Page 12 - Human Inspired Dexterity in Robotic Manipulation

6 Human Inspired Dexterity in Robotic Manipulation

1.2.5 Kinematics and Statics of Robots
m
Let r 2R be the vector representing the position and orientation of the
robot at the task frame (m is the dimension of the task space), and q 2 R n
be the joint vector with the n dimension. r is the function of q:
r ¼ rqðÞ (1.6)
It should be noted that the problem to derive r from the given q is
referred to as forward kinematics while the problem to derive q from the
given r is referred to as inverse kinematics. By differentiating Eq. (1.6) with
respect to time, we get

∂r
_ r ¼ _ q ¼ JqðÞ_q (1.7)
∂q
d
where J(q) is the Jacobian matrix. Let w 2R be the wrench whose compo-
nents are constructed by force and moment exerted at the task frame, and τ 2
n
R be the joint torque vector. From Eq. (1.7) and the principle of virtual
work, we have

(1.8)
T
τ ¼ JqðÞ w
For more details, please see robotic text books such as [2].

1.2.6 Dynamics of Robots

The dynamics of robots can be approximately represented by the model of
mechanical impedance while accurately derived as the equation of motion.
The equation of motion for robotics is given by

τ ¼ M€+ Cq, _qÞ_q + gqðÞ (1.9)
ð
q
where M is the inertia matrix, Cq, _qÞ_q is Coriolis and centrifugal forces, and
ð
g(q) is a gravitational term. The equation of motion can be derived by uti-
lizing Lagrange’s equations or Newton’s and Euler’s equations. See robotic
text books for a detailed deviation. If considering the w at the task frame, Eq.
(1.9) becomes

ð T (1.10)
τ ¼ M€+ Cq, _qÞ_q + gqðÞ + JqðÞ w
q
where statics term is added.
For more details, please see robotic text books such as [2].