Velocitat del efector per un robot planar 2DOF

Dades del problema

Braços del robot: \(a_1 = 3\), \(a_2 = 2\)
Angles: \(q_1 = 30^\circ = \pi/6 \, \text{rad}\), \(q_2 = 45^\circ = \pi/4 \, \text{rad}\)
Velocitats articulars: \(\dot{q}_1 = 1 \, \text{rad/s}\), \(\dot{q}_2 = 0.5 \, \text{rad/s}\)

1️⃣ Posició del efector

\[ x = a_1 \cos q_1 + a_2 \cos(q_1 + q_2), \quad y = a_1 \sin q_1 + a_2 \sin(q_1 + q_2) \]

2️⃣ Jacobiano

\[ \begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix} = \begin{bmatrix} \frac{\partial x}{\partial q_1} & \frac{\partial x}{\partial q_2} \\ \frac{\partial y}{\partial q_1} & \frac{\partial y}{\partial q_2} \end{bmatrix} \begin{bmatrix} \dot{q}_1 \\ \dot{q}_2 \end{bmatrix} \] \[ \frac{\partial x}{\partial q_1} = -a_1 \sin q_1 - a_2 \sin(q_1 + q_2), \quad \frac{\partial x}{\partial q_2} = - a_2 \sin(q_1 + q_2) \] \[ \frac{\partial y}{\partial q_1} = a_1 \cos q_1 + a_2 \cos(q_1 + q_2), \quad \frac{\partial y}{\partial q_2} = a_2 \cos(q_1 + q_2) \]

3️⃣ Substituïm valors

\[ q_1 + q_2 = 0.5236 + 0.7854 = 1.309 \, \text{rad} \] \[ \sin(q_1) = 0.5, \quad \cos(q_1) \approx 0.866 \] \[ \sin(q_1+q_2) \approx 0.966, \quad \cos(q_1+q_2) \approx 0.259 \]

4️⃣ Calcul del Jacobiano numèric

\[ \frac{\partial x}{\partial q_1} = -3(0.5) - 2(0.966) = -3.432 \] \[ \frac{\partial x}{\partial q_2} = -2(0.966) = -1.932 \] \[ \frac{\partial y}{\partial q_1} = 3(0.866) + 2(0.259) = 3.116 \] \[ \frac{\partial y}{\partial q_2} = 2(0.259) = 0.518 \] \[ J = \begin{bmatrix} -3.432 & -1.932 \\ 3.116 & 0.518 \end{bmatrix} \]

5️⃣ Calcul de la velocitat del efector

\[ \begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix} = \begin{bmatrix} -3.432 & -1.932 \\ 3.116 & 0.518 \end{bmatrix} \begin{bmatrix} 1 \\ 0.5 \end{bmatrix} \] \[ \dot{x} = -3.432 - 0.966 = -4.398, \quad \dot{y} = 3.116 + 0.259 = 3.375 \]

✅ Resultat final

\[ \boxed{\dot{x} \approx -4.40 \, \text{unitats/s}, \quad \dot{y} \approx 3.38 \, \text{unitats/s}} \]