Manipulator Kinematics
Figure 1.
Manipulator kinematic modeling with static and motion parts
Figure 1 shows a kinematic model of the manipulator under this study. The method presented in (Xi, 2009; Lin, Xi, Mohamed, & Tu, 2010) is used here for kinematic modeling. This method formulates the manipulator kinematics through two parts. The first is a static part to represent the initial configuration of each link and the second is a motion part to represent the movement of each joint. For the static part, a set of initial configuration set-up (ICSU) are defined including a static rotation matrix 
and a static body vector 
, for each link, that is
(1)(2) where
,
,
are three rotation matrices about X, Y, and Z axis of local frame
relative to local frame
, and
,
,
are the three unit vectors of local frame
attached to the
th joint.
is a vector representing the
th link at the initial configuration. In this chapter, a bold vector is expressed with respect to the base frame, and a bold vector with an apostrophe is expressed with respect to a local frame.
The total translation 
and total rotation 
of the 
th link is expressed by including the motion part as
(3)(4) where
and
are the motional rotation matrix and motional body vector, respectively. In practice,
corresponds to a rotational joint driven by a rotary motor and
corresponds to a prismatic joint driven by a linear motor. In case of revolution joint,
is equal to
, and
is the rotation angle of the
th joint.
To this end, the end-effector’s position 
and orientation 
, can be expressed with respect to the base frame of the mobile platform as
(5)(6) where
represents the rotation matrix of link
with respect to the base frame, which is multiplied sequentially by a number of
. In eqn. (5), subscript
indicates the tip of the last link where the end-effector is located, and the end-effector’s orientation coincides with the last link, expressed by
.
Taking the time derivative of eqn. (5) and (6) leads to the following forward recursive velocity equations
(7)(8) where
is the velocity vector at the
th joint and
is the angular velocity vector of the
th link. Note that the following holds
(9)(10)Furthermore, taking the time derivative of eqn. (7) and (8) results in the following forward recursive acceleration equations
(11)(12) where
is the acceleration vector at the
th joint and
is the angular acceleration vector of the
th link.