A. System Identification (Week 1) last check

Introduction

The Quanser Aero Experiment can be configured as a conventional dual-rotor helicopter, as shown in Figure 1. The front rotor that is horizontal to the ground predominantly affects the motion about the pitch axis while the back or tail rotor mainly affects the motion about the yaw axis (about the shaft).

Fig. 1: Quanser Aero Experiment

The tail rotor in helicopters is also known as the anti-torque rotor because it is used to reduce the torque that the main rotor generates about the yaw. Without this, the helicopter would be difficult to stabilize about the yaw axis. Because the rotors on the Quanser Aero Experiment are the same size and equidistant from each other, the tail rotor also generates a torque about the pitch axis. As a result, both the front and back/tail rotors generate a torque on each other.

Background

Equations of Motion

The free-body diagram of the Quanser Aero Experiment is illustrated in Figure 2.

Fig. 2: Simple free-body diagram of Quanser Aero Experiment

The following conventions are used for the modeling:

• The helicopter is horizontal and parallel with the ground when the pitch angle is zero, i.e., $\theta=0$ .
• The pitch angle increases positively, $\dot{\theta}(t)>0$ , when the front rotor is moved upwards and the body rotates clockwise (CW) about the Y axis.
• The yaw angle increases positively, $\dot{\psi}(t)>0$ , when the body rotates counter-clockwise (CCW) about the Z axis.
• Pitch increases, $\dot{\theta}>0$ , when the front rotor voltage is positive $V_\theta>0$ .
• Yaw increases, $\dot{\psi}>0$ , when the back (or tail) rotor voltage is positive, $V_{\psi}>0$ .

When voltage is applied to the pitch motor, $V_\theta$ , the speed of rotation results in a force, $F_\theta$ that acts normal to the body at a distance $r_\theta$ from the pitch axis. The rotation of the propeller generates a torque about the pitch rotor motor shaft which is in turn seen about the yaw axis. Thus, rotating the pitch propeller does not only cause motion about the pitch axis but also about the yaw axis. As described earlier, that is why conventional helicopters include a tail, or anti-torque, rotor to compensate for the torque generated about the yaw axis by the large, main rotor.

Similarly, the yaw motor causes a force $F_\psi$ that acts on the body at a distance $r_\psi$ from the yaw axis as well as a torque about the pitch axis.

The equations of motion can be approximated after neglecting reaction torques from the rotors:

J_\theta\ddot{\theta}+D_\theta\dot{\theta}+K_\theta\theta=\tau_\theta \tag{1}

J_\psi\ddot{\psi}+ D_\psi\dot{\psi}=\tau_\psi \tag{2}​

where the torques acting on the pitch and yaw axes are

\tau_\theta = K_{\theta \theta} V_{\theta} + K_{\theta \psi} V_{\psi} \tag{3}

\tau_\psi = K_{\psi \theta} V_{\theta} + K_{\psi \psi} V_{\psi} \tag{4}​

The parameters used in the EOMs above are:

• $J_\theta$ : the total moment of inertia about the pitch axis
• $D_\theta$ : the damping about the pitch axis
• $K_\theta$ : the stiffness about the pitch axis
• $J_{\psi}$ : the total moment of inertia about the yaw axis
• $D_{\psi}$ : the damping about the pitch axis
• $K_{\theta\theta}$ : torque thrust gain from the pitch rotor
• $K_{\psi\psi}$ : torque thrust gain from the yaw rotor
• $K_{\theta\psi}$ : cross-torque thrust gain acting on the pitch from the yaw rotor
• $K_{\psi\theta}$ : cross-torque thrust gain acting on the yaw from the pitch rotor
• $V_\theta$ : voltage applied to the pitch rotor
• $V_\psi$ : voltage applied to the yaw rotor motor

The total moment of inertia acting about the pitch and yaw axes are

J_\theta=J_{\rm body}+2J_{\rm prop} \tag{5}

J_\psi=J_{\rm body}+2J_{\rm prop}+J_{\rm yoke} \tag{6}

Expressing the rotor as a single-point mass, the inertia acting about the pitch or yaw axis from a single rotor is $J_{\rm prop}=m_{\rm prop}r^2$ . Modeling the helicopter body as a cylinder rotating about its center, the inertia is $J=m_{\rm body} \displaystyle \frac{L^2_{\rm body}}{12}$. Finally, the forked yoke that rotates about the yaw axis can be approximated as a cylinder rotating about its center as well and expressed as $J_{\rm yoke}=m_{\rm yoke}\displaystyle \frac{r^2_{yoke}}{2}$. Evaluating the moment of inertia using the parameters listed in the Quanser Aero Experiment User Manual gives:

$$J_\theta=0.0219 \space \rm kg \cdot m^2$$

$$J_\psi=0.0220\space \rm kg \cdot m^2$$

First-Order Response

The step response of a first-order transfer function

Y(s) = \displaystyle \frac{K}{\tau s+1}U(s) \tag{7}

where K is the DC or steady-state gain and τ is the time constant as illustrated in Figure 3. This is for a system with $K = 1$ and $\tau = 0.05$.

To obtain the time constant from the response, find the time it takes to reach $1-e^{-1}$or 63% of its final steady-state value:

y(t_1)=y_1=(1-e^{-1})(y_{\rm ss}-y_0) +y_0 \tag{8a}

The time constant is $\tau=t_1-t_{0 }$, where $t_0$ is the start time of the step and $t_1$ is the time it takes to reach 63% of the final value, as illustrated in Figure 3.

Fig. 3.a : First-order step response

To find the time constant from first-order response from an impulse (or short step) as depicted in Figure 4, find the time it takes for the response to reach $e^{-1}$ or 37% of its final steady-state value.

In this case we need to find

y(t_1)=y_1=e^{-1}(y_0 - y_{\rm ss})+ y_{ss} \tag{8b}

and the time constant is $\tau=t_1-t_{0 }$

Figure 3.b: First-order impulse decaying response

Second-Order Response

The free-oscillatory equation of motion of a second-order system is described by

J\ddot{\alpha}+D\dot{\alpha}+K\alpha =0 \tag{9}

is shown in Figure 4. Assuming the initial conditions $\alpha(0-)=\alpha_0$, the Laplace transform of Equation (9) is \alpha(s)=\displaystyle{\frac{\alpha_0/J}{s^2+D/Js+K/J}} \tag{10}

The prototype second-order equation is defined

s^2+2\zeta\omega_ns+\omega_n^2 \tag{11}

where $\zeta$ is the damping ratio and $\omega_n$ is the natural frequency. Equating the characteristic equation in Equation (10) to this gives

\omega_n^2=\displaystyle\frac{K}{J}\tag{12}​

2\zeta\omega_n=\displaystyle\frac{D}{J} \tag{13}

Finding the Natural Frequency

The period of the oscillations in a system response can be found using the equation

T_{\rm osc}=\displaystyle\frac{t_n-t_1}{n-1} \tag{14}

where $t_n$ is the time of the $n^{th}$ oscillation, $t_1$ is the time of the first peak, and $n$ is the number of oscillations considered. From this, the damped natural frequency (in radians per second) is

\omega_d=\displaystyle \frac{2\pi}{T_{\rm osc}} \tag{15}

and the undamped natural frequency is

\omega_n=\displaystyle\frac{\omega_d}{\sqrt{1-\zeta^2}} \tag{16}

Finding the Damping Ratio

The damping ratio of a second-order system can be found from its response. For a typical second-order underdamped system, the subsidence ratio (i.e. decrement ratio) is defined as

\delta = \displaystyle\frac{1}{n-1}\ln\frac{O_1}{O_n} \tag{17}

Fig. 4: Free Oscillation Response

where $O_1$ is the peak of the first oscillation and $O_n$ is the peak of the $n^{th}$ oscillation. Note that $O_1$ > $O_n$ , as this is a decaying response. The damping ratio can then be found using

\zeta= \displaystyle\frac{\delta}{\sqrt{4\pi^2+\delta^2}} \tag{18}

Estimating the Viscous Damping Coefficients

The viscous damping coefficients acting about the pitch and yaw axes, $D_\theta$ and $D_\psi$ in Equation (1) and Equation (2), can be found from the free-oscillation response. The free-oscillation response about the pitch and about the yaw are different, however.

Pitch Axis: By locking the yaw axis (using the Allen key supplied), this allows us to focus on the 1 DOF pitch-only system. Apply a short step voltage to mimic an impulse and get the free-oscillation response of the pitch. Remark that the impulse response is second-order free-oscillation response. The resulting 1 DOF pitch-only equations of motion is

J_\theta\ddot{\theta}+D_\theta\dot{\theta}+K_\theta\theta=0 \tag{19}

Taking its Laplace transform gives

J_\theta\left(\Theta(s)s^2-\theta(0^-)-\dot{\theta}(0^-)\right)+D_\theta\left(\Theta(s)-\theta(0^-)\right)+K_\theta\Theta(s)=0 \tag{20}

Assuming the initial velocity is zero, $\dot{\theta}(0^-)=0$, and solving for position we get

\Theta(s)=\displaystyle\frac{J_\theta}{J_\theta s^2+D_\theta s+K_\theta}\theta(0^-)=\frac{J_\theta D_\theta}{s^2+D_\theta/J_\theta +K_\theta/J_\theta}\theta(0^-) \tag{21}

The pitch free-oscillation transfer function matches the prototype second-order transfer function in Equation (10) Based on the measured damping ratio and natural frequency of the response, the friction (or stiffness) of the system is

K_\theta=J_\theta\omega_n^2 \tag{22}

and the viscous damping is

D_\theta=2\zeta\omega_nJ_\theta \tag{23}

Yaw Axis: The 1 DOF yaw-only equations of motion is

J_\psi\ddot\psi+D_\psi\dot\psi=0 \tag{24}

In terms of angular rate, the equation becomes

J_\psi\dot\omega_\psi(t)+D_\psi\omega_\psi(t)=0 \tag{25}

where $\omega_\psi(t)=\dot{\psi}(t)$ . Taking its Laplace transform

J_\theta(\Omega_\psi(s)s-\omega_\psi(0^-))+D_\psi\Omega_\psi(s)=0 \tag{26}

and solving for the speed we get

\Omega_\psi(s)=\displaystyle \frac{J_\psi}{J_\psi s+D_\psi}\omega_\psi(0^-)=\frac{J_\psi/D_\psi}{J_\psi/D_\psi s+1}\omega_\psi(0^-) \tag{27}

The yaw free-oscillation transfer function matches the prototype first-order transfer function in Equation (7). Based on the measured time constant of the response, its damping can be found with

D_\psi=\displaystyle\frac{J_\psi}{\tau} \tag{28}

Estimating the Thrust Parameters

By locking the yaw axis, this allows us to focus on the 1 DOF pitch-only system, i.e., eliminating any motion introduced in the yaw axis when applying a voltage to the pitch rotor. The equations of motion for the 1-DOF actuated system is

J_\theta\ddot\theta+D_\theta\dot\theta+K_\theta\theta=K_{\theta\theta}V_\theta \tag{29}

Solving for the thrust gain we get

K_{\theta\theta}=\displaystyle \frac{J_\theta\ddot\theta+D_\theta\dot\theta+K_\theta\theta}{V_\theta} \tag{30}

Remark that this is the thrust torque gain parameter. To force thrust gain would be $K_{\theta\theta}/r_p$, where $r_p$is the distance between the helicopter pivot and the center of the pitch rotor. Similarly, to find the thrust gain acting on the yaw axis only system, lock the pitch axis, and apply a voltage to the tail rotor. This system is represented by

J_\psi\ddot\psi+D_\psi\dot\psi=K_{\psi\psi}V_\psi \tag{31}

or,

J_\psi\dot\omega_\psi+D_\psi\omega_\psi=K_{\psi\psi}V_\psi \tag{32}

where $\omega_\psi=\dot\psi$ is the angular rate of the yaw axis. The yaw torque thrust gain is

K_{\psi\psi}=\displaystyle\frac{J_\psi\dot\omega_\psi+D_\psi\omega_\psi}{V_\psi} \tag{33}

The cross-torque thrust parameters, $K_{\theta\psi}$and $K_{\psi\theta}$in Equation (3) and (4), represent the coupling between the axes. To find the cross-torque acting on the pitch axis from a torque applied to the tail rotor, unlock both pitch and yaw axes such that it is free to move in 2 DOF, apply a voltage to the tail rotor, and examine the response of the pitch. The equations representing these dynamics when $V_\theta=0$ , are

J_\theta\ddot\theta+D_\theta\dot\theta+K_\theta\theta=K_{\theta\psi}V_\psi \tag{34}

Putting this in terms of angular rate, $\omega_p=\dot\theta$, and solving for the gain we get

K_{\theta\psi}=\displaystyle\frac{J_\theta\dot\omega_\theta+D_\theta\omega_{\theta}}{V_\psi} \tag{35}

Similarly, to identify the cross-torque gain parameter that is generated about the yaw axis from a pitch torque (i.e. voltage applied to the front rotor), we have the equation

J_\psi\ddot\psi+D_\psi\dot\psi=K_{\psi\theta}V_\theta \tag{36}

and the gain can be found using

K_{\psi\theta}=\displaystyle\frac{J_{\psi}\dot\omega_\psi+D_\psi\omega_{\psi}}{V_\theta} \tag{37}

Week 1: System Identification

Experimental Steps for Finding System Parameters

First download the zip file below and extract it in the desktop:

177KB

quanser_aero.zip

archive

1. Finding Pitch Damping, $D_{\theta}$ and pitch trust gain $K_{\theta\theta}$

1) Lock the yaw axis to enable motions about the pitch axis only. (place the allen key to lock yaw axis and make sure that the pitch axis is free to rotate, loosen two screws)

2) Open the q_aero_free_osc_response_pitch SIMULINK file.

3) Select a (-20) V impulse

4) Select simulation time 30 sec.

5) To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

6) Click Connect button under Monitor & Tune and then run SIMULINK by clicking Start .

7) Copy aero_pitch_free_osc_rsp.mat to your folder. Data is saved in following order: 1: Time (s) 2: Pitch input (V) 3: Pitch position (rad) 4: Pitch speed (rad/s)

8) Close the SIMULINK model. DO NOT SAVE THE CHANGE

3. Finding Yaw Damping , $D_{\psi}$

1) Unlock the yaw axis

2) Lock the pitch axis

3) Open the q_aero_free_osc_response_yaw SIMULINK file.

4) Apply an impulse of 20V to the tail rotor

5) Select simulation time 10 sec.

6) To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

7) Click Connect button under Monitor & Tune and then run SIMULINK by clicking Start .

8) Copy aero_yaw_free_osc_rsp.mat to your folder Data is saved in following order: 1: Time (s) 2: Yaw input (V) 3: Yaw position (rad) 4: Yaw speed (rad/s) 5: Yaw acceleration (rad/s^2)

9) Close the SIMULINK model. DO NOT SAVE THE CHANGE.

Fig. 7: Sample free response about yaw

4. Finding Yaw thrust gain, $K_{\psi\psi}$

1) Open the q_aero_step_response_yaw SIMULINK file.

2) Select a step voltage 15V

3) Select simulation time 40 sec.

4) To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

5) Click Connect button under Monitor & Tune and then run SIMULINK by clicking Start .

6) Copy aero_yaw_step_rsp.mat to your folder Data is saved in following order: 1: Time 2: Yaw input (V) 3: Yaw position (rad) 4: Yaw Speed (rad/s)

7) Close the SIMULINK model. DO NOT SAVE THE CHANGE.

Fig. 8: Sample yaw Step Response

5. Identifying the cross-torque gain parameter $K_{\theta\psi}$

1) Unlock both the pitch and yaw axes to enable the full 2 DOF motion.

2) Open the q_aero_step_response_pitch_from_yaw SIMULINK file.

3) Select yaw step voltage to 12V and pitch to 0V.

4) Select simulation time 40 sec.

5) To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

6) Click Connect button under Monitor & Tune and then run SIMULINK by clicking Start .

A pitch up motion must be seen due to torque from the tail rotor.

7) Copy aero_pitch_from_yaw_step_rsp.mat to your folder Data is saved in following order: 1: Time (s) 2: Yaw input (V) 3: Pitch Position (rad) 4: Pitch Speed (rad/s)

8) Close the SIMULINK model. DO NOT SAVE THE CHANGE.

Fig. 9: Sample Pitch Step Response from Yaw Voltage

6. Identifying the cross-torque gain parameter $K_{\psi\theta}$

1) Open the q_aero_step_response_yaw_from_pitch SIMULINK file.

2) Select pitch step voltage to 12V and yaw to 0V.

3) Select simulation time 40 sec.

4) To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

5) Click Connect button under Monitor & Tune and then run SIMULINK by clicking Start .

A yaw motion must be seen due to torque from front motor.

6) Copy aero_yaw_from_pitch_step_rsp.mat to your folder Data is saved in following order: 1: Time (s) 2: Pitch input (V) 3: Yaw Position (rad) 4: Yaw Speed (rad/s)

7) Close the SIMULINK model. DO NOT SAVE THE CHANGE.

Fig. 10: Sample Yaw Step Response from Pitch Voltage

Analysis (should be done before week 2 and Controller design)

1. Find the natural frequency and damping ratio from the impulse response of the pitch axis (HINT: first find the peaks and use equation 16, 17, and 18). Then find the stiffness and viscous damping coefficient about the pitch axis (Hint: equation 22,23).

2. Find the time constant from the yaw pulse response and obtain the damping $D_\psi$. Use First-order impulse decaying response. To find the time constant, examine the decaying response that starts at an initial maximum speed and settle down to 0 rad/s (use equation 8b). If data obtained does not fully shows the 37% try estimate the value by curve fitting.
3. Find the thrust and cross-torque thrust gain parameters $K_{\theta\theta}, K_{\psi\psi}, K_{\theta\psi}, K_{\psi\theta}$ by performing discrete derivative to peak values or from the steady state of step response.