Page 58 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

DYNAMICS 33

has only two degrees of freedom, angles θ 1 and θ 2 , two DOF that fully define its
configuration in space. All such sets (θ 1 ,θ 2 ) define the arm’s two-dimensional
C-space.

2.2 STATICS

As formulated in Section 2.1, statics describes (a) the relationships between
forces and torques that the arm exerts on the surrounding objects and (b) the
relationships between internal forces and torques at the links. Statics analysis is
done on an isolated link, taking into account the forces and torques contributed
by neighboring links. For link i, the result of analysis is the net force f i ,net
torque n i ,and

m i g Force of gravity
f i−1,i Force exerted on link i by link i − 1
n i−1,i Torque exerted on link i by link i − 1
The force balance is

f i = f i−1,i − f i,i+1 + m i g (2.7)

The minus in front of the second term above is due to the changed direction of
the exerted force. In Figure 2.5, we have

∗
p − r ∗ Vector from the link i center of gravity to the joint i
i i
r ∗ Vector from the link i center of gravity to the joint i + 1
i
The torque balance is (see Figure 2.5)

∗ ∗ ∗
n i = n i−1,i − n i,i+1 − (p + r ) × f i−1,i + r × f i,i+1 (2.8)
i
i
i
2.3 DYNAMICS

As defined above, dynamics describes relationships between kinematics and stat-
ics. For example, the relation between torques at the arm joints and link positions
represents the arm’s dynamics. Dynamics is typically a final step in deriving joint
torques. Given various forces acting on the arm, one needs joint torques to realize
the desired trajectory. Then by Newton–Euler equations we relate the D’Alamber
force f i and torque n i (from the static equations) to the acceleration of link i [7].
Let r i be the vector from the arm base to the center of mass of link i
(Figure 2.5). Take two links, link 1 and link 2, of masses m 1 and m 2 , respec-
tively. Then the net forces f 1 and f 2 acting upon the links 1 and 2 relate to