Chantal Morley

Martina Mcdonnell

The gendering of

Keywords: Computer programming, gendering process, Europe, discrimination

This chapter documents the role that women played in the computing field in three different European countries from the late 1960s into the early 1990s: Finland, a late-comer to the computer industry which was then deemed of national importance; France which boasted several computer manufacturing companies, and where IT service companies played an important role in the early history of computing; and the United Kingdom, also involved in computer manufacturing, but where the public sector played a major role. We will see that despite national differences, similarities exist concerning the role women played in the computer industry and that the masculinisation of the profession can be attributed to similar causes. Initially jobs were considered unskilled and marked out as women's work. When women acquired the necessary skills to play a more important role, various forms of discrimination slowly discouraged them from staying in computer science. The study of these three countries at the moment when computing was introduced into the public and private sectors and became a major tool for management and strategic decisions, shows how software activities were socially constructed as masculine.

Introduction

The history of computer science in the United States clearly shows that women played active roles in its beginnings. They were largely responsible for operating the first American electronic calculators, and several of these women took part in major developments in computer software and computer languages [START_REF] Gürer | Pioneering Women in Computer Science[END_REF][START_REF] Breton | Women in computing: historical roles, the perpetual glass ceiling, and current opportunities[END_REF]. Starting in the 1960s and during the two following decades an ever-increasing number of women entered the new field of computer programming [START_REF] Ensmenger | The computer boys take over: Computers, programmers, and the politics of technical expertise[END_REF][START_REF] Abbate | Recoding gender -Women`s Changing Participation in Computing[END_REF]. In Europe as well computer programming jobs from the late 1960s into the early 1990s were largely held by women. In this chapter we will document this trend by contrasting the situation in three different countries: Finland, a late-comer to the computer industry which was then deemed of national importance; France which boasted several computer manufacturing companies, and where IT service companies played an important role in the early history of computing; lastly the United Kingdom, also involved in computer manufacturing, but where the public sector played a major role. We will see that despite national differences, similarities exist concerning the role women played in the computer industry and that the masculinisation of the profession can be attributed to similar causes.

The role of women in the institutionalisation of computing in Finland

This section draws heavily on the work of Vehvilaïnen (1997Vehvilaïnen ( , 1999)), who explored the culture and practices of computing in Finland by examining the archives of the Finnish Information Technology Society and the autobiographies of Finnish pioneers in computer science. Women in Finland withdrew from the information technology field in the early 1990s as they did in most Western countries. However, for twenty years, from the 1960s to the 1980s, Finland presented a paradox: on the one hand many women were involved in key activities of IT such as programming and analysis, but on the other hand they were almost completely absent from the decision-making process which contributed to the institutionalisation of IT in Finland. In this section we will first show how women were kept out of the spheres of decision-making despite Finland's reputation as a pioneer in female/male equality (in 1906 it was the first country in the world to introduce women's right to vote and equal access to public office). We will then track women's involvement in the field of Information technology in Finland.

The participants in the development of IT in Finland

Compared with other Western countries Finland is considered to be a latecomer to computing. The first Finnish computer, the ESKO, based on the German computer G1, was not built until the late 1950s. After World War II the aim was twofold -reduce Finland's dependence on German technology and at the same time strengthen national culture. Nonetheless the first computer in operation in Finland was an IBM 650 bought by the Finnish Postal Bank in 1958, and the ESKO, already obsolete, was donated to the University of Helsinki. Despite this late start there was a rapid increase in the number of computers, and ten years later more than 160 computers were in operation in Finland. Although initially dependent on foreign technical knowledge and expertise, Finland soon gained its technological independence. Three different institutions played a major role in this national and patriotic project [START_REF] Paju | National projects and international users: Finland and early European computerization[END_REF]. Women however were almost totally absent in this early stage.

The first institution involved was the Punched Card Association (renamed The Computer Association in 1960). It published a journal aimed at disseminating knowledge of electronic machines. Membership was selective and members had to be considered experts (either those with considerable experience or those who had worked in top management positions). Only 3% of the members were women, none held a leadership or editorial position, and working groups were allmale. At the beginning of the 1970s, the Association modified its rules and encouraged computer professionals, which included many women, to become members. The participation of these women in discussions on professional practices has been widely documented in the Association's journal. Nevertheless, no articles were written by women in this journal until as late as the 1980s.

The Association of Data Processing (ADP), which also played a key role, was founded in 1961 by Otto Karttunen, a former World War II army officer. Its aim was to organize training in computer science across Finland. It was run rather like a "secret society" (Vehvilaïnen 1999: 46) -membership was limited to those who held leadership or management positions in computing. Until the end of the 1960's, no women had joined the "club". ADP members believed they had a mission to promote national progress, and its most influential members wrote to each other as "my good brother" (id.). The ADP and the Computer Association were actively involved in setting up computer departments and services in both the public and private sector. Both groups also worked on defining the ADP training programs and on developing the methods that organized work and practices in computing.

The third institution that influenced the development of computing in Finland was IBM. This company dominated the computer market in Finland in the 1950s and was also the unique source of computer training in Finland for several years (end 1950s to beginning 1960s). IBM like many other companies was male dominated, and women long remained in a minority despite government measures in favour of gender equality.

The place of women in IT professions in Finland

In Finland the male -dominated founding bodies and the absence of critical debate among decisionmakers resulted in the Information Technology (IT) world becoming a closed culture implicitly linked to men. Nevertheless, decision-makers increased the number of training programs that were open to all, regardless of sex or social class, in order that Finland reach a level enabling it to be a player on the international scene. Women benefited greatly from this opportunity. Between 1960 and 1980, the percentage of female programmers rose from 7% to over 30%, and the number of female analysts from 9% to 30%. However, the entry of women into computer professions must also be seen as part of a wider societal evolution. As early as the 1960s, female/male equality gave rise to much debate and led to measures being taken, notably in the field of education and childcare. Women entered the job market on a broad scale, and rapidly the number of women with high school diplomas had exceeded the number of men. By 1980, 30% of all computer professionals were women.

Computer programmers were recruited on the basis of tests. Many women applied and were hired. Nevertheless in the 1960s women's expertise was undervalued. They had little chance of promotion, and the experts continued to be men (Vehvilaïnen 1997: 441). The situation evolved during the 1970s when universities began to offer courses in computer science. Soon 30% of students in these courses were women, and these women went on to hold senior positions in companies and institutions. In many universities between 1980 and 1990, almost 50% of those studying Information Technology were female. This figure began to decline from 1990 onwards. Moreover technical universities continued to harbour a predominantly male culture, and few female students applied. In the 1980s only 8 to 10% of students in these schools were women and this figure decreased during the 1990s.

The case of Finland: Concluding remarks

To sum up, computer science in Finland, introduced and developed in the 1950s, was considered to be strategic and in the national interest. Only men and often old comrades-in-arms were involved in its promotion and development. Women were absent not only in the national decision-making spheres but also from associations and publications. On the other hand, as soon as they were given the opportunity they chose to study information technology and to work in computer-related jobs. The case of Finland illustrates that the determining factor in branding the field of computing as masculine was not so much men and women's relationships to technology but rather the fact that a small group of socially dominant men occupied the terrain in terms of both decision-making and visibility. Furthermore the increasing number of women in university IT courses, in contrast to the almost total absence of women in similar courses in technical institutions, makes it clear that it is not the discipline that is at issue here (IT courses being similar in both types of institutions) but rather a difference of culture. Technical universities retained practices that discouraged women from entering the field. To illustrate the role of gender in limiting access to certain areas, Vehvilaïnen reported in reference to Otto Karttunen "In Karttunen's world, there were three kinds of women: 1) those related to machines within the routines of scientific management, 2) the attractive secretaries, and 3) one token woman who was able to make men laugh during the lunch break" (Vehvilaïnen (1999: 48).

The role of women in the development of computing in France

France lagged behind Finland and other European countries in granting political equality to women; women's right to vote and to hold public office was not won until after World War II. However, women were admitted to study sciences at university as early as the second half of the 19 th century. Nevertheless, the more renowned and highly respected French engineering schools continued to maintain a male-only policy well into the 20 th century (Marry 2003 : 19). In 1917, the loss of a great number of young men during World War I convinced leaders of the need to allow women access to engineering jobs. Thus four engineering schools opened their doors to women. In 1925, a female engineer Marie-Louise Paris, frustrated that few women were being admitted into these schools, set up an engineering school exclusively for women, the Ecole Polytechnique Féminine (EPF). However, women would have to wait until 1972 for all engineering schools to be co-educational, notably the highly prestigious Ecole Polytechnique. A second institutional mechanism contributed to prevent women from accessing positions of power in computing: the 'engineering corps' [START_REF] Canepa | Mission d'étude sur l'avenir des corps d'ingénieurs de l'Etat[END_REF]). Since the 18 th century the French State has recruited engineers on a very selective basis. These engineers become government employees and are organized into 'engineering corps' according to their area of expertise -mining; civil engineering; telecommunications engineering, etc. The majority of these engineers are former students of the Ecole Polytechnique. They then follow additional courses at the other engineering schools (Mines, Ponts et Chaussées, Télécom, etc.) During the period 1960 to 1990, this 'engineering corps', notably the graduates from the Ecole Polytechnique played an important role in industrial development in France (Blazy 2009), acting as key decisionmakers in government bodies and state-owned companies (in the fields of electricity, nuclear energy, telecommunications, armaments, aeronautics, transport, etc.). In addition, the Ecole Polytechnique alumni represented an influential network in French politics and economics [START_REF] Kosciusko-Morizet | La 'Mafia' polytechnicienne[END_REF]. This was notably true in the development of computing in France.

The major players in the development of computing in France.

The French State played a major role in the early days of the development of computer science and the computer industry in France. In 1954, SEA (Société d'Electronique et d'Automatisme) secured a military contract thus allowing France to join the "club" of manufacturers of large computers. The French State continued to intervene throughout the turbulent history of computing, both politically and economically in particular in the creation and mergers of computer manufacturing firms -CII (Compagnie Internationale pour l'Informatique), Bull, Unidata, CII-Honeywell-Bull, R2E, SEMS, etc. (cf. [START_REF] Griset | Aux origines de l'informatique française : entre Plan Calcul et UNIDATA, 1963-1975[END_REF]. A government programme, the 'Plan Calcul', was launched in 1966 with the aim of developing a national computer manufacturing industry. In addition to the construction of large computers, the world's first micro-computer (Micral) was developed in France in 1972 thanks to a government contract. The State continued to support the development and commercial distribution of mini-computers. As a result the total number of computers in use increased considerably. While in 1962 there were only 600 computers installed in France, by 1970 there were 1,300. Other major players in the computerisation of both companies and public sector bodies were the IT services companies. Their mission was to provide qualified personnel and turn-key solutions. They flourished between 1958 and 1973, and were more numerous in France than elsewhere (Lacombe and Rosé 2011: 10). Several were set up as subsidiaries of banks or major industrial firms (Crédit Lyonnais, Société Générale, Crédit du Nord, France Telecom, Schlumberger, Péchiney, and Alcatel). Thousands of companies specialised in programming were formed between 1965 and 1969 but more than half had folded ten years later (Lacombe and Rosé 2011: 31). However, the majority of the firms which were to play a decisive role in IT services at an international level (Sema, SESA, SOPRA, CGI, STERIA, Sligos) were started by engineering graduates of Ecole Polytechnique [START_REF] Bret | L'histoire des 40 premières années des SSII en France à travers leurs hommes et leurs activités[END_REF].

The women involved in the growth of IT in France.

Between 1960 and 1975 women were almost entirely absent from decision-making and entrepreneurial spheres with regard to the development of computing. Firstly, they were late in gaining access to the various "engineering corps", notably those which dominated the history of IT as shown with the case of the Ecole Polytechnique, and it was not until the 1980s that a few female engineers were admitted to the influential "corps des mines" or "corps des telecommunications". Despite this, women were interested and active in computer science. We can see this by examining three different areas: computer design, computer science studies and IT professions.

In computer design, in the second half of the 1950s two female engineers, Alice Recoque (ESPCI-Ecole Supérieure de Physique et de Chimie Industrielles, graduate engineer) and Françoise Becquet (EPF-Ecole Polytechnique Féminine, graduate engineer) designed a small conversational machine, the CAB 500, a forerunner of the personal computer and SEA's biggest commercial success. Following the merger of SEA with the newly-formed company CII in 1966, Alice Recoque was put in charge of the design of CII's Mitra range of mini-computers, notably the Mitra 15 at the end of the 1960s. The latter was also very successful both technically and commercially (Mounier-Kuhn 1990). These minicomputers were not within the scope of the 'Plan Calcul' which may explain why a woman was placed in charge of research in what was then termed 'small computers'. Despite her achievements, Alice Recoque never reached a position of responsibility either in the public or private sector.

Concerning higher education, women in France as in Finland were attracted to computer science studies. In 1972 women represented less than 5% of students in engineering schools, but the highest percentage of women (9%) were in engineering schools specialising in computer science (Marry, 2004: 109). Ten years later, in 1983, the percentage of female engineering students in schools specialising in computer science peaked at over 20%. Then numbers began to fall, with a sharp decrease at the end of the 1980s, despite the fact that the overall number of women entering engineering schools continued to increase during this period. The case of the INSA engineering school in Rennes is noteworthy. For three consecutive years male/female parity was more or less achieved (percentage of women in 1978: 47%; 1980: 55%; 1981: 50%) (Collet 2005: 29). In a study carried out on ten engineering schools in France, Isabelle Collet showed that in the 1970s the number of female computer science graduates was relatively high in comparison to the number of male graduates, given the higher percentage of male students in engineering schools overall. However, she pointed out that at the end of the 1980s the figures for female graduates in computer science remained stable whereas the number of male graduates soared. In 1979, male female parity was reached, with approximately 100 students overall, whereas in 1989 there were about 120 female students and 400 male students (Collet 2005: 31).

Lastly many women clearly chose to work in the field of computer science. A study conducted in 1969 [START_REF] De Peslouan | Qui sont les femmes ingénieurs en France ? PUF[END_REF] showed that 30% of EPF (Ecole Polytechnique Féminine) graduates worked in computing. For other engineering schools the trend was less pronounced. However, the author noted that 17% of all female engineering graduate students aged thirty and under chose to work in computing. At the beginning of the 1980s a government report on the range and variety of jobs held by women, noted that the computer science field was evidently not marked by gender stereotypes as were fields traditionally dominated by men. The authors of the report were optimistic: "We may therefore hope that the advent of new technologies and the ensuing changes in work organization will allow women, if they are able to seize the opportunity, to find their place in the new technical fields," [START_REF] Appert | Diversification de l'emploi féminin: insertion professionnelle des femmes formées dans des métiers traditionnellement masculins[END_REF], quoted by Collet 2005: 55, our translation). In the 1980s many IT services firms trained graduates, and recruitment was very open. In 1982, 35% of computer scientists were women. However, by the end of the 1980s the trend began to revert: in 2002, women represented 20% of the computer science workforce (DARES 2004) and by 2011 the IT engineering profession figured among the 20 professions which contributed most to job segregation (DARES 2013). Concerning entrepreneurship, few women set up IT services firms. For example, in 1974 Marie-Therese Bertini developed a structured programming method that encountered some success. Nevertheless, MBT, the company she set up in 1985, never reached international scope although it is still operating. Despite her expertise in methodology she was not invited to participate in the 'Merise' project i.e. the development of a national information system development methodology.

The case of France: Concluding remarks

Women had little access to certain positions -decision-makers or key players -during the early period where there was a rapid development of IT. This was due to their late entry to the institutions that played a major role in this development. As regards setting up a business, women encountered more obstacles than men for two major reasons. First, they did not belong to the alumni network of the 'Grandes Ecoles', considered a major advantage in France in terms of securing partners and clients. Second, until 1965 according to French law married women were not entirely independent legally. They required their husbands' permission to open a bank account, to manage family property, their own personal property, or even to hold a job. Nevertheless, early on female engineers were interested in the new and exciting area of computers and computing. To women at that time the field appeared very open, as Françoise Becquet who started working for the manufacturer SEA in the late 1950s explained: "It didn't matter if you were a man or a woman, a French national or a foreigner as long as you worked diligently and did the job well" (Becquet 2010; our translation).

At the beginning of the 1980s, more and more women moved into the area of software. Nonetheless, a detailed study [START_REF] Stevens | The professional fate of woman engineers in the computer sciences: Unexpected reversals[END_REF] clearly showed that women were the first, voluntarily or not, to leave the field as soon as the sector was hit by a crisis. IT once again became a male activity in the early 1990s. "It is certain that IT jobs and careers are open to women. Neither their knowledge, interest or competence in the field can be questioned. Women are present in IT. However, it will be difficult for them to say they are computer engineers. Often they have the impression of being guests in the profession and of being welcome for their subsidiary 'feminine' qualities" concluded a study of female scientists some of whom had spent their entire career in IT services companies [START_REF] Collet | L'informatique a-t-elle un sexe? Hackers, mythes et réalités[END_REF]: 79, our translation).

Women and IT in the public sector in the United Kingdom

In the United Kingdom as early as the 19 th century the struggle for equal rights for women gave rise to diverse social movements. At the turn of the 20 th century most British universities were open to women, although the most prestigious universities were the last male-only strongholds: Oxford (1920) and Cambridge (1948).In 1928 women obtained the right to vote. 1954 saw the introduction of a policy of equal salaries for men and women in the civil service. Between the years 1955 and 1985, the public sector occupied an ever-increasing position in the country's economy (Clark and Dilnot 2002: 2) with the welfare state and the wave of nationalisations (banks, industry, transport, etc.). Along with Germany and the United States, the United Kingdom was a pioneer in the computer industry (e.g. Colossus Mark1 in 1944;EDSAC and Manchester Mark in 1949;Elliott 152 in 1950;LEO 1 in 1951;Ferranti Mark 1 in 1951 ;DEUCE in 1955 ;Pegasus in 1956). Computers were widely used to support growth in the public sector. Contrary to what one might think women did not receive equal treatment with men in this new area of activity. The 'meritocracy' ideal was to suffer as work in the field of computing required higher qualifications and was better paid. The obstacles notwithstanding, many women worked as operators and programmers. However faced with discrimination, certain pioneering women decided to set up their own IT businesses.

Female computer scientists in the civil service and the nationalized industries

This section draws heavily on the work of Hicks which focused on gender in the history of computerisation in British government from 1950 (2010; 2011). Before 1960, operating calculators required trained staff for command entry, fine-tuning of cables and replacing faulty tubes. These jobs were called 'machine positions' in the civil service job classification system and were almost exclusively held by women. In 1954, it was decided that these positions would not be covered by the equal salaries legislation. Furthermore, the few existing male operators were shifted to higher paid jobs with better prospects of promotion. Electronic computers began to be installed in civil service and government departments at the end of the 1950s. This required the staff to acquire new technical capabilities but also management skills as the number of computer related projects increased.

Women who had been hired to write code for the calculators progressed to programming computers. By 1963, 70% of programmers were women. Nevertheless, it was considered inappropriate for women to hold management positions, especially as the ever-increasing need for programmers meant attracting more men into the profession. Moreover, it was customary for young women to give up working when they married. Initially programmers were recruited internally from middle management within the civil service which was heavily male dominated (90% men). Senior women programmers, although required to train the new young male recruits, had no prospects of promotion. This wave of recruitments was relatively unsuccessful, partly because the aptitude tests were ill-adapted and did not guarantee that recruits would be competent programmers. Yet despite the preference for recruiting men, women for a variety of reasons did not abandon the programming field. In the first place women increasingly chose to continue working after marriage, and secondly there was a growing need for programmers. Also following the 1965 economic crisis, wages in the civil service were frozen, notably those of the IT staff which made the jobs less attractive. Young men did not rush to join a sector of activity which was predominantly female and thus with low prospects. In the end the government decided to recruit externally, targeting those under twenty-five. No qualifications were required and no reference was made to gender in the job advertisements. Many women applied and were hired to be operators and programmers and the principle of equal pay was better respected. Nevertheless female programmers and operators gradually left the profession due to discrimination encountered in their career path. For example, at the end of the 1960s the high salaries women earned were considered an anomaly and they were refused promotion while their male colleagues were promoted. Women became increasingly rare in computer departments (Hicks 2011: 161).

In her research on women in computing Abbate (2012: 117-119) pointed out the difficulty, in both Great Britain and the United States, for a qualified female IT engineer to come back to a salaried job after maternity leave. She describes the case of two female British computer engineers who had been working for computer manufacturing companies -Bobby Hersom at Elliott Brothers and Mary Berners-Lee (the mother of Tim Berners-Lee) at Ferranti -and who regretted having to quit their jobs after the birth of their first child. They then had to be content with working freelance.

But discrimination also led two British women, both of whom held degrees in mathematics and were enthusiastic about the fledgling computer industry, to set up their own software companies.

Two pioneering women in the software and IT service industry

Dina St Johnston, née Vaughan, began her career in 1947 working in a research institute while continuing to follow evening classes at Croydon Polytechnic [START_REF] Lavington | An Appreciation of Dina St Johnston 1930-2007 Founder of the UK's First Software House[END_REF]Porter 2008). She passed the entrance exams to two major universities (Bedford and Royal Holloway), but 90% of the places were reserved for World War II veterans and the remaining 10% were allotted to men. She finally enrolled in London University and obtained a degree in mathematics. However the company she was working for refused her a promotion (which was systematically granted to men under similar circumstances). She therefore changed jobs and went to work for Elliott Brothers, a computer manufacturing company. There she learned computer programming and soon proved to be brilliant. She married a colleague at Elliott Brothers and in 1959 left the firm to set up her own company. She founded the first private sector IT services company in the United Kingdom (Vaughan Systems and Programming). It specialised in software for industrial control and automation systems (warehouse automation, flight reservation systems, railway signalling systems). The company even built small dedicated computers such as the Vaughan 4M which introduced an automatic system for locating trains. Dina Vaughan continued programming until 1996 when she sold her company to retire. Three years after Dina St Johnston, Stephanie Shirley also set up her own company. She had joined the research department of the London Postal Services in 1951 (Abbate 2012: 125 -143). She took evening classes and obtained an honours degree in mathematics in 1956. She discovered computer science and became enthusiastic. She worked on several important computer related projects for the Post Office. Although she was eligible for promotion having received a degree in mathematics, she was not promoted. The members of the promotion panel refused to consider appointing a woman to that higher grade. She left the Postal Services in 1959 and went to work for a software development company that was a subsidiary of two computer manufacturing companies GEC computers and ICT. She led a team carrying out tests on a new computer. She was very well paid however the "glass ceiling" once again reappeared. When she made suggestions to the project steering committee, she was asked not to intervene in the project management meetings on non-technical subjects and to keep to her role as technician. She thus understood that she would never be able to move up the ladder in the company, and she decided to become her own boss (Abbate 2012: 126). She was also hoping to have children and wanted to continue on working afterwards. In 1962, at the age of 29 she set up a software development company (Freelance Programmers) using the name Steve Shirley (Id.: 138). She adopted the name Steve so that potential clients would not know she was a woman until the first face-to-face meeting. She recruited women who wanted to work part-time in order to look after their children. All the analysts and programmers were women who worked freelance from home. The company soon became a British success story: in 1965 there were 65 employees, almost exclusively female until the 1975 Sex Discrimination Act required the company to hire more male employees. By 1986 16% of employees were men. In an interview Stephanie Shirley quotes one of her female employees who had difficulty persuading the Inland Revenue Service that she had earned her money honestly working at home and looking after two children [START_REF] Shirley | A company without offices[END_REF]! By 2009 the company, then known as F International, boasted 1,000 employees in three different countries (the U.K., Denmark and the Netherlands), with sales of $10 million and an annual growth rate of 30%. The name of the company was subsequently changed to Xansa and was bought by the French IT services company Steria in 2007. Stephanie Shirley was honoured with the title of 'Dame' in 2000 for her contribution to the British computer software industry.

The case of the United Kingdom: Concluding remarks

After World War II, the United Kingdom invested in computers to compensate for its decline as a world power. The public sector adopted IT early on. The civil service equality policies notwithstanding, computer coding was initially considered as women's work, badly paid and with no prospects for promotion. These jobs were then voluntarily redefined as male because it became imperative to reposition computing as a male field to meet the challenges of computerisation. On the other hand stereotypes concerning women's technical incompetence were unknown. Later, in the years 1960 to 1980, other factors contributed to preventing women from establishing themselves in computer science. The first factor was low pay, as male/female work categories were maintained thus circumventing the 1954 law on equal pay. Finally higher salaries were offered because of the need to recruit more staff in computer departments, women employees were soon considered to be over paid and their career advancement was blocked, but this was not true for male employees. The second factor was the question of careers. Despite a 1969 government campaign to encourage young women to join the workforce, women were not really supposed to have a career, in particular if they had children. Employers were not encouraged to train or promote women and most believed women would leave employment early on. The last and perhaps most important factor was, by not allowing female computer scientists, regardless of their abilities, to hold management positions, most of these women remained in subordinate positions. Nevertheless, women showed a real interest and enthusiasm for the computer industry. This is clearly illustrated by the successful pioneer female start-ups in IT services which expanded employing only women but also paid decent wages.

Differences and common traits

Historical studies of women in IT in Europe are as yet rare or incomplete, and it is virtually impossible to establish a strict comparison of the situations of Finland, France and the United Kingdom. It is interesting nonetheless to examine the three countries during the period between 1960 and 1990 despite the fact that the data for each country focused on different aspects of the situation. In the first place, in all three cases there was a strategic commitment to invest in the manufacturing of computers and the State played a role in this. On the other hand, the countries differed in terms of female/male equality. Finland had a pioneering role, the United Kingdom was well known for women's movements and equal rights and in France women only gained access to certain educational institutions and economic independence in 1960. Despite these differences, common traits can be identified concerning women in computing professions. First we see that women took advantage of any opportunity that arose to enter the field: In Finland, young women applied in great numbers to study the new field of computer science; In France, computer engineering became the most popular job choice among young female engineering graduates; In the UK great numbers of young, unqualified women took up programming jobs in the civil service. We then see that the stereotype of women's presumed technical incompetence, which later became so pervasive, rarely came into play until the late 1980s. At the outset technical incompetence was not identified with women in this relatively new field. One possible explanation is that the countries considered as pioneers in the field of computing, notably the United States and the United Kingdom, employed women to operate and code the early computers. This implies that women in these countries were familiar with and worked in proximity with computers. Nevertheless, it must be noted that as early as the 1950s advertisers used images of women to convey the simplicity of using computers and thus to promote sales. Such advertising campaigns may have given rise to the idea that women working in the field of computer science were less skilled [START_REF] Hicks | Only the Clothes Changed: Women Operators in British Computing and Advertising, 1950-1970[END_REF]. Moreover, we observe that in the early days of computer technology certain structural barriers kept women not only out of the decision-making spheres and out of the highly visible positions in the field -for example, the all male composition of the founding bodies of Finnish computing, i.e., the army, IBM, etc., and the gendered practices such as the old boys' club mentality; in France women prevented from accessing both the 'engineering corps' and entrepreneurship; and in the UK the barriers to women's career advancement both in the public sector and in the computer construction industry. The study of these three countries at a moment when computing was introduced into the public and private sectors and became a major tool for management and strategic decisions, shows how software activities were socially constructed as masculine. Lastly, we note, a turning point at the end of the 1980s. Although the number of women studying computer science or working in computing continued to increase, the trend was beginning to reverse: the percentage of women declined slowly as large numbers of men hastened to join a promising sector that provided well-paid employment. And when the computer manufacturing industry entered a crisis, women's withdrawal from the field continued, as they were nowhere firmly established in decision-making spheres or positions of power and responsibility. This appears to have particularly affected female computer scientists in France. Many of them chose to move into non-technical professions. And in the United Kingdom the civil service recruitment and career advancement policies favoured men thereby pushing women out of computing in the public sector.

The over-representation of men in computer science in the 1990s has also been documented in the United States (Klawe and Levenson 1995) and elsewhere in Europe, notably in Germany [START_REF] Oechtering | Situations and advancement measures in Germany[END_REF]. Starting in the mid 1980s and increasingly throughout the following decade, technological innovations and social practices around computers raised new and additional issues. The diffusion of home computers contributed to creating new gendered spheres and practices in households [START_REF] Miquel | Mythologies modernes et micro-informatique. La puce et son dompteur. L'Harmattan[END_REF][START_REF] Jouët | Le vécu de la technique[END_REF][START_REF] Grundy | Women and computers[END_REF]. The hacker phenomenon, recognized as an emerging culture [START_REF] Turkle | The second self. Computers and the human spirit[END_REF], and the communities which emerged from the free-software movement led by Richard Stallman in the United States in 1985 followed by Linus Torvalds in Finland in 1991, contributed to a widespread representation of computing as a male activity [START_REF] Collet | L'informatique a-t-elle un sexe? Hackers, mythes et réalités[END_REF]. Thus as computers and computing moved beyond the professional sphere, computer technology emerged as a masculine culture (Wacjman 1991: 141-142). Nevertheless our study shows that women had already begun to leave computing before the advent of micro-computers and network technology. Thus women in IT should be studied within the broader context of the long-term evolution of computing technology, industry and communities.


Abstract-The sit-to-stand movement is a key feature for wide adoption of powered lower limb orthoses for patients with complete paraplegia. In this paper we study the control of the ascension phase of the sit-to-stand movement for a minimally actuated powered lower limb orthosis at the hips. A particularly important objective of the design is to ensure robustness to variations in parameters of the users, such as weight. First, we generate a pool of finite horizon Linear Quadratic Regulator feedback gains, designed under the assumption that we can control not only the torque at the hips but also the loads at the shoulders that in reality must be applied by the user. Next, we conduct reachability analysis to define a performance metric measuring the robustness of each controller against parameter uncertainty, and choose the best controller from the pool with respect to this metric. Of course in practice the loads at the shoulder are ultimately provided by the user, while the computer control is only applied to the hips. Therefore in the second part of the paper, we remove the assumption of computer control for the shoulder loads, and study how the shoulder actions could be learned by a human during rehabilitation and physical therapy. As an abstraction of this process we choose an Iterative Learning Control algorithm to replace the nominal shoulder control. Indeed this algorithm obtains torque and forces at the shoulders that result in successful simulations of the sit-tostand movement, regardless of parameter uncertainty and factors deliberately introduced to hinder learning. Thus it is reasonable to expect that the superior cognitive skills of real users will enable them to cooperate with the hip torque controller through training. Index Terms-Assistive devices, orthotics, rehabilitation robotics, exoskeletons, optimal control, uncertain systems, reachability analysis, sensitivity, iterative learning control, learning, optimization.

I. I

P OWERED Lower Limb Orthoses (PLLOs) are medical devices worn in parallel to the legs to assist standing and/or walking. State of the art PLLOs for people with paraplegia (≈ 115, 000 individuals in the USA [1]) are commercially known as medical exoskeletons. Their users must have healthy enough skeletal, cardiovascular, vestibular, and visual systems to tolerate standing, as well as mobility in hands, arms, and shoulders to interact with the ground by means of crutches. The majority of exoskeletons are equipped with actuation at the hips and knees [START_REF] Cyberdyne | HAL for medical use[END_REF]- [START_REF]An Exo created for you[END_REF], but the most affordable [START_REF] Suitx | PhoeniX Medical Exoskeleton[END_REF] uses a minimally actuated architecture where O. Narvaez-Aroche and A. Packard are with the Department of Mechanical Engineering, University of California, Berkeley, CA, 94720 USA (e-mail: ocnaar@berkeley.edu, apackard@berkeley.edu). P-J. Meyer, S. Tu, and M. Arcak are with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720 (pjmeyer@berkeley.edu, stephent@berkeley.edu, arcak@berkeley.edu).

Manuscript received July 6, 2019; revised September 22, 2019. torque is exclusively applied at the hips [START_REF] Mckinley | Design of lightweight assistive exoskeletons for individuals with mobility disorders[END_REF]. In addition gait cycles on level ground with this design look more natural than those of its competitors. However, it is more difficult for the users to perform the sit-to-stand (STS) movement, which is the sequence of actions for rising from a chair. The STS movement is executed in three distinctive phases: preparation, ascension, and stabilization, as illustrated in Figure 1. The ascension phase (Figure 1b) starts at seat-off and ends when the links of the shanks and thighs segments almost align with the vertical, and the torso has a slight forward tilt, with all angular velocities close to zero in order to facilitate stabilization about the standing position. It is the most challenging phase because of the greater ranges of joint motion, torques, and forces involved. It also requires precise coordination between the actuators of the PLLO and the loads applied by the upper limbs of the user to avoid sit-back or step failures [START_REF] Eby | Modeling and control considerations for powered lower-limb orthoses: A design study for assisted STS[END_REF].

This paper makes two main contributions. First, it provides a performance metric to quantify the robustness against parameter uncertainty of a controller for the ascension phase of a PLLO. Second, it assesses through numerical simulation if a controller would be suitable for clinical implementation, using an Iterative Learning Control (ILC) algorithm as a proxy for the loads applied by the user at the shoulders during rehabilitation and physical training with the PLLO.

Section II reviews the Euler-Lagrange equations of the threelink robot model of the system, and our procedure for planning the motion of the PLLO, which is based on the desired angular position and velocity of the links for the thigh segment, and the kinematics of the center of mass (CoM). It also derives finite horizon LQR controllers to track the reference trajectories under the provisional assumption that the controller has authority not only over the torque of the actuators at the hips of the PLLO, but also over the torque and forces at the shoulders of the user.

In Section III we select from a pool of finite horizon LQR controllers the one that optimizes a performance metric that measures robustness against parameter uncertainty. This metric is defined using a suitable reachability analysis when the parameters vary within given intervals.

Since the PLLO can only drive the actuators at the hips, in Section IV we propose an ILC algorithm to simulate the loads at the shoulders that would be applied by a user, when being trained to perform the ascending phase of the STS movement in closed-loop with the optimal controller. To avoid identification experiments that would expose the user to non validated controllers, we tune the internal gains of the ILC with a derivative free optimization approach. We confirm that this simple proxy for the user achieves successful STS movements after a reasonable number of iterations despite considerable weight fluctuations and factors hindering learning.

Companies producing PLLOs for people with complete paraplegia are moving towards stand-alone mobility solutions that can be operated outside of rehabilitation centres, and without the supervision of a specially trained physical therapist. This calls for extensive clinical trials for certifying the safety and feasibility of their designs to stand up and walk under a wide variety of conditions, as was done in [START_REF] Baunsgaard | Gait training after spinal cord injury: safety, feasibility and gait function following 8 weeks of training with the exoskeletons from Ekso Bionics[END_REF], [START_REF] Zeilig | Safety and tolerance of the rewalk™ exoskeleton suit for ambulation by people with complete spinal cord injury: A pilot study[END_REF] to certify the potential benefits on gait function and balance. Even though our simulations cannot replace such tests, they can be valuable tools for improving both the mechanical design and control strategies of the devices prior to a comprehensive training protocol for the STS movement.

Notation: Coordinate aligned boxes play an important role in this study. For a, b ∈ R n we use the notation a ≤ b to mean This section describes the three-link robot model used for motion planning, control design, and reachability analysis. The contents of this section and Section III-A that follows were published in [START_REF] Narvaez-Aroche | Finite time robust control of the Sit-to-Stand movement for powered lower limb orthoses[END_REF] and [START_REF] Narvaez-Aroche | Reachability analysis for robustness evaluation of the Sit-to-Stand movement for powered lower limb orthoses[END_REF]. They are included here for selfcontainment of the paper.

a i ≤ b i ∀i, define an interval of R n as [a, b] := {ξ ∈ R n |a ≤ ξ ≤ b} ⊆ R n , and compute its volume vol([a, b]) ∈ R as: vol([a, b]) = i ∈ {1,...,n} (b i -a i ). For matrices Λ, Λ, Λ ∈ R n×m we write Λ ∈ Λ, Λ if Λ i j ∈ Λ i j , Λ i j ∀(i, j) ∈ {1, . . .,

A. Model of the Powered Lower Limb Orthosis and its User

Assuming sagittal symmetry, no movement of the head relative to the torso, and that the feet are fixed to the ground, we model the user, crutches and PLLO as a three-link planar robot with revolute joints coaxial to the ankles, knees and hips, as shown in Figure 2. θ 1 is the angular position of link 1 (shanks) measured from the horizontal, θ 2 is the angular position of link 2 (thighs) relative to link 1, and θ 3 is the angular position of link 3 (torso) relative to link 2. The system parameters are the masses of the links m 1 , m 2 , and m 3 ; the moments of inertia about their respective CoMs I 1 , I 2 , and I 3 ; their lengths l 1 , l 2 , and l 3 ; and the distances of their CoMs from the joints l c 1 , l c 2 , and l c 3 . The actuators of the orthosis exert torque τ h about the hips; while torque τ s , horizontal force F x and vertical force F y capture the inertial and gravitational forces of the arms and loads applied on the shoulders of the user. There is no actuation at the knees. In terms of the joint angles vector θ := [θ 1 ; θ 2 ; θ 3 ], input u := τ h ; τ s ; F x ; F y , and parameter p := m 1 ; m 2 ; m 3 ; I 1 ; I 2 ; I 3 ; l 1 ; l 2 ; l 3 ; l c 1 ; l c 2 ; l c 3 , the Euler-Lagrange equations of the three-link planar robot in Figure 2 can be written as

M (θ (t), p) θ (t) + F θ (t), θ (t), p = A τ (θ (t), p) u (t), (1)
where M (θ, p) ∈ R 3×3 , M (θ, p) 0 is the symmetric mass matrix of the system, F θ, θ, p ∈ R 3 is the vector of energy contributions due to the acceleration of gravity and Coriolis forces, and A τ (θ, p) ∈ R 3×4 is the generalized force matrix. Their entries are shown explicitly in Appendix A. Three rigid link dynamic models like this have been used to accurately describe the STS movement of human subjects [START_REF] Eby | Modeling and control considerations for powered lower-limb orthoses: A design study for assisted STS[END_REF].

B. Sit-To-Stand Motion Planning

Biomechanical studies measure the kinematics of the CoM of the human body instead of joint angles to classify and assess dynamic balance of the STS movement [START_REF] Fujimoto | Dynamic balance control during Sit-to-Stand movement: An examination with the center of mass acceleration[END_REF]. Therefore, considering θ 2 , and the position coordinates of the CoM of the three-link planar robot in its inertial frame (x CoM , y CoM ), we define z := [θ 2 ; x CoM ; y CoM ] and plan the STS motion over the finite time horizon t ∈ t 0 , t f with reference trajectories

θ2 (t) = θ2 (t 0 ) + θ2 t f -θ2 (t 0 ) Θ t, t f , xCoM (t) = xCoM (t 0 ) + xCoM t f -xCoM (t 0 ) Θ t, t f , ŷCoM (t) = ŷCoM (t 0 ) + ŷCoM t f -ŷCoM (t 0 ) Θ t, t f . (2)
For a rest-to-rest maneuver from ẑ (t 0 ) to ẑ t f , define Θ t, t f := -2

t 3 t 3 f + 3 t 2 t 2 f
; which is the only cubic polynomial satisfying Θ t 0 , t f = Θ t f , t f = 0, Θ t 0 , t f = 0, and Θ t f , t f = 1.

Relying on kinematic equations, it was shown in [START_REF] Narvaez-Aroche | Motion planning of the Sit to Stand movement for powered lower limb orthoses[END_REF] that for feasible and realistic STS movements excluding the vertical position (θ 1 = π /2, θ 2 = θ 3 = 0), a transformation of the form

θ (t) ; θ (t) ; θ (t) = h ẑ (t), ẑ (t), ẑ (t), p (3)
exists; so that the reference trajectories for the ascending phase in the z space from (2) can be mapped into the θ space with the nominal value of the parameter p. The derivation of this mapping is included in Appendix B. We use the computed torque method [START_REF] Slotine | Applied Nonlinear Control[END_REF] to get reference trajectories û (t). However, as the system of equations in (1) is underdetermined, at every t ∈ t 0 , t f a control allocation problem [START_REF] Johansen | Control allocation -a survey[END_REF] is solved to minimize the 2-norm of the input weighted by W u ∈ R 4×4 in the presence of box constraints u, u ∈ R 4 :

û (t) = arg min ξ ∈R 4 1 2 W u ξ 2 2 (4) subject to A τ θ (t), p ξ = M θ (t), p θ (t) + F θ (t), θ (t), p u ≤ ξ ≤ u.
We take W u := diag ([1, 1, 10, 1]) to ensure that the contributions from τh (t), τs (t) and Fy (t) outweigh Fx (t). To account for the limits of torque for the actuators at the hips, as well as the ones for the loads applied by the user at the shoulders, we set u : Different values for W u , u, and u might lead to different û (t) for the same θ(t), θ(t), θ(t), and p. Since a subject pushes the crutches down to propel upwards during a STS movement, the constraint F y (t) ≥ 0 must always be imposed.

C. Finite Time Horizon LQR Controller

Defining the state x ∈ R 6 as x := θ; θ , we first note from (1) that the dynamics of the three-link planar robot are

x (t) = θ (t) M -1 (θ (t), p) A τ (θ (t), p) u (t) -F θ (t), θ (t), p =: f (x (t), p, u (t)) .
(

) 5
Next we linearize (5) to design a finite horizon LQR controller for tracking the reference state trajectory x (t) := θ (t), θ (t) obtained from ( 2) and (3). The state deviation variable δ x (t) := x (t) -x (t) satisfies

δ x (t) = f (x (t), p, u (t)) -f ( x (t), p, û (t)),
which can be approximated with a first order Taylor series expansion of f (x (t), p, u (t)) about x (t), p and û (t):

δ x (t) ≈ ∂ f (x, p, u) ∂ x x = x (t) p = p u = û (t) (x (t) -x (t)) + ∂ f (x, p, u) ∂p x = x (t) p = p u = û (t) (p -p) + ∂ f (x, p, u) ∂u x = x (t) p = p u = û (t) (u (t) -û (t)) = A(t) δ x (t) + B 1 (t) δ p + B 2 (t) δ u (t) . (6)
From [START_REF] Athans | Optimal Control: An Introduction to the Theory and its Applications[END_REF], for unconstrained δ u (t), symmetric matrices Q, S 0, and R 0, the optimal control of the stabilizable linear time varying system in [START_REF]An Exo created for you[END_REF] with quadratic cost

J LQR = 1 2 δ x t f Sδ x t f + 1 2 ∫ t f t 0 δ x (t) Qδ x (t) + δ u (t) Rδ u (t) dt
exists, and is unique, given by the time varying formula

δ u (t) = -R -1 B 2 (t) P (t) δ x (t) =: -K LQR (t) δ x (t) . (7)
With the boundary condition P t f = S, P (t) is the solution of the Riccati matrix differential equation

P (t) = -P (t) A(t) -A (t) P (t) + P (t) B 2 (t) R -1 B 2 (t) P (t) -Q. ( 8
)
The nonlinear dynamics of the three-link robot under state feedback control with the time varying matrix gain III. R P M U P U There is no guarantee that a choice of Q, R, S in the design of the finite time horizon LQR controller of the previous section will achieve a safe STS movement in the presence of parameter uncertainty, i.e., when p is an unknown constant lying within an interval, due to manufacturing variability of the links of the PLLO and weight fluctuations of its user. To properly evaluate the robustness, we define a performance metric for assessing the worst-case deviations of the state x(t), output y(t) := [x CoM (t); y CoM (t); x CoM (t); y CoM (t)], and input u(t) from their desired trajectories x(t), ŷ(t), and û(t), based on over-approximations of their reachable sets at particular instants of time. We then identify the controller that optimizes the performance metric over a pool of candidates, as the most suitable for implementation.

K LQR (t) ∈ R 4×6 , become x (t) = f x (t), p, û (t) -K LQR (t) (x (t) -x (t)) =: ϕ (t, x, p) . ( 9
) TABLE I B P U S [p, p] Link m i [kg] I i kg • m 2 l i [m] l c i [m] 1 [9.2, 10.2] [1.
Since ascension starts from rest, with the shanks and torso segments parallel to the vertical, and the thighs segment parallel to the horizontal, we set 3) and (4) with t 0 = 0[s] and t f = 3.5[s], and a final configuration that places the CoM directly above the origin of the inertial frame with the values θ2

x (t 0 ) = [90[ • ]; -90[ • ]; 90[ • ]; 0[ • /s]; 0[ • /s]; 0[ • /s]] .
t f = -5[ • ], xCoM t f = 0[m], and ŷCoM t f = 0.97[m]. The desired output ŷ(t) is determined with the mapping ζ : R 6 × R 12 → R 4 in Appendix C, so that ŷ(t) = ζ ( x(t), p).
The numerical computation of the first order Taylor series expansion in [START_REF]An Exo created for you[END_REF] gives the time varying matrices A(t), B 1 (t) and B 2 (t) required in ( 7)-( 8) to get K LQR (t) as a function of the weight matrices Q, R and S.

It is assumed that the unknown parameter of the system lies within the interval p, p ⊆ R 12 in Table I, which was calculated for a fluctuation of ±5% of the nominal weight of the user with anthropometric data from [START_REF] Bartel | Orthopaedic biomechanics: mechanics and design in musculoskeletal systems[END_REF].

A. Sensitivity-based Reachability Analysis

Consider a continuous-time, time varying system η = g(t, η, ρ) with state η ∈ R n η , uncertain parameter ρ ∈ ρ, ρ ⊆ R n ρ , and continuously differentiable vector field g : R × R n η × R n ρ → R n η . Denoting the state reached by this system at time t ≥ t 0 from fixed initial state η 0 as Φ(t; t 0 , η 0 , ρ), we write the reachable set under parameter uncertainty as

Reach(t, [ρ, ρ]) := {Φ(t; t 0 , η 0 , ρ) | ρ ∈ [ρ, ρ]} ⊆ R n η ,
and the sensitivity function of the state trajectories with respect to the parameter as

S(t; t 0 , η 0 , ρ) := ∂Φ(t; t 0 , η 0 , ρ) ∂ ρ ∈ R n η ×n ρ .
We take the following lemma from [START_REF] Meyer | Sampled-data reachability analysis using sensitivity and mixed-monotonicity[END_REF].

Lemma 1: Assume that there exist S, S : [t 0 , +∞) → R n η ×n ρ such that S(t; t 0 , η 0 , ρ) ∈ S(t), S(t) for all t ≥ t 0 , and ρ ∈ ρ, ρ . Let the functions r, r : [t 0 , +∞) → R n η be defined as

r i (t) := Φ i (t; t 0 , η 0 , π i (t)) -d i (t)(π i (t) -π i (t)), r i (t) := Φ i (t; t 0 , η 0 , π i (t)) + d i (t)(π i (t) -π i (t)), (10)
where the j th elements of the parameter values π i (t), π i (t) ∈ ρ, ρ and row vector d i (t) ∈ R n ρ are determined according to the sign of the entries of the center of the interval matrix S(t), S(t) as

π i j (t) := ρ j , π i j (t) := ρ j , d i j (t) := min 0, S i j (t)        if Ŝij (t) ≥ 0; π i j (t) := ρ j , π i j (t) := ρ j , d i j (t) := max 0, S i j (t)        if Ŝij (t) < 0.
Then r(t), r(t) is an interval over-approximation of the reachable set of states at time t ≥ t 0 , so that Reach t, ρ, ρ ⊆ r(t), r(t) .

Given fixed x 0 := x(t 0 ) for the configuration of the links at seat-off, the sensitivity function S x (t; t 0 , x 0 , p) for the state trajectory Φ x (t; t 0 , x 0 , p) of ( 9), with zero initial condition S x (t; t 0 , x 0 , p) = 0 6×12 , satisfies the differential equation

S x (t; t 0 , x 0 , p) = ∂ϕ(t, x, p) ∂ x x=Φ x (t;t 0 , x 0 , p) S x (t; t 0 , x 0 , p) + ∂ϕ(t, x, p) ∂p x=Φ x (t;t 0 , x 0 , p) , (11)
over t ∈ t 0 , t f [START_REF] Khalil | Nonlinear Systems[END_REF].

The successors of y and u are determined from the static mappings of x and p that describe the kinematics of the CoM, and the control input with LQR state feedback:

Φ y (t; t 0 , x 0 , p) :=ζ (Φ x (t; t 0 , x 0 , p), p), Φ u (t; t 0 , x 0 , p) := û(t) -K LQR (t) (Φ x (t; t 0 , x 0 , p) -x(t)) .
Therefore, their sensitivity functions can be obtained from the solution of [START_REF] Zeilig | Safety and tolerance of the rewalk™ exoskeleton suit for ambulation by people with complete spinal cord injury: A pilot study[END_REF] as

S y (t; t 0 , x 0 , p) := ∂ζ (x, p) ∂ x x=Φ x (t;t 0 , x 0 , p) S x (t; t 0 , x 0 , p) + ∂ζ (x, p) ∂p x=Φ x (t;t 0 , x 0 , p) ∈ R 4×12 , S u (t; t 0 , x 0 , p) := -K LQR (t) S x (t; 0, x 0 , p) ∈ R 4×12 . ( 12
)
We showed in [START_REF] Narvaez-Aroche | Reachability analysis for robustness evaluation of the Sit-to-Stand movement for powered lower limb orthoses[END_REF] that tight over-approximation functions for the reachable sets of the state Reach x t, p, p ⊆ r x (t), r x (t) , output Reach y t, p, p ⊆ r y (t), r y (t) , and input Reach u t, p, p ⊆ r u (t), r u (t) , can be obtained from Lemma 1 through a sampling approach. This consist in randomly drawing a set of 500 parameters P b ⊂ p, p from a Latin Hypercube [START_REF] Mckay | Comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF], and numerically solving [START_REF] Zeilig | Safety and tolerance of the rewalk™ exoskeleton suit for ambulation by people with complete spinal cord injury: A pilot study[END_REF] for each of them. The sensitivity bounds S x (t), S x (t) are directly estimated by minimizing/maximizing the i j entries of the solutions for S x (t; t 0 , x 0 , p) at time t for all p ∈ P b . The estimates for S y (t), S y (t) , and S u (t), S u (t) require first plugging the solutions sampled for the state sensitivity in [START_REF] Narvaez-Aroche | Finite time robust control of the Sit-to-Stand movement for powered lower limb orthoses[END_REF], and then computing the extremal values for their respective entries.

B. Robust Performance Metric

For

[a, b] ⊆ R n , and c ∈ R n , let ν ([a, b], c) ∈ R be ν ([a, b], c) := i={1,...,n} a i + b i 2 -c i .
To evaluate the worst-case performance of different controllers for tracking x(t), û(t), and ŷ(t), we propose the metric:

J P := t ∈T P w v               vol r x (t), r x (t) vol r y (t), r y (t) vol r u (t), r u (t)               + t ∈T P w o               ν r x (t), r x (t) , x(t) ν r y (t), r y (t) , ŷ(t) ν r u (t), r u (t) , û(t)               , ( 13
)
where T P ⊆ t 0 , t f is the set of time instants where the overapproximation functions are computed, w v ∈ R 3 weighs the volumes enclosed by the intervals defined by such functions, and w o ∈ R 3 weighs the volumes of the offsets between the center of the intervals and their reference trajectories.

To have a baseline value for J P , we choose T P := {0, 0.875, 1.75, 2.625, 3.5}[s], and compute

t ∈T P vol r x (t), r x (t) , t ∈T P ν r x (t), r x (t) , x(t) , t ∈T P vol r y (t), r y (t) , t ∈T P ν r y (t), r y (t) , ŷ(t) , t ∈T P vol r u (t), r u (t) , t ∈T P ν r u (t), r u (t) , û(t) ,
for the system in ( 9) under the action of the finite horizon LQR controller from [START_REF] Narvaez-Aroche | Finite time robust control of the Sit-to-Stand movement for powered lower limb orthoses[END_REF]; which causes undesired variations of the loads at the shoulders [START_REF] Narvaez-Aroche | Reachability analysis for robustness evaluation of the Sit-to-Stand movement for powered lower limb orthoses[END_REF]. Using the reciprocals of these values, we set the weights in [START_REF] Narvaez-Aroche | Reachability analysis for robustness evaluation of the Sit-to-Stand movement for powered lower limb orthoses[END_REF] to so that the performance metric for this baseline controller is J P = 6. The large difference in the order of magnitude of the weight entries is due to the units and dimensionality of the hypercubes from which they are calculated.

w v := [6.
Computing the over-approximation functions in ( 13) is too expensive to implement a derivative free optimization method, such as the one used in Section IV-B to tune the ILC gains. Hence here we opt for a brute force approach where we construct sets of 300 diagonal, positive definite matrices of LQR weight candidates Q ⊂ R 6×6 , R ⊂ R 4×4 , and S ⊂ R 6×6 . Their entries are randomly drawn from a Latin Hypercube of 300 samples on 16 variables, with the values for Q and S in (0, 100), and the ones for R in (0, 0.01). Each of the sampled triplets of weights are plugged into [START_REF] Mckinley | Design of lightweight assistive exoskeletons for individuals with mobility disorders[END_REF], which is solved with tools documented in [START_REF] Moore | Finite horizon robustness analysis using integral quadratic constraints[END_REF] to obtain their corresponding time-varying matrix gain K LQR (t) from [START_REF] Suitx | PhoeniX Medical Exoskeleton[END_REF]. Then the technique described in Section III-A is applied to find the overapproximation functions and calculate J P for all the controllers. The triplet of weight matrices Their matrix gain K LQR (t) leads to J P = 1.31. The significant improvement in tracking the input reference using this controller over the baseline is illustrated in Figure 3, which exhibits the vertical force at the shoulders for simulations of the ascension phase for a set P s ⊆ p, p of 500 parameters from a Latin Hypercube sampling (note that this set is different from P b ). When K LQR (t) controls the system (9) (green lines) the deviations of the trajectories from the reference (red dashed line) are smaller than the ones achieved under the baseline controller (blue lines). Although this behavior is only expected at the time instants in T P , it gracefully holds along the entire horizon. Similar improvement is registered for tracking x(t), and ŷ(t).

Q , R , S = arg min Q ∈Q, R ∈R,S ∈S J P (Q, R, S) (14)

IV. I L C A P U A

Although the feedback controller

u(t) = û(t) -K LQR (t)(x(t) -x(t))
Fig. 3. Comparison of the vertical force obtained for 500 randomly sampled parameter values, when the system is under the action of the baseline controller (blue), and under the optimal controller with respect to the performance metric J P (green).

obtained in Section III-B is the optimal choice from a batch of candidates, actual execution of the STS movement relies on the interaction of two agents driving different control inputs of the system. More precisely, of the 4-dimensional control input u = [τ h ; τ s ; F x ; F y ], only the torque at the hips τ h is executed by the actuators of the PLLO under the authority of its onboard computer, while the three other controls (torque τ s , horizontal F x and vertical F y forces at the shoulders) are to be applied solely by the user interacting with the ground through crutches. Thus, unlike the accurate computer implementation at the hips, the human implementation at the shoulders will rely on a limited perception of the state of the system due to paraplegia, and no preconceived knowledge of reference trajectories. Therefore, a controller which is optimal in simulations for [START_REF] Eby | Modeling and control considerations for powered lower-limb orthoses: A design study for assisted STS[END_REF], when assuming perfect state feedback and actuation by the user, is not guaranteed to work experimentally. The purpose of this section is thus to assess whether a proxy for the user actions can learn to cooperate with the LQR controller designed in Section III-B through repeated trials and achieve a safe STS movement. For this, we choose to represent the interaction of the user with the PLLO using an Iterative Learning Control (ILC) algorithm. We believe that evaluating the performance of a controller for the hips combined with an ILC controller for the shoulders is a reasonable test prior to actual implementation of the PLLO.

A. Iterative Learning Control Algorithm

We start by rewriting the dynamics in [START_REF]ReWalk personal 6[END_REF] to better encompass the separate actions of the controller of the PLLO and the user. This is done by plugging the closed-loop expression for the input at the hips obtained from the finite horizon LQR controller in Section II-C, and leaving the input of the user in open-loop. For a more realistic representation, this model also incorporates saturation of the inputs; so far, constraints on their values have only been taken into account while solving the control allocation in (4) for û(t). We assume the user has healthy enough vestibular and visual systems, and adequate proprioception of the upper body to know the angular position θ 3 (t) and velocity θ 3 (t) of the torso; and that the PLLO is instrumented to display in a monitor the real time trajectories of the position (x CoM (t), y CoM (t)) and velocity ( x CoM (t), y CoM (t)) of the CoM, together with their references. This is in similar fashion as for the Robot Suit HAL, where users can see plots of the desired center of pressure and its actual position during training of the STS movement, in order to achieve proper synchronization with the device [START_REF] Tsukahara | Sit-to-stand and stand-to-sit transfer support for complete paraplegic patients with robot suit HAL[END_REF].

Let The input applied at the hips with state feedback from the finite horizon LQR controller, and within the limits of operation of the PLLO actuators, is

τ h (t, x) := sat D 1 û (t) -K LQR (t) (x -x (t)) , D 1 u, D 1 u . ( 15
)
Denote the loads at the shoulders as µ := [τ s ; F x ; F y ] ∈ R 3 , and the output measured by the user as Υ := [θ 3 ; x CoM ; y CoM ; θ 3 ; x CoM ; y CoM ] ∈ R 6 . Taking D 2 ∈ R 3×4 as D 2 := [0 3×1 I 3 ], where I 3 is the identity matrix, we define μ(t) := D 2 û(t), µ := D 2 u, and µ := D 2 u. Υ can be determined from the state x and parameter p with a mapping denoted as Ψ : R 6 × R 12 → R 6 using the kinematic equations of the CoM of the three-link robot in Appendix C. Plugging τ h (t, x) into (5), the nonlinear dynamics of the system with user input µ and output Υ are

x (t) = f x (t), p, τ h (t, x) µ (t) =: Ξ (t, x, p, µ) Υ (t) = Ψ (x(t), p) . ( 16
)
The desired trajectory for Υ (t) during the ascension phase of the STS movement is computed as Υ (t) = Ψ ( x (t), p). The algorithm to emulate the loads applied at the shoulders by a user, over N ascension attempts with the PLLO, is built upon the general current-iteration ILC referred to in [START_REF] Bristow | A survey of iterative learning control[END_REF]. Translating such control strategy to our problem, for successive iterations indexed by j ∈ {1, . . ., N }, the user input µ j (t) at t ∈ [t 0 , t f ] is given by

µ j (t) := γ j µ j-1 (t) + L Υ(t) -Υ j-1 (t) + K Υ(t) -Υ j (t) , (17)
where γ j ∈ R 3×3 , L, K ∈ R 3×6 , and Υ 0 (t) := Υ(t). The block diagram in Figure 4 shows that the feedforward component of this basic learning algorithm consists of two terms relying on the memory of the user about the past-iteration j -1. The feedforward gain L modifies the input µ j (t) according to the error of the output Υ(t) -Υ j-1 (t), while the recalling matrix γ j is inspired by [START_REF] Arif | Iterative manual control model of human operator[END_REF] and is used to capture the ability of the user to remember and execute µ j-1 (t). If γ j = I 3 then µ j-1 (t) is perfectly incorporated into µ j (t), but if γ j I 3 we will interpret the mismatch between the values of γ j µ j-1 (t) and µ j-1 (t) to be the consequence of either a memory flaw, an imperfect execution of the required loads at the shoulders, or a combination of both. The feedback component changes the input of the user by multiplying the error of the outputs at the current-iteration j by the feedback gain K. Although at the beginning of training there is no preconceived notion of the input that needs to be exerted to attempt a STS movement with the PLLO, we consider that the values μ(t 0 ) and μ(t f ) are known to the user, so that the ILC can be initialized with the linear interpolation

µ 0 (t) = μ(t f ) -μ(t 0 ) t f -t 0 (t -t 0 ) + μ(t 0 ). (18)
When the input µ j (t) from the basic ILC algorithm in (17) acts on ( 16), the state trajectory x(t) might lead to a configuration of the links of the PLLO which is harmful for the user, or even mechanically impossible to reach. To elaborate on this situation, we delimit the feasible ranges of motion for the user and the PLLO with the state bounds

x := [80 [ • ] ; -120 [ • ] ; 0 [ • ] ; -20 [ • /s] ; -5 [ • /s] ; -70 [ • /s]], x := [120 [ • ] ; 0 [ • ] ; 130 [ • ] ; 10 [ • /s] ; 60 [ • /s] ; 20 [ • /s]] ;
and let t j s ∈ [t 0 , t f ] be the maximum value such that the state at iteration j satisfies x(t) ∈ x, x for all t ∈ [t 0 , t j s ]. To account for situations where the user would need to abort execution of the STS movement due to safety concerns, we stop the ongoing iteration when x(t) goes out of bounds, reset the state of the system to the initial condition x 0 , and proceed to the next one.

Suppose the human input caused the STS movement in the past-iteration j -1 to stop prematurely, so that t 0 < t j-1 s < t f . In this scenario, the error Υ(t) -Υ j-1 (t) and µ j-1 (t) in ( 17) only exist for t ∈ t 0 , t j-1 s . In order to compute µ j (t) for t > t j-1 s , define Γ j (t) ∈ R 3 for j ∈ {1, . . ., N } as

Γ j (t) := γ j µ j-1 (t) + L Υ(t) -Υ j-1 (t) + K Υ(t) -Υ j (t) , t < t j-1 s α j t + β j , t ≥ t j-1 s , (19)
where

α j := μ(t f )-µ j-1 (t j-1 s ) t f -t j-1 s
and β j := μ(t f ) -α j t f implement a linear extrapolation of time for the human input, between µ j-1 (t j-1 s ) and μ(t f ). This is to simulate the torque and forces that the user would begin to apply at t j-1 s while attempting to counteract the negative effects of the past-iteration input on the ascension. For initialization purposes, t 0 s := t f .

Adding saturation to the extremal loads that the user is physically capable of exerting at the shoulders, we complete our ILC algorithm as: µ j (t) := sat Γ j (t), µ, µ .

(20)

B. Tuning of the ILC gains

As humans have tuned their muscles for precise movements of the upper limbs since birth, it is reasonable to assume that a human possesses a more complex learning process than the ILC algorithm. We thus reason that if we can find ILC gains for simulating a realistic ascension phase, a real user would be able to coordinate with the controller for the hips of the PLLO through training, and complete a successful STS movement. We resort to a derivative free optimization approach to numerically search for the values of the feedforward L and feedback K gains in [START_REF] Bartel | Orthopaedic biomechanics: mechanics and design in musculoskeletal systems[END_REF].

Define the j th iteration cost

J j L := ∞, t j s < t f ∫ t f t 0 Υ(t) -Υ j (t) 2 + w µ µ j (t) 2 dt, t j s = t f , (21)
where Υ(t) -Υ j (t) 2 penalizes the deviation from the reference trajectories for the ascension phase according to the output measured by the user, µ j (t) 2 enforces that the loads applied by the ILC do not undergo sudden changes, and the weight w µ := 10 -4 is used to account for the different units of Υ j (t) and µ j (t). The torque at the hips τ h (t, x) is not included in the iteration cost, since it is commanded by the LQR controller obtained from 14, and thus it is not modified in the tuning of the ILC gains.

If a particular choice of gains causes the final iteration N to stop the STS movement prematurely t N s < t f , it must be discarded for modeling the behavior of the user. Otherwise (if t N s = t f ), the quality of its corresponding ILC algorithm should be assessed, based on the deviation of the output in ( 16) from its desired trajectory Υ(t), and the rate of change of the input µ N (t) ∈ R 3 . With the nominal parameter p = p in ( 16), a time step of 4 [ms] for computing a discrete version of the iteration cost J j L , γ j = I 3 for all j ∈ {1, . . ., N }, and N := 30 iterations, we select the gains in [START_REF] Bartel | Orthopaedic biomechanics: mechanics and design in musculoskeletal systems[END_REF] as:

K , L = arg min K, L ∈R 3×6 J N L (K, L) . ( 22
)
There are two major difficulties for computing the solution to [START_REF] Mckay | Comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. The first is that the problem is in general non-convex and hence global minimization is intractable. The second is that computing derivatives with respect to K and L is cumbersome. The first issue is typically dealt with in practice via heuristic local search; we find this to be effective for our problem. To deal with the second issue, we apply a standard technique from the optimization literature for minimizing a function using only black-box function calls, which we describe briefly.

Suppose we want to minimize g : R n → R over η ∈ R n . For σ > 0, we define a smoothed function G σ (η) := E ξ {g (η + σξ)}, where ξ ∼ N (0, I n ) is an isotropic Gaussian random vector, and E denotes the expectation. Under reasonable regularity conditions on g, a standard calculation shows that the gradient of G σ is given by

∇G σ (η) = E ξ g(η + σξ) -g(η -σξ) 2σ ξ .
That is, we can differentiate G σ (η) by only using function calls of g. We can interpret this as a finite-difference method applied in a random direction. Furthermore, it is clear that as σ → 0, G σ (η) approaches g(η). Hence, optimizing G σ is a reasonable proxy for optimizing g; this is made formal in [START_REF] Nesterov | Random gradient-free minimization of convex functions[END_REF]. The most basic way to apply derivative free optimization is to run stochastic gradient descent:

η k+1 = η k -ρ k g(η k + σξ k ) -g(η k -σξ k ) 2σ ξ k , ( 23
)
where {ρ k } k ≥0 is an appropriate sequence of step sizes and {ξ k } k ≥0 is an independent and identically distributed sequence of N (0, I n ) random vectors. We apply a slightly modified version of ( 23) as described in [START_REF] Mania | Simple random search provides a competitive approach to reinforcement learning[END_REF]. First, at every iteration k we draw n d random directions {ξ i k } n d i=1 . We then sort the indices i = 1, ..., n d in ascending order with the value assigned to each index given by min g(η k + σξ i k ), g(η k -σξ i k ) , and compute the update direction as:

η k+1 = η k - ρ k n t n t i=1 g η k + σξ (i) k -g η k -σξ (i) k 2σ n t ξ (i) k . ( 24
)
Here, n t ≤ n d , ξ (i) k denotes the sorted directions, and σ n t denotes the empirical standard deviation of the 2n t costs used in the update.

We run this method for 10,000 iterations with g = J N L , n d = 30, n t = 10, σ = 0.01, and ρ k ≡ 0.04 on a machine with 72 physical cores running at 2.10 [GHz]. After approximately two days of computation, we obtain the gains

K =       -100.
     
, with a cost of J L := 8.66. As a reference, plugging Υ N (t) = Υ(t) and µ N (t) = μ(t) into [START_REF] Khalil | Nonlinear Systems[END_REF] gives ĴL := 8.75.

Figures 5678910111213show in black the phase planes for the state x, position and velocity of the CoM in the sagittal plane, and input trajectories u(t) of the three-link robot model in [START_REF] Slotine | Applied Nonlinear Control[END_REF] simulated under µ 30 (t).

The ILC algorithm with gains K and L achieves almost perfect tracking of Υ(t); as a consequence, Figures 7,8, and 9 show the trajectories in black essentially overlapping with it. Deviations from the reference in the phase planes θ 1 -θ 1 (Figure 5) and θ 2 -θ 2 (Figure 6) are expected, since they are not penalized in [START_REF] Khalil | Nonlinear Systems[END_REF]. Nevertheless, θ 1 is off the vertical at the end of the ascension just by 0.5[ • ], with the absolute values of both angular velocities less than 1.2[ • /s], which should not compromise the ability of a controller for the stabilization phase to reach the standing position with ease. As θ 2 (t) remains less than zero for the entire trajectory, there is no hyperextension of the knees, and thus the input µ 30 (t) should not pose a threat to the physical integrity of the user.

Even though the tracking errors for the angular position and velocity of the shank and thigh links do not directly affect the computation of µ j (t) in (20), they do impact (together with the tracking errors of the angular position and velocity of the torso) the value of the torque at the hips through the state feedback of the LQR, hence causing it to differ from τh (t) in Figure 10. It is especially interesting that although we did not consider μ(t) in [START_REF] Khalil | Nonlinear Systems[END_REF], both the torque and horizontal force at the shoulders in Figures 11 and12 follow their reference trajectories reasonably well, within errors of 15[N.m], and 10[N], respectively. The absolute value of the torque applied at the hips in Figure 10 is in general greater than τh (t), which compensates for lower vertical forces attained by µ 30 3 (t) in Figure 13 relative to Fy (t). From the rate of change µ 30 3 (t) observed in Figure 13, we can infer that J L < ĴL is mostly due to the difference of its values over time with respect to Fy (t). Although Fy (t) remains constant for about 1[s], it decreases

C. Robustness Evaluations

The results discussed above indicate that the ILC in ( 20) is able to successfully coordinate with the LQR controller driving the actuators at the hips in [START_REF] Narvaez-Aroche | Motion planning of the Sit to Stand movement for powered lower limb orthoses[END_REF] to complete the desired ascension phase with no risk of sit-back or step failures [START_REF] Eby | Modeling and control considerations for powered lower-limb orthoses: A design study for assisted STS[END_REF]. Moreover, it does so exhibiting input trajectories that could be realistically executed by both the PLLO and the user after 30 learning iterations. Since the gains K and L that make this behavior possible were found considering a constant recalling matrix γ j = I 3 in (19) across every iteration, and the nominal value of the parameter p in [START_REF] Slotine | Applied Nonlinear Control[END_REF], the purpose of this section is to evaluate the robustness of the ILC algorithm to perform the STS movement in two scenarios: imperfect recalling and execution of µ j-1 (t), and parameter uncertainty.

For the imperfect recalling and execution of µ j-1 (t) we plug the iteration-varying matrix γ j = I 3 + q j-1 ϑ j in [START_REF] Johansen | Control allocation -a survey[END_REF], with q := 0.8, and randomly sample the entries of ϑ j ∈ R 3×3 at every iteration j ∈ 1, . . ., N within the interval [-0.05, 0.05]. While the off-diagonal entries in ϑ j couple the loads at the shoulders, the decay to zero of the power function q j-1 as the number of trials increases, captures the idea that a user would eventually recall and apply the appropriate values of µ j-1 (t). With the nominal parameter value p, starting the learning algorithm from the linear interpolation in [START_REF] Athans | Optimal Control: An Introduction to the Theory and its Applications[END_REF], and applying 30 iterations, we obtained the behavior of the system [START_REF] Slotine | Applied Nonlinear Control[END_REF] shown in blue in Figures 5678910111213. The degraded tracking of Υ(t) and x(t) is evident in Figures 56789. However, the trajectory of the CoM position in Figure 8 shows that it is still possible to complete the ascension. Furthermore, Figure 6 affirms that the integrity of the knee joints will be preserved. Judging from the deviation from the reference in the phase plane for the angular position and velocity of the shanks in Figure 5, and most importantly the non-zero final velocity of the CoM observed in Figure 9, we predict that the stabilization phase of this STS movement would be more challenging than the one obtained for µ 30 (t) in the previous section, when γ j = I 3 . The control inputs of the PLLO (Figure 10) and those obtained in the ILC algorithm remain between the bounds [u, u], verifying that K and L still attain realistic trajectories after the same number of iterations with γ j I 3 , although their oscillations and sudden changes do lead to an increased value for the cost in (21) of J 30 L = 40. We now study the effect of parameter uncertainty after the ILC algorithm has completed 30 iterations under the nominal value of the parameter p, and constant recalling matrix γ j = I 3 . For this purpose, we set the new µ 0 (t) in [START_REF] Bartel | Orthopaedic biomechanics: mechanics and design in musculoskeletal systems[END_REF] equal to the trajectories in black from Figures 111213, and simulate the system in [START_REF] Slotine | Applied Nonlinear Control[END_REF] under the action of [START_REF] Meyer | Sampled-data reachability analysis using sensitivity and mixed-monotonicity[END_REF] for two different parameter values: All entries match the bounds in Table I, except for the ones representing the lengths of the links, which come from subtracting (for p L ), and adding (for p U ) 1[mm] to the nominal lengths in p. This choice puts more emphasis on studying changes in performance stemming from fluctuations of the total mass of the user, rather than from variations on the length of the links, since, in practice, the latter are only expected to occur due to wear of mechanical components after an extended period of use, while the former are bound to happen several times through the day. Keeping track of the cost J j L over j = {1, . . ., 30}, we identify the iterations where the ILC algorithm attains the minimum values, and present their corresponding simulations in Figures 5678910111213. The results for p L after 13 iterations are in magenta, while the ones for p U after 18 are in green. The associated costs are J 13 L := 27.9 and J 18 L := 29.1. Since x(t) and Υ(t) are determined based on the nominal value p, the reference for the position of the CoM in Figure 8 cannot be perfectly tracked with the mismatch in the parameter values. Their simulations run approximately parallel to the reference, with the lower bounds for l c i in p L causing its trajectory to be below it, and the upper bounds for l c i in p U putting its trajectory above. Although the velocities of the CoM for both parameter values in Figure 9 also deviate from the reference, zero velocity is achieved at the end of the movements, which together with the behavior observed in the phase plane in Figure 6, proves that the proxy for the user action can safely complete the ascension phase. According to the larger offsets from the final desired state that exhibit the trajectories for p U with respect to p L in Figures 567, we predict that the stabilization phase for a situation where the total mass of the user increases by 5% of its nominal value, might be more challenging than the one where the total mass decreases by the same amount. Figures 10 and11 show that an increased mass of the wearer also requires larger contributions from the torque executed by the PLLO at the hips and the torque applied by the user at the shoulders. The oscillations and sudden changes of the force profiles (in magenta and green) in Figures 12 and13, together with the deviations from Υ(t) in Figures 789, contribute to the larger values of J 13 L and J 18 L relative to J L . It is interesting that the most abrupt changes of µ 13 (t) happen when the velocity of the CoM achieves its maximum (approximately at 1.77[s]), and in the last 0.1[s] of the ascension; where µ 18 (t) also experiences its most significant changes. To further analyze the effect of parameter uncertainty on the performance of our proposed ILC algorithm in [START_REF] Meyer | Sampled-data reachability analysis using sensitivity and mixed-monotonicity[END_REF], we also considered 500 random parameter values p ∈ [p L , p U ] from a Latin Hypercube sampling to perform analogous evaluations to the ones for p L and p U . All simulations exhibited safe, realistic, and successful ascending STS movements.

V. C

The first part of the paper presented a procedure to choose, from a pool of candidates, the best finite time horizon LQR controller to be implemented in a minimally actuated PLLO to perform the ascension phase of a STS movement. To do so, a robust performance metric that accounts the error incurred at tracking the reference trajectories in the presence of parameter uncertainty was defined. The second part of the paper evaluated if a user would be able to coordinate with the torque commanded to the actuators at the hips by the chosen LQR controller. This was done by using an ILC algorithm to simulate the forces, and torques applied at the shoulders by the user, during rehabilitation and physical therapy sessions for performing the STS movement with the medical device. To model our assumption that the human acts optimally in learning, the gains of the ILC algorithm are tuned via a derivative free optimization under nominal conditions, and perfect recalling/execution of the input trajectory from previous trials.

The procedures can be custom tailored to the actuators available in the PLLO, and the height and weight of the user. Nevertheless, since they rely completely on the three-link robot model, future work might include exploring control schemes that also take into account model uncertainties in addition to the parameter uncertainties already considered in this paper.

Given the complexity of the closed-loop dynamics of the system, estimating the sensitivity bounds for computing the over-approximation functions in the performance metric required a time consuming sampling of the parameter interval. Improving this procedure is another appealing research direction.

Our most important conclusion is that the best controller with respect to the robust performance metric works in harmony with the ILC algorithm abstracting the human learning of the shoulder actions. It is remarkable that we incur only limited performance degradation, even in the presence of extreme parameter uncertainties, and under scenarios of flawed memory and lack of coordination at the shoulders.

These favorable conclusions encourage further steps towards using the optimal LQR controller relative to the robust perfor-mance metric in human testing. We believe that our procedures can set a good benchmark to systematically choose actuators of PLLOs to fit a larger variety of users, estimate the time and effort required for training, and develop a protocol for better assessing the robustness of the STS movement in clinical trials. This would then help to close the gap between PLLOs and standing wheelchairs, which still remain the most reliable mobility solution for patients with complete paraplegia.

A

A T E -L E For notational convenience in the following sections, denote c i := cos θ i , c i j := cos θ i + θ j , c i jk := cos θ i + θ j + θ k , and similarly for sin (•). Also, define the coefficients

k 0 (p) := (m 1 + m 2 + m 3 ) -1 , k 1 (p) := l c 1 m 1 + l 1 m 2 + l 1 m 3 , k 2 (p) := l c 2 m 2 + l 2 m 3 , k 3 (p) := l c 3 m 3 ,
whose explicit dependence with respect to the parameter p will be omitted onwards for compactness. The entries of the symmetric mass matrix M (θ, p) ∈ R 3×3 in (1) are

M 11 =I 1 + I 2 + I 3 + l 2 c 1 m 1 + m 2 l 2 1 + 2l 1 l c 2 c 2 + l 2 c 2 + m 3 l 2 1 + 2l 1 l 2 c 2 + 2l 1 l c 3 c 23 + l 2 2 + 2l 2 l c 3 c 3 + l 2 c 3 M 12 =I 2 + I 3 + l c 2 m 2 l 1 c 2 + l c 2 + m 3 l 1 l 2 c 2 + l 1 l c 3 c 23 + l 2 2 + 2l 2 l c 3 c 3 + l 2 c 3 M 13 =I 3 + l c 3 m 3 l 1 c 23 + l 2 c 3 + l c 3 M 22 =I 2 + I 3 + l 2 c 2 m 2 + m 3 l 2 2 + 2l 2 l c 3 c 3 + l 2 c 3 M 23 =I 3 + l c 3 m 3 l 2 c 3 + l c 3 M 33 =I 3 + l 2 c 3 m 3 .
The vector of energy contributions due to the acceleration of gravity g := 9.81 [ m /s 2 ] and Coriolis forces F θ, θ, p ∈ R 3 is This section reviews the derivation of the transformation from the z to the θ space, detailed in [START_REF] Narvaez-Aroche | Motion planning of the Sit to Stand movement for powered lower limb orthoses[END_REF]. The position of the CoM of the three-link robot can be expressed as a sum of three vectors whose geometric representation is shown in Figure 14:

F θ, θ, p = Ω (θ, p)        θ 2 1 θ 1 + θ 2 2 θ 1 + θ 2 + θ 3 2        + g       k 1 c 1 + k 2 c 12 + k 3 c 123 k 2 c 12 + k 3 c 123 k 3 c 123       , with Ω (θ, p) ∈ R 3×3 as Ω (θ, p) =       l 1 (k 2 s 2 + k 3 s 23 ) -k 2 l 1 s 2 + k 3 l 2 s 3 -k 3 (l 1 s 23 + l 2 s 3 ) l 1 (k 2 s 2 + k 3 s 23 ) k 3 l 2 s 3 -k 3 l 2 s 3 l 1 k 3 s 23 k 3 l 2 s 3 0       . The generalized force matrix A τ (θ, p) ∈ R 3×4 is A τ (θ, p) =       0 -1 -l 1 s 1 -l 2
x CoM y CoM = k 0 k 1 c 1 s 1 + k 0 k 2 c 12 s 12 + k 0 k 3 c 123 s 123 = r 1 + r 2 + r 3 . (25)
According to the angles drawn in Figure 14, we can establish the following relationships:

α = θ 2 -π, β = arctan y CoM x CoM , (26)
θ 1 = β -φ + ϕ, (27)
θ 3 = β + ψ -(θ 1 + θ 2 ), (28)
where for feasible and realistic STS movements φ ≥ 0, and ϕ, ψ ∈ [-π /2, π /2].

Applying the law of cosines to the triangle formed by vectors r 1 + r 2 , r 3 , and [x CoM ; y CoM ], as well as using the trigonometric identity c 2 = c 12 c 1 + s 12 s 1 , we have x CoM , and according to expressions [START_REF] Arif | Iterative manual control model of human operator[END_REF] and [START_REF] Mania | Simple random search provides a competitive approach to reinforcement learning[END_REF] this condition will hold if and only if ψ = 0; which requires vectors r 1 + r 2 and r 3 to be aligned. In the case of feasible and realistic STS movements, this will only occur in the vertical position. For all other configurations, we calculate the angular velocities of links 1 and 3 as

φ (z) = arccos r 3 2 -r 1 +
θ 1 θ 3 = -y CoM -k 0 k 3 s 123 x CoM k 0 k 3 c 123 -1 × x CoM y CoM -k 0 θ 2 -k 2 s 12 -k 3 s 123 k 2 c 12 + k 3 c 123 =: V (z, z)
so that the transformation h 2 : z, z → θ is defined

θ =       1 0 0 0 0 1       V (z, z) +       0 θ 2 0       =: h 2 (z, z) .
The acceleration of the CoM is

Robust

  Control of the Sit-to-Stand Movement for a Powered Lower Limb Orthosis Octavio Narvaez-Aroche, Member, IEEE, Pierre-Jean Meyer, Stephen Tu, Andrew Packard, Fellow, IEEE, and Murat Arcak, Fellow, IEEE.

Fig. 1 .

 1 Fig. 1. Phases of a dynamic sit-to-stand movement for a powered lower limb orthosis. a) Preparation. b) Ascension. c) Stabilization.

Fig. 2 .

 2 Fig. 2. Three-link planar robot for modeling a powered lower limb orthosis (PLLO) during a sit-to-stand (STS) movement.

  = [-200 [N • m]; -175 [N • m]; -40 [N]; 0[N]] and u := [200 [N • m]; 50 [N • m]; 40 [N]; 650 [N]].

  With nominal parameter p := [9.68 [kg] ; 12.59 [kg] ; 44.57 [kg] ; . . . 1.16 kg • m 2 ; 0.52 kg • m 2 ; 2.56 kg • m 2 ; . . . 0.53 [m] ; 0.41 [m] ; 0.52 [m] ; . . . 0.27 [m] ; 0.21 [m] ; 0.26 [m]], the corresponding initial position of the CoM of the three-link robot is [ xCoM (t 0 ) ; ŷCoM (t 0 )] = [0.31; 0.67][m]. We calculate x (t) and û(t) from (

  characterizes the best LQR gain for tracking the desired STS movement from the pool of candidates, with respect to the performance metric. The values found after 8.2 days of computation with a workstation of 4 cores at 2.7[GHz] running Matlab Parallel Toolbox, are Q = diag ([80, 95, 95, 68, 90, 83]), R = diag 1.0 × 10 -3 , 2.0 × 10 -4 , 6.0 × 10 -4 , 4.4 × 10 -3 , S = diag ([30, 37, 19, 29, 92, 82]) .

  D 1 := [1 0 0 0], and define the saturation of c ∈ R n over the interval [a, b] ⊆ R n as the element-wise min/max operation: sat (c, [a, b]) := min (b, max (a, c)) .

Fig. 4 .

 4 Fig. 4. Basic ILC algorithm with feedback gain K, feedforward gain L, and recalling matrix γ j .

Fig. 5 .

 5 Fig. 5. Phase plane of trajectories θ 1 (t) and θ 1 (t).

Fig. 6 .

 6 Fig. 6. Phase plane of trajectories θ 2 (t) and θ 2 (t).

Fig. 7 .

 7 Fig. 7. Phase plane of trajectories θ 3 (t) and θ 3 (t).

Fig. 8 .

 8 Fig. 8. Trajectories for the position of the CoM in the sagittal plane.

Fig. 9 .

 9 Fig. 9. Trajectories for the velocity of the CoM in the sagittal plane.

Fig. 10 .

 10 Fig. 10. Torque applied at the hips by the LQR controller of the PLLO.

Fig. 11 .

 11 Fig. 11. Torque applied at the shoulders by the ILC algorithm.

Fig. 12 .

 12 Fig. 12. Horizontal force applied at the shoulders by the ILC algorithm.

Fig. 13 .

 13 Fig. 13. Vertical force applied at the shoulders by the ILC algorithm.

p

  L := [9.2 [kg] ; 11.2 [kg] ; 42.3 [kg] ; . . . 1.10 kg • m 2 ; 0.49 kg • m 2 ; 2.40 kg • m 2 ; . . . 0.529 [m] ; 0.409 [m] ; 0.519 [m] ; . . . 0.23 [m] ; 0.17 [m] ; 0.24 [m]], p U := [10.2 [kg] ; 13.2 [kg] ; 46.8 [kg] ; . . . 1.21 kg • m 2 ; 0.54 kg • m 2 ; 2.65 kg • m 2 ; . . . 0.531 [m] ; 0.411 [m] ; 0.521 [m] ; . . . 0.30 [m] ; 0.23 [m] ; 0.28 [m]].

  s 12l 3 s 123 l 1 c 1 + l 2 c 12 + l 3 c 123 0 -1 -l 2 s 12l 3 s 123 l 2 c 12 + l 3 c 123 1

Fig. 14 .

 14 Fig. 14. Geometric representation of vectors and angles used for expressing θ 1 and θ 3 as a function of [x CoM ; y CoM ], and θ 2 .

2 1 + k 2 2 + 2k 1 k 2 c 2 ,ψ 3 = arcsin k 2 1 + k 2 2 += k 0 θ 2 -k 2 s 12 -k 3 s 123 k 2 c 12 + k 3 c 123 + -y CoM -k 0 k 3 s 123 x CoM k 0 k 3 c 123 θ 1 θ 3 . k 0 k 3 s

 2223222121233 of sines, for the triangle of vectors r 1 , r 2 , and r 1 + r 2 , we knowϕ (z) = arcsin r 2 sin α r 1 + r 2 = arcsin k 2 sin (θ 2 -π) k (z) = arcsin r 1 + r 2 sin φ (z) r 2k 1 k 2 c 2 sin φ (z) k 3 . (31)Plugging (26), (29), and (30) into (27), as well as[START_REF] Arif | Iterative manual control model of human operator[END_REF], and (31) into (28), we define the transformation h 1 : z → θ asθ = CoM x CoM -φ (z) + ϕ (z) θ 2 arctan y CoM x CoM + ψ (z) -(θ 1 (z) + θ 2 ) h 1 (z) .Because this transformation relies on the triangulation of the vectors in equation (25), it does not hold in the vertical position, where θ 1 = π /2 and θ 2 = θ 3 = 0. The velocity of the CoM of the three-link robot is x CoM y CoM 123 x CoMk 0 k 3 c 123 y CoM = 0 if θ 1 + θ 2 + θ 3 = arctan y CoM

xCoM k 0 k 3 c 123 θ 1 θ 3 ,θ 1 θ 2 k 2 c 12 + k 3 c 123 k 2 s 12 + k 3 s 123 -2k 0 k 3 θ 1 + θ 2 θ 3 c 123 s 123 + k 0 θ 2 -k 2 s 12 -k 3 s 123 k 2 c=: h 3

 3121212321223 CoM y CoM = a (h 1 (z), h 2 (z, z), z) + -y CoM -k 0 k 3 s 123 x where a θ, θ, z := -x CoM k 0 (k 2 c 12 + k 3 c 123 ) k 0 k 3 c 123 y CoM k 0 (k 2 s 12 + k 3 s 123 ) k 0 k 3 s 123 12 + k 3 c 123 .Thus, we can define the transformationh 3 : z, z, z → θ as CoM y CoM a (h 1 (z), h 2 (z, z), z) (z, z, z) .For compactness, we useh (z, z, z) := [h 1 (z) ; h 2 (z, z) ; h 3 (z, z, z)](32)to denote the transformation from z coordinates to θ coordinates. velocity coordinates of the CoM of the three-link planar robot in Figure2are computed from the kinematic equations derived in[START_REF] Narvaez-Aroche | Motion planning of the Sit to Stand movement for powered lower limb orthoses[END_REF], with the mapping ζ : R 6 × R 12 → R 4 defined as:k 1 c 1 + k 2 c 12 + k 3 c 123 ) k 0 (k 1 s 1 + k 2 s 12 + k 3 s 123 ) -θ 1 y CoM -θ 2 k 0 (k 2 s 12 + k 3 s 123 ) -θ 3 k 0 k 3 s 123 θ 1 x CoM +θ 2 k 0 (k 2 c 12 + k 3 c 123 ) + θ 3 k 0 k 3 c 123 ζ (x, p) .

  n} × {1, . . ., m}. The center of the interval matrix Λ, Λ is represented by Λ.

	Acronyms:
	STS sit-to-stand.
	PLLO powered lower limb orthosis.
	CoM center of mass.
	LQR Linear Quadratic Regulator.
	ILC Iterative Learning Control.
	II. D	T	C
	P	L	L	O

  98 × 10 7 ; 9.67 × 10 -7 ; 9.71 × 10 4 ],w o := [1.85 × 10 18 ; 7.24; 1.07 × 10 13 ],

  r 2 2x 2 CoM + y 2

CoM

-2 r 1 + r 2 x 2 CoM + y 2 CoM = arccos (k 0 k 3 ) 2k 2 0 k 2 1 + k 2 2 + 2k 1 k 2 c 2x 2 CoM + y 2 CoM -2k 0 k 2 1 + k 2 2 + 2k 1 k 2 c 2 x 2 CoM + y 2

A

The first author would like to thank the Consejo Nacional de Ciencia y Tecnología (CONACYT), the Fulbright-García Robles program, and the University of California Institute for Mexico and the United States (UC MEXUS) for the scholarships that have made possible his Ph.D. studies, as well as Dr. Maria Vrakopoulou for the drawings in Figure 1, and Matthias Hirche for his comments on the manuscript. The authors gratefully acknowledge support from the National Science Foundation under grant ECCS-1405413. Andrew Packard also acknowledges the generous support of the FANUC Corporation. R