ∆h x(t), t = −Kσ − κΛ(σ) − S¯ g x(t), t ˆ −θ Φ x(t), u(t), t θ(t) −S¯ g x(t), t η x(t), t χ x(t), t +S¯ g x(t), t ∆h x(t), t (48)
As a result, the derivative of V is T ˆ − θ Γ−1 θ(t) ˆ˙ V˙ = σ T σ˙ + θ(t)
= −σT Kσ − σ T κΛ(σ) ˆ −θ −σT S¯ g x(t), t Φ x(t), u(t), t θ(t) −σT S¯ g x(t), t η x(t), t χ x(t), t T ˆ −θ +σT S¯ g x(t), t ∆h x(t), t + θ(t) T σ S g¯ x(t), t Φ x(t), u(t), t = −σT Kσ − σ T κΛ(σ) ˆ −θ −σT S¯ g x(t), t Φ x(t), u(t), t θ(t) −σT S¯ g x(t), t η x(t), t χ x(t), t +σT S¯ g x(t), t ∆h x(t), t
ˆ −θ g x(t), t Φ x(t), u(t), t θ(t) +σT S¯
= −σT Kσ − σ T κΛ(σ) −σT S¯ g x(t), t η x(t), t χ x(t), t +σT S¯ g x(t), t ∆h x(t), t
T T
T T
g x(t), t = −σ Kσ − σ κΛ(σ) − σ S¯
T η x(t), t + σ S¯ g x(t), t ∆h x(t), t (49) where the Schwarz inequality is used, namely, V˙ ≤ −σT Kσ − σ T κΛ(σ)
T T
η x(t), t − σ S g¯ x(t), t
T T
· ∆h x(t), t + σ S g¯ x(t), t
≤ −σT Kσ − σ T κΛ(σ)
T T
η x(t), t ¯ σ S g x(t), t −
T T
η x(t), t ¯ σ +
S g x(t), t
≤ −σT Kσ − σ T κΛ(σ) σ1 = − σ1 σ2 σ3 K σ2 σ3 sgn (σ1 ) − σ1 σ2 σ3 κ sgn (σ2 ) sgn (σ3 ) σ1 a0 0 = − σ1 σ2 σ3 0 b 0 σ2 σ3 0 0 c sgn (σ1 ) −d σ1 σ2 σ3 sgn (σ2 ) sgn (σ3 )
(50)
(51)
(52)
= −(aσ12 + bσ22 + cσ32 )
−d(σ1 + σ2 + σ3 )
(53)
a0 0 where K = 0 b 0 is positive define. Since 0 0 c ˙ V becomes negative definite for σ, the stability of the system with the equivalent control (21) is ensured. So far, it is not clear if the control using u1 and u2 can also guarantee the system stability, where, u1 and u2 are obtained by (42) and (43), From the relationship shown in (41), (42) and (43), based on the above proof, u1 , u2 and x∗1 are bounded provided that z2 + e2 x5 + e3 sin x2 = 0 and x1 = π2 . The, we can control the helicopter system by using u1 and u2 [3,4]. In the real control, for the obtained u1 and u2 , by using the relationship shown in (3) ∼ (6), right rotor and left rotor are controlled.
4. SIMULATION
s
Simulations by using the design method and the PD controller in [3] are performed. In the simulation, as an example, we change the desired reference value of yaw angle at 30[Sec], the desired control result is obtained (see Fig. 3). In this case, using PD controller can also obtain satisfied result, where we omitted it. However, when uncertainties exist, the proposed controller shows a desired control result. In the simulation, we change the physical parameters of the system (10) as the above mentioned uncertainties, where a1 = 16.4059, a2 = 1.02117, b1 = 12.4411, b2 = 1.02117, b3 = 26.1264, c1 = 12.4411, c2 = 1.83941. The other related values are shown in Tables 1 and 2. Ir Ip Iy La Lm ma /2
0.06[kg·m2 ] 0.07[kg·m2 ] 0.07[kg·m2 ] 0.195[m] 0.3[m] 0.15[kg]
0
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Fig. 3. System outputs by using the proposed controller (angles)
)
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(
Table 1. Parameters of the experimental system
Role angle 0 x∗1 x∗1
Initial value [rad] Reference 1[rad] Reference 2[rad]
Pitch angle 1.13 1.57 1.57
Yaw angle 0 0 0.62
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Table 2. Values of angles
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The design parameters of the PD controller are given in Table 3. Kp Kd
Role angle 1.8 4.4
Pitch angle 3.5 9.2
Yaw angle 1.5 28.8
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Fig. 4. System output using PD controller
Table 3. Parameters of the PD controller
Simulation results by changing the parameters are shown in Figs. 4 and 5. The desired control result is obtained by using the proposed controller. In the two simulation, the initial values are same, but the desired reference x∗1 is different based on the different control algorithm.
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5. CONCLUSION
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In this paper, combined attitude control scheme for an underactuated helicopter experimental system is designed. Comparing with the existed result by PD controller, the proposed method is effective for the existence of uncertainties. REFERENCES [1] C. Y. Su, and Y. Stepanenko, Adaptive variable set-point control of underactuated robots,
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Fig. 5. System output using the proposed controller IEEE Trans. on Automatic Control, Vol.44, No.11, 2090-2093, 1999. [2] T. Henmi, M. Deng, A. Inoue, N. Ueki, and Y. Hirashima, Swing-up control of a serial double inverted pendulum, Proc. of the 2004 American Control Conference, pp. 3992-3997, Boston, 2004.
[3] M. Deng, A. Inoue, T. Kishida, and N. Ueki, Modeling and control of an underactuated helicopter experimental system, Proc. of the 26th Chinese Control Conference (International), pp. 648-651, Zhangjiajie, 2007. [4] M. Saeki, Y. Wada, and J. Imura, Flight control design and experiment of a twin rotor helicopter model via 2 sep exact linearization, Proc. of the IEEE Conference on Control Applications, pp. 146-131, USA, August 1999. [5] A. Inoue, and M. Deng, Framework of combined adaptive and non-adaptive attitude control system for a helicopter system, International Journal of Auto. and Compu., Vol. 3, No. 3, pp. 229-234, 2006. [6] A. C. Bajpai, L. R. Mustoe, and D. Walker, Engineering Mathematics, John Wiley & Sons, 1982. [7] A. Inoue, M. Deng, T. Shimizu, and T. Harima, Experimental study on attitude control system design of a helicopter experimental system, Proc. of International Conference Control 2006(UKACC06), Glasgow, August 2006.
where ma : W eight of motor 2 g : Gravity acceleration r : Role angle p : P itch angle y : Y aw angle Ir : M oment on direction of role angle Ip : M oment on direction of pitch angle Iy : M oment on direction of yaw angle Dr : F riction coef. on direction of role angle Dp : F riction coef. on direction of pitch angle Dy : F riction coef. on direction of yaw angle La : Distance f rom role angle axis to motor Lm : Distance f rom pitch angle axis to motor
Appendix Based on Figs. 6 and 7, the dynamic equations on roll angle, pitch angle and yaw angle of the helicopter are shown as follows [6].
L
L
m
m
m
p
y
a
Fig. 6. Model on pitch angle m 2
a
L
L 2
m
m
a b L
1 (58) Ip = Iap + ma L2m + 2Ib + mb L2m 2 1 Iy = Iap + ma L2m + 2Ib + mb (Lm sin p)2 (59) 2
o
b
y
a
m 2
a
Fig. 7. Model on yaw angle Ir r¨ + Dr r˙ = f˜2
(54)
Ip p¨ + Dp p˙ + ma Lm g sin p = Lm f˜1 cos r
(55)
Iy y¨ + Dy y˙ = Lm f˜1 sin r
(56)
Ir = ma L2a
(57)
