User:Prof McCarthy/analysis/analysis draft

Analysis of kinematic chains
A kinematic chains is modeled using rigid transformations [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the dimensions of each link. For example, the constraint equations for a serial open chain are the sequence of rigid transformations alternating joint and link transformations from the base of the chain to end link, which is equated to the specified position for the end link. A chain of n links connected in series has the the kinematic equations,
 * $$[T] = [Z_1][X_1][Z_2][X_2]\ldots[X_{n-x}][Z_n],\!$$

where [T] is the transformation locating the end-link---notice that the chain includes a "zeroth" link which is the ground frame. These equations are called the forward kinematics equations of the serial chain.

The constraint equations for kinematic chains of a wide range of complexity are obtained bey equation the kinematics equations of serial open chains to form loops within the kinematic chain. These equations are often called loop equations.

The complexity (in terms of calculating the forward and inverse kinematics) of the chain is determined by the following factors: Explanation:- Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system.
 * Its topology: a serial chain, a parallel manipulator, a tree structure, or a graph.
 * Its geometrical form: how are neighbouring joints spatially connected to each other?

The term kinematic pairs actually refers to kinematic constraints between rigid bodies. The kinematic pairs are divided into lower pairs and higher pairs, depending on how the two bodies are in contact: Formula: L=2P-4 Where, L=No. of links, P=No. of pairs.