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Useful applications of closed-loop signal shaping controllers
ARTICLE IN PRESS
Control Engineering Practice 16 (2008) 836–846 www.elsevier.com/locate/conengprac
Useful applications of closed-loop signal shaping controllers John R. Huey, Khalid L. Sorensen, William E. Singhose Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 28 March 2007; accepted 21 September 2007 Available online 26 November 2007
Abstract Input shaping is a well-established open-loop technique used for reducing the vibratory response of dynamic systems. Some researchers have investigated the stability and utility of using this technique within a feedback control loop. The main contribution of the prior investigations was to identify stable configurations of in-the-loop input shaping systems. This paper identifies three promising applications of the stable controllers. Performance comparisons are made between the in-the-loop input shaping systems and more conventional feedback control strategies. Experimental results from a 10-ton industrial bridge crane, a portable bridge crane, and a portable tower crane are used to demonstrate the utility of the closed-loop input shaping control architecture. r 2007 Elsevier Ltd. All rights reserved. Keywords: Input shaping; Closed-loop input shaping; Feedback control; Nonlinear; Human operation; Crane
1. Introduction The control of flexible systems is an immense field of research. Many control strategies have been developed to mitigate undesired oscillation. These include feedback control, open-loop filtering methods, zero-phase error tracking control, and other combinations of feed-forward, open-loop, and closed-loop approaches. One particularly effective form of vibration suppression is input shaping (Singer & Seering, 1990; Smith, 1957). Input shaping is a command modification technique that causes a system to cancel out its own motion-induced oscillation. It has been used to reduce transient and residual oscillation in cranes (Lewis, Parker, Driessen, & Robinett, 1999; Singer, Singhose, & Kriikku, 1997; Singhose, Porter, Kenison, & Kriikku, 2000), coordinate measuring machines, (Jones & Ulsoy, 1999; Singhose, Singer, & Seering, 1996), flexible spacecraft (Gorinevsky & Vukovich, 1998; Singh & Vadali, 1993a; Tuttle & Seering, 1997), and long-reach manipulators (Kwon, Hwang, Babcock, & Burks, 1994; Magee & Book, 1995).
Corresponding author. Tel.: +1 404 385 0668.
E-mail addresses: [email protected] (K.L. Sorensen), [email protected] (W.E. Singhose). 0967-0661/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2007.09.004
An input shaper is a sequence of impulses. A general, n-impulse input shaper can be expressed in the time domain as ISðtÞ ¼
n X
Ai dðt ti Þ;
0pti otiþ1 ; Ai a0,
(1)
i¼1
where dðtÞ is the Dirac delta function, Ai is the amplitude of the ith impulse, and ti is the time of the ith impulse. Input shaping is implemented by convolving an input shaper with a reference command. The convolution product, instead of the original command, is then issued to a plant. For reference commands that reach a steadystate value, and for correctly designed input shapers, a linear system can exhibit zero residual oscillation in response to the modified command. This scenario is illustrated in Fig. 1(a) for a reference step command and a two-impulse input shaper. A block diagram representing a general input-shaped system is shown in Fig. 1(b). IS is the input shaper, and H is the linear plant. The two-impulse input shaper used in the preceding example is called a zero-vibration (ZV) shaper (Singer & Seering, 1990) because it results in zero residual system vibration when accurate estimates of system frequency and damping are available. The ZV shaper is defined as ISðtÞ ¼ A1 dðtÞ þ A2 dðt t2 Þ.
(2)
ARTICLE IN PRESS J.R. Huey et al. / Control Engineering Practice 16 (2008) 836–846
Step Command
ZV Shaper
W
A1
A2
Shaped Command W A1W
0
R
H
t2
Closed-Loop Transfer Function, H
Response W
* 0
Linear Plant
837
0
t2
0
IS
+-
C
G
Y
t2
Fig. 2. Outside-the-loop input shaping control architecture (OLIS).
r(t)
IS
H
y(t)
Fig. 1. Input shaping process. (a) Shaped command actuating a linear plant; (b) input shaping block diagram.
The input shaper parameters are functions of z and on , the damping ratio and natural frequency of H, respectively: A1 ¼
epzon =od , 1 þ epzon =od
A2 ¼ 1 A1 , t2 ¼
p p pffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 o d on 1 z
(3) (4) (5)
If the frequency and damping ratio of a system cannot be estimated accurately, then higher-order input shapers that are robust to modeling errors can be used (Singhose, Porter, Tuttle, & Singer, 1997; Singh & Vadali, 1993b). The penalty associated with increased robustness is that shaper duration is lengthened. Subsequently, rise time also increases. The vibration-reducing properties of an input shaper can be conceptually understood in the Laplace domain. The transfer function of the shaped system in Fig. 1(b) is Y IS H n IS H n ¼ IS H ¼ ¼ , R Hd H dr H di
(6)
where H n and H d are the numerator and denominator of H, respectively. For an undamped second-order system, H d defines the two imaginary poles of H. In the more general case, where H is an nth-order transfer function, H d can be decomposed into two polynomials: H dr and H di . The real (non-oscillatory) poles of H are defined by H dr . The imaginary (oscillatory) poles of H are defined by H di . For correctly designed input shapers, the input shaping parameters are selected so that the oscillatory poles of H (specified by the polynomial, H di ) are canceled by the zeros of IS (Bhat & Miu, 1990; Singh & Vadali, 1993b, 1994). In many industrial implementations of input shaping control, the plant, H, is comprised of a feedback controller, C, and a linear block, G. This scenario is shown in the block diagram of Fig. 2. By utilizing input shaping in this serial configuration, outside of a feedback loop, motioninduced oscillation of the closed-loop system can be reduced by the input shaper. The input shaper parameters are selected so that the oscillatory poles of the closed-looptransfer function are canceled by the zeros of the input
shaper. Other sources of system oscillation, such as disturbances, non-zero initial conditions, and actuator saturation are addressed by the feedback control block. This type of control architecture is referred to as outside-the-loop input shaping (OLIS). While the structure and implementation of input shaping resembles conventional filtering techniques, the design of input shaping filters is fundamentally different. The impulse sequence used in the shaping process is derived by solving a set of constraint equations that enforce a specified upper limit on residual vibration amplitude, even in the presence of modeling errors. Conventional filtering techniques do not usually directly impose constraints on vibration amplitude. They generally seek to minimize an energy cost function, or suppress frequencies in the commanded signal. Furthermore, virtually all conventional filters have pass bands where the filter attempts to pass frequencies without attenuation. This requirement imposes significant additional constraints that input shapers do not need to satisfy. The subtle design differences between input shaping and conventional filters have a substantial influence on system performance. In Singhose, Singer, and Seering (1995), input shaping was compared with several common lowpass and notch filters. The comparison was made by measuring the residual vibration amplitude of a harmonic oscillator in response to filtered step commands. The commands were filtered either by an input shaper or a conventional lowpass/notch filter. The systems using input shaping exhibited lower levels of vibration and faster rise times then those using conventional filters, even when significant modeling errors were present. Some key results from this study are summarized in Fig. 3. The bar graph in Fig. 3(a) represents the residual vibration amplitude for the various input shapers and filters that were tested. The bar graph in Fig. 3(b) represents the duration of the input shapers and filters. Filter/shaper duration is important because it provides a lower bound on rise time. These results were obtained for the case when a 15% modeling error in system frequency was present. Ordinarily, input shaping is used in an open-loop manner previously illustrated in Fig. 2. Fig. 4 shows a different control architecture where an input shaper is located within a feedback loop. This ‘‘in-the-loop-shaping’’ architecture is referred to as closed-loop signal shaping (CLSS). Given the advantages of input shaping over traditional filtering techniques at reducing oscillation,
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838
Filters
Vibration Amplitude (% Caused by Step)
Shapers
60
A) ZVD B) ZVDD C) ZVDDD D) 1H EI E) 2H EI F) 3H EI G) Neg. ZVD H) 1H Neg. EI I) 2H Neg. EI J) 3H Neg. EI
K) Butterworth Lowpass L) Chebyschev Lowpass M) Elliptical Lowpass N) Hamming Lowpass (short) O) Hamming Lowpass (long) P) Park-McClellan Lowpass Q) Chebyschev. Notch R) 256-P. Park-McClellan Notch S) Butterworth Notch T) Hamming Notch U) Elliptical Notch
40
20
A B C D E F G H
I
J K L M N O P Q R S T U
Input Shapers
Conventional Filters
off scale
Filter/Shaper Duration (Vibration Cycles)
10
5
A B C D E F G H
I
J K L M N O P Q R S T U
Fig. 3. Performance comparison between input shapers and conventional filters. (a) Vibration amplitude of a harmonic oscillator in response to shaped/ filtered step commands; (b) shaper/filter duration measured in cycles of vibration.
R
+-
C
IS
G
Y
Fig. 4. Closed-loop input shaping control architecture (CLSS).
many authors have speculated that this type of architecture can be used advantageously to address disturbances, effects of modeling errors, and other secondary sources of oscillation. However, only a few authors have numerically or experimentally verified these speculations. Kapila, Tzes, and Yan (2000) compared OLIS to CLSS when errors in the timing of shaper impulses was considered. Zuo and Wang (1992), Zuo, Drapeau, and Wang (1995) and Drapeau and Wang (1993) designed a CLSS controller for flexible manipulators. They experimentally compared it to an OLIS architecture that utilized conventional PID feedback control. They demonstrated the ability of the CLSS architecture to reject sensor disturbances.
The objective of this paper is to expand and unify this prior work by investigating some useful applications of the CLSS architecture. Potential advantages of this architecture over more conventional PID control and OLIS control are also discussed. The control issues addressed in the context of presenting these applications and performance comparisons are: (1) (2) (3) (4)
force disturbances; sensor disturbances; discontinuous nonlinearities; human-operated systems.
Fig. 5 shows the OLIS and CLSS architectures that will be considered in this paper. These block diagrams also depict some of the control problems addressed: force disturbances ðDf Þ, sensor disturbances ðDs Þ, and system nonlinearities (NL). An important issue of any control system is stability. CLSS controllers present a potential instability problem
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Df R
IS
+-
C
NL
Df G
Y
R
+-
C
IS
G
Y
Ds Df R
+-
C
IS
NL
G
Y
Fig. 6. CLSS controller with actuator disturbances.
Ds
because input shapers partially delay signals passing through them. The literature contains several stability analyses of these systems, in addition to the CLSS papers mentioned previously (Huey, 2006; Smith, 1958; Sorensen, Singhose, & Dickerson, 2007; Staehlin & Singh, 2003). A significant contribution made by these studies was to show that stability can be achieved for an identifiable range of controller gains. Thus, because the scope of this work is focused on applications and performance comparisons with conventional architectures, stability is not discussed at length, but it is achieved in the control problems presented here. Section 2 discusses force and sensor disturbances in the context of CLSS. Section 3 discusses the utility of CLSS on systems with hard nonlinearities. In Section 4, a CLSS controller is used to improve the performance of humanoperated cranes. Experimental results are presented throughout Sections 3 and 4. 2. Disturbance rejection Force and sensor disturbances are common sources of undesirable system dynamics. This section discusses the capacity of input shapers within a CLSS architecture to reduce oscillation caused by these sources. It will be shown that force disturbances cannot be rejected by CLSS controllers, but CLSS can be used to reject sensor disturbances. Contrasts are made between the performance of the CLSS architecture and conventional feedback control. A general principle is proposed for determining the utility of input shapers within feedback loops. 2.1. Force disturbances Fig. 6 shows a typical CLSS control architecture subjected to force disturbances acting on the plant. An important characteristic to note in this block diagram is that the disturbance force enters the plant, G, directly, and therefore is not filtered by the input shaper. This suggests that the input shaper will not contribute to eliminating this source of oscillation. The oscillatory effects of force disturbances on the CLSS system can be conceptualized by considering the transfer
0.5 Response
Fig. 5. OLIS and CLSS block diagrams with nonlinear element and disturbances. (a) OLIS block diagram; (b) CLSS block diagram.
CLSS PD Control
0
-0.5 0
5
10
15
20
Time (s)
Fig. 7. Actuator disturbance responses.
function between Df and Y: Y G Gn ¼ . ¼ Df 1 þ IS G C ðGd þ IS G n CÞ
(7)
Gn and Gd are the numerator and denominator of G, respectively. The oscillatory poles of this transfer function are specified by the parenthetical term in (7). When this term is evaluated at the oscillatory poles of G, the result is equal to zero. Thus, the oscillatory poles of G are contained in the oscillatory poles of Y =Df . Therefore, if G is lightly damped, then one may rightly anticipate that the system will respond to Df in a highly oscillatory manner. Fig. 7 demonstrates this situation. A simulated response of this system to an impulse force disturbance is shown with the solid line. For this example, the block, C, is a PD controller with K p and K d both equal to 1. The natural frequency and damping ratio of G are 1 and 0.05, respectively. The input shaping block is a ZV shaper tuned to cancel the oscillatory poles of G. For comparison purposes, the response of the system when input shaping is not used is also shown (dashed line). Without the input shaping block, the system reduces to a conventional PD feedback control architecture. The results shown in Fig. 7 illustrate the concept embodied in (7). Namely, that the oscillatory poles of G are not canceled by the input shaper. This can be attributed to the fact that the disturbance force is not filtered by IS prior to entering G. Therefore, the frequency content of the system response will contain the resonant frequency of the plant, G. In contrast, the conventional PD controller can provide significant damping to the system by varying the proportional and derivative gains.
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2.2. Sensor disturbance Fig. 8 shows a CLSS controller experiencing sensor disturbances. In this configuration, the disturbance signals pass through the input shaper before arriving at the plant, and are therefore filtered by the input shaper. This suggests that the CLSS controller is capable of rejecting oscillations caused by this disturbance. The capability of the CLSS controller to reject sensor disturbances is revealed in the transfer function between Ds and Y: Y IS G C IS G n C ¼ . ¼ Ds 1 þ IS G C ðGd þ IS G n CÞ
(8)
Eq. (8) is similar to (7). The oscillatory poles specified by the parenthetical term contain the oscillatory poles of G. However, in (8) the IS term in the numerator serves to cancel these poles. Therefore, the oscillatory dynamics of G are not exhibited by the system. Instead, any additional poles of the characteristic equation will describe the system’s oscillatory behavior. Therefore, suitable controller gains, in conjunction with the input shaper, can yield desirable sensor disturbance rejection. Simulation results are shown in Fig. 9 to validate this assertion. The solid line in this figure illustrates the system response to an impulse sensor disturbance. For this example, the block, C, was a proportional controller with unity gain. The input shaper and plant, G, remained the same as in the previous example of Section 2.1. The dashed line in Fig. 9 illustrates the system response when input shaping is not used. As demonstrated by this oscillatory response, the lack of derivative control prevents the conventional feedback controller from damping out the induced oscillation. These results are significant because they demonstrate the capability of CLSS to eliminate oscillation caused by sensor disturbances. They also reveal that CLSS can reduce sensor disturbances more effectively than proportional R
+-
C
IS
G
Y Ds
Fig. 8. CLSS controller with sensor disturbances.
CLSS P Control
Response
0.5
0
-0.5
control alone. Although a proportional and derivative control can yield results similar to those exhibited by the CLSS control, the capability to dampen out sensor-induced oscillation without derivative control is especially useful. Pure derivative control is acausal, therefore, when used, it is usually approximated by using numerical backward difference techniques. These techniques are effective for well behaved signals. However, the same techniques can produce erroneous results in the presence of noise. Numerical techniques can accommodate noise, but at a cost of utilizing more data points, which can introduce undesirable latency into the signal. CLSS eliminates the need for derivative control when rejecting sensor disturbances, and is less affected by noise than conventional derivative control. The force and sensor disturbance case studies provide insight into the utility of CLSS controllers. This architecture is not suitable for eliminating force disturbances, but can be very effective for reducing sensor-induced oscillation. A guiding principle for determining whether or not in-the-loop input shaping is beneficial for a given application is to consider the paths of the system signals: if the in-the-loop architecture permits signals to be issued to the input shaper prior to entering the plant, then the closed-loop response can be well behaved. In contrast, if signals filtered by the input shaper are altered, interrupted, or augmented, then the closed-loop response can exhibit oscillatory dynamics. 3. Hard nonlinearities All physical systems exhibit discontinuous dynamic effects, particularly saturation and rate limiting. These nonlinearities exist in all physical plants because real systems cannot be actuated with infinite effort (saturation), or be accelerated instantaneously (rate limit). Furthermore, in many industrial systems saturation and rate limiting are intentionally programmed into motor drives to prevent excessive speeds, and sudden starting and stopping. A programmed saturator truncates incoming signals that are outside the saturation threshold range. Similarly, a programmed rate limiter places a bound on the maximum allowable slew rate of incoming signals so that the time derivative of signals exiting a rate limiter are constrained to be within the rate limit threshold range. Systems with these types of hard nonlinearities can be modeled in the manner depicted in Fig. 10 (Sorensen et al., 2007), where a nonlinear element, NL, is serially coupled to a linear plant, G. When this type of nonlinear plant is incorporated into an OLIS or CLSS architecture, the resulting block diagrams look like those previously shown in Fig. 5 (without the Nonlinear Plant
0
5
10
15
Time (s)
Fig. 9. Sensor disturbance responses.
20
NL
G
Fig. 10. Serially coupled nonlinear element and linear plant.
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force and sensor disturbances). The presence of the nonlinearity in either architecture can be problematic because intended control signals are potentially corrupted by the nonlinearity prior to entering G. Consequently, the vibration-reducing properties of either of these architectures can be degraded. The objective of this section is to describe how the detrimental effects caused by saturation and rate limiting can be mitigated on CLSS systems more easily than on OLIS systems. To achieve this objective, some important mathematical properties of input shapers are identified. Then, these properties are used in a mitigation strategy for the CLSS architecture. The difficulty of implementing a similar mitigation technique on an OLIS system is discussed. Finally, experimental results obtained from an industrial 10-ton bridge crane using a CLSS architecture are presented. 3.1. Time derivative property Consider the class of n-impulse input shapers conforming to (1). For any finite valued function, xðtÞ, where xðtÞ is equal to zero for time tp0, the convolution relationship between the input shaper, the input signal, and the shaped signal, can be expressed in the time domain as ys ðtÞ ¼ xðtÞ ISðtÞ.
(9)
Given the constraints on xðtÞ, it may be shown that the _ input/output relationship of this convolution when xðtÞ, instead of xðtÞ, is convolved with the input shaper is _ ISðtÞ. y_ s ðtÞ ¼ xðtÞ
(10)
Thus, if a signal, xðtÞ, is convolved with a given input shaper to yield ys ðtÞ, then the time derivative of xðtÞ convolved with the same input shaper will yield the time derivative of ys ðtÞ. That is, (9) implies (10). 3.2. Boundedness property Eq. (9) may be expanded by substituting (1) into the convolution, and taking the absolute value to obtain jys ðtÞj ¼ jA1 xðtÞ1ðtÞ þ þ An xðt tn Þ1ðt tn Þj.
(11)
If the impulse amplitudes of the input shaper are constrained to be positive numbers, and m is equal to maxjxðtÞj, then the following inequality may be established: jys ðtÞjpmjA1 1ðtÞ þ þ An 1ðt tn Þj.
(12)
Further, because (10) holds, it follows that ^ 1 1ðtÞ þ þ An 1ðt tn Þj, jy_ s ðtÞjpmjA
^ jy_ s ðtÞjpm.
(15)
3.3. Mitigation technique The CLSS system previously shown in Fig. 5(b) can be modified to mitigate the detrimental effects of saturation and rate limiting. The modified CLSS architecture is shown in Fig. 11 for the case of a saturation element with threshold p. An artificial saturator with the same threshold parameter as the actual saturator has been added to the feed forward ~ path to modify xðtÞ. Then, maxjxðtÞjpp. It follows from (14) that if the impulse amplitudes of the input shaper are both positive and sum to unity, then jys ðtÞjpp. As a result, the shaped signal will be unmodified by the actual saturator because it is within the saturation threshold range. Therefore, the shaped signal acts directly on G, and can cancel its oscillatory poles. A similar modification can be made for the rate limiter. An artificial rate limiter with the same threshold parameters as the actual rate limiter can be placed in the feed forward path to modify xðtÞ. If the threshold parameters ^ for the actual and artificial rate limiters are specified by p, _~ ^ If given a positive impulse shaper, where then maxjxðtÞjp p. the amplitudes sum to unity, then it follows from (15) that ^ Again, the shaped signal will then be unmodified jy_ s ðtÞjpp. by the nonlinearity because the slope of the shaped signal does not exceed the rate limit threshold range. Therefore, by augmenting the feed forward path of a CLSS system with artificial nonlinear elements, shaped commands are permitted to act directly on G. Similar results may be obtained even when saturation and rate limit parameters of the artificial nonlinear elements are not identically equal to the parameters of the actual elements. All that is essential for the mitigation technique to be effective is for the artificial parameters to be more conservative than the actual parameters. In this way, knowledge of the exact values of p and p^ is not necessary (Sorensen & Singhose, 2007). In contrast to the straightforward method of mitigating saturation and rate limiting on CLSS systems, accomplishing a similar objective on OLIS systems is more difficult. Consider, for example the OLIS system shown in Fig. 12 subjected to an inherent saturation limit. To ensure that the vibration-reducing properties of this system are not degraded, one must ensure the control signal, xðtÞ, remains within the saturation threshold range. Meeting this condition, however, can be difficult because the range of this signal depends on the controller gains, the state of the
(13)
^ is equal to maxjxðtÞj. _ where m Thus, the right-hand sides of (12) and (13) represent bounding curves that the shaped outputs, ys ðtÞ and y_ s ðtÞ, will not exceed. In the special case where the impulse amplitudes sum to unity, (12) and (13) reduce to jys ðtÞjpm and
841
(14)
x(t) R
+-
~ x(t)
C
ys(t)
ynp(t)
IS Artificial Threshold, p
G Actual Threshold, p
Fig. 11. Modified CLSS architecture.
Y
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842
x(t) R
IS
+-
C
G
Y
Actual Fig. 12. OLIS control system with inherent saturation.
plant, and the reference signal. Similar difficulties exist for rate limiting. In contrast, the mitigation strategy for the CLSS architecture ensures that shaped commands are always issued to G, regardless of the controller gains, the state of the plant, or the reference command. Therefore the CLSS architecture is more suited for addressing saturation and rate limiting than the OLIS architecture. This advantage is the primary reason why a 10-ton industrial bridge crane at the Georgia Institute of Technology (Georgia Tech) utilizes a CLSS control (Sorensen et al., 2007).
Fig. 13. 10-Ton bridge crane.
Vtrolley Vreference
DM
G
Vpayload
Fig. 14. Crane actuation block diagram.
3.4. 10-Ton industrial bridge crane using CLSS Fig. 13 shows a photograph of the 10-ton industrial bridge crane. A block diagram of the crane actuation process is shown in Fig. 14. The block, DM, represents the nonlinear functionality of the system drives and motors. This plant accepts velocity commands issued to the crane and converts the reference velocity to the actual velocity of the overhead trolley. Several noticeable nonlinearities are inherent to this plant, including saturation and rate limiting. The block, G, is a linear transfer function relating the velocity of the overhead trolley to the velocity of the suspended payload: G¼
V payload 2zon s þ o2n ¼ 2 . V trolley s þ 2zon s þ o2n
(16)
This model reveals that the linear portion of the system behaves like a damped harmonic oscillator. The damping ratio is approximately 0.01. To enable oscillation-free positioning of the payload, the CLSS control scheme shown in Fig. 15 was implemented on the crane (Sorensen et al., 2007). Stable proportionalderivative control gains that achieved desirable system performance were obtained by using a root locus approximation technique presented in Huey (2006). These results were verified using numerical methods presented in Sorensen et al. (2007). Given that the payload eventually comes to rest directly beneath the overhead support unit, final positioning of the trolley is equivalent to final positioning of the payload. Therefore, the positioning of the payload is accomplished by using feedback to control the position of the trolley. Pr , Pt , and Pe are the desired trolley position, actual trolley position, and positioning error, respectively. In response to the positioning error, a proportional-derivative control block, C, generates a desired velocity signal. If this signal were issued directly to the drives and motors, then the
Pe Pr
+ Pt
-
Vr C
Vs IS
Vt DM
G
Vp
Artificial Saturator Fig. 15. CLSS control scheme on bridge crane.
objective of trolley positioning would be accomplished, however, noticeable cable sway would be exhibited. To achieve both precise positioning and oscillation suppression, an input shaper is introduced into the system to cancel the oscillatory dynamics of the payload plant. Additionally, an artificial saturator truncates velocity commands so that saturation inherent to the drives and motors does not degrade the properties of the input shaper. The response of the 10-ton crane to a typical position command while using the CLSS controller is shown in Fig. 16(a). For this experiment, the trolley was commanded to move 2-m. The position response of the trolley is shown with the solid line, while the position response of the payload is shown with the dashed line. The proportional and derivative gains were set to 1 and 0.5, respectively. A ZV input shaper was tuned to cancel the oscillatory poles of the payload. For comparison purposes, Fig. 16(b) shows the crane response from an identical experiment, except with the artificial saturator removed from the control architecture. The detrimental effects of saturation on the shaped signals are exhibited by the oscillatory response of the payload. Fig. 16(c) shows the results when both the artificial saturator, as well as the input shaper are removed from the architecture. This scenario reduces the system to a conventional proportional-derivative feedback controller. In this system there is no mechanism for eliminating the
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Position (m)
Operator Command
2
Transfer Objective
1
Trolley Payload 10
20
Crane Interface
Shaped Command
Input Shaper
Crane
Response
30 Fig. 17. CLSS architecture with human operator.
Time (s)
Position (m)
Reference Command
Sensory Feedback
0 0
843
2 1
Trolley Payload
0 0
10
20
30
Position (m)
Time (s)
2 1
Trolley Payload
0 0
10
20
30
Time (s) Fig. 18. Portable bridge crane. Fig. 16. Positioning and oscillation suppression capability of different controllers. (a) CLSS with artificial saturation; (b) CLSS without artificial saturation; (c) conventional PD-control.
oscillation of the payload, as demonstrated by the dashed line. The experimental results of Fig. 16 demonstrate that CLSS can be used to achieve the dual objectives of positioning and vibration suppression, even on systems with hard nonlinearities. 4. Human-centered CLSS When a human operator manipulates a crane, the operator’s sensory feedback is used to make real-time command decisions. This operational scenario constitutes a high-level form of feedback control, where the operator acts as a control element within a feedback structure. If the crane is equipped with input shaping, then the operatorgenerated reference commands are modified before being issued to the crane. This type of human-centered CLSS architecture is shown in Fig. 17. The input shaper serves to cancel the flexible modes of the crane so that cable sway will be reduced. The intention of reducing the cable sway is to increase the efficiency and safety performance of the crane. However, because input shaping modifies and partially delays an operator’s intended commands, an operator may become confused or annoyed at the unanticipated crane response. A natural question arising from this scenario is: Does the modification of human operator commands by input shaping lead to more
efficient and safer performance than if input shaping were not used? This section addresses this question by comparing the performance of human-centered CLSS with conventional human-controlled crane operation, where input shaping is not used. A human-operator study is presented that evaluated the performance of several operators that used a portable bridge crane (Lawrence & Singhose, 2005) and portable tower crane (Lawrence et al., 2006), both with and without input shaping. Two metrics were used to assess the performance of an operator: (1) the completion time to maneuver the payload from a starting region to a target region, and (2) the number of collisions occurring between the payload and workspace obstacles. 4.1. Operator studies on a portable bridge crane A photograph of the portable bridge crane is shown in Fig. 18. Operators were instructed to use the crane to manipulate the payload quickly through three different cluttered workspaces while avoiding obstacles. The workspaces are illustrated in the overhead views of Fig. 19. As illustrated, the path length and number of corners in each path increases from workspace-A to workspace-C. Operators drove the crane along each path twice using ZV input shaping and twice without input shaping. Completion time and collisions were recorded for each trial.
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80
Unshaped Shaped
Obstacle
Obstacle
Excess Maneuver Time (s)
Goal
Goal 60
Unshaped Shaped
40 20 0
Start
Start
Workspace-A
Workspace-B
Goal
Obstacle
Average Collisions
8
Workspace-C
Unshaped Shaped
6 4 2 0
Start Fig. 19. Portable bridge crane workspace paths. (a) Workspace-A; (b) workspace-B; (c) workspace-C.
A typical shaped and unshaped maneuver for workspace-A is shown in Fig. 19(a). For unshaped motion, the region where the nominal path changes direction generated the most disruptive payload oscillations. This was generally true for all of the workspace paths. This is because when the trolley stops traveling in one direction and begins traveling in a different direction, oscillation exhibited in the initial path plane contributes to out-of-plane oscillation in the new trolley path. Thus, multiple path direction changes can result in significant payload oscillation. For this reason, workspaces with more corners and constricted pathways are more difficult to traverse. Accordingly, one may anticipate that longer, more difficult paths require a greater time duration to complete. Therefore, to compare path completion time for each workspace, the completion times were normalized by subtracting the optimal completion time inherent to each nominal path. The nominal paths were characterized by simple straight line motions centered within the spacing between obstacles. The optimal time for each course was then calculated by adding the time required to traverse each of the straight lines at maximum velocity. The excess maneuver time on a course was then obtained by subtracting the optimal time from the actual completion time. The average excess maneuver time per trial is shown in Fig. 20(a) for each workspace. It is evident from this figure that input shaping noticeably reduces excess maneuver time. This result may be attributed to reduced oscillation exhibited when input shaping was used. The time ordinarily required by an operator to manually eliminate
Workspace-A
Workspace-B
Workspace-C
Fig. 20. Performance comparison for shaped and unshaped crane motion. (a) Excess maneuver time per trial; (b) number of collisions per trial.
Fig. 21. Portable tower crane.
oscillation without the use of input shaping is not expended when input shaping is implemented. Fig. 20(b) shows the average number of collisions that occurred per trial. It is evident that shaped motion greatly assisted operators in avoiding obstacles. 4.2. Operator studies on a portable tower crane A photograph of the portable tower crane is shown in Fig. 21. Experiments were conducted on this crane, similar to those conduced on the portable bridge crane. Operators were instructed to navigate through different workspaces
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benefit of input shaping, in terms of manipulation efficiency and collision avoidance, is significant. 5. Conclusions Start Goal
Start Goal
Crane Base
Crane Base
Fig. 22. Portable tower crane workspace paths. (a) Workspace-A; (b) workspace-B.
Average Maneuver Time (s)
60 Unshaped Shaped 40
20
Acknowledgments 0 Workspace-A
Average Collisions
The utility of closed-loop signal shaping (CLSS) was investigated for force and sensor disturbance rejection, hard nonlinearity accommodation, and human-in-the-loop control scenarios. A guiding principle was established to aid in determining whether or not the CLSS architecture could benefit other applications as well. The capacity of the CLSS architecture to reject sensor disturbances was demonstrated. Physical implementation of a CLSS controller on a 10-ton industrial bridge crane validated the utility of the architecture for controlling systems with saturation and rate limiting. Benefits of the architecture were also validated through two human-centered CLSS applications. Operators using the architecture manipulated cranes more efficiently and safer than operators using conventional control.
2
Workspace-B
Unshaped Shaped
References
1
0 Workspace-A
This project would not have been possible without the generous support of CAMotion and Siemens Energy & Automation.
Workspace-B
Fig. 23. Performance comparison for shaped and unshaped crane motion. (a) Average completion time per trial; (b) number of collisions per trial.
quickly while avoiding obstacles. Half of the trials were conducted with ZV input shaping and half of the trials did not use input shaping. Overhead illustrations of the two workspaces used for these experiments are shown in Fig. 22. They are identical to one another with the exception that workspace-B contains additional barriers that restrict the path width. The average completion time for all trials is shown in the bar graph of Fig. 23(a). These results indicate that CLSS architecture assisted operators in manipulating the tower crane more quickly. Likewise, the collision results shown in the bar graph of Fig. 23(b) indicate that the CLSS architecture reduced the number of payload collisions. The portable bridge and tower crane case studies demonstrate the utility of the human-centered CLSS architecture. Although input shaping modifies and partially delays an operator’s intended commands, the overall
Bhat, S. P., & Miu, D. K. (1990). Precise point-to-point positioning control of flexible structures. Journal of Dynamic Systems, Measurement, and Control, 112(4), 667–674. Drapeau, V., & Wang, D. (1993). Verification of a closed-loop shapedinput controller for a five-bar-linkage manipulator. In IEEE conference on robotics and automation (pp. 216–221), Atlanta, GA. Gorinevsky, D., & Vukovich, G. (1998). Nonlinear input shaping control of flexible spacecraft reorientation maneuver. Journal of Guidance, Control, and Dynamics, 21(2), 264–270. Huey, J. R. (2006). The intelligent combination of input shaping and PID feedback control. PhD Dissertation, Georgia Institute of Technology. Jones, S., & Ulsoy, A. G. (1999). An approach to control input shaping with application to coordinate measuring machines. Journal of Dynamic Systems, Measurement, and Control, 121(June), 242–247. Kapila, V., Tzes, A., & Yan, Q. (2000). Closed-loop input shaping for flexible structures using time-delay control. Journal of Dynamic Systems, Measurement, and Control, 122, 454–460. Kwon, D.-S., Hwang, D.-H., Babcock, S. M., & Burks, B. L. (1994). Input shaping filter methods for the control of structurally flexible, longreach manipulators. In IEEE conference on robotics and automation (Vol. 4, pp. 3259–3264), San Diego, CA. Lawrence, J., Fatkin, B., Singhose, W., Huey, J., Weiss, R., Erb, A., et al. (2006). An internet-driven tower crane for dynamics and controls education. In IFAC symposium on advances in control education, Madrid, Spain. Lawrence, J., & Singhose, W. (2005). Design of a minicrane for education and research. In International workshop on research and education in mechatronics (pp. 254–259), Annecy, France. Lewis, D., Parker, G. G., Driessen, G., & Robinett, R. D. (1999). Comparison of command shaping controllers for suppressing payload sway in a rotary boom crane. In IEEE international conference on control applications (pp. 719–724), Hawaii, USA.
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Magee, D. P., & Book, W. J. (1995). Filtering micro-manipulator wrist commands to prevent flexible base motion. In American control conference (pp. 924–928), Seattle, WA. Singer, N., Singhose, W., & Kriikku, E. (1997). An input shaping controller enabling cranes to move without sway. In ANS 7th topical meeting on robotics and remote systems (Vol. 1, pp. 225–231), Augusta, GA. Singer, N. C., & Seering, W. P. (1990). Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement, and Control, 112(March), 76–82. Singh, T., & Vadali, S. R. (1993a). Input-shaped control of threedimensional maneuvers of flexible spacecraft. Journal of Guidance, Control, and Dynamics, 16(6), 1061–1068. Singh, T., & Vadali, S. R. (1993b). Robust time-delay control. Journal of Dynamic Systems, Measurement, and Control, 115(June), 303–306. Singh, T., & Vadali, S. R. (1994). Robust time-optimal control: A frequency domain approach. AIAA Journal of Guidance, Control and Dynamics, 17(2), 346–353. Singhose, W., Porter, L., Kenison, M., & Kriikku, E. (2000). Effects of hoisting on the input shaping control of gantry cranes. Control Engineering Practice, 8(10), 1159–1165. Singhose, W., Singer, N., & Seering, W. (1995). Comparison of command shaping methods for reducing residual vibration. In European control conference (pp. 216–221), Rome, Italy. Singhose, W., Singer, N., & Seering, W. (1996). Improving repeatability of coordinate measuring machines with shaped command signals. Precision Engineering, 18(April), 138–146.
Singhose, W. E., Porter, L. J., Tuttle, T. D., & Singer, N. C. (1997). Vibration reduction using multi-hump input shapers. Journal of Dynamic Systems, Measurement, and Control, 119(June), 320–326. Smith, O. (1958). Feedback control systems. New York: McGraw-Hill Book Co., Inc. Smith, O. J. M. (1957). Posicast control of damped oscillatory systems. Proceedings of the IRE, 45(September), 1249–1255. Sorensen, K., & Singhose, W. (2007). Effects of common hard nonlinearities on input shaped signals. In American control conference, New York City, NY. Sorensen, K. L., Singhose, W. E., & Dickerson, S. (2007). A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Engineering Practice, 15(7), 825–837. Staehlin, U., & Singh, T. (2003). Design of closed-loop input shaping controllers. In American control conference (pp. 5167–5172), Denver, CO. Tuttle, T. D., & Seering, W. P. (1997). Experimental verification of vibration reduction in flexible spacecraft using input shaping. Journal of Guidance, Control, and Dynamics, 20(4), 658–664. Zuo, K., Drapeau, V., & Wang, D. (1995). Closed loop shaped-input strategies for flexible robots. International Journal of Robotics Research, 14(5), 510–529. Zuo, K., & Wang, D. (1992). Closed loop shaped-input control of a class of manipulators with a single flexible link. In IEEE international conference on robotics and automation (pp. 782–787), Nice, France.
Control Engineering Practice 16 (2008) 836–846 www.elsevier.com/locate/conengprac
Useful applications of closed-loop signal shaping controllers John R. Huey, Khalid L. Sorensen, William E. Singhose Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 28 March 2007; accepted 21 September 2007 Available online 26 November 2007
Abstract Input shaping is a well-established open-loop technique used for reducing the vibratory response of dynamic systems. Some researchers have investigated the stability and utility of using this technique within a feedback control loop. The main contribution of the prior investigations was to identify stable configurations of in-the-loop input shaping systems. This paper identifies three promising applications of the stable controllers. Performance comparisons are made between the in-the-loop input shaping systems and more conventional feedback control strategies. Experimental results from a 10-ton industrial bridge crane, a portable bridge crane, and a portable tower crane are used to demonstrate the utility of the closed-loop input shaping control architecture. r 2007 Elsevier Ltd. All rights reserved. Keywords: Input shaping; Closed-loop input shaping; Feedback control; Nonlinear; Human operation; Crane
1. Introduction The control of flexible systems is an immense field of research. Many control strategies have been developed to mitigate undesired oscillation. These include feedback control, open-loop filtering methods, zero-phase error tracking control, and other combinations of feed-forward, open-loop, and closed-loop approaches. One particularly effective form of vibration suppression is input shaping (Singer & Seering, 1990; Smith, 1957). Input shaping is a command modification technique that causes a system to cancel out its own motion-induced oscillation. It has been used to reduce transient and residual oscillation in cranes (Lewis, Parker, Driessen, & Robinett, 1999; Singer, Singhose, & Kriikku, 1997; Singhose, Porter, Kenison, & Kriikku, 2000), coordinate measuring machines, (Jones & Ulsoy, 1999; Singhose, Singer, & Seering, 1996), flexible spacecraft (Gorinevsky & Vukovich, 1998; Singh & Vadali, 1993a; Tuttle & Seering, 1997), and long-reach manipulators (Kwon, Hwang, Babcock, & Burks, 1994; Magee & Book, 1995).
Corresponding author. Tel.: +1 404 385 0668.
E-mail addresses: [email protected] (K.L. Sorensen), [email protected] (W.E. Singhose). 0967-0661/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2007.09.004
An input shaper is a sequence of impulses. A general, n-impulse input shaper can be expressed in the time domain as ISðtÞ ¼
n X
Ai dðt ti Þ;
0pti otiþ1 ; Ai a0,
(1)
i¼1
where dðtÞ is the Dirac delta function, Ai is the amplitude of the ith impulse, and ti is the time of the ith impulse. Input shaping is implemented by convolving an input shaper with a reference command. The convolution product, instead of the original command, is then issued to a plant. For reference commands that reach a steadystate value, and for correctly designed input shapers, a linear system can exhibit zero residual oscillation in response to the modified command. This scenario is illustrated in Fig. 1(a) for a reference step command and a two-impulse input shaper. A block diagram representing a general input-shaped system is shown in Fig. 1(b). IS is the input shaper, and H is the linear plant. The two-impulse input shaper used in the preceding example is called a zero-vibration (ZV) shaper (Singer & Seering, 1990) because it results in zero residual system vibration when accurate estimates of system frequency and damping are available. The ZV shaper is defined as ISðtÞ ¼ A1 dðtÞ þ A2 dðt t2 Þ.
(2)
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Step Command
ZV Shaper
W
A1
A2
Shaped Command W A1W
0
R
H
t2
Closed-Loop Transfer Function, H
Response W
* 0
Linear Plant
837
0
t2
0
IS
+-
C
G
Y
t2
Fig. 2. Outside-the-loop input shaping control architecture (OLIS).
r(t)
IS
H
y(t)
Fig. 1. Input shaping process. (a) Shaped command actuating a linear plant; (b) input shaping block diagram.
The input shaper parameters are functions of z and on , the damping ratio and natural frequency of H, respectively: A1 ¼
epzon =od , 1 þ epzon =od
A2 ¼ 1 A1 , t2 ¼
p p pffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 o d on 1 z
(3) (4) (5)
If the frequency and damping ratio of a system cannot be estimated accurately, then higher-order input shapers that are robust to modeling errors can be used (Singhose, Porter, Tuttle, & Singer, 1997; Singh & Vadali, 1993b). The penalty associated with increased robustness is that shaper duration is lengthened. Subsequently, rise time also increases. The vibration-reducing properties of an input shaper can be conceptually understood in the Laplace domain. The transfer function of the shaped system in Fig. 1(b) is Y IS H n IS H n ¼ IS H ¼ ¼ , R Hd H dr H di
(6)
where H n and H d are the numerator and denominator of H, respectively. For an undamped second-order system, H d defines the two imaginary poles of H. In the more general case, where H is an nth-order transfer function, H d can be decomposed into two polynomials: H dr and H di . The real (non-oscillatory) poles of H are defined by H dr . The imaginary (oscillatory) poles of H are defined by H di . For correctly designed input shapers, the input shaping parameters are selected so that the oscillatory poles of H (specified by the polynomial, H di ) are canceled by the zeros of IS (Bhat & Miu, 1990; Singh & Vadali, 1993b, 1994). In many industrial implementations of input shaping control, the plant, H, is comprised of a feedback controller, C, and a linear block, G. This scenario is shown in the block diagram of Fig. 2. By utilizing input shaping in this serial configuration, outside of a feedback loop, motioninduced oscillation of the closed-loop system can be reduced by the input shaper. The input shaper parameters are selected so that the oscillatory poles of the closed-looptransfer function are canceled by the zeros of the input
shaper. Other sources of system oscillation, such as disturbances, non-zero initial conditions, and actuator saturation are addressed by the feedback control block. This type of control architecture is referred to as outside-the-loop input shaping (OLIS). While the structure and implementation of input shaping resembles conventional filtering techniques, the design of input shaping filters is fundamentally different. The impulse sequence used in the shaping process is derived by solving a set of constraint equations that enforce a specified upper limit on residual vibration amplitude, even in the presence of modeling errors. Conventional filtering techniques do not usually directly impose constraints on vibration amplitude. They generally seek to minimize an energy cost function, or suppress frequencies in the commanded signal. Furthermore, virtually all conventional filters have pass bands where the filter attempts to pass frequencies without attenuation. This requirement imposes significant additional constraints that input shapers do not need to satisfy. The subtle design differences between input shaping and conventional filters have a substantial influence on system performance. In Singhose, Singer, and Seering (1995), input shaping was compared with several common lowpass and notch filters. The comparison was made by measuring the residual vibration amplitude of a harmonic oscillator in response to filtered step commands. The commands were filtered either by an input shaper or a conventional lowpass/notch filter. The systems using input shaping exhibited lower levels of vibration and faster rise times then those using conventional filters, even when significant modeling errors were present. Some key results from this study are summarized in Fig. 3. The bar graph in Fig. 3(a) represents the residual vibration amplitude for the various input shapers and filters that were tested. The bar graph in Fig. 3(b) represents the duration of the input shapers and filters. Filter/shaper duration is important because it provides a lower bound on rise time. These results were obtained for the case when a 15% modeling error in system frequency was present. Ordinarily, input shaping is used in an open-loop manner previously illustrated in Fig. 2. Fig. 4 shows a different control architecture where an input shaper is located within a feedback loop. This ‘‘in-the-loop-shaping’’ architecture is referred to as closed-loop signal shaping (CLSS). Given the advantages of input shaping over traditional filtering techniques at reducing oscillation,
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Filters
Vibration Amplitude (% Caused by Step)
Shapers
60
A) ZVD B) ZVDD C) ZVDDD D) 1H EI E) 2H EI F) 3H EI G) Neg. ZVD H) 1H Neg. EI I) 2H Neg. EI J) 3H Neg. EI
K) Butterworth Lowpass L) Chebyschev Lowpass M) Elliptical Lowpass N) Hamming Lowpass (short) O) Hamming Lowpass (long) P) Park-McClellan Lowpass Q) Chebyschev. Notch R) 256-P. Park-McClellan Notch S) Butterworth Notch T) Hamming Notch U) Elliptical Notch
40
20
A B C D E F G H
I
J K L M N O P Q R S T U
Input Shapers
Conventional Filters
off scale
Filter/Shaper Duration (Vibration Cycles)
10
5
A B C D E F G H
I
J K L M N O P Q R S T U
Fig. 3. Performance comparison between input shapers and conventional filters. (a) Vibration amplitude of a harmonic oscillator in response to shaped/ filtered step commands; (b) shaper/filter duration measured in cycles of vibration.
R
+-
C
IS
G
Y
Fig. 4. Closed-loop input shaping control architecture (CLSS).
many authors have speculated that this type of architecture can be used advantageously to address disturbances, effects of modeling errors, and other secondary sources of oscillation. However, only a few authors have numerically or experimentally verified these speculations. Kapila, Tzes, and Yan (2000) compared OLIS to CLSS when errors in the timing of shaper impulses was considered. Zuo and Wang (1992), Zuo, Drapeau, and Wang (1995) and Drapeau and Wang (1993) designed a CLSS controller for flexible manipulators. They experimentally compared it to an OLIS architecture that utilized conventional PID feedback control. They demonstrated the ability of the CLSS architecture to reject sensor disturbances.
The objective of this paper is to expand and unify this prior work by investigating some useful applications of the CLSS architecture. Potential advantages of this architecture over more conventional PID control and OLIS control are also discussed. The control issues addressed in the context of presenting these applications and performance comparisons are: (1) (2) (3) (4)
force disturbances; sensor disturbances; discontinuous nonlinearities; human-operated systems.
Fig. 5 shows the OLIS and CLSS architectures that will be considered in this paper. These block diagrams also depict some of the control problems addressed: force disturbances ðDf Þ, sensor disturbances ðDs Þ, and system nonlinearities (NL). An important issue of any control system is stability. CLSS controllers present a potential instability problem
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Df R
IS
+-
C
NL
Df G
Y
R
+-
C
IS
G
Y
Ds Df R
+-
C
IS
NL
G
Y
Fig. 6. CLSS controller with actuator disturbances.
Ds
because input shapers partially delay signals passing through them. The literature contains several stability analyses of these systems, in addition to the CLSS papers mentioned previously (Huey, 2006; Smith, 1958; Sorensen, Singhose, & Dickerson, 2007; Staehlin & Singh, 2003). A significant contribution made by these studies was to show that stability can be achieved for an identifiable range of controller gains. Thus, because the scope of this work is focused on applications and performance comparisons with conventional architectures, stability is not discussed at length, but it is achieved in the control problems presented here. Section 2 discusses force and sensor disturbances in the context of CLSS. Section 3 discusses the utility of CLSS on systems with hard nonlinearities. In Section 4, a CLSS controller is used to improve the performance of humanoperated cranes. Experimental results are presented throughout Sections 3 and 4. 2. Disturbance rejection Force and sensor disturbances are common sources of undesirable system dynamics. This section discusses the capacity of input shapers within a CLSS architecture to reduce oscillation caused by these sources. It will be shown that force disturbances cannot be rejected by CLSS controllers, but CLSS can be used to reject sensor disturbances. Contrasts are made between the performance of the CLSS architecture and conventional feedback control. A general principle is proposed for determining the utility of input shapers within feedback loops. 2.1. Force disturbances Fig. 6 shows a typical CLSS control architecture subjected to force disturbances acting on the plant. An important characteristic to note in this block diagram is that the disturbance force enters the plant, G, directly, and therefore is not filtered by the input shaper. This suggests that the input shaper will not contribute to eliminating this source of oscillation. The oscillatory effects of force disturbances on the CLSS system can be conceptualized by considering the transfer
0.5 Response
Fig. 5. OLIS and CLSS block diagrams with nonlinear element and disturbances. (a) OLIS block diagram; (b) CLSS block diagram.
CLSS PD Control
0
-0.5 0
5
10
15
20
Time (s)
Fig. 7. Actuator disturbance responses.
function between Df and Y: Y G Gn ¼ . ¼ Df 1 þ IS G C ðGd þ IS G n CÞ
(7)
Gn and Gd are the numerator and denominator of G, respectively. The oscillatory poles of this transfer function are specified by the parenthetical term in (7). When this term is evaluated at the oscillatory poles of G, the result is equal to zero. Thus, the oscillatory poles of G are contained in the oscillatory poles of Y =Df . Therefore, if G is lightly damped, then one may rightly anticipate that the system will respond to Df in a highly oscillatory manner. Fig. 7 demonstrates this situation. A simulated response of this system to an impulse force disturbance is shown with the solid line. For this example, the block, C, is a PD controller with K p and K d both equal to 1. The natural frequency and damping ratio of G are 1 and 0.05, respectively. The input shaping block is a ZV shaper tuned to cancel the oscillatory poles of G. For comparison purposes, the response of the system when input shaping is not used is also shown (dashed line). Without the input shaping block, the system reduces to a conventional PD feedback control architecture. The results shown in Fig. 7 illustrate the concept embodied in (7). Namely, that the oscillatory poles of G are not canceled by the input shaper. This can be attributed to the fact that the disturbance force is not filtered by IS prior to entering G. Therefore, the frequency content of the system response will contain the resonant frequency of the plant, G. In contrast, the conventional PD controller can provide significant damping to the system by varying the proportional and derivative gains.
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2.2. Sensor disturbance Fig. 8 shows a CLSS controller experiencing sensor disturbances. In this configuration, the disturbance signals pass through the input shaper before arriving at the plant, and are therefore filtered by the input shaper. This suggests that the CLSS controller is capable of rejecting oscillations caused by this disturbance. The capability of the CLSS controller to reject sensor disturbances is revealed in the transfer function between Ds and Y: Y IS G C IS G n C ¼ . ¼ Ds 1 þ IS G C ðGd þ IS G n CÞ
(8)
Eq. (8) is similar to (7). The oscillatory poles specified by the parenthetical term contain the oscillatory poles of G. However, in (8) the IS term in the numerator serves to cancel these poles. Therefore, the oscillatory dynamics of G are not exhibited by the system. Instead, any additional poles of the characteristic equation will describe the system’s oscillatory behavior. Therefore, suitable controller gains, in conjunction with the input shaper, can yield desirable sensor disturbance rejection. Simulation results are shown in Fig. 9 to validate this assertion. The solid line in this figure illustrates the system response to an impulse sensor disturbance. For this example, the block, C, was a proportional controller with unity gain. The input shaper and plant, G, remained the same as in the previous example of Section 2.1. The dashed line in Fig. 9 illustrates the system response when input shaping is not used. As demonstrated by this oscillatory response, the lack of derivative control prevents the conventional feedback controller from damping out the induced oscillation. These results are significant because they demonstrate the capability of CLSS to eliminate oscillation caused by sensor disturbances. They also reveal that CLSS can reduce sensor disturbances more effectively than proportional R
+-
C
IS
G
Y Ds
Fig. 8. CLSS controller with sensor disturbances.
CLSS P Control
Response
0.5
0
-0.5
control alone. Although a proportional and derivative control can yield results similar to those exhibited by the CLSS control, the capability to dampen out sensor-induced oscillation without derivative control is especially useful. Pure derivative control is acausal, therefore, when used, it is usually approximated by using numerical backward difference techniques. These techniques are effective for well behaved signals. However, the same techniques can produce erroneous results in the presence of noise. Numerical techniques can accommodate noise, but at a cost of utilizing more data points, which can introduce undesirable latency into the signal. CLSS eliminates the need for derivative control when rejecting sensor disturbances, and is less affected by noise than conventional derivative control. The force and sensor disturbance case studies provide insight into the utility of CLSS controllers. This architecture is not suitable for eliminating force disturbances, but can be very effective for reducing sensor-induced oscillation. A guiding principle for determining whether or not in-the-loop input shaping is beneficial for a given application is to consider the paths of the system signals: if the in-the-loop architecture permits signals to be issued to the input shaper prior to entering the plant, then the closed-loop response can be well behaved. In contrast, if signals filtered by the input shaper are altered, interrupted, or augmented, then the closed-loop response can exhibit oscillatory dynamics. 3. Hard nonlinearities All physical systems exhibit discontinuous dynamic effects, particularly saturation and rate limiting. These nonlinearities exist in all physical plants because real systems cannot be actuated with infinite effort (saturation), or be accelerated instantaneously (rate limit). Furthermore, in many industrial systems saturation and rate limiting are intentionally programmed into motor drives to prevent excessive speeds, and sudden starting and stopping. A programmed saturator truncates incoming signals that are outside the saturation threshold range. Similarly, a programmed rate limiter places a bound on the maximum allowable slew rate of incoming signals so that the time derivative of signals exiting a rate limiter are constrained to be within the rate limit threshold range. Systems with these types of hard nonlinearities can be modeled in the manner depicted in Fig. 10 (Sorensen et al., 2007), where a nonlinear element, NL, is serially coupled to a linear plant, G. When this type of nonlinear plant is incorporated into an OLIS or CLSS architecture, the resulting block diagrams look like those previously shown in Fig. 5 (without the Nonlinear Plant
0
5
10
15
Time (s)
Fig. 9. Sensor disturbance responses.
20
NL
G
Fig. 10. Serially coupled nonlinear element and linear plant.
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force and sensor disturbances). The presence of the nonlinearity in either architecture can be problematic because intended control signals are potentially corrupted by the nonlinearity prior to entering G. Consequently, the vibration-reducing properties of either of these architectures can be degraded. The objective of this section is to describe how the detrimental effects caused by saturation and rate limiting can be mitigated on CLSS systems more easily than on OLIS systems. To achieve this objective, some important mathematical properties of input shapers are identified. Then, these properties are used in a mitigation strategy for the CLSS architecture. The difficulty of implementing a similar mitigation technique on an OLIS system is discussed. Finally, experimental results obtained from an industrial 10-ton bridge crane using a CLSS architecture are presented. 3.1. Time derivative property Consider the class of n-impulse input shapers conforming to (1). For any finite valued function, xðtÞ, where xðtÞ is equal to zero for time tp0, the convolution relationship between the input shaper, the input signal, and the shaped signal, can be expressed in the time domain as ys ðtÞ ¼ xðtÞ ISðtÞ.
(9)
Given the constraints on xðtÞ, it may be shown that the _ input/output relationship of this convolution when xðtÞ, instead of xðtÞ, is convolved with the input shaper is _ ISðtÞ. y_ s ðtÞ ¼ xðtÞ
(10)
Thus, if a signal, xðtÞ, is convolved with a given input shaper to yield ys ðtÞ, then the time derivative of xðtÞ convolved with the same input shaper will yield the time derivative of ys ðtÞ. That is, (9) implies (10). 3.2. Boundedness property Eq. (9) may be expanded by substituting (1) into the convolution, and taking the absolute value to obtain jys ðtÞj ¼ jA1 xðtÞ1ðtÞ þ þ An xðt tn Þ1ðt tn Þj.
(11)
If the impulse amplitudes of the input shaper are constrained to be positive numbers, and m is equal to maxjxðtÞj, then the following inequality may be established: jys ðtÞjpmjA1 1ðtÞ þ þ An 1ðt tn Þj.
(12)
Further, because (10) holds, it follows that ^ 1 1ðtÞ þ þ An 1ðt tn Þj, jy_ s ðtÞjpmjA
^ jy_ s ðtÞjpm.
(15)
3.3. Mitigation technique The CLSS system previously shown in Fig. 5(b) can be modified to mitigate the detrimental effects of saturation and rate limiting. The modified CLSS architecture is shown in Fig. 11 for the case of a saturation element with threshold p. An artificial saturator with the same threshold parameter as the actual saturator has been added to the feed forward ~ path to modify xðtÞ. Then, maxjxðtÞjpp. It follows from (14) that if the impulse amplitudes of the input shaper are both positive and sum to unity, then jys ðtÞjpp. As a result, the shaped signal will be unmodified by the actual saturator because it is within the saturation threshold range. Therefore, the shaped signal acts directly on G, and can cancel its oscillatory poles. A similar modification can be made for the rate limiter. An artificial rate limiter with the same threshold parameters as the actual rate limiter can be placed in the feed forward path to modify xðtÞ. If the threshold parameters ^ for the actual and artificial rate limiters are specified by p, _~ ^ If given a positive impulse shaper, where then maxjxðtÞjp p. the amplitudes sum to unity, then it follows from (15) that ^ Again, the shaped signal will then be unmodified jy_ s ðtÞjpp. by the nonlinearity because the slope of the shaped signal does not exceed the rate limit threshold range. Therefore, by augmenting the feed forward path of a CLSS system with artificial nonlinear elements, shaped commands are permitted to act directly on G. Similar results may be obtained even when saturation and rate limit parameters of the artificial nonlinear elements are not identically equal to the parameters of the actual elements. All that is essential for the mitigation technique to be effective is for the artificial parameters to be more conservative than the actual parameters. In this way, knowledge of the exact values of p and p^ is not necessary (Sorensen & Singhose, 2007). In contrast to the straightforward method of mitigating saturation and rate limiting on CLSS systems, accomplishing a similar objective on OLIS systems is more difficult. Consider, for example the OLIS system shown in Fig. 12 subjected to an inherent saturation limit. To ensure that the vibration-reducing properties of this system are not degraded, one must ensure the control signal, xðtÞ, remains within the saturation threshold range. Meeting this condition, however, can be difficult because the range of this signal depends on the controller gains, the state of the
(13)
^ is equal to maxjxðtÞj. _ where m Thus, the right-hand sides of (12) and (13) represent bounding curves that the shaped outputs, ys ðtÞ and y_ s ðtÞ, will not exceed. In the special case where the impulse amplitudes sum to unity, (12) and (13) reduce to jys ðtÞjpm and
841
(14)
x(t) R
+-
~ x(t)
C
ys(t)
ynp(t)
IS Artificial Threshold, p
G Actual Threshold, p
Fig. 11. Modified CLSS architecture.
Y
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842
x(t) R
IS
+-
C
G
Y
Actual Fig. 12. OLIS control system with inherent saturation.
plant, and the reference signal. Similar difficulties exist for rate limiting. In contrast, the mitigation strategy for the CLSS architecture ensures that shaped commands are always issued to G, regardless of the controller gains, the state of the plant, or the reference command. Therefore the CLSS architecture is more suited for addressing saturation and rate limiting than the OLIS architecture. This advantage is the primary reason why a 10-ton industrial bridge crane at the Georgia Institute of Technology (Georgia Tech) utilizes a CLSS control (Sorensen et al., 2007).
Fig. 13. 10-Ton bridge crane.
Vtrolley Vreference
DM
G
Vpayload
Fig. 14. Crane actuation block diagram.
3.4. 10-Ton industrial bridge crane using CLSS Fig. 13 shows a photograph of the 10-ton industrial bridge crane. A block diagram of the crane actuation process is shown in Fig. 14. The block, DM, represents the nonlinear functionality of the system drives and motors. This plant accepts velocity commands issued to the crane and converts the reference velocity to the actual velocity of the overhead trolley. Several noticeable nonlinearities are inherent to this plant, including saturation and rate limiting. The block, G, is a linear transfer function relating the velocity of the overhead trolley to the velocity of the suspended payload: G¼
V payload 2zon s þ o2n ¼ 2 . V trolley s þ 2zon s þ o2n
(16)
This model reveals that the linear portion of the system behaves like a damped harmonic oscillator. The damping ratio is approximately 0.01. To enable oscillation-free positioning of the payload, the CLSS control scheme shown in Fig. 15 was implemented on the crane (Sorensen et al., 2007). Stable proportionalderivative control gains that achieved desirable system performance were obtained by using a root locus approximation technique presented in Huey (2006). These results were verified using numerical methods presented in Sorensen et al. (2007). Given that the payload eventually comes to rest directly beneath the overhead support unit, final positioning of the trolley is equivalent to final positioning of the payload. Therefore, the positioning of the payload is accomplished by using feedback to control the position of the trolley. Pr , Pt , and Pe are the desired trolley position, actual trolley position, and positioning error, respectively. In response to the positioning error, a proportional-derivative control block, C, generates a desired velocity signal. If this signal were issued directly to the drives and motors, then the
Pe Pr
+ Pt
-
Vr C
Vs IS
Vt DM
G
Vp
Artificial Saturator Fig. 15. CLSS control scheme on bridge crane.
objective of trolley positioning would be accomplished, however, noticeable cable sway would be exhibited. To achieve both precise positioning and oscillation suppression, an input shaper is introduced into the system to cancel the oscillatory dynamics of the payload plant. Additionally, an artificial saturator truncates velocity commands so that saturation inherent to the drives and motors does not degrade the properties of the input shaper. The response of the 10-ton crane to a typical position command while using the CLSS controller is shown in Fig. 16(a). For this experiment, the trolley was commanded to move 2-m. The position response of the trolley is shown with the solid line, while the position response of the payload is shown with the dashed line. The proportional and derivative gains were set to 1 and 0.5, respectively. A ZV input shaper was tuned to cancel the oscillatory poles of the payload. For comparison purposes, Fig. 16(b) shows the crane response from an identical experiment, except with the artificial saturator removed from the control architecture. The detrimental effects of saturation on the shaped signals are exhibited by the oscillatory response of the payload. Fig. 16(c) shows the results when both the artificial saturator, as well as the input shaper are removed from the architecture. This scenario reduces the system to a conventional proportional-derivative feedback controller. In this system there is no mechanism for eliminating the
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Position (m)
Operator Command
2
Transfer Objective
1
Trolley Payload 10
20
Crane Interface
Shaped Command
Input Shaper
Crane
Response
30 Fig. 17. CLSS architecture with human operator.
Time (s)
Position (m)
Reference Command
Sensory Feedback
0 0
843
2 1
Trolley Payload
0 0
10
20
30
Position (m)
Time (s)
2 1
Trolley Payload
0 0
10
20
30
Time (s) Fig. 18. Portable bridge crane. Fig. 16. Positioning and oscillation suppression capability of different controllers. (a) CLSS with artificial saturation; (b) CLSS without artificial saturation; (c) conventional PD-control.
oscillation of the payload, as demonstrated by the dashed line. The experimental results of Fig. 16 demonstrate that CLSS can be used to achieve the dual objectives of positioning and vibration suppression, even on systems with hard nonlinearities. 4. Human-centered CLSS When a human operator manipulates a crane, the operator’s sensory feedback is used to make real-time command decisions. This operational scenario constitutes a high-level form of feedback control, where the operator acts as a control element within a feedback structure. If the crane is equipped with input shaping, then the operatorgenerated reference commands are modified before being issued to the crane. This type of human-centered CLSS architecture is shown in Fig. 17. The input shaper serves to cancel the flexible modes of the crane so that cable sway will be reduced. The intention of reducing the cable sway is to increase the efficiency and safety performance of the crane. However, because input shaping modifies and partially delays an operator’s intended commands, an operator may become confused or annoyed at the unanticipated crane response. A natural question arising from this scenario is: Does the modification of human operator commands by input shaping lead to more
efficient and safer performance than if input shaping were not used? This section addresses this question by comparing the performance of human-centered CLSS with conventional human-controlled crane operation, where input shaping is not used. A human-operator study is presented that evaluated the performance of several operators that used a portable bridge crane (Lawrence & Singhose, 2005) and portable tower crane (Lawrence et al., 2006), both with and without input shaping. Two metrics were used to assess the performance of an operator: (1) the completion time to maneuver the payload from a starting region to a target region, and (2) the number of collisions occurring between the payload and workspace obstacles. 4.1. Operator studies on a portable bridge crane A photograph of the portable bridge crane is shown in Fig. 18. Operators were instructed to use the crane to manipulate the payload quickly through three different cluttered workspaces while avoiding obstacles. The workspaces are illustrated in the overhead views of Fig. 19. As illustrated, the path length and number of corners in each path increases from workspace-A to workspace-C. Operators drove the crane along each path twice using ZV input shaping and twice without input shaping. Completion time and collisions were recorded for each trial.
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80
Unshaped Shaped
Obstacle
Obstacle
Excess Maneuver Time (s)
Goal
Goal 60
Unshaped Shaped
40 20 0
Start
Start
Workspace-A
Workspace-B
Goal
Obstacle
Average Collisions
8
Workspace-C
Unshaped Shaped
6 4 2 0
Start Fig. 19. Portable bridge crane workspace paths. (a) Workspace-A; (b) workspace-B; (c) workspace-C.
A typical shaped and unshaped maneuver for workspace-A is shown in Fig. 19(a). For unshaped motion, the region where the nominal path changes direction generated the most disruptive payload oscillations. This was generally true for all of the workspace paths. This is because when the trolley stops traveling in one direction and begins traveling in a different direction, oscillation exhibited in the initial path plane contributes to out-of-plane oscillation in the new trolley path. Thus, multiple path direction changes can result in significant payload oscillation. For this reason, workspaces with more corners and constricted pathways are more difficult to traverse. Accordingly, one may anticipate that longer, more difficult paths require a greater time duration to complete. Therefore, to compare path completion time for each workspace, the completion times were normalized by subtracting the optimal completion time inherent to each nominal path. The nominal paths were characterized by simple straight line motions centered within the spacing between obstacles. The optimal time for each course was then calculated by adding the time required to traverse each of the straight lines at maximum velocity. The excess maneuver time on a course was then obtained by subtracting the optimal time from the actual completion time. The average excess maneuver time per trial is shown in Fig. 20(a) for each workspace. It is evident from this figure that input shaping noticeably reduces excess maneuver time. This result may be attributed to reduced oscillation exhibited when input shaping was used. The time ordinarily required by an operator to manually eliminate
Workspace-A
Workspace-B
Workspace-C
Fig. 20. Performance comparison for shaped and unshaped crane motion. (a) Excess maneuver time per trial; (b) number of collisions per trial.
Fig. 21. Portable tower crane.
oscillation without the use of input shaping is not expended when input shaping is implemented. Fig. 20(b) shows the average number of collisions that occurred per trial. It is evident that shaped motion greatly assisted operators in avoiding obstacles. 4.2. Operator studies on a portable tower crane A photograph of the portable tower crane is shown in Fig. 21. Experiments were conducted on this crane, similar to those conduced on the portable bridge crane. Operators were instructed to navigate through different workspaces
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benefit of input shaping, in terms of manipulation efficiency and collision avoidance, is significant. 5. Conclusions Start Goal
Start Goal
Crane Base
Crane Base
Fig. 22. Portable tower crane workspace paths. (a) Workspace-A; (b) workspace-B.
Average Maneuver Time (s)
60 Unshaped Shaped 40
20
Acknowledgments 0 Workspace-A
Average Collisions
The utility of closed-loop signal shaping (CLSS) was investigated for force and sensor disturbance rejection, hard nonlinearity accommodation, and human-in-the-loop control scenarios. A guiding principle was established to aid in determining whether or not the CLSS architecture could benefit other applications as well. The capacity of the CLSS architecture to reject sensor disturbances was demonstrated. Physical implementation of a CLSS controller on a 10-ton industrial bridge crane validated the utility of the architecture for controlling systems with saturation and rate limiting. Benefits of the architecture were also validated through two human-centered CLSS applications. Operators using the architecture manipulated cranes more efficiently and safer than operators using conventional control.
2
Workspace-B
Unshaped Shaped
References
1
0 Workspace-A
This project would not have been possible without the generous support of CAMotion and Siemens Energy & Automation.
Workspace-B
Fig. 23. Performance comparison for shaped and unshaped crane motion. (a) Average completion time per trial; (b) number of collisions per trial.
quickly while avoiding obstacles. Half of the trials were conducted with ZV input shaping and half of the trials did not use input shaping. Overhead illustrations of the two workspaces used for these experiments are shown in Fig. 22. They are identical to one another with the exception that workspace-B contains additional barriers that restrict the path width. The average completion time for all trials is shown in the bar graph of Fig. 23(a). These results indicate that CLSS architecture assisted operators in manipulating the tower crane more quickly. Likewise, the collision results shown in the bar graph of Fig. 23(b) indicate that the CLSS architecture reduced the number of payload collisions. The portable bridge and tower crane case studies demonstrate the utility of the human-centered CLSS architecture. Although input shaping modifies and partially delays an operator’s intended commands, the overall
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