Binding Subcomponents with Degrees of Freedom

Subcomponents whose attachment is only partially constrainted (such as the lowerarm ASSEMBLY of a PUMA robot) also need to be aggregated. As the subcomponent has been recognized previously, its coordinate frame must be consistent with the object's, given the degrees-of-freedom inherent in the modeled relationship between their respective coordinate frames. At the same time, the test also binds the values for the remaining degrees-of-freedom in the coordinate relationship. This results in numerical values being bound in the particular hypothesis context for the symbolic variables used in the model definition (Chapter 7). Figure 9.21 illustrates the partially constrained subcomponent attachment process.

Figure 9.21: Partially Constrained Subcomponent Aggregation (in 2D)

The matching and binding is by a "numerical" unification process:

Let:
	$G$ be the global reference frame transformation for the object
	$S$ be the global reference frame transformation for the subcomponent
	$A(\vec{x})$ be the transformation from the subcomponent's reference frame to the object's reference frame with $\vec{x}$ as the unbound variables
Then:
	Compare $G^{-1}S$ to $A(\vec{x})$
	Where $A(\vec{x})$ has bound variables, then the values must match (i.e. parameter estimate ranges overlap)
	Where $A(\vec{x})$ has unbound variables, then the variables are set to the corresponding parameter ranges from $G^{-1}S$

The results for the correct partially constrained subcomponent aggregations are summarized in Table 9.13. All correct bindings were made, and the table shows that the connection parameters were estimated well. Most of the incorrect combinatorial subcomponent groupings were eliminated during the binding process because of inconsistent reference frames, so this was also a benefit.

In the test scene, one binding occurred between inappropriate hypotheses, when constructing the armasm ASSEMBLY. Because of symmetry, each of its two subcomponents (upperarm and lowerarm) had two hypotheses. Hypothesis pairing produced four pairs, of which two passed the above test. Only one should have passed, but because of the tolerances on reference frame estimates, it was not possible to eliminate the grouping that placed the lower arm behind the upper arm solely by using geometric constraints. Self-occlusion analysis during verification (Chapter 10) did eliminate the extra hypothesis, however.

Table 9.13: Correct Partially Constrained Subcomponent Attachments

Object	Subcomponents	Modeled Parameter	Measured Value	Estimated Value
robot	link, robbody	jnt1	2.24	2.18
link	armasm, robshould	jnt2	2.82	2.79
armasm	upperarm, lowerarm	jnt3	4.94	4.56