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Numerically determinate the manipulator Jacobian

In classic kinematics, the Jacobian is used to solve the inverse velocity kinematics.

The forward kinematics equation is given by \( \quad x = f(\theta) \)and the Jacobian matrix is a linear approximation to \(f\).

$$J(\theta) = \begin{bmatrix} \frac{\partial p_x}{\partial \theta_1} & \frac{\partial p_x}{\partial \theta_2} & … & \frac{\partial p_x}{\partial \theta_n} \\[0.3em] \frac{\partial p_y}{\partial \theta_1} & \frac{\partial p_y}{\partial \theta_2} & … & \frac{\partial p_y}{\partial \theta_n} \\[0.3em] … & … & … & … \\[0.3em] \frac{\partial a_z}{\partial \theta_1} & \frac{\partial a_z}{\partial \theta_2} & … & \frac{\partial a_z}{\partial \theta_n} \end{bmatrix}$$

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