IEEE Robotics & Automation Magazine - June 2013 - 92

Common Error 1: Logic Errors in Geometric Relation Calculations
Many logic errors can occur during geometric relation
calculations. For instance, the inverse of Position ^e C, f D h is
Position ^f D, e C h (point and reference point switch) while
the inverse of LinearVelocity ^e C, D h is LinearVelocity ^e D, C h
(point does not change). A second example appears when
composing the relations involving three rigid bodies: in order
to get the geometric relation of body C with respect to body
D, one can compose the geometric relation between C and a
third body E with the geometric relation between body E and
body D (and not the geometric relation between body D and
body E for instance).
The C++ code in Algorithm S1 and the program output in
Algorithm S2 show how the software proposed in this tutorial
prevents each of the above two logic errors.

Algorithm S1. Logic Errors in Geometric
Calculations-C++
// Inversion
PositionSemantics pos("e","C","f","D");
PositionSemantics pos_inverse = pos.inverse();
LinearVelocitySemantics linVel("e","C","D");
LinearVelocitySemantics linVel_inverse = 9
linVel.inverse();
// Composition
PoseSemantics pose1("g","g","C","h","h","D");
PoseSemantics pose2("i","i","E","h","h","D");

calculations with the geometric relations between rigid bodies on the top of existing geometric libraries, which only
work on specific coordinate representations. To our knowledge, the proposed software is the first to offer a semantic
interface for geometric operation software libraries.
It is assumed that the readers are familiar with the notation
and basic concepts of the semantic representation of the geometric relations and the corresponding semantic operations
introduced in [3].

{b}

{o1}

{o2}

{e}

Figure 1. The running example of a robot performing a spraypainting operation.

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PoseSemantics pose_composition = compose 9
(pose1, pose2);

Algorithm S2. Logic Errors in Geometric
Calculations-Output
// Inversion
Result: Inverse of Position(e|C,f|D) is 9
Position(f|D,e|C)
Result: Inverse of LinearVelocity(e|C,D) is 9
LinearVelocity(e|D,C)
// Composition
Semantic output: Composition of 9
Pose((g,[g])|C,(h,[h])|D) and 9
Pose((i,[i])|E,(h,[h])|D) is NOK since:
* Either the reference point, reference 9
orientation frame, and reference body of 9
Pose((g,[g])|C,(h,[h])|D) have to be equal to 9
the point, orientation frame, and body of 9
Pose((i,[i])|E,(h,[h])|D) respectively
OR
the point, orientation frame, and body of 9
Pose((g,[g])|C,(h,[h])|D) have to be 9
equal to the reference point, reference 9
orientation frame, and reference body of 9
Pose((i,[i])|E,(h,[h])|D) respectively.

Running Example
In this tutorial, we use a running example from robotics to
illustrate how the software helps to perform the semantic
checking for the geometric relations between rigid bodies.
The running example involves a robot that spray paints a
cylindrical object, as illustrated in Figure 1. The cylindrical
object is fixed in the environment, while the robot holds
the spray gun.
The running example does not attempt to show all the
possible errors the geometric semantics software can prevent, but rather to give a robotics showcase of the software.
See "Common Errors in Geometric Rigid-Body Relations
Calculations in Robotics" for more examples of how the
presented software can prevent common errors in geometric calculations.
To complete the painting task, the robot program has to
determine the joint angles of the robot holding the spray gun
such that a predefined relative pose between the spray gun
and cylindrical object is obtained.
The first step for solving the geometry problems is identifying the rigid bodies and the frames attached to them. In the
running example, we have: " b , attached to the base B of
the robot, " e , attached to the end-effector E of the robot,
" o 2 , attached to the spray gun O 2, and " o 1 , attached to
cylindrical object O 1 . In the example, the following poses are
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