Electronic cam motion involves velocity tracking control of the master motor and trajectory generation of the slave motor. Special concerns such as the limits of the velocity, acceleration and jerk are beyond the considerations in the conventional electronic cam motion control. This study proposes the curve-fitting of a Lagrange polynomial to the cam profile, based on trajectory optimization by cubic B-spline interpolation. The proposed algorithms may yield a higher tracking precision than conventional master-slaves control method does, providing an optimization problem is concerned. The optimization problem contains three dynamic constraints including velocity, acceleration and jerk of the motor system.
2.1 Prerequisite of Electronic Cam (ECAM) Tracking Control
Electronic cam (ECAM) control is a well-known master-slaves system. Figure 2.1(a) schematically depicts a block diagram of the proposed ECAM control system for mathematical representation. The variables and symbols in the figure are defined in the following sections. In Fig. 2.1(a), the motion of the slave motor clearly depends on the estimated slave position command, , which is generated by cubic B-spline interpolation, combined with an optimization algorithm. Such optimization is performed to meet the demands of limited performance - the constraints of velocity, acceleration and jerk. The method of cubic B-spline curve-fitting is based on substituting the estimated master position
pk+1
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into the cam trajectory. It is established using Lagrange’s interpolation formula to generate a Lagrange polynomial curve. However, the predicted master position is estimated by the electronic gearing (E-gearing) process.
2.1.1 External disturbance estimator
External disturbances (or loads) applied to the master may directly impact the efficiency of E-gearing. Therefore, disturbances must be suppressed. A disturbance estimator, depicted in Fig. 2.1(a), is used to estimate and suppress the external loads of the master motor.
However, Fig. 2.1(b) is one practical embodiment for the proposed disturbance suppressed control.
In Fig. 2.1(b), the external load (τ ) is estimated from the input current (i ) and the L angular velocity (
a
ω ), where Ka, Lˆf , Rˆf , Kˆ , Jˆ and Bˆ represent the nominal back
electromotive force constant, the nominal armature current inductance, the nominal armature current resistance, the nominal torque constant, the nominal moment of inertia and the nominal damping coefficient of the motor, respectively. Furthermore, , , , K, J
and B represent the reference voltage input, the actual armature current inductance, the actual armature current resistance, the actual (uncertain) torque constant, the actual (uncertain) moment of inertia and the actual (uncertain) damping coefficient of the motor, respectively.
Consider the dynamics of a DC motor:
Vref Lf Rf
According to Fig. 2.2(a), this estimator cannot be realized because of the differential term ( ) of angular velocity. The estimator depicted in Fig. 2.2(a) is also very numerically sensitive to the measurement noise because it yields high gains in the high-frequency field. Accordingly, a first-order low pass filter is used to estimate the disturbance,
s
Rearranging this external disturbance estimator in Fig. 2.3 yields no differential term. The estimated disturbance (τˆL) is then fed back to the current loop, and the external disturbance is
suppressed. In practice, due to the current-loop’s bandwidth is much larger than the speed-loop’s bandwidth, the electric dynamics (1/(Lfs+Rf)) can be neglected in Fig. 2.1(b)
for analysis.
2.1.2 Suppressing external disturbance
According to Fig. 2.3, the pole of the disturbance estimator equals the pole of the low pass filter, specified by Eq. (2.2). Thus, the estimated value for low delay time is obtained by reducing the time constant ( ) of the low pass filter. However, the small time constant trades off the estimated precision and robustness because it suffers more on measurement noise and modeling uncertainty.
ℑ
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Figure 2.2(b) is equivalently transformed to Fig. 2.2(c) to elucidate the effect of the external disturbance (τ ). According to Fig. 2.2(c), the effect of L τ is that of passing L τ L through the filterℑs/(ℑs+1). Accordingly, the external disturbance can be suppressed when the disturbance frequency is less than 1/ℑ rad/s. Thus, the smaller time constant yields better efficiency for suppressing high-frequency disturbances. However, a trade-off exists between estimated precision and robustness, as described in the above paragraph.
ℑ
Due to considerations of robustness, the measurement noise and the modeling uncertainty must also be considered in determining the time constantℑ. Appendix A discusses the sensitivities, , and to the uncertainties, where , and are the sensitivities of the current loop transfer function to the uncertain parameters
Gc
SK SGJc SBGc SGKc SJGc SBGc
Gc K , J and
B , respectively. Moreover, the effect of measurement noise is discussed with reference to a numerical simulation in Section 2.4.1.
2.2 Electronic Gearing (E-Gearing) Process
The electronic gearing (E-Gearing) differentiating itself from the mechanical gearing for that the E-Gearing system employed only electronic means to achieve the constant input/output velocity ratio. It is assumed that the output velocity control system is stiff and the main issue for the electronic gearing is to predict the future master velocity from its past. The velocity of the slave (output) motor is controlled according to the velocity of the master (input)
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motor.
The velocity of the master motor varies when loads or other external disturbances are applied. Therefore, the master velocity is not usually constant and may exhibit harmonics.
Even the amplitudes of the harmonic velocity are greatly reduced by using the proposed disturbance estimator, there still exists velocity variations. The procedure for estimating the master position and/or velocity is an important step for E-gearing. Methods of tracking control have been developed in various fields, and include radar tracking control [40] and others. This study proposes an order polynomial tracking method to perform the E-gearing process.
Nth
According to the order polynomial, the master velocity at time t can be expressed as,
Nth
= ∑
= N i
i it c
0
ω (2.4)
To determining the above coefficients ( ) in real-time, two procedures are proposed.
cN
c c0, 1,...,
(I) Initial procedure, t = kT, 1≤k≤ N +1, is the various order ( order) polynomial extrapolation, where the symbol k is a real-time counter of time base, T is the PC-based programming sampling time and kT denotes the present time over all this dissertation.
k 1)th
( −
l k i
i∑−ci l⋅T =ω
=1 ( )
0 , l = 0 to k-1 (2.5) Here use the assumption of00 =1.
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(II) Main procedure, t = kT, k > N+1, is the fixed Nth order polynomial extrapolation: Similarly, the symbol k is a real-time counter of time base, T is the PC-based programming sampling time. Where ωl( ) are the measured angular velocities during the interval [lT, (l+1)T], are the recorded positions of the master measured from the encoder at the past time lT. Furthermore, Figure 2.4 shows the temporal relations of the two proposed procedures.
Rewriting Eq. (2.6) in matrix form yields,
Ω matrix of polynomial coefficients and the matrix of measured angular velocities, respectively.
Moreover, the element of M in the row and column can be expressed as,
In Eq. (2.8), M is a constant matrix and exists; the computation involves no numerical degeneracy. Then the estimated velocity
−1
However, the estimated initial angular velocity may be chosen as the reference master velocity (ω ) which is the desired velocity of the master, i.e. ωˆ0 =ω .
Then, the estimated position of the master is, T
x
xˆk+1 = k +ωˆk ⋅ (2.11) where and are the measured position of the master at the present sample time kT and the estimated position at the next sample time (k+1)T, respectively.
xk xˆk+1
2.3 Predicting the Position of Slaves
This study uses Lagrange’s interpolation formula to establish piecewise cam trajectories.
If the piecewise reciprocal master-slave’s coordinates, , obtained from the given cam profile table, specify n+1 points, where i = 0 to n, and
)
Lagrange polynomial is,
= ∑
are the Lagrange interpolation coefficients. Table 2.1 is an example of a cam profile table.
Substituting Eq. (2.11) into Eq. (2.12), yields the next ideal cam profile position of the slave:
However, in Eq. (2.14), as the order n increased, ripples and oscillations may occur [9].
Furthermore, the design of the cam profile may not consider the dynamic capability of the control plant in advance. Some dynamic limitations that degrade the slave motion generally apply; for example, a cutting machine tool may chatter due to over-large jerk, so the jerk has to be limited during the cutting process. Restated, maximal velocity and acceleration are limited by the motor and servo drive system. Consequently, the actual trajectory of slave motion may not be fulfilled Eq. (2.14), but must be close to the ideal trajectory provided fitting the specified constraints. Given its low sensitivity, the piecewise trajectory of the actual slave motion with respect to time is proposed to follow a cubic B-spline curve of fourth degree [9], as shown in Fig. 2.5:
denotes the curve segment number and u = 0 to 1 within each curve segment.
are the control points of the spline. are the blending
functions.
The fourth degree cubic B-spline, as shown in Fig. 2.5, exhibits second-order continuity.
All the variables of the B-spline are defined below.
(I) denotes the initial control point of the time interval, where is the previous position command of the slave at time (k-4)T and equivalently the fifth
)
control point of the (k−4)th time interval.
(II) denotes the 1st control point of the time interval, where
is the previous position command of the slave at time (k-3)T and equivalently the fifth control point of the time interval. is the previous position of the slave at time (k-2)T and equivalently the fifth control point of the time interval. is the previous position of the slave at time (k-1)T and equivalently the fifth control point of the time interval.
the previous position of the slave at time kT and equivalently the fifth control point of the time interval.
k 1)th
( + pk ,5
kth
(VI) denotes the position command of the slave motor yet to be determined, and is equivalently the fifth control point of the time interval.
5 ,
pk+1
k 1)th
( +
(VII) denotes the sixth control point of the time interval, where is derived from the cam profile position at time (k+2)T, as indicated in Eq. (2.14).
ˆ ))
Statements (I) ~ (VII) include a total of seven unknowns and six independent equalities. There is an extra degree of freedom left for the following optimization problem:
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The slave’s position error between the next unknown position command and the ideal cam profile position command at time (k+1)T can be expressed as,
5
The objective error function is defined in quadratic form as,
2
To ensure that the velocity, acceleration and jerk do not exceed the maximal values,
posed on the optimi
(2.18a)
Minimizing the objective error function su and
(Vmax ,AmaxandJerkmax) allowed for the motor’s system, three inequality constraints are im zation. The first-, second- and third-differentiation of the cubic B-spline curve at the start, u = 0, of the fourth segment, can be expressed as follows [9].
5
bject to the constraints on velocity, acceleration
jerk, yields the one dimensional constrained optimization problem:
2
ization pro quadratic cost fu n
max (2.19b) to | 1 ,4(0) | max (2.19c) | 1 ,4(0)| max (2.19d)
blem of a nctio has an easy-to-find The constrained optim
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optimal solution, p*k+1 ,5 = fL(xˆk+1) with zero cost, when none of the constraints is violated.
Acc
ization problem
ording to equation (2.19a) ~ (2.19d), the optimization problem may be reformulated as an
unconstrained minim as follows:
2
1, 5 ˆ 1 2 v v 1, 5 a a 1, 5 J J 1, 5
pk L( k ) W g (pk ) W g (pk ) W g (pk )
Minimize + − f x + + + + + + + (2.20) whereW , v W and a W are the weighting factors of velocity constraint, acceleration J
constraint and jerk constraint, respectively, and
⎨ ≥
In an extreme case that , the minimization problem implies a constraint violation priority that s much more important than
ng techniques to is highly nonlinear, existi find the global optimization is not guaranteed. It needs to enumerate all the possible cases for the global solution. Figure 2.7 shows all the possible optimal solution for the extreme case thatWv >>Wa >>WJ. The bounds of pk+1 ,5for each of the constraints may be easily calculated from equations (2.19b), (2.19c) and (2.19d) by substituting the inequality sign into equality sign
1,3
1,6 1,4
1,3 p
3 p 1
pk+ + k+ + k+ (2.22c)
max 4
, 1 5
,
1 3
)) 1 0 ( 3 (
pk+ =−1sign rkuuu+ ⋅Jerk −
The optimal solution process may be depict
According to the flow chart, the solution of the optimization
s the velocity constraint, as shown in Figs. 2.7 (a) ~ (l);
(ii)
.4.1 Simulation of disturbance estimator
external disturbance is assumed to be a square wave ed in the flow chart as shown in Fig. 2.6.
problem is unique, and thus guarantees to be the global optimum.
In Fig. 2.6, all possible cases are enumerated and categorized as follows. (i) The ideal cam profile position command violate
the ideal cam profile position command violates acceleration constraint and does not violate velocity constraint, as shown in Figs. 2.7(m) and 2.7(n); (iii) the ideal cam profile position command violates only the jerk constraint, as shown in Figs. 2.7(o) ~ (q); (iv) the ideal cam profile position command satisfies all of the constraints, as shown in Fig. 2.7(r).
2.4 Simulation and Experimental Results 2
For simulation purposes, the nominal function.
L =
τ Amp (2.23) The torque amplitude Amp of the square wave is set to 4.8773 (N⋅m) and the frequency of the squa
velocity obtained using the proposed disturbance estimator feedback control and without re wave is 1 Hz. Figures 2.8(a) and (b) present the master’s simulated angular
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using the disturbance estimator. The nominal parameters of ster motor defined in Section 2.1.1, are Kˆ =0.55( N⋅m/A) , )Jˆ =0.093( kg⋅m2 , )Bˆ =0.008( N⋅m⋅s/rad ,
H 046 .
=0
Lf ,
the ma
Ω
= 1
Rf and Ka =0.55 V⋅s /rad. The sampling time of the current loop is set to 0.001s in the simulation. Furthermore, the amplitude of the disturbance load torque is 4.8773 (N⋅m) and the torque constant 0.55 is (N⋅m/A), that s power loss is around 78.6 (W) (provided by the paper reviewer).
However, the power loss of 78.6 (W) is not that serious for the applications with motors up to several kW
Figures 2.9(a) and (b) show the maximum errors between the fed torque ( L
is, the operating current i about 8.9 (A), the ia2Rf
.
τ ) and the
estimated torque (τˆL) for various time constants (ℑ). In Fig. 2.9(a), a smaller ℑ yields a smaller mean tor error. However that a lower yield a larger mea
d
s set to ten times the current loop
que Fig. 2.9(b) reveals ℑ s
surement noise. Furthermore, the measurement noise was assumed to be a zero-mean, normally (Gaussian) distributed random signal in the simulation.
Both a larger mean torque error and a larger measurement noise re uce the tracking performance of the master, so the time constant must be neither too small nor too large. In the experiment, the time constant (ℑ) of the disturbance estimator wa
sampling time. As depicted in Fig. 2.8(b), the time constant (ℑ) and the current loop sampling time are set to 0.01 sec and 0.001 sec, respectively.
2.4.2 Experimental results for tracking performance of the electronic gearing process
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Table 2.2 lists the parameter settings of the ECAM control. The accuracy of the tracking of the master’s velocity is characterized by the maximum error between the actual position and the estimated position. Figures 2.10(a) ~ 2.10(d) show that the maximum tracking error of the master’s position, using the fifth order polynomial tracking control method, is zero when the master’s nominal mean speed is 10π rad/s. Table 2.3 shows the maximum tracking error of the master’s position for polynomial tracking control methods of various orders (N = 0 to 5).
2.4.3 Performance of the electronic c rocess
Figure 2.11(b) shows an example of a reference trajectory that corresponds to the elec
am p
tronic cam motion. According to a constant master speed of 10π rad/s and a maximum slave travel distance of 40π rad, the reference trajectory yields a 100π rad/s maximum slave speed, 630π rad/ s2 maximum acceleration and 4060π rad/ s3 maximum jerk.
The master’s speed is generally not constant and may be harmonic, as shown in Figs. 2.8(a) and (b). The speed will exhibit the actual position of the master and the ideal cam trajectory, as shown in F nd 2.11(c), respectively. This p is ajectory contains 191 points. Three performance indices are used to quantify the accuracy and consistency. The tracking accuracy of the slave motion is characterized by the maximum error and the root mean square (RMS) error. The consistency of the cam tracking – that is, the cycle-to-cycle variation - is characterized by the RMS difference between the particular error response and
igs. 2.11(a) a iecew e cam tr
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the error response averaged over a number of cycles. Fifty cycles of tracking error data were collected. Figures 2.12(a) ~ 2.12(d) summarize the results of slave position. Table 2.4 lists the maximum tracking errors of the slave’s position in encoder counts, using the N order th polynomial tracking control method and the pure Lagrange polynomial curve-fitting method.
Furthermore, Fig. 2.13 shows the partial results of the slave’s tracking velocity, acceleration and jerk, according to Lagrange polynomial curve-fitting with or without the aforem tioned optimization. Similarly, Table 2.5 indicates the tracking control performance, also for the Lagrange polynomial curve-fitting method with or without the aforementioned optimization.
2.4.4 Computational load on the CPU of the proposed ECAM tracking control
The selection of N depends on the accuracy demanded. As stated above, tracking using a higher order polynomial yields higher precision; however a tradeoff exists between th
en
e computational tim
“order” of the polynomial used and the CPU time required. In practice, the
e of the proposed algorithm (fifth-order tracking) is about 0.02 ms in a programming cycle on an Intel Pentium III 900MHz CPU. The computational time of a programming cycle is much less than the PC-based sampling time, 10 ms.
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