IEEE Robotics & Automation Magazine - September 2010 - 93
positive semidefinite). Expressed in Pl€
ucker coordinates, the
spatial inertia of a rigid body is
I þ m c 3 c 3 T
I¼ c
m c3T
m c3
,
m1
(25)
where m is the body's mass, c is a 3-D vector locating the
body's center of mass, and Ic is the body's rotational inertia
about its center of mass. Observe that a rigid-body inertia is a
function of ten parameters: one in m, three in c, and six in Ic .
More general kinds of spatial inertia, such as articulated-body
and operational-space inertia, do not have the special form
shown in (25), and they can be functions of up to 21 independent parameters.
All spatial inertias, whether rigid or not, obey the following
coordinate-transformation rule:
B
I ¼ B X ÃA A I A X B :
(26)
A constraint force does no work
in any direction of motion permitted
by the constraint.
expressed in generalized coordinates. Second, it offers the
opportunity to split f into a known part and an unknown part,
and incorporate the former into p. For example, if the forces
acting on the body consisted of an unknown force and a gravitational force, you could define f in (30) to be the unknown
force and define p as follows:
p ¼ v 3Ã Iv À fg ,
where f g is the gravitational force. Incidentally, if ag is the
acceleration due to gravity (in a uniform gravitational field),
then the force of gravity acting on a rigid body with inertia I is
This formula is valid for any dual coordinate system, not only
Pl€
ucker coordinates. If a rigid body has a velocity of v and an
inertia of I, then the time derivative of its inertia is
fg ¼ Iag :
Motion Constraints
d
I ¼ v 3Ã I À Iv 3 :
dt
(27)
Another useful equation is
E¼
1
v Á Iv,
2
(28)
In the simplest case, a motion constraint between two rigid
bodies, B1 and B2 , restricts their relative velocity to a vector
subspace S M 6 , which can vary with time. If r is the dimension of S, then the constraint allows r degrees of relative
motion freedom between the two bodies, and consequently
imposes 6 À r constraints. If s1 Á Á Á sr are any set of vectors that
span S (i.e., they form a basis on S), then the relative velocity
can be expressed in the form
which gives the kinetic energy of a rigid body.
Equation of Motion
Expressed in spatial form, the equation of motion for a rigid
body having a velocity of v and an inertia of I is
f ¼
d
(Iv) ¼ Ia þ v 3Ã Iv,
dt
(29)
(30)
where p 2 F 6 is called a bias force. There are two main reasons
why you might want to do this. First, the algebraic form of this
equation is identical to the algebraic form of several other
important equations of motion, such as the articulated-body
equation of motion and the equation of motion of a rigid body
SEPTEMBER 2010
r
X
si q_ i ,
i¼1
where q_ i are a set of velocity variables. However, we usually
collect the vectors together into a single 6 3 r matrix S, and
express the relative velocity as follows:
v2 À v1 ¼ Sq_ ,
where f is the total force acting on the body, and a is the
resulting acceleration. [Can you verify this equation using (27)?]
In words, it says that the total force acting on a rigid body
equals its rate of change of momentum. This equation incorporates both Newton's equation applied to the center of mass
and Euler's equation for the rotation of the body about its center of mass.
It is often useful to write the equation of motion in the following simplified form:
f ¼ Ia þ p,
vrel ¼ v2 À v1 ¼
(31)
where q_ is an r-dimensional coordinate vector containing the
velocity variables. To obtain a constraint on the relative acceleration, we simply differentiate this equation, giving
a2 À a1 ¼ S_ q_ þ S€q:
(32)
_ q_ , and S would
In a typical dynamics problem, the quantities S,
all be known, and €q would be unknown. Expressions for S and
S_ will depend on the type of constraint. If the component
vectors of S are fixed in body i (i ¼ 1 or 2), then S_ ¼ vi 3 S.
Motion constraints are implemented by constraint forces,
and constraint forces all have the following special property: a
constraint force does no work in any direction of motion permitted by the constraint.
This is simply a statement of D'Alembert's principle of virtual
work, or Jourdain's principle of virtual power, depending on
IEEE Robotics & Automation Magazine
93
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