7.1 Quadcopter Asoslari — Masalalar

Jami: 25 ta | Yechim bilan: ✅

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Asosiy Masalalar (1-10)

Masala 1 ⭐⭐

Quadcopter massasi 2 kg. Hovering uchun har bir motorning thrust kuchi?

Yechim

$T_{total} = mg = 2 \times 10 = 20$ N

$T_{motor} = \frac{T_{total}}{4} = \frac{20}{4} = 5$ N

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Masala 2 ⭐⭐

Motor: $k_T = 1.5 \times 10^{-7}$ N/(rad/s)². Hovering uchun motor RPM? (m = 2 kg)

Yechim

$T_{motor} = 5$ N

$\omega^2 = \frac{T}{k_T} = \frac{5}{1.5 \times 10^{-7}} = 3.33 \times 10^7$

$\omega = 5774$ rad/s

$RPM = \frac{\omega \times 60}{2\pi} = \frac{5774 \times 60}{6.28} = 55130$ RPM

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Masala 3 ⭐⭐

3S LiPo batareya (11.1V, 5000mAh). Hovering quvvati 150W. Uchish vaqti?

Yechim

Energiya: $E = 11.1 \times 5 = 55.5$ Wh

Vaqt: $t = \frac{E}{P} = \frac{55.5}{150} = 0.37$ h = 22 daqiqa

(Real: 80% efficiency → ~18 min)

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Masala 4 ⭐⭐

Roll burchagi 10°. Horizontal tezlanish? (m = 2 kg, hovering thrust)

Yechim

Horizontal kuch: $F_x = T \sin\theta = 20 \times \sin 10° = 20 \times 0.174 = 3.47$ N

Tezlanish: $a = \frac{F_x}{m} = \frac{3.47}{2} = 1.74$ m/s²

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Masala 5 ⭐⭐

Propeller: diametr 10", pitch 4.5". Static thrust taxmini? (Formula: $T \approx \frac{D^{3.5} \times pitch \times RPM^2}{10^{12}}$ g)

Yechim

D = 10", pitch = 4.5", RPM = 10000 deb faraz qilsak:

$T \approx \frac{10^{3.5} \times 4.5 \times 10000^2}{10^{12}}$

$= \frac{3162 \times 4.5 \times 10^8}{10^{12}} = 1.42 \times 10^{12} / 10^{12} = 1423$ g ≈ 1.4 kg = 14 N

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Masala 6 ⭐⭐

Roll moment kerak: $\tau_\phi = 0.5$ Nm. Motor arm L = 0.2 m. Thrust farqi?

Yechim

$\tau_\phi = L \times \Delta T \times 2$ (2 ta motor har tomonda)

$\Delta T = \frac{\tau_\phi}{2L} = \frac{0.5}{0.4} = 1.25$ N

Motor 1,4 va 2,3 orasidagi farq.

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Masala 7 ⭐⭐

Barometer: dengiz sathida P₀ = 101325 Pa. 100 m balandlikda bosim?

Yechim

Taxminiy formula: $\Delta P \approx \rho g h = 1.2 \times 10 \times 100 = 1200$ Pa

$P = P_0 - \Delta P = 101325 - 1200 = 100125$ Pa

(1 hPa ≈ 8.4 m)

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Masala 8 ⭐⭐

GPS: 5 Hz update, 1 m accuracy. 10 m/s tezlikda pozitsiya xatosi?

Yechim

Update interval: $\Delta t = 0.2$ s

Xato: 1 m + harakat = 1 + 10 × 0.2 = 3 m (worst case)

Shuning uchun IMU fusion kerak!

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Masala 9 ⭐⭐

Accelerometer: 0.5g kuch o'lchadi. Dron qaysi burchakda (hovering)?

Yechim

Hoverda accelerometer faqat g ni o'lchaydi (vertikal).

0.5g o'lchash — bu egilish burchagi:

$\theta = \arcsin(0.5) = 30°$

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Masala 10 ⭐⭐

Gyroscope drift: 0.01°/s. 1 daqiqada qancha drift?

Yechim

$drift = 0.01 \times 60 = 0.6°$ per minute

10 daqiqada: 6° — sezilarlı xato!

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O'rtacha Masalalar (11-18)

Masala 11 ⭐⭐⭐

Attitude controller: Roll error 5°, $K_p = 5$, $K_d = 0.5$. Roll rate 2°/s. PD chiqishi?

Yechim

$e_\phi = 5°$, $\dot{\phi} = 2°/s$ (error rate = $-\dot{\phi}$ if trying to reduce)

$u = K_p e - K_d \dot{\phi} = 5 \times 5 - 0.5 \times 2 = 25 - 1 = 24$

(Birlik: moment yoki rate command)

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Masala 12 ⭐⭐⭐

Mixer: T = 22N, $\tau_\phi$ = 0.5 Nm, $\tau_\theta$ = 0, $\tau_\psi$ = 0.1 Nm. Motor thrustlari?

Yechim

import numpy as np

# Parameters
L = 0.2  # arm length
kT = 1.5e-7
kQ = 1e-8

# Simplified mixer (X configuration)
# T1 + T2 + T3 + T4 = T
# L(-T1 + T2 + T3 - T4) = tau_phi
# L(T1 + T2 - T3 - T4) = tau_theta
# (T1 - T2 + T3 - T4) * kQ/kT = tau_psi

T = 22
tau_phi = 0.5
tau_theta = 0
tau_psi = 0.1

# Solve (simplified, ignoring drag moment)
T1 = T/4 - tau_phi/(4*L) + tau_theta/(4*L) + tau_psi*kT/(4*kQ)
T2 = T/4 + tau_phi/(4*L) + tau_theta/(4*L) - tau_psi*kT/(4*kQ)
T3 = T/4 + tau_phi/(4*L) - tau_theta/(4*L) + tau_psi*kT/(4*kQ)
T4 = T/4 - tau_phi/(4*L) - tau_theta/(4*L) - tau_psi*kT/(4*kQ)

print(f"T1 = {T1:.2f} N")  # ~4.9
print(f"T2 = {T2:.2f} N")  # ~6.1
print(f"T3 = {T3:.2f} N")  # ~6.1
print(f"T4 = {T4:.2f} N")  # ~4.9

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Masala 13 ⭐⭐⭐

Complementary filter: $\alpha = 0.98$. Gyro = 45°, Accel = 44.5°. Filtered angle?

Yechim

$\theta_{filt} = \alpha \times \theta_{gyro} + (1-\alpha) \times \theta_{accel}$

$= 0.98 \times 45 + 0.02 \times 44.5 = 44.1 + 0.89 = 44.99°$

Gyro dominant — high frequency movements.

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Masala 14 ⭐⭐⭐

Thrust-to-weight ratio (TWR) = 2. Maksimal vertikal tezlanish?

Yechim

$TWR = \frac{T_{max}}{mg} = 2$

$T_{max} = 2mg$

$a_{max} = \frac{T_{max} - mg}{m} = \frac{2mg - mg}{m} = g = 10$ m/s²

Pastga ham: $a_{down} = \frac{0 - mg}{m} = -g$ (erkin tushish)

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Masala 15 ⭐⭐⭐

Position hold: GPS pozitsiya error = 2m (sharq). Kerakli roll burchak? ($K_p = 0.5$ 1/m)

Yechim

$\phi_{cmd} = K_p \times e_y = 0.5 \times 2 = 1$ rad?

Yo'q, bu juda katta. Real systemda:

$\phi_{cmd} = K_p \times e_y = 0.1 \times 2 = 0.2$ rad = 11.5°

(Saturatsiya bilan ~15° max)

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Masala 16 ⭐⭐⭐

Motor inertia: $I_{motor} = 5 \times 10^{-6}$ kg·m². 0 dan 10000 RPM ga 50ms da. Kerakli moment?

Yechim

$\omega_f = 10000 \times \frac{2\pi}{60} = 1047$ rad/s

$\alpha = \frac{\omega_f - 0}{t} = \frac{1047}{0.05} = 20940$ rad/s²

$\tau = I \alpha = 5 \times 10^{-6} \times 20940 = 0.105$ Nm

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Masala 17 ⭐⭐⭐

Yaw control: 4 ta motor, ikki juft CW/CCW. Yaw tezligi 30°/s kerak. $I_{zz} = 0.05$ kg·m². Moment?

Yechim

Agar barqaror yaw rate kerak (tezlanishsiz), $\tau = 0$ (ideally).

Lekin 0 dan 30°/s ga 0.5s da:

$\dot{\psi} = 30 \times \pi/180 = 0.524$ rad/s

$\ddot{\psi} = 0.524 / 0.5 = 1.05$ rad/s²

$\tau_\psi = I_{zz} \times \ddot{\psi} = 0.05 \times 1.05 = 0.052$ Nm

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Masala 18 ⭐⭐⭐

ESC protocol: PWM 1000-2000μs. 50% throttle = ?

Yechim

PWM range: 1000μs (0%) - 2000μs (100%)

50%: $1000 + 0.5 \times (2000 - 1000) = 1500$ μs

Dshot/OneShot — raqamli, PWM emas.

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Murakkab Masalalar (19-25)

Masala 19 ⭐⭐⭐⭐

Altitude controller: Setpoint 10m, actual 8m, vertical speed -0.5m/s. PID output? ($K_p=5, K_i=1, K_d=3$, dt=0.01, integral=2)

Yechim

# Error
e = 10 - 8 = 2  # m

# P
P = 5 * 2 = 10

# I (assuming integral already has some value)
I = 1 * (2 + 2*0.01) = 2.02  # approximately

# D (derivative of error = -velocity)
D = 3 * (-(-0.5)) = 3 * 0.5 = 1.5

# Total
u = P + I + D = 10 + 2 + 1.5 = 13.5

# This is thrust adjustment (N or %)

Thrust += 13.5 (gravity compensation ustiga)

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Masala 20 ⭐⭐⭐⭐

Quadcopter simulyatsiya: 1 sekund hovering keyin roll 10°. Trajectory?

Yechim

import numpy as np

def simulate_quadcopter(duration=5, dt=0.01):
    m = 2  # kg
    g = 10
    
    # State: [x, y, z, vx, vy, vz, phi, theta, psi]
    state = np.zeros(9)
    state[2] = 10  # Start at 10m
    
    t = 0
    trajectory = []
    
    while t < duration:
        x, y, z, vx, vy, vz, phi, theta, psi = state
        
        # Control
        if t < 1:
            T = m * g  # Hover
            phi_cmd = 0
        else:
            T = m * g / np.cos(np.radians(10))  # Compensate
            phi_cmd = np.radians(10)
        
        # Simple dynamics
        phi = phi_cmd  # Instant attitude (simplified)
        
        ax = T * np.sin(phi) / m
        ay = 0
        az = T * np.cos(phi) / m - g
        
        # Update
        vx += ax * dt
        vy += ay * dt
        vz += az * dt
        x += vx * dt
        y += vy * dt
        z += vz * dt
        
        state = np.array([x, y, z, vx, vy, vz, phi, theta, psi])
        trajectory.append([t, x, y, z])
        
        t += dt
    
    return np.array(trajectory)

traj = simulate_quadcopter()
print(f"Final position: x={traj[-1,1]:.2f}m, z={traj[-1,3]:.2f}m")

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Masala 21 ⭐⭐⭐⭐

Sensor fusion: Kalman filter for altitude (barometer + accelerometer).

Yechim

class AltitudeKalman:
    def __init__(self):
        # State: [altitude, vertical_velocity]
        self.x = np.array([0.0, 0.0])
        
        # Covariance
        self.P = np.eye(2) * 100
        
        # Process noise
        self.Q = np.array([[0.01, 0], [0, 0.1]])
        
        # Measurement noise
        self.R_baro = 0.5  # m²
        self.R_accel = 0.1  # (m/s²)²
    
    def predict(self, accel, dt):
        # State transition
        F = np.array([[1, dt], [0, 1]])
        B = np.array([0.5*dt**2, dt])
        
        self.x = F @ self.x + B * accel
        self.P = F @ self.P @ F.T + self.Q
    
    def update_baro(self, z_baro):
        H = np.array([[1, 0]])
        y = z_baro - H @ self.x
        S = H @ self.P @ H.T + self.R_baro
        K = self.P @ H.T / S
        
        self.x = self.x + K.flatten() * y
        self.P = (np.eye(2) - np.outer(K, H)) @ self.P
    
    def get_state(self):
        return self.x[0], self.x[1]

# Usage
kf = AltitudeKalman()
kf.predict(accel=0.5, dt=0.01)
kf.update_baro(z_baro=10.2)
alt, vel = kf.get_state()

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Masala 22 ⭐⭐⭐⭐

Waypoint navigation: 3 ta waypoint, ETA hisoblash.

Yechim

import numpy as np

def navigate_waypoints(waypoints, max_speed=5, current_pos=np.array([0,0,0])):
    total_distance = 0
    pos = current_pos.copy()
    
    for wp in waypoints:
        wp = np.array(wp)
        distance = np.linalg.norm(wp - pos)
        total_distance += distance
        pos = wp
        print(f"To {wp}: {distance:.1f}m")
    
    eta = total_distance / max_speed
    print(f"\nTotal: {total_distance:.1f}m, ETA: {eta:.1f}s")
    return eta

waypoints = [
    [10, 0, 5],
    [10, 10, 10],
    [0, 10, 5]
]

navigate_waypoints(waypoints)

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Masala 23 ⭐⭐⭐⭐

Battery estimation: Current draw vs remaining capacity.

Yechim

class BatteryEstimator:
    def __init__(self, capacity_mah=5000, voltage=11.1):
        self.capacity = capacity_mah
        self.remaining = capacity_mah
        self.voltage = voltage
    
    def update(self, current_a, dt_s):
        # mAh consumed
        consumed = current_a * 1000 * (dt_s / 3600)
        self.remaining -= consumed
        
        # Voltage drop model (simplified)
        self.voltage = 12.6 - (1 - self.remaining/self.capacity) * 3.6
        
        return self.remaining, self.voltage
    
    def time_remaining(self, current_a):
        return (self.remaining / 1000) / current_a * 3600  # seconds
    
    def percent(self):
        return (self.remaining / self.capacity) * 100

# Simulation
bat = BatteryEstimator(5000, 12.6)
current = 15  # Amps

for t in range(0, 1200, 60):  # 20 minutes
    remaining, voltage = bat.update(current, 60)
    print(f"t={t//60}min: {bat.percent():.1f}%, {voltage:.2f}V, "
          f"remaining: {bat.time_remaining(current)/60:.1f}min")

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Masala 24 ⭐⭐⭐⭐

Geofence: Doira ichida yoki tashqarisini aniqlash.

Yechim

import numpy as np

class Geofence:
    def __init__(self, center, radius, max_altitude=100):
        self.center = np.array(center)
        self.radius = radius
        self.max_alt = max_altitude
    
    def check(self, position):
        pos = np.array(position)
        
        # Horizontal distance
        h_dist = np.linalg.norm(pos[:2] - self.center[:2])
        
        violations = []
        
        if h_dist > self.radius:
            violations.append(f"RADIUS: {h_dist:.1f}m > {self.radius}m")
        
        if pos[2] > self.max_alt:
            violations.append(f"ALTITUDE: {pos[2]:.1f}m > {self.max_alt}m")
        
        if pos[2] < 0:
            violations.append(f"BELOW GROUND: {pos[2]:.1f}m")
        
        return len(violations) == 0, violations
    
    def clamp(self, position):
        """Return safe position"""
        pos = np.array(position)
        
        # Clamp horizontal
        h_vec = pos[:2] - self.center[:2]
        h_dist = np.linalg.norm(h_vec)
        if h_dist > self.radius:
            h_vec = h_vec / h_dist * self.radius
            pos[:2] = self.center[:2] + h_vec
        
        # Clamp vertical
        pos[2] = np.clip(pos[2], 0, self.max_alt)
        
        return pos

# Test
fence = Geofence([0, 0, 0], radius=50, max_altitude=100)

test_positions = [
    [30, 40, 50],   # Inside
    [60, 0, 50],    # Outside radius
    [0, 0, 120],    # Too high
]

for pos in test_positions:
    ok, violations = fence.check(pos)
    print(f"{pos}: {'OK' if ok else violations}")

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Masala 25 ⭐⭐⭐⭐

Auto-land: Vertikal tezlikni cheklash bilan qo'nish.

Yechim

def auto_land_controller(altitude, velocity, target_velocity=-0.5):
    """
    Safe landing controller
    
    target_velocity: desired descent rate (negative = down)
    """
    Kp = 5
    Kd = 2
    
    # Near ground - slow down
    if altitude < 2:
        target_velocity = -0.2
    if altitude < 0.5:
        target_velocity = -0.1
    
    # Velocity error
    vel_error = target_velocity - velocity
    
    # Control (simplified)
    # Negative = reduce thrust, positive = increase
    thrust_adjust = Kp * vel_error
    
    # Safety limits
    thrust_adjust = np.clip(thrust_adjust, -5, 5)
    
    return thrust_adjust, target_velocity

# Simulation
altitude = 10
velocity = 0
g = 10
m = 2
dt = 0.01

print("Auto-landing simulation:")
for i in range(2000):
    adj, target = auto_land_controller(altitude, velocity)
    
    # Base hover thrust + adjustment
    thrust = m * g + adj
    
    # Dynamics
    accel = thrust/m - g
    velocity += accel * dt
    altitude += velocity * dt
    
    if i % 100 == 0:
        print(f"t={i*dt:.2f}s: alt={altitude:.2f}m, vel={velocity:.2f}m/s")
    
    if altitude <= 0:
        print(f"LANDED at t={i*dt:.2f}s, impact vel={velocity:.2f}m/s")
        break

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✅ Tekshirish Ro'yxati

• 1-10: Asosiy hovering va thrust
• 11-18: Kontrol va sensorlar
• 19-25: Murakkab flight control

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Keyingi Qadam

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