ISSN 02805316ISRN LUTFD2/TFRT--7596--SE

Modelling of the ETH HelicopterLaboratory Process

Magnus Gfvert

Department of Automatic ControlLund Institute of Technology

November 2001

1. Introduction

This report contains derivation of the dynamics of the ETH helicopter lab-oratory process, see Figure 1, using the Euler-Lagrange approach. The pro-cess is designed at the Automatic Control Laboratory at ETH in Zrich,see Mansour and Schaufelberger (1986) and Schaufelberger (1990). A de-scription of the setup is found in Morari, M., W. Schaufelberger and A.Glattfelder (1995). The process is of MIMO type with nonlinear dynam-ics, and static input nonlinearities, as will be shown below. The presentmodel is derived with the purpose of accurate simulation of the helicopterprocess. It may also prove helpful for nonlinear controller design. Identi-fication of a linear model, and a linear controller design is presented inkesson, Gustafson and Johansson (1996).

Figure 1 The ETH helicopter laboratory process.

A schematic picture of the process is found in Figure 2. The helicopterconsists of a vertical axle (A), on which a lever arm (L) is connected by acylindric joint. The helicopter has two degrees of freedom: the rotation ofthe vertical axle (angle ) with respect to the fixed ground, and the pivotingof the lever arm (angle ) with respect to the vertical axle. Two rotors aremounted on the lever arm: R1 and R2, with the resultant aerodynamicforces giving rise to moments in the and directions respectively. Thevoltages u1 and u2 to the rotor motors are the inputs to the process. Aweight is mounted at an adjustable position on the lever arm towardsrotor R2.

2. Kinematics

It is assumed that the mass distribution on the lever arm is restricted to astraight line between the rotors, a distance h from the pivot point. Denoteby O an origo on this line. Let [rx(R), ry(R), rz(R)] denote a point P onthe lever arm parameterized in the distance R from O , expressed in anearth fixed reference system with origo in O and oriented with ez along the

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PSfrag replacements

ex

eyezOO

O

R1

R2 h

l1

l2

L

A

Figure 2 Helicopter process configuration.

center axle. Then

rx(R) = R cos cos h sin cosry(R) = R cos sin h sin sinrz(R) = R sin + h cos

(1)

The corresponding velocities are obtained from differentiation of (1) withrespect to time:

vx(R) = R sin cos R cos sin h cos cos + h sin sinvy(R) = R sin sin + R cos cos h cos sin h sin cosvz(R) = R cos h sin

(2)

The squared magnitude of the velocity of P is then given by v2(R) = v2x(R)+v2y(R) + v2z(R):

v2(R) = R2 (2 + cos2 2)+ h2 (2 + sin2 2) 2hR cos sin 2 (3)3. Energy expressions

The kinetic and potential energies are derived from

T = 12

v2(R) dm(R) (4)

V = k

rz(R) dm(R) (5)

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where k is the acceleration of gravity. With (3) inserted this yields for leverarm

TL = 12[2 + cos2 2] JL h cos sin2mlc + 12h2 [2 + sin2 2]m

(6a)VL = mk sin lc +mkh cos (6b)

and for the axle

TA = 12 JA2 (6c)

VA = 0 (6d)

where the inertia of the lever arm JL*= R2 dm(R), the center of gravity

of the lever arm lc*= 1m

R dm(R), the lever arm mass m *= dm(R),

and the center axle inertia JA have been introduced. The total kinetic andpotential energies are

T = TL + TA (7)V = VL + VA (8)

4. Equations of motion

Forming the Lagrangian

L = T V (9)

the equations of motion are given by

ddt

(V LV) V LV =

ddt

(V LV

) V LV =

(10)

Inserting (6) in (10) gives[2 cos sin + cos2 ] JL+2h [(sin2 cos2 ) cos sin]mlc+ h2 [2 sin cos + sin2 ]m + JA = (11a)[

+ cos sin2] JL + [h ( sin2 + cos2 ) 2 + k cos]mlc+ [h2 h2 sin cos2 kh sin]m = (11b)

These equations may be expressed on matrix form as

D( , )[

]+ C( , , , )

[

]+ k( , ) = (12)

The fundamental property N( , , , ) = D( , )2C( , , , ) is fulfilledwith the skew symmetric matrix N( , , , ). The matrices D(), C(), k(),and N() are defined as in Figure 3.

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D(

,)

* =[ cos

2J

L

2hco

ssi

nm

l c+

h2

sin2

m+

JA

0

0J

L+

h2m

]

C(

,,

,)

* =[ c

os

sin

J

L+

h( sin2

cos2

) m

l c+

h2

sin

co

sm

co

ssi

n

JL+

h( sin2

cos2

) m

l c+

h2

sin

co

sm

cos

sin

J

L+

h( si

n2+

cos2

) m

l c

h2

sin

co

sm

0

]

k(,

)* =[

o

kcos

ml c

mkh

sin

]

N(

,,

,)

* =[

02

cos

sin

J

L

2h( sin2

cos2

) m

l c

2h2

sin

co

sm

2co

ssi

n

JL

2h( si

n2

+

cos2

) m

l c+

2h2

sin

co

sm

0

]

Figure 3 Matrices for Equation (12).

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5. Rotors

The rotors are driven by DC-motors without current-control. The motoroperation is described by

Laddt

ia = Raia k + u (13a)

Jddt

= Td TL (13b)

where Ra and La are the rotor-circuit resistance and inductance respec-tively, and k the motor constant. The driving moment is described by

Td = kia (14)and the motor load is described by

TL = D e e (15)where D is the aerodynamic torque coefficient, according to propeller BladeElement Theory, see e.g. Weick (1926) or, if preferred, a modern textbookon theory of flight. Combining (1315) in steady-state yields

u0 = RaDk 0e0e+k0 k0 (16)

The approximation is found valid by examining experimental results inMorari et al. (1995). The second order motor dynamics may then be ap-proximated by first order dynamics as

R1 :ddt

1 = 1T1 1 +1

k1T1u1 (17a)

R2 :ddt

2 = 1T2 2 +1

k2T2u2 (17b)

where T1 and T2 are the time constants of the motors.The resulting aerodynamic drag forces are given by F1 = C11e1e and

F2 = C22e2e with C1 and C2 being aerodynamics drag coeffients (Weick1926). Each rotor affects the helicopter with a moment resulting from theaerodynamic force, and with a moment that is the reaction moment fromthe driving torque of the rotor motor. (The sign of the reaction momentsdepend on the configuration of the rotor blades.) Gyroscopic effects of therotors are assumed to be small and are neglected. (Any gyroscopic mo-ments resulting from the rotor rotations would mainly include componentsperpendicular to the and rotation axis.) For R1:

1, = D11e1ecos (18a)1, = l1C11e1e (18b)

and for R2:

2, = l2 cos C22e2e (19a)2, = D22e2e (19b)

These moments are combined to form

= 1, + 2, (20a) = 1, +2, (20b)

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6. Simulation model

The complete set of equations describing the helicopter process is given by(11) together with (17)(20). Rewriting these on state-space form gives

ddt

= [cos2 JL 2h cos sin mlc + h2 sin2 m + JA]1

[2 cos sin JL 2h

(sin2 cos2 ) mlc 2h2 sin cos m

+D11e1ecos + l2 cos C22e2e] (21a)ddt

= (21b)ddt

= [JL + h2m]1 [ cos sin2JL h ( sin2 + cos2 ) 2mlck cos mlc + h2 sin cos2m + mkh sin + l1C11e1e+D22e2e

](21c)

ddt

= (21d)ddt

1 = 1T1 1 +1

k1T1u1 (21e)

ddt

2 = 1T2 2 +1

k2T2u2 (21f)

7. Equilibrium points

Equations (11), (17) (20)may be solved for stationary points (0, 0, u1,0, u2,0)by setting = = = = 1 = 2 0:

0 = D11,0e1,0ecos + l2 cos C22,0e2,0e (22a)mk (lc cos0 h sin0) = l1C11,0e1,0e+D22,0e2,0e (22b)

For the unforced system with = 0 then = 0, = 0 areequilibrium points, with arbitrary 0 and tan0 = lc/h. Since the tan1function is periodic there are infinitely many equilibrium points 0. In par-ticular there is one 0 [pi/2, pi/2), and one 0 [pi ,pi/2)

[pi/2, pi ).Stability for the lever arm dynamics around (0, 0) may be investigatedby regarding the resulting simplification and Taylor expansion of (11b):

= mkJL

[h sin lc cos ] = mkJL[h cos0 + lc sin0] +O ( 2)

= mk cos0hJL

(h2 + l2c

) +O ( 2) (23)

with *= 0. Thus the stationary point (0, 0) with 0 [pi/2, pi/2)is stable for h < 0, and unstable for h > 0, and the stationary point with0 [pi ,pi/2)

[pi/2, pi ) is unstable for h < 0, and stable for h > 0. Theresulting dynamics is a pendulum equation.

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8. Parameters

Parameters for a real helicopter process are presented in Morari et al.(1995). Geometric and interial parameters are presented directly. Motorand rotor properties are presented in graphs resulting from experiments.The corresponding parameters presented here are computed from the graphs.

Description Parameter Value Unit

Arm length to R1 l1 0.1995 [m]Arm length to R2 l2 0.1743 [m]Mass of lever arm bar ml 0.280 [kg]Pivot height h 0.0298 [m]Mass of weight mw 0.158 [kg]Distance to weight (nominal1) lw 0.090 [m]Mass of rotor R1 m1 0.3792 [kg]Mass of rotor R2 m2 0.1739 [kg]Time constant for rotor R1 T1 1.1 [s]Time constant for rotor R2 T2 0.33 [s]Motor constant for rotor R1 k1 1.00 102 [Vs/rad]Motor constant for rotor R2 k2 1.39 102 [Vs/rad]Aerodynamic drag for rotor R1 C1 2.50 105 [Ns2/rad2]Aerodynamic drag for rotor R2 C2 1.58 106 [Ns2/rad2]Aerodynamic torque for rotor R1 D1 2.90 107 [Nms2/rad2]Aerodynamic torque for rotor R2 D2 1.76 107 [Nms2/rad2]

Table 1 Helicopter model parameters.

The total mass of the lever arm is

m = ml +m1 +m2 +mw (24)

The moment of inertia for the lever arm is the sum of the moment of inertiafor the solid lever bar and for the point masses of the rotors and the weight:

JL =ml3

l31 + l32l1 + l2 +m1l

21 +m2l22 +mwl2w (25)

The moment of inertia for the vertical axle may be neglected:

JA 0 (26)

The center of gravity is

lc = ml (l1 l2) + m1l1 m2l2 mwlwm (27)1May be varied in the range 0.0705 0.119 [m].

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9. Bibliography

kesson, M. and E. Gustafson and K. H. Johansson (1996): Control Designfor a Helicopter Lab Process, Preprints 13th World Congress of IFAC,San Francisco, California.

Mansour, M. and W. Schaufelberger (1989): Software and laboratory ex-periments using coomputers in control education. IEEE Control Sys-tems Magazine, 9:3, pp. 1924.

Morari, M., W. Schaufelberger and A. Glattfelder (1995): Klassische Regelungeines Helikoptermodells, Fachpraktikumversuch A50, Institut for Au-tomatik, ETH, Zrich, Rev. 6, Nov. 1996.

Schaufelberger, W. (1990): Educating future control engineers. Proc. ofIFAC World Congress. Tallin, Estonia, pp. 3951.

Weick, F. E. (1926): Propeller design I: practical application of the bladeelement theory, NACA TN 235, Langley Memorial Aeronautical Labo-ratory, Washington.

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