Manipulator Kinematics

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Chapter 3 - Manipulator Kinematics

Next: Robot Dynamics and Control

The kinematics of a robot manipulator describes the relationship between the motion of the joints of the manipulator and the resulting motion of the rigid bodies which form the robot. This chapter gives a description of the kinematics for a general <math>n</math> degree of freedom, open-chain robot manipulator using the tools presented in Chapter 2 - Rigid Body Motion. We also present a brief treatment of redundant and parallel manipulators using this same framework.

Chapter Summary

The following are the key concepts covered in this chapter:

1. The forward kinematics' of a manipulator is described by a mapping which describes the end-effector configuration as a function of the robot joint variables. For open-chain manipulators consisting of revolute and prismatic joints, the kinematics can be written using the product of exponentials formula:
   

   where is the twist corresponding to the \th{i} joint axis in the reference () configuration.

2. The (complete) workspace of a manipulator is the set of end-effector configurations which can be reached by some choice of joint angles. The reachable workspace defines end-effector positions which can be reached at some orientation. The dextrous workspace defines end-effector positions which can be reached at any orientation.
3. The inverse kinematics of a manipulator describes the relationship between the end-effector configuration and the joint angles which achieve that configuration. For many manipulators, we can find the inverse kinematics by making use of the following subproblems:

   Subproblem 1:		rotate one point onto another
   Subproblem 2:		rotate about two intersecting twists
   Subproblem 3:		move one point to a specified distance from another

   To find a complete solution, we apply the manipulator kinematics to a set of points which reduce the complete problem into an appropriate set of subproblems.

4. The manipulator Jacobian relates the joint velocities to the end-effector velocity and the joint torques to the end-effector wrench :
   

   If the manipulator kinematics is written using the product of exponentials formula, then the manipulator Jacobians have the form:

   
5. A configuration is singular if the manipulator Jacobian loses rank at that configuration. Examples for a general six degree of freedom arm include:
   • Two collinear revolute joints
   • Three parallel, coplanar revolute joint axes
   • Four intersecting revolute joint axes
   The manipulability of a robot provides a measure of the nearness to singularity.
6. A manipulator is kinematically redundant if it has more than the minimally required degrees of freedom. The self-motion manifold describes the set of joint values which can be used to achieve a desired configuration of the end-effector. Internal motions correspond to motions along the self-motion manifold and satisfy
   
7. A parallel manipulator has multiple kinematic chains connecting the base to the end-effector. For the case of two chains, the kinematics satisfies the structure equation
   

   where is twist for the the \th{j} joint on the \th{i} chain. The Jacobian of the structure equation has the form

   

   where . A kinematic singularity occurs when the dimension of the space of admissible forces drops rank. Other singularities can occur when the set of end-effector forces which can be generated by the actuated joints drops rank.

Additional Information