Malvina A Apostolou
email: malvina.apostolou@gmail.com

Phenomenal Transparency in Architecture : The case of Victor Horta

Keywords: Victor Horta, architecture, phenomenal transparency

The object of this paper is to demonstrate the means of implementation of phenomenal transparency in the architecture of Victor Horta. More specifically, the buildings studied are the Hôtel Tassel (1893-1894), the Hôtel van Eetvelde (1895), and the personal residence of Victor Horta (1898)(1899)(1900)(1901), in the city of Brussels. According to the definitions that have been attributed to the concept of phenomenal transparency (originally by Colin Rowe and Robert Slutzky in Transparency : Literal and Phenomenal) the paper seeks to understand the term and its connection with the architecture of the three works of Victor Horta, after the study of each of them. The elaboration of work is based on field research, study, recording and mapping held after visiting these three projects in the city of Brussels. After having studied the notion of this metaphorical transparency a conclusion is drawn: the unfolding or the multiplication of the perceived size of the space, and finally the creation of the image (Eidolon). This is illustrated by the creation of new plans on the basis of the collected data that, to my point of view, intensify the existence of phenomenal transparency. These new, virtual plans, are produced after the capture of multiple viewings, interpenetrations and perspectives that were recorded.

PHENOMENAL TRANSPARENCY

Transparent 1. Having the property of transmitting light, so as to render bodies lying beyond completely visible, that can be seen through. b. Penetrating, as light c. Admitting the passage of light through interstices open, candid, ingenuous easily seen through, recognized, or detected; manifest, obvious By the dictionary definition the quality or state of being transparent is a material condition -that of being pervious to light and air, the result of an intellectual imperative-of our inherent demand for that which should be easily detected, perfectly evident (...) 'Simultaneity', 'interpenetration', 'superimposition', 'ambivalence', 'space-time', 'transparency': in the literature of contemporary architecture these words, and others like them, are often used as synonyms.(1) (...) this twin concept of actual and apparent transparency appears above all to be a precise tool for the study of architecture. It distinguishes between Essence and Appearance in the concept of transparency, and refers to the relationship between Content and Form in architecture -and to the still enormous question of whether a building is, or whether it means. [START_REF] Hoesli | Commentary in Transparency[END_REF] Colin Rowe, Robert Slutzky and Bernard Hoesli define the concept of phenomenal transparency and distinguishes it from the state of the literal one. An interpretation of phenomenal transparency as a means of implementation in a work of art as well as a declaration of the ambiguity of transparency is defined by Georgy Kepes in The Language of Vision :

If one sees two or more figures partly overlapping one another, and each of them claims for itself the common overlapped part, then one is confronted with a contradiction of spatial dimensions. To resolve this contradiction, one must assume the presence of a new optical quality. The figures are endowed with transparency ; that is, they are able to interpenetrate without an optical destruction of each other. Transparency however implies more than an optical characteristic ; it implies a broader spatial order. Transparency means a simultaneous perception of different spatial locations. Space not only recedes but fluctuates in a continuous activity. The position of the transparent figures has equivocal meaning as one sees each figure now as the closer, now as the further one. [START_REF] Kepes | The Language of Vision[END_REF] In Vision in Motion, Laszlo Moholy-Nagy indicates that (...) superimpositions overcome space and time fixations and unite strange and diverging subjects into new entities. They transpose insignificant singularities into meaningful complexities; banalities into vivid illumination. The transparent quality of the superimpositions often suggest transparency of content as well, revealing unnoticed structural qualities in the object. [START_REF] Moholy -Nagy | Vision in Motion[END_REF] We therefore observe that the transparency can be an inner quality of the materiality, or an inherent organisational characteristic/quality. Thus we distinguish literal and phenomenal transparency. Il its latter situation, as a quality of the organization, transparency is the sense of continuity of the void, a visible continuity through successive opaque volumes [START_REF] Andreadou | Transparency a quality of matter, transparency a quality of organization[END_REF]. The depth could be visible, or the volumes could be overlapped in one of their parts. A characteristic of transparency in its metaphorical sense is that the position of the individual forms in space is ambiguous: the eye is experiencing an infinite sequence of larger and smaller organizations within the whole. In general, transparency is generated where there are parts in the space that can be attributed to two or more systems of reference-where the classification is undetermined.

The realities of the deep and narrow space are constantly in contrast. Gyorgy Kepes, while studying the representation of depth by overlapping figures notes that if one spatial form obstructs our view of another form, we do not assume that the second ceases to exist because it is hidden. We recognize, as we look at such overlapping figures, that the first or uppermost has two spatial meanings -itself and beneath itself. The figure which intercepts the visible surface of another figure is perceived as nearer. We experience spatial differences or depth. Representation of overlapping indicates depth. It creates a sense of space. Each figure appears parallel with the picture-plane and tends to establish a receding spatial relationship.(6) A typical characteristic is the insertion of the axis, the most powerful means for capturing the spatial depth: a linear axis penetrating the composition. Transparent organizations implicate spatial transitions and indicate the existence of possible directions. Therefore, the observer may perceive himself in relation with the both reference systems. The choice for one or the other path also means an entry in one or the other layout. Thus, he does not remain an uninvolved observer but becomes a part of the composition, through his participation he introduces a dialogue/relation and through his imagination he is involved in the reading of the composition.

A continuation of the void, enabling the vision in the depth, and in a differentiated way as the observer moves (...) That is, the space, with that quality of organization of volumes enters into it and makes part of it, the time, which has to do with the movement of the observer. [START_REF] Andreadou | Transparency a quality of matter, transparency a quality of organization[END_REF] Finally, as Colin Rowe and Robert Slutzky point out, (...) the intimations of depth (...) are consistently retracted. (...) a displacement, and a sliding sideways occur along the line of its minor axis; (...), an argument between a real and deep space and an ideal and shallow one.(8) (...)these stratifications, devices by means of which space becomes constructed, substantial, and articulate, are the essence of that phenomenal transparency (...) [START_REF] Rowe | Transparency: Literal and Phenomenal[END_REF] Transparency therefore creates multiple readings of spatial relationships and their connections. It allows a flexibility in the organization of the composition. The connection between the space of two separate levels/planes result not only in a visual increase of the size of small spaces but also produces ambiguous spatial relationships: the spatial zones are differentiated and at the same time combined, united. Consequently, transparency as a means bears the possibility of optical spatial groupings. According to Bernard Hoesli, transparency is defined as a state of relationships and may be considered and used as a means of spatial organization. As soon as we see and understand solid and void as equally participating in or equally constituent of a figure-ground continuity it is no longer necessary to insist on their perceptually antithetical nature. [START_REF] Hoesli | Excursus on the concept of architectural space in Transparency[END_REF] This limit is the dividing line, the dividing plane, the separating volume beyond of which something new begins. It is the border. (...) While passing from one area to another the user acquires different stimuli because of the different properties that govern each area. (...) The most common type of limit is (...)the opaque, which does not allow the light to pass through it. (...) The absence of a compact, opaque barrier does not imply the absence of limit. The void, the absence of matter can produce an equally strong limit (...) Unlike opaque boundaries, the transparent ones does not clearly define the relationship of the in / out, the up / down, but generate interesting commentary, which depends on the perception and the position of the observer in relation to them. [START_REF] Papaioannou | The perception of limits, transparency-opacitytranslucency in Transparency and Architecture Limits and Challenges[END_REF] The spaces that are separated by glass partitions can be perceived as single (...) Because of transparency, these limits have as distinctive characteristic the sense of absence of matter as well as the unhindered diffusion of light. (...) (...) The separation in / out and is blunted and a sense, illusion of continuity, extension of space is created. [START_REF] Papaioannou | The perception of limits, transparency-opacitytranslucency in Transparency and Architecture Limits and Challenges[END_REF] As three different, complementary ways of implementation of transparency, which does not always refer to the use of glass, can be considered the fluidity, the porosity and the glare. 'These three ways refer to multiple complementary concepts of transparency, permeability, perforation and translucency. [START_REF] Konstantopoulos | The blur, the fluid and the porous: Building with Transparency in Transparency and Architecture Limits and Challenges[END_REF] Translucency operates creatively in the space. It gives the illusion of an open or respectively enclosed space, depending on the reading made by the user. (...) A characteristic of translucency is its function as a filter. The information diffused through the translucent boundaries are susceptible to different readings, since they are not clear but blurred. [START_REF] Papaioannou | The perception of limits, transparency-opacitytranslucency in Transparency and Architecture Limits and Challenges[END_REF] The element that is behind the other side activates the imagination and generates multiple readings. The combination of the concepts of inside, outside, up, down, forward, backward, diagonally, closed, open, full, empty is redefined, as well as the relation between line -point -level and surface and volume.

READING THE PROJECTS OF HORTA THROUGH THE NOTION OF PHENOMENAL TRANSPARENCY

After the comprehension of the organisational structure of Victor Horta's three projects I intend to attribute them notions included in the Phenomenal Transparency-as described by Colin Rowe and Robert Slutzky in their article Transparency: Literal and Phenomenal as well as the characteristics attributed to transparency by L. Moholy Nagy and G.Kepes mentioned above. I attempt the finding of elements in the interior as well as the exterior part of the buildings, that are related to the superimposition of visual depths, the layering of levels and the fluidity of spatial parties. Thus, meanings of the term of phenomenal transparency are being attributed to the projects selected; the research attempts to find elements that are associated with the notions of superimposition, interpenetration of visual field, the overlay of levels of depths as well as the successive layering. Aiming at comprehending the characteristics that bear the impression that the building is of a bigger surface and amplitude, I am led to a classifications of the categories that intensify the relation with the spatial cognition after the in situ study of the houses and the architectural promenade in their interior. After a visit of the three selected projects, that are in situ meticulously observed, studied and registered I grouped the following elements that intensify the state of phenomenal transparency:

-Framings -superimpositions / Layering / Overlays of levels and depth / stratifications / interpenetrations -Semi -Levels/planes (optical axes) -Mirrors -Surfaces of glass in the interior space that reflect -Form or figure in a background (stained glass, skylights, drawings in the ceiling/floor/side walls, sculptures) -Light (omnipresent in various types of manifestations) Superimpositions / Overlays We understand parts of the space as extracts, as even if they are defined they cannot be detached from the whole. The totality of the composition remains present and simultaneous with the reading of its parts. A sequential stratification (layering) of elements (such as fine bearing columns or side walls) ultimately generate a single composition of alternating depth. There are many levels of depth; their creation depends on our position, their existence or not on our interpretation. While traversing the area, moving away or approaching an element, these images consecutively change, highlighting every time another frame. Contents-sections which although they may be perceived as independent, do not lose their integration in the totality to which they belong. When moving elements (such as doors, windows, sliding walls) close or open the field of view changes again, new superimposed views are added to the previous ones. The longitudinal axis which traverses the composition helps to maintain a stable trajectory around of which (through the diagonal axes or the elements that added or removed during the movement in the space) successive views are generated, consecutive frames that when superposed bear varying interpretations and readings. The space presents the characteristics of a fluid, where the inside, outside, the right, the left are alternated, diffused, appear or disappear gradually. Now the space can be read in many ways. The limits, although clear, no longer operate as strict points of a start or an end; sometimes they mingle, they blur, they are erased or signify new unities -spaces. The foreground, the intermediate and the background: the sightings of the scene generate narratives; the pictorial grids create various gradients of depth, frontality, perspective. It is as if the same spatial canvas imply pluralities of overlays. Transparency as a quality of organization produces overlays of smaller organizations inside the whole.

Axes -semi levels

The optical axes, whether longitudinal and traversing the composition from one end to another, whether diagonal, upside, downside or lateral, play a major role in the existence of the phenomenon of phenomenal transparency. Through the semi-planes, the gaps (voids), the vertical circulations, the axes alter the angles of views, and thus provide multiple simultaneous sightings (views). As a result, while moving or standing in the space, we perceive more than one situations, we witness many zones of the space, lateral, horizontal, vertical; we perceive more than one depths, we have simultaneous parallel supervisions of sections of space. The space in its whole becomes more fluid, boundless, plenty of points / elements that can be read, penetrated, crossed, continued. Even if someone is in the same place, the perspectives change according to the point of his view; the space is perceived and discovered differently. The spiral path of the body, the turn, the stop, the movement, the rest, the twist. The axis around which someone moves is his own self, while other perspectives as well, lateral, horizontal, are hooked like a joint, and the gaze follows them.

Mirrors

The attribution of large parts or entire walls to mirrors contributes again to an annihilation of the limits. The space loses its clear definition from the side walls, and the wideness of the space is doubled. The folding of space gives a sense of continuity, by provoking a virtual size to the space. When two mirrors are placed opposite to each other, the image of the space is multiplied to infinity, the limits are erased, giving their position to the endless, the continuous, the intangible.

Translucent glass

The use of translucent or reflective glass enables the occupant of the space, during both his movement and stance, to perceive again an increased wideness of the space, by feeling present and visible the image he just passed behind. The area is not multiplied here, but, being reflected gently, the limits are diffused, resulting in the expression of phenomenal transparency. The architectural analog thereof can be found in the material qualities of the glass, overlays that are generated by the reflections of light in translucent surfaces.

Form in a background

A form in a background removes the limit. It annihilates it. As if the gaze does not stand, but penetrates. The painted walls negate the side wall, they break the side bonds and continue as parts of the fluid composition, as parts of the spaces. Now there is no beginning nor an end. Everything coexist as a whole, one proceeds to another, gets into the other, comes out of the other, diffused in all directions and blended with the facade, with the openings, the staircase, the floor, the ceiling. Here as well each form in a background is a separate abstract and can be read as such, but at the same time is integrated and inseparable from the rest. It is important to note that the state of a form in a background creates an endless impression.

Light

The light can be perceived here as a unifying factor of the space. Being omnipresent in the house, moving from the side on the road to the back one and the garden, following vertical paths through the skylights on the roofs, horizontal through the glass surfaces of the interior partitions, lateral through the semi-levels, carrying a variety of qualities through reflections or light spillovers due to materials, it flows with the space. It is born in the heart of the home, but is consolidated with the outside. Its different expressions through close and open levels define different realities in each subspace.

Phenomenal Transparency at the Hôtel Tassel (1893-1894, 6, rue Paul-Emile, Brussels)

At the Hôtel Tassel, depths are alternated within the limits of outlines; sequentially the depth is diffused to the outside. The depth is present on each alternation. It is recorded in the successive overlays as part of the longitudinal axis that penetrates the plan and defines the composition -as well as vertical ones.

Superimpositions

As entering the house after passing through the entrance hall the visitor is located on the octagonal lobby. From here, through the main longitudinal axis, the hole depth of the house is visible, until the dining room and the garden. At the same time, through the two openings with stained glass on the right and left side, the visual field is widened through the diagonal axes.

After ascending the seven steps, the view of the depth remains present, through the longitudinal axis ; through the two diagonal axes the width of the house is perceived, from one to the other side wall. Here the visitor is located at the intermediate space which is the preamble of the living room.

In the sequel, while situated in the living room, through the longitudinal axis the one continues to perceive the depth, which gradually diminishes. This space is at the same time the preamble of the dining room while here the entire width of the house is perceptible.

Finally, in the dining room and the end of the longitudinal axis, the width narrows, the depth is delimited to the garden. (figure 1)

Following the reverse path of the longitudinal axis, from the dining room to the entrance, we find four such overlays:

In the living room, while looking towards the entrance of the house, the visitor distinguishes three levels of depth on the longitudinal axis (the perforated limit of the living room, the smoking room, the entrance) and through the diagonal axes the wideness of space is expanded.

In the steps that lead to the level of the ground floor the depth gradually subsides. Diagonal perspectives also exist.

In the widthwise axis at this point, the visitor has an enlarged view through the diagonal perspectives.

During the ascent to the first floor, at the level of the doorstep, a lateral view leads to the smoking room and, while being in it, the longitudinal axis and the two diagonals are dominants.

Axes

In the section, on the levels of the ground floor and the elevated ground floor the longitudinal axis that penetrates the house is dominant. By inserting the semi level of the smoking room, two diagonal axes are being created between the semi level, the elevated ground floor and the ground floor. (figure 2a,2b)

Mirrors

The side wall of the greenhouse is partially covered by a mirror which starts from the visual level of a human occupying the entire width of the greenhouse. As a result the width of the house is multiplied ; the visitor perceives it as a folded space, as well as a part of the living room because of its glass limit with the intermediate space between the greenhouse and the staircase.

Reflections

The surface of perforated glass which defines the living room creates reflections of the previous space, by projecting him in the following one; as a result the perception of the space is slightly enlarged.

Form in a background

When entering the house, the reception hall is defined by a double door with stained glass. The area of the octagonal vestibule has on its both sides openings with color glass. Also, the smoking room is surrounded by stained glass, as the wall of the greenhouse in the first floor. On the ground, the hall and the octagonal vestibule, the forms can be read autonomously by defining each area, but also as a unity, emphasizing the fluidity and interpenetration of the spaces. These forms remain in the background of the side walls, the walls of the staircase, the greenhouse, the side walls of the floors. The two spaces that carry the light in the heart of the house (greenhouse and staircase) are covered with slightly colour glass. Finally we highlight the existence of the statue of David, at the level of the doorstep of the main staircase.

Light

As far as the movement of the light in the building is concerned, it is implemented through the longitudinal axis, vertically through the axis of the staircase and through smaller transversal axes. On the ground floor, the light circulates in the space through the stained glass of the entrance hall, the glass wall of the living room and the dining room and through the ceilings of the greenhouse and the staircase. In the first floor, through the office door, the stained glass of the greenhouse and the staircase's roof. At the last levels, the light is present via the axis defined by the greenhouse, the intermediate space, the staircase.

Phenomenal Transparency at the Hôtel Van Eetvelde (1895, avenue Palmerston 4, Brussels)

At the Hôtel Van Eetvelde the standstill at different points of the space defines levels / parts : the back, the intermediate, the front. However, at the same time one can read the depth, which is present in each case. Every level has a contour -it is clearly defined, but also read as a whole. The eye stops in smaller unities -one being inside the other -without interrupting the whole. It is in fact successively superimposed excerpts that are located behind or in front of others. The foreground, the intermediate and the background are simultaneously superimposed.

Superimpositions

The movement in a diagonal axis during the gradual approach of the greenhouse generates alternating layers of depth. In addition, in the level of the first floor, because of the permeability of the longitudinal axis and the existence of the semi level of the greenhouse, successive superimpositions of levels are created, depending on our position.

As someone enters the house, he finds himself in front of three depth levels: this that leads to the service areas, the intermediate of the greenhouse, and the foreground.(figure 3a)

In The middle of the diagonal path he approaches the level of the greenhouse.(figure 3b)

In The middle of the staircase he gradually acquires a view of the width and depth of the house. (figure 3c)

In the greenhouse's level, the full overlap and interpenetration of the spaces on the first floor and part of the ground floor is perceived. (figure 3d)

Gradients of layers and depths are visible in the living room where the gaze has contact and penetrates into intermediate layerings while having simultaneously contact with the depth.(figure [START_REF] Moholy -Nagy | Vision in Motion[END_REF] In the intermediate space before the dining room, and at the landing of the main staircase. Furthermore, a double row of columns in the living room and the dining room create small entities/unities within the whole.

Axes

The longitudinal axis penetrates the house from one edge to edge another. In the section also, due to the existence of the semi-level of the greenhouse, diagonal axes are created (ground floor, greenhouse, first floor). Furthermore, the vertical axis is present in the centre of the octagon.

Mirrors

In the Hotel van Eetvelde mirrors that occupy the spaces from one edge to another are absent. However, two mirrors of significant size are situated on the side wall of the living room and the dining room on the first floor, contributing again to the unfolding of the space the strengthening of phenomenal transparency that characterizes him.

Reflections

Glass walls defining the living room of the first level slightly reflect the area contributing to its penetration in the area of the octagon that follows, while also increasing the size of the living room by erasing its limits.

Form in a background

The stained glass of the dining room's door and the spaces enclosed around the octagon, the floor and the walls of the greenhouse, the ceiling and the walls of the living room and the dining room unify the parts of the space by implying their fluid relations . At the same time, the limits are annihilates the sense of the wideness of the space increases.

Light

The existence and propagation of light takes place through the longitudinal axis, as well as through three vertical ones. On the ground floor, through the glass walls; in the semi-level through the ceiling; on the first floor through the glass walls of the living room, the stained glass of the dining room, the ceilings of the two staircases; on the second floor through the stained glass situated around the central octagon and the transparent parts of the door. Finally, on the third floor blurred glass surfaces around the octagon and crystal parts in doors contribute to the unobstructed flow of the light in the house. The horizontal and vertical flow of the light from one edge to another in the interior of the house joins with the exterior, through the openings to the garden and the road. The inside and the outside are consolidated through the lighting and the ventilation. In the personal residence and atelier of Victor Horta the visitor is part of the vertical axis that leads to the end of the route, of the promenade in the interior of the building. The light is omnipresent during the movement. Beyond the iron thin frame and the light shades, he finds the sky, endless, deep.

Superimpositions

In this house the successive superimpositions of levels are generated during the ascent through the staircase that plays the role of an articulation of the individual spaces. Here there are not simultaneous penetrations of depths but individual chapters -extracts diffused through each other; as a result the space is perceived as continuous and fluid.

Axes

The basic axis that constitutes the phenomenal transparency at the residence of Horta, is the vertical spiral axis of movement. (figure 5) As the body follows the axis at every turn or stop, new views are added to previous horizontal or lateral ones, revealing at the same time many spaces. The longitudinal axis is also important, while penetrating the composition. Here we do not have floors, but semi-levels : this results at a constant expansion of the optical perspectives and reveals multiple views. (figure 6)

Mirrors

In his personal residence Horta has predicted mirrors in almost every room, through which the wideness is intensified. However, significant multiplications of the surface or unfoldings do not occur in every room that has a mirror. In the bedroom of Horta on the second floor and in the dressing room, the mirror occupies the whole defined space, thus creating an illusionary doubled space. Furthermore, on the third floor, in the bedroom of his Horta's daughter, Simone, a similar space unfolding is taking place. Finally, the most important virtual increase of size occurs in the last level of the house, with the two mirrors that are placed opposite each other, under the roof of the skylight. They reflect the image to infinity, producing an endless virtual unfolding.

Reflections

In the dining room of the first floor and in Horta's bedroom, the glass walls that define the rooms reflect slightly parts of the space, helping to create a sense of liquidity between the annihilated limits.

Light

The light, an integral component of the phenomenal transparency of this resident as a unifying factor of its individual spaces, is circulating through the staircase which like a backbone diffuses light in all the parts/semi levels with which is being articulated.

CONCLUSIONS

Victor Horta was a pioneer of his times. He redefined the use and combination of new materials, the architectural solutions in the ordinary plots of Brussels and re-approached the classicist teaching of the School of Fine Arts of Paris; his architecture was a comment upon a changing society.

His architecture was not part of a style or a trend; it was integrated into the movement that was reflecting the research of the artists of the time: it had a structure, expressed through logic and the feeling, through a comprehensive treatment, understanding and practice of architecture, construction and decoration art. The curves of the external skin of his buildings are interwoven with those of the internal; the structural elements and construction details with the charm of the forms.

Through the research on the three projects of Victor Horta, the Hotel Tassel, the Hotel van Eetvelde, and his personal residence with the atelier (Maison Personnelle et Atelier), the following conclusions can be drawn concerning the structure and the laws governing his architecture and their connection with the phenomenal transparency -from the synthetic solutions and the function, until the doubling of space and the creation of Eidolon within the limits of an oblong plot.

The staircase as a source of light and articulation of the semi-levels/planes, the light and ventilation, that is the literal transparency of the building, is a basic idea of Horta compositions.

The unity and continuity of the whole is also a vital component of the architecture of Victor Horta: the transition and fluidity from inside to outside, from outside to inside, from the part/section to the entity, from the individual to the part and the part to the whole, from the detail to the total.

Through the ongoing interpenetrations, superimpositions, the side walls and the semi-levels Horta annihilate the boundaries and lead to the passage from reading the composition via the literal transparency, to experiencing it through phenomenal transparency. The perceived size of the space is doubled or multiplied endlessly, through its virtual complementary and finally lead to the creation of the Eidolon, based on plans drawn up by the on-site mapping of various views, alterations of depths, interpenetration and reflections.

2. 3 .

 3 Phenomenal Transparency at the Residence and Atelier of Victor Horta (nowadays the Horta Museum) (1898-1901, 23-25 rue Américaine, Saint-Gilles, Brussels)

Figures

  Figures

  figure 2b : The axes that penetrate the Hôtel Tassel are longitudinal (in each mezzanine), diagonals (due to the smokehouse's semi level, and the vertical axis of the staircase. These axes create horizontal, vertical and lateral sightings as well as superimpositions and successions of depths.

figure 1 :

 1 figure 1 : the superimpositions through the optical viewings, during the promenade in the elevated ground floor of Hotel Tassel. Because of the side walls/partitions the gaze escapes through the corners.

figure 5 :

 5 figure 5 : the main axis constituting the phenomenal transparency is the vertical axis. Various sightings, horizontal and lateral are interpenetrating during the twist of the body.


Samer Riachy

Yuri Orlov

Thierry Floquet

Raul Santiesteban

Jean-Pierre Richard

Second Order Sliding Mode Control of Underactuated Mechanical Systems I: Local Stabilization with Application to an Inverted Pendulum

Second order sliding mode control synthesis is developed for underactuated mechanical systems, operating under uncertainty conditions. In order to locally stabilize an underactuated system around an unstable equilibrium, an output is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Then, the desired stability property of the closed-loop system is provided by applying a quasihomogeneous second order sliding mode controller, driving the system to the zero dynamics manifold in finite time. Although the present synthesis exhibits an infinite number of switches on a finite time interval, it does not rely on the generation of first order sliding modes, while providing robustness features similar to those possessed by their standard sliding mode counterparts. A second order sliding mode appears on the zero dynamics manifold which is of co-dimension greater than the control space dimension. Performance issues of the proposed synthesis are illustrated in numerical and experimental studies of a cart-Pendulum system.

Introduction

Stabilization of underactuated systems, forced by fewer actuators than degrees of freedom, presents a challenging problem [START_REF] Fantoni | Energy Based Control of the Pendubot[END_REF][START_REF] Ortega | Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment[END_REF]. As well known (see, e.g., [START_REF] Berkemeier | Tracking Fast Inverted Trajectories of the Underactuated Acrobot[END_REF][START_REF] Zhang | Hybrid control of the Pendubot[END_REF]), these systems possess nonholonomic properties, caused by nonintegrable differential constraints, and therefore, they cannot be stabilized by means of smooth feedback. With this in mind, the present investigation on stabilization of underactuated mechanical systems is focused on variable structure control methods. The problem of eliminating undesirable high-frequency oscillations (typically, referred to as the chattering problem [START_REF] Bartolini | Chattering avoidance by second-order sliding mode control[END_REF]), caused by fast switching in the discontinuous control signal, is left for further investigation. While being non-trivial, this problem is however well-understood from the existing literature (see, e.g., [START_REF] Hirschorn | Generalized sliding mode control for multi-input nonlinear systems[END_REF] and references therein) and hopefully general methods of chattering reduction apply here as well.

The control synthesis to be developed relies on a nonlinear transformation of an underactuated system into a canonical form similar to that of [START_REF] Bartolini | Second order sliding mode control of container cranes[END_REF][START_REF] Utkin | Sliding modes in Electromechanical Systems[END_REF]. In order to locally stabilize the underlying system around an unstable equilibrium, an output of the system is specified to ensure that the corresponding zero dynamics is locally asymptotically stable. Once such an output is chosen, the desired stability property of the closed-loop system is provided by applying a variable structure controller that drives the system to the zero dynamics manifold in finite time. Synthesis of such a controller that maintains an underactuated system on the zero dynamics manifold, which is of co-dimension greater than the control space dimension (a singular case in sliding mode control), is a contribution of the paper.

The variable structure controller constructed is inspired from the quasi-homogeneous controller of [START_REF] Orlov | Extended invariance principle for nonautonomous switched systems[END_REF]- [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF], stabilizing a one-link manipulator in finite time. Although that controller exhibited a so-called Zeno behavior with an infinite number of switches on a finite time interval (see [START_REF] Liberzon | Switching in Systems and Control Birkhauser[END_REF][START_REF] Lygeros | Dynamical properties of hybrid automata[END_REF] for Zeno modes), it did not rely on the generation of first order sliding modes, while providing robustness features similar to those possessed by their standard sliding mode counterparts. A Second Order Sliding Mode (SOSM) appeared in the equilibrium point only (see the original work [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] for second order sliding modes and [START_REF] Bartolini | Chattering avoidance by second-order sliding mode control[END_REF][START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF][START_REF] Fridman | Higher order sliding modes, in Sliding mode control in engineering[END_REF] for advanced results in the area).

In contrast to standard sliding mode control algorithms which are capable of providing the closed-loop manipulator with the ultimate boundedness property only [START_REF] Lu | Robust sliding mode control of uncertain nonlinear systems[END_REF], the afore-mentioned SOSM-based controller stabilized the manipulator in finite time, thus constituting an interesting alternative to standard sliding mode controllers. In applications to fully actuated friction mechanical systems [START_REF] Orlov | Extended invariance principle for nonautonomous switched systems[END_REF]- [START_REF] Orlov | Global position regulation of friction manipulators via switched chattering control[END_REF] these controllers were demonstrated to provide the desired system performance in spite of significant uncertainties in the system description as it is typically the case in control of electromechanical systems with complex hard-to-model nonlinear phenomena (examples are dry friction and backlash).

In the present work the SOSM synthesis procedure is developed for underactuated systems, operating under uncertainty conditions. Capabilities of the procedure are illustrated in numerical and experimental studies of a cart-pendulum. Inverted pendulums have always served as a test bed in control laboratories. They were initially used to illustrate ideas in linear control such as stabilization of non-minimum phase unstable systems. Due to their nonlinear nature, they have also maintained their usefulness to highlight many ideas in the nonlinear control field. A recent application to smart two-wheel vehicles has demonstrated a practical interest to underactuated inverted pendulums.

A classical control problem for a cart-pendulum is to swing it up and stabilize it around an unstable equilibrium position. Many papers, such as [START_REF] Chatterjeea | Swing-up and stabilization of a cart-pendulum system under restricted cart track length[END_REF][START_REF] Lozano | Stabilization of the inverted pendulum around its homoclinic orbit[END_REF][START_REF] Spong | The Swing Up Contol Problem for the Acrobot[END_REF][START_REF] Wei | Nonlinear control for an inverted pendulum having restricted travel[END_REF][START_REF] Zhao | Hybrid control for global stabilization of the cart pendulum system[END_REF] to name a few, have been devoted to this problem. The traditional approach is to develop two controllers. The first controller is based on the nonlinear model and it is dedicated to the swing-up phase. The second controller is derived from the linearized model and it is a locally stabilizing controller around the unstable equilibrium position. A commutation (hybrid) strategy is then determined to switch the swinging controller to the balancing controller when the system achieves the attraction domain of the latter controller. While being a linear static feedback, local balancing controllers, used in all the afore-mentioned works, are of limited practical utility because of the size of their attraction domains. Apart from this, the frictional influence has been ignored in these works that may severely limit the achievable performance of such a controller.

The SOSM-based balancing controller, deduced in the present paper for an underactuated inverted pendulum located on a cart, does not suffer from the above drawbacks. The locally stabilizing synthesis that does not take into account the swing-up phase is based on the nonlinear equations, thus providing a large domain of attraction, efficient for the hybrid strategy developed in our companion paper [START_REF] Santiesteban | Second Order Sliding Mode Control of Underactuated Mechanical Systems II: Orbital Stabilization with Application to Swing Up/Balancing Control of an Inverted Pendulum[END_REF]. In addition, the controller developed is shown to be robust with respect to friction forces.

The rest of the paper is outlined as follows. Section 2 reviews some basic results on the SOSM-based stabilization of a simple one degree-of-freedom manipulator, operating under uncertainty conditions. In Section 3, these results are extended to the SOSM synthesis of underactuated mechanical systems. Performance issues of the proposed synthesis are then illustrated in Section 4 in an experimental study of balancing an underactuated cart-pendulum system. Section 5 presents numerical and experimental results. Section 6 finalizes the paper with some conclusions.

Background: robust finite time stabilization of a onelink manipulator

The quasihomogeneous synthesis, to subsequently be defined, is first illustrated with a simple one degree-of-freedom mechanical manipulator, operating under uncertainty conditions. The dynamics of the manipulator is governed by

ÿ = ω(y, ẏ, t) + u ( 1
)
where y is the position of the manipulator, ẏ is the velocity of the manipulator, u is the controlled input, ω(y, ẏ, t) is a piece-wise continuous nonlinearity that captures all forces (viscous and Coulomb frictions, gravitation, etc.), affecting the manipulator. Solutions of the differential equation ( 1) with piece-wise continuous right-hand side, are defined in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] as that of a certain differential inclusion with a multi-valued right-hand side. Operating under uncertainty conditions implies imperfect knowledge of the nonlinearity ω(y, ẏ, t). This possibly destabilizing term ω(y, ẏ, t) = ω nom (y, ẏ, t) + ω un (y, ẏ, t)

typically contains an a priori known nominal part ω nom (y, ẏ, t) to be handled through nonlinear damping and an uncertainty ω un (y, ẏ, t) to be rejected. It is assumed that ω un (y, ẏ, t) is globally bounded i.e. the upper estimate

|ω un (y, ẏ, t)| ≤ N (3)
holds for almost all t ≥ 0 and (y, ẏ) ∈ IR 2 and for some a priori known constant N > 0. Apart from this, both functions ω nom (y, ẏ, t) and ω un (y, ẏ, t) are assumed to be piece-wise continuous.

The following control law

u = -ω nom (y, ẏ, t) -asign(y) -bsign( ẏ) -hy -p ẏ, (4)
N < b < a -N, h, p ≥ 0 (5)
with the standard notation sign(.) for the signum function, appears to stabilize the uncertain system (1)-( 3) in finite time. Apparently, the above controller ( 4), ( 5) consists of the nonlinear damping -ω nom (y, ẏ, t), the linear gain -hy -p ẏ, and the homogeneous relay part ϕ(y, ẏ) = -asign(y) -bsign( ẏ) such that ϕ(cy, c ẏ) = ϕ(y, ẏ) for all c > 0. Such a controller is further referred to as quasihomogeneous.

Definition 1 A controller is said to be quasihomogeneous iff it is representable as a combination of a homogeneous switched controller and a continuous controller, vanishing in the origin of the state space.

Until recently, finite time stability of asymptotically stable homogeneous systems has been well-recognized for only continuous vector fields [START_REF] Bhat | Finite-Time Stability of Homogeneous Systems[END_REF][START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF]. Extending this result to variable structure systems has required proceeding differently [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF][START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF] because a smooth homogeneous Lyapunov function, whose existence was proven in [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] for continuous asymptotically stable homogeneous vector fields, can no longer be brought into play.

The novel technique that was developed in [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF] has established that the finite time stability of a variable structure homogenous system is conserved regardless of some inhomogeneous perturbations. In particular, it has been shown that the inhomogeneous system (1)-( 5) is globally finite time stable regardless of whichever external disturbance (2) subject to (3) affects the system.

The qualitative behavior of the one-link manipulator (1)-( 3), driven by the variable structure controller (4), [START_REF] Borne | Vector Lyapunov functions nonlinear, time-varying, ordinary and functional differential equations[END_REF], is as follows. While approaching the origin y = ẏ = 0, the system trajectories rotate around it. As well-known from [START_REF] Orlov | Finite-time stability and robust control synthesis of uncertain switched systems[END_REF], the closed-loop system is locally finite time stable and the switching times of the controller have a finite accumulation point. Thus, system (1)-( 5) does exhibit Zeno behavior with an infinite number of switches in a finite amount of time. This system does not generate sliding motions everywhere except the origin. If a trajectory starts there at any given finite time, the so-called sliding mode of the second order appears [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. In a particular case, when the uncertainty ω(y, ẏ, t) = ω un (y, ẏ, t) has no nominal part and the control gains h, p are set to zero, the proposed control law (4) degenerates to the well-known homogeneous twisting algorithm [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF][START_REF] Fridman | Higher order sliding modes, in Sliding mode control in engineering[END_REF].

Stabilization of underactuated mechanical systems

In the present section, the quasihomogeneous synthesis is developed for stabilization of underactuated systems of the form

q = M -1 (q)[Bτ -C(q, q) q -G(q) -F ( q)]. (6)
In the above equation, q ∈ IR n is the joint position vector, τ ∈ IR m , m < n is the input torque, q and q are the velocity and acceleration vectors, respectively, M (q) ∈ IR n×n is the inertia matrix, C(q, q) q represents centrifugal and Coriolis terms, G(q) is the gravity vector, F ( q) ∈ IR n is the friction force and B is the input matrix of rank m. Under certain conditions, system (6) can locally be represented by means of a nonlinear change of state coordinates and a feedback transformation in the form

η = g(η, η, ξ, ξ) ξ = f (η, η, ξ, ξ) + u. (7)
If in addition, this system is locally minimum phase and sufficiently smooth, it can locally be stabilized by a quasihomogeneous controller similar to (4). Throughout, the following assumptions are made.

1. The functions g(η, η, ξ, ξ) and f (η, η, ξ, ξ) are piece-wise continuous in all the arguments and, in addition, g(η, η, ξ, ξ) is continuous in (ξ, ξ) locally around (ξ, ξ) = 0 for almost all (η, η).

The system

η = g(η, η, ξ(t), ξ(t)) (8)
is input-to-state stable.

3. The system η = g(η, η, 0, 0)

has 0 as a locally asymptotically stable equilibrium.

Solutions of the state differential equations ( 7)-( 9) with piece-wise continuous righthand sides, are defined in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. Under Assumption 1 the existence of such a solution (possibly non-unique) of each equation with an arbitrary initial condition and piece-wise continuous control input is guaranteed by Theorem 8 of [10, p. 85].

Other assumptions are made for technical reasons. Assumption 2 is introduced to avoid the destabilizing effect of the peaking phenomenon (a detailed treatment of the peaking phenomenon in the continuous setting can be found in [START_REF] Sussman | The peaking phenomenon and the global stabilization of nonlinear systems[END_REF]). Assumption 3 means that [START_REF] Canudas De | A new model for control of systems with friction[END_REF], specified with the output ξ, is a locally minimum phase system. The role of this notion is well-known from the theory of smooth fields [START_REF] Byrnes | Asymptotic stabilization of minimum phase nonlinear systems[END_REF] and it is now under study for variable structure nonautonomous systems.

As in the manipulator case (1), system ( 7) is operating under uncertainty conditions. The nonlinear vector field g cannot destabilize the closed-loop system because of the input to state stability and the minimum phase hypotheses, which is why no more information is required for this vector field. The possibly destabilizing term

f (η, η, ξ, ξ) = f nom (η, η, ξ, ξ) + f b (η, η, ξ, ξ) (10)
is partitioned into a nominal part f nom , known a priori, and an uncertain bounded gain f b whose components f b j , j = 1, . . . , m are globally upper estimated

|f b j (η, η, ξ, ξ)| ≤ N j (11)
by a priori known constants N j > 0. Apart from this, both functions f nom and f b are assumed to be piece-wise continuous. Being inspired from the quasihomogeneous controller (4), ( 5), the following variable structure control law

u(η, η, ξ, ξ) = -f nom (η, η, ξ, ξ) -αsign ξ -βsign ξ -Hξ -P ξ (12)
H = diag{h j }, P = diag{p j }, ( 13
) α = diag{α j }, β = diag{β j } (14)
with

N j < β j < α j -N j h j , p j ≥ 0, j = 1, . . . , m (15)
is proposed to locally stabilize the uncertain system ( 7), ( 10), [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF] whose state (η, η, ξ, ξ) is available for measurements. The notation diag is used to denote a diagonal matrix of an appropriate dimension; sign ξ with a vector ξ = (ξ 1 , . . . , ξ m ) T stands for the column vector (sign ξ 1 , . . . , sign ξ m ) T .

In what follows, the variable structure control law ( 12), ( 15) is shown to drive the uncertain system [START_REF] Canudas De | A new model for control of systems with friction[END_REF] to the zero dynamics manifold ξ = ξ = 0 in finite time thereby yielding desired stability properties of the closed-loop system.

Theorem 1 Let Assumptions 1-3 be satisfied and let the uncertain system (7), [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF] be driven by the state feedback [START_REF] Fridman | Higher order sliding modes, in Sliding mode control in engineering[END_REF] such that condition [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF] holds. Then the closed-loop system (7), ( 12)-( 15) is locally asymptotically stable, uniformly in the admissible uncertainties [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF].

Proof: The closed-loop system [START_REF] Canudas De | A new model for control of systems with friction[END_REF] driven by ( 12) is represented as follows

η = g(η, η, ξ, ζ) (16) ξj = ζ j , ζj = f b j (η, η, ξ, ζ) -α j sign ξ j -β j sign ζ j -h j ξ j -p j ζ j , j = 1, . . . , m. (17)
Due to Assumption 1, Theorem 8 of [10, p. 85] is applicable to system ( 16), [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], and by applying this theorem, the system has a local solution for all initial data and globally bounded uncertainties [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF]. Let us demonstrate that each solution of ( 16), ( 17) is globally continuable on the right. First, let us note that given j ∈ (1, . . . , m), no motion appears on the axes ξ j = 0 and ζ j = 0 except their intersection ξ j = ζ j = 0. Indeed, if ξ j (t) = 0 on a trajectory of ( 16), [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] then it follows from (16) that ζ j (t) = 0 along the trajectory. In turn, if ζ j (t) = 0 on a trajectory of ( 16), [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] then due to the parameter subordination [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF], the second equation of (17) fails to hold for ξ j = 0.

Next, let us compute the time derivative of the function V j (ξ j , ζ j ) = α j |ξ j | + 1 2 (h j ξ 2 j + ζ 2 j ), j = 1, . . . , m along the trajectories of [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. Taking into account [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF], one derives that

Vj (ξ j (t), ζ j (t)) = α j ζ j sign ξ j + h j ξ j ζ j + ζ j {f b j (η, η, ξ, ζ) -α j sign ξ j -β j sign ζ j -h j ξ j -p j ζ j } = -[β j -f b j (η, η, ξ, ζ)sign ζ j ] × |ζ j | -p j ζ 2 j ≤ -(β j -N j )|ζ j (t)| (18
) everywhere but on the axis ξ j = 0 where the function V j (ξ j , ζ j ) is not differentiable. Since no sliding motion appears on the axis ξ j = 0 except the intersection ξ j = ζ j = 0 where Vj (ξ j (t), ζ j (t)) = 0, inequality [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] remains in force for almost all t.

By virtue of ( 15), it follows that each solution of subsystem [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] subject to ( 11) is uniformly bounded in t. Coupled to Assumption 2, this ensures that all possible solutions of the over-all uncertain system (11), ( 16), ( 17) remain bounded on any finite time interval and by the property B of Theorem 9 of [10, p. 86], these solutions are globally continuable on the right. Now let us observe that due to the global boundedness [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF] of the uncertain terms f b j (η, η, ξ, ζ), j = 1, . . . , m and by virtue of the parameter subordination [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF], each subsystem ( 16), ( 17) is similar to that considered in Section II, and it is therefore globally finite time stable, uniformly in the admissible uncertainties [START_REF] Fridman | Higher order sliding modes as a natural phenomenon in control theory[END_REF]. It follows that starting from a finite time moment, the overall uncertain system (11), ( 16), ( 17) evolves on the zero dynamics manifold ξ = ζ = 0 where its behavior is governed by the zero dynamics equation [START_REF] Fantoni | Energy Based Control of the Pendubot[END_REF].

In order to complete the proof, it remains to note that by Assumption 3, the zero dynamics ( 9) is locally asymptotically stable. Coupled to the local uniform finite time stability of [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], this ensures that the closed-loop system ( 16), ( 17) is locally uniformly asymptotically stable. The proof of Theorem 1 is thus completed.

Summarizing, the following local Quasihomogeneous Synthesis Procedure is proposed for underactuated systems. First, an output of the system is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Once such an output has been chosen, the underactuated system is locally transformed into the normal form [START_REF] Canudas De | A new model for control of systems with friction[END_REF], whose stabilization is achieved by applying the quasihomogeneous controller ( 12), [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF].

In the sequel, the capabilities of the proposed synthesis procedure are illustrated in numerical and experimental studies of a cart-pendulum system of two degrees of freedom with n = 2 and m = 1.

Application to a Cart-Pendulum System 4.1 Model of the cart-pendulum

By applying Newton's second law of motion, the system dynamical equations are derived:

(M + m)ẍ + ml sin θ θ2 -ml cos θ θ = τ -ψ( ẋ) + ω 1 (t), 4 3 ml 2 θ -ml cos θ ẍ -mgl sin θ = -ϕ( θ) + ω 2 (t). In the above relations, x and θ are the generalized coordinates (see Fig. 1), M and m are the cart and pendulum mass respectively, l is the distance from the pendulum Figure 1: The cart-pendulum in generalized coordinates x and θ and the experimental testbed center of mass to its pivot, g is the gravitational acceleration, τ is the input force, ω 1 (t) and ω 2 (t) are unknown bounded perturbations which may account for model uncertainties and external disturbances. ψ( ẋ) and ϕ( θ) are cart and pendulum friction forces respectively. Coulomb and viscous frictions are used, and modeled as follows:

ψ( ẋ) = ψ Dahl ( ẋ) + ψ v ẋ and ϕ( θ) = ϕ Dahl ( θ) + ϕ v θ
ψ v and ϕ v denote viscous friction coefficients. As seen hereafter, the control design requires that the unknown perturbations and the friction forces are differentiable. Thus, it is assumed here that the time evolution of the Coulomb friction is described by the Dahl model and is governed by:

ψDahl ( ẋ) = σ x ẋ - ψ Dahl ( ẋ) ψ c | ẋ| and φDahl ( θ) = σ θ θ - ϕ Dahl ( θ) ϕ c | θ|
σ x and σ θ are stiffness coefficients, whereas ψ c and ϕ c represent cart coulomb friction force (linear motor) and rod coulomb friction torque respectively. The system equations can be put into the form (6) with:

M (q) = M + m -ml cos θ -ml cos θ 4 3 ml 2 ; C(q, q) = 0 ml sin θ θ 0 0 ; G(q) = 0 -mgl sin θ ; B = 1 0 ; F ( q) = ψ( ẋ) -ω 1 (t) ϕ( θ) -ω 2 (t) .
After some mathematical manipulations the following state space representation is obtained:

(ϕ( θ) -ω 2 (t)) (19)
with D = l(4M +m+3m sin 2 θ) > 0. By examining [START_REF] Liberzon | Switching in Systems and Control Birkhauser[END_REF], it can be seen that ϕ( θ) and ω 2 (t) are unmatched friction and perturbation terms. We will see that the developed controller provides attenuation of their perturbing effect.

Control law design

In this section, a quasi-homogeneous SOSM controller is derived to locally stabilize the cart-pendulum system around its unstable equilibrium point. The key idea is to find a diffeomorphic state space transformation to bring the cart-pendulum system into its regular form [START_REF] Canudas De | A new model for control of systems with friction[END_REF]. For this purpose, let us first define the following local change of variable:

η = x - 4 3 l (θ) (20)
with

(θ) = ln 1 + sin θ cos θ , |θ| < π 2 . (21)
Then, system ( 19) is transformed into two chains of integrators with the control input acting on the second one only:

- " g + 4 3 l θ2 cos θ " tan θ θ 3((M +m)g-ml cos θ θ2 ) sin θ D 3 7 7 7 5 + 2 6 6 2 6 6 4 η ηθ θ 3 7 7 5 = 2 6 6 6 4 η
0 0 0 3 cos θ D 3 7 7 5 (τ -ψ+ω1)- 2 6 6 4
0 1 ml cos θ 0 3(M +m) mlD 3 7 7 5 (ϕ-ω2) 4
We can see that the unmatched terms ϕ and ω 2 influence the (η, η) subsystem of the regular form [START_REF] Lygeros | Dynamical properties of hybrid automata[END_REF]. These terms are not treated in the theoretical Section 3, but we will see that the developed controller provides attenuation of their effects. Now a fictitious output ξ is to be chosen to ensure that the (η, η) subsystem with ϕ = ω 2 = 0 is minimum phase with respect to this output. Let us demonstrate that the required output ξ can be locally chosen as:

ξ = tan θ -λ 1 η -λ 2 η, (23)
with λ 1 and λ 2 > 0.

Taking into account ( 20), [START_REF] Lu | Robust sliding mode control of uncertain nonlinear systems[END_REF], and ( 23), both θ and x prove to converge to zero whenever η, η and ξ tend to zero. Thus, enforcing a sliding motion on ξ = 0 solves the problem of the asymptotic stabilization of x and θ under the assumption that the (η, η) subsystem is minimum phase. The problem is now to design a switching controller which guarantees the finite time convergence of ξ to zero. By differentiating ξ twice, one gets:

ξ = µ + z + u, (24)
with

µ " θ, θ" = 2 tan θ cos 2 θ θ2 + » 1 cos 2 θ + 8lλ2 θ tan θ 3 cos θ - λ2ϕv ml cos θ - " 3[(M + m)g -ml cos θ θ2 ] sin θ -3 cos θψ( ẋ) -3 M +m ml ϕ( θ) D # +λ1 " g + 4 3 l θ2 cos θ « tan θ + λ2 " g + 4 3 l θ2 (1 + sin 2 θ) cos θ « θ cos 2 θ - " λ1 ml cos θ + λ2 θ tan θ ml cos θ « ϕ( θ) - λ2 ml cos θ φDahl ( θ), u = 3ml + 8ml 2 λ 2 θ sin θ -3λ 2 cos θϕ v mlD cos θ τ, (25)
and the uncertain term

z = 1 ml cos θ 0 @ 2 4 ml cos θ + 8ml 2 λ 2 θ tan θ 3 -λ 2 ϕv 3 5 " 3 cos θ D ω 1 + 3(M + m) mlD ω 2 # + " λ 1 + λ 2 θ tan θ " ω 2 + λ 2 ω2 1 A .
Due to physical constraints, λ 2 can be chosen small enough to ensure that 3ml + 8ml 2 λ 2 θ sin θ -3λ 2 cos θϕ v = 0 in (25). The system described by

η = -g + 4 3 l θ2 cos θ (ξ + λ 1 η + λ 2 η) + 1 ml cos θ (ϕ -ω 2 ). (26)
together with [START_REF] Orlov | Finite time stability of homogeneous switched systems[END_REF] is in a regular form similar to (7) 1 , when ϕ = ω 2 = 0. As ( 24) is similar to (1), let us set the control law u as follows:

u = -µ θ, θ -α 1 sign(ξ) -β 1 sign( ξ) -hξ -p ξ,
which gives:

τ = mlD cos θ h 3ml + 8ml 2 λ 2 θ sin θ -3λ 2 cos θϕv i " -µ " θ, θ" -α 1 sign(ξ) -β 1 sign( ξ) -hξ -p ξ" . (27)
Assume that ω 1 , ω 2 , ω2 and θ are bounded. Then z is a matched disturbance, uniformly bounded by some positive ∆, |z| < ∆ for almost all t, θ and θ. Thus, as shown in Section 2, setting

∆ < β 1 < α 1 -∆ h, p ≥ 0, (28)
the quasihomogeneous controller ( 27), [START_REF] Ortega | Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment[END_REF] ensures the uniform finite time stability of the (ξ, ξ) system. The equivalent dynamics on the manifold ξ = 0 is given by

η η = 0 1 -λ 1 ρ -λ 2 ρ η η + 0 1 ml cos θ (ϕ -ω 2 ). ( 29
) with ρ = g + 4 3 l θ2 cos θ > 0, ∀θ ∈ -π 2 , π 2 .
Finally, it remains to investigate the asymptotic stability of [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], when ϕ = ω 2 = 0. Note that there exists a change of basis:

Ω = Ω 1 Ω 2 = P η η , with P = 1 0 κ 1 , κ > 0,
that transforms system (29) into the form

Ω = -κ 1 κσ δ Ω, (30)
σ = -κ 2 + λ 2 ρκ -λ 1 ρ κ and δ = κ -λ 2 ρ. Taking V (Ω) = |Ω| = |Ω 1 | |Ω 2 |
as a vector Lyapunov function [START_REF] Borne | Vector Lyapunov functions nonlinear, time-varying, ordinary and functional differential equations[END_REF], [START_REF] Lj | General aggregation of large-scale systems by vector Lyapunov functions and vector norms[END_REF] the right-hand side Dini derivative of V (Ω) gives

D + V (Ω) ≤ ΓV (Ω), with Γ = -κ |1| |κσ| δ . If λ 2 2 > 4λ 1 ρ , (31)
then there exists κ > 0 such that κ 2 -λ 2 ρκ + λ 1 ρ < 0 and

Γ = -κ 1 κσ δ . (32)
Note that the non-constant terms in [START_REF] Spong | Control of Underactuated Mechanical Systems Using Switching and Saturation[END_REF] appear in the last row only, and Γ is a Metzler matrix (-M matrix). This allows one to apply the linear stability theory (see [START_REF] Borne | Vector Lyapunov functions nonlinear, time-varying, ordinary and functional differential equations[END_REF], [START_REF] Lj | General aggregation of large-scale systems by vector Lyapunov functions and vector norms[END_REF]). Therefore, under condition [START_REF] Spong | The Swing Up Contol Problem for the Acrobot[END_REF], Ω = 0 is an asymptotically stable equilibrium for the nonlinear system [START_REF] Santiesteban | Second Order Sliding Mode Control of Underactuated Mechanical Systems II: Orbital Stabilization with Application to Swing Up/Balancing Control of an Inverted Pendulum[END_REF]. Now, by taking into account pendulum friction ϕ( θ) and the non-matched perturbation ω 2 (t), it is obvious that we loose the asymptotic stability property, but it is proven by the linear stability theory that the perturbed linear system (29) converges into a ball around the origin, thus obtaining practical stability. The radius of this ball can be decreased by the tuning of λ 1 and λ 2 . This will be supported by experimental results.

Numerical and experimental verification

In order to illustrate the effectiveness of the above controller, numerical simulations and experiments were conducted on a cart-pendulum system, installed in the LAGIS Laboratory of Ecole Centrale de Lille (see Fig. 1). The following numerical values are used: M = 3.4Kg m = 0.147Kg; l = (0.351/2)m; λ 1 = 0.2m -1 ; λ 2 = 0.25 sec /m.rad K 1 = 40rad/sec 2 ; K 2 = 20rad/sec; h = 30rad/sec 2 ; p = 0rad/sec

ψ v = 8.5N sec/m; ϕ v = 0.0015N msec/rad; σ x = 10000N/m; σ θ = 50N m/rad ψ c = 6.5N ; ϕ c = 0.00115N m
The simulation runs were conducted without any unmatched frictions and unknown perturbations, and with the following initial conditions:

(x, ẋ, θ, θ) T (0) = (0.2m, 1m/sec, 0.8rd, 2rd/sec).

The initial condition used for θ in the simulation was chosen as θ(0) = 0.8rd = 45.8 • in order to illustrate the reasonably large size of the attraction domain of the controller. Results are depicted in Fig. 2. As it is theoretically expected, both cart and pendulum position are asymptotically stabilized. In addition, an experimental test has been conducted on the real-life cart-Pendulum, having both unmatched pendulum joint and matched cart frictions. To clearly carry out the controller robustness, a significant disturbance was also manually applied to the pendulum during the time interval from about 6 sec to 10 sec. Under the parameters tuned to λ 1 = 0.2 and λ 2 = 0.25, practical stability and good performance were obtained (see Fig. 3).

Conclusions

The quasihomogeneous synthesis is developed to asymptotically stabilize underactuated mechanical systems, operating under uncertainty conditions. The proposed synthesis presents an interesting alternative to the standard sliding mode control techniques. Although resulting quasihomogeneous controllers do exhibit Zeno modes with an infinite number of switches on a finite time interval, however, they do not rely on the generation of standard sliding motions on the switching manifolds. Second order sliding motions appear on their intersections only. Compared to standard sliding mode controllers, the SOSM controllers exhibit additional attractive features recently illustrated in applications to fully actuated friction mechanical systems [START_REF] Orlov | Extended invariance principle for nonautonomous switched systems[END_REF]- [START_REF] Orlov | Global position regulation of friction manipulators via switched chattering control[END_REF]. In those publications the SOSM controllers were demonstrated to be capable of providing the static position stabilization and the desired system performance in spite of significant uncertainties in the system description as is typically the case in control of electromechanical systems with complex hard-to-model nonlinear phenomena.

In the present work, the quasihomogeneous SOSM control synthesis is extended to underactuated mechanical systems. In order to locally stabilize such a system around an unstable equilibrium, an output of the system is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Then, the desired stability property of the closed-loop system is provided by applying a SOSM-based controller, driving the system to the zero dynamics manifold in finite time.

Capabilities of the proposed synthesis and its performance issues are illustrated in numerical and experimental studies of a 2 degrees-of-freedom cart-pendulum system. The proposed local stabilization has been involved into a hybrid controller design which aimed to swing up the pendulum from the stable downward position and stabilize it about the unstable upright position. This work is presented in the second part of the paper.

Figure 2 :

 2 Figure 2: Numerical simulation without unmatched friction: (left) cart position and rod angle, (right) control input.

Figure 3 :

 3 Figure 3: Experimental results: (left) cart position and rod angle, (right) control input.

Samer Riachy, Thierry Floquet, Jean-Pierre Richard: Projet ALIEN, INRIA FUTURS and Equipe SyNeR, LAGIS UMR CNRS 8146, Ecole Centrale de Lille, P.O. Box: 48, Cité Scientifique, 59651 Villeneuve d'Ascq, France. Yuri Orlov, Raul Santiesteban: CICESE Research Center, P.O. Box 434944, San Diego, CA, 92143-4944.

[START_REF] Bartolini | Second order sliding mode control of container cranes[END_REF] This work is supported by the regional council Nord-Pas-de-Calais and the FEDER (European Funds of Regional Development) under the projects ARCir RoboCoop and TAT31 as well as by CONACYT under grant number 45900

In order to put the cart-pendulum system into its regular form given by[START_REF] Canudas De | A new model for control of systems with friction[END_REF], one has to represent the state equations in terms of η and ξ. The full regular form is not explicitly given here because it is rather lengthy from a computational point of view and because it is not essential for the design of the controller. As a matter of fact, (24) and (26) constitute the regular form since the variables θ and θ are implicit functions of the new state variables η, η, ξ and ξ.


João B Pinto Neto

Nathalie Mitton
email: nathalie.mitton@inria.fr

Miguel Elias
email: miguel@gta.ufrj.br

Mitre Campista

Henrique Maciel Luís

Kosmalski Costa

Henrique M K Luís

Costa

Dead Reckoning Using Time Series Regression Models

Keywords: Networks → Mobile networks, Network reliability, • Computing methodologies → Model verification and validation, • Computer systems organization → Embedded systems, Redundancy, Robotics, Dead-reckoning, connected vehicles, time series, robot

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

INTRODUCTION

Connected vehicles mean applications, services, and systems that enable vehicle connectivity to its vicinity including safety applications, road side assistance, and autonomous self-driving, among others. Connected vehicle technologies are a promise to reduce vehicle accidents and casualties, producing numerous positive economic and societal impacts. By limiting the number of crashes, thousands of lives will be saved, millions of injuries prevented and billion dollars will be saved annually [START_REF] Harding | Vehicle-to-Vehicle Communications: Readiness of V2V Technology for Application[END_REF].

However, these applications must rely on an accurate vehicle localization, perception of the environment, and wireless communication means [START_REF] Jiménez | Advanced Driver Assistance System for Road Environments to Improve Safety and Efficiency[END_REF]. Concerning vehicle localization, automotive navigation systems usually employ Global Positioning System (GPS) receivers, combined with other sensors, to estimate the vehicle position. Sensor fusion techniques can overcome the unreliability of GPS when the vehicle drives in urban canyons, tunnels, and under foliage [START_REF] Zair | A-Contrario Modeling for Robust Localization Using Raw GNSS Data[END_REF]. In extreme situations however, where GPS is completely out of service, vehicle position can still be estimated using sensors and the last reliable position. This blind navigation is called Dead Reckoning (DR).

There are numerous works exploring DR navigation using different fusion techniques combined with various sensor devices. A Dead Reckoning With Dynamic Errors (DRWDE) using Kalman Filter with dynamic covariance matrix merged with an Interacting Multiple Model (IMM) is proposed by [START_REF] Chandrasekaran | A survey of multisensor fusion techniques, architectures and methodologies[END_REF] to predict vehicle position 3 s ahead of time. A single-sensor device dynamic model for DR navigation was proposed in [START_REF] Laftchiev | Vehicle Localization Using In-Vehicle Pitch Data and Dynamical Models[END_REF]. The vehicle kinetic data acquired by an Inertial Measurement Unit (IMU) is modeled using a set of AutoRegressive models with eXogenous input (ARX). The GPS independent localization terrain based system with lane level accuracy proposed in [START_REF] Ahmed | Terrain Based GPS Independent Lane-Level Vehicle Localization Using Particle Filter and Dead Reckoning[END_REF], aims to avoid collisions due the lane change maneuvers. With the aid of a terrain map, the system uses information of an IMU and vehicle odometer to perform the localization algorithm. A particle filter is responsible to estimate the location, using data provided by the IMU while the vehicle drives in the same lane. [START_REF] Zhang | A new method of seamless land navigation for GPS/INS integrated system[END_REF] introduced a Wavelet Neural Network (WNN) based DR systems. The WNN is carried out when GPS is reliable where the localization is estimated with the information of a Strapedown Inertial Navigation System (SINS), i.e., an IMU device mounted rigidly to the body axes of the vehicle, a magnetometer, and GPS data fused into a Kalman filter. Authors of [START_REF] Belhajem | A Hybrid Machine Learning Based Low Cost Approach for Real Time Vehicle Position Estimation in a Smart City[END_REF] proposed a novel hybrid approach based on neural networks (NN) and AutoRegressive Integrated Moving Average (ARIMA) models to improve the accuracy of vehicle position estimation. During the training mode, NN is trained with GPS, odometer, and gyroscope data fused by an Extended Kalman Filter (EKF), while ARIMA works over the residuals of non-linear structures learned by NN. The aforementioned works are state-of-art DR systems. Different strategies and sensor combinations are used to improve accuracy of vehicle positioning and tracking. Regarding the sensors, only one system ( [START_REF] Laftchiev | Vehicle Localization Using In-Vehicle Pitch Data and Dynamical Models[END_REF]) acquires DR data from a single sensor (the IMU), while the remaining use two or more sensors to perform DR navigation. Regarding the applied technique, Kalman Filter was adopted by almost all works followed by neural networks. Our proposal differs from those presented by using a single data source native in all vehicles to implement DR navigation. This allows use of the system by any vehicle regardless of brand, cost or age.

In this context, we aim to support a significant number of vehicles not protected by safety systems. We propose a DR navigation system, called Time Series Dead Reckoning System (TedriS) that uses a conventional GPS as navigation device and the data of the rear wheels of the vehicle, provided by the native ABS speed wheel sensors through the interface with the Controller Area Network (CAN), i.e., it does not require any additional sensor. TedriS comprises a training phase and a predicting phase working in a collaborative way to estimate the vehicle location when GPS fails. During the training phase, the system collects data from the GPS and CAN interface to feed a time series regression model to generate a database with the relationship between the two variables in small blocks of data. This data is later used in the predicting phase to estimate the vehicle location, only using a simple implementation of a time series regression model and CAN interface data. TedriS performance was evaluated with real car traces and experiments with a robot. The results show that TedriS accuracy during a 80 s interval is situated within the range achieved by the state-of-art DR systems. The results also show better accuracy with robot experiments, confirming that TedriS can also be applied to robot DR navigation.

This work is organized as follows. Section 2 describes the system modeling. Section 3 describes TedriS training and predicting phases and time series regression model. Section 4 evaluates the performance of TedriS . Finally, Section 5 provides closing remarks and future work directions.

SYSTEM MODELING

The design of a DR system faces several challenges to achieve accurate vehicle localization. This section details the tools and techniques employed in the system modeling.

Dynamic Model

We assume that dynamic model state variables are referenced from car-like front wheel drive with no lateral slipping of the tires. We define a coordinate system whose origin is attached to the center of gravity of the vehicle (CG) as the body frame where x axis is aligned with the vehicle longitudinal axis. We also define an east-north-up coordinate system as navigation frame where the heading φ is the angle from east to north, the same GPS heading rule (Fig. 1). The relationship between front ( φ F ) and rear ( φ R ) yaw rates and wheels speeds is [START_REF] Kiencke | Automotive Control Systems: For Engine, Driveline and Vehicle[END_REF]:

φ F = (V F R -V F L )R F b F cos δ and φ R = (V RR -V RL )R R b R , (1)
where Front wheel drive vehicle dynamic model with the body frame coordinate system attached in the Center of Gravity (CG) and the navigation frame coordinate system adopted in Dead Reckoning System design. The heading angle φ follows the same GPS heading rule (from [START_REF] Kiencke | Automotive Control Systems: For Engine, Driveline and Vehicle[END_REF]).

V F R , V F L , V RR , V RL
wheel radius, respectively. We use φ R to estimate the vehicle's CG yaw rate without the need of the steering angle δ . Thus, the kinematic model of the vehicle is x = -V sin φ and y = V cos φ, where V is the instantaneous speed. There are many sources of errors that can compromise the efficiency of the dynamic model, such as speed sensors noise, difference of wheels radii, and position of the GPS antenna in relation to CG. Several studies to estimate the radius of wheels [START_REF] Boufadene | Vehicle online parameter estimation using a nonlinear adaptive observer[END_REF][START_REF] Özkan | A Bayesian approach to jointly estimate tire radii and vehicle trajectory[END_REF] confirm the impact of this factor over the position accuracy. Thus, our focus is to overcome this source of error using a technique to estimate this difference dynamically. Fig. 2 shows the typical behavior of the difference of rear wheels speeds in relation to time by a vehicle describing a straight-line trajectory. Noise amplitude ϵ delimited by W L and W H is represented by the black dashed lines and the mean µ is represented by the dashed line. Suppose that the noise is a white noise like and the wheels have exactly the same radius, the mean µ would be zero. This does not occur in this case due to tire different pressure, unbalanced tire wearing and electronic circuitry issues, including sensors misalignment. In order to reduce these issues effects, TedriS evaluates periodically the predicted position accuracy. A dynamic correction factor generated at each evaluation point is used to improve DR navigation performance.

Time Series Regression Models (TSRM)

Time series data is characterized by the correlation displayed by observations and their temporal sequence [START_REF] Gasparrini | Time Series Regression Analysis[END_REF]. TedriS explores TSRM in order to determine the relationship between the heading angle rate ( θ ) acquired from the GPS and the vehicle yaw rate ( φ) acquired from rear wheels speeds information through the CAN interface. These two independent collections of values, sampled within a uniform time interval, fit the time series definition [START_REF] Mahalakshmi | A survey on forecasting of time series data[END_REF]. Thus, the use of time series regression will give the temporal relation between these series both in the matter of delay and amplitude. The design of TSRM starts by the discretization of the series to investigate their relationship. Assume φ t and θ t are the discrete values of φ and θ at time t, we define ω t = diff(φ t ) and γ t = diff(θ t ), where diff is the difference operator.

To know whether past values of ω t can be used to predict γ t and how this relationship happens, sample Cross Correlation Function (CCF) is used to generate a set of sample correlations, or lags, of ω t . These values are used to predict γ t as input of a transfer function model that consists of an ordinary linear regression model. Hence, according to [START_REF] Robert | Time Series Analysis and Its Applications[END_REF], the relationship between ω t and γ t is:

γ t = γ 0 + α 0 ω t + α 1 ω t -1 + α 2 ω t -2 + α 3 ω t -3 ...α n ω t -n + η t , (2)
where γ 0 is the mean value of γ t when ω ( th) = 0, h = 0...n, α 1 , α 2 , α 3 ... α n are the weights of each correlated lag and η t is the error. The analysis of this error or residual will help to decide if we have to correct the model using an Autoregressive (AR) or a Moving Average (MA) approach [START_REF] Robert | Time Series Analysis and Its Applications[END_REF].

TEDRIS DESCRIPTION

TedriS is composed of two distinct stages. The first one, the training phase, collects, treats and processes the input data of GPS receiver and CAN interface and in addition it generates and stores the data that will be used later on. The second one defined as DR navigation phase uses the data generated by the training phase to predict the trajectory of vehicle, based only on CAN data.

System Training Phase

TedriS training phase must be performed a priori when GPS receiver data is reliable. The dynamic characteristics between GPS and CAN are estimated by TSRM while the vehicle performs a road course. Nevertheless, instead of applying the linear regression on the entire set of data captured after the training phase, TedriS applies it in small consecutive blocks of data in order to capture the non-linearities more accurately. The training phase is depicted in Fig. 3. The CAN yaw rate (ω t ) and GPS heading (θ ) are acquired after the synchronization to overcome the differences of sample rates. Controlled by the fastest sample rate clock, the buffer data feeds a First-in-First-out queue. Once the queue is full, the data is transferred to the sample register at each clock pulse, i.e. new data available in the buffer. As θ varies from 0 to 360 degrees, it is necessary to remove its discontinuity when a transition occurs (0 to 360 or 360 to 0). After that, the GPS yaw rate (γ t (τ )) is obtained by applying a difference operator to θ . The relationship between γ t and ω t is carried out using a set of k ordinary linear regressions whose inputs are τ consecutive values of ω t as in: where t 0 is the start time and γk is the predicted variable. Thus, at each block of τ samples of ω t , a linear regression is made to predict the value of γ t . The coefficients a mn are calculated using Eq. 2 with the highest correlation lag of ω t . During all the training phase long, the tuple (Med(ω t ), Med( γt )) is stored to be used in the DR navigation phase. The use of the median rather than the average value to estimate the central tendency is justified by the small number of samples and to avoid the influence of outliers.

γk = a 0k + a 1k • median(ω t k -1 , ω t k , ω t ( k +1 , ..., ω t k +τ -1 ), (3)
In addition, in this phase the values of ϵ, µ and the upper (W H ) and lower (W L ) bound of wheel speeds difference noise (Fig. 2) are estimated by the Straight Line Trajectory (SLT) detector. Defining [t s , t e ] as a SLT time interval and ω se the set of all values of ω t where t ∈ [t s , t e ], we have:

µ = mean(ω [se] ), W H = max(ω [se] ), W L = min(ω [se] ), and ϵ = W H -W L .
The tuples ( γt , ω t ) must be selectively stored to provide fast retrieval of data when requested by the DR phase. In order to meet the dynamic range of ω t and limit the size of the databases, the storage is divided into small repositories corresponding to intervals of ω t values delimited by W L and W H . Thus, for values of ω t belonging to the interval [W L ,W H ], the tuples ( γt , ω t ) will be stored into the repository R0, and for values of ω t greater than W H they will be stored in Rn for ω t ∈ (W H + (n -1)ϵ,W H + nϵ].

The same applies for values of ω t smaller than W L to generate R -1 , R -2 ,..., R -n . Generalizing, if ω t ∈ [-k,k], the number of repositories will be 2k/ϵ. Hence, the total storage required for the training phase can be easily estimated by delimiting a minimum and a maximum amount of data for each repository. The training phase completes when all repositories are filled at least with the minimum amount of data. TedriS executes the training phase periodically updating the repositories in conformity with eventual changes in the vehicle dynamical model.

System Dead Reckoning (DR) Phase

TedriS DR phase (or predicting phase) is the main focus of the system. All previous techniques and procedures are designed to predict vehicle position with the highest accuracy. This phase is performed when the GPS receiver data is unreliable and periodically after the training phase to update the dynamic correction factor (D C F ). DR phase uses the difference of rear wheels, speeds acquired from the CAN interface and the data stored selectively by the training phase to predict the vehicle heading angle as shown in Fig. 4. The input sequence is similar to the training phase where τ consecutive block samples of ω t are acquired after the full charge of the FIFO queue. In the same way that the training phase data was computed, the median value of ω t (τ ) ( ωt ) is used to predict γdr t . In order to allow a fast recovery of stored data, TedriS determines from which repository the set of tuples (γm t , ωm t ) is used to apply TSRM. If ωt ∈ [W L ,W H ], the repository data R0 will predict γt (τ ) according to:

γdr t = α 0 + α 1 Med(ωm t -h )) + D C F , (4)
where α 0 , α 1 are the TSRM coefficients, h is the lag of ωm t corresponding to the highest correlation coefficient of CCF(γm t , ωm t ) and D C F is the dynamic correction factor. For values of ωt greater than W H , γdr t is calculated also using Eq. Once TSRM is applied, γdr t is calculated by making D C F = µ. This is the dynamic correction factor initial value that will be periodically updated by the system. Based on the value of ωt , the appropriate data is retrieved from the repository and γdr is obtained using Eq. 4. Hence, applying the cumulative sum operator (CS), the DR heading is given by:

θ DR k = θ 0 + m k =1 γdr k . ( 5
)
During the DR update mode, TedriS adjusts the D C F to minimize the DR navigation error (E DR ) over a fixed time interval k. Considering θ k and θ DR k , GPS and DR heading at instant k, we have:

E DR M in = min(RMSE(θ k -θ DR k )). ( 6
)
The value of D C F that minimizes E DR is used in the DR phase predicting mode until a new update is performed.

TedriS Algorithm Sequence

DR functions are performed depending on GPS heading data status. When the GPS is operational, the training phase is executed, producing the data needed for the next phase. When all repositories of training data have enough data, the DR update phase is executed, generating D C F . This cycle runs continuously until a GPS failure occurs and triggers the DR predicting phase.

TEDRIS PERFORMANCE EVALUATION

The evaluation of TedriS performance was carried out in two distinct environments: a vehicle and a robot.

Vehicle Performance

The data used in this evaluation comes from real traces captured at the campus of Federal University of Rio de Janeiro. The vehicle's on-board unit was equipped with a U-blox M8 GNSS device ([15]) that provides the heading angle at 4 samples per second and with a CAN interface that provided the 4 wheels speed at 25 samples per second. The input data of the training phase was acquired while the vehicle was traveling on the road stretch shown in Fig. 5. The black solid line is the GPS heading data and the white dashed line is the training phase estimate heading. Temporal data analysis reveals a strong correlation between ωm t and γm t registering peak values at lags -2 and -3, highlighted in Fig. 8a (these values are determined dynamically using CCF). An analysis of residuals for regression correction using the AR and MA models revealed significance which is not desirable, confirmed with the Box-Pierce statistic test. Thus, choosing lag -2 the Time Series regression model is shown in Table 1.

Table 1: TedriS vehicle regression model. The coefficients α 0 and α1 satisfies the relation

γm t = α 0 + α 1 ωm t -2 + η t .
Estimate Standard Error t.value p.value α 0 0.068529 0.003772 18.17 0.0000 α 1 2.251480 0.022151 101.64 0.0000

Nevertheless, TedriS applies this model not in the entire set of the tuple (γm t , ωm t ), but in τ blocks of data. The best τ value was found empirically by varying τ until the training phase error below 1 m threshold level is achieved. The study shows that small values of τ increases training phase performance. The output data of this phase, provided dynamically by the data set (γm t , ωm t ) is τ = 6, max( ωt ) = 0.281 rd/s, min( ωt ) = -0.607 rd/s, µ = -0.0352 m, W L = -0.0581 m, W H = -0.01 m and ϵ = 1.5(W H -W L ) = 0.0721 m Fig. 6 shows both the update and predicting phases applied to two distinct road stretches. The corresponding DR Error can be evaluated against the 1 m threshold red line on the bottom. This threshold level is the desired performance of DR navigation systems.

The DR update phase has produced a dynamic correction factor, D C F = -0.03755, slightly different from the value of the mean, µ = -0.0352, that guarantees the smallest E DR (Fig. 6a). This value of D C F has resulted in the best algorithm response and was used in both road stretches of the DR predicting phase. For the first road segment where TedriS predicted the position of the vehicle, the mean DR error is close to 3 m for little more than 80 s, as shown in Fig. 6b. The second DR navigation test where the vehicle drives more than 80 s presented a mean DR error close to 4 m (Fig. 6c).

Robot Performance

Experiments with a Turtlebot2 robot were carried out in the dependencies of Inria at Villeneuve d'Ascq, France. The Turtlebot2 is an open source hardware platform and mobile base equipped with a 110 degrees/s factory-calibrated gyroscope [START_REF]Turtlebot2 Robot[END_REF]. Software packages to control the Turtlebot2 were developed using the Indigo distribution of the Robot Operating System (ROS) platform [START_REF] Goebel | ROS By Example[END_REF]. With Turtlebot2, TedriS worked with gyroscope and wheel encoder ticks. An advantage is that the two signals come from the same source at 10 Hz sample rate and no synchronization was needed. All the trajectories the robot speed was maintained constant at 0.3 m/s. To provide data for the training phase, the trajectory shown in Fig. 8b was generated in the same way as in previous experiments. Black solid line is the gyroscope heading data and the white dashed line is the training phase heading. The average system error was below 0.16 m. Note that the trajectory was shaped to provide a wide range of yaw rate values. It initiates with light curves and gradually the curves become sharper. The correlation between γm t and ωm t , showed a peak value at lag -3, is highlighted in Fig. 8c.

Analyzing the residuals, we identify a moving average 2 (MA2) process which resulted in the correction of the original model. The new coefficients and parameters that characterize the MA2 structure for the residuals are shown in Table 2. performance evaluation (Section 4.1), the entire DR phase, including the update phase and the predicting phase with two different robot pathways is depicted in Fig. 7. Proportionally, the threshold red line at 0.159 m respects the rate between robot and vehicle rear track width (0.25/1.57). All the DR phase experiments carried out with the robot, show more stable behavior and better DR navigation accuracy than vehicle DR phases, as depicted in Fig. 7a where the robot performs DR update phase trajectory keeping E DR below the red mark. This is due to better training data, a controlled environment, and constant speed that we can achieve with robot platforms but bit easily with a car. The minimum E DR was yielded with a D C F = 0.00005, quite different of µ = -0.0047. To simulate a common vehicle trajectory, the first DR predicting phase path was designed with straight paths, light and sharp curves. TedriS kept the DR error below the red mark during almost 60 s (Fig. 7b. The second DR predicting phase trajectory is an inverse version, but exhibited a slightly lower accuracy (Fig. 7c).

CONCLUSION AND FUTURE WORK

This work presented TedriS , a dead reckoning navigation system designed to enable any vehicle, independent of brand, model, cost or age, to have a DR navigation system, a gap left open by current DR systems that only address new high-end car models. To achieve this goal, TedriS uses an ordinary GPS as navigation device (or a gyroscope) and vehicle rear speed wheel sensors captured from CAN interface combined with a novel absolute position prediction technique. The core of TedriS is the relationship estimation between GPS and CAN yaw rate by a time series regression model using small blocks of data. This approach combined with parameters acquired dynamically during the DR update phase is responsible to set TedriS performance within the range achieved by state-of-art DR navigation systems, confirmed by the error margin acquired in the two road stretches of the DR predicting phase.

The experiments with the robot, attest the TedriS efficiency to predict localization in DR navigation. Analyzing proportionally, the results with the robot presented error margins below the results with the vehicle. The controlled environment, the smaller number of embedded devices and the constant speed, contributed that the system reaches out these levels. Nevertheless, experiments with a robot proved that TedriS methodology can be applied to any type of vehicle on wheels. As future work, we intend to use other techniques like multiple linear regression models and autoregressive models with exogenous input in order to improve TedriS performance. Our main goal is to achieve sub-metric error margins for longer times.

Figure 1 :

 1 Figure1: Front wheel drive vehicle dynamic model with the body frame coordinate system attached in the Center of Gravity (CG) and the navigation frame coordinate system adopted in Dead Reckoning System design. The heading angle φ follows the same GPS heading rule (from[START_REF] Kiencke | Automotive Control Systems: For Engine, Driveline and Vehicle[END_REF]).

Figure 2 :

 2 Figure 2: Typical wheels speeds difference acquired when a vehicle describes a straight line trajectory where ϵ is the estimated noise amplitude interval, µ the mean value, W L and W H are the lower and the upper bounds.

Figure 3 :

 3 Figure 3: Block Diagram of TedriS training phase. Each consecutive τ data block of GPS heading (θ t ) and CAN yaw rate (ω t ) are processed by the TSRM to generate the tuple (γm t , ωm t ) database. Straight Line Trajectory (SLT) provides parameters related to the difference in wheels speeds data.

Figure 4 :

 4 Figure4: TedriS dead reckoning (predicting) phase. The median ( ωt ) of consecutive τ data blocks of ω t is the index to retrieve data from the repositories where tuples (γm t , ωm t ) are stored. All the data generated by the training phase feeds the appropriate TSRM to generate the DR heading.

  4 and the index number n of the repository Rn is n = ⌈( ωt -W H )/ϵ⌉ and the corresponding lower (a) and upper (b) bounds are a = W H + (n -1)ϵ and b = a + ϵ. Similarly for values of ωt lower than W L the index number n of the repository Rn is n = ⌊( ωt -W L )/ϵ⌋ and the corresponding lower (c) and upper (d) bounds are c = dϵ and d = W L -nϵ.

Figure 5 :

 5 Figure 5: TedriS vehicle training phase road stretch with straight sections, light and sharp curves and roundabout. GPS is the black solid line, TedriS is the white dashed line.

  DR update phase error.

  First DR phase road stretch error.

  Second DR phase road stretch error.

Figure 6 :

 6 Figure 6: Vehicle DR phase performance. DR update, first and second predicting phases and their corresponding DR errors. The DR mean error below 5 meters was held by 80 s. The Red line is 1m DR threshold mark error.

  Second predicting phase Error.

Figure 7 :

 7 Figure 7: Robot DR phase performance (DR update, first and second predicting phases with DR errors).

  TedriS robot training path. It includes a wide yaw rate range. Gyro position is the black line, TedriS training position is the white dashed line. Temporal correlation between ωm t and γ m t from the robot training phase. Like in the vehicle training phase, a strong correlation is revealed at lag -3.

Figure 8 :

 8 Figure 8: Vehicle and Robot Temporal correlation (a,c) and TedriS Robot training phase (b).

Table 2 :

 2 TedriS robot MA model. The new coefficients α 0 and α1 was used in the TSRM DR predicting phase.Previous tests with this new model produced better performance without selective data storage, i.e., ϵ = ∞, thus the remaining data provided by this phase is τ = 6, max( ωt ) = 0.482 rd/s, min( ωt ) = -0.526 rd/s, µ = -0.0047m. Using the same pattern of the vehicle Temporal correlation between ωm t and γ m t from the vehicle training phase. One can observe a strong correlation with peak values at lags -2 and -3.

		Estimate Standard Error t.value p.value
	MA2	0.6333	0.0137	46.3037	0.0000
	α 0	0.0000	0.0001	-0.2277	0.8199
	α 1	0.0492	0.0004	140.3375 0.0000