Multi-axis coordinated trajectory following is important in CNC machines and metal cutting tools. Recently, flight simulators with electrical actuators have been in increasing demand. However, the coordinate control scheme affects the accuracy of the motion because motors have an insufficient load capacity relative to the hydraulic actuators. The electronic cam (ECAM) is typically used to perform coordinated control. However, selection of the master may determine potentially very different characteristics of motion. This study proposes an automatic master switching method. The conditions and results of the master switching method for electronic cam are detailed. The robustness and stability of the proposed control system is also demonstrated using the well-known structured perturbation analysis tool, µ.
4.1 Method of Building Cam Profiles (Master-slaves Trajectories) 4.1.1 Polynomial curve-fitting
A polynomial curve-fitting method is proposed to build a continuous curve in order to fit a known discrete signal, and the established curve is treated as the piecewise-continuous cam profile (master-slaves trajectories). As presented in Fig. 4.2, T is the sampling time of the driving system and is the period of motion planning. The predictive planned N points are the known discrete commands for which equals N times T; the cam profile of each axis can be expressed as a function of time index t, which describes the common relationship
t VR
tVR
43
between master and slaves, for 0≤t≤ N⋅T, and in which q is the numbe aster and all slaves. By expanding Eq. (4.1), then
rs of all axes including m
,0 (0)
where )fi(t is the position of the ith axis motion planning with respect to time index t,
which normally equals the planning time kT, unless an external equivalent force acts aon n axis exceeds the critical value, and further, Tmatrix, Ci and F are the constant time matrix, i positions of the ith axis, is constant and nonsingular so exists. Adequately estim
the polynomial parameters of the ith axis and the predicted
x
1 matrix
T−
respectively. The matrix Tmatri
ating the master’s next position f t enables the above equation to be used to m( ) determine the time index t, and then the estimated positions of all of the slaves f t (i = 1 i s, ( )
to q-1) are determined by substituting t into Eq. (4.1). Note that the subscript m and i,s over all this chapter mean the axis of the master and the ith axis of the slaves, respectively. The algorithm includes the following steps.
1. Estimating the next position of the master is an electronic gearing process, and the proper
44
estimate is expressed follows. the ma , and is the velocity estimated during the process of motion planning.
2. Substituting the estimated position of the master into Eq. (4.1) yields,
1 This equation generally has N-1 solutions, and only one real rational solution is correct. A proper constraint is added to Eq. (4.4) to limit the region in which the solution ma two solutions satisfy this constraint, but identifying the correct one is not dif to the properties of the polynomial curve and the planned velocities of the m , the sign of the slope of the curve plotted against the time index must be the same as that of the ideal velocity . For example, in Fig. 4.4, the solution near (k+1)T is the correct one.
The master velocity in terms of the time index is expressed as,
ˆ 1 y be found. Sometimes,
ficult. According
1, as ( ) is po
3. The time index is estimated in the preceding steps, and the estimated position of the ith slave is represented as,
⎩
4.1.2 Poly-line Curve-Fitting
The poly-line curve-fitting method is used to fit the signal of higher frequency according to the viewpoint of Nyquist frequency. And then yields a poly-line curve as shown in Fig. 4.3.
If the number of motion planning points equals N, then as in the section 4.1.1, the cam profile can be expressed as a function of the time index t.
∑ ⋅ −
where the parameters in Eqs. (4.8) and (4.9) are all defined as in the above section. Similarly, (4.9)
⇒
46
matrix is constant and nonsingular; thus, exists.
The next time index is properly determined by substituting the estimated position of the master into Eq. (4.8) and considering the following conditions.
Case 1:
Under these conditions, the general formulation is as follows.
matrix
This equation is solved first by determining whether the value of ( ) is positive or negative. Restated, the probable region of must be determined correctly. The region is the correct choice, where is the actual time index obtained by substituting the actual master’s position into Eq. (4.10) at time kT. Multi solutions may be in this region, so the correc f Eq. (4.10) must
aforementioned, the sign of the slope of the poly-line function of the time index must be
−1 N 1
t solution o next be identified. As
ˆk+1
t the same as that of the ideal velocity vˆ . That is, k
47
ˆ )
From the above analysis, the time index t can be estimated; then, the estimated ˆk+1 tion of the ith slave can be represented as,
,
ity Norm of the Master Switching ECAM Controller
By rearranging Eq. (4.12a) in a ma the control input of the ith axis of the ma switching method can be expressed as
,
i s t i t matrix i
f ( ) = Γt ⋅C =Γ ⋅T− , (4.12b)
where t T−
M control scheme, the actual speed of each axis theoretically does not exceed its reference sp
disp
Then, from the characteristics of the master switching ECA
eed. Therefore, the control lacement y is confined by c,i |y | c, r, i ≤ |y |i , where yr, i is reference displacement input
4.3 A
M control scheme is applied to the control system of multi-axes mechanism
DOF motion simulator SP-120 (Fig. 3.4) is used to implement the generalized ECAM loaded axis reaches its critical valu
ather, the cockpit may sometimes leave its nominal workspace. Accordingly, the ma
master fixed ECAM method to this application.
hea f the six DO
su .
4.3.1 Inverse Kinematics
Inverse kinematics of the simulator SP-120 is stated in section 3.1.
4.3.2 Jacobian formul
yi xi
xi
pplying the Proposed Control Scheme to a Six DOF Motion Simulator This proposed master switching ECA
s to demonstrate its advantages. In this dissertation, the six
tracking technique. If the current (force) of the most heavily
e, then the cockpit cannot easily execute its planned motion easily by directly feeding individual, planned commands to each axis. R
ster switching ECAM control scheme is better suited than the
The master of the flight simulator is predetermined the heaviest loaded axis, so the Jacobian matrix [41] of the simulator must be calculated and updated from time to time. To find the most vily loaded axis o F motion simulator, SP-120, the Jacobian of the simulator should be calculated by the following b-sections
ation of simulator SP-120 From Eq. (3.1),
0 /d dq q )/d p d(q )
pyi yi yi zi zi
(q /d dq
q ⋅ t+ − ⋅ − t+ ⋅ t= (4.14)
49
6
transpose of the matrix and all the parameters are considered in the Si coordinate frame. From Eq. (3.2)
(4.16)
where is the partial derivative of
T
is the partial derivative of
⎥⎥
is the partial derivative of
⎥⎥
T
are the Euler angles measured in the body
embedded coordinate fram kpit coordinate system. When dealing with angular velocity, the inertial frame st be the reference frame. Let be the
cockpit angular velocity measured in the inertial frame X-Y-Z-O. Then,
c -s c d /d d /d
Substituting Eq. (4.22) into Eq. (4.20) yiel
t
} (4.24)
According to the definition of Jacobian matrix J, the jo space, such that,
(4.25)
int space is converted into Cartesian
d / dX t= ⋅J dΘ/dt
As mentioned above, the angular velocity in the inertial fr
ed in the body ame. Form
ame is more meaningful than that measur embedded fr Eq. (4.26), the elements of J−1 can be summarized directly as follows.
The first part of Eq. (4.26) comprises the first three columns of J −1
4.3.3 Calculate the loaded torque of each joint using
elationship between the 6×1 joint torque vector τ, τ = [τ1, τ2, τ3, τ4, τ5, τ6]Τ, and the 6×1 equivalent Cartesian force-moment vector , = [ma,
Jacobian matrix The r
F F I3 3× α]Τ, acting at the mass center of the upper plate, can be written in the form [41]
(4.29) where m denotes the mass of the simulator’s cockpit in this section; and
τ = J F . T
∈ 3
a R α∈ R3 are the linear acceleration vector and the angular acceleration
4.4 Analysis of Stabilit
he dynamics of each slider of the SP-120 motion sim (Figs. 3.4 and 3.5) can be
modeled by parametric uncertai ies t l for
vector of the cockpit, respectively;
3 3×
I represents the moment of inertia of the cockpit.
y and Robustness
T ulator
nt , using he inear fractional trans mation (LFT)
52
representation. An equivalent mass, m, and a nominal damping coefficient, c, are introduced to simplify the dynamics of the slider motion and to decouple the components of the system’s nonlinear terms, to explicate the stability and the robust performance of the system. Thus, a
c c simplified dynamic model of each slider is,
p/ 2 n
u s K K E
τ = ⋅ π = − θ&+ , (4.30a)
and
π θ sp/2
x= ⋅ (4.30b) where x is the displacement of each slider in this section; s is the lead screw pitch; u p represents the force applied to slider; θ& is the angular velocity of motor’s shaft;K is the AC c servo motor constant (here is 0.0529); K is hen C serv otor constant (here is 0.00242552); E is the input voltage of the servo-motor c
t A o m
2
f n f c c
u= −K K x K K E&+ (4.31) where 2 /Kf = π sp and x& represent the machine constant and th r’s line ocity,
respectively. As presented in Fig. 4.5, the slider’s linear acceleration can be expressed as m
u x m c
x&&=−( / )&+ /
e slide ar vel
(4.32) uppose that the physical parameters m and c are not known exactly, but are believed to lie in S
known intervals. Assume,
c c m
m c c
m
m= +∆ δ , = +∆ δ (4.33) where the nominal mass is m =(mH +mL)/2 , and the nominal damping is
53
2
wer bounds of the slider’s damping, respectively.
Figure 4.6 presents the system’s block according to the foregoing dynam equations. S the control input is [ ,w w y1 2, c, i]T and the output is [z1,z2 ,y]T. Then, using the Doyle’s representation, the transformation matrix can be represented as below.
2 2 2
and the ction is
12
Ka ortional gain in position loop, and the system including the perturbations δm and δc, can be represented using LFT. That is,
uncertainty. Stability is often not the only property of a closed-loop system that must be
54
µ robust to perturbations. The most well-known use of a
omain. Figures 4
s a robustness analysis tool is in the frequency d .8 and 4.9 show the singular value frequency responses of
)
1(jω
G and the structured singular values, µ (G1(jω)), respective
ng the theorem of
∆ ly, for each frequency
with ∆ C , obtained by adjusting the proportional gain, ∈ 2×2 K . These figures are obtained a by programmi µ [2]. Moreover, the bounds of µ∆(G1(jω)) are formulated within the reference book [39] as presented in appendix B. In Figs. 4.8 and 4.9, the maximum singular value of G1(jω) by decreasing the al gain, and the maximum structured singular value is increased by increasing the proportional
is increased proportion
gain.
Table 4.1 presents the maximum singular values G1(jω) ∞ , the maximum structured singular values supµ ( 1( ω))
ω G j
R ∆
∈
, m structured si
and the bandwidth of the control system for various
proportional gains. Moreover if the upper bound of the nominal mass exceeds a critical value, then the maximu ngular value will be larger than unity, possibly causing the ormance to be unsatisfied. Table 4.2 presents the critical upper bounds of for various proportiona
e constant of the system responses is higher. Thus, a tradeoff exists between the robustness and th
system’s response. Nevertheless, by carefully considering this tradeoff, the most suitable proportional gain can be conveniently adjusted to fit the specific demands of the control. In requirement for robust perf
l gains, K . The critical upper bound increases as the a mH
proportional gain decreases. Combining Table 4.1 and Table 4.2 reveals that the system is more robustly stable at a lower proportional gain, but the tim
e performance of the
55
this paper, mH is estimated to be around 250 kg by transforming the maximum torque of each joint of the motion simulator SP-120 to the equivalent mass. The maximum torque is by applying the critical velocity and t tolerable acceleration to drive the slider of the flight simulator provided traveling most the nomi workspace of the simulator.
Moreover, for example, if the damping ratio is set to 0.707, then the proportional gain must be adjusted to 6.3, and the maximum structured singular value is then calculated as 0.801358.
Clearly, the sufficient and necessary condition for robust performance is satisfied. That is, the maximum structured singular value must be less than unity. Consequently, according to the theorem of
obtained he maximum
nal
µ and µ-synthesis, the system is well-d internally stable under the
structured perturb
efined and ation, ∆ ∞ <1.
By combining Eq. (4.13c) with the above results, the maximum structured singular value
of the entire system, G1G2, is confined by the following inequality.
1 )) ( ( sup )) ( ) ( (
sup 1 2 ≤ ∆ 1 <
∆ ∈
∈ µ ω ω µ ω
ω
ω G j G j G j
R R
(4.37)
Restated, the master switching control system is ore robustly stable than the original stable system.
4.5 Numerical Method for the Forward Kinematics of Six DOF Flight Simulator
The cockpit trajectories obtained using conventional tracking control and the proposed tracking control, are compared to demonstrate the precision of the proposed control scheme.
m
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Therefore, the six sliders must be transformed into the cockpit positions off-line; that is, forward kinematics will be used to transform the six axis coordinates into the cockpit’s coordinates, including translation components and rotation components (and representing a transformation from J to S). However, direct forward kinematics is difficult to formulate for a six DOF flight simulator. Therefore, this study proposes the use of a numerical method, such as Newton’s method to execute the transformation (J to S) indirectly. The following iterative steps describe the numerical, steepest descent approach [11, 42].
1. Set k =0, and set the initial cockpit position, x , to the cockpit home position. 0 2. Calculate the present Jacobian matrix J , according to the algorithm presented in the k
Appendix A.
3. Calculate the estimated errors in the positions of the six sliders as,
αk = py −pest,k∈R6 (4.38) where p y is the actual positions of six slider, pest,k is the estimated positions of the six sliders, calculated by inverse kinematics, and αk is the chosen step size.
4. Calculate the next estimated cockpit position,
k 1
k k k
x + =x +J ⋅α (4.39) where the Jacobian J matrix is the equivalent gradient matrix. k
5. If ||xk+1−xk ||2<ς or ||αk ||2<ξ, terminate the iteration; the approximate cockpit position is x , where k+1 ς and are the set maximum tolerable errors. ξ
57
6. Set k = k+1; repeat steps 2 to 5.
The convergence of this algorithm takes about two to three iterative loops, given the setting 12
1 −
ς = e and ξ = e1 −12.
4.6 Experimental Results and Comparisons
In this study, the proposed ECAM tracking scheme is used on the SP-120 simulator to simulate ground earthquake signal received at Shui-Li Primary School on September 21, 1998.
Figur show wer
spect uen
the signal are not all less than one-tenth of the Nyquist frequency (here is 50 Hz). Accordingly,
poly-F res 4.12 ~ 4.14 co
angle e sp and
earthqua e must
not in above, the six axes may mutually pull
and drag each other, caus
the root ) e ed ECAM tracking
schem the master fix AM trac
this sim profile and
e 4.10 s a part of this ground earthquake signal. Figure 4.11 presents the po rum density of this signal at various freq cies. As aforementioned, the frequencies of
line curve-fitting method is used in the proposed control scheme.
igu mpare the Euler’s roll angle errors, pitch angle errors and yaw rrors, re ectively, between the conventional proposed method. This ground
ke signal involves only the translation; restated, the simulator’s output attitud clude a rotational component. However as stated
ing rotational motion during this pure translation. Table 4.3 presents mean square (RMS rrors of Euler angles for using the propos
e and ed EC king method executed on the simulator SP-120. In ulation, the poly-line curve-fitting method is used to establish the ECAM
58
the p ing accuracy depends on the system sampling time: a smaller sampling time yields
sampling f y. For example, with a calculation time of around 0.5 ~ 1 ms, the system ampling frequency may be set to 100 Hz. Therefore, some small errors still occur (as shown ching tracking control is applied to the simulator syst
osition
greater accuracy. However, a tradeoff exists between the calculation time and the system requenc
s
in Figs. 4.12 ~ 4.14) even if the master swit
em. Thus, higher performance computers clearly track more precisely.
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