INRIA, BP 93, 06902 Sophia-Antipolis, France

E-mail: Jean-Pierre.Merlet@sophia.inria.fr

In the past recent years parallel robots have drawn a lot of interest in the robotics community and in many applicative domains: medical, machine-tools, pick-and-place,etc, where the advantages (e.g. accuracy, rigidity, high velocity) of closed-loop chains may be useful. The purpose of this paper is to identify important open theoretical problems in this field.

Another interesting aspect of parallel robots is that they enable to
use unconventional actuators like, for example, wire with
winches [1] or *binary* linear actuators (having only
two states, fully extended or retracted) [7]. This
enable to extend the application area of parallel robot but at the
same time induces additional constraints on the theoretical problems
to be solved (e.g. computing the workspace not only taking into
account the limits on the stroke of the actuators but also the
preservation of the tension in the wires). Another promising field is
the study of *reconfigurable* robots in which the location of the
joints may be changed at will to obtain the best robot for the task at
hand [30,50], a problem that we will address in the *synthesis* section.

Parallel robots may use complex joints like multiple ball-and-socket and universal joints, whose restricted motions have a large influence on the workspace volume. New joint designs have been proposed: with large motion capabilities [20], flexible for parallel micro-robot [41]. Still improvements of these joints have received scant attention.

This is clearly an area where a cooperative work with mathematicians has produced in the last 10 years the most beautiful results. If we measure only the articular coordinates the direct kinematics will have, in general, multiple solutions. As we are interested mainly in the current pose of the platform we may rely on an iterative scheme, e.g. Newton-Raphson, to calculate this pose. These schemes need an initial estimate of the solution (which is in general available) but may experience convergence problem or, worse, may lead to a solution which does not correspond to the current pose. Alternate scheme has been proposed to solve this problem [4,11,13], but without improvement in the computation time.

Another approach is to first determine all the possible
solutions of the problem and, then, to sort the solutions in order to
determine the current pose. The complexity of finding all the
solutions increase, in general, with the number of DoF of the
end-effector. Note that we may encounter
numerical problems even for simple robot as shown by
Guglielmetti [19] for the *Delta* robot.
For complex robots the solutions may be determined
either by using a pure numerical method like the
*continuation method* [42] or by
*elimination* [14], i.e. by manipulating the equations
of the inverse kinematics
in order to reduce the problem to the solution of an univariate polynomial,
which
real roots enable to determine all
the possibles poses of the end-effector.
Although the former method was able to solve the problem in some cases,
its efficiency seems to be lower than the latter method, on which we
will focus.
A drawback of the elimination method is that it can be performed in
many different ways, not all of them leading to the same degree for
the resulting polynomial.
Therefore to determine the best univariate polynomial, i.e. the one
having the
lowest possible degree, it is necessary to determine a bound on the
number of real solutions of the direct kinematics. Such bounds have been
obtained using either classical, but often forgotten, geometrical
theorems or the most recent results in algebraic geometry. Basically
these bounds are obtained by using analysis theorems, like
Bezout's theorem, which enable to determine
the maximum number of roots of a given system just by inspection
and subtracting from this number all the roots that are at
infinity. This number gives an upper bound of the number of real and
complex roots of the system. Strangely, in many cases it has
always been possible to find a configuration of the robot such that
the number of real solutions of the direct kinematics is exactly the
bound. This is exemplified by the case of the general Gough platform
for which a bound of 40 has been found in 1992 [45]
while an example with
40 real solutions has been found in 1998 [12].

In the elimination method, finding the univariate polynomial
is either tedious and/or mathematically complex. Fortunately
the calculation done for a particular mechanical architecture may
sometime be used for another architecture (e.g. the direct kinematics
of the *Hexa* robot may be solved by using the algorithm for the
Gough platform).

Still there is some work to be done in this area as the theoretical algorithms are difficult to use in practice: for example in the case of the Gough platform finding the 40th order polynomial for any geometry and any leg lengths is still a difficult job [25]. Furthermore the manipulation leading to this polynomial are so complex that they prevent any symbolic factorization, which will lead to a simplified, faster solution. But this simplification may be possible as shown for the Stewart platform: the elimination method leads to a 12th order polynomial but this polynomial is (at least) the product of two 6th order polynomials [32]. Clearly this area has not been sufficiently investigated.

Up to now we have considered only the first step of the solution of
the direct kinematics. Indeed as
we are interested mainly in the *current* pose of the end-effector, we have to sort the set of
solutions. Possible sorting criterion are that the solution should be
reached from the *initial assembly mode*, i.e. the pose of the
end-effector when it was first assembled, without crossing a
singularity and without links interference.
At one time it was thought that the former criteria was sufficient to
determine an unique solution. It is now known that this is false
as shown by Innocenti for planar robot [28] and by
Wenger for spatial robot [49]: there exist singularity-free
trajectories joining different solutions of the direct
kinematics. Still singularity analysis
may help to eliminate solutions, but showing that the singularity and
interference criterion will lead to an unique solution is an open
problem, together with an algorithm implementing these criterion.

Another approach to solve the direct kinematics is to add extra sensors to the
robot. Indeed the *n* articular sensors provide a system of equations
in the *n* pose parameters: hence each extra sensor will provide an
additional equation, leading to an over-constrained system which,
hopefully will have an unique solution.
The problem is here to determine the
minimal number of sensors and their location in order to have an
unique solution with the simplest analytic form and quite robust with
respect to the sensor errors. Some of these problems have been
addressed in [2,6,21,36,46], but
this issue is far from being solved. Note that adding sensors may play
also an important role in the robot calibration (see the next
section).

Another approach is the *auto-calibration*
methods [10]. In that case
either extra sensors are used or mechanical constraints are imposed on
the legs of the robot (e.g. by clamping a leg so that its direction
remains fixed during the calibration). These methods seem to have a
large potential but have received little attention.

The main difficulty of workspace analysis for parallel robot is that,
as the reachable locations of the end-effector are dependent on its
orientation, a
complete representation of the workspace should be
embedded
in a 6-dimensional workspace
for which there is no possible graphical illustration. Only
subsets of the workspace may therefore be represented.
The most investigated workspace is the 3D *constant orientation
workspace*, which describe the possible location of the origin of the
end-effector for a constant orientation:
geometric or algebraic approaches may be
used [22,17,31], the former being faster and the
latter more general. But many other types of workspace are of
interest, for example the *reachable workspace* (all the locations
that can be
reached by the origin of the end-effector), the *orientation
workspace* (all the orientations of the end-effector for a given
location of the origin of the end-effector) or the *inclusive orientation
workspace* (all the locations
that can be
reached by the origin of the end-effector with every orientation in a
given set). Although the determination of these types of workspace has been
addressed for planar robot [36], they remain largely
ignored for spatial robots. Furthermore they can be complexified at
will by adding constraints (e.g. singularity-free workspace or
workspace with a lower bound on the transmission factor). A related
problem is to find the volume swept by an object lying on the
end-effector [22,34].

A companion problem to workspace analysis is the trajectory planning problem. This may be understood as to determine first if a given trajectory between two poses fully lie in the workspace of the robot and is singularity-free, and, if the answer is negative, find an alternate trajectory that join the two poses. An interesting variant of this problem for robots having more DoF than necessary (e.g. for a 6 DoF milling machine where the rotation around the normal of the end-effector is not used) is to determine the possible ranges of the extra DoF which ensure that a given trajectory lie in the workspace of the robot, with the further problem of determining the value in these ranges which optimize another criteria (e.g. for which the maximal value of the articular forces over the trajectory is minimal).

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J-P. Merlet