Controlling a contactless planar actuator with manipulator
Citation for published version (APA):Gajdusek, M. (2010). Controlling a contactless planar actuator with manipulator. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR673132
DOI:
10.6100/IR673132
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Controlling a Contactless Planar Actuator
with Manipulator
Controlling a Contactless Planar Actuator
with Manipulator
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor Promoties in het openbaar te verdedigen
op dinsdag 11 mei 2010 om 16.00 uur
door
Michal Gajdušek
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. P.P.J. van den Bosch en
prof.dr. E.A. Lomonova MSc
Copromotor:
dr.ir. A.A.H. Damen
This work is part of the IOP-EMVT program (Innovatiegerichte onderzoeksprogramma’s - Elektromagnetische vermogenstechniek). This program is funded by SenterNovem, an agency of the Dutch Ministry of Economic Affairs.
This dissertation has been completed in fulfillment of the requirements of the Dutch Institute of Systems and Control DISC.
Copyright © 2010 by Michal Gajdušek
A catalogue record is available from the Eindhoven University of Technology Library. Controlling a contactless planar actuator with manipulator / by Michal Gajdušek. – Eindhoven: Eindhoven University of Technology, 2010.
Proefschrift. - ISBN: 978-90-386-2228-6 NUR 959
vii
Contents
Chapter 1
Introduction 1
1.1 Background 2
1.2 Contactless Planar Actuator with Manipulator 3
1.2.1 Project Description 3
1.2.2 Initial Experimental Setup 3
1.3 Thesis Goal 6
1.4 Organization of the Thesis 7
Chapter 2
Commutation 9
2.1 Introduction 9
2.2 Norm-Based Commutation Algorithms 13 2.2.1 l2-norm Based Commutation (L2C) 13 2.2.2 l∞-norm Based Commutation (LiC) 14 2.2.3 Clipped l2-norm Based Commutation (CL2C) 16 2.2.4 Example on a Model of the COPAM 20
2.2.5 Conclusions 26
2.3 Full Yaw Rotation of Planar Actuator 27 2.3.1 Full Yaw Rotation Investigation 28
2.3.2 Concept Design for Full Rotation 31
2.3.3 Conclusions 37
2.4 Commutation Techniques and Full Rotation on the Model of COPAM 37
2.4.1 Maximum Current 39
2.4.2 Maximum Voltage 44
2.4.3 Dissipated Power 47
2.4.4 Error Sensitivity 51
2.4.5 Lower Bound of the Maximum Acceleration and Velocity 55
2.5 Summary and Conclusions 60
Chapter 3
Errors in Commutation 63
3.1 Introduction 63
3.2 Software-related Errors 64
3.2.1 Numerical Precision of the Modeled Coupling 64 3.2.2 Look-Up-Table-Based Description of the Coupling 64
3.3 Hardware-related Errors 67
3.3.1 Coil Position 67
3.3.2 Coil Dimensions 70
3.3.3 Magnet Array Bending 70
3.3.4 Magnets 72
3.3.5 Measurement System 73
3.3.6 Current amplifiers 74
3.4 Comparison and Summary 74
3.5 Auto-Alignment of the Measurement System 75 3.5.1 Consequence of Offsets in the Measurement System 76 3.5.2 Auto-Alignment Procedure 78 3.5.3 Investigation of Stability of the Auto-Alignment Procedure 79 3.5.4 Measurements on the Experimental Setup 82 3.5.5 Discussion and Extension 83
3.6 Conclusions 85
Chapter 4
Control 87
4.1 Introduction 87
4.3 Trajectory Generation and Feedforward Design 96 4.4 Control of the Planar Actuator 100
4.4.1 PID Control 100
4.4.2 Sliding-Mode Control 105
4.4.3 Iterative Learning Control 113 4.5 Control of Planar Actuator with Manipulator 119
4.5.1 Multi-body Model 120
4.5.2 Simulations 122
4.6 Summary and Conclusions 124
Chapter 5
Wireless Communication 129
5.1 Introduction 129
5.2 Desired Specifications of the Wireless Link 129 5.3 Overview of Standardized Wireless Solutions 131 5.4 Infrared Wireless Link for Real-Time Motion Control 133
5.4.1 Data Processing 133
5.4.2 Custom Communication Protocol 134 5.5 IR Wireless Link Performance 135
5.6 Conclusions 141
Chapter 6
Experiments 143
6.1 Introduction 143
6.2 Improvements of the Experimental Setup 144
6.2.1 Measurement System 144
6.2.2 Contactless Energy Transfer 147
6.2.3 Wireless Control 150
6.3 Identification 151
6.4 Control 157
6.4.1 Control of the Planar Actuator 157 6.4.2 Control of the Manipulator 165 6.4.3 Control of the Planar Actuator with Manipulator 165 6.4.4 Clipped l2-norm based commutation 172
Chapter 7
Conclusions and Recommendations 177
7.1 Commutation 177 7.2 Errors in Commutation 179 7.3 Control 179 7.4 Wireless Communication 180 7.5 Experimental Verification 181 7.6 Thesis Contributions 183
7.7 Recommendations for Future Development 184
Appendix A
Dimensions of the Experimental Setup 187
Appendix B
List of Symbols and Abbreviations 191
Appendix C
Additional Simulation Data 195
Appendix D
(3+2)-DOF Experimental Setup 199
D.1 Introduction 199
D.2 Experimental setup 200
D.3 Model 203
D.4 Identification 206
D.5 Control 211
D.6 Summary and Conclusions 211
Bibliography 213
Summary 223
Samenvatting 225 Acknowledgements 227
1
Introduction
If, in a hundred years’ time, somebody looks into a book about technological development of the end of the 20th and the beginning of the 21st century, he or she will
definitely see adjectives like smaller, cheaper, faster or more powerful. In order to manufacture devices with these, ever increasing, requirements, the processing machines must have higher precision and performance. Sometimes, the improvement of the machine performance hits its physical or economical limits and a new approach must be found. The new-principle machine is useless without proper control, which helps the new device to outperform the old one. Magnetically levitated planar actuators with moving magnets [62, 51, 114, 108, 26, 95, 56, 76, 13] are an example of such a technological improvement. These machines use a new approach to achieve high-precision planar movement: magnetic levitation without cable connection to the translator assures movement without friction, which otherwise degrades the device precision. Moreover, the planar movement provided by a single 6-degree-of-freedom (DOF) actuator reduces the weight of the moving mass of the machine. However, without proper control these planar actuators are unstable, as was proven by Samuel Earnshaw in 1842 [35]. A novel control strategy for the decoupling of the forces and torques acting on the planar actuator was developed recently [95, 76] and extended in this thesis.
This thesis describes the application of this novel control strategy for a novel planar actuator that combines three contactless technologies: 6-DOF electromagnetic suspension and propulsion, contactless energy transfer and wireless motion control.
1.1 Background
High-precision positioning machines with emphasis on planar movement are required in many industrial devices, e.g. for wafer scanners in semiconductor lithography, for pick-and-place machines in assembly lines, or for inspection systems.
These machines usually consist of stacked long- and short-stroke linear and rotary actuators, which are supported by ball or air bearings. Stacking of the actuators leads to high moving mass or reduced stiffness. In extreme situations, the moving mass of the machine can be even a thousand times heavier than the mass of the moved object. On the other hand, ball bearing causes friction and neither ball nor air bearing can be used in high vacuum. Moreover, a cable slab for power and communication cables, and possibly cooling liquid connected to the moving part causes disturbances such as vibrations and friction, and acts as an additional load.
The idea of combining several 1-DOF actuators into one magnetically levitated multi-DOF planar actuator has been studied for some time [62, 51, 24, 10, 114, 25, 65, 108, 26, 95, 56, 76]. Such a solution reduces the moving mass of the machine while the effective force can remain the same. Moreover, the magnetic bearing is frictionless and can be used in vacuum. Although mostly only a 2-DOF planar motion is desired, these actuators need to be controlled in all six DOF’s to obtain a stable active magnetic bearing (stabilizing the three rotations and the levitation). Two main principles for long-stroke horizontal movement have been studied: 1) moving coils with stationary magnets [25] and 2) moving magnets with stationary coils [51, 26, 56, 76].
The first principle needs many more magnets than coils for long-stroke planar movement. This has benefits in a lower number of amplifiers or switches necessary for powering the coils and a simpler, analytical commutation algorithm (dq0-transformation [31-33, 91, 92] can be applied [25]). However, these planar actuators require power cables and cooling tubes to the moving coils.
On the other hand, moving-magnet planar actuators are really contactless actuators with no cable connection to the translator. Since these actuators are more difficult to control [95, 76], because dq0-transformation cannot be applied [76], only a few prototypes have been developed in recent years [26, 56, 76].
It often happens that for an application some sensors, electronics or an additional manipulator (e.g. micromanipulator) need to be placed on the translator. Then the problem with the cables between the stator and translator appears again. The solution is a magnetically levitated planar actuator with built-in contactless energy transfer and wireless communication. Although a combination of contactless energy transfer with wireless communication has been studied in the past, a combination with contactless actuator has not been studied yet [13].
1.2 Contactless Planar Actuator with Manipulator
1.2.1 Project Description
The project concerning the Contactless Planar Actuator with Manipulator (COPAM) consists of two parts. In the first part, described in [13], the experimental setup was designed and built, and simple control was applied. The setup combines three contactless technologies:
6-DOF electromagnetic suspension and propulsion Contactless energy transfer
Wireless motion control
The electromagnetic suspension and propulsion were realized in a planar actuator with moving magnets. The contactless energy transfer is for powering a manipulator on top of the translator, which is wirelessly controlled to keep the whole system without any cable connection between the stator and translator.
As the control designed in the first part of this project was only used for verification of the principle of the system, the second part of the project, this thesis, focuses on the control of the prototype.
1.2.2 Initial Experimental Setup
The planar actuator uses an array of permanent magnets in the translator and an array of coils in the stator. The dimensions of the magnets and coils have been optimized to utilize the same coils also for the contactless energy transfer. The coils have two functions. When the coil is in the magnetic field of the magnet array, it is connected to the low-frequency current power amplifier and the force between the coil and the translator is generated. On the other hand, when the coil is under the moving secondary coil (out of the magnet-array magnetic field) it is driven by a high-frequency amplifier and the coil is used as the primary coil for the contactless energy transfer. Due to high frequencies, the coils have to be made of litz wire. Litz-wire coil suffers from a lower filling factor compared to the standard solid-wire coil. The lower number of turns leads to higher current needed to produce the same force and, consequently, more expensive amplifiers to be used. In order to optimize the costs of the setup, only the coils that are really used for the contactless energy transfer are made of litz wire (see Appendix A, Figure A.1). The other (solid wire) coils differ only in the number of the turns (and consequently the impedance).
The relation between the current applied to one particular coil and the produced force and torque vector acting on the translator is called coupling (also known as force constant). This coupling is dependent on the actual position of the coil with respect to the magnet array. The calculation of the coupling developed in [13] together with utilization of the rounded coils allow for the control of the translator in any orientation about the vertical z-axis. Hence, apart from the long-stroke translational xy-movement, the setup was
theoretically capable of full yaw rotation. Therefore, the measurement system for the planar actuator was designed with the goal of exhibiting the rotational capability of the setup. Three eddy-current sensors and three laser triangulation sensors were used. The eddy-current sensors have limited range of 2 mm and they were used for measuring the vertical z-position and two titling angles (q y) of the planar actuator. The laser , triangulation sensors were used for measuring the x- and y-position and (yaw) f-orientation (see Figure 1.1). The laser triangulation sensors were chosen for their ability to measure position on a surface tilted by several degrees. With a tilt angle of 5º, the error is less than 0.12 % of the measuring range of the laser triangulation sensors.
Figure 1.1: Drawing of the experimental setup
The manipulator on top of the translator has two three-phase ironless linear actuators connected with a beam (like a small H-bridge). In the center of the beam, a three-phase rotary actuator was assembled with an arm attached to it. The tip of this arm can be positioned anywhere in the xy-plane between the two horizontal linear legs by combining the translation of the beam and the rotary movement of the arm. Because of its shape, the manipulator is sometimes called an H-drive. Next to the manipulator, there are additional electronics on the translator: a contactless energy transfer unit, three power amplifiers for the actuators, a wireless communication module and a module with a Field-Programmable Gate Array (FPGA) (see Figure 1.2). The FPGA communicates with the wireless link and computes the commutation for the manipulator’s actuators based on the position readings from the encoders. Via the wireless link, the current setpoints are received from the controller and actual position and orientation of the manipulator is transmitted back. With the controllers for the planar actuator and the manipulator in one place (in the fixed world), feedforward disturbance compensation can be easily realized. The wireless communication link, developed for the setup, is based on 2.4 GHz radio-frequency (RF) modules with custom packet and data processing [13].
x,θ y,ψ z,f Eddy-current sensors Laser-triangulation sensors Secondary coil Manipulator
FPGA and wireless communication unit
Moving magnet array
Stationary coil array
Figure 1.2: Photo of the experimental setup
The manipulator was not designed to achieve the highest accuracy, but to show the principle of the wirelessly powered and controlled machine on top of the magnetically levitated platform. Therefore, the resolution of the linear and rotary actuator is several times worse than the resolution of the PA measurement system based on the eddy-current sensors. The dimensions and properties of experimental setup are listed in Appendix A.
Since the prototype is a complex system, which for the first time integrates three contactless technologies into one multi-level system, a few problems arose during the building and testing of the setup.
The first item to be solved is in the custom-made RF wireless communication, which has quite high packet-loss average ratio of about 10-2. The ratio is increased even
more when the contactless energy transfer is active.
The second issue is with the electromagnetic interference produced by the contactless energy transfer unit. The interference is mostly visible in the increased packet-loss ratio of the wireless link and in the increased noise levels of the eddy-current sensors.
The third problem is caused by the laser-triangulation sensors in the measurement system. The problem is the sensitivity of the sensors to the perpendicular movement of the measured targets. The sensors measure distance by capturing a laser beam reflected on the measured surface. From the principle of these sensors, not the directly reflected but diffused light is captured by the sensor. Therefore, the sensor is sensitive to the roughness of the surface. On one hand, the surface must be rough (otherwise no light is diffused and captured by the sensor and hence no mirror surface can be used). On the other hand, variation in roughness causes uncertainty of the measurement. The problem is not in the actual position error, which is within the linearity specifications of those sensors, but in the fast variation of the error. Even a 10 μm step in the perpendicular direction could cause a measurement error of about 20 μm in the measured direction. Therefore, these sensors are causing vibrations of the levitated and multi-DOF controlled planar actuator. To remove
Secondary coil CET electronics
Motor amplifiers Manipulator
FPGA and wireless communication unit
the vibrations, two laser triangulation sensors were temporarily replaced by two short-stroke eddy-current sensors. As a result, a long-short-stroke translational movement of the PA is only possible in one direction [13].
1.3 Thesis Goal
The goal of this thesis is to systematically explore the opportunities for improving the performance of the existing contactless planar actuator with the manipulator (COPAM) by developing new control algorithms, a new wireless communication strategy, and by proposing some new electromechanical hardware alternatives.
The control of the system can be split into two parts. In the first part, the multi-DOF system will be decoupled. For the planar actuator consisting of many active coils this means the development of a proper commutation method, which decouples the applied forces and torques. In the second part, once the system has been decoupled, the actual controllers can be designed for the separate DOF’s.
As several problems were discovered during building and operation of the final setup, which could not be solved within the time given for the first part of the project, improvement of the setup is also considered in this thesis. In particular, the measurement system, which caused vibrations in the setup, has to be improved. The wireless RF communication link with the high packet-loss ratio has also to be improved or replaced to guarantee reliable control of the manipulator.
The thesis objectives are summarized in the following recommendations:
Commutation: Research new commutation techniques for the decoupling of the forces and torques applied on the planar actuator that can help to achieve higher performance of the system. Since this planar actuator is theoretically capable of full yaw rotation, research on this feature can be performed.
Errors in commutation: Research the bottlenecks that limit the accuracy of the forces and torques decoupling, as the precision of the decoupling is dependent on the accuracy of the commutation.
Control: Research and develop high-performance control strategies suitable for the multi-DOF planar actuators with moving magnets, which exceeds the performance of present solutions.
Wireless communication: Design and implement a reliable, high-performance wireless communication strategy for the final prototype.
Experimental verification: Design and build a suitable pre-prototype of the COPAM on which the control strategies for the final prototype can be tested. Implement the derived control software, wireless communication and hardware improvements on the final prototype to validate the obtained performance improvements.
1.4 Organization of the Thesis
Chapter 2 is devoted to commutation for ironless, magnetically levitated planar actuators. Here, development of novel commutation methods and their comparison with the up-to-date commutation method [76] is presented. Since full rotation of the planar actuator is, from the control point of view, related to the commutation, it is also included in this chapter.
Comparison and quantification of different kinds of causes that degrade the performance of the commutation is presented in Chapter 3. In [13], an interesting procedure for estimating and removing the offsets in the measurement system of the COPAM was presented. Proof of the correctness of the procedure is given in this chapter. Moreover, an extension to a larger class of planar actuators with moving magnets is shown.
Control of the multi-DOF system is discussed in Chapter 4. At first, a 6-DOF model of the planar actuator with the position-dependent disturbances and cross-coupling based on the available data is derived. Then, several control techniques are described and applied on the model of the planar actuator. The results from the simulations are compared. Later on, the model is extended by the manipulator and its disturbing movement. The simulation results with the different controllers are again compared.
Wireless communication for real-time control of a fast motion system is discussed in Chapter 5. Commercially available wireless solutions are compared and a novel infrared-light based communication link with excellent packet-loss ratio and radically reduced delay is presented.
Chapter 6 is devoted to the experimental verification. As the experiments on the pre-prototype are described in Appendix D, this chapter focuses on the final prototype. Here, the improvements to the hardware are presented as well as the results from identification. The controllers presented in Chapter 4 are applied and the results are shown and compared.
9
Commutation
2.1 Introduction
This chapter focuses on commutation algorithms for a class of ironless planar actuators (PA) with moving magnets that can be described by a linear relation between a current i applied to the jj th coil in the stator and a wrench vector
j
w produced by that coil
and acting on the center of mass of a moving magnet array (e.g. [86, 95, 56, 76, 13]): ( )
j j j
w =K q i (2.1)
where the coupling K is an m ´ vector mapping the current to the wrench vector and 1 dim( )
m= w . The coupling vector K is dependent on location and orientation of the coil
with respect to the magnet array, [ , , , , , ]T
j j j j j j j
q = x y z q y f . The wrench vector w is the composition of forces and torques: [ , , , , , ]T
x y z x y z
w= F F F T T T acting on the center of mass of the magnet array. Consequently, the coupling vector consists of six components (m = ): 6
T
[ , , , , , ]
x y z x y z
F F F T T T
K = K K K K K K . The relation (2.1) can be applied on ironless actuators assuming that they do not suffer from cogging force and reluctance effects due to the permanent magnets can be neglected [56]. Therefore, relation (2.1) is linear and it is dictated only by the Lorentz force acting on a piece of wire carrying an electrical current in a magnetic field [34]. Moreover, the linear relation is valid also during transient operation assuming that [56]: 1) the magnetic fields are quasi-static 2) the force caused by eddy currents is negligible 3) the wire diameter of the coil is smaller than skin-depth.
The coupling vector K is highly nonlinear with the position of the coil q [95, 56, j
76, 13] (see Figure 2.1 and Figure 2.2). Accurate computing is difficult if closed-loop sample rates of several kHz are required [76, 56, 13]. One approach is to use a simplified model of the real system as in [56], where the magnetic flux density of the planar actuator
Figure 2.1: Position dependent coupling KFz( )qj .
Figure 2.2: Position dependent coupling KTx( )qj .
is represented as a sum of spatial harmonics [60, 112]. Moreover, the coils are modeled as a number of filament surfaces for which it is possible to find an analytical solution for the Lorentz force and the corresponding torque [60]. Although the analytical expressions provide a good insight into the mechanisms that determine the forces and torques on the planar actuator, the simplifying assumptions compromise the accuracy of the model [13]. To allow real-time wrench calculation, only the first spatial harmonic of the magnet array has been included [56] neglecting the end-effects. Therefore, the coils on the boundary of the magnet array cannot be used and the current in these coils must be forced to zero [77, 79]. An addition of multiplicative edges to the model that slightly increases the range in which the coils can be used effectively is discussed in [3]. However, the coils in the model are still reduced to current sheets [56].
Another approach, which uses a look-up table (LUT), was developed in [18]. First, the magnetic flux density distribution is calculated using the surface charge model of the
magnet array [56, 16]. Magnetic flux density distribution of a single cuboidal permanent magnet is calculated as:
( )
(
)
( )
(
)
( )
1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 log , 4 1 log , 4 1 atan2 , 4 i j k r x i j k i j k r y i j k i j k r z i j k B B R T B B R S B ST B RU + + = = = + + = = = + + = = = = - -p = - -p æ ö÷ ç ÷ = - çç ÷÷ ç p è øå å å
å å å
å å å
(2.2)where atan2 is the four quadrant arctangent function, B is the remanent magnetization of r
the permanent magnet and:
(
) ( )
(
) ( )
(
) ( )
2 2 2, 1 , 1 , 1 , i m j m k m R S T U S x x a T y y b U z z c = + + = - -= - -= - (2.3) where [ , , ]T m m mx y z is center position of the magnet with sizes 2a, 2b and 2c in the x-, y- and
z-direction, respectively. The total magnetic flux density generated by the magnet array is
calculated by superposition [120].
The coil is modeled as a bundle of filamentary wires, where for a wire filament dl with current i in the magnetic field of the permanent magnet the produced force vector is c
dFc =i lcd ´Bm, (2.4) where [ , , ]T
m x y z
B = B B B is the magnetic flux density vector. The same force in opposite direction is applied on the magnet array (Newton’s third law): dFm= -dFc. The torque exerted on the magnet array by the wire filament equals:
dTm=rc´dFm = - ´rc i lcd ´Bm, (2.5) where r is the vector from the center of the mass of the magnet array to the wire filament. c
The position-dependent coupling K is pre-computed using this analytical-numerical Lorentz force model [56] for a high-density grid of positions of one coil in the space of the magnetic field of the magnet array [16]. In the last step, the look-up tables are generated from the calculated coupling [18]. The main benefit of a LUT is that the end-effects and higher harmonics of magnetic flux density can easily be included and therefore the LUT can be very accurate as the time-consuming calculations are done offline whereas accessing a LUT is quite fast. Using the coupling calculated for relevant space of the magnetic field with high resolution would require extreme memory requirements. Therefore, symmetries and periodicity of the magnetic field are used to reduce the look-up table size to just a few hundreds of kilobytes [18]. For that reason, the LUT approach is preferred.
When assuming rigid body behavior, superposition of the coil currents can be applied to obtain the total wrench on the center of mass of the translator [76]:
( )j j j w=
å
K q i (2.6) or in matrix form: 1 1 ( ), , (n ), ( )n ( ) w=éêK q K q- K q ùúi = q i ë û K , (2.7)where i is an n´ vector of all coil currents and ( )1 Kq is an m n´ coupling matrix varying with the position and orientation q of the translator with the respect to the stator with coil
array.
Magnetically levitated planar actuators are usually over-actuated actuators. That means that the number of active coils is always greater than the number of mechanical degrees of freedom (DOF’s). In a mathematical sense, the set of equations (2.7) is under-determined or dim( )i >dim( )w . The rank of the matrix K must be equal to the amount of DOF’s for " Îq Sadm, where S is set of admissible coordinates, for the system of equations adm
to be consistent. To linearize and decouple the system, an inverse mapping of (2.7) is necessary*:
des
( , )
i = K-q w , (2.8)
where w is the desired wrench vector and des K is a mapping from the desired wrench
-vector to the current -vector i at the actual position q . For brushed DC machines, the
commutator/brush system provides commutation, which can be seen as a transformation that transforms armature voltages to the field-winding reference frame [39]. The process of the coordinate transformation provided by the commutator is analogous to the inverse mapping (2.8) when considering the class of synchronous brushless AC actuators that can be described by (2.1) [76]. Therefore, the transformation provided by the inverse mapping is defined as commutation in this thesis.
The inverse mapping K is not necessarily linear in w, because over-actuation
-brings freedom in the choice of the current vector. If K is linear in w, the inverse mapping,
-being a vector function of w, can be written as a position-dependent matrix K-( )q
independent of w:
( , )q w ( )q w
-
-K » K . (2.9)
This chapter is organized as follows. Novel commutation methods and their comparison with the up-to-date commutation method is presented in Section 2.2. Full rotation of the planar actuator, from the control point of view related to the commutation, is also presented in this chapter, in Section 0. The findings from the both sections are applied on a model of the contactless planar actuator with the manipulator (COPAM) and
* Please note that there is a distinction between the symbols for the coupling vector K defined in (2.1), the coupling matrix K defined in (2.6), the inverse coupling vector mapping K-- (Greek capital kappa) and the
several aspects (e.g. maximum current and velocity, dissipated power, error sensitivity and worst-case acceleration and velocity) are studied in Section 2.4.
2.2 Norm-Based Commutation Algorithms
In this section, three norm-based commutation methods will be described of which two novel methods are presented as alternatives for over-actuated actuators that can be described by the linear relation between the applied currents and the produced wrench (2.7).
For most rotational (and linear) actuators, this inverse mapping is achieved by using dq0- or Park’s transformation [31-33, 91, 92]. For linear and planar actuators, this transformation can be used directly to derive a commutation that decouples only the force components. In moving-coil planar actuator, such as [24, 22], it is possible to use design symmetries, which reduce the complexity of the torque equations. Then an additional transformation can be derived, which allows for decoupling of the torques [25]. Moving-magnet planar actuators with integrated Moving-magnetic bearing [51, 26, 59, 18] have complex torque equations [77]. As a result, the dq0-transformation is not convenient for them. In literature, attempts can be found to decouple torque using additional transformation after applying dq0-transformation [108, 114]. Nevertheless, the resultant disturbance torque is still significant. The algorithm was further improved by Binnard et al. [10, 11] resulting in commutation for 6-DOF planar actuator, but the full torque equations are still not included.
2.2.1 l
2-norm Based Commutation (L2C)
To overcome the problem of torque decoupling, a method for direct wrench-current decoupling has been developed independently and in parallel by [78] and [95]. The under-determined set of equations (2.7) offers the possibility to impose extra constraints while creating an inverse mapping. An interesting additional constraint is to minimize the sum of the Ohmic losses in the coils (or dissipated power). This can be done by minimizing the l2-norm of the current vector:
des 2 2 des2
( )q i wmin i ( )q w
-= =N
K K . (2.10)
The matrix K , which minimizes the l-2 2-norm, is a reflexive generalized inverse of K, also
known as a pseudo-inverse of K [96], and can be calculated explicitly:
(
)
1 1 T 1 T 2( )q ( )q ( )q ( )q -- = - -K N K K N K , (2.11)where N is a positive-definite weighting matrix. The main advantage of this method is the minimum dissipated power if NºR, where R is a diagonal weighting matrix of which the
elements correspond to the resistances of each active coil [78]. If all the coils have the same resistance, the weighting matrix can be omitted. The second benefit is that the commutation is obtained in a single step (no iterations are necessary). The third benefit is the short calculation time due to the single step solution and the simplicity of calculating
the pseudo-inverse. The inverse in (2.11) is calculated only from a m m´ matrix, where dim( )
m= w and therefore for any over-actuated actuator m £ . The last, but very 6 important property is the continuous current characteristic with the position q or the
wrench w variation. For commutation on a real over-actuated actuator, continuous des
current change is important, because too high di/dt causes peaks in the terminal voltage uj
of the jth-coil [40]: 1 d d d d d d n j k j j k k k j i i x u i R L M t = t x t ¹ ¶L = + + + ¶
å
, (2.12)where R (W), L (H) and M (H) are resistance, inductance and mutual inductance, respectively, of the coil, ¶L ¶ is a change of the flux linkage of the permanent magnets x
with the coil.
The drawback is that the l2-norm based commutation cannot imply any
constraints on the current maximum. The currents calculated by using the pseudo-inversion (2.11) can easily be over the physical limits of the amplifiers or coils. In such a situation amplifiers saturate the current or, even worse, they turn themselves off due to overload/overheat. In any case, forces and torques of the actuator are no longer linearized and decoupled. For that reason, it might be better to use different criteria for minimization.
2.2.2 l
∞-norm Based Commutation (LiC)
The infinity-norm is the only norm that directly puts constraints on the maximum current. The criterion for minimization is given as:
des des ( )q i wmin i ( ,q w ) -¥ ¥ ¥ = = K K . (2.13)
In mathematical literature, this problem is known under term: Chebyshev solution
of an underdetermined system of linear equations [1, 6]:
min
{
|}
x x ¥ Ax=y , (2.14)
where x¥ =sup
(
x1,x2, , xn)
is called l∞- or Chebyshev norm. From the nature of the problem, the inversion mapping-¥
K is not affine in w anymore, and, consequently, -¥
K cannot be written as a matrix as in the l2-norm situation. The main difficulty of solving
this mathematical problem is that it is not possible to obtain the solution explicitly. All the known methods reach the solution of (2.14) iteratively. The methods differ in memory requirements and in the necessary calculation time, which is mostly dependent on the number of iterations. An extensive summary of the methods for calculation of underdetermined system of linear equations can be found in Abdelmalek [1]. However, one additional, fast and effective algorithm is discussed by Ascher [6]. The Ascher’s algorithm employs a linear programming algorithm for the solution of a set of over-determined linear equations in the l1-norm to obtain a minimum l∞-norm solution to the set of consistent
normalized first. Then the solution is iteratively obtained via the dual l1 problem. The
main benefit is a lower number of necessary iterations, which depends on the number of equations and less on the number of variables. Thus, the algorithm is supposed to lead to a shorter calculation time. This is in contrast to a similar principle for the calculation that can be found in [1] where the linear programming is used to obtain only the initial solution. After this step, the iteration process employs a slightly modified simplex method.
The effectiveness of both algorithms has been verified by the following test. Test sets of equations with different dimensions were filled with random numbers. The test was evaluated ten thousand times with each set of equations always filled with new random values. Mean, maximum number of iterations and time were captured as shown in Table 2.1. The calculation time in Table 2.1 is, for obvious reasons, scaled. The results in Table 2.1 clearly confirm that the number of iterations in the Ascher’s algorithm [6] depends more on the number of equations (m) whereas the Abdelmalek’s algorithm [1] is dependent on the number of variables (n). Both algorithms obtain the same solution whereas the Ascher’s is always about twice as fast regardless the dimensions of the matrix A. This is
not what one would expect when comparing only the number of iterations. For a higher dimension of x, the Abdelmalek’s algorithm needs several times more iterations than the first one to obtain the solution, but it needs less calculation time per iteration. Still, the Ascher’s algorithm is more time-effective in obtaining the solution. The probability distribution of the number of necessary iterations for different number of variables and equations is plotted in Figure 2.3. The probability Pr (-) of obtaining the solution in it iterations is defined as ( )Pr it =pit /k, where p the number of solutions in it iterations and it k is the total number of experiments. Hence, SitPr it( )= . 1
The advantage of the l∞-norm minimization is a direct minimization of the
maximum currents. This goes so far that na-m+ variables have the same minimized 1 absolute value [6], where n is number of variables that have nonzero coefficient in at least a
one of the equations (number of the active coils). From a physical point of view, it means
Table 2.1: Comparison of two approaches for LiC calculation
Iterations (-) Calculation time (scaled) Ascher Abdelmalek Ascher Abdelmalek
A(m,n)
Avg Max Avg Max Avg Max Avg Max
4 6 4 6 7 12 1 2 4 8 4 10 5 9 8 15 2 3 6 10 5 10 6 10 10 15 3 5 7 10 4 50 7 14 23 33 10 14 20 32 4 100 8 14 46 72 27 35 58 75 5 100 11 19 50 81 30 45 70 100 6 100 14 23 57 92 42 55 83 110
0 10 20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Iterations (-) P robab ilit y ( -) Ascher 4x10 Ascher 6x10 Ascher 4x100 Ascher 6x100 Abdelmalek 4x10 Abdelmalek 6x10 Abdelmalek 4x100 Abdelmalek 6x100
Figure 2.3: Probability distribution of the number of necessary iterations calculated for both
methods and different matrix A sizes.
that as many coils (na-m+ ) will be energized to the same level even if their 1 contribution to the final wrench is very small. In consequence, although the maximum current is reduced, the actuator needs much more power (so also more heat production) than the l2-norm solution. In addition, the energized coils will produce more force, which
will cancel each other (the final wrench must be the same); hence, the sensitivity to coupling matrix errors will be higher. Another drawback, from an application point of view, is a discontinuous current-variation with a change of position q .
2.2.3 Clipped l
2-norm Based Commutation (CL2C)
So far, two limit solutions for commutation of an over-actuated actuator have been discussed in this section. The first solution minimizes the l2-norm of the current
vector and, therefore, the dissipated power with no constraints on the maximum current. The second one minimizes the l∞-norm of the current vector or the maximum currents whereas the power usage increases rapidly. Obviously, neither of the solutions is perfect. The compromise is to minimize the used power whereas the maximum current can be limited if necessary: des clip 2 ( )q i wmin,i i i ¥ = £ K , (2.15)
where iclip is the limit on the current. Minimization of l2-norm of constrained system of
linear equations can be solved by quadratic programming (e.g. [87]) where a cost function ( )
f x is minimized subject to inequality and equality constraints:
T T 1 ( ) , 2 , . f x x x c x x b x d = + < = Q A E (2.16)
By using the notation of (2.15), the quadratic programming problem (2.16) can be defined as:
T
clip clip des
1 ( ) , 2 , , f i i i i i i i w = - < < K = (2.17)
where i is the clip n´ vector of the current limit values. The large calculation time, for 1 solving this quadratic programming problem, prohibits real-time commutation. A fast analytical solution for one equality constraint was presented in [21]. However, since K has
m rows for m-DOF system, the method is not applicable on this problem. For that reason
a novel, fast commutation method has been developed that, as well, puts constraints on the maximum current whereas the current vector is still minimized in l2-norm. The calculation
process itself is also iterative, but the number of iterations is low and each iteration step is calculated fast.
The algorithm is based on the fact that if the currents are calculated with L2C and are exceeding the limit (| |ij >iclip), they will the most probably have the limit value
clip
: sgn( )
j j
i =i i when solved according to (2.15). The algorithm has the following steps: 1. Calculate the pseudo-inversion K (2.11) of the coupling matrix K, which 2
-minimizes the current vector in l2-norm (2.10).
2. If any of the values exceeds the clipping limit ij >iclip,j= , continue, 1, ,n
otherwise go to end.
3. All exceeding currents are saturated, a vector of clipped (saturated) currents ci is
generated : clip clip clip sgn( ) if [ ] , 1,..., 0 if j j c j j i i i i i j n i i ìï > ïï =íï = £ ïïî . (2.18)
4. A new matrix l2K is created from K, where the columns corresponding to nonzero
elements of vector ci are set to zero. A new desired wrench vector 2 des lw is calculated: 2 des des lw =w - K ci (2.19)
5. By using the pseudo-inversion (2.11) a new current vector l2i is calculated from
the reduced system:
2 2 2
des
l Kl i =lw . (2.20)
6. The final current vector cl2i is the summation of the clipped values ci and the
vector of l2-norm minimized values l2i (because for each row j either c 0 j i = or 2 0 l j i = ): 2 2 cl i =l i + . (2.21) ci
7. The process loops back to the step 2 (with i :=cl2i) while any of the current
values exceeds the clipping limit.
It is important to mention that the solution of this heuristic algorithm is not necessarily optimal. It might happen that by saturating one of the currents some other, which was also saturated, could be lowered to obtain the optimal solution, but it is not lowered. The optimal solution can be obtained for example by quadratic programming. The main advantage of this approach in comparison to the quadratic programming is in the calculation time, which is just k-times the calculation time of l2-norm solution, where k is
number of iterations needed.
The number of iterations mostly depends on the clipping value iclip for the current limitation. If the clipping value is higher than the actual maximum current obtained with
l2-norm, only one iteration step is needed. If the value is lower than the maximum current,
at least one additional iteration step is required. The number of iterations increases as the clipping value decreases down to its lowest limit, which is equal to the value of the minimized l∞-norm of the current vector min i
¥. The maximum theoretical number of
iterations is thus na-m+ , where 2 n is number of the active coils. This situation occurs a
only when in each iteration step one (extra) current is saturated.
Although the algorithm is heuristic, the obtained solution shows an exact match with the solution obtained with quadratic programming in the majority of tests (see the next section). From simulations, the same solution as from quadratic programming was obtained with higher probability if the clipping value was further from the minimum l∞ -norm solution (fewer currents had to be saturated). Even if the obtained solution is suboptimal, it still satisfies given constraints and the non-saturated currents are minimized in l2-norm. Therefore, the dissipated power will always be lower than those calculated by
LiC, so:
2 2
2 2 2
l cl l
i £ i £ ¥ i . (2.22)
The proof of (2.22) is trivial. Since L2C minimizes 2-norm of the current vector,
2 2 2 2 l cl i £ i and 2 2 2 l l
i £ ¥ i is automatically true. The third part, 2
2 2
cl l
i £ ¥ i , is true because CL2C also minimizes 2-norm of the current vector with the inequality
constraint on the maximum current (see (2.15)). Therefore, if there exists any
2
min i solution with i ¥<iclip, it will be calculated with CL2C.
Moreover, the current steps caused by the CL2C are not as severe as in the LiC, because only the currents close to i value can make the step to/from the saturation limit. clip
Other currents (further form the boundary) will also make small steps in consequence, to satisfy the desired wrench production. With known parameters of an ironless actuator and by using (2.12), one can calculate how large a current step the current amplifiers can handle. For example, with a coil inductance of 10 mH, sampling frequency of 1 kHz, clipping limit iclip=2 A and a current step of 10 % of i (= 0.2 A), the peak in the voltage clip
will have amplitude of 2 V. Current amplifiers can usually handle much higher voltage peaks.
Continuous current-change is guaranteed if the consecutive solutions are the same as those obtained by quadratic programming and hence optimal. Continuity of solution obtained via quadratic programming is shown e.g. in [98, 49, 28].
In a real application, current amplifiers should not be the limiting factor during the whole trajectory of the actuator. On the contrary, current limitation via commutation should be considered as a safety layer for unexpected circumstances (increased load, disturbing forces etc.) or just for several spots in the trajectory of the actuator. If a significant number of coils need to be limited for most of the time of the motion, the actuator will have increased dissipated power. Obviously, this is not a good design. If only several currents are limited, the obtained solution is with high probability optimal and hence continuous. Therefore, applicability of CL2C on a real actuator should be possible.
For a predefined trajectory of the actuator, the current continuity and the peak voltages in the case of the discontinuity can be tested in simulation in advance.
Reduction of the Number of Iterations
It might happen that several currents are very close to the clipping limit, but just one or a few of them are actually exceeding the limit. Then the algorithm saturates the exceeding currents, which will cause an increase of the rest of the currents including the ones close to the border. Consequently, one or more of the currents under the limit might cross the limit and needs to be saturated again. This causes another iteration step. Clearly, the situation can repeat for several times. To prevent such a situation, we changed the third step of the proposed algorithm:
3. Set the clipping limit in the condition to the a-times the original value with (0,1]
a Î . All currents exceeding the lowered boundary value a.iclip are saturated - a vector of clipped currents ci is generated :
clip clip clip sgn( ) if . [ ] , 1,..., 0 if . j j c j j i i i i i j n i i ìï > a ïï =íï = £ a ïïî . (2.23)
In this way if any of the currents in step 2 exceeds i , the algorithm preventively saturates clip
all the values over a.iclip to the i value. The proposed value of gain a is between 0.8 and clip
0.95, but it can be tuned as it is presented in the next section. In this way, all currents close to the clipping limit will be preventively saturated. As a result, the number of iteration steps will be reduced. The price for the decreased calculation time is a possibly suboptimal solution and hence increased dissipated power and current discontinuity. This trade-off can be tuned by the gain a according to the application. Tuning of a can lead to a maximum of two iteration steps (calculations of the l2-norm) if the clipping value is not
too close to the min i
¥ for all q. This condition becomes clear if one imagines that if iclip is
close to the min i
¥ limit, then a large number of currents must be limited at once. Hence
prediction of which currents need to be limited if the boundary is even lowered (a.iclip), leads to a wrong choice.
In Table 2.2, all three norm-based commutation methods are compared. The calculation time is based on a model of an actuator with n = 100 coils and m = 6 degrees
Table 2.2: Comparison of the norm-based commutation methods Commutation method l 2-norm (L2C) l∞-norm (LiC) clipped l2-norm (CL2C) Current limitation – ++ + Current continuity ++ – +/– Power losses ++ – +
Calculation timea 7b 45 iter 7c
Overall Performance + 0 ++
a Scaled by the same factor as in Table 2.1.
b Real calculation time in dSpace system D1005 with n = 99 and m = 6 is about 60 ms.
c Calculation time is approximately equal to the calculation time of the L2C multiplied by the
number of iterations.
of freedom. From the comparison, clipped l2-norm based commutation comes out as the
best trade-off between the algorithm performance and the constraint satisfaction.
Commutation methods based on minimization of other ln-norms with 2< < ¥ n
have no practical sense, since they cannot limit the peak current, nor they minimize the dissipated power.
2.2.4 Example on a Model of the COPAM
The presented norm-based commutation algorithms have been tested on a model of the COPAM. Since the number of active coils is always greater than number of DOF’s, the system satisfies the definition of over-actuated actuator. Because the commutation is a static mapping between the desired wrench vector w and the calculated current vector i, simulation of the commutation gives exactly the same results as we would obtain from the real setup. The dimensions of the coupling matrix K from (2.7) are n = 99 and m = 6. The
dimensions and properties of the PA are in Appendix A.
Comparison of L2C, CL2C and LiC
Figure 2.4 shows the energized coils example for one position of the PA when the commutation is calculated with the L2C and LiC for wdes= [ , , , , , ]T
x y z x y z
F F F T T T =
T
[200 N, 200 N, 200 N, 0 Nm, 0 Nm, 0 Nm] . The difference is visible immediately. With the L2C, only a few coils are energized to the significant level (those that have the highest effect on the commutation). On the other hand, with the LiC all the coils with nonzero effect on commutation are energized. Moreover, most of them are energized to the same absolute value, which is less than the maximum value reached by the L2C.
The current values are also plotted in Figure 2.5 where all three types of commutation are used: L2C, CL2C and LiC with l2i , cl2i , and l¥i current vectors,
i1 i9 i91 i99 i1 i9 i91 i99 3.5 A -3.5 A
Figure 2.4: Distribution of currents in the coils calculated with the L2C and LiC; the solid square
represents actual position of the PA; the dashed rectangle is working area of the PA.
0 10 20 30 40 50 60 70 80 90 100 -4 -3 -2 -1 0 1 2 3 Coil number (-) C u rre n t ( A ) l2i cl2i, i clip= 2 A li
Figure 2.5: Distribution of currents in the coils calculated with the L2C, CL2C and LiC for the
position of the PA in Figure 2.4.
1.5 2 2.5 3 3.5 4 550 600 650 700 750 800 850 900 950
Clipping current iclip (A)
D is si p at ed p ow er ( W ) L2C CL2C LiC
The dissipated power, calculated as P=iTR , where R is a diagonal matrix of i
the coil resistances, for different clipping currents of the CL2C is presented in Figure 2.6. From the figure, one can conclude that reducing the maximum current moderately will not cause significant power increase, however trying to get close to the minimum l∞-norm
current limit, where most of the coils are limited, will increase the power demands substantially. In this example, it is more than 50 %. For i > clip 2
max
li , the dissipated power
is the same as in the l2-norm solution.
Investigation of Optimality of the Solution
The solution obtained by the CL2C is not always optimal (on the other hand, the solution always satisfies the constraints and the non-clipped currents are minimized in l2
-norm). Therefore, a test has been performed on the model.
First, let’s define a relative clipping value r Îc [0,1]. Then the actual clipping current is calculated as
(
2)
clip max max max
l l l
c
i = ¥i +r i - ¥i
. (2.24)
If 0r =c , the actual clipping limit is equal to min i
¥ solution and for r =c 1 there is no
clipping because iclip is equal to the maximum current calculated with l2-norm. The test was
performed for a dense grid of positions q of the PA. The probability of achieving the optimal solution (the same current vector as from commutation calculation with quadratic programming: cl2i ºqpi) depending on the relative clipping value
c
r is presented in
Figure 2.7 left. Already for r ³c 0.2 the probability of an optimal solution was > 97 % and for 0.8r ³c the optimal solution was found in 100 % of the cases. The question is: how far from the optimal solution can the suboptimal one be? To get a notion about that, the maximum relative l2-norm difference over all positions q between the suboptimal and the
optimal solution:
(
2)
max adm cl qp qp q S i i i Î (2.25)for different r c is plotted in Figure 2.7 right. The maximal relative difference is less than 0.01 already for r =c 0.1 (very close to l∞ limit value) which means that the CL2C will
require less than 1 % more power than necessary. At the LiC limit value the difference is at most 10 %. For the increasing r the maximum power difference decreases exponentially. c
If both findings are combined, it is possible to get an impression of how accurate the solution is. It is also important to emphasize that the maximal difference occurs only for a limited amount of cases.
Current Discontinuity
If the continuous current-change is required, the optimal solution must be obtained always. Unfortunately, we cannot guarantee this. On the other hand, with increasing rc value, the probability of the optimal solution increases as well. In the presented example, for r ³c 0.8 the optimal solution was found in 100 % of cases.
min Linf 70 0.2 0.4 0.6 0.8 min L2 75 80 85 90 95 100 P robab ility o f o p tim a l s o lu tio n ( % )
Relative clipping value rc (-) min Linf 0.2 0.4 0.6 0.8 min L2
10-16 10-12 10-8 10-4 100 M ax . r el at ive di ffer nce ( -)
Relative clipping value rc (-)
Figure 2.7: Left: Probability of obtaining the optimal solution with CL2C, the same solution as
from quadratic programming; Right: Worst-case relative l2-norm difference.
The current (dis)continuity was investigated for the change of the wrench w and des
change of the position q of the PA. Figure 2.8 top shows an example of the current in the coils when the desired vertical force Fz increases for one position of the PA. The current values are clipped at 2 A. For the force Fz around 500 N, most of the currents are already clipped. Steps in the current caused by the increase of the wrench are shown in Figure 2.8 bottom. The force step D was chosen very small (0.03 N, while max(d dFz i Fz) < 0.2 A/N) so only the current steps due to discontinuity of the solution are clearly visible in the figure. The biggest discontinuity is about 0.2 A. With the sampling frequency of 3 kHz and the coil inductance of 8.1 mH, the voltage peak has amplitude of less than 5 V, while the used current amplifiers can operate with up to 50 V.
An example of variation of the currents in the coils as the PA moves at the maximum velocity of 0.5 m.s-1 is shown in Figure 2.9 top. The number of the limited
currents varies from many (at 0.05 s) to almost none (at 0.12 s). Change of the currents in one time step (0.33 ms) is shown in Figure 2.9 bottom. The discontinuity of the solution is visible as the steps or pulses. From the figure, one can see that the discontinuity of the solution does not cause bigger steps in the current than the steps that are required for the continuous change in heavily saturated situation.
Reduction of Iteration Steps
Several tests were performed to verify the effectiveness of the proposed reduction of iteration steps. In the first test, the influence of gain a on the number of iteration steps was investigated for different clipping values, for the same position q and wrench vector w as in Figure 2.4 and Figure 2.5. The results that are presented in Figure 2.10 left show that it is possible to reduce the number of iteration steps. For example already for gain a = 0.9 with clipping current iclip = 2.2 A, the commutation will still be calculated in two steps. Of course, this works only if the clipping current is not too close to the min i ¥ limit value. The plot with a = 1.0 is equal to the original CL2C.
200 250 300 350 400 450 500 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Desired vertical force (N)
C u rr ent in the co ils ( A ) 200 250 300 350 400 450 500 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Desired vertical force (N)
C ur ren t s tep i (A )
Figure 2.8: Influence of the wrench variation on the current continuity calculated with CL2C;
calculated for one position of the PA; Top: Current in the coils; Bottom: Discontinuity Di in the current.
Another aspect is increased dissipated power. The situation for different gains a and clipping currents is shown in Figure 2.10 right. As expected, gain a results in the coise of a wrong set of currents and, therefore, a suboptimal solution is obtained. On the other hand, in the presented example for a = 0.9 the dissipated power does not increase much whereas two iteration steps are sufficient for calculation, which facilitates real-time implementation.
The last performed test checked the maximum number of iterations on the whole workspace of the planar actuator again for the different a and iclip (see Figure 2.11). Although the clipping value, for which two iterations are sufficient, is higher, it is still possible to guarantee just two iteration steps for a significantly reduced maximum current.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time (s) C u rr en t in th e co ils ( A ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -0.1 -0.05 0 0.05 0.1 Time (s) C ur rent s tep i (A )
Figure 2.9: Influence of the PA position variation on the current continuity calculated with
CL2C; calculated for a constant wrench and v = 0.5 m.s-1; Top: Change of the current in the
coils; Bottom: Steps in the current.
1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6
Clipping current iclip (A)
Iter at io ns ( -) = 1.00 = 0.90 = 0.80 = 0.70 1.5 2 2.5 3 3.5 4 550 600 650 700 750 800
Clipping current iclip (A)
Di ssi pa ted pow e r ( W ) = 1.00 = 0.90 = 0.80 = 0.70
Figure 2.10: Left: Number of iteration steps; Right: Dissipated power calculated with the CL2C
2.2 2.4 2.6 2.8 3 3.2 3.4 0 2 4 6 8 10
Clipping current iclip (A)
M a x. Ite ra tio n s (-) = 1.00 = 0.90 = 0.80 = 0.70
Figure 2.11: Worst-case number of iteration steps of the CL2C depending on clipping current iclip
and gain a; over all positions of the PA.
2.2.5 Conclusions
In this section, three norm-based commutation methods have been compared. The first one, l2-norm based commutation (L2C), was developed and successfully used in the
past for control of magnetically levitated planar actuators [78, 95]. Its main benefit is in the minimization of dissipated power. Moreover, the commutation is calculated very fast by using pseudo-inversion of the original coupling matrix. The drawback is its inability to constrain the maximum current, which can cause saturation of the amplifiers and, consequently, the difference between the desired and the real forces and torques. This problem can be solved by two other norm-based commutation techniques. Although the minimal l∞-norm solution for underdetermined system of linear equations has existed in the mathematical literature for some time [1, 6], application on commutation for an over-actuated actuator has never been proposed before. The benefit of this kind of commutation is that the maximal current is always minimized. The negative aspects are power losses, which are not optimized. Because almost all active coils are set to the same maximum current level, although they can have minimal influence on the final force/torque, the power usage increases dramatically. Therefore, a novel, third kind of norm-based commutation is proposed which puts constraints on the maximal current whereas the non-saturated coils are minimized in the l2 norm. By varying the clipping (saturation) value,
the obtained solution is closer to either the l∞- or the l2-norm based commutation. The
applied algorithm for the clipped l2-norm based commutation (CL2C) does not necessarily
lead to the optimal solution, but the performed tests indicate that the optimal solution in
l2-norm sense can be obtained in the majority of cases whereas the suboptimal solution is
still very close to the optimal one and practically indistinguishable in the real system. The benefit of this algorithm is in calculation time, which is reduced considerably in comparison to the LiC or optimal CL2C obtained via quadratic programming. Continuous current-variation with position and wrench current-variation can be guaranteed in the L2C and can be achieved in the CL2C if the solution is optimal. Nevertheless, as was shown in the example, the current discontinuity caused by the CL2C will not introduce high voltage peaks in the current amplifiers.
