Exemple Numèric de Dinàmica de Braç Robòtic

1. Paràmetres del Sistema

• Massa del segment 1 ($m_1$): 1 kg
• Massa del segment 2 ($m_2$): 0.5 kg
• Longitud del segment 1 ($l_1$): 1 m
• Longitud del segment 2 ($l_2$): 0.5 m
• Acceleració gravitatòria ($g$): 9.81 m/s²
• Angle del segment 1 ($\theta_1$): $\frac{\pi}{4}$ rad
• Angle del segment 2 ($\theta_2$): $\frac{\pi}{6}$ rad
• Velocitat angular del segment 1 ($\dot{\theta_1}$): 1 rad/s
• Velocitat angular del segment 2 ($\dot{\theta_2}$): 0.5 rad/s
• Acceleració angular del segment 1 ($\ddot{\theta_1}$): 0.1 rad/s²
• Acceleració angular del segment 2 ($\ddot{\theta_2}$): 0.05 rad/s²

2. Energia Cinètica

2.1. Posicions:

$$x_1 = l_1 \sin(\theta_1) = 1 \cdot \sin\left(\frac{\pi}{4}\right) = 0.707$$

$$y_1 = l_1 \cos(\theta_1) = 1 \cdot \cos\left(\frac{\pi}{4}\right) = 0.707$$

$$x_2 = l_1 \sin(\theta_1) + l_2 \sin(\theta_1 + \theta_2)$$ $$= 1 \cdot \sin\left(\frac{\pi}{4}\right) + 0.5 \cdot \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ $$= 0.707 + 0.5 \cdot \sin\left(\frac{5\pi}{12}\right)$$ $$= 0.707 + 0.5 \cdot 0.965 = 0.707 + 0.483 = 1.19$$

$$y_2 = -l_1 \cos(\theta_1) - l_2 \cos(\theta_1 + \theta_2)$$ $$= -1 \cdot \cos\left(\frac{\pi}{4}\right) - 0.5 \cdot \cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ $$= -0.707 - 0.5 \cdot 0.258 = -0.707 - 0.129 = -0.836$$

2.2. Velocitat:

$$\mathbf{v}_1 = \begin{bmatrix} l_1 \cos(\theta_1) \dot{\theta_1} \\ -l_1 \sin(\theta_1) \dot{\theta_1} \end{bmatrix} = \begin{bmatrix} 1 \cdot \cos\left(\frac{\pi}{4}\right) \cdot 1 \\ -1 \cdot \sin\left(\frac{\pi}{4}\right) \cdot 1 \end{bmatrix} = \begin{bmatrix} 0.707 \\ -0.707 \end{bmatrix}$$

$$\mathbf{v}_2 = \begin{bmatrix} l_1 \cos(\theta_1) \dot{\theta_1} + l_2 \cos(\theta_1 + \theta_2) (\dot{\theta_1} + \dot{\theta_2}) \\ -l_1 \sin(\theta_1) \dot{\theta_1} - l_2 \sin(\theta_1 + \theta_2) (\dot{\theta_1} + \dot{\theta_2}) \end{bmatrix}$$ $$= \begin{bmatrix} 0.707 + 0.5 \cdot \cos\left(\frac{5\pi}{12}\right) \cdot (1 + 0.5) \\ -0.707 - 0.5 \cdot \sin\left(\frac{5\pi}{12}\right) \cdot (1 + 0.5) \end{bmatrix}$$ $$= \begin{bmatrix} 0.707 + 0.5 \cdot 0.258 \cdot 1.5 \\ -0.707 - 0.5 \cdot 0.965 \cdot 1.5 \end{bmatrix}$$ $$= \begin{bmatrix} 0.707 + 0.194 \\ -0.707 - 0.723 \end{bmatrix} = \begin{bmatrix} 0.901 \\ -1.43 \end{bmatrix}$$

2.3. Energia Cinètica:

$$K_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \cdot 1 \cdot \left(0.707^2 + (-0.707)^2\right) = \frac{1}{2} \cdot (0.5 + 0.5) = 0.5$$

$$K_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \cdot 0.5 \cdot \left(0.901^2 + (-1.43)^2\right)$$ $$= 0.25 \cdot (0.812 + 2.044) = 0.25 \cdot 2.856 = 0.714$$

$$K = K_1 + K_2 = 0.5 + 0.714 = 1.214$$

3. Energia Potencial

3.1. Energia Potencial:

$$U = - (m_1 + m_2) g l_1 \cos(\theta_1) - m_2 g l_2 \cos(\theta_1 + \theta_2)$$

$$U = - (1 + 0.5) \cdot 9.81 \cdot 1 \cdot \cos\left(\frac{\pi}{4}\right) - 0.5 \cdot 9.81 \cdot 0.5 \cdot \cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ $$= -1.5 \cdot 9.81 \cdot 0.707 - 0.5 \cdot 9.81 \cdot 0.258$$ $$= -10.41 - 1.27 = -11.68$$

4. Lagrangiana

4.1. Lagrangiana:

$$L = K - U$$

$$L = 1.214 - (-11.68) = 1.214 + 11.68 = 12.894$$

5. Equacions de Lagrange

5.1. Equacions de Lagrange:

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta_i}} \right) - \frac{\partial L}{\partial \theta_i} = \tau_i$$

$$\frac{\partial L}{\partial \dot{\theta_1}} = \frac{\partial}{\partial \dot{\theta_1}} (0.5 \cdot (0.707^2 + (-0.707)^2)) = 0.5 \cdot \left(0.707 \cdot 1 + 0.707 \cdot 1\right) = 0.707$$

$$\frac{\partial L}{\partial \theta_1} = - \frac{\partial}{\partial \theta_1} (10.41 \cdot 0.707) = -10.41 \cdot (-0.707) = 7.36$$

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta_1}} \right) = \frac{d}{dt}(0.707) = 0$$

$$\tau_1 = 0 - 7.36 = -7.36$$

$$\frac{\partial L}{\partial \dot{\theta_2}} = \frac{\partial}{\partial \dot{\theta_2}} (0.714) = 0.714$$

$$\frac{\partial L}{\partial \theta_2} = - \frac{\partial}{\partial \theta_2} (1.27) = -1.27$$

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta_2}} \right) = \frac{d}{dt}(0.714) = 0$$

$$\tau_2 = 0 - (-1.27) = 1.27$$

6. Forces i Moments

6.1. Força de Reacció:

$$F_{r1} = m_1 g \sin(\theta_1) = 1 \cdot 9.81 \cdot \sin\left(\frac{\pi}{4}\right) = 9.81 \cdot 0.707 = 6.93$$

$$F_{r2} = m_2 g \sin(\theta_2) = 0.5 \cdot 9.81 \cdot \sin\left(\frac{\pi}{6}\right) = 0.5 \cdot 9.81 \cdot 0.5 = 2.45$$

6.2. Moment de Reacció:

$$M_{r1} = m_1 g l_1 \cos(\theta_1) = 1 \cdot 9.81 \cdot 1 \cdot \cos\left(\frac{\pi}{4}\right) = 9.81 \cdot 0.707 = 6.93$$

$$M_{r2} = m_2 g l_2 \cos(\theta_2) = 0.5 \cdot 9.81 \cdot 0.5 \cdot \cos\left(\frac{\pi}{6}\right) = 0.5 \cdot 9.81 \cdot 0.433 = 2.12$$