tras_man

Inteco Krakow

Two Rotor Aero-dynamical System

Message or faulty action	Solution
Board not detected	Check mounting of the board. Check if driver is installed
Angles measurements failed	Check the Enc socket and wiring
Propellers do not rotate	Check M socket, Mains and ON switch
Velocities are not measured	Check T socket and wiring

3. TRAS Control Window

The user has a rapid access to all basic functions of the TRAS control system from TRAS Control Window. It includes tests, drivers, models and application examples.

TRAS Control Window shown in Fig. 2.1 contains four groups of the menu items:

• Tools -Basic Test, Manual Set-up, Reset Encoders and Stop Experiment,
• Drivers -RTWT Device Driver,
• Simulation Models: Pitch , Azimuth and 2-DOF model,
• Identification -Steady State Characteristics,
• Demo Controllers – PID azimuth, PID pitch and cross-coupled PID controller

3.2 Basic test

The Basic Test tool was described in the previous section.

3.3 TRAS Manual Setup

The TRAS Manual Setup program gives access to the basic parameters of the laboratory Two Rotor Aerodynamical System setup. The most important data transferred from the RT-DAC4/PCI board and the measurements of the TRAS may be shown. Moreover, the control signals may be set.

The application contains four frames (see Fig. 3.1):

• RT-DAC4/PCI board,
• Encoders,
• Control and
• Tacho.

All the data accessible from the TRAS Manual Setup program are updated 10 times per second.

RT-DAC4/PCI board frame

The RT-DAC4/PCI board frame presents the main parameters of the PCI board.

No of detected boards

Reads the number of detected RT-DAC4/PCI boards. If the number is equal to zero it means that the software has detected none of the RT-DAC4/PCI board. When more then one board is detected the Board list must be used to select the board that communicates with the program.

3.3.1.1 Board

Contains the list applied to the selected board currently used by the program. The list contains a single entry for each RT-DAC4/PCI board installed in the computer. A new selection executed at the list automatically changes values of the remaining parameters.

Bus number

Displays the number of the PCI bus where the current RT-DAC4/PCI board is plugged-in. If more then one board is used this parameter may be useful to distinguish the boards.

Slot number

The number of the PCI slot in which the current RT-DAC4/PCI board is plugged-in. If more then one board is used this parameter may be useful to distinguish the boards.

Base address

The base address of the current RT-DAC4/PCI board. The RT-DAC4/PCI board occupies 256 bytes of the I/O address space of the microprocessor. The base address is equal to the beginning of the occupied I/O range. The I/O space is assigned to the board by the computer operating system and may be different for various computers. The base address is given in the decimal and hexadecimal forms.

Logic version

The number of the configuration logic of the on-board FPGA chip. A logic version corresponds to the configuration of the RT-DAC4/PCI boards defined by this logic.

Application

The name of the application the board is dedicated for. The name contains four characters.

I/O driver status

The status of the driver that allows the access to the I/O address space of the microprocessor. The status has to be OK string. In the other case the driver HAS TO BE INSTALLED.

Encoders frame

The state of the encoder channels is given in the Encoder frame. The encoders are applied to measure the azimuth and pitch angles.

Azimuth, Pitch

The values of the encoder counters, the angles expressed in radians and the encoder reset flags are listed in the Azimuth and Pitch rows.

Value

The values of the encoder counters are given in the respective columns. The values are 16bit integer numbers. When an encoder remains in the reset state the corresponding value is equal to zero.

Angle [rad]

The angular positions of the encoders expressed in radians are given in the respective columns. If the encoder remains in the reset state the corresponding angle is equal to zero.

Reset

When the checkbox is selected the corresponding encoder remains in the reset state. The checkbox has to be unchecked to allow the encoder to count the position.

Control frame

The Control frame allows to change the control signals. DC drives are controlled by PWM signals.

Azimuth and Pitch edit fields and sliders

The control edit boxes and the sliders are applied to set a new control values of the corresponding DC drives. The control value may vary from –1.0 to 1.0.

STOP

The pushbutton is applied to switch off the control signals. If it is pressed then both the azimuth and pitch control values are set to zero.

Azimuth and Pitch PWM prescaler

The divider of the PWM reference signal is given. The frequency of the corresponding PWM control is equal to:

FPWM = 40000/1023/(1+PWMPrescaler) [kHz]

Azimuth and Pitch Thermal flag / status

The thermal flags and the thermal statuses of the power amplifiers. If the thermal status box is checked the corresponding power interface is overheated. If the power interface is overheated and the corresponding thermal flag is set the RT-DAC4/PCI board switches off the PWM control signal corresponded to the overheated power amplifier.

Tacho frame

The Tacho frame displays two measured analog signals generated by the tachogenerators. The voltages and the corresponding velocities of the propellers are displayed.

Azimuth and Pitch Voltage [V]

Displays the voltage at the outputs of the tacho generators.

Azimuth and Pitch Velocity [RPM]

Displays the velocity of the propellers. The velocities are calculated based on the corresponding voltages and are given in RPM.

3.4 RTWT Device Driver

The driver is a software go-between for the real-time MATLAB environment and the RTDAC4/PCI I/O board. The control and measurements are transerred. Click the TRAS Device Driver button and the driver window opens (Fig. 3.2).

When one wants to build his own application one can copy this driver to a new model. The Reset Encoder input can be used in the real-time mode only.

Do not do any changes inside the original driver. They can be introduced only inside its copy!!! Make a copy of the installation CD

The device driver has two inputs: control u(t) ⊂ [−1+1] and signal Reset. If signal Reset changes to one the encoders are reset and do not work. If signal Reset is equal to zero encoders normally work. It is important that Reset switch works only if the real-time code is executed. It means that changing the state of the switch, when real time mode is not running, is not effective. However when switching occurs while the real time is running, the encoder resets and starts measure when the switch returns to the zero (normal) position.

The mask of this block (shown Fig. 3.3) contains the base address of the RT-DAC4/PCI board (detected automatically) and sample time with the default value set to 0.002 s. If one needs to change the default sampling time he has to use the same mask of the device driver.

The details of the device driver are depicted in Fig. 3.4. The driver uses functions which communicates directly with a logic stored at the RT-DAC4/PCI board.

Parameters Measurements

3.5 Simulation Models

There are three simulation models available for the TRAS system. The first one is a 1-DOF (degree of freedom) azimuth model. This model simulate behaviour of the system in the horizontal plane only. Click the 1-DOF Azimuth Simulation Model button to open the model shown in Fig. 3.5. Next, click the subsystem block to see details of the model.

TRAS azimuth model

Scope

Abs

A 1-DOF pitch is the second model. It describes behaviour of the system in the vertical plane. Click the 1-DOF Pitch Simulation Model button and click the subsystem block to see the 1-DOF pitch model and its interior (see Fig. 3.6)

TRAS pitch

Scope

The third one is the complete simulation model. It describes movements in both planes with an interaction between the pitch and azimuth axes. Click the 2-DOF Simulation Model button and the subsystem block to see the model and its interior (see Fig. 3.7)

TRAS 2_dof model

DCP-azimuth

4. Model and parameters

Modern methods of design and adaptation of real time controllers require high quality mathematical models of the system. For high order, non-linear cross-coupled systems classical modelling methods (based on Lagrange equations ) are often very complicated. That is why a simpler approach is often used, which is based on block diagram representation of the system which is very suitable for the SIMULINK environment. The relations between the block diagram and mathematical model of the TRAS are explained in sections 4.2 – 4.5.

Fig. 4.1. shows an aero-dynamical system considered in this manual. At both ends of a beam, joined to its base with an articulation, there are two propellers driven by DC-motors. The articulated joint allows the beam to rotate in such a way that its ends move on spherical surfaces. There is a counter-weight fixed to the beam and it determines a stable equilibrium position. The system is balanced in such a way, that when the motors are switched off, the main rotor end of beam is lowered. The controls of the system are the motor supply voltages.

The measured signals are: position of the beam in the space that is two position angles and angular velocities of the rotors. Angular velocities of the beam are software reconstructed by differentiating and filtering measured position angles of the beam.

The block diagram of the TRAS model is shown in Fig. 4.2. The control voltages U_h and U_v are inputs to the DC-motors which drive the rotors (PWM mode). A rotation of the propeller generates an angular momentum which, according to the law of

conservation of angular momentum, must be compensated by the remaining body of the TRAS beam. This results in the interaction between two transfer functions, represented by the moment of inertia of the motors with propellers khv and kvh (see Fig. 4.2). This interaction directly influences the velocities of the beam in both planes. The forces F_hand

F multiplied by the arm lengths l (α ) and l are equal to the torques acting on the arm.

vhvv

Uh

Uv

The following notation is used in Fig. 4.2: α_h -horizontal position (azimuth position) of TRAS beam [ rad]; Ω_h -angular velocity (azimuth velocity) of TRAS beam [rad/s]; -horizontal DC-motor PWM control input ;

^Uh

ω_h -rotational speed of tail rotor [rad/s] -non-linear function

�

=H (U ,t) [rad/s] ;

h hh

F_h-aerodynamic force from tail rotor -non-linear function Fh=Fh(wh) [N]; l_h -effective arm of aerodynamic force from tail rotor lh=lh(av)[m]; J_h -non-linear function of moment of inertia with respect to vertical axis,

Jh = Jh(av) [kg m²]; M_h -horizontal turning torque [ Nm]; K_h -horizontal angular momentum [N m s]; f_h -moment of friction force in vertical ax-[N m]; α_v -vertical position (pitch position) of TRAS beam [ rad]; Ω_v -angular velocity (pitch velocity) of TRAS beam [rad/s]; U_v -vertical DC-motor PWM voltage control input;

�

ω -rotational speed of main rotor -non-linear function =H (U ,t) [rad/s];

v vvv

�

F_v-aerodynamic force from main rotor -non-linear function Fv = Fv(_v) [N]; l_v -arm of aerodynamic force from main rotor [m]; J_v -moment of inertia with respect to horizontal ax-[kg m²];

M_v -vertical turning moment [ Nm]; -vertical angular momentum [Nms];

^Kv f_v -moment of friction force in horizontal ax-[Nm];

�

R -vertical returning moment R = f + f = R(�,) [Nm];

v hcf g hvh

J_hv -vertical angular momentum from tail rotor [Nms]; J_vh -horizontal angular momentum from main rotor [Nms]; H -differential equation ω= H (U ,t);

v vvv

H_h -differential equation ω_h = H_h (U_h ,t) ;

G -aerodynamical dumping torque from main rotor G (ω ,Ω) ;

vvvv G_h-aerodynamical dumping torque from tail rotor G_h(ω_h ,Ω_h ).

Controlling the system consists in stabilising the TRAS beam in an arbitrary (within practical limits) desired position (pith and azimuth) or making it track a desired trajectory. Both goals may be achieved by means of appropriately chosen controllers. The user can select between two types of PID controllers and a state feedback controller (see section 6).

4.2 Non-linear model

The mathematical model is developed with some simplifying assumptions. First, it is assumed that the dynamics of the propeller subsystem can be described by first order differential equations. Further, it is assumed that friction in the system is of the viscous type. It is assumed also that the propeller-air subsystem could be described in accord with postulates of the flow theory.

The above assumptions allow us to define the problem clearly. First, consider the rotation of the beam in the vertical plane i.e. around the horizontal axis. Having in mind that the driving torque’s are produced by the propellers the rotation can be described in principle as the motion of a pendulum. From the second dynamics law of Newton we obtain:

2�

d_v

M = J (1)

vv 2

dt where: M_v -total moment of forces in the vertical plane, J_v -the sum of moments of inertia relative to the horizontal axis, α _v -the pitch angle of the beam.

Then:

6 8

M_v = _ƒ M _vi , J_v = _ƒ Ji i=1 i=1

To determine the moments of forces applied to the beam and making it rotate around the horizontal axis consider the situation shown in Fig. 4.3 .

return torque which determines the equilibrium position of the system.

M

v1

=

g

ÀÃÕ

ÿŸ⁄

’

’

»

m m

b

»

^lt

−

−

cb

sin

�

v

l

ml

^lb cd

+ _tr +

++

�

+

÷◊

÷◊

cos

m m

ts

m m

mr m v

…

…

2

2

2

M_v1 = g[(A-B ) cos _�v − C sin _�v ] where:

’

A

l

+

tr ⁺

=

÷◊

m m

ts

t

2

m

’

B

l

+ m +

=

÷◊

m

mr m

2

m

b

2

»

…

C

l_b + m_cd l

=

cb

where: M_v1 -is the return torque corresponding to the forces of gravity, m_mr-is the mass of the main DC-motor with main rotor, m_m -is the mass of the main part of the beam, m_tr-is the mass of the tail motor with tail rotor,m_t-is the mass of the tail part of the beam,

-is the mass of the counter-weight,

^mcb m_b -is the mass of the counter-weight beam, m_ms -is the mass of the main shield, m_ts-is the mass of the tail shield, l_m -is the length of the main part of the beam, l_t-is the length of the tail part of the beam,

¤

‹›

ÿ

≈Δ «

m

tm _Ÿ⁄

^ms ÿ

≈Δ «≈^m Δ«≈Δ«

t

m ms

^Ÿ_⁄

TRAS User’s Manual -21

l_b-is the length of the counter-weight beam, l_cb-is the distance between the counter-weight and the joint. g -is the gravitational acceleration,

M = lF (^�)

v2 mv m

M_v2 -is the moment of the propulsive force produced by the main rotor, ω_v -is angular velocity of the main rotor, F_v (^� _v ) -denotes the dependence of the propulsive force on the angular velocity of

the rotor; it should be measured experimentally (see section 4.5).

�

ÿŸ⁄

≈Δ «

or in the compact form:

2

M =−^�(A+B+C ) sin �cos �

v3 h vv

M_v3-is the moment of centrifugal forces corresponding to the motion of the beam around the vertical axis,

�d�

and: = ^h (2)

^h dt

� _h -is the angular velocity of the beam around the vertical axis, α_h -is the azimuth angle of the beam.

�

M =− f

v4 vv

≈Δ «

’

’

»

m m

t

2

m

2

b

=−

sin

M

l l ml

cb cb

l

^m + m +

+

++

+

+

cos

��

÷◊

÷◊

m m m

3

h btr ts t

v

mr ms m v

…

2

2

M_v4 -is the moment of friction depending on the angular velocity of the beam around the horizontal axis.

�d�

where: _v = ^v (3)

dt

�

_v -is the angular velocity around the horizontal axis,

f_v -is a constant

M -is the cross moment from U ; M = Uk

v5 hv5 h hv

k -is constant

hv _��

M -is the dumping torque from rotating propeller M =−a abs()

v6 v61 vv

a₁-is constatnt

According to Fig. 4.4 we can determine components of the moment of inertia relative to the horizontal axis. Notice that this moment is independent of the position of the beam.

_l 2 2 m 2

J = ml , J = m , J = ml

v1 mr m v2 mv3 cb cb

3

_l 2 _l 2 b 2 t

J = m , J = ml , J = m

v4 bv5 tr t v6 t

33

m

22 22

J = ^ms r + ml , J = mr + ml

v7 ms ms m v8 ts ts ts t

2

r_ms-is the radius of the main shield,

r_ts-is the radius of the tail shield Similarly we can describe the motion of the beam around the vertical axis. Having in mind that the driving torque’s are produced by the rotors and that the moment of inertia depends on the pitch angle of the beam the horizontal motion of the beam (around the vertical axis) can be described in principle as rotative motion of a solid:

_d 2

�h

M = J (4)

hh 2

dt

where: M_h -is the sum of moments of forces acting in the horizontal plane,

J_h -is the sum of moments of inertia relative to the vertical axis.

Then: M_h = _ƒ⁴ M _hi , J_h = _ƒ⁸ J_hi i=1 i=1

To determine the moments of forces applied to the beam and making it rotate around the vertical axis consider the situation shown in Fig. 4.5.

M

ω _t

propulsive experimentally, see section 4.5)

�

M =− f

h2 hh

M_h2 -is the moment of friction depending on the angular velocity of the beam

around the			vertical axis,
hf -is constant.
h3M		-is the cross moment from vU ,		vh vh U kM = 3
vh k -is constant h4M -is the dumping torque from rotating propeller,					24h � aM = −	habs(�	)h
a2	-is constant

According to Fig. 4.5. we can determine components of the moment of inertia relative to vertical axis (it depends on pitch position of the beam).

J

h1

-

=

m

2 m 2

≈Δ «’≈Δ «’

t

� �

l

J

l

m

=

cos cos

÷◊÷◊

,

,

h1 h2 tm v v

3

3

m

b

22

≈Δ «’≈Δ «’

�

�

sin

J

l

J

l

=

=

cos

m

÷◊÷◊

,

,

4

h3 b h tr tv v

3

2 2

≈Δ «’≈Δ «’

�

sin�

J

l

J

l

=

=

cos

m m

÷◊÷◊

,

5 6

h h cb cb

mr ts m v v

m

222 2

≈Δ «

l

’≈Δ «

l

’

� �

J

J

+

m

ts

+

m

=

=

cos cos

r

ts

mr

ms

÷◊÷◊

,

7 8

h ht v ms ms m v

2

or in the compact form: J = D cos ² �+ E sin ² �+ F

h vv

TRAS User’s Manual -24

where: D,E,F are constants: m

b 22

D = l + ml ,

b cb cb

3

m

2 ts 2

= _ms r +

^{F m} ms ^rts

2 M_v1 = g {(A-B ) cos _�v − C sin _�v }

Using (1)-(4) we can write the equations describing the motion of the system as follows:

��

d_v l_m F_v(^� _m) − _v k_v + g((A − B) cos _�v − C sin _�v )

= ....

dt J_v

1 �2 �� (5)

≈Δ «

− _h (A + B + C)sin 2_�vU_hk_hv +U_hk_hv − a_1 vabs(_v )

2

...

^Jv

d�

v �

= _v , (6)

dt

��

�

dK M lF(�) cos − k +Uk − a abs(�)

h h tht vhh vvh 2 hh

== (7)

2 �2 �

dt J D sin + E cos + F

h vv

d�

h ��^Kh

= _h , _h = (8)

dt J_h ()

�

v

and two equations describing the motion of rotors:

dd

�h-1 �v-1

I_h = U_h − H_h (_�h ) and I_v = U_v − H_v (_�v )

dt dt

I_h -moment of inertia of the tail rotor I_v -moment of inertia of the main rotor.

The above model of the motor-propeller dynamics is obtained by substituting the nonlinear system by a serial connection of a linear dynamic system and static non-linearity.

≈Δ «

’

’

m m

t

3

2 2

E

l l

^m + m +

+

++

=

÷◊

÷◊

m m m

ts

,

tr tmr ms m

3

4.3 State equations

Finally, the mathematical model of the TRAS (compare Fig. 4.2) becomes as the set of four non-linear differential equations with two linear differential equations and four non-linear functions.

Ω

^Kh

ÿ

ÿŸŸŸŸŸŸŸ

»

»

h

α

α

…

…

h h

…

…

ω

K

ω

ÿŸ⁄

u

h

»

……

…

h

is the input,

X=

is the state, and

Y=

is the output vector.

U=

t

…

…

Ω

u

v

v v

…

…

α

α

…

…

v v

ω

ω

…

…

ŸŸŸŸŸŸŸ⁄

⁄

v v

In order to apply the model for control of TRAS the parameters and non-linear functions should be determined first. They can be divided in to three groups:

• physical parameters,
• non-linear static characteristics,
• time constants of the linear part of the model.

It is described in details in the next section.

4.4 Physical parameters

To obtain the values of model coefficients it is necessary to perform some measurements. First, geometrical dimensions and moving masses of TRAS should be measured. Following are the results of measurements for a given laboratory TRAS set-up.

m_tr = 0.154 [kg] l_t = 0.216 [m] m_mr = 0.199 [kg]

[m]

l = 0.202 m = 0.024 [kg]

^m [m] ^cb

l = 0.15 m = 0.031 [kg]

^b [m] ^t

l = 0.15 m = 0.029 [kg]

cb [m] m r_ms = 0.145 _[m] m_b = 0.011 [kg] r_ts = 0.10 [m] m_ts = 0.061 [kg] m_ms = 0.083 [kg]

Using the above measurements the moment of inertia about the horizontal axis can be calculated as:

2

J = _ƒ⁸ J = 0.02421 [kg m].

v iv i

The terms of the sum are calculated from elementary physics laws: J = ml ² = 0.00718 [kg m²]

v1 tr t

J = ml ² = 0.00054[kg m²]

v2 cb cb

J = m_mr l² = 0.00811 [kg m²]

v3 m

J = ml ² / 3 = 0.00049 [kg m²]

v4 tt

J = ml ² / 3 = 0.00040 [kg m²]

v5 mm

J = ml ² / 3 = 0.00008 [kg m²]

v6 bb

22 2

J = m(r / 2 + l) = 0.00426 [kg m ]

v7 ms ms m

22 2

J = m(r / 2 + l) = 0.00315 [kg m ]

v8 ts ts t

The calculated moment of inertia about the vertical axis is:

9

J_h = _ƒ J _hi , i

where the terms of the sum are:

22 2

J_h1 = m_t (l_t cos _�v ) / 3 = 0.000482 cos _�v [kg m ]

22 2

J_h2 = m_m (l_m cos _�v ) / 3 = 0.000394 cos _�v [kg m ]

22 2

J_h3 = m_b (l_b sin a_v ) / 3 = 0.000082 sin _�v [kg m ]

22 2

J_h4 = m_mr (l_m cos _�v )= 0.008119 cos _�v [kg m ]

22 2

J_h5 = m_tr (l_t cos _�v )= 0.007185 cos _�v [kg m ]

22 2

J_h6 = m_cb (l_cb sin _�v )= 0.00054 sin _�v [kg m ]

J = 0.00063 [kg m²]

222 22

J ^h_h8⁷ = m_ts (r_ts / 2 + l_t cos _�v )= 0.00031 + 0.00284 cos _�v [kg m ]

222 22

J_h9 = m_ms (r_ms / 2 + l_m cos _�v )= 0.00087 + 0.00387 cos _�v [kg m ]

giving finally (Fig. 4.6):

22 22

J = _ƒ⁹ J =D cos �+ E sin �+F = 0.022893 cos �+ 0.0006225 sin �+ 0.001267

hhi vv v v i

.

r ri i

and its components are given by: M = -9.81 ml cos ^�/ 2 =-0.0287 cos ^�[N m]

r1 mmv v

M = -9.81 ml cos ^�=-0.3943 cos ^�[N m]

r 2 mr mv v

M = -9.81 ml cos ^�=-0.16 cos ^�[N m]

r3 ms mv v

M = + 9.81 ml cos ^�/ 2 = 0.0328 cos ^�[N m]

r 4 ttv v

M = + 9.81 ml cos ^�= 0.3263 cos ^�[N m]

r5 tr tv v

M = + 9.81 ml cos ^�= 0.12925 cos ^�[N m]

r6 ts tv v

M = -9.81 ml sin ^�/ 2 = 0.01618 sin ^�[N m]

r7 bbv v

M = -9.81 ml sin ^�= 0.03531 sin ^�[N m]

r8 cb cb v v

giving finally (Fig. 4.7)

��

M = _ƒ⁸ M =(-0.0947 cos + 0.05149 sin ) [N m] .

rri vv i

The moment of centrifugal forces is:

M = _ƒ⁶ M ,

cf cfi i

where

� �

22 2

M = (m +m )l cos sin =0.01003 cos sin [N m]

cf 1 tr ts th �v �v h �v �v

� �

22 2

M = ml cos sin / 4 = 0.00144 cos sin [N m]

cf 2 tt h �v �v h �v �v

� �

22 2

M = ml cos sin / 4 =0.00025 cos sin [N m]

cf 3 bb h �v �v h �v �v

� �

22 2

M = ml cos sin =0.00054 cos sin [N m]

cf 4 cb cb h �v �v h �v �v

� �

22 2

M = ml cos sin / 4 =0.00118 cos sin [N m]

cf 5 mm h �v �v h �v �v

� �

22 2

M = (m +m )l cos sin =0.01150 cos sin [N m]

cf 6 mr ms mh �v �vh �v �v

giving finally (Fig. 4.8)

�

2 ��

M _cf = _ƒ ⁶ M _cfi = 0.024946 _h cos _v sin _v [N m] . i

4.5 Static characteristics

It is necessary to identify the following functions:

• Two non-linear input characteristics determining dependence of the DC-motor

��

rotational speed on the input voltage (RPM characteristics): = H(U ) , = H(U )

vvv hhh

To measure the characteristics double click the Static characteristics button in TRAS Control Window. The window given in Fig. 4.9 opens. In this window one defines the minimal and maximal control values and a number of measured points. The control order can be set as: Ascending, Descending or Reverse. Also one can choose the pitch or azimuth static characteristic. Note, that the control signal is normalised and changes in the range [-1, +1] what corresponding to the input voltage range [-24V, +24V] .

Choose Azimuth axis (tail rotor) and click the Run button. The constant value of control activates the DC motor so long as is required to obtain a steady state of the shaft angular velocity. Then, the velocity is measured and the control value is changed to the next constant value and DC motor is activated again. These steps are repeated to the end of the control range. This action should be repeated for pitch axis (main rotor) to obtained the both characteristics. Examples of the measured static characteristic for the main and tail rotors are shown in Fig. 4.10.

RPM vs. PWM

Rotor velocity [RPM]

8000

6000

4000

2000

0 -2000 -4000 -6000 -8000

PWM control value

Fig. 4.10 Main and tail rotor static characteristics

If the characteristics is measured in Reverse mode (the control has been changed from –1 to +1 and reverse), there are two slightly different plots.

• Two non-linear characteristics determining dependence of the propeller thrust on DC

��

motor rotational speed (thrust characteristics): F=F () ,F =F().

hhh vvv

The thrust static characteristics of the propellers should be measured in the case when the propellers were changed by a user. In this case a proper electronic balance (no delivered with the system) is necessary to measure the force created by rotational movements of the propellers. The characteristics included in the TRAS Toolbox and shown in this section were obtained by the manufacturer of the TRAS.

4.5.1 Main rotor thrust characteristics

To perform measurements correctly block the beam so that it could not rotate around the vertical axis, place the electronic balance under the beam in such a way that it is pulled by the propeller straight up. To balance the beam in the horizontal position attach a weight to the beam (as in Fig. 4.11) .

For further applications the measured characteristics should be replaced by their polynomial approximations. For this purposes one can use the MATLAB polyfit.m function. An example is given in Fig.3.6. The obtained polynomials have the form:

~ -18 5 -16 4 -11 3 -8 2 -5

F = -1.8 ⋅10 ω -7.8 ⋅10 ω+ 4.1 ⋅10 ω+ 2.7 ⋅10 ω+ 3.5 ⋅10 ω -0.014

vvv vvv

~ 3726 45 24 332 3

ω= -5.2 ⋅10 U -1.1 ⋅10 U +1.1 ⋅10 U +1.3 ⋅10 U -9.2 ⋅10 U -31 U + 6.1 ⋅10 U -4.5

v vvvvvvv

4.5.2 Tail rotor thrust characteristics

Fig. 4.14 shows laboratory set-up for measuring thrust of the tail rotor.

To measure the static thrust characteristics one should to rearrange the laboratory set-up as shown in Fig. 4.14 and the electronic balance should be used.

The measured by the producer of the TRAS thrust static characteristics of the tail motor are given in Fig. 4.15.

For further applications the characteristics can be replaced by their polynomial approximations. For this purposes one can use the MATLAB polyfit.m function. The obtained polynomials are as follows:

~ -20 �5 -17 �4 -12 �3 -9 �2 -5 �

F = -2.6 ⋅10 + 4.1⋅10 + 3.2 ⋅10 -7.3⋅10 + 2.1⋅10 + 0.0091

h h hhhv

�~ 3524 33 22 3

= 2.2 ⋅10 U-1.7 ⋅10 U-4.5 ⋅10 U + 3⋅10 U + 9.8 ⋅10 U-9.2

h hvvv v

5. RTWT model

In this section the process of building your own control system is described. The Real Time Windows Target (RTWT) toolbox is used. An example how to use the MSS software is shown later in section 5.3. In this section we give indications how to proceed in the RTWT environment.

Before start, test your MATLAB configuration and the compiler installation by building and running an example of real-time application. Real-time Windows Target includes the model rtvdp.mdl. Running this model will test the installation by running Real-Time Workshop, your third-party C-compiler, Real-Time Windows Target, and the Real-Time Windows Target kernel. In the MATLAB Command Window, type

rtvdp

Next, build and run the real-time model. For details refer to the Real-Time Windows Target help, section Installation and Configuration.

To build the system that operates in the real-time mode the user has to:

• create a Simulink model of the control system which consists of TRAS Device Driver and other blocks chosen from the Simulink library,
• build the executable file under RTWT,
• start the real-time code from the Simulation/Start real-time code pull-down menus; in this way the system runs in real-time.

5.2 Creating a model

The simplest way to create a Simulink model of the control system is to use one of the models included in the Tras Control Window as a template. For example, click on the PID Azimuth button and save it asMySystem.mdl name. The MySystem Simulink model is shown in Fig. 5.1.

Now, you can modify the model. You have absolute freedom to develop your own controller. Remember to leave the Tras Device driver model in the window. This is necessary to work in RTWT environment. Though it is not obligatory, we recommend you to leave the scope. You need a scope to watch how the system runs. The saturation blocks are built in the Tras driver block. They limit the currents to the DC motors for safety reasons. However they are not visible for the user who may amaze at the saturation of controls. Other blocks remaining in the window are not necessary for our new project. Creating your own model on the basis of an old example ensures that all-internal options of the model are set properly. These options are required to proceed with compiling and linking in a proper way. To put the Tras Device Driver into the real-time code a special make-file is required. This file is included to the TRAS software. You can apply most of the blocks from the Simulink library. However, some of them cannot be used (see RTW or RTWT references manual). The scope block properties are important for an appropriate data acquisition and watching how the system runs. The Scope block properties are defined in the Scope property window (see Fig. 5.2). This window opens after the selection of the Scope/Properties tab. You can gather measurement data to the Matlab Workspace marking the Save data to workspace checkbox. The data is placed under Variable name. The variable format can be set asstructure or matrix. The default Sampling Decimation parameter value is set to 1. This means that each measured point is plotted and saved. Often we select the Decimation parameter value equal to 5 or

10. This is a good choice to get enough points to describe the signal behaviour and to save the computer memory.

When the Simulink model is ready, click the Tools/External Mode Control Panel option and next click the Signal Triggering button. The window presented in Fig. 5.3 opens. Select Select All check button, set Sourceas manual, set Duration equal to the number of samples you intend to collect, and close the window.

5.3 Code generation and the build process

Once a model of the system has been designed the code for real-time mode can be generated, compiled, linked and downloaded into the processor. The code is generated by the use of Target Language Compiler (TLC) (see description of Simulink Target Language). The make-file is used to build and download object files to the target hardware automatically.

At the beginning you have to specify the simulation parameters of your Simulink model in the Simulation parameters dialog box. The RTW page appears when you select the RTW tab (Fig. 5.4). The RTW page is used to set the real-time build options and then to start the building process of the RTW.DLL executable file.

The system target file name is rtwin.tlc. It manages the code generation process. The tras_win_vc.tmf template make-file is devoted to C code generation using the Microsoft Visual C++ 6.0 compiler.

_Û Template make-file must correspond to the compiler in use.

The Solver page appears when you select the Solver tab (Fig. 5.5). The Solver page is used to set the simulation parameters. Several parameters and options are available in the window. The Fixed-step size editable text box is set to 0.002 (this is the sampling period given in seconds).

The Fixed-step solver is obligatory for real-time applications. If you use an arbitrary block from the discrete Simulink library or a block from the driver

library remember that different sampling periods must have a common divider.

The Start time has to be set to 0. The solver has to be selected. In our example the fifth-order integration method − ode5 is chosen.

If all parameters are set properly you can start the DLL executable building process. For this purpose press the Build push button on the RTW page (Fig. 5.4). Successful compilation and linking processes generate the following message:

### Created Real-Time Windows Target module MySystem.rwd. ### Successful completion of Real-Time Workshop build procedure for model: MySystem

Otherwise, an error massage is displayed in the MATLAB command window.

6. Controllers and real-time experiments

In the following section we propose three PID controllers. It is possible to tune the parameters of the controllers without analytical design. Such approach to the control problem seems to be reasonable if a well identified model of the TRAS is not available. The effectiveness of the PID controllers discussed here is illustrated by control experiments.

PID controllers

One degree of freedom (1-DOF) control problem is the following. Design a controller that will stabilise the system, or make it follow a desired trajectory in one plane (one degree of freedom) while motion in the other plane is blocked mechanically or being controlled by another controller. If TRAS is free to move in both axes we refer to the control as two degree of freedom (2DOF). The four PID controllers for TRAS: PIDvv, PIDvh, PIDhvand PIDhh (h-horizontal (azimuth), v-vertical (pitch)) are considered. The subscripts indicates the source-sink relation for the controller. Each control signal (Uv and Uh) is the sum of two controller outputs. For example, vertical control denoted later as Uv is the sum of two output signals: PIDvv and PIDhv. The internal structure of each PID controller is shown in Fig. 6.15b. There are three parameters to be set for every controller: KP, Kiand Kd. The TRAS control in the vertical and horizontal planes requires setting altogether 12 (3×4) controller parameters. Saturation blocks introduce four additional Isat parameters: Ivvsat, Ivhsat, Ihhsat and Ihvsat, which are the limits of absolute values of the integrals of errors, and two: Uhmax and Uvmax parameters, which are the limits of absolute value of controls. These 18 (12+4+2) parameters have their default values.

6.2 1-DOF controllers

The task of the one-degree-of-freedom (1-DOF) controllers is to move the TRAS to an arbitrary position in the selected plane and to stabilise it there.

6.2.1 Vertical 1-DOF control

At the beginning we restrict our control objective to stabilising the system in the vertical plane only (using the included clamp). We reduce the original system to the 1-DOF system by mechanically blocking its freedom to move in the horizontal plane. A corresponding block diagram of the PID control system is presented in Fig. 6.1.

 vd -desired pitch Fig. 6.1 1-DOF pitch control system

The block diagram below shows the system in a more detailed form (Fig. 6.2). Notice, that only the vertical part of the control system is considered.

6.2.2 Real-time 1-DOF pitch control experiment

Fix the TRAS in the horizontal plane using the special plastic clamps delivered with TRAS. Set it in the neutral vertical position and wait until the all oscillations are finished. In the Tras Control Window double click theReset Encoders block. Click the PID Pitch controller button and the model shown in Fig. 6.3 opens. Set all PID controller coefficients as: K = 0.6784 K = 0.4415 and K = 1.31196 . Also set saturation

pi d

of the integral part of the controller to 1.43. Build the model and click on the Simulation/Connect to target option and Start real-time code option.

TRAS 1-DOF PID Pitch

The results of the experiment are shown in Fig. 6.4. Notice, that control changes with high frequency. This phenomena appears due to the quantization effects of the signal caused by the differential part of the controller. For this reason the control signal is filtered. It is shown in the upper part of Fig. 6.4.

The details of the above experiment are shown in Fig. 6.5, Fig. 6.6 and Fig. 6.7.

time [s]

time [s] Pitch position and reference signal

0.3

0.2

0.1

0 -0.1 -0.2 -0.3 -0.4 -0.5

time [s]

Fig. 6.7 The pitch position and reference signal

6.2.3 Horizontal 1-DOF control

In the next experiment we will apply stabilising PID controller in the horizontal plane. We block the system in one axis so that it cannot move in the vertical plane (using the included fixing rectangle).. A corresponding block diagram of the control system is shown in Fig. 6.8, and in a more detailed form in Fig. 6.9.

 hd -desired azimuth Fig. 6.8 1-DOF control closed-loop system (azimuth stabilisation)

Notice that only the ‘horizontal’ part of the control system is considered.

6.2.4 Real-time 1-DOF azimuth control experiment

Fix the TRAS in the vertical plane using the special fixing rectangle delivered with TRAS. Set it in the zero position and click on the Reset Encoders block in Tras Control Window. Click PID Azimuth controller and the model shown in Fig. 6.10 opens. Set all PID controller coefficients as K = 4.9395 K = 0.0022 and K = 5.1898 . Also set saturation

pi d

of the integral part of the controller to 1.0. Build the model and click on the Simulation/Connect to target and Start real-time code options.

The results of the experiment are shown in Fig. 6.11. Notice, that control similar to the pitch control changes with a high frequency. This phenomena appears due to the quantization effects of the signal caused by the differential part of the controller. For this reason the control signal is filtered. It is shown in the upper part of Fig. 6.11.

time [s] PID controller for azimuth position -control

time [s]

Azimuth position and reference signal

0.5

0

-0.5

0 5 10 15 20 25 30 time [s]

Fig. 6.14 The azimuth position and reference signal

6.3 2-DOF PID controller

The structure of the cross-coupled multivariable PID controller is shown in Fig. 6.15.

b)