Biomedical Engineering Reference
In-Depth Information
Fig. 6.4 The torque
controller of the iCub. See
text for details
6.3.1
iCub Dynamics
We start by considering an open (single or multiple branches) kinematic chain with
n
DoF composed of
n
+ 1 links. Adopting the Denavit-Hartenberg notation
(Sciavicco and Siciliano
2005
), we define a set of reference frames
h
0
i
,
h
1
i
,
...
,
, attached at each link. The
i
th
link of the chain is described by a vertex
v
i
(sometimes called node), usually represented by the symbol
h
n
i
. A hinge joint
between the link
i
and the link
j
(i.e., a rotational joint) is represented by an oriented
edge
e
i,j
connecting
v
i
with
v
j
:
i j
;. In an
n
DoF open chain, each vertex
(except for the initial and terminal,
v
0
and
v
n
, respectively) has two edges. There-
fore, the graph representation of the
n
-link chain is an oriented sequence of nodes
v
i
,
connected by edges
e
i
1,
i
. The orientation of the edges can be either chosen
arbitrarily (it will be clear later on that the orientation simply induces a convention)
or it can follow from the exploration of the kinematic tree according to the
regular
numbering scheme
(Featherstone and Orin
2008
), which induces a parent-child
relationship such that each node has a unique input edge and multiple output edges.
We further follow the classical Denavit-Hartenberg notation, we assume that each
joint has an associated reference frame with the z-axis aligned with the rotation
axis; this frame will be denoted
i
h
e
i
,
j
i
. In kinematics, an edge
e
i,j
from
v
i
to
v
j
is fixed in the
i
th
link. In dynamics,
e
i,j
represents the
fact that the dynamic equations will compute (and make use of)
w
i,j
, i.e., the wrench
that the
i
th
link exerts on the
j
th
link, and not the equal and opposite reaction—
w
i,j
,i.
e., the wrench that the
j
th
link exerts on the
i
th
link. In order to simplify the
computations of the inverse dynamics on the graph, kinematic and dynamic mea-
surements have been explicitly represented. Specifically, the graph representation
has been enhanced with a new set of graphical symbols: a triangle to represent
kinematic quantities (i.e., velocities and acceleration of links—
represents the fact that
h
e
i
,
j
i
ˉ
,
ˉ
,
p
,
_
€
p
) and a
rhombus for wrenches (i.e., force sensors measurements on a link
). More-
over these symbols have been further divided into known quantities to represent
sensors measurements and unknown to indicate the quantities to be computed, as in
the following:
f,
ʼ
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