3RPR : the forward kinematic
problem
A 3RPR planar manipulator consists of a platform moving in a
plane and linked to the base via three legs A_{i}B_{i}
, each one consisting of:
 a passive revolute joint (R) centered at A_{i}
,
 an actuated prismatic joint (P) controlling the length
of the leg A_{i}B_{i} ,
 a passive revolute joint (R) centered at B_{i }.
If the position of the platform B_{1}B_{2}B_{3} is known, then the
lengths l_{i}
of the legs A_{i}B_{i} are known. The forward kinematic
problem is the following : how many positions of the platform
correspond to given (l_{1}, l_{2}, l_{3}) ?
If the lengths of the
legs A_{1}B_{1} and A_{2}B_{2} are fixed to l_{1} and l_{2} and the leg A_{3}B_{3} is cut, see what
happens:
The third vertex
of the moving platform
describes a curve which is a 3cyclic sextic (3cyclic means that the
cyclic points
belong to the curve with multiplicity 3). On the animation, the leg A_{1}B_{1 }rotates. To each
position of this leg correspond at most two possible positions of
the leg A_{2}B_{2} : one with "knee B_{2} up" (this mode is
indicated in green) and the other with "knee B'_{2} down" (in red). The
corresponding points B_{3}
and B'_{3} are
actually on the
same sextic, which has two connected components in the default setting
of the applet. But you can move the point A_{2} to see how the shape
of the curve evolves and obtain eventually a situation where the sextic
has only one connected component (with two segments of different
colors).
The number of intersections of the sextic with the circle of
center A_{3}
and radius l_{3} is
the number of solutions to the forward kinematic
problem. You can move the point A_{3},
change the radius l_{3}
and check
that the number of intersections is generically 0, 2, 4 or 6.
Reference : JP. Merlet, Parallel
Robots, Springer
Michel Coste, created avec GeoGebra
