6.1 Robot Kinematikasi — Nazariya

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Kirish

Robot kinematikasi — robotning geometrik harakatini o'rganadi. Motorlar qancha aylansa, robot qo'li qayerga yetadi?

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1. Asosiy Tushunchalar

Darajalar Erkinligi (DOF)

DOF (Degrees of Freedom) — robot mustaqil harakat qila oladigan yo'nalishlar soni.

• 6 DOF — to'liq fazoviy harakat (3 translatsiya + 3 rotatsiya)
• 3 DOF — tekislikda harakat

Bo'g'in Turlari

Turi	Belgi	DOF	Misol
Revolute	R	1	Aylanish
Prismatic	P	1	Siljish
Spherical	S	3	Sharli

Kinematika Turlari

• Forward (To'g'ri): Bo'g'in burchaklari → End effector pozitsiyasi
• Inverse (Teskari): End effector pozitsiyasi → Bo'g'in burchaklari

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2. 2D Robot Qo'li (2-DOF)

Geometriya

      L₂
   ●-----● End Effector (x, y)
  /θ₂
 / 
●-----● Joint 2
 \θ₁   L₁
  \
   ● Base (0, 0)

Forward Kinematika

$$x = L_1\cos\theta_1 + L_2\cos(\theta_1 + \theta_2)$$ $$y = L_1\sin\theta_1 + L_2\sin(\theta_1 + \theta_2)$$

Inverse Kinematika

Maqsad: $(x, y)$ berilgan, $\theta_1, \theta_2$ topish.

1-qadam: $\theta_2$ ni toping (cosine qoidasi):

$$\cos\theta_2 = \frac{x^2 + y^2 - L_1^2 - L_2^2}{2L_1L_2}$$ $$\theta_2 = \pm\arccos\left(\frac{x^2 + y^2 - L_1^2 - L_2^2}{2L_1L_2}\right)$$

2-qadam: $\theta_1$ ni toping:

$$\theta_1 = \arctan2(y, x) - \arctan2(L_2\sin\theta_2, L_1 + L_2\cos\theta_2)$$

Ikki Yechim

$\theta_2$ uchun $+$ yoki $-$ — bu "tirsak yuqorida" yoki "tirsak pastda" konfiguratsiya.

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3. Denavit-Hartenberg (DH) Parametrlari

Robotning har bir bo'g'inini 4 ta parametr bilan ifodalash:

Parametr	Belgi	Ta'rif
Link uzunligi	$a_i$	$x_{i-1}$ bo'ylab masofa
Link twist	$\alpha_i$	$x_{i-1}$ atrofida burchak
Link offset	$d_i$	$z_i$ bo'ylab masofa
Joint burchak	$\theta_i$	$z_i$ atrofida burchak

Transformatsiya Matritsasi

$$T_i^{i-1} = \begin{bmatrix} \cos\theta_i & -\sin\theta_i\cos\alpha_i & \sin\theta_i\sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i\cos\alpha_i & -\cos\theta_i\sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Umumiy Transformatsiya

$$T_n^0 = T_1^0 \cdot T_2^1 \cdot T_3^2 \cdots T_n^{n-1}$$

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4. Jacobian Matritsasi

Tezliklar orasidagi bog'lanish:

$$\dot{\mathbf{x}} = J(\mathbf{q}) \cdot \dot{\mathbf{q}}$$

• $\dot{\mathbf{x}}$ — end effector tezligi (linear + angular)
• $\dot{\mathbf{q}}$ — bo'g'in tezliklari
• $J$ — Jacobian matritsa

2-DOF uchun Jacobian

$$J = \begin{bmatrix} -L_1\sin\theta_1 - L_2\sin(\theta_1+\theta_2) & -L_2\sin(\theta_1+\theta_2) \\ L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2) & L_2\cos(\theta_1+\theta_2) \end{bmatrix}$$

Singularlik

$\det(J) = 0$ bo'lganda — singularlik. Robot bu holatda ba'zi yo'nalishlarda harakat qila olmaydi.

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5. Workspace (Ish Hududi)

Robot yeta oladigan barcha nuqtalar to'plami.

Reachable Workspace

End effector yeta oladigan barcha nuqtalar.

Dexterous Workspace

End effector istalgan orientatsiyada yeta oladigan nuqtalar.

2-DOF uchun

$$|L_1 - L_2| \leq r \leq L_1 + L_2$$

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6. Trajectory Planning

Point-to-Point

Boshlang'ich va oxirgi nuqta berilgan:

Cubic polynomial:

$$\theta(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3$$

Chegaralar: $\theta(0) = \theta_0$, $\theta(T) = \theta_f$, $\dot{\theta}(0) = 0$, $\dot{\theta}(T) = 0$

Linear Interpolation with Parabolic Blends (LSPB)

Tezlanish va sekinlashish bilan tekis harakat.

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7. Differential Drive Robot

Mobil robot (2 g'ildirak + castor):

Kinematika

$$v = \frac{v_R + v_L}{2}$$ $$\omega = \frac{v_R - v_L}{L}$$

• $v$ — linear tezlik
• $\omega$ — burchak tezlik
• $L$ — g'ildiraklar orasidagi masofa

Inverse

$$v_R = v + \frac{\omega L}{2}$$ $$v_L = v - \frac{\omega L}{2}$$

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8. Odometriya

G'ildirak aylanishidan pozitsiyani hisoblash:

$$\Delta s = \frac{\Delta s_R + \Delta s_L}{2}$$ $$\Delta \theta = \frac{\Delta s_R - \Delta s_L}{L}$$ $$x_{new} = x + \Delta s \cdot \cos\left(\theta + \frac{\Delta\theta}{2}\right)$$ $$y_{new} = y + \Delta s \cdot \sin\left(\theta + \frac{\Delta\theta}{2}\right)$$

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Xulosa

Tushuncha	Formula/Ta'rif
Forward Kinematika	$\theta \to (x, y)$
Inverse Kinematika	$(x, y) \to \theta$
Jacobian	$\dot{x} = J\dot{q}$
Diff Drive	$v = (v_R + v_L)/2$
Odometriya	Encoder → pozitsiya

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Keyingi Qadam

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