Flexible Robots - A Survey

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FLEXIBLE ROBOTS - A SURVEY K. Desoyer, P. Kopacek, P. Lugner and I. Troch TI'{'/lIIim/ L"llil'Prsit.\" of" I"inlllfl IIl/d Cl/il 'I' nit.\" of I.il/ : . AI/strill

Abstract. Industrial robots and handling devices will be of great importance in future . Today robots are used for various purposes in different industries. Main disadvantages of the robots used today are the very heavy construction, the relatively slow speed and the "unintelligence". Therefore the next generation of robots will be more "intelligent " . Such robots have to be equipped with external sensors giving them additional information about their surroundings. In addition these robots will be faster and therefore they have to be lightweight constructions. The last two features will lead to socalled "flexible" robots which are very complicated to describe mathematically and to control. The paper starts with a short description of the main parts of an industrial robot . After some general considerations on the construction of lightweight robots, some methods for the mechanical modelling of flexible robot structures are discussed and compared. Based on these various models for the dynamic behavior an overview of possible control methods and strategies is given . Keywords. Flexible robots; dynamics; control; literature survey .

These parts are the arm, the drives including the gears, the control computer, the gripper including the gripping device, internal sensors or the position measurement system and external sensors.

INTRODUCTION Today, robots are used for various purposes in large manufacturing companies specialized in distinct fields (automotive industry, electrical industries) as well as in small companies. on of be an in

Arm geometrY, For reaching each point in the working space with a prescribed orientation of the gripper at least six degrees of freedom are necessary. Three of them are usually realized by the arm. These can be either translatorial (T) or rotatorial (R):

Work supported by the Austrian Fund for the Promotion of Scientific Research.

- Arm with three translatorial degrees of freedom, type TTT, cartesian robot. - Arm with one rotational and two translatorial degrees of freedom, type TTR, TRT or RTT, cylindrical robot.

Before starting with considerations "flexible " robots the main parts industrial robots available today should shortly described. The main parts of industrial robot are shown in Figure 1 form of a block diagram.

position measuring system

~ controller I--

t

drives

I

-

T arm

,-----

I---

I objec grippt!r f--Jen viro unit I L_

J

sensors

Fig. 1. Main parts of an industrial robot

23

-

K . Dcs()\"(>r 1'1 al.

- Arm with two r o tational and one translatorial degree of freedom, type TRR, RTR or RRT, polar or spherical robot. - Arm with three rotational degrees of freedom, type RRR, revolute or anthropomorphic robot. For different applications different configurations may be appropriate. A revolute arm (RRR) might be the best for reaching into a tub, whereas a cylindrical arm (TRT or RTT ) might be the best suited to execute a straight thrust between the dies o f a punch press. Gripper and gripping devices. The gripper usually realizes three degrees of freedom. These can also be either translatorial o r An important part of an r otati o nal. industrial robot is the gripping device o r end effector. Senso r equipped "ha nds " are being devel oped.

goals f o r the next generati o n o f r o bots will be a rati o in this dimensi o n. This might be possible o nly by developing lightweight r obots. Their arms are made o f new materials (e.g. fibreglass reinf o rced plasti c) and therefore elastic deformations occ ur during the work o f the r obo t as a consequence of the static and dynamic forces. Nevertheless, the handpoint of the robot has to rea c h each point in the working space with adequate accuracy and without overshooting. This requires new " advanced " control alg o rithms which can be developed only on the basis o f appropriate mo dels for the dynamic behavior. Therefore first of all such mo del s based o n me chanical equati ons h ave to be available. GENERAL CONSIDERATI ONS ON FLEXIBILITY

Driving systems. Each robot articulation requires its o wn driving system.

" Flexibility " in using has two meanings:

Pneumatic drives use compressed air, are of light weight, fast and relatively inexpensive. Unf o rtunately these drive systems are very difficult to control. - Hydraulic drives wo rk with fluids, are more expensive than pneumatic drives, but are much stronger. - Electric drives are used in mos t robots. Typi cal forms are serv o ~)tors, stepping motors, pulse motors , linear soleno ids and rotati o nal solenoids. They are strong, and compa red to other drives they consume litt le energy . But they are expensive The control is easy and accurate.

a) flexibility in perf o rming tasks a wanted property, and b) elastic flexibility of the links and other substructures a disturbing property.

Control system . The control system for a robot is extremely important and usually quite complicated. The main functions of the control system are position control of each link (servocontrol) , - trajectory computati o n, -

sensor processing,

- program interpretati on. Internal sensors . Internal sensors are mounted on each joint f o r giving the actual position in rob o t coo rdinates. Encoders, resolvers or other devices on each robot axis provide current position information . External sensors " Intelligent" robots of the future have to be equipped with sensors. These sensors give the control computer of the robot additional information about the surroundings. External sensors available today may be classified into four main groups: - tactile sensors are used f o r measuring of forces and moments, usually between the gripping devi ce and the ob ject or between the gripper and th e end effector. - proximity sensors prov ide information about presence , distance, speed of approximation They work without contact . visual sensors are of great importance for tasks in connection with assembly operations ( part identification, orientation ... ). They work mainly with TV cameras. - auditive sensors are being developed and serve for speech rec o gnition . This paper deals with one feature of the robots of the next generation, namely their structural flexibility . Today the ratio of payload to weight of an industrial robot ranges from 1:10 to 1:30 and less . Compared with man it is very low - a person is able to have a ratio of 1:3 or more. One of the

industrial

robots

Kinematics, dynamics and control of industrial r o bots have been studied extensively under the assumption that the links can be modelled as " rigid bodies " . This assumpti on i s warranted in the great maj or ity of manipulato rs in use today, socalled " heavy const ru ct i ons" - in order to avoid unt ol erable p os itioning inaccuracies that may be c aused by elastic defle c tions and vibrations. The advantage of such a c onstructi on is that angular encoders at the a ctuat o r shafts or at the joints can be used to get information on the actual position of the end-effector in wo rld coordinates in a purely geometric manner . Theref o re the co ntrol device can use this informati o n directly. But, that is not enough if the deflections of the substructures of a manipulator cannot be neglected anymore . robots Fo r assembling tasks, f o r example, should be faster and therefore lightweight constructions are necessary. Nevertheless, they should have better positioning accuracy. Therefore investigations are made all over the world to gain mechanicalmathematical models and advanced control algorithms for the " next generation" of robots considering the flexibility of their substructures . They will be equipped with internal and external sensors giving them additional information about their own status and their surroundings . Thus, lightweight con structions and flexibility of the components will be consequences of compromises between - higher acceleration and speed, - higher ratio payload mass/moved

total

mass,

- smaller actuators, lower consumption and safer operation reduced masses, - positioning accuracy, - range of working space, - complexity of control system , - total costs, etc . There are three examples for the of moved mass and kineti c energy:

energy due to

reduction

1) the links are characterized by a) their geometric dimensions b) their material (Young's modulus , mass density, permitted maximum stress).

Flexible Robots - A Sun'ey

Both items determine the torsional and bending stiffness on the one hand, and the total mass on the other hand . Considerations in optimizing are presented e.g. in Book and Majette (1983), Kiedrzynski (1986) and Weck, Stave (1986) . 2) Even with todays heavy constructions the rotors of the driving devices with large gear ratios contain a deal of the total kinetic energy of the manipulators (e . g. Springer, Lugner, Desoyer, 1985 , Fig.3) . Here, it seems appropriate to combine rotors and gears so that the kinetic energy of the rotors is reduced to a minimum (for a given maximum of power needed). With the high gear ratios used today the fast rotating rotors attached to moving vibrating links may cause periodic gyroscopic moments which produce additional disturbing vibrations transversal to these vibrations of the links . That are further reasons for keeping the moment of momentum of the rotors low (Desoyer , Kopacek, Troch, 1985 ; Desoyer, Lugner, Springer, 1985). 3) The question of distributing the driving devices to the links or some of them in the foundation, in connection with the driving shafts which are more or less likely to vibrate seems important too. In order to obtain good compromises as mentioned above theoretical and experimental studies in the field of dynamics and control of flexible robots are imperative in consideration of the vast variety of construction possibilities . As computer aided methods are being developed fast , such studies seem quite promising now. This is also shown by the great number of previous publications - and those presented at this symposium - on dynamics and control of manipulators with flexible parts. These remarks now are followed by a short survey of the present trends of research . MECHANICAL MODELLING The robot as a flexible system consists of a number of flexible substructures - links, joints, drives - in most cases of complex shapes and geometrical configuration . For an accurate mathematical description of flexibility more precisely the time variant elastic deformations - of such a structure partial differential equations are responsible together with kinematic and dynamic boundary and initial conditions . Only approximate solutions of this problem can be found using modelling techniques to simulate a more or less idealized substitute . Based on well-known approaches to analyse the elastic and dynamic behavior of flexible structures, especially beams, plates , shells, a variety of different methods are applied in the field of flexible robots. Generally the methods can be classified into the main categories : - kinetostatic methods - dynamic methods : vibration mode approach finite element methods - other or combined methods . The reviewed literature*) is based on recent publications related mainly to the problem of modelling and controlling flexible robots or robot components, but also other structures with flexible parts, like satellites, are included . Some general papers complete the survey. An attempt will be made to characterize the different methods outlining their essential features and their range of application .

25

Kinetostatic methods . The considerations leading to the kinetostatic methods start with the, motion of the robot as a system of rigid bodies . Due to inertial forces and loads small elastic deformations are superimposed to the basic rigid body motion . E . g . the angle of relative motion of a link with respect to the preceding link may be composed of two parts : a part which may vary over a wide range and a second part that is always considered to be small (including its derivatives with respect to time) . The elastic deformations of a link or a drive correspond to those deformations induced by external loads applied at proper positions . As an example the elasticities of links and drives are reduced into the joints . So the revolutionary joint between two links provides an additional degree of freedom ad jointed to its elastic property represented by a torsional spring . For a given basic relative motion of the two links this small additional relative angle can be cal c ulated using the equations of motion for the system. Equally , the elastic deflection of a link, a drive train or a part of complex structure due to a static loading may be used to derive a substitutive spring . With respect to a spatial arrangement , an elastic transfer matrix can be established. In a similar way substitutive dampers may be defined or assumed . For sensor positions coinciding with joints or positions used for determining substitutive spring constants the position coordinates, appro ach vectors, etc . can be provided directly by the numerical calculation . For other positions interpolations have to be done for example by means of functions corresponding to the static deflections. The frequency range encountered with this method depends on the masses attached to the substitutive springs and may vary over a large range . The kind and precision of work that has to be done by the robot will determine the modelling of rigid and flexible parts and so will lead to a problem-oriented restriction of the frequency range . When lightweight constructions of robot components are to be used at high working velocities the vibrations of the components in themselves may become essential. A long robot link formed by a beam can be excited to vibration modes in between the two adjacent joints . The effects of those vibrations cannot be modelled and simulated by kinetostatic methods. But the use of vibration modes or finite element methods can provide the information needed for the control of such systems . Naturally , a higher calculation effort will be necessary for that more detailed information . A comparison of these two different dynamic methods can also show their essential features . Finite element methods , The block diagram of Figure 2 gives an overview of how the modelling of a flexible robot structure can be done by FE-methods . The flexible parts of the robot are modelled by a finite number of elements, their number depending upon the shape of the part and the properties attached to an element . of a beam as a A subdivision , e . g . substitute for a robot link, can be done with finite beam elements. Such an element presents the mathematical description of *)

The authors thank P. Plockinger and Hohenbichler for their assistance .

G.

K. Oesover

26

fl

al.

I. fl exi bI e robot structure I Ilink i+1

, I

I

decomposition into a finite number of elements with appropriate mechanical properties

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.~"O

~



equations of motion for the elements in the local ( eI emen t) frames

QJ ....

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ICO

geometriclinearization: local frames equal to link fixed frame

no transformation into >-------+link fixed frame considering the I oca I elements attitudes

4- -_ en 4- Vl~

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compilation of all link matri ces

I

----4~~----.) iteration

of nonlinear

Fig . 2 . Scheme of a FE-modelling of a flexible robot structure the connection of the displacements and angular deflections and the forces and moments of both ends, the nodes to the next finite elements . In order to be able to describe the vibrations of the link structure, a mass or better a mass matrix has to be attached to this finite beam element . This mass matrix may for example be found using the kinetic energy of the element described by a homogeneous mass distribution and local displacements calculated by the nodal displacements and static shape functions . The mathematical description of such a finite element is generally available and can be included in a more or less computerized generation of the equations of motion. Principally, with the help of other forms of finite elements providing a more or less complicated modelling of local flexibilities and massdistributions, each complex link structure or also a joint can thus be presented in a mathematical formulation based on a sometimes great number of finite elements. The compilation of the finite elements of a link can be done in a geometrically linearized form . In a chain of beam elements the local frames do not take into account the deformation angles caused by the preceding element. The consideration of such influences of the local angular deformations will lead, as a result of the time derivation of the necessary transformation matrices, to a nonlinear matrix representation of the properties of the whole flexible link.

With a further transformation of the link matrices into the reference frame for the global robot motion and their compilation a generally large . set of equations can . be established, the mathematical descript10n of the robot model - a model composed of small elements , where deflections or forces can be given at their nodes . As shown in Figure 2 part of the work to establish such a set of equations can be done off-line with the assistance of computerized algorithms . It seems appropriate to include the transformation from the link frames to the reference frame for the global robot motion - generally an inertial frame into the numerical evaluation for each time step. The solution of the linearized system can be found directly, in the nonlinear case an iteration can be used . In Figure 2 a connection between the two branches is shown in brackets trying to indicate that the linear solution may be used as a first iteration of the nonlinear system. Vibration mode approach . Figure 3 shows an overview of how to use the vibration mode approach to model a system with flexible parts . An analytical description of the possible vibration modes or eigenfunctions of a flexible part can be derived directly from the partial differential equations and the boundary conditions, presupposed the shape

27

Flexible Robots - A Sun'ev

of the part and the boundary conditions are relativelY simple. This is indicated in the second level of the scheme in Figure 3 assuming that the considered links may approximately be modelled by beams.

Avoiding the employment of the partial differential equation for the elastic part, a set of generally orthogonal shape functions may be used which suit the geometric and any given dynamic boundary conditions. This approximation is based on the fact that this set of functions can present a series expansion of the possible vibration modes . The quality and the number of shape functions used in the simulation will determine the differences with respect to the eigenfunction approach.

Using the partial differential equations for the description of the time-spatial bending and/or torsional characteristics and a separation of the variables a set of eigenfunctions with time dependent amplitudes representing the deflections and vibrations of the modelled beam can be found. The mass distribution of the beam is already included .

Each eigenfunction with its time dependent amplitude represents an additional degree of freedom for the robot system. So, together with the rigid body motion - which provides the basic location of links and joints the kinematics of the .whole flexible structure can now be establ~shed . With this kinematic information the equations of motion for each flexible link will be derived directly with respect to the inertial system whereas the FE-methods use local frames first.

The boundary conditions are essential for the shapes of the eigenfunctions - but they present a major problem. For simple geometric and dynamic boundary conditions, e . g. a beam with a fixed and a free end, the eigenfunctions can be stated in an analytic form. More complex conditions, as they will occur in a moving robot structure as a result of the forces and moments transmitted in the joints, are complicated to include if at all.

In the mathematical formulation of the equations of motion of a flexible link in a chain of other flexible links, the generalized coordinates due to the elastic angular deformations of the preceding li~ks will also appear in the highest order t~me derivative . If these influences are neglected by assuming that the rigid body motion and the own flexibility DOF are dominant, the coupling of the highest order derivatives of the flexibility DOF of the

Thus an often made approach uses the approximation of the beam with fixed-free ends . In a chain structure the spatial and angular deflection of the joint at the end of the beam can be calculated providing the position and attitude for the fixed end of the next beam. Of the infinite number of eigenfunctions only those corresponding to the 2 or 3 lowest eigenfrequencies will be taken into account.

Ifl exi b1e

robot s tructurel

single structural element beam shaped link i

I

-1beam shaped link i+11

t

~

eigenfunctions for the deflections with amplitudes as functions of time

I

1 t

) kinematics of robot structure including the OOF of the

1

I

I motion equations of for link

Iwith respect to iinertial system ===----of angular deflections

--

preceding links neglected

I

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deflections~

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+,

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compilation of 1i nk equations: partially decoupling of rigid body motion and elastic deformations

compil ation of link equations leads to coupled nonlinear equations for the system

W

r -·----t----------±---~~ ~ I I solution

I

solution

I

l

~

.....

~ s::

::> 0 c~

'" s::

to ::>

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Fig. 3 . Scheme of a modelling of flexible structures with eigenfunctions

28

K. Desoyer et al.

whole system can be avoided partially. This will lead to an increasing advantage with respect to the evaluation time for increasing numbers of connected flexible links. Assembling the complete equations of motion for the system will provide a set of implicit differential equations, compare e.g. equations (7). Thus, for the numerical evaluation an inversion of a large matrix or an equivalent solution algorithm for a set of linear equations has to be employed at each time step. Some comparative considerations, A comparison of these two dynamic methods with respect to quality of information, calculation and evaluation efforts will always be a problem. The FE-method has a greater flexibility to model links of geometrically complex shapes . But this will make it necessary to increase the number of finite elements and thereby the evaluation time. On the other hand, eigenfunctions or shape functions for complex shapes can be precalculated by FE-methods, analytically approximated and then used in the vibration mode methods. In principle , this method has the advantage that a greater part can be done off-line and the on-line evaluation time is shorter . Using the FE-method no problems occur connecting the links, whereas using shape functions which violate boundary conditions introduces a systematic incoherence into the system . Increasing the accuracy means more shape functions and more precise modelling of the boundary conditions and a higher number of finite elements, or finite elements with more sophisticated properties . That naturally always results in higher evaluation times. Higher accuracy will especially be achieved for lower frequency vibrations. The fact that the application of these methods in the field of robotics with the intention to integrate robot models into the control loops is just at the beginning is shown by the examples investigated in the literature . Caused by the complexity of such flexible structures not more than two or three flexible links, mainly in planar movements, are considered . Modelling of flexibility with shape functions is done using only those corresponding to the lowest two frequencies. But with the conceiveable increase in computer capacity these methods including static, dynamic deflections and vibrations between joints will become very important . At the basis of the kinetostatic method and both dynamic methods there are greater or smaller elements with flexible properties or elements providing flexibility by the kind of connexions. So there is also a transition in the methods applied for modelling flexibilities from simple elements to such with improved properties and to an increasing number of elements. Most investigations which can be appointed to the group "other or combined methods" may also be appointed to one of the other mentioned methods, because they generally also use some segmentation for the robot elements . So e.g. a method models a flexible beam-like structure by rigid body subelements that are connected by hinges or spherical joints and springs and dampers . The off-line preparation for the calculation can then use already established symbolical computer algorithms for rigid body systems . A major problem which, on the one hand, helps but, on the other hand, imposes uncertainties should also be mentioned . The structural damping in each system

influences especially the shapes, frequencies and amplitudes of possible vibrations or the system responses to rapid changes in forces and moments . Generally, it is assumed that for low frequency vibrations the structural damping may be neglected which may not hold for a simplified modelling of links with different internal structures or additional parts - and that high frequency vibrations are suppressed, - which is confirmed by experience. An increase of the structural damping can also be achieved by internal linings of links with vibration insulating material. This may be an improvement which can be utilized systematically with a minimum of additional mass . Especially for longer flexible chains partially used simplifications, e . g . shown in Figures 2,3 , or mathematically suitable but simplified subelements have to be seen more critically. Neglected nonlinearities or dynamic coupling can cause increasing deviations for the position of the end effectors and their approach vectors between the model simulation in the control loop and reality . The components of increasing computer capacity , design configuration of links, joints, drives and the possibilities of control algorithms and system modelling specially for flexible systems have to be optimized to achieve the required high performance of future robots .

CONTROL CONCEPTS As usual. the term 'c ontro l ' has a multiple meaning: It c an c har ac t e rize an o pen-loo p strategy based o n a mo r e o r l ess ac c urate model of the system und e r investigation in this specific cas e the rob o t - or . it refers to a feedb ac k law making use of the control deviation. In s o me c ases. it may also characterize a combinati on of these two major categori e s. i.e. an o pen-l oo p strategy for the gro ss moti o n with an underlying feedback acco unting for small deviations. Except for s ome pil ot pro jects at universities and r e searc h laboratories. r o bot c ontro l of t oday has t o be considered always as an o pen-l oo p c ontro l . Thi s is mainly due to the l ac k o f sensors which are able to measure positi o n and o rientation with respect to the task space with reasonable speed and precision and which are yet robust and c heap enough. Consequently . in practi c al applicati o ns measurements are performed only with respe c t to the configurati o n space. as e . g . j oint angles or joint accelerations . From this follows that the contro l deviation is equally built with respect to the c o nfiguration space . From this may result a closed-loop control in the c onfiguration space but an open-l oo p c o ntro l in the task space. the latter being obv i o usly the more important one. When considering the c ontrol of flexible links one proceeds in many cases quite similar to the rigid body case. Yet , one can distinguish two main approches : The more simpler one proc eeds exactly in the same way as in the rigid b ody case in order to compute the necessary geometric , kinematic or kinetic quantities except for using a mathematical model which acco unts also for deviations c aused by the elastic properties of the arm . This is done mainly by considering the first eigenmodes of each link where each link is considered to be an ideal beam. When proceeding in this way. the question has to be investigated whether all parameters of the robot as e . g . the payload can be identified with satisfactory

29

Flexible Robots - A Sun·ey accuracy. Unly if this is possible the use of such a model is meaningful. Otherwise not only the movement of the joints has to be measured but additional measurements are necessary in order to get information about the deflections. The second - and more involved approach requires additional measurements e.g. by strain gauges or by optical sensors . These measurements are then used to improve the performance ot the control and to compensate deviations caused by the elasticities. But. as these additional measurements in most cases are again performed within the configuration space an open-loop control with respect to task space will result. Well-known and established concepts (Paul. 1982. Whitney. 1969. Luh. IYBO. Wu and Paul. 1982) such as resolved position/rate/force / acceleration control (Rl'lPC. RMRC. RMF'C . are adapted to the new situation by RMAC). accounting for flexibilty of links or joints. Bases for the methods just mentioned are the geometri c relations between task space coordinates collected in a vector x and the corresponding configuration space coordinate vector p. Following the notation of Uenavit-Hartenberg ( 19f,5 )

,r'm,u.

I

I

tor r

rn.

the rigid body case with .. position of a point on link m in terms of the coordinate trame i betore (after) deflection Ix . ) ... position of the origin at m.l frame m in terms of frame i before lafter) deflection

.

1

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ill

1 }

1

;1

number of degrees of freedom IOOF)

...

Book (1979) adds a further transformation which accounts for the deviations due to the flexibility of the links:

x ::: flP) + y instead of 141. These supplementary terms are calculated and the following problems are solved. 1i ) I ii )

I i ii

I

for given p ::: Pit) calculate x ::: X(tl . For given drive moments and forces collected in a vector k Or = kOrIP.t) this inclUdes feedback IkDr(p) I as well as open-loop IkOrlt}) control compute the movement in the configuration space. Pit). as well as the one in the task space. xlt). For given movement x ::: x(tl compute kOrl t I.

During the last years a lot of work was devoted and is still gOlng on to what can be called basic research. Here questions of modelling and controlling flexible one-link arms are investigated and various control concepts are presented. The common lines are that this work is based on the use of the drive moments as control variables and usually no external or additional measurements are considered to get information about the deformation of course. exceptions exist. Among these lines. the work by Judd and Falkenburg (1983). Henrichfreise (19851. Alberts et al. (1985) . Cannon and Schmitz (1984). Hastings and Book (1985). Nelson and Mitra (1986). Sakawa et al.(1985) and de Wit and van den Bossche (1986) are to be mentioned who suggest controls based on optimization concepts. Of course. decoupling plays also an important role. especially the decoupling of 'rigid' and 'elastic' coordinates. Further work to be mentioned along these lines is the one by Walendy and Weber (1985) and by Ohkawa et al.(1986). The latter (which is contained in these Proceedings) differs insofar from related research as the investigations are based directly on a discrete model. Now. the design of feedback controls for flexible arms shall be discussed in some more detail. Consider the flexible link sketched in Fig.4. The deviation due to bending can be described approximately by

I

I

I

x·

.

1.1-

(6 )

1i

i

(2)

11 This results formula ,

I

iI r m.o ir

11

in a modified transformation

;r )m.m I

11

I

l

(3 )

A different approach to this problem is found in Chernous'ko (1981) . Let the geometric relation

x = fIP~)

r

(4)

between task space coordinates x and configuration space coordinates p be given for the case of a rigid body arm. U~der certain assumptions like that the Ilnks are elastic bodies whose deformations 1n the course of the motion are small and can be described by linear theory Chernous'ko (1981) can show that th~ elastic properties can be treated within ~he fra~ework of a rigid model body by lntroduclng extra terms in the equations of motion leading to TOR-a-

Fig.4.

:~()

K. DeSmTr 1'1 Ill.

.i . ,0,. the expansion att",r shap", functions which is olten retered to as finite version of a (generalized) Ritz approach. The spatial func:t ions a l l are either taken to be the eigentunctions ot the eigenvalue problem associated with a linear partial differential equation for the flexible link. e.g. Singh and Schy 11984. 19B!:>. 1 9861. Chassiakos and Bekey (1985). Balas (1978) or. a more general Ritz approach is performed. e.g. Truckenbrodt (1982). using a suitable set of functions which meet th~ boundary conditions. This approach leads to a model for the dynamic behaviour of the robot (which is sometimes directly established) of the torm

Mlz)z + fIZ.2)

1'10.)

where p

z

l"Tb)

q

is a vector collecting the vectors p and q of rigid and elasti~ coordInates in the configuration space. In case not only p but also quantities describing the deformations

Judd and falkenburg 11~831 and - to soma extent also K~rkk~inen and Halme (198~) -. also start with a linearization but perform it around the tra~ectory of the corresponding rigid arm as the reference trajectory. Judd and ¥alkenburg 11983) suggest a linear state feedback controller which is designed via the solution at an algebraic Riccati equation. As example cl one-link arm is considered and it would be interesting to know a bit mare about the computational burden in compar1son with similar methods as e.g. the one suggested by K~rkk~inen and Halme (198~) . especially for arms with more than c·ne 1 ink. A further approach starts with a separation of equations 17) in those describing the rigid body motion. p. and in those describing the flexibility effects given by q. Then. the latter equations. i.e. those for q are linearized. This is baSed on the reflection that only small deviations from the rigid body motion will occur and are to be consl.dered. :;)ingh and Schy (198:'. 1980) apply then nonlinear decoupling to the equations tor the rigid coordinates and develop a stabilizing linear state feedback control for the overall system.

can be measured an output equation

An

V

=:

I"IIZ)

is added.

Then.

control system

(8

J

a control for the nonlinear (·I.~J

in state space forol is

designed by standard control techniques without taking into account the speclal structure of the model. Linear and especially linear. autonomous models are quite attractive in control technique because there exists a well developed theory for analysis and controller design. Une way to derive such a model which is used quite frequently consists in linearization of (7.8) around a constant reference position. This is done e.g. by Henrichfreise (1986) who uses the instrumental design method based on vector optimization. Book et a l .(19751 start also with a linear model but suggest pole assignment as presented by Simon and Mitter 119ti81 or pole sensitivity as design tool for the required control. In connection with these ideas a work by Moritz et al. (1985) should be emphasized. The authors consider linear state feedback based on sensitivity considerations for the eigenvalues and give interesting comments on the limitations due to this linearization. idea of a group of approaches as The main the three discussed in the following e.g. system structure as given in leads to a Fig. 5.

Controller for ngid body motion

Lnterestlng approach was presented by ana Halme (198~1 who also start with linearization of that part ot equations (1) which describes the elasticities Afterwards. a change of coordinates is performed in such a way that a system of decoupled Ilnear oscillators results for ql I. The latter is then discretized with respect to time and a control law is designed by minim1zing a quadratic criterion over each individual time interval. By this. the solution of a Riccati equation can be avoided and a linear time-invariant 'one-step-ahead' controller results for the control of the flexible states. K~rkk~inen

approach to derive a linear model in nonlinear decoupling and techniques. Freund and Hoyer related ( 198U 1 . F'reund (1982).

Another

consists

After performing this decoupling a controller can be designed for the resulting linear system by optimal control techniques or adaption algorithms etc .. Fig. 6. MUller and Ackermann (198bl suggest to use after such a decoupling a multi - level control consisting of linear complete state feedback in connection with observers for the state and the disturbances at the lower level and an adaptive model following control technique at the upper level .

A quite interesting work was presented by Marino and Nicosia (1985) who suggest the use of an adaptive rigid body controller based on model reference for the nonlinearly decoupled system. Then they investigate carefully the elasticity properties by singular perturbation techniques after rewriting equations (7) in the form

i-o.

f 1 (p·p·q·U)

(91

f .. (P.p.q.u) L.

~

..

p

Flexible Arm

q

!

Their investigations yield a controller u

i

Compensator for f deviations due to flexibility :' Fig.5.

= UsIP.P.tl + Uf(P.P.q.q)

where Us controls the slow i.e . the rigid body motions and u f the fast modes i.e. the elastic modes. A fundamental research in this area. see again Fig.b. was carried out by de Luca et al. (19851 who show that for robotic arms

Flexible Robots - .-\ Sun'e,'

-,

z

Flexible Arm

"'1

31

Measurements

... I

,y

Nonlinear

Observer

z

;..

[Jecoupling

t ·

lif necessary I

Gontruller tor .Linear system

Hg.b.

with elasticit1es under certain assumptions a dynamic nonlinear state feedback can be assigned in such a way that a prescribed Jynamic behaviour characterized by a linear r~ference model is achieved. In certain ~ases decoupled chains of integrators or prescribed linear autonomous systems can be obtained in this way. For the latter a controller can be designed by using conventional or modern design methods for linear autonomous systems.

resulting equations (7) a quadratic performance index is added which measures the deviation of the shape of the link from a given time history for this shape. i.e. wdesired(r.tl - Wactuallr.t) The use of the state augmentation technique leads in the usual way to a Riccati equation. After its solution the location of the points is optimized where the point actuators will act.

Sometimes a linear decoupled model can be arrived directly. as it is done e.g. by Jacubasch and Kuntze (19851. But their work ditfers insofar from related investigations as they model also the dynamic behaviour of the actuators, i.e. they use indeed the control voltage and not ~he drive moments as controls. Fig.7. In order to compensate for elastic deformatiuns caused by widerange heavy payloads. they use a control law where not o nly position and velocity are fed back but also position and speed of the motor shaft and the motor torque.

To some extent. this work can be regarded as a continuation of earlier investigations by Hemami 1198b) and especially by Balas (1978) who equal ly used point force actuators. By using a modal decomposition in 161 he designed a linear state feedback controller being aware of the fact that a Luenberger observer might become necessary in order to be able to feed back the complete state. Balas investigates also the differences in the resulting linear state feedback when pole assignment or a quadratic performance criterion are used as design tools. A further quite interesting problem treated by Balas (1978) is concerned with the investigation of the influence of the number N of eigenfunctions considered in (61.

All the approaches discussed so far are based on the assumptions that as controls only joint torques as in the rigid body case can be applied. In addition to this, research is going on about using additional point actuators to influence the deformations due to elasticity. Among these. the work by Chassiakos and Bekey (198bl should be mentioned first. Their investigations start also with the Ritz method, (61. which is applied to the partial differential equation describing the motion of an elastic arm. To the

1- - - - -

I

Truckenbrodt (1982) goes a step further and investigates the error which results from the finiteness of the sum (61 in comparison to the exact representation of w(r,t) by an infinite series.

---I 1*"---

Controller ;--. .--

-_ ··· .. --1

~

... 1

I

--.6-

control voltages

i

- !

Actuators

I

i

! Flexible

-- " --r,

1..

Gears

1- '

[--.1 . ..j [ . .I i

Arm

-1

i !

Moments of load (?ropor~ _~~~a~ __ kQr I

Fig.7.

i

_______ _i

32

K. Desoyer 1'1 al.

SUMMARY After a short descript i on of the main parts of industrial rob o ts available today the field of flexible robots is reviewed . Starting with some considerations about flexibility approaches to the modelling are discussed which are used today in the literature. Three differ e nt approaches the kinet o static method the method of finite elements the method of vibration modes as well as combinations of these methods are co nsidered. Afterwards. various control co ncepts are presented . Starting with approaches which c an be termed 'c lassical ' in the field of r o botics. and discussing further linearization methods. more modern methods are reviewed. There main idea consists in finding first a feedback whi c h leads to a linear system whi c h can be controlled by state space methods. An extended but inevitably incomplete list of references concludes th e survey.

REFERENCES Alberts, Th.E., St. Dickerson and W.J . Book (1985). Modeling and control of flexible manipulators. Proc. Robots 9 ~, Detroit, Michigan, June 1985 , p.1 / 1-59 . Austin, F. and H. H. Pan (1970). Planar dynamics of free rotating flexible beams with tip masses. AIAA Journal, Vol . 8, No.4, April 1970, p.726-733. Bagci, C . and S. Kalaycioglu (19?9). Elastodynamics of planar mechan~sms using planar actual finite l~ne elements, lumped mass systems, matr~x­ exponential method , and the method of "C ritical- GeometryKineto- ElastoStatics". Journal o f Mec hanical Design. Vol.101, July 1979, p.417 427 . Balas, M.J. (1978). Feedback control of flexible systems . IEEE Trans . on Automatic Control. Vol.AC 23, Nr . 4, Aug.1978 , p . 673 679 . Barraco, A.. B. Cuny and G. Ishiomin (1985). Dynamics of flexible systems. IUTAM/IFToMM-Symposium, "Dynamics of Mul tibody Systems " , Udine. (Springer Verlag 1986), p.1-16. and G. Bogelsack, G., E. Kallenbach Linnemann (1984) . Roboter in der GerateBerlin technik . VEB-Verlag Technik, 1984. Book, W. J ., W. Maizza-Neto and D. E. Whitney (1975). Feedback control of two beam, two joint systems with distributed flexibility . Journal of Dyn. Systems. Meas. and Contro l, Dez 1975, p . 424 431 . Book , W.J. (1979). AnalysiS of massless elastic chains with servo controlled joints. Journal of Dyn. Syst .. Meas. and Control, Vol.101, p . 56 98. Book. W. J. and M. Majette (1983) . Controller design for flexible distributed parameter mechanical arms via combined state space and frequency domain techniques . Journal of Dyn. Systems. Meas. and Control. Vol . 105 . p .2 45-254. Bremer. H. (1977) . Bewegungsgleichungen hybrider Systeme beliebiger Anordnung. ZAMH 57, T57-T59. Bremer. ij . (1978). Zur Dynamik hybrider Mehrkorpersysteme. Diss . TU MUnchen . Bremer, H. (1980). Dampfungsdurchdringungen bei hybriden Mehrkorpersystemen. ZAMM .2Q, T51-T53 . Bremer. H. (1981). Bewegungsgleichungen hybrider Mehrkorpersysteme: Analytische und synthetische Verfahren. ZAMH 61, T27-T29 .

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Flexible Robots - :\

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SlIrH'\·

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K. lksOlt'r 1'1 Ill.

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