INTERNATIONAL
CARPATHIAN CONTROL CONFERENCE
ICCC’2010
Eger, Hungary
May 26-29, 2010
OPTIMAL KINEMATIC DESIGN OF PARALLEL SPHERICAL WRIST
MANIPULATOR
1
M. ŠVEJDA1
Department of Cybernetics, UWB, Pilsen, Czech Republic, msvejda@kky.zcu.cz
Abstract: The paper deals with the optimal kinematic design of 3 rotation DoF parallel Spherical
Wrist Manipulator (SWM). The kinematics of the parallel SWM is described and workspace
determination is based on the Stratified Workspace Boundary Determination Methodology
(SWBDM). The Parallel SWM Optimization Editor is introduced to manage and visualize the
optimization process. Finally, some results are illustrated.
Key words: parallel spherical wrist manipulator, workspace boundary determination, optimization
1. Introduction
Parallel SWM, see Fig. 1, is often used in an
industry as the last part of the industrial
manipulators and adds 3 rotation DoF.
Parallel manipulators have a number of
advantages in comparison with classical serial
manipulators due to their special kinematic
architecture, e.g. higher stiffness, higher
accuracy, lower moving mass. On the other
hand there are some drawbacks which
complicate the design of the parallel
manipulators, e.g. more complex kinematic
structure,
more
complex
input-output
relationship, an irregular shape of the
workspace.
Optimal design of the parallel SWM plays a
crutial role if the manipulator is supposed to
perform a given task. But the optimization of
the parallel manipulators except the very simple
architectures leads to the nonlinear optimization
problem with nonlinear constraints. Many
researchers have dealt with this problem. The
atlases (map) of the global (local) stiffness and
dexterity index are considered in [1] to analyse
performance of the 3DoF parallel SWM with
rotation joints and the optimization are done
manually. The optimal design presented in [2] is
based on the interval analysis. The boxes of the
design parameters complying
Fig. 1: Parallel spherical wrist manipulator in a
home position (mutual rotation of the base and
end-effector is denoted by )
compulsory requirements (stroke of the actuator
and accuracy) are found for the prescribed shape
of the workspace. A brief summary of the
parallel robots optimization problem and many
methods proposed by the researchers to solve
this problem can be found in [3].
2. Kinematics of parallel SWM
Parallel SWM consists of three independent
kinematic chains which are mounted to the base
by the weld joints and to the end-effector by the
spherical joints. The links of each kinematic
chain are connected by the universal joint. The
parallel SWM is actuated by three prismatic
actuators. A passive stabilization element
ensures only three rotation degrees of freedom
of the end-effector.
Let
then
the
constant
length
can be expressed as
vector
(6)
Consequently the inverse kinematic
mapping of the parallel SWM is derived
from (6) as follows
(7)
(1)
are the actuated joint coordinates and
Time differentiation of the constant length
vector
leads to the relationship
between the actuated joint velocities
and
the end-effector velocity vector .
(2)
(8)
are the generalized end-effector coordinates
where
are XYZ Euler's angles
representing the consecutive rotations of the
end-effector about X-axis, Y-axis and Z-axis.
The set of the design parameters of the parallel
SWM is given by the vector
(3)
It holds for i-th kinematic chain
(4)
where
,
and
are the position
vectors of the vertices of the equilateral triangle
in the base
and end-effector
coordinate systems respectively
and which depend only on the design
parameters (3). The rotation matrix between
these coordinate systems depends on the
generalized coordinates and it is given by
where
is the inverse kinematic jacobian. For more
information see [4].
2.1 Workspace
A definition of the orientation workspace of
the parallel SWM includes two main parts:
1. Mechanical constraints: The extension of
the actuators, the slope angle at the universal
and spherical joints and the minimal distances
between the links lie within a given interval.
2. Workspace quality requirements: The local
dexterity index
is grater than
a given threshold.
Then the orientation workspace of the
parallel SWM for given design parameters is
defined as
(5)
(9)
If we denote the known vector in (4) as
where
is the i-th row of the vector
function (7),
returns the slope angle at
the universal joints ( ) and spherical joints ( ,
),
returns the minimal distances
between the links
,
.
The algorithm of the workspace
determining is based on the SWBDM algorithm
firstly presented in [5]. We suppose the angle
(rotation of the end-effector about Z
axis), where is a constant parameter. So we
can cover the plane
by the rectangular grid
with a given accuracy. Then we can say that the
point (node) of the rectangular grid
belongs to the
orientation workspace if it satisfies the
definition (9). Contrary to the classical
discretization methods the SWBDM is based on
an idea not to test all points (nodes) of the given
grid but to search and test only a few points
around the workspace boundary (point of the
workspace layer is denoted by
). We can
say that the point
is an exterior point of
the layer if all of the -neighbourhood points
lie out of the orientation workspace, the point
is an interior point if all of the neighbourhood points lie within the orientation
workspace. Else
is the boundary
point. We defined -neighbourhood points of
as
. It is worth noticing that the
3D orientation workspace is not suitable for
practical purposes due to its irregular shape.
Therefore we will consider a maximum cylinder
with a longitudinal axis parallel to zaxis and passing the central point
which is
inscribed into the 3D orientation workspace .
The inscribed cylinder can be easily found by
assuming the boundary points
. More
information and complete SWBDM algorithm
can be found in [4].
Fig. 2: SWBDM principle
2.2 Optimal design of the parallel SWM
The optimal design problem of the parallel
SWM is defined as
Fig. 2 briefly shows the principle of the
SWBDM. The search direction vectors are given
as
The workspace layer area
is determined
as the sum of the areas of the triangles
. If we denote a direction along
as a stratified direction of the orientation
workspace we can divide the 3D orientation
workspace into
layers
, where
and the central point
. Then a volume of
the 3D orientation workspace
is
(10)
where
is a volume of the
maximum inscribed cylinder
.
Note that the 3D orientation workspace is
independent of the design parameter and the
home position of the actuators
is linear
dependent on
through the inverse
kinematics (7).
The solution of the term (10) leads to the
nonlinear optimization problem with nonlinear
constraints, see (9). Generalized Pattern Search
algorithm (GPS) in Matlab is used to find
solution of (10). GPS is method for solving
optimization problems without the need of the
knowledge of the objective function gradient.
GPS is described in detail in [5].
A GUI Parallel SWM Optimization Editor,
see Fig. 3, was developed for easy managing of
the optimization process.
V
W
= 1.5717 V
inCyl
=1.0816
Maximum inscribed cylinder: R = 0.47908, H = 1.5
0.5
0
 [rad]
-0.5
-1
-1.5
-2
0.5
-0.4
-0.2
0
 [rad]
Fig. 3: Parallel SWM Optimization Editor
3. Results and conclusion
The optimization process of the parallel
SWM was performed for different initial design
parameters and the sets of the resulting optimal
design parameters
might be further sorted by
the other relaxable constraints.
The workspace constraints (9) were chosen
as
We consider a normalization of the design
parameters to be
which
ensures a limitation on the overall manipulator‘s
size. So we get the Optimal Normalized Parallel
SWM (ONPSWM) design parameters. Fig. 4
shows 3D orientation workspace with the
maximum inscribed cylinder for one set of the
optimal design parameters
Similar parallel SWMs are given by the
multiplying of the ONPSWM´s design
parameters and workspace constraints which
have
a
metric
dimension
(
) by a given constant.
It can be proven that these similar manipulators
will have the same shape and volume of 3D
orientation workspace.
0
0.2
0.4
0.6
-0.5
 [rad]
Fig. 4: 3D orientation workspace of parallel
SWM
References
[1.] LIU, X-J, JIN, Z-L, GAO, F. Optimum
design of 3-DoF spherical parallel
manipulators with respect to the conditining and stiffness indices. Mechanism and Machine Theory 2000. pp.
1257-1267
[2.] HAO, F, MERLET, J.-P. Multi-criteria
optimal design of parallel manipulators based on interval analysis. Mechanism and Machine Theory 2005. pp.
157-171
[3.] Wan, Y., Wang, G., Ji, S., Liu, L. A
survey on the parallel robot optimization. Second International Symposium
on Inteligent Information Technology
Application IEEE 2008
[4.] SVEJDA, M. Kinematic analysis of
parallel spherical wrist manipulator.
Technical report 2010, FR-TI1_174_8
[5.] Wang, Z., Ji, S.,Sun, J., Ou, C., Wan,
Y. A metodology for Determining the
reachable and dexterous workspace of
parallel manipulators. International
Conference on Mechatronics and Automation, IEEE 2007