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Australian Defense Force Academy. University of .... The force experienced by the helicopter is a sum of .... Bd. 0.71. Kr. 3.1. Zcol. 10.6. Lw. 0. Za. -3.0523. Krfb. -9.5. Nped. -103. Lb. 166 .... “Vision Guided Landing of an Unmanned Air Vehicle”,.
Proceedings of the 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April 2005

Autonomous Helicopter Landing on a Moving Platform Using a Tether Hemanshu Roy Pota† , Professor Matt Garrett‡ , Lecturer

So-Ryeok Oh, Graduate Student Kaustubh Pathak, Graduate Student Sunil K. Agrawal, Professor

School of Electrical Engineering† School of Aerospace and Mechanical Enginnering‡ Australian Defense Force Academy University of New South Wales Canberra ACT 2600 Australia h-pota,[email protected]

Mechanical System Lab Department of Mechanical Engineering University of Delaware Newark DE, 19716, USA oh,pathak,[email protected]

Abstract— In this paper, we address the design of an autopilot for autonomous landing of a helicopter on a rocking ship, due to rough sea. The deck is modeled to have a sinusoidal motion. The goal of the helicopter is to land on it during motion. In this work, we use a tether to help in target tracking. Based on the measurement of the angle between the cable and the helicopter/ship, a novel hierarchical two time-scale controller has been proposed to ensure landing of the helicopter on the ship. The system is demonstrated by computer simulation. Currently, work is under progress to implement the algorithm using an instrumented model of a helicopter using a tether.

I. I NTRODUCTION In recent years, considerable research has been performed on the design, development, and operation of autonomous helicopters. Helicopters can perform low-speed tracking maneuvers and operate under situations where runways are not available for take-off and landing, such as on the deck of a ship. A problem of importance for autonomous helicopters is the design of autopilots for landing on moving decks, subject to disturbances such as in rough sea. The control problem for landing of autonomous helicopters is challenging since the vehicle dynamics is highly nonlinear and coupled with unknown motion of the sea. Furthermore, helicopters are underactuated systems, i.e., have smaller number of control inputs than the number of generalized coordinates. The control of a helicopter in hover has been dealt with from different points of view: (i) Linear control design includes the use of adaptive controllers [1], LQG [2], H2 [3], H∞ [4], µ-synthesis ([5],[6]), and dynamic inversion methods [7]. These methods are based on linearized helicopter models around hover and trim conditions. (ii) Nonlinear control designs include sliding mode [8], nonlinear H∞ [9], neural network based controller [10], fuzzy control ([11],[12]), approximate input-output linearization [13], differential flatness [14], and backstepping [15].

0-7803-8914-X/05/$20.00 ©2005 IEEE.

Fig. 1. A tethered helicopter with an autopilot to land on the deck of a ship in rough sea.

The landing problems for autonomous vehicles is typically attempted using vision and global positioning systems. Vision guided landing uses the assumption that the target’s shape is known and the target is moving slowly ([16]-[18]). In this paper, we deal with the problem of landing of a helicopter on a ship’s deck using a tether. The problem of landing contains two key elements : (i) Detection of the motion of the deck using an instrumented tether with angle sensors at the two ends, namely the helicopter and the ship’s deck. (ii) Regulation of the location of the helicopter relative to the deck during its motion. One method to control underactuated systems is to partition the degrees-of-freedom and use some of these as

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control variables. This division is such that the resulting system is fully actuated. In practice, this scheme is realised using two-time scale control—fast dynamics for the controlled variables and slow for the independent variables. A helicopter with tether is subjected to tether moments as a function of helicopter position. This establishes a coupling between position and orientation variables which is normally absent in a free helicopter. In this paper, position degrees-of-freedom are used as controlled variables leaving the freedom to choose the orientation variables arbitrarily. This freedom is utilised in aligning the orientation of the helicopter with the tossing and turning ship-motion. As a result of this selection of controlled and independent set, the position variables become fast dynamics and orientation slow dynamics. This is different from what is normally done in helicopters where orientation is controlled to achieve a change in position. The results of this paper show that this scheme works for a tethered helicopter. Our approach gives an ability to this underactuated system to achieve an arbitrary orientation and align with the ship. Our control scheme uses a hierarchical architecture with two different time scales. The x, y translational dynamics is considered to be faster and the available inputs are used to closely track their set points, computed by a higher level slower time-scale controller. The set-points are computed such that the remaining degrees-of-freedom consisting of translation z and rotations φ, θ, ψ track the motion of the deck precisely. Even though the navy adopts this protocol for landing manned helicopters on ships in rough sea, in our knowledge, this is the first time that a tether is used for unmanned helicopter landing problem. The rest of the paper is organized as follows: Section II shows the nonlinear dynamic model used in this paper. This model was suggested to us by our collaborators at the Australian Defense Force Academy (ADFA). In Section III, the model of the helicopter is used to convey the concepts of the proposed control scheme. Section IV describes simulation results. These are followed by conclusions of the work. II. H ELICOPTER M ODEL A helicopter dynamic model is presented in this section. The model considers the fuselage of the helicopter as a rigid body attached to the main rotor and a tail rotor. R ∈ SO(3) is a rotation matrix between the body axes relative to a spatial coordinate frame. We parameterize R by ZY X Euler angles, with φ, θ, ψ along x, y, z axes respectively and define Θ = [φ θ ψ]T . The position of the center of mass of the helicopter is given by x = [x, y, z]T in the inertial frame and the forward, sideways, and downward velocities are given by u, v, w in the body frame, respectively. The angular rates for roll, pitch, and yaw in the body frame are ˙ q = θ, ˙ r = ψ. ˙ ˙ = [p, q, r]T , where p = φ, given by Θ The angular velocity is defined as ω. Other states are the main rotor lateral and longitudinal flapping angles a and

b. These are related to the longitudinal and lateral cyclic pitch as follows. a = δlon , b = δlat .

(1)

Here, we assume that the response of the rotor blades to rolling cyclic and pitching cyclic δlon , δlat is instantaneous [19]. The five control inputs are the main rotor lateral and longitudinal flapping angles a and b, collective pitch angle for the main rotor δcol , the tail rotor collective pitch angle δped , and the cable tension F . In this paper, the flexibility of the rotors and the fuselage, the dynamics of engine and actuators are ignored. Fig. 2 shows a schematic of a helocopter, which includes the rotary wing dynamics, force and moment generation process, and rigid body dynamics. In Fig. 2, TM and TT stand for the forces and torques generated by the main and tail rotor.

Fig. 2.

A schematic of Helicopter Dynamics.

A. Rigid Body Dynamics The equations of motion for a rigid body subject to an external wrench F b = [f b , τ b]T applied at the center of mass is given by Newton-Euler equations. These can be written as        ¨ mI 0 0 Rf b x + = , (2) ωb × Iωb 0 I ω˙ b τb where I is the inertia matrix. The kinematics is given by ˙ = Πωb , where Π is defined as Θ 

1 sinφ tanθ Π = 0 cosφ 0 sinφ secθ

 cosφ tanθ −sinφ  . cosφ secθ

(3)

Note that this mapping has singularities at θ = ±π/2. For the following discussions, we assume that the pitch angle of the helicopter does not reach these singularities.

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are functions mapping the control inputs to the forces and moments. Note that Mδped , Mabδcol are determined by the geometry of the helicopter.     0 Xu u Xa 0  , J =  0 Ya fu =  Yv v 0  , (8) Zw w + Zr r Za Zb Zcol 

 Ixx (Lu u + Lv v + Lw w) , Iyy (Mu u + Mv v + Mw w) fw =  Izz (Nv v + Np p + Nw w + Nr r + Nrf b rf b ) (9)   0 0 fg = RT  0  , Mabδcol =  Iyy Ma mg 0 

Ixx Lb 0 0

 0 0 , Izz Ncol (10)



Mδped

Fig. 3. Geometric Modeling of the Helicopter: the helicopter frame H, the ship frame S, and the inertial frame N .

˙ = Πωb into Eq. (2), one can rewrite On substituting Θ these as        ¨ mI 0 x Rf b 0 (4) ¨ + ΓΘ = τΘ 0 I Θ where ΓΘ and τΘ are defined as ˙ + ΠI −1 Π−1Θ ˙ ×Θ ˙ ΓΘ = −Π˙ Π−1 Θ −1 b τΘ = ΠI τ

(5)

The superscript b represents a vector in the body frame.

 0 , 0 = Izz Nped

The cable force and moment are given by   cos(α) cos(β) fcable = Fˆl(α, β) = F  cos(α) sin(β)  , sin(α) mcable = rHC × fcable ,

(11)

(12)

(13)

where α and β are the elevation and azimuth angle measured from a cable angle sensor attached to the helicopter (see Fig. 3), rHC is a position vector from the center of mass of the helicopter to the cable attachment point, expressed in the local frame. ˆl(α, β) is a unit vector of the cable, shown in Fig. 3. The additional parameters are listed in Table I.

B. Force and Moment Generation

C. Measuring the ship motion

The force experienced by the helicopter is a sum of forces generated by the main and tail rotors, aerodynamic forces from the fuselage, and gravitational force. The torque is composed of the torques generated by the main rotor, tail rotor, and the fuselage. In hover or forward flight with slow velocity, we can ignore the drag contributed from the fuselage. So, resultant force f b and moment τ b can be written as [1]   a f b = fu + J  b  + fcable + fg (6) δcol

Using the loop closure, as shown in Fig. 3, the ship translation motion xS satisfies the following equation:



 a τ b = fw + Mabδcol  b  + Mδped δped + mcable (7) δcol where fu and fw are aerodyanmics terms and fg is the gravitational force. The remaining terms, J, Mδped , Mabδcol

x + R(Θ)rHC + l R(Θ)ˆl(α, β) − xS = 0.

(14)

The orientations Θs of the ship is calculated from the following relation: R(ΘS )ˆl(αS , βS ) = R(Θ)ˆl(α, β),

(15)

where αS and βS are the elevation and azimuth angles measured from the cable angle sensor attached to the ship. ΘS stands for the Euler angles of the ship. From Eqs. (14), the ship position can be computed if the following variables are measured: helicopter position via GPS, its orientation using attitude sensors, the cable length l, and the cable angles α, β with respect to the helicopter. In order to know the partial orientation of the ship, we need additional cable angle sensors on the ship’s deck αS , βS . This will provide two angles that characterize the orientation of the ship.

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TABLE I

III. T WO T IME -S CALE C ONTROL

S YSTEM PARAMETERS IN MKS UNIT

We now proceed to design a two time-scale controller for this system. The proposed controller design follows several steps as outlined below: Step1: Grouping The slow dynamics variables are selected to be z, φ, θ, ψ, while x, y constitute the fast dynamics variables. This grouping is chosen to keep the helicopter safe from collision with the ship as long as the slow dynamics variables track well the corresponding ship motion zs , φs, θs , ψs. The slow Dynamics (SD) runs at a sampling time-period ∆Th , an order of magnitude slower than the fast dynamics (FD) which has sampling period ∆TL . Therefore, SD assumes that the reference signals generated by its controller for FD variables, at any sample time, will already have been achieved due to the faster response of FD variables. Step2: Design of Lower Level Control x, y are controlled using the three inputs a, b, δcol . Since there are three inputs, we add z to the control variables of LLC. The structure of LLC is selected as 

 a  b  = J −1 ( −fu − fcable − RT g + mRT v ) δcol On substituting Eq. (16) into Eq. (2), we get   x ¨d − kdx (x˙ − x˙ d ) − kpx (x − xd ) x = v =  y¨d − kdy (y˙ − y˙d ) − kpy (y − yd )  ¨ z¨s − kdz (z˙ − z˙s ) − kpz (z − zs )

(16)

(17)

Note that xd , yd are generated by HLC, while zs (t) is the ship motion along z axis. Step3: Design of Higher Level Control Due to the fast dynamics of FD variables achieved by LLC, HLC expects the following conditions to be satisfied   x = xd , x˙ = 0  y = yd , y˙ = 0  (18)  . v1 = v2 = 0 z = zs (t) Let us define τ b as a virtual input for HLC. The feedback linearizing control law with respect to the virtual input τ b and outputs Θ can be obtained as τ b = IΠ−1(ΓΘ + w),

(19)

which leads to  φ¨s − kdφ (φ˙ − φ˙ s) − kpφ (φ − φs ) ¨ = w =  θ¨s − kdθ (θ˙ − θ˙s ) − kpθ (θ − θs )  . (20) Θ ψ¨s − kdψ (ψ˙ − ψ˙ s ) − kpψ (ψ − ψs ) 

We can equate the control law in Eq. (19) with Eq. (7), which provides three algebraic equations in terms of x, y, F, δped.

Par. m Xu Xa Yv Yb Lu Lv Lw Lb Mu Mv

Val. 69 -0.06 -9.8 -0.03 9.8 0.023 -0.7 0 166 1.3 -1.2

Par. Ixx Ma Mw Ab Ac Ba Bd Za Zb Zw Zr

Val. 5.98 82.3 0 -0.189 0.644 0.368 0.71 -3.0523 -15.063 -1.3456 0.2222

Par. Iyy Nv Np Nw Nr Nrf b Kr Krf b τf τs Yped

Val. 9.99 0 -0.01 1.1 -2.9 -22 3.1 -9.5 0.04 0.34 0

Par. Izz Mcol Alat Alon Blat Blon Zcol Nped Ncol Clon Dlat

Val. 10 0 0.01 0.05 0.07 0 10.6 -103 4.5 0.13 0.13

fw +Mabδcol uf ast(x, y, F )+Mδped δped +Mcable (x, y, F ) = τ b, (21) where   a uf ast =  b  = J −1 ( −fu − fcable − RT g + mRT v ) , δcol (22)  Note that fcable = F ˆl = Fl RT (Θ)(x+ − xs ) − rHC from Eq. (14), where x+ = [x, y, z]T . In other words, uf ast and Mcable become a function of xd , yd , F . Although we have 3 equations and 3 control inputs, there may not be any feasible solution δped , x, y to satisfy Eq. (21), since the pseudo inputs x, y appear weakly through the unit cable vector ˆl(x, y). Hence, we design the cable tension F as an additional control input. IV. S IMULATION R ESULTS We evaluate the effectiveness of the two time-scale controllers proposed in Sections III. For the simulations, the inertial, geometric and aerodynamic parameters are listed in Table I. We consider a specific move of the ship, xs = [0, 0, 0.2 sin(π t), 5o sin(π t), 5o sin(π t), 0]T . All signals are in MKS unit. The piece-wise constant reference signals xd , yd are generated by HLC. The helicopter’s error for the ship’s deck ([xs, ys ] = [0, 0]) is bounded by ±0.4m and ±0.2m, respectively. LLC achieves a fast convergence every sample time 0.2 [sec] as shown in Fig. 4. The input trajectories are shown in Fig. 5, which are kept in reasonable bound throughout the simulation. V. C ONCLUSIONS This paper presented an autopilot for a tethered helicopter to solve the problem of autonomous landing on a moving ship platform. The problem was presented as a nonlinear tracking control problem for an underactuated system with a two time-scale controller. Through simulations, the proposed approach was demonstrated to be effective for the problem. In future, the tracking controller will be implemented and tested on a model helicopter developed by Australian Defense Force Academy.

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Fig. 4.

The state trajectories : actual one(dotted), desired one(solid)

Fig. 5.

ACKNOWLEDGMENT The authors appreciate financial supports of NSF Award No. IIS-0117733.

[9]

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[10]

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The input trajectories

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