ON DYNAMICS AND CONTROL OF MULTI-LINE; FLEXIBLE SPACE MANIPULATORS W. Gawronski, C.-H.C. Ih, and S.J. Wang Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 Jet Propulsion Laboratory, California Institute of Technology Pasadena, CA 91109 Abstract. In this paper dynamics, inverse dynamics, and control problems for multi-link flexible space manipulators are presented. In deriving the flexible manipulator dynamics the following are assumed: flexible deformations are relatively small; angular rates of the links are much smaller than their fundamental frequencies; nonlinear terms (centrifugal and Coriolis forces) in the flexible manipulator model are the same as those in the ri id body model. These assumptions are reasonable or large space manipulators, such as the space crane. Flexible displacements are measured with respect to the rigid body configuration, for which a linear time-varying system is obtained. The inverse dynamics problem consists of determination of joint torques, given tip trajectory, such that joint angles in flexible configuration are equal to the angles in the rigid body configuration. The manipulator control system consists of the feedforward compensation and feedback control loops. Simulation results of a two-link space crane with large payload show that the performance of this linearized dynamics and control approach is reasonable and robust subject to parameter variations during slew operations. the rigid body configuration. Our simulation results, which include Coriolis and centrifugal forces, show that the solution of the first inverse dynamics problem is unstable and sensitive to model parameter variations. On the other hand, the solution of the second problem is stable, and robust to model errors. The flexible manipulator position control presented in this paper is motivated by the rigid robot control approaches, see Ref. 16. Two types of control loops are implemented, the feedforward control loop and the feedback control loop. Feedforward control torques are precomputed from the flexible inverse dynamics model and applied to the joints. The manipulator tip is expected to closely follow the prescribed trajectory. The error, due to modeling inaccuracy and external disturbances, is compensated by the feedback loops with collocated sensors and actuators. The feedback loops consist of joint control loops and member stiffening loops. A pole mobility factor is introduced in the paper and used for determination of feedback gain. ! Simulation results show good performance in terms of tip accuracy and robustness of the manipulator system. 1. Introduction The study of dynamics of flexible manipulators has been reputed in several papers, see Refs.1-11. In this paper a linear time-varying model of a manipulator is derived. The flexible displacements are determined with respect to the rigid body configuration. Assuming constant parameter values within an appropriately small time interval, a time invariant model is obtained. The parameter values are updated for each time interval. 2. Flexible manipulator dynamics A finite element approach is used to derive the flexible body dynamics. Let qi be the displacement relative to a time-varyin coordinate frame attached to the rigid body coniguration of the manipulator, see Figla. Given the tip a trajectory (and additional constraints for redundant manipulator) the rigid body configuration is unique for every time instant, hence the flexible deformation is uniquely defined. Tbe following assumptions are introduced: 1. Amplitudes of flexible deformations are small relative to the link length. 2. Articulation rates of manipulator links are much smaller than the links fundamental frequencies. 3. Rigid body inertia, centrifugal and Coriolis forces and torques in the flexible manipulator model are determined from the corresponding rigid body model. The inverse dynamics problem for a flexible manipulator is the problem of the determination of joint torques such that the end-effector of the manipulator follows a prescribed trajqory. This problem was presented by Bayo et al. for robot manipNator in frequency domain, by Das, Utku and Wada for trusf, structures in time domain, and by Aasada and Ma , for manipulators in time domain. These fpethods are of iteratwe in nature. Bay0 and Moulin solved linear inverse dynamics problem with Coriolis and centrifugal forces neglected. In this paper we consider two types of flexible inverse dynamic problems. In the first one, joint torques are determined such that the tip trajectory error is annihilated. In the second one joint torques are determined such that the joint angles in the flexible configuration are equal to those in The first two assumptions can be met for many space manipulators by design. For example, the space crane command torque profiles can be designed such that amplitudes of flexible motion are 0 Copyright American Institute of Aeronautics and Astronautics, Inc., 1990. All rights reserved. 725 small"; also, as studied in Ref. 18 and Ref. 19, for 300 ft crane, the rotation rate of each link is less then 0.0024Hz while the link fundamental frequency is 0.8Hz. When a link rotates, its natural frequencies are shifted; when the rotation rates are small, the changes in frequencies can be ne lected. Hence, natural frequency shifts due to lin rotation rates are ignored in this paper. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 'k Nonlinear effects are due to large manipulator articulations and to the presence of Coriolis and centrifugal forces. The effects of large articulations are removed from the derivation of equations of motion with respect to the manipulator rigid body configuration. Elastic deformations measured with respect to the rigid body configuration are linear; hence, linear time-varying system is obtained. For small vibrations, Coriolis and centrifugal forces in flexible configuration are very close to these of rigid body configuration. In recent studies and linF flexible numerical simulations of two manipulators, Padilla and von Flotow presented that for small rotation rates one can model a system with reasonable accuracy by keeping the ';near equations together with Coriolis and centrifugal forces determined from the rigid body motion. The manipulator dynamics is described by the following equations of motion +~p~,t). xe(o)=xe0, ;,co) M(XJ;: = T ~ According to the second assumption, the centrifugal and Coriolis forces in the flexible body configuration are approximately the same as those in the rigid body configuration F,cXo5J F,cx,,i) (4) Introducing (3) and (4) to (1) one obtains M(x);* +C(x); 0. +K(x)x=f(xr,xr,xr) (5) x(0)=xo, i(0)=vo, where In f(xr,xr,xr)=Te 0 0 +FC(x,,;)-M(x);:. q.(5), M(xJ=M(x) is used due to small elastic deformations. The vector f is determined fully from the rigid body configuration. It can be decomposed into vectors of joint torques T and the nodal forces and torques F f=T+F wbere T = P T 0IT, F=[O F:].' equation (5) becomes With the above notation M(xl)i* +C ( X ) +K(xl)x ~ =T(xr,xr,xr) m o +F(xr,xr,xr) 0 0 0. (6) x(O)=xa, Z(0)=vo, =vo0, (1) wbere xe is the generalized displacement vector, To is a vector of external torques, M(xJ is a nonlinear mass matrix, Fe is a vector of nodal forces and torques caused by centrifu al and Coriolis forces as well as elastic de ormation forces. The generalized displacement vector is of dimension n =nr+ne consisting of manipulator joint angles ao, of dimension nr and elastic link displacements q,, of dimension ne, For a given tip trajecto7, the inverse kinematics relates uniquely rigid configuration to time. Matrices in time dependent, and eq.(6) is a linear differential equation f rigid body manipulator eq.(6) am time-variant (74 M(t);O+C(t);+K(t)x=T(t)+F(t) x(0)=xo, i(0)=vo. Finally, with the generalized displacement vector divided into joint and nodal components XT=[eT, qT] Let xr be the rigid body displacement vector, the elastic displacement vector with and x=x;xr respect to the rigid body configuration. For small elastic deformations, the link deformation forces are linear in x. Thus eq. (7a) is re-written with the dropped for clarity, {eT(o) where K, C are the stiffness and dampin matrices, respectively, and Fc is the centrikgal and Coriolis forces vector. The above flexible model, unlike the rigid body model, has two independent angles BIiand e,, and two independent torques, Tfi and Tfi at each joint i, see Fig.lb. The angles i satisfy the relationship: at joint a01.-aA =e,-efi. (2b) T T T qT(o))T={eoqo) , argument t is {BTco) 4T(o))T=(~T,f>' (7b) For free rotating joints, the stiffness matrix is positive semi-definite, Similar statement can be applied to the damping matrix, that is, zero friction results in positive semi-definite damping matrix. The mass matrix, however, is always positive definite since rotary inertia always has non-zero value. 726 I Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 3. Flexible manipulator inverse dynamics A solution of the forward dynamics problem is obtained from Eq.(7): flexible deflections of the tip and links, as well as joint angles are determined given the applied forces and torques. For most applications, however, is desirable to command the tip or other parts of a manipulator to follow a prescribed trajectory. This forms the inverse dynamics problem. In this problem, one determines the required forces and torques such that the tip or other parts of the system follow a specified trajectory. The most obvious way to define the inverse dynamic problem is to keep the tip elastic displacements zero, and to determine joint torques to meet this requirement. Our investigations and simulations show that the solution of this problem is unstable. Thus, another way to solve the flexible body inverse d amics problem is needed. The new approach is t at the joint torques are determined such that the joint angles of the flexible model equal the joint angles of the rigid body model. This approach yields a stable and robust results, and at the same time the tip displacement error is relatively small. In the following we will concentrate on the latter approach. f Consider a manipulator with joint 1 connectin link 1 to the base, and each of the remaining joints connecting two separate links together. Let e,, TI be the angle and torque at joint 1. In the manipulator model, each joint that is connected to the base is associated with iodependent angles Ofi and e,, and independent torques, Tfi, T, not two two 1 0 0 0 1 0 O 1 0 ne where Ik is an identity matrix of dimensions k x k . Note that PIF=F, thus the left multiplication of eq. (7) by P1 gives P~MY +P~C;+P~KX =T+F The equality of joint angles in the rigid (ari in Fig.lb) and flexible configuration (aei in Fig.1b) requires that left and right angles are the same, say 8, (8,=0 for i = l ) Bfi=8,=0~, and Od=0 (9a) Manipulators joint torques are t ically internally reacting with equal and opposite epects on the two attachment points, therefore one independent torque per joint is sufficient, say Tai, such that With conditions (9), vector is now the generalized displacement To 1 In this cas the displacement and force vectors in eq47) are where Next, the transformation of the torque vector is introduced, such that and eq.(8) .- ~. fix +ex +gx =T+F where fi=PIMP2, where e! =P1CP2, =PIKPz, ?' =[TTO0IT(11b) and TT=[TT TT]. In the above equation x, and T, arc unknown variables. Defining the unknown vector p:' x;lT, the forcing vector FT.=[O Fz], and 727 6. Compute T,(r> from q.(l2b). 4. Flexible manipulator control q.(ll) can then be transformed into the following The flexible manipulator is controlled by joint torques T, and forces u, as shown in Fig.2. Joint torques consist of feedforward torque T, pre-computed from the inverse dynamics and feedback torque Tb which is proportional to the error signal a;ar, (deviation of flexible body joint angle from the rigid body joint angle). Flexible displacements of links are damped through the member-stiffening loop, by sensing member bending angles and actuating against the bending torques, in which sensors and actuators are collocated. form Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 or, equivalently and T = = M ~ ~+cl ~ ; : 21;1 +K, From eq.(12), the manipulator equation motion can be rewritten in the state-space form z ~ s Z =AX + B ~ U+ B ~ F ~ , From eq.(l2a) one obtains the flexible body displacements, velocities and accelerations of the links; eq.(l2b) gives the required control torques. of y=cx where u is the control force, and F, represents the disturbance forces (Coriolis and centrifugal). With a feedback loop, let Note that for stable system described by eq.(ll), the control torques of eq.(l2) exist and their amplitudes are bounded. u=Gy. Solutions of the forward and inverse flexible dynamics problems are computationally intensive, since their differential equations are time-variant (e.g. mass matrix is joint an le dependent). A solution procedure is proposed %elow. Since both forward and inverse dynamics problems are described by a similar set of equations (cf. eqs.(7a) and (12a)), only a procedure for solving the inverse dynamic problem eq.(12) is described here. One has to determine the gain matrix G such that the tip trajectory satisfies specified performance criteria. he root locus methodm is used for determining the closed-loop gain G. The gain is computed for both the full system as well as thr reduced order model. The ain for the reduced order model is determined as kllows. For time interval ti, let (A,B1,C) be the state space representation of the system, and its balanced representation be (Ab,Bb,Cb). Limited-time balancing method is used for balancing the system in the finite time interval ti(see Ref.21 and 22). The manipulator model is a slowly time-varying one, the link rotation rates are small, compared with their fundamental frequencies. Thus, the solution of eq.(l2a) can be obtained by assuming piece wise constant values for the matrices in the time-varying equations. Let At be a small time interval, and let i=O,1, ...,m, such that mdt=r, r is the tip fly-time. Denote ti=idt, and time interval ri=[ti, ti+J. For the time interval ri one solves eq.(l2a) with constant matrices fi.(t>, “t), ti), with the initial condition x$t> determined from previous interval, and x$O)=O. Then from eq.(l2b) one finds Tx(7>. For lightly damped flexible structures with separate poles, the matrix Ab is almost block-diagonal with 2 x 2 blocks, see Refs. 23,24,21 Ab= llAKj 112 ~ ~ A b i i ~ ~ z ~IIz9 ~Abjj .. The procedure for solving the inverse flexible dynamics problem is summarized as follows: 1. For gwen tip trajectory compute the rigid body joint angles (which form a rigid body configuration) and torques by a standard rigid body inverse kinematics and dynamics procedure. 2. Use the joint angles obtained in step 1 for time ti to compute the mass, damping and stiffness matrices. 3. Assemble matrices to obtain (12).4. Com Ute the rigid body Coriolis and cenmxgal forces b m the rigid body configuration for time interval ri, and apply them to the flexible model, q.(l2a). 5. Solve q.(l2a) for the time interval ri with the initial condition x,(t>. 728 i j = 1,. ,n2, i # j, n2 =n/2, 11. llz-spctral norm. Matrix Bb consists of two-row blocks BE, and matrix C, consists of two-column blocks CK associated with blocks AM wtl,..., BTb d 9 cb=ICbl .,cml 9. The system pole shift due to output feedback is studied by means of pole mobility factor, scc Appendix. Partition the balanced state variable x,, into two parts: x, and xt, thus x ~ = [ x ~ , x ~where ], x, is a k x 1 vector, and xt is a (n-k)x 1 vector. Partition Ab, Bb,Cb accordingly - - r to the rigid body configuration. with res*t Applying the feedforward torques, the simulated manipulator elastic tip elastic displacements .ire sb;wn in Fig.4, with solid line for the x-Comtx\nmt and dashed line for the z-component. The feedforward torques are shown in Fig.5. One can notice small oscillations during the transition. This is due to the small incompatibility between the models of two sequential time instants. The smaller the time intervals is, the smaller discrepancies will appear. The subsystem (A,,B,,C,) consists of components with high mobility poles, and the subsystem(At,Bt,Ct) consists of components with low . With collocated sensors and actuators t e pole mobility is characterized by the mobility factor pi, defined by eq.(A7) in the Appendix res Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 p,=up, Note that the feedforward torques act on the joints, lowering natural frequencies of the manipulator, especially for the low fr uency modes, see Fig.6. As a result, the crane w% the feedforward torques is softer then the crane without feedforward compensation. (14) With output feedback for (Ab,Bb,Cb) The robustness of the feedforward compensation algorithm to system parameter variations is checked as follows. Let Ma, KO, Co be the nominal values of the mass, stiffness and dampin matrices, respectively; let their actual values %e M=aMo, K=aKo, C=aCo, where a is a scalar. The compensation torques are determined for the nominal parameters, while the manipulator dynamics is obtained for the actual parameters. Let s,=[x,, z,] be the tip elastic displacement of a manipulator for actual parameters, with the applied feedforward torques determined for the nominal parameters. Let sto=[xm, z,,] be the tip elastic displacement for nominal parameters, then p= IIs,II Jlls Jl Qo is a measure of robustness of feedforward compensation. The feedforward control is considered robust if psi1 (tip displacement for the deviated parameters is about the same as for the nominal parameters). The robustness indicator p is determined for various values of the multiplier a ran@ng from 0.5 to 2. The plots of p ( a ) , corresponding to the deviation of the mass, stiffness and damping matrices are shown in Fig.7. The plots indicate that the feedforward control algorithm is fairly robust to stiffness and damping variations (with 0.998<p<1.001 for 100% variations from their nominal values), and less robust to mass matrix variations (with 0.5 < p < 2 for 100% matrix variations from the nominal value, and 0.909 < p < 1.11 1 for 10% variations from the nominal value). u=k C x cl b b diagonally dominant and BtCt small one obtains the following closed loop matrix Ac b‘ A =Ab+kqBbCbsdiag(Ar+kqBrCr,At). This expression shows that eigenvalues of Ar are shifted about the same amount as the related eigenvalues of Ac, while eigenvalues of At remain nearly unchanged. Therefore, the gain obtained. for the reduced order model is nearly the same as that of the full model. Note that the feedback loop not necessarily moves poles of the most observable and controllable components, or components with the highest cost, but those with the highest mobility factor, which is a ratio of square of the component cost and the joint measurez of controllability and observability, that is u,ly,. This will be illustrated in the next section. 5. Applications In-plane dynamics of a two-link flexible manipulator, which is considered as a two-link version of the space crane, Refs. 18, 19, is considered. The two links are assumed identical, see Fi .3a, with the following physical properties: length f = l l 8 l . l in, cross section area A=54.72 in , bending moment of inegia I=14510 in4, modulus of elasticity E=33.8 x 10 lb/in2, shear modulus G=13 x 10 lb/in*, linear mass density mo=0.009631 lb sec2/in. The payload at the tip weighs 40000 lbs. The finite element model of each link consists of 3 beam elements. The mani ulator’s initial is defined by joint angles 4=n14 rad, and rad. The manipulator performs lifting maneuver, with tip vertical movement h=800 inches in 20 sec. The simulation results presented here were obtained for no external disturbances applied to the model. Hence the error a;ar--0 and the member stiffening feedback loop is the only loop that is active. The gain of this feedback loop is determined by the root-locus method for the full and reduced order model. Assuming the output feedback u=ky, where k is a scalar, the plot of the root locus of the full system is shown in Fig.8a. One can scc that middle-frequency poles are significantly moved, while low and high frequency poles remain almost unchanged. The manipulator dynamics and inverse dynamics were simulated for 40 sec time interval. This interval was divided into 21 smaller intervals, the first 20 of them are of length 1 sec, while the last one of length 20 sec (no update is necessary in the last interval, during which the rigid body configuration of the crane is constant). The rigid body configurations, for which the crane model is updated are shown in Fig.3b. As mentioned earlier, the manipulator clastic displacement is measured In the reduced order root-locus technique the system Hankel singular values and component costs are determined and shown in Fig.9a,b. The plots indicate that the low fr uency modes are the most observable and controllab e, and have the highest cost as well. The plot of Fig.8a indicates, cost and high however, that those high 7 729 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 controllability and observability modes are h o s t unchanged when output feedback is applied. Instead, the middle frequency modes are significantly moved. The pole mobility factor defined as a ratio of the square cost to the Hankel singular value, eq.(14), is shown in Fig.9c. The plot indicates that the middle frequency components 5, 6, 7, 8, and 9 have the highest pole mobility ratio. These components have been chosen for the reduced order model. The root-locus plot for the reduced 5 component (10 states) model is shown in Fig.8b. The root trajectories are similar to those of the full order model. 'R.P. Judd, a d D.R. Falkenbw? "Dynamics of Nonrigid Articulated Robot Linkages , ZEEE Trans. Autom. Control, vol.AC-30, 1985, pp.499-502. 7 G. Naganathan, and A.H. Sod: "Coupling Effects of Kinematics and Flexibility in Manipulators", Znt. J. Robotics Research, ~01.6, 1987, pp.75-84. 8 K.H. Low: "Solution Schemes for the System Equations of Flexible Robots", J. Robotic Systems, VO1.6, 1989, pp.383-405. 9Y. Huang, C.S.G. Lee: "Generalization of Newton-Euler Formulation of Dynamic Equations to Dynamic System, Nonrigid Manipulators", J. ~01.110, 1988, Measurement, and Control, pp.308-3 15. The open-loop response (yo), and closed-loop response (y) are compared in Fig.10, for k=2*ld. The performance, defined as p= ((yc((2/((yo((z is good, namely p < 0.01. lop. Tomei, and A. Tornambe: "Approximate Modeling of Robots Having Elastic Links", ZEEE Trans. System,Man, Cybernetics, vol.CAS-18, 1988, pp. 831-840. 6. Conclusions I1 M.A. Serna, and E. Bayo: "A Simple and Efficient Computational Approach for the Forward Dynamics of Elastic Robots", J. Robotic System, ~01.6,1989, pp.363-382. A novel linearized approach for solving flexible manipulator dynamics is proposed in the paper. This approach has greatly reduced the complexity of tbe control design and simulation costs. Based on this approach, the inverse dynamics problem is defined and solved. The forward compensation torques are determined with the joint angles in the flexible body configuration matching the angles in the rigid body configuration. The combined feedforward compensation and feedback control is robust to the model parameter uncertainties and resulted 111 satisfactory manipulator performance. 12 E. Bayo, M.A. Serna, P.Papadopoulus, J. Stubbe: "Inverse Dynamics and Kinematics of Multi-Link Elastic Robots. An Iterative Frequency Domain Approach", Report UCSB-ME-87-7, 1987. I3S.K. Das, S. Utku, and B.K. Wada: "Inverse Dynamics of Adaptive Structures Used as Space Cranes", JPL Znternal Document 0-6489, 1989. 14H. Asada and 2.-D. Ma: "Inverse Dynamics of Flexible Robots ", Proc. I989 American Control Conference, Pittsburgh, 1989, pp.2352-2359. ACKNOWLEDGEMENT This research was performed at the Jet Laboratory, California Institute of Tec ology, under contract with the National Aeronautics and Space Administration. I5 Pr- E. Bayo, and H. Moulin: "An Efficient Computation of the Inverse Dynamics of Flexible Manipulators in the Time Domain", ZEEE Robotics ana' Automation Conference, pp.7 10-715, 1989. 16 J.J Craig, Introduction to Robotics. Mechanics and Control. Addison-Wesley, Reading, 1988. REFERENCES 'C.E. Padilla, A.H. von Flotow: "Nonlinear Strain-Displacement Relations in the Dynamics of a Two-Link Flexible Manipulator", Report SSL #6-89, MIT, 1989. 'W. Gawronski: "Detection of Flexible Deformations in Space Crane Dynamics", JPL Znt. Document, Engineering Memo, EM 343 -1149, 1989. 'M.A. %ma, and E. Bayo: "A Simple and Efficient Computational Approach for the Forward Dynamics of Elastic Robots", Journal of Robotic System, vo1.6, no.4, 1989, pp.363-382. "W. Gawronski, C.H. Ih: "The Space Crane Kinematics, Dynamics and Parametric Study for Control Design (2D Rigid Body Model), JPL Znternal Document 0-6657, 1989. 3P.B. Usoro, R. Nadira, and S.S. Mahil "A Finite Element/Lagrange Approach to Modeling Lightweight Flexible Manipulators", Journal of Dynamic System, vol. 108, 1986, Measurement, and Control, pp. 198-205. "W. Gawronski, C.H. Ib: "3D Rigid Body Dynamic Modeling of Space Crane for Control Design and Analysis", JPL Znternal Document 0-6878, 1989. 'OL. Meirovitch, Dynamics and Structures, Wiley, New York 1990. 4 X. Cyril, J. Angels, and A.K. Misra: "Flexible-Link Robotic Manipulator Dynamics", Proc, I989 American Control Conference, Pittsburgh, 1989, pp.2346-235 1. Control of 21 W. Gawronski, and J.-N. Juang: "Model Reduction for Flexible Structures". In: Control and Dynamic System, ed. C. Leondes, vo1.35, Academic Press, New York 1990. 5 M. Benati, and A. MOKO: "Dynamics of Chain of Flexible Links", J . Dynamic System, Measurement, and Control, ~01.110, 1988, pp.410-415. 22 W. Gawronski, and J.-N. Juang, "Model Reduction in Limited Time and Fr uency Intervals", Int. J . System Science, v01.21, 1 9 8 , pp.349-376. 730 23 C.Z. Gregory, Jr., "Reduction of Large Flexible Spacecraft Models Using Internal Balancing Theory", J. Guidance, Control and Llynamics, vo1.7, 1984, ~p.725-732. Hankd singular values y,, see Ref.25, characterize joint controllability and observability of system components, while coats, ai, see Ref.26, characterize the participation of the components in the system output. They may be considered as reasonable measures of pole mobility, therefore we relate them to the pole mobility factor. For flexible structures, see Refs.21 or 24, "W. Gawronski, and T. Williams: "Model Reduction for Flexible Space Structures", J . Guidance, Control and Dynamics, vo1.14, 1991. 2s K. Glover: AU Optimal Hankel Norm Approxinbfbtions of Linear Multivariable systems and Their L Error Bound", Znt. JoumZ of Control, ~01.39, 1984, pp. 1115-1193. where Ai=2Cioi is a half power frequency, hence, from (A5) and (A6) one obtains Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 %.E.Skelton, and A. Yousuff "Component Cost Analysis of Large Scale Systems", Int. Journal of Control, vo1.37, 1983, pp.285-304. Pi =<IYi=yiAi =uidd =0.5YiAi =YiCioi where Yi=IIH(o,)l(, is a spectral norm of the output at frequency o=oi. The determination of Y, and A , for a single-input single-output system is shown in Fig. 11. APPENDIX. Pole mobility for symmetric flexible structures Consider a linear system represented by a triplet (A,B,C), where A is n x n , B is n x p , C is qxn, and n is an even integer. A linear system with matrix A having separated complex poles with small real parts exhibip characteristics of a flexible structure. If C=B S, where S=diag(l,-1, ...,1,-1) is a sign matrix, then the system (A,B,C) is said to be a symmetric one. For example, any flexible structure with collocated sensors and actuators is a symmetric system. Furthermore, let (A,B,C) be a balanced representation of a symmetric flexible structure. In this case, matrices A and BC are almost block-diagonal with 2 x 2 blocks, compare Ref.21 and 24, A 1diag(Ai), . BC Idiag(B,Ci), i = 1,. . $2 One can see clearly that the pole mobility depends neither on system controllability and observability properties, nor component cost. It rather depends on their ratio, or combination of Hankel singular value and half power frequency, M combination of cost and half power frequency. Although the above properties were derived for symmetric flexible structures, they are also true for broader class of flexible structures, as long as the product BC, in the balanced representation, is almost block diagonal. For single-input single-output system satis es ta9y hs requirement, and many non-symmetric systems as well. exmte, (Al) where n2=n12, and Bi is two-row block of B, Ci is two-column is block of C, oi is the i-th natural frequency, i-th modal damping coefficient. In the balanced ral norms of those blocks are representation q u a l to each ot er, lIBil12=IlCiIIz. ci SF The output feedback loop with a scalar gain k results in the closed-loop matrix Ac=A+kBC, or A Idiag(A,), where The almost diagonal property of A and BC allows one to characterize the shift of the i-th pole, per unit gain by the pole mobility factor pi Using (A3), (A4) becomes Pi - (A7) llBiCiIl2 731 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 1200 joint 2 Fig.1. aentrrlizod manipulator coordinates of a flexible Fig.3. Two link space crane and its configurations during slew maueuver Fi.2. Plexibk manipulator control 0013CCpt 732 1.8 - ., 8 1.6..... c) 0 ;a” 1.4- mass A cl) a 12- stiffness and damping ... : 1 1- 0 a -20Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 .r( c) 7.8 - 0.6 - .......... ... ... ..... ...... ................. 0.4 -30 time, stc Fig.4. Tip elastic displacements Fig.7. Robustness indicator xi07 I, full order model; 1500 loo0 ....... ..... .......... . -1sOo I 0 5 10 15 20 25 30 35 40 --2OOo -1600 time, SeC -1400 l -1200 o -lo00 -500 0 o -800 W 0 Re Fig.5. Feedforward joint torques 1500 - loo0 - lo’ reduced order m~Ie1. without torques ............. ... - .... -............ J Fig.6. Crane natural frequencies Fig.8. Root-loci for models 733 full and reduced order Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3396 ‘ -6 0 -30; 5 10 15 20 2.5 30 35 30 5 10 15 20 25 30 35 40 time, x10’ 25, 2 I I Sac 1 I n Fig.10. Tip elastic displacement for feedforward control only (solid line) and for feedforward and feedback control (dashed line) component no Fig.9. Costs, HanLel singular mobility for tbe crane model values, and 2 pole I I 3 4 5 I I I l l 6 7 8 9 1 0 20 0 Fig. 11. Determining Hankel singular values from a SISO transfer function plot. 7 34

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