ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

B Proof of (G C DU ε ∆X M ) being I.I.D and dimensionless C An example application requiring high positioning performance along selected axes of the robot's platform D Simulation study to assess the influence of error in applied force/moment during stiffness identification on compensation accuracy for a hexapod positioning system D.1 Simulation description . . . . . . . .

V. E. Gough (right) [START_REF] Stewart | A platform with six degrees of freedom[END_REF] Errors in estimated stiffness parameters due to errors in applied forces/moments during stiffness identification ε ∆F M DU ε ∆X M DU ε ∆X M scaled using G DU ε ∆X C Component of error in predicted deflection ∆X C as a result of DU ε c 8 LIST OF SYMBOLS FE ε ∆X C Component of error in predicted deflection ∆X C as a result of FE ε c ε ∆F M Errors in applied forces/moments during stiffness identification DU ε X A Errors in pose attained after compensation as a consequence of errors in measured deflections during stiffness identification FE ε X A Errors in pose attained after compensation as a consequence of errors in forces/moments applied during stiffness identification FE,t ε X A Translational components of FE ε X A FE,r ε X A Rotational components of FE ε X A DU ε X C Component of error in predicted command pose ( X C ) as a consequence of errors in measured deflections during stiffness identification FE ε X C Component of error in predicted command pose ( X C ) as a consequence of errors in forces/moments applied during stiffness identification DU,t ε X A Vector containing the translational components of DU ε X A DU,r ε X A Vector containing the rotational components of DU ε X A Matrix which is function of J, J and ap K F in Initial load vector applied at the end-effector/platform of a robot during stiffness identification as F in Initial load vector assumed to have been applied at the endeffector/platform of a robot during stiffness identification ac F in Initial load vector actually applied at the end-effector/platform of a robot during stiffness identification

F fi

Final load vector applied at the end-effector/platform of a robot during stiffness identification as F fi to respective elements of Cov( DU ε X A )

X in

Pose vector of a robot's end-effector/platform when the load vector ap-

plied on it is F in X fi
Pose vector of a robot's end-effector/platform when the load vector ap-

plied on it is F fi X A
Pose attained by a robot after elastostatic error compensation when there are no errors in identified parameters

X A
Pose attained by a robot after elastostatic error compensation when there are errors in identified parameters

X C

Pose vector that is entered into the controller of a robot that doesn't have the compliance error model embedded in it (command pose)

X C

Predicted command pose with error

X D

Pose at which best positioning is desired Rows of Z corresponding to FE,t ε X A r Z

Rows of Z corresponding to FE,r ε X A Firstly, I would like to express my gratitude to François Pierrot, my thesis director, for all his help during the course of this research. More importantly, he deserves special praise for inspiring engineers and researchers like me to strive for excellent research.

My sincere thanks also goes to Sébastien Krut, firstly, for giving me the opportunity to work on this project and secondly, for encouraging and aptly advising me to perform high-quality research. His valuable advices have shaped the course of this work to a great extent.

I would like to express my deepest gratitude to Olivier Company for being a wonderful guide and for being there whenever I needed him the most. His ability to have a good helicopter view of the problem at hand and his vast knowledge in robotics have been instrumental in shaping the course of this work.

I would also like to thank Alain Vissière for his help throughout the course of this research. I thank him for introducing me to the work of POSILAB, for his active involvement in this work and for helping me in the process of writing papers and this thesis.

13 their profound knowledge of hexapods has been instrumental in sculpting this work.

A special thanks goes to other colleagues from Symétrie who have been very helpful for this research with their knowledge about hexapods and by doing everything necessary to ensure that I could have ample "play time" with their hexapods.

I would also like to thank the reporters of this thesis, Stéphane Caro and Pierre Renaud, for taking time to read this thesis despite their extremely busy schedules.

I would like to thank all (past and present) colleagues from LIRMM for the stimulating discussions and for maintaining an exceptional atmosphere for performing research. In particular, Marc Gouttefarde deserves special praise for sharing his knowledge about the world of research.

Lastly, I would like to thank Agence Nationale de la Recherche (ANR) for funding this work.

Vinayak Kalas

Montpellier, April 2020

General introduction Context

The work presented in this thesis was carried out within the framework of POSILAB project. This project was carried out in a joint laboratory, POSILAB, between an academic laboratory called LIRMM (Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier) and a private enterprise named Symétrie. LIRMM is a multi-partner multi-disciplinary research institution which conducts research in computer science, microelectronics and robotics. The DEXTER team within LIRMM is involved in the POSI-LAB project and this team specializes in, among other things, development of robotic manipulators, especially, parallel robots (example: hexapods). Symétrie is an enterprise that specializes in hexapods for precise-positioning and motion applications. This project was funded by French National Research Agency (ANR). The main aim of this project was to find innovative solutions to achieve better positioning performance, than what is currently possible, with high-precision hexapod positioning systems.

Hexapods (see figure 1), commonly known as Gough-Stewart platforms, are parallel robots with six actuated legs. Parallel robots have multiple serial chains connecting the base to the end-effector/platform. In contrast, serial robots have a single chain connecting the base to the end-effector/platform (see Figure 2). The first hexapods were developed 15 in 1950's and 1960's. Notably, the first known hexapods were developed by V. E. Gough for testing tyres and D. Stewart for flight simulation [START_REF] Stewart | A platform with six degrees of freedom[END_REF] (see figure 3). Hence, the name Gough-Stewart platform. Today, they are extensively used for two main class of applications: high-precision 6-DOF positioning (see figure 4) and 6-DOF motion generation (see figure 5). Hexapods are attractive for high-precision positioning applications due to their high stiffness 1 and due to the fact that they are statically determinate or isostatic structures. The advantage of hexapod being statically determinate is that the platform is not susceptible 2 to unwanted internal stresses that deform the platform.

GENERAL INTRODUCTION
Figure 1: A hexapod 1. This is a general advantage of parallel robots over their serial counterparts. 2. Consider a septapod (hexapod with one extra leg). This robot is statically indeterminate. Assuming that only position control is used, the platform will be subjected to (even if it is minimal) internal stresses. This is because there are more legs constraining the platform than that are absolutely required. Consequently, the constraining legs fight with each other and induce stresses in the hexapod and its platform, unless the legs are machined and controlled extremely precisely. Furthermore, temperature changes can worsen the internal stresses in the structure [START_REF] Soemers | Design principles for precision mechanisms[END_REF]. These internal stresses can deform the structure and these are not easily predictable (since static equilibrium equations can't be solved), thereby making them unattractive for high-precision positioning applications.

GENERAL INTRODUCTION

Motivation

A robot's static positioning performance is affected by many factors such as geometric, errors, thermal deflections, friction, compliance of robot's components, etc. Depending on the application at hand and the components used in a robot, some of these factors dominate positioning errors over others. Symétrie's positioning hexapods are designed to achieve high repeatability. Various techniques -such as geometric calibration -are used to ensure high accuracy too (see figure 6 for difference between repeatability and accuracy of positioning). However, as new challenging applications emerge, problems that could be neglected before must be taken into consideration. One such problem is the deflection of the hexapod when a heavy load is placed on the platform. The accuracy of these hexapods deteriorate when heavy payload is mounted on their platforms, as a result of compliance of their components. Consequently, Symétrie was interested in understanding the influence of compliance of hexapod's components on its accuracy when a heavy payload is mounted on its platform. Subsequently, the goal was to improve the accuracy of hexapods with heavy payload mounted on their platforms. This was important because a growing number of their clients have been demanding for positioning with high payload and high accuracy.

Figure 6: Illustration of repeatability and accuracy of positioning [START_REF] Alexander | Precision machine design[END_REF] Accuracy of loaded robots can be increased in two ways. The robot can be designed MOTIVATION 21 to be more stiff 3 or the errors in positioning due to payload on the platform can be compensated for by prescribing extra displacements in actuators. The former option is not attractive as it is expensive to achieve this and deflections can't be completely eliminated.

Robot calibration is a concept that deals with compensating for robot's positioning errors due to various factors. Robot elastostatic calibration [START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF][START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF] deals with compensating for positioning errors of loaded robots due to compliance of their components. The general 4 manner of performing robot elastostatic calibration involves (in the order mentioned): (a) modelling stiffness of the robot using a parametric model, (b) measuring deflections of the robot when known loads are applied, (c) identifying/estimating the parameters of the stiffness model, and (d) compensating for the positioning errors of the loaded robot. It is known that in this elastostatic calibration method, the choice of end-effector/platform poses (position and orientation) and forces/moments used for performing the deflection measurements affects the quality of estimated parameters [START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF][START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF]. This, in turn, influences the quality of compensation. In the literature, best poses and forces/moments were chosen for stiffness identification using criteria which minimized the influence of deflection measurement uncertainty 5 on stiffness identification. Many such criteria exist and they can be broadly classified into the ones that minimize parameter errors and the ones that minimize pose error after compensation, for a given uncertainty of deflection measurement.

For applications concerning many of Symétrie's customers and for many other robotic positioning applications, precise positioning is often required only at some predetermined poses in the workspace, along predetermined axes and with a predetermined payload.

It is, therefore, important to have criterion for selecting poses and forces/moments for stiffness identification that can maximize positioning accuracy after compensation under the given conditions. Furthermore, preliminary study 6 revealed that realistic differences between forces/moments actually applied during stiffness identification experiment and those assumed to have been applied can have considerable impact on compensation qual-3. Using stiffer materials and/or designing a stiffer geometry. 4. Other methods also exist for performing robot elastostatic calibration. However, this method is more common and advantageous due to reasons explained in section 1.1.2.

5. Uncertainty of deflection measurements exists as a consequence of uncertainty of pose measurement instrument used for performing deflection measurements.

6. Presented later in this thesis.

GENERAL INTRODUCTION

ity. Hence, the criterion for selecting poses and forces/moments for stiffness identification must also minimize the influence of this error on compensation quality.

The criteria present in the literature for selecting best poses and forces/moments for stiffness identification can't ensure best possible positioning performance at predetermined poses, along predetermined axes and with predetermined forces/moments on endeffector/platform. Furthermore, these criteria do not minimize the influence of errors in forces/moments applied during stiffness identification on compensation quality. Hence, new criteria had to be developed to satisfy the requirements stated above.

Thesis goals

In the view of above mentioned requirements, the following goals were defined for this thesis:

• Thesis goal 1: Development of new criterion or set of criteria for selection of poses and forces/moments for stiffness identification which:

-Sub-goal 1: minimize the influence of deflection measurement uncertainty and errors in forces/moments applied during stiffness identification experiment on compensation quality.

-Sub-goal 2: maximize positioning accuracy after compensation at predetermined pose(s), along predetermined axe(s) of end-effector/platform and with predetermined force(s) & moment(s) applied to the end-effector/platform.

• Thesis goal 2: Implementing elastostatic calibration of a high-precision hexapod positioning system and using the developed criterion/criteria for optimizing its stiffness identification.

Thesis outline

To address the above defined goals of this thesis, the contents of this thesis are organized as follows:

Chapter 1 introduces the state of the art of robot elastostatic calibration and its optimization. It presents the concept of robot elastostatic calibration and type of robot elastostatic calibration chosen to accomplish the goals of this thesis. The necessary mathematical background of elastostatic calibration and its optimization is also presented. This chapter concludes with the presentation of existing criteria for stiffness identification optimization, their limitations and subsequent requirements from new criterion/criteria to be developed.

Chapter 2 presents the framework to formulate stiffness identification optimization criterion/criteria that counters the limitations of the existing criteria for the same.

The necessary mathematical formulations of the said framework are derived and the ways to use them are discussed.

Chapter 3 is devoted to validation of efficacy of the presented stiffness identification optimization framework using simulated elastostatic calibrations of a bipod. These simulation studies were performed on a bipod to facilitate ease of analysis of results as this mechanism is simple.

Chapter 4 documents the experimental studies on elastostatic calibration of a highprecision positioning hexapod. The presented studies validate the efficacies of elastostatic calibration of hexapods and the stiffness identification optimization framework presented in chapter 2.

Finally, the conclusions of this thesis are presented which highlights the main contributions of this thesis and presents some recommendations for future work. Different factors, from the ones listed, dominate the total positioning error depending on the robot, the control method and the application. In order to achieve good positioning performance, these errors are kept minimal (relative to required positioning accuracy and repeatability) by using high-quality mechanical components. Alternatively or additionally, appropriate compensation techniques are used to minimize or eliminate the influence of these error sources on the robot's positioning performance.

The concept of robot calibration deals with compensation for robot's positioning error due to different errors [START_REF] Zvis Roth | An overview of robot calibration[END_REF]. Different types of robot calibrations seek to minimize the influence of different types of errors on the static positioning performance of a robot: geometric calibration [START_REF] Hayati | Robot geometry calibration[END_REF] to minimize the influence of geometric errors, elastostatic calibration [START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF][START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF] to minimize the influence of compliance errors, thermal calibration [START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF] to minimize the influence of thermal errors, and so on.

Robot geometric calibration is a widely studied and applied concept. However, robot elastostatic calibration has only recently garnered serious attention. Robot elastostatic calibration becomes necessary for robots that are used for applications that require highaccuracy positioning in the presence of heavy loads at the end-effector/platform. One application where the need for this has been demonstrated is high-accuracy machining [START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF]. In these high-accuracy machining applications, machining forces induce considerable deformations within the robot that reduce the accuracy of the end-effector to unacceptable levels [START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF]. This in turn affects the quality of the machined product. Similarly, for other high-accuracy positioning applications, positioning accuracy can deteriorate to unacceptable levels when relatively heavy loads are applied on the robot.

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

In the context of high-precision positioning hexapods, figure 1.1 illustrates the problem of accuracy deterioration due to compliance of robot's components and the required solution. A hexapod which is designed for and is capable of accurate positioning, without heavy load mounted on the platform, will deliver less positioning accuracy when a heavy load is placed on it. This decrement in accuracy increases with increase in applied load.

Furthermore, the resulting accuracy deterioration is dependent on the hexapod design, configuration/pose of the robot and the nature & magnitude of applied forces/moments. In order to investigate this problem, a preliminary test was performed on a highprecision hexapod positioning system from Symétrie. The repeatability of this hexapod is ±0.75 µm along translational coordinates and ±3.25 µrad along rotational coordinates.

More details about this product can't be disclosed due to confidentiality reasons. In this test, the deflection of the hexapod's platform due to mounting of load was measured. The hexapod was loaded using a series of weights with the platform in zero pose 1 . These weights were placed in a manner that made sure that the force applied was (approximately)

purely along the Z-axis of the platform. The 6-DOF pose of the platform was measured 2

with different (and no) weights placed on the platform. These measurements were in turn used to calculate the 6-DOF deflections of the robot's platform. Figure 1.2 shows the test setup used in this test. The loads applied during this test were below the maximum allowable payload of this system. Figure 1.3 shows the results of this test. As can be seen from these results, mounted weights cause considerable deflections (relative to the repeatability) of the hexapod's platform. For example, a mass of 26.5 kg (≈260 N) mounted on the platform causes deflections of upto 11 µm in translations and 21 µrad in rotations. These deflections will also degrade the accuracy of this hexapod by the same amounts. Hence, there is considerable room for improvement of this hexapod's positioning accuracy when payload is mounted on its platform.

1. For Symétrie's hexapods, pose (position & orientation) of the platform is defined using a coordinate frame fixed at the center of the platform (platform frame). The pose of the platform with the hexapod in any arbitrary configuration is defined with respect to the platform frame that exists when all of the hexapod's legs are locked at the center of their strokes. Zero pose is the pose of the platform in which all the six pose parameters (defining the 3 translations & 3 rotations) are zero. All of the hexapod's legs are locked at the center of their strokes in zero pose.

2. Pose measurement performed using method described in Appendix A. , 2015;[START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF][START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF][START_REF] Kammerer | High accuracy patient positioning system: geometric and elastic error calibration of a flexible model[END_REF][START_REF] Khalil | Geometric calibration of robots with flexible joints and links[END_REF][START_REF] Lightcap | Improved positioning accuracy of the pa10-6ce robot with geometric and flexibility calibration[END_REF]Abele et al., 2007;[START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF], or (b) geometric parameters dependent on load and robot's configuration [START_REF] Marco A Meggiolaro | Geometric and elastic error calibration of a high accuracy patient positioning system[END_REF][START_REF] Chalfoun | Calibration using generalized error matrices of a long reach articulated carrier[END_REF].

Therefore, in the parametric approach to elastostatic calibration, a parametric model is used to predict and correct the pose error due to applied load at the end-effector/platform.

Non-parametric elastostatic calibration, on the other hand, doesn't use parametric relationships to compensate for pose error due to applied load. This could be achieved using:

(a) measured relationships between the platform pose errors and joint variables when the platform is subjected to load(s) 3 [Shamma and Whitney, 1987], or (b) compensation of pose error by measuring it in real-time. Most of the work present in the literature on robot elastostatic calibration is based on the parametric approach. However, non-parametric approaches to robot elastostatic calibration have found their use in some companies.

Non-parametric elastostatic calibrations have been used by some companies to improve the accuracy of their loaded robots because they can be very easy to implement when: (a) the requirements are not very challenging (for example: high accuracy at some selected poses), or (b) budget allows use of expensive sensors. When an application doesn't allow this leniency, non-parametric elastostatic calibration isn't a very attractive option.

This is because the mentioned non-parametric elastostatic calibration approaches (listed before) have disadvantages such as: (a) large number of measurements required to achieve compensation in rather small workspace volume, or (b) necessity of extra sensors. The parametric approach, on the other hand, do not pose these problems. The identified parameters can be used to predict and correct pose error of a loaded robot throughout its workspace without any extra sensors. Within the parametric approach, using stiffness parameter model is better than using load and configuration dependent geometric parameters. This is because the identification of load and configuration dependent geometric parameters requires large number of measurements as compared to the ones required for identification of stiffness parameters.

3. These relationships can be measured at discrete pose(s) in the workspace with certain load(s) applied to the platform, in order to compensate at those pose(s) and load(s). Furthermore, polynomial interpolation can be used to predict these relationships for positioning at different poses and using different loads.

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

Efficient parametric elastostatic calibration needs the chosen parametric (stiffness) model to make accurate predictions. This requires accurate estimation of stiffness parameters. Stiffness parameter estimation has been approached in the literature along the following directions: (a) experimental estimation of stiffness model parameters [START_REF] Dumas | Development of methods for metal and composite parts trimming with a robot[END_REF][START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF]Abele et al., 2007;[START_REF] Carbone | A procedure for experimental evaluation of cartesian stiffness matrix of multibody robotic systems[END_REF]Alici and Shirinzadeh, 2005;Bonnemains et al., 2009;Zhou and Kang, 2015;[START_REF] Chalfoun | Calibration using generalized error matrices of a long reach articulated carrier[END_REF][START_REF] Marco A Meggiolaro | Geometric and elastic error calibration of a high accuracy patient positioning system[END_REF][START_REF] Kammerer | High accuracy patient positioning system: geometric and elastic error calibration of a flexible model[END_REF][START_REF] Lightcap | Improved positioning accuracy of the pa10-6ce robot with geometric and flexibility calibration[END_REF][START_REF] Ruggeri | Kinetostatic calibration of a scara robot[END_REF][START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF], and (b) analytical estimation of stiffness model parameters [START_REF] Majou | Parametric stiffness analysis of the orthoglide[END_REF][START_REF] Charles M Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF][START_REF] Li | Stiffness analysis of a stewart platformbased parallel kinematic machine[END_REF][START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF][START_REF] Chen | Instantaneous stiffness analysis and simulation for hexapod machines[END_REF][START_REF] Rebeck | A method for evaluating the stiffness of a hexapod machine tool support structure[END_REF][START_REF] Klimchik | Cad-based approach for identification of elasto-static parameters of robotic manipulators[END_REF]. Analytical estimation of stiffness parameters, as the name suggests, relies on analytically estimating the stiffness of components of the robot (such as using FEM). These analytically estimated stiffness of components are then used to estimate the stiffness parameters of the chosen stiffness model. In contrast, experimental estimation of stiffness parameters involves their estimation by means of measurements carried out in suitably designed experiments.

Analytical estimation of stiffness parameters can be more computationally expensive, more time consuming and more complicated as compared to their experimental estimation. Also, analytical estimation needs to be performed again when any small design change is implemented on the robot. Furthermore, it is known from experience that two robots with same design and components exhibit different stiffnesses at the endeffector/platform to a level that is unacceptable for applications such as precise positioning. This behavior can be difficult or impossible to capture using an analytical estimation approach since the reasons for this behavior are not understood. This behavior can, however, be captured using the experimental estimation approach.

For the application in focus in this thesis, precise elastostatic error compensation is desired throughout the workspace of the robot at low cost. From the characteristics of available approaches to elastostatic calibration presented above, parametric elastostatic calibration using experimental stiffness parameter identification seems ideal for our application. Hence, this approach was chosen.

INTRODUCTION TO ROBOT ELASTOSTATIC CALIBRATION

35

Stiffness modelling

As mentioned in section 1.1.2, parametric elastostatic calibration using stiffness parameters is desired. This type of model relates the deflection of the platform/end-effector to the load applied on it. This model can then be used to predict and correct the pose error due to load applied on the platform/end-effector. The modelling technique differs based on the type of robot under consideration. The modelling technique needs to take into account if: (a) the robot is over-constrained or not, (b) the compliance of links are negligible or not, etc., and (c) the arms/links of the robot are heavy enough to cause considerable deflections of robot's components or not [START_REF] Klimchik | Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings[END_REF]. [START_REF] Pashkevich | Enhanced stiffness modeling of manipulators with passive joints[END_REF] and [START_REF] Klimchik | Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings[END_REF] have presented good overview of all the stiffness modelling methods available for all sorts of robotic manipulators. It must be noted here that robot of interest in this thesis, hexapod, is not over-constrained. Furthermore, the hexapods studied in this project had light (not heavy) legs. Therefore, the modelling approaches relevant only to such robots will be discussed. Stiffness modelling of such robotic manipulators can be classified based on two characteristics: parametric model being used and stiffness mapping method.

Based on parametric model being used, stiffness modelling approaches can be can be classified into the ones based on: (a) finite element analysis (FEA) [START_REF] Corradini | Evaluation of a 4-degree of freedom parallel manipulator stiffness[END_REF][START_REF] Nagai | A systematic approach to stiffness analysis of parallel mechanisms and its comparison with fem[END_REF][START_REF] Bc Bouzgarrou | Rigidity analysis of t3r1 parallel robot with uncoupled kinematics[END_REF][START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF], (b) matrix structural analysis (MSA) [START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF][START_REF] Li | Stiffness analysis of a stewart platformbased parallel kinematic machine[END_REF], and (c) virtual joint modelling (VJM) [START_REF] Klimchik | Cad-based approach for identification of elasto-static parameters of robotic manipulators[END_REF] or lumped stiffness modelling. FEA based modelling uses the classical finite element theory to discretize the components of the robot and evaluate the stiffness at the end-effector/platform using computed stiffnesses of the discrete components.

The stiffnesses of these discrete elements are computed by making use of the known material properties of the components of robots. This computation is executed completely analytically (on the computer) and no experiments are involved. This method is very computationally expensive and time consuming. MSA is based on the idea similar to FEA but considers larger elements (trusses, beams, etc). This reduces the computation effort and time as compared to FEA. VJM or lumped stiffness modelling is based on extension of the conventional rigid model of the robot by considering virtual springs to describe elastic deformations of links, joint and actuators. The number of parameters is largely reduced in comparison to FEA and MSA in this method. The stiffness parameters need to be analytically computed in FEA and MSA whereas VJM allows for experimental stiffness parameter identification. Many versions of the VJM modelling technique can be found in literature and they differ in modelling assumptions. One of the simplest versions of VJM is where each actuated joint is replaced by virtual spring [START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF]. This is used when the compliance of parts other than the actuated joints are negligible. This modelling technique is very simple and largely reduces complications in an experimental parameter identification framework. Such a modelling technique has also been used successfully for elastostatic calibration of serial robots for precise machining [START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF].

Stiffness mapping deals with mapping the influence of stiffness parameters of a robot to the stiffness experienced at the end-effector/platform. Based on stiffness mapping method, two types of stiffness modelling techniques exist: one based on conservative stiffness mapping and the other based on non-conservative stiffness mapping. When the endeffector/platform of a robot is loaded, the configuration of the robot changes and consequently, the stiffness experienced at the end-effector/platform changes. Conservative stiffness mapping considers the impact of change in robot's configuration, when loaded, on stiffness at the end-effector/platform whereas non-conservative stiffness mapping does not. Salisbury [START_REF] Salisbury | Active stiffness control of a manipulator in cartesian coordinates[END_REF] first introduced the non-conservative stiffness mapping for a robotic manipulator and the conservative stiffness mapping was later introduced by

Griffis & Duffy [START_REF] Griffis | Global stiffness modeling of a class of simple compliant couplings[END_REF]. The magnitude of the difference in stiffness computed by the conservative and non-conservative methods depends on the stiffness of the robot and the load applied at the end-effector/platform. Therefore, non-conservative stiffness mapping is sufficient for robots which are very stiff and don't experience large deflections when loaded.

For the application concerned to this thesis, a parametric stiffness model was needed that could be used for experimental parameter identification (due to reasons stated in section 1.1.2) and that could make precise deflection predictions. Also, the robots under consideration in this thesis are very stiff and experience very small deflections relative to the size of the robot (see figure 1.3). Furthermore, compliance along actuated joints in these hexapods was expected to be dominate the compliance experienced at their plat-

INTRODUCTION TO ROBOT ELASTOSTATIC CALIBRATION

37

forms since other components of these hexapods are relatively rigid. Therefore, a simple VJM based stiffness model (like the one in [START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF]) along with non-conservative stiffness mapping was used.

Outline of robot elastostatic calibration framework used in this thesis

The robot elastostatic calibration framework that will be used in this thesis was chosen as a result of the choices that were presented in sections 1.1.2 and 1.1.3. These choices were made considering the intended application: elastostatic calibration of a high-precision hexapod. As follows from those choices, a parametric robot elastostatic calibration framework shall be used that employs experimental stiffness identification. An elastostatic calibration framework of this nature consists of the following steps (in the order mentioned):

(a) stiffness modelling, (b) measurement of pose deflections caused by application of a known forces/moments, (c) stiffness parameter identification, and (d) pose error compensation of the loaded robot. These steps can be described as follows:

(a) Stiffness modelling: This is the first step in this process. The stiffness properties of the robot must be suitably modelled. This stiffness model must facilitate ease of experimental parameter identification while describing the system's stiffness properties accurately. As follows from the choices made in sections 1.1.2 and 1.1.3, the stiffness model to be used here will be based on virtual joint modelling (VJM) that will only consider stiffness in actuated joints. This stiffness modelling will also employ non-conservative stiffness mapping to map the influence of stiffness of actuated joints on the stiffness experienced at the end-effector/platform. Section 1.2 deals with the mathematical details of this modelling methodology.

(b) Measurement of pose deflections: This is the second step in this process. A known load (or set of loads) must be applied at the end-effector/platform and the resultant pose deflections must be measured. (d) Compensation: This is the fourth and the last step in this process. The desired pose to be achieved and the desired load to be applied on the end-effector/platform must be known. The set of stiffness parameters identified in the previous step can be used to predict the deflection of the end-effector/platform under desired loading. This prediction model can then be used to obtain the right set of actuator displacements that let the robot achieve the desired pose after undergoing deflection under the influence of the desired load.

Mathematical framework for experimental stiffness identification in robot elastostatic calibration

This section presents the mathematical framework that is required for experimental stiffness identification in robot elastostatic calibration. This mathematical framework is as per the choices described in section 1.1. Section 1.2.1 presents the mathematical equations concerning the chosen stiffness modelling method and the stiffness identification technique. The stiffness parameter identification relies on least squares technique and this requires appropriate scaling of measured deflections and parameters, to facilitate good identification of stiffness parameters [Siciliano and Khatib, 2008, Chapter 14]. Section 1.2.2 presents the necessary appropriate techniques for scaling the measurements and parameters for good identification of these stiffness parameters.

Experimental stiffness identification framework

Let F in and F fi be the initial and final load vectors applied at the end-effector/platform of the robot. In a 3D case, these vectors have the components of force in the first three elements followed by the components of 3D moment in the next three elements. Let X in and X fi be the corresponding initial and final pose vectors of the end-effector/platform.

The relationship between change in force/moment on the platform (∆F) and its resultant deflection (∆X) is given by [START_REF] Merlet | Parallel robots[END_REF]]

∆F = K C ∆X (1.1)
Here, ∆F and ∆X are given by

∆F = F fi -F in (1.2) ∆X = X fi -X in (1.3)

MATHEMATICAL FRAMEWORK FOR EXPERIMENTAL STIFFNESS IDENTIFICATION IN ROBOT ELASTOSTATIC CALIBRATION 41

K C is called the Cartesian stiffness matrix. In order to obtain the expression for K C , we need to use the equivalence between work done by the wrench applied at the endeffector/platform and the work done by resultant forces in the actuators of the robot. This relationship can be written as

η T F = qT τ (1.4)
Here, F is any wrench applied at the end-effector/platform and η is its resulting velocity vector. τ is the force in each actuator as a consequence of F and q is the vector with actuator positions/lengths. η and q are related by means of the Jacobian matrix [START_REF] Khalil | Modeling, identification and control of robots[END_REF] as

η = J q (1.5)
Equations 1.4 and 1.5 give us

F = J -T τ (1.6)
Differentiating equation 1.6, we get

dF = J -T dτ + d(J -T ) τ (1.7)
When non-conservative stiffness mapping is employed, i.e., the influence of change in geometry of the robot (due to applied load) on the stiffness experienced at the endeffector/platform is neglected, the second term of equation 1.7 is neglected [START_REF] Chen | Conservative congruence transformation for joint and cartesian stiffness matrices of robotic hands and fingers[END_REF]. Equation 1.7 can then be rewritten as

dF = J -T dτ (1.8) CHAPTER 1.

STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

Using equation 1.8, we can write

∆F = J -T ∆τ = J -T (τ fi -τ in ) (1.9)
Here, τ fi and τ in are the forces along actuators corresponding to the endeffector/platform wrenches F fi and F in , respectively. We also have

∆τ = K ∆q (1.10)
Here, ∆q is the change in actuator lengths as a result of change in forces along actuators ∆τ. This type of modelling assumes virtual springs along each actuator (as was mentioned in section 1.1.3). K is a diagonal matrix with the stiffness of actuators as its diagonal elements.

K = diag(k 1 ,k 2 ,...,k n p ) (1.11)
Here, n p denotes the number of stiffness parameters which is equal to the number of actuators in this case. Substituting for ∆τ from equation 1.10 in equation 1.9, we get ∆F = J -T K ∆q

(1.12)

We also have

∆q = J-1 ∆X (1.13)
Here, J is the modified Jacobian matrix. When a robot has less than two rotational degrees of freedom at the end-effector/platform, J = J. Using equations 1.12 and 1.13, we can write

∆F = J -T K J-1 ∆X (1.14)

MATHEMATICAL FRAMEWORK FOR EXPERIMENTAL STIFFNESS IDENTIFICATION IN ROBOT ELASTOSTATIC CALIBRATION 43

Comparing equations 1.1 and 1.14, we can write

K C = J -T K J-1 (1.15)
Equation 1.14 can be rearranged to get

∆X = J K -1 J T ∆F (1.16)
When elements of equation 1.14 are rearranged, we obtain A c = ∆X

(1.17)

A in equation 1.17 is a function of ∆F and the Jacobians and is called the observation matrix. It is given by

A ij = Jij n p r=1 J rj ∆F r (1.18)
Here, A ij , J ij and Jij denote the j th element of the i th row of matrices A, J and J, respectively. ∆F r denotes the r th element of vector ∆F. c is the vector containing the compliance parameters.

c = 1 k 1 1 k 2 ... 1 k n p T (1.19)
The goal is to estimate the parameter vector c by measuring pose deflections ∆X M at some poses X M under the influence of force/moment applied ∆F M . However, measurements are always accompanied with errors. All previous works on this subject have considered the error due to uncertainty of pose measurement system. It is then necessary to ensure that all the systematic errors of the measurement instrument are corrected. Taking

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

the random errors (due to uncertainty of pose measurements) into account, equation 1.17 can be rewritten as

A M (c + DU ε c ) = ∆X M + DU ε ∆X M (1.20)
Here, A M is the mn × n p observation matrix corresponding to ∆F M and X M , where m is the number of measurements and n is the number of elements in a single deflection vector (DOFs of the end-effector/platform). DU ε ∆X M is a mn × 1 vector containing the errors in measurement due to uncertainty of the measurement system. The left superscript "DU" stands for deflection uncertainty. DU ε i ∆X M is the i th measurement vector (of size

n × 1) of DU ε ∆X M vector. The expectations of DU ε i ∆X M and DU ε ∆X M , E( DU ε i ∆X M ) and E( DU ε ∆X M )
, are zero vectors. DU ε c is the error in the estimated parameter set due to DU ε ∆X M . The parameters that give the best fit are then generally estimated using least squares approach.

Scaling deflections and parameters for proper stiffness identification

Appropriate scaling of measurements and parameters is necessary to ensure good parameter estimation. This topic has been studied very well in the context of robot geometric calibration [START_REF] Siciliano | Springer handbook of robotics[END_REF][START_REF] Schröer | Theory of kinematic modelling and numerical procedures for robot calibration[END_REF]. Since parameter identification framework for robot geometric calibration is similar to that of elastostatic calibration, same problems (and solutions) regarding scaling exist.

Two types of scaling need to be performed here [Siciliano and Khatib, 2008, Chapter 14]:

(a) task variable scaling, and (b) parameter scaling. Task variable scaling is performed to ensure that: (a) the elements of the measured deflection error vector are independent 4 and identically distributed 5 , and (b) the units of measurements being used for least squares fitting are same. Parameter scaling is done to improve the conditioning of the regressor matrix which in turn improves the identification quality. The regressor matrix is generally illconditioned when the parameter vector contains entities of varying magnitudes [Schröer,4. meaning the elements of the vector are uncorrelated 5. meaning the elements of the vector have same standard deviation CALIBRATION 45 1993]. Taking these scaling recommendations into account, equation 1.20 can be rewritten as

MATHEMATICAL FRAMEWORK FOR EXPERIMENTAL STIFFNESS IDENTIFICATION IN ROBOT ELASTOSTATIC

G A M H H -1 (c + DU ε c ) = G ∆X M + G DU ε ∆X M (1.21)
In equation 1.21, G and H are the task variable and parameter scaling matrices, respectively. It must be noted that the parameter scaling matrix, H, is generally an identity matrix for elastostatic calibration of the kind being used in this work. This is because the stiffness parameters (the stiffnesses of actuated joints) of the robot have approximately same values.

The method to obtain G is well known when elements of DU ε ∆X M are independent but don't have identical distribution (different standard deviations). In this case, G must contain the inverse of standard deviations of the corresponding elements of DU ε ∆X M as its diagonal elements. Let this resulting task variable scaling matrix be called G U .

G U = mn×mn                1 M 11 0 • • • 0 0 1 M 12 . . . 0 . . . . . . 1 M 1n 0 1 M 21 . . . 0 0 • • • 0 1 M mn                (1.22)
In equation 1.22, M i1 ...M in are the standard deviations of elements of DU ε i ∆X M (the i th measurement of the DU ε ∆X M vector). When G U is used for task variable scaling in least squares estimation, the method is also referred to as weighted least squares estimation [START_REF] George | Linear regression analysis[END_REF]. When the elements of DU ε ∆X M are correlated, it is usually ignored [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF]. However, this ignorance is not necessary. Correlated measurements can be dealt with using the generalized least squares method [START_REF] George | Linear regression analysis[END_REF]

G C = mn×mn        1 S -1 0 • • • 0 0 2 S -1 • • • 0 . . . . . . . . . 0 • • • 0 m S -1        (1.23) i S is related to Cov DU ε i ∆X M as Cov DU ε i ∆X M = i V = i S i S T (1.24)
To obtain i S, eigen value decomposition of Cov DU ε i ∆X M needs to be performed. As mentioned earlier, task variable scaling is done so that the resulting measured deflection vector becomes I.I.D (independent and identically distributed) and their elements possess same units. Appendix B presents the proof of the resulting deflection error vector being I.I.D and dimensionless.

Stiffness identification optimization

This section presents the concept and state of the art of the methods to optimize experimental stiffness parameter identification for robot elastostatic calibration. Section 1.3.1 presents the mathematical background behind parameter identification optimization while using a least squares technique. Section 1.3.2 presents the various methods that exist in the literature for optimizing parameter identification and provides an in-depth explanation of each one of them. Section 1.3.3 lists and elaborates on the limitations of the existing stiffness identification optimization methods.

STIFFNESS IDENTIFICATION OPTIMIZATION

Optimizing experiment design

In experimental parameter identification, a linear regression model is used to describe the relation between measurements and parameters to be estimated by means of a regressor. Equation 1.25 shows the general form of this regression model.

A X = A (X + ε X ) = B + ε B = B (1.25)
Here, parameter vector X needs to be identified when B is measured. These two are assumed to be related using the regressor matrix A. Many redundant measurements are performed in order to find the parameter set that best fits the measurement data. Measurement is always accompanied with error vector ε B which are assumed to be a consequence of random measurement errors from the measurement system. This measurement error leads to errors in identified parameters, ε X . X is obtained using

X = A + A T A -1 A T B (1.26)
where A + is the pseudo-inverse of A. It can be seen clearly in equation 1.26 that the regressor matrix A controls the propagation of errors from measurements to identified parameters. Hence, choice of A is crucial for good parameter identification. Theory of experiment design [Atkinson et al., 2007] focuses this phenomenon and approaches to obtain better parameter estimates when measurements have the said random errors.

To understand the significance of optimization of experiment design in the context of experimental stiffness parameter identification, let us first reformulate equation 1.21 to obtain

A M H -1 c = A M c = G(∆X M + DU ε ∆X M ) (1.27) CHAPTER 1.

STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

Here, A M = G A M H, c = c + DU ε c and c = H -1 c. The stiffness parameter estimate is then obtained as per

c = H A M T A M -1 A M T G(∆X M + DU ε ∆X M ) (1.28)
It is apparent from equation 1.28 that A M controls the propagation of error (due to measurement uncertainty) from measurements to identified parameters. Consequently, A M must be chosen carefully for stiffness parameter identification. It is important to minimize this propagation and identify best set of parameters because they directly affect the quality of compensation achieved. Figure 1.6 shows an illustration of error propagation from measured deflection to accuracy attained after compensation in a one dimensional case when an identification experiment is performed many number of times. Many previous works have also demonstrated the importance of good experiment design by choosing a good regressor matrix for robot geometric calibration [START_REF] Menq | Identification and observability measure of a basis set of error parameters in robot calibration[END_REF][START_REF] Morris | Significance of observation strategy on the design of robot calibration experiments[END_REF][START_REF] Joubair | Comparison of the efficiency of five observability indices for robot calibration[END_REF]. In the context of robot geometric calibration, the regresor matrix is a function of pose at which the identification experiment is performed.

Consequently, best set of poses are chosen for performing geometric parameter identification. In our case, as A M depends on the choice of measurement pose (X M ) and the effective force/moment applied at the platform (∆F M ), it is necessary to find the best set of X M and ∆F M .

Existing methods for optimizing parameter identification

As mentioned earlier, the topic of finding the best regressor matrix in order to obtain the best set of parameter estimates has been a topic of interest in various fields. Many criteria exist in the theory of experiment design that are used to optimize experiment design in order to obtain best parameter estimates [Atkinson et al., 2007]. This topic has also been extensively studied to find the best set of poses for measurement in robot geometric calibration. Due to similarity of the general mathematical framework of parameter estimation, the parameter identification optimization criteria proposed in theory of experiment design and robot geometric parameter identification can be directly applied to robot stiffness pa- do not explicitly consider the specific requirements of the positioning application that the robot will be used for. In contrast, criteria focused on minimizing pose error after compensation considers (to varying extents, depending on the criterion) the intended application's specification. This allows the user to identify the set of parameters best suited for the particular application. Some of these criteria are listed in table 1.1. This classification defines the difference between application-oriented and non-application-oriented robot elastostatic calibrations (see figure 1.7).

The reasoning behind different previously proposed criteria for optimizing parameter identification, explained in the context of stiffness parameter identification optimization, are as follows:

(a) A-optimality: This criterion aims at minimizing the total variance of parameter es- Menq, 1989] [START_REF] Morris | Significance of observation strategy on the design of robot calibration experiments[END_REF]] [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF] µ min → max O 4 [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF]] [START_REF] Sun | Observability index selection for robot calibration[END_REF]] shown. The regressor matrix, as already known, controls the scaling and rotation of 6. Error ellipses (or confidence ellipse/ellipsoid) are used to visualize the boundary within which a certain number of samples of randomly varying variables lie. These elliptical/ellipsoidal boundaries are defined by axes dependent on standard deviations of the variables. For example, for a bi-variate (two-dimensional) Gaussian distribution in which both variables have same variance, 99.7% of the samples lie within the circular boundary defined by a radius that is three times the standard deviation of the variables' distributions [START_REF] George | Linear regression analysis[END_REF].

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

→ max G -optimality [Atkinson et al., 2007] max diag Cov(c) → min O 1 [Borm and
(µ 1 µ 2 µ 3 ....µ np ) 1/np m → max O 2 [Driels
µ min µ max → max O 3 [
(µ min ) 2 µ max → max O 5 [
1 1 µ 1 + 1 µ 2 +...+ 1 µn p → max O TP [Wu, 2014] tr A D H A M T A M -1 H T A T D → min;l = 1 O MV [Carrillo et al., 2013] 1 l l i=1 tr i A D H A M T A M -1 H T i A T D → min 'l'

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

these error ellipses. D-optimal design aims at minimizing the area (or volume for an ellipsoid) of the parameter error ellipse for a given measurement error ellipse. (µ 1 µ 2 µ 3 ....

µ n p ) 1/n p m → max (1.34)
Here, µ i are the singular values of A M . From a geometrical viewpoint (of error ellipse propagation as in figure 1.8), it can be shown that the volume of parameter error ellipsoid is proportional to the product of the singular values of the inverse of regressor It is given by

µ min µ max → max (1.35)
Here, µ min and µ max are the minimum and maximum singular values of A M , respectively. In the context of the illustration in figure 1.8, criterion aims at minimizing the eccentricity of the parameter error ellipsoid.

(g) O 3 : Nahvi et al. [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF] proposed this index and it aims at maximizing the minimum singular value of the regressor matrix.

µ min → max

(1.36)

Nahvi et al. [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF] showed that the minimum singular value of the regressor matrix acts as a de-amplifier in the propagation of the norm of measured deflection errors to norm of estimated parameter errors. In the context of the illustration in figure 1.8, this criterion aims at minimizing the largest semi-axis of the parameter error ellipsoid.

(h) O 4 : Nahvi and Hollerbach [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF] proposed the following optimization problem to optimize parameter identification:

(µ min ) 2 µ max → max (1.37)
This observability index is also called the noise amplification index. This is because Nahvi and Hollerbach [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF]

tr A D H A M T A M -1 H T A T D → min;l = 1 (1.39)
In this, A D is a function of the desired set of poses and forces/moments at and with which best performance positioning is required. l is the number of pose-force sets at which best positioning performance is required. This criterion is useful when the end-effector pose coordinates have translational coordinates only 7 .

(k) O MV : Carillo et al. [START_REF] Carrillo | On task-oriented criteria for configurations selection in robot calibration[END_REF] proposed a criterion similar to O TP . In this, the average (over desired high performance pose-force sets) of the RMS of Euclidean norms of possible end-effector pose errors after calibration is minimized.

1 l l i=1 tr i A D H A M T A M -1 H T i A T D → min (1.40)
From all the stiffness parameter identification criteria listed above, all except O TP and O MV focus on minimizing parameter errors. It is worth noting here that Imoto et al. [START_REF] Bibliography Junichi Imoto | Optimal kinematic calibration of robots based on maximum positioning-error estimation (theory and application to a parallel-mechanism pipe bender)[END_REF] were the first ones to present a parameter identification optimization criterion focused on minimizing pose error after compensation. This criterion was developed for 7. Alternatively, the end-effector's orientations can be expressed using Cartesian coordinates of several reference points, as done by Wu [START_REF] Symétrie | Optimal pose selection for the identification of geometric and elastostatic parameters of machining robots[END_REF].

STIFFNESS IDENTIFICATION OPTIMIZATION

55 robot geometric calibration. However, this criterion was designed considering their specific method of measuring the robot's accuracy which is different from how accuracy of robots are generally specified or measured.

Limitations of existing stiffness identification optimization methods

Having robots perform their assigned positioning task as best as possible is very desirable. In many cases, the specification of these positioning tasks are pre-defined. It is very desirable in these cases to identify parameters which are best suited for the intended application. It must be noted that parameter estimation is just a stepping stone to pose error compensation. Parameter errors can not be eliminated completely but can be acquired in a way that is most favorable for the intended compensation. Thus, it is advantageous to find poses and forces/moments for stiffness identification that minimize the influence of errors influencing stiffness identification on the accuracy after compensation. Moreover, some applications can demand best positioning performance along certain selected axes of the end-effector/platform only. In such a case, it is best to identify parameters that minimize the pose errors after compensation along those axes only as the user does not need best positioning performance along the other axes 8 . This is especially important in the context of the project corcerned to this thesis. This is because the partner company, Symétrie, often receives demands of this nature from their customers. Appendix C presents the details of an application where best positioning was required along certain selected axes of the platform of the robot. None of the methods present in the literature can be used in this case to obtain best possible results. Furthermore, one aspect of stiffness parameter identification optimization has not been given any attention as yet. It is the fact that the force/moment actually applied during identification experiment will not be the same as the assumed value. This can have significant impact on compensation accuracy, especially for high-precision positioning robots.

8. Not obtaining best positioning performance along certain axes does not imply that the positioning performance along the those axes will be extremely bad. The robot will achieve high positioning performance along "non-best-performance axes" as well when a high-precision pose measurement instrument is used However, it won't achieve the best possible positioning performance along these non-best-performance axes.

CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION

Appendix D presents a study in which the impact of error in applied force/moment on the compensation accuracy has been studied for a hexapod positioning system. Results of this study show that error in applied force/moment indeed has a non-negligible impact on compensation accuracy.

The limitations of existing stiffness parameter identification criteria highlighted above define the requirements for a new criterion/criteria to be developed. Therefore, this new method should allow the user to optimize stiffness parameter identification such that:

1. best possible positioning performance can be achieved: (a) at desired poses, (b) along desired axes of the robot, and (c) with desired loads on the robot's platform.

2. the influence of error sources impacting stiffness identification, deflection measurement uncertainty and errors in forces/moments applied, on compensation quality can be minimized.

Conclusion

This chapter presented the state of the art and background of robot elastostatic calibration and its optimization. The concept of elastostatic calibration was first introduced along with the justification for its need for high-precision positioning hexapods. Different existing choices to perform elastostatic calibration were presented. This was followed by presentation and justification of the chosen method: parametric elastostatic calibration with experimental stiffness identification. The necessary mathematical background for experimental stiffness identification and its optimization was then presented. The existing criteria for optimizing stiffness identification for robot elastostatic calibration were presented and their limitations were discussed. These limitations dictated the requirements for a new criterion/set of criteria to be developed for optimizing stiffness identification.

This set of requirements have been presented and discussed.

INTRODUCTION 59

Introduction

Requirements and proposed solutions

As mentioned in chapter 1, parametric elastostatic calibration with experimental stiffness identification was chosen to improve the accuracy of high-precision positioning hexapods when their platforms are loaded. This involves modelling the stiffness properties of the robot using a parametric stiffness model. The parameters of this model are estimated with the help of least squares method using measured deflections of the robot due to known applied loads. It was also shown that stiffness identification can be optimized by choosing the best set of poses and forces/moments used for the identification experiment.

Several works proposing different criteria for stiffness identification optimization exist in the literature. These can be classified into the ones focused on minimizing the errors in estimated parameters and the ones focused on minimizing the compensation error after calibration. The latter approach was considered suitable for the work in this thesis. However, the existing criteria to achieve this have two main disadvantages and they define the requirements for the new criterion/criteria to be developed. In light of the requirements mentioned above, a two step strategy is used here to find the criterion/criteria to optimize stiffness identification. First, the stiffness identification problem (equation 1.21) is reformulated to consider the influence of applied force/moment error on estimated stiffness parameters. Following this, equations are derived that relate the input 3 and the output 4 errors. These relationships allow a user to formulate criterion/criteria to choose best set of poses and forces/moments for stiffness identification based on a given application's specifications.

Outline of this chapter

Section 2.2 presents the reformulation of the stiffness identification problem. Section 2.3 presents the relationship between uncertainty of measured deflections and the uncertainty of resultant pose errors after compensation. It also discusses ways to formulate criteria to minimize the influence of uncertainty of measured deflections on the pose error after compensation in the application at hand. Section 2.4 presents the relationship between errors in applied forces/moments during stiffness identification and the resultant pose error after compensation. Subsequently, it discusses ways to formulate criterion/criteria to minimize the influence of errors in forces/moments applied for stiffness identification on the pose error after compensation in the application at hand. Finally, section 2.5

presents the conclusions of this chapter.

3. Errors in applied forces/moments and uncertainty of deflection measurement errors. 4. Resultant pose errors after compensation.

Reformulation of the stiffness identification problem

The need for reformulating the stiffness identification problem as in equation 1.21 is necessary in order to accommodate the influence of error in applied forces/moments during stiffness identification experiment. To do this, let us first mathematize the origin of this problem.

The origin of the problem is the difference (no matter how small) between the assumed and actual forces/moments applied during stiffness identification experiments. As a consequence of this, matrix A M of equation 1.21 will be formulated using assumed applied loads and not the actual loads applied. Let the matrix A M formulated using assumed loads be called as A M . as A M is a function of X M and the assumed differential force vector applied at the platform ( as ∆F M ). as ∆F M is given by

as ∆F M = n f m×1 as 11 ∆F M T as 12 ∆F M T .... as 21 ∆F M T .... T (2.1)
Here, n f is the number of elements in a single force vector. as ij ∆F M is the assumed differential force vector corresponding to the j th measurement at i th measurement pose

X i M . as ij ∆F M is given by as ij ∆F M = as ij F fi M -as ij F in M (2.2)
Here, as ij F fi M and as ij F in M are the final and initial loads that are assumed to have been applied at the end-effector/platform during the corresponding measurement, respectively. These are, however, different from the loads that are actually applied at the endeffector/platform. The actual differential load applied, ac ij ∆F M , is given by

ac ij ∆F M = ac ij F fi M -ac ij F in M (2.3)
ij ε ∆F M = ac ij ∆F M -as ij ∆F M (2.4) ε ∆F M = 11 ε ∆F M T 12 ε ∆F M T ... 21 ε ∆F M T .. T (2.5)
The error in applied force/moment, ε ∆F M , will result in an extra error in measured deflection 5 . Let this error in measured deflection due to ε ∆F M be called FE ε ∆X M . The left superscript "FE" stands for force error. The stiffness identification problem can then be reformulated to include the influence of applied force/moment error in the following manner:

G as A M H H -1 (c + DU ε c + FE ε c ) = G (∆X M + DU ε ∆X M + FE ε ∆X M ) (2.6)
In equation 2.6, errors in estimated stiffness parameters are a consequence of errors in applied forces/moments and errors in deflection measurements. In this, FE ε ∆X M , which is a consequence of ε ∆F M , leads to the corresponding error in estimated parameter, FE ε c .

It must be noted here that matrix G must be appropriately computed in this case. This is because FE ε ∆X M can also influence the variance of the total measured deflection vector.

As mentioned before, DU ε ∆X M is a consequence of the uncertainty of pose measurement system. Consequently, the variance of measurements of pose measurement system decides the variance of DU ε ∆X M . This in turn contributes to the variance of positioning errors after compensation. Hence, a relationship relating the variance of DU ε ∆X M to the corresponding component of variance of compensation errors can be useful in minimizing the influence of the former on the latter when the former is given. Section 2.3 presents this relationship and the ways to use it to optimize stiffness identification. The stiffness identification optimization criterion developed using this approach shall be called DUIR criterion, which stands for deflection measurement uncertainty influence reduction criterion.

5. In addition to the error in measured deflection due to uncertainty of the pose measurement system.

REFORMULATION OF THE STIFFNESS IDENTIFICATION PROBLEM

63

Similarly, a relationship relating FE ε ∆X M and the corresponding component of compen- sation error can be useful in minimizing the influence of the former on the latter. Section 2.4 presents this relationship and the ways to use it to optimize stiffness identification.

The stiffness identification optimization criterion developed using this approach shall be called FEIR criterion, which stands for force error influence reduction criterion.

Relationship between deflection measurement uncertainty and the uncertainty of resultant positioning error after compensation

In order to derive the necessary relationship, the expression for the estimated parameter set can be first derived. To simplify this derivation, equation 2.6 can be rewritten in a simpler form as

as A M H -1 c = G ∆X M (2.7) In equation 2.7, as A M = G as A M H, c = c + DU ε c + FE ε c and ∆X M = ∆X M + DU ε ∆X M + FE ε ∆X M .
The estimated parameter set c can then be estimated using the formula

c = H ( as A M T as A M ) -1 as A M T as A M + G ∆X M (2.8)
In equation 2.8, ( as A M T as A M ) -1 as A M T is nothing but the pseudo-inverse of as A M , as A M + . From equations 2.8 and 2.6, the following expression can be written:

DU ε c = H as A M + ( as A M T as A M ) -1 as A M T N G DU ε ∆X M DU ε ∆X M (2.9) Let H ( as A M T as A M ) -1 as A M T
in equation 2.9 be replaced by N for simplifying its further analysis. Furthermore, let G DU ε ∆X M be replaced by DU ε ∆X M for the same reason.

CHAPTER 2. APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

Now, the influence of deflection measurement uncertainty on the uncertainty of estimated parameter set can be evaluated using the expression

Cov( DU ε c ) = N Cov( DU ε ∆X M ) N T
(2.10)

In equation 2.10, Cov( DU ε ∆X M ) can be written as

Cov( DU ε ∆X M ) = ( DU σ ε ∆X M ) 2 I (2.11)
Here, DU σ ε ∆X M

is the standard deviation of elements 6 of DU ε ∆X M and I is an identity matrix. The expression of equation 2.11 follows from the fact that DU ε ∆X M is independent and identically distributed 7 . By substituting for Cov( DU ε ∆X M ) from equation 2.11 in equation 2.10, the following expression can be obtained:

Cov( DU ε c ) = N ( DU σ ε ∆X M ) 2 I N T
(2.12) Equation 2.12 can then be expanded to get

Cov( DU ε c ) = ( DU σ ε ∆X M ) 2 H ( as A M T as A M ) -1 as A M T H ( as A M T as A M ) -1 as A M T T = ( DU σ ε ∆X M ) 2 H ( as A M T as A M ) -1 as A M T ( as A M T as A M ) -1 as A M T T H T = ( DU σ ε ∆X M ) 2 H ( as A M T as A M ) -1 as A M T as A M I ( as A M T as A M ) -T H T = ( DU σ ε ∆X M ) 2 H ( as A M T as A M ) -1 H T (2.13)
Equation 2.13 presents the relationship that governs the scaling of uncertainty of measured deflections to uncertainty of resultant errors in estimated parameters. The goal is, 6. Note that all the elements of DU ε ∆X M have the same standard deviation. Therefore, DU σ ε ∆X M is a scalar value. 7. See section 1.2.2 to know more about this.

MINIMIZING THE INFLUENCE OF DEFLECTION MEASUREMENT UNCERTAINTY ON POSITIONING ERROR AFTER COMPENSATION (DUIR CRITERION) 67

however, to find the relationship governing scaling of uncertainty of measured deflections to uncertainty of resultant errors in compensation. Hence, it is necessary to understand the propagation of errors in estimated parameters to compensation errors.

The parameters identified are used to predict the necessary actuator displacements, q C , that ensures that the end-effector/platform reaches the desired pose X D after undergoing deflection due to an applied load F D . Let X C be the pose array corresponding to q C considering zero deflection due to load. X C is the pose array that needs to be entered into the control interface of a robot's controller that doesn't have the compliance error model embedded in it. Let X C be called "command pose". X C is estimated using the formula

X D = X C + ∆X C (2.14)
Here, ∆X C is the deflection at pose X C due to a load ∆F D . The initial load on the end-effector/platform, which is the load on the platform for which the X D = X C , is considered zero. The solution to X C in equation 2.14 needs to be found so that the robot's platform/end-effector reaches (close to) the desired pose.

In reality, however, the predicted command pose will have errors due to errors in estimated parameters. This can be expressed as

X C = X D -∆X C (2.15)
where X C is the predicted command pose with error and ∆X C is the predicted deflection with error at pose X C and load ∆F D . X C and ∆X C are given by

X C = X C + DU ε X C + FE ε X C (2.16) ∆X C = ∆X C + DU ε ∆X C + FE ε ∆X C
(2.17)

∆X C = A C c (2.18) CHAPTER 2.

APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

Here, DU ε ∆X C and FE ε ∆X C are the components of error in predicted deflection ∆X C as a result of DU ε c and FE ε c , respectively. DU ε X C and FE ε X C are the components of error in predicted command pose as a consequence of DU ε ∆X C and FE ε ∆X C , respectively. Matrix A C is a function of X C and ∆F D .

The robot's end-effector/platform attains a pose X A after elastostatic error compensation and this is equal to X D when there are no parameter errors. Therefore, the following expression can be written:

X A = X C + ∆X C = X D (2.19)
In the presence of errors in estimated parameters, however, the platform reaches a different pose X A . The following can then be written:

X A = X C + ac ∆X C (2.20)
Here, ac ∆X C is the actual deflection of the end-effector/platform at the pose X C as a result of actual stiffness parameters (c) and applied load ∆F D . Now, the following expression can be obtained using equations 2.20 and 2.15:

X A = X D -∆X C + ac ∆X C
(2.21) Equation 2.21 can be expanded to obtain

X A + DU ε X A + FE ε X A = X D -∆X C + DU ε ∆X C + FE ε ∆X C + ac ∆X C (2.22)
In equation 2.22, DU ε X A and FE ε X A are the errors in pose attained after compensation as a consequence of DU ε ∆X C and FE ε ∆X C , respectively. DU ε ∆X C and FE ε ∆X C are, in turn,

MINIMIZING THE INFLUENCE OF DEFLECTION MEASUREMENT UNCERTAINTY ON POSITIONING ERROR AFTER COMPENSATION (DUIR CRITERION) 69

a consequence of DU ε c and FE ε c , respectively. DU ε ∆X C and FE ε ∆X C can be expressed as

DU ε ∆X C = A C DU ε c
(2.23)

FE ε ∆X C = A C FE ε c (2.24)
Furthermore, the following can also be written:

∆X C = A C c ≈ A C c = ac ∆X C (2.25) Note that A C is a function of X C and F D while A C is a function of X C and F D . X C and X C
need to be very close in the workspace for equation 2.25 to be true. X C and X C will indeed be very close in the workspace unless: (a) the errors in measurements are very high relative to the level of positioning precision required 8 , and/or (b) the force/moment assumed to have been applied at the end-effector/platform during positioning is very different from the force/moment actually applied. These two reasons are assumed to be untrue. Hence, X C and X C can be assumed to be close and consequently, equation 2.25 can be assumed to be valid. Simlarly, X D and X C will also be very close in the workspace. The following can then be written:

∆X C = A C c ≈ A D c = ∆X D (2.26)
Here, A D is a function of X D and F D . Now, using equation 2.22 and that X A = X D (see equation 2.19) and ∆X C ≈ ac ∆X C (see equation 2.25), the following expression can be obtained: 27) 8. When, for example, the required positioning precision is in microns and the errors in measurements are in centimeters.

DU ε X A ≈ -DU ε ∆X C (2.

CHAPTER 2. APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

Finally, the covariance matrix of DU ε X A can be derived which will be useful in evalu- ating the influence of uncertainty of measured deflections on the uncertainty of resultant compensation errors. Using equation 2.27, the following can be obtained:

Cov( DU ε X A ) ≈ Cov( DU ε ∆X C ) ≈ Cov(A C DU ε c ) (2.28)
Since, A D and A C are approximately equal, equation 2.28 can be further expanded as

Cov( DU ε X A ) ≈ Cov(A D DU ε c ) ≈ A D Cov( DU ε c ) A T D (2.29)
Finally, equations 2.29 and 2.13 lead to the following expression:

Cov( DU ε X A ) ≈ ( DU σ ε ∆X M ) 2 A D H ( as A M T as A M ) -1 H T A T D U (2.30)
Equation 2.30 presents the relationship that governs the propagation of uncertainty in measured deflections to uncertainty of resultant errors in pose attained after compensation. In this, matrix U contains the scaling factors that scale DU σ 2 ε ∆X M to respective elements of Cov( DU ε X A ). U is a function of the poses and forces/moments used for stiff- ness identification. Hence, U can be used as a medium to evaluate the influence of poses and forces/moments used for stiffness identification on the propagation of uncertainty from deflection measurements to resultant compensation errors. Section 2.3.2 discusses about the appropriate usage of the relationship of equation 2.30 to identify the set of poses and forces/moments for stiffness identification which are best suited for the application at hand.

MINIMIZING THE INFLUENCE OF DEFLECTION MEASUREMENT UNCERTAINTY

ON POSITIONING ERROR AFTER COMPENSATION (DUIR CRITERION) 71

Formulation of DUIR criterion

As seen in equation 2.30, matrix U controls the propagation of uncertainty in measured deflections to uncertainty of resultant compensation errors. Hence, relevant components of U must be minimized in order to optimize stiffness identification in a way that is best for a given application at hand. The diagonal elements of U scale DU σ 2 ε ∆X M

to the variances of the respective elements of DU ε X A . Minimizing the first diagonal element of U, for example, will ensure that variance of first element of DU ε X A will be the least possible for a given DU σ ε ∆X M

. When best positioning performance is needed along multiple axes of the robot's end-effector/platform, multiple diagonal elements of U must be minimized.

This can be done using a multi-objective optimization approach or by minimizing some combination of the diagonal elements of U.

When the robot's end-effector/platform pose coordinates have both translations and rotations, the diagonal elements of U correspond to translational and rotational elements.

Consequently, if the diagonal elements of U need to be compared in an optimization procedure, the issue of non-homogeneous comparison arises. This is because comparing the diagonal elements of U corresponding to translational coordinates of DU ε X A with the di- agonal elements of U corresponding to rotational coordinates of DU ε X A is not valid. To solve this issue, equation 2.30 can be split into two: one concerned only to the translational elements of DU ε X A and the other concerned only to rotational elements of DU ε X A . They can be written as

Cov( DU,t ε X A ) ≈ ( DU σ ε ∆X M ) 2 t A D H ( as A M T as A M ) -1 H T t A T D (2.31) Cov( DU,r ε X A ) ≈ ( DU σ ε ∆X M ) 2 r A D H ( as A M T as A M ) -1 H T r A T D (2.32)
In equation 2.31, DU,t ε X A contains the translational components of DU ε X A . Similarly, in equation 2.32, DU,r ε X A contains the rotational components of DU ε X A .

A D t A D r A D c = ∆X D ∆X D t ∆X D r (2.

Minimizing the influence of applied force/moment

error on positioning error after compensation (FEIR criterion)

Relationship between applied force/moment error and resultant positioning error after compensation

In order to derive the necessary relationship, the expression relating errors in forces/moments applied during stiffness identification and the resultant errors in estimated pa-

MINIMIZING THE INFLUENCE OF APPLIED FORCE/MOMENT ERROR ON POSITIONING ERROR AFTER COMPENSATION (FEIR CRITERION) 73

rameters can be obtained first. Equation 2.6 can be used to obtain this expression and it can be written as

G as A M H H -1 FE ε c = G FE ε ∆X M (2.34)
Furthermore, the following expression can also be written:

as ij ∆F M = i J -T K i J-1 ∆X M ij (2.35)
In equation 2.35, ∆X M ij is the j th deflection measurement performed at the i th measurement pose, X i M . i J and i J are the Jacobians corresponding to X i M .

Any error in the load applied on the platform leads to an error in the measured deflection. Therefore, the following can be written:

as ij ∆F M + ij ε ∆F M = i J -T K i J-1 (∆X M ij + FE ij ε ∆X M ) (2.36)
In equation 2.36, FE ij ε ∆X M is the error corresponding to ∆X M ij due to the applied force/moment error ij ε ∆F M . Equations 2. 36 and 2.35 give

ij ε ∆F M = i J -T K i J-1 FE ij ε ∆X M (2.37)
Equation 2.37 can be rewritten as

FE ij ε ∆X M = i J K -1 i J T i D ij ε ∆F M (2.38) 74 CHAPTER 2.

APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

Equation 2.38 leads to the following expression:

FE ε ∆X M = D             1 D 1 D 0 . . . 2 D 0 . . . . . .             ε ∆F M (2.39)
Substituting for FE ε ∆X M from equation 2.39 in equation 2.34 leads to

G as A M H H -1 FE ε c = G D ε ∆F M (2.40)
Equation 2.40 can be rewritten as

FE ε c = H as A M + G D ε ∆F M (2.41)
Equation 2.41 can be used to evaluate the effect of ε ∆F M on FE ε c . However, matrix D is not known a priori since K (matrix containing stiffness parameters to be estimated) is not known a priori. Consequently, a preliminary test must be performed to obtain the matrix containing approximate values of stiffness parameter values, ap K. The following expressions can then be written:

ap i D = i J ap K -1 i J T (2.42) ap D = diag( ap 1 D ap 1 D ... ap 2 D...) (2.43)
ap D is the matrix D formulated using ap K. Equation 2.41 can then be rewritten as

FE ε c = H as A M + G ap D ε ∆F M (2.44)

MINIMIZING THE INFLUENCE OF APPLIED FORCE/MOMENT ERROR ON POSITIONING ERROR AFTER COMPENSATION (FEIR CRITERION) 75

Equations 2.15, 2.16 and 2.17 give

X C + DU ε X C + FE ε X C = X D -(∆X C + DU ε ∆X C + FE ε ∆X C ) (2.45)
Following from the definitions of DU ε X C , FE ε X C , DU ε ∆X C and FE ε ∆X C and equations 2.45 and 2.14, the following can be written:

FE ε X C = -FE ε ∆X C (2.46)
Furthermore, equations 2.46, 2.18, 2.25, 2.26 and 2.6 give

FE ε X C = -A C FE ε c ≈ -A D FE ε c
(2.47) Also, equations 2.47 and 2.44 give

FE ε X C ≈ -A D H as A M + G ap D ε ∆F M (2.48)
Additionally, equation 2.21 gives

X A + DU ε X A + FE ε X A ≈ X C + DU ε X C + FE ε X C + ∆X D (2.49)
In equation 2.49, DU ε X A and FE ε X A are the errors in pose attained after compensation as a consequence of DU ε X C and FE ε X C , respectively. DU ε X C and FE ε X C are a consequence of DU ε ∆X C and FE ε ∆X C (see equation 2.45), respectively. Equation 2.49 is true because ∆X D ≈ ac ∆X C (see equation 2.25 and 2.26). Now, using equations 2.49 and 2.19 and from definitions of DU ε X A , FE ε X A , DU ε X C and FE ε X C , the following can be written:

FE ε X A ≈ FE ε X C (2.50) CHAPTER 2.

APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

Finally, equations 2.48 and 2.50 yield the necessary expression:

FE ε X A ≈ -A D H as A M + G ap D ε ∆F M (2.51) ≈ -A D as A M + ap D Z ε ∆F M
Equation 2.51 presents the relationship that governs the propagation of errors in applied force/moment during stiffness identification to the resultant errors in pose attained after compensation. The choice of poses and forces/moments for stiffness identification experiment changes matrix "Z" which in turn influences propagation of ε ∆F M to FE ε X A .

Z can be used as a medium to evaluate the influence of poses and forces/moments used for stiffness identification on the propagation of errors in applied force/moment during stiffness identification to the resultant errors in pose attained after compensation. Section 2.4.2 discusses about the appropriate usage of the relationship of equation 2.51 to identify the set of poses and forces/moments for stiffness identification which are best suited for an application at hand.

Formulation of FEIR criterion

As can be seen in equation 2.51, matrix Z controls the propagation of errors in forces/moments applied for stiffness identification to the resultant errors in pose attained after compensation. Hence, relevant components of Z must be minimized in order to optimize stiffness identification in a way that is best for a given application at hand. For example, when best performance is required along the first axis of the end-effector/platform (first element of X D ) at the desired target pose, the first element of FE ε X A must be minimized.

When no prior information about the errors in applied forces/moments is available, for example when force/moment is manually applied without measuring it, it is best to minimize all the elements of first row of Z. One way to do this is to minimize the 2-norm of the first row of Z. Sometimes, even when the load is applied manually without measuring it, the user can have an idea as to which element(s) of ε ∆F M might acquire high values. This

MINIMIZING THE INFLUENCE OF APPLIED FORCE/MOMENT ERROR ON POSITIONING ERROR AFTER COMPENSATION (FEIR CRITERION) 77

can be evident due to the nature of loading 9 . Consequently, the corresponding elements of matrix Z can be minimized.

When the applied load is measured using some force measurement instrument, the uncertainty of these measurements are usually specified by the instrument's supplier. This information can be helpful in optimizing stiffness identification. Equation 2.51 can be used to get the following expression:

Cov FE ε X A ≈ Z Cov ε ∆F M Z T (2.52)
When the uncertainties of force measurements are known, Cov ε ∆F M can be de- duced. The concerned elements of Z Cov ε ∆F M Z T can then be minimized as per the requirement of the application. Z Cov ε ∆F M Z T has a structure similar to any covariance matrix. The variances of the individual elements of FE ε X A are along its diagonal while the off-diagonal elements correspond to correlations between elements of the vector FE ε X A . Therefore, the diagonal elements of Z Cov ε ∆F M Z T decide the variances along the corre- sponding elements of FE ε X A . Minimizing the first diagonal element of Z Cov ε ∆F M Z T , for example, will ensure that variance of first element of FE ε X A will be the least possible for a given Cov ε ∆F M . When best positioning performance is needed along multiple axes of the robot's end-effector/platform, multiple diagonal elements of Z Cov ε ∆F M Z T must be minimized using a multi-objective optimization approach or by minimizing some combination of its diagonal elements.

When the robot's end-effector/platform pose coordinates have both translations and rotations, care must be taken while comparing elements of Z or Z Cov ε ∆F M Z T . This is because comparing variances of rotational coordinates (or scaling factors corresponding to thereof ) with their translational counterparts is not valid. To solve this issue, separate relationships corresponding to the rotational and translational coordinates of the 9. For example, in the loading case for stiffness identification in Appendix D, moment errors about the X and Y axes of the robot's platform can be expected due to the nature of loading.

CHAPTER 2. APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION

end-effector/platform pose vector can be used. Equations 2.51 and 2.52 can be used to get the following expressions:

FE,r ε X A ≈ - r Z r A D as A M + ap D ε ∆F M (2.53) FE,t ε X A ≈ - t Z t A D as A M + ap D ε ∆F M (2.54) Cov FE,r ε X A ≈ r Z Cov ε ∆F M r Z T (2.55) Cov FE,t ε X A ≈ t Z Cov ε ∆F M t Z T
(2.56)

In the above equations, FE,t ε X A and FE,r ε X A are the translational and rotational com- ponents of FE ε X A , respectively. t Z and r Z are the rows of Z corresponding to FE,t ε X A and FE,r ε X A , respectively. Furthermore, some applications demand for minimization of positioning error after compensation in terms of translational distance (Euclidean norm). In such cases, another set of expressions can be useful to formulate FEIR criterion. When the variance of expected force/moment error is not known, the relationship between the norm of FE,t ε X A and the error in applied force/moment can be useful. This expression can be obtained using equation 2.54 in the following form 10 :

|| FE,t ε X A || 2 ≈ || -t Z ε ∆F M || 2 ≤ || -t Z|| 2 ||ε ∆F M || 2 (2.57)
It can be seen in equation 2.57 that minimizing || -t Z|| 2 minimizes the influence of errors in applied forces/moments on the resultant distance error after compensation. When the variance of the expected force/moment error is known 11 , the expression derived in Appendix F can be useful. In this, the relationship between the RMS value of possible Euclidean norms of FE,t ε X A , denoted as FE ρ, and Cov( FE,t ε X A ) is shown. In Appendix F, it can be seen that 12 tr t Z Cov(ε ∆F M ) t Z T is equal to FE ρ and the value of tr t Z Cov(ε ∆F M ) t Z T is dependent on the measurement conditions (poses and forces/moments used for stiffness identification). Consequently, choosing poses and forces/moments for stiffness identification that minimize tr t Z Cov(ε ∆F M ) t Z T will minimize the RMS value of possible Euclidean norms of FE,t ε X A . When positioning at multiple poses and/or multiple loads needs to be optimized, multiple values of tr t Z Cov(ε ∆F M ) t Z T are obtained (one corresponding to each positioning case). Consequently, multiples values of tr t Z Cov(ε ∆F M ) t Z T must be minimized. This can be accomplished using a multiobjective optimization approach or by minimizing some combination of those values.

Conclusion

This chapter presented a stiffness identification optimization framework that counters the limitations of the existing criteria for the same, which were presented in chapter 1.

To accomplish this, a reformulated stiffness identification problem was firstly presented. This reformulated stiffness identification problem considers the influence of two error sources impacting stiffness identification: the errors in forces/moments applied during stiffness identification experiment and the uncertainty of deflection measurements performed. This is unlike previous works as they ignored the former source of error. This was followed by derivation of relationships that relate the errors impacting stiffness identification to the resultant pose errors after compensation. Discussion was then provided on the usage of the presented relationships to formulate necessary criterion/criteria for best stiffness identification depending on the specifications of the application at hand. 11. For example, when the force/moment applied is measured using an instrument and the uncertainty of the measurements made using this instrument is known.

12. Note here that tr() denotes the trace of the corresponding matrix.

Chapter Abstract

This chapter aims at validating the efficacy of stiffness identification optimization framework presented in chapter 2. This is achieved using simulated elastostatic calibrations of a bipod. Three simulation studies are presented which validate the efficacy of using: (a) DUIR criterion 1 , (b) FEIR criterion2 , and (c) both criteria together, to select best poses and forces for stiffness identification. Results of these simulation studies confirm the efficacy of the presented stiffness identification optimization framework.

Introduction

This chapter presents simulation studies performed on a planar bipod to evaluate the efficacy of the presented stiffness identification optimization framework. This mechanism was chosen for these simulation studies to facilitate ease of analysis of results as this mechanism is simple.

Three simulation studies are presented in this chapter. In these studies, elastostatic calibrations of a bipod are simulated. In the first simulation study, elastostatic calibrations are simulated with only deflection measurement uncertainty influencing stiffness identifications. The performances of stiffness identification measurement conditions (poses and forces) selected using DUIR criterion and various previously proposed criteria 3 are then compared. In the second simulation study, elastostatic calibrations are simulated in which stiffness identifications are influenced only by errors in forces applied. These are used to evaluate the ability of FEIR criterion to recommend stiffness identification measurement conditions which minimize the influence of these errors on compensation quality. In the third and the last simulation study, errors from both sources 4 influence the simulated stiffness identifications. These are then used to evaluate the efficacy of using DUIR and FEIR criteria together to select best set of poses and forces for stiffness identification.

This chapter is organized as follows: section 3.2 describes the assumed mechanism, its kinematics and its stiffness model. Sections 3.3, 3.4 and 3.5 present the details and results of the three simulation studies. Finally, section 3.6 presents the conclusion of this chapter.

Mechanism description, kinematics and stiffness model

Figure 3.1 shows the bipod under study and its assumed dimensions. This bipod is assumed to be driven using two prismatic actuators as shown in the figure. This bipod is assumed to have a simple workspace, as shown in figure 3.1, for ease of analysis. The compliance is assumed to exist only in the actuated joints and the rest of the structure 3. For minimizing the influence of deflection measurement uncertainty on stiffness identification. 4. Deflection measurement uncertainty and errors in forces applied. ELASTOSTATIC CALIBRATIONS OF A BIPOD is assumed to be rigid. The stiffness of each of the two actuated joints is assumed to be 10 N/µm. Also, the joints of this mechanism are assumed to be frictionless and the legs are assumed to have negligible mass. The end-effector position vector, B X, defines the position of the end-effector with respect to its position when the bipod has the configuration as in figure 3.1. B q contains the lengths of the prismatic links and is written as

B q = [ B q 1 B q 2 ] T (3.1)
The relationship between the velocity of the end-effector and those of the actuated joints is given by

B η = B J Ḃq (3.2)
Here, B η is the end-effector's velocity vector, Ḃq contains the velocities of the actuators and B J is the Jacobian matrix. The equation relating the differential position vector ( B ∆X)

MECHANISM DESCRIPTION, KINEMATICS AND STIFFNESS MODEL

85

and the differential actuator position vector ( B ∆q), as in equation 1.13, can be written for this case as

B ∆q = B J -1 B ∆X (3.3)
Note that the relationship in equation 3.3 contains the normal Jacobian matrix B J instead of a modified Jacobian matrix as in equation 1.13. This is because the end-effector pose coordinates have less than two rotational degrees of freedom (zero in this case).

A simple lumped stiffness model (shown in figure 3.2) was used to model the static stiffness characteristics of this mechanism. One spring is used to model the stiffness of each leg ( B k i=1,2 ). This modelling method is consistent with the method described in chapter 1. The Cartesian stiffness matrix (as in equation 1.15), B K C , for this bipod can be written as

B K C = B J -T B K B J -1 (3.4)
where matrix B K is diagonal matrix with leg stiffnesses forming its diagonal elements.

It is given by

B K = B k 1 0 0 B k 2 (3.5)

CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

The Cartesian stiffness matrix B K C relates the differential force vector ( B ∆F) and the differential position vector ( B ∆X) as

B ∆F = B K C B ∆X (3.6)
Rearrangement of this equation yields the relationship similar to equation 1.17:

B A B c = B ∆X (3.7)
Here, B A is the observation matrix and it is given by

B A ij = B J ij 2 r=1 B J rj B ∆F r (3.8)
where B A ij and B J ij denote the j th element of the i th row of matrices B A and B J, respectively. B ∆F r denotes the r th element of vector B ∆F. B c is the vector with compliance parameters and is given by

B c = 1 B k 1 1 B k 2 T
(3.9)

Simulation study 1: validation of DUIR criterion

Stiffness identification optimization

The stiffness identification equation ( equation 2.6) can be rewritten for this case as

B1 G U as B1 A M B1 H B1 H -1 ( B1 c + DU B1 ε c ) = B1 G U ( B1 ∆X M + DU B1 ε ∆X M ) (3.10)
Here, the left subscript "B1" specifies that the corresponding matrices/arrays of equation 2.6 have been formulated for this case. Since there is no applied force error, the corresponding terms of equation 2.6 do not exist in equation 3.10. B1 G U is a 6 × 6 matrix with inverse of deflection measurement standard deviations along its diagonal elements and is formulated as shown in equation 1.22. B1 H is a 2 × 2 identity matrix because the expected magnitudes of estimated parameters are same. B1 ∆X M and DU B1 ε ∆X M are 6 × 1

vectors which together constitute the measured deflections. DU B1 ε ∆X M contains the errors in measured deflections due to deflection measurement uncertainty. as B1 A M is a 6×2 matrix which is a function of the forces assumed to be applied and the position used for stiffness identification. Parameter set of equation 3.10 is estimated using least squares method.

5. The criteria from theory of experiment design (ex: A-optimality, D-optimality, etc.) are not mentioned here because each of them has an equivalent observability index [START_REF] Sun | Observability index selection for robot calibration[END_REF].

CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

Let B1 X A be the position attained after compensation in this case. U from equation 2.30 can be rewritten for this case as

B1 U = B1 A D B1 H as B1 A M T as B1 A M -1 B1 H T B1 A T D ( 3

Simulation study 2: validation of FEIR criterion

Consider a case where elastostatic calibration of the bipod (figure 3.1) must be performed to position its end-effector precisely at a position B2 X D = [0;0] mm with a force B2 F D = [0;-200] N applied on it. Assume that the measurement system used for deflection measurements has zero (relatively negligible) uncertainty. Also suppose that identification can only be done at one position using one deflection measurement and using a force B2 F M = [0;-200] N applied at the end-effector. Furthermore, assume that the forces applied along the X and Y axes at the end-effector are measured using two independent force gauges which have standard uncertainty 7 of 10 N. Therefore, the force assumed to be applied at the end-effector will have an error due to the uncertainty of the force measurement instrument used. The best position for stiffness identification must, therefore, be selected to minimize the influence of error in force applied during stiffness identification on compensation quality, depending on the specifications of desired positioning task 8 .

Since only uncertainty of force applied affects stiffness identification, only FEIR criterion had to be formulated to evaluate the suitability of positions for stiffness identification.

Here, FEIR criteria were formulated for the case where best positioning would be desired along the X-axis or Y-axis of the end-effector after compensation. In order to validate the proposed criteria, Monte-Carlo simulations were performed in which elastostatic calibrations were simulated many times. In these Monte-Carlo simulations, stiffness identifications were simulated at different positions across the workspace. The performance of these identification positions were then compared with the indication of the proposed FEIR criteria. Section 3.4.1 presents the formulation and indications of FEIR criteria for this case.

The details and results of the validation simulations are then presented in section 3.4.2.

7. Uncertainty of measurement expressed in terms of standard deviation. 8. Depending on whether best positioning accuracy is desired : (a) along the X-axis only, or (b) along the Y-axis only, or (c) along both axes, or (d) in terms of distance errors, of the end-effector after compensation.

CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

Formulation of FEIR criterion

The stiffness identification equation (equation 2.6) can firstly be rewritten for this case as

B2 G as B2 A M B2 H B2 H -1 ( B2 c + FE B2 ε c ) = B2 G ( B2 ∆X M + FE B2 ε ∆X M ) (3.12)
Here, the left subscript "B2" specifies that the corresponding matrices/arrays of equation 2.6 have been formulated for this case. Since there are no errors in measured deflections due to uncertainty of deflection measurement system, the corresponding terms of equation 2.6 do not exist in equation 3.12. B2 G is a 2 × 2 identity matrix since there is only one deflection measurement. B2 H is a 2 × 2 identity matrix because the expected magnitudes of estimated parameters are same. B2 ∆X M and FE B2 ε ∆X M are 2 × 1 vectors which together constitute the measured deflection. FE B2 ε ∆X M contains the error in measured de- flection due to error in force applied during stiffness identification experiment. as B2 A M is a 2 × 2 matrix which is a function of the force assumed to be applied and the position used for stiffness identification. Parameter set of equation 3.12 is estimated using least squares method.

Matrix Z from equation 2.51 can be written for this case first as it relates the errors in force applied during stiffness identification to the resultant errors in position attained after compensation. It is given by

B2 Z = B2 A D as B2 A M + ap B2 D (3.13) where B2 Z is a 2 × 2 matrix. 2 × 2 matrix B2 A D is
Cov FE B2 ε X A ≈ B2 Z Cov B2 ε ∆F M B2 Z T (3.14)
Here, FE B2 ε X A is the error in position attained after compensation due to the error in force applied during stiffness identification ( B2 ε ∆F M ). Cov B2 ε ∆F M is known from data about uncertainty of the force gauges. Equation 3.14 can then be written as

Cov FE B2 ε X A ≈ B2 Z v I B2 Z T ≈ v B2 Z B2 Z T (3.15)
v in equation 3.15 is a scalar which is determined by the diagonal elements of Cov B2 ε ∆F M . The diagonal elements of Cov B2 ε ∆F M contain the variances of each el- ement of B2 ε ∆F M and they are equal to 9 100. Consequently, v is equal to 100. Cov FE B2 ε X A is a 2 × 2 matrix and its first diagonal element corresponds to the variance along the X-axis of end-effector after compensation. Consequently, the position that minimizes the first diagonal element of Cov FE B2 ε X A must be found for best positioning along end-effector's Xaxis. Similarly, its second diagonal element must be minimized for best positioning along end-effector's Y-axis. Since B2 Z B2 Z T is only scaled with v to get Cov FE B2 ε X A , the diagonal elements of B2 Z B2 Z T can also be minimized. Let ζ contain the diagonal elements of

B2 Z B2 Z T . ζ = diag B2 Z B2 Z T

Simulation study 3: validation of use of both criteria together

Consider a case where elastostatic calibration of the bipod (figure 3.1) has to be performed to position its end-effector at position B3 X D = [0;0] mm with a force B3 F D = [0;-200] N applied to it. Assume that best positioning accuracy is required only along its Xaxis. Also suppose that stiffness identification can be done at only one position using three deflection measurements and using force B3 F M = [0;-200] N applied at the end-effector.

Let the uncertainty of position measurement system lead to independent deflection measurement uncertainty of 10 µm standard deviation along end-effector's X and Y axes. Furthermore, assume that the applied force can't be determined precisely due to the setup being used. Let there be an error in applied force described by the vector [-75;-75] N every time force is applied using this apparatus. Note that this applied force error is not known to the user. Best position for stiffness identification must, therefore, be selected to identify optimal parameter set.

Stiffness identification is bound to be affected by uncertainty of deflection measurements and errors in forces applied in this case. Therefore, both DUIR and FEIR criteria must be used together to find the best position for stiffness identification. For the sake of comparison, best position for stiffness identification can also be found using only DUIR criterion and only FEIR criterion. To compare the performances of these identification positions, Monte-Carlo simulations can be performed to simulate elastostatic calibrations

CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

of the bipod many times. Section 3.5.1 presents the details and results of stiffness identification optimization. The details and results of the validation simulations are shown in section 3.5.2.

Stiffness identification optimization

The stiffness identification equation (equation 2.6) can be written for this case as

B3 G U as B3 A M B3 H B3 H -1 ( B3 c + DU B3 ε c + FE B3 ε c ) = B3 G U ( B3 ∆X M + DU B3 ε ∆X M + FE B3 ε ∆X M ) (3.17)
Here, the left subscript "B3" specifies that the corresponding matrices/arrays of equation 2.6 have been formulated for this case. B3 G U is a 6 × 6 matrix with inverse of deflection measurement uncertainties along its diagonal elements and is formulated as shown in stiffness identification, respectively. as B3 A M is a 6×2 matrix which is a function of the forces assumed to be applied and the position used for stiffness identification. Parameter set of equation 3.17 is estimated using least squares method.

To formulate a DUIR criterion, matrix U must first be formulated for this case. Using equation 2.30, matrix U for this case can be written as

B3 U = B3 A D B3 H as B3 A M T as B3 A M -1 B3 H T B3 A T D (3.18)
Here, B3 A D is a 2 × 2 matrix which is a function of the target position and the force applied at the end-effector during the intended positioning. B3 U is a 2×2 matrix and it relates the uncertainty of deflection measurements to uncertainty of resultant errors in positioning after compensation (see equation 2.30). The first element of B3 U, B3 U 11 , is the scaling factor corresponding to the variance of X-coordinate of errors in position attained after 3.5. SIMULATION STUDY 3: VALIDATION OF USE OF BOTH CRITERIA TOGETHER 101 compensation. Hence, the identification position that leads to minimum value of B3 U 11 minimizes the impact of uncertainty of measured deflections on the desired compensation.

To formulate a FEIR criterion for this case, matrix Z needs to be formulated first. Using equation 2.51, Z can be written for this case as

B3 Z = B3 A D as B3 A M + ap B3 D (3.19)
Here, B3 Z is a 2 × 6 matrix. ap B3 D is a 2 × 2 matrix which is a function of the identification position and the approximate values of stiffness parameters (see section 2.4.1). The assumed actual values of stiffness parameters (10 N/µm for each spring) were used as approximate stiffness parameters. Given that prior information is not available about the error in applied force, one way to minimize its impact on the desired compensation is to minimize the norm of first row of B3 Z. The reason for this is that the values of the first row of B3 Z scale the error in applied force to give the X component of error in position attained after compensation.

Since both DUIR and FEIR criteria need to be considered to find the best position for stiffness identification, a multi-objective optimization needs to be done. The multiobjective optimization problem for this case can be written as min

B3 X M {f 1 ,f 2 } s.t. Workspace limits where, f 1 = B3 U 11 f 2 = norm B3 Z 11 B3 Z 12 ... B3 Z 16 (3.20)
Here, B3 X M is the identification position which minimizes both f 1 and f 2 . B3 Z ij is the j th element of i th row of B3 Z. The optimization problem of equation 3.20 leads to a number of Pareto optimal solutions. These are shown in figure 3.11. 50 solutions were considered on the Pareto front. Many methods exist to choose the optimal solution from these Pareto solutions. One of these methods, called the method of global criterion [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF],

was used to find the optimal solution in this case. For this, the ideal objective vector must be

CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

obtained first, which is a solution that would be ideal but not reachable. The ideal objective vector, denoted here as w id , is defined by the lower bounds of the Pareto front. It is given by

w id = min f 1 ( PF B3 X i M ) ,min f 2 ( PF B3 X i M ) ,i = 1...50 (3.21)
where PF B3 X i M is the identification position corresponding to the i th solution on the Pareto front. The solution on the Pareto front that is closest to the ideal objective vector is then usually chosen as the optimal solution. However, when the objective functions have different units and magnitudes, the function values of Pareto solutions must be scaled before choosing the closest point [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF]. The new scaled objective function values are given by

f 1 r = f r 1 -w id 1 w nad 1 (3.22) f 2 r = f r 2 -w id 2 w nad 2 (3.23)
In equations 3.22 and 3.23, f 1 r , f 2 r , f 1 r and f 2 r are the values of f 1 , f 2 , f 1 and f 2 corresponding to the r th Pareto solution (out of 50), respectively. w id i and w nad i refer to the i th components of w id and w nad , respectively. w nad is the Nadir objective vector [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF] and it is defined by the upper bounds of Pareto front. It is given by

w nad = max f 1 ( PF B2 X i M ) ,max f 2 ( PF B2 X i M ) ,i = 1...50 (3.24)
Ideal and Nadir objective vectors are shown in figure 3.11. The scaled Pareto front and the corresponding ideal and Nadir objective vectors are shown in figure 3.12 . The solution on the scaled Pareto front closest to the scaled ideal objective vector was then chosen as the optimal solution. The optimal solution is shown in figures 3.11 and 3.12.

The best position for stiffness identification were found using:

Validation simulations

To Figure 3.14 shows the probability density functions of X-coordinates of position errors after compensation, obtained using the Monte-Carlo simulations described above. Table 3.4 lists the corresponding standard deviations and means. These results show that the standard deviation of X-coordinates of position errors is least when only DUIR criterion was used to choose the best identification position. This result was expected because the variance of position errors after compensation was dependent only on the variance of deflection measurements in this case and DUIR criterion was framed to minimize this propagation. Furthermore, the mean of X-coordinates of position errors after compensation is least when only FEIR criterion was used to select the best identification position. This too is as expected because the constant error in forces applied in the Monte-Carlo simulations lead to a constant offset in positions attained after compensation and the FEIR criterion formulated here minimizes this propagation. When the best identification position

Conclusion

This chapter presented three simulation studies performed on a bipod to evaluate the efficacy of stiffness identification optimization framework presented in chapter 2. In the first simulation study, elastostatic calibrations were simulated in which only deflec-3.6. CONCLUSION 107 tion measurement uncertainty affected stiffness identification. In this, stiffness identification measurement conditions (poses and forces) suggested by DUIR criterion and various previously proposed criteria were used for stiffness identification. The measurement conditions suggested by DUIR criterion lead to best compensation quality as compared to the ones suggested by previously proposed criteria. Furthermore, results also showed that DUIR criterion could precisely predict the influence of stiffness identification measurement conditions on compensation quality, in the presence of deflection measurement uncertainties. In the second simulation study, elastostatic calibrations were simulated in which stiffness identification was affected only by errors in applied forces. In this, elastostatic calibrations were simulated using different measurement conditions for stiffness identification. The positioning performances achieved using the different stiffness identification measurement conditions were compared to the indications of FEIR criteria. Results showed that FEIR criterion can precisely predict and allow to minimize the influence of errors in forces applied during stiffness identification on the compensation quality.

In the last simulation study, elastostatic calibrations were simulated in which errors from both sources 11 were made to influence stiffness identification. In these simulated elastostatic calibrations, three sets of measurement conditions were selected for stiffness identification: one using DUIR criterion, another using FEIR criterion and the last one using both criteria. Results confirmed that using both criteria for selecting measurement conditions for stiffness identification gives best compensation quality in this case.

11. Deflection measurement uncertainty and errors in forces applied during stiffness identification.

Introduction

This chapter presents experimental studies on elastostatic calibration of a highprecision hexapod positioning system. These studies are aimed at: The experimental studies presented in this chapter were performed on a hexapod used for high-precision positioning applications from Symétrie [Symétrie, d] (see figures 4.1 and 4.2). This hexapod has a repeatability of ±0.5 µm in translations and ±2.5 µrad in rotations. More details about this hexapod can't be disclosed due to confidentiality. These experiments required an apparatus for measuring the pose of the hexapod's platform and an apparatus for applying necessary forces/moments on it. The pose measurement apparatus consisted of a coordinate measuring machine (LK-METRIS CMM with a RENISHAW SP25M scanning probe [Renishaw]) and three precision balls. The uncertainty of points measured using this CMM, quantified using the MPE P value [ISO, 2000], is about ±2 µm..

The apparatus available for applying force/moment consisted of a set of weights that could be only placed on the platform (see figure 4.2). This loading setup lead to the constraint that load could be applied only along the Z-axis of the hexapod's platform. Furthermore, the hexapod couldn't be rotated about its X and Y axes with the mass mounted on it as it would lead to sliding off of the weights from the platform.

This chapter is organized as follows: section 4.2 presents the kinematic and stiffness modelling of hexapods. Section 4.3 presents the details and results of the experimental study performed to validate elastostatic calibration of the hexapod. This is followed by H q contains the lengths of the legs and is given by H q = [ H q 1 H q 2 H q 3 H q 4 H q 5 H q 6 ] T (4.

Kinematic and stiffness modelling of hexapod

2)

The velocity of the end-effector and those of the actuated joints are related as per the following equation:

H η = H J Ḣq (4.3)
Here, H η is the end-effector's velocity vector, Ḣq contains the velocities of the actuators and H J is the Jacobian matrix. The equation relating the differential pose vector ( H ∆X) and the differential actuator position vector ( H ∆q), as in equation 1.13, can be written for this case as

H ∆q = H J-1 H ∆X (4.4)
Here, H J is the modified Jacobian matrix. H J-1 is given by

H J-1 = H J -1 1 0 0 B (4.5)
where I is a 3 × 3 identity matrix and B is given by J-1 relates the derivatives of components of H X to

B =     c Rz .c

KINEMATIC AND STIFFNESS MODELLING OF HEXAPOD

115 the derivatives of components of H q whereas H J -1 relates the end-effector velocity vector to the actuator velocities. Consequently, J -1 must be modified to get H J-1 as per equation 4.5 to take into account the relationship between angular velocities and angle derivatives.

The reader is referred to [Ardakani and Bridges, 2010] for the derivation of the expression for B.

A simple lumped stiffness model can be used to model the static stiffness characteristics of this hexapod (see figure 4.3). This model uses one linear spring2 to model the stiffness of each leg ( H k i=1..6 ). This modelling method is consistent with the method described in chapter 1, i.e., only stiffnesses along actuated joints are considered. The Cartesian stiffness matrix (as in equation 1.15), H K C , for this stiffness model of the hexapod can be written as

H K C = B J -T H K H J-1 (4.7)
where matrix H K is diagonal matrix with leg stiffnesses forming its diagonal elements.

It is given by

H K =             H k 1 0 0 0 0 0 0 H k 2 0 0 0 0 0 0 H k 3 0 0 0 0 0 0 H k 4 0 0 0 0 0 0 H k 5 0 0 0 0 0 0 H k 6             (4.8)
Matrix H K C relates the differential force vector ( H ∆F) and the differential position vector ( H ∆X) as

H ∆F = H K C H ∆X (4.9) CHAPTER 4.

EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEM

Rearranging equation 4.9 yields a relationship similar to equation 1.17:

H A H c = H ∆X (4.10)
In equation 4.10, H A is the observation matrix and it is given by

H A ij = H Jij 6 r=1 H J rj H ∆F r (4.

11)

H A ij , H J ij and H Jij in the above equation denote the j th element of the i th row of matrices H A, H J and H Jij , respectively. H ∆F r is the r th element of vector H ∆F. H c is the vector containing compliance parameters and is given by

H c = 1 H k 1 1 H k 2 1 H k 3 1 H k 4 1 H k 5 1 H k 6 T (4.12)
Appendix G presents a preliminary study performed on another high-precision positioning hexapod to validate the efficacy of the stiffness model (for hexapods) presented here to predict the deflections of a hexapod with loaded platform.

Experimental study 1: validation of elastostatic calibration of hexapod

This section presents the details and results of an experimental study performed to validate elastostatic calibration of the hexapod shown in figures 4.1 and 4.2. This elastostatic calibration was performed to achieve best possible positioning performance along all the axes (Tx-Rz) of the platform and throughout its workspace, with a mass of 121.25 kg (close to the maximum payload allowed for the given hexapod) mounted on it. The setups used for pose measurement and loading for this study are described in section 4.1.

One of the factors 3 defining the possible pose-force sets for stiffness identification was 3. Other factors defining the possible pose-force sets were: (a) the joint and actuator limits, and (b) payload limit.

EXPERIMENTAL STUDY 1: VALIDATION OF ELASTOSTATIC CALIBRATION OF HEXAPOD 117

the apparatus available for applying forces/moments on the platform. As shown in figure 4.2, this setup consisted of a set of weights which could be placed on the hexapod's platform. Due to this loading constraint, force could be applied only along the Z-axis of the platform and the poses for stiffness identification needed to have Rx and Ry to be zero. The magnitude of mass to be placed on the platform for stiffness identification experiment also had to be chosen. The maximum possible mass (121.25 kg -close to the hexapod's payload limit) was chosen for this purpose. This was desirable because higher magnitude of measured deflections leads to lesser impact of measurement uncertainty on identified parameters. Furthermore, choice was made to perform stiffness identification at just one pose for the sake of simplicity. Therefore, the best pose had to be found for stiffness identification of this hexapod and maximum possible number of deflection measurements had to be performed at this pose. The factor limiting the maximum number of deflection measurements was the thermal deflection of the hexapod during pose measurements. It was necessary to keep the thermal deflection of the hexapod as less as possible so that it wouldn't adulterate the load-deflection measurements. From past experience, 30 minutes was deemed as a good estimation of the maximum time until which thermally stable readings could be made. Approximately 5 deflection measurements (5 pose measurements before and after loading) could be made in these 30 minutes using the method described in appendix A.

Section 4.3.1 presents the details and results of stiffness identification and its optimization in this case. This is followed by details and results of the experiments performed to validate the compensation using estimated stiffness parameters in section 4.3.2.

Stiffness identification optimization

For performing stiffness identification as per choices described above, the stiffness identification equation can be written for this case first. Equation 2.6 can be written for this case as A M is a 30×6 matrix which is a function of the forces/moments assumed to be applied and the pose used for stiffness identification. Parameter set of equation 4.13 is estimated using least squares method. Best pose had to be found using the DUIR and FEIR criteria to ensure best possible compensation.

H1 G C as H1 A M H1 H H1 H -1 ( H1 c+ DU H1 ε c + FE H1 ε c ) = H1 G C ( H1 ∆X M + DU H1 ε ∆X M + FE H1 ε ∆X M ) (4.
For the formulation of a DUIR criterion, matrix H1 G C must be known beforehand (see section 2.3). Formulation of matrix H1 G C requires knowledge of variance and correlation of deflection measurements. However, the variance and correlation of measured deflections is extremely difficult to predict in this case. Among other factors such as complexity of CMM measurements and the measurement method (as in appendix A), minor thermal deflections of the hexapod complicate the estimation of deflection measurement uncertainty beforehand. Consequently, H1 G C could not be determined beforehand in this case and a DUIR criterion could not be formulated. Hence, the best pose for stiffness identification was found using FEIR criterion only.

For the formulation of FEIR criterion, matrix Z must be formulated first. Since the platform pose coordinates have both translations and rotations, two separate Z matrices must be formulated here as in equations 2.53 and 2.54. They can be written as H1 ε X A and FE,r H1 ε X A . r H1 Z st and t H1 Z st can be used to optimize positioning after compensation at the 3409 target poses in the workspace. The sizes of FE,r H1 ε X A and r H1 Z st are 10227 × 1 and 10227 × 30, respectively. FE,t H1 ε X A and t H1 Z st have sizes similar to FE,r H1 ε X A and r H1 Z st , respectively. An information important for formulating FEIR criterion for this case is that the errors in applied forces/moments are due to the offset of the center of mass from its assumed position. This offset produces undesired moments about the X and Y axes of the platform.

This implies that H1 ε ∆F M can have non-zero terms in its fourth and fifth elements. Due to the redundant nature of H1 ε ∆F M , the fourth and fifth elements also repeat themselves at the appropriate spots within this array 6 . The elements of r H1 Z st and t H1 Z st that scale these elements of H1 ε ∆F M had to be minimized. Furthermore, the elements of r H1 Z st and t H1 Z st that couple with the fourth and fifth elements of H1 ε ∆F M also repeat themselves at appropriate spots in these matrices. Consequently, minimizing the terms of r H1 Z st and 6. For instance, the 4 th element of H1 ε ∆F M repeats itself at 10 th , 16 th , 22 nd and 28 th spots. H1 X M in equation 4.18 is the identification pose. E t i and E r i are the i th elements of E t and E r , respectively. t H1 Z st i,j and r H1 Z st i,j are the j th elements of the i th rows of t H1 Z st and r H1 Z st , respectively. The optimization problem of equation 4.18 leads to a number of Pareto solutions. The best solution was obtained using the method of global criterion [START_REF] Miettinen | Nonlinear multiobjective optimization[END_REF], similar to the manner in which the optimal solution was chosen for the optimization problem of equation 3.20 of chapter 3. Subsequently, the best pose obtained for stiffness identification for this case was [14.3 mm -1.8 mm -4.3 mm 0 • 0 • -3.67 • ].

Stiffness identification experiment was performed at the best pose obtained for this case. To accomplish this, the hexapod's platform was commanded to the identified best pose for stiffness identification. The platform's pose was measured 7 five times without the mass (121.25 kg) mounted on its platform first. Following this, the mass was mounted and the platform's pose was measured five times again. The variance and correlation of the deflection measurements were computed and matrix H1 G C was formulated as per equation 1.23. The set of six stiffness parameters was then identified using least squares method.

Evaluation of compensation efficiency

In order to assess the efficiency of compensation using estimated stiffness parameters (table 4. 8. The pose measurement method outlined in appendix H was used instead of the one in appendix A because latter is susceptible to thermal deflections of the hexapod when pose measurements need to be performed for long duration, unlike the former one. The pose measurements to be performed for the validation study in section 4.3.2 was bound to take long time. Consequently, pose measurement method outlined in appendix H was used for pose measurement in this case. 4.4 and4.5 present some quantitative metrics to assess the level of improvement in positioning accuracy with the use of elastostatic error compen-9. This is the coordinate frame with respect to which all poses were measured.

CHAPTER 4. EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEM

sation. It can be seen that the maximum of differences between measured pose errors of the loaded hexapod with compensation and the hexapod without load are less than 5.48

µm for translations and 23.9 µrad for rotations, as compared to 31.25 µm and 90.27 µrad for loaded hexapod without compensation. Furthermore, the RMS values of differences between measured pose errors of the loaded hexapod with compensation and the hexapod without load are less than 3.28 µm for translations and 12.86 µrad for rotations, as compared to 26.08 µm and 49.49 µrad for loaded hexapod without compensation.

-60 -30 0 30 60

X-axis position

Experimental study 2: validation of FEIR criterion

In this section, an experimental study is presented which was performed to validate the FEIR criterion. This study was performed on the hexapod shown in figures 4.1 and 4.2, and using the measurement and loading apparatuses shown in them.

In order to validate FEIR criterion, a simple scenario of elastostatic calibration was considered. In this, elastostatic calibration of the hexapod was required to achieve best possible positioning along its Tz axis and at poses listed in table 4.3, with a mass of 121.25 kg mounted on its platform. The force/moment to be applied for stiffness identification was fixed. Stiffness identification experiments were then performed at: (a) best poses for stiffness identification as per FEIR criterion 10 , (b) respective target poses, and (c) the zero pose. Each of these stiffness identification experiments was performed with a high error in force/moment applied at the platform. The compensation qualities using stiffness parameters identified at each of these identification poses were then compared.

For stiffness identification at the various poses listed above, the force/moment was applied on the platform by mounting a mass of 121.25 kg (as shown in figure 4.2) on it.

10. One pose for stiffness identification was chosen for each target pose.

EXPERIMENTAL STUDY 2: VALIDATION OF FEIR CRITERION

127 However, the force/moment that was assumed to have been applied 11 had an extra moment of 1000 Nm about the X-axis (M x ). This big error in applied load was introduced to clearly show the problem and improvement (using the proposed optimization). The errors in identified stiffness parameters would, in this case, be dominated by errors (in thereof ) due to applied force/moment error. Owing to this, only one deflection measurement was enough for each stiffness identification experiment.

The remaining part of this section is organized as follows: section 4.4.1 presents the details about stiffness identification optimization using FEIR criterion. This is followed by section 4.4.2 which presents the details of experiments performed to validate the improvement in compensation quality when FEIR criterion are used to optimize stiffness identification.

Stiffness identification optimization

For performing stiffness identifications as per choices described above, the stiffness identification equation can be written for this case first. Equation 2.6 can be rewritten for this case as is the 6 × 1 array containing errors in forces/moments applied during stiffness identification and FE H2 ε X A is the 6 × 1 array containing the resultant errors in poses attained after compensation. Since the applied force/moment error is a moment about the X-axis and best positioning performance was desired along Tz axis, identification pose that minimizes the fourth element of the third row of H2 Z, H2 Z 34 , had to be found. This is because H2 Z 34

H2 G as H2 A M H2 H H2 H -1 ( H2 c + DU H2 ε c + FE H2 ε c ) = H2 G ( H2 ∆X M + DU H2 ε ∆X M + FE H2 ε ∆X M ) ( 4
scales the element of H2 ε ∆F M corresponding to the moment about X-axis to contribute to the element of large number of starting points were supplied so that the best solution could be found.

The deflection measurements were then performed at the identified best poses, the target poses and the zero pose. The mass weighing 121.25 kg (as in figure 4.2) was mounted on the platform during these stiffness identification experiments. The measured deflections and the "false assumed applied load" were used to estimate the stiffness parameters.

Evaluation of compensation efficiency

For validation of optimal parameter sets identified using FEIR criterion, positioning experiments were performed at poses listed in table 4.3. The platform was commanded to position at these poses with a mass of 121.25 kg mounted on its platform using command poses that were generated using: (a) the identified optimal parameter sets, (b) parameter set identified when stiffness identification was performed at zero pose, and (c) parameter sets identified when stiffness identification was performed at respective target poses.

Errors in poses attained were then measured 12 . For the sake of comparison, errors in attained poses were also measured with the platform commanded to those poses without load. These positioning experiments were conducted multiple times and the results were found to be very repeatable (< ±1µm for translations and < ±2µrad for rotations). Owing to this, results of just one trial are presented here for analysis. of hexapod without load). One is that the matrix H2 Z possible under the given workspace constraints can't completely eliminate the impact of the applied load error 13 considered in this case on the positioning task intended in this case. The second reason could be that the best solution was not found in the optimization routine. The second possibility is highly unlikely because optimization was carefully performed using a large number of starting points.

-60 -30 0 30 60 tification is presented. The parametric calibration framework is similar for robot elastostatic and geometric calibrations. Also, one error source, uncertainty of deflection measurements performed, impacts parameter identification in both these calibration routines. Consequently, the framework for formulating criteria for minimizing the influence of this error on elastostatic calibration performance, the DUIR criterion, can also be used for minimizing the influence of this error on the performance of robot geometric calibration. Appendix I discusses this in further detail.

X-axis position

Further investigations can be carried out on this.

Evaluating uncertainty of pose measurement:

Methods to evaluate the uncertainty of 6-DOF pose measurement methods, documented in appendixes A and H, must be developed. Alternatively, a new 6-DOF pose measurement method for hexapods can be developed in which the uncertainty of pose measurement can be evaluated easily. This is necessary to exploit the developed parameter identification optimization framework completely. Currently, the criterion to minimize the influence of deflection measurement uncertainty on compensation quality, DUIR criterion, can't be used with the pose measurement methods documented in appendices A and H. This is because, in order to use DUIR criterion, it is necessary to know the uncertainty of measured deflections. This, in turn, is dependent on the uncertainty of pose measurements performed.

Elastostatic calibration of large and heavy hexapods:

The method documented in this thesis to perform elastostatic calibration of hexapods must be tested for large and heavy hexapods. The stiffness modelling method for hexapod used in this thesis assumes that the legs are light enough to cause negligible deflections of the hexapod's components. In large and heavy hexapods, this assumption might not hold true [START_REF] Klimchik | Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings[END_REF]. If this assumption does indeed not hold true for large and heavy hexapods, the stiffness model needs to be amended. S can then be obtained using the method outlined by [Arun et al., 1987].

This method requires i to be at least three. As a result, at least three precision balls are needed to perform pose measurement as per the method outlined here. The elements of the j th column of i S -1 possess a unit which is the inverse of that of the j th element of DU ε i ∆X M . Consequently, the resulting measurement vector after scaling,

G C

DU ε ∆X M , is dimensionless. In order to check this, let us consider a simple case where DU ε i ∆X M has two coordinates, one translational and the other rotational. Matrices i V and i S (refer equation 1.24), in this case, will have the following structure and units: Here, i V pq and i S pq are the p th elements the q th column of matrices i V and i S, respectively. i S -1 , for this case, can be written as The resulting i th measured deflection error vector is then given by

F

Relation between Cov( FE,t ε X A ) and the RMS value of possible Euclidean norms of FE,t ε X A Here, the derivation of relation between Cov( FE,t ε X A ) and the RMS value of possible Euclidean norms of FE,t ε X A is shown. Equation 2.54 can be used to get the following expression: E FE,t ε X A T FE,t ε X A = FE ρ 2 = E ( t Z ε ∆F M ) T ( t Z ε ∆F M ) (F.1)

Here, E(e) denotes expectation of e. In equation F.1, FE ρ is the RMS value of possible Euclidean norms of translational components of FE ε X A , FE,t ε X A . Equation F.1 can be further expanded to get Before performing loading experiments for stiffness identification, practical constraints needed to be taken into account. This constraint was a result of the loading setup available. This loading setup consisted of weights that could be placed on the platform of the hexapod (see figure G.2). This setup demanded the platform to not be rotated about its X and Y axes. This constraint was incorporated to make sure that the mounted mass does not slide off of the platform. Furthermore, using this setup for loading also came with the constraint that the force could only be applied along the Z axis of the platform. The number of poses and number of deflection measurements per pose also had to be chosen. High number of measurements leads to better accuracy of identified parameters but time limits the number of measurements. Hence, compromising between timeefficiency and accuracy, three poses and three measurements per pose were chosen. Furthermore, it is also assumed that the legs of the hexapod exhibit linear stiffness behavior.

FE ρ 2 = tr E t Z ε ∆F M t Z ε ∆F M T = tr E t Z ε ∆F M ε ∆F M T t Z T = tr t Z E(ε ∆F M ε ∆F M T ) t Z T
Consequently, just one load vector should be enough to identify the stiffness in each leg.

Figure G.4 shows the the results of an experiment that shows the hexapod's linear stiffness behavior. Also, it is best to apply maximum possible force/moment for better identifiability of parameters 2 . Due to the above reasons, choice was made to use just one load vector of maximum possible magnitude. This was 34.5 kg in the given case. Also, it was necessary to place an initial load of 12.2 kg on the platform to suppress the play in actuators. Hence, the effective force using which the platform deflection was to be measured was 22.3 kg along the Z-axis of the platform.

Considering the choices listed above, it was necessary to choose three poses at which stiffness identification had to be performed. Since the goal here was only to validate the stiffness model, these three identification poses were chosen arbitrarily. The chosen iden-2. More force/moment leads to more deflection of the platform and this is good for identifiability of parameters. This is because the errors in deflection measurements have less impact on the identified parameters when the magnitude of platform deflections are high. Figures G.5 and G.6 show the comparison between the predicted and measured 6-DOF deflections of the platform at different poses along the X and Y axes of the hexapod, respectively. Table G.3 shows the RMS values of errors in prediction of these 6-DOF deflections.

These results show that the predicted and measured deflections of the platform are very close. As seen in table G.3, the RMS values of prediction errors are under 3.1 µm for translational deflections and 8.8 µrad for rotational deflections. Therefore, it can be concluded that the stiffness model and its estimated parameters are effective in predicting the deflections of the loaded hexapod. In this experimental study, the pose to be measured (refered to as measurement pose from here), called S 1 in sections H.1 and H.2, was the zero pose. This pose was chosen to be measured in order to facilitate the ease of understanding results as the hexapod is symmet- necessary measurements were made to perform the pose measurement as per the thermal deflection decoupled method (see section H.2). Note that (a part of) these measurements can also be used for performing pose measurements as per the conventional method of appendix A. Ten trials of measurements were performed and the hexapod's legs were heated during this using the electric heating mats. The measurements were then post-processed as per the conventional (appendix A) and proposed (section H.2) methods.

In the proposed pose measurement method, the thermal expansion of the legs with the platform in measurement pose had to be predicted. This had to be done using the measured thermal expansions of the legs at the reference pose (see section H.2). The following logic was used for this: the legs of the hexapod used in this study could be divided length- Tz mes using the conventional method is similar to the trend of the change in temperature of all legs. This behaviour is logical given the orientation of all legs in zero pose. Also, deviation seen in Tx mes using conventional method increases with every consecutive trial until the end. This can be explained by the temperatures measured in legs 2, 3, 4 and 5. The temperatures of legs 2 and 5 are higher than those of legs 3 and 4 during the test and this difference increases with every consecutive trial until the end. Consequently, legs 2 and 5 push the platform more in positive X-direction as compared to legs 3 and 4 pushing it in the opposite direction. Furthermore, deviation seen in Rx mes using conventional method also increases with every consecutive trial until the end. This can be explained by the difference in temperatures of legs 3 and 5 (with leg 5 heating more than leg 3) which follows a similar trend. Consequently, leg 5 pushes the platform more about the X-axis as compared to leg 3 and results in a positive rotational deviation about the X-axis with every consecutive trial.

The pose parameters measured using the proposed method do not deviate with change in temperature of hexapod's legs, unlike the ones measured using conventional method.

It is, therefore, clear that the proposed method is effective in eliminating the influence of thermal deflection of the hexapod on the measured pose parameters.

  Figures showing the first designs of hexapod proposed by D. Stewart (left) and

6of Tables 4 . 4

 44 figure 1.2) due to applied loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Classification of robot calibration approaches . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of elastostatic calibration used in this thesis (illustrated on a bipod) . . . 1.6 Error propagation from measured deflection to attained compensation in a one dimensional case when an identification experiment is performed many number of times (with the assumption that the stiffness model is perfect) . . . . . . . 1.7 Application-oriented and non-application-oriented stiffness identifications . . 1.8 Propagation of error ellipses in stiffness parameter identification routine (shown for a two dimensional case) . . . . . . . . . . . . . . . . . . . . . . . . . . .

A

  Also called observation matrix, it is a function of ∆F, J & J and relates ∆X & c A D Matrix A formulated using the set of poses (X D ) and forces/moments (∆F D ) at which best positioning performance is desired r A D Rows of A D corresponding to the rotational elements of ∆X D t A D Rows of A D corresponding to the translational elements of ∆X D A M Observation matrix (A) corresponding to ∆F M and X M A M Matrix A M scaled using matrices G and H as A M Matrix A M formulated using pose X M and the assumed differential force vector applied at the platform ( as ∆F M ) during stiffness identification c Vector containing inverses of k i c Estimated stiffness parameters that are corrupted by DU ε c c Estimated stiffness parameters that are corrupted by DU ε c & FE ε c D Matrix which is function of J, J and K ap D

Finalq

  load vector assumed to have been applied at the endeffector/platform of a robot during stiffness identification ac F fi Final load vector actually applied at the end-effector/platform of a robot during stiffness identification ∆F Difference between the initial and final load vectors applied at the endeffector/platform of the robot during stiffness identification ∆F D Forces/moments with which best positioning performance is desired at X D ∆F M Difference between the initial and final load vectors applied at the endeffector/platform of the robot during stiffness identification with the end-effector/platform at pose X M as ∆F M Difference between the initial and final load vectors assumed to have been applied at the end-effector/platform of the robot during stiffness identification ac ∆F M Difference between the initial and final load vectors actually applied at the end-effector/platform of the robot during stiffness identification G Task variable scaling matrix G U Task variable scaling matrix when elements of DU ε ∆X M are independent but don't have identical distribution (different standard deviations) G C Task variable scaling matrix when elements of DU ε ∆X M are not indepen- Stiffness along i th actuator of a robot K A diagonal matrix containing the stiffness along robot's actuators (k i ) Vector containing actuator positions/lengths ∆q Vector containing differentials of actuator positions/lengths U Matrix containing scaling factors that scale DU σ 2 ε ∆X M

Z

  a robot's end-effector/platform at a pose where stiffness identification is performed ∆X Deflection of a robot's end-effector/platform due to a change in load ∆F on it ∆X C Deflection at pose X C due to a load ∆F D ∆X C Predicted deflection with error at pose X C and load F D ∆X M Deflection measured at pose X M under the influence of force/moment applied ∆F M ∆X M Deflections measured during stiffness identification and corrupted by DU ε ∆X M & FE ε ∆X M ac ∆X C Actual deflection of the end-effector/platform at the pose X C as a result of actual stiffness parameters (c) and applied load ∆F D ∆X D Deflection at pose X D due to an applied load ∆F D Matrix containing scaling factors that scale ε ∆F M to FE ε X A t Z

Figure 2 :Figure 4 :

 24 Figure 2: Serial robots (leftmost) connect the base to the end-effector/platform (red) using single chain with all actuators (blue) in series, as opposed to parallel robots (second and third robots)

  to robot elastostatic calibration . . . . . . . . . . . . . . . . . 27 1.2 Mathematical framework for experimental stiffness identification in robot elastostatic calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.3 Stiffness identification optimization . . . . . . . . . . . . . . . . . . . . . . . 46 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The static positioning performance of a robotic manipulator is deteriorated due to a number of factors. Some of the major factors include geometric errors, thermal deflections of robot's components, compliance of robot's components, friction between mating components of the robot, clearance in joints and play/backlash in gears, ball screws, etc.

Figure 1 . 1 :

 11 Figure 1.1: Phenomenon of accuracy deterioration of a hexapod positioning system with mounted payload and the required solution

CHAPTER 1 .Figure 1 . 3 :

 113 Figure 1.2: Setup to measure the deflection of the platform of a high-precision hexapod positioning system due to load mounted on it's platform

Figure 1 . 4 :

 14 Figure 1.4: Classification of robot calibration approaches

Figure 1 .

 1 Figure 1.5 illustrates this procedure applied to a bipod.

Figure 1

 1 Figure1.6: Error propagation from measured deflection to attained compensation in a one dimensional case when an identification experiment is performed many number of times (with the assumption that the stiffness model is perfect)

  Figure 1.7: Application-oriented and non-application-oriented stiffness identifications

Figure 1

 1 Figure 1.8: Propagation of error ellipses in stiffness parameter identification routine (shown for a two dimensional case)

  the product of singular values of the regressor matrix minimizes the product of the singular values of the inverse of regressor matrix. This will effectively minimize the volume of parameter error ellipsoid for any given measurement error ellipsoid. (f ) O 2 : Driels and Pathre [Driels and Pathre, 1990] introduced this observability index.

  The first disadvantage of existing stiffness identification optimization criteria is their inability in ensuring best possible compensation no matter what the specifications of the positioning application are. Robot positioning applications can be of varied natures. An application might require the robot to be able to achieve best possible accuracy: (a) at some chosen poses or throughout the workspace, (b) along some chosen axes of the robot or along every axis, and (c) using some fixed load or a set of loads. Therefore, a truly application-oriented elastostatic calibration method must be able to deliver best possible positioning accuracy no matter what the application's specifications are. Consequently, the stiffness identification optimization criterion/criteria must be able to ensure best compensation quality in any specified positioning application. As explained in chapter 1, none of the stiffness identification optimization criteria present in the literature can satisfy this requirement. Hence, the new criterion or set of criteria for stiffness identification optimization must be able to satisfy this requirement.The second disadvantage of existing stiffness identification optimization criteria is CHAPTER 2. APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION their inability to minimize the influence of errors in forces/moments applied during stiffness identification on compensation quality. It was shown that the errors in applied forces/moments during stiffness identification experiment can have considerable impact on the positioning accuracy after compensation of a loaded robot. No work exists in the literature that considers the influence of this while choosing the best set of poses and forces/moments for stiffness identification experiment. Hence, the new criterion/criteria to be developed must minimize the influence of: (a) the errors in forces/moments applied during stiffness identification experiment, and (b) the uncertainty of measured deflections, on the positioning accuracy after compensation of the loaded robot.

  Figure 2.1presents the flowchart of procedure to find best set of poses and forces/moments for stiffness identification using this method.

Figure 2 . 1 :

 21 Figure 2.1: Flowchart of procedure to find best set of poses and forces/moments for stiffness identification as per the proposed method

  r A D and t A D are the rows of A D corresponding to the rotational and translational elements of ∆X D , re-CHAPTER 2. APPLICATION-ORIENTED ROBOT ELASTOSTATIC CALIBRATION OPTIMIZATION spectively. t A D and r A D are related to their respective elements of ∆X D ,

  10. This follows from the inequality ||A X|| 2 ≤ ||A|| 2 ||X|| 2 for any m × n matrix A and n-vector X (refer chapter 9 of[START_REF] George | Linear regression analysis[END_REF]).

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2 Mechanism description, kinematics and stiffness model . . . . . . . . . . . 83 3.3 Simulation study 1: validation of DUIR criterion . . . . . . . . . . . . . . . 86 3.4 Simulation study 2: validation of FEIR criterion . . . . . . . . . . . . . . . . 93 3.5 Simulation study 3: validation of use of both criteria together . . . . . . . . 99 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 82 CHAPTER 3. VALIDATION OF DEVELOPED TECHNIQUES USING SIMULATED ELASTOSTATIC CALIBRATIONS OF A BIPOD

Figure 3 . 1 :

 31 Figure 3.1: Schematic of the bipod under study

Figure 3 . 2 :

 32 Figure 3.2: Lumped stiffness model of the bipod under study

  Consider a case in which elastostatic calibration of the bipod (figure3.1) must be performed to position its end-effector precisely at a position B1 X D = [250;-250] mm with a force B1 F D = [0;-200] N applied on it. Assume that best possible positioning accuracy is desired only along the X-axis of the end-effector and that the positioning accuracy along its Y-axis is inconsequential. Also suppose that stiffness identification can be done at only one position using three deflection measurements and with a force of B1 F M = [0;-200] N applied at the end-effector. Let the uncertainty of position measurement system lead to independent deflection measurement uncertainties of 10 µm and 20 µm standard deviations along the end-effector's X and Y axes, respectively. Furthermore, let the errors in 3.3. SIMULATION STUDY 1: VALIDATION OF DUIR CRITERION 87 forces applied during stiffness identification be zero in this case. The best position for stiffness identification must, therefore, be selected to minimize the influence of deflection measurement uncertainty on positioning accuracy after compensation.Since only uncertainty of measured deflections affects stiffness identification quality, DUIR criterion must be formulated to find the best position for stiffness identification in this case. For the sake of comparison, other previously proposed criteria 5 (O 1 -O 5 , O TP and O MV ) can be used to find the best position for stiffness identification. To compare the performances of different identification positions, Monte-Carlo simulations were performed in which elastostatic calibrations of the bipod were simulated many times. Section 3.3.1 presents the details and results of stiffness identification optimization. The details and results of the validation simulations are then presented in section 3.3.2.

  .11) Here, B1 U and B1 A D are 2 × 2 matrices. B1 A D is a function of the target position and force applied at the end-effector during the intended positioning. The first diagonal element of B1 U, B1 U 11 , is the scaling factor corresponding to variance along the X-axis of B1 X A . Hence, the identification position which minimizes B1 U 11 must be found. The plot of DUIR criterion formulated for this case, B1 U 11 , varying across the allowed workspace is shown in figure 3.3. Also, figure 3.4 shows values of other position selection criteria (O 1 -O 5 , O TP and O MV ) at positions across the workspace of the bipod. Note thatO TP and O MV yield same values in this case. This is because these two criteria have same expressions when best positioning is desired at just one position (see table 1.1). Figure3.5 shows the best position as per each criterion for this case.

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Plot showing values of DUIR criterion ( B1 U 11 ) for the elastostatic calibration of simulation study 1 (position having the lowest value of B1 U 11 is the best position for stiffness identification as per DUIR criterion in this case)

Figure 3 .

 3 Figure3.7 shows the probability density functions of errors in positions attained after simulated compensations using parameters identified at different identification positions mentioned above. Table3.1 lists the standard deviations of these distributions along with the corresponding values of B1 U 11 . These results show that the best identification position 6. Suggested by different criteria.

  Figure 3.6: Flowchart of the Monte-Carlo simulations of simulation study 1

Figure 3 . 7 :

 37 Figure 3.7: Probability density functions of errors in positions attained after simulated compensations in Monte-Carlo simulations of simulation study 1

  a function of the target position and force applied at the end-effector during the intended positioning.

apB2D

  is also a 2×2 matrix which is a function of the identification position and the approximate values of stiffness parameters (see section 2.4.1). The assumed actual values of stiffness parameters were used as approximate stiffness parameters.In this case, the relation between covariance matrices of applied force error and the corresponding error in position attained after compensation (see equation 2.52) can be used 3.4. SIMULATION STUDY 2: VALIDATION OF FEIR CRITERION 95 to optimize stiffness identification. The relation between covariance matrices of applied force error and the corresponding attained position error in this case is given by

  position that minimizes the first element of ζ, ζ 1 , must be found for best positioning along X-axis of the end-effector. Similarly, the second element of ζ, ζ 2 , must be minimized for best positioning along its Y-axis. Figures 3.8 and 3.9 show the values of ζ 1 and ζ 2 for different identification positions across the workspace of the bipod. 9. Since the standard deviations of each element of B2 ε ∆F M is 10 N.

Figure 3

 3 Figure 3.8: ζ 1 values for different identification positions across the workspace of the bipod

  (a) both FEIR and DUIR criteria (optimal solution on the Pareto front), (b) DUIR criterion only (minimizing only f 1 3.5. SIMULATION STUDY 3: VALIDATION OF USE OF BOTH CRITERIA TOGETHER 103 from 3.20), and (c) FEIR criterion only (minimizing only f 2 from 3.20). The best positions for stiffness identification as per all three criteria mentioned above are shown in table 3.3.

Figure 3 .

 3 Figure 3.11: Pareto front for the multi-objective optimization problem of equation 3.20

  compare performances of different identification positions, Monte-Carlo simulations of simulated elastostatic calibrations of the bipod were performed with the positioning specifications of this simulation study. Stiffness identification experiment was simulated 10000 times at each identification position with different set of errors in measured deflections every time. The errors in measured deflections had errors due to: (a) uncertainty of measured defections (10 µm standard deviation along each axis), and (b) error in force applied ([-75; -75] N). The estimated set of stiffness parameters in each trial was used to simulate position compensation. The mean and standard deviations of errors in positions attained after simulated compensations can then be used to evaluate the performance of the three identification positions. Figure 3.13 shows the flowchart for this Monte-Carlo simulation.

  Figure 3.13: Flowchart for the Monte-Carlo simulations of simulation study 3

Figure 3 .

 3 Figure 3.14: Probability density functions of errors in positions attained after compensation in the Monte-Carlo simulations described in section 3.5.2 (dashed lines show the mean of each distribution)

  (a) experimentally validating elastostatic calibration of hexapod, and (b) experimentally validating the FEIR criterion. Experimental validation of DUIR criterion was not performed because of impracticality of this endeavour with the setup available. This is because this study would require repetition of positioning experiments large number of times (>> 1000) and this was not practical with the experimental setup available. Large number of positioning experiments are required because obtaining reliable values of variance of poses attained after compensation is necessary for validation of DUIR criterion. This is because DUIR criterion minimizes the variance of possible errors in poses attained after compensation (see section 2.3).

CHAPTER 4 .Figure 4

 44 Figure 4.1: Hexapod (without mass mounted on the platform) along with the pose measurement apparatus

Figure 4 .

 4 Figure 4.2: Hexapod (with mass mounted on the platform) along with the pose measurement apparatus

Figure 4 .

 4 Figure 4.3 shows the kinematic scheme of the UPS (universal-prismatic-spherical) hexapod of figures 4.1 and 4.2. Each leg consists of a UPS chain that connects the base to the platform. Poses of the platform of this hexapod are defined by a coordinate frame fixed to the platform at its center (platform frame).

Figure 4 . 3 :

 43 Figure 4.3: Kinematic scheme and the lumped stiffness model of the hexapod under study

  c • = cos(•), s • = sin(•) and R is the rotation matrix. The operation of equation 4.5 is necessary because H

  that the deflection measurements are correlated in this case and consequently, task variable scaling matrix must be formulated as per equation 1.23. The deflection measurements are correlated because the individual measured pose parameters (Tx, Ty,...,Rz) are correlated. This is a consequence of the pose measurement method (refer appendix A).

  subscript 'H1' specifies that the corresponding matrices/arrays of equations 2.53 and 2.54 have been formulated for this case. t H1 A D and r H1 A D are the matrices containing first and last three rows of H1 A D , respectively. H1 A D is a 6 × 6 matrix which is a function of the target pose and the force/moment applied at the end-effector during the intended positioning. To get ap H1 D, some preliminary approximate stiffness estimations must be made 5 . The approximate stiffness parameter values were found by performing a set of three deflection measurements at zero pose by mounting 121.25 kg mass on the platform.

  H1 ε ∆F M to posi- tioning errors at poses throughout the workspace must be found. To do this, the workspace was discretized using uniformly distributed poses in the allowed workspace. The positioning errors at these poses were then minimized. This is one way to ensure best positioning performance throughout the workspace. 3409 uniformly distributed poses were chosen for this purpose. The equations relating H1 ε ∆F M to the errors in poses attained (at 3409 uniformly distributed poses) after compensation are given by , the right superscript 'st indicates that the corresponding matrices/arrays from equations 4.14 and 4.15 are stacked row-wise and they correspond to 3409 5. See section 2.4 for more explanation. 120 CHAPTER 4. EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEM arrays of FE,r

  3.1), positioning errors of the hexapod's platform were measured: (a) without load, (b) with load and without elastostatic error compensation, and (c) with load and with elastostatic error compensation. These positioning experiments were performed at different poses along the X and Y axes of the hexapod. These poses are listed in table 4.3 and visually described in figure 4.4. The pose measurements were performed using the method 8 outlined appendix H. The mass mounted on the hexapod during these positioning experiments was 121.25 kg. Table 4.3: Poses of hexapod's platform at which positioning experiments were performed Poses along Pose parameters Tx (mm) Ty (mm) Tz (mm) Rx (deg) Ry (deg) Rz (deg)

Figure 4 . 4 :

 44 Figure 4.4: Visual description of platform frame poses at which compensation efficiency was evaluated

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Measured errors in poses attained by the hexapod's platform at poses along X-axis

Figure 4 .

 4 Figure 4.7 shows the measured errors in poses attained by the hexapod's platform at different poses. It can be seen that the best compensations were achieved using stiffness parameters identified at poses as per FEIR criterion. Two data points stand out in the obtained results (in figure 4.7): (a) pose error at pose [30 mm 0 mm 0 mm 0 • 0 • 0 • ] using parameters identified at the same pose, and (b) pose error at pose [0 mm -60 mm 0 mm 0 • 0 • 0 • ] using parameters identified at the best pose as per FEIR criterion. The reader is referred to figure 4.8 to gain an understanding of the reason behind (a). This figure shows values of H2 Z 34 for each of the poses along X-axis (listed in table 4.3) when stiffness identification is performed at the respective poses for positioning 12. Pose measurement method described in appendix H was used for this.

CHAPTER 4 .

 4 EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEMat the same pose. These values scale the error in forces/moments applied during stiffness identification to the Tz-axis error in pose attained after compensation. As can be seen in figure4.8, the large error in (a) is due to the scaling provided by H2 Z 34 for this particular case. The magnitude of H2 Z 34 is approximately 11.7 times larger for the pose [30 mm 0 mm 0 mm 0 • 0 • 0 • ] as compared to the other poses in figure4.8. This correlates with the observation in figure4.7, i.e., the magnitude of positioning error is approximately 11.7 times larger for pose [30 mm 0 mm 0 mm 0 • 0 • 0 • ] as compared to the other poses along X-axis when stiffness identification is performed at those respective poses. Regarding (b), two possibilities exist for the reason behind the error in pose attained for the best case scenario (using FEIR criterion) not reaching the ideal level (measured accuracy

  with compensation using parameters identified at poses as per FEIR criterion (one pose for stiffness identification per target pose) With load and with compensation using parameters identified at zero pose With load and with compensation using parameters identified at respective target poses Without load

Figure 4 . 7 :

 47 Figure 4.7: Measured errors in poses attained by the hexapod's platform, along its Tz axis, at poses listed in table 4.3

Figure 4 . 8 :

 48 Figure 4.8: Values of H2 Z 34 for each of the poses along X-axis listed in table 4.3 when stiffness identification is performed at the respective poses for positioning at the same pose

146

  Figure A.1: Test setup used for pose measurement of hexapod

Figure A. 2 :S

 2 Figure A.2: Illustration of the measurement method

  M ) is independent and identically distributed, it needs to be shown that Cov( i S -1 DU ε i ∆X M ) is a diagonal matrix containing same numbers along its diagonal (equal variances). Equations 1.23 and 1.24 give usCov( i S -1 DU ε i ∆X M ) = i S -1 Cov( DU ε i ∆X M ) i S-T seen from equations B.1, B.2 and B.3 that (G C DU ε ∆X M ) is independent and identically distributed.

  (m 2 ) i V 12 (mrad) i V 21 (mrad) i V 22 (rad 2 )

  i S -1 DU ε i ∆X M = i S 22 i S 11 i S 22i S 12 i S 21 (m -1 ) -i S 12 i S 11 i S 22i S 12 i S 21 (rad -1 ) -i S 21 i S 11 i S 22i S 12 i S 21 (m -1 ) i S 11 i S 11 i S 22i S 12 i S 21 (rad -1 ) the j th element of DU ε i ∆X M . As can be seen from equation B.8, i S -1 DU ε i ∆X M is dimensionless. Consequently, G CDU ε ∆X M is also dimensionless. This can be seen in equation B.9.

  Figure D.1: Hexapod positioning system with 120 kg mass mounted on the platform

  Figure G.1: Hexapod (without mass mounted on the platform) along with the pose measurement apparatus, used in this study

  Figure G.3: Kinematic scheme and the lumped stiffness model of the hexapod under study

  Figure G.5: Plot of predicted and measured 6-DOF deflections of the loaded hexapod at poses along X-axis

Figure H. 3 :

 3 Figure H.3: Flowchart of procedure to post-process the measured data to obtain the required pose vector in the thermal deflection decoupled pose measurement method

Figure H. 4 :

 4 Figure H.4: Test setup

  Figure H.5: Hexapod with platform in [0 mm 0 mm 0 mm 0 • 0 • 0 • ] pose (top view)

  wise into an Aluminium part of fixed length and a Steel part of variable length. When the platform is moved from one pose to another, the Steel parts of legs change their lengths to achieve the new required lengths. When the thermal expansion of legs at the reference pose were measured, the corresponding thermal expansions of the Aluminium and the Steel parts could be determined. This could be done because the lengths and the thermal expansion coefficients of the two parts were known. The length of each leg and the corresponding length of the Steel part, with the hexapod in the measurement pose, were also known. The thermal expansion of the Steel part of each leg measured in reference pose was then appropriately scaled to estimate the thermal expansion of the Steel part of each leg in measurement pose. The thermal expansion of the Aluminium part was same for the reference and measurement poses as this part does not change its length. The total H.3. EXPERIMENTAL VALIDATION OF THE PROPOSED POSE MEASUREMENT METHOD 179 thermal expansion of each leg at the measurement pose was then obtained by adding the corresponding thermal expansions of the Steel and Aluminium parts.

Figure H. 6

 6 Figure H.6 shows the pose parameters of the measurement pose measured by using the conventional and proposed methods. Tx mes , Ty mes and Tz mes are the components of measured pose vector corresponding to translations along X, Y and Z axes of the hexapod, respectively. Rx mes , Ry mes and Rz mes are the components of measured pose vector corresponding to rotations about X, Y and Z axes of the hexapod, respectively. Figure H.7 shows the temperatures measured at different locations during this test.

Figure H. 6 :

 6 Figure H.6: Measured pose parameters using conventional and thermal deflection decoupled methods with the platform in zero pose

  in which CHAPTER 1. STATE OF THE ART AND BACKGROUND OF ROBOT ELASTOSTATIC CALIBRATION AND ITS OPTIMIZATION task variable scaling is dealt with differently. The task variable scaling matrix in this case,

G C , is given by

Table 1 .

 1 1: Existing criteria to select best robot calibration experiment design

	Criterion	Objective function
	A -optimality [Atkinson et al., 2007]	tr(Cov(c)) → min
	D -optimality [Atkinson et al., 2007]	det Cov(c)	-1 → max
	E -optimality [Atkinson et al., 2007]	min eig Cov(c)	-1

  33) Furthermore, some applications can demand for minimization of positioning error after compensation in terms of translational distance (Euclidean norm). Appendix E shows the relationship between the RMS value of possible Euclidean norms of DU,t ε X A , denoted as DU ρ, and Cov( DU,t ε X A ). In equation E.7 of Appendix E, it can be seen that DU Ω from E.6 is a scalar that is directly proportional to DU ρ and it is dependent on the measurement conditions (poses and forces/moments used for stiffness identification). Therefore, choosing appropriate poses and forces/moments for stiffness identification that minimizes DU Ω

will minimize the RMS value of possible Euclidean norms of DU,t ε X A . It must be noted here that this criterion is similar to O TP and O MV (see table 1.1). When positioning at multiple poses and/or multiple loads needs to be optimized, multiple values of DU Ω are obtained (one corresponding to each positioning case). A multi-objective optimization to minimize the multiple values of DU Ω can then be done. Alternately, some combination of the values of DU Ω can be minimized.

Table 3 .

 3 2: Standard deviations of errors in positions attained after compensations, in the Monte-Carlo simulations described in figure3.10

	Position label	Identification position (mm)	Standard deviation of FE B2 ε x X A (µm)	Standard deviation of FE B2 ε y X A (µm)
	P1	[0;250]	0.4	1.3
	P2	[0;0]	0.7	1.3
	P3	[0;-250]	1.9	1.3
	P4	[-250;-250]	1.9	2.5
	P5 Note: FE B2 ε x X A and FE [250;-250] B2 ε y X A are the X and Y coordinates of FE 1.9 B2 ε X A , respectively. 2.5

  equation 1.22. B3 H is a 2×2 identity matrix because the expected magnitudes of estimated parameters are same. B3 ∆X M , DU B3 ε ∆X M and FE B3 ε ∆X M are 6×1 vectors which together con-

	stitute the measured deflections. DU B3 ε ∆X M and FE B3 ε ∆X

M contain the errors in measured deflections due to deflection measurement uncertainty and errors in force applied during

Table 3 .

 3 3: Best position for stiffness identification as per different criteria discussed in section 3.5.1

	Best position as per	Position (mm)
	DUIR and FEIR criteria	[25.4;134.4]
	DUIR criterion only	[0;-100]
	FEIR criterion only	[300;300]

  Ry -s Rz .c Rx + c Rz .s Ry .s Rx s Rz .s Rx + c Rz .s Ry .c Rx s Rz .c Ry c Rz .c Rx + s Rz .s Ry .s Rx -c Rz .s Rx + s Rz .s Ry .c Rx

	-s Ry	c Ry .s Rx	c Ry .c Rx

  13) CHAPTER 4. EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEM Left subscript "H1" of the variables in the above equation specifies that the corresponding matrices/arrays of equation 2.6 have been formulated for this case. H1 G C is a 30 × 30 task variable scaling matrix 4 and is formulated as shown in equation 1.23. Matrix H1 H in this case is a 6 × 6 identity matrix because the stiffness parameters are expected to have the same order of magnitude. This is because every leg of the given hexapod has the same design and materials. H1 ∆X M , DU H1 ε ∆X M and FE H1 ε ∆X M are 30 × 1 vectors which together

	constitute the measured deflections. DU H1 ε ∆X M and FE H1 ε ∆X M contain the errors in mea-
	sured deflections due to deflection measurement uncertainty and errors in force/moment
	applied during stiffness identification, respectively. as H1

Table 4 .

 4 1 shows the approximate values of these stiffness parameters. r H1 Z and t H1 Z are 3 × 30 matrices. H1 ε ∆F M is a 30 × 1 array in which the first six elements repeat themselves five times due to presence of redundant measurements. It must be noted that equations 4.14 and 4.15 consider only one pose at which best positioning is desired. Since best possible positioning is desired throughout the workspace, equation that relates

Table 4 .

 4 1: Approximate values of stiffness parameters evaluated for the hexapod studied in section 4.3.1

	Approximate stiffness parameter values (N/µm)
	ap H1 k 1	ap H1 k 2	ap H1 k 3	ap H1 k 4	ap H1 k 5	ap H1 k 6
	10.51	12.11	12.59	10.86	11.57	11.03

Table 4 .

 4 2 lists the identified optimal stiffness parameter set.

	7. Using the pose measurement method outlined in appendix A.

Table 4 .

 4 4: Comparing measured pose errors of loaded hexapod (with and without compensation) with those of the hexapod without mounted load, for poses measured along its X-axis Axis ξ max,WC ξ max,WoC ξ RMS,WC ξ RMS,WoC WC and ξ RMS,WC are the maximum and RMS values of absolute differences between measured pose errors of hexapod without load and those of hexapod with load & with compensation, respectively.; ξ max,WoC and ξ RMS,WoC are the maximum and RMS values of absolute differences between measured pose errors of hexapod without load and those of hexapod with load & without compensation, respectively.

	Tx	2.25 µm	30.24 µm	1.53 µm	19.26 µm
	Ty	1.79 µm	2.95 µm	1.17 µm	2.09 µm
	Tz	1.29 µm	31.25 µm	0.75 µm	26.08 µm
	Rx	23.92 µrad	20.47 µrad	11.55 µrad	11.35 µrad
	Ry	13.65 µrad	90.27 µrad	9.44 µrad	49.49 µrad
	Rz	13.90 µrad	6.14 µrad	7.49 µrad	4.54 µrad
	ξ max,

Table 4 .

 4 5: Comparing measured pose errors of loaded hexapod (with and without compensation) with those of the hexapod without mounted load, for poses measured along its Y-axis Axis Ξ max,WC Ξ max,WoC Ξ RMS,WC Ξ RMS,WoC max,WC and Ξ RMS,WC are the maximum and RMS values of absolute differences between measured pose errors of hexapod without load and those of hexapod with load & with compensation, respectively.; Ξ max,WoC and Ξ RMS,WoC are the maximum and RMS values of absolute differences between measured pose errors of hexapod without load and those of hexapod with load & without compensation, respectively.

	Tx	5.48 µm	2.75 µm	3.28 µm	1.48 µm
	Ty	3.44 µm	25.74 µm	2.41 µm	17.77 µm
	Tz	3.55 µm	28.58 µm	1.99 µm	25.15 µm
	Rx	9.33 µrad	70.69 µrad	6.79 µrad	48.84 µrad
	Ry	17.54 µrad	23.18 µrad	12.86 µrad	13.66 µrad
	Rz	13.90 µrad	6.34 µrad	9.73 µrad	4.25 µrad

Ξ

  .19)Left subscript "H2" of the variables in the above equation specifies that the corresponding matrices/arrays of equation 2.6 have been formulated for this case. Matrix H2 G is a 6 × 6 identity matrix. This is because only one measurement is performed in each stiffness identification experiment in this case, resulting in zero variance of measured deflections. is the observation matrix which is a function of the identification pose used in each stiffness identification experiment and the forces/moments assumed to have been applied in them. Matrix H2 H is a 6 × 6 identity matrix since the stiffness parameters are expected to have the same order of magnitude. This is because every leg of the given hexapod has the same design and materials. H2 ∆X M , DU H2 ε ∆X M and FE H2 ε ∆X M are 6 × 1 vectors which 11. as ∆F M from equation 2.1.

	CHAPTER 4. EXPERIMENTAL VALIDATION OF DEVELOPED TECHNIQUES USING
	Matrix as together constitute the measured deflections. DU ELASTOSTATIC CALIBRATIONS OF A HEXAPOD POSITIONING SYSTEM H2 ε ∆X M and FE H2 ε ∆X M contain the errors in measured deflections due to deflection measurement uncertainty and errors in force/mo-ment applied during stiffness identification, respectively. Parameter set of equation 4.19 is estimated using least squares method. The errors in these estimated parameters were bound to be (mostly) a consequence of errors in forces/moments applied during stiffness identification. This is a consequence of the large error in force/moment applied for stiff-ness identification. In order to formulate a FEIR criterion for this case, matrix Z (from equation 2.51) must be formulated first. This is because this matrix relates the errors in forces/moments ap-plied during stiffness identification to the resultant errors in poses attained after compen-sation. Equation 2.51 can be rewritten for this case as FE H2 ε X A ≈ -H2 Z H2 A D as H2 A M + ap H2 D H2 ε ∆F M (4.20) In equation 4.20, H2 Z and ap H2 D are 6 × 6 matrices. ap H2 D was computed (as per equation H2 A M 128 2.43) using the approximate values of stiffness parameter values listed in table 4.1. H2 ε ∆F M

  FEH2 ε X A corresponding to platform's Tz axis. The minimization problem to find an identification pose for each of the target poses listed in table 4.3 can be written as

	4.4. EXPERIMENTAL STUDY 2: VALIDATION OF FEIR CRITERION	129
	used to solve the optimization problem and obtain the best identification poses. These
	optimization routines are very sensitive to the starting point supplied. To tackle this, a
	min H2 X M	H2 Z 34	(4.21)
	s.t. Workspace constraints
	where H2 X M is the identification pose. H2 Z 34 from equation 4.21 was minimized to
	obtain one identification pose for each target pose. MATLAB optimization toolbox was

  S 11 i S 22i S 12 i S 21 ) (mrad) -1 i S 22 (rad)i S 12 (m) i S 21 (rad) i S 11 (m) S 21 i S 11 i S 22i S 12 i S 21 (m -1 ) i S 11 i S 11 i S 22i S 12 i S 21 (rad -1 )

		(B.6)
	i S 22
	=	(B.7)

i S -1 = 1 ( i i S 11 i S 22i S 12 i S 21 (m -1 ) -i S 12 i S 11 i S 22i S 12 i S 21 (rad -1 ) -i

Table G .

 G 3: Error in deflection prediction(RMS values)

	∆Tx	2.7 µm
	∆Ty	3.1 µm
	∆Tz	2.2 µm
	∆Rx 6.4 µrad
	∆Ry 8.3 µrad
	∆Rz 8.8 µrad

This criterion can be formulated using the presented stiffness identification optimization framework. It minimizes the influence of errors in forces applied for stiffness identification on the compensation quality.

The applied force error varied as per the assumed standard deviations of force gauge uncertainty: 10 N along X and Y coordinates of the applied force.

This is the configuration of the hexapod in which all of the platform pose parameters are zero. In this configuration, all the legs are locked in the center of their strokes and have the same lengths.

This follows the assumption that each of the legs exhibit linear stiffness behavior.

This is the pose of the platform in which all its pose parameters (defining the 3 translations and 3 rotations) are zero. In this pose, all the legs are locked at the center of their strokes and have the same length.

This is the configuration of the hexapod in which all of the platform pose parameters are zero. In this configuration, all the legs have the same length.

Deflection measurement uncertainty and errors in forces/moments applied during stiffness identification experiment. 181

Acknowledgements

The work presented in this thesis would not have been possible without the support of many people, whom I would like to thank here.

ACKNOWLEDGEMENTS

I would like to thank Thierry Roux, Pierre Noiré and Olivier Lapierre from Symétrie for their continuous support in the past three years. They have been incredibly helpful and

Chapter Abstract

In this chapter, experimental studies on elastostatic calibration of a high-precision positioning hexapod are documented. These studies were aimed at experimentally validating the efficacies of elastostatic calibration of hexapods and the stiffness identification optimization framework presented in chapter 2. Results of these studies confirm the efficacies of both.

Conclusions

Here, general conclusions of this thesis are presented. The prime contributions of this thesis are highlighted first. This is followed by some recommendations for future work.

Validation simulations

Monte Carlo simulations were performed to validate the efficacy of the proposed FEIR criteria. Figure 3.10 shows the flowchart for this simulation. In these Monte Carlo simulations, stiffness identification experiments were simulated at different positions. These are: P1 = [0;250] mm, P2 = [0;0] mm, P3 = [0;-250] mm, P4 = [-250;-250] mm and P5 = [250;-250] mm. At each of these identification positions, stiffness identification experiments were simulated 10000 times and the error in applied force was different in each trial 10 . The estimated parameter set in each trial was used to simulate the compensation at the target position B2 X D with a force [0;-200] N at its end-effector. To validate the proposed FEIR criteria, the compensation quality achieved (in terms of compensation errors along the X and Y axes) using the different identification positions can be cross-checked with the indication of FEIR criteria (figures 3.8 and 3.9). Table 3.2 lists the standard deviations of errors in positions attained after performing the Monte-Carlo simulations described above. Comparing the indication of FEIR criteria

Contributions of this thesis

The contributions of this thesis can be summarized as follows:

1. A new approach to optimize stiffness identification for robot elastostatic calibration: This is a framework to formulate criteria to choose the best set of poses and forces/moments for stiffness identification of non-over-constrained robots in which compliance can be considered only in actuated joints. The parameters identified under experimental conditions (poses and forces/moments) suggested by these criteria ensure minimum impact of deflection measurement uncertainty and errors in forces/moments applied during stiffness identification on compensation quality.

Furthermore, it also maximizes positioning accuracy at desired pose(s), along desired axe(s) of the end-effector/platform and with desired forces/moments on the end-effector/platform. Validation studies documented in this thesis confirm the efficacy of this framework. This stiffness identification optimization framework was developed to enable best possible compensation of positioning errors due to com-

133

CONCLUSIONS

pliance of robots in predefined applications. This aspect is very important in highprecision applications in which the robot's positioning specifications are predefined.

Elastostatic calibration of hexapod:

Elastostatic calibration of a high-precision hexapod positioning system was accomplished. The stiffness of this hexapod was modelled using a lumped stiffness model. The parameters of this model were identified at best poses and forces/moments as per the criteria formulated using the proposed stiffness identification optimization framework. The identified parameters were then used to compensate for the pose errors (due to compliance) of the loaded hexapod. Results showed that the loaded hexapod with compensation using identified stiffness parameters can achieve the level of accuracy of the unloaded hexapod. Elastostatic calibration of hexapod is necessary to facilitate high-accuracy 6-DOF positioning when a heavy payload is mounted on a high-precision positioning hexapod.

Thermal deflection decoupled 6-DOF pose measurement method for hexapods:

A method was developed to eliminate the influence of thermal deflection of a hexapod on the measured 6-DOF pose of its platform. This was validated experimentally using pose measurements of a high-precision hexapod using a CMM. This method is necessary to ensure that the thermal deflections of the hexapod do not impact pose measurement of hexapods. This was crucial in some tests that were performed to validate methods developed in this thesis. Furthermore, this method is also beneficial for robot geometric calibration which requires pose measurements at constant temperature, which is difficult and expensive to achieve.

Recommendations for future work

The work documented in this thesis solves many problems. Additionally, it also opens up new research directions that can be explored. Some of them are:

1. Using presented parameter identification optimization method for optimizing robot geometric calibration: In this thesis, a framework for formulating criteria to choose best measurement conditions (poses and forces/moments) for stiffness iden-

APPENDIX

A A method to measure the 6-DOF pose of hexapod's platform

Here, a method is presented which can be used to perform 6-DOF pose measurements of hexapod's platform using a coordinate measurement machine (CMM). Section A.1 provides the details of the measurement setup required for this measurement and section A.2 presents the measurement method.

A.1 Measurement setup

A.2 Measurement method

Pose measurements are always performed by measuring several points using a measurement system. Here, this measurement system is the CMM. Let M be the coordinate frame of the CMM. A coordinate frame fixed to the platform defines the pose of the plat-

C

An example application requiring high positioning performance along selected axes of the robot's platform Here, a simulation study is presented which was performed to assess the influence of error in applied force/moment during stiffness identification on compensation accuracy, for a high-precision hexapod positioning system. Section D.1 presents the details of this simulation study and section D.2 presents the results of this study.

D.1 Simulation description

This simulation study consists of a Monte-Carlo simulation in which elastostatic calibrations of a high-precision hexapod positioning system (shown in figure D.1) was simulated many times. These simulated elastostatic calibrations were assumed to be influenced by realistic errors in applied forces/moments during stiffness identification. The deflection measurement errors were assumed to be negligible in these simulations. These APPENDIX D.

elastostatic calibrations were simulated 10,000 times. Tx, Ty and Tz denote the translational coordinates of this hexapod's platform and this hexapod possesses a repeatability of 0.5 µm along these coordinates. Rx, Ry and Rz denote the rotational coordinates of this hexapod's platform and this hexapod possesses a repeatability of 2.5 µrad along these coordinates. These simulated realistic stiffness identification experiments are performed at the zero pose of the hexapod. Also, the desired pose to reach was the zero pose in the simulated positioning experiments. The load was assumed to be applied by placing a mass of 120 kg on the platform of this hexapod (as in figure D.1). This is similar to the manner in which stiffness identification is performed in the experimental study performed in chapter 4. The error in applied force/moment during identification experiments is due to the difference between the assumed position of the center of mass (CoM) and its real value. Realistic value of CoM position errors are assumed. These CoM position errors are assumed:

(a) to be distributed normally (Gaussian distribution), and (b) to have maximum values of approximately 3mm along X-, Y-and Z-coordinates (99.7 % confidence interval). Also, the pose measurement system is assumed to be perfect, i.e., deflection measurement errors are assumed to be zero. The resulting compensation errors are, therefore, a consequence of the errors in applied forces/moments (during stiffness identification) only. tual spring in each leg). Stiffness parameters were then estimated using the said stiffness model and the simulated measured deflections (with errors due to applied force/moment errors). Subsequently, command pose 1 was generated such that the platform reaches the desired target pose (zero pose). The pose attained after compensation using estimated stiffness parameters was then computed and subsequently, the difference between the at-1. Command pose is the pose entered in the control interface of a robot's controller that doesn't have the compliance error model embedded in it. When the robot's platform is commanded to reach this command pose, the robot's platform reaches (close to) the desired target pose due to the platform's deflection under the given load. ures show the corresponding 99.7% confidence intervals. As can be seen from the output probability functions, the 99.7% confidence interval boundaries of compensation errors along Rx and Ry axes are considerably more than the repeatability of this hexapod along the respective axes (2.5 µrad). Since, the goal is to reach the level of repeatability of the hexapod, these compensation errors are unacceptable. Hence, it can be seen in this example that the influence of errors in applied forces/moments during stiffness identification on compensation accuracy is indeed not negligible in a realistic case. Here, the derivation of relationship between Cov( DU,t ε X A ) and the RMS value of pos- sible Euclidean norms of DU,t ε X A is shown. Firstly, using equation 2.27, the following can be written:

In equation E.1, superscript t indicates that they are the translational components (or elements corresponding to translational components) of the respective array/matrix corresponding to them 1 . The following can then be written:

Here, E(e) denotes expectation of e. In equation E.4, DU ρ is the RMS value of possible Euclidean norms of translational components of DU ε X A . This equation can be further expanded to get

Here, tr(W) denotes trace of matrix W. However, E DU ε c DU ε T c = Cov( DU ε c ).

Therefore, using equations 2.13 and E.5, the following can be written:

Comparing equations E.6 and 2.31, the following relationship can be obtained:

Equation E.7 describes the relationship between Cov( DU,t ε X A ) and the RMS value of possible Euclidean norms of DU,t ε X A , DU ρ.

APPENDIX F.

Here, tr(W) denotes trace of matrix W.

Consequently, equation F.2 can be written as

Finally equations 2.56 and F.3 give the following expression:

Equation F.4 describes the relationship between Cov( FE,t ε X A ) and the RMS value of possible Euclidean norms of FE,t ε X A , FE ρ.

G

Preliminary results of stiffness identification of a hexapod using the stiffness model presented in chapter 4

Here, the details and results of a preliminary study performed to validate the efficacy of stiffness model for hexapods, presented in section 4.2 of chapter 4, is presented. In this study, stiffness identification of a hexapod was performed using the said stiffness model.

Following this, the estimated stiffness parameters of this stiffness model were used to predict the deflections of the loaded hexapod. These predictions were compared with the measured deflections to study the efficacy of this stiffness model and the estimated parameters. This study was performed on another high-precision positioning hexapod 1 from Symétrie and this is shown in figures G.1 and G.2. The setup available for performing pose measurements and for applying force/moment on the platform were same as described in section 4.1 of chapter 4.

Section G.1 presents the details and results of stiffness identification performed on this hexapod. This is followed by details and results of the study performed to validate the efficacy of the stiffness model and its estimated parameters in section G.2.

G.2 Validation of stiffness model efficacy

To validate the efficacy of the stiffness model and the estimated parameters, platform deflection measurements were performed at different poses along the X and Y axes of the hexapod's platform. The poses at which these pose measurements were performed are 3. Pose measurement method presented in appendix A was used in this case.

G.2. VALIDATION OF STIFFNESS MODEL EFFICACY

167 listed in table G.2. The load applied during these deflection measurements was same as the one used for for stiffness identification. The method presented in appendix A for 6-DOF pose measurement of hexapods is susceptible to thermal deflections of the hexapod. In this appendix, a new method is presented which is not susceptible to thermal deflections of the hexapod. Section H.1 presents a deeper understanding of the aforementioned problem with the conventional pose measurement method (like the one on appendix A). Section H.2 presents the thermal deflection decoupled pose measurement method. This is followed by details and results of an experimental study performed to validate the presented method in section H.3.

H.1 Conventional pose measurement method and its drawback

Pose measurements are always performed by measuring points using a measurement system which has a coordinate frame (M) attached to it. O t 1 . Let the corresponding measured pose be denoted as

. 1. This is the pose of the platform in which all the pose parameters are zero. In this pose, all the legs have the same length.

2. O is measured with respect to M by measuring different points on the platform of the hexapod. Check appendix A for more description where a similar measurement is described.

H.2. THERMAL DEFLECTION DECOUPLED POSE MEASUREMENT METHOD 173

From the description presented above, it can be easily seen that the measured transformation would have been different if the legs of the hexapod would have had the temperature set t 1 . The platform frame in this case (S t 1 1 ) would have a different pose vector

. This is due to the thermal deflection of the legs of the hexapod with the change in their temperatures from set t 1 to set t 2 . Temperature change also affects other dimensions of the hexapod. However, for most hexapods, the thermal deflection of legs is much higher than that of the other parts because: (a) the legs generally have larger dimensions (length) as compared to the other parts, and (b) driving motors are mounted on/near the legs which heat the legs more than the other parts.

To understand the drawback of this measurement method, consider a case in which pose measurements need to be performed for a long duration of time 3 . When conventional pose measurement is used in this case, different platform poses will be measured with legs at different temperatures. This can happen due to heating supplied by motors or the surrounding air. Consequently, different measured poses have the influence of different magnitudes of thermal deflections of legs. This can be problematic, for example, in the case of pose measurements performed in the experimental studies in sections 4.3.2 and 4.4.2. In these cases, the accuracy of positioning of the unloaded robot and the loaded robot with error compensation are compared to evaluate the efficiency of compensation.

These two measurements are made with legs possibly at different temperatures. The thermal deflections in the legs due to the aforementioned temperature difference can cause considerable thermal deflections, thereby making it difficult to evaluate the efficiency of compensation. Therefore, a new pose measurement method is necessary that is insusceptible to thermal deflections of the hexapod.

H.2 Thermal deflection decoupled pose measurement method

Figure H.2 illustrates the proposed method to measure the 6-DOF pose of the platform frame with the platform at an arbitrary pose S 1 , with respect to frame O. In this method, 3. Like in pose measurements needed in validation studies described in sections 4.3.2 and 4.4.2 METHOD FOR HEXAPODS frame O t 1 must be measured first (with the legs having temperature set t 1 ). Immediately after this, the platform must be moved to a reference pose R. The transformation between the platform frame R and O t 1 must be measured quickly such that the measurement hap- pens with the legs having the temperature set t 1 . Let this measured frame be called R t 1 .

The platform can then be moved to any arbitrary pose (frame S 1 ). The transformation between this frame and O t 1 can be measured with the legs having a temperature set t 2 . This measured frame is S t 2

1 . An additional measurement, transformation between frames R and O t 1 , must be performed quickly before/after measuring S t 2 1 . This measurement must be performed with the hexapod's legs having the temperature set t 2 (measured coordinate frame: R t 2 ). using the measurement data obtained from the measurements outlined above. The measurement procedure described can be used to obtain three transformations:

H.3. EXPERIMENTAL VALIDATION OF THE PROPOSED POSE MEASUREMENT METHOD 175

Consequently, the corresponding 6-DOF pose vectors,

can be obtained. These pose vectors can be used to get the corresponding leg lengths of the hexapod by using the inverse geometric model (IGM) of the hexapod. q R t 1 , q R t 2 and q S t 2 1 are the arrays containing the leg lengths of the hexapod corresponding to pose vectors

, respectively. q R t 1 and q R t 2 can then be used to compute the thermal deflection the hexapod's legs corresponding to temperature change from set t 1 to set t 2 , with the platform at reference pose R. Let the array containing these leg deflections be called ∆q R t 1 -R t 2

and let ∆q i R t 1 -R t 2 be the deflection of the i th leg. The thermal deflection due to temperature change of legs from set t 1 to set t 2 of the i th leg of the hexapod at the arbitrary pose

, can then be estimated easily. The task here is to find the thermal deflection of the legs with lengths q S 1 t 2 , when the temperature of these legs change from set t 1 to set t 2 , when the thermal deflection of the same legs with lengths q R t 1 are known. The method to perform this computation must respect the dimensions and material properties of the components of the leg assembly. ∆q S t 1 1 -S t 2 1 can then be subtracted from q S 1 t 2 to obtain q S 1 t 1 . q S 1 t 1 is the array containing the leg lengths when the platform is at the arbitrary pose S 1 and the legs have temperature set t 1 . Finally, the necessary pose vector X O t 1 S t 1 1 can be obtained by using forward geometric model (FGM) of the hexapod corresponding to q S 1 t 1 . When multiple platform poses shall be measured using this method while leg temperatures change, the measured poses will not have the influence of different magnitudes of thermal deflections of legs. Hence, the drawback of the conventional method can be overcome using this method.

H.3 Experimental validation of the proposed pose measurement method

This section presents the details of an experimental study performed to compare the conventional and proposed methods for 6-DOF pose measurement of hexapods. The legs of the hexapod were heated during this experiment to control and slightly exaggerate heating in legs.This was done to clearly show the advantage of the proposed pose measurement method over the conventional method. This thesis presented a framework for formulating criteria for choosing the best set of poses and forces/moments for stiffness identification. These formulated criteria minimize the impact of errors influencing stiffness identification 1 on the poses attained after positioning compensation. As was mentioned in section 1.3.2, the parametric calibration framework used for robot elastostatic calibration is similar to the one used for robot geometric calibration. Furthermore, one error source impacts parameter identification in both these robot calibrations: uncertainty of deflection measurements performed. As a result, the framework for formulating criterion for minimizing the influence of this error on elastostatic calibration performance, the DUIR criterion, can also be used for minimizing the influence of this error on the performance of robot geometric calibration. The manner of formulating DUIR criterion for optimizing geometric parameter identification in robot geometric calibration is discussed below.

APPENDIX I.

Equation 1.17 can be rewritten for the case of geometric calibration as W p = ∆X (I.1)

In equation I.1, A and c from equation 1.17 are replaced by W and p, respectively. W and p are the equivalents of A and c in the case of geometric calibration, respectively. W is a function of actuator positions and the assumed geometric parameter set while p contains the difference between the actual and assumed geometric parameters' values [START_REF] Sun | Observability index selection for robot calibration[END_REF]. ∆X in this case is the difference between the measured and expected poses. Let all variables have the same names in the case of geometric calibration, except for the ones related to A and c (A M , A D ,.. etc replaced by W M , W D ,.. etc and c, DU ε c ,...etc replaced by p, DU ε p ,...etc). Equation 2.30, which is used for formulating DUIR criterion in the case of robot elastostatic calibration, can be written for the case of geometric calibration as

Here, U g is a function of poses used for geometric parameter identification. It controls the propagation of uncertainty in measured pose deflections to uncertainty of resultant errors in poses attained after compensation. Equation I.2 can be used in ways similar to the ways described in section 2.3.2 to find best poses for geometric parameter identification.

List of publications

Abstract

Hexapods are increasingly being used for high-precision 6-DOF positioning applications such as for positioning mirrors in telescopes and for positioning samples in synchrotrons. These robots are designed and controlled to be very repeatable and accurate.

However, structural compliance of these positioning systems limits their positioning accuracy. As accuracy requirements become more stringent in emerging applications, compensating for inaccuracy due to structural compliance becomes necessary.

In this regard, firstly, a method for elastostatic calibration of hexapods is presented. This method uses a lumped stiffness parameter model to parametrize the relationship between the platform deflections and the force/moment applied on it. These parameters can be estimated using deflection measurements performed using known forces/moments applied on the platform. The estimated parameters can then be used to predict and correct hexapod's positioning errors due to compliance.

Secondly, a new approach is presented to optimize stiffness identification for robot elastostatic calibration. In this, a framework is proposed to formulate criteria to choose best set of poses and forces for stiffness identification experiment. The parameters identified under experimental conditions (poses and forces) suggested by these criteria ensure minimum impact of errors influencing stiffness identification (uncertainty of deflection measurements and errors in forces applied) on compensation quality. Additionally, it also maximizes accuracy after compensation at desired pose(s), along desired axe(s) of the platform and with desired forces/moments on the platform. This stiffness identification optimization framework ensures best compensation for positioning errors due to compliance as per the positioning requirements of the application at hand.

Lastly, a method is presented to eliminate the influence of thermal deflection of a hexapod on the measured 6-DOF pose of its platform. This method is necessary when thermal deflections of the hexapod are large enough to impact results of a study, which was the case with some tests performed to validate methods developed in this thesis.

The efficacy of presented methods have been validated by means of simulation studies on a bipod and experimental studies on a high-precision hexapod positioning system.

Keywords