Proceedings of the euspen International Conference – Zurich - May 2008
3 DOF Scanning Probe using Parallel Link Mechanism T. Harada1 and N.Adachi1 1 Kinki University, Osaka, Japan
[email protected] Abstract A 3-dof (XYZ) scanning probe using parallel kinematic mechanism (PKM) is proposed. Link mechanisms of the proposed PKM are directory driven by force controlled voice coil motors. They are suitable for fast and soft scanning motion. First, an optimum design of the link mechanisms was proposed as positional measurement and touching force sensitivity are equal for all directions. Next, experimental mechanism by using the proposed design method was introduced. Working range of the mechanism was ± 1 mm cubic volume, and positional resolution became less than 1μm. In order to keep stability of the scanning motion, mechanical impedance control was applied. For practical usages, advanced control algorithm which compensates modeling error or friction of the joints by using disturbance observer was proposed. Introduction Conventional researches about parallel kinematic mechanism (PKM) for coordinate measurement have been intended for a fine positioning device in dozens of centimeters area [1-2]. For actual usage of the coordinate measurement, sensing devices at the tip of the CMM are more important. Recently, advanced sensing probes have been proposed, such as Laser trapping probe [3], fiber deflection probe [4], vibration probe [5], and so on. They are intended for small touching force in sub micro meters working area. However, these probes are still categorized into go-stop touch probes. Scanning probe continuously detects numerous points of coordinate. The scanning probing has advantage over the touch probing in shorter measurement time and accurate measurement with larger points of the coordinate [6-7]. For stable scanning motion, it is important to control not force but mechanical impedance, mass, viscosity and stiffness, at the tip of the probe. Active scanning probe achieves the desired mechanical impedance by servo control of actuators [8]. However, it is not suitable for the on-machine measurement because of the large size of the device and not proof against ill environment of the on-machine measurement. In this paper, a 3-dof active scanning probe for the on-machine measurement using the PKM is proposed. PKMs for position control in sub millimeters area using spring hinges have been proposed [9-10]. Our proposed PKM extends the working area to over millimeters size by using universal joints and intended for not position but impedance controlled active scanning device. In this paper mechanical design and control method of the scanning probe are introduced. Calibration of the mechanical parameter [1-2, 11] and uncertainty analysis [12] of the PKM are important to achieve an accurate positioning. In these researches, kinematic derivative equations of the PKM were analytically used for estimating the mechanical parameter. We instead synthetically use the derivative equations for optimum design of the PKM.
Proceedings of the euspen International Conference – Zurich - May 2008
Concept Design of the Scanning Probe Fig. 1 shows concept design of the PKM scanning probe. In order to drive end plate along xyz direction while fixing its orientation, 3-PRRRR link mechanisms are employed. Prismatic pair of the link is driven by a linear actuator. Four Rotational pairs of the link are constructed by universal joints. The linear actuators are allocated all together at the base of the PKM. It is useful to achieve a compact casing design for dust and drip proof of the on-machine measurement. The link mechanisms are directory driven by small force controlled voice coil motors. Mechanical impedance control is installed for active scanning motion. Kinematics of the scanning probe Kinematic parameters of the PKM scanning probe model are shown in fig. 2. Valuables in the figure are defined as follows, p : central position of the end plate. pbi : vector from central position to i th pair of the base plate. pei : vector from central position to i th pair of the end plate. rb : radius of the base plate. re : radius of the end plate. zi : unit vector from pair of i the actuator to i th pair of the end plate. a : unit vector from the actuator to the end plate pair. li : length of rod. ci : controlled value of the i th actuator. Direct kinematics of the PKM is expressed as T Li = l xi l yi l zi = p + pei − pbi = ci a + lc z i (1). Inverse kinematics is easily derived using geometrical relations of the model as
[
]
T
T
T
2
ci = Li a − (( Li a ) 2 − Li Li + lc ) = l zi − lc 2 −l xi 2 −l yi 2
(2).
Derivative of the kinematics is given as J1δp = J 2 δc
(3).
Optimum design for isotropy of measurement sensitivity The base plate pairs are assigned equivalent angles of 120 degrees as shown in the Fig. 2. Consequently, the eq. (1) becomes Li = p + pei − pbi = p − (rb − re )[cos(ϕi ) sin(ϕi ) 0]
T
(4). It means that the kinematic parameter Li is function of difference between the radius of the base plate rb and the end plate re. Here after, the difference is described as rbe. The inverse kinematic equation (2) and the difference equation (3) are functions of the Li. So, those kinematic characteristics are determined by the rbe. force sensor
stylus end plate z
y
y 3
link
lc zi
x base plate linear actuator
z p
re
x
y pb
x
end plate
rb base plate
Figure 1 Schematic view of the PKM
1
ci Li a
acutuator casing
pe
2 link arrangement rbe re lc γ rb link angle
Figure 2 Parameters of the PKM
Proceedings of the euspen International Conference – Zurich - May 2008
Next, the derivative equation (3) is evaluated by the manipulability ellipsoid [13]. Small motions of the actuators δc are transferred to small motion of the end δp as −1
δp = ( J1 J 2 )δc = J12 δc (5). The PKM is controlled that scanning motion be executed around the central position of the end effecter, i.e. px = px = 0. Singular value decomposition of the J12 at the central position is derived by symbolical mathematic manipulation as J12 = UΣV T U = I , Σ = diag ( 2 / 3 (lc / rbt ) 2 − 1, 2 / 3 (lc / rbt ) 2 − 1,1 / 3 )
(6).
γ = tan −1 ( (lc / rbe ) 2 − 1 )
(7).
At the central position, link angle γ in the fig.2 is given as
Optimum design of the PKM is defined as the manipulatabilities of the mechanism be equal for all directions. It means that the sensitivity of forces and positions of the mechanism become isotropic. Manipulatability ellipsoid (M.E.) is shown in fig. 3. The optimum condition is realized when relation between the lc and the rbe is given as rbe / lc = 2 / 3
(8).
At that time, the link angle γ yields to
γ = tan −1 1 / 2 = 35.26(deg)
(9).
Prototype of the scanning probe Prototype of the scanning probe was designed as shown in the fig. 4. Non-backlash universal joint with rod length lc of 8.1mm, and voice coil motor with stroke length of ± 2.5 mm , rated force of 1.3N, positional resolution of 1μm were used as mechanical elements. From the eq. (8), the optimum value of the rbe equals 6.61mm. In order to achieve the condition, radius of the base plate rb and radius of the end plate re were designed as 18.61mm and 12.00mm, respectively. Working range of the mechanism was ± 1 mm cubic volume as shown in fig. 5. Impedance control for active scanning motion Impedance control which can set arbitrary mechanical characteristics at the measurement point was installed. Dynamical equation of the PKM is given as f t = M ( p) p + h( p, p )
(10).
ft is generalized force of the end effecter. M(p) and h( p, p ) are inertia matrix and vector express the nonlinear paragraphs including centrifugal, Corioli, and gravity forces, respectively. A dynamical behavior with the impedance control becomes f e = M I e + B I e + K I e, e = p − pr (11). MI, BI and KI are desired mass, viscosity and stiffness, respectively. Force command fc which realize the desired impedance (11) to the dynamical system (12) is given as T f c = J12 ( M ( p) p + h ( p, p ) − f e ) (12). Where M ( p) and h( p, p ) are approximate values of M and h in the eq. (10). If the
fc is applied to the eq. (10), actual behavior of the PKM becomes
Proceedings of the euspen International Conference – Zurich - May 2008
0
y
z mm
-2
-2 -2
20
actuator
0 y mm
Δx μm
x mm
without observer with observer desired response
0.1 0 0
0.2
0.4 t
s 0.6
0.8
1
mm
8 4 0 -4 -8 0
2
-2
y
0
mm
2
z x
y x
0
-2
Figure 3 M.E. of the PKM Figure 4 Prototype design 0.2
mm
-2
0 x mm
-20 -20
0
z
base plate
20
x
2
universal joint
0 20
0
z
mm
2
mm
2
end plate 40 Manipulatability ellipsoid
x
0
mm
2
Figure 5 Working range
without observer with observer 0.2
0.4
t
0.6 s
0.8
1
Figure 6 Step force response with/without disturbance observer f e = M I e + BI e + K I e + δ f
(13).
δf expresses an impedance error, which is occurred by modeling errors and disturbances. Applying minimum dimensions observer [14], the impedance error δf is estimated and compensated as follows, T
f c = J12 ( M ( p) p + h ( p, p ) − f e + δ f )
(14).
Fig. 6 shows an example of numerical simulation of a step force response. The response error becomes 1/2 with the proposed control law (14). Conclusion A 3-dof scanning probe for on-machine measurement using parallel kinematic mechanism was proposed. Optimum design of the mechanism and impedance control method of the scanning motion with disturbance observer were introduced. References: [1] T. Oiwa, Proc. 6th ICMT 2002, (2002) 433. [2] M. Abbe, K. Takamasu and S. Ozono, IMEKO-XVI World Congress, (2000), 180. [3] Y. Takaya, et. al, Annals of the CIRP, 54, 1 (2005), 467. [4] B. Muralikrishnan, et. al., Precision Engineering, 30, (2006), 154. [5] N. Yamamoto, et. al., Proc of 13th Annual Int. Conf. on MEMS, (2007), 217. [6] R. Knebel, American Machinist, 144, 11, (1999), 76 [7] A. Wozniak, IEEE Transactions on Instrumentation and Measurement, (2007), 2767 [8] Newsletter, 13.02, (2004), Carl Zeiss IMT GmbH. [9] H. H. Pham and I. M. Chen, Precision Engineering, 29, (2005), 467. [10] Q. Yao, J. Dong and P,M. Ferreira, Int. J of Machine Tools & Manufacture, (2006), 946. [11] Y. Ting, H. C. Jar and C. C. Li, Precision Engineering, 31, (2007), 226. [12] B. J. Jr., J. C. Ziegert and L. Bieg, Precision Engineering, 25, (2001), 48. [13] T. Yoshikawa, J. RSJ, 2, 1 (1984) 63 (in Japanese). [14] T. Harada, et. al, Robotics, Mechatronics and Manufacturing Systems, (1993) 339.
