Since many metrics and measures are possible, it may sometimes seem that there is no ``correct'' choice. This can be frustrating because the performance of sampling-based planning algorithms can depend strongly on these. Conveniently, there is a natural measure, called the Haar measure, for some transformation groups, including . Good metrics also follow from the Haar measure, but unfortunately, there are still arbitrary alternatives.
The basic requirement is that the measure does not vary when the sets are transformed using the group elements. More formally, let represent a matrix group with real-valued entries, and let denote a measure on . If for any measurable subset , and any element , , then is called the Haar measure5.2 for . The notation represents the set of all matrices obtained by the product , for any . Similarly, represents all products of the form .
An invariant metric can be defined from the Haar measure on . For any points , let , in which is the shortest length (smallest measure) interval that contains and as endpoints. This metric was already given in Example 5.2.
To obtain examples that are not the Haar measure, let represent probability mass over and define any nonuniform probability density function (the uniform density yields the Haar measure). Any shifting of intervals will change the probability mass, resulting in a different measure.
Failing to use the Haar measure weights some parts of more
heavily than others. Sometimes imposing a bias may be desirable, but
it is at least as important to know how to eliminate bias. These
ideas may appear obvious, but in the case of and many other
groups it is more challenging to eliminate this bias and obtain the
Steven M LaValle 2012-04-20