Thank you for your comments on our last technical report about constraint
singularities. Firstly, I would like to point out that, as we just learned, the
constraint singularity in the 3-UPU orientational parallel mechanism has been
already correctly identified by Professor Di Gregorio. His results have been
presented a few days ago at the 2001 IEEE/ASME International Conference at
Como, Italy. Secondly, I would like to bring to your attention one common
misinterpretation of the Jacobian matrix of parallel mechanisms with less than
I was contacted by Sameer Joshi, Prof. Tsai's Ph.D. student, who was naturally
wondering why the 3x3 (constrained) Jacobian matrix of the 3-RPS parallel
mechanism is not singular at the configuration shown in Fig. 2(b) of our last
report (). That configuration
is clearly a Type 2 singularity. To construct the 3x3 constrained Jacobian, one
needs to select three independent output coordinates, and that choice is not
unique. Hence, I quickly assumed that it should be the original 3x6 Jacobian
that should be rank deficient, which was probably wrong.
As Dr. Dimiter Zlatanov explained in a private communication, it might be the
full 6x6 Jacobian (?) matrix that is to be inspected and not portions of it.
What follows is a brief summary of his explanation.
. . .
Let us see the necessary conditions for the platform to move when the actuators
are locked. With that assumption, each leg imposes two linearly independent
constraints on the motion of the platform. One is the constraint wrench which
is present even when the actuator is not locked; in the 3-RPS this is a force
through the S joint and parallel to the R joint. The second is due to the
locking of the P joint and is most naturally chosen as the force along the leg.
The vector-space sum of these three 2-systems gives the constraint system of
the platform with locked actuators. When it is of dimension six, then the
motion system has dimension zero and there is no uncontrollable platform
motion. Otherwise, if the six constraint wrenches span a system of dimension
five or less, the platform can move and there is a singularity.
So the full 6x6 matrix that will correctly show all singularities should be
composed of the line coordinates of the described six forces. The six forces
can be divided in two groups: the 3 constraint wrenches present even when the
P joints are not locked; and the three forces along the legs. The whole system
of six degenerates, if and only if, either (i) one of these two subsystems
degenerates or (ii) the two subsystems have a non-zero intersection.
In Fig. 2(b), each of the two subsystems is of full rank, i.e., of dimension 3.
However, the two systems intersect and the intersection is of dimension 1,
spanned by a pure force. As a result, the sum of the two systems is a 5-system
and the reciprocal system (the system of uncontrollable platform motions) is a
1-system of rotations about line BC.
Since in this configuration there is no constraint singularity, and the
platform does not gain a fourth DOF, the uncontrollable motion is an RO
(redundant output) motion, and this is an RO (or Type 2) singularity.
. . .
I will be on vacation until August 7. ParalleMIC will resume normal operation
with a new technical report focused on a peculiar parallel mechanism that is
full of singularities. I was just pleasantly surprised with a plastic model of
that mechanism, built by rapid prototyping.
July 17, 2001
* Mark Rossheim is confident that his Omniwrist III, 2-DOF orientational
parallel mechanism will be the Next Big Thing in machine design
* Luc Rolland from Loria contributed a review on his 4-DOF parallel robot Kanuk
* A spin-off medical robotics company of the Aachen University of Technology