DOI 10.1007/s00170-004-2119-2 O R I G I N A L A R T I C L E
Chih-Wei Lue · Yuan-Ming Cheng · Jih-Hua Chin
System structure and contour tracking for a hybrid motion platform
Received: 6 October 2003 / Accepted: 28 January 2004 / Published online: 8 December 2004 ©Springer-Verlag London Limited 2004
Abstract This work investigates the spatial contour tracking on a new kind of motion platform that is constructed by extending the parallel platform with a proven planar motion table. Multi-axis cross-coupled tracking control schemes were developed on the basis of this hybrid structured Stewart platform, which formed a 5-axis working machine. A framework for the functions of this system was established. Roadmap for further development that leads to a highly versatile and efficient precision motion plat-form was made.
Keywords 5-axis machine tool· Contouring · Cross-coupled tracking· Stewart platform · Trajectory tracking
C.-W. Lue · Y.-M. Cheng · J.-H. Chin (u) Department of Mechanical Engineering, National Chiao Tung University,
1001, Tashue Road, Hsinchu 300, Taiwan, R.O.C. E-mail: [email protected]
Tel.: +886-3-5712121 ext. 55105 Fax: +886-3-5727485
Fig. 1. Proposed hybrid motion plat-form. Left: computer model, right: prototype
1 Introduction
A parallel machine tool is featured by its high structural rigidity and precision, while traditional serial machine tool is proven for its mature performance of continuous path contouring. But both suffer from inherited disadvantages. Serial structure has long load-bearing chains and hence limited stiffness and dynamics, while the parallel structure of a Stewart platform only prevails in a limited motion space that contains threatening singular points.
The purpose of this work is to pioneer a novel hybrid plat-form that integrates the advantages of both parallel and serial ma-chines. The framework for the conceived hybrid motion platform was established and the multi-axis 3-D precision contour track-ing was investigated. Figure 1 shows the novel system studied in this work.
In the trajectory tracking, Koren [1] first proposed the cross-coupled biaxial control, which has become the state-of-the-art in precision contour tracking [2–7]. Chin and Lin [8] enhanced the cross-coupling operation with a velocity precompensation. The
precompensated cross-coupled control proved to have the best contour precision in circumstances of high feed rates and great trajectory curvatures [9, 10].
In the field of Stewart platform, mechanism design and an-alysis [11–13], accuracy analyses [14–17], workspace and sin-gular point [18–21] were investigated. Portman et al. [22] con-structed a 6-DOF parallel platform, which enabled displacements in the micron and submicron ranges.
Since the proposed novel system works like a 5-axis ma-chine, some latest development in 5-axis concerning kinemat-Fig. 2. Block diagram of the MCCPM
controlled hybrid motion platform
ics [23], path generation [24–26] and real-time interpolation [27, 28] were of interest.
2 System construction
Figure 2 shows the system block diagram of the constructed mo-tion platform. The upper part of Fig. 2 that produces (xa, ya)
pro-duces (L1· · · L6) represents the control of leg lengths of parallel platform. The cross-coupled, precompensation control (CCPM), mainly planar techniques, was chosen and extended to multi-axis CCPM (MCCPM) in this work.
2.1 Six-DOF Stewart platform 2.1.1 Inverse kinematics
Figure 3 shows the upper and the lower platform of a Stewart platform. The six joint locations of the upper platform in the base coordinates can be expressed as:
b Ei =bT e ∗eE i , i = 1 ∼ 6, (1) where [eEi] = [LeCφi, LeSφi, 0, 1]T, i= 1 ∼ 6 and [bTe] is transformation matrix.
The six joint locations of base platform in the base coordi-nates are
b Bi
= [LbCθi, LbSθi, 0]T, i = 1 ∼ 6. (2)
The leg lengths can be obtained by calculating:
|Li| =bEi−bBi, i= 1 ∼ 6. (3)
2.1.2 Forward kinematics
Contrary to the simple calculation for leg lengths, six non-linear equations are to be solved to get the required centroid
Fig. 3. The locations of 12 joints of the platform shown in Fig. 1
(x, y, z; α, β, γ) for a given set of Li=1∼6:
Fi(x, y, z, α, β, γ) =bE i−bBi 2 x+ b Ei−bBi 2 y+ b Ei−bBi 2 z − L2 i. (4)
2.1.3 Singular points and work space
The advantages of parallel platform are won at the cost of so-phisticated determination and exclusion of singular points. This becomes more difficult when a continuous spatial trajectory is to be tracked. The finding of singular points can be performed by differentiating equations Fi(x, y, z, α, β, γ) as follows:
dFi dt = ∂Fi ∂L ∂L ∂t + ∂Fi ∂P ∂P ∂t = W(L) • L+S(P )P•= 0. (5) Case 1: Inverse kinematics singularities
When det W(L) = 0,P•= 0 satisfies the equation dFi/dt = 0,
which corresponds to the workspace boundaries. Case 2: Forward kinematics singularities
When det S(P ) = 0,L• = 0 satisfies the equation dFi/dt = 0,
which corresponds to the singular points.
2.2 Cross-coupled precompensation method (CCPM)
CCPM [8–10] was an extended version of cross-coupled control. The motion platform shown in Fig. 2 is already in CCPM control, which can be identified by the blocks “contour error calculator” and the “velocity calculator.”
The speed precompensation requires that the feeding velocity be modified as:
−→V = −→V
t + −→Vk. (6)
3 New system developments
Based upon the knowledge stated above, the following function groups were developed.
3.1 Trajectory planning in the sense of 5-axis motion 3.1.1 Trajectory construction
A continuous spatial surface S(u, v) is shown in Fig. 4, upon which a trajectory Pi(u, vc) can be constructed by setting v
con-stant and u variable.
If bi-cubic patch [29] is chosen to construct the surface, then S(u, v) = 3 i=0 3 j=0 aijuivj, (0u1, 0v1). (7)
Fig. 4. A trajectory on surface
Fig. 5. The mapping of trajectory to position and orientation sets
The normal to the surface at the machining point can be obtained by: nx, ny, nz = ∂S ∂v× ∂S ∂u / ∂S ∂v× ∂S ∂u .
The tool position and orientation (Pt, Ot)TM correspond to (xt, yt, zt; 0, 0, −1)T in base coordinates. The trajectory and its
normal are denoted by [S(u, v); n(u, v)]Teffector, which can be transformed to base coordinates as follows:
[S(u, v); n(u, v)]T
M= [T.M.]baseM [S(u, v); n(u, v)]Tbase, (8)
where[S(u, v)]Tbase= [T.M.]effectorbase [S(u, v)]effectorT , and[n(u, v)]Tbase= [R.M.]effectorbase [n(u, v)]Teffector.
The transformation leads toα, β and γ as well as the follow-ing position relationships:
xxy+ xc= xt−Sxn1+ Syo1+ Sza1, yxy+ yc= yt− Sxn2+ Syo2+ Sza2 , (9) zc= zt− Sxn3+ Syo3+ Sza3 .
The demand leg lengths for a Stewart platform can be ob-tained by inverse kinematics:
Li= 1 ∼ 6 = Inv-kinematics (xc, yc, zc, α, β, γ) . (10)
3.1.2 Trajectory generator (real-time interpolator)
Equations 7, 8 are too complicated to be used in real-time path generation. Instead, the trajectory information[S(u, v); n(u, v)] could be mapped onto two separate sets of position Pi(u, vc) and
orientation Oi(u, vc) by Hermite curves and be stored in memory
as a database. Figure 5 describes the mapping of centroid of the upper platform. Hermite curves: P(x) = [Fi(x)] P0 P1 P0 P1 T , (0x1). (11)
Fig. 6. The flowchart of real-time trajectory gener-ation
Tangent vector is obtained as Pk(x) = ∂F k(x) ∂x P0 P1 P0 P1 T , (0x1). (12) Figure 6 describes the operation of the real-time trajectory generation. At each sampling time the MCCPM control provides Pk−1and uk−1to interpolator, which calculates parameter uk
ac-cording to Eq. 13 [10]. Information Pi, Pi+1, Pi, Pi+1, Qi, Qi+1,
Qi, Qi+1corresponding to ukare then recalled from the database
and are used by interpolator to produce the trajectory information Pk, Ok, Pk, Ok according to Eqs. 11, 12:
uk= uk−1+ VbT
xk−12+ yk−12+ zk−12
, (13)
where Vbis the feedrate.
3.2 Multi-axis trajectory tracking 3.2.1 Contour errors
Position control is the basics in CNC tracking. When tracking continuous contour the contour error become a primary target to be diminished.
In Fig. 7, the distance E between the actual position Paand
the desired position Piis position error. The shortest distance
be-tween the actual position and the desired contour is contour error Er, which can be found to be the following:
Er=
Ex, Ey, Ez
T
=E−E· VV∗−→i , −→j , −→k T. (14) Analogously to the trajectory contour, as shown in Fig. 8, an “orientation contour” Eorcan be built inα, β, γ coordinates:
Eor= Eor,α, Eor,β, Eor,γ T = [Eo− (Eo· ω) ω] ∗ −→α , −→β , −→γ T . (15)
The trajectory and orientation contour error, respectively, can be obtained as ⎛ ⎜ ⎝ Erx Ery Erz ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ Vy ExVy− EyVx + Vz ExVz− EzVx VxEyVx− ExVy+ VzEyVz− EzVy VxEzVx− ExVz+ VyEzVy− EyVz ⎞ ⎟ ⎠ (16) ⎛ ⎜ ⎝ Eor,α Eor,β Eor,γ ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ ωβEo,αωβ− Eo,βωα+ ωγEo,αωγ− Eo,γωα ωαEo,βωα− Eo,αωβ + ωγEo,βωγ− Eo,γωβ ωαEo,γωα− Eo,αωγ+ ωβEo,γωβ− Eo,βωγ ⎞ ⎟ ⎠ (17)
Among these six error components Erx, Eryshall be distributed
to the x-y table and Stewart platform according to some al-gorithms or optimization process to be developed in the fu-ture. The Stewart platform determines the six error components,
Erx , Ery, Erz, Eor,α, Eor,β, Eor,γ
, such that these components must be the inverse of their leg-length-referred counterparts: ELi=1∼6 = Inv-KinematicsErx, Ery, Erz, Eor,α, Eor,β, Eor,γ , (18) Erx(X−Y )= Erx− Erx; Ery(X−Y )= Ery− Ery. (19)
Fig. 7. Position error and contour error
3.2.2 Precompensation
The concept of precompensation is a velocity modification, which gives the machine a better adaptation to higher trajectory curvature.
Based on the experience in [9], better results could be achieved by using −→V =V x Vy Vz T = Vt−→t + Kv−→Er+ Ki t 0 −→E rdt. (20)
Similarly, the precompensation of angular velocity can be ex-tended to −→ω =α β γT = ωb−→t0 + Ko,v−→Eo,r+ Ko,i t 0 −→E o,rdt. (21) The velocity components Vx, Vyshall be distributed to x-y table
(Vx(X−Y ), Vy(X−Y )) and Stewart platform (Vx, Vy). In order to
execute the required velocity precompensation by Stewart plat-form, the following inverse kinematics is necessary:
˙Li=1∼6= Inv-Kinematics
Vx, Vy, Vz, ωa, ωb, ωγ
. (22)
3.2.3 Coordination between the Stewart platform and the x-y table
The x-y table provides additional degrees of freedom that can be used to detour singular points or pursue optimization of motion. Assume that at (n− 1) discrete time point the centroid of plat-form is at Pa,n−1= [x, y, z, α, β, γ]a,n−1, and the command feed
at time n is∆Pnamong which the componentsµx∆xn, µy∆yn
shall be assigned to the Stewart platform whereµx, µyare the
ra-tios of distribution. This makes the distributed feed of the Stewart platform and x-y table, respectively, at time n to be as follows: ∆Ps,n= µx∆xn− xa,n−1, µy∆yn− ya,n−1, ∆zn, ∆αn, ∆βn, ∆γn, (23) ∆Pxy,n= (1 − µx) ∆xn+ xa,n−1, 1− µy ∆yn+ ya,n−1 . (24) The feeds of Stewart platform in terms of axis movement are
∆Li,n = J Pa,n−1+∆Ps,n 2 ∗∆Ps,n i=1∼6, and Ps,n= Pa,n−1+ ∆Ps,n. (25)
Equation 25 issues the command feed of each axis for the Stewart platform: ∆Li,n= (aix+ bi) µx∆xn− xa,n−1 + (ciy+ di)µy∆yn− ya,n−1+ ei∆zn + ( fiy+ gi) ∆αn+ (hix+iiy+ ji) ∆βn + (kix+liy+ mi) ∆γn, (26) where x= xa,n−1+ (µx∆xn− xa,n−1)/2; y= ya,n−1+ (µy∆yn− ya,n−1)/2. It gives ∆Li,n= Aiµ2x+ Biµ2y+ Ciµx+ Diµy+ Ei, (i = 1 ∼ 6). (27) The stroke of motion axes of Stewart platform can be written as
6 i=1 Li,n= Aµ2x+ Bµ2y+ Cµx+ Dµy+ E , (28) 6 i=1 Li,n= A (µx+ C/2A) + Bµy+ D/2B+ EE. (29)
Substituting µx= −C/2A, µy= −D/2B into Eqs. 28, 29
yields the respective command feed. 3.2.4 Multi-axis CCPM
The extension of CCPM to multi-axis (MCCPM) is featured by generating tracking command for x-y axis and the axes of Stew-art platform as follows:
xf(X−Y )(n) = xf(X−Y )(n − 1) + TVx(X−Y ), (30)
yf(X−Y )(n) = yf(X−Y )(n − 1) + TVy(X−Y ), (31)
Lf,i(n) = Lf,i(n − 1) + T ˙Li, i = 1 ∼ 6. (32)
The feeding errors of respective motion axis are calculated by Ex(X−Y )(n) = Ex(X−Y )(n − 1) +xf(X−Y )(n) − Xt(X−Y )(n),
(33) Ey(X−Y )(n) = Ey(X−Y )(n − 1) +yf(X−Y )(n) − yt(X−Y )(n),
(34) EL,i(n) = EL,i(n − 1) + Lf,i(n) − Lt,i(n) , i = 1 ∼ 6. (35) The control signals for respective motion axis in MCCPM are generated as follows
Ux(X−Y )(n + 1) = KexEx(X−Y )(n) + Kεr,xEr,x(X−Y )(n), (36)
Uy(X−Y )(n + 1) = KeyEy(X−Y )(n) + Kεr,yEr,y(X−Y )(n), (37)
UL,i(n + 1) = KeL,iEL,i(n) + Kεr,LiEr,Li(n) , i = 1 ∼ 6.
(38) Figure 9 describes the operation of the proposed hybrid platform under multi-axis CCPM control.
4 Simulation for the hybrid platform under MCCPM
4.1 Virtual axes
Despite its simple appearance the proposed system (Figs. 1, 2) is highly sophisticated. Inverse kinematics shall be performed three times before the platform can do anything in terms of pursuing a contour (Eqs. 10, 18, 22).
An approach using a virtual axis is proposed and shown in Fig. 10 to simplify the whole control process. In Fig. 10 P repre-sents virtual axes X, Y , Z and O reprerepre-sents virtual axesα, β, γ.
The inverse kinematics calculation is only performed at the final stage to find the control signals: UX−Y & Li=1∼6= Inv-kinematics(UP, UO), where U denotes control signals.
Fig. 10. Control process using “virtual axes”
Fig. 11. The trajectory of free-form surface (S(u, v), vc= 0.6)
Fig. 12. Desired position trajectory Pi(u, vc) transformed from S(u, vc) of Fig. 11
Fig. 13. Desired position trajectory Oi(u, vc) transformed from S(u, vc) of Fig. 11
Fig. 14. The position contour error (MCCPM)
4.2 System simulation
Figure 11 shows the shape of a computer mouse for which the trajectory S(u, vc) is to be tracked. The hardware
specifica-tions are:φ = 15◦,θ = 15◦, Le= 50, Lb= 70, Ob= [0, 0, 20],
Pt= [0, 0, 150] (unit: mm) (see Fig. 3). The trajectory is first
transformed to position and orientation information according to Sect. 3.1.2, as shown in Figs. 12 and 13, respectively, and then the tangent vector is computed. The results of overall tracking are shown in Fig. 14. The position IAE for the case in Fig. 14 is 0.019 mm, and the IAE orientation is 0.0004 rad. It is seen that the proposed hybrid motion platform is capable of precision tracking of continuous 3-D trajectory.
5 Conclusion and future work
This work pioneered a novel hybrid motion platform, which in-tegrated the precision and the stiffness of parallel structure and the continuous contour tracking ability of serial machine. Frame-work for the multi-axis cross-coupled tracking control schemes
based on this hybrid motion platform was established. Topics including trajectory construction, trajectory generation, forward and inverse kinematics, coordination of serial table and paral-lel platform, multi-axis tracking and system simulation were addressed.
In order to do real-time interpolation, the trajectory, which was derived from surface model, was mapped into position and orientation sets and stored as database. Real-time trajectory gen-eration was performed by the calculation of trajectory parameter and retrieval of information from database. In order to cope with the five-axis feature, the planar CCPM scheme was extended to include both position and orientation precompensation. And the motion coordination between the x-y table and Stewart plat-form was explored. Finally, a simulation system was developed in which virtual axes were used to avoid repeated computation of inverse kinematics.
It was shown that this new kind of motion platform is capa-ble of doing precision 3-D contour tracking under the proposed system structure. Extensive implementation of the system struc-ture on a prototype platform is currently underway and new aspects of the technological advantages will be reported in future analyses.
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