Nicolas Lutz

Basile Sauvage

Frédéric Larue

Jean-Michel Dischler

Anisotropic Filtering for On-the-fly Patch-based Texturing

Keywords: CCS Concepts, Computing methodologies → Rendering, Texturing, Antialiasing

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Introduction

Adding detail to 3D surfaces in virtual scenes is often performed through the mapping of textures, as they are able not only to represent a simple color (albedo), but also complex reflectance parameters, or relief. Since very large scenes can nowadays be modelled and visualized, very large textures are required too, leading to storage problems due to graphics hardware limitations. On-the-fly texture synthesis methods have arisen to solve these issues, by generating the textures on demand at run-time, thus leading to virtually unbounded textures. Among these methods, tile-based approaches are very efficient and popular. They consist in pre-computing a set of tiles which are assembled on a regular grid at run-time. However, a reduced set of tiles generally leads to repetition artifacts due to the fact that the same large pieces of contents are reused too often. Conversely, using a high number of tiles may help to remove such artifacts, but is not tractable, due to memory consumption. To get round this issue, on-the-fly patch-based approaches [VSLD13, KCD16] have proposed to pre-compute patches, namely contiguous regions with irregular shapes, in a repeatable tile. Then, every patch occurrence is filled-in at run-time with a content selected randomly (see Figure 1). This has proved to be effective for irregular and homogeneous textures with structured patterns (i.e. with sharp features), like the ones presented in Figure 2.

Texture filtering is mandatory to make real-time texturing techniques useful in practice, as it removes under-sampling artifacts which occur when viewing a textured surface from far away or at a grazing angle. In the case of on-the-fly texture synthesis, render-1: Content selection process. At rendering, each occurrence of the patch p is filled with a content Ip,c randomly chosen by C among a set of pre-defined candidates.

ing requires to compute every screen-space pixel in constant time, whatever its footprint in texture space. This makes the filtering difficult: one cannot afford computing the high-resolution texture, and then averaging over an arbitrarily large footprint. Instead, filtering and synthesis have to be coupled: only the useful texture resolution must be computed. As far as we know, no such solution has been proposed for the aforementioned patch-based methods.

In this paper, we propose a technique for anisotropic filtering of [VSLD13]. This solution consists in storing explicitly low resolutions of the contents, each one being pre-filtered independently. The patch-map (which indicates the patch each texel belongs to) is also MIP-mapped using a special bitwise-OR boolean operation. We show how our method allows for simple filtering at every resolution while remaining relatively cheap in terms of both memory and computational costs. After a review of the literature (Section 2), our method is detailed in Section 3 and 4, and evaluated in Section 5. We then provide some perspectives in the concluding Section 6.

Related work

In this paper, we focus on real-time texture synthesis, and specifically tile-based methods. The quality of the results of the methods which fall in this category is not meant to be competitive against the best offline synthesis results, but rather to provide a good balance between quality, size, and speed. These methods generate an output by putting tiles next to each other, in a way such that there is no visible seam between adjacent tiles. To achieve this, both Wang Tiling [CSHD03, Wei04, LD05] and Corner Tiling [LD06, SD10] indicate the compatibility of the edges or corners of square tiles, and then assemble them in an aperiodic tiling. The filtering problem for these methods has been solved by [START_REF] Wei | Tile-based texture mapping on graphics hardware[END_REF], by packing the tiles as a correct tiling in a texture which is then filtered, allowing borders of adjacent tiles to blend into each other at low resolutions without introducing visual artifacts.

A drawback of these methods is that repeating the same set of squared tiles introduces visually unpleasant repetition artifacts, unless the amount of different edges (resp. corners) is increased, which exponentially increases the corresponding amount of tiles. Both [VSLD13] and [KCD16] solve this issue by partitioning the tiles into regions called patches, whose contents can be exchanged for different, pre-determined alternative contents taken from the tile itself, or from a larger input. While [VSLD13] segments the entire tile with a single set of patches, at the risk of introducing visible seams near the patch borders, [KCD16] limits seams by computing different sets of non-overlapping patches, but at the cost of reintroducing possible alignments, especially at the tile corners. For these methods, the filtering solution proposed by [START_REF] Wei | Tile-based texture mapping on graphics hardware[END_REF] is impractical, since it would require to generate tiles representing every possible combination of contents, yielding a memory complexity which grows exponentially with the number of patches.

Even if the decomposition of the input tile into patches may seem to be related to the problem of multi-chart texture filtering, approaches like [RNLL10] are not adapted to our case, because contents that must be blended are only known at rendering time, thus preventing an a priori filtering of texture seams. To our knowledge, no solution has been proposed yet for the filtering of on-thefly patch-based texturing.

Problem statement

Content exchange

The algorithm of [VSLD13] works as follows. For clarity, we consider a single periodic tile. The tile is partitioned into P patches numbered 1 p P. They are encoded by a patch-map P such that P(x x x) = p if the texel x x x belongs to patch p. Because the tile is assumed to be periodic, the patch-map is also repeated periodically. Let Ip,c be the c-th content of patch p: Ip,c(x x x) yields a texel value (Figure 1). Without loss of generality, we assume that every patch is given the same number C of contents, numbered 1 c C. At each tile repetition, each patch p is repeated, but its content changes: a function C(p) yields a random content number c. The synthesized texture is then defined, for any position x x x, as I(x x x) = I P(x x x),C(P(x x x)) (x x x).

(1)

Filtering

Once the texture I is mapped to a 3D scene, every screen-space pixel has a projected footprint in texture space. The filtering consists in approximating the average of I over this footprint. In a standard pipeline with pre-computed I, it relies on a MIP-map I ( ) of the texture: as the level increases, I is down-sampled by averaging over 2 × 2 square regions (decreasing resolution). In our context, the challenge consists in computing I ( ) on-the-fly.

Tiling techniques solve this problem by MIP-mapping the tiles offline and by assembling the appropriate resolution at runtime [START_REF] Wei | Tile-based texture mapping on graphics hardware[END_REF]. This does not work in our case because the patches have irregular shapes. So, at low resolution, the square regions may overlap several patches whose contents are known only at runtime. Pre-computing all possible combinations of contents is not tractable either: it would produce C P tiles, which cannot be stored and pre-filtered.

Filtering content exchange

Our method follows the same intuition as [START_REF] Wei | Tile-based texture mapping on graphics hardware[END_REF]: since we cannot assemble I offline and pre-filter it as I ( ) , we pre-filter the data so as to compute I ( ) on demand at run-time. We achieve this by:

• Pre-filtering the patch-map as a MIP-map P ( ) in such a way that the overlap of patches can be recovered at every level . • Pre-filtering all contents I ( ) p,c and packing them in a texture atlas.

• Assembling I ( ) at run-time from P ( ) and the content atlas.

Pre-filtering the patch-map

The patch-map is encoded as a bitmask of size P: at the higher resolution the p-th bit of P (0) (x x x) is set to 1 if x x x belongs to the patch p. Then the MIP-map is computed by using a bitwise-OR instead of an average. Namely, the value P ( ) (x x x) at texel x x x is the bitwise-OR of its four parents in the previous level P ( -1) .

As a consequence, the p-th bit of P ( ) (x x x) equals to 1 iff the patch p overlaps the texel x x x which corresponds to a 2 × 2 square at high resolution. These patches are the ones whose contents contribute to the final color I ( ) (x x x).

Pre-filtering the contents

To compute the pre-filtered contents I ( ) p,c , we proceed as follows.

For each content c of each patch p : • The tile is filled with zeros, except for the patch p, which is filled with content Ip,c. • The tile is MIP-mapped using four-to-one standard averaging.

Boundary texels are thus blended with the null-valued background (see Figure 3). • At every level ≥ 1, the content I ( ) p,c is cut out and packed in an atlas (see Section 4.4).

Assembling the filtered output

When rendering the texture, we need to compute I on demand for any level and at any texel x x x. We compute it as

I ( ) (x x x) = ∑ p∈P ( ) (x x x) I ( ) p,C(p) (x x x), (2)
where the sum blends all the contents I ( ) p,C(p) overlapping x x x at level . The patches p involved are recovered from the pre-filtered patch-map P ( ) . Thus, equation ( 2) is an exact evaluation of the MIP-map of I, without the need to store it explicitly. Then, standard filtering operations, such as trilinear and anisotropic filtering, can be applied as usually done with a MIP-map.

Storage

Our method requires storing the patch-map at all resolutions as an array of bitmasks (of P bits for P patches), as well as every resolution of the contents. To limit the impact of irregular patch shapes on the memory, we build an atlas (one for each content identifier c) p,c of every patch ( ≥ 1, p ∈ P), as shown in Figure 4. Note that we store only isotropic MIP-map levels, from which anisotropic filtering can be computed. Since contents of the highest resolution do not present the same overlapping issues as those of lower resolutions, they do not need to be stored in this atlas, but can be accessed directly from the input tile using translation offsets, like in [VSLD13, KCD16]. To use this atlas, it is necessary to store the origins of the content in the atlas and of the related patch in the input tile, for each resolution. It takes up a few more kilo-bytes, which is negligible.

Blending the contents

Patch-based texturing tends to introduce visible seams at patch boundaries. They can be hidden by blending adjacent contents. Our technique can filter the blended contents by making the following changes:

• The patches are dilated, which sets several bits to 1 in P (0) (x x x) when x x x is near a boundary. • Blending weights are defined in overlapping areas. In our tests, we used a simple decreasing distance to the patch boundaries. • Ip,c are weighted accordingly to get the high-resolution contents

I (0) p,c .
Then our pipeline (Sections 4.1, 4.2, and 4.3) is unchanged, generating the lower resolutions which automatically integrate the blending weights in the filtered contents. This enables to get the blending almost for free: a slight additional cost of memory is due to the storage of blending weights and more terms are involved in the sum of equation (2).

Results

Figure 5 presents the results of our filtering technique : it is indistinguishable from the ground truth (middle), while unfiltered results (left) are noisy at low resolution. The accompanying video shows how the filtering reduces flickering artifacts in dynamic scenes. Note that some flickering remains, which is not due to pre-filtering: it is also present in the ground truth and can be further reduced with supersampling or temporal anti-aliasing. Figure 6 shows the improvement of a simple blending, based on the texel distance to the patch boundaries. In regions where seams between different contents remain visible, blending reduces their visual impact. This blending has been enabled for all the results presented in the paper. Although the blending we used is very simple, more complex, content-dependant weights may be used as well.

Table 1 lists the memory usage, which grows linearly with the number N of pixels and the number C of contents. For typical use cases, with a tile of 1024 2 texels and 6 contents, the additional cost is about 5×, which makes it possible to generate high-resolution and unbounded textures with less than 20Mb. We rendered our results on a steep surface covered by 2.6G texels (51.2K × 51.2K) using a 800 × 700 pixels viewport. We enabled trilinear filtering, linear magnification, and anisotropic filtering. Rendering was running at 90 frames per second on an NVidia GTX 1060. Note that these performances could be improved by bypassing equation (2) at very low resolutions (i.e. far away), by lowering the degree of anisotropic filtering (we used 16 samples at most), and by improving the implementation to reduce cache misses.

Conclusion and future works

We presented in this paper a filtering technique for on-the-fly patchbased texturing approaches, which deals with the difficult challenge of managing different contents coming from non-adjacent regions of the input texture and randomly chosen at rendering time. Prefiltering both the contents and the patch-map allows for a runtime evaluation of the resulting texture MIP-map. To achieve this, our method requires to store explicitly the pre-filtered low-resolution contents, as well as the patch-map at each resolution. Yet, the total memory storage still remains acceptable, typically a few tens of Mb for unbounded textures. Moreover, we showed that our al-gorithm is able to perform blending at patch boundaries by only a slight modification of our pre-processing. It reduces the visual impact of seams potentially appearing in on-the-fly patch-based texturing. This technique performs well on a modern GPU, and can be implemented with standard shader languages, leading to real-time performances.

We are interested in investigating how our filtering technique can be extended. Indeed, it has been especially designed to work for [VSLD13], but cannot be applied directly to [KCD16] due to differences in data representation. We would like to extend the algorithm so as to be able to filter the latter as well. Following the same direction, we think that our algorithm is able to filter tilebased approaches too, by considering the set of tiles as the different contents of a unique square patch, and we would like to conduct experiments to validate this assumption. Finally, we want to investigate the possibility of using our method for more complex data that cannot be linearly filtered, such as normal maps or shadow maps.

Figure 2 :

 2 Figure 2: Examples of irregular, homogeneous and structured textures, particularly suitable for content exchange.

Figure 3 :

 3 Figure 3: Each content Ip,c is pre-filtered at different levels / resolutions. Top: fixed resolution. Bottom: fixed size.

Figure 4 :

 4 Figure 4: Packing of the contents. Left: the high resolution ( = 0) does not need special storage. Right: for each content id c, all lowresolution ( ≥ 1) contents {I ( ) p,c } c, are packed in an atlas.

Figure 5 :

 5 Figure 5: Results for various examples. Our filtering method (right) is compared to the ground truth (middle) and no filtering (left). The ground truth is computed by an exact filtering of the high resolution. The leftmost view indicates the MIP-map levels used.

Figure 6 :

 6 Figure 6: Rendering of a texture with (bottom / right) and without (top / left) blending at patch boundaries. Seams between different contents are less visible when blending is enabled.

Table 1 :

 1 Memory footprint analysis for different input sizes N and different numbers of contents C. The additional cost for the patchmap is fixed (+433%). The cost for the contents is +50% per content, which must be compared to +33% if the packing were perfect.

	Data (N/C)	No filtering (Mb) contents patch-map	Our filtering (Mb) contents patch-map
	512 2 / 3	0.79	0.26	2.0 (+150%) 1.4 (+433%)
	512 2 / 6	0.79	0.26	3.2 (+300%) 1.4 (+433%)
	1024 2 / 3	3.1	1.05	8.1 (+150%) 5.6 (+433%)
	1024 2 / 6	3.1	1.05	13 (+300%) 5.6 (+433%)
	Asymptotic	Θ(N)		Θ(NC)

which encompasses the low resolution contents I ( )

c 2019 The Author(s) Eurographics Proceedings c 2019 The Eurographics Association.


Katie N Kang

Dustin R Klein

Misha I Kazi

Francois Guerin

Vincent Cattoir

Jennifer S Brodbelt

Joseph M Boll
email: joseph.boll@uta.edu

E Cloacae

François Guérin

Colistin heteroresistance in Enterobacter cloacae is

Keywords: heteroresistance, antimicrobial resistance, lipid A, outer membrane, colistin, Enterobacteriaceae K heteroresistance, antimicrobial resistance, lipid A, outer membrane, colistin, Enterobacteriaceae

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

Introduction

Gram-negative bacteria assemble a highly conserved outer membrane (OM) barrier.

Glycerophospholipids comprise the periplasmic monolayer of the asymmetric lipid bilayer, while the surface-exposed monolayer is enriched with lipopolysaccharide (LPS). The LPS glycolipid is organized into three domains; an O-antigen carbohydrate repeat, core oligosaccharide, and the membrane anchor, lipid A [START_REF] Whitfield | Biosynthesis and export of bacterial lipopolysaccharides[END_REF]. The lipid A domain is initially synthesized as a β-1',6-linked glucosamine disaccharide that is both phosphorylated and fatty acylated. Lipid A is the bioactive portion of LPS and robustly activates the human Toll-like receptor 4 (TLR-4) and myeloid differentiation factor 2 (MD-2) immune complex to induce immune reactivity [START_REF] Whitfield | Biosynthesis and export of bacterial lipopolysaccharides[END_REF][START_REF] Needham | Modulating the innate immune response by combinatorial engineering of endotoxin[END_REF][START_REF] Boll | A penicillin-binding protein inhibits selection of colistin-resistant, lipooligosaccharidedeficient Acinetobacter baumannii[END_REF][START_REF] Boll | Reinforcing Lipid A Acylation on the Cell Surface of Acinetobacter baumannii Promotes Cationic Antimicrobial Peptide Resistance and Desiccation Survival[END_REF]. Gram-negative pathogens encode highly conserved regulatory mechanisms that modify lipid A to prevent TLR-4/MD-2 recognition and to fortify the OM against immune effectors and antimicrobials, which promotes survival in the host [START_REF] Raetz | Lipid A modification systems in gram-negative bacteria[END_REF].

Lipid A modification enzymes are transcriptionally regulated by two-component systems (TCS) [START_REF] Prost | The Salmonellae PhoQ sensor: mechanisms of detection of phagosome signals[END_REF][START_REF] Gunn | The Salmonella PmrAB regulon: lipopolysaccharide modifications, antimicrobial peptide resistance and more[END_REF]. The PmrAB and PhoPQ TCSs are well-studied phosphorelay signaling systems that regulate lipid A modifications in response to specific environmental signals [START_REF] Véscovi | Mg2+ as an extracellular signal: environmental regulation of Salmonella virulence[END_REF][START_REF] Bader | Recognition of antimicrobial peptides by a bacterial sensor kinase[END_REF][START_REF] Gunn | Recognition and integration of multiple environmental signals by the bacterial sensor kinase PhoQ[END_REF]. PmrAB and PhoPQ are highly conserved among pathogenic Enterobacteriaceae [START_REF] Needham | Fortifying the barrier: the impact of lipid A remodelling on bacterial pathogenesis[END_REF].

PmrAB responds to high Fe 3+ concentrations, cationic antimicrobial peptides (CAMPs), and slightly acidic pH to directly activate eptA (also known as pmrC) and arn operon expression [START_REF] Herrera | Activation of PmrA inhibits LpxTdependent phosphorylation of lipid A promoting resistance to antimicrobial peptides[END_REF][START_REF] Wösten | A signal transduction system that responds to extracellular iron[END_REF][START_REF] Perez | Acid pH activation of the PmrA/PmrB two-component regulatory system of Salmonella enterica[END_REF], which encode phosphoethanolamine (pEtN) and 4-amino-4-deoxy-L-arabinose (L-Ara4N) transferases, respectively [START_REF] Trent | An inner membrane enzyme in Salmonella and Escherichia coli that transfers 4-amino-4-deoxy-L-arabinose to lipid A: induction on polymyxin-resistant mutants and role of a novel lipid-linked donor[END_REF][START_REF] Gunn | PmrA-PmrB-regulated genes necessary for 4-aminoarabinose lipid A modification and polymyxin resistance[END_REF][START_REF] Zhou | Lipid A modifications in polymyxin-resistant Salmonella typhimurium: PMRA-dependent 4amino-4-deoxy-L-arabinose, and phosphoethanolamine incorporation[END_REF][START_REF] Lee | The PmrA-regulated pmrC gene mediates phosphoethanolamine modification of lipid A and polymyxin resistance in Salmonella enterica[END_REF]. Cationic amine addition to the lipid A domain of LPS neutralizes the surface charge to protect the cell from CAMP-mediated lysis [START_REF] Trent | An inner membrane enzyme in Salmonella and Escherichia coli that transfers 4-amino-4-deoxy-L-arabinose to lipid A: induction on polymyxin-resistant mutants and role of a novel lipid-linked donor[END_REF][START_REF] Zhou | Lipid A modifications in polymyxin-resistant Salmonella typhimurium: PMRA-dependent 4amino-4-deoxy-L-arabinose, and phosphoethanolamine incorporation[END_REF].

PhoPQ is activated in response to depletion of divalent cations such as Mg 2+ and Ca 2+ and the presence of CAMPs [START_REF] Véscovi | Mg2+ as an extracellular signal: environmental regulation of Salmonella virulence[END_REF][START_REF] Gunn | Recognition and integration of multiple environmental signals by the bacterial sensor kinase PhoQ[END_REF][START_REF] Bader | Recognition of antimicrobial peptides by a bacterial sensor kinase[END_REF]. PhoPQ phosphotransfer directly activates transcription of genes encoding PagL (only in Salmonella [START_REF] Raetz | Lipid A modification systems in gram-negative bacteria[END_REF]) and PagP, which add or remove acyl chains from lipid A, respectively [START_REF] Véscovi | Mg2+ as an extracellular signal: environmental regulation of Salmonella virulence[END_REF][START_REF] Miller | A two-component regulatory system (phoP phoQ) controls Salmonella typhimurium virulence[END_REF][START_REF] Richards | Cationic antimicrobial peptides serve as activation signals for the Salmonella Typhimurium PhoPQ and PmrAB regulons in vitro and in vivo[END_REF][START_REF] Kawasaki | Inhibition of Salmonella enterica serovar Typhimurium lipopolysaccharide deacylation by aminoarabinose membrane modification[END_REF]. Additionally, it directly activates A c c e p t e d M a n u s c r i p t arn expression in Klebsiella and Yersinia spcs. [START_REF] Mitrophanov | Evolution and dynamics of regulatory architectures controlling polymyxin B resistance in enteric bacteria[END_REF][START_REF] Winfield | Transcriptional regulation of the 4amino-4-deoxy-L-arabinose biosynthetic genes in Yersinia pestis[END_REF]. While the PmrAB and PhoPQ TCSs each regulate distinct subsets of genes, the independent signaling pathways also converge through the connector protein, PmrD [START_REF] Rubin | PmrD is required for modifications to Escherichia coli endotoxin that promote antimicrobial resistance[END_REF][START_REF] Gunn | PhoP-PhoQ activates transcription of pmrAB, encoding a two-component regulatory system involved in Salmonella typhimurium antimicrobial peptide resistance[END_REF][START_REF] Soncini | Two-component regulatory systems can interact to process multiple environmental signals[END_REF]. PmrD binds phospho-PmrA, which prevents PmrB-mediated dephosphorylation [START_REF] Rubin | PmrD is required for modifications to Escherichia coli endotoxin that promote antimicrobial resistance[END_REF][START_REF] Kato | Connecting two-component regulatory systems by a protein that protects a response regulator from dephosphorylation by its cognate sensor[END_REF][START_REF] Luo | Structural basis of a physical blockage mechanism for the interaction of response regulator PmrA with connector protein PmrD from Klebsiella pneumoniae[END_REF][START_REF] Kox | A small protein that mediates the activation of a two-component system by another two-component system[END_REF]. Constitutive PmrA-dependent gene expression increases pEtN and L-Ara4N lipid A modifications.

The Enterobacter cloacae complex (ECC) is composed of thirteen closely-related Gram-negative bacterial clusters (designated C-I to C-XIII) [START_REF] Morand | Specific distribution within the Enterobacter cloacae complex of strains isolated from infected orthopedic implants[END_REF]. ECC are typically associated with the host microbiota. However, many clusters have been associated with hospital-acquired infections, especially in immunocompromised patients [START_REF] Sanders | Enterobacter spp.: pathogens poised to flourish at the turn of the century[END_REF]. Infections manifest in a wide range of host tissues with symptoms including skin, respiratory tract, urinary tract, wound and blood infections [START_REF] John | Enterobacter cloacae: bacteremia, epidemiology, and antibiotic resistance[END_REF]. ECC have increasingly emerged in nosocomial settings and are problematic because they harbor multidrug resistance (MDR) mechanisms, which limit treatment options [START_REF] Sanders | Enterobacter spp.: pathogens poised to flourish at the turn of the century[END_REF][START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF][START_REF] Mezzatesta | Enterobacter cloacae complex: clinical impact and emerging antibiotic resistance[END_REF][START_REF] Pagès | Enterobacter aerogenes and Enterobacter cloacae; versatile bacterial pathogens confronting antibiotic treatment[END_REF]. Alternative last-line therapeutics used to treat MDR Gram-negative infections include the CAMP, colistin (polymyxin E), which binds the lipid A portion of LPS to perturb the outer membrane and lyse the bacterial cell. Despite success as a last-line therapeutic [START_REF] Carlet | Antibacterial agents: back to the future? Can we live with only colistin, co-trimoxazole and fosfomycin?[END_REF][START_REF] Nation | Colistin in the 21st century[END_REF], many ECC clusters demonstrate heteroresistance, where a subset of the clonal population is colistin resistant [START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF][START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF][START_REF] Napier | Colistin Heteroresistance in Enterobacter cloacae Is Associated with Cross-Resistance to the Host Antimicrobial Lysozyme[END_REF][START_REF] Kuper | Antimicrobial susceptibility testing: a primer for clinicians[END_REF]. We do not fully understand the underlying molecular mechanism(s) that regulate colistin heteroresistance in ECC; further characterization will advance our understanding of antimicrobial resistance and could help inform new treatment strategies.

A previous report showed that colistin heteroresistance naturally occurs within clonal ECC clusters [START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF]. Moreover, colistin heteroresistance in E. cloacae was induced by innate immune defenses within a murine infection model, which led to treatment failure [START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF].

Transcriptional analysis of susceptible and resistant populations suggested that pEtN and L-Ara4N lipid A modifications contribute to heteroresistance [START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF] and PhoPQ contributed to A c c e p t e d M a n u s c r i p t regulation [START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF][START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF], as described in other Enterobacteriaceae [START_REF] Needham | Fortifying the barrier: the impact of lipid A remodelling on bacterial pathogenesis[END_REF]. However, it was not established that the lipid A modifications actually occur, nor has PhoPQ-dependent, PmrABindependent regulation of colistin heteroresistance been fully described in E. cloacae or other ECC isolates.

Herein, we demonstrate that E. cloacae colistin heteroresistance involves PhoPQ Ecl , which regulates L-Ara4N modification of lipid A. The PhoP Ecl response regulator directly binds to the promoter region of arnB Ecl , which is the first gene of a seven-gene operon (arnBCADTEF Ecl ). Transcriptomics analysis supports a model of PhoPQ-dependent, PmrABindependent arn Ecl regulation. Furthermore, L-Ara4N modification of lipid A increased in response to growth in limiting Mg 2+ , which amplified colistin resistance in a PhoPQ Ecldependent manner. Lastly, sequencing of twelve ECC colistin-susceptible isolates pinpointed mutations within the PhoPQ-dependent lipid A modification pathway that promoted colistin susceptibility. This study advances our understanding of the molecular mechanisms that mediate colistin heteroresistance in ECC.

Results

Colistin heteroresistance in E. cloacae is regulated by PhoPQ Ecl , but not PmrAB Ecl .

To elucidate the underlying mechanisms that regulate colistin heteroresistance in ECC, we analyzed a collection of E. cloacae subsp. cloacae strain ATCC 13047 genetic mutants by calculating the colony forming units (CFUs) during exponential growth in the absence and presence of colistin (Fig 1A). While wild type and all mutant E. cloacae strains grew in standard growth media, ∆phoPQ Ecl was not viable when 10 µg/ml of colistin was added to the media. Clinical resistance to colistin is defined as >4 µg/ml [START_REF] Reis | Polymyxin-resistant Acinetobacter spp. isolates: what is next?[END_REF]

Since lipid A modifications induce colistin resistance in pathogenic

Enterobacteriaceae [START_REF] Needham | Fortifying the barrier: the impact of lipid A remodelling on bacterial pathogenesis[END_REF], we analyzed wild type and mutant E. cloacae lipid A for modifications. 32 The lipid A anchor of LPS is a pathogen associated molecular pattern (PAMP) that is bound with high affinity by the mammalian host TLR-4/MD-2 complex [START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF], which activates a proinflammatory response to clear the bacterial infection [START_REF] Kawai | Pathogen recognition with Toll-like receptors[END_REF]. Structural alterations to lipid A can dramatically alter TLR-4/MD-2-dependent host immune activation (2) and a previous report nicely demonstrated that E. cloacae colistin heteroresistance was induced by innate immune effectors [START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF]. Therefore, we examined if E. cloacae containing modified or unmodified lipid A would differentially activate TLR-4/MD-2 in a human embryonic kidney reporter cell line (HEK-blue) [START_REF] Needham | Modulating the innate immune response by combinatorial engineering of endotoxin[END_REF]. Wild type and phoPQ Ecl mutant strains stimulated TLR- Thus, while PhoPQ Ecl -dependent lipid A modifications contribute to CAMP resistance in E. cloacae, they do not significantly affect innate immune recognition and reactivity.

A c c e p t e d M a n u s c r i p t

Determination of E. cloacae lipid A modifications.

In order to define outer membrane modifications, we isolated lipid A from wild type E. cloacae grown in media supplemented with 10 µg/ml of colistin and from ∆phoPQ Ecl , which was grown without colistin. Purified lipid A was analyzed by direct infusion nanoESI. The Based on transcriptomics studies, a previous report suggested that E. cloacae adds pEtN and L-Ara4N to lipid A to develop colistin heteroresistance [START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF]. However, our genetic and high resolution mass spectrometry analysis demonstrate that only L-Ara4N modifies the 1).

L-Ara4N lipid A modifications are dependent on PhoPQ Ecl , but not PmrAB Ecl .

To further characterize lipid A modifications in the ∆pmrAB Ecl mutant, we analyzed purified lipid A from wild type and mutant E. cloacae using MALDI-TOF mass spectrometry. Wild type produced a lipid A mixture, which included L-Ara4N modified lipids (Fig S7A andB).

In contrast, analysis of ∆phoPQ Ecl and ∆arn Ecl lipid A indicated that L-Ara4N modified lipids were not present. Expression of PhoPQ Ecl in trans from an IPTG-inducible promoter restored L-Ara4N modified lipid A in the phoPQ Ecl mutant. Furthermore, ∆pmrAB Ecl produced the L-Ara4N modification, similar to wild type (Fig S7A). The m/z of each prominent peak in our MALDI-MS analysis corresponded with the exact mass of an expected structure with only the L-Ara4N-containing structures demonstrating colistin resistance (Fig S7B). Here, we confirmed that L-Ara4N modification of lipid A in E. cloacae is not dependent on PmrAB Ecl .

PhoP Ecl directly binds to the arnB Ecl promoter.

The arn operon is composed of seven genes and expression is driven by a promoter upstream of arnB [START_REF] Gunn | PmrA-PmrB-regulated genes necessary for 4-aminoarabinose lipid A modification and polymyxin resistance[END_REF]. This genetic organization is conserved in E. cloacae as illustrated in Fig 3A .   phoP expression is autoregulated in Enterobacteriaceae, where PhoP binds to the PhoP box to interact with RNA polymerase, which induces transcription [START_REF] Soncini | Transcriptional autoregulation of the Salmonella typhimurium phoPQ operon[END_REF] Importantly, PhoP Ecl does not bind to E. coli P arnB , which does not encode the PhoP box motif (Fig 3C). Furthermore, the PhoP Ecl -arnB Ecl promoter interaction was abrogated when unlabeled E. cloacae P arnB was added in increasing amounts, as a competitive inhibitor. We also show that the interaction is specific because addition of noncompetitive DNA (poly(dI-dC)) did not reduce the PhoP Ecl and E. cloacae P arnB interaction (Fig 3D). Lastly, PhoP Ecl bound E. cloacae and E. coli P phoP , which both encode the nucleotide sequence specific to the PhoP box (Fig 3E). Together, these findings suggest that E. cloacae encodes a mechanism that enables L-Ara4N biosynthesis to respond directly to PhoPQ Ecl .

RNA-sequencing analysis of the phoPQ Ecl and pmrAB Ecl mutants.

To better understand PhoPQ Ecl and PmrAB Ecl transcriptional regulation, we isolated and sequenced total RNA from wild type and mutant E. cloacae strains. A heat map illustrates the fold change of arn Ecl , phoPQ Ecl , and pmrAB Ecl gene expression in the TCS mutants relative to wild type (Fig 4). Expression of the arn Ecl genes were significantly down regulated in ∆phoPQ Ecl compared to wild type, suggesting that activation of the pathway is dependent on PhoPQ Ecl . In contrast, arn Ecl gene expression was not significantly altered in the ∆pmrAB Ecl mutant relative to wild type. A complete list of the significant up-and down-regulated genes (P <0.05) is included in Table S4. Together, these analyses indicate colistin heteroresistance in wild type E. cloacae (Fig 1A and Table 1) is mediated by L-Ara4N modification of lipid A, which is regulated in a PhoPQ Ecl -dependent manner under standard growth conditions. In E. coli and Salmonella, PhoPQ is activated by various signals, including low Mg 2+ and CAMPs [START_REF] Véscovi | Mg2+ as an extracellular signal: environmental regulation of Salmonella virulence[END_REF][START_REF] Bader | Recognition of antimicrobial peptides by a bacterial sensor kinase[END_REF][START_REF] Richards | Cationic antimicrobial peptides serve as activation signals for the Salmonella Typhimurium PhoPQ and PmrAB regulons in vitro and in vivo[END_REF]. Here we analyzed if PhoPQ Ecl responds to similar physiological cues to induce colistin resistance in E.

Inactivation of colistin heteroresistance in ECC clinical isolates.

A previous report showed that while many ECC clinical isolates were colistin heteroresistant, some were susceptible to colistin-mediated lysis. Interestingly, ECC colistin heteroresistance was cluster dependent [START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF]. To determine the genetic basis for colistin sensitivity in ECC clusters, we examined twelve colistin susceptible clinical isolates (CI) and compared them to heteroresistant wild type ATCC 13047. We confirmed colistin susceptibility in all isolates using the BMD method to determine MICs (Table S3) and sequenced each genome. We obtained ~70 X 10 6 reads for each susceptible isolate and the wild type strain, which were mapped to the annotated ATCC 13047 genome [START_REF] Ren | Complete genome sequence of Enterobacter cloacae subsp. cloacae type strain ATCC 13047[END_REF]. The coverage region of each isolate varied between 82% to 99%, where most coverage variation occurred in plasmid DNA regions. Initial analysis of our wild type strain confirmed it did not encode mutations in the phoPQ Ecl , pmrAB Ecl , or the arn Ecl coding sequences (Table S3), consistent with the published annotation. In contrast, multiple mutations were found in the phoPQ Ecl and arn Ecl operons of the colistin-susceptible isolates. All of the amino acid changes in these coding regions are listed in Table S3. Interestingly, ten of the twelve isolates encoded single nucleotide polymorphisms (SNPs) in the phoPQ Ecl promoter sequence immediately upstream of the

Discussion

E. cloacae and other ECC members encode PmrAB Ecl and PhoPQ Ecl homologs. We hypothesized these TCS regulatory systems functioned together in a pathway to control L-Ara4N and pEtN modification of lipid A, based on previous transcriptomics analysis of resistant and susceptible populations [START_REF] Band | Antibiotic failure mediated by a resistant subpopulation in Enterobacter cloacae[END_REF] and because these lipid A modifications are highly conserved among Enterobacteriaceae [START_REF] Needham | Fortifying the barrier: the impact of lipid A remodelling on bacterial pathogenesis[END_REF]. However, our genetic and high-resolution mass spectrometry analysis of E. cloacae lipid A determined that colistin heteroresistance in E. cloacae was mediated by PhoPQ Ecl -dependent, PmrAB Ecl -independent L-Ara4N lipid A modification. Therefore, we identified a mechanism of ECC colistin heteroresistance that involves the PhoPQ system.

E. cloacae and other ECC members do not encode a PmrD homolog, which couples

PhoPQ signal transduction to regulation of PmrA-dependent genes in many Enterobacteriaceae [START_REF] Needham | Fortifying the barrier: the impact of lipid A remodelling on bacterial pathogenesis[END_REF] . Moreover, PmrA Ecl shares only 52% identity with E. coli PmrA and PmrB Ecl shares only 57% identity with E. coli PmrB, suggesting the L-Ara4N lipid A modification pathway in E. cloacae diverged from E. coli and Salmonella. We confirmed direct binding of PhoP Ecl to the arnB Ecl promoter, which supports a model where L-Ara4N addition to lipid A and colistin heteroresistance in E. cloacae is dependent on PhoPQ Ecl , but not PmrAB Ecl .

A c c e p t e d M a n u s c r i p t

hypothesize that uncoupling PmrAB Ecl regulation from L-Ara4N modification bypasses an important regulatory checkpoint, which likely promotes misregulated PhoPQ Ecl -dependent arn Ecl expression. Furthermore, colistin heteroresistance has also been associated with Klebsiella pneumoniae [START_REF] Jayol | Heteroresistance to colistin in Klebsiella pneumoniae associated with alterations in the PhoPQ regulatory system[END_REF], another Enterobacteriaceae family member that activates arn expression independently of PmrAB [START_REF] Mitrophanov | Evolution and dynamics of regulatory architectures controlling polymyxin B resistance in enteric bacteria[END_REF]. Since selection has driven ECC and other opportunistic pathogens to maintain an altered lipid A modification signaling network, we predict that it is advantageous to maintain a CAMP resistant subpopulation in some environments. Presumably, the alternative regulatory mechanism promotes bacterial fitness in environments specific to their commensal and pathogenic niches.

Colistin heteroresistance is not well-understood at the molecular level in Enterobacteriaceae. Our study indicates that PhoPQ Ecl signal transduction contributes to heteroresistance in ECC. However, additional studies are necessary to understand if there is a genetic determinant within the heteroresistant subpopulation that promotes resistance.

Alternatively, colistin heteroresistance could be a byproduct of promoter noise-induced bimodality, which has been linked to heteroresistance in other bacteria [START_REF] Hung | Heteroresistance to cephalosporins and penicillins in Acinetobacter baumannii[END_REF][START_REF] Van Merode | Influence of culture heterogeneity in cell surface charge on adhesion and biofilm formation by Enterococcus faecalis[END_REF]. Despite these two possibilities, our studies demonstrated that PhoPQ Ecl phosphotransfer (PhoQ H227 and PhoP D56 ) is required for colistin heteroresistance (Table 1), suggesting the phenotype is regulated by PhoPQ Ecl .

Experimental Procedures

Bacterial Strains and Growth

E. cloacae subsp. cloacae ATCC 13047 and ECC strains were initially grown from freezer stocks on Luria-Bertani (LB) agar. Isolated colonies were used to inoculate LB broth or N A c c e p t e d M a n u s c r i p t minimal medium (0.1M Bis-Tris, pH 7.5 or 5.8, 5 mM KCl, 7.5 mM (NH 4 ) 2 SO 4 , 0.5 M K 2 SO 4 , 1 mM KH 2 PO 4 , 0.10% casamino acids 0.2% glucose, 0.0002% thiamine, 15 M FeSO 4 , 10 M or 10 mM MgSO 4 ) at 37 C. Strains were grown into mid-logarithmic growth (OD 600 = 0.6) before analysis. Kanamycin was used at 25 g/ml for selection and colistin was used at 5 g/ml or 10 g/ml where indicated.

All strains and plasmids used in this study are listed in Table S1. Briefly, E. cloacae subsp. cloacae 13047 mutant strains were constructed as previously described using recombineering with the plasmid pKOBEG [START_REF] Guérin | Complex Regulation Pathways of AmpC-Mediated β-Lactam Resistance in Enterobacter cloacae Complex[END_REF]. Linear PCR products were introduced in to the E. cloacae ATCC 13047/pKOBEG strain by electroporation and plated on selective media. Selected clones were transformed with pCP20 to cure the antibiotic resistance cassette.

To complement E. cloacae mutants, the coding sequence from phoPQ Ecl was cloned into the SalI and KpnI sites in pMMBKn (3). To generate point mutants in PhoQ H277A and PhoQ D56A , site directed mutagenesis was performed using Pfu Turbo using primers that incorporated the associated alanine-encoded nucleotide replacements. All constructs were validated using Sanger sequencing. IPTG inducible constructs were transformed into the phoPQ mutant and grown in 2.0 mM IPTG to induce expression.

Broth Microdilution assays

MICs of colistin were determined in triplicate by the broth microdilution (BMD) method.

Briefly strains were inoculated from overnight cultures when OD 600 = 0.1. Various concentrations (0 -256 g/ml) of colistin were added to each well and cultures were incubated overnight. Growth was measured by reading the OD 600 . The lowest concentration A c c e p t e d M a n u s c r i p t at which growth was inhibited was recorded as the MIC. E. coli W3110 and WD101 were used as control strains. In some cases, 'skip wells' were observed suggesting a heteroresistance phenomenon and the MIC was determined disregarding the clear wells [START_REF] Guérin | Cluster-dependent colistin hetero-resistance in Enterobacter cloacae complex[END_REF].

Population Analysis

Population analysis profiling was performed by plating 1 X 10 10 CFU onto LB agar containing 1 to 64 g/ml colistin (in 2-fold increments). Plates were incubated overnight at 37 C and frequency the subpopulation was determined by dividing by the total number of cells [START_REF] El-Halfawy | Antimicrobial heteroresistance: an emerging field in need of clarity[END_REF].

Isolation of Lipid A

Isolation of lipid A for TLC analysis involved 32 P-radiolabeling of whole cells was performed as previously described with slight modifications [START_REF] Hankins | Elucidation of a novel Vibrio cholerae lipid A secondary hydroxyacyltransferase and its role in innate immune recognition[END_REF]. In brief, 12.5 ml of E. cloacae was grown at 37 C to OD 600 = 1.0. Bacteria were harvested by centrifugation at 10,000 X g for 10 min. Lipid A extraction was carried out by mild-acid hydrolysis as previously described [START_REF] Zhou | Lipid A modifications characteristic of Salmonella typhimurium are induced by NH4VO3 in Escherichia coli K12. Detection of 4-amino-4-deoxy-L-arabinose, phosphoethanolamine and palmitate[END_REF].

Mass Spectrometry

MS1 spectra of lipid A in Figure S7 were collected on a MALDI-TOF/TOF (Axima Performance, Shimadzu) mass spectrometer in the negative mode. All other spectra were collected in the negative mode on a Thermo Scientific Orbitrap Fusion Lumos mass spectrometer (San Jose, CA, USA) modified with a Coherent ExciStar XS ArF excimer laser (Santa Clara, CA), as previously described [START_REF] Klein | Shotgun Analysis of Rough-Type Lipopolysaccharides Using Ultraviolet Photodissociation Mass Spectrometry[END_REF]. HCD was performed with the normalized collision energy (NCE) of 25%. UVPD was performed with the laser emitting 193 nm photons at 5 mJ per laser pulse with 5 pulses per scan. The laser pulse repetition rate was 500 A c c e p t e d M a n u s c r i p t

TLR-4 Signaling Assays

HEK-Blue hTLR4, cell line was maintained according to the manufacturer specifications (Invivogen). Overnight bacterial cultures in stationary phase were serial diluted for assays as previously described [START_REF] Needham | Modulating the innate immune response by combinatorial engineering of endotoxin[END_REF][START_REF] Boll | A penicillin-binding protein inhibits selection of colistin-resistant, lipooligosaccharidedeficient Acinetobacter baumannii[END_REF]. At least two biological replicates were each done in triplicate and one representative set was shown.

Colony Forming Unit Counts and Colistin Survival Assays

For colony forming unit counts (CFUs), E. cloacae subsp. cloacae 13047 and mutant strains were initially grown from freezer stocks on Luria-Bertani (LB) agar. Isolated colonies were resuspended and used to inoculate LB broth with 10 g/ml or without colistin at an OD 600 = 0.01. Cells were plated at designated time points on LB agar. Plates were grown overnight at 37 C and colony forming units (CFU) were counted and reported.

Colistin survival assays were performed as previously described with slight modifications [START_REF] Rubin | PmrD is required for modifications to Escherichia coli endotoxin that promote antimicrobial resistance[END_REF]. Briefly, wild type and mutant E. cloacae strains were grown overnight on LB agar. The following day, N minimal media pH = 7.5 containing either 10 µM MgSO 4 (low Mg 2+ ) or 10 mM MgSO 4 (high Mg 2+ ) were inoculated at OD 600 = 0.05 with bacteria from overnight cultures after cells were washed with N minimal media without Mg 2+ .

Cultures were grown until OD 600 = 0.6, when they were split and treated with 0, 5 or 10 µg/ml of colistin (Polymyxin E). Cultures were incubated for 1 h at 37 C and then colony-A c c e p t e d M a n u s c r i p t forming units were plated on LB, grown, and calculated. Percent survival was calculated by dividing the number of bacteria after treatment with colistin relative to those incubated in the absence of colistin and then multiplied by 100.

Protein Purification

To purify the PhoP Ecl protein, the coding sequence was cloned into pT7-7Kn, as previously described [START_REF] Boll | A regulatory checkpoint during flagellar biogenesis in Campylobacter jejuni initiates signal transduction to activate transcription of flagellar genes[END_REF]. Briefly, the phoP Ecl coding sequence was amplified from E. cloacae cDNA with primers that added a C-terminal His 8X tag. From an overnight starter culture, 1 Liter of LB broth containing 25 µg/ml of kanamycin was inoculated at 1:50 and grown at 37 C until the OD 600 = 0.5. IPTG was added to a final concentration of 1mM, and the culture was incubated at 37 C for an additional 4 h. Bacteria were recovered by centrifugation at 10,000

x g for 10 min, and the bacteria were resuspended in lysis buffer. Bacteria were lysed using sonication and the soluble fraction was recovered by centrifugation at 10,000 x g for 30 min. PhoP Ecl -His 8X was purified on a Ni-nitrilotriacetic acid (NTA) beads according to the manufactures instructions (Qiagen).

Electrophoretic Mobility Shift Assay

PhoP Ecl -His 8X proteins were purified as described above. EMSAs were performed based on a modified protocol (56). 250-bp DNA fragments of phoP Ecl and arnB Ecl spanning -230 to +20 relative to the translational start site were amplified from E. cloacae or E. coli cDNA using 5'-biotinylated primers. PhoP Ecl -His 8X proteins were incubated with biotinylated DNA at 25° C for 20 min. For competition experiments, unlabeled E. cloacae P arnB and poly(dI-dC) were added at 1:1, 2:1, or 5:1 ratios relative to biotin-labeled P arnB DNA. 0.1 -10 μM of PhoP Ecl -His 8X proteins were used. After electrophoresis at 4° C, protein/DNA was transferred onto a A c c e p t e d M a n u s c r i p t positively charged nylon membrane. Blots were blocked in 5% milk in TBS for 20 min and streptavidin conjugated HRP was used at a 1:300 dilution.

Nucleic Acid Extraction

Total RNA was extracted using the Direct-Zol RNA MiniPrep Kit (Zymo Research) from E. cloacae grown to a final OD 600 = 0.6. Isolated RNA was treated with DNA-free DNA removal kit (Thermo-Fisher Scientific) to eliminate genomic DNA contamination. DNasedepleted RNA was used for qRT-PCR and RNA-seq.

RNA-sequencing

RNA-sequencing was performed as previously described (57). Briefly, DNA-depleted RNA was processed for Illumina sequencing using the NEB Next Ultra Directional RNA Library Prep kit for Illumina as described by the manufacturer (NEB). Sequencing was performed using Illumina HiSeq. Sequencing data was aligned to the E. cloacae subs. cloacae ATCC 13047 published genome annotations (47) using CLC genomic workbench software (Qiagen) and RPKM expression values were determined. The weighted proportions fold change of expression values between samples was determined and a Baggerley's test on proportions was used to generate a false discovery rate corrected P-value. We then used a cut-off of 2fold weighted proportions absolute change with a false-discovery rate corrected P-value of ≤ 0.05 to identify significantly differentially regulated genes between samples. The sequencing data for the clinical isolates has been deposited in the Nation Center for Biotechnology's Gene Expression Omnibus (GSE127802).

  P-radiolabelled lipid A was isolated and chromatographically separated based on hydrophobicity. As controls, we also analyzed lipid A from E. coli strain W3110 (Fig 1B, lane 1), which does not significantly modify its lipid A, and E. coli strain WD101 (Fig 1B, lane 9), which constitutively expresses pmrA to produce modified lipid A (15). Thin layer chromatography (TLC) analysis indicated that wild type E. cloacae produced a mixture of lipid A consistent with modified and unmodified species (Fig 1B, lane 2). ∆phoPQ Ecl and the ∆arn Ecl strains did not produce modified lipid A (Fig 1B, lanes 3 and 8). PhoPQ complementation fully restored production of modified lipid A in the phoPQ mutant (Fig 1B,

4 /

 4 MD-2-dependent activation equally (Fig S1B), suggesting that lipid A modifications do not significantly alter host immune recognition. Reporter activation by E. cloacae lipid A was attenuated compared to E. coli lipid A at higher cell densities, suggesting differential recognition by the human TLR-4/MD-2 complex. The Gram-positive Staphylococcus aureus, which does not produce lipid A, did not stimulate the TLR-4/MD-2 complex (Fig S1B).

  photodissociation (UVPD) MS/MS spectra were obtained for the ions of m/z 1042.68 and

A c c e p t e d M a n u s c r i p t 1 -

 1 and 4'-phosphates of lipid A in a PhoPQ Ecl -dependent manner (Fig 2A and B) and this amine-containing modification correlates with colistin heteroresistance (Fig 1A and Table

  A c c e p t e d M a n u s c r i p t conserved within E. coli P arnB , suggesting regulatory mechanisms that control promoter activation are different (Fig 3B). We performed electrophoretic mobility shift assays (EMSAs) using E. cloacae P arnB to determine if PhoP Ecl directly binds the promoter to activate arn Ecl transcription. Increasing concentrations of purified PhoP Ecl (Fig S8) induced a shift of the biotinylated arnB Ecl promoter fragment, which contains the putative PhoP box binding motif (Fig 3C).

A c c e

  p t e d M a n u s c r i p t E. cloacae colistin resistance is amplified in response to limiting Mg 2+ .

  cloacae. Wild type and mutant E. cloacae were grown in N minimal medium with high (10 mM) or low (10 M) Mg 2+ levels. All cultures were exposed to colistin at mid-logarithmic growth. Wild type and complemented phoPQ Ecl mutant strains grown in high Mg 2+ demonstrated some susceptibility to 5 and 10 g/ml of colistin (Fig 5A, High Mg 2+ ), suggesting colistin-susceptible and -resistant populations were present, which is indicative of heteroresistance. When grown under limiting Mg 2+ conditions, E. cloacae cells were more resistant (Fig 5A, Low Mg 2+ ). In contrast, phoPQ Ecl demonstrated a fitness defect in either Mg 2+ concentration when exposed to colistin (Fig 5A). These data suggest that E. cloacae PhoPQ Ecl amplifies colistin resistance in response to limiting Mg 2+ growth conditions. PhoPQ Ecl responds to limiting Mg 2+ conditions by inducing L-Ara4N lipid A modification. To determine if increased colistin resistance was dependent on L-Ara4N modification of lipid A, we isolated lipid A after growth in either low or high Mg 2+ . TLC analysis demonstrated that wild type and the complemented phoPQ Ecl mutant primarily produced L-Ara4N-modified lipid A when Mg 2+ concentrations were limiting (Fig 5B, Low Mg 2+ ). In contrast, the same strains grown in excess Mg 2+ , produced a mixture of modified and unmodified lipid A, which is indicative of heteroresistance (Fig 5B, High Mg 2+ ). Interestingly, growth in excess Mg 2+does not completely shut-off production of PhoPQ Ecl -dependent lipid A modification in E.

A c c e

  p t e d M a n u s c r i p t cloacae, as was previously shown in E. coli[START_REF] Rubin | PmrD is required for modifications to Escherichia coli endotoxin that promote antimicrobial resistance[END_REF]. Together, these studies suggest that a subset of the clonal E. cloacae population activates PhoPQ Ecl -dependent L-Ara4N modification of lipid A under standard growth conditions to promote heteroresistance. However, depletion of Mg 2+ amplifies L-Ara4N modification (Fig 5B) and colistin resistance (Fig 5A) throughout the population.

phoP

  Ecl translational start codon (within 8 nucleotides), which indicated the SNPs likely disrupted the ribosome binding site (Fig 6A). To determine if the SNPs attenuated PhoPQdependent colistin resistance, clinical isolate-3 (CI-3) (ECC cluster III), which did not encode

A c c e

  p t e d M a n u s c r i p t amino acid changes in the phoPQ Ecl or arn Ecl genetic coding regions, was transformed with pPhoPQ Ecl . Complementation restored colistin resistance, where the MIC increased from 0.5 in CI-3 to >256 g/ml after PhoPQ Ecl signaling was restored (Fig 6B). Complementation suggested that SNPs in the phoPQ Ecl ribosome binding sites contributed to colistin susceptibility. SNP mutations, which abrogate translation of PhoPQ Ecl , support a model where colistin heteroresistance is dependent on PhoPQ Ecl in diverse ECC isolates.

  Genomic sequences were analyzed as previously done[START_REF] Boll | A penicillin-binding protein inhibits selection of colistin-resistant, lipooligosaccharidedeficient Acinetobacter baumannii[END_REF]. Briefly, samples were processed for Illumina sequencing using the NEB Next Ultra DNA Library Prep kit (NEB). Sequencing was performed using Illumina HiSeq. Reads were aligned to E. cloacae strain ATCC 13047 published genome annotations using CLC genomic workbench software (Qiagen) with 90% length fraction and 90% similarities parameters. Mapped reads were locally realigned and fixed ploidy detection identified low and high frequency variants. E. cloacae variant tracks were compared to identify mutations. Mutations not present in strain ATCC 13047 were called if 95% of aligned reads contained the variant. The sequence data have been submitted to the GenBank under accession number SUB4176618.

A c c e

  p t e d M a n u s c r i p t 56. Boll JM, Hendrixson DR. (2011). A specificity determinant for phosphorylation in a response regulator prevents in vivo cross-talk and modification by acetyl phosphate. Proc Natl Acad Sci U S A. 108:20160-20165. 57. Davies BW, Bogard RW, Young TS, Mekalanos JJ. 2012. Coordinated regulation of accessory genetic elements produces cyclic di-nucleotides for V. cholerae virulence.Cell. 149:358-370.

  Analysis Profile using an initial culture of 10 10 CFU/mL b BMD : Broth microdilution method * Presence of skip wells ** Presence of squatter colonies inside the zone of inhibition A c c e p t e d M a n u s c r i p t

Figure 1 :

 1 Figure 1: Survival of colistin heteroresistant E. cloacae is dependent on PhoPQ Ecl , but

Figure 2 :

 2 Figure 2: Expanded MS1 spectra of lipid A isolated from (A) wild type E. cloacae grown

Figure 3 :

 3 Figure 3: PhoP Ecl binds to the arnB promoter of E. cloacae (Ecl), but not E. coli (Ec). (A)

A c c e p t e d M a n u s c r i p t

  concentrations of PhoP Ecl . PhoP Ecl was used at concentrations of 0, 0.1, 1.0, 5.0 and 10.0 µM. EMSA using Ec P arnB in the absence or presence of PhoP Ecl, respectively. (D) EMSA competition experiments where increasing concentrations (1:1, 2:1, 5:1) of unlabeled P arnB competes with biotin-labeled P arnB , but nonspecific unlabeled poly(dI-dC) (2:1, 5:1) does not.(E) PhoP Ecl binds to both the Ecl and Ec phoP promoters.

Figure 4 :

 4 Figure 4: RNA-sequencing analysis of E. cloacae genes. Heat map illustrating the altered

Figure 5 :

 5 Figure 5: PhoPQ Ecl -dependent activation of L-Ara4N addition induces colistin resistance in low Mg 2+ . (A) Wild type and mutant E. cloacae strains were grown in N minimal medium with high (10mM, top) or low (10 μM, bottom) Mg 2+ . Strains were

Figure 6 :

 6 Figure 6: PhoP Ecl single nucleotide polymorphisms that attenuate colistin

A c c e

  p t e d M a n u s c r i p t triangles indicate single nucleotide insertion mutations. (B) E-test strips of CI-3 and CI-3/pPhoPQ Ecl to visualize heteroresistant colonies. The black arrow indicates colistin resistant colonies. A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t

Table 1 )

 1 . The decrease in ∆phoPQ Ecl cell viability suggested that PhoPQ Ecl contributes to colistin heteroresistance. Furthermore, wild type E. cloacae grown in colistin demonstrated approximately ten-fold less CFUs at hour two (P <0.05), suggesting a survival defect in early logarithmic growth phase. However, the fitness defect was no longer significant at hour three. By hour four, CFUs were equivalent to growth without colistin (Fig 1A). Ecl /pPhoPQ Ecl , and ∆pmrAB Ecl strains, but not ∆phoPQ Ecl . We confirmed . Wild type, ∆phoPQ Ecl /pPhoPQ Ecl , and ∆pmrAB Ecl E. cloacae all demonstrated MICs >256 µg/ml, while the ∆phoPQ Ecl , ∆phoPQ Ecl /pPhoPQ H277A , ∆phoPQ Ecl /pPhoP D56A Q and ∆arn Ecl (arnBCADTEF Ecl ) MIC was 0.5 µg/ml. Together, these studies confirm that PhoPQ Ecl signal transduction and the arn Ecl biosynthetic operon (L-

	M a n u s c r i p t
	A c c e p t e d

  ). Furthermore, site-directed mutagenesis to substitute H277 in PhoQ Ecl or D57 in PhoP Ecl with alanine limited lipid A assembly to only unmodified species (Fig 1B lanes 5 and 6). These results confirm that PhoPQ Ecl phosphotransfer and L-Ara4N biosynthesis are essential for lipid A modification in E. cloacae. Interestingly, the pmrAB Ecl mutant assembled

	a modified lipid A, similar to wild type (Fig 1B, lane 7), and exhibited colistin
	heteroresistance (Fig 1A, Table 1, Fig S1A), suggesting that PmrAB Ecl does not regulate
	colistin heteroresistance in E. cloacae.

  . The putative PhoP box in the phoP promoter region (P phoP ) is conserved in E. coli, Salmonella, and E. cloacae (Fig 3B). Alignment of the E. cloacae arnB promoter region (P arnB ) with E. coli, Salmonella, and E. cloacae P phoP suggested a putative PhoP box region. Importantly, E. cloacae P arnB , which contains a putative PhoP box, is highly conserved among ECC. However, this feature was not

TableTable 1 . MICs of colistin and PAP analysis for each E. cloacae mutant Frequency of appearance of subpopulations (PAPs a ) Concentration of colistin

 1

		MIC of	MIC of
	Isolate	colistin (µg/ml) by BMD b	by Etest colistin (µg/ml)	1 µg/ml	2 µg/ml	4 µg/ml	8 µg/ml	16 µg/ml	32 µg/ml	64 µg/ml
	Wild type	≥256*	0.125**	9.5 X 10 -3	6.3 X 10 -2	5.2 X 10 -

Acknowledgements

This work was supported by NIH GM103655 (J.S.B), and Welch Foundation F-1155 (J.S.B.).

Funding from the UT System for support of the UT System Proteomics Core Facility Network is gratefully acknowledged.

We would like to thank Cara Boutte and Mark Pellegrino for thoughtful review of the manuscript.


Zane Zake
email: zane.zake@ls2n.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Adolfo Suarez

François Chaumette
email: francois.chaumette@inria.fr

Nicolò Pedemonte
email: nicolo.pedemonte@irt-jules-verne.fr

Stability Analysis of Pose-Based Visual Servoing Control of Cable-Driven Parallel Robots

Keywords: cable-driven parallel robots, visual servoing, stability

Cable-driven parallel robots are robots with cables instead of rigid links. The use of cables introduces advantages such as high payload to weight ratio, large workspaces, high velocity capacity. Cables also bring drawbacks such as bad accuracy when the robot model is not accurate. In this paper, a visual servoing control is proposed in order to achieve high accuracy no matter the robot model precision. The stability of the solution is analyzed to determine the tolerable perturbation limits. Experimental validation is performed both in simulation and on a real robot to highlight the differences.

Introduction

In cable-driven parallel robots (CDPRs) rigid links are substituted by flexible cables. This substitution leads to advantages such as: (i) large workspace (WS), (ii) reconfigurability [START_REF] Gagliardini | Discrete Reconfiguration Planning for Cable-Driven Parallel Robots[END_REF], and (iii) high payload to weight ratio. However, the accuracy of CDPRs is to be improved. Previously, the following methods to improve accuracy have been studied: (i) increasing the complexity of the CDPR model in model-based control [START_REF] Schmidt | Modeling Techniques and Reliable Real-Time Implementation of Kinematics for Cable-Driven Parallel Robots using Polymer Fiber Cables[END_REF]; using proprioceptive sensors, such as (ii) force sensors to measure cable tensions [START_REF] Picard | Pulleys and Force Sensors Influence on Payload Estimation of Cable-Driven Parallel Robots[END_REF] or (iii) angular position sensors to measure cable angles [START_REF] Fortin-Côté | Improving Cable-Driven Parallel Robot Accuracy Through Angular Position Sensors[END_REF]; using exteroceptive sensors, namely vision sensors [START_REF] Dallej | Vision-Based Modeling and Control of Large-Dimension Cable-Driven Parallel Robots[END_REF] [6] [START_REF] Ramadour | Grasping Objects With a Cable-Driven Parallel Robot Designed for Transfer Operation by Visual Servoing[END_REF].

The increasing popularity of vision-based control is due to its high robustness to unexpected change in the environment. The two main approaches of such control are: image-based visual servoing (IBVS) and pose-based visual servoing (PBVS) [START_REF] Chaumette | Visual servo Control I. Basic Approaches[END_REF]. The latter is used when information from the image and some additional knowledge about the object (usually its model) allows us to estimate the 3D pose of the object in the camera frame F c . The control consists of minimizing the difference between the acquired object pose and the desired one at each iteration. IBVS is used when estimation of the 3D pose is not possible. Here, visual features such as 2D image coordinates or image moments are used instead. The control then consists of minimizing the error in the image space by comparing the desired and the current visual features.

Vision-based control on CDPRs is not yet well researched. It is surprising given the challenge to achieve accurate model-based control, since it requires to predict complex aspects like cable elongation and sagging, and pulley effect. Furthermore, to find the solutions to the Forward Kinematic problem for CD-PRs is a tedious task that requires a good knowledge of actual cable lengths and tensions, acquired by proprioceptive sensors. Although the addition of proprioceptive sensors can increase the robot accuracy, vision-based control is inherently robust to modeling errors and uncertainties, and avoids the need of computing the Forward Kinematic problem. This is attained by actively perceiving either the MP with a static camera in the so-called eye-to-hand configuration [5] [6], or the object of interest with a moving camera mounted on MP in the eye-in-hand configuration [START_REF] Ramadour | Grasping Objects With a Cable-Driven Parallel Robot Designed for Transfer Operation by Visual Servoing[END_REF]. The robot is then actuated according to what is perceived. Dallej et al. used multiple static cameras facing the MP as well as observing cable exit points to determine their sag [START_REF] Dallej | Vision-Based Modeling and Control of Large-Dimension Cable-Driven Parallel Robots[END_REF]. Multiple control schemes were proposed. Similarly, Chellal et al. used 6 infra-red sensors to determine the MP pose with high accuracy [START_REF] Chellal | A Kinematic Vision-Based Position Control of a 6-DoF Cable-Driven Parallel Robot[END_REF]. Remy et al. used the eye-in-hand configuration with a single camera [START_REF] Ramadour | Grasping Objects With a Cable-Driven Parallel Robot Designed for Transfer Operation by Visual Servoing[END_REF]. Their task was to control a spatial CDPR with three translational degrees of freedom. Finally, a control scheme was proposed in [START_REF] Zake | Vision-based Control and Stability analysis of a Cable-Driven Parallel Robot[END_REF] to control all six degrees of freedom while still using only one camera mounted on the MP.

One of the main characteristics of any control scheme is its stability. It was found in [START_REF] Zake | Vision-based Control and Stability analysis of a Cable-Driven Parallel Robot[END_REF] that the PBVS control of a CDPR is highly robust to modeling errors. However due to the wide range of such errors and their combined effect on the system it was not possible to determine their maximum values. In this paper, possible sources of errors are studied separately to find their maximum values for different MP working range. Such a separated analysis does not allow us to evaluate the interaction between different errors. For this reason, a second evaluation is done, assuming a constant non-negligible perturbation. These values are experimentally validated both in simulation and on a CDPR prototype.

This paper is organized as follows. Section 2 presents the control scheme. Stability condition is established in Section 3. A case study is shown in Section 4. Finally, the conclusions are drawn in Section 5.

Vision-Based Control of a CDPR

CDPR Kinematics

The schematic of a spatial CDPR in a suspended configuration is shown in Fig. 1.

Given that the camera is mounted on the MP, the transformation matrix p T c between the camera frame F c and the MP frame F p is constant. On the contrary, the transformation matrices b T p and c T o change with time. The length l i of the ith cable is the 2-norm of the vector # » A i B i pointing from cable exit point A i to cable anchor point B i , namely,

l i = # » A i B i 2 (1)
with l i b u i = b # » A i B i = b b i -b a i = b R p p b i + b t p -b a i (2)
where b u i is the unit vector of b # » A i B i that is expressed as:

b u i = b # » A i B i b # » A i B i 2 = b b i -b a i b # » A i B i 2 = b R p p b i -b a i + b t p b # » A i B i 2 (3)
The cable velocities li are obtained upon differentiation of Eq. ( 2) with respect to (w.r.t.) time:

l = A b v p (4)
where b v p is the Cartesian velocity of the MP expressed in F b , l is the cable velocity vector, and A is the Forward Jacobian matrix of the CDPR, defined as [START_REF] Pott | Cable-Driven Parallel Robots: Theory and Application[END_REF]:

A =    b u T 1 ( b R p p b 1 × b u 1 ) T . . . . . . b u T m ( b R p p b m × b u m ) T    (5)

Pose-Based Visual Servoing

The control scheme proposed in this paper is shown in Fig. 2. An image is retrieved from the camera and processed with a computer vision algorithm, that returns the current pose of the object s. It is compared to a previously known desired pose s *5 and an error vector e is defined as e = [e T t e T ω ] T , where To decrease the error e, an exponential decoupled form is selected ė = -λe with a positive adaptive gain λ, that is computed at each iteration, depending on the current ||e|| 2 [START_REF] Ramadour | Grasping Objects With a Cable-Driven Parallel Robot Designed for Transfer Operation by Visual Servoing[END_REF].

e t = c t o -c * t o * = [
The relationship between ė and the Cartesian velocity of the camera c v c , expressed in F c , is defined as:

ė = L s c v c (6)
where L s is the interaction matrix and it is defined in [START_REF] Chaumette | Visual servo Control I. Basic Approaches[END_REF]: Finally, the instantaneous velocity of the camera in its own frame is expressed as a function of the pose error as follows:

c v c = -λ L -1 s e (7)
where L -1 s is the inverse of the estimation of the interaction matrix L s . Note that the inverse is directly used, because for PBVS L s is a (6 × 6)-matrix that is of full rank [START_REF] Chaumette | Visual servo Control I. Basic Approaches[END_REF].

Kinematics and Vision

To combine the modeling shown in Sects. 2.1 and 2.2, the MP twist b v p is expressed as a function of camera velocity c v c :

b v p = A d c v c ( 8
)
where A d is the adjoint matrix that takes the following form [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]:

A d = b R c b t c × b R c 0 3 b R c ( 9
)
where b t c = b R p p t c and p t c is the vector pointing from O p to O c .

Stability Condition

One of the main characteristics of any system is its stability. It is a measure to assess the effects of estimation quality. That is, how coarse can the estimation be for the system to still converge to its goal [START_REF] Khalil | Nonlinear systems[END_REF]. Lyapunov analysis is used to determine the stability of the closed-loop visual servoing system. From Eqs. ( 4), ( 6) and ( 8) the model is the following:

ė = L s A -1 d A † l ( 10
)
Upon injecting ( 8) and ( 7) into ( 4), the output of the control scheme, that is, the cable velocity vector takes the form:

l = -λ A A d L -1 s e (11)
where A and A d are the estimations of A and A d , respectively.

The following closed-loop equation is obtained from ( 10) and ( 11):

ė = -λ L s A -1 d A † A A d L -1 s e (12)
From ( 12), the system stability criterion is defined as:

Π = L s A -1 d A † A A d L -1 s > 0, ∀t (13)
Π > 0 is a sufficient condition to obtain global asymptotic stability (GAS). It can be seen from the closed-loop equation ( 12) that if Π is positive definite, then the control scheme will ensure an exponential convergence of the error e to 0. However, if it is negative then the error e will increase and the system may diverge from the goal. Indeed, ( 13) is only a sufficient condition, therefore the stability of the system is uncertain once the condition is not held.

Estimated Parameters

Given the closed loop equation [START_REF] Khalil | Nonlinear systems[END_REF], the following variables are estimated and can therefore affect the system stability:

ŝ -object pose in F c is computed from image features, so it will not be the exact pose s p T c -the pose of the camera in the MP frame F p . We have an idealistic model, however due to manufacturing imprecisions, the actual camera pose will be a little bit different from the modeled pose. -p B i -the Cartesian coordinates of cable anchor points to the MP, expressed in F p . Due to mechanical solution of the anchor points, the actual point is a point located on a sphere around the nominal point. -b A i -the Cartesian coordinates of cable exit points, expressed in the base frame F b . We are using the simplified CDPR model, that does not include pulleys actually located at the exit points.

( b T p ) t+∆t = ( b T c ) t+∆t c T p = ( b T c ) t exp ( c vc,∆t) c T p (14)
Since velocity c v c , necessary for exponential mapping, is computed from the object pose measurement ŝ, which we admit to be different from s, then b T p = b T p . Furthermore, initial b T p is only coarsely known 6 .

Case Study

The stability condition ( 13) is applicable to any CDPR with a PBVS control.

However, this analysis includes model-related parameters, thus it should be done for each CPDR separately. Hence, in the following sections the chosen CDPR is presented and results of stability analysis are shown.

ACROBOT and Simulation in V-REP

ACROBOT For this paper, a CDPR prototype, named ACROBOT and shown in Fig. 3(a), is used. Its WS is a 1 m 3 cube. The robot is assembled in a suspended configuration. A simple webcam AUTOPIX MT4018 is mounted on the MP in the eye-in-hand configuration facing the ground.

To simplify the computer vision part, AprilTags [START_REF] Olson | AprilTag: A robust and flexible visual fiducial system[END_REF] are used instead of real objects. They are especially convenient to use in combination with ViSP library [START_REF] Marchand | ViSP for visual servoing: a generic software platform with a wide class of robot control skills[END_REF], because the latter contains functions that allow to recognize the tags and retrieve their 3D pose. V-REP Model The V-REP simulation environment [START_REF] Rohmer | V-REP: A versatile and scalable robot simulation framework[END_REF] was used in order to create a dynamic model of ACROBOT including the vision sensor. This gives us the capacity to use the same software to control the real and the virtual hardware. As a result we can speed up the development time, test and debug our algorithms in simulation (with a perfectly known ground truth), then use the real robot for final verifications.

To create a dynamical simulation, the pulleys and cables are modeled as a sequence of joints and mass objects, as shown in Fig 4 . The current model does not take into account the pulley diameter and the cable sag. The pulleys are represented as a vertical revolute passive joint followed by a small spherical mass and a horizontal revolute passive joint. The cables are modeled as a sequence of prismatic joint, cylindrical mass, prismatic joint, cylindrical mass and a final spherical joint attaches the cable to the MP. The first prismatic joint is used to change the cable length. The second prismatic joint is responsible for the cable behavior through a specific joint control callback script, which models the cable forces as either an elastic spring, when in tension, or an element transmitting zero force, when in compression.

To have a stable simulation some model design rules need to be considered [START_REF]Designing Dynamical Simulations[END_REF]. In our tests, the Vortex physical engine was used.

Numerical Analysis

It is not possible to express analytically the pseudo-inverse of the Jacobian A † , thus stability analysis is only possible in numerical form. In this paper, we will only study two parameters, namely p T c and b A i .

The Cartesian coordinates of b A i are known, we need to find out the range of perturbation that does not destabilize the system. In Fig. 5(a) the perturbation range is defined by the radius r Ai of the sphere centered at point A i . We want to find the maximum value r Ai,max so that for any point within a sphere of radius r Ai ≤ r Ai,max the system is stable.

Unlike cable exit points, camera pose in F p could in fact be changed, if necessary. Therefore, here we first define the range, where the camera could Then for any camera position, the perturbation range is defined by the small sphere radius r pc . Our goal is to find r pc,max so that if r pc ≤ r pc,max and camera position is within the sphere defined by radius R pc , then the system will be stable. Finally, we transform p R c into axis-angle representation θ pc u. We define the perturbation as δθ pc , that can be made about any unit vector u δ . Therefore, our goal is to find δθ pc,max so that |δθ pc | ≤ δθ pc,max for any u δ , u and θ pc .

To find the range of perturbation, we define the system as stated in Table 1. Then we portray two distinct cases: (i) The system is assumed to be ideal, no perturbations other than the one we are studying; (ii) the system has small perturbations in all parameters. The perturbation values are chosen either based on the mechanical errors (such as p B i and b A i ) or as 5 to 10 percent of the actual parameter value. For the latter, we also distinguish the results depending on the desired range of motion of the MP in F b expressed as R bp .

The perturbation limits that we have established based on condition (13) are shown in Table 2. Here are the observations:

-None of the parameters are affected by the actual value of p t c or p R c . This is especially surprising for the perturbations on these parameters defined by r pc and δθ pc . -To keep the full motion range R bp and the full rotational range b R p , perturbations on b A i must not be larger than 0.01 m. -In the ideal case, the perturbation values for camera pose in F p are large. In fact these perturbations do not appear separately. Indeed, all the considered perturbations are affecting the system at the same time, not independently. For this reason it was important to do the second part of this study while considering perturbations in all parameters. -When the perturbations, defined in Table 1, are added to the system, the latter remains stable for all tested motion range of MP. -However, once the MP motion range is reduced, it is possible to increase the perturbation levels without making the system unstable. -As soon as R bp is reduced, r pc increases significantly, as well as δθ pc . This allows to conclude that perturbations in the respective variables have little effect on the stability of the system.

-There is a considerable increase in r Ai as well. This means that the modeling accuracy of exit points is not important. Even at WS borders the tolerated perturbation easily covers the corresponding model inaccuracies. That is, even though pulleys that are present on the cable exit points are not modeled, the small changes in the actual exit point location on the pulley are covered by the perturbation r Ai = 0.01 m and do not affect the system stability.

Experimental Validation

Some experiments have been conducted to validate the theoretical results 7 . First, we assume that the rotational error of the camera in F p and about Z axis is out of bounds of stability, i.e. δθ pc = 85 • . As shown in Fig. 6(a) the robot diverges from its goal position and error oscillates and slowly increases (Fig. 6(b)), the system is not stable.

If δθ pc = 55 • , the V-REP model will successfully reach the target as shown in Fig. 6(c), though the trajectory is far from optimal (a straight line). Some error oscillations can be observed in Fig. 6(d). The same rotational error on the actual robot is shown in Fig. 6(e). The initial behavior is similar, but AprilTag leaves the image and the task is failed. This is not surprising, given the additional uncertainties of the actual robot and the lower image quality.

Finally, δθ pc was set to its maximum for a noisy system with range of motion reduced to R bp = 0.3 m, i.e. δθ pc = 16 • . Once implemented on ACROBOT, we see that though the trajectory is perturbed, the MP reaches its targeted pose (Fig. 6(g)). Indeed, error e converges to zero without oscillations (Fig. 6(h)).

To summarize, for an ideal robot the range of perturbation on a single parameter is very large. On the contrary, as soon as we acknowledge that all the system parameters are noisy, then each individual perturbation has quite a limited range within the bounds of stability.

Conclusions

This paper proposed a method to analyze the stability of Pose-Based Visual Servoing (PBVS) control of Cable-Driven Parallel Robots (CDPRs). A general stability criterion was introduced. The stability of ACROBOT, a CDPR prototype located at IRT Jules Verne, was analyzed. Two CDPR model-related parameters were studied and their maximum perturbation range was found both for ideal robot and for a noisy one. A dynamic CDPR model in V-REP was presented.

It was found that for an ideal system any one parameter could be highly perturbed without making the system unstable. This result was successfully validated in simulation. When adding a large perturbation (validated in simulation) to ACROBOT, the system does not converge. However, if the perturbation is kept within the corresponding (noisy system) range, the robot will be able to complete its task.

For ACROBOT, the tolerated perturbation on cable exit points is rather large. This is beneficial, because it allows to avoid adding pulley kinematics to the model. Indeed, cable exit point variations on the pulley are smaller than the tolerated perturbation, which does not affect the stability of the system.

The added perturbations affect the trajectory to the goal and the system's ability to actually reach it. However, perturbations in the robot model do not affect the final accuracy. This is because the final pose is considered to be reached only when the error between the current and the desired object pose becomes smaller than the threshold defined in the visual servoing loop.

Future work will deal with the implementation of PBVS control on a large semi-industrial CDPR prototype as well as its stability analysis. It is also of interest to consider real objects instead of AprilTags to validate this approach for more realistic use cases.

Fig. 1 .

 1 Fig.1. Schematic of a spatial CDPR with eight cables, a camera mounted on its MP and an object in the WS.

Fig. 2 .

 2 Fig. 2. Control scheme for pose-based visual servoing of a CDPR

Fig. 3 .

 3 Fig. 3. CDPR prototype: (a) ACROBOT; (b) V-REP model of ACROBOT

Fig. 4 .

 4 Fig. 4. The representation of a cable and its pulley in V-REP

Fig. 5 .

 5 Fig. 5. Perturbation ranges for: (a) cable exit points Ai; (b) camera pose in Fp

Fig. 6 .

 6 Fig. 6. CDPR behavior depending on added perturbations. (a) and (b): the AprilTag trajectory and error e over time in V-REP simulation with δθ = 85 • ; system is not stable (only center-point trajectory shown). (c) and (d): the AprilTag trajectory and error e over time in V-REP with δθ = 55 • ; system is stable; (e) and (f ): the AprilTag trajectory and error e over time on ACROBOT with δθ = 55 • ; robot does not converge because AprilTag leaves the camera field of view. (g) and (h): the AprilTag trajectory and error e over time on ACROBOT with δθ = 16 • ; system is stable.

Table 1 .

 1 Variable and perturbation ranges

	variable family	variable value	perturbation value
	object pose in Fc	Rs t = 0.5 m |θω| ≤ 50 •	rs t = 0.05 m |δθω| ≤ 2.5 •
	camera pose in Fp	-0.032 -0.026 -0.01 180 • 0 • 180 •	rpc = 0.01 m δθpc = 3 •
	cable anchor points p Bi	In eight corners of MP of size 0.1 × 0.1 × 0.05 m	rBi = 0.008 m
	cable exit points b Ai	Two exit points at each top corner of ACROBOT	rAi = 0.01 m
		R bp is equal to 0.5 m, 0.3 m and 0.1 m, resp.	r bp = 0.02 m
	MP pose in F b	Rotation about global axes: 45 • about Z,	5 • about Z axis, 3 •
		20 • about Y, 20 • about X	about Y and X axes

Table 2 .

 2 Perturbation change depending on MP motion range

	Condition	Camera pose in Fp	Cable exit points b Ai
	Ideal robot, no other perturba-tion in the system, R bp = 0.5 m	rpc = 0.5 m	|δθpc| ≤ 55 •	rAi = 0.01 m
	Minimal perturbations from Table 1, R bp = 0.5 m	rpc = 0.03 m	|δθpc| ≤ 3 •	rAi = 0.01 m
	Minimal perturbations from Table 1, R bp = 0.3 m	rpc = 0.5 m	|δθpc| ≤ 16 •	rAi = 0.09 m
	Minimal perturbations from Table 1, R bp = 0.1 m	rpc = 0.8 m	|δθpc| ≤ 24 •	rAi = 0.18 m

In this paper, a superscript * denotes the desired value, e.g. desired object pose s * and c * in c * to * refers to desired camera frame Fc * in which the object is in the desired pose s *

For this CDPR, the initial pose was defined at the center of the WS, which itself was measured by hand with a measurement error of ±2 cm along X and Y axes, resp.

Please also see the accompanying video at https://youtu.be/tfiTDlp1ZIY

Acknowledgment This work is supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing) in the framework of the PERFORM project.