Reducing Vibration and Providing Robustness with Multi-Input Shapers

29 Aerican Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -2, 29 WeA6.4 Reducing Vibration and Providing Robustness with Multi-Input Shapers Joshua Vaughan and Willia Singhose Abstract Input shaping is a coand generation ethod that creates coands that resuln low levels of residual vibration. This paper introduces new ethods to design input shapers for ulti-input systes. To justify the new ethods, the ulti-input vector diagra is used. Multi-input shaping design procedures are introduced that resuln shapers that use secondary actuators to reduce vibration and increase robustness to odeling errors. Siulations of a nonlinear obile tower crane odel are used to illustrate the proposed ethod. *! I. INTRODUCTION Input shaping reduces vibration by intelligently shaping the reference signals such that the vibratory odes are not excited [], [2]. To ipleent this ethod, the reference signals are convolved with a sequence of ipulses, called an input shaper. The tiing and aplitudes of the ipulses are deterined using estiates of the syste frequencies and daping. This process is deonstrated with a step coand and a two-ipulse Zero Vibration (ZV) input shaper in Fig.. Although applying input shaping to ulti-input systes can be straightforward, a few researchers have attepted to optiize the extension of input shaping to the ulti-input doain. Soe of the first published ethods for optiized ulti-input shaping relied on a zero-placeent ethod to solve for ultiple ipulse sequences that are used with each input [3], [4]. Later wors presented iproveents on this initial algorith [5], [6]. The original ethod, and the subsequent iproveents, need soewhat accurate inforation about the influence of each input on the vibratory odes of the syste. The firsproveent to this algorith reoved the ipulse aplitude constraints fro the original solution procedure, then scaled the resulting solution according to rigid body constraints [6]. This ethod generated arginally faster shapers than the original approach. An alternative iproveent included additional robustness constraints, entioned in the original wor, but nopleented [5]. This wor also proposed an adaptive ulti-input shaping routine to account for situations when one or ore inputs becoe zero. Liitations of this approach included possible actuator saturation and the ability to find a solution that eets both vibration and ipulse aplitude requireents. Other researchers have approached the proble by reforulating the proble as a quasi-convex optiization [7] [9]. In each of these cases, the proble is transferred to the digital doain. Once in the digital doain, constraints All authors are with The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA singhose@gatech.edu Fig. The Input Shaping Process on vibration and ipulse aplitudes are created to for the optiization proble. Additional constraints can be added to the forulation to increase robustness and/or satisfy transient response requireents [9]. Only two previous papers have presented experiental results [9], []. The ethods published thus far present several difficulties for practical ipleentation. The firss that the ajority of the ethods do not account for situations where one or ore of the inputs is not used; the solutions require all actuators to be acting at all ties. A second liitation is the requireent that each input shaper utilize an equal nuber of ipulses, spaced equally in tie. This liits the solution space. Finally, there is no explicit consideration of actuator liits in the ethods published to date. In addition to these liitations, the solution procedures are typically uch less intuitive than those for single input shapers. This paper presents ulti-input shaping ethods that use ultiple inputs to achieve perforance thas not possible with a single input. Multi-input shapers are generated using a ulti-input vector diagra, an extension of the vector diagra design technique used for single input systes []. These graphical design ethods are discussed in Sections II and III. This paper will exaine several classes of ultiinput shapers. The first case uses secondary, copensating inputs to overcoe the structural liitations of one priary input. Section IV outlines the design of such shapers using a ulti-input vector diagra. The second class of ulti-input shapers, introduced in Section V, use additional actuators to iprove the robustness of the control syste. A obile crane is used as the priary application exaple 978--4244-4524-/9/$25. 29 AACC 84

A A 2 A 2!t2 2 A 2 =.5 2 A = -.5 t 2 A A =. Fig 2. Representing Ipulses on a Vector Diagra A 2 A R! R A Fig 3. Resultant Vibration Vector fro Adding Ipulses of ulti-input shaping in this paper. However, the ethods can be applied to other ulti-input systes and are particularly well suited to overactuated systes, such as ultistage positioning systes or hard dis drives. However, the ethods presented in this paper assue a linear (or near linear) syste. Extensions of the ethods to systes that are highly nonlinear or non-stationary ay require additional odifications. II. VECTOR DIAGRAMS Due to the nonlinear nature of the constraints used to for input shapers, finding a solution to a coplex forulation can be difficult. One tool that sees to siplify the tas is the vector diagra []. The vector diagra represents the vibration caused by an ipulse as a vector. The vibration induced by an ipulse sequence is represented by the su of the sequence s representative vectors. The vector diagra can serve as both an input shaper analysis tool, as well as an input shaper design tool. The process of representing an ipulse sequence on a vector diagra is illustrated in Figure 2. Each ipulse is plotted on the vector diagra in polar coordinates. The agnitude of each vector on the plos siply the ipulse agnitude. The angle of the i th vector is: θ i = ω () where ω is the vibration frequency, and is the tie of the i th ipulse. The tie of the firspulse is always zero, so the resulting angle is zero as well. To plot a negative ipulse, the vector siply points inward to the origin instead of outward. The angle is plotted in a anner just lie positive ipulses. Negative ipulses are typically indicated with dashed lines. To calculate the residual vibration caused by a sequence of ipulses, the representative vectors are sued, as shown in Figure 3. The agnitude of the resultant vector, A R, is Fig 4. Multi-Input Vector Diagra proportional to the agnitude of the residual vibration caused by the ipulse sequence. The angle of the resultant, θ R, is equivalent to the phase shift of the residual vibration relative to vibration fro a single ipulse applied at tie zero. The vector diagra can also be used as an input shaper design tool. For exaple, a third vector can be added to the two shown in Figure 3 to produce zero vibration. This third vector is placed opposite of A R. When is placed this way, the three vectors will su to zero, indicating the ipulse sequence will excite zero vibration at the design frequency. III. MULTI-INPUT VECTOR DIAGRAMS This section presents an extension of the vector diagra to the ulti-input doain. For clarity on ulti-input vector diagras, vectors ust not only be designated according to their place in the ipulse sequence, but also according to the input they are used to shape. Vectors are labeled A i, where represents the input they apply to and i represents their place in the ipulse sequence for thanput. Figure 4 shows an exaple vector diagra for a ultiinput shaper. The firsnpus applied to the syste without any odification. This can be represented on the vector diagra as a unity agnitude vector at tie zero, labeled A. To for an input shaper that results in zero vibration, vectors 2 A and 2 A 2 are added to the diagra. Vector 2 A is located at tie zero and has an aplitude of.5( A ). Vector 2 A 2 is located at tie ωt 2 = π and has an aplitude of.5( A ). The su of vectors A, 2 A, and 2 A 2 is a resultant vector of zero length, indicating that the su of these two ipulse sequences will resuln zero vibration. The vectors in Fig. 4 were chosen to iic the effect of a ZV shaper. Any input shaper utilizing ultiple actuators will be qualified with MI to indicate is a ulti-input shaped coand. The shaped coand created fro the su of inputs will be included after the MI designation. For exaple, if ultiple actuators are cobined to create a ZV shaped coand, as in Fig. 4, it will be labeled MI-ZV. Just as with a single-input vector diagra, there are an infinite nuber of choices to create ipulse sequences that resuln low levels of vibration. Fig. 5 shows another such choice. In this case, the vectors were all chosen to be unity agnitude (UM), and the sequence was chosen to atch a UM-ZV shaper [2], [3]. Due to the large nuber of possible solutions conceived using vector diagras, it aes an excellent tool to develop 85

2 A 2 2 A Force (N).5.5 -.5 u u 2 - MI-ZV Shaped u +u 2 - MI-ZV Shaped A Fig 5. Multi-Input Vector Diagra Unity Magnitude Ipulse Constraints x x 2 u u M M 2 2 c Fig 6. Siple Multi-Input Multi-Output Model ulti-input shapers. It provides a graphical representation of the ability to change ipulse aplitudes and tie locations to atch physical syste requireents. It should be noted that ulti-input vector diagra exaple above assued that each ipulse sequence was shaping a coand that equally affected the vibratory ode. Proper scaling of the vector solutions and/or the inclusion of additional vectors allows the diagra to represenpulses applied to ultiple inputs with unequal excitation effects. Also, the exaples in this paper assue zero daping. Daping can be incorporated into ulti-input vectors diagras using the sae ethods as single-input vector diagras []. IV. DESIGN USING MULTI-INPUT VECTOR DIAGRAMS This section presents design ethods utilizing the ultiinput vector diagra that can develop ulti-input shapers for a wide variety of applications. The ethods also provide the necessary tools to create shapers that are able to account for varying contributions between inputs and result in low vibration coands. A two-input, single-vibratoryode syste will be used first to deonstrate the use of the ulti-input vector diagra. Exaples fro a full, nonlinear obile tower crane odel are then presented. A. Copensating Inputs One category of ulti-input shapers is the class of shapers where the priary input driving the syste cannot be properly shaped to eliinate vibration. However, a secondary input exists and can be properly shaped to cancel the vibration induced by the priary input. This ethod is particularly applicable to systes that are redundantly actuated, such as obile cranes. For exaple, one priary actuator ay be used to position the base of the syste, while the secondary input(s) copensate for the vibration caused by the base otion. To deonstrate the ulti-input shaping principles, the two-ass-spring-daper odel in Fig. 6 is utilized. The Displaceent () - -.5 2 4 6 8 Tie (s).8.7.6.5.4.3.2. (a) Inputs Bang-Coast-Bang (u ) MI-ZV Shaped 2 4 6 8 Tie (s) (b) Response of x 2 Fig 7. Responses to Unshaped and MI-ZV Coands inputs are u and u 2 and the outputs are the positions of M and M 2, x and x 2, respectively. The two asses are connected via a spring with spring constant and daper with daping coefficient c. Is easy to see that the inputs affect the outputs identically. A bang-coast-bang input fro u, shown as a solid line in Fig. 7(a), will excite significant vibration in the syste. This can been seen in the response of x 2, as shown by a solid line in Figure 7(b). Suppose that u cannot be converted fro an on-off function to a properly shaped function that produces low vibration. This could result fro actuator liitations or liitations on the actuator force resolution. The ulti-input vector diagra can be used to properly design the shaper for the second input, u 2, that copensates for the liitations of u. If each inpus shaped according to the ulti-input vector diagra shown in Fig. 4, then the su of the two inputs, u + u 2, is a properly shaped input. Note that u is convolved with a single, unity agnitude ipulse, which eans is unchanged. The bang-coast-bang function is then convolved with the ipulses 2 A and 2 A 2 to get the u 2 input. As seen in Fig. 7(a), the cobination of u + u 2 gives a properly shaped coand. The response shown as a dashed line in Figure 7(b) deonstrates the effectiveness of this new Multi-Input ZV (MI-ZV) shaped coand. B. Scaling for Input Contributions This section presents a ethod to scale the ulti-input vector diagra solutions to create ipulse sequences that account for the influence of each actuator. This procedure is 86

L x 2 x 3 x Displaceent () 2.5 2.5.5 Unshaped MI-ZV - Unscaled MI-ZV 2 3 4 5 Tie (s) Fig 9. The Effects Multi-Input Shaping Ipulse Scaling Fig 8. Siple Rotational Model first deonstrated on an exaple syste. Then, the general ethod is outlined. The odel in Fig. 8 represents a case where inputs have different effects on the output. It consists of a translational input, x, attached to a assless bea of length L. The second input to the syste is the rotation of this bea, θ. A spring of stiffness is attached to the end of the assless bea. The other side of the spring is attached to a ass,, only capable of translational otion, x 3. This odel is very siilar to a obile tower crane in which the centripetal effects of jib rotation are ignored. The state-space for of the linearized equations of otion for the syste is: x = A x + Bū (2) [ ] x x = x + L (3) θ The odel has two inputs, x and θ, that do not equally affect the vibratory response of the syste. The coefficients of the B atrix reveal how each input affects the syste. As a result, they also provide insighnto how ulti-input shaper ipulse sequences should be scaled. The design of a ulti-input shaper for this syste begins with the ulti-input vector diagra. One choice of ipulses was shown in Fig. 4. For this syste, the two sequences of ipulses fro this vector diagra are: Ai x : = (4) Ai.5.5 θ : = π (5) If these sequences are used directly on the syste, without scaling, then the ulti-input shaped coands actually excite ore vibration than the unshaped, as seen in Fig. 9. This effect occurs because the vector diagra in Fig. 4 assued that each inpunduces the sae aplitude of vibration, but, in this case, they do not. To properly account for each input s influence on the output, the ipulse aplitudes of each sequence are scaled according to the inverse of its corresponding entry in the B atrix. When this scaling is perfored, the sequences becoe: [ ] [ ] Ai x : = (6) Ai.5 L.5 L θ : = (7) π The sequences now properly account for the influence of each input on the vibratory dynaics of the syste. However, the ipulse aplitudes no longer su to one. This eans that the DC gain of the shaping process is not one. To correct the DC gain for this case, each ipulse is ultiplied by, the inverse of the su of the ipulses fro the two sequences. The scaled ipulse sequences becoe: Ai x : = (8) Ai.5 L.5 L θ : = π (9) The response of the syste, using these two ipulse sequences is shown by the dashed line in Fig. 9. The MI-ZV shaped response now exhibits no residual oscillation. For systes with two inputs, this process is easily copleted by scaling one of the ipulse sequences by the ratio of coefficients fro the B atrix. For this exaple, the θ ipulse sequence could be scaled by: B(2, ) B(2, 2) = L = L () To generally apply the ethods described above, the syste ust be represented in bloc diagonal for: x = A x + Bū () ȳ = C x where, [ A = blocdiag[a l ]=blocdiag B = [... bloccol b l b l 2... b l ] ωl 2 (2) 2ζω l ], l =,..., p 87

Trolley Jib Base v base Fig. Multi-Input Shaping Exaple Configuration Base Displaceent ().35.3.25.2.5..5 Base 9 88 86 84 82 8 -.5 78 2 4 6 8 Tie (s)! Jib Angle (deg) Fig. Multi-Input Shaping Coands for Full Tower Model where there are inputs to the syste, p is the nuber of odes, and ω l and ζ l represent the frequency and daping ratio of the l th ode, respectively. Placing the syste in this for allows the influence of each input on each vibratory ode to be deterined. To eliinate vibration for ultiple odes of vibration, vector diagras representing each ode ust be exained siultaneously. Using this ethod, ulti-input coands were generated for a full, nonlinear obile tower crane odel, using a cobination of base otion and jib rotation [4]. To establish a benchar coand and response, only base otion was used with the jib held perpendicular to the base velocity, as shown in Fig.. For the MI-ZV shaped case, the sae base coand was used, but the jib was rotated to eliinate the vibration caused by the base otion. For both cases, the suspension cable length, l, was set to.245 and the trolley position, r, at.8. Both cable length and trolley position were held constant during the siulations. The natural frequency of oscillation is approxiately.22hz in this configuration. The base and slewing (θ) coands are shown in Fig.. The unshaped and MI-ZV shaped payload responses are shown in Fig. 2. The unshaped base otion alone causes significant vibration in the β-direction, tangential to the jib, as seen in Fig. 2. Because the base otion is exactly perpendicular to the jib, no radial vibration is excited. The MI-ZV shaped case, utilizing jib slewing, is also shown in Fig. 2. The vibration in the β-direction, excited by the base otion, is eliinated. However, the centripetal effects of rotating the jib have excited a sall level vibration in the radial direction. Despite this, the total aount of vibration is draatically reduced. The vector diagra is a powerful tool to visualize the vibration caused by an ipulse sequence. However, it presents only one ethod to satisfy the vibration constraints. A ore Angle (deg.) 5 5-5 Unshaped! (Tangential) Shaped! (Tangential) Shaped " (Radial) 2 4 6 8 Tie (s) Fig 2. Full Tower Model Responses to Unshaped and MI-ZV Coands general ethod ay do so via an optiization routine. To suarize, the generalized algorith for the procedure detailed in this section is: ) Model the syste in bloc diagonal for 2) Siultaneously solve for ipulse sequences to satisfy vibration constraints for each vibratory ode 3) Scale resulting ipulses according to the inverse of corresponding B-atrix entries 4) Chec that the ipulse sequences satisfy the rigid body constraints, and, if needed, apply unifor scaling to satisfy In this generalized extension, the ethod becoes siilar to those presented previously in the literature [3] [9]. V. ADDING ROBUSTNESS VIA SECONDARY ACTUATION In addition to cobining inputs to create low-vibration coands when one input cannot be properly shaped, ultiinput shapers can also be used to increase the robustness to odeling errors. The illustration of these ethods will utilize the two-ass spring daper syste shown in Fig. 6. Suppose that the firsnput, u, is only capable of the nonrobust ZV shaped coand. This scenario could result fro a nuber of issues, naely a liited nuber of possible actuation states or other actuator liitations. In this case, the second input, u 2, can be designed such that the nonrobust ZV coand of u is augented by the secondary input, u 2, to create a ore robust coand. This process is shown in Fig. 3 for an initial ZV-shaped coand thas converted to a robust Zero Vibration and Derivative (ZVD) [2] coand, u s, by the addition of a negative pulse in u 2. The vibration at the design frequency will still reach a theoretical iniu of zero, as seen in Figure 4, but the coand signal is ore robust to errors in natural frequency. The benefit of the added robustness is shown in Figure 5, which shows the response of x 2 when there is a 2% odeling error in the frequency. The vibration resulting fro the MI-ZVD coand reains well below that of the ZVshaped case, illustrating thas ore robust to errors in frequency. VI. CONCLUSION This paper presented ethods to design ulti-input shapers that utilize design techniques siilar to those for 88

u ZV us u 2 ZVD Fig 3. Copensation for Non-Robust Coands Utilizing Redundant Actuation Displaceent ().8.7.6.5.4.3.2. Unshaped ZV Shaped MI-ZVD Shaped 2 4 6 8 Tie (s) Fig 4. x 2 Responses to Unshaped, Base ZV-Shaped, and MI-ZVD coands single input shapers. The ulti-input vector diagra was introduced and used to design ulti-input shapers. Methods of scaling these ulti-input vector diagra solutions to account for the influence of each input on the vibratory dynaics were shown. In addition to creating ulti-input shaped coands, a ethod was introduced that uses secondary actuators to increase the robustness of a single inputshaped coand. Siulations of a nonlinear obile tower crane illustrated ey aspects of the proposed ethod. Displaceent ().8.6.4.2 Unshaped MI-ZV -2% Error MI-ZVD -2% Error 2 4 6 8 Tie (s) Fig 5. x 2 Responses to MI-Shaped Coands with 2% Frequency Error [6] C. F. Cutforth and L. Y. Pao, A odified ethod for ultiple actuator input shaping, in Proceedings of the 999 Aerican Control Conference, vol., 2-4 June 999, pp. 66 7. [7] M. D. Baugart and L. Y. Pao, Discrete tie-optial coand shapers and controls for ulti-input ulti-output systes, in Proceedings of 22 Aerican Control Conference, vol. vol.3, May 22, pp. 2279 84. [8], Discrete tie-optial coand shaping, Autoatica, vol. 43, no. 8, pp. 43 49, 27. [9] S. Li, H. Stevens, and J. How, Input shaping design for ultiinput flexible systes, Journal of Dynaic Systes, Measureent and Control, vol. 2, no. 3, pp. 443 7, 999. [] J. Vaughan, W. Singhose, P. Debenest, E. Fuushia, and S. Hirose, Initial experieents on the control of a obile tower crane, in ASME International Mechanical Engineering Congress and Exposition, Seattle, Washington, 27. [] W. Singhose, W. Seering, and N. Singer, Residual vibration reduction using vector diagras to generate shaped inputs, ASME J. of Mechanical Design, vol. 6, pp. 654 659, June 994. [2] W. Singhose, N. Singer, and W. Seering, Tie-optial negative input shapers, J. of Dynaic Systes, Measureent, and Control, vol. 9, pp. 98 25, June 997. [3] S. S. Gurleyu, Optial unity-agnitude input shaper duration analysis, Archive of Applied Mechanics, vol. 77, no., pp. 63 7, 27. [4] J. Vaughan and W. Singhose, Modeling and control of a obile crane syste, in The Third International Conference for Advances in Mechanical Engineering and Mechanics, Haaet, Tunisia, 26. ACKNOWLEDGEMENTS The authors would lie to than Sieens Energy and Autoation, Boeing Phanto Wors, and the Hirose- Fuushia Laboratory at the Toyo Institute of Technology for their support of this wor. REFERENCES [] O. Sith, Feedbac Control Systes. New Yor: McGraw-Hill Boo Co., Inc., 958. [2] N. C. Singer and W. P. Seering, Preshaping coand inputs to reduce syste vibration, Journal of Dynaic Systes, Measureent, and Control, vol. 2, pp. 76 82, March 99. [3] L. Pao, Input shaping design for flexible systes with ultiple actuators, in Proceedings of the 3th World Congress International Federation of Autoatic Control, San Francisco, CA, USA, 997, pp. 267 72. [4] L. Y. Pao, Multi-input shaping design for vibration reduction, Autoatica, vol. 35, pp. 8 9, 999. [5] M. D. Baugart and L. Y. Pao, Cooperative ulti-input shaping for arbitrary inputs, in Proceedings of 2 Aerican Control Conference, vol., June 2, pp. 275 28. 89