Sensorimotor Coordination in a Humanoid Robot: Building Intelligence on the iCub - Bioinspired Approaches for Human-Centric Technologies - page 162

• ∇ : Unknown kinematic information

• : Known (e.g., measured) kinematic information

▼

• : Unknown dynamic information

◊

• ♦ : Known (e.g., measured) dynamic information

In general, kinematic variables can be measured by means of gyroscopes,
accelerometers, or simply inertial sensors. When attached on link i th , these sensors
provide angular and linear velocities and accelerations

ð ˉ ˉ p , and p Þ

at the
specific location where the sensor is located. We can represent this measurement
in the graph with a black triangle ( ▼ ) and an additional edge from the proper link
where the sensor is attached to the triangle. As usual, the edge has an associated
reference frame, in this case corresponding to the reference frame of the sensor. An
unknown kinematic variable is represented by a white triangle (

, ,

) with an asso-
ciated edge going from the link (where the unknown kinematic variable is attached)
to the triangle. Similarly, we introduce two new types of nodes with a rhomboidal
shape: black rhombus (

∇

) to represent known (i.e., measured) wrenches and white

♦

rhombus ( ) to represent unknown wrenches which need to be computed. The

reference frame associated to the edge will be the location of the applied or
unknown wrench. The complete graph for the iCub is shown in Fig. 6.5 .
From the graph structure, we can define the update rule that brings information
across edges, and by traversing the graph, we therefore compute either dynamical or
kinematic unknowns (

◊

and ∇ , respectively). For kinematic quantities this is

◊

ˉ iþ 1 ﾼ ˉ i þ ʸ iþ 1 z i ,
ˉ iþ 1 ﾼ ˉ i þ ʸ iþ 1 z i þ ʸ iþ 1 ˉ i z i , ð 6 : 8 Þ

p iþ 1 ﾼ p i þ ˉ i r i , iþ 1 þ ˉ iþ 1 ˉ iþ 1 ð r i , iþ 1 Þ

,

where z i is the z -axis of , i.e., we propagate information from the base to the

end-effector visiting all nodes and moving from one node to the next following the
edges. The internal dynamics of the manipulator can be studied as well: if the
dynamical parameters of the system are known (mass m i , inertia I i , center of mass
C i ), then we can propagate knowledge of wrenches applied to, e.g., the end-effector
( f n +1 and

h i i

ʼ n +1 ) to the base frame of the manipulator so as to retrieve forces and
moments f i ,

ʼ i :

m i €

f i ﾼ f iþ 1 þ p C i ,

ð 6 : 9 Þ

ʼ i ﾼ ʼ iþ 1 f i r i 1, C i þ f iþ 1 r r i , C i þ I i ˉ i þ ˉ i I i ˉðÞ ,

where

€ p C i ﾼ € p i þ ˉ i r i , C i þ ˉ i ˉ i ð r i , C i Þ , ð 6 : 10 Þ

noting that these are the classical recursive Newton-Euler equations. Knowledge of
wrenches enables the computation of w i as needed in ( 6.2 ) or the corresponding
joint torques from

i z i 1 .

τ i ﾼ ʼ