Manipulator Kinematics
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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:
- 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:
<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>where <amsmath>\xi_i</amsmath> is the twist corresponding to the \th{i} joint axis in the reference (<amsmath>\theta = 0</amsmath>) configuration.
- 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.
- 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: <amsmath>e^{\widehat{\xi}\theta} p = q</amsmath> rotate one point onto another Subproblem 2: <amsmath>e^{\widehat{\xi}_1\theta_1} e^{\widehat{\xi}_2\theta_2} p = q</amsmath> rotate about two intersecting twists Subproblem 3: <amsmath>\|q - e^{\widehat{\xi}\theta} p\| = \delta</amsmath> 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.
- 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>:
<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>If the manipulator kinematics is written using the product of exponentials formula, then the manipulator Jacobians have the form:
<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> - 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
- 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
<amsmath> J_{st}(\theta) \dot\theta = 0.
</amsmath> - 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
<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>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
<amsmath> V_{st}^s = J_1^s(\Theta_1) \dot\Theta_1 = J_2^s(\Theta_2) \dot\Theta_2,
</amsmath>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.