MOTION CONTROL OF A HYDRAULIC STEWART PLATFORM
WITH COMPUTED FORCE FEEDBACK
Chih-Chin Hsu and I-Kong Fong* Department of Electrical Engineering
National Taiwan University Taipei, Taiwan 106, R.O.C.
Key Words: Stewart platform, force feedback control, gyroscope.
ABSTRACT
On a Stewart platform with six hydraulic links, we mount three gyroscopes to measure the angular motion rates of its upper platform. Using these rate signals and the link lengths measured by LVDTs, we compute the orientation and position of the upper platform, as well as the static force acting on each link. These forces are put into a dy-namic model of the hydraulic links based on which we determine the feedback gains for controlling the link lengths. Experiments on the platform with an unbalanced payload show that utilizing gains deter-mined this way generates better results than utilizing a constant gain for all links.
*Correspondence addressee
I. INTRODUCTION
A Stewart platform is a parallel manipulator with a movable upper platform and a fixed base connected by six variable length links. When the length of each link is controlled properly, the upper platform is ca-pable of performing six degree-of-freedom motion within its working space. With high stiffness, simple inverse kinematics and good dynamic response characteristics, the Stewart platform has been the sub-ject of many studies including kinematics and dynam-ics analysis (Cheok et al., 1993; Do and Yang, 1988), practical design considerations (Fichter, 1986), work-ing space and swork-ingularities determination (Liu et al., 1993), controller synthesis (Charles et al., 1993; Kosuge et al., 1996; Wen et al., 1998a, 1998b), and many application developments (Grant and Reid, 1997; Salcudean et al., 1994). Figure 1 is a picture of the hydraulic Stewart platform assembled in the Advanced Control Laboratory of the Department of Electrical Engineering, National Taiwan University. In this paper we investigate the motion control
problem of the hydraulic Stewart platform shown in Fig. 1. A similar problem is studied in (Charles et
al., 1993) for a much smaller platform, with links
driven by DC servo motors, which has a different set of dynamic characteristics from the platform consid-ered here. The work of Kosuge et al. (1996) involves a hydraulic Stewart platform, but the control purpose is mainly force exertion. With the parallel structure of the Stewart platform, it is easy to comprehend, that, when one tries to control its link lengths, one has to deal with interactions among all links. We see in our dynamic model of the hydraulic links that such inter-actions are primarily through forces acting on the links. In (Wen et al., 1998a, 1998b) we treat these interactions as uncertain parameters of the dynamic model, and successfully find robust controllers for handling trajectory commands that are known in advance. Here we adopt a more explicit approach and seek to decide the interactions for the controller to cancel, at least partially. We mount three gyro-scopes to measure the angular motion rates of the upper platform, which together with the link lengths
measured by LVDTs enable us to compute the orien-tation and position of the upper platform, as well as the static force acting on each link. These static forces are only approximation of the true interactions, which are dynamic forces. However, due to the fact that the hydraulic platform is not a fast moving apparatus, and according to our experience in (Wen et al., 1998a, 1998b), we believe that such an approximation is suf-ficiently good. Results from experiments performed in this work also support this usage, which deliber-ately worsen the interactions by putting a heavy un-balanced payload on the platform. It is worth noting that our use of three gyroscopes along with six LVDTs offers an alternative way of solving the forward ki-nematics problem to that proposed in (Cheok et al., 1993), where a total of nine LVDTs are wisely used to find the position of the upper platform. Our method lets us compute both position and orientation of the upper platform quickly enough for the purpose of real-time control.
The rest of this paper is organized as follows. In Section II, we derive the kinematics and static force equations, which will be used to determine the posi-tion of the upper platform and the static forces acting on the links. The orientation of the upper platform is obtained in Section III, where a system of three rate gyroscopes, set to serve our purpose, is described. In Section IV, we introduce the dynamic model for a single link with its hydraulic actuator. In addition, we propose a controller with an independently adjust-able gain for each link based on the force the link withstands. Experimental results are given in Sec-tion V to compare the performance of the proposed controller with that of a constant gain controller. Finally, conclusions are given in Section VI.
II. KINEMATICS AND STATIC FORCE EQUATIONS
Before we start, some remarks regarding the
notational convention adopted in this paper. We use boldface upper and lower cases letters to denote ma-trix and vector variables, respectively. In addition, we have three dimensional Cartesian space vectors like AB, which is the vector from point A to point B. Also, in three dimensional Cartesian space, we may have different coordinate frames. For example, frame {F} is the one that has the origin OF and rectangular
coordinate XYZF with the right-hand perpendicular
axes XF, YF, ZF and corresponding unit vectors iF, jF, kF. For vectors v and AB, vF and AB
F
denote their 3×1 representation column vectors with respect to iF, jF, kF. However, the superscript T is reserved for the
transpose operation of matrices and column/row vectors. Finally, || . ||2 stands for the Euclidean norm of a vector. In particular, it is equal to the length of its argument vector in three dimensional Cartesian space.
1. Kinematic Equations
The basic geometry of our Stewart platform and associated kinematic equations are described in this subsection. Moreover, we shall show how to deter-mine the position of the upper platform quickly if its orientation and all link lengths are known.
Figure 2(a) is a simplified sketch of the Stewart platform to be discussed, where we attach frames {P} and {B} to the upper platform and the lower base, respectively. We also attach frame {Ji} to the ith link,
which is connected to the platform and the base through two universal ball joints at points Pi and Bi,
respectively. Marked in Fig. 2(b), which is a top view, these joints lie on circles centered on centroids OP
and OB with radius RP and RB, respectively. The angle
between P1 and P2 is θP, and the angle between B1
and B2 is θB. Other points distribute, symmetrically,
along the circles with a 2
3π span. The XB axis of frame {B} is perpendicular to the line connecting the two joints B1 and B6, and the XB-YB plane coincides with
the base plane. The ZB axis is determined by the right
hand rule. Fixed to the upper platform, the reference coordinate XYZP of frame {P} is similarly defined.
As to the ith link frame {Ji}, i=1, ..., 6, the origin OJi
is at the point Bi, the XJi-axis points toward Pi, the YJi
axis is parallel to the cross product of BiPi and -kB,
and the ZJi axis is determined by the right hand rule.
For the sake of simplicity, hereafter we write OJi as Oi, XJi as Xi, and iJi as ii, etc.
Using the notations defined above, we consider the relative position and orientation of frame {P} with respect to frame {B} first. The relative position can be described by OBOP
B
=[x y z]T
, while the orienta-tion can be described by three Euler angles α, β, and
γ, obtained by performing the following sequence of rotations to make the inertial frame {B} become frame Fig. 1 A hydraulic Stewart platform
{P}: first rotate frame {B} about the ZB-axis a (yaw)
angle α to get a frame {B′}, then rotate {B′} about its
YB′-axis a (pitch) angle β to get another frame {B′′},
and finally rotate {B′′} about its XB′′-axis a (roll) angle γ to obtain the frame {P}. This orientation represen-tation also gives an Euler orienrepresen-tation matrix RP
B
equal to
cos(α) cos(β) cos(α) sin(β) sin(γ) – sin(α) cos(γ) sin(α) cos(β) sin(α) sin(β) sin(γ) + cos(α) cos(γ)
– sin(β) cos(β) sin(γ)
cos(α) sin(β) cos(γ) + sin(α) sin(γ) sin(α) sin(β) cos(γ) – cos(α) sin(γ)
cos(β) cos(γ)
≡ rr1121 rr1222 rr1323 r31 r32 r33
,
which also serves as the coordinate transformation matrix from the frame {P} to the frame {B }. Now if we denote the angle between OBBi and the XB-axis by Λi, and the angle between OPPi and the XP-axis by λi, i=1, 2, ..., 6, then we have
Λi= 13π⋅i – θB 2 , i=1, 3, 5, λi= 13π⋅i – θP 2 , i=1, 3, 5, (2) Λi=Λi−1+θB, i=2, 4, 6, λi=λi−1+θP, i=2, 4, 6. Thus OPPi P = [RPcos(λi) RPsin(λi) 0] T ≡[Pix Piy 0] T , i=1, 2, ..., 6, (3) OBBi B = [RBcos(Λi) RBsin(Λi) 0] T ≡[Bix Biy 0] T , i=1, 2, ..., 6. (4)
With these preparations, it is easy to see from Fig. 2(a) that the link vector BiPi expressed with respect to the frame {B} is the sum of three vectors
BiPi B = OBOP B + RP B⋅ OPPi P – OBBi B . (2)
Hence the length li of the ith link can be computed
from
li= BiPi B
2. (6)
Eq. (6) represents the closed-form solution to the in-verse kinematics problem. More explicitly, it can be written as li2=x2+y2+z2+RP2+RB2 +2(r11Pix+r12Piy)(x−Bix) +2(r21Pix+r22Piy)(y−Biy) +2(r31Pix+r32Piy)z −2(Bixx+Biyy). (7)
Subtracting (7) with a general index i from the same equation with the index i set to1 leads to
Fig. 2 (a) Basic geometry of the Stewart platform and the attached frames. (b) Top view of the ball joints arrangement on the base and upper platform.
(b) (a) Link Platform Base {B} OBBi OPOB OBOP XB B4 B3 P3 P2 B2 B1 B6 B5 P5 P4 RB RP P6 P1 P ZB ZP OP OB XP {P} YP YB Yi Zi Xi BiPi OPPi θ B θ
[r11(P1x−Pix)+r12(P1y−Piy)−(B1x−Bix)]x
+[r21(P1x−Pix)+r22(P1y−Piy)−(B1y−Biy)]y
+[r31(P1x−Pix)+r32(P1y−Piy)]z =[(r11P1x+r12P1y)B1x−(r11Pix+r12Piy)Bix +(r21P1x+r22P1y)B1y−(r21Pix+r22Piy)Biy] + 1 2(l1 2− li2). (8)
From (8) with i=2, 3, ..., 6, we have a system of five linear equations about the variables x, y, and z, which can be put into the matrix form
M⋅N xy z = b≡ b1 b2 b5 (9) with M = (P1x– P2x) (P1y– P2y) (B2x– B1x) (B2y– B1y) (P1x– P3x) (P1y– P3y) (B3x– B1x) (B3y– B1y) (P1x– P4x) (P1y– P4y) (B4x– B1x) (B4y– B1y) (P1x– P5x) (P1y– P5y) (B5x– B1x) (B5y– B1y) (P1x– P6x) (P1y– P6y) (B6x– B1x) (B6y– B1y) (10) N = r11 r21 r31 r12 r22 r32 1 0 0 0 1 0 (11) bj=1 2(l1 2 – lj + 1 2 )+(r11P1x+r12P1y)B1x −[r11P(j+1)x+r12P(j+1)y]B(j+1)x +(r21P1x+r22P1y)B1y -[r21P(j+1)x+r22P(j+1)y]B(j+1)y, j=1, 2, ..., 5. (12)
Note that M is a constant matrix decided by the geometric parameters RP, RB, θP, and θB. In our case
it has full column rank. The matrix N also has full column rank unless r31 and r32 become zeros at the
same time, but since we shall only use (9) to find x and y, the first two rows of N will play no roles. As
to the vector b, besides the platform geometric parameters, we must know the orientation angles of the upper platform and the link lengths. For the lat-ter we have six LVDTs mounted on the links, and for the former we use the method to be described in Sec-tion III. Given all the needed informaSec-tion, we can multiply the pseudo-inverse of M to both sides of (9) to form N xy z = (MTM)– 1MTb≡b = b1 b2 b3 b4 (13)
Due to the simple structure of the lower half of the matrix N, it is easy to conclude that
x = b3, (14)
y = b4. (15)
Note that to accelerate the computation, we only need to multiply the last two rows of (MT
M)−1
MT
to b, since b1 and b2 are useless to us. With x and y in hand, we
turn back to (7) for determining z. Adding (7) with
i=1, 2, ..., 6 together, dividing the sum by 6, and
us-ing some identities of trigonometric functions, we get z = {1 6 li 2
Σ
i = 1 6 + 1 3i = 1Σ
[(r11Pix+ r12Piy)Bix 6 + (r21Pix+ r22Piy)Biy] – x2– y2– RP 2 – RB 2 }1/2. (16) Thus, we solve “half” of the complicated forward ki-nematics problem by assuming that in addition to the link lengths, we know the orientation of the upper platform. Moreover, the required computation time to get the unique solution is short enough that it can be utilized in our real-time feedback system.2. Static Force Equations
The Xi-axis of the ith link coincides with BiPi,
so the unit vector ii can be expanded as
ii= BiPi BiPi 2
≡iixiB+ iiyjB+ iizkB, (17)
where iix, iiy, and iiz are exactly the three components
of BiPi B
in (5) divided by ii. Furthermore, from the
ji= ii×( – kB) ii×( – kB) 2 ≡jixiB+ jiyjB+ jizkB, (18) ki=ii×ji≡kix iB+kiy jB+kiz kB. (19)
Note that by the rule of cross product, jiz=0 for i=1,
2, ..., 6, in (18). With these quantities, the coordi-nate transformation matrix from frame {B} to frame {Ji} can be constructed as RB Ji = [ii B ji B ki B ]T≡ iix iiy iiz jix jiy jiz kix kiy kiz . (20)
Hence, the coordinate transformation matrix Ti P
from the frame {Ji} to the frame {P} can be constructed
by combining two transformations as
Ti P = RB P ⋅RJi B = RP BT ⋅RB JiT ≡ ti11 ti12 ti13 ti21 ti22 ti23 ti31 ti32 ti33 . (21) Now we derive the equations for the static forces acting on the links. Because we are concerned with the forces that the hydraulic actuators must generate, we shall focus on the forces along the direction of ii.
Suppose the Stewart platform is a rigid body in a bal-anced and still status, with its base fixed to the earth. Then the net forces and moments acting on the upper platform and all links are zeros. Let us consider the free-body diagrams of the ith link and the upper form separately. Assume the force that the upper plat-form exerts on the ith link is fPi, and the weights of
the upper platform and the ith link are wP and wi,
respectively. Additionally, the center of mass of the upper platform is assumed to be its center position, and the center of the mass of each link is assumed to be its middle position. In our practical situation, the link is a hydraulic cylinder, so its center of mass de-viates from the middle position a little when its length changes, but this detail can be ignored because the resultant error is small. As to the upper platform, when there is only symmetric payload, the assump-tion about the posiassump-tion of its center of mass is valid, but, when there is unbalanced payload, we have to estimate the position of the mass center, and adjust the term OPPi in (28) below, accordingly. Fortunately,
the system of equations derived this way has similar form to that obtained in the following derivation.
(i) The net moment due to the weight vector wi=
−wi.kB of the ith link and the force fPi acting on the ith link about point Bi is zero. Thus
2ri×fPi+ri×wi =2ri×fPi+ri×(wix ii+wiy ji+wiz ki)=0, (22) where fPi=fPix ii+fPiy ji+fPiz ki, (23) ri= 1 2liii, (24) wix wiy wiz = RB Ji⋅ 0 0 – wi . (25) Solving (22), we get fPiy= 1 2jizwi= 0 , (26) fPiz= 1 2kizwi. (27)
(ii) The net moment acting on the upper plat-form about its center of mass is zero. Thus
OPPi
Σ
i = 1 6 ×( – fixiP– fiyjP– fizkP) = 0 , (28) where fix fiy fiz = Ti P⋅ fPix fPiy fPiz = Ti P⋅ fPix 0 kizwi/2 . (29)(iii) The net force due to the weight of the plat-form −wPkB and the forces fPi, i=1, 2, ..., 6, acting on
the upper platform is zero. Thus
0 0 – wP – RJi B
Σ
i= 1 6 ⋅ fPix 0 kizwi/ 2 = 0 . (30)Rearranging (28) and (30), we have six equa-tions in a compact matrix form
D.fp=p, (31)
where the variable vector fP=[fP1x fP2x...fP6x]T, the
p = 1 2 (ti 23Pixkizwi– ti13Piykizwi) – ti 33Pixkizwi ti 33Piykizwi kixkizwi kiykizwi kiz 2 wi + 0 0 0 0 0 wP
Σ
i = 1 6 , (32)the coefficient matrix D=[d1 d2 ... d6], and
di= ( – Pixti21+ Piyti11) Pixti31 – Piyti31 – iix – iiy – iiz . (33)
The system of equations (31) can be uniquely solved provided D is nonsingular. Note that D plays the same role as a force transformation matrix in (Bhaskar and Mruthyunjaya, 1998) and (Bhaskar and Mruthyunjaya, 2000), which transforms three forces and three moments of end-effector into linear actua-tor forces of each link. When D is singular, the trans-formation degenerates and it means that some loads on the platform can not be supported by the actuator forces fp. Note, also, that the inverse velocity
kine-matics are essentially described by DT
(Tsai, 1999), which connects the velocity of each link to the rates of position and orientation of the upper platform. The singularity condition of D is characterized by zero-ing its determinant, but findzero-ing the closed form of the roots is very difficult, and regarded as an open problem in (Bhaskar and Mruthyunjaya, 2000). In order to avoid the singularity condition, we take two measures. The first one is carried out when a trajec-tory command is given in advance. We can, off-line, examine the condition numbers of D to decide whether the platform will go through a singularity-free path. The second one is to, on-line, calculate the condition number of D and give a warning sign whenever it in-creases too fast, or halt the system when it reaches some pre-determined threshold. This way the plat-form is prevented from being operated in an ill-con-ditioned mode in which excessively large actuator forces are needed. In our experimental experience, we encounter no trouble solving (31). Thus the six forces fPix, i=1, 2, ..., 6, are available to us.
Of course, as we explained in Section I, these forces are only approximations to the true interac-tions between the upper platform and the links. However, the analysis in (Wen et al., 1998a) reveals
that the differences between dynamic and static forces are quite small for a typical motion. Furthermore, for our heavy hydraulic Stewart platform, operated at a moderate speed, the static forces are indeed the main interactions.
III. ON-LINE ORIENTATION MEASUREMENT
Our orientation measurement system comprises three solid-state rate gyroscopes manufactured by the Systron Donner Inertial Division of the BEI Sensors & Systems Company. Figure 3 is a picture of the hardware setup. It is seen, therein, that we employ the strapdown method and attach the sensors to the upper platform, with sensing directions of the three gyroscopes parallel to the three axes of the frame {P}, respectively. Since the three gyroscopes sepa-rately measure the roll, pitch, and yaw angular rates of the moving upper platform with respect to the base frame, the orientation information we need must be transformed from the measured signals. The outputs of the three gyroscopes are denoted as P (roll rate),
Q (pitch rate), and R (yaw rate). From the
defini-tions of the frames introduced in Section II, if we define ωωP =αkB+βjB′+γiB′′, then we have (Lin,
1991)
ω
ωP= [γ–αsin(β)] iB+ [αcos(β) sin(γ) +βcos(γ)] jB
+ [αcos(β) cos(γ) –βsin(γ)]kB. (34)
Thus
ω ωPB=
γ–αsin(β)
αcos(β) sin(γ) +βcos(γ)
αcos(β) cos(γ) –βsin(γ) = P Q R , (35) or equivalently
α β γ = 1
cos(β)[Qsin(γ) + Rcos(γ)]
Qcos(γ) – Rsin(γ)
P + tan(β)[Qsin(γ) + Rcos(γ)]
. (36)
Note that for common Stewart platforms, including the one discussed in this paper, the angle β is well limited inside the range of -90 degrees to 90 degrees. Therefore cos(β) is never zero and tan(β) is always finite in the above expression.
In our practical application, the continuous-time relationship (36) is discretized by the backward rect-angular rule to the difference equations
αn βn γn = αn – 1 βn – 1 γn – 1 + α′ β′ γ′ ∆ t , (37)
where αn, βn, γn are the values of α, β, γ at the nth
s a m p l i n g i n s t a n t , α′= 1 cos(βn – 1)
[ Qns i n (γn−1) + Rncos(γn−1)], β′=Qncos(γn−1)−Rnsin(γn−1), γ′=Pn+
tan(βn−1) [ Qnsin(γn−1)+Rncos(γn−1)], Pn, Qn, Rn are the
values of P, Q, R at the nth sampling instant, and ∆t
is the sampling time. To generate the orientation in-formation of the upper platform iteratively, this set of difference equations can be programmed on a com-puter in a straightforward manner, but to actually use it the bias problem of the rate gyroscopes must be taken care of first. Our experience shows that not only does the gyroscope have a bias in its outputs, but also the bias is time-varying, especially during the period the unit is warming up from a “cold start”. Therefore, at the start of each experiment, we need to estimate the steady-state bias by averaging the outputs of the gyroscope over a period. The result is then deducted from Pn, Qn, and Rn subsequently. This
procedure is called initial testing.
Figure 4 is a block diagram of the overall system.
Most blocks in the diagram have been explained by now, except the hydraulic actuator system and the controller, which are the subjects of the next Section.
IV. A SIMPLE HYDRAULIC ACTUATOR MODEL AND THE CONTROLLER DESIGN
In the Stewart platform of Fig. 1, each link is a single-rod (or asymmetric) hydraulic cylinder ac-tuated by a Bosch proportional valve. Since all six cylinder-valve systems are identical, only the me-chanical structure of the ith one is shown by the sche-matic diagram of Fig. 5, where a spool controlled by the electrical currents in the coils alters the oil flow direction and effective orifice sizes. The currents in the coils are the outputs of a compatible Bosch am-plifier which responds linearly (within its operational range) to a voltage control signal. In the hydraulic system, a pump driven by a 10 HP induction motor sends oil with a constant pressure ps to all six valves,
and at the end of a cycle the oil flows out to a tank with a negligible pressure.
Fig. 4 Block diagram of the overall system
Fig. 5 The mechanical structure of the ith cylinder and its asso-ciate proportional valve
In Fig. 5, qi1, qi2, and pi1, pi2, are respectively
oil flow rates and pressures at the designated areas.
a1 and a2 are respectively the head and rod side areas
of the piston. ξi is the displacement of the ith piston/
rod, which is equal to the link length li minus the fixed
length of the cylinder at its reset (shortest) position.
ui is the control signal, i.e., the input voltage of the
amplifier whose outputs are the currents in the coils. Finally −fPix is the static force exerted by the upper
platform on the ith link, which is derived in Section II. Ignoring the leakage and compression effects of the working oil, as well as the inertial and frictional forces generated by the piston and rod, we can estab-lish a dynamic model (Merritt, 1967; Wen et al., 1998b) for the cylinder-valve combination based on the orifice equations, flow continuity equations, and load equation:
qi1= cw⋅ ps– pi1⋅ui= a1ξi, (38)
qi2= cw⋅ p2⋅ui= a2ξi, (39)
for the “push” operation (ui≥0), or
qi1= cw⋅ pi1⋅ui= a1ξi, (40)
qi2= cw⋅ ps– pi2⋅ui= a2ξi, (41)
for the “pull” operation (u≤0), where cw is the
pro-portional constant of the actuator, and
pi1a1−pi2a2=−fPix. (42)
F r o m t h e s e c o n d e q u a l i t i e s o f ( 3 8 ) a n d ( 3 9 ) respectively, expressions for pi1 and pi2 can be
for-mulated and substituted into (42) to form
ξi=
cw psa1+ fPix a13+ a23
⋅ui (43)
for ui≥0. Similarly, from (40), (41), and (42), it is
straightforward to form ξi= cw psa2– fPix a13+ a23 ⋅ui (44) for ui≤0.
The expressions (43) and (44) imply that the in-teraction between the upper platform and the ith link is via the force fPix only. Without it the length of each
link could be controlled independently. Thus we are motivated to cancel the effect of the force term by adjusting the control signal u accordingly.
Equipped with the information derived in the previous Sections, we propose to use the simple
feedback control
ui(t)=ki(t).[ri(t)−ξi(t)] (45)
to cancel the effect of the force term, where ri(t) is a
reference signal for the ith link length change, and ki(t) = a13+ a23 τcw psa1+ fPi x(t) ri(t) –ξi(t)≥0 , a13+ a23 τcw psa2– fPi x(t) ri(t) –ξi(t) < 0 , (46)
is a time-varying gain. Then the closed-loop dynamic equation for the link becomes
ξi= – 1τ ⋅ ξi+ 1τ ⋅ri, (47)
which is a first-order system with the time constant τ and unity d.c. gain. Of course, in practical applica-tions values of the system parameters like cw, ps, etc.,
are not known precisely, and there are measurement noises as well as computation errors that hinder the perfect estimation of fPix. Therefore, neither is it
pos-sible to make a link have the ideal linear time-invari-ant characteristic formulated in (47), nor it is pos-sible to make all links have the identical, decoupled dynamic characteristic in practice. However, even the simplest constant gain feedback system has some robustness, so our goal is not to make an exact can-cellation by k(t), but only to make a reasonably accu-rate cancellation. The remaining question is how to choose the time constant τ, which should not be too small due to the limited driving capability of the actuators. The question will be answered in the next Section.
V. EXPERIMENT RESULTS
Table 1 below summarizes physical param-eter values of the Stewart platform and the hydraulic actuator system in the experiments. Though the sys-tem is essentially the same one in (Wen et al., 1998a, 1998b), the hydraulic actuator part has been slightly modified. Hence some parameter values are differ-ent in the presdiffer-ent setup.
From Table 1 it is easy to see that ξi=li−74.68
when the unit is cm. The six LVDTs used to measure
ξi, i=1, 2, ..., 6, convert a centimeter of length change
to 0.2 V of output voltage. The three rate gyroscopes described in Section III have a measurement range of -50°/sec to +50°/sec that corresponds to the output range of −2.5 V to +2.5 V. Signals from the LVDTs and gyroscopes are sent to a Pentium PRO 180 PC,
which performs all necessary computations to imple-ment the control law with a sampling rate of 100 Hz, and then sends out six control signals to the amplifi-ers of the proportional valves.
Before the control law (45) with the time-vary-ing gain (46) is tested, a 150 kgf payload is put on the upper platform at the center OP. Then all links
are commanded to lengthen equally, lifting the upper platform upwards with neither rotations nor horizon-tal translations. The control law adopted in this pre-paratory experiment is (45) with a constant gain ki(t)
=k for all links. Because the operation is symmetric in the sense that every link experiences the same force from the upper platform and the payload, tuning a single constant gain to use in all links is justifiable. From several trials the gain k=k*
=1.8 producing the most satisfactory response is recorded for later use.
In order to evaluate the performance of the pro-posed control scheme, we arrange two experiments. The first one tests only the basic upward motion, while the second one tests the other five degrees of freedom simultaneously. In both experiments, the control law (45) with a constant gain ki(t)=k*=1.8 for
all links is applied first. Then the same law (45) with the time-varying gain (46) is applied for the ith link,
i=1, 2, ..., 6. The results are compared. To make the
distribution of forces on the links uneven, we change the payload into two graduate students, who weigh 150 kgf in total approximately, and sit at the edge of the upper platform, on top of link 1. The picture in Fig. 6 depicts the arrangement, which affects the com-putation of static forces. For convenience, it is as-sumed that near the joint P1, there is a point mass of
150 kg. Thus the mass center of the upper platform and the payload can be determined accordingly and used to modify (28), as explained in Section II.
Experiment 1.
Let the motion trajectory command be [xc(t) yc(t) zc(t) αc(t) βc(t) γc(t)]=
[0 0 65 + 6t 0 0 0] for 0≤t≤5 ,
[0 0 95 0 0 0] for 5 < t≤6 , (48)
where units for [xc(t) yc(t) zc(t)] are cms, for [αc(t) βc(t) γc(t)] are degrees, and for t is seconds. This
means that the upper platform and the unbalanced payload should be lifted from 65 cm to 95 cm at a speed of 6 cm/sec, without making any horizontal movements or rotations.
Up to now almost all formulas and data for calculating the time-varying gain (46) are avail-able, except the value of τ. As the ideal system characteristic (47) must be feasible and realistic, τ is chosen to satisfy k*= a13+ a23 τcw psa1+ fPi x a , (49) where fPi x a
=57.4245 kgf is the static force every link withstands when the upper platform carries a 150 kgf payload at OP, and is held still at [x y z α β γ]=[0 0 80
cm 0 0 0]. Note that 80 cm is exactly the half way between 65 cm and 95 cm. This choice of τ attempts to match the ideal characteristic to that of the system controlled by the constant gain in a representative operation situation. The resultant value of τ=0.1629, and the time-varying gain for the ith link is
ki(t) = 24.02 235.62 + fPi x(t) for ri(t) –ξi(t)≥0 , 24.02 151.01 – fPi x(t) for ri(t) –ξi(t) < 0 , (50) Fig. 6 Two graduate students act as the unbalanced payload
Table 1 Various parameter values of the Stewart platform system
Parameter Value
base radius (RB): 84 cm
upper platform radius (RP) 56 cm
θB in Fig. 2(b): 2
9π radian
θP in Fig. 2(b): 4
9π raian
weight of the upper 150 kgf
platform (wP):
weight of each link (wi): 25 kgf
length variation range 74.68 cm≤li≤124.68 cm
of each link (li):
piston head side area (a1): 19.635 cm2
piston rod side area (a2): 12.584 cm2
proportional constant (cw): 25 cm4sec−1volt−1kgf−1/2
where ri(t) is the reference signal for the ith link. In
principle, for the above command (48) only “push” action is needed, unless there are overshoots, which are unlikely to happen. For the sake of safety, the gain formula for ri(t)−ξi(t)<0 in (50) is still
pro-grammed into the controller.
In response to the command (48) all links’ lengths li should increase uniformly from 74.68 cm
( c o r r e s p o n d i n g t o z ( t ) = 6 5 c m ) t o 1 0 1 . 8 6 c m (corresponding to z(t)=95 cm), but due to the unbal-anced load, some links will respond faster than others. In Fig. 7, the dotted line is the reference signal ri(t)
for all links to follow, the circles are the response of the ideal system (47) with τ=0.1629, the solid lines
are the measured responses of the fastest and slowest links controlled by the time-varying gain (50), and the dashed lines are the measured responses of the fastest and slowest links controlled by the constant gain k*=1.8. First we see that all link responses are slower than that of the ideal system. This is attrib-uted mainly to neglected frictional forces. Second, we see that the maximal difference between the fast-est response and the slowfast-est one with the constant gain is 2.393 cm, while the difference with the time-varying gain is only 0.952 cm. Obviously, letting each link adjust its own gain can cope with the un-balanced load better. This can also be seen in Fig. 8, where the control signals for link 6 are displayed. Fig. 7 Reference signal for the link length (dotted line), response
of the ideal system (circles), fastest and slowest responses of the links controlled by the constant gain (dashed lines), and by the time-varying gain (solid lines)
Fig. 8 Controls signals for link 6 generated by the constant gain controller (dashed line) and time-varying gain controller (solid line)
Fig. 9 Position trajectories x(t), y(t) and tracking errors of z(t) obtained from using the constant gain (dashed lines) and the time-varying gain (solid lines) respectively
Fig. 10 Orientation trajectories α(t), β(t), γ(t) obtained from us-ing the constant gain (dashed lines) and the time-varyus-ing gain (solid lines) respectively
Contrary to our intuition, it is link 6 that withstands the largest force in our arrangement, not link 1. This is due to the moment that link 6 has to produce. Fig. 8 tells us that the control signal from the constant gain controller is not large enough at first, and has to increase later on so that the enlarging tracking error can be reduced.
Figures 9 and 10 are respectively the position trajectories of x(t), y(t), tracking error of z(t), and the orientation trajectories of α(t), β(t), γ(t). These tra-jectories are obtained through off-line computations made after the experiment. The measured link lengths are inserted on the left sides of the kinematic Eq. (7), and the Newton-Raphson method is used to iteratively solve for the position and orientation variables. From the trajectories we see that ignoring unbalanced pay-load does not result in significant tracking error in the vertical displacement, but does cause the upper platform to move more, horizontally, and to rotate more.
It is worth mentioning that off-line analysis and computation reveal that the largest difference between static and dynamic forces along the trajectory in this experiment is within 10−5%. This justifies the use of static forces approximation in the current experiment.
Experiment 2.
In this experiment, the upper platform carrying the unbalanced payload is brought to the state [x y z
α β γ]=[0 0 90 cm 0 0 0] first. Then the time is reset to t=0 and the motion trajectory command for the system to follow is
[xc(t) yc(t) zc(t) αc(t) βc(t) γc(t)]
=[5t 5t 90 -2.5t -5t 5t] ∀0≤t≤2, (51)
where the units are the same as those in (48). Clearly, this command is to test the responses of the system with respect to all degrees of freedom other than ver-tical displacement. Furthermore, it worsens the un-even force distribution among all links caused by the unbalanced payload, because it asks the upper plat-form to move outward, taking the payload farther from the original mass center of the entire system. For this command, the largest difference between static and dynamic forces along the trajectory is within 0.1%, according to off-line simulation and analysis. Here we re-select the ideal system time constant
τ by requiring it to satisfy k*= a1 3 + a23 τcw psa1+ fPi x b , (52) where fPi x b
=55.94 kgf is the static force every link withstands when the upper platform carries a 150 kgf payload at OP, and is held still at [x y z α β γ]=[0 0
90 cm 0 0 0], the initial state of the command (51). The resultant value of τ=0.1621, and the time-vary-ing gain for the ith link is
ki(t) = 24.02 235.62 + fPi x(t) for ri(t) –ξi(t)≥0 , 24.02 151.01 – fPi x(t) for ri(t) –ξi(t) < 0 , (53) Similar to those of Experiment 1, the results of this experiment show that with the time-varying gain Fig. 11 Tracking errors of x(t) obtained from using the constant
gain (dashed lines) and the time-varying gain (solid lines) respectively
Fig. 12 Tracking errors of β(t) obtained from using the constant gain (dashed lines) and the time-varying gain (solid lines) respectively
(53), the tracking errors are smaller than that with the constant gain k*
=1.8. For the sake of brevity, only Figs. 11 and 12 are given to show the tracking errors of the position variable x(t) and those of the Euler angle β(t), respectively.
In both Figs. 11 and 12 there are two diagrams each. The upper diagrams are obtained by using off-line computations, like Figs. 9 and 10, while the lower d i a g r a m s a r e f r o m d a t a r e c o r d e d d u r i n g t h e experiments. More specifically, the lower diagram of Fig. 11 is from (14), and that of Fig. 12 is the out-puts of the difference equations (37). Because only outputs of gyroscopes are used for on-line determi-nation of the orientation variables, the mismatch between the upper diagram and the lower diagram in Fig. 12 is easier to detect. In contrast, for on-line determination of the translational variables, the link lengths are also used. Hence the lower diagram of Fig. 11 is closer to its off-line counterpart, which re-lies completely on the link lengths.
VI. CONCLUSIONS
In this study, we explicitly formulate the inter-actions among different links of a Stewart platform through the forces acting on the links, and try to cancel, at least partially, these interactions by using a feedback control scheme which takes these interac-tions into consideration. To actually implement the scheme, we propose to use three gyroscopes to mea-sure the orientations of the upper platform, and use the outputs along with link lengths to compute the static forces that the links must generate in the con-trol loop. Our experiments show that the results are better than those obtained by using a single constant feedback gain for all links. In the future we shall investigate the effects of actuator saturation, which do not happen in the experiments described here be-cause we limit the operation speed of the Stewart platform.
ACKNOWLEDGEMENT
This research is supported in part by the National Science Council of the R.O.C. under Grants NSC 86-2623-D-002-025 and NSC 89-2213-E-002-088.
NOMENCLATURE
a1 piston head side area
a2 piston rod side area
cw proportional constant
fPix static force exerted by the upper platform
on the ith link
ki(t) controller gain for the ith link li length of the ith link
P, Q, R outputs of rate gyroscopes
ps supply pressure
pi1, pi2 oil pressure of the ith link qi1, qi2 oil flow rate of the ith link
RB base radius
RP upper platform radius
ui control signal for the ith link wi weight of the ith link
wP weight of the upper platform
x, y, z position variables of the upper platform {B} coordinate frame attached to the base {Ji} coordinate frame attached to the ith link
{P} coordinate frame attached to the upper plat-form
iP, jP, kP three unit vectors of the frame {P} iB, jB, kB three unit vectors of the frame {B} ii, ji, ki three unit vectors of the frame {Ji} RP
B
the Euler orientation matrix
RBJ i, Ti P
coordinate transformation matrices
Greek symbols
α, β, γ Euler angles of the upper platform
τ time constant
ξi displacement of the ith rod REFERENCES
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Manuscript Received: Nov. 20, 2000 Revision Received: Feb. 26, 2001 and Accepted: May 21, 2001
