geometry-doc.sgml

机器人开源项目orocos的源代码

is connected with through one of its ports.
</para>
<note>
<para>
This text collects all of the above-mentioned data structures in 
<emphasis>one single</emphasis> symbolic class, to be used as an
aggregate component of a
<link linkend="geometric-classes">geometric class</link>. This is to
some extent an arbitrary decision!
</para>
</note>
<para>
However, the decision to use all of the above-mentioned data
structures looks reasonable for a
<link linkend="geometric-database">geometric database</link>, because
of the involved level of complexity.
While in the case of a static, efficient and specific-purpose library
(<emphasis>e.g.</emphasis> to support the kinematics of one particular
robot), all of the symbolic information can be eliminated without many
problems.
</para>

</section>


<section id="physical-properties">
<title>Physical properties</title>
<para>
Most geometric objects have a set of operations that one can perform
on them, and that have <emphasis>physical</emphasis> meaning,
<emphasis>i.e.</emphasis> the result is independent of the data
structures and the coordinates used to represent the object. For
example, generating the normal direction to a given plane is a
physically uniquely defined operation, but that normal direction can
have many mathematical coordinate representations. Other examples of
physical operations are: finding the common perpendicular of two
lines; inverting the direction of a velocity; generating the
&ldquo;exponential&rdquo; curve generated by a given velocity; etc. 
</para>
<para>
Typically, every physical method can be implemented with various
mathematical representations. Developers must take care not to mix the
variability of the representations with the non-variability of a
physical method. Ideally, the mathematical representation should be
&quot;hidden&quot; in the method call of a physical property.
<note>
<para>
So, when developing a <emphasis>general</emphasis> geometric software
library, one is confronted with the practical problem that the
geometric object can store its state in one particular mathematical
representation, while the implementations of its physical methods
internally use another representation. This flexibility is only useful
in general-purpose, &ldquo;component-based&rdquo; systems
(<emphasis>e.g.</emphasis> as defined in the context of CORBA
components), but not for small-scale, &ldquo;real-time&rdquo;
applications. The design of the library interfaces in this document is
such that the application builders can select what level of generality
they want to implement; the most important factor in this flexibility
is that the mathematical coordinate representations and algorithms of
the physical properties can be used independently from any symbolic
&ldquo;meta&rdquo; properties.
</para>
</note>
</para>
<para>
Many (but not all) physical methods apply to similar geometric
objects in different dimensions (1D, 2D and 3D). For example, the
<emphasis>inversion</emphasis> of the velocity of an object. It is
advisable to use the same names for the same physical properties in
all different spaces.
</para>
<para>
In contrast to their symbolic properties, geometric objects do not
have many physical properties in common; hence, they are described
below in the sections of each individual geometric class.
</para>

</section>


<section id="geometric-classes">
<title>Geometric classes</title>
<para>
This Section describes the <emphasis>basic</emphasis> geometric
classes, without discussion of &ldquo;advanced&rdquo; concepts, such
as for example spline surfaces. Each geometric class has its own set
of <link linkend="physical-properties">physical properties</link>
(encoded as <emphasis role="strong">method calls</emphasis> in the
geometric class), coordinate representation
<emphasis role="strong">data structures</emphasis>, and possibly
aggregates a
<link linkend="symbolic-class">symbolic class</link> (encoded
as a <emphasis role="strong">data structure</emphasis>).
</para>

<section id="point">
<title>Point</title>
<para>
A &ldquo;Point&rdquo; represents a &ldquo;position&rdquo;, 
<emphasis>i.e.</emphasis> an element in the space in which the robot
or machine tool works.  Most of its properties are covered by the
generic <link linkend="symbolic-class">symbolic properties</link>: 
<parameter>name</parameter>.
<parameter>dimension</parameter>, and the various lists of
<parameter>Ports</parameter>.  The numeric coordinate representation
consists of three floating point numbers, in a given reference
<link linkend="frame">frame</link> (which is encoded by a 
<link linkend="symbolic-port">geometricPort</link>).
</para>

</section>


<section id="vector">
<title>Vector</title>
<para>
A Vector has many different meanings, hence, it has an extra
<link linkend="symbolic-class">symbolic property</link>:
<variablelist>
 <varlistentry>
 <term>
  <anchor id="vector-type">
  <parameter>type</parameter>:
 </term>
  <listitem>
  <para>
  <parameter>freeVector</parameter>,
  <parameter>lineVector</parameter>,
  <parameter>unitVector</parameter>,
  <parameter>pointVector</parameter>,
  <parameter>segmentVector</parameter>.
  </para>
  <para>
 A <parameter>freeVector</parameter> and a
 <parameter>unitVector</parameter> don't need extra symbolic
information; a <parameter>lineVector</parameter> and
<parameter>pointVector</parameter> need a
<link linkend="geometric-port">geometricPort</link> to connect it to
a geometric object with the corresponding type
(<emphasis>i.e.</emphasis> 
<link linkend="line">line</link> and
<link linkend="point">point</link>, respectively). A 
<parameter>segmentVector</parameter> is the vector that links two 
<link linkend="point">Points</link>, so it needs to be connected
to at least one, and in general, to both of these Points.
  </para>
  </listitem>
 </varlistentry>

</variablelist>
The numeric coordinate representation for all
<parameter>Vector</parameter>s consists of three floating point
numbers in a given reference <link linkend="frame">frame</link>.
</para>
<para>
A Vector has the following
<link linkend="physical-properties">physical properties</link>:
<variablelist>
 <varlistentry>
 <term>
<parameter>addition</parameter>, <parameter>inversion</parameter>,
<parameter>scaling</parameter>.
 </term>
  <listitem>
   <para>
The exact result of these operations can depend on the
<parameter>type</parameter> of the <parameter>Vector</parameter>
(<emphasis>e.g.</emphasis> <parameter>inversion</parameter> and
<parameter>addition</parameter> are meaningless for a
<parameter>lineVector</parameter>).
   </para>
  </listitem>
 </varlistentry>
</variablelist>
</para>

</section>


<section id="frame">
<title>Frame</title>
<para>
A Frame is the combination of a <link linkend="point">Point</link> and
an ordered list of 1, 2 or 3 (depending on the dimension of the space)
(unit) <link linkend="vector">Vector</link>s.
The geometric importance of a Frame is two-fold: (i) it is the vehicle
in which <emphasis role="strong">numeric coordinates</emphasis> are
defined; and (ii) it marks positions and orientations in space.
</para>
<para>
<variablelist>

<title>Symbolic properties</title>
 <varlistentry>
 <term>
  <anchor id="frame-type">
  <parameter>type</parameter>:
 </term>
  <listitem>
  <para>
  <parameter>rightHanded</parameter>, <parameter>leftHanded</parameter>,
  <parameter>orthogonal</parameter>,
  <parameter>&hellip;</parameter>,
plus the <link linkend="geometric-port">geometricPorts</link> to the 
<link linkend="point">Point</link> and the unit
<link linkend="vector">Vector</link>s that make up the frame.
  </para>
  </listitem>
 </varlistentry>

</variablelist>

<variablelist>
<title>Physical properties</title>
 <varlistentry>
 <term>
  <anchor id="frame-move">
  <parameter>displacement</parameter>
  </term>
  <listitem>
   <para>
The displacement of a frame is again a frame. 
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="frame-rotation">
  <parameter>rotation</parameter>,
  <parameter>position</parameter>
  </term>
  <listitem>
   <para>
These operations extract the rotation and position parts of the frame.
   </para>
   <para>
(Is this a <emphasis>physical</emphasis> operation?)
   </para>
  </listitem>
 </varlistentry>
</variablelist>

<variablelist>
<title>Numeric properties:</title>
 <varlistentry>
 <term>
  <anchor id="htmatrix">
  <parameter>homogeneousTransformationMatrix</parameter>,
  (<parameter>HTMatrix</parameter>) 
 </term>
  <listitem>
   <para>
This is the data structure that encodes the position and orientation
of a Frame with respect to another &ldquo;world Frame&rdquo;.
   </para>
   <para>
   4-by-4 matrix (in 3D), with the 
<link linkend="rotation-matrix">rotationMatrix</link> of direction cosines 
   as top-left sub-matrix, the position coordinate vector of the 
<link linkend="point">Point</link>
   as top-right 3-by-1 matrix, and (0 0 0 1) as fourth row.
   </para>
   <para>
   In 2D, a 3-by-3 matrix, similar to the 3D case, but without the
   third row and column.
   </para>
   <para>
	In 1D, a <link linkend="point">Point</link> and a direction
   <link linkend="vector">Vector</link>.
   </para>
  </listitem>
 </varlistentry>
 <varlistentry>
 <term>
  <anchor id="frame-move-method">
  <parameter>move</parameter>
 </term>
  <listitem>
   <para>
This is the method call that displaces a Frame from its current
configuration to another configuration, with respect to the
&ldquo;world Frame&rdquo;. The argument of the method
call is a <link linkend="displacement">Displacement</link>.
   </para>
  </listitem>
 </varlistentry>
</variablelist>
</para>

</section>


<section id="line">
<title>Line</title>
<para>
A <parameter>Line</parameter> can geometrically be defined in various
ways: the &ldquo;join&rdquo; of two points, the
&ldquo;intersection&rdquo; of two planes( only in 3D), the combination
of a <link linkend="point">Point</link> and a
<link linkend="vector">Vector</link>, etc. It also comes in different
&ldquo;flavours&rdquo;:
<variablelist>
 <varlistentry>
 <term>
  <anchor id="line-type">
  <parameter>type</parameter>:
 </term>
  <listitem>
  <para>
<parameter>directedLine</parameter>, 
<parameter>unboundedLine</parameter>,
<parameter>lineSegment</parameter>,
etc.
  </para>
  </listitem>
 </varlistentry>

 <varlistentry>
 <term>
  <anchor id="line-coordinate-type">
  <parameter>coordinateType</parameter>:
 </term>
  <listitem>
  <para>
  the type of the line coordinates:
  <parameter>PluckerAxisCoordinates</parameter> (the
<parameter>Line</parameter> is the &ldquo;meet&rdquo; of two 
<link linkend="plane">planes</link>),
  <parameter>PluckerRayCoordinates</parameter> (the
<parameter>Line</parameter> is the &ldquo;join&rdquo; of two 
<link linkend="point">points</link>),
  <parameter>pointVectorCoordinates</parameter> (the
<parameter>Line</parameter> is represented by one
<link linkend="point">point</link> on the <parameter>Line</parameter>
and one <link linkend="vector">line vector</link>), 
  <parameter>vectorVectorCoordinates</parameter> (the
<parameter>Line</parameter> is represented by one
the <emphasis>moment vector</emphasis> of a
<link linkend="point">point</link> on the <parameter>Line</parameter>,
and one <link linkend="vector">line vector</link>),
etc.
  </para>
  </listitem>
 </varlistentry>

</variablelist>
The <link linkend="physical-properties">physical properties</link> of
a Line are:
<variablelist>
 <varlistentry>
 <term>
<parameter>inversion</parameter>
 </term>
  <listitem>
   <para>
 of a <parameter>directedLine</parameter>.
   </para>
  </listitem>
 </varlistentry>
</variablelist>
</para>
<para>
The numeric coordinate representations
for <parameter>Line</parameter>s are a number of floating point
values, expressed in a given reference
<link linkend="frame">frame</link>, but with an interpretation that
depends on the type of the <parameter>Line</parameter> coordinates:
<itemizedlist>

<listitem>
<para>
  <parameter>PluckerAxisCoordinates</parameter>: &hellip;
</para>
</listitem>

<listitem>
<para>
  <parameter>PluckerRayCoordinates</parameter>: &hellip;
</para>
</listitem>

<listitem>
<para>
  <parameter>pointVectorCoordinates</parameter>: &hellip;