IEEE Robotics & Automation Magazine - September 2010 - 87

3-D vectors, and "Solving a Two-Body Dynamics Problem
Using Spatial Vectors" shows how to solve the same problem
using spatial vectors. We shall refer to them as the 3-D example and the spatial example, respectively. The 3-D example
employs the methods of 3-D vectorial dynamics, and the spatial
example employs the methods of spatial vector algebra. Even
the briefest glance reveals that the spatial example is simpler in
every aspect: the problem statement is shorter, the diagram is
simpler, and the solution is much shorter. Let us now examine
these examples in more detail.
Starting with the 3-D example, eight quantities are needed
to describe the rigid-body system: the mass, center of mass,
and rotational inertia of each body, plus the quantities P and s
to define the joint axis. A further two quantities are needed to
describe the forces acting on B1 , and a further two to describe
its resulting acceleration. Furthermore, it is not enough merely
to state that f , n, a1 , and x_ 1 describe the forces and accelerations-a complete description requires that we also identify
the line of action of f and the particular point in B1 to which
a1 refers. Having introduced the three points C1 , C2 , and P as
a necessary part of describing the problem, it becomes desirable to show these points on the diagram, as they will play a
major role in the solution process.
In the terminology of the 3-D-vector approach, f , n, a1 , and
x_ 1 are said to be referred to (or expressed at) C1 , meaning that C1
serves as the reference point for these quantities. Equations (S1) and
(S2) are likewise referred to (expressed at) C1 . This need to define
various points in space, and to refer various vectors and equations to
these points, is a characteristic feature of the 3-D-vector approach
to solving a rigid-body problem. It accounts for a large part of the
algebraic complexity, and it forces the analyst to think explicitly in
terms of which point will be used to express which equation and
which quantities will have to be transferred from one reference
point to another. A poor choice of reference points can render a
complicated solution procedure even more complicated.
In the 3-D example, we can see that the equations of
motion of each body have been expressed at their respective
centers of mass, and that the equations of constraint (S14)-
(S16) have been expressed at P. These are good choices, but
they require us to define an extra eight quantities (Pa1 to 1 n2 )
and an extra seven equations (S5)-(S11) to manage all the necessary transfers of vectors from one reference point to another.
At the highest level, the solution strategy is this: express the
acceleration of B2 in terms of a1 , x_ 1 , and a, and then use the
force-constraint equation (S16) to obtain an expression for a
in terms of a1 and x_ 1 . At this point, every force and acceleration in the system can be expressed in terms of a1 and x_ 1 ; so,
the solution is obtained by expressing f and n in terms of a1
and x_ 1 , and then inverting the equations to express the accelerations in terms of the forces.
Let us now examine the spatial-vector example. In this case,
only three quantities (I 1 , I 2 , and s) are required to describe the
rigid-body system; only one quantity ( f ) is required to
describe the forces acting on B1 ; and only one quantity (a1 ) is
required to describe its acceleration. Furthermore, the solution
procedure introduces only another three quantities (a2 , f J , and
a). So, the whole problem now involves only eight quantities.
SEPTEMBER 2010

Observe that there is no mention of any 3-D point anywhere in this example. The problem has been stated and
solved without reference to any point in space. This absence
of reference points is a characteristic feature of the spatial-vector approach (and some other 6-D formalisms) and is a key
aspect of thinking in 6-D.
Referring back to the 3-D example, it is clearly possible to
pair up corresponding 3-D vectors ( f with n, a1 with x_ 1 , and
so on) to make 6-D vectors, and this would result in some reduction in the volume of algebra. However, the points C1 , C2 , and
P would still be an essential part of the problem statement and
the solution process. Thus, the stacking of pairs of 3-D vectors is
purely a notational device: the resulting vectors are 6-D, but the
concepts, methods, and thought processes are all still 3-D.
Returning to the spatial-vector example, the diagram is
clearly simpler, but the arrows now have different meanings.
The arrow associated with f , which points from empty space
to B1 , indicates only that f is an external force acting on B1 . It
does not convey any geometrical information (such as the line
of action of a force). Likewise, the arrow associated with f J ,
which points from B1 to B2 , indicates only that f J is a force
transmitted from B1 to B2 , whereas the arrow associated with
a1 , which points out of B1 , indicates only that a1 is the acceleration of B1 . From the directions of the arrows (and knowledge
of spatial vectors), we can immediately deduce that the net
force on B1 is f À f J and the net force on B2 is f J . The arrow
associated with s, which is aligned with the joint's rotation
axis, is the only one with any geometrical significance.
The reason why there are no 3-D points in the spatial-vector example and why most of the arrows have no geometrical
significance is because all the necessary positional information
is intrinsic to the relevant spatial quantities. The inertias I 1 and
I 2 implicitly locate the centers of mass of the two bodies; the
line of action of f (if it has one) can be deduced from its value;
and s defines both the direction and the location of an axis of
rotation in 3-D space. Acceleration is a little more complicated
and will be discussed in a later section. Nevertheless, a1 does
provide a complete description of a body's acceleration and
does not need to be referred to any point.
The high-level solution strategy in the spatial-vector example is the same as that in the 3-D example: express a2 in terms
of a1 and a; then, substitute into the force-constraint equation
(S36) to obtain an expression for a in terms of a1 ; then, express
f in terms of a1 and invert to express a1 in terms of f . Using
spatial vectors, the analyst is able to follow this high-level strategy
directly, without having to think about the messy details associated with the 3-D-vector approach. Observe how the expression for a is obtained almost immediately, after just two simple
substitutions, and the desired expression for f is obtained after
just three more simple substitutions.
Another benefit of spatial vectors, which is not evident
from this example, is that it is quite easy to prove that the
expression in parentheses in (S39) is a symmetric, positive-definite matrix, and therefore invertible. The same is also true of
the 6 3 6 matrix in (S27) of the 3-D example, but the proof
(using 3-D vectors) is relatively complicated. Yet another
advantage of spatial vectors is that a person who is new to the
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Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2010