Editing Robot kinematics (section)

==Robot Jacobian==
The time derivative of the kinematics equations yields the [[Jacobian matrix and determinant|Jacobian]] of the robot, which relates the joint rates to the linear and [[angular velocity]] of the end-effector.  The principle of [[virtual work]] shows that the Jacobian also provides a relationship between joint torques and the resultant force and torque applied by the end-effector.  Singular configurations of the robot are identified by studying its Jacobian.

===Velocity kinematics===

The robot Jacobian results in a set of linear equations that relate the joint rates to the six-vector formed from the angular and linear velocity of the end-effector, known as a [[Twist (screw theory)|twist]].  Specifying the joint rates yields the end-effector twist directly.

The '''inverse velocity''' problem seeks the joint rates that provide a specified end-effector twist.  This is solved by inverting the [[Jacobian matrix]]. It can happen that the robot is in a configuration where the Jacobian does not have an inverse.  These are termed singular configurations of the robot.

===Static force analysis===

The principle of [[virtual work]] yields a set of linear equations that relate the resultant force-torque six vector, called a [[screw theory|wrench]], that acts on the end-effector to the joint torques of the robot.  If the end-effector [[Wrench (screw theory)|wrench]] is known, then a direct calculation yields the joint torques.

The '''inverse statics''' problem seeks the end-effector wrench associated with a given set of joint torques, and requires the inverse of the Jacobian matrix.  As in the case of inverse velocity analysis, at singular configurations this problem cannot be solved. However, near singularities small actuator torques result in a large end-effector wrench. Thus near singularity configurations robots have large [[mechanical advantage]].