[1][2] Kinematics equations are used to analyze and design articulated systems ranging from four-bar linkages to serial and parallel robots.
[3][4] This convention positions the joint frame so that it consists of a screw displacement along the Z-axis and it positions the link frame so it consists of a screw displacement along the X-axis, The kinematics equations are obtained using a rigid transformation [Z] to characterize the relative movement allowed at each joint and separate rigid transformation [X] to define the dimensions of each link.
The first called forward kinematics uses specified values for the joint parameters to compute the end-effector position and orientation.
The second called inverse kinematics uses the position and orientation of the end-effector to compute the joint parameters values.
Remarkably, while the forward kinematics of a serial chain is a direct calculation of a single matrix equation, the forward kinematics of a parallel chain requires the simultaneous solution of multiple matrix equations which presents a significant challenge.