Reasoning about operations on physical objects requires some way of expressing the sizes of objects, the sizes of object features, and distances between objects; this is the aspect of scene representation in robot reasoning systems which normally involves a large amount of quantitative information. The quantitative information available is essentially a set of measurements of the scene. For a qualitative representation to be useful, some size information must be retained, while reducing the reliance of the representation on numeric information.
What kind of size information should be retained in a qualitative representation for use by a robot? The alternatives include a quantity space with distinguished points, such as that used by Forbus [For84a], a set of fuzzy values, as defined by Zadeh [Zad79], a combination of the two techniques as proposed by D'Ambrosio [D'A87], or a set of relative sizes.
Typical tasks involving size judgement which a robot might encounter are:
The size of object components can be described relative to characteristic dimensions of the overall object. In two dimensions, for example, there are two characteristic size references which are easy to determine for any object. These are the widest extent of the object (along which I define the ``major axis'' in the first two-dimensional representation described below), and the narrowest extent (the ``minor axis'').
If some characteristic size is chosen as a local size reference value, it could be used as a distinguished point in the size quantity space, so that qualitative size distinctions are made by comparison to this value. There are however two disadvantages to such an approach. One of these is that there are qualitative size distinctions which cannot be made in terms of the reference. If a bolt were described by reference to its length and the width of its head, there would be no way of discriminating between the thread diameter and thread pitch. The second disadvantage is that the reference sizes may have no relevance at all to the operation being planned. If a collection of bolts are being matched to appropriate nuts, the length of each bolt is quite incidental.
The first of these disadvantages can be alleviated by further dividing the quantity space, in order to discriminate between sizes that are smaller than any of the reference values. This can be done by defining fractions of the reference values as new ``distinguished'' points. One of the representations described below uses this approach, and simply defines binary orders of magnitude as appropriate fractions, so that something more than half the size of the reference value can be contrasted with something less than quarter the size.
The second disadvantage is more difficult to relieve, and is actually exaggerated by the binary orders of magnitude approach, because of the effect of surplus distinguished points in the quantity space. These extra points can arbitrarily produce a perceived distinction between close values that are actually qualitatively equal, but happen to fall on either side of a binary order of magnitude point. The object-relative size representation described below does not attempt to solve this problem, but other researchers have attempted to do so - notably Connell and Brady, who used a Gray coded representation for the ordering of feature size ranges when measured as a proportion of object axis size [CB87]. The use of Gray coding allowed values to fall within overlap regions, where they share the properties of both neighbouring size ranges. Alternatively, magnitude comparison can be performed on a ``fuzzy'' basis, as described by D'Ambrosio [D'A87].
There are numerous ways of extending the local reference value approach so that the size criteria used are less arbitrary than simple comparison to overall object size. One of these is to allocate special values according to a statistical analysis of the distribution of feature sizes in the scene. It would then be possible to describe the sizes of individual shape features by reference to these statistically compiled values in the same way as for comparison to axis sizes.
Another approach to size comparison is to use some simple technique to broadly distinguish size across the whole scene, then provide a size-on-enquiry ability to determine relative sizes of objects only when that information is needed. This allows an initial simple distinction to be refined if necessary. Such an approach avoids the accumulation of superfluous and even misleading size classifications, but it does incur computational overhead, since a program acting on a scene description must decide when it is necessary to find further information.
Relative size representation is the first facility to be considered in designing a qualitative spatial reasoning system. It provides a natural way to describe size dependent operations without reference to any specific measurement system, and it provides facilities that can operate with inexact data. There are a number of options to follow in achieving relative size description. This type of capability is an immediately apparent requirement of a ``qualitative'' spatial reasoning system, since linear measurement information is the obvious candidate for application of the quantity space. The first of the two representations presented later in this chapter uses a system of distinguished points in the size quantity space that are derived from local shape axes, while the second uses a partial ordering of size on a global basis.