IEEE Robotics & Automation Magazine - June 2013 - 93

Different Use Cases for Running Example
Coordinate Calculations
"e , E
T
"o 2 , O2

"e , E
T
"o 1 , O1
"e , E
T
"b ,B

=

=

=

" o 2 , O 2 -1
T ,
"e , E
"o 2 , O2
T
"o 1 , O1

"o 1 , O1
T
"b , B

(S1a)
"e , E
T
"o 2 , O2

(S1b)

"e , E
T.
"o 1 , O1

(S1c)

Semantics Checking
Pose ^" e , E, " o 2 , O 2 h

=

inverse(Pose ^" o 2 , O 2 " e , E h ),

(S2a)

Pose ^" e , E, " o 1 , O 1 h

=

compose (Pose ^" e , E, " o 2 , O 2 h, Pose ^" o 2 , O 2, " o 1 , O 1 h ),

(S2b)

Pose ^" e , E, " b , B h

=

compose (Pose ^" e , E, " o 1 , O 1 h, Pose ^" o 1 , O 1, " b , B h ).

(S2c)

Coordinate Semantics Checking
PoseCoord ^" e , E, " o 2 , O 2,6o 2@h =

inverse2(PoseCoord ^" o 2 , O 2, " e , E,6e @h ),

(S3a)

PoseCoord ^" e , E, " o 1 , O 1,6o 1@h

=

compose (PoseCoord ^" e , E, " o 2 , O 2,6o 2@h , PoseCoord ^" o 2 , O 2, " o 1 , O 1,6o 1@h ),

(S3b)

PoseCoord ^" e , E, " b , B,6b@h

=

compose (PoseCoord ^" e , E, " o 1 , O 1,6o 1@h , PoseCoord ^" o 1 , O 1, " b , B,6b@h ).

(S3c)

Coordinate Calculations and Coordinate Semantics Checking
Coordinate calculations using (S1a)-(S1c) and coordinate semantics checking using (S3a)-(S3c). Additionally, the semantic
constraints imposed by particular coordinate representations are checked (see [3]).

available: PoseCoord ^" o 1 , O 1, " b , B, 6b@h determined by
the mounting of the cylindrical object with respect to the
robot base, PoseCoord ^" o 2 , O 2, " e , E, 6e @h determined by
the mounting of the spray gun on the robot end-effector, and
PoseCoord ^" o 2 , O 2, " o 1 , O 1, 6o 1@h determined by the
desired relative spray-painting pose.
To find the joint angles of the robot, the robot programmer has to find the pose of the robot end-effector with respect
to the base, and then use the inverse kinematics of the robot.
The solution to this problem has the following outline: (1)
invert Pose ^" o 2 , O 2, " e , Eh to obtain the pose of the endeffector with respect to the spray gun, Pose
^" e , E, " o 2 , O 2 h; (2) compose Pose ^" e , E, " o 2 , O 2 h ,
and Pose ^" o 2 , O 2, " o 1 , O 1h to obtain the pose of the robot
end-effector with respect to the cylindrical object, Pose
^" e , E, " o 1 , O 1 h; and (3) compose Pose ^" e , E, " o 1 , O 1 h
and Pose ^" o 1 , O 1, " b , B h to obtain the pose of the robot
end-effector with respect to the robot base, Pose
^" e , E, " b , B h . Finally, the inverse kinematics of the robot
is used to calculate the joint angles from the end-effector pose
with respect to the base, Pose ^" e , E, " b , B h .

Coordinate Calculations
The first use case is the one that involves numerical calculations using particular coordinate representations, but without
the semantic checking. This is the level that all available
geometry libraries, such as Kinematics and Dynamics Library
(KDL) [5], Robot Operating System (ROS) geometry stack
[4], and Robotics Library (RL) [1], work on.
For the running example, an option is to use the 4 # 4
" ,
homogeneous transformation matrix [6] "hg , CD T as a coordinate repres ent ation for t he p os e Pos eC o ord
^" g , C, " h , D, 6h@h . In
this case, the inverse and
The software still has to
the compositions needed
in the solution outline (See
allow for efficient code
the "Running Example"
section) are obtained by
at runtime, which is of
doing a matrix inversion
and matrix multiplications,
particular importance in
respectively, as shown in
"Different Use Cases for
robotics applications.
Running Example."

Use Cases
Regarding calculations with the geometric relations between
rigid bodies, four use cases are relevant that are explained and
illustrated using the running example below.

Semantics Checking
The second use case is the one that involves semantic checking, but that abstracts from the actual coordinate frames and
of the particular coordinate representations. In this case, the

jUNE 2013

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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93
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2013
