Accuracy of Hexapods Evaluation of Influencing Criteria and Methods for a Model-Based Compensation Michael Schwaar, Reimund Neugebauer ABSTRACT. Accuracy of hexapods is dominantly determined by interaction of mechanical components taking into account manufacturing imprecisions and control functions. Modelling of necessary control algorithms is based on models of kinematic errors, elastic deformation, thermodynamic response of individual components and machine dynamics. Developing relatively simple and well structured machine models is enabled by the hexapods' modularity. These models, in turn, are the base to work out compensation routines as well as functions for diagnosis support. As a result, all components are assembled to a mechatronic system which shows the advantages of the concept even at higher accuracy demands. KEY WORDS: Parallel kinematic accuracy, Model based control, Compensating control

Introduction Having studied an abundance of implemented parallel kinematic structures, the authors have recognised, that the construction and planning engineers of all machine types are to a great or less extent faced with the following problems: -

Machine accuracy is frequently insufficient. Accuracy is additionally impaired by thermal displacement. The machine's dynamic behaviour does not conform with the initial expectations. Concerning their effect, all machine features are mutually related to a large degree. For that reason, it is mostly complicated to detect the causes of certain phenomena.

Diagnostics and follow-up development of compensation measures realised by control functions is impinged, in particular, by the interdependence mentioned above.

Are there any specific approaches to be applied for the typical problems of parallel kinematic link mechanisms? With respect to construction, parallel link mechanisms follow a highly modular structure. This modularity implies the possibility to create relatively simple models which represent the static and dynamic behaviour. Associated examples are explained in the following. As a goal of research, we have to find out if and to what extent these models can be employed to separate those variables influencing the machine accuracy and to enhance the machine features by means of model-based compensation methods. Domain knowledge has to be integrated into the control to make the CNC's more intelligent (so-called systems with integrated intelligence). These intelligent functions are expressed by the control's ability to find out each current machine state based on models and measuring values and further influence the manipulated variables upon the knowledge manifested by the models.

1.

Models to represent the hexapods' static and dynamic accuracy

Which models are required to specify the parameters relevant for accuracy? To represent and enhance the hexapod parameters, in particular, the following models were proved as necessary.

1.1. Kinematic error model The relationship between position deviations at the working position and geometric deviations of the individual machine components is described in the kinematic error model. The transformation model which represents the transition from the Cartesian into the real drive axes' coordinate system is an associated submodel. As an advantage, the kinematic model of hexapods has actually few parameters, only, whereas the feasible movements are really very complex. Assuming that the joints are designed in a manner enabling that all axes of rotation cross at one point, the kinematic model is entirely determined by the position of the joint points at the frame - related to the absolute coordinate system- , the location of the joint points at moving effector - referred to the coordinate system of this effector - as well as the strut length values are completely defined. This assumption is based on ideal joints and infinite machine stiffness as starting premises. Following these assumptions, kinematic error models ought to describe the relationship between the joint points' dislocations or alterations of certain strut length offsets which are known as the kinematic machine parameters, on the one hand, and the end effector's dislocations resulting therefrom, on the other hand. As a starting point, a machine of indefinite stiffness and ideal joints with constant pivot is assumed. The kinematic error model also embraces how manufacturing accuracy acts on the machine parameters. This way, it is also possible to represent how non-ideal joints influence the machine accuracy.

1.2. Modelling of elastic deformations In the elastic deformation model, the static machine dislocations caused by the empty weight, especially of the mobile carrier, are depicted. In traditional machines of higher accuracy requirements, these static dislocations conform with machine sag models which are employed for compensation inside the CNC. As a typical feature of parallel mechanisms, all axes are highly interdependent. However, when making use of symmetry assumptions, simple and complete mathematical notations can be found. A dislocation which significantly depends on the current carrier position may result from the hexapod's moving carriers weight. Let the carrier's weight be m*g, whereas m is the carrier weight and g the acceleration due to gravity. In controlled machine state, flexibility is caused by the elastic machine elements outside the system. In parallel link mechanisms, the necessary joints are frequently located outside the drive axes' measurement systems and are a dominant reason of flexibility. In the model, it is assumed to a first approximation that all struts with joints are acting as a longitudinal spring. The spring rate c can be immediately measured in a simple experiment. Let X be the Cartesian and L the drive axes' coordinates. We also introduce another coordinate system whose origin is the same as the moving carrier's centre of gravity and whose coordinate axes position referred to the carrier is constant. If transforming the position X into this coordinate system, the coordinate vector Xd is arising. All angular coordinates of Xd are equal to zero. The translational coordinates correspond with the carrier's centre of gravity. Following these definitions, two Jacobians J and Jd which are in turn named as equation [1] are calculated.

dL =

J * dX

dL =

J d * dX d

∂L( 1 )   ∂L( 1 )   ∂X ( 6 )   ∂X ( 1 )   •   •  =    •   •   ∂L( 6 )   ∂L( 6 ) ∂X ( x ) ∂X ( 6 )   

*

dX

[1]

As a next step, the calculation of sag ∆X resulting from the carrier's weight can be described by equation [2]: ∆X

=

Jd

−1

* c − 1 * (J' ) * m * g −1

[2]

Formulae of higher complexity but still to be written in a simple notation are created if considering further reasons of dislocation such as empty weight.

1.3. Modelling of thermodynamic behaviour As any other machine tool, parallel link mechanisms are subject to thermal dislocations, too. Learning from model reasoning and experience, it could be found out, that it is not necessarily more complicate to keep the hexapods' thermal influences under control than in any other conventional machine. Conversely, even some advantages which are founded in the concept itself can be used. Two benefits are explained below. Due to strut structure, most machine elements have low thermal time constants which are additionally similar to each other. As a result, all component react on temperature influences from the outside in a uniform manner - although truly quickly. In other words, if all bar-shaped components are changing homogeneously and in proportion to their length, no dislocation will appear at the working position. In all thermal alterations including internal heat sources, bar-shaped components are mainly altered according to their length. Assuming that these length alterations especially those of the struts whose heating due to manufacturing cannot be excluded are known, the machine's kinematic models can be corrected in an easy way. To determine the amount of length each bar has been altered is much more easier than to control thermal dislocations in compact parts. The strut's thermal state can be adequately determined by a sufficient quantity of sensors. Alteration of length can be abstracted from this thermal state by means of simple algorithms. For reasons of costs, it won't be possible to install a high number of actuators. However, the number of necessary sensors can be drastically reduced by knowledge available in various models. The methodology is focused on a thermodynamic model making use of a state variable vector T which represents temperature values in a point-wise or integral manner, that is along a bar. The model follows the general system equation dT dt ∆l

=

A* T

=

C*T

+

B* u

[3]

Matrix A describes the thermodynamic system's characteristic dynamics, whereas u is a vector of measurable variables from the outside which influence the thermal behaviour. For example, the speed of a strut acts on its temperature. Matrix B specifies how these influencing variables effect the temporal temperature gradient whereas matrix C defines the relationship between temperature field and alteration of length. As a starting point, the matrices A, B and C can be abstracted from the general onedimensional equation of heat conduction by discretization based on a theoretical algorithm. To determine the thermodynamic boundary conditions, the model should be balanced with simple experiments. Based on this model, each current thermal state T can be determined by only few measuring values analysed over time and the input data u(t) . Therefore, a monitoring algorithm has been derived from system equation [3]. The associated principle is illustrated in Figure 1. lin

.

.

Strut

. Joint

li2 li1 Joint

Temperatures

Machine Control

T 1(t), T 2(t) Temperature Probes Work spindle

∆l T(l 1, . . . , ln) (Temperature field) State estimation procedure

∆l = C * T

Figure 1. Making use of a state estimation function to find out the current temperature field

1.4. Modelling the dynamic behaviour of machine and drives Dynamic machine models are a must to evaluate measuring results, detect faults within machine elements and efficiently adjust the closed-loop and control parameters. In chapter 4, the authors demonstrate by means of an example, that

classical methods of controller design provide insufficient dynamic characteristics, only. To develop generally new methods, dynamic models which sufficiently adequately represent the non-linear behaviour of these machines have to be employed. Having to decide upon the type of model to depict the parallel link mechanism's dynamic behaviour, multi-body models and FE models consisting of a number of degrees of freedom are frequently in use. Both approaches are less suitably to be employed within CNC's: The model's expressivity as well as its feasibility for machine and control suppliers is tightly limited only by the efforts necessary to determine and verify the high number of free parameters. Models can be employed for structural design of control algorithms basing on the premise that significant machine and control parameters can be described by using the model in its simplest expression. If it is possible to create a reasonable and feasible model of the machine's dynamic behaviour, it should be based on symmetry regularities. A clearly structured dynamic hexapod model can be achieved considering the following premises: -

All struts are characterised by analogous dynamic response. Only the longitudinal strut movements are paid attention. Frame deformations are neglected.

In principle, even the strut's lateral movements and possible frame deformations could be integrated into this model. However, it depends on the specific machine features, whether this extension is necessary for sufficiently precise description. A methodology on how to create simple dynamic models for hexapods has already been described in [NEU 98]. The idea of this approach is to decompose the entire structure into the following components: -

Frame Struts with variable length (position control) Spindle body (moving carrier) and CNC.

These elements are assembled according to positional dependency of their interaction to form the entire machine. The controlled strut is represented as a oneaxis multi-body model. Hereby, the submodel's complexity can be varied. As a rule, from the strut models, both following operator equations can be generated independently of the given structure of the strut submodel. The transfer functions Gs(s) and GZ (s) are defined by these equations. Transfer function Gs (s) expresses the signal curve li (t) manifesting the response of a separate strut i to alteration of nominal value lseti (t) . Hereby, variable s stands for the differential operator. In the representation, the variation of the signals with s was taken if the corresponding

signal had been transformed from the time into frequency domain. Transfer function GZ (s) implies the strut length alteration against a longitudinal force FZ (t): i l i ( s ) = G s ( s ) * l set (s) +

G Z ( s ) * F Li ( s )

[3]

Linear behaviour was assumed for reasons of an easier mathematical representation. Nevertheless, it is no problem to transfer this notation into a non-linear model and/or models which are partially discrete in time. The carrier movement is expressed by the following equation:

m  m   m   Θω    

Θθ

    2 2 * d X d = M * d X d = J T * F + F d L Z  dt 2 dt 2   Θϕ  

[4]

with M: weight matrix. The weight matrix' moments of inertia refer to the centre of gravity and a coordinate system which is invariant related to the carrier. Making use of equation [1], the dynamic behaviour of a hexapod can now be described by the following set of equations. Here, the time derivatives of the Jacobians were neglected. As shown by corresponding estimations, in case of machine tools, these derivatives can also be neglected if these machines are travelled at higher speeds. This strategy is based on the premise, that the machine's transfer behaviour does not strongly react on small position alterations. G Z * ( J' ) − 1 * M * J − 1 * s 2 * L + L

=

G s * Lset + J'* F Z

[5]

Starting out from these equations and taking a simulation system for common differential equations, a very clearly structured model can be built. Hereby, it is favourable to employ vector and matrix functions as possible in the system MATLAB/ SIMULINK. The arising calculation times are sufficiently low for interactive use. The model is engineered to determine the machine behaviour along selected parts of NC programs. However, imprecision resulting from manufacturing can be integrated into the model. Thus, the imprecisions' influence on the dynamic behaviour can be analysed. This function also allows to particularly effectively support diagnostic processes. Last but not least, this model is fitted to evaluate various closed-loop structures and to parameterise closed loops.

2.

Calibration

The manufacturing tolerances of several machine elements are so high that, if not calibrated, they would result in machine imprecision totally unacceptable at all. Without any knowledge of the model, in general, calibration could be aimed as follows: a sufficiently dense grid of spatial points is approached whereby all coordinate axes have been paid attention. On these spatial points, all coordinate values have to be found out. Oppositely to machines whose axes are in vertical position to each other, the individual coordinate directions cannot be analysed separately, that is one independently of the others. For a 5-axis machine, during calibration routine, all five coordinates had to be measured at a number of N5 points, with N : number of axis graduations. For N = 10 , all coordinates should be sampled at 10, 000 spatial points. This demand cannot be realized in practice. Concluding, entirely alternative approaches are demanded. A first approach can be derived from the kinematic error model. Exactly 42 values summarized in a parameter vector P have to be found out, since the entire kinematic behaviour is determined by 12 joint points, that is 36 scalar values and also 6 strut length values or their offsets. Related to the complexity of expression which can be achieved, this quantity is quite small, but to determine these 42 mainly correlating variables is also a considerable numerical problem. All previous experience and thinking has taught us, that calibration only makes sense if the machine has already been mounted before. Consequently, all records can only be gained upon mediate measurements. The fundamental calibration procedure can be characterised by the following steps: -

-

-

-

First, a plan of experiments is carried out to decide on the machine's spatial positions the records are to be taken from. The indicated values of all those six strut axes are sampled at each spatial point. Additionally, at least one Cartesian coordinate is exactly recorded. The records can be obtained by various methods. To determine translational coordinates, a specimen can be put into the machine. This specimen is fitted with contact areas which were gauged very exactly before. The specimen is aligned to machine table. A probe is travelled to the scanning surface via machine control until it contacts it. At this moment, the probe position is absolutely defined in this one coordinate. At the same time, the measuring values of all strut lengths are stored. Angles can be taken at certain positions by means of an electronic bubble level. The position is approached and the measured angle as well as the measured strut length values are stored. To determine all 6 coordinate values of a certain position, also a specimen with a special set of sensors and additionally one iterative position control can be employed.

-

Starting out from the kinematic model, sensitivity vector sj is calculated for each measuring position j in order to evaluate the calibration measurements. This sensitivity vector consists of the same number of elements as also parameter vector P. Let coordinate x have been determined in measurement i , then sj should look like the following: sj

-

= (

... ,

∂x ) ∂p42

Let ∆X be the vector of all recorded deviations of the Cartesian coordinates at individual tests. Furthermore, let S be a matrix whose lines are formed by the sensitivity vectors sj. Then, the parameter vector P is quantified by solving the equation : S *

-

∂x , ∂p1

P

= ∆X

As a recommendation, this set of equations should contain more equations than unknown quantities. Concluding, as a rule, it is not possible to solve all equations in a consistent manner. A parameter vector P is determined whose mean squares of the resulting residual deviations are minimised. Then, the real residual error acts as an indicator of model quality. The experiments ought to be planned enabling a condition of the set of equations which is as good as possible. However, this plan can frequently not be realized due to the limits in the practical performance of tests. Additional equations which limit the dispersion of results enhance the algorithm's numerical robustness.

Figure 2 illustrates the calibration sequence.

Measured Data: L = (L1, L2, ..., L6) probe

Xi : (x,y,z,A,B,or C)

specimen

P

Kinematic Model Elastic Deformation Model

(updated)

Machine Control

Calculation S*P=∆X

Figure 2: Sequence of calibration The numerical behaviour can be enormously enhanced by decoupling the measurements by means of appropriate test strategies and plans. The measurements can only be insulated, if all the six position values can be found out in the machine at the same time. Therefore, corresponding methods were elaborated. As an essential secondary advantage of decoupling, individual manufacturing errors can be separated. Machine diagnosis is efficiently supported by this functionality.. As a rule, other hexapod layout variants differing from the described result in complicated calibration strategies. For example, the number of parameters necessary for kinematic model and consequently the quantity of values to be determined by calibration is increased in case of joints whose axes have an offset. In order to master calibration even in cases like these, to reduce the number of fault parameters is frequently necessary, but also possible. The application of an ideal kinematic model inevitably results in residual errors which can be always evaluated if the number of independent experimental results exceeds the quantity of parameters to be determined. For that reason, as a rule, a test plan should foresee redundant records in any case. The following case study should demonstrate the consequences on residual error which might occur if the assumptions of the model approach have been infringed.

100 µm

without calibr. with calibr.

0 µm

1400 mm 300 mm

800 mm

-300 mm

Figure 3. Calibration results in case of joint displacement In Figure 3, we see the result of a calibration procedure in case of joint displacement. The upper graph shows the displacement before calibration. After calibration, no displacement occurs (lower graph). Incorrect model assumptions result in residual errors. In case of Figure 4, an elastic deformation model with a wrong mass is used. For that reason, a residual error can be observed after calibration.

100 µm without calibr. with calibr.

0 µm

1400 mm 300 mm

800 mm

-300 mm

Figure 4. Calibration results when using an incorrect elastic deformation model The following Figure 5 illustrates the residual errors which had been observed at the same machine before and after having exchanged damaged joints. The values represent deviations from the coordinate x on different spatial points which would occur if the calibration measurement would be analogously repeated with the newly calculated calibration values without taking into consideration stochastic errors. According to the left Figure, for this machine, there is no set of parameters whose deviations at the test plan positions are smaller than the value of 10 micrometers which is given in the Figure. The values indicate an important model fault. Second Figure shows that the values conform with the model assumptions very well. The residual errors are less than 1 micrometre. This conformity could only be achieved by adding the model approach an elastic deformation model. The graphs exclusively represent measuring values which have also been taken for the calculation of the correction parameters. In case of repeated measurements, greater deviations would result due to dispersion or drift of measuring values. An absolute evaluation of residual errors at various test plans may cause misinterpretations since the test plans themselves strongly effect the residual error.

Figure 5. How faulty joints influence the residual error after calibration Assuming a rigid kinematic structure, the minimal residual error of the right Figure cannot be obtained. The model approach has also to involve elasticity feature whose model representation is expressed by equation [2]. In the present case, the experiment could be executed at a speed accelerated so much that thermal dislocations had not been paid attention at all. However, in principle, a thermodynamic compensation method can be linked with a calibration procedure.

3.

Model-based compensation

3.1. Compensation of thermal dislocations Kinematic and thermodynamic models can be linked in an easy way. For a hexapod type machine tool, thermal dislocations could be sufficiently compensated by the following limitations: -

External temperature variations with limited gradients cause few effects on the working position. Temperature variations at the main spindle are recorded via sensor. From the measuring value, alteration of length in tool direction is calculated. The amount the strut temperature differs from the frame's one is recorded. The difference is taken to compute the strut length offsets making use of a state estimation routine according to Figure [1].

The determined correction values can immediately be linked with the kinematic model's parameters. Further comprehensive calculations are not necessary. Figure 6 illustrates an example how thermal dislocations were compensated when the strut temperature had been increased due to manufacturing. Starting out from the

"cold" state, the machine was run at different speed values over a period of 2.5 hours which resulted in a difference between frame and strut temperature. The strut expansion was calculated upon a dynamic supervision routine and the result was taken care of in the transformation algorithm. Position z was inspected in cycles.

1,1

0,01 0,005 0

0,8

-0,005

0,7

-0,01

0,6

-0,015

0,5

-0,02

0,4

-0,025

0,3

-0,03

0,2

-0,035

temperature difference between strut and frame delta z without compensation

0,1

-0,04

0 -0,1 00:00

∆z, mm

temperature difference, K

1 0,9

-0,045 delta z - compensation with dynamic model

00:30

01:00

01:30

02:00

-0,05 02:30

time, h:min

Figure 6. Results of compensation of thermal displacement The graph indicates the following curves: Difference between strut and frame temperature (lower increasing function) Resultant dislocation in z direction without correction (strongly decreasing function) Residual error in z direction, with compensation As proven by the example, the method is principally suitable to compensate thermal dislocations in hexapods.

3.2. Controller design Based on the assumption, that closed loops of hexapods are adjusted according to the same rules as traditional machines, only few circular amplifications can be used. The machine's limiting frequencies does not come up to expectations. Consequently, the machine's benefits founded by its structure are mainly lost. What are the causes for this result ? -

The entire structure is only slightly mechanically damped.

The many slightly damped machine elements are tightly mutually linked. Therefrom result a lot of sources for vibrations. Vibrations, in turn, can be incited in a number of frequency ranges. The machine's dynamic features clearly depend on position. As a rule, each controller is to be setup in a way enabling stability even at worst position.

-

-

When occurring simultaneously, the features mentioned above mutually amplify. The dynamic model provides a structural approach to enhance the controller characteristics. Strut damping can be increased by making use of a number of feedbacks inside the closed loop at the same time. As a result, the engineered structure acts analogous to a state controller. Figure 7 illustrates the possible manifestations of step response. The machine was subject to a nominal value jump of 50 µ step response which arises if the hexapod drives have been adjusted according to the rules valid for traditional machines and if the demand for stability in all positions of space is fulfilled, is shown at the left. Results of a new controlling concept are given in the right graph.

[mm]

Conventional Controller

Controller adapted to Parallel Kinematics

[mm]

0.2

0.2

0.1

0.1

0.0 0.0

100

[ms]

200.0

0.0 0

100

[ms]

200

Figure 7. Step response using different controller structures The second graph involves feedback via main control in addition to the closed loops in the drive controllers. This results in a considerable time lag of 1.5 ms mainly determined by the cycle time of the drive bus (here: SERCOS interface). Further evident enhancements are expected due to modified drive actuators and the integrated circuit of higher speed which is going to be made available this year. As a result of this controller change-over, dynamic behaviour could be enhanced within the entire working area. Regardless of this, the resultant characteristics are still significantly dependent upon the centre of linearization. Possible approaches to compensation have already been mentioned in [NEU 98]. However, even these algorithms can only be realized immediately inside the control, and the desired effect is tightly restricted by idle time. As shown by simulation results, this task can be solved much better by means of the new SERCOS interface.

3.3. Control functions An appropriate controlling hierarchy is necessary to integrate various compensation functions into control. Commonly, transformation of nominal values out of the machine coordinate system into the coordinate system of the real driving axes which is commonly mentioned first, manifests only a small percentage of additional functions. Conversely, the following specific functions become more and more important: -

Coordinate transformation into both directions as wide as possible at the end of data flow (that is in controlled cycle) Complex supervision of non-linear kinematic limits incl. monitoring of the multi-dimensional joint angles Supervise non-linear dynamic limits via look-ahead function Compensation algorithms to correct the machine in a model-based manner Functions to support machine calibration Making available diagnosis modules to separate influencing variables

In general, it is possible to transform the Cartesian into another coordinate system at various positions of the data flow. However, the later transformation the more functionalities of the original control are maintained without any change. For that reason, in case of the used control type andronic 400 HEX , this transformation was placed immediately before the (Sercos-) interface to drive bus . In this example, the coordinate systems were transformed at a bus cycle of 0.5 ms.

4.

Conclusions

Hexapod accuracy is determined by manufacturing accuracy at defined machine elements and new control functions. Both components are characterized by much more interaction than traditional machines have. Here, attainable accuracy is decisively influenced by the capability of mastering mechatronic interaction. The hexapod potentials cannot be opened up by traditional control and compensation solutions. In the opposite case, this approach could even imply the risk of creating entire solutions whose quality is insufficient no matter how advantageous the concept seems to be. Relatively simple models describing kinematics incl. error estimation, elastic deformations, thermal behaviour and dynamic response of machine and drive can be generated due to the modular and symmetric machine structure. Compensation strategies to be integrated into CNC functionality are based on those models. Concerning their complexity, all presented models are able to be integrated into the controlling algorithm.

All explained models represent essential tools for a diagnosis system. Due to strong coupling in parallel link mechanisms, diagnosis procedures have to be unconditionally supported by corresponding software. The want of solutions for practice should be noted. Making available controlling solutions and support systems - especially to set machines into operation - , hexapod machines can then be applied even at higher precision requirements. As a premise to be taken into account, the entire mechatronic behaviour should already be investigated in the developmental phase.

5.

References

[NEU 98] NEUGEBAUER, R., SCHWAAR, M., WIELAND, F.,.: Accuracy of parallelstructured machine tools, International Seminar on Improving Machine, Proceedings of the International Seminar on Improving Machine Tool Performance, p. 521 – 533, 1998 [NI 98] NI, J., JINGXIA, Y., Recent advancement of real-time error compensation systems for machine tool accuracy enhancement, International Seminar on Improving Machine, Proceedings of the International Seminar on Improving Machine Tool Performance, p. 327 – 538, 1998 [HON 97] HONEGGER, M., COUDOUREY, A., BURDET, E. Adaptive Control of the Hexaglide, a 6 dof Parallel Manipulator, Proceedings IEEE International Conference on robotics and Automation, 1997