For example, we can imagine when we try to move the whole hand. The first thing that we do is to calculate the minimum distance that could be reached by our hand only using the wrist joint. After that, we then move backwards and try manipulate the elbow joint and again calculate the minimum distance that could be reached by our hand. We do this until the last joint is manipulated, for instance, the shoulder joint, or even our backbone joint. Each human body joint are capable to rotate in one to three different axis (3-Dimension). This ability to rotate is called Degree of Freedom of a joint. So each joint could have 1 to 3 Degree of Freedom (DOF). Using CCD method, we try to find the best joint angle for x-axis, y-axis and z-axis for a joint given the coordinate of the destination point (i.e. the best joint angle of each axis that could make the end-effector closer to the destination point). Suppose a joint cannot move in an axis, the best joint angle for that axis is zero. After calculating both three joint angles for each axis, we then calculate the rotation matrix for each axis, using the following rotation matrix:
Where RotX is the rotation matrix for x-axis, RotY is for y-axis and RotZ is for z-axis. We then combine these rotation matrixes into a single rotation matrix:
Where R matrix is the rotation matrix of a joint. Applying rotation matrix of Joint(i) towards the end-effector will cause all the part from Joint(i) up to the end-effector to rotate. After that, the calculation will move backwards to Joint(i-1) and again we calculate the joint angle for Joint(i-1) and calculate the rotation matrix. In order to move end-effector given the rotation of Joint(i-1), besides rotation matrix of the current joint, we also need rotation matrix of the previous joint and the translation matrix of the previous link. The following is the translation matrix:
Where dX, dY, dZ is the length of the link (i.e. the distance between two joints). For transforming the end-effector given the rotation of Joint(i-1), we need the matrix of R(i-1)*T(i)*R(i), where T(i) is the translation matrix from the current joint to the previous joint (i.e. length of the link) and R(i) is the rotation matrix from the previous joint. Keep in mind that the calculation is done backwards starting from the end-effector towards the parent joint.This calculation will continue until we reached the parent joint where there will be no more joints that could rotate after this joint. After reaching the parent joint, the calculated distance between current end-effector location and the desired location should be as minimum as possible.