Fractional Motion Control: Application to an XY Cutting Table B. ORSONI, P. MELCHIOR, Th. BADIE*, G. ROBIN* and A. OUSTALOUP LAP-UMR 5131 CNRS - Université Bordeaux 1-ENSEIRB - 351 cours de la Libération - 33405 TALENCE cedex - France Phone: +33 (0)556 846 607 - Fax: +33 (0)556 846 644 - Email:
[email protected] - URL: www.lap.u-bordeaux.fr * Lectra Systèmes SA - 23 chemin de Marticot - BP 34 - 33611 Bordeaux Cestas Cedex - France
Abstract: In path tracking design, the dynamic of actuators must be taken into account in order to reduce overshoots appearing for small displacements. A new approach to path tracking using fractional differentiation is proposed with its application on a XY cutting table. It permits the generation of optimal movement reference-input leading to a minimum path completion time, taking into account both maximum velocity, acceleration and torque and the bandwidth of the closed-loop system. Fractional differentiation is used here through a Davidson-Cole filter. A methodology aiming at improving the accuracy especially on checkpoints is presented. The reference-input obtained is compared with spline function. Both are applied to an XY cutting table model and actuator outputs compared. Keywords: Path Tracking, Splines Functions Optimization, Fractional Calculus, Davidson-Cole Filter, XY Cutting Table.
1. Introduction Much work has been carried out in path tracking design. The polynomial approach [1] respects the limitations imposed by the maximal velocity and acceleration, but permit neither maximal values of speed and acceleration to be kept, nor minimal path completion time to be reached. The Bang Bang approach [1] takes into account the same physical constraints but does provide a minimal path completion time. However, as for the polynomial approach, the dynamics (bandwidth) of the control loop are not taken into account, so overshoots can appear on the end actuator. Although popular, these approaches are now being abandoned by some manufacturers as they not able to limit overshoots appearing for small displacements. A simple digital filter is often preferred, as it is easy to implement and to adapt for killing overshoots. It reduces the high frequency energy of the path planning signal using a low-pass filter with trial-and-error determined parameters. This type of path tracking, based on position step filtering, does not permit separate control over maximal values of velocity and acceleration, which stay proportional to the amplitude of the step applied. When the control loop is perfectly defined, algorithms of Shin and Mac Kay [2,3] or Bobrow [4] allow the synthesis of the optimal actuator control that takes into account constraints on the inputs and the details of the dynamics manipulator. The dynamic model of the process must be designed by applying Lagrange formalism. The use of curvilinear abscissa allows reduction of the number of variables without loss of information. The minimal path time is determined from the phase curve using the Pontryagyn maximum principle. However, this is fastidious and must be done for each trajectory. On the other hand, this method provides no connection to tracking accuracy, except for some particular case [5]. These disadvantages limit the development of these algorithms in industry. When k points on it define the trajectory, the form of the curve between each point has to be determined, while respecting nominal speed and acceleration. A polynomial of interpolation can be calculated, but the degree of this polynomial increases with the number of points, and can be unmanageable. Cubic spline functions (order 3 piecewise polynomials) overcome this difficulty. They are now widely used in robotics and can be considered as the present industrial reference method for motion control in robotic. They are minimal curvature curves [1] that eases the design of smooth industrial curves. Cubic spline functions, made up of one jerk step per point, the duration of each step can be optimized using Lin's algorithm [6], or De Luca's algorithm [7]. Based on the nonlinear simplex optimization algorithm [8], optimized cubic splines offer a complete-path reference solution. However, as in the polynomial or Bang Bang approach, the dynamics of the control loop are not taken into account: overshoots on the end actuator appear for small displacements. To avoid overshoots, preshaping command inputs [9] could be used to reduce vibration appearing for small displacement. This technique is an extension of the posicast technique developed by Smith [10]. Vibrations are eliminated by convolving a sequence of impulses, the input shaper, with a desired system command to produce a shaped input. By adding more impulses in the shaper, multiple vibration modes and robust elimination of vibration can be performed. It is a practical and efficient method, well adapted for point to point motions. The minimum duration required to reach the target without residual vibrations directly depends on the dynamic of the
plant. Using the preshaping technique, even if the error is null after a finite time, ∆t, the integral of the error tends towards a constant; the second integral of the error increase in t; the third integral of the error increase in t2. The finite time, ∆t is here necessary to eliminate vibrations but not sufficient to control the error between input and output. A new approach to path planning design using fractional differentiation [11,12,13] applied to non-timevarying plants is developed by Melchior [14]. It permits the generation of optimal movement reference-input leading to a minimum path completion time, taking into account both the maximum velocity, acceleration and jerk and the bandwidth of the closed-loop system. Motion control using a Davidson-Cole filter is easy to implement, as it is simply a numerical filter, and it efficiently kills overshoots, especially for small displacement [15,16]. As spline functions, made up of one jerk step per point, are the present industrial reference in robotics, we decided to take and adapt Lin's algorithm [6] to define a fractional filter with jerk input, considering both, the maximum velocity, acceleration and jerk and the bandwidth of the closed-loop system. The article is based on the principle of fractional filter with jerk input described in the congress [15]. It develops and specifies the interest of the method and provides a study of the error on output, and particularly on checkpoints. The remainder of this paper is divided into four sections. Section 2, gives the principle of the approach. The model of the XY cutting table is briefly presented in Section 3. In Section 4, splines functions and fractional filter with jerk input are both applied to an XY cutting table and actuator outputs are compared.
2. Davidson-Cole filters 2.1. Davidson-Cole filter Polynomial interpolation and Bang Bang laws have a bandwidth that varies with the length of the displacement. Overshoots observed for small displacements are due to these variations. Numerical filters have a fixed bandwidth, allowing optimization in the frequency domain once and for all, and for all displacements, to limit end actuator vibration. A low-pass filter described by the transmittance: 1 , (1) F (s) = 1 + τ f s nf
(
or F ( s ) =
)
1
, (2) n f s 1 + ωf uses real poles and prevents frequency resonance. The choice of identical poles allows the greatest possible energy on a given bandwidth (Figure 1). The filter given by expression (1), where parameter n is real and no longer restricted to being an integer, is a Davidson-Cole filter [17]. A Davidson-Cole filter reduces energy of the signal at high frequencies. It continuously controls not only the bandwidth (time constant τ) but also the selectivity (real order n) as can be seen in Figure 2. |F (j ω )|
ω [rd /s] o rder 1 p o le o rder 4 p o le
Figure 1. Pole assignment for a maximum energy in a given pass-band. 1/τf
––
.....
Control Filter
ωr
nf
Figure 2. Power spectral density assignment of the Davidson-Cole filter compared to resonance frequency placement of the control loop to which it is applied.
A Davidson-Cole filter permits continuous optimization on its two constitutive parameters to limit resonance modes and to minimize the completion time. This filter can be implemented as either analog or digital filters. For position control, such a filter, used as prefilter, is efficient to reduced overshoots. However, maximal values of speed, and acceleration are reached only punctually, so generation is under-optimal. Moreover, maximal values of speed and acceleration are linked as they are both proportional to the amplitude of the step applied. Then, only null intermediate speeds are possible for a reference-input sequence. A Davidson-Cole filter can be implemented in the form of a digital filter one for all as it is independent of checkpoints and it only depends on the dynamic of the plant. 2.2. Optimization of the Davidson-Cole filter The Davidson-Cole filter has, at its corner frequency, a magnitude: F ( jω f ) = −3 n f dB .
(3)
The resonance Q at the pulsation ω r , can thus be eliminated by the Davidson-Cole filter for which the corner frequency is: 1 ωf = = ωr . (4)
τf
and for which the selectivity is: Q n f = dB , (5) 3 Then, Equations (4) and (5) define completely the Davidson-Cole filter by fixing its fractional order of selectivity and its corner frequency. Others optimizations or criterions of optimization could be used but the easy of use method described above is efficient to cut the resonance and it highlights the interest of using a fractional order of integration. 2.3. Definition of Davidson-Cole filter with jerk input Using Davidson-Cole filter with position inputs of step type, speed and acceleration in the filter output are proportional to the amplitude of the steps applied. To separate speed and acceleration control, a Davidson-Cole with speed input has been developed allowing intermediate speed control for path tracking [14,15]. As the spline function, made up of one jerk step per point, is a reference in robotics, we decided to develop a Davidson-Cole filter with jerk input. The principle of motion control by Davidson-Cole filter with jerk input is described in Figure 3.
q0(t)
q(t)
(
1
s3 1+τ
Jerk reference input
f
Filter
s
)n
f
y(t) H ( p)
Control
Figure 3. Principle of motion control by Davidson-Cole filter with jerk input. As for spline functions, the duration of each step can be optimized using Lin's algorithm [6] or De Luca [7]. The optimization algorithm is a non-linear simplex one [8]. It determines minimal time intervals between each point, by taking into account nominal values of speed, acceleration and jerk. Increasing Vnom, Anom and Jnom respectively by a factor λ, λ2 and λ3, the final time of the optimal solution is reduced by this same factor λ. 1 s Given: L { f (λ t ) u(t ) }= F . (6) λ λ The more static the magnitude Vnom, Anom et Jnom, the shorter the final time, but the greater the actuator bandwidth must be. Static magnitudes, Vnom, Anom and Jnom , have no direct link with the bandwidth of the system: it is possible to find two actuators with the same static characteristics but with very different bandwidths. However, as indicated by property (6), the static magnitudes fix the reference-input bandwidth for spline functions. This bandwidth is not necessarily inferior to the actuator bandwidth. This corroborates empirical verifications and simulations: cubic spline functions, as polynomial methods, do not reduce overshoots for small displacements. It can be noted that the nominal value of jerk can serve arbitrarily and empirically to limit the bandwidth of the reference-input, and is thus misused. The main objective is to improve the spline function to avoid overshoots observed for small displacements.
The fixed bandwidth of the Davidson-Cole filter is used within its two constitutive parameters, to obtain that result. 2.4. Dynamic constraint For the design of the control loop reference input, the dynamic of the actuators is also taken into account by introducing a specific supplementary constraint: the relaxation time. It is the duration required for the system to rejoin its asymptote (for a given precision). Each jerk step is also controlled independently if a minimum duration is insured. However, after the Davidson-Cole filter and three integrations, errors on checkpoints occur on the filtered spline. The relaxation time, which is the duration required for the system to rejoin its asymptote, is now evaluated. The duration required by the step response of the Davidson-Cole filter to reach its asymptote with a given relative precision, arbitrarily fixed here at 0.1% , is computed. As the step response of a Davidson-Cole is an ever increasing function, the relaxation duration is the minimum value of time for which the effective error is inferior to the given relative error: t rx = min ((1 − gammainc(n, t ) ) < ε ) . t >0
(7)
2.5. Error estimation of the filter effect By filtering the control loop reference input, errors appear on checkpoints. Using a Davidson-Cole filter, this error is reduced by approximating the filter by a delay. So, the resulting error is now studied. Three integrations provide the position from jerk. The final value theorem can thus provide the error after three integrations between the jerk step and the output of the Davidson-Cole filter. A truncated series of order 3 is then used: s 1 1 nτ a (8) ~ 3 − 2 + 2 − a3 , n 3 s 0 s s s (1 + τ s ) n (n + 1) 2 τ , 2 n (n + 1)(n + 2 ) 3 τ . a3 = 6 a2 =
where: and:
(9) (10)
The asymptotic behavior of the output of the Davidson-Cole filter with jerk input f(t) can thus be deduced from Equation (8): t t3 t2 (11) f (t ) ~ − nτ + a2 t − a3 . 2 ∞ 6 The output of jerk step integrated three times is: t t3 . (12) g (t ) ~ ∞ 6 Thus, the error in position e(t), between the function g(t) and the function f(t) is: (13) e(t ) = g (t ) − f (t ) , t2 (14) − a2 t + a3 , 2 ∞ where a2 and a3 are given by Equations (9) and (10). The error between f(t) and its order three approximation (Equation (11)) tends towards zero for the asymptotic behavior and thus the order 3 approximation (14) represents the error e(t) when: (15) t > t rx . So the error should be anticipated by feedforward in the initial spline. However, such a modification implies a new optimization as the checkpoints are modified and the convergence of such iterations is not proved. A less accurate but more practice estimation of the error is then used. t
e(t ) ~ n τ
2.6. Approximation of the filter effect by a delay The truncated series development of the Laplace transform of a delay ∆t, is: s
D ( s ) = e − ∆t s ~ 1 − ∆t s . 0
(16)
The truncated series development of the Laplace transform of the Davidson-Cole filter is: s 1 ~ 1 − nτ s . (1 + τ s )n 0
(17)
The low-frequency behavior of the Davidson-Cole transfer function (17) and the delay (16) are also identical when: ∆t = n τ . (18) The Davidson-Cole transfer function, on which the spline is applied, can thus be approximated by a delay given by Equation (18). After three integrations, the order three truncated series development of the Laplace transform of the Davidson-Cole filter is: s 1 1 nτ a (19) ~ 3 − 2 + 2 − a3 . n 0 3 s s s s (1 + τ s ) After three integrations, the order three truncated series development of the delay ∆t = n τ is: e − ∆t s s 1 n τ (n τ )2 (n τ )3 . − + − ~ 2s 6 s3 0 s3 s 2 The difference between Equations (20) and (19) is: s 1 1 nτ 2 − nτ s − + b3 , ~ e − 0 2s s 3 (1 + τ s )n
(20)
(21)
n (3n + 2 )τ 3 . 6 It can thus be deduced from (21) that for a unity jerk step input and after three integrations, the Laplace transform of the error in position between the delay and the Davidson-Cole filter is:
where b3 =
nτ 2 t + b3 , ∞ 2 The maximum area of a jerk step is limited by the maximum admissible acceleration. If J k and t k are respectively the amplitude and the duration of the jerk step, the area of this jerk step must be inferior to the maximum admissible excursion of the acceleration: J k t k ≤ 2 Anom . (22) From Equation (21), the asymptotic error in position ε (t ) of a non-unit jerk step is deduced: t
e(t ) ~ −
nτ 2 t k − b3 . 2 Using inequality (22) in (23), it is deduced that the asymptotic error is bounded:
ε (t ) = J k
(23)
lim ε (t ) ≤ n τ 2 Anom − b3 .
(24)
t →∞
The approximation of the Davidson-Cole filter by a delay can be used easily in the optimization algorithm. It does not require any modifications of the checkpoints in order to feedforward the error generated. The delay introduced is sufficient to obtain a bounded error and a more accurate output. 2.7. Improvements of Davidson-Cole filter with jerk input The Davidson-Cole filter takes into account the dynamic of the plant by fixing the bandwidth of the control loop reference input. The control signal is smoother and the resonance of the plant is reduced. Moreover, the relaxation time of the filter can be deduced and each jerk step can be controlled independently. The relaxation time is the duration required for the system to rejoin its asymptote. Lin's algorithm is thus modified considering a dynamic constraint: each jerk step must have a duration superior to the relaxation time. This insures that first, each jerk step effect is decoupled, and second, the asymptotic and bounded error is reduced by introducing the delay ∆t = n τ which approximates the filter effect. The coordination problem for axes with different dynamics is often a difficult one. Here, the delay given by Equation (18) is calculated for each axis. These delays can thus be anticipated separately so that each checkpoint is reached more precisely at the predicted instant. Thus, different delays for different dynamics insure coordination and an improvement of the accuracy at the output. The Davidson-Cole filter with jerk input is designed by taking into account the dynamic of the plant in order to reduce the resonance. The reference input is also designed by taking into account the limit bandwidth of the Davidson-Cole filter on which it is applied throught the relaxation time which provides a minimum duration of jerk step.
3. Modelization and identification of an XY cutting table 3.1. Description of the plant The study plant is a cutting table for leather or fabric, using laser or high-pressure water as cutting tools. Several types of tables exist, H-shaped, T-shaped, cross-shaped or XY tables, each with a specific field of application. The table considered here is an XY cutting table, whose operating principle is illustrated in Figure 4. This XY design determines the specific mechanical features of this type of table, which results in two study properties. First, the two degrees of freedom are independent. Second, the mechanical differences between the two degrees of freedom are not structural but parametric, in particular: carried mass: tool mass (Y) 2-20 kg, beam plus tool mass (X) 30-60 kg; stiffness of the driving belts; characteristics of the driving motors Mx and My . DC Motor MX + Pulley
θX
0
Transmission Shaft for the X-axis
Y
X
Belt (stiffness Kx)
Belt (stiffness Ky))
θY
M
Tool Conveyor
Y axis support Beam
Cutting area DC Motor MY + Pulley
Figure 4. Schematic of the XY cutting table. Only the degree of freedom on the Y-axis is modeled, as the X-axis is deduced from it by simply changing the parameter values. Jp1y
Jp2y y(t)
Jry bry
bp1y
K1y
bp2y
K2y Meqy fy
ry Jmy
My
bmy
Ly Ry Uy
Figure 5. Functional design of the Y axis. 3.2. Dynamic model design The dynamic model design of the Y axis is given in Figure 5. As the belts are not perfectly rigid, the Y-axis system has two degrees of freedom θy(t)and y(t). By applying the Laplace transform on the Lagrange Equations of the system, we obtain a system of differential Equations. a) mechanical part (25) J θ s 2 + bθ s + kθ Θ my (s ) = K eqy r y R py Y ( s) + C mY (s)
(
(M
)
)
(26)
b) electrical part U y (s) = E y ( s) + ( L y s + R y ) I y (s) ,
(27)
y
s 2 + b y s + K eqy Y ( s ) = K yθ Θ my ( s ) − f y ( s )
c) electro-mechanic part E y ( s ) = s K ey θ my( s ) ,
(28)
C my ( s ) = K cy I y ( s ) .
(29)
2
with: J θ = ry J p1y + J my , bθ = bm + bry + r y
2,
2
2
kθ = ry R py K eqy , M y = M eqy + J p 2 y / R py 2 , b y = b p 2 y / R py 2 and
K yθ = K eqy r y R py .
3.3. Identification Dynamic identification determines numerical parameter values of the model, and their variation range according to functioning conditions. We are not permitted to disclose these values. SANYO DC motors are used. The modeling of the belt stiffness is based on two assumptions: - the belt is sufficiently pre-stressed for stretching not to create a sagging phenomenon; - stiffness calculation only takes into account the elasticity of the steel cables forming the belt, as the polyurethane support adds negligible stiffness in parallel with the cable stiffness (combination law of two springs mounted in parallel). Hooke’s law allows expression of the various stiffness constants versus the features of the belt considered, and to the straight positions of the tool conveyor. Hence, the stiffness of the belt along the Y degree of freedom: 1 1 K eqy (Y ) = K y + (30) −Y Y 2 . L where Ky is the constant characteristic of the cable material, called Young modulus. To validate the model more easily, a functioning point is defined, at the point of least variation of the stiffness. The identification of inertia in rotation, and masses in translation, is determined from the nature and dimension of each element. Dry friction values are measured on the cutting table, using a balance. Current consumption of the DC motors provides the viscous friction coefficients, which are proportional to angular speed of axle, and displacement tests at constant speed along each axis can be carried out. However, the influence of viscous friction on the system is weak and difficult to measure. Evaluations have been defined by sameness with others processes. The approximations used in the model, notably the viscous and dry friction values, are under-estimated. However, the model gives a good evaluation of the end-actuators and end-manipulators outputs. 3.4. Frequency responses As can be seen in the Bode diagrams (Figure 6), for the frequency responses of the X and Y control axes, the resonance values and the resonance pulsation values are nearly identical: 5 dB and 110 rd/s respectively. These values are parameters that will be used to define the reference input for the Davidson-Cole filter with jerk input. Bode Diagrams From: Input Point
0
Y axis
-40
0 To: Output Point
Phase (deg); Magnitude (dB)
X axis -20
-100 -200
X axis Y axis
-300 -400
101
102
Frequency (rad/sec)
Figure 6. Frequency response of the X and Y axes. 3.5. Model validation From the identified model of the plant, time and frequency responses of the model can be simulated, to compare with real readings from the table. A 100mm diameter circle of cotton dress material to be cut (in our test drawn on paper by a pencil, replacing the cutting head) is taken as a validation example. The nominal value for acceleration is 10m/s2 and the nominal speed value is 1m/s. The reference input, the controller signals, and the actuator outputs are digitized. The reference input is used as input for our simulation model, and the resulting simulated controller signals and actuator outputs compared to those of the real cutting table (Figures 7-8).
These results show that the model is partially validated. The approximations used in the model, notably the viscous and dry friction values, are under-estimated. However, the model gives a good evaluation of the endactuators and end-manipulators outputs. 50 0 -50 -100
Y axis input [V]
X axis input [V]
100
0.05
0.1
0.15
0.2
0.25
0.3 0.35 time [s]
0.4
0.45
0.5
0.55
0.05
0.1
0.15
0.2
0.25 0.3 0.35 time [s]
0.4
0.45
0.5
0.55
50 0 -50
Figure 7. Plant inputs for 0.1m circle (joint space). position X [m]
0.05 0 -0.05 -0.1 -0.15
0
0.1
0.2
0.3
0.4
Reference0.6 input 0.5 Real output Simulated output
0.7
0.4
0.5
0.7
time(s)
position X [m]
0.1 0.05 0 -0.05 -0.1
0
0.1
0.2
0.3
0.6
time(s)
Figure 8. Axes positions for 0.1m circle. 3.6. Davidson-Cole Jerk Filter parameters In order to filter the resonance Q of 5 dB at 110 rd/s, the corner frequency of the filter is the resonance frequency (Equation (4)): ω f = 110 rd/s (31) and the selectivity is deduced from Q and Equation (5): n = 1.66 . (32) The Davidson-Cole filter is then completely defined by its corner frequency provided by Equation (31) and by its fractional order provided by Equation (32). Considering real values for parameter n allows continuous optimization. It is possible to use the nearest integer for implementation but it is no more difficult to implement a non integer filter. 3.7. Dynamic constraint computation The relaxation time is the duration required for the system to rejoin its asymptote. In the particular case of the XY cutting table studied, the delay required on the two axes is the same. There is no need to consider a coordination parameter, except to keep in mind that the checkpoints will be reached after the delay introduce by the prefilter and the dynamic of each axis. For a corner frequency and a fractional order of the Davidson-Cole filter given by Equations (31) and (32), the relaxation duration computation provides by (7) is: ∆t min = t rx = 0.0774 s , (33) which is thus the minimum duration of a jerk step or even the minimum duration between two checkpoints.
4. Comparisons Davidson-Cole filter with jerk input and spline function are both applied as reference inputs and compared using the CRONE toolbox [18]. Comparisons are made in the two dimensions of the operational plane. The comparison protocol is based on two classical figures (square and circle) of small and medium dimensions for the cutting table. Both methods consider constraints on speed, acceleration and jerk: x y Vmax = Vmax = 1 m/s ,
(34)
x y Amax = Amax = 10 m/s 2 .
(35)
The maximum admissible value for jerk is not known. A value adapted for a given trajectory can be used but it would be a misused as this value must not depend on the trajectory to follow. Thus, this value is not constrained for the optimization. Simulations are shown in the Figures 9-16 and discussed below.
♦ Square 0.2 x 0.2 m (Figures 9, 10): - For the spline function, the follow-up is good, except in a corner where an overshoot appears. The path-time is 1.39 s. - For the Davidson-Cole filter with jerk input, the follow-up is very good without significant overshoot. The path-time is almost equal with 1.41 s. ♦ Square 0.01 x 0.01 m (Figures 11, 12): - For the spline function, the follow-up is not good, especially in corners. The path-time is of 0.36 s. - For the Davidson-Cole filter with jerk input, the follow-up is better with overshoots lower than 0.2 mm but the path-time is longer 1.2 s. ♦ Circle ∅=0.1 m (Figures 13, 14): - For the spline function, the follow-up is good. The path-time is 1.45 s. - For the Davidson-Cole filter with jerk input, the trajectory is smoother. The path-time is almost equal with 1.55 s. ♦ Circle ∅=0.01 m (Figures 15, 16): - For the spline function, the follow-up is very bad. The path-time is 0.46 s. - For the Davidson-Cole filter with jerk input, the follow-up is quite good the path-time is longer with 1.58 s. 200
Y [mm]
160
120
80
40
0
0
40
80
120
160
200
X [mm]
Figure 9. Square 0.2m; spline; : checkpoint; dotted line: reference input; solid line: output.
200
Y [mm]
160
120
80
40
0
0
40
80
120
160
200
X [mm]
Figure 10. Square 0.2m; Davidson-Cole filter; : checkpoint; dotted line: reference input; solid line: output. 12
10
Y [mm]
8
6
4
2
0
-2 -2
0
2
4
6
8
10
12
X [mm]
Figure 11. Square 0.01m; spline; : checkpoint; dotted line: reference input; solid line: output.
12
10
Y [mm]
8
6
4
2
0
-2 -2
0
2
4
6
8
10
12
X [mm]
Figure 12. Square 0.01m; Davidson-Cole filter; : checkpoint; dotted line: reference input; solid line: output.
100
Y [mm]
80
60
40
20
0 -60
-40
-20
0
20
40
60
X [mm]
Figure 13. Circle 0.1m; Spline; : checkpoint; dotted line: reference input; solid line: output. 100
Y [mm]
80
60
40
20
0 -60
-40
-20
0
20
40
60
X [mm]
Figure 14. Circle 0.1m; Davidson-Cole filter; : checkpoint; dotted line: reference input; solid line: output. 10
Y [mm]
8
6
4
2
0 -6
-4
-2
0
2
4
6
X [mm]
Figure 15. Circle 0.01m; spline; : checkpoint; dotted line: reference input; solid line: output.
10
Y [mm]
8
6
4
2
0 -6
-4
-2
0
2
4
6
X [mm]
Figure 16. Circle 0.01m; Davidson-Cole filter; : checkpoint; dotted line: reference input; solid line: output. The Davidson-Cole filter with jerk input takes into account a supplementary dynamic constraint. Trajectories are better followed than the spline function method in all examples, but do take longer. The fractional method calculates the optimal extra time required to ensure a more accurate output. It requires negligible computation time by comparison with Lin's algorithm. In the particular case of the 0.01 m circle, output and control signals on axes X and Y are also compared (Figures 17-20). The control signal obtained by using the Davidson-Cole filter with jerk input is three times lower than the one which uses the spline (Figures 17 and 18). Overshoots appearing for the 0.01 m circle are also studied in the joint space (X and Y axes) through Figures 19 and 20. For the spline function reference input (Figure 19), the error between input and output oscillates. For the output obtained using the Davidson-Cole filter with jerk input, the error is smoother. Moreover the outputs match the delayed inputs. 15
Control axis X [V]
10 5 0 -5 -10 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.4
0.5
0.6
0.7
time [s] 10
Control axis Y [V]
5
0 -5
-10 0
0.1
0.2
0.3 time [s]
Figure 17. Circle 0.01m; spline: control signal. 4
Control axis X [V]
2
0
-2
-4
0
0.2
0.4
0.6
0.8
1 time [s]
1.2
1.4
1.6
1.8
2
0
0.2
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0.6
0.8
1 time [s]
1.2
1.4
1.6
1.8
2
3
Control axis Y [V]
2 1 0 -1 -2 -3
Figure 18. Circle 0.01m; Davidson-Cole filter with jerk input: control signal.
-3
x 10
6
position X [m]
4 2 0 -2 -4 -6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.4
0.5
0.6
0.7
time(s) -3
x 10
15
position Y [m]
10
5 0
-5
0
0.1
0.2
0.3 time(s)
Figure 19. Circle 0.01m; spline: X and Y axes output signal (joint space) dotted line: reference input; solid line: output. -3
position X [m]
5
x 10
0
-5 0
0.2
0.4
0.6
0.8
1 time(s)
1.2
1.4
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position Y [m]
8 6 4 2 0 -2 0
Figure 20. Circle 0.01m; Davidson-Cole filter with jerk input: X and Y axes output signal (joint space) dotted line: reference input; solid line: output.
5. Conclusion Motion control by a Davidson-Cole filter with jerk input allows the reduction of overshoots and overvoltages by fixing a fixed bandwidth for the control loop reference inputs. The Davidson-Cole filter with jerk input is simply defined by only two synthesis parameters, and can be implemented in the form of a classical digital filter. The filter effect of the Davidson-Cole filter can be anticipated and the overshoots and overvoltages reductions are no more balanced by a reduction of the accuracy on checkpoints. The relaxation time allows a decoupled control of each jerk step. Lin's algorithm is also modified to have a minimum duration for each jerk step. The asymptotic and bounded error is thus reduced by introducing the delay ∆t = n τ which approximate the filter effect. The delays for each axis can thus be anticipated separately so that each checkpoint is reached more precisely at the predicted instant. Thus, different delays for different dynamics insure coordination and an improvement of the accuracy in the output. The reference input obtained with the Davidson-Cole filter with jerk input takes into account a dynamic constraint. The Lin's algorithm adaptation doesn't increase the required time for computation. Moreover, trajectories are better followed than the spline function method. The fractional method calculates the optimal extra time required to ensure a more accurate output. Comparisons are not made with the same duration because it is the methodology which takes into account the dynamic of the plant that provides the minimum practical duration time. This information is not available when the dynamic of the plant is not considered. Experience shows that how many path points are chosen, and where they are placed, has a significant effect on the quality of the resulting path. This needs further investigation.
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