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Unity Arm Robot Inverse Kinematics -analytical approach

Explaining analytical approach to inverse kinematics for robot arm. This method is applyed only for robots that have a spherical wrist (axis of joints P4,P5,P6 meets at one point). And robot should not have joint offsets. reference : textbook “Robotics” by Hirose Vector $(a,b)$ represents the direction and rotation of end effector. \begin{align*} (a,b) = R^Y R^P R^R\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{bmatrix} \end{align*} \begin{align*} \begin{bmatrix} a_{x} & b_{x} \\ a_{y} & b_{y} \\ a_{z} & b_{z} \end{bmatrix} = \begin{bmatrix} C_{Y} C_{P} & C_{Y} S_{P} C_{R} + S_{Y} S_{R} \\ S_{Y} C_{P} & S_{Y} S_{P} C_{R} – C_{Y} S_{R} \\ – S_{P} & …