EXPERIMENTAL RESULTS IN THE VIBRATION CONTROL OF A HIGHLY FLEXIBLE MANIPULATOR

Transcription

1 EXPEIMENTA ESUTS IN THE VIBATION CONTO OF A HIGHY FEXIBE MANIPUATO Andrew Plummer School of Mechanical Engineering, University of eeds, eeds S 9JT, UK. A..Plummer@leeds.ac.uk oland P Sutton School of Mechanical Engineering, University of eeds, eeds S 9JT, UK. David A Wilson Department of Electronic and Electrical Engineering, University of eeds, eeds S 9JT, UK. George D Halikias Department of Electronic and Electrical Engineering, University of eeds, eeds S 9JT, UK. ABSTACT The aim of this study is to investigate the motion of a two-ais single-link fleible manipulator. In particular the motion in the vertical plane is modelled, and a mied sensitivity H controller is designed. This controller is found to give a better tracking response than a classical controller, and can be designed to be robust to payload variation. However, careful choice of weighting functions is required to ensure good performance. INTODUCTION A robot manipulator which has been designed to reduce weight and improve speed, or to have a particularly long reach, is liable to ehibit significant structural fleibility in one or more links of the arm. If neglected this will cause vibration and static deflection. Thus the fleibility must be modelled and controlled if the manipulator design is to be viable. Many researchers have addressed the problem of controlling fleible manipulators over the last twenty years. Initial studies concentrated on the motion of a single link fleible arm driven in the horiontal plane. atterly multi-link manipulators have been investigated, still largely in the horiontal plane; this presents a comple non-linear vibration control problem. In the present study a two-ais single link manipulator is considered, driven about a revolute hub in both vertical and horiontal planes. The objectives are: to model the manipulator, including any interactions between horiontal and vertical motions to design a robust H-infinity controller to tackle the particular difficulties encountered in controlling the manipulator in the vertical plane under the influence of gravity. This paper focusses on motion in the vertical plane, and follows on from simulation results contained in [1]. A number of H controller design approaches have been proposed for fleible manipulators. en et al [] developed a mied sensitivity H controller that showed considerable promise. The majority of the work that has been conducted has concentrated on single payload loading conditions. However, Cashmore [] used an uncertainty model to take into account some variation in payload conditions so utilising the robustness of the H controller. ecent work on H controllers for fleible manipulators has been published by Matsuno & Tanaka [], Estiko et al [5], andau et al [6], and Jovik & ennartson [7]. All of these authors highlight the difficulties associated with choosing the weighting functions. EXPEIMENTA FEXIBE MANIPUATO An eperimental single link two degree of freedom fleible manipulator has been constructed (Figures 1 and ). The two degrees of freedom allow the end effector to be moved in the horiontal and vertical planes. Table 1 contains component specifications. FIGUE 1: Eperimental fleible manipulator

2 Fleible link θ fle Hub θ tot θ hub FIGUE : Fleible manipulator schematic FIGUE : The components of link tip position TABE 1: Eperimental manipulator specifications Fleible link ength () 1. m Mass (M b ). kg Offset from motor aes().6 m Fleural rigidity (EI) 7. Nm Payloads Mass (M p ) variable from.75 kg to.75 kg Hub Horiontal inertia (J hy ).68 kgm Vertical inertia (J h ).9 kgm Torque motors Maimum torque Nm otor inertia.1 kgm Torque constant 1. Nm/A Maimum control signal ± 8.5V Encoders ine count 6 (incremental) esolution ( counting). 1 - rad Potentiometers inearity ±.% inearity wrt link tip position ± 1 - rad Interface A/D (-1V differential inputs) 1bit D/A ( ±1V outputs) 16 bit Encoder interface card bit The fleible link is an homogeneous, cylindrical, aluminium rod with constant properties along its length. The motion of the link is not artificially constrained in any plane. The link is symmetrical about its longitudinal ais, and this ais intersects both horiontal and vertical drive aes at the hub. Each of the interchangeable payloads is symmetrical about the longitudinal ais of the link, with centre of gravity positioned coincident with the link tip. The vertical and horiontal displacements of the tip of the link are measured. Each measurement is composed of two parts as shown in Figure. Firstly, the position of the link tip in relation to the hub ais is measured using a precision potentiometer (θ fle ). This is combined with the angle of the hub relative to earth aes, measured by an incremental optical encoder (θ hub ). These signals are interfaced to a PC which is used to implement control algorithms. MODEING ink element model The Newton-Euler method for modelling transverse vibrations in beams will be used. Only motion in the vertical plane will be considered. The approach broadly follows that of Meirovitch [8], with etensions to accommodate gravitational effects. A three dimensional free body diagram of a small element, length Δ, of the fleible link is shown in Figure. In the figure, shear forces are denoted by V, and bending moments by M. Neglecting angular inertia and shear deformation of the element, and neglecting second order effects, the following link element equation for vertical motion can be derived: Zt (, ) mg EI = m Zt (, ) n (1) where Z(,t) is the vertical displacement of the element in earth aes, and g n is the gravitational component normal to the ais of the element. The angular motion of the link can be represented by linear motion of its elements if the angular deviation is small. M Z V Δ mδg n V V + Δ M M + Δ FIGUE : ink element (vertical plane)

3 Boundary conditions Equation (8) represents a second order system with ero Four boundary conditions can be derived. There is a damping and natural frequency ω. Equation (9) geometric boundary condition at the root of the link: describes the amplitude variation along the link, i.e. the Zt (, ). Zt (, ) mode shape. The boundary conditions for this = () differential equation are derived by substituting equation (6) into () to (5): Taking moments about the hub ais gives a second boundary condition at the root of the link: S ds ( ) ( ) = (1) Zt (, ) Zt (, ) T () t EI + EI T g EI ds + EI d ( ) S ( ) () (11) Zt (, ) = J ds h = J ( ) hω where T is the motor torque, T g is the torque required to support the manipulator weight, and J h is the hub EI d S ( ) = Mpω S (1) inertia in the vertical plane. Two boundary conditions are associated with the payload. The centre of gravity of the payload lies on EI d S ( ) ds = J pω (1) the link ais at. esolving forces acting on the payload vertically: Zt (, ) Zt (, ) EI Mp. gn = M p. () where M p is the mass of the payload. Finally, considering the moments acting about the centre of gravity of the payload: Zt (, ) Zt (, ) EI = J p (5) where J p is the moment of inertia of the payload. Mode shapes A set of mode shapes will be determined by considering the free response of the system. These will then be used to develop the solution to the forced vibration problem. The free response is defined by element equation (1) and boundary conditions () to (5), with the motor torque and gravity terms set to ero. A solution to these equations is assumed to have the form: Zt (, ) = S( Q ) ( t) (6) Substituting this solution into equation (1), and neglecting the gravity forcing term, allows the variables to be separated: 1 d Q () t EI 1 d S. Q () t dt m S. ( ) For the two sides of the equation (7) to be equal for all values of and t, they must be constant.. let this constant be -ω, where ω is real. Hence: d Q () t +ω Q() t = (8) dt d S m ω S = (9) EI Solving equation (9) with these boundary conditions gives a set of mode shapes. Note that mode can be interpreted as a rigid mode, i.e. the solution of these equations obtained with ω =. State space model The general solution of equation (1) with boundary conditions () to (5) is a sum of modal terms: Zt (, ) = S( Q ) ( t) Substituting equation (1) into (1): ms d Q t EI d i() S i ( ) i( ) + Qi() t mg = dt i i (1) d Qi () t + ω i Qi () t msi mg = n (16) dt d Qi () t + ω i dt n (15) Qi () t msi Sj (17) = mg S n j = (7) From equations (9) to (1) it can be shown that the following orthogonality condition holds [9]: dsi dsj m Si Sj + Jh + (18) M S S J ds ds p i j i j + p = δ = = ij ( δ = for i j and δ = 1 for i = j ). ij ij

4 In addition, substituting equation (1) into the boundary From equation (5) the time responses for the individual conditions () to (5) respectively gives: modes are independent. It is epected that in general T T EI ds i EI d S i the higher the frequency the less significant the mode s g + + Q i contribution to the overall response. Hence a finite number of the lower frequency modes, i = to n, will (19) provide a good approimation. So equation (5) can be dsi = Jh Q i used to construct a state space model of finite dimension: = A + B u ( ) EI ds (8) i Qi Mpgn = MpSi Q ( ) i () where T = [ Q Q Q1 Q 1... Qn Q = n EI d S i Q J ds i Q ] (9) i p i (1) 1... Then substituting (11) into (19) gives: ω C dsi dsj... Jh ( Q i + ω i Qi ) = 1... () ( ) T T ds j = ( g) A = ω 1 C1... () : : : : : : and substituting (1) into () gives:... : : : : : : MpSi S ( Q Q ) j ( )... 1 = i + ω i i ()... ω n Cn = MpgnSj and finally substituting (1) into (1) gives: J ds S ds Mg i j p ( Qi + i Qi ) S h g S Mpg mgs ω = () = S B Mg S 1( ) 1( ) h g S Mpg mgs = 1( ) 1( ) (1) Substituting the orthogonality condition (18), and equations () to (), into (17) gives: d Q dt i () t Q t T t T S i ( ) + ω i i () = ( () g ) (5) M g S mg S p n i n i The influence of the weight of link and payload depends on the relative orientation of the gravity vector. The following approimation will be used: gn = g cos( θ tot ) (6) The angle θ tot is the angular displacement of the tip, or payload, above the horiontal. This is a good angle to use as in many configurations of the eperimental manipulator it is the payload weight which is by far the most significant. If the centre of gravity of the hub is at angle θ g above the hub angle, and displaced g from the hub ais, then: T = M g cos( θ + θ ) (7) g h g hub g Sn( ) n Mg S ( ) h g Sn( ) Mpg mgsn u T () t = cos( θhub + θ g ) cos( θ tot ) () and where C yi are damping coefficients included to represent the low levels of structural and mechanical damping which will be present. The following output equation defines three system outputs: the total angular deflection of the link, the hub angle, and the angular deflection due to the fleible motion of the link alone, respectively. y = C () [ ] y tot hub fle = θ θ θ () T

5 S( ) Sn( ).. ( + ) ( + ) S Sn C = ( ) ( ).. S S Sn S ( ) n( ).. ( + ) ( + ) (5) MODE VEIFICATION Simulated and actual responses can be compared to verify this model. As the open-loop plant is only marginally stable, a comparison of open-loop responses is impractical. Hence closed-loop control is required, and a PIV (proportional plus integral plus velocity feedback) controller is used for this purpose. A comparison between simulated and eperimental link tip position responses in the vertical plane is shown in Figure 5. Note that three modes (n =,1,) are simulated, and the payload is.1 kg. A good correlation between the simulated and eperimental responses is seen. It is found that including more modes in the model gives no improvement in the simulated response [9], and so it can be deduced that a three mode model will be adequate for controller design. controller, using the method of Zhou et al [1]. The controller is designed for a nominal payload. Thus a known modelling error results when other payloads are used. The control sensitivity weighting is chosen to be greater than the maimum additive modelling error at any frequency across the whole frequency range. This enables closed-loop stability to be guaranteed across the whole payload range. The design of the controller is discussed fully by Sutton [9]. Figure 6 presents the step response results with the actual plant being the same as the nominal. The results show that the system has a fast response and a small settling time. The performance of the controller is significantly better than that achieved using the PIV controller. The integral action is seen to give ero steady state error after the initial position offset is eliminated. The results presented in Figure 7 are the step responses with the whole range of payloads (see Table 1). The payload varies by a factor of ten, so the plant variation is very significant. Although the H controller ehibits stability robustness across this range as designed, there is distinct variation in the achieved level of performance. Eperimental Simulated FIGUE 5: eal and simulated step responses H CONTOE DESIGN The PIV controller has been designed, by trial and error, to give the best achievable compromise between speed of response, settling time, and steady-state error. However it is apparent that the response is far from acceptable. Thus an H controller has also been designed for this manipulator, with the objective of improving the tracking performance whilst in addition guaranteeing stability despite payload variation A mied sensitivity H controller is used. The cost function consists of weighted sensitivity function, weighted complementary sensitivity function, and weighted control sensitivity function. The sensitivity weighting function is chosen to give an integral action FIGUE 6: Eperimental H control response CONCUSIONS The need for stiff structures in precision dynamic machines has a significant impact on performance. The ability to use lightweight, fleible structures would improve dynamic response, reduce power consumption, and also reduce machine sie, weight and cost. obot manipulators are just one application to which such comments can be applied.

6 Acknowledgements The authors would like to thank both BNF (British Nuclear Fuels plc) and EPSC and for their financial support of this project. FIGUE 7: H controller, varying payload The degree of sophistication required in the controller for a fleible manipulator is dramatically greater than its rigid counterpart. As part of the development towards a control scheme for general fleible manipulators, an eperimental manipulator has been constructed, with a single link which can be driven in horiontal and vertical planes. The effect of gravity, almost always ecluded from other studies, is included in the modelling of this system. As shown for the vertical plane, a close correlation between the response of the model and that of the eperimental system is found. A proportional integral controller with velocity feedback, tuned eperimentally, will give a stable closed loop response. However, with this manipulator, the best balance of performance that can be achieved still gives a very oscillatory response. Hence the motivation to design a model-based controller. A mied sensitivity H controller not only gives the freedom to manipulate the dynamic response for the nominal plant, but robustness to parameter variation can be eplicitly incorporated. In this work the variation of the payload within a known range of masses is considered. The eperimental results show the improved performance with the H controller, and verify that stability is maintained with payload mass variation. eferences 1. Sutton,.P., Halikias, G., Plummer, A.. & Wilson D.A. obust control of a fleible manipulator under the influence of gravity 6th IEEE Conference on Control Applications, Connecticut, October en, K, Obay, H., Tannebaum, A., Turi, J. & Morton, B. Frequency domain analysis and robust control design for an ideal fleible beam Automatica, 1991, Vol. 7, pp Cashmore,. D. Adaptive and obust Control of a Fleible Manipulator System. Ph.D. Thesis, 199, University of eeds.. Matsuno, F. & Tanaka, M. obust force and vibration control of a fleible arm without a force sensor Proceedings of the th Conference on Decision and Control, 1995, pp Estiko,., Tanaka, H. Moran, A. & Hayase, M. H control of fleible arm considering motor dynamics and optimum sensor location IPEC 95, andau, I. D., anger, J., ey, D. & Barnier J obust control of 6 fleible arm using the combined pole placement/sensitivity function shaping method. IEEE Transactions on Control Systems Technology, 1996, Vol., No., pp Jovik, I. & ennartson, B On the choice of criteria and weighting functions of an H / H design problem 1th Triennial World Congress, 1996, pp Meirovitch,. Elements of Vibration Analysis McGraw Hill Book Company td, Sutton, P The control and modelling of a highly fleible single link robot arm. PhD Thesis, University of eeds, Zhou, K., Doyle, J. C. & Glover, K. obust and Optimal Control, Prentice Hall, 1996.