Sep 20, 2017 - Impulse response cancellation to form a Zero Vibration input shaper. 8 ..... lighting conditions, such measurement is impractical in many cases.
Attenuating Vibration through Input Shaping and Feedback Control Techniques
A Thesis Presented to the Graduate Faculty of the University of Louisiana at Lafayette In Partial Fulfillment of the Requirements for the Degree Masters of Science
Daniel Newman Spring 2018
c Daniel Newman
2018 All Rights Reserved
Attenuating Vibration through Input Shaping and Feedback Control Techniques Daniel Newman
APPROVED:
Joshua E. Vaughan, Chair Assistant Professor of Mechanical Engineering
Terrence Chambers Associate Professor of Mechanical Engineering
Raju Gottumukkala Assistant Professor of Mechanical Engineering
Mary Farmer-Kaiser Dean of the Graduate School
To mom and dad, who taught me to never give up.
“I don’t measure a man’s success by how high he climbs but how high he bounces when he hits bottom.” — George S. Patton
Acknowledgments First of all, I would like to thank my advisor, Dr. Joshua Vaughan, for his mentorship through my graduate studies. His guidance has been invaluable to me through this part of my career, and I am profoundly grateful for the opportunities that he has facilitated. One such opportunity was a two-month stay in Korea in the summer of 2017 through the East Asia Pacific Summer Institutes Program. I would like to thank the National Science Foundation and the Korean National Research Foundation for providing funding for this amazing opportunity. Furthermore, Professor Seong-Wook Hong at the Kumoh National Institute of Technology generously hosted and mentored me through this program. He and his students made my stay a wonderful experience. I will forever cherish my time in Korea. I wish to also thank my labmates in the CRAWLAB for making this an enjoyable experience. Namely, Gerald Eaglin, Daniel Ashkeboussi, Joe Fuentes, and Forrest Montgomery. I also worked with a fantastic group of undergraduates, including Lane Elder and Joshua Keller who helped me with experimental results for this thesis. Best of luck to each of you. Lastly, I’d like to thank my girlfriend Vallerie for her support, encouragement, and occasional reminders to sleep.
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Table of Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Epigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Goals and Methods . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 1
List of Tables
I
II Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Input Shaping . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Input shaper design . . . . . . . . . . . . . . . . . 2.1.2 Tools for evaluating input shaper performance 2.1.3 Specified insensitivity input shaping . . . . . . . 2.2 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 2.3 Comparison of Input Shaping and Feedback Control 2.3.1 Comparison of settling time . . . . . . . . . . . . 2.3.2 A simple example . . . . . . . . . . . . . . . . . . 2.4 Combining Input Shaping and Feedback Control . . . 2.4.1 Closed-loop input shaping . . . . . . . . . . . . . 2.4.2 Outside-the-loop input shaping . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 6 7 11 12 14 15 15 17 24 25 25 26
III Initial Condition Input Shaping . . . . . . . . 3.1 Frequency-Domain Design . . . . . . . . . 3.2 Time-Domain Design . . . . . . . . . . . . 3.2.1 Incorporating actuator limitations 3.2.2 Robustness considerations . . . . . 3.3 Impulse Response Example . . . . . . . . 3.4 Experimental Verification . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . .
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27 28 30 32 35 38 40 46
IV An Application in Boom Crane Control . . . . . . . . . . . . 4.1 Control of an Undisturbed Boom Crane . . . . . . . . . 4.1.1 Stationary dynamic model . . . . . . . . . . . . . . 4.1.2 Approximating impulse response characteristics 4.1.3 Initial condition input shaper design . . . . . . . 4.2 Boom Crane Subject to a Harmonic Disturbance . . .
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47 47 47 49 53 54
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4.2.1 Dynamic model . . . . . . . . . . . 4.2.2 Validity of linear approximations 4.2.3 Attenuation of the free-vibration 4.3 Free-Response Attenuation Simulation 4.4 Experimental Verification . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . .
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55 58 59 61 63 70
Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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71 71 73 74 77 80 81 84
VI Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85 85 86
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
V Concurrent Design of Input Shaping and Linear Feedback 5.1 Control Approach . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Choosing control gains for concurrent design . 5.1.2 Command shaping . . . . . . . . . . . . . . . . . . 5.1.3 Comparison to smooth command generation . 5.2 Control of a Planar Crane . . . . . . . . . . . . . . . . . 5.2.1 Crane control simulation . . . . . . . . . . . . . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Tables
Table 1.
Simulation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Table 2.
Benchmark System Experimental Parameters . . . . . . . . . . . . .
41
Table 3.
Simulation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
Table 4.
Experimental Parameters . . . . . . . . . . . . . . . . . . . . . . . .
64
Table 5.
Planar Crane Simulation Parameters . . . . . . . . . . . . . . . . . .
81
Table 6.
Controller Weight Results . . . . . . . . . . . . . . . . . . . . . . . .
82
Table 7.
Concurrent Design Response Characteristics . . . . . . . . . . . . . .
84
ix
List of Figures
Figure 1. Street blocked by downed trees on September 20, 2017 [1] . . . . . .
4
Figure 2. U.S.S. Kearsarge amphibious assault ship [2] . . . . . . . . . . . . . .
5
Figure 3. Mobile harbor concept [3] . . . . . . . . . . . . . . . . . . . . . . . .
5
Figure 4. Convolution of an input shaper with a reference command resulting in zero vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Figure 5. Impulse response cancellation to form a Zero Vibration input shaper
8
Figure 6. Sensitivity plot of a ZV, ZVD, and EI shaper . . . . . . . . . . . . .
11
Figure 7. Residual vibration summation of vector diagram impulses . . . . . .
12
Figure 8. Vibration suppression through frequency sampling . . . . . . . . . . .
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Figure 9. Block diagram of feedback control . . . . . . . . . . . . . . . . . . . .
14
Figure 10. Settling time of shaping versus feedback control methods . . . . . . .
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Figure 11. Planar crane model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 12. Crane payload response . . . . . . . . . . . . . . . . . . . . . . . . .
19
Figure 13. Crane velocity command . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 14. Comparison of pole locations . . . . . . . . . . . . . . . . . . . . . .
21
Figure 15. Sensitivity curve of a multi-mode shaper . . . . . . . . . . . . . . . .
23
Figure 16. Step response of a crane under LQR and ZV-shaped control . . . . .
24
Figure 17. Payload swing response of a crane under LQR and ZV-shaped control
24
Figure 18. Block diagram of an input shaper . . . . . . . . . . . . . . . . . . . .
25
Figure 19. Block diagram of a closed-loop input shaper . . . . . . . . . . . . . .
25
Figure 20. Block diagram of an input shaper outside a control loop . . . . . . .
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Figure 21. Initial condition impulse cancellation . . . . . . . . . . . . . . . . . .
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Figure 22. Simple flexible system . . . . . . . . . . . . . . . . . . . . . . . . . .
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x
Figure 23. Response characteristics of an impulse vs a pulse . . . . . . . . . . .
33
Figure 24. Vector Diagram of an input shaper subject to actuator limitations . .
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Figure 25. Phase shift of a pulse response relative to an impulse response . . . .
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Figure 26. Amplitude shift of a pulse response relative to an impulse response .
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Figure 27. Sensitivity of a ZV-IC shaper to Ae and θe . . . . . . . . . . . . . . .
37
Figure 28. ZV-IC shaper sensitivity . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 29. SI-IC shaper sensitivity . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 30. ZV-IC shaped pulse response . . . . . . . . . . . . . . . . . . . . . .
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Figure 31. SI-IC shaped pulse response . . . . . . . . . . . . . . . . . . . . . . .
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Figure 32. Sensitivity curves of a ZV-IC and SI-IC shaper . . . . . . . . . . . .
40
Figure 33. Benchmark experimental system
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Figure 34. Typical unshaped response . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 35. ZV-IC shaped response at the designed frequency . . . . . . . . . . .
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Figure 36. SI-IC shaped response at the designed frequency . . . . . . . . . . . .
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Figure 37. Experimental sensitivity to changes in natural frequency . . . . . . .
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Figure 38. Experimental sensitivity to Ae . . . . . . . . . . . . . . . . . . . . . .
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Figure 39. Experimental sensitivity to θe . . . . . . . . . . . . . . . . . . . . . .
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Figure 40. Absolute vibration amplitude of each command . . . . . . . . . . . .
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Figure 41. Planar boom crane model . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 42. Phase shift for an undamped linear system . . . . . . . . . . . . . . .
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Figure 43. Amplitude shift for an undamped linear system . . . . . . . . . . . .
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Figure 44. Boom crane linearized approximation . . . . . . . . . . . . . . . . . .
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Figure 45. Amplitude shift of a boom crane based on varying γ0 and τacc . . . .
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Figure 46. Phase shift of a boom crane based on varying γ0 and τacc . . . . . . .
52
Figure 47. Boom crane simulation subject to nonzero initial conditions . . . . .
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Figure 48. Dynamic crane model . . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 49. Deviation in natural frequency from a simple pendulum . . . . . . . .
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Figure 50. Steady-state payload swing and approximation
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Figure 51. Full nonlinear response vs. superposed approximation . . . . . . . . .
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Figure 52. FFT of full nonlinear response vs. superposed approximation
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Figure 53. Full nonlinear response vs. superposed approximation . . . . . . . . .
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Figure 54. Payload swing angle . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Figure 55. FFT of a ZV shaped response vs. an ZV-IC shaped response . . . . .
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Figure 56. Small scale experimental boom crane . . . . . . . . . . . . . . . . . .
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Figure 57. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Figure 58. Simulated luff angle for experiments . . . . . . . . . . . . . . . . . .
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Figure 59. Simulated and experimental unshaped swing response . . . . . . . . .
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Figure 60. Unshaped trajectory following . . . . . . . . . . . . . . . . . . . . . .
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Figure 61. SI-IC experimental responses . . . . . . . . . . . . . . . . . . . . . .
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Figure 62. ZV-IC experimental responses . . . . . . . . . . . . . . . . . . . . . .
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Figure 63. Experimental and theoretical sensitivity of the SI-IC shaper . . . . .
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Figure 64. Experimental and theoretical sensitivity of the ZV-IC shaper . . . . .
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Figure 65. Block diagram of the proposed control method . . . . . . . . . . . . .
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Figure 66. Actuator effort of an unshaped and ZV shaped command under PD control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Figure 67. ZV shaper with multiple pairs of impulses . . . . . . . . . . . . . . .
76
Figure 68. Step input manipulation with a modified ZV shaper . . . . . . . . . .
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Figure 69. Actuator effort of a shaped S-Curve and ZV-LE shaper . . . . . . . .
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Figure 70. Sensitivity of a ZV shaper, shaped S-Curve, and ZV-LE shaper . . .
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Figure 71. Simple planar trolley crane
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Figure 72. Pole locations for concurrently designed crane controllers . . . . . . .
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Figure 73. Bode plot of proposed control methods . . . . . . . . . . . . . . . . .
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Figure 74. Simulated step response . . . . . . . . . . . . . . . . . . . . . . . . .
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I Introduction The past several decades have seen an explosion in the capabilities of robotic manipulators in a wide range of industries. In parallel with this improvement, substantial research has been conducted in advancing the control methods which are used to precisely position robotic systems such as cranes and serial manipulators. However, significant advancements are needed to accommodate the ever-increasing task complexity for robotic systems. 1.1
Research Goals and Methods This thesis advances the state-of-the-art in the control of flexible systems. The
proposed control techniques are applied to a linearized trolley crane as well as a planar boom crane. Each system exhibits unique characteristics which must be accounted for in the design of their respective controllers. For example, boom crane dynamics are characterized by the oscillation of a pendulum-like payload in the presence of mild nonlinearities due to vertical and rotational motion of the boom. This platform serves as a good benchmark for the capability of the proposed methods to accomodate these moderate nonlinearities. The linearized trolley crane serves as a simple testbed for analyzing the performance of the linear optimal control method presented in this thesis. 1.2
Thesis Contributions In the following chapters, several new techniques for the control of flexible
systems are introduced, representing contributions to the current state-of-the-art. These contributions are: 1. Development of an input shaping control technique which eliminates nonzero initial states in a flexible system – Chapter 3 A thorough analytical and numerical development of the input shaping 1
technique is presented. Consideration is given to factors such as robustness to plant uncertainty and actuator bandwidth limitations. Experimental results support the presented findings. 2. An application of initial condition cancellation to boom crane control – Chapter 4 The initial condition input shaping technique is applied to a boom crane to demonstrate a application of this control method. Experimental results validate the use of initial condition input shaping on a planar boom crane undergoing luff commands. Further analysis is performed on a harmonically excited boom crane to simulate the effects of ocean waves. The proposed control technique is capable of attenuating unforced vibrations in the presence of this external disturbance. 3. Development of a technique to concurrently design input shaping and linear feedback control – Chapter 5 Because input shaping and feedback control have complementary strengths, implementing them both as part of a controller is a reasonable choice. This thesis presents a technique for concurrently designing both the input shaper parameters and feedback control gains in order to yield desired performance. This technique will be shown to result in faster settling time and better low-frequency disturbance rejection than a similarly-tuned controller that was sequentially designed. The next chapter will introduce the motivation and relevant background for this work. Two fundamental control paradigms – command shaping and feedback control – will be discussed and compared. Additionally, methods for intelligently combining these control methods will be presented. In Chapter 3, a specific input shaping control method is proposed as a means to counteract existing oscillation in flexible systems.
2
While not a true feedback control method, initial condition input shaping can facilitate point-to-point motion of human-operated flexible systems in the presence of moderate disturbances. Chapter 4 applies initial condition input shaping to eliminate the radial payload swing of a boom crane through luff commands. This approach is extended to a ship-mounted boom crane to attenuate the free oscillation frequency. Next, Chapter 5 presents a combined control approach through which input shaping is concurrently designed with linear feedback control. The complementary traits of input shaping and feedback control are intelligently combined for a simple crane. Finally, Chapter 6 provides concluding remarks and suggests future extensions of this work.
3
II Background On September 20th , 2017, hurricane Maria made landfall on the island of Puerto Rico, decimating most of its infrastructure. Its remote location mandated the use of sea and air transport to facilitate relief efforts and distribute emergency aid. However, due to the unprecedented damage to the internal infrastructure of the island, such as road blockages like the one shown in Figure 1, shipping to large, deep-water ports such as San Juan did little to provide immediate relief. Although supplies were quickly shipped to these hubs, no means existed to truck them inland. As part of the U.S. military relief effort, an amphibious assault ship, the U.S.S. Kearsarge, shown in Figure 2, was activated. This vessel has the capability to offload supplies and equipment in shallow-draft vessels. The transferring of cargo from large, deep-water vessels to smaller, lighter craft has been pursued by multiple research groups to facilitate access to ports which may be unable to accommodate large ships. In a disaster relief effort such as the response to Hurricane Maria, such access could have allowed necessary supplies to be brought to
Figure 1. Street blocked by downed trees on September 20, 2017 [1]
4
Figure 2. U.S.S. Kearsarge amphibious assault ship [2]
Figure 3. Mobile harbor concept [3]
citizens instead of sitting in the large port of San Juan, waiting for internal infrastructure to be restored. The concept of a mobile harbor capable of unloading and ferrying cargo from large vessels to shallow harbors is one which gained significant research attention in South Korea due to concerns that the increasing size of cargo ships may exceed the capacity of its ports. A smaller vessel – the “mobile harbor” – would meet the container ship in deeper water and utilize a crane to unload the necessary cargo before bringing it into port, as shown in Figure 3 [3]. The proposed control systems for these
5
cranes favor nonlinear feedback methods [4–7]. For example, a sliding-mode controller has been developed based on payload accelerometer data [3]. In another method for acquiring payload motion, machine vision algorithms can be used [8]. If the system states are fully known, a Lyapunov control law can stabilize the motion of the crane spreader [9]. In the more complex, three-dimensional case, full nonlinear control has been used to stabilize the mobile harbor crane system [10]. Optimal linear control has also been proposed to control the offshore crane system [5, 6]. Similar technology has also been pursued by the U.S. Navy and other research entities to transfer cargo between ships while at sea [11–13]. This type of cargo transfer typically involves the use of boom cranes, which exhibit significantly different dynamics from bridge and gantry cranes. A planar, linear analysis has been used to limit the swing angle [12, 14, 15]. Further analysis accounting for ship orientation and ocean dynamics has also been presented [16]. Finally, a nonlinear sliding-mode control law has been developed to enable precise payload tracking for a planar boom crane [17, 18]. 2.1
Input Shaping Input shaping is a command shaping method wherein a series of impulses is
designed to eliminate oscillatory modes of a flexible system. This sequence of impulses, when convolved with an arbitrary reference command, results in a command profile which significantly reduces the excitation to the designed mode. This process is demonstrated in Figure 4. Originally presented as “Posicast Control” in 1957 [19], input shaping has been implemented in a wide range of mechanical systems since the 1990’s [20, 21]. It has been shown to be exceptionally useful in human-operated systems such as cranes, where feedback control methods can frustrate the operator [22]. Because human operators essentially serve as feedback controllers in an attempt to drive the system to a desired
6
Command
Shaped Command
Response
* 0
1 Time (s)
Shaped Response
0
2
0
1 Time (s)
2
Figure 4. Convolution of an input shaper with a reference command resulting in zero vibration
position, adding another feedback controller can cause the crane to behave in unexpected ways, degrading performance. Furthermore, because input shaping is an open-loop method, feedback sensors are not required. This allows for cost-efficient and simple implementation. In many cases, accurate sensing of the desired states can pose a nontrivial problem for control formulation [23]. For instance, computer vision techniques are often employed to sense crane payload deflection. However, given that many crane systems exhibit doublependulum dynamics and cranes can be employed in a wide range of environments and lighting conditions, such measurement is impractical in many cases. The efficiency and effectiveness of input shaping has made it an excellent control method in a wide range of systems and industries. As mentioned, its use in the control of cranes is thoroughly documented [24–42]. Input shaping has also been used in the orientation of flexible spacecraft [43–49], coordinate measurement machines [50, 51], high-speed positioning stages [52], and flexible manipulators [53–57]. This technique has also been employed to compensate for nonlinearities due to friction [58], on multimodal systems [24, 59], to systems with limited actuator bandwidth [60], and to limit perceived overshoot in human-operated systems [61].
7
0.6
A1
Position
0.4
A1 Response A2 Response Total Response
A2
0.2 0 -0.2 -0.4 0
0.5
1
1.5
2
2.5
3
Time Figure 5. Impulse response cancellation to form a Zero Vibration input shaper
2.1.1
Input shaper design. The input shaping process is based on the linear
superposition of impulse responses to become self-canceling, as shown in Figure 5. In this plot, an initial impulse, A1 , causes an oscillatory response shown in blue. By choosing an appropriate amplitude and time to apply impulse A2 , the superposition of both resulting responses yields zero residual vibration. In the frequency domain, input shaping can be formulated as a series of impulses which cancels the poles of a second-order plant. The Laplace transform of an under-damped, second-order system is: Y (s) =
s2
1 , + 2ζωs + ω 2
(1)
where ζ is the damping ratio and ω is the natural frequency. An input shaper is formulated by creating an impulse sequence which, when multiplied by (1), cancels its poles. In this simple case, the poles of (1) are given by: s0 = −ζω ± jω
p
1 − ζ 2.
(2)
To eliminate residual vibration, a series of impulses must therefore be designed to place zeros at s = s0 . In order to avoid any unnecessary delay in the rise time of the shaped command, the first impulse should occur at time t = 0. In the context of operator-inthe-loop control, this design choice ensures that the system immediately responds to a 8
user input. Because a single impulse will result in an oscillatory response, at least one time-delayed impulse is required to cancel the initial oscillation. The simplest, twoimpulse input shaper is expressed by: M (s)|s=s0 = A1 + A2 e−s0 t2 = 0.
(3)
If the set-point of the shaped command should be the same as the unshaped command, the sum of the impulses should be equal to one: A1 + A2 = 1.
(4)
Given (3) and (4), the impulse amplitudes and times can be solved analytically to form the Zero Vibration (ZV) input shaper: 1 Ai K+1 ZV = = 0 ti
K K+1
ω
√π
where:
K = exp
−ζπ p 1 − ζ2
1−ζ 2
!
,
.
(5)
(6)
Alternatively, an input shaper can be formed in the time domain. The residual vibration amplitude of a second-order system subject to a series of impulses is given by:
where:
p ω A= p e−ζωtn [C(ω, ζ)]2 + [S(ω, ζ)]2 , 1 − ζ2 C(ω, ζ) = S(ω, ζ) =
n X
i=1 n X i=1
Ai e
ζωti
p 2 cos ω 1 − ζ ti ,
p Ai eζωti sin ω 1 − ζ 2 ti ,
(7)
(8) (9)
and Ai and ti are the ith impulse amplitude and time, respectively. When (7) is normalized by the response amplitude of a unity magnitude impulse at time t = 0, ω A↑ = p , 1 − ζ2 9
(10)
the resulting equation gives percentage residual vibration of a shaped command relative to an unshaped command: P RV (ω, ζ) =
A × 100%. A↑
(11)
An important performance characteristic of input shapers is robustness to parametric uncertainty. To determine this metric, (11) is evaluated at frequencies and damping ratios away from the modeled values. Practically speaking, errors in natural frequency are much more significant than those in damping ratio and are therefore most frequently considered when analyzing shaper robustness [62]. While the ZV shaper is the shortest duration positive amplitude input shaper, it is also the least robust to deviations in natural frequency. A number of methods have been developed to create input shapers which exhibit greater robustness to plant uncertainty [62, 63]. One such approach is to take partial derivatives of (11) with respect to ω and force these equations to equal zero [20]. The result is the Zero Vibration and Derivative (ZVD) shaper: 1 Ai 1+2K+K 2 ZVD = = 0 ti
2K 1+2K+K 2
K2 1+2K+K 2
√π ω 1−ζ 2
√2π ω 1−ζ 2
.
(12)
Because some level of modeling error is expected to exist, relaxing the vibration
constraint to some tolerable level, Vtol , at the designed parameters can yield an input shaper which provides acceptable vibration reduction over a maximum range of frequencies [64]. Maximizing the range of variation in natural frequency over which (11) is less than Vtol results in the Extra Insensitive (EI) shaper [65]: A3 A1 1 − (A1 + A3 ) EI = √2π 0 t2 2 ω
10
1−ζ
(13)
0.40 0.29 0.06
Figure 6. Sensitivity plot of a ZV, ZVD, and EI shaper where, A1 = 0.24968 + 0.24962Vtol + 0.80008ζ + 1.23328Vtol ζ + 0.49599ζ 2 + 3.17316Vtol ζ 2 , (14) A3 = 0.25149 + 0.21474Vtol − 0.83249ζ + 1.41498Vtol ζ + 0.85181ζ 2 − 4.90094Vtol ζ 2 , p ω (1 − ζ 2 ) t2 = 0.49990 + 0.46159Vtol ζ + 4.26169Vtol ζ 2 + 1.75601Vtol ζ 3 2π 2 2 2 2 3 +8.57843Vtol ζ − 108.644Vtol ζ + 336.989Vtol ζ ,
(15)
(16)
and Vtol is frequently chosen to be 5%. 2.1.2
Tools for evaluating input shaper performance. A number of
tools exist which facilitate quantifying the performance of an input shaper. When determining the robustness to parametric insensitivity, a sensitivity curve is commonly used [62]. This plot illustrates the residual vibration as a function of normalized frequency error
ω , ωm
where ωm is the modeled natural frequency of the system. Such a
curve is shown in Figure 6, which compares the ZV, ZVD, and EI shapers. The variation in normalized frequency across which the input shaper yields vibration below Vtol is called insensitivity, I. In this plot, the performance increase of the various 11
y
v
Figure 7. Residual vibration summation of vector diagram impulses
robustness methods is clearly demonstrated. While the duration of the ZV shaper is shortest at
τd , 2
its Insensitivity of I = 0.06 is the worst of the three. The ZVD and EI
shapers have the same duration, τd , but their Insensitivities are I = 0.29 and I = 0.40, respectively. Vector diagrams can also be used to graphically analyze input shaper performance [65]. Because input shapers are a sequence of impulses, their summation can be visualized as a series of vectors, as shown in Figure 7. Here, the first impulse, A1 , is separated from the second impulse, A2 , by the angle, θ2 = ωt2 . The residual vibration amplitude of the resulting shaper, AR , is proportional to the sum of the two impulses. 2.1.3
Specified insensitivity input shaping. The most general method of
generating an input shaper is through Specified Insensitivity (SI) [66]. With SI shaping, input shaper robustness is defined by specifying a desired range of natural frequencies over which the shaper will limit vibration below Vtol and finding impulse times and amplitudes of the minimum-time impulse sequence resulting in the desired level of robustness. The ZV and EI shapers are specials case of this method where I = 0.06 and I ≈ 0.4, respectively.
12
Limit Vibration at Specific Frequencies in the Desired Range
Figure 8. Vibration suppression through frequency sampling
SI shaping is accomplished by enforcing the vibration constraint (11) to less than Vtol at a finite number of frequencies. This process is shown in Figure 8. By discretely sampling frequencies in this way, the computational burden of performing the resulting optimization is reduced. Because (11) is continuous everywhere, a moderate number of sampling frequencies, typically n = 50 in practice, is sufficient to ensure a residual vibration below the desired level. Formally, a positive amplitude SI shaper is found by performing the following numerical optimization: minimize t
tn
subject to P RV (ωk , ζ) ≤ Vtol , ∀ωk ∈ [ω1 , ω2 ], Ai > 0, n X
(17)
Ai = 1,
i=1
where ω1 and ω2 are the frequency bounds dictated by the designer, and ωk is the k th sampled frequency in this range. Note that the transcendental nature of (11) results in a non-convex optimization problem with nonlinear constraints. While (17) will yield the Karush-Kuhn-Tucker
13
Figure 9. Block diagram of feedback control necessary conditions for a local optima, global optimality is not assured. Therefore, it is important to choose a reasonable first guess when performing this optimization. Because the EI shapers give closed-form, time-optimal solutions for given insensitivity levels [62, 64], they can be used to form the initial guess for SI shapers. 2.2
Feedback Control Another method of reducing vibration is to compare the current system states to
the desired states and make corrections to the control signal based on weights given to the error of each state. Generally known as feedback control, the most basic implementation is shown in Figure 9. The desired states, xd , are passed into the feedback loop where the controller, C, modifies the command signal to the plant, G. The output from the plant is then compared to the desired states. The resulting error signal is fed through the controller until the system states match the desired states. This control loop requires sensors which allow for the states to be measured. However, an intelligently designed feedback control system will be capable of rejecting disturbances to the system due to its closed-loop nature. A linear time invariant (LTI) system is given by: x˙ = Ax + Bu,
(18)
where A, B and x are the state transition matrix, input matrix, and current states, respectively. A feedback control law dictates the magnitude of the input, u, is based on the difference between the desired and current system states: u = K(xd − x), 14
(19)
where K is the gain matrix which weights the magnitude and direction of the input signal according to the deviation of the state variables, x, from the desired state vector, xd . An easy-to-tune, linear control method for an LTI system is Linear Quadratic Regulation (LQR). This control method finds optimal control gains, K, which minimize the cost function: J(u) =
Z
∞
0
xT Qx + uT Ru dt,
(20)
where Q and R are the state and input weighting matrices, respectively. For a fully observable and controllable system, large values in the Q matrix penalize the states for deviating from their desired values, while large values in the R matrix penalize high actuator effort. 2.3
Comparison of Input Shaping and Feedback Control Input shaping and feedback control represent two fundamentally different
control approaches. While input shaping modifies a command signal to produce self-canceling vibrations, feedback control generally relies on sensor data to produce control inputs which drive the system to a set of desired states. When designing a controller, a number of response characteristics must be balanced against one another based on the desired performance of the system. Specifically, rise time, settling time, overshoot, actuator requirements, and disturbance rejection must all be considered. For either feedback control or input shaping alone, compromises must be made. 2.3.1
Comparison of settling time. One distinction between input shaping
and feedback control is the time it takes to eliminate oscillation in the system. Because feedback control generally creates a damping effect on the output, its settling time is related to the time constant of the system by: ts,f eedback
p ln 0.05 1 − ζ 2 , = −ζω 15
(21)
5% Settling Time (s)
Feedback ZV Shaping
101
100 0.2
0.4 0.6 Damping Ratio ζ
0.8
Figure 10. Settling time of shaping versus feedback control methods
where ts,f eedback is the 5% settling time. As a result of this relationship, feedback control techniques rely on creating highly-damped modes to improve settling time. This relationship starkly contrasts against the settling time for an input shaped command. The Zero Vibration shaper has a duration of
τd . 2
Therefore, this time readily
serves as an approximation of the settling time of a shaped command. Although increasing robustness through ZVD or EI constraints will modify this relationship, the ZV shaper yields a fair comparison of non-robust control methods. The settling time of a ZV shaped command is: ts,shaping =
π p . ω 1 − ζ2
(22)
Note that this duration explicitly refers to settling time from command-induced vibration. In contrast, (21) refers to general damping of an oscillatory response, independent of a command signal. Comparing (21) to (22), it is clear that ts,f eedback and ts,shaping benefit from high damping and low damping, respectively. This comparison is quantified in Figure 10. As shown, the two control methods have an inverse relationship to the system damping ratio. Furthermore, there exists a point at ζ ≈ 0.75 where using input shaping increases the settling time of the system. Recalling that this graph more generally applies to any
16
m Figure 11. Planar crane model
damped system, it provides the further insight that input shaping should only be used for lightly-to-moderately damped modes of oscillation. 2.3.2
A simple example. To show a comparison of input shaping and
feedback control, consider the simple system shown in Figure 11. This represents the pendulum dynamics of a planar bridge crane, where the control input x moves the attachment point of the rigid, massless cable of length l. The payload is modeled as a point mass m. After linearizing about θ θ˙ 0 1 g θ¨ − 0 l = x˙ 0 0 0 0 x¨
= 0, its state space representation is given by: θ 0 0 0 ˙ 1 0 0 θ l (23) + u(t). 0 1 x 0 0 0 x˙ 1
If the position of x can be completely controlled, as (23) indicates, the input to
the system can be governed by a velocity input, x. ˙ This method of control is commonly used in cranes [67]. If a maximum acceleration and velocity, Amax and Vmax , are specified, the resulting point-to-point command can be presented as a trapezoidal 17
velocity command. If no feedback control is enabled, and the reference trapezoidal velocity command is unshaped, a large amount of residual vibration will persist after the command. However, by generating an input shaper and applying it to the unshaped reference command, the residual vibration can be eliminated. To create an input shaper for (23), the natural frequency and damping ratio must be determined. The natural frequency of a simple pendulum is given by the equation: ω=
r
g . l
(24)
If a controller which eliminates command-induced vibration is desired, and ω is accurately known, a single-mode ZV shaper will sufficiently perform the desired task. Given ω from (24) and ζ = 0, the ZV shaper for this system is: 1 1 2 Ai 2 = . ZV = q 0 π gl ti
(25)
To move x in a time-optimal manner subject to acceleration and velocity
constraints, a bang-coast-bang acceleration command can be issued. This command type accelerates the system to maximum allowable velocity, Vmax , by accelerating at the maximum allowable value, Amax . In order to stop at the desired location, maximum negative acceleration brings the system back to rest. The resulting velocity profile is trapezoidal. The response of the system subject to a ZV-shaped bang-coast-bang acceleration command is given in Figure 12. The unshaped response is also given as a comparison. The unshaped command excites vibration in the system. Because the system is undamped this oscillation will continue indefinitely. On the other hand, the ZV-shaped command excites no residual vibration. The input shaper also excites less transient vibration.
18
20
Unshaped ZV
Angle (deg)
10 0 −10 −20
0
1
2
3 Time (s)
4
5
Figure 12. Crane payload response
x Velocity ( m s)
Unshaped ZV
0.4
0.2
0.0
0
1
2
3 Time (s)
4
5
Figure 13. Crane velocity command
The velocity commands for both cases are shown in Figure 13. Here, a noteworthy drawback of input shaping is clearly seen. The unshaped command immediately accelerates the system to its maximum velocity, 0.5 m s , while the ZV shaper breaks the reference acceleration command into two pieces. To cover the same distance, the ZV shaper results in a command duration increase of τ2 , or t2 from (25). This increase in command duration is usually a small price to pay for the drastically reduced vibration in the payload response. Clearly, input shaping offers significantly better performance when compared to
19
no control at all. This comparison is relevant when considering a system with no sensor feedback to mitigate unwanted vibration in the absence of input shaping. Because the sensors and additional engineering time required to implement such control cause increased complexity and cost for the system, this shaped-vs-unshaped comparison holds valid for systems where state feedback is not readily available. However, it is important to quantify the performance between input shaping and feedback control if such sensing is available. To demonstrate the comparison between feedback control and input shaping, an LQR controller serves as a benchmark for performance. Instead of the bang-coast-bang command from the previous example, a step in the desired position of x is given for the reference command. This change in reference command generation allows for direct comparison of input shaping to feedback control. To drive the system to the desired states, a shaped step command is implemented and proportional control on the trolley position, x, is used for the input shaped response. The resulting gains for the input shaped case are:
T
K= 0 0 1 0
,
(26)
where the unity magnitude gain on the trolley position, x, is sufficient to follow the desired shaped trajectory. Note that implementing this control over the trolley position would not require any sensing beyond the previously assumed exact control over the location of this point. However, this formulation results in oscillatory dynamics of the previously rigid mode. The closed-loop state transition matrix is given by: Acl = A − BK. By evaluating the closed-loop eigenvalues of this matrix, the new dynamics can be determined. An LQR controller will be used as the benchmark for control performance comparison. Although LQR yields a mathematically optimal gain matrix, K, for a
20
(27)
⌘ 2
20 10 Real ( ⇣!)
⇣ p Imaginary ! 1
30
⇣2
Open Loop Shaped LQR
0
2
Figure 14. Comparison of pole locations
given cost function, the state and input weights do not represent any true physical optimality. In order to generate desired system performance, the state and input weights must be translated into real performance metrics such as actuator effort, phase margins, rise time, and settling time. The 1000 0 Q= 0 0
state weight matrix used is: 0 0 0 10 0 0 , 0 1000 0 0 0 0
(28)
and the input weight is R = 0.01. Intuitively, this choice of weights strongly penalizes ˙ The both θ and x from deviating from their desired values and moderately penalizes θ.
resulting gains are:
T
KLQR = 225.50 −77.52 316.23 123.17
.
(29)
When the closed loop pole locations of the LQR controlled system and the input shaped system under proportional control are compared, the benefit of feedback control is clear. The pole locations for (23) are shown in Figure 14. This plot graphically demonstrates the dynamic behavior of the system under the different control methods. Pole locations in the negative real plane are asymptotically stable, while poles along the 21
imaginary axis are marginally stable. As shown, proportional control simply adds an oscillatory element to the rigid mode. This plot also shows the open-loop pole locations for reference. Note that the point at (0, 0) represents the rigid mode without control. When proportional control is added, the rigid mode develops a magnitude on the imaginary axis, indicating that this mode develops oscillatory behavior, while the flexible mode remains unchanged. Importantly, because the system is modeled without damping, each shaped pole lies along the imaginary axis. As a result, these poles are not asymptotically stable. Adding the LQR controller moves the poles into the negative real domain, resulting in increased stability and additional damping to the response. The closed-loop pole locations for the LQR controller changes the characteristics of the flexible and rigid modes. The existence of multiple modes requires a modified approach to input shaping. Because a closed-form ZV, ZVD, or EI shaper is only designed for one natural frequency and damping ratio, successfully eliminating vibration from multiple modes requires convolving multiple input shapers. Although robust SI shapers could also provide a desired level of vibration, a two-mode convolved ZV shaper provides a simple demonstration for this example. For the system under proportional control, the natural frequencies are ωi = (1, 3.13)Hz. The resulting single-mode ZV shapers are: 0.50 0.50 ZV1 = , 0.00 3.14 and
0.50 0.50 ZV2 = . 0.00 1.00
Convolving (30) and (31) results in a two-mode ZV shaper: 0.25 0.25 0.25 0.25 ZV2mode = . 0.00 1.00 3.14 4.14 22
(30)
(31)
(32)
Percentage Vibration
ZV1 ZV2
ZV2mode
100
50
0 0.0
0.2
0.4 0.6 Frequency (Hz)
0.8
Figure 15. Sensitivity curve of a multi-mode shaper
The vibration-reducing capability of the shapers given in (30), (31), and (32) is shown in Figure 15. Each single mode shaper eliminates ω1 and ω2 , respectively, and the convolved shaper combines the vibration reduction of both shapers and completely eliminates both modes. This method of forming multi-mode input shapers is not time-optimal [68], but provides a simple example for this case. A step response comparison of the two control methods is given in Figure 16. The large gains result in a rapid rise time for the LQR controlled response. Although the chosen gains result in some overshoot, this response settles before the shaped response. A force disturbance of 20N at t = 7.5s is applied to the payload, resulting in the brief correction in the x coordinate of the LQR controlled response. Because the shaped controller has no means of detecting the disturbance to the payload, no correction is issued to its position. The swing angle response of the payload for the given control methods is presented in Figure 17. The aggressive LQR gains result in a large swing angle at the beginning of the command, which is quickly eliminated to achieve a settling time equal to the shaped command. Although the input shaping controller results in nearly identical settling time for the payload and dramatically reduced transient vibration, the
23
1.5 Angle (deg)
Position x (m)
LQR
ZV
40
1.0 0.5 0.0 0.0
ZV LQR
2.5
5.0
7.5 10.0 Time (s)
12.5
Figure 16. Step response of a crane under LQR and ZV-shaped control
20 0 −20
15.0
0.0
2.5
5.0
7.5 10.0 Time (s)
12.5
15.0
Figure 17. Payload swing response of a crane under LQR and ZV-shaped control
lack of feedback on the flexible mode results in an inability to dampen the oscillation resulting from the force disturbance at t = 7.5s. 2.4
Combining Input Shaping and Feedback Control While the input shaping and LQR controllers from the previous example are not
optimized for performance, some intuition for the behavior of these control methods can be gained. The input shaping method is an excellent means to form rest-to-rest commands which do not excite oscillatory motion. However, its open-loop nature strictly prohibits it from from being capable of rejecting disturbances or parametric deviations in the plant. The disturbance response in Figure 17 demonstrates this deficiency. This open-loop design is demonstrated in Figure 18. If disturbance rejection is desired, feedback methods must be used. However, state feedback control methods are suboptimal for performing rest-to-rest motion. As Figure 16 shows, the LQR controller excites unwanted oscillation which must be damped before the system reaches its settling point. From these examples, an inherent performance trade-off is evident between rapid, efficient commanded motion and disturbance damping capabilities for input shaping and feedback control. However, an intelligent combination of the two methods should result in a controller with better open and closed-loop performance than either method alone.
24
Figure 18. Block diagram of an input shaper
Figure 19. Block diagram of a closed-loop input shaper
2.4.1
Closed-loop input shaping. When combining input shaping and
feedback control, the shaper is sometimes located within the feedback loop [35, 69–75], as shown in Figure 19. Commonly called closed-loop input shaping, this approach permits the shaper to act directly on the plant. This implementation ensures that the control signal is always fully shaped. However, closed-loop input shaping mandates that the control gains be chosen to ensure stability subject to the time-delay caused by the input shaper [76]. The effectiveness and stability of such methods has been thoroughly analyzed in the application of closed-form input shapers used within the control loop of PD and PID control [71, 73, 77]. Additionally, a similar approach has been analyzed in the use of H∞ control and compared against the combination of input shaping and LQ control [72]. 2.4.2
Outside-the-loop input shaping. In each cited example of
closed-loop input shaping, the input shaper and feedback gains were designed sequentially. Recent work indicates that placing an input shaper outside the feedback loop and concurrently designing the input shaper with the feedback gains shows promise as a control approach [78–81]. However, [78, 79], and [80] only consider the control of a point mass. A flexible system was considered in [81], where robustness to parametric uncertainty was improved by utilizing full-state feedback control optimized
25
Figure 20. Block diagram of an input shaper outside a control loop
concurrently with input shaping. 2.5
Conclusion This chapter has presented the background and motivation for the contributions
made in this thesis. Although a large breadth of research has been conducted in the input shaping and feedback control of flexible systems, significant advancements are still needed. While the synthesis of command shaping and feedback control are individually well-understood concepts, elements from each method can be used to compensate for the shortcomings of the other in new ways to improve system performance. In the following chapters, this topic will be thoroughly addressed through advancements in both input shaping and combined control techniques.
26
III Initial Condition Input Shaping The vast majority of input shaping research focuses on motion where the system begins from rest. If a system begins with nonzero initial states, traditional input shaping is unable to eliminate these initial conditions. Recently, some research has been performed to create impulse-based controllers which counteract nonzero initial conditions. The general approach for this control can be visualized in a vector diagram such as Figure 21 [65, 82]. Here, the initial condition is modeled as an impulse of magnitude A0 and phase θ0 = ωd t0 . The the sum of the controller impulses, A1 and A2 , yields a resultant impulse, As , which is equal in magnitude and directly out of phase with A0 , resulting in zero residual vibration. This approach has been proposed to reject vibration due to: a change in the desired setpoint of a time-optimal trajectory for a flexible system [82]; a known force on a long-reach telescopic handler [83]; the transient sway of a harmonically excited boom crane [84]; and the point-to-point motion of cranes at nonzero initial conditions [85–87].
Figure 21. Initial condition impulse cancellation
27
The general procedure for constructing these impulse sequences is called initial condition input shaping. The following chapter describes the analytical development of initial condition input shaping and provides simulated and experimental validation for this approach. The next section provides a computationally efficient solution in the frequency domain suitable for real-time implementation of initial condition shaping. Then, Section 3.2 presents a formulation in the time domain. This section also proposes a Specified Insensitivity method of improving robustness to parametric uncertainty. Actuator bandwidth constraints create nonlinearities in system responses which must be quantified in impulse-based control such as Initial Condition (IC) shaping. Section 3.2.1 provides a discourse on modifications to the IC shaping technique based on known actuator bandwidth limitations. A holistic evaluation of shaper robustness is presented in Section 3.2.2. Finally, example simulation results are provided in Section 3.3, followed by experimental validation in Section 3.4. 3.1
Frequency-Domain Design A simple, second order system is pictured in Figure 22. The position of mass m
is given by x and is based on the force interactions between the position input, y, and the connected spring, k, and damper, c. In order perform the desired motion while eliminating the initial conditions, a sequence of impulses will be used. If this system
x
y k m
c
Figure 22. Simple flexible system
28
exhibits nonzero initial displacement, x0 , and velocity, x˙ 0 , its frequency domain formulation is given by: Y (s) =
1 M (s), s2 + 2ζωs + ω 2
(33)
where M (s) = x0 s + x˙ 0 + ω 2 (A1 + A2 e−t2 s ),
(34)
and t1 = 0. The impulse amplitudes, A1 and A2 , and second impulse time, t2 must be solved to eliminate the initial conditions. The pole cancellation of (33) dictates that its zeros are located by: M (s)|s=s0 = s0 x0 + x˙ 0 + ω 2 (A1 + A2 e−s0 t2 ) = 0, where s0 = −ζω ± jω
(35)
p 1 − ζ 2.
If the shaped command and the reference command have equal endpoints, the sum of the shaper amplitudes must equal one: A1 + A2 = 1.
(36)
These amplitudes can be found by solving: 1 A1 1 1 = −s0 t2 − (s0 x0 + x˙ 0 ) 12 A2 1 e ω resulting in impulse amplitudes of: A1 = A2 =
C −1− 1+
ζx0 ω
ζx0 ω
−
x˙ 0 ω2
C+ S−
x0 ω
p
C2 + S2 p + ωx˙ 02 C + xω0 1 − ζ 2 S , C2 + S2
,
1 − ζ2 S
(37)
,
(38) (39)
where C = 1 − eζωt2 cos ωd t2 ,
(40)
S = eζωt2 sin ωd t2 .
(41)
29
In a damped system, numerical methods must be used to solve for the second impulse time, t2 , which satisfies the constraint that the imaginary component of (37) be zero. For the undamped case, the Zero Vibration, Initial Condition (ZV-IC) shaper can be solved analytically [88]:
where:
1 Ai 2 (1 − α) ZV-IC = = 0 ti α=
3.2
1 (1 2
2 ω
tan−1
+ α)
ω 2 +x˙ 0 x0 ω
,
x˙ 0 x20 . + ω 2 ω 2 − x˙ 0
(42)
(43)
Time-Domain Design An identical solution can be reached using a time-domain formulation. In this
approach, the initial conditions are modeled as an impulse and directly incorporated into the residual vibration equation. The amplitude and phase of the impulse representing the initial conditions are: v u u A0 (x0 , x˙ 0 ) = tx20 + θ0 (x0 , x˙ 0 ) = tan−1
!2 + ζx0 p , x0 1 − ζ 2 ! p x0 1 − ζ 2 . x˙ 0 + ζx0 ω x˙ 0 ω
(44) (45)
The components of the initial condition impulse are: C0 = C0 (ω, ζ, x0 , x˙ 0 ) = A0 cos(−θ0 ),
(46)
S0 = S0 (ω, ζ, x0 , x˙ 0 ) = A0 sin(−θ0 ).
(47)
The resulting residual vibration amplitude due to the input shaper as well as the initial conditions can then be written as: p ω V (ω, ζ, x0 , x˙ 0 ) = p e−ζωtn [C0 + C]2 + [S0 + S]2 . 1 − ζ2 30
(48)
In order to normalize the residual vibration by the vibration amplitude resulting from a unity magnitude impulse at t = 0, the effects of the initial condition must be incorporated into the normalization term. Specifically: ω V↑ (ω, ζ, x0 , x˙ 0 ) = p e−ζωtn 2 1−ζ
q (1 + C0 )2 + S02 .
(49)
The percentage residual vibration resulting from the input shaper impulses is therefore: p [C0 + C]2 + [S0 + S]2 p P RVIC = × 100%. (50) (1 + C0 )2 + S02 . The time-domain solution for the ZV-IC shaper can then be solved through
numerical optimization by: minimize t
tn
subject to P RVIC = 0, Ai > 0, n X
(51)
Ai = 1.
i=1
The Specified Insensitivity method can be used to generate IC shapers which are robust to deviations in natural frequency. In order to generate such Specified Insensitivity, Initial Condition (SI-IC) input shapers, negative amplitudes must be allowed. A property of negative amplitude input shapers is that they can generate larger amplitude residual vibrations than their positive amplitude counterparts [63]. When an IC shaper is desired, this trait promotes a larger available solution space for the input shaper which yields a resulting amplitude A0 while maintaining the desired
31
robustness. As a result, robust input shapers which cancel a wide range of initial conditions can be generated. The expression for an SI-IC shaper is therefore: minimize t
tn
subject to P RVIC ≤ Vtol , ∀ωk ∈ [ω1 , ω2 ], |Ai | ≤ 1, n X
(52)
Ai = 1,
i=1 k−1 X A i ≤ 1, ∀k ∈ [0, n]. i=1
Although the robust SI-IC shaper solution is presented here in the time domain, an analogous formulation could be presented in the frequency domain through minimax optimization [64]. 3.2.1
Incorporating actuator limitations. In the previous analysis, it is
assumed that the actuators can exactly follow the commands resulting from the convolution of the reference command with the shaper impulses. Obviously, no actuator can exert an infinite force, so pulses of finite amplitude must be used. Although this distinction is unnecessary for typical input shaping implementation due to the principles of linearity, the differences between the impulse and pulse response must be quantified for an initial-condition-canceling input shaper. Figure 23 demonstrates the phase and amplitude shifts, and δ respectively, of a pulse response compared to an impulse response. These shifts affect the performance of an IC shaper as demonstrated on the vector diagram in Figure 24. The phase shift, , corresponds to an apparent lag in the resulting input shaper impulse while the amplitude shift, δ, corresponds to a modified vector amplitude. The phase and amplitude shifts of the pulse versus impulse response will depend on the system natural frequency, damping ratio, and normalized acceleration time,
32
Position (m)
10
Pulse
Impulse
5 0 5 0.00
0.25
0.50
0.75 1.00 Time (s)
1.25
1.50
Figure 23. Response characteristics of an impulse vs a pulse
Figure 24. Vector Diagram of an input shaper subject to actuator limitations
defined by: τacc =
tacc . τd
(53)
These relationships can be determined by iteratively fitting a shifted response, ω y = δp e−ζωt sin(ωd t − ), 2 1−ζ
(54)
to the numerically integrated pulse response of a second-order system. Figures 25 and 26 show the phase and amplitude shifts, respectively, as a function of normalized 33
Phase Shift ✏
5
5
4
4 3
3
2
2
1
1
1.0
0.8
0.6
⌧acc
0.6 0.4
0.4 0.2
0.0
0.2 0.0
⇣
Figure 25. Phase shift of a pulse response relative to an impulse response
5
Amp. Shift
5
4
4 3
3
2
2
1
1
1.0
0.8
0.6
0.4
⌧acc
0.2
0.4 0.0
0.2 0.0
0.6
⇣
Figure 26. Amplitude shift of a pulse response relative to an impulse response acceleration time, τacc , and damping ratio, ζ. In Figure 25, a linear relationship between and τacc is evident for each damping ratio value. This slope increases as a function of ζ. Figure 26 shows that for low damping levels, the amplitude shift trends toward zero as τacc increases. Conversely, the amplitude shift becomes extremely large when both the damping ratio and normalized acceleration time are high. The data in this figure are clipped at δ = 5 to improve clarity. This trend towards high δ values occurs because the phase shift of the pulse command delays the response beyond the point at which the impulse response is settled.
34
Because the pulse response exhibits these shifts, an IC shaper must be designed to cancel the shifted initial conditions. These new values are: A0 sin(−θ0 + ), δ A0 = ωd cos(−θ0 + ). δ
y,δ =
(55)
y˙ ,δ
(56)
These shifted values are substituted into (44) and (45) in order for the corrected shaper to be determined. 3.2.2
Robustness considerations. The effectiveness of an input shaper is
typically measured by using sensitivity curves [62]. These plots show the percentage residual vibration of an input shaped response normalized by that of a unity magnitude impulse at time t = 0. For a system at rest, this expression simplifies to (7). Because the system under consideration exhibits nonzero initial conditions, the sensitivity curve takes a form similar to (50). Shaper performance in this case will be degraded by inaccurately timing the shaped command or designing for an incorrect initial condition amplitude. Therefore, the full expression of the sensitivity of the input shaper is: q [C0,e + C(ω, ζ))]2 + [S0,e + S(ω, ζ)]2 , VIC = q 2 2 [δ cos( − θe ) + C0,e ] + [δ sin( − θe ) + S0,e ]
(57)
where:
A0 + Ae cos(θ0 + ), δ A0 + Ae = sin(θ0 + ), δ
C0,e =
(58)
S0,e
(59)
and Ae and θe are errors in the designed shaper amplitude and phase, respectively. Equations (58) and (59) represent the oscillation amplitude due to the initial condition, subject to these measurement errors. An initial condition input shaper can be made robust to deviations in ω and ζ, but not Ae and θe . However, quantifying the degradation of shaper performance subject to these errors provides valuable insight into how accurate the state estimation must be for this technique to function properly. 35
Because (57) is normalized by the response amplitude of a unity magnitude impulse at t = 0, subject to nonzero initial conditions, the exact shape of the shaper sensitivity curve will depend on the initial conditions. However, the sensitivity function is smooth and continuous near the constraints in most cases. For (57) to be discontinuous, the following conditions must be met: A0 = δ 2 ,
(60)
θ0 = 0.
(61)
While problematic for graphically demonstrating IC shaper sensitivity, these conditions indicate that the unshaped input would completely eliminate the initial conditions. Barring these exact initial conditions, for which a shaper would be unnecessary, (57) is continuous everywhere and will therefore be well-behaved in regions near constraints. Regardless of the enforced robustness constraints, IC shapers exhibit nearly identical sensitivity to errors in initial condition estimation. This performance measure can be quantified on a sensitivity plot like the one shown in Figure 27. Here, perfect modeling of the system dynamics, ω and ζ, is assumed. The residual vibration is zero at exactly one point, and quickly degrades as errors in initial condition phase and amplitude increase. For a shaper with Specified Insensitivity constraints, this curve will be shifted slightly off-center due to the non-zero vibration resulting at the designed conditions. Figure 28 demonstrates the combined sensitivities of a ZV-IC shaper to initial conditions and normalized frequency error. Because the shaper is designed to result in responses with zero residual vibration at the modeled frequency, a single point exists which minimizes this sensitivity along each dimension. Taking a cross-section normal to the
ωn ωm
axis for each plot results in a standard sensitivity curve for the given initial
condition values. The insensitivity of an SI-IC shaper is significantly different, as shown in Figure 29, due to the substantially increased robustness along the
36
ωn ωm
axis.
Percent Vibration
50
50 40 30 20 10
40 30 20 10
60 30 0 −30
θe
−0.2 −0.4
−60
0.0
0.2
0.4
Ae
Percent Vibration
Percent Vibration
Figure 27. Sensitivity of a ZV-IC shaper to Ae and θe
80 40
40
20
20
60
1.2
30
60
60
40
40
20
20 0.4
1.1
0 −30
θe
80
80
60
60
100
100
80
1.0 −60
0.9 0.8
0.2
0.0 −0.2 −0.4 Ae
ωn ωm
(a) Sensitivity to θe and
ω ωm
1.2 1.1 1.0 0.9 0.8
ω ωm
(b) Sensitivity to Ae and
ω ωm
Percent Vibration
Percent Vibration
Figure 28. ZV-IC shaper sensitivity
70 60
60
50
40
40 30
20
20 10
60
1.2
30
1.1
0
θe
−30
1.0 −60
0.9 0.8
40
40
30 20
20 10 0.4
ω ωm
1.2
0.2
0.0 −0.2 −0.4 Ae
ωn ωm
(a) Sensitivity to θe and
50
1.1 1.0 0.9 0.8
ωn ωm
(b) Sensitivity to Ae and
Figure 29. SI-IC shaper sensitivity
37
ω ωm
Table 1. Simulation Values
3.3
Variable
Value
ω ζ tacc y˙ 0 y0
2π rad s 0.1 0.1s −8.47 m s −1.5m
Impulse Response Example The proposed shaping methods were simulated using the values given in Table 1.
Both ZV-IC and SI-IC input shapers were designed to eliminate the specified initial oscillation. The range of frequencies to be suppressed by the SI-IC shaper was chosen to be [0.8ω, 1.2ω], resulting in an Insensitivity of I = 0.4. The resulting input shapers are: Ai 0.62 0.38 (62) ZV-IC = = , 0 0.59 ti
and
Ai 1.0 −0.98 .67 −0.46 0.72 −0.29 0.35 SI-IC = = . 0 0.14 0.40 0.76 0.93 1.28 1.39 ti
(63)
Figure 30 shows the response of the system subject to a ZV-IC-shaped pulse input. As expected, the shaped command completely eliminates the residual vibration. A shaped command that was designed without consideration to the actuator limitations is also given for comparison. The importance of the phase and amplitude shifts resulting from these limitations is evident in the residual vibration resulting from the unshifted command. Finally, a modeling error of
ωn ωm
= 1.2 is introduced to demonstrate
the effect of such an error on the shaped response. Because the ZV-IC shaper does not incorporate any robustness constraints, the performance suffers as a result of this change. Similarly, the SI-IC-shaped responses are shown in Figure 31. Because a tolerable level of vibration is permitted at the design frequency, a small amount of 38
Position (m)
4
Shaped Unshifted Error: ωωmn = 1.2
2
0
−2
0
1
2 3 Time (s)
4
5
Figure 30. ZV-IC shaped pulse response
Position (m)
4
Shaped Unshifted Error: ωωmn = 1.2
2 0 −2 0
1
2 3 Time (s)
4
5
Figure 31. SI-IC shaped pulse response residual vibration exists after completing the shaped command. Furthermore, the robustness constraints result in a longer duration of the shaped command. This robustness is evident in the response subject to the modeling error of
ωn ωm
= 1.2, the
upper limit of the suppressed frequency range. The residual vibration subject to this error is significantly lower than the ZV-IC-shaped case. The response of the system subject to an SI-IC shaper designed without considering the actuator limitations results in an increased level of vibration, as expected. The robustness of these shapers to modeling uncertainty is given by the sensitivity curve in Figure 32. While the ZV-IC shaper is designed to yield no residual
39
SI-IC (I = 0.4) ZV-IC
Percent Vibration
60
40
20
0 0.8
0.9 1.0 1.1 Normalized Frequency ωωmn
1.2
Figure 32. Sensitivity curves of a ZV-IC and SI-IC shaper Mass Flexible Rod Laser Scanner
Measurement Controller Track
Figure 33. Benchmark experimental system
vibration at the designed frequency, it rapidly loses effectiveness when the actual natural frequency deviates from this value. The SI-IC shaper, on the other hand, maintains a low level of vibration across the entire range of frequencies. 3.4
Experimental Verification To further validate this command shaping method, an experimental platform at
the Kumoh National Institute of Technology in Korea, pictured in Figure 33, was used. In this system, the flexible rod is commanded to move along the track, resulting in oscillation of the mass. This oscillation is measured by a laser scan micrometer.
40
Table 2. Benchmark System Experimental Parameters Variable
Value
ω ζ tacc Vmax y˙ 0
14.28 rad s 0.01 0.17s 0.15 m s 5.19 mm s
y0
0.0mm
The values used in this experimental analysis are summarized in Table 2. The natural frequency and damping ratio were determined experimentally by measuring the free response of the system for a specified mass height. The acceleration time, tacc , and maximum velocity, Vmax were determined by analyzing the step response. A typical system response is presented in Figure 34. Here, the experimental response is compared to simulation predictions based on a linear model. A known impulse generates oscillation, prior to the beginning of the command. In this unshaped case, the command-induced vibration increases the oscillation beyond the level introduced through the initial conditions. The residual vibration is measured by the amplitude of oscillation after the completion of the commanded motion. This amplitude is used to compare the performance of the shapers to the unshaped case. In order to minimize experimental error in this analysis, the end of the command is shaped using a standard ZV shaper based on the calculated natural frequency and damping ratio of the system. This ZV shaper introduces approximately no additional vibration into the system while simultaneously bringing the rigid mode to rest. This allows for consistent measurement of the residual vibration due to the shaping methods under consideration. The ZV-IC and SI-IC input shapers were designed to eliminate the given initial conditions. The SI-IC shaper was designed to suppress vibration in a range of frequencies, [0.9ω, 1.1ω]. These vibration constraints on the SI-IC shaper, shown in 41
Position (mm)
1.5 Command Duration
1.0
Simulation Experimental
0.5 0.0 0.5 0
1
2 Time (s)
3
Figure 34. Typical unshaped response
Position (mm)
0.50
Simulation Experimental
0.25 0.00 −0.25 −0.50
0
1
2 Time (s)
3
Figure 35. ZV-IC shaped response at the designed frequency
Figure 36, result in larger residual vibration than the ZV-IC case, shown in Figure 35, at the frequency used to design the shapers. In addition, the complex impulse sequence of the SI-IC shaper yields higher transient vibration while the system completes the motion. In each case, the experimental trials closely resemble the simulated results. Both shaping methods were tested for robustness to modeling uncertainty by determining the residual vibration amplitude of the shaped and unshaped responses at various natural frequencies. The results for the ZV-IC shaper are summarized in Figure 37a. This plot compares the theoretical residual vibration amplitude subject to
42
Position (mm)
Simulation Experimental
0.5
0.0
−0.5 0
1
2 Time (s)
3
80
Theoretical Experimental
200 Percent Vibration
Percent Vibration
Figure 36. SI-IC shaped response at the designed frequency
60 40 20 0.8
Theoretical Experimental
150 100 50
1.0 1.2 Normalized Frequency ωωmn
0.8
(a) ZV-IC shaper
1.0 1.2 Normalized Frequency ωωmn
(b) SI-IC shaper (I = 0.2)
Figure 37. Experimental sensitivity to changes in natural frequency deviations in natural frequency, given by (57), to those found by experimental trials. Each experimental data point in this figure is the mean of three trials, where the variance between each trial is approximately zero. Here, the data closely match the predicted values, particularly near the designed natural frequency. Because the natural frequency of the experimental system is experimentally estimated and assumed to vary linearly based on these estimates, some modeling error due to nonlinear dynamics of the system is to be expected. Note that in this plot, the residual vibration remains below the unshaped case for all sampled frequencies. Similar results for the SI-IC shaper are shown in Figure 37b. A trend similar to
43
Theoretical Experimental
Percent Vibration
Percent Vibration
60
40
20
0
−0.2
0.0 0.2 Amplitude Error Ae
Theoretical Experimental
40
20
−0.2
0.4
(a) ZV-IC shaper sensitivity to Ae
0.0 0.2 Amplitude Error Ae
0.4
(b) SI-IC shaper (I = 0.2) sensitivity to Ae
Figure 38. Experimental sensitivity to Ae the ZV-IC results is evident for these trials; the data increasingly deviate from the theoretical prediction at frequencies significantly different from the modeled frequency. In the suppressed range, however, the experimental data show that the shaper effectively limited residual vibration. As a result of the negative amplitudes in this shaper, the residual vibration percentage increases more quickly than the ZV-IC shaper as the natural frequency deviates from the suppressed range. The shaped command results in greater residual vibration than the unshaped command at greater than approximately 20% natural frequency error. Further experiments were performed subject to errors in initial condition amplitude, Ae . In these experiments, the same IC shapers were used, but the actual initial condition amplitude was varied by an error factor, Ae . The results are summarized in Figure 38. Both shapers exhibit nearly identical sensitivity to Ae , where the SI-IC shaper in Figure 38b is skewed to the left and the ZV-IC shaper in Figure 38a is centered around the nominal condition. In both cases, the experimental results closely follow the theoretical prediction, validating the sensitivity equation with respect to Ae . Experiments to evaluate the shaper sensitivity to θe were also performed in a similar fashion. These results are summarized in Figure 39. Similar to the sensitivity to Ae , both shapers exhibit nearly identical sensitivity to phase error. These experiments
44
Theoretical Experimental
100 Percent Vibration
Percent Vibration
100 75 50 25 −60
−40
0 20 −20 Phase Error θe (deg)
40
75 50 25 −60
60
(a) SI-IC shaper (I = 0.2) sensitivity to θe
Theoretical Experimental
−40
0 20 −20 Phase Error θe (deg)
40
60
(b) ZV-IC shaper sensitivity to θe
Vibration Amp. (mm)
Figure 39. Experimental sensitivity to θe
Unshaped ZV-IC
0.6
SI-IC
0.4 0.2 0.0
0.8
1.0 1.2 Normalized Frequency ωωmn
Figure 40. Absolute vibration amplitude of each command
were performed by activating the shaped command at a known time away from the designed phase. Because the SI-IC shaper does not have zero residual vibration at the nominal condition, its theoretical and experimental sensitivity is shifted by a small amount. Figure 40 provides additional insight into the performance of the shaped commands relative to the unshaped command. This plot shows the residual vibration amplitude of the shaped commands as well as the unshaped command. An approximately linear decrease in vibration levels at higher frequencies is visible for the unshaped command. Because the residual vibration is measured at the end of the
45
command, the effects of damping are apparent in this vibration amplitude measurement. While the natural frequency increases and the command duration is constant, more oscillation is damped during the command, resulting in this measurement trend. Although the effect of this damping is significant on the measured residual vibration amplitudes of each command shaping method, all three are affected approximately equally. 3.5
Conclusion This chapter has introduced multiple methods of designing input shapers which
are capable of eliminating initial oscillation in a flexible system. The frequency domain solution can be solved in closed-form for an undamped system, while it requires a simple optimization for the more general, damped case. A time domain design procedure can be used to generate IC shapers which are robust to modeling uncertainty. Additionally, these shapers can be modified based on the actuator constraints of the system. Experimental results validated the proposed shaping methods. The experimental system responses closely matched those predicted by simulation. Furthermore, the robustness of the ZV-IC and SI-IC shaper were measured subject to varying natural frequencies. These experimental results support the effectiveness of each shaping method, while demonstrating the increased robustness of the SI-IC shaping approach.
46
IV An Application in Boom Crane Control The input shaping method presented in Chapter 3 can be readily applied to any linearizable dynamic system. One such example can be found in boom crane control [84, 85]. The dynamic properties of boom cranes have been thoroughly documented by multiple researchers [30, 39]. The moderate nonlinearities of this type of system make it a good candidate for demonstrating the efficacy of initial condition input shaping. This chapter will illustrate the use of initial condition-canceling input shaping in boom cranes and highlight specific challenges in linearizing this system. 4.1
Control of an Undisturbed Boom Crane 4.1.1
Stationary dynamic model. A planar boom crane is depicted in
Figure 41. Beam R and cable l are assumed to be massless and rigid. Payload m is modeled as a point mass at the end of the cable. All motion is assumed to occur in the xy plane and is described by luff angle, γ, and radial payload swing angle, φ. The beam
B y
R
l
A
Figure 41. Planar boom crane model
47
and cable are assumed to rotate without frictional losses around points A and B, respectively. The equation of motion that governs the payload swing angle is: ¨ = −¨ φl γ R sin(φ − γ) + Rγ˙ 2 cos(φ − γ) − g sin(φ).
(64)
Notice that (64) is nonlinear due to the trigonometric functions in the γ¨ and γ˙ 2 terms. However, this system will be controlled about the equilibrium point, φ = 0. It is therefore possible to linearize about this datum. Furthermore, the time it takes for a boom crane to accelerate to a maximum velocity, γmax , is short relative to the pendulum dynamics. As a result, γ¨ contributes minimally to the free response of the system, and the assumption γ¨ = 0 can be made. Equation (64) then simplifies to: R g − Rγ˙ 2 sin(γ) φ = γ˙ 2 cos(γ). φ¨ + l l
(65)
This linearized equation still has multiple nonlinear terms. Particularly, the Rγ˙ 2 sin(γ) term on the left side of the equation prohibits (65) from being written as a linear system. However, noting that a constant luff rate permits luff angle γ to be written as γ = γt ˙ + γ0 , the boom crane can be approximated as a pendulum with a natural frequency of: ω=
r
g − Rγ˙ 2 sin(γt ˙ + γ0 ) . l
(66)
The natural frequency is time-varying over the course of a luff motion, as it depends on γ and γ. ˙ However, this nonlinearity tends to have a minimal effect on the dynamics of the crane. Because actuator limitations generally limit γ˙ well below 1 rad , its square s typically becomes inconsequentially small. As a result, g >> Rγ˙ 2 sin(γ), allowing the luff-angle-varying term to be safely ignored. The further linearized equation of motion is: φ¨ + ωφ = F (γ, γ), ˙
48
(67)
where the natural frequency is simply that of a pendulum: r g ω= l
(68)
and the nonlinear forcing function is: F (γ, γ) ˙ =
R 2 γ˙ cos(γt ˙ + γ0 ). l
(69)
Equation (69) indicates that the forcing function depends on the current luff angle. As a result, (69) is expressed as a harmonic function with a frequency of γ. ˙ A thorough discussion of these assumptions can be found in [30]. 4.1.2
Approximating impulse response characteristics. In Chapter 3, it
was observed that a system subject to actuator limitations exhibits phase and amplitude shifts based on the normalized acceleration time required to bring the system to its terminal velocity. This relationship, repeated here for clarity, is: y = δp
ω 1 − ζ2
e−ζωt sin(ω
p
1 − ζ 2 t − ).
(70)
Because the boom crane is assumed to be undamped, ζ disappears from (70). For an undamped system, the phase and amplitude shifts, and δ, can be numerically determined to simply be functions of normalized acceleration time, τacc . These relationships are: (τacc ) = πτacc ,
(71)
and δ(τacc ) =
3 2 0.65τacc − 1.31τacc − 0.34τacc + 1 . 2 0.35τacc − 0.34τacc + 1
(72)
These relationships are determined by using the same methods presented in Chapter 3, after eliminating ζ. The near-perfect fit of these functions relative to their true values can be seen in Figures 42 and 43. With (70), (71), and (72), the pulse response of a linear second-order system can be accurately characterized. However, a boom crane is not exactly linear. For the 49
Phase Shift
3
2
1 Actual Curve Fit
0.2
0.4 0.6 0.8 Norm. Acceleration τacc
Figure 42. Phase shift for an undamped linear system
Actual Curve Fit
Amp Shift δ
1.00 0.75 0.50 0.25
0.2 0.4 0.6 0.8 Norm. Acceleration τacc
Figure 43. Amplitude shift for an undamped linear system
linearized pulse response to be approximated, a boom crane can be considered as shown in Figure 44. When a short-duration pulse in γ¨ is given to the system, the tip of the boom accelerates along a curved trajectory which can be decomposed into horizontal and vertical acceleration, ax and ay . This causes a displacement in the payload, mx . When the boom crane starts from γ = 0, the ax component is approximately zero. This amplitude varies sinusoidally as a function of γ, where it reaches its maximum value when γ = 90◦ . This relationship between the displacement of the payload and a luffing
50
y
Figure 44. Boom crane linearized approximation acceleration, γ¨ can be seen by simply linearizing (64) about φ ≈ 0: g − Rγ˙ 2 sin γ + γ¨ R cos γ γ¨ R sin γ + Rγ˙ 2 cos γ φ¨ + φ= . l l
(73)
If the system is accelerating from rest, the assumption γ˙ 20◦ . 4.1.3
Initial condition input shaper design. In order to design a shaper
which eliminates the nonzero initial states φ0 and φ˙ 0 , the pulse response must be once again compared to the impulse response for an analogous linear system. The approach for the boom crane is nearly identical to that of a linear, damped, second-order system. If a boom crane has known actuator limitations which dictate the maximum luff angular velocity, γ˙ max , and acceleration time, tacc , the swing response subject to a luff command can be approximately characterized by a linear system with a natural frequency: ω=
r
g − Rγ˙ max sin γ0 , l
(78)
where γ0 is the initial luff angle. To normalize the initial conditions of the boom crane, the commanded pulse is scaled by the radial velocity of the suspension cable attachment point at the beginning of the command. This scaling factor is configuration dependent, resulting in dynamics which differ from a simple, linear system. Therefore, a second amplitude shift, δbc and phase shift, φbc must be computed and used to shift the target initial conditions in the same way as demonstrated in Section 3.2.1. Once the initial conditions are shifted in this manner, the ZV-IC and SI-IC shaper can be solved using the procedures presented in this work. A representative simulated response is shown in Figure 47. As shown in Figure 47b, the boom crane begins at luff angle γ = 60◦ and is commanded to move to γ = 30◦ through a ZV-IC and SI-IC shaped command. The acceleration phase is shaped by the IC shapers while the deceleration phase of the command is shaped by a
53
ZV-IC SI-IC
60 Luff Angle (deg)
Swing Angle (deg)
ZV-IC SI-IC
2
0
−2
50 40 30
0
2
4 Time (s)
6
0
(a) Swing angle response
2
4 Time (s)
6
(b) Luff angle command
Figure 47. Boom crane simulation subject to nonzero initial conditions ZVD shaper [20] to clearly demonstrate the performance of each IC shaper. Figure 47a demonstrates the vibration cancellation of the IC shapers while performing a downward luff command. The moderate nonlinearity of the boom crane results in a small level of oscillation in the ZV-IC response, as is expected. The SI-IC shaper results in slightly more residual vibration due to the relaxed vibration constraint at the designed frequency. 4.2
Boom Crane Subject to a Harmonic Disturbance Beyond the simple planar boom crane, initial condition input shaping can be
used to mitigate payload swing in a boom crane subject to a harmonic disturbance. Although a number of researchers have developed effective control laws to ensure precise payload positioning of such cranes for the application of offshore crane control [10, 16, 17], it may be difficult to implement such control on full scale boom cranes with actuator limitations. For example, sliding mode control has been used to stabilize the crane system [17]. This control method relies on rapid actuation and large peak actuator effort. As a result of these requirements, actuator bandwidth constraints in ship-mounted cranes limits the feasibility of this control approach. Offshore cranes are greatly impacted by disturbances caused by ocean waves. In moderate-to-severe sea states, the payload swing of such a crane easily exceeds
54
m
Figure 48. Dynamic crane model
acceptable levels as a result of harmonic excitation from the ocean waves [89]. If a crane is subjected to a known harmonic disturbance, input shaping can be used to counteract the transient dynamics due to the disturbance. This method can reduce transient payload deflection while the crane is in motion and allow the payload to come briefly to rest shortly after the end of the command. As a result, the peak actuator effort of a closed-loop controller can be significantly reduced. 4.2.1
Dynamic model. The dynamic model of a planar, ship-mounted boom
crane used in this work is shown in Figure 48. The ship reference frame, S, has its origin at the center of gravity on the ship and is aligned such that its y-axis runs parallel to the deck, normal to its z-axis. This frame is rotated about the inertial frame, G, by angle α. The boom crane attachment point, C, is displaced from origin O by the vector RC . Luff angle γ and payload swing angle φ are expressed in frame S. The boom crane is modeled as two rigid, massless members R and l, where the payload is assumed to be a point mass, m, swinging from cable l about point A. This swing is assumed to be undamped. The ship is assumed to undergo a known harmonic disturbance. Because the
55
crane is oriented perpendicularly to the longitudinal, x-axis of the ship, the primary contributor to unwanted radial swing, φ, is the ship roll angle, α. Assuming that this angle is known for all time, the disturbance and its derivatives can be expressed as harmonic functions: αS (t) = a sin(ω0 t),
(79)
ωS (t) = aω0 cos(ω0 t),
(80)
S (t) = −aω02 sin(ω0 t),
(81)
where ω0 is the disturbance frequency and a is the ship roll amplitude. Functions αS (t), ωS (t), and S (t) represent the roll angle, velocity, and acceleration of the ship, respectively. This representation of the disturbance is based on the assumption that a fully developed sea state can be modeled as a series of harmonic waves of varying amplitudes, frequencies, and phase shifts. These waves are then superimposed, and the resulting forces can be calculated [90]. Although (79), (80), and (81) ignore any dynamic interaction between wave disturbance and the ship to which the boom crane is mounted, they serve as a baseline to test the validity of the command-shaping methods proposed in this work. The equation of motion governing the payload swing angle, φ, has been previously found in [17]. Modifying the original equation for the simplified assumptions in this model, the following expression is derived: lφ¨ = γ¨ R sin(φ − γ) − S R sin(φ − γ) + R(γ˙ + ωS )2 cos(φ − γ) − S l − g sin(φ − αS ) + (S dz + ωS2 dy ) cos(φ) + (−S dy + ωS2 dz ) sin(φ), (82) where dy and dz represent the y and z component of vector RC , respectively. This vector is defined in the reference frame S. Making the further assumptions that the swing angle is small (φ ≈ 0) and the 56
luff velocity is zero (γ˙ = 0), (82) can be expressed as undamped, forced oscillation: φ¨ + ω 2 φ = F1 + F2 ,
(83)
where ω is the approximate natural frequency of the crane, subject to forcing functions F1 and F2 . This equation describes the steady-state forced response of the boom crane. The full expression of the natural frequency is: ω 2 l = g − aω02 R cos(γ0 ) + aω0 dy sin(ω0 t)
− 2aRω0 γ0 sin(γ0 ) cos(ω0 t)
− a2 Rω02 sin(γ0 ) + a2 ω02 dz cos2 (ω0 t). (84)
The natural frequency of the system is time-varying due to the influence of the harmonic disturbance. In practice, the time-varying effects due to the rolling motion result in a minimal change in natural frequency. This effect is evident in Figure 49, which shows the percent variation of the modeled crane natural frequency with respect to that of a simple pendulum. The frequency remains within approximately 4% of the pendulum frequency in the presence of the harmonic disturbance. For the purposes of this analysis, the deviation can be ignored. The natural frequency can therefore be approximated as that of a simple pendulum: ω=
r
g . l
(85)
With the given disturbance functions, (79), (80), and (81), the forcing functions can be expressed as: sin(ω0 t) F1 = −aRω02 sin(γ0 ) + aω02 l + ga − aω02 dz , l cos(2ω0 t) + 1 F2 = R cos(γ0 )a2 ω02 + dy a2 ω02 . 2l
(86) (87)
These equations show that a harmonically disturbed boom crane that is not undergoing luff is subject to two excitation frequencies: ω0 and 2ω0 . The steady-state 57
Figure 49. Deviation in natural frequency from a simple pendulum solution to (83), φss (t), can be found by summing the responses subject to (86) and (87): φss (t) = φ1 (t) + φ2 (t). The constituent equations are: −aRω02 sin(γ0 ) + aω02 l + ga − aω02 dz sin(ω0 t), φ1 (t) = l(ω 2 − ω02 )
(88)
(89)
and φ2 (t) =
4.2.2
R cos(γ0 )a2 ω02 + dy a2 ω02 2l(ω 2 − 4ω02 )
cos(2ω0 t) +
R cos(γ0 )a2 ω02 + dy a2 ω02 . 2lω 2
(90)
Validity of linear approximations. Figure 50 compares the
linearized approximations to the full solution of (83). Because (88) assumes the superposition principle applies, this approximation is sensitive to increases in ship roll amplitude, a, and ω0 . In the neighborhoods of ω and ω2 , ω0 will drive (88) to infinity. Furthermore, in the range
ω 2
< ω0 < ω, resonance will begin to occur. Therefore, this
approximation is valid strictly for frequencies significantly lower than the natural frequency of the crane, ω0 < 0.4ω, where the linear fit is good (r2 ≈ 0.95) for amplitudes up to a = 10◦ . Figure 50 shows an example comparison of the approximation and full nonlinear response in this lower frequency region. 58
Figure 50. Steady-state payload swing and approximation
4.2.3
Attenuation of the free-vibration response. Using the same basic
procedure in Section 4.1.3, IC shaping can be used to attenuate frequencies near ω if some disturbance has caused the payload to oscillate at this free-response frequency. Given that the forcing functions are known and the payload steady-state response is given by (88), superposition can be used to determine the contribution due to the free-vibration response: φ(t) = φss (t) + φf ree (t).
(91)
Using this information, an approximation of the free response can be used to create an IC shaper which brings the crane to steady-state. As with the previous linearization in (88), the accuracy of this approximation is dependent on the degree to which the linear assumptions hold. Figure 51 demonstrates an example of this procedure for a case where ω0 = 0.4ω. Because this disturbance frequency is near the upper bound of the previously defined range of validity, there are noticeable errors in the approximation. However, the fit is still good enough to allow for a shaped command to be generated which can attenuate the free response. Further examination of the accuracy of this approximation can be performed by analyzing Fast Fourier Transform (FFT) of the full nonlinear response and the superposed approximation. Figure 52 gives this comparison. Because the approximation is defined 59
Swing Angle (deg)
Full Response
Superposed
20 10 0 −10
0
2
4 6 Time (s)
8
10
Figure 51. Full nonlinear response vs. superposed approximation
FFT Magnitude
Full Response
Superposed
0.2
0.1
0.0 0.0
0.2
0.4 0.6 Frequency (Hz)
0.8
1.0
Figure 52. FFT of full nonlinear response vs. superposed approximation
by the linearized equation (91), it will only have components at ω, ω0 , and 2ω0 . The contribution of 2ω0 is significantly lower than any other frequency for these conditions and is therefore ignored. In this case, eliminating the term associated with this frequency results in a more accurate approximation. The response-revised approximation is compared to the full nonlinear model in Figure 53, demonstrating the improvement.
60
Swing Angle (deg)
Full Response
Superposed
20 10 0
0
2
4 6 Time (s)
8
10
Figure 53. Full nonlinear response vs. superposed approximation Table 3. Simulation Values Variable
Simulation Value
Boom length 8m Cable length 2.4m dy 4m dz 2m Roll amplitude 8◦ Roll frequency 0.808 rad s
4.3
Free-Response Attenuation Simulation Computer simulations were performed to test the vibration-attenuating
capabilities of the initial condition shaper. The values used in the simulations are given in Table 3. The crane geometry constants were chosen to represent a large-scale version of the system presented in [17]. Disturbance amplitude, a, and disturbance frequency, ω0 , were chosen to be large values within the permissible range defined earlier. Figure 54 shows the swing response of the modeled boom crane. The reference command in this simulation is a bang-coast-bang acceleration profile from initial luff angle, γi = 30◦ , to final luff angle, γf = 80◦ . A standard ZV-shaped response is compared with a ZV-IC-shaped response. Although the ZV-shaped command does not
61
Swing Angle (deg)
ZV-IC
ZV
20
10
0
0
5
10
15 20 Time (s)
25
30
Figure 54. Payload swing angle
ZV-IC
FFT Magnitude
ZV
0.10
0.05
0.00 0.0
0.1
0.2 0.3 Frequency (Hz)
0.4
0.5
Figure 55. FFT of a ZV shaped response vs. an ZV-IC shaped response
excite any vibration at ω, it permits the initial oscillation to continue. The ZV-IC-shaped command attenuates this free response. Figure 55 compares the Fast Fourier Transform (FFT) of these responses. In this figure, the benefit of the ZV-IC input shaper is clearly demonstrated by the reduction in amplitude at ω. Note that the ZV-IC-shaped command does not generate a response that completely eliminates the free-vibration due to the approximations made in the construction of the command. In this example, the ZV-IC was only used on the acceleration portion of the luff command, while the deceleration portion was shaped with a standard ZV shaper.
62
Figure 56. Small scale experimental boom crane Boom Crane
Crane Controlling PC
Camera Video Processing PC
Figure 57. Experimental setup
4.4
Experimental Verification In order to demonstrate the effectiveness of initial condition input shaping on
boom cranes, experimental trials were performed. The experimental system is shown in Figure 56. This cherrypicker [91] was re-purposed to serve as a small-scale boom crane for these experiments. Its major structural components are highlighted in this figure. Its motion was limited to luff, γ, exciting radial swing, φ. Out-of-plane motion was ignored. The full experimental setup is shown in Figure 57. The boom crane is controlled
63
Table 4. Experimental Parameters Variable
Value
l R
0.35m 0.81m deg 17.4 s 0.1s 0.00 m s 0.02m 60◦ 30◦
γ˙ max tacc y˙ 0 y0 γi γf
by uploading velocity data from the Crane Controlling PC. A camera, connected to the Video Processing PC, records the payload swing. The parameters for this experiment are given in Table 4. A desired luff command from γi = 60◦ to γf = 30◦ is issued to the boom crane for a ZV-IC, SI-IC, and unshaped command and the residual vibration is measured by computer vision with the Python bindings of the OpenCV library [92]. In order to consistently create the desired initial condition, an initial luff from γ0 = 30◦ to γi is issued. The change in γ can be seen in Figure 58. For each command type, the first three seconds are identical to create identical initial conditions. From this point, the three command shapes diverge based on the input shaper used. At approximately t = 2s, each shaping method accelerates the crane downwards. To bring the system back to rest at γf , a ZV shaper is used. This ZV shaper is designed based on the actual cable length for the given trial to allow the performance of the desired shapers to be compared directly. Based on the given initial conditions, the ZV-IC and SI-IC input shaper were created using the methods presented in this chapter. The ZV-IC shaper which cancels y0 and y˙ 0 is:
Ai 0.21 0.79 ZV-IC = = . ti 0 0.78
64
(92)
ZV-IC SI-IC Unshaped
Luff Angle (deg)
60 50 40 30 0
2
4 Time (s)
6
8
Swing Angle (deg)
Figure 58. Simulated luff angle for experiments
Simulation Experimental
Command Duration
20 10 0 10 0
5
10 Time (s)
15
Figure 59. Simulated and experimental unshaped swing response
The SI-IC shaper with an insensitivity of I = 0.2 is given by: Ai 1.0 −1.0 −1.0 0.54 1.0 0.46 −0.09 −0.91 1.0 SI-IC = = . ti 0.0 0.16 0.26 0.38 0.48 0.58 0.58 0.78 1.0
(93)
A representative response for the unshaped command is shown in Figure 59.
Beginning from rest, oscillation is excited by the command. Because this command is unshaped, a large amplitude residual oscillatory response persists after the command terminates. In order to compare the desired and actual trajectory of the boom crane, 65
Luff Velocity (deg/s)
Luff Angle (deg)
20
Simulation Experimental
60 50 40
Simulation Experimental
10 0 −10
30 0
2
4 6 Time (s)
8
0
(a) Luff angle
2
4 6 Time (s)
8
(b) Luff velocity
Figure 60. Unshaped trajectory following encoder data was recorded over the course of the commanded motion. Figure 60 shows the simulated and actual trajectory data for the unshaped command. The sampling rate of the experimental system is 25Hz, which is substantially lower than the available sampling rate for the simulations on which the command is based. Therefore, the experimental luff angle in Figure 60a is much less smooth than the baseline simulation. Although the maximum velocity commands issued to the experimental system were approximately 70% of the maximum allowable values, some discrepancies between the experimental and simulated values are visible on the velocity plot in Figure 60b. Regardless, the experimental and desired trajectories align quite well. Representative command and response data from the SI-IC shaped trials are given in Figure 61. These plots show the experimental and simulated data from a trial where the actual cable length and modeled cable length were identical. As Figure 61a shows, the experimental response matches closely with what was predicted through simulation. The SI-IC shaper correctly limits residual vibration to a level consistent with the simulated response. This matching response data is driven by the experimental system closely following the simulated command, as shown in Figure 61b. Although the experimental system reaches a slightly higher luff angle than the simulation, the changes in velocity are very similar. This fact is most clearly
66
Simulation Experimental
Swing Angle (deg)
10 5 0 −5 0
5
10 Time (s)
15
(a) Swing response Luff Velocity (deg/s)
Luff Angle (deg)
20
Simulation Experimental
60 50 40
Simulation Experimental
10 0 −10
30 0
2
4 6 Time (s)
8
0
(b) Luff angle
2
4 6 Time (s)
8
(c) Luff velocity
Figure 61. SI-IC experimental responses demonstrated in Figure 61c. The experimental system minimally deviates from the desired velocity profile over the course of the trial. Similar response data for the ZV-IC shaped command is presented in Figure 62. Through Figure 62a, it is immediately apparent that the experimental system does not match the simulation very well. While vibration was reduced by the ZV-IC shaper, its actual performance is much worse than expected. The unmodeled nonlinearities of this experimental system introduce transient deviations in natural frequency which degrade the performance of the sensitive ZV-IC shaper. In order to properly characterize the performance of the shapers, sensitivity curves [62] were used. The residual vibration of the unshaped, ZV-IC, and SI-IC shaped responses were recorded for a series of trials for varying frequencies. For both shaping methods, the residual vibration was normalized by the corresponding unshaped
67
Simulation Experimental
Swing Angle (deg)
10
5
0
−5
0
5
10 Time (s)
15
(a) Swing response Luff Velocity (deg/s)
Luff Angle (deg)
20
Simulation Experimental
60 50 40
Simulation Experimental
10 0 −10
30 0
2
4
6 Time (s)
8
10
0
(b) Luff angle
2
4
6 Time (s)
8
10
(c) Luff velocity
Percent Vibration
Figure 62. ZV-IC experimental responses Theoretical Experimental
100
50
0
0.8
1.0 1.2 Normalized Frequency ωωmn
Figure 63. Experimental and theoretical sensitivity of the SI-IC shaper
vibration response and plotted in Figures 63 and 64. In these plots, the mean of three trials is shown as a diamond, where the minimum and maximum values are given by the error bars above and below it. Figure 63 demonstrates the resulting sensitivity of 68
Percent Vibration
150
Theoretical Experimental
100
50
0
0.8
1.0 1.2 Normalized Frequency ωωmn
Figure 64. Experimental and theoretical sensitivity of the ZV-IC shaper
the SI-IC shaper. The shaper correctly limits vibration in the designed region, and deteriorates in performance at frequencies significantly away from the nominal value. Similarly, Figure 64 shows the performance of the ZV-IC shaper. Although some error was introduced into the ZV-IC command, its performance is still below the unshaped command for all tested values. A number of sources of experimental error are present in this set of experiments. First, the experimental system exhibits some nonlinearities and unaccounted high-mode excitation due to the flexibility of the boom. Because the boom is modeled as a rigid link and the payload is modeled as a point mass, these discrepancies between the model and the experimental system account for some error. Next, there is some uncertainty in the measurement of the initial condition. Because the trials were designed based on the linearized model, the difference between the modeled and actual system behavior can cause the quality of the initial condition approximation to deteriorate. Furthermore, as discussed in this chapter, the discretization of the desired command and implementation in the experimental system causes some inaccuracies in the resulting command profile. Most notably, this affected the ZV-IC shaper and its low-velocity motion. In spite of these sources of error, the experimental results presented in this
69
section provide an initial demonstration for the effectiveness of initial condition input shaping on a boom crane. 4.5
Conclusion This chapter has presented a method for analyzing the behavior of the
moderately nonlinear boom crane in such a way that initial condition input shaping can be applied. By comparing the luff dynamics of the boom crane to a simple linear system, the appropriate adjustments can be made to the standard initial condition input shaping process, resulting in minimal vibration in the presence of nonzero initial states. Robustness can be increased through SI-IC shaping as well. Further analysis shows that a ship-mounted boom crane subject to harmonic excitation can be appropriately linearized to within a < 10% margin of error and initial condition input shaping can be used to attenuate the free response of the system. Experimental trials validate the use of initial condition input shaping on the boom crane as well.
70
V Concurrent Design of Input Shaping and Linear Feedback Control Because input shaping generates commands which do not excite oscillatory plant dynamics, it will result in more energy-efficient rest-to-rest commands than feedback control alone. Furthermore, by delaying a portion of the reference command, input shaping can reduce the peak actuator effort required to reach the desired states. If the closed-loop control gains are intelligently designed to maximize the advantages that input shaping provides, a control system with better performance than either method alone can be created. This chapter identifies a method of concurrently designing an optimal linear feedback controller alongside an input shaper to yield low settling time in the presence of actuator effort limitations. While the following discussion utilizes simple step inputs as the reference command, this work serves as a baseline for investigating more complex trajectory tracking controllers. A new input shaping approach is proposed to minimize the necessary actuator effort to achieve this motion while yielding minimal vibration. Representative examples in crane and flexible manipulator control are given to support this approach. 5.1
Control Approach When considering the concurrent design of input shaping and feedback control,
both elements of the controller must be designed in such a way to yield optimal performance based on the desired criteria. Previous work demonstrated control of an inertial element through proportional-derivative (PD) control combined with input shaping [79, 80]. While this research motivates the following analysis, substantial improvement can be made to the command shaping approach in order to improve performance. This work will consider a simple application of concurrently-designed input
71
shaping and linear feedback control. A desired step input is issued to a dynamic system and the control input is given by: u = K(xd − x),
(94)
and xd represents the desired system states which are chosen through input shaping. To expand the concurrent design of input shaping and feedback control to second-order plants, Linear Quadratic Regulation (LQR) will be chosen as the means of tuning the control gains. If a Linear Time Invariant (LTI) system: x˙ = Ax + Bu,
(95)
y = Cx should be subject to the control law (94), the state and input weight matrices Q and R need to be chosen such that desired performance is achieved. Importantly, the system is subject to limited observability through the output matrix C. Therefore, output weighting dictates that: Q = C T Q0 C
(96)
where the modified state weight matrix Q0 is chosen based on the desired performance. This choice of control will yield optimal control gains based on the quadratic cost function: J(u) =
Z
0
∞
xT Qx + uT Ru dt,
(97)
where the solution to (97) can be found by solving the steady-state Algebraic Riccati Equation: AT P + P A − (P B)R−1 B T P + Q = 0,
(98)
and the solution to (98), P , is used to determine the control gains K: K = R−1 B T P.
72
(99)
Figure 65. Block diagram of the proposed control method
Several metrics are available for quantifying the performance of dynamic systems. This work will focus on rise time, settling time, overshoot, maximum actuator effort, and disturbance rejection. Incorporating input shaping to a control system inherently reduces settling time and overshoot at the cost of an increased rise time due to the convolution process. By intelligently choosing the control gains, this rise time can be minimized while limiting maximum actuator effort and providing acceptable disturbance rejection. 5.1.1
Choosing control gains for concurrent design. In the concurrent
design of input shaping and feedback control, a controller is desired which provides fast rise time subject to actuator constraints. This controller will utilize input shaping to modify the reference command outside of the feedback loop while the feedback gains and input shaper parameters are concurrently designed. The block diagram for this controller is shown in Figure 65. Forming the controller in this way as opposed to a shaper-in-the-loop approach eliminates the need to consider possible instability caused by the time delay from the shaper impulses. For the desired controller, the state and actuator weight matrices Q and R will be tuned by minimizing a cost function associated with the shaped step response properties. The cost function is given by: J=
αt2s
+β
Z
ts
t0
1 u(t)
2
dt.
(100)
This function penalizes the shaper duration and the inverse of the shaped actuator effort, respectively. The weights α and β can be modified to penalize these values 73
differently. These terms were selected based on their contribution to the characteristics of the desired response. The 5% settling time of the output, ts , is chosen to represent the command duration. For a shaped command, this duration is approximately equal to the time of the final impulse of the input shaper, tn . In order to further ensure that the optimized command pushes the system to its desired states as quickly as possible, the second term penalizing the inverse of the actuator effort is added. Equation (100) is minimized subject to the constraint that the peak actuator effort must not exceed the allowed value: umax ≤ uallow .
(101)
For the shaped case, an additional constraint is enforced to keep the damping ratio of the shaped mode below an allowable level: ζk,opt ≤ 0.7,
(102)
which is approximately the crossover point where ZV shaping requires more time to eliminate vibration than damping alone. This constraint ensures that the settling time of the shaped mode is governed by the input shaper. 5.1.2
Command shaping. The discontinuous control input of a desired step
input to a flexible system represents, in some ways, a worst-case in terms of required actuator effort. If the reference command is unmodified, a large jump in the control input will be required in an attempt to reach the desired states. For a Single-Input-Single-Output linear system, this initial increase will generally represent the maximum required actuator effort, umax , over the course of the issued command. If the controller is designed without input shaping in mind, the proportional control gain associated with the desired state will necessarily be reduced in order to limit actuator saturation resulting from the step. One direct benefit of using input shaping in this instance is the modification of the reference step into a series of smaller steps. In doing so, the magnitude of the 74
Unshaped ZV Shaped
Input Force (N)
20 10 0 −10 0.0
0.5
1.0
1.5 2.0 Time (s)
2.5
3.0
Figure 66. Actuator effort of an unshaped and ZV shaped command under PD control
instantaneous increase in actuator effort is reduced. This characteristic is demonstrated in Figure 66. A simple inertia element is subject to a step response under Proportional Derivative control. By breaking the step into smaller elements, the shaped command results in lower peak actuator effort. If the controller gains are chosen to result in identical peak actuator effort for each command, the ZV shaper will allow larger proportional gains. In general, the use of large proportional gains will result in higher closed-loop natural frequencies for an oscillatory system. Because the settling time of a shaped command is directly related to its associated oscillation period, this increase in frequency can reduce the duration of the shaper, thereby improving the settling time of the concurrently-designed system. It is therefore reasonable to design a shaped command in such a way that peak actuator effort is naturally minimized. In doing so, the controller gains can be made significantly more aggressive while meeting the actuator effort constraint. A number of methods exist which facilitate smooth trajectory generation and therefore continuous command profiles. In its classical form, input shaping does not generate such commands. However, the basic formulation of the ZV shaper includes two impulses A1 and A2 , which result in zero residual vibration. These impulses can be
75
Figure 67. ZV shaper with multiple pairs of impulses
replaced by pairs of opposing impulses which meet the same vibration requirement while smoothing the convolved command and thereby reducing the maximum required actuator effort to follow it. Figure 67 demonstrates this decomposition of the ZV shaper into a series of smaller impulse pairs. These impulse pairs are separated by θ=
π , m
where m is the number of pairs utilized. The amplitude of the impulses prior to
θ = π will sum to A1 , resulting in
A1 m
per impulse. Similarly, the amplitude of impulses
after θ = π will sum to A2 . As the number of pairs increases, the impulses and their time spacing decrease. As the number of pairs increases, the resulting input shaper resembles a continuous modification to the reference command. The maximum duration of such an input shaper is τd , which is equivalent to the duration of a ZVD shaper. The result of such a shaped input, shown in Figure 68, modifies a reference step command into a simple ramp. When m = 1, the solution is identical to a ZV shaper. Successive additions of impulse pairs bring the shaped command closer to the limiting solution of a ramp. The fully-realized shaper with the maximum number of impulses can be called the Zero Vibration, Low Effort (ZV-LE) shaper and is expressed in the frequency
76
Position (m)
1.00 0.75 0.50
m=1 m=2 m=5 m=∞
0.25 0.00 0.00
0.25
0.50 0.75 Time (s)
1.00
1.25
Figure 68. Step input manipulation with a modified ZV shaper domain by: n X
Ai e−isT ,
(103)
i=1
where T is the sampling time of the system. The time of the final impulse for this shaper is tn = τd , which is identical to the duration of a ZVD shaper. To determine the appropriate amplitudes and times of the ZV-LE shaper, the corresponding digitized ZV shaper must first be found [93]. The ZV-LE shaper has n impulses: n=
tn . T
(104)
Because the ZV-LE shaper has twice the duration of the ZV shaper, its length will also be twice that of the digital ZV shaper. The impulse amplitudes are given by: N X AZV,j Ai = N j=i−N
∀i ∈ {1, . . . n}
(105)
where AZV,j gives the j th impulse and N is the number of samples for the discretized ZV shaper. 5.1.3
Comparison to smooth command generation. The primary
motivation behind forming the Zero Vibration, Low Effort (ZV-LE) shaper is to reduce the peak actuator effort required to transition a system from one state to another while 77
S-Curve ZV-LE
Input Force (N)
10 5 0 −5 −10 0.00
0.25
0.50 0.75 Time (s)
1.00
1.25
Figure 69. Actuator effort of a shaped S-Curve and ZV-LE shaper
under closed-loop control. A secondary benefit of this formulation is that the ZV-LE shaper naturally attenuates high frequency signals. This characteristic is important when considering that such a shaper can be designed for the lowest mode in a multi-modal system, and high-mode excitation can be simultaneously eliminated without directly designing for it. In a different approach which accomplishes similar objectives, a smooth, S-Curve trajectory can be convolved with a ZV shaper to allow for smooth actuator effort transitions while attenuating the shaped mode and high-mode oscillation [94]. This smooth trajectory results in similarly reduced actuator effort for rest-to-rest motion. However, if state feedback is used, the S-Curve trajectory will result in higher actuator effort for the same command duration. This phenomenon is demonstrated in Figure 69. Once again, a simple inertia plant is subjected to a simple point-to-point motion for each command shaping method. Under Proportional control, the ZV-LE method results in a symmetric, perfectly sinusoidal actuator effort curve while the peak effort of the shaped S-Curve of the same duration is noticeably higher. When optimizing the proportional gain, the ZV-LE shaped command will be capable of having a higher gain and therefore a shorter rise time than the S-Curve command. This trend occurs
78
1.0
ZV S-Curve ZV-LE
Gain |X(jω)|
0.8 0.6 0.4 0.2 0.0
1 2 3 Normalized Frequency ωωmn
Figure 70. Sensitivity of a ZV shaper, shaped S-Curve, and ZV-LE shaper
because the proportional gain of a controller operates similarly to a physical spring. In the same way that higher spring stiffness corresponds to higher oscillation frequency, higher proportional gain also forces the closed-loop frequencies higher. Although the ZV-LE shaping method effectively attenuates high-mode oscillation, it is not as effective as the S-Curve trajectory generation method. Figure 70 compares the magnitude of the shaper gain for these two methods as well as a ZV shaper for an undamped system. The ZV shaper simply places zeros at multiples of the design frequency and is equally sensitive to each mode. The S-Curve command acts as a low-pass filter which almost completely attenuates all frequencies above the shaped frequency. While the ZV-LE shaper does not perform as well as the S-Curve at attenuating all high frequencies, it performs substantially better than the ZV shaper. This trait will prove favorable in the context of concurrent design. The settling time of the lowest mode is limited by the available actuator effort and command duration. It is therefore of the utmost importance that this mode is fully attenuated by the shaped command. Higher modes will generally be damped by the feedback controller. As a result, their moderately attenuated vibration levels will not degrade the overall system settling time.
79
m Figure 71. Simple planar trolley crane
5.2
Control of a Planar Crane As an initial investigation into this control approach, consider the point-to-point
control of a simple planar crane. This crane consists of a trolley of mass, mt , which is located by position, x; a cable of length, l, which rotates about its attachment point on the trolley by angle θ; and a payload of mass mp . The force input u acts upon the trolley, causing it to move. There are assumed to be no losses due to friction, the cable is modeled as a rigid, massless rod, and mp and mt are considered point masses. The linearized dynamic x˙ 0 x¨ 0 = ˙ θ 0 0 θ¨
model for this system is given by: x 0 1 0 0 p 1 g 0 0 −m x˙ mt mt + u(t). 0 0 1 θ 0 1 0 − mt +mp g 0 θ˙ mt
l
mt l
The desired output for control is the horizontal position of the payload.
80
(106)
Table 5. Planar Crane Simulation Parameters Variable
Value
l mt mp τmax xd α β
1m 2kg 1kg 10N 1m 1 100
Therefore, the output is given by: x x˙ y = 1 0 −l 0 . θ θ˙
(107)
This is a Linear, Time-Invariant (LTI) system. Because this system is fully observable and controllable, it can be stabilized using state feedback. 5.2.1
Crane control simulation. Using the values in Table 5, the
concurrent design control approach was compared to smooth trajectory generation and optimal linear control with no command shaping. To ensure a fair comparison between the S-Curve profile and ZV-LE shaping, both methods were designed to have the same duration in the presence of equal control gains. Therefore, because the ZV-LE shaper has a duration of τd , the S-Curve has a rise time of
τd 2
and is convolved with a ZV
shaper. In order to determine the state and input weight values, the step response of the system is iteratively solved, and the cost function (100) and constraints (101) and (102) are evaluated. The optimization parameters, Q and R, are used to generate the control gains, K, which drive the closed-loop dynamics. At each iteration, the shaped 81
7.5
⌘ 5
5.0
0
2.5
⇣ p Imaginary ! 1
10.0
⇣2
ZV-LE S Curve Unshaped
5 Real ( ⇣!)
Figure 72. Pole locations for concurrently designed crane controllers Table 6. Controller Weight Results
Q0 R
Unshaped
S-Curve
ZV-LE
0.9932 0.0089
0.9974 0.0003
9.0243 0.0005
command, and therefore the desired states, xd (t), are updated based on the new closed-loop eigenvalues. The optimal control gains and input shaper parameters therefore yield the tuned LQR controller and input shaper which exhibits low settling time while limiting maximum actuator effort. The results from the simulated system are given in Table 6. The outputfeedback-weighted LQR controller which minimizes the cost function (100) for each control method has the state and input weights as shown. Immediately, it is clear that the ZV-LE shaper enables an order of magnitude increase on the state weight while being comparable to the S-Curve result in terms of input weighting. The resulting pole locations for the optimized control methods are shown in Figure 72. Due to the limit on available actuator effort, the unshaped LQR controller results in the least-damped pole locations. While the S-Curve allows higher control gains to push the poles further into the left-half plane, the proposed ZV-LE command shaping method results in significantly more-highly damped pole locations than either 82
0
−20
−20
Gain (dB)
Gain (dB)
0
−40 −60 −80 −100 −1 10
ZV-LE S Curve Unshaped
100 101 Frequency (Hz)
−40 −60 −80 −100 −1 10
102
(a) Sensitivity to disturbance in x
ZV-LE S Curve Unshaped
100 101 Frequency (Hz)
102
(b) Sensitivity to disturbance in θ
Figure 73. Bode plot of proposed control methods other method. Simultaneously, the natural frequencies of the ZV-LE and S-Curve methods are higher than the unshaped methods, resulting in faster dynamics. The magnitude Bode plots shown in Figure 73 show the sensitivity of the payload position to force disturbances. As both Figures 73a and 73b indicate, the inclusion of command shaping yields lower DC gain for low frequency disturbances to both the trolley and payload while each method responds identically to high frequency excitation. The shaped commands both exhibit slightly higher gains near the corner frequency in Figure 73b, indicating increased sensitivity at the closed-loop natural frequency. Because the proposed control methods do not explicitly attenuate these frequencies in the pursuit of the previously mentioned control objectives, this behavior is expected. If such attenuation is desired, the control law can readily be modified. Figure 74 shows the step response for the simulated system subject to the three control methods presented here. As expected, the settling time of the ZV-LE shaped command is significantly lower than both the S-Curve and Unshaped commands. While the Unshaped command has a similar rise time to the S-Curve, it also exhibits overshoot and therefore has a longer settling time. Figure 74b shows the required input force for each command. While the Unshaped command has a single, sharp spike in actuator effort at t = 0 due to the step command, the shaped trajectories exhibit peak effort near the middle of the command duration. This results in higher energy 83
Input Force (N)
Payload Position (m)
0.75 0.50 ZV-LE S Curve LQR
0.25 0.00
ZV-LE S Curve LQR
10
1.00
0
1
2 Time (s)
5 0 −5 −10
3
0
(a) Payload position response
1
2 Time (s)
3
(b) Actuator effort
Figure 74. Simulated step response Table 7. Concurrent Design Response Characteristics
Settling Time (s) Energy Consumption (N · s)
consumption, defined by
R ts 0
Unshaped
S-Curve
ZV-LE
2.56 5.23
1.68 7.28
1.39 8.93
udt, for the shaped commands. The settling times, ts , and
overall actuator effort of each control method are compared in Table 7. As this table shows, the ZV-LE shaper results in a settling time 54% as long as the unshaped case in exchange for a 71% increase in overall energy consumption. 5.3
Conclusion This chapter has presented a method for concurrently designing input shaping
and feedback control to achieve low settling time in the presence of actuator limitations. To do so, a new shaping approach which limits actuator effort and attenuates high-mode oscillation was introduced. This ZV-LE shaper can be used to shape the low frequency mode of a multi-modal system while Linear Quadratic Regulation is used to provide desired closed-loop dynamics. A proposed example in crane control demonstrated the effectiveness of this method.
84
VI Conclusions and Future Work This thesis presented new tools for improving the control of flexible systems through input shaping and feedback control. First, initial condition input shaping was discussed and experimentally tested. This approach can be used to counter existing oscillation in a system. Considerations were given to actuator bandwidth limitations and robustness to parametric uncertainty. The experimental data support the simulation and mathematical theory for a simple harmonic oscillator. This input shaping approach was also successfully adapted for the moderately nonlinear boom crane system. Additional considerations for the dynamics of the luffing motion of the boom crane were simulated and tested experimentally. This approach was also proposed to attenuate the free response of a ship-mounted boom crane under a known harmonic disturbance. Additionally, preliminary work identifying a concurrent design method for input shaping and feedback control was presented. Because input shaping provides point-to-point commands which do not excite oscillatory plant dynamics, it can be used in conjunction with feedback control to yield more time and energy efficient commands, while still taking advantage of the closed-loop properties of feedback methods. 6.1
Thesis Contributions In this thesis, several new techniques for the control of flexible systems were
introduced, representing contributions to the current state-of-the-art. These contributions are: 1. Development of an input shaping control technique which eliminates nonzero initial states in a flexible system. – Chapter 3 A thorough analytical and numerical development of the input shaping technique was presented. Consideration was given to factors such as
85
robustness to plant uncertainty and actuator bandwidth limitations. Experimental results support the presented findings. 2. An application of initial condition cancellation to boom crane control – Chapter 4 The initial condition input shaping technique was applied to a boom crane to demonstrate the applications of this control method. Experimental results validate the use of initial condition input shaping on a planar boom crane undergoing luff. Further analysis was performed on a harmonically excited boom crane to simulate the effects of ocean waves. The proposed control technique is capable of attenuating unforced vibrations in the presence of this external disturbance. 3. Development of a technique to concurrently design input shaping and linear feedback control – Chapter 5 Because input shaping and feedback control have complementary strengths, a number of researchers have combined these methods. This thesis presented a technique for concurrently designing both the input shaper parameters and feedback control gains in order to yield desired performance. This technique resulted in faster settling time and better low-frequency disturbance rejection than a similarly-tuned controller that was sequentially designed. 6.2
Future Work The contributions of this thesis provide the motivation for performing additional
research into advanced input shaping design. First, the preliminary investigation into the concurrent design of input shaping and feedback control can be used to motivate a more theoretically rigorous and generalizable solution to yield optimal control performance. This area of research could benefit significantly from experimental results verifying the simulated performance gains. Furthermore, the linear control law used in 86
conjunction with input shaping should be compared with more advanced, nonlinear control strategies used to control challenging systems. By distributing the burden of controlling the system across the input shaper and feedback controller, a simpler control strategy which still provides excellent vibration reduction could be achieved.
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Newman, Daniel. Bachelor of Science, University of Louisiana at Lafayette, Spring 2015; Masters of Science, University of Louisiana at Lafayette, Spring 2018 Major: Engineering, Mechanical Engineering Concentration Title of Thesis: Attenuating Vibration through Input Shaping and Feedback Control Techniques Thesis Director: Dr. Joshua E. Vaughan Pages in Thesis 110; Words in Abstract: 141
Abstract The control of flexible systems is a broad and active field of research due to the prominence and innate challenge of controlling complex systems in demanding environments. One well-proven technique for eliminating vibration in such systems is the implementation of a command shaping technique called input shaping. By designing a command which quickly performs a desired move without exciting oscillatory dynamics, input shaping is an excellent means of efficiently controlling vibration. This approach is also useful in mitigating vibration due to known force disturbances to a system if they are considered as initial conditions. Although unable to account for unknown disturbances by itself, input shaping can be implemented alongside a feedback controller to account for such externalities. This thesis presents new techniques to advance the state of the art in input shaping control to account for both known and unknown disturbances.
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Biographical Sketch Though originally a Midwest native, Daniel Newman currently calls Lafayette, Louisiana home. An unrepentant nerd, his only question when deciding collegiate field of study was, “which engineering degree?” In a desire to be around like-minded nerds, he chose mechanical engineering and received his Bachelor of Science degree in Spring of 2015. After one year in industry, he made the choice to pursue his Master of Science Degree with a focus on system dynamics and controls under Dr. Joshua Vaughan.
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