Franck Luthon
email: franck.luthon@univ-pau.fr

Mihai-Dan Steriu
email: dansteriu@yahoo.com

OPEN ARCHITECTURE FOR SIGNAL PROCESSING LAB DISTANCE LEARNING

Keywords: Distant learning, Lab work, LabVIEW, EEE engineering, Cybernetics education, Signal processing Communication Baseline Web Pages LabVIEW Programs CGI Communication Baseline

  L'archive ouverte pluridisciplinaire

INTRODUCTION

This work describes the design and implementation of a generic architecture (software and hardware) dedicated to electronics and signal processing lab distance learning. The laboratory is accessible via TCP/IP protocol network.

All technology resources in the nature (energy, materials and information) are taken into account in the learning ojectives of engineering education by remote control or monitoring (observation) [START_REF] Feisel | The role of the laboratory in undergraduate engineering education[END_REF].

Fig. 1.

Client-server open architecture : the client uploads experimental parameters and downloads results. The server executes client queries, manages user accesses and experiment lists for an optimal use of the hardware system The student at home can not only do simulations of signals and virtual instrumentation as with J-DSP [START_REF] Spanias | J-DSP Java Digital Signal Processing and On-line Laboratory[END_REF] (interactive online simulations for signal processing), but also do real (practical) measurements in order to characterize physical circuits and systems remotely.

Five types of client plug-ins are available for:

• experiments,

• simulations,

• web pages (start web browser with URL address),

• demos (for illustration during on-line courses) and

• others (e.g. help for report editing and production of deliverables, advanced representation and visualisation of data, quizzes).

As regards server plug-in, a template is also available that helps any authorized tutor to plug in easily a new experiment inside the menu (Fig. 2).

Fig. 2. Template plug-in

Applications are numerous: electronics lab [START_REF] Luthon | Instrumentation programmée pour TP d'électronique distant[END_REF], data acquisition and signal processing lab, image processing lab, control system and automation, supervision of controllable / programmable logic devices (PLD), standard communication interfaces (serial, parallel, USB).

SOFTWARE ARCHITECTURE OVERVIEW

Server application description

The main part is the G web server that works with different programs like in Fig. 3:

Fig. 3. Server application structure

The simplified actions of the server are presented in Fig. 4.

We use a G web server (HTTP server) with Internet Developper Toolkit and communicate through CGI.

The management of users access (multi-session, multiuser, multi-experiment) is handled by the control center on the server side, that is located near the real devices, systems and power sources. The server manages the user query list so that the hardware system is used in an optimal way. The main functions of the control center are (Fig. 5): • start remote panel server,

• start watchdog,

• install the experiment configuration,

• manage users (login, password),

• manage the list of pluggable experiments,

• visualization of logs, history, statistics of use,

• manage experiment queue,

• remote administration.

Client application description

A CDROM containing all the pedagogical content and the runtime version of the application is furnished to each remote client (student at home). The client application ("Virtual Laboratory", see Fig. 6) consists in an HMI for enabling connection and selection of learning mode (simulation vs. real experiment, off-line vs. online) and parameter setting.

The activity diagram of the client application is shown in Fig. 7.

The client sends to the server (uploads) the choices about the experiment, with configuration parameters. After execution of the experiment by the server, the remote client downloads the results of measurements. Then the student does analysis and interpretation of experimental data in connection with the theory and models in agreement with learning objectives: plot curves, validate theoretical models, and generation of the report that will be evaluated by the teacher.

Server plug-ins

To make easier the programming of the server plug-ins, we propose a structure which permits to work in two situations: Local (the settings for experiments are made on the server) and Client (the settings come from a remote client, Fig. 8). Also it is possible to obtain the results from a real experiment (switch on Manip) or from a simulation module (switch on Sim). Indeed it is easier to program and verify a server plugin working without a client connection. The current working situation (request by client → real experiment on server → results to client) is presented in Fig. 9.

Fig. 9. Server plug-in diagram in the current working situation

Also, it is possible to answer to a client's request with the results of a simulation, using precise mathematical models of the devices. It is very useful when there is maintenance running on the equipments.

PROTOTYPE IMPLEMENTATION

The prototype is based on LabVIEW which is a graphical programming environment for measurement and testing. It offers easy to use programming by wiring graphical objects togheter and human-friendly interfaces, yielding fast prototyping cycle in the development of applications [START_REF] Travis | Distance learning solutions guide[END_REF].

The application runs on PC with Windows system, Internet access, web browser and local GPIB bus (IEEE 488 standard for instrumentation). The measurement devices (waveform generator, programmable power supply, oscillscope, multimeter, data acquisition card) and the switching matrices (multiplexing for connection configuration and power on/off) are controlled and run remotely by the student connected through Internet to the server that is located at our institute (Fig. 10).

APPLICATION EXAMPLE AND RESULTS

An operationnal multi-user, multi-experiment prototype was implemented and tested successfully: 5 to 30 simultaneous connections, plug-ins validated for various experiments among which data acquistion with signal filtering, and automatic plot of Bode curves (gain and phase) to characterize the transfer function of a linear system (electronic circuit, see diagram schematics on Fig. 11).

The measurement setup for Bode plots is shown in Fig. 12. The estimation of the phase requires the use of an oscillscope for the measurement of the time delay between the input and output signals. Note that such a device is much more complicated to program than a simple multimeter. That is one reason why we choosed to implement this experiment, in addition to the fact that the study of the transfer function of 2nd order After connection, the student chooses the experiment (Bode plot) and the parameter settings (minimal and maximal frequencies f min , f max , input voltage V in , number of samples N s , display mode (linear, dB, decade). The watchdog allows to inform the client of data ready available for download (Fig. 13). In addition to our two current experiments (Bode curve plotting and Data Acqusition and Filtering) taht are already plugged in the prototype, we intend to plug new experiments that we have already developped separately [START_REF] Luthon | Instrumentation programmée pour TP d'électronique distant[END_REF]:

• charaterization of an audio LF power amplifier (Fig. 14), Presently, the prototype is validated from a technical point of view (remote real experimentation and network communication) but we still have to calibrate the learning objectives and to evaluate the prototype with many more students (to assess the pros and cons of real experimentation vs. simple simulation as regards learning goals).

For experiments requiring real-time (for example live video showing in real-time the physical phenomenon in motion or action), we are currently investigating what choices have to be changed in the proposed architecture (web control by CGI vs. ActiveX vs. Java ; DataSocket vs. VI Server vs. TCP VIs).

Future work will also concern the evaluation process (student evaluation form, quizzes, constructivist approach [START_REF] Ferreira | The MARVEL EU project: A social constructivist approach to remote experimentation[END_REF] etc), and the use of the new high speed standard HS-488. Comparison and better positioning with respect to other interesting methods and works [START_REF] Müller | Online labs and the MARVEL experience[END_REF][START_REF] Garcia-Zubia | Towards a canonical software architecture for multi-device weblabs[END_REF][START_REF] Garcia-Zubia | Evolving towards better architectures for remote laboratories: a practical case[END_REF][START_REF] Bouhadada | SAPIENCE: A simulator of interactive pedagogical activities for distance learning environment[END_REF][START_REF] Masseux | Modélisation d'une interaction didactique distante individuelle synchrone (IDIS)[END_REF][START_REF] Bonivento | A web-based laboratory for control engineering education[END_REF][START_REF] Chiculita | A web remote control laboratory[END_REF] is also under investigation.

Fig. 4 .

 4 Fig. 4. Flowchart of server actions

Fig. 5 .

 5 Fig. 5. Control center on server side: a) Left: webserver start ; Middle: watchdog start ; Right and bottom: configuration and management ; b) HTTP server running

Fig. 6 .Fig. 7 .

 67 Fig. 6. Remote Laboratory Client Application

Fig. 8 .

 8 Fig. 8. Server plug-in diagram in verification phase of the experiment: Local=False (i.e. remote client) and Manip=True (i.e. real experiment, not a simulation)

Fig. 10 .Fig. 11 .Fig. 12 .

 101112 Fig. 10. Experimental setup in server room: server PC on the left, GPIB instruments on top and connection board with DUT (device under test) on the table

Fig. 13 .

 13 Fig. 13. Download of results with display on the client side: a) Bode curve of the Sallen-Key 2nd order filter: Gain in dB; b) Plot of the Phase variation of the filter

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Characterization of a LF audio power amplifier: a) Experimental setup ; b) Graphical user interface that displays the power and efficiency curves

ACKNOWLEGMENT

The authors thank Thierry Nodenot and Philippe Lopistéguy from University of Pau, Bayonne, France, for fruitful discussion and references.

This work was supported financially by the University of Pau, with an invited professor position for three months in Bayonne, France.


domain: math math math.math-ho shs

text:
Mariam Haspekian
email: mariam.haspekian@u-paris.fr

Cédric Fluckiger
email: cedric.fluckiger@univ-lille.fr

Analyzing teachers' views on the integration of computer science into mathematics teaching

Keywords: teaching practices, computer science, instrumental distance, didactical landmark

This contribution addresses the issue of understanding teachers' difficulties in integrating computer science and programming software into their practices. To do so, we previously used the theoretical concepts of distance/landmarks to analyze the development of such practices and show the importance of the teacher's personal component. We use this lens here to analyze some interviews of teachers' views on this integration. The results show that the barriers of ICT integration situate mostly on the changes that ICT introduce at cognitive and mediative levels. Conversely, institutional, and social components rather seem to play in favor of integration.

INTRODUCTION

The last changes in mathematics curricula at French primary and secondary levels strongly bring in computer science into mathematics teaching, in association with several software. Our research question is to understand teacher's difficulties in integrating computer science and programming software into their practices. To do so, our previous work analyzed the practices of teachers, first time using computer-related technologies as Bee-Bot robots, or Scratch [START_REF] Haspekian | Informatique, Scratch et robots: De nouvelles pratiques enseignantes en mathématiques ? STICEF[END_REF]. The results concerned the didactic references [START_REF] Haspekian | Computer science in mathematics new curricula at primary school: New tools, new teaching practices[END_REF] that the teachers built in these new situations in terms of both knowledge and tool use: these teachers built some sessions that allowed them to remain not too distant from what they already knew, while, at the same time, they gain new reference marks [START_REF] Haspekian | Informatique, Scratch et robots: De nouvelles pratiques enseignantes en mathématiques ? STICEF[END_REF]. These ideas of distance/ landmarks, together with other theoretical concepts of the Instrumental Approach (detailed below), allowed to study the instrumental geneses of the teachers, both personal (to master Scratch or robots) and professional (to teach mathematical knowledge via these instruments). The results lead to reflections on teaching practices: how do teachers develop new didactic references, and how do they use previous situations to manage new ones, as long as these are not too distant. In this paper, using these same frames, we are interested this time in the teachers' opinions and representations about these tools and this new teaching. The following sections present the theoretical frames we use, the first results, and a discussion of these.

THEORETICAL FRAMES FOR THE STUDY

Our work integrates 2 frames briefly describe below.

The five components of the Double Approach of Robert & Rogalski (2002)

This frame approaches the teachers' activity through the ideas of constraints and room for action. Indeed, the teachers have some liberty in their didactic choices, as learning goals of each session, tasks given to students (these are part of the cognitive component), or still scenarios, class organization, etc. (part of the mediative component). These choices are very personal, depending on the teacher's history, vision of what is teaching, learning... (this is the personal component). Yet, these choices are made within the limits of some constraints: official instructions, curricula, time duration of the lessons, profile of the students… (the institutional and social components).

Our work focusses on the personal component and considers the four others from the teacher's point of view. Indeed, our previous research used this component to explain teachers' practices, considering that it contains knowledge on cognitive, mediative, social and institutional ones themselves [START_REF] Haspekian | Computer science in mathematics new curricula at primary school: New tools, new teaching practices[END_REF]. The daily cognitive/ mediative choices are imprinted of various didactic knowledge, which pre-exist in this personal component. This diversified knowledge, covering these fours dimensions, acts as didactic landmarks (ibid.) guiding the activity. These are needed to perform new cognitive and mediative choices. The next section presents this tension distance/ landmarks.

The instrumental genesis and instrumental distance of the Instrumental Approach

Since the activity is instrumented by new technological tools, initially, our work is also framed by the Instrumental Approach in didactics [START_REF] Artigue | Learning mathematics in a cas environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF]Guin et al., 2005;[START_REF] Lagrange | L'intégration d'instruments informatiques dans l'enseignement: Une approche par les techniques[END_REF][START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF] with its key issue of instrumental genesis [START_REF] Verillon | Cognition and artifacts: A contribution to the study of though in relation to instrumented activity[END_REF]. This is the psychological process, through which the subject uses and transforms an artefact so that it becomes an instrument for the activity and, through which, conversely, using the artefact affects the subject's conceptualizations. Applying this concept to the teacher, we divided it into two processes: a professional instrumental genesis (for the didactic activity of the teacher) and a personal one (for the mathematical activity of the teacher) [START_REF] Haspekian | Évolution des usages du tableur[END_REF].

The Instrumental Approach also focusses the attention on how tools affect mathematical concepts. The idea of instrumental distance [START_REF] Haspekian | An "Instrumental Approach" to study the integration of a computer tool into mathematics teaching: The case of spreadsheets[END_REF] has been initially introduced to study this impact of a new artefact on concepts and conceptualizations. For example, the spreadsheet introduces a distance regarding the usual concept of variable in mathematics.

On the teacher's side, this distance affects the teacher's didactic landmarks. For example, teaching algebra with a spreadsheet causes a loss of these marks. Thus, the instrumental distance has been then extended in a more general idea of distance to practices and to its usual didactic landmarks [START_REF] Haspekian | Computer science in mathematics new curricula at primary school: New tools, new teaching practices[END_REF], that is the impact and deviation from the practices produced by the introduction of a new tool, a new domain, or still a new discipline as computer science into mathematics teaching.

A theoretical grid to analyze the didactic distance/ landmarks

To theoretically structure the factors that contribute to didactic distance and landmarks, we used the 5 components described above, isolating more specifically in the personal one the epistemology and representations of teachers [START_REF] Haspekian | Computer science in mathematics new curricula at primary school: New tools, new teaching practices[END_REF]. These elements may play in favor of integration, or in disadvantage because of the distance that they create. For example, teaching algebra with spreadsheet presents a distance at the cognitive, personal, and institutional levels.

The possible new practices that may develop result from an equilibrium between elements of these components that legitimate/ support the integration of the newness, and elements that create some tensions and distance to the pre-existent didactic marks (at institutional, social, cognitive, and mediative levels). The Table 1 shows this reading grid for teaching practices in innovative situations.

Table 1: A grid for analyzing teaching practices in innovative situations

Applying this categorization for analyzing emergent teaching practices with Scratch and robots, we showed that the integration of these novelties depends on two conditions on each one of the 5 components I, S, C, M, P -a condition on legitimacy and a condition on the didactic landmarks:

1. A certain legitimacy must be perceived/conferred by the teacher to this object at the institutional (I, S), didactic (C, M) and personal (E and R) levels 2. This legitimacy alone is not enough, the "newness" should not create (on each components I, C or M) a too distant situation to the teacher's former landmarks, i.e. that the integration of new can be done on landmarks close to those already acquired.

The Table 2 details the theoretical elements in the different dimensions. On the Institutional and Social dimensions: legitimacy is given by curricula, inspection, assessments, schoolbook; and by societal developments. On counterpart, this requires the creation of new landmarks, even if curricula give some.

On the Didactic (cognitive and mediative) dimension: research studies, professional training, and literature, legitimize the contributions and benefits to cognitive and mediative levels, but, a priori, for an ordinary teacher, the newness introduces a loss of cognitive and mediative marks. Instrumental professional geneses are to develop in terms of orchestration [START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations[END_REF], particularly to manage students' instrumental geneses.

On the Personal dimension: the epistemology of the teacher and his/ her representations may legitimate/ foster or hinder the integration, according to teachers. It depends on the person, her very knowledge of the disciplines at stake, her epistemology of the mathematics to be taught, her representation in general on teaching and learning (not specifically disciplinary).

Finally, the distance to usual practices is problematic if too few landmarks remain (I, C, M) (negative factors). This loss is counterbalanced by the perceived/ conferred legitimacies at the levels (S, I, C, M) (positive factors), and by the personal component, particularly the teachers' representation and epistemology in the concerned domain (P: R/ E) (factor positive or negative according to the person).

In IeCare project, we use these theoretical concepts to analyze interviews of primary and secondary school teachers. The collecting of these data is still in progress, we share, here, our first analyzes: the barriers of ICT integration situate mostly on the changes that ICT introduce at cognitive and mediative levels. Conversely, institutional, and social components rather seem to favor integration.

METHODOLOGY

The data come from the ongoing French National Research Agency project IeCARE, where we made some questionnaires and interviewed primary school teachers' feelings, beliefs, or knowledge about these new curricula. In these data, we try to understand the determinants of the activity of the teacher regarding computer science and the integration of new technological artefacts: What contents are identified (mathematics? computer science? instrumental only?)? Through which functionalities and modalities? What goals are assigned to these contents? How will students construct these? What knowledge do they have to integrate these new goals? What resources do they use? What motivates the teachers deep down? Where are their main difficulties?

We used a methodology of semi-directives interviews (De Ketele & Roegiers, 1996, p. 172), that we transcribed then in series of speaking turns. Each round of speech is numbered and coded in the following way: "3.TDP12" means the 12 th round of speech, in the interview of Teacher 3.

The theoretical grid above provided a lens of reading through which to analyze the transcripts. The analyze consisted of reading, highlighting the salient facts in different colors according to the associated dimension, summarizing them superficially, then in a more exhaustive way in the grid.

The collection, transcription and analyses of these interviews are still ongoing. Up to now, we analyzed three of them and obtained some interesting results: each of the interviews presents on one side, some elements that play in favor of integrating computer science in mathematics teaching, on the other side, some deeper elements revealing resistances and playing against this integration. These elements are identified according to the dimension they belong to cognitive, mediative, institutional, social or personal. We thus obtained a picture of how each dimension of the profession weighs. The picture is discussed in the last section; we summarize here the salient facts for the three interviews.

TEACHERS' OPINIONS ABOUT COMPUTER SCIENCE AND ICT INTEGRATION WITHIN MATHS CURRICULA Interview 1 -Salient facts:

-An opposition between openly stated opinions that refers mainly to institutional and social components, and a deeper opinion rather conflicting with the first one. -A cognitive component very little present -A personal component opposed to what is openly expressed Indeed, Teacher 1 clearly expresses a positive attitude at the beginning of the interview:

-She is "for" the integration of computer science in mathematics teaching at primary school levels.

The arguments that she gives are situated here at institutional, mediative and social levels. There is not any "pro" argument at cognitive level (except very superficially, and not for mathematics, but to evoke the benefits of some word processing software for the teaching and learning of French language, thanks to typing texts and formatting).

-She uses "politically correct" arguments (institutional) when she needs to explain why she does not use these software despite the favorable position she stated: lack of training, lack of equipment, lack of quality connection, old school… Resorting to institutional and social components (she is in a very particular environment), she admits reticently her "lack of interest"

Cognitive component: in her discourse, the cognitive component is not very present. Basically, she has the following representations: "computer science" means "hardware, computer, tools", and "teaching computer science" means essentially having an interactive video projector and teaching students to use some tools" (essentially Office tools). Let us note that the word "programming" was used once (to say that the teachers would be interested in it)

Personal component: As the interview progresses, an "opposing" view to the displayed acceptance reveals. We access to what she thinks deeply (which is "not correct" according to her):

-Regarding students: she is against screens, against children always being on them. These opinions conflict with the institution (programs) and what is promoted in society (modern aspects, etc.). Therefore, she has difficulty saying these opinions, even asking "do I have the right to say that?"… -Regarding her own person: as to her person, she finally admits that she is not "hyper motivated by", that she is not "interested" in these novelties (computer science, Scratch, robots…)

-It is also interesting to note that she explicitly expresses her need of didactic reference points: she says that she feels not comfortable with this teaching, that she, would like to ask her colleague, who does it, how she does it, she lacks resources that would help her "at the beginning". The precision about the "beginning" makes us reasonably think that the didactic reference points are missing. Once acquired, she would be able to go on without help.

Interview 2-Salient facts:

Here, the "political correctness" is present too and refers to institutional and social components.

The cognitive component is much more present than in interview 1 as for mathematics:

-The link here to mathematics teaching goes with a more developed professional instrumental genesis of computer tools. The teacher explicitly exploits ICT to do math sessions.

-The teacher 2 is conscious that "computer science" means also "learning to program", but he says he is not comfortable in this field.

Regarding the personal component: teacher 2 has personal interest in computers. He likes learning about this field; he even mentions quantum computers... The main obstacle here is not his personal opinion. What he feels as an obstacle is the time to solve hardware problems plus the time of the students' instrumental geneses, essentially because of the teacher's usual orchestrations: his practices consist of letting students explore during several sessions. This indicates some needs in training, not at technical level, but at didactic (in particular mediative) level in order to gain different approach and see different orchestrations allowing saving time on the students' instrumental geneses.

Besides, he adds that the lack of training on technological abilities is not a hindrance for him.

Interview 3-Salient facts:

Teacher 3 is specialized in the use of digital technology for sports.

Related to institutional and social components: the "politically correct" is present here again.

Regarding the cognitive component: Teacher 3 says his sessions always mix aims that are both numerical skills and disciplinary contents: mathematics or French language. The sessions are thus orchestrated in a way that allows the students' instrumental genesis to develop at the same time numerical skills and content knowledge (a point that was difficult for Teacher 2).

Besides, he also distinguishes the learning of computer science from the learning of using tool to serve other disciplines. Yet, he reduces the place of computer science in the curricula to "digital" aspects, i.e. knowing how to use digital tools, particularly how to use a computer.

The personal component: compared to Teacher 2, the interview shows more than interest from the part of Teacher 3: he states "I like it. (…) I like to use digital" and repeats this several times.

-His professional instrumental geneses with digital technologies seem quite developed. For instance, he is conscious of the need to homogenize the different instrumental geneses of the students, or still the fact that some tools require a smaller personal instrumental genesis (L 151).

He also explains that it is necessary to have a minimum of personal and professional instrumental geneses before starting in class (3.TDP44, 96). Regarding the orchestrations: he never does sessions aiming at "techniques for techniques". However, he mentions the need for a free exploration/discovery session (3.TDP120) if the tool is new, but in the following sessions, afterwards the technical learning is always mixed with other disciplinary aims.

-He also evokes (3.TDP30) the question of the distance to old practices, stating that he starts from his usual practices, then sees what and how it is possible to go on (3.TDP96). Some tools are less easy and, as Teacher 2, he explicitly states the lack and need of landmarks in some cases (3.TDP118). For instance, he needs references for the different orchestrations with the robots:

3.TDP153 there are many people, who would like to start with robots but do not know how, where to start with the students. There are lots of questions" (...) it's difficult to start in front of 25 children without being ready. You can plan a session and it can go in all directions [laughs]. I've been preparing sessions for quite a few years now and there is never a session that goes the way we imagined. So, we must be able to bounce back and that's also what must scare some teachers, they wonder if they'll be able to react to a technical problem, able to answer all the questions… This is a concern of our profession.

FIRST RESULTS AND DISCUSSION

The previous section gives an overview of the kind of analyses that we are undertaking, using the theoretical concepts in the form of the grid of analysis.

Reporting the statements in the grid allows showing where the positive and the negative arguments are respectively situated. For example, the Tab.3 shows the grid for the teacher 1.

Analogous pictures are obtained with the teacher 2 and 3. Synthetizing these analyses gives a picture of how these 3 teachers perceive the "newness" and its distance to their usual practices. For questions of space, here we give a summary of the results obtained in this final picture. Some convergences in the teachers' discourses Regarding these three interviews, the theoretical grid shows some convergences:

It is less the arguments "officially" announced by the teachers (the "political correctness": training, tools, resources...) than personal arguments (vision/conception of the disciplines, of the teaching, of the learning etc.) that play in the teachers' integration of technology and computer science in mathematics teaching.

These 2 "poles" of arguments are sometimes opposed (personal versus institutional component) such as positive and negative arguments clearly draw two separated sets.

The positive arguments are situated mainly on institutional and social components, whereas negative ones pertain to the teacher's personal dimensions: personal representations of their own abilities (in computer science and with technology), personal representation of what is computer science, personal interests, and personal epistemology and ethical considerations about their social and educative role and the meaning of teaching.

Regarding the cognitive and mediative dimensions, they mainly play as barriers: the didactic landmarks are crucial in these innovative situations. These latter move the teachers away from their didactic landmarks formerly built. New marks, guiding the teacher in her action, must be created. If former references are too much disrupted without new ones being considered, the teacher will not integrate the novelty: the teachers clearly express the need for gaining didactic landmarks on cognitive and mediative levels. In some rare cases, they play positively, the teachers mainly seeing the benefit of using ICT to teach transdisciplinary aims (work in pairs, investigating procedures…).

We meet here the results of previous research [START_REF] Haspekian | Informatique, Scratch et robots: De nouvelles pratiques enseignantes en mathématiques ? STICEF[END_REF], where we explained that this choice is not fortuitous: choosing transdisciplinary aims provides well-known landmarks, easily transferable to new situations because without underlying disciplinary concepts.

Discussion and perspectives for the research

In conclusion, drawing on the two theoretical frameworks of the Instrumental Approach and the Double Approach, we used concepts such as instrumental geneses, distance/landmarks, or personal component, to understand what is at stake when teachers implement new practices involving digital tools and computer science. This allowed identifying what is critical in this implementation:

-it is less the arguments "officially" announced by the teachers (the "political correctness": training, tools, resources...) than more personal arguments (vision/conception of the disciplines, of their teaching, of learning etc.) that play.

-these 2 "poles" of arguments are sometimes opposed (personal versus institutional component)

These concepts have been used before to analyze the practices observed with robots and Scratch. In both cases, they appear helpful to analyze the data. Yet, in the case of interviews, we access more deeply to the personal component. Therefore, we chose this type of data to explore teachers' personal opinions, which is not accessible if only observing effective practices. Conversely, the cognitive and mediative components are more accessible with observations; they are only indirectly caught (and may present some bias) with interviews.

It could be interesting to compare this approach with others, such as TPACK [START_REF] Mishra | Technological Pedagogical Content Knowledge: A framework for teacher knowledge[END_REF], or the Mathematics Knowledge for Teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. Some of the concepts used here, as the double instrumental genesis, have already been used to find some connections between the Instrumental approach and TPACK [START_REF] Tabach | The knowledge and skills that mathematics teachers need for ict integration: The issue of standards[END_REF] and we have previously evoked too these connections (Haspekian, 2018[START_REF] Haspekian | Teaching practices in digital era: Some theoretical and methodological perspectives[END_REF], but this work is still on-going. Yet, some more general reflexions revisiting theories that frame research on teaching mathematics with digital technology can be found in Sinclair et al. (to come).

Lastly, the analyzes should be pursued at a larger scale, interviewing more teachers. The idea would also be to interview different categories and compare the pictures obtained for each category: primary school teachers, secondary mathematics teachers and secondary technology teachers. One could reasonably presume that the pictures will be different, especially in terms of the epistemological component, due to the due to their different professional identities and specific background.

Table 2 : The possible positive or negative impacts on practices of a newness at different dimensions

 2

Table 3 : Use of the grid to analyze the teacher 1 Interview

 3


Marc Fabritius

Guillermo Rubio-Gómez

Christoph Martin

João Cavalcanti Santos

Werner Kraus

Andreas Pott

Guillermo Rubio Gomez

A nullspace-based force correction method to improve the dynamic performance of cable-driven parallel robots

Keywords: cable-driven parallel robots, force control, redundancy resolution, wrench-feasible workspace

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Cable-Driven Parallel Robots (CDPRs) belong to a family of parallel robots in which cables replace rigid links that hold the platform. The position and orientation of most CDPRs' platforms are controlled by varying the lengths of their cables. A common way to do this is coiling them onto winches [START_REF] Pott | Cable-Driven Parallel Robots: Theory and Application[END_REF]. Since this works well even for long cables, which represent relatively lightweight links, CDPRs can have a large workspace, high dynamic performance, and a good payload-to-weight ratio. These benefits set them apart from other types of robots. Another difference is that the portion of a CDPR's workspace accessible in practice not only depends on its geometric parameters but also on the cable force distribution induced by the controller. The cable forces must be kept between a lower and an upper limit to prevent malfunctioning or damaging the robot due to excessive forces or coiling errors. The wrench-feasible workspace of a CDPR is defined as the set of platform positions and orientations (this tuple is called the platform pose hereafter) where its wrench equilibrium can be satisfied for a given set of external wrenches using cables forces within predefined limits [START_REF] Verhoeven | Analysis of the Workspace of Tendon-based Stewart Platforms[END_REF]. CDPRs with more cables than degrees-of-freedom of their platform are classified as redundantly-constrained CDPRs [START_REF] Ming | Study on multiple degree-of-freedom positioning mechanism using wires. I: Concept, design and control[END_REF]. For such CDPRs, finding feasible cable force distributions is an underdetermined problem with multiple solutions.

Email address: marc.fabritius@ipa.fraunhofer.de (Marc Fabritius ) An example from this group of CDPRs is the IPAnema Mini [1, page 398]. It is used to conduct the simulations and experiments in this work and is depicted in Figure 1. In practice, CDPRs often cannot fully access their theoretical wrench-feasible workspace. This can be caused by errors in the model parameters or unmodelled effects like cable creep, preventing CDPR controllers from obtaining a feasible force distribution, even though one exists. This makes designing controllers for CDPRs challenging, as it requires weighing between two main objectives: guaranteeing the feasibility of the cable force distribution or controlling the platform pose. For controlling the platform pose, the different methods found in CDPR literature can be broadly divided into two main strategies: control in the operational space (i.e. the platform pose) and control in the joint space (i.e. the cables' lengths). For control strategies in the operational space, the error between the actual and desired platform pose is used as input to the controller. Its output is applied to the platform through separate force controllers for each joint. Multiple works combine this control architecture with different types of controllers such as PID [START_REF] Khosravi | Robust PID control of fully-constrained cable driven parallel robots[END_REF], adaptive dual-space controllers [START_REF] Lamaury | Dual-space adaptive control of redundantly actuated cable-driven parallel robots[END_REF], H ∞ controllers [START_REF] Chellal | Model identification and visionbased H ∞ position control of 6-dof cable-driven parallel robots[END_REF], sliding-mode controllers [START_REF] Aguas | A sliding mode control for a planar 4-cable direct driven robot[END_REF][START_REF] Ren | Sliding mode set point control of a six-dof cablesuspended parallel robot with tension constraints and uncertain disturbances[END_REF][START_REF] Wang | Adaptive super-twisting nonsingular fast terminal sliding mode control for cable-driven manipulators using time-delay estimation[END_REF], or exact linearization techniques [START_REF] Hamann | Model-based control of a planar 3-dof cable robot using exact linearization[END_REF]. To enable control in the operational space, the actual platform pose must be measured by means of cameras [START_REF] Dallej | Vision-based modeling and control of large-dimension cable-driven parallel robots[END_REF][START_REF] Chellal | Model identification and visionbased H ∞ position control of 6-dof cable-driven parallel robots[END_REF][START_REF] Zake | Control stability workspace for a cable-driven parallel robot controlled by visual servoing[END_REF][START_REF] Khalilpour | Robust cascade control of a deployable cable-driven robot[END_REF], lasers [START_REF] Zi | Dynamic modeling and active control of a cable-suspended parallel robot[END_REF][START_REF] Hamann | Model-based control of a planar 3-dof cable robot using exact linearization[END_REF], or estimated from the effective cable lengths processed through the forward kinematic model of the robot as in [START_REF] Kraus | Force control of cable-driven parallel robots[END_REF][START_REF] Lamaury | Control of a large redundantly actuated cable-suspended parallel robot[END_REF][START_REF] Khosravi | Robust PID control of fully-constrained cable driven parallel robots[END_REF][START_REF] Abdelaziz | Control of cable-driven manipulators in the presence of friction[END_REF][START_REF] Bieber | Motor current based force control of simple cable-driven parallel robots[END_REF][START_REF] Santos | A nonlinear model predictive control for the position tracking of cable-driven parallel robots[END_REF][START_REF] Santos | Redundancy resolution integrated model predictive control of cdprs: concept, implementation and experiments[END_REF]. Joint space controllers map the desired platform pose into the joint space using an inverse kinematic model. The resulting commands are fed to controllers that move the winches accordingly and receive feedback on the actual joint positions from their encoders. Some control approaches used in the joint space include proportional-differential controllers [START_REF] Reichenbach | Velocity based hybrid position-force control of cable robots and experimental workspace analysis[END_REF], wave-based control [START_REF] Khalilpour | Robust dynamic sliding mode control of a deployable cable driven robot[END_REF], impedance control [START_REF] Reichert | Robust internal force-based impedance control for cable-driven parallel robots[END_REF], synchronization [START_REF] Shang | Synchronization control in the cable space for cable-driven parallel robots[END_REF] and adaptive control [START_REF] Lamaury | Dual-space adaptive control of redundantly actuated cable-driven parallel robots[END_REF]. Due to the constantly changing payloads applied to each winch and the non-linear dynamics of CDPRs, feed-forward linearization of the non-linear terms in the dynamic model, is a common technique used in the joint space control of CDPRs (e.g. [START_REF] Fang | Motion control of a tendon-based parallel manipulator using optimal tension distribution[END_REF][START_REF] Oh | Cable suspended planar robots with redundant cables: Controllers with positive tensions[END_REF][START_REF] Zheng | Feedback linearization control of a wire-driven parallel support system in wind tunnels[END_REF][START_REF] Shang | Synchronization control in the cable space for cable-driven parallel robots[END_REF][START_REF] Wang | Practical robust control of cable-driven robots with feedforward compensation[END_REF]). Most of the previously mentioned approaches perform dynamic control of the torque generated by the winches. Some of them rely on the knowledge of the dynamic models of the CDPR's platform and winches to perform some compensations through dynamic inversion or feed-forward linearization techniques. Alternatively, servomotors can be used in many applications, as they provide robust and high-performance motor position control. In such settings, the dynamics of the actuators can be ignored as the motors can reliably control the cable lengths in the joint space. This simplifies the control scheme of CDPRs, as only an inverse kinematic model has to be implemented. A drawback of this simple approach is that it cannot explicitly control the cable forces. Its ability to access a CDPR's entire wrench-feasible workspace in practice is therefore often limited. This phenomenon and a possible solution are discussed in the authors' prior work [START_REF] Fabritius | A practical force correction method for over-constrained cable-driven parallel robots[END_REF]. Regardless of whether the position control is performed in the operational or in the joint space, the cable forces have to be kept within their limits. A common approach for redundantly restrained CDPRs is to employ hybrid position-force controllers. Hereby, feasible force distributions are obtained through redundancy resolution techniques based on the robot's structure matrix and its nullspace (e.g. [START_REF] Oh | Cable suspended planar robots with redundant cables: Controllers with positive tensions[END_REF][START_REF] Oh | Generation of feasible set points and control of a cable robot[END_REF][START_REF] Khosravi | Robust PID control of fully-constrained cable driven parallel robots[END_REF][START_REF] Qi | Decoupled modeling and model predictive control of a hybrid cable-driven robot (hcdr)[END_REF][START_REF] Taghirad | An analytic-iterative redundancy resolution scheme for cable-driven redundant parallel manipulators[END_REF][START_REF] Reichenbach | Velocity based hybrid position-force control of cable robots and experimental workspace analysis[END_REF]). The obtained force distribution is then employed along with a dynamic con-trol strategy to control the platform position and guarantee cable force feasibility simultaneously. Due to the non-linear nature of the underlying mathematical problem, most of these approaches employ iterative algorithms as in [START_REF] Hassan | Optimization of actuator forces in cablebased parallel manipulators using convex analysis[END_REF][START_REF] Agahi | Redundancy resolution of wire-actuated parallel manipulators[END_REF][START_REF] Gouttefarde | A versatile tension distribution algorithm for n-dof parallel robots driven by n+2 cables[END_REF] and some of them [START_REF] Oh | Cable suspended planar robots with redundant cables: Controllers with positive tensions[END_REF][START_REF] Lamaury | Control of a large redundantly actuated cable-suspended parallel robot[END_REF][START_REF] Gouttefarde | A versatile tension distribution algorithm for n-dof parallel robots driven by n+2 cables[END_REF][START_REF] Gouttefarde | Geometry selection of a redundantly actuated cable-suspended parallel robot[END_REF] are only able to deal with CDPRs with a maximum degree of redundancy of two. Mattioni et al. resolve the redundancy of CDPR by splitting their cables into two sets that are position and force controlled, respectively [START_REF] Mattioni | Force-distribution sensitivity to cabletension errors in overconstrained cable-driven parallel robots[END_REF]. The choice of this partition is based on a force sensitivity index to minimize the errors in the cable force distribution. Santos et al. introduce model predictive control (MPC) methods that compute the set of desired cable tensions as the solution of an optimal control problem considering a dynamic model of the CDPR [START_REF] Santos | A nonlinear model predictive control for the position tracking of cable-driven parallel robots[END_REF][START_REF] Santos | Redundancy resolution integrated model predictive control of cdprs: concept, implementation and experiments[END_REF]. The goal of their controllers is to minimize a combination of the error in the platform pose, the variation of the cable forces, and their norm. The feasibility of the resulting cable forces is formulated as an explicit constraint to the optimization problem. All previously discussed control methods from the literature, which achieve high performance, either rely on the direct feedback of the platform pose or use complex models, which depend on the knowledge of various model parameters. The drawbacks of these approaches are that suitable sensors are often prohibitively expensive, complex models have high computational costs, and their parameters can be difficult to measure with the required precision or may be subject to change in practice, e.g. the cables' stiffness or the platform's mass. In prior research leading up to this work, [START_REF] Fabritius | A practical force correction method for over-constrained cable-driven parallel robots[END_REF], a novel force correction method for redundantly-constrained CDPRs, which does not require knowledge of any cable or platform weight parameters, is presented. It relies on servomotors for control in the joint space and modifies the cable forces within the nullspace of the CDPR's structure matrix to keep them inside their limits. A systematic study of its performance throughout the static workspace of the IPAnema 3 CDPR [1, page 395] shows that the method can significantly extend the workspace portion that can be accessed in practice when compared to a kinematic control method. This work introduces a further development of this method, which additionally aims to keep the cable forces close to a specified level. This new feature could be used to adjust the stiffness of the platform. A new practical tuning procedure enables the method to achieve a high performance in disturbance rejection and on dynamic trajectories. The new method is compared to the MPC approach from [START_REF] Santos | Redundancy resolution integrated model predictive control of cdprs: concept, implementation and experiments[END_REF] in simulation and to a purely kinematic controller in experiments on the IPAnema Mini CDPR. These comparisons cover the full range of the CDPR's dynamic capabilities, showing its ability to reject disturbances and its behavior on three trajectories for various velocities. The rest of this work is organized as follows. Section 2 introduces the notation and a dynamic model of CDPRs. The new control method and a practical tuning procedure are presented in Section 3. In Section 4, it is evaluated through simulations and experiments and compared to state-of-the-art controllers in terms of disturbance rejection and on dynamic trajectories with various platform velocities. Finally, the conclusion and outlook are given in Section 5.

Laser Tracker

Notation and Modelling of CDPR

Let m denote the number of cables and n the degrees-offreedom of a CDPR. Each cable i ∈ {1, ..., m} of a CDPR originates from a proximal anchor point a i ∈ R 3 , given in the fixed robot coordinate system K 0 , and connects to a distal anchor point on the moving platform, which is denoted as b i ∈ R 3 in the platform coordinate system K p . The CDPR's n degrees-offreedom usually consist of its platform position r ∈ R 3 and rotation matrix R ∈ SO(3) expressing its orientation for n = 6. As a tuple, they are called the pose (r, R) ∈ SE(3) of the platform and determine the mapping between the coordinate systems K 0 and K p . The location of b i can be expressed in K 0 as r + Rb i . The state of the platform within its n = 6 degrees-of-freedom is parameterized by a vector x ∈ R 6 , whose first three components are the platform position r and the remaining three are a parameterization of the rotation matrix R. The cables of the CDPR are assumed to be straight lines, and their mass is neglected. Their lengths are collected in the vector

l = l 1 • • • l m ∈ R m .
For each cable i ∈ {1, ..., m}, the vector connecting its endpoints d i ∈ R 3 , its direction u i ∈ R 3 and length l i ∈ R are calculated as

d i = a i -r -Rb i , u i = d i ∥d i ∥ , l i = ∥d i ∥ . (1)
Figure 2 shows the relationships of these vectors. Elements in the platform coordinate system K p are colored blue. The cable directions u i are used to assemble the structure matrix A T ∈ R n×m of the CDPR. Each column A T i of this matrix encapsulates the direction in which the i-th cable exerts a force and moment on the platform

K p b i K 0 d i u i a i r + R b i r
A T i = u i Rb i ×u i ∈ R 6 . (2)
The vector of cable forces

f = f 1 • • • f m ∈ R m ,
which pull on the platform, is multiplied with A T to formulate the wrench equilibrium equation

A T f + g = 0 . (3)
The platform's wrench due to gravity is denoted as g ∈ R 6 .

Cable force distributions f are called feasible if they satisfy the wrench equilibrium in Equation ( 3) and the predefined limits

f min ≤ f i ≤ f max for all i ∈ {1, ..., m} . (4)
The upper limit f max is usually imposed by the capabilities of the CDPR's motors and winches. The lower limit f min > 0 is chosen sufficiently large to prevent cable sagging or coiling errors on the winches. Redundantly-constrained CDPRs (m > n) are usually designed such that the structure matrix A T has full rank n within the desired workspace. As a result, the linear system in Equation ( 3) is underdetermined and has infinitely many solutions f * which can be split into two components [START_REF] Verhoeven | Analysis of the Workspace of Tendon-based Stewart Platforms[END_REF] f * = -A T+ g + Nα .

(

) 5
The first term -A T+ g is the least-squares solution, which is calculated using the pseudo-inverse A T+ of the structure matrix. The second term Nα represents the position α ∈ R (m-n) of the solution f * within an arbitrary basis N ∈ R m×(m-n) for the nullspace of the structure matrix. By definition, any value of α fulfills the wrench equilibrium in Equation ( 3), but only those α that also satisfy Equation ( 4) are considered feasible. This property is exploited by the new force control method presented in Section 3.

Dynamic Model

To validate the proposed control approach in simulation, a dynamic CDPR model is constructed by augmenting the wrench equilibrium from Equation (3). The platform's Newton-Euler equation of motion is given by

M ẍ + C ẋ = A T f + g , (6)
where M ∈ R 6×6 is the inertia matrix of the platform, that is multiplied by its acceleration ẍ. The matrix C ∈ R 6×6 represents the influence of Coriolis and centripetal forces. The cables are modeled as a Kelvin-Voigt material. This dynamic model is often used for the cables of CDPRs, as it offers a good representation of their short-term dynamics [38, page 159]. The model consists of a linear spring and a damping element arranged in parallel. The force f i in each cable i ∈ {1, ..., m} is calculated as a function of the cable length l i from Equation ( 1), the rotational positions of the motors θ ∈ R m , and their derivatives

f i = k c l i r w θ i -1 + η li r w θi . (7)
Hereby, k c is the stiffness of the spring, η is the viscosity of the damping element, and r w is the effective radius of the winch.

The torque τ ∈ R m generated by the winches is calculated based on their rotational state θ and the cable forces f

τ = J θ + ν θ + r w f . (8)
The dynamic properties of the winches are captured in the diagonal inertia and viscous friction matrices J, ν ∈ R m×m , respectively.

Force Control in the Nullspace

The presented force control in the nullspace (FCN) for redundantly-constrained CDPRs is designed as an add-on to a simple control scheme based on an inverse kinematic model (IK). The latter calculates desired cable lengths l corresponding to the desired platform pose (r, R). In the joint space, built-in controllers of the servomotors that drive the CDPR's winches are responsible for reaching these desired cable lengths.

When the FCN method is activated, it adds offsets ∆l to the desired cable lengths l before they are passed to the servomotors in the winches. The goals of the FCN method, in order of increasing priority, are:

1. The cable forces f should be kept as close as possible to the force level f ref , which can be freely chosen within the limits from Equation (4). 2. The cable forces f should be within the limits of Equation (4). 3. The platform pose (r, R) should not be altered.

Figure 3 shows the structure of the proposed method as a block diagram. All blocks are repeatedly executed by the control system of the CDPR.

A T IK l CDPR + Nullspace Calculation Force Correction PID Offset Calculation f f ref (r, R) N N α β ∆l N old Z -1 f Figure 3: Structure of the Proposed Force Control Method
The FCN method first calculates a basis N for the nullspace of the structure matrix A T . This basis is then used for the calculation of α and the offsets ∆l in order to satisfy the third goal of the FCN method. The desired force correction α is calculated within the nullspace basis N in pursuit of the first two goals of the FCN method. The proportional-integral-derivative (PID) controller tries to move α to zero by means of its output β, which is also interpreted as a position in the nullspace. This output is translated to cable length offsets ∆l, which are added to the cable lengths l calculated by the inverse kinematic.

The following sections explain the function of the blocks in Figure 3 in detail.

Inverse Kinematics

The inverse kinematic block IK maps the desired platform pose (r, R) to corresponding cable lengths l. It uses the pulley kinematic model developed in [START_REF] Pott | Influence of pulley kinematics on cable-driven parallel robots[END_REF], which is applicable to the IPAnema Mini CDPR. The cables are considered to be straight lines originating from pulleys pivoting around proximal anchor points a i . Their elasticity and weight are neglected. The model also calculates the structure matrix A T at the current platform pose and passes it to the nullspace calculation block.

Nullspace Calculation

The nullspace calculation block computes a nullspace basis N for the structure matrix A T , which is a continuous function of the pose (r, R). The essential property of continuity is not guaranteed when using standard nullspace calculation algorithms, as the calculated basis can be arbitrarily oriented within the nullspace. This free choice on behalf of the algorithm can introduce discontinuous jumps when the parameters of A T change.

The necessary steps to ensure continuity are described as pseudocode in Algorithm 1.

Algorithm 1: Nullspace Calculation

Input: structure matrix A T ∈ R n×m , previous nullspace basis N old ∈ R m×(m-n) 1 Ñ = Nullspace A T ∈ R m×(m-n) 2 U, Σ, V = SVD ÑT N old 3 N = ÑUV Output: continuous nullspace basis N
The algorithm receives the nullspace basis from the previous cycle N old as an input from the delay block which is labeled Z -1 in Figure 3. In Line 1, a nullspace basis of A T is calculated. To remove possible discontinuities in the basis' orientation, a coordinate transformation between the nullspace basis from the previous cycle N old and the current one Ñ is calculated. If the nullspace itself changes between two executions of Algorithm 1, the transformation ÑT N old is not an orthogonal matrix and has singular values smaller than 1. Applying N = Ñ ÑT N old would avoid discontinuities in N, but the resulting matrix is not an orthonormal basis of the nullspace as the norms of its column vectors would be smaller than 1. To correct this, Algorithm 1 instead uses the nearest orthogonal matrix of this transformation. This approximation is calculated using the singular value decomposition UΣV = ÑT N old in Line 2.

In practice, the nullspace changes only in small steps between two executions of Algorithm 1, even when the platform is moving at a high velocity. Therefore, all singular values of ÑT N old are always close to 1 and UV is a sufficiently close orthogonal approximation of UV ≈ ÑT N old . The matrix N, as calculated in Line 3, is an orthonormal basis of the nullspace and a continuous function of the platform pose (r, R).

Force Correction

The force correction block in Figure 3 calculates the desired change in the cable forces in order to achieve the three goals of the FCN method. To address the first goal, the cable forces f are shifted

f = f -f ref 1 , (9)
such that the objective can be stated as f = 0. Hereby, 1 ∈ R m denotes the vector whose components are all 1. To move towards its first goal, the FCN method changes the cable force in the direction ∆ f with

∆ f = -f . (10)
If the forces are not within their limits from Equation ( 4), achieving this state becomes a higher priority than the force level f ref .

The direction of the force changes ∆ f is subsequently skewed towards the cable which violates its limits the most. This shift between the first two goals is implemented by raising the vector f to the power p ≥ 1 while preserving its sign

f p = sign f 1 f 1 p • • • sign f m f m p T . ( 11
)
The power p ∈ R is a function of the largest cable force limit violation

p = 1 + p scale • max              0, max i=1,...,m ( f i -f max ) , max i=1,...,m ( f min -f i )              . ( 12
)
The slope parameter p scale determines how quickly the controller changes its priority from the first to the second objective when cable forces start to violate their limits. While the vector from Equation (11) determines the direction in which the cable forces are changed, the magnitude of this change is scaled linearly with the distance ∥ f ∥ from f ref .

Finally, the desired change of the cable forces ∆ f can be calculated as

∆ f = - f f p f p . ( 13
)
Notice that if the second goal is satisfied, p = 1 according to Equation ( 12) and the force changes ∆ f from Equations ( 10) and ( 13) are the same. Therefore, an explicit distinction between the cases p > 1 and p = 1 is not necessary.

To gain a better understanding for the behavior of Equation (13), ∆ f is visualized in Figure 4 as a vector field for the two-dimensional case with m = 2. Hereby, f ref is set to be the average of the force limits. This visualization shows how ∆ f is a continuous function of the cable forces f . To satisfy the third goal of not altering the platform pose (r, R), the desired force change ∆ f is projected onto the nullspace of the structure matrix A T at the current pose

α = N T ∆ f . ( 14
)
The resulting vector α ∈ R (m-n) is the position of the desired force changes within the nullspace basis N ∈ R m×(m-n) . The previously described steps of the force correction block are summarized in Algorithm 2 as pseudocode.

PID Controller

The PID block in Figure 3 contains a proportional-integralderivative controller, whose goal is to drive α ∈ R (m-n) to zero

f min f ref f max f min f ref f max Figure 4: Two-Dimensional Visualization of ∆ f ∈ R 2 for All Cable Forces f ∈ R 2 Algorithm 2: Force Correction Input: cable forces f ∈ R m , force level f ref , nullspace basis N ∈ R m×(m-n) 1 f = f -f ref 1 2 p = 1 + p scale • max              0, max i=1,...,m ( f i -f max ) , max i=1,...,m ( f min -f i )              3 ∆ f = - f f p f p 4 α = N T ∆ f
Output: desired nullspace position α by adjusting its output β ∈ R (m-n) . The controller has a diagonal structure and takes the form R (s) = R (s) I in the Laplace domain. Hereby, I ∈ R (m-n)×(m-n) denotes an identity matrix and R(s) is the transfer function

R(s) = K p + K i s + K d s , (15)
where the proportional K p , integral K i , and derivative K d gains are chosen using the following procedure.

The FCN method's first goal of bringing all cable forces as close to f ref as possible also implies that their average f moves to this level. For a fixed platform pose (r, R), and only for controller tuning purposes, let us consider all blocks from Figure 3 as a single-input single-output time-invariant linear system M(s) with the input f ref and output f . This is assumed to be equivalent to a simple unity-feedback system with f ref being the setpoint variable and f the controlled variable yielding

M(s) = R(s)G(s) 1 + R(s)G(s) (16)
where R(s) represents the PID controller and G(s) is the collection of all other blocks in Figure 3. The first step in tuning the controller is to approximate the transfer function M(s) based on its time response. To obtain this data, the controller is restricted to contain only a proportional term R(s) = K p with an appropriate value such that f exhibits several oscillations after a step increase of f ref .

Although the dynamics of CDPRs are known to be highly non-linear, modeling M(s) as a linear system allows using a black box identification technique [START_REF] Ljung | Black-box models from input-output measurements[END_REF], which simplifies the tuning procedure. From the identified system M(s), which includes the proportional controller, the transfer function G(s) can be calculated using Equation [START_REF] Lamaury | Control of a large redundantly actuated cable-suspended parallel robot[END_REF]. The well-known frequency response tuning technique is employed to obtain a PID controller R(s), yielding an open loop transfer function H(s) = R(s)G(s) with desired properties. Based on definitions from [41, page 464], the following relationships are established for the system's gain crossover frequency ω gc , phase crossover frequency ω pc , phase margin ϕ m , and gain margin

K g ℜ R jω gc = K p , (17)
ℑ R jω gc = K d ω gc - K i ω gc . (18)
Hereby, R( jω gc ) = -e jϕ m /G( jω gc ) is the value of the controller in the frequency domain. While the value of K p is determined by Equation ( 17), one degree of freedom remains for K d and K i whose relationship is established in Equation ( 18). This can be resolved by setting a convenient value for the pair (K g , ω pc ).

Offset Calculation

The offset calculation block in Figure 3 translates the nullspace position β ∈ R (m-n) into cable length offsets ∆l ∈ R m . It assumes that the cables behave similarly to linear springs when changes in the cable forces Nβ are converted to changes in the cable lengths. The stiffness of the cables is assumed to be proportional to their lengths l divided by a specific stiffness constant k c that depends on the cables' material and diameter. The cable length offsets ∆l ∈ R m are calculated as

∆l = 1 k c diag (l) Nβ , (19)
where diag (l) is the diagonal matrix whose entries are the components of l. The assumptions underlying Equation ( 19) are simplifications and might not be true in practice. Nevertheless, this equation provides a sufficient approximation of the cables' behavior for the FCN method to work as shown in Section 4.

Modeling errors concerning the cables' behavior and elasticity (e.g. a wrong value of k c ) can be compensated in practice by tuning the gains of the controller.

CDPR

The CDPR block in Figure 3 represents the robot's behavior which follows the commanded cable lengths l + ∆l using the servomotors in its winches. The resulting cable forces f are measured and passed to the force correction block. For the purpose of tuning the PID controller as described in Section 3.4, the average cable force f ∈ R is calculated and provided as an output

f = 1 m m i=1 f i . (20)

Evaluation in Simulations and Experiments

The simulations and experiments in this work are performed with the IPAnema Mini at Fraunhofer IPA [1, page 398]. It is a redundantly-constrained CDPR with m = 8 cables and n = 6 degrees-of-freedom, which is shown in Figure 1. Its geometry parameters a i and b i are listed in Table 1. These restrictive limits are purposefully chosen to investigate the behavior of the control methods when the platform leaves the wrench-feasible workspace defined by these limits. The IPAnema Mini's platform weighs 0.25 kg. Its control system is based on the TwinCAT 3.1 software by Beckhoff and runs on a programmable logic control (PLC) with a cycle time of 1 millisecond.

For the experiments in Section 4.3, the platform's position is measured with a Leica AT960 laser tracker and a target as shown in Figure 1. This laser tracker can measure the target's position with an absolute accuracy of ± 15 µm + 6 µm m , which decreases by 6 µm per meter distance from the target. For the static experiments in this work, the laser tracker was placed three meters away from the center of the workspace of the IPAnema Mini, which implies an absolute measurement accuracy of ±33 µm. The sampling rate of these measurements is 1000 Hz. The experiments and simulations in this work are conducted at three different platform positions and on three linear trajectories dispersed throughout the IPAnema Mini's workspace. The trajectories start at the origin r 0 = 0 in the fixed coordinate system K 0 , in the center of the frame, and end at the points

r 1 =           180 mm 140 mm 100 mm           , r 2 =           180 mm 0 mm 0 mm           , r 3 =           -140 mm 150 mm -35 mm           . ( 21
)
The platform orientation remains the same R = I ∈ SO(3) for all simulations and experiments.

Figure 5 shows renderings of the IPAnema Mini and these trajectories from two perspectives created with the open-source CDPR simulation software WireX [START_REF] Pott | Wirex -an open source initiative scientific software for analysis and design of cable-driven parallel robots[END_REF]. The volume of its static wrench-feasible workspace is depicted in red. It is calculated for the force limits from Equation (4), using the force distribution method from [START_REF] Mikelsons | A real-time capable force calculation algorithm for redundant tendon-based parallel manipulators[END_REF]. The platform wrench g = -A T f , used for this computation, is calculated from the measured cable forces f at the position r 0 .

Figure 5 shows that the point r 2 lies within this workspace, while r 1 and r 3 do not. On the straight lines from r 0 through the points r 1 , r 2 , r 3 , the workspace ends after 155 mm, 182 mm, 127 mm, respectively. The following sections discuss the tuning of the FCN controller for the IPAnema Mini and its comparison to state-of-the-art controllers in simulation and experiments.

Controller Tuning

The PID controller within the FCN method is tuned according to the procedure from Section 3.4. The parameter p scale , which controls how quickly the FCN method shifts between its first two goals, is set to p scale = 2. This choice is a compromise between how quickly the controller can shift its priorities and its robustness at the borders of the workspace. The design values used for the tuning are ω gc = 17.6 rad/s and ϕ m = 80 • . The value of ω gc is selected such that the controlled system is 14 times faster than the open-loop system, based on the approximate formula ω gc ≈ π/t s , where t s is the system's settling time. This is a compromise between the speed and robustness of the controller. Experiments show that larger values of ω gc cause the system to be unstable at the borders of the workspace when p > 1.

A value of K i = 14.3 is selected as a trade-off between speed and stability, yielding values of K g = 15.2 dB and ω pc = 194 rad/s. The resulting values for the remaining controller gains by means of Equations ( 17) and ( 18) are K p = 0.206 and K d = 0.0063.

Comparison to a MPC Method

The MPC method introduced by Santos et al. [START_REF] Santos | Redundancy resolution integrated model predictive control of cdprs: concept, implementation and experiments[END_REF] is a stateof-the-art CDPR control method capable of pursuing multiple objectives at once. Following an optimal control approach, it specifies a quadratic program that minimizes a weighted sum of the CDPR's pose error, variation of its cable forces, and their norm ∥ f ∥. The feasibility of the cable forces from Equation ( 4) is mandated as an explicit constraint to the optimization problem. Therefore, the CDPR always remains within the wrench-feasible workspace, even when the commanded trajectory leaves it. This is a significant difference to the FCN method, which always follows the commanded trajectory even when the force limits are violated. To ensure a fair comparison between these methods within the wrench-feasible workspace of the CDPR, the third objective of the MPC method is altered to minimize ∥ f ∥ as defined in Equation ( 9) instead of ∥ f ∥. This comparison is performed in simulation instead of experiments since implementing the MPC method on the IPAnema Mini CDPR would be very laborious, requiring the implementation of complex numerical solvers within its PLC. Furthermore, it is easier to ensure a fair comparison between the methods when using the dynamic model of the IPAnema Mini from Section 2.1 in MATLAB Simulink. Simulations show that the methods behave similarly on all three trajectories from Figure 5. Therefore, only the third trajectory is analyzed in detail for a velocity of 1 m/s. Figure 6 shows the evolution of the cable forces f , force limit violations p, and deviations from the nominal trajectory of the FCN and MPC methods over time, during and after this trajectory. The dotted vertical lines indicate the acceleration and deceleration phases of the trajectory. Between the first two vertical lines, the CDPR moves with a constant platform velocity of 1 m/s. The first plots show that both controllers produce similar cable forces during the trajectory. The FCN method is slightly better at keeping them closer to their desired level f ref , which can be observed from the cable force averages f , that are shown as thick lines. The second plot shows that the FCN method can also keep the cable forces feasible (i.e. p = 1) slightly longer than the MPC method. In terms of platform position error, the FCN is better than the MPC method due to the different priorities of the two methods and the fact that the FCN method only alters the cable forces within the nullspace. Once the trajectory has ended after 0.23 seconds, the MPC method moves the platform to a position where it can find feasible cable forces again. The computation times of the FCN and MPC methods during these simulations, performed in MATLAB Simulink, are measured and averaged over 50 executions. The FCN method is 79% faster than the MPC method. This significant difference in computational costs could be exploited to run the FCN controller at faster cycle times than MPC, enabling it to react more quickly to changes in the cable forces.

Disturbance Rejection

The ability of the FCN method to reject disturbances in the cable forces is evaluated in experiments at the center r 0 , near the border r 2 , and outside r 1 of the wrench-feasible workspace shown in Figure 5.

For each pose, the same experiment is conducted with the IK and the FCN method. The platform is moved to one of the poses, and the cable forces settle into a stationary state. A mass of w = 2.916 kg is manually placed on the platform. After the system reaches a new disturbed stationary state, the additional weight is removed again. In these experiments, the robot shows qualitatively the same behavior at all three positions. Therefore, it is only analyzed in detail for r 0 in Figure 7.

The force distribution plot shows the cable forces and their averages (thick lines) for the FCN and IK methods. The average of the FCN method shows its reaction to the disturbance. For the first 200 ms, it decreases similar to the IK method, but then it increases and stabilizes slightly above the desired level The plots of the controller's input and output magnitudes show how quickly the FCN method reacts to the disturbance of the weight being placed on the platform. It takes around one second before the controller regains its desired state of ∥α∥ = 0 after the weight is placed on the platform. When it is removed again, this only takes half a second. This difference might be caused by the fact that the weight is manually placed and removed from the platform. Thus, the resulting wrench acting on the platform does not change instantaneously as the weight transitions between the hands of the experimenter and the platform. While the weight is resting on the platform, the magnitude of the controller output ∥β∥ slowly increases to counteract creep effects in the cables.

The platform displacement plot shows the relative change in the platform position ∆r as measured by the laser tracker. We observe an identical behavior in ∆r for both methods. This shows that the FCN method does not influence the stiffness of the platform at r 0 , as it does not significantly change the cable forces.

The fourth plot shows the relative changes in the first three components of the platform's gravity g xyz ∈ R 3 due to the weight. This force is calculated based on the cables' measured forces f and directions u i from Equation ( 1)

g xyz = - m i=1 u i f i . (22)
Its plot shows an identical behavior for both methods. This is expected, as the force corrections of the FCN method are designed to be inside the nullspace of the structure matrix A T .

The experimental results at all three platform positions from Equation ( 21) are summarized in Table 2. For each position and control method, the columns f and f w contain the average cable force in the stationary state before and after the weight is placed on the platform, respectively. In all cases, the FCN method brings the average force closer to the desired level f ref when compared to the IK method. This effect is stronger at the positions near the border r 2 or outside r 1 of the workspace. The columns p and p w show the force limit violation defined in Equation ( 12) for the stationary states before and after the weight is placed on the platform. For r 1 and r 2 , the FCN method significantly reduces the value of p relative to the IK method. The column ∥∆r∥ contains the total deviation of the platform's position due to the weight. This deviation depends on the stiffness of the platform, which in turn depends on the cable forces and their average level f . A high cable force average implies a high stiffness of the platform and therefore reduces the deviation ∥∆r∥ induced by the weight. The experiments in this section show that the FCN method can quickly reject disturbances and keeps the cable forces close to their desired level f ref .

By adjusting this level, the FCN method could be used to indirectly modify the stiffness of the platform.

Workspace Investigation

The behavior of the IK and FCN methods is evaluated on the three trajectories presented in Figure 5 for the platform veloci-ties 1/3 m/s, 2/3 m/s, and 1 m/s. The behavior is also investigated in the static workspace, i.e. at velocity 0 m/s. To measure this, the three trajectories are discretized into 50 poses, where measurements are taken after the CDPR has reached a stationary state. On each dynamic trajectory, the platform quickly accelerates from its resting position at r 0 to reach one of the three previously mentioned velocities. It travels with this constant velocity for the majority of the trajectory until it decelerates to stop at r 1 , r 2 , or r 3 . The length of the acceleration and deceleration phases of the trajectories increases for higher velocities, with a maximum of 13.55 mm necessary for reaching 1 m/s. Hereby, the platform experiences a maximal acceleration of 27 m/s 2 , reaching the limits of the IPAnema Mini's capabilities.

Figure 8 shows the influence of the platform's velocities on the average cable force f , force limit violation measure p, and the magnitudes of the controller's input ∥α∥ and output ∥β∥. These values are plotted as a function of the distance along the three trajectories. After the platform stops at the endpoint of each trajectory, the values are recorded for an additional second to show how they reach a stationary state.

Comparing the two methods for the same velocity shows that the FCN method can keep f closer to the desired level f ref and p closer to 1 than the IK method. For the velocities 1/3 m/s, 2/3 m/s, and 1 m/s, the behavior of the IK method is almost identical, and only differs in the static experiments. The behavior of the FCN method changes with increasing velocities as its objectives become more challenging and its distance to them increases. After the platform stops at the end of the trajectories, the FCN method brings f and p to a stationary state which is independent of the platform's previous velocity. The plots of p in Figure 8 show how much of the IPAnema Mini's workspace is accessible to the controllers in practice. Similar to the observations in the prior work [START_REF] Fabritius | A practical force correction method for over-constrained cable-driven parallel robots[END_REF] on the IPAnema 3 CDPR, the static experiments show a good correspondence between the extent of the wrench-feasible workspace shown in Figure 5 and the portion of the trajectories in Figure 8 where p = 1 for the FCN method. On the first and third trajectories, the accessible workspace of the FCN method in the static case ends 26 mm before and 5 mm after the wrench-feasible workspace, respectively. The second trajectory is contained within the wrench-feasible workspace and the accessible one of the FCN method in the static experiments. With increasing velocities, only smaller portions of the trajectories are within the accessible workspace of the controllers. Nevertheless, the FCN significantly decreases the force limit violation p compared to the IK method. The plots of the magnitudes of the FCN method's controller input ∥α∥ look similar for the velocities 1/3 m/s, 2/3 m/s and 1 m/s. Once the platform stops at the end of each trajectory, it takes around half a second until the FCN method brings the controller input to its desired state ∥α∥ = 0. In the static experiments, ∥α∥ = 0 at every point of the trajectories because the measurements are recorded once the CDPR reaches a stationary state.

The plots of the magnitude of the controller output ∥β∥ show that it steadily increases along the trajectories. This increase is quicker for slower velocities as the controller has more time to work towards its objectives.

Conclusion and Outlook

This work presents a force control method for redundantlyconstrained CDPRs that can keep the cable forces within their limits and close to a specified force level. It does not require knowledge of model parameters that are difficult to measure or might change during the operation of a CDPR. The performance of the new method is evaluated when disturbances are applied to the cable forces and on different trajectories with platform velocities of up to 1 m/s on the IPAnema Mini CDPR.

In simulations, the method shows similar behavior in the cable forces as a state-of-the-art MPC method while being simpler and requiring 79% less computational effort. The experiments in this work show that the new method can quickly react to disturbances and can significantly improve the feasibility of the cable forces, especially for high platform velocities, compared to a purely kinematic control method. Future research could investigate the behavior of this method on CDPRs with different geometric properties or sagging cables to analyze its performance and applicability for redundantlyconstrained CDPRs in general. The implications of modifying the desired force level f ref during the operation of the new control method could also be analyzed with regard to the CDPR's stiffness.

Figure 1 :

 1 Figure 1: IPAnema Mini CDPR and Laser Tracker

Figure 2 :

 2 Figure 2: Illustration of the vectors associated with cable i

  uses Dyneema cables with a diameter of 1.5 mm and a specific stiffness of k c = 28500 N [1, page 398]. The viscosity of the cables in the dynamic simulation model Section 2.1 is set to η = 200 N. The winches are dimensioned for cable forces of up to 40 N. For the purpose of this work, the cable force limits are set to f min = 10 N and f max = 25 N.

Figure 5 :

 5 Figure 5: Visualization of the IPAnema Mini with its wrench-feasible workspace (red) and trajectory endpoints r 1 , r 2 , r 3 from two perspectives.

Figure 6 :

 6 Figure 6: Simulation of the FCN and MPC Methods on the Third Trajectory with a Velocity of 1 m/s

Figure 7 :

 7 Figure 7: Measurements of the Disturbance Rejection Experiment in the Workspace Center r 0

Figure 8 :

 8 Figure 8: Behavior of the IPAnema Mini for Various Velocities During and After the Three Trajectories

Table 1 :

 1 IPAnema Mini Geometry

Table 2 :

 2 Static Experiments ResultsPosition Method f [N] f w[N]

	p	p w	∥∆r∥ [mm]

Acknowledgment

The authors wish to thank the University of Castilla-La Mancha and the European Social Fund for their financial support through the pre-doctoral funding [2019/451] and mobility grant for doctoral students 2020.