7.1 Quadcopter Asoslari — Nazariya

Hafta: 2 | Masalalar: 25 | Qiyinlik: ⭐⭐⭐

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Kirish

Quadcopter — 4 ta propeller bilan uchuvchi dron. Oddiy mexanika, murakkab kontrol.

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1. Asosiy Tuzilish

     Motor 1 (CW)          Motor 2 (CCW)
         ◯━━━━━━━━━━━━━━━━━━━━◯
          \                  /
           \      ┌──┐      /
            \     │  │     /
             ━━━━━│  │━━━━━
            /     │  │     \
           /      └──┘      \
          /                  \
         ◯━━━━━━━━━━━━━━━━━━━━◯
     Motor 3 (CCW)         Motor 4 (CW)

Motor Yo'nalishlari

• CW (Clockwise): Motor 1, 4
• CCW (Counter-Clockwise): Motor 2, 3

Qarama-qarshi aylanish — yaw momentni bekor qiladi.

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2. Koordinata Sistemasi

Body Frame (B)

• X: oldinga
• Y: o'ngga
• Z: pastga

World Frame (W)

• X: shimol
• Y: sharq
• Z: pastga (NED) yoki yuqoriga (ENU)

Euler Burchaklari

Burchak	Belgi	Ta'rif
Roll	$\phi$	X atrofida aylanish
Pitch	$\theta$	Y atrofida aylanish
Yaw	$\psi$	Z atrofida aylanish

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3. Kuchlar va Momentlar

Thrust (Itarish Kuchi)

Har bir motor:

$$T_i = k_T \cdot \omega_i^2$$

• $k_T$ — thrust koeffitsienti
• $\omega_i$ — motor burchak tezligi (rad/s)

Umumiy thrust:

$$T_{total} = T_1 + T_2 + T_3 + T_4$$

Drag Moment

Motor aylanishidan moment:

$$Q_i = k_Q \cdot \omega_i^2$$

Roll Moment

$$\tau_\phi = L \cdot (T_2 + T_3 - T_1 - T_4)$$

Pitch Moment

$$\tau_\theta = L \cdot (T_1 + T_2 - T_3 - T_4)$$

Yaw Moment

$$\tau_\psi = Q_1 - Q_2 + Q_3 - Q_4$$

$L$ — markaz va motor orasidagi masofa.

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4. Hovering (Muallaq Turish)

Hovering sharti:

$$T_{total} = mg$$ $$T_1 = T_2 = T_3 = T_4 = \frac{mg}{4}$$

Quvvat

$$P_{hover} = \frac{(mg)^{3/2}}{\sqrt{2\rho A}}$$

• $\rho$ — havo zichligi (~1.2 kg/m³)
• $A$ — propellerlar yuza

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5. Harakat Tenglamalari

Newton-Euler

Chiziqli tezlanish (body frame):

$$m\dot{\mathbf{v}} = R^T \mathbf{F}_g + \mathbf{T} - m\boldsymbol{\omega} \times \mathbf{v}$$

Burchak tezlanish:

$$I\dot{\boldsymbol{\omega}} = \boldsymbol{\tau} - \boldsymbol{\omega} \times I\boldsymbol{\omega}$$

Soddalashtirilgan Model

Kichik burchaklar uchun:

$$\ddot{x} = (\cos\phi\sin\theta\cos\psi + \sin\phi\sin\psi)\frac{T}{m}$$ $$\ddot{y} = (\cos\phi\sin\theta\sin\psi - \sin\phi\cos\psi)\frac{T}{m}$$ $$\ddot{z} = \cos\phi\cos\theta\frac{T}{m} - g$$ $$\ddot{\phi} = \frac{\tau_\phi}{I_{xx}}$$ $$\ddot{\theta} = \frac{\tau_\theta}{I_{yy}}$$ $$\ddot{\psi} = \frac{\tau_\psi}{I_{zz}}$$

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6. Mixer (Kuch Taqsimoti)

Kerakli kuchlar va momentlardan motor tezliklariga:

$$\begin{bmatrix} T \\ \tau_\phi \\ \tau_\theta \\ \tau_\psi \end{bmatrix} = \begin{bmatrix} k_T & k_T & k_T & k_T \\ -Lk_T & Lk_T & Lk_T & -Lk_T \\ Lk_T & Lk_T & -Lk_T & -Lk_T \\ k_Q & -k_Q & k_Q & -k_Q \end{bmatrix} \begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix}$$

Teskari:

$$\begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix} = M^{-1} \begin{bmatrix} T \\ \tau_\phi \\ \tau_\theta \\ \tau_\psi \end{bmatrix}$$

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7. Kontrol Arxitekturasi

┌─────────────────────────────────────────────────────────────┐
│                    POSITION CONTROLLER                       │
│   (x,y,z)_ref ──> PID ──> attitude_ref + thrust              │
└──────────────────────────┬──────────────────────────────────┘
                           │
┌──────────────────────────▼──────────────────────────────────┐
│                   ATTITUDE CONTROLLER                        │
│   (φ,θ,ψ)_ref ──> PID ──> rate_ref                           │
└──────────────────────────┬──────────────────────────────────┘
                           │
┌──────────────────────────▼──────────────────────────────────┐
│                     RATE CONTROLLER                          │
│   (p,q,r)_ref ──> PID ──> τ_φ, τ_θ, τ_ψ                     │
└──────────────────────────┬──────────────────────────────────┘
                           │
┌──────────────────────────▼──────────────────────────────────┐
│                         MIXER                                │
│   (T, τ_φ, τ_θ, τ_ψ) ──> Motor PWM                          │
└─────────────────────────────────────────────────────────────┘

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8. Sensorlar

IMU (Inertial Measurement Unit)

• Gyroscope: Burchak tezlik (p, q, r)
• Accelerometer: Chiziqli tezlanish

Barometer

Balandlik (havo bosimidan):

$$h = \frac{T_0}{L}\left[\left(\frac{P}{P_0}\right)^{-\frac{LR}{gM}} - 1\right]$$

GPS

• Pozitsiya (lat, lon, alt)
• Tezlik
• 1-5 Hz update rate

Magnetometer

Yaw burchagi (kompas).

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9. Uchish Rejimlari

Rejim	Kontrol	Foydalanuvchi
Stabilize	Attitude	Roll/pitch/yaw/throttle
Alt Hold	Altitude	Roll/pitch/yaw
Position	Position	Waypoints
Auto	Mission	-
Acro	Rate	To'liq manual

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10. Xavfsizlik

Failsafe

• Batareya past → RTL (Return to Launch)
• GPS yo'qolishi → Land
• RC signal yo'qolishi → Hover/RTL

Geofence

Belgilangan hududdan chiqmaydi.

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Xulosa

Tushuncha	Formula
Thrust	$T = k_T \omega^2$
Hovering	$T = mg$
Roll moment	$\tau_\phi = L(T_2 + T_3 - T_1 - T_4)$
Kontrol	Position → Attitude → Rate → Mixer

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Keyingi Qadam

📝 Masalalar — 25 ta masala yeching!