IEEE Robotics & Automation Magazine - September 2014 - 101

Read Mt and Mc

P

Read Mp

Identify the configuration
(link modules, linkages,
joint modules, VJs).

Find the port positions
of all the link modules
within Mp.

Mark the sequence of all
link modules in each linkage.

Mark the port positions
of VJs.

56~64
55

44~45

Find the center of mass of all
link modules in each linkage.

Get the D-H coordinate
systems based on
kinematic modeling.

43
31~42
30

Get the D-H dynamic
parameters based on
dynamic modeling.

20~29

19
Output all the D-H
kinematic parameters
and dynamic parameters.

10~18
9

Figure 5. The procedure of the automated kinematic and
dynamic model [9].

2~8

Center of Mass of Link i
Link i is assembled with n modules, so the coordinate ri of
the center of mass of link i is
ri = ^m i,1 ri,1 + m i,2 ri,2 + g + m i, n ri, nh /
(m i,1 + m i,2 + g + m i, ni),

(8)

Moment of Inertia of Link i
We ignore the product of inertia of the link and only develop
the representation of the moment of inertia. Note that the
moment of inertia must be with respect to a local coordinate
system that takes the center of mass as its origin and is further
parallel to Fi = " O i + 1,1, X i, Yi, Z i ,, as previously discussed.
The following describes the equation to compute the
moments of inertia of all modules in link i. For details, see [21].
=I x i,0 x i,0,1 + m i,1 ^^Yi,0 $ ^ri - ri,1hh2 + ^Z i,0 $ ^ri - ri,1hh2h,
2
2
i, 0 y i, 0,1 =I y i, 0 y i, 0,1 + m i,1 ^^ X i,0 $ ^ ri - ri,1 hh + ^ Z i,0 $ ^ ri - ri,1 hh h,
2
2
i, 0 z i, 0,1 =I z i, 0 z i, 0,1 + m i,1 ^^ X i,0 $ ^ ri - ri,1 hh + ^ Yi,0 $ ^ ri - ri,1 hh h,
(9)
i, 0 x i, 0,1

O

Z
Y

where ri is the center of mass of link i and ri, j is the center of
mass of each module in link i.
Note that the center of mass of each basic MPM module
(i.e., ri and ri, j ) should refer to a reference coordinate system
that is the coordinate system of link i, denoted by
Fi = " O i + 1,1, X i, Yi, Z i , (Figure 4). Note that this coordinate
system was determined in the kinematic modeling discussed
in the "Moment of Inertia of Link i" section.

rI x
rI y
rI z

1

O

X

Figure 6. An MPM system built with LEGO modules. The
backbone is marked with the curves, where dark (blue circle)
curves denote links and light (pink circle) curves denote joints.
Each link module is marked with a dot and is named sequentially.

where rI j1, j2, j3 represents the principal inertial moment of the
mass component j 3 at the mass center of link i with respect
to the inertial axes j 1 and j 2; I j1, j2, j3 represents the principal
inertial moment of the mass component j 3 at its mass center
with respect to the inertial axes j 1 and j 2 .
The moments of inertia of the other modules in link i can
also be obtained through a similar procedure. Therefore, the
moments of inertia of link i are described by
j =n

j =n

j =n

j =1

j =1

j =1

I x i x i = / rI xx, j I y i y i = / rI yy, j I z i z i = / rI zz, j ,

(10)

where / j = 1 rI xx, j represents the moments of inertia of link i
at its mass center with respect to the principal axes.
A procedure is used to generate the kinematic and
dynamic models for MPM systems automatically, as shown
j=n

september 2014

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

101
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2014
