CARDAN JOINTS WITH AXIS OFFSET IN SINGULARITY FREE
HEXAPOD STRUCTURES FOR NANOMETER RESOLUTION
Rainer Gloess
Advanced Mechatronics
Physik Instrumente (PI) GmbH & Co. KG
Karlsruhe, GERMANY
INTRODUCTION
The main advantages of hexapod systems are
user-friendly compact shipment design and high
stiffness.
An improvement in the accuracy of hexapod
systems can be achieved via better
actuator/sensor, cable outlet and joint design.
For most applications the use of external 6 DOF
sensor systems at the platform is not suitable
due to the sensor performance, the size or the
price of such sensor systems. Therefore, sensor
systems inside the struts have to be used as a
reference for each single axis of the hexapod.
The platform position will be determined by the
strut-sensor systems, the joints – and the
mechanical
tolerances
(which
can
be
calibrated).
This paper describes different joints used in
nanometer
repeatable,
parallel-kinematic
systems. Such joint designs require large
computing power of the controller.
conditions or special scan routines demand
different structure design. [1]
Two different parallel-kinematic designs are
realized at PI:
Systems with constant length of the struts:
ƒ vertical motion of the lower joints
ƒ horizontal motion of the lower joints
FIGURE 2. a: Vertical moving joints – for small
and high load and vacuum flange mounted
hexapods. b: Horizontal moving joints – for low
height of the position platform
Systems with changeable length of the struts
ƒ Stewart Gough platforms
HEXAPOD STRUCTURES
M-850K114
M-850K102
FIGURE 3. Fixed mounted lower joints for small
and high load systems with different shaping
M-810
F-206
M-824
M-840
M-850
FIGURE 1. Hexapod systems for payload
capacity between 10 N and 10.000 N (Physik
Instrumente (PI), product examples www.pi.ws)
The multi-axis application defines the structure
of the hexapod parallel kinematics. Mostly the
customer requirements determine size, height
and shaping of the systems. Also environmental
The advantage of systems with changeable strut
lengths is that no additional linear guiding parts
influence the positioning performance. Such
systems should be used preferably for highaccuracy applications. However, systems with
movable joints could have better dynamic
properties because the drives are better
decoupled from the platform. All parasitic forces
from the drives can be decoupled with the linear
guiding.
Sphere
Flexure
FIGURE 4. a: Example of a cardan joint with
axes in one plane, b: Example of a cardan joint
with axis offset
DIFFERENT JOINT DESIGNS
The cardan joint with axis offset features a
compact joint part between the two links. The
two axes can be used as inseparable cylinder
parts. In case of high accuracy applications,
both bearings inside the joint part (stone) can be
realized with long needles. Systems with cardan
joints with axis offset provide twice as much
stiffness as systems using cardan joint design
with crossed axes.
During assembly the cardan joint bearings have
to be preloaded or fine adjustment modules
should be used for preload. For small angular
ranges below 10 degree in each joint, the out-ofplane motion and the roundness of highprecision bearings can reach values better than
50 nm. Hysteresis effects are of high
importance. They are caused by the friction and
different rolling lines for forward and backward
motion. These hysteresis effects can be
minimized by medium preload and specially
designed spindle bearings.
Sphere joints facilitate the calculation of the
inverse kinematics. Our sphere joints consist of
ceramics for fully nonmagnetic hexapod systems
with piezo actuators. The drawback of such
joints is the hysteresis effect due to friction. A
very
thin
lubricant
layer
is
strongly
recommended achieving high-precision joints.
There are also sphere joints with balls between
the joint sockets, but their disadvantage is that
boring forces occur during rotation.
Cardan
crossed axis
Fig. 4a
Cardan
axis offset
Fig. 4b
Simple
mathematics
for
calculation of the inverse
kinematics,
friction
and
hysteresis effects, also boring
effects for ball bearing
spherical joints
Excellent
hysteresis-free
designs
are
possible,
problems with non-fixed pivot
point, pivot point location
depends on the angular
motion and load, single part
flexures and cardan flexures
are integrated
High-precision design with
preload,
medium
load
conditions
High-precision design with
preload, high-load designs
are possible
Flexure design for joints has the advantage of
very small hysteresis effects. It can be used for
small angular motion of the joints up to 10
degrees. For bigger rotation angles special
super-elastic materials are appropriate. The
flexure design can also be used for medium load
conditions at the joints. The wire flexure is a
simple joint which can take in bending as well as
torque forces. The drawback of such structures
is the unstable pivot point.
INVERSE KINEMATICS OF HEXAPODS WITH
CARDAN JOINTS
The hexapod with cardan joints with axis offset
can be described as a linkage of six 6R serial
linkages systems placed between the base plate
and the moving platform. Many different
solutions were developed in the 80s and 90s of
the last century for the solution of inverse
kinematics of a general 6R linkage system.
Raghavan and Roth [3] introduced a general
algorithm for solving 6R manipulator and related
linkages.
D = T1 ⋅ G1 ⋅ T2 ⋅ G2 ⋅ T3 ⋅ G3 ⋅ T4 ⋅ G4 ⋅ T5 ⋅ G5 ⋅ T6 ⋅ G6 (1)
Where the transformation
geometric properties are:
0
0
⎡1
⎢0 cos(u ) − sin(u )
i
i
Ti = ⎢
⎢0 sin(ui ) cos(ui )
⎢
0
0
⎣0
0⎤
0⎥⎥
0⎥
⎥
0⎦
matrixes
and
(2)
⎡1
⎢a
Gi = ⎢ i
⎢0
⎢
⎣di
⎤
⎥
1
0
0
⎥
0 cos(α i ) − sin(α i )⎥
⎥
0 sin(α i ) cos(α i ) ⎦
0
0
0
(3)
A very interesting paper was presented by
M. Husty [4] in which he showed that the general
problem could be solved by numerical
calculation without iteration. The general 6R
kinematics of each strut-platform module was
cut into two 3R serial chains. The rotation angles
ui of the revolute joints have to be computed.
Input: The lower plane e collects the spindle
r
axes, the direction vector of joint u and the
center point of joint P’. In an equal manner plane
r
d collects direction vector r , point P and the
center point of joint Q’
Output: All joint angles, the distance P’ – Q’ and
the angle between both planes (spindle/nut).
For the upper part of such 3R-chain systems the
direct kinematics can be described as
U 1 = T1 ⋅ G1 ⋅ T2 ⋅ G2 ⋅ T3 ⋅ G ⋅31
(4)
⎡ 1
⎢α 2
⎢ 3
G31 = ⎢⎢ 0
⎢
⎢ d
⎢⎣ 3
0
⎤
⎥
0
0
⎥
⎛α ⎞
⎛α ⎞
0 cos⎜ 3 ⎟ − sin⎜ 3 ⎟⎥⎥
⎝ 2 ⎠
⎝ 2 ⎠
⎥
⎛ α3 ⎞
⎛ α3 ⎞ ⎥
0 sin⎜ ⎟ cos⎜ ⎟
⎝ 2 ⎠
⎝ 2 ⎠ ⎥⎦
0
1
0
(5)
The lower part of the separated 3R-chain can be
set to a just equivalent term. A numeric example
in the paper shows that the equation can be
solved via a simplified chain structure. This
method produces sixteen “solutions” for the
inverse kinematics.
In our first application in 1993 we used a very
similar philosophy but we established an
iteration algorithm to solve the two equation
parts for the upper and lower linkage system of
each strut. Our method provides exactly one
solution for each pose. In the past the first
algorithm did not rely on the Jacobian matrix for
acceleration of the iteration. In the iteration we
found a very linear dependence. Also for non
pre-settings not more than 3 iteration steps are
necessary (typically two steps). We need less
than 60 µs for the direct transformation of all six
struts and about 4 ms for the inverse
transformation.
It is necessary to indicate that in most of our
hexapod systems we use the rotation between
th
spindle and nut for the 5 DOF in each strut.
Otherwise an additional precise bearing should
be inserted between both cardan joints at
platform and base-plate.
FIGURE 5. Description of variables at general
axes configuration
Two equations describe the angular relations of
the upper und lower part of the 3R system.
(6), (7)
r
r
r
r
r
r
r
q + b ( v cos β + w sin β ) = p + fr + g ( s cosα + t sin α )
r
r
r
r
r
r
r
q + du + e ( v cos β + w sin β ) = p + a ( s cosα + t sin α )
The values a and b represent the axis offset. It is
assumed that the linkage part axes are
rectangular.
With two additional vectors and the scalar
product of e. g. p*w six equations are derived.
Each iteration step calculates the distance
between e. g. point P’ and plane e. The distance
has to be minimized.
F1 (α , β ) = ( qps + svb cos β + swb sin β ) sin α −
( qpt + tvb cos β + twb sin β ) cos α
F2 (α , β ) = ( pqv + sva cos α + tva sin α ) sin β −
( pqw + swa cos α + twa sin α ) cos β
=0
=0
(8)
(9)
Equations (8) and (9) lead to two independent
constraints for the angles alpha and beta.
⎛ ∂F1
⎜ ∂α
J (α , β ) = ⎜
⎜ ∂F2
⎜ ∂α
⎝
∂F1 ⎞
∂β ⎟
⎟
∂F2 ⎟
∂β ⎟⎠
(10)
Jacobian matrix for the two searched angles
⎛ α n+1 ⎞ ⎛ α n ⎞
⎛ F1 (α n , β n ) ⎞
−1
⎜
⎟ = ⎜ ⎟ − J (α n , β n ) ⎜
⎟
⎝ F2 (α n , β n ) ⎠
⎝ β n +1 ⎠ ⎝ β n ⎠
(11)
Equation of the iteration algorithm to receive the
unknown angles alpha and beta
SOFTWARE
SIMULATION
HEXAPOD SYNTHESIS
TOOL
FOR
The sensor consists of the optical head and a
small industrial computer with three A/D
channels. All three deviations from the initial
position are reported with command structure.
This sensor can also be used for calibration
tasks.
The static and dynamic performance of hexapod
systems is measured with Zygo laser
interferometers in single axes at the platform.
FIGURE 7. Step test protocol for hexapod
platform M-810 (repeatability is better than
0.2 µm)
SUMMARY
FIGURE 6. Software simulation tool
The simulation program is based on identical
routines which also used in our controller. It
provides hexapod synthesis, working space
analysis, load conditions and joint angle
calculation and some scanning features. All
types of PI’s hexapods with different joint
structures are automatically recognized.
MEASUREMENT RESULTS
An optical sphere sensor was developed for the
measurement of pivot point stability. A small
precise ball is placed at the desired pivot point
and fixed at the platform. The 3D sensor
housing is placed around this position.
FIGURE 8. Sphere position sensor (3D optical
contactless) with resolution of less than 0.1µm
Major
improvements
in
resolution
and
repeatability of hexapod systems can be
achieved by optimizing the joint design. It is
shown that the design of cardan joints with axis
offset improve the repeatability of hexapod
systems and provide high stiffness. Although
cardan joints require much more computing
power than simple spherical joints, with standard
CPU’s the transformation time can be reduced
to values less than 1 ms.
REFERENCES
[1] Physik Instrumente (PI) GmbH & Co.KG
Katalog, Piezo Nano Positionierung, 2009,
Pages:4.3 -4.18
[2] R.Gloess.
Hexapod-Strukturen
mit
Mikrometer-Genauigkeit,
Chemnitzer
Parallelstruktur-Seminar,
April
1998,
Page(s).:63-67
[3] M.Raghavan, B.Roth, Inverse kinematics of
the general 6R manipulator and related
linkages, Trans. ASME, Journal of
Mechanical Design 115 (1990) 228-235
[4] M.Husty, M.Pfurner, H.-P.Schröcker. A new
and efficient algorithm for the inverse
kinematics of a general 6R manipulator,
Robotics
and
Automation,
IEEE
Transactions on Volume 10, Issue 5, Oct
1994 Page(s): 648 - 657