0amax; 0 t
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7
Optimal Vibratory Stick-Slip Transport Paul Umbanhowar and Kevin M. Lynch
x
Fig. 1. A point fixed to the surface is at relative to a fixed point in the world, and the part is at . Gravity holds the part to the surface.
x
Abstract—We describe a vibratory part transport mechanism that utilizes both static and dynamic friction to linearly transport parts in a horizontal direction. We derive a horizontal driving profile for the feeder surface that maximizes the steady-state velocity of parts on the surface subject to acceleration limits on the surface. Experiments verify the predicted transport behavior. We then derive the optimal feeding motion for a surface that can move in both the horizontal and vertical directions. Note to Practitioners—One type of vibratory conveyance, popular in industrial parts feeding and presentation, makes use of friction and sliding between the part and the support surface. This paper gives the surface vibratory motions that maximize the speed of transport. It also describes an experimental implementation and a control algorithm for achieving the desired vibratory motion. Index Terms—Friction, vibratory parts feeding.
I. INTRODUCTION Vibratory transport is utilized in a number of industrial parts feeding mechanisms, including bowl feeders and linear conveyance systems [1]. Linear conveyance is typically achieved by some combination of vertical and horizontal vibration, and the forces to transport the parts arise from two phenomena: impacts with the support surface (“hopping”) and frictional sliding [2], [3]. In this paper, we focus on the case that parts always remain in contact with the support, so that transport is due to frictional sliding. Transport by sliding can be achieved by vibratory motion of the surface in the longitudinal direction only [4]–[6] or by some combination of normal and longitudinal oscillation [1], [7], [8]. This paper derives the oscillatory motions of a surface that maximize the speed of transport of parts, given a fixed period of oscillation and an upper bound on the acceleration of the surface. The surface is horizontal, and it either oscillates in the horizontal direction only (Sections II–III) or in both the horizontal and vertical directions (Section IV). In the case of horizontal-only motion, this work builds on the “Coulomb pump” of Reznik and Canny [5]. They observed that transport occurs for asymmetric horizontal vibration of the surface because the friction force ff felt by the part during slipping contact with the surface satisfies
jff j = k mg
In their analysis, Reznik and Canny made the simplifying assumption that the part always slips relative to the surface, and they chose a particular driving profile for the surface [5]. Quaid [6] chose a different asymmetric driving profile and demonstrated parts transport using a full stick-slip analysis. In this paper, we design the forcing of the surface to take advantage of knowledge of the static (stick) and kinetic (slip) coefficients of friction between parts and the surface. In Section II, we use a full stick-slip analysis to find the surface motion that maximizes the speed of part transport, subject to bounds on the surface acceleration. We then demonstrate the improvement of this optimal driving over optimized versions of the motion profiles proposed in [5] and [6]. The theoretical results are validated by experiments in Section III. Our vibratory transport mechanism uses iterative learning control to achieve high-accuracy control of bang-bang acceleration profiles. Okabe et al. [4] studied the similar problem of finding the periodic longitudinal waveform of an inclined surface that maximizes a slightly different measure of the transport speed. They used a finite parameterization of the waveform and conducted a numerical study to develop guidelines for the choice of these parameters. In this paper, we incorporate acceleration (control) constraints and solve exactly for the optimal driving waveform, without parameterization. In Section IV, we extend our analysis to the case of both vertical and horizontal oscillation. We derive the optimal driving motion for this case, which yields a maximum transport velocity of up to four times what is possible with horizontal-only oscillation. II. OPTIMAL STICK-SLIP TRANSPORT A. Stick-Slip Transport Let the position of points fixed to the surface and the part be xS and xp , respectively, with respect to a stationary reference (Fig. 1). As a ! is C 1 and periodic function mapping time to position, xS : with period T , xS (t) = xS (t + T ). In steady state, the velocity of the part is also periodic, x_ p (t) = x_ p (t + T ), with a net displacement over the period of D = xp (t + T ) 0 xp (t), yielding an average part velocity vave = D=T . We assume the part moves on the surface according to dry Coulomb k be the static and kinetic friction coefficients, friction. Let s respectively, with associated maximum accelerations as = s g and ak = k g. When the part is initially at rest with respect to the surface (x_ S = x_ p ) and the surface moves with an acceleration x S [ as ; as ], p = x S and the part remains fixed to the surface. If x then x S > as , p = ak sgn( xS ), i.e., it begins to slip on the acceleration of the part is x the surface. If the part is already slipping (x_ S = x_ p ), the acceleration p = ak sgn(x_ S x_ p ). of the part is x A simple two-stage periodic surface motion that induces steady-state stick-slip transport of a part is
where k is the kinetic friction coefficient, m is the mass of the part, and g is the gravitational constant. In other words, the force magnitude is independent of the relative slipping speed: the part is subject to a force k mg when the signed surface velocity is greater than the part velocity and a force k mg when the surface velocity is less than the part velocity. Therefore, if we arrange the asymmetric periodic motion of the surface so that it spends more time with velocity greater than the part, the part feels a net force in the positive direction over the cycle.
0
Manuscript received August 19, 2006; revised August 23, 2007. This paper was recommended for publication by Associate Editor H. Ding and Editor M. Wang upon evaluation of the reviewers’ comments. This work was supported in part by the NSF under Grants IIS-0308224 and CMMI-0700537. The authors are with the Mechanical Engineering Department, Northwestern University, Evanston, IL 60208 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASE.2008.917021
20 j j
6
0
xS (t) =
6
0amax; a1 ;
t < T1 T1 t < T
0
(1)
where amax are control limits on the surface acceleration, 0 < a1 < amax , and T1 = a1 T=(a1 + amax ).1 Choosing a1 as , we obtain the stick-slip driving profile considered in [6]. As shown in Fig. 2, in steady-state the part slips forward with respect to the surface when the 1Throughout the paper, we assume slipping.
1545-5955/$25.00 © 2008 IEEE
a
> a
to ensure transport by
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The adjoint equation
_ = 00_ = 1 +0 indicates that (t) is constant and (t) is a linear function of time. ; a ] and all t 2 [t ; t ], we have By Pontryagin, for all u 2 [0a H(z3 (t); 3 (t); u3 (t)) H(z3 (t); 3(t); u(t)) 3 (t)u3 (t) 3 (t)u(t) indicating that u3 (t) = 0a for 3 (t) > 0 and u3 (t) = a for 3 (t) < 0. In other words, the optimal surface motion during slipping is bang-bang. As (t) is linear in t, there is at most one switch during the time interval. Employing (3), and defining 1t = t 0 t , we get 0a ; t t < t + u3 (t) = (4) a ; t + tt @H @z @H @z
1 2
1
1
2
max
Fig. 2. Part and surface velocity with the simple two-phase driving motion (1).
max
2
surface accelerates backwards in phase 1, then eventually matches the velocity of the surface and sticks again for part of phase 2. The slope of the part velocity is ak when slipping and a1 when sticking. The area between the part velocity and the surface velocity is the amount that the part is transported. The optimal driving problem is to maximize that area subject to the bounds on surface acceleration.
0
B. Optimal Transport The problem we will solve is the following. Given a period T and amax , find xS ;T bounds on the surface acceleration xS amax ; amax that maximizes vave in steady state and satisfies the pexS T ; xS xS T . In other words, riodicity conditions xS the problem is to maximize
j j
]
[0
: [0 ] !
(0) = ( ) _ (0) = _ ( )
T 0
Ldt =
T
(_x (t) 0 x_ (t))dt: p
0
0
When the part slips backward, we have L < , and when it slips forward, we have L > . The optimal surface trajectory, steady-state periodic part velocity, and average part velocity are written x3S t ; x3p t , 3 , respectively. and vave We make the following two observations. 1) Observation 1: During any periodic steady-state motion, there ; T such that xS t xp t . exists a t x3p t for all 2) Observation 2: For any optimal solution, x3S t t ;T . The first observation follows from the fact that, at steady state, xp xp T . The surface velocity is a continuous function, and if it were never equal to the part velocity (i.e., always larger or smaller), ak then the part would have a constant acceleration with xp during the cycle, a contradiction. The second observation follows from the fact that any part velocity xp t that can be realized for xp t with some xS t > xp t can also be realized for xS t a larger value of L. These observations indicate that the optimal surface trajectory has the part either sticking to the surface or slipping forward, xp > xS . Only during the slipping phase is L > . Let t1 ; t2 ; T represent a time interval during which the part is slipping forward, with the boundary conditions
() _ ()
2 [0 ]
_ ()= _ ()
_ () _ ()
2 [0 ]
_ (0) = _ ( )
j j =
_ ()
_ ()
_ ()
6= 0
0
x_ p (t1 ) = x_ S (t1 ) x_ p (t2 ) = x_ S (t2 ):
(2) (3)
: [
t1 ; t2 We would like to find the surface acceleration xS amax ; amax satisfying these constraints and maximizing
[0
J=
t
] (_x (t) 0 x_ (t))dt = p
t
S
t t
=[
] !
(_x (t ) 0 a t 0 x_ (t))dt: p
1
S
k
] =[ _]
Setting the state vector to be z z1 ; z2 T xS ; xS T and the surface acceleration control to be u, the state equations are written z z2 ; u T . We write the Hamiltonian as
_=[
]
H = 0x_ (t ) + a t + (1 + )z + u: p
1
k
1
2
2
max
max
2
2
2
2
max
1
max
=(
1
1
1
2
)
+ )1 (2
amax ak t= amax . The net displacement of the part where relative to the surface during this motion is d(1t) =
a2max 0 a2k 2 4amax 1t :
(5)
The preceding analysis applies to a particular subinterval [t ; t ] of [0; T ]. Now by the requirement that the net acceleration of the part 1
Tslip =
2
2
as T as + ak
and this is achieved if the part is sticking to the surface accelerating as during the rest of the cycle. Therefore, the sum of the at xS slipping times (the only times that contribute to the objective function J ) during the optimal motion is Tslip . It remains only to determine how this slipping time is distributed in slipping intervals over the cycle. t2 for any From (5), it is apparent that d t1 d t2 < d t1 t1 ; t2 > . From this we conclude that there is only one slipping interval during an optimal cycle, of duration Tslip . Theorem 1 summarizes the discussion above. Theorem 1: The surface motion that maximizes a part’s average transport speed is given by
=
1 1
_ () = _ ()
_ _ ] [0 ]
[
2
during the entire cycle be zero at steady state, the maximum time a part can slip forward is
S
0
1
(1 )+ (1 )
0
x3S (t) =
(1 +1 )
as ; 0 t < T1 0amax ; T1 t < T2 amax ; T2 t < T
(6)
T1 = ak T =(as + ak ) and T2 = T1 + (amax + ak )as T=((2amax )(as + ak )). The average velocity of the part
where is
3 = vave
d(Tslip ) T
= 4aa a (a +0 aa ) 2 s
2 max
max
2 k
s
k
2
T:
(7)
Example surface and part motions are illustrated in Fig. 3. The average part velocity is given in Fig. 4 as a function of ak =as and amax =as for as ,T .
=1 =1
C. Comparison to Other Driving Functions In [5], Reznik and Canny proposed the three-stage “bang-coastbang” driving waveform
xS (t) =
amax ; 0; 0amax ;
0 0ztT
(amax there is a sticking phase during the coasting period, and the average velocity is actually given by
0
=
v1
amax (z
0 1)
2
2
amax
2
+ ak
(amax +
2 ak )
02
ak amax
T:
When driven in the stick-slip regime, the transport velocity decreases. The optimal driving is attained for z = (amax ak )=(2amax ), and the optimal transport velocity by this driving waveform is
0
3
The ratio driving
K1
=
(amax
=
v1
0
2
ak )
16amax
T:
3 measures the improvement of our optimal
3
vave =v1
2
K1
=
4as (amax + ak ) (ak + as )2 (amax for all amax
0
>
ak )
> as
ak >
1 0:
The performance of the optimized bang-coast-bang driving profile (8) approaches that of the optimal driving of Theorem 1 when amax is large as . and ak In [6], Quaid proposed the two-stage driving waveform
x S (t)
If a1 is
as
0
amax ;
=
a1 >
0;
0
(a
a T
+a )
t < T
(9)
:
, there is a sticking phase, and the resulting average velocity
v2
If a1
(a
+a )
2
Fig. 3. Optimal motion of the part (dashed) and surface (solid) over one cycle, for a = 1, a = 0:5, a = 5, and T = 1. Top: acceleration, middle: velocity, and bottom: position.
t
as
=
a1 (amax
0
ak )
2(amax + a1 )(a1 + ak )
(10)
T:
, the part is always slipping, and v2
=
ak (amax
0
a1 )
4(amax + a1 )
T:
The optimal case is a1 = as and is used for comparison to our driving. Because the part sticks to the surface for part of the second stage, we refer to the optimized waveform (9) as “bang-stick” driving. The ratio 3 3 K2 = vave =v2 measures the improvement of our optimal driving K2
=
(amax + ak )(amax + as ) 2amax (as + ak )
>
1 for all amax
> as
ak >
0:
The performance of the optimized bang-stick driving approaches that of the optimal driving of Theorem 1 when amax is small. Fig. 5 shows the improvement ratios K1 and K2 as functions of amax =as and ak =as . III. EXPERIMENTS
Fig. 4. Average transport velocity of the part relative to the surface over one cycle, for T = 1 and a = 1. The performance increases with increasing a and decreasing a .
where z is the fraction of the cycle during which the surface coasts with zero acceleration. Assuming that the part always slips relative to the surface, they derived the average velocity equation v1
=
1 4
2
amax z T
Fig. 6 is an image of our experimental setup. The horizontally oscillated support surface is a 0.5 7 25 cm3 glass plate attached to an air slide (NewWay Air Bearings) with the 25-cm side oriented along the motion axis. The slide is connected to a voice coil actuator (VTS VG100) driven by an audio amplifier. Acceleration along the principal axis is measured by a high resolution accelerometer (PCB Piezotronics 352B) while the uncontrolled orthogonal horizontal and vertical components are measured by a two-axis accelerometer (Analog Devices ADXL320). Accelerometer signals are acquired by a PC running Labview using a National Instruments DAQ board, and the control waveform is sent to the amplifier by the same board. Input and output to the
2 2
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Fig. 7. Target (solid line) and measured ( ) acceleration for one period of driving at f = 17:2 Hz using 50 modes. Note the ringing in the target drive waveform due to the finite Fourier series approximation of the theoreticallyideal waveform. The target waveform is obtained from the optimal driving wave=g = 1. The form of Theorem 1 using a =g = 0:16, a =g = 0:11, and a inset shows the difference (rms = 2 10 g) between the target and the measured acceleration.
2
A. Iterative Learning Control
Fig. 5. Improvement ratios K and K of the part equilibrium velocity for optimal driving compared to other driving waveforms as a function of the nondimensional accelerations a =a and a =a . Top: Compared to the optimal bang-coast-bang driving in (8). Bottom: Compared to the optimal bang-stick driving in (9).
Acceleration control of the oscillating support surface is implemented using a spectral decomposition learning control method appropriate for linear systems. A numeric representation of a single period of the ideal driving waveform is generated. This waveform is then discrete Fourier transformed and the resulting amplitudes and phases recorded. By choosing N of the Fourier components, each represented by a frequency fk , amplitude Atk , and phase kt , k N , the target waveform is formed from the components, N At t k=1 kc c fk t k . The initial control waveform, represented by fk ; Ak ; k , is obtained from the target waveform by the approximate inverse linear dynamics of the system. The measured acceleration of the surface when driven by the control waveform is recorded and m decomposed into fk ; Am k ; k . Differences between the target and t m Ak Ak and measured amplitudes and phases are formed, A k t m k k . The control waveform components are updated k c A Ak k and by the iterative learning control laws Ack;new c c ; with k;new k . In our experiments, this control : k typically converges to errors of less than 1% rms in the acceleration profile. Fig. 7 shows an example of target and measured acceleration . Typically, good agreement can profiles for a waveform with N be achieved using a relatively small number of modes. However, when any interval of the driving waveform is short relative to the period, good agreement will not be obtained unless a sufficient number of modes is used to resolve the shortest interval.
= 1. . . sin 2 + f
g
f
1 =
0
= +1
Fig. 6. Experimental setup.
PC is sampled at four times the highest frequency component in the drive signal. A camera (Sony XCD-X700), synchronized to the driving signal, images the experiments from directly above. A modified U.S. one-cent coin (penny) was used as our sliding part. To ensure that plate contact was well defined, three 0.2 cm diameter 316 stainless steel balls were glued to one face of the penny near the outer diameter and spaced at 120 . The top face of the penny was covered with black paper so that the penny was the darkest object in the image. Images were thresholded and the penny position determined as a function of time by computing its centroid. The penny velocities were found by applying a linear fit to the position versus time data. Each velocity measurement reported is the average of eight complete transits of the support surface, four in each direction.
g
= 0 05
1 = 0 = + 1
= 50
B. Parameter Estimation To achieve optimal driving, ak and as must be determined. Two distinct methods were employed that gave the same quantitative values. In the first, which uses the bang-stick driving (9), as was found by increasing amax with a1 < amax until the part began to move relative to the plate. Then, ak was obtained by fitting (10). In the second method, velocity measurements were made for provisional values of ak and as , and the actual values were taken as those where the optimal driving profile of Theorem 1 maximized the part velocity. This second proce: where dure was carried out at relatively low values of amax vertical vibration and horizontal vibration orthogonal to the primary
( 0 5 g)
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Fig. 8. Part velocity is maximum at a 0:16 g for f = 17:2 Hz, a = 0:5 g, and a = 0:16 g, when driven with the optimal driving waveform given by Theorem 1.
axis were small. As an example, Fig. 8 shows that as the working value of as , which we call astk , is increased at constant f , ak , and amax , the penny has a maximum velocity near astk = 0:16 g. For both methods ak 0:16 g. As is evident from Fig. 5, the benwe found that as efit of optimal driving is especially apparent when as is larger than ak , which was not the case in our experiment. In our actual experiments, we chose to use a value astk = 0:11 g as a conservative approximation to as to generate the optimal driving waveforms. This value was used to find the driving of Theorem 1 as well as the optimized bang-stick waveform (9). (The value of as does not impact the optimized bang-coast-bang waveform (8).) This was done to accommodate the unmodeled parasitic vibrations of the plate in the vertical and undriven horizontal directions, which tended to cause the part to slip when the plate was accelerated near as . This conservative approximation also accommodates some ringing in the drive motion (Fig. 7). The estimated value of ak = 0:16 g is also used in generating the optimized drive waveforms of Theorem 1 and bang-coastbang driving (8); it does not impact the optimized bang-stick driving (9).
11
Fig. 9. Experimental (open circle) and theoretical (line) part velocity with opat a = 0:11 g, a = 0:16 g, f = 17:2 Hz, and timal driving versus a N = 50. The agreement with theory (upper inset) is good below a 1:7 g. Above this value horizontal oscillations orthogonal to the driving become large enough that the total horizontal acceleration exceeds a during the expected sticking phase (lower inset). Consequently, the part slips and the average velocity is less than predicted by theory.
C. Results The penny velocity was measured under the driving of Theorem 1 with ak = 0:16 g and astk = 0:11 g for a range of amax at fixed f and for a range of f at fixed amax (Figs. 9 and 10, respectively). Figs. 9 and 10 show good agreement between the measured velocity and the theory, verifying the friction model in our work. Note that in Fig. 9 (velocity versus amax ) agreement between experiment and theory breaks down when the maximum plate acceleration becomes large, as relatively large unmodeled horizontal vibrations orthogonal to the driving direction cause the part to slip during phases when it is presumed to be sticking. Fig. 11 compares the performance of the three driving waveforms we have discussed as a function of amax . At low amax , the bang-stick driving (9) outperforms the bang-coast-bang driving (8). As amax increases, however, the situation reverses. Because we chose astk < as to implement the optimal driving of Theorem 1 (due to unmodeled sideways vibrations of the plate), theoretically the optimized bang-coast-bang driving (8) actually outperforms the approximated optimal driving of Theorem 1 for large amax , as indicated by the intersec1. In the experiment, howtion of the theoretical curves near amax ever, the performance of the bang-coast-bang driving stays below that of the driving of Theorem 1. We do not yet completely understand this
Fig. 10. Measured part velocity (open circles) decreases as 1=f with increasing = 0:11 g, a = frequency as predicted by Theorem 1 (solid line) with a 0:16 g, a = 0:5 g, and N = 50. The inset shows the same data on a log-log plot; the solid line (from theory) has slope 1.
0
discrepancy. Apart from this discrepancy, the experimental data match the theoretical predictions well. D. Discussion We found that maintaining a constant and spatially uniform friction coefficient was necessary for consistent results. To achieve this, we cleaned the glass and the penny’s three supports with acetone before each experiment. Our original experiments were conducted with an unmodified penny. We found, however, that the contact points were not well defined and time-varying, causing changing values of ak . The ringing in the drive waveform (Fig. 7) can be eliminated by lowpass filtering the ideal waveform before taking the Fourier transform. We found this procedure increased the time where the plate acceleration was greater than as , however, reducing the part velocity. Ideally the
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Fig. 12. Optimal surface and part motion for = 0:3, = 0:2, and a = = 2 g. Top: the horizontal velocity of the part and surface. Bottom: the effective gravity g = y + g .
a
Fig. 11. Measured (symbol) and theoretical (line) part velocity versus a for the optimal driving of Theorem 1 ( , solid line), optimized bang-coast-bang a )=(2a )), and optimized driving (8) (+ and dashed line with z = (a bang-stick driving (9) ( and dash-dot line with a = a ). Drive parameters = 0:11 g, a = 0:16 g, a = 0:5 g , and N = 50. are a
0
4
finite bandwidth of the actuator would be taken into account during the optimization, rather than during a post-processing step. The biggest weakness of our experimental implementation is the unmodeled vibration of the plate. This can be mitigated by a stiffer structural design. Finally, the advantage of the optimized driving of Theorem 1 is most clearly evident in materials with a significant difference between kinetic and static friction, such as glass on glass, unlike the materials in our experiment.
IV. EXTENSION: VERTICAL VIBRATION By adding controlled vibration of the surface in the vertical direction, it is possible to further increase the transport velocity by decreasing the normal force (or effective gravity) when the plate is accelerating backward and increasing the effective gravity when it accelerates forward. This idea is embodied in a device built by Frei et al. [8]. A surface performs small translational circular motions in a horizontal plane while it is sinusoidally oscillated in the vertical direction. The phasing of these motions determines the motion of parts on the surface, as parts move in the direction of horizontal-plane velocity when the effective gravity is largest. Defining the vertical acceleration of the surface as yS , Theorem 2 gives the optimal driving profile in both the vertical and horizontal directions. Theorem 2: Let the vertical acceleration yS of the surface be subject yS ay;max , where the lower bound to the control constraint g is enforced to prevent the surface from breaking contact with the part and impacting later. (Implicit in this lower bound is the assumption that the plate is capable of accelerating downward at 1g .) Assuming amax s (g + ay;max ), the optimal transport driving is given by
0
yS3 (t) = and
x3S (t) =
0g;
0 t < Tdown ay;max; Tdown t < T
0amax ;
0 t < T1 amax ; T1 t < T 0 Tup;2 s (g + ay;max ); T 0 Tup;2 t < T
where Tdown = ay;max T=(g + ay;max ), Tup = gT =(g + ay;max ), Tup;1 = s Tup =(s + k ), Tup;2 = k Tup =(s + k ), and
T1 =
g(ay;max + g)k s + amax (gs + ay;max (k + s )) T: 2amax (ay;max + g)(k + s )
Proof: See the Appendix. Fig. 12 illustrates the optimal driving. The part slips over the plate with zero effective gravity, ge = yS +g = 0, as the surface accelerates backward maximally (0 t < T1 ). Then the surface accelerates forward maximally, still with zero effective gravity (T1 t < T Tup ). While still maximally accelerating in a horizontal direction, the plate then implements maximal acceleration ay;max in the upward direction, creating a large effective gravity (T Tup t < T Tup;2 ). Friction forces slow the part until it matches the speed of the surface at t = T Tup;2 , at which point the surface switches to a forward acceleration of s (g + ay;max ), the maximum possible without slipping. At t = T , the process repeats. The expression for the optimal transport velocity is quite complex, so we do not present it here. It is simply the area between the x_ p and x_ S trajectories over a cycle, divided by T . As ay;max grows large, the horizontal surface velocity in Fig. 12 approaches a triangle waveform and the average part velocity approaches the theoretical limit
0
0
0
0
amax T 4
which is also the maximum horizontal velocity a periodically-moving as , this gives surface can attain. For large values of amax and ak a speedup over purely horizontal driving of approximately a factor of four.
V. CONCLUSION This paper has presented the periodic motion waveforms that maximize the friction-induced transport velocity of parts sliding on the surface. Experiments confirm the theoretical results for horizontal-only forcing. We have shown that adding control of the vertical vibration allows a maximum transport velocity of up to four times that attainable by horizontal-only forcing. Applications of vibratory transport extend beyond simple linear conveyance. Reznik and Canny showed that three-degree-of-freedom vibration in the horizontal plane can be used to create a large family of effective velocity fields on the surface of the plate [9], [10]. One limitation of planar vibration is that the generated fields are divergence-free:
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13
the flow of the field into any region is equal to the flow out. This restriction is eliminated by allowing the surface a full six degrees of freedom of vibratory motion [11]. Fields generated by such a surface can be used for sensorless parts feeding, sorting, planar assembly, and simultaneously controlling the motions of multiple parts along independent trajectories. The design of six-degree-of-freedom vibratory motions to create maximally strong fields, beyond the simple uniform fields described in this paper, is a topic of ongoing research.
To maintain periodic motion in the vertical direction, however, the maximum time the surface can accelerate at yS = g during a cycle of duration T is Tdown = ay;max T=(ay;max + g). If t2 t1 > Tdown , then u2 must switch from g to ay;max no later than t1 + Tdown , and J is maximized if the switch occurs exactly at t1 + Tdown . Finally, as in the proof of Theorem 1, it is straightforward to show that the total slipping distance over one cycle is maximized if there is only one period of slip during the cycle. The result in Theorem 2 is obtained by observing that the time of maximum upward acceleration is Tup = T Tdown , which can be separated into the times Tup;1 and Tup;2 where the part is slipping and sticking to the surface, respectively. The duration T1 is obtained by enforcing the constraint that the surface’s horizontal velocity be periodic
[0; T ] be the duration of a slipping phase with Let [t1 ; t2 ] boundary conditions x_ p (t1 ) = x_ S (t1 ) and x_ p (t2 ) = x_ S (t2 ). The displacement of the part relative to the plate during slipping is given by
J=
t t
= t
(x_ p (t)
0 x_ S (t))dt
(x_ p (t1 )
(T2
0 T1 )amax + Tup 2 s (g + a
where T1 + T2 + Tup;2 =
0 k (yS (t) + g) 0 x_ S (t))dt:
0
;
y;max )
=0
T. REFERENCES
Choosing the state vector [z1 ; z2 ; z3 ; z4 ]T = [xS ; x_ S ; yS ; y_S ]T and the xS ; yS ]T , the Hamiltonian is control vector [u1 ; u2 ]T = [
H = 0x_ p (t1 ) + k (u2 + g) + (1 + 1 )z2 + 2 u1 + 3 z4 + 4 u2 : 0
The adjoint equation @H=@z = _ indicates that 1 and 3 are constants and 2 and 4 are linear functions of time. By Pontryagin, we have
H(z3 (t); 3(t); u3 (t)) H(z3 (t); 3 (t); u(t)) 32 (t)u13(t) + 34 (t)u23(t) 32 (t)u1 (t) + 34 u2 (t):
(11)
Equation (11) indicates that the controls u13 , u23 always take their extreme values when 32 , 34 are nonzero. Because both 32 , 34 are linear functions of time, they each have at most one switch in sign. We use this to enumerate all possible cases, where each of 32 , 34 is initially positive, negative, or zero, and either switches signs or retains the same sign over the interval [t1 ; t2 ]. A simple case analysis shows that 32 (t1 ) > 0, 34 (t1 ) > 0, i.e., u13 (t1 ) = amax and u23 (t1 ) = g. The boundary condition x_ p (t2 ) = x_ S (t2 ) dictates that u13 (t2 ) = amax . In the absence of any other constraints, then, we use the boundary condition x_ p (t2 ) = x_ S (t2 ) to solve for the switch time for u1 and u2 = g over the interval [t1 ; t2 ].
0
0
0
APPENDIX
t
0
0
0
[1] G. Boothroyd, Assembly Automation and Product Design. New York: Marcel Dekker, 1992. [2] G. H. Lim, “On the conveying velocity of a vibratory feeder,” Comput. Struct., vol. 62, no. 1, pp. 197–203, Jan. 1997. [3] P. Wolfsteiner and F. Pfeiffer, “Modeling, simulation, and verification of the transportation process in vibratory feeders,” Z. Angew. Math. Mech., vol. 80, no. 1, pp. 35–48, Jan. 2000. [4] S. Okabe, Y. Kamiya, K. Tsujikado, and Y. Yokoyama, “Vibratory feeding by nonsinusoidal vibration—optimum wave form,” ASME J. Vibr., Acoust., Stress, Rel. Design, vol. 107, pp. 188–195, Apr. 1985. [5] D. Reznik and J. Canny, “The Coulomb pump: A novel parts feeding method using a horizontally-vibrating surface,” in Proc. IEEE Int. Conf. Robot. Autom., 1998, pp. 869–874. [6] A. E. Quaid, “A miniature mobile parts feeder: Operating principles and simulation results,” in Proc. IEEE Int. Conf. Robotics Autom., 1999, pp. 2221–2226. [7] W. A. Morcos, “On the design of oscillating conveyers—Case of simultaneous normal and longitudinal oscillations,” ASME J. Eng. For Ind., vol. 92, no. 1, pp. 53–61, 1970. [8] P. U. Frei, M. Wiesendanger, R. Büchi, and L. Ruf, “Simultaneous planar transport of multiple objects on individual trajectories using friction forces,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 49–64. [9] D. Reznik and J. Canny, “A flat rigid plate is a universal planar manipulator,” in Proc. IEEE Int. Conf. Robotics Autom., 1998, pp. 1471–1477. [10] D. Reznik and J. Canny, “C’mon part, do the local motion!,” in Proc. IEEE Int. Conf. Robotics Autom., 2001, pp. 2235–2242. [11] T. H. Vose, P. Umbanhowar, and K. M. Lynch, “Vibration-induced frictional force fields on a rigid plate,” in Proc. IEEE Int. Conf. Robotics Autom., 2007, pp. 660–667.
7
Optimal Vibratory Stick-Slip Transport Paul Umbanhowar and Kevin M. Lynch
x
Fig. 1. A point fixed to the surface is at relative to a fixed point in the world, and the part is at . Gravity holds the part to the surface.
x
Abstract—We describe a vibratory part transport mechanism that utilizes both static and dynamic friction to linearly transport parts in a horizontal direction. We derive a horizontal driving profile for the feeder surface that maximizes the steady-state velocity of parts on the surface subject to acceleration limits on the surface. Experiments verify the predicted transport behavior. We then derive the optimal feeding motion for a surface that can move in both the horizontal and vertical directions. Note to Practitioners—One type of vibratory conveyance, popular in industrial parts feeding and presentation, makes use of friction and sliding between the part and the support surface. This paper gives the surface vibratory motions that maximize the speed of transport. It also describes an experimental implementation and a control algorithm for achieving the desired vibratory motion. Index Terms—Friction, vibratory parts feeding.
I. INTRODUCTION Vibratory transport is utilized in a number of industrial parts feeding mechanisms, including bowl feeders and linear conveyance systems [1]. Linear conveyance is typically achieved by some combination of vertical and horizontal vibration, and the forces to transport the parts arise from two phenomena: impacts with the support surface (“hopping”) and frictional sliding [2], [3]. In this paper, we focus on the case that parts always remain in contact with the support, so that transport is due to frictional sliding. Transport by sliding can be achieved by vibratory motion of the surface in the longitudinal direction only [4]–[6] or by some combination of normal and longitudinal oscillation [1], [7], [8]. This paper derives the oscillatory motions of a surface that maximize the speed of transport of parts, given a fixed period of oscillation and an upper bound on the acceleration of the surface. The surface is horizontal, and it either oscillates in the horizontal direction only (Sections II–III) or in both the horizontal and vertical directions (Section IV). In the case of horizontal-only motion, this work builds on the “Coulomb pump” of Reznik and Canny [5]. They observed that transport occurs for asymmetric horizontal vibration of the surface because the friction force ff felt by the part during slipping contact with the surface satisfies
jff j = k mg
In their analysis, Reznik and Canny made the simplifying assumption that the part always slips relative to the surface, and they chose a particular driving profile for the surface [5]. Quaid [6] chose a different asymmetric driving profile and demonstrated parts transport using a full stick-slip analysis. In this paper, we design the forcing of the surface to take advantage of knowledge of the static (stick) and kinetic (slip) coefficients of friction between parts and the surface. In Section II, we use a full stick-slip analysis to find the surface motion that maximizes the speed of part transport, subject to bounds on the surface acceleration. We then demonstrate the improvement of this optimal driving over optimized versions of the motion profiles proposed in [5] and [6]. The theoretical results are validated by experiments in Section III. Our vibratory transport mechanism uses iterative learning control to achieve high-accuracy control of bang-bang acceleration profiles. Okabe et al. [4] studied the similar problem of finding the periodic longitudinal waveform of an inclined surface that maximizes a slightly different measure of the transport speed. They used a finite parameterization of the waveform and conducted a numerical study to develop guidelines for the choice of these parameters. In this paper, we incorporate acceleration (control) constraints and solve exactly for the optimal driving waveform, without parameterization. In Section IV, we extend our analysis to the case of both vertical and horizontal oscillation. We derive the optimal driving motion for this case, which yields a maximum transport velocity of up to four times what is possible with horizontal-only oscillation. II. OPTIMAL STICK-SLIP TRANSPORT A. Stick-Slip Transport Let the position of points fixed to the surface and the part be xS and xp , respectively, with respect to a stationary reference (Fig. 1). As a ! is C 1 and periodic function mapping time to position, xS : with period T , xS (t) = xS (t + T ). In steady state, the velocity of the part is also periodic, x_ p (t) = x_ p (t + T ), with a net displacement over the period of D = xp (t + T ) 0 xp (t), yielding an average part velocity vave = D=T . We assume the part moves on the surface according to dry Coulomb k be the static and kinetic friction coefficients, friction. Let s respectively, with associated maximum accelerations as = s g and ak = k g. When the part is initially at rest with respect to the surface (x_ S = x_ p ) and the surface moves with an acceleration x S [ as ; as ], p = x S and the part remains fixed to the surface. If x then x S > as , p = ak sgn( xS ), i.e., it begins to slip on the acceleration of the part is x the surface. If the part is already slipping (x_ S = x_ p ), the acceleration p = ak sgn(x_ S x_ p ). of the part is x A simple two-stage periodic surface motion that induces steady-state stick-slip transport of a part is
where k is the kinetic friction coefficient, m is the mass of the part, and g is the gravitational constant. In other words, the force magnitude is independent of the relative slipping speed: the part is subject to a force k mg when the signed surface velocity is greater than the part velocity and a force k mg when the surface velocity is less than the part velocity. Therefore, if we arrange the asymmetric periodic motion of the surface so that it spends more time with velocity greater than the part, the part feels a net force in the positive direction over the cycle.
0
Manuscript received August 19, 2006; revised August 23, 2007. This paper was recommended for publication by Associate Editor H. Ding and Editor M. Wang upon evaluation of the reviewers’ comments. This work was supported in part by the NSF under Grants IIS-0308224 and CMMI-0700537. The authors are with the Mechanical Engineering Department, Northwestern University, Evanston, IL 60208 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASE.2008.917021
20 j j
6
0
xS (t) =
6
0amax; a1 ;
t < T1 T1 t < T
0
(1)
where amax are control limits on the surface acceleration, 0 < a1 < amax , and T1 = a1 T=(a1 + amax ).1 Choosing a1 as , we obtain the stick-slip driving profile considered in [6]. As shown in Fig. 2, in steady-state the part slips forward with respect to the surface when the 1Throughout the paper, we assume slipping.
1545-5955/$25.00 © 2008 IEEE
a
> a
to ensure transport by
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The adjoint equation
_ = 00_ = 1 +0 indicates that (t) is constant and (t) is a linear function of time. ; a ] and all t 2 [t ; t ], we have By Pontryagin, for all u 2 [0a H(z3 (t); 3 (t); u3 (t)) H(z3 (t); 3(t); u(t)) 3 (t)u3 (t) 3 (t)u(t) indicating that u3 (t) = 0a for 3 (t) > 0 and u3 (t) = a for 3 (t) < 0. In other words, the optimal surface motion during slipping is bang-bang. As (t) is linear in t, there is at most one switch during the time interval. Employing (3), and defining 1t = t 0 t , we get 0a ; t t < t + u3 (t) = (4) a ; t + tt @H @z @H @z
1 2
1
1
2
max
Fig. 2. Part and surface velocity with the simple two-phase driving motion (1).
max
2
surface accelerates backwards in phase 1, then eventually matches the velocity of the surface and sticks again for part of phase 2. The slope of the part velocity is ak when slipping and a1 when sticking. The area between the part velocity and the surface velocity is the amount that the part is transported. The optimal driving problem is to maximize that area subject to the bounds on surface acceleration.
0
B. Optimal Transport The problem we will solve is the following. Given a period T and amax , find xS ;T bounds on the surface acceleration xS amax ; amax that maximizes vave in steady state and satisfies the pexS T ; xS xS T . In other words, riodicity conditions xS the problem is to maximize
j j
]
[0
: [0 ] !
(0) = ( ) _ (0) = _ ( )
T 0
Ldt =
T
(_x (t) 0 x_ (t))dt: p
0
0
When the part slips backward, we have L < , and when it slips forward, we have L > . The optimal surface trajectory, steady-state periodic part velocity, and average part velocity are written x3S t ; x3p t , 3 , respectively. and vave We make the following two observations. 1) Observation 1: During any periodic steady-state motion, there ; T such that xS t xp t . exists a t x3p t for all 2) Observation 2: For any optimal solution, x3S t t ;T . The first observation follows from the fact that, at steady state, xp xp T . The surface velocity is a continuous function, and if it were never equal to the part velocity (i.e., always larger or smaller), ak then the part would have a constant acceleration with xp during the cycle, a contradiction. The second observation follows from the fact that any part velocity xp t that can be realized for xp t with some xS t > xp t can also be realized for xS t a larger value of L. These observations indicate that the optimal surface trajectory has the part either sticking to the surface or slipping forward, xp > xS . Only during the slipping phase is L > . Let t1 ; t2 ; T represent a time interval during which the part is slipping forward, with the boundary conditions
() _ ()
2 [0 ]
_ ()= _ ()
_ () _ ()
2 [0 ]
_ (0) = _ ( )
j j =
_ ()
_ ()
_ ()
6= 0
0
x_ p (t1 ) = x_ S (t1 ) x_ p (t2 ) = x_ S (t2 ):
(2) (3)
: [
t1 ; t2 We would like to find the surface acceleration xS amax ; amax satisfying these constraints and maximizing
[0
J=
t
] (_x (t) 0 x_ (t))dt = p
t
S
t t
=[
] !
(_x (t ) 0 a t 0 x_ (t))dt: p
1
S
k
] =[ _]
Setting the state vector to be z z1 ; z2 T xS ; xS T and the surface acceleration control to be u, the state equations are written z z2 ; u T . We write the Hamiltonian as
_=[
]
H = 0x_ (t ) + a t + (1 + )z + u: p
1
k
1
2
2
max
max
2
2
2
2
max
1
max
=(
1
1
1
2
)
+ )1 (2
amax ak t= amax . The net displacement of the part where relative to the surface during this motion is d(1t) =
a2max 0 a2k 2 4amax 1t :
(5)
The preceding analysis applies to a particular subinterval [t ; t ] of [0; T ]. Now by the requirement that the net acceleration of the part 1
Tslip =
2
2
as T as + ak
and this is achieved if the part is sticking to the surface accelerating as during the rest of the cycle. Therefore, the sum of the at xS slipping times (the only times that contribute to the objective function J ) during the optimal motion is Tslip . It remains only to determine how this slipping time is distributed in slipping intervals over the cycle. t2 for any From (5), it is apparent that d t1 d t2 < d t1 t1 ; t2 > . From this we conclude that there is only one slipping interval during an optimal cycle, of duration Tslip . Theorem 1 summarizes the discussion above. Theorem 1: The surface motion that maximizes a part’s average transport speed is given by
=
1 1
_ () = _ ()
_ _ ] [0 ]
[
2
during the entire cycle be zero at steady state, the maximum time a part can slip forward is
S
0
1
(1 )+ (1 )
0
x3S (t) =
(1 +1 )
as ; 0 t < T1 0amax ; T1 t < T2 amax ; T2 t < T
(6)
T1 = ak T =(as + ak ) and T2 = T1 + (amax + ak )as T=((2amax )(as + ak )). The average velocity of the part
where is
3 = vave
d(Tslip ) T
= 4aa a (a +0 aa ) 2 s
2 max
max
2 k
s
k
2
T:
(7)
Example surface and part motions are illustrated in Fig. 3. The average part velocity is given in Fig. 4 as a function of ak =as and amax =as for as ,T .
=1 =1
C. Comparison to Other Driving Functions In [5], Reznik and Canny proposed the three-stage “bang-coastbang” driving waveform
xS (t) =
amax ; 0; 0amax ;
0 0ztT
(amax there is a sticking phase during the coasting period, and the average velocity is actually given by
0
=
v1
amax (z
0 1)
2
2
amax
2
+ ak
(amax +
2 ak )
02
ak amax
T:
When driven in the stick-slip regime, the transport velocity decreases. The optimal driving is attained for z = (amax ak )=(2amax ), and the optimal transport velocity by this driving waveform is
0
3
The ratio driving
K1
=
(amax
=
v1
0
2
ak )
16amax
T:
3 measures the improvement of our optimal
3
vave =v1
2
K1
=
4as (amax + ak ) (ak + as )2 (amax for all amax
0
>
ak )
> as
ak >
1 0:
The performance of the optimized bang-coast-bang driving profile (8) approaches that of the optimal driving of Theorem 1 when amax is large as . and ak In [6], Quaid proposed the two-stage driving waveform
x S (t)
If a1 is
as
0
amax ;
=
a1 >
0;
0
(a
a T
+a )
t < T
(9)
:
, there is a sticking phase, and the resulting average velocity
v2
If a1
(a
+a )
2
Fig. 3. Optimal motion of the part (dashed) and surface (solid) over one cycle, for a = 1, a = 0:5, a = 5, and T = 1. Top: acceleration, middle: velocity, and bottom: position.
t
as
=
a1 (amax
0
ak )
2(amax + a1 )(a1 + ak )
(10)
T:
, the part is always slipping, and v2
=
ak (amax
0
a1 )
4(amax + a1 )
T:
The optimal case is a1 = as and is used for comparison to our driving. Because the part sticks to the surface for part of the second stage, we refer to the optimized waveform (9) as “bang-stick” driving. The ratio 3 3 K2 = vave =v2 measures the improvement of our optimal driving K2
=
(amax + ak )(amax + as ) 2amax (as + ak )
>
1 for all amax
> as
ak >
0:
The performance of the optimized bang-stick driving approaches that of the optimal driving of Theorem 1 when amax is small. Fig. 5 shows the improvement ratios K1 and K2 as functions of amax =as and ak =as . III. EXPERIMENTS
Fig. 4. Average transport velocity of the part relative to the surface over one cycle, for T = 1 and a = 1. The performance increases with increasing a and decreasing a .
where z is the fraction of the cycle during which the surface coasts with zero acceleration. Assuming that the part always slips relative to the surface, they derived the average velocity equation v1
=
1 4
2
amax z T
Fig. 6 is an image of our experimental setup. The horizontally oscillated support surface is a 0.5 7 25 cm3 glass plate attached to an air slide (NewWay Air Bearings) with the 25-cm side oriented along the motion axis. The slide is connected to a voice coil actuator (VTS VG100) driven by an audio amplifier. Acceleration along the principal axis is measured by a high resolution accelerometer (PCB Piezotronics 352B) while the uncontrolled orthogonal horizontal and vertical components are measured by a two-axis accelerometer (Analog Devices ADXL320). Accelerometer signals are acquired by a PC running Labview using a National Instruments DAQ board, and the control waveform is sent to the amplifier by the same board. Input and output to the
2 2
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Fig. 7. Target (solid line) and measured ( ) acceleration for one period of driving at f = 17:2 Hz using 50 modes. Note the ringing in the target drive waveform due to the finite Fourier series approximation of the theoreticallyideal waveform. The target waveform is obtained from the optimal driving wave=g = 1. The form of Theorem 1 using a =g = 0:16, a =g = 0:11, and a inset shows the difference (rms = 2 10 g) between the target and the measured acceleration.
2
A. Iterative Learning Control
Fig. 5. Improvement ratios K and K of the part equilibrium velocity for optimal driving compared to other driving waveforms as a function of the nondimensional accelerations a =a and a =a . Top: Compared to the optimal bang-coast-bang driving in (8). Bottom: Compared to the optimal bang-stick driving in (9).
Acceleration control of the oscillating support surface is implemented using a spectral decomposition learning control method appropriate for linear systems. A numeric representation of a single period of the ideal driving waveform is generated. This waveform is then discrete Fourier transformed and the resulting amplitudes and phases recorded. By choosing N of the Fourier components, each represented by a frequency fk , amplitude Atk , and phase kt , k N , the target waveform is formed from the components, N At t k=1 kc c fk t k . The initial control waveform, represented by fk ; Ak ; k , is obtained from the target waveform by the approximate inverse linear dynamics of the system. The measured acceleration of the surface when driven by the control waveform is recorded and m decomposed into fk ; Am k ; k . Differences between the target and t m Ak Ak and measured amplitudes and phases are formed, A k t m k k . The control waveform components are updated k c A Ak k and by the iterative learning control laws Ack;new c c ; with k;new k . In our experiments, this control : k typically converges to errors of less than 1% rms in the acceleration profile. Fig. 7 shows an example of target and measured acceleration . Typically, good agreement can profiles for a waveform with N be achieved using a relatively small number of modes. However, when any interval of the driving waveform is short relative to the period, good agreement will not be obtained unless a sufficient number of modes is used to resolve the shortest interval.
= 1. . . sin 2 + f
g
f
1 =
0
= +1
Fig. 6. Experimental setup.
PC is sampled at four times the highest frequency component in the drive signal. A camera (Sony XCD-X700), synchronized to the driving signal, images the experiments from directly above. A modified U.S. one-cent coin (penny) was used as our sliding part. To ensure that plate contact was well defined, three 0.2 cm diameter 316 stainless steel balls were glued to one face of the penny near the outer diameter and spaced at 120 . The top face of the penny was covered with black paper so that the penny was the darkest object in the image. Images were thresholded and the penny position determined as a function of time by computing its centroid. The penny velocities were found by applying a linear fit to the position versus time data. Each velocity measurement reported is the average of eight complete transits of the support surface, four in each direction.
g
= 0 05
1 = 0 = + 1
= 50
B. Parameter Estimation To achieve optimal driving, ak and as must be determined. Two distinct methods were employed that gave the same quantitative values. In the first, which uses the bang-stick driving (9), as was found by increasing amax with a1 < amax until the part began to move relative to the plate. Then, ak was obtained by fitting (10). In the second method, velocity measurements were made for provisional values of ak and as , and the actual values were taken as those where the optimal driving profile of Theorem 1 maximized the part velocity. This second proce: where dure was carried out at relatively low values of amax vertical vibration and horizontal vibration orthogonal to the primary
( 0 5 g)
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Fig. 8. Part velocity is maximum at a 0:16 g for f = 17:2 Hz, a = 0:5 g, and a = 0:16 g, when driven with the optimal driving waveform given by Theorem 1.
axis were small. As an example, Fig. 8 shows that as the working value of as , which we call astk , is increased at constant f , ak , and amax , the penny has a maximum velocity near astk = 0:16 g. For both methods ak 0:16 g. As is evident from Fig. 5, the benwe found that as efit of optimal driving is especially apparent when as is larger than ak , which was not the case in our experiment. In our actual experiments, we chose to use a value astk = 0:11 g as a conservative approximation to as to generate the optimal driving waveforms. This value was used to find the driving of Theorem 1 as well as the optimized bang-stick waveform (9). (The value of as does not impact the optimized bang-coast-bang waveform (8).) This was done to accommodate the unmodeled parasitic vibrations of the plate in the vertical and undriven horizontal directions, which tended to cause the part to slip when the plate was accelerated near as . This conservative approximation also accommodates some ringing in the drive motion (Fig. 7). The estimated value of ak = 0:16 g is also used in generating the optimized drive waveforms of Theorem 1 and bang-coastbang driving (8); it does not impact the optimized bang-stick driving (9).
11
Fig. 9. Experimental (open circle) and theoretical (line) part velocity with opat a = 0:11 g, a = 0:16 g, f = 17:2 Hz, and timal driving versus a N = 50. The agreement with theory (upper inset) is good below a 1:7 g. Above this value horizontal oscillations orthogonal to the driving become large enough that the total horizontal acceleration exceeds a during the expected sticking phase (lower inset). Consequently, the part slips and the average velocity is less than predicted by theory.
C. Results The penny velocity was measured under the driving of Theorem 1 with ak = 0:16 g and astk = 0:11 g for a range of amax at fixed f and for a range of f at fixed amax (Figs. 9 and 10, respectively). Figs. 9 and 10 show good agreement between the measured velocity and the theory, verifying the friction model in our work. Note that in Fig. 9 (velocity versus amax ) agreement between experiment and theory breaks down when the maximum plate acceleration becomes large, as relatively large unmodeled horizontal vibrations orthogonal to the driving direction cause the part to slip during phases when it is presumed to be sticking. Fig. 11 compares the performance of the three driving waveforms we have discussed as a function of amax . At low amax , the bang-stick driving (9) outperforms the bang-coast-bang driving (8). As amax increases, however, the situation reverses. Because we chose astk < as to implement the optimal driving of Theorem 1 (due to unmodeled sideways vibrations of the plate), theoretically the optimized bang-coast-bang driving (8) actually outperforms the approximated optimal driving of Theorem 1 for large amax , as indicated by the intersec1. In the experiment, howtion of the theoretical curves near amax ever, the performance of the bang-coast-bang driving stays below that of the driving of Theorem 1. We do not yet completely understand this
Fig. 10. Measured part velocity (open circles) decreases as 1=f with increasing = 0:11 g, a = frequency as predicted by Theorem 1 (solid line) with a 0:16 g, a = 0:5 g, and N = 50. The inset shows the same data on a log-log plot; the solid line (from theory) has slope 1.
0
discrepancy. Apart from this discrepancy, the experimental data match the theoretical predictions well. D. Discussion We found that maintaining a constant and spatially uniform friction coefficient was necessary for consistent results. To achieve this, we cleaned the glass and the penny’s three supports with acetone before each experiment. Our original experiments were conducted with an unmodified penny. We found, however, that the contact points were not well defined and time-varying, causing changing values of ak . The ringing in the drive waveform (Fig. 7) can be eliminated by lowpass filtering the ideal waveform before taking the Fourier transform. We found this procedure increased the time where the plate acceleration was greater than as , however, reducing the part velocity. Ideally the
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Fig. 12. Optimal surface and part motion for = 0:3, = 0:2, and a = = 2 g. Top: the horizontal velocity of the part and surface. Bottom: the effective gravity g = y + g .
a
Fig. 11. Measured (symbol) and theoretical (line) part velocity versus a for the optimal driving of Theorem 1 ( , solid line), optimized bang-coast-bang a )=(2a )), and optimized driving (8) (+ and dashed line with z = (a bang-stick driving (9) ( and dash-dot line with a = a ). Drive parameters = 0:11 g, a = 0:16 g, a = 0:5 g , and N = 50. are a
0
4
finite bandwidth of the actuator would be taken into account during the optimization, rather than during a post-processing step. The biggest weakness of our experimental implementation is the unmodeled vibration of the plate. This can be mitigated by a stiffer structural design. Finally, the advantage of the optimized driving of Theorem 1 is most clearly evident in materials with a significant difference between kinetic and static friction, such as glass on glass, unlike the materials in our experiment.
IV. EXTENSION: VERTICAL VIBRATION By adding controlled vibration of the surface in the vertical direction, it is possible to further increase the transport velocity by decreasing the normal force (or effective gravity) when the plate is accelerating backward and increasing the effective gravity when it accelerates forward. This idea is embodied in a device built by Frei et al. [8]. A surface performs small translational circular motions in a horizontal plane while it is sinusoidally oscillated in the vertical direction. The phasing of these motions determines the motion of parts on the surface, as parts move in the direction of horizontal-plane velocity when the effective gravity is largest. Defining the vertical acceleration of the surface as yS , Theorem 2 gives the optimal driving profile in both the vertical and horizontal directions. Theorem 2: Let the vertical acceleration yS of the surface be subject yS ay;max , where the lower bound to the control constraint g is enforced to prevent the surface from breaking contact with the part and impacting later. (Implicit in this lower bound is the assumption that the plate is capable of accelerating downward at 1g .) Assuming amax s (g + ay;max ), the optimal transport driving is given by
0
yS3 (t) = and
x3S (t) =
0g;
0 t < Tdown ay;max; Tdown t < T
0amax ;
0 t < T1 amax ; T1 t < T 0 Tup;2 s (g + ay;max ); T 0 Tup;2 t < T
where Tdown = ay;max T=(g + ay;max ), Tup = gT =(g + ay;max ), Tup;1 = s Tup =(s + k ), Tup;2 = k Tup =(s + k ), and
T1 =
g(ay;max + g)k s + amax (gs + ay;max (k + s )) T: 2amax (ay;max + g)(k + s )
Proof: See the Appendix. Fig. 12 illustrates the optimal driving. The part slips over the plate with zero effective gravity, ge = yS +g = 0, as the surface accelerates backward maximally (0 t < T1 ). Then the surface accelerates forward maximally, still with zero effective gravity (T1 t < T Tup ). While still maximally accelerating in a horizontal direction, the plate then implements maximal acceleration ay;max in the upward direction, creating a large effective gravity (T Tup t < T Tup;2 ). Friction forces slow the part until it matches the speed of the surface at t = T Tup;2 , at which point the surface switches to a forward acceleration of s (g + ay;max ), the maximum possible without slipping. At t = T , the process repeats. The expression for the optimal transport velocity is quite complex, so we do not present it here. It is simply the area between the x_ p and x_ S trajectories over a cycle, divided by T . As ay;max grows large, the horizontal surface velocity in Fig. 12 approaches a triangle waveform and the average part velocity approaches the theoretical limit
0
0
0
0
amax T 4
which is also the maximum horizontal velocity a periodically-moving as , this gives surface can attain. For large values of amax and ak a speedup over purely horizontal driving of approximately a factor of four.
V. CONCLUSION This paper has presented the periodic motion waveforms that maximize the friction-induced transport velocity of parts sliding on the surface. Experiments confirm the theoretical results for horizontal-only forcing. We have shown that adding control of the vertical vibration allows a maximum transport velocity of up to four times that attainable by horizontal-only forcing. Applications of vibratory transport extend beyond simple linear conveyance. Reznik and Canny showed that three-degree-of-freedom vibration in the horizontal plane can be used to create a large family of effective velocity fields on the surface of the plate [9], [10]. One limitation of planar vibration is that the generated fields are divergence-free:
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. UMBANHOWAR AND LYNCH: OPTIMAL VIBRATORY STICK-SLIP TRANSPORT
13
the flow of the field into any region is equal to the flow out. This restriction is eliminated by allowing the surface a full six degrees of freedom of vibratory motion [11]. Fields generated by such a surface can be used for sensorless parts feeding, sorting, planar assembly, and simultaneously controlling the motions of multiple parts along independent trajectories. The design of six-degree-of-freedom vibratory motions to create maximally strong fields, beyond the simple uniform fields described in this paper, is a topic of ongoing research.
To maintain periodic motion in the vertical direction, however, the maximum time the surface can accelerate at yS = g during a cycle of duration T is Tdown = ay;max T=(ay;max + g). If t2 t1 > Tdown , then u2 must switch from g to ay;max no later than t1 + Tdown , and J is maximized if the switch occurs exactly at t1 + Tdown . Finally, as in the proof of Theorem 1, it is straightforward to show that the total slipping distance over one cycle is maximized if there is only one period of slip during the cycle. The result in Theorem 2 is obtained by observing that the time of maximum upward acceleration is Tup = T Tdown , which can be separated into the times Tup;1 and Tup;2 where the part is slipping and sticking to the surface, respectively. The duration T1 is obtained by enforcing the constraint that the surface’s horizontal velocity be periodic
[0; T ] be the duration of a slipping phase with Let [t1 ; t2 ] boundary conditions x_ p (t1 ) = x_ S (t1 ) and x_ p (t2 ) = x_ S (t2 ). The displacement of the part relative to the plate during slipping is given by
J=
t t
= t
(x_ p (t)
0 x_ S (t))dt
(x_ p (t1 )
(T2
0 T1 )amax + Tup 2 s (g + a
where T1 + T2 + Tup;2 =
0 k (yS (t) + g) 0 x_ S (t))dt:
0
;
y;max )
=0
T. REFERENCES
Choosing the state vector [z1 ; z2 ; z3 ; z4 ]T = [xS ; x_ S ; yS ; y_S ]T and the xS ; yS ]T , the Hamiltonian is control vector [u1 ; u2 ]T = [
H = 0x_ p (t1 ) + k (u2 + g) + (1 + 1 )z2 + 2 u1 + 3 z4 + 4 u2 : 0
The adjoint equation @H=@z = _ indicates that 1 and 3 are constants and 2 and 4 are linear functions of time. By Pontryagin, we have
H(z3 (t); 3(t); u3 (t)) H(z3 (t); 3 (t); u(t)) 32 (t)u13(t) + 34 (t)u23(t) 32 (t)u1 (t) + 34 u2 (t):
(11)
Equation (11) indicates that the controls u13 , u23 always take their extreme values when 32 , 34 are nonzero. Because both 32 , 34 are linear functions of time, they each have at most one switch in sign. We use this to enumerate all possible cases, where each of 32 , 34 is initially positive, negative, or zero, and either switches signs or retains the same sign over the interval [t1 ; t2 ]. A simple case analysis shows that 32 (t1 ) > 0, 34 (t1 ) > 0, i.e., u13 (t1 ) = amax and u23 (t1 ) = g. The boundary condition x_ p (t2 ) = x_ S (t2 ) dictates that u13 (t2 ) = amax . In the absence of any other constraints, then, we use the boundary condition x_ p (t2 ) = x_ S (t2 ) to solve for the switch time for u1 and u2 = g over the interval [t1 ; t2 ].
0
0
0
APPENDIX
t
0
0
0
[1] G. Boothroyd, Assembly Automation and Product Design. New York: Marcel Dekker, 1992. [2] G. H. Lim, “On the conveying velocity of a vibratory feeder,” Comput. Struct., vol. 62, no. 1, pp. 197–203, Jan. 1997. [3] P. Wolfsteiner and F. Pfeiffer, “Modeling, simulation, and verification of the transportation process in vibratory feeders,” Z. Angew. Math. Mech., vol. 80, no. 1, pp. 35–48, Jan. 2000. [4] S. Okabe, Y. Kamiya, K. Tsujikado, and Y. Yokoyama, “Vibratory feeding by nonsinusoidal vibration—optimum wave form,” ASME J. Vibr., Acoust., Stress, Rel. Design, vol. 107, pp. 188–195, Apr. 1985. [5] D. Reznik and J. Canny, “The Coulomb pump: A novel parts feeding method using a horizontally-vibrating surface,” in Proc. IEEE Int. Conf. Robot. Autom., 1998, pp. 869–874. [6] A. E. Quaid, “A miniature mobile parts feeder: Operating principles and simulation results,” in Proc. IEEE Int. Conf. Robotics Autom., 1999, pp. 2221–2226. [7] W. A. Morcos, “On the design of oscillating conveyers—Case of simultaneous normal and longitudinal oscillations,” ASME J. Eng. For Ind., vol. 92, no. 1, pp. 53–61, 1970. [8] P. U. Frei, M. Wiesendanger, R. Büchi, and L. Ruf, “Simultaneous planar transport of multiple objects on individual trajectories using friction forces,” in Distributed Manipulation, K. F. Böhringer and H. Choset, Eds. Norwell, MA: Kluwer, 2000, pp. 49–64. [9] D. Reznik and J. Canny, “A flat rigid plate is a universal planar manipulator,” in Proc. IEEE Int. Conf. Robotics Autom., 1998, pp. 1471–1477. [10] D. Reznik and J. Canny, “C’mon part, do the local motion!,” in Proc. IEEE Int. Conf. Robotics Autom., 2001, pp. 2235–2242. [11] T. H. Vose, P. Umbanhowar, and K. M. Lynch, “Vibration-induced frictional force fields on a rigid plate,” in Proc. IEEE Int. Conf. Robotics Autom., 2007, pp. 660–667.
