View source for Manipulator Kinematics

{{chapter header|Rigid Body Motion|Manipulator Kinematics|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 {{ch: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:
<ol>
<li>The ''forward kinematics' of a manipulator is described by a
mapping <amsmath>g_{st}:Q \to \mbox{\it{SE}}(3)</amsmath> 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'':
<center><amsmath>
  g_{st}(\theta) = e^{\widehat{\xi}_1 \theta_1} e^{\widehat{\xi}_2 \theta_2} \cdots 
  e^{\widehat{\xi}_n \theta_n} g_{st}(0),
</amsmath></center>
where <amsmath>\xi_i</amsmath> is the twist corresponding to the \th{i} joint axis in
the reference (<amsmath>\theta = 0</amsmath>) configuration.
</li>

<li>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.
</li>

<li>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:
{| align=center
|- 
| align=left   | Subproblem 1: &nbsp;&nbsp;
| align=center | <amsmath>e^{\widehat{\xi}\theta} p = q</amsmath>
| align=left   | &nbsp;&nbsp; rotate one point onto another
|-
| align=left   | Subproblem 2:
| align=center | <amsmath>e^{\widehat{\xi}_1\theta_1} e^{\widehat{\xi}_2\theta_2} p = q</amsmath>
| align=left   | &nbsp;&nbsp; rotate about two intersecting twists
|-
| align=left   | Subproblem 3:
| align=center | <amsmath>\|q - e^{\widehat{\xi}\theta} p\| = \delta</amsmath>
| align=left   | &nbsp;&nbsp; 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.
</li>

<li>The ''manipulator Jacobian'' relates the joint velocities
<amsmath>\dot\theta</amsmath> to the end-effector velocity <amsmath>V_{st}</amsmath> and the joint
torques <amsmath>\tau</amsmath> to the end-effector wrench <amsmath>F</amsmath>:
<center><amsmath>
  \alignedat 3
    V_{st}^s &= J_{st}^s(\theta) \dot\theta &\qquad 
      \tau &= (J_{st}^s)^T F_s &\qquad
      &\text{(spatial)} \\
    V_{st}^b &= J_{st}^b(\theta) \dot\theta &\qquad 
      \tau &= (J_{st}^b)^T F_t &\qquad
      &\text{(body)}.
  \endalignedat
</amsmath></center>
If the
manipulator kinematics is written using the product of exponentials
formula, then 
the manipulator Jacobians have the form:
<center><amsmath>
  \alignedat 2
  J_{st}^s(\theta) &= 
    \bmatrix \xi_1 & \xi_2' & \cdots & \xi_n' \endbmatrix
    &\qquad
    \xi_i' &= \operatorname{Ad}_{\bigl(
      \displaystyle
      e^{\widehat{\xi}_1 \theta_1} \cdots e^{\widehat{\xi}_{i-1} \theta_{i-1}}
    \bigr)} \xi_i \\
  J_{st}^b(\theta) &=
    \bmatrix 
      \xi_1^\dagger & \cdots & \xi_{n-1}^\dagger & \xi_n^\dagger 
    \endbmatrix &\qquad 
    \xi_i^\dagger &= \operatorname{Ad}^{-1}_{\bigl(
      \displaystyle 
      e^{\widehat{\xi}_i \theta_i} \cdots e^{\widehat{\xi}_n \theta_n} g_{st}(0)
    \bigr)} \xi_i.
  \endaligned
</amsmath></center>
</li>

<li>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.  
</li>

<li>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
<center><amsmath>
  J_{st}(\theta) \dot\theta = 0.
</amsmath></center>
</li>

<li>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
<center><amsmath>
  g_{st} =
  e^{\widehat{\xi}_{11}\theta_{11}} \cdots e^{\widehat{\xi}_{1n_1}\theta_{1n_1}} g_{st}(0) =
  e^{\widehat{\xi}_{21}\theta_{21}} \cdots e^{\widehat{\xi}_{2n_2}\theta_{2n_2}} g_{st}(0),
</amsmath></center>
where <amsmath>\xi_{ij}</amsmath> is twist for the the \th{j} joint on the \th{i} chain.
The Jacobian of the structure equation has the form
<center><amsmath>
  V_{st}^s = J_1^s(\Theta_1) \dot\Theta_1 = J_2^s(\Theta_2) \dot\Theta_2,
</amsmath></center>
where <amsmath>\Theta_i = (\theta_{i1}, \dots, \theta_{in_i})</amsmath>.  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.
</li>
</ol>

== Additional Information ==