Marie Candito
email: marie.candito@u-paris.fr

Auxiliary Tasks to Boost Biaffine Semantic Dependency Parsing

 show modest but systematic performance gains on a near state-ofthe-art baseline using transformer-based contextualized representations. This provides a simple and robust method to boost SDP performance.

Introduction and related work

Semantic dependency parsing is the task of producing a dependency graph for a sentence. Depending on the datasets, these dependencies may correspond to predicate-argument relations, with labels numbering semantic arguments (as in or dependencies with intermediate status between syntax and semantics, with labels being canonical grammatical functions that normalize syntactic alternations (e.g. in Figure 1-bottom, the clitic l' (him) is the canonical object of the passive verb form sollicité (solicited)).

If one views each dependency as a decision to make, both dependency parsing (DP, outputing a syntactic tree) and semantic dependency parsing (SDP) are known to exhibit high interdependence of decisions. For instance in DP, when parsing a question answering machine, choosing machine as root is linguistically coherent with the machine She went back and spoke to the desk clerk. [START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF]. Bottom: French Deep syntactic graph as defined by [START_REF] Candito | Deep syntax annotation of the sequoia French treebank[END_REF].

→ answering → question analysis only, whereas (wrongly) choosing question as root is syntactically coherent with the question → answering → machine analysis.

In DP though, the interdependence between arcs is partially solved by the tree constraint: choosing one head for a given token amounts to ruling out all other heads. This structural interdependence is absent in SDP. Complex structural, lexical and semantic factors control whether a given dependent should be attached to zero, one or several heads.

Several approaches exist in the literature to capture interdependence of arcs in SDP, which often derive from proposals made for DP. One is to use a higher-order graph-based parser. [START_REF] Wang | Second-order semantic dependency parsing with endto-end neural networks[END_REF] achieve state-of-the-art results (without pretrained LM) on the English part of the SemEval2015-Task18 data, by using second-order factors to score the graphs, yet at the cost of a O(n 3 ) complexity.

Another main approach is to use sequential decisions, and hence take advantage of previous decisions by encoding the previously predicted arcs. This is the case in the transition-based parser of [START_REF] Fernández | Transition-based semantic dependency parsing with pointer networks[END_REF], or in the system of [START_REF] Kurita | Multi-task semantic dependency parsing with policy gradient for learning easy-first strategies[END_REF], which selects a new head for certain tokens at each iteration, using reinforcement learning to order this selection of heads. Both models have a O(n2 ) complexity (when used without cycle detection), and in both cases, sequential decisions benefit from the encoding of previously predicted arcs, yet at the cost of error propagation. For that reason, [START_REF] Bernard | Multiple tasks integration: Tagging, syntactic and semantic parsing as a single task[END_REF] propose a system close to that of [START_REF] Kurita | Multi-task semantic dependency parsing with policy gradient for learning easy-first strategies[END_REF], yet allowing the system to overwrite previous decisions and hopefully correct itself.

On the contrary, the biaffine system of Dozat and Manning (2018) (hereafter DM18) performs a simultaneous scoring of all candidate arcs, decides to predict an arc independently of the other ones. This results in a highly parallelizable O(n 2 ) inference, with surprisingly high performance albeit below second-order parsing.

As for most NLP tasks, SDP performance increases when integrating transformer-based contextual representations when encoding input tokens. On the English dataset from the SemEval 2015 Task 18 [START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF], [START_REF] Fernández | Transition-based semantic dependency parsing with pointer networks[END_REF] (hereafter FG20) report a +0.7 and +2.0 increase for the in-domain (ID) and out-of-domain (OOD) test sets respectively. 1 In this work, we retain the simple O(n 2 ) biaffine architecture of DM18, and we investigate how simple auxiliary tasks can introduce some interdependence between arc decisions, in a multi-task learning setting [START_REF] Caruana | Multitask learning[END_REF]. We show modest but statistically significant improvements on the three English datasets of the widely used SemEval2015-Task18 data [START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF]. We also test another appealing property of the biaffine architecture, which is the absence of formal constraints on the output graphs. Experiments on French deep syntactic graphs [START_REF] Ribeyre | Semi-Automatic Deep Syntactic Annotations of the French Treebank[END_REF], which are highly cyclic, also demonstrate the effectiveness of our auxiliary tasks for SDP.

1 Using the biaffine DM18 architecture, [START_REF] He | Establishing strong baselines for the new decade: Sequence tagging, syntactic and semantic parsing with BERT[END_REF] report a +2 and +3 point increase in ID and OOD. Yet, these results are not comparable: the authors have used a different pre-processing, which adds orphan dependencies from the root to orphan tokens, resulting in an easier task (p.c. with the authors and https://github.com/emorynlp/bert-2019/is sues/1).

The baseline biafine graph parser

We reuse the computation of the arc and label scores of the DM18 model, which we modernized by using contextual representations: input sequence w 1:n is passed into a pretrained language model. We represent a word-token w i by concatening the contextual vector of its first subword 2 h (bert) i and a word embedding e (word) i

.

v i = h (bert) i ⊕ e (word) i (1)
For some of the experiments, we also concatenate a lemma and a POS embedding.

v i = h (bert) i ⊕ e (word) i ⊕ e (lemma) i ⊕ e (POS) i (2)
The sequence of word-tokens representations is passed into several biLSTM layers:

r 1:n = biLSTM(v 1:n ).
The recurrent representation r i is then specialized according to two binary features: head versus dependent, and arc versus label score:

h (arc-head) i = MLP (arc-head) (r i ) h (lab-head) i = MLP (lab-head) (r i ) h (arc-dep) i = MLP (arc-dep) (r i ) h (lab-dep) i = MLP (lab-dep) (r i ) (3)
We use a simplified biaffine transformation for arc scores, and a per-label one for label scores:

s (arc) i→j = h (arc-dep) j U (arc) h (arc-head)⊤ i + b (arc) s (l) i→j = h (lab-dep) j U (l) h (lab-head)⊤ i + b (l)
(4) For each position pair i, j, a binary cross-entropy loss is used for the existence of arc i → j, and a cross-entropy loss is used for the labels of gold arcs. At inference time, any candidate arc with positive score s (arc) i→j is predicted, and receives the label with maximum score for this arc.

Auxiliary tasks targeting sets of arcs

Preliminary experiments on English semantic dependency graphs [START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF] and on French deep syntactic graphs [START_REF] Candito | Deep syntax annotation of the sequoia French treebank[END_REF] have shown us that the biaffine graph parser gives good results, but with inconsistencies that are clearly related to the locality of decisions. In particular, a quick error analysis revealed incompatible Here is a good rule of thumb : ... arc combinations. More precisely, we noticed impossible sets of labels for the set of heads of a given dependent. For example in the DM part of the SemEval2015-Task18 dataset, tokens are sometimes attached with a mwe label (for a component of a multi-word expression) and attached to another head with a non-mwe label, as shown for the rule token in Figure 2. This incorrect situation actually never happens in the training set, but this impossibility is not captured by the model. In the predicted French deep syntactic graphs, we noticed punctuation tokens wrongly attached to two different heads with the specific punct dependency label.

A second observation is a tendency in some of the datasets to predict disconnected tokens (i.e. with no incoming nor outgoing arcs) too frequently. More generally, when counting the number of predicted heads for each token in the predicted graphs, the accuracy is about 95% in the English datasets, and below 92% for the French one.

Auxiliary tasks

Hence the idea of using auxiliary tasks taking into account all the heads (resp. dependents) of a given token. More precisely, we experiment multi-task learning on the two target tasks (tasks A and L for arc and label prediction), plus the following auxiliary tasks, which predict for each token w j :

• tasks H and D: the number of governors and number of dependents. For instance in top Figure 1, spoke has two governors (went and to) and one dependent (She);

• the labels of the incoming arcs, either as:

task S: the concatenated string of the incoming arcs labels, in alpha-betic order (e.g. for the spoke token, AND_C+ARG1) task B: or the "bag of labels" (BOL) sparse vector, whose components are the numbers of incoming arcs to w j bearing each label. For the spoke token, this would give a 1 for the AND_C label component and 1 for the ARG1 label, and zeros for all other labels.

Technically, for each auxiliary task, a specific MLP is used to specialize the recurrent representation r j of each word-token w j . Tasks H and D are regression tasks, which use MLPs with a single output neuron and a squared error loss.3

nbh j = MLP (H) (r j ) nbd j = MLP (D) (r j )
The S task is a classification task into categories corresponding to multi-sets of labels encountered in the training set. 4 For w j , the vector of scores of all the label multi-sets is s j = MLP (S) (r j ), and cross-entropy loss is used at training.

For the B task, we use a MLP with final layer of size |L|: BOL j = MLP (B) (r j ). The component for label l, BOL jl , is interpreted as 1+ log of the number of l-labeled incoming arcs to w j . The loss we use is the L2 distance between gold and predicted BOL vectors.

The two example inconsistencies cited above are indirectly captured by these auxiliary tasks. Firstly, in case of a token w j that is a component of a multi-word expression, the recurrent representation r j for this token will be optimized to lead to a single mwe incoming label for the S or B tasks, and a value 1 for the H task (cf. a single governor for w j ). Hopefully, when used for the A and L tasks, r j will favor incoming arcs from close next tokens (cf. e.g. for the DM format, mwe components are attached to the right, and quite locally). The other mentioned problem of predicting disconnected tokens too frequently is captured by the H and D tasks. Predicting no incoming nor outgoing arc for a given token will only be coherent with H=0 and D=0 for this token. Hence, hopefully, predicting more than 0 for the H task for a token w j , will lead to higher scores for arcs pointing to w j .

Combining sublosses

At training time, for each batch, we seek to minimize a weighted sum of the losses for all the tasks, whether main or auxiliary. Manually tuning these weights is cumbersome and suboptimal. We use the notion of task uncertainty and the approximation proposed by [START_REF] Kendall | Multi-task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics[END_REF], who introduce a parameter σ t for each subtask t, to be interpreted as its "uncertainty". Noting T as the set of tasks, the overall loss for a batch is t∈T

1 σ 2 t L t + ln(σ t ).
The parameters σ t are initialized to 1 and modified during the learning process. The first term of the sum ensures that the more uncertain the task, the less its loss will count, while the second term prevents arbitrarily augmenting the σ t values, thus reducing the loss weights.

Stack propagation

We test two multi-task learning configurations: first, simple parameter sharing up to the biLSTM layers (all specialization MLPs applied on the recurrent token representations). Second, we test the technique of "stack propagation", which Zhang and Weiss (2016) experimented for the POS tagging and parsing tasks. In our case, it amounts to using the dense layers of the auxiliary tasks MLPs to score the arcs and their labels.

For example, to use the H task in stack propagation mode, let hidden (H) j be the hidden layer of MLP (H) for the dependent j, and c (H) a coefficient hyperparameter. The computation of s (arc) i→j (cf. equation 4) is modified as follows:

sp (arc-dep) j = h (arc-dep) j ⊕ c (H) hidden (H) j s (arc) i→j = sp (arc-dep) j U (arc) h (arc-head)⊤ i + b (arc)
Similarly, to use task B in stack propagation mode, we modify the score of each label:

sp (lab-dep) j = h (lab-dep) j ⊕ c (B) hidden (B) j s (l) i→j = sp (lab-dep) j U (l) h (lab-head)⊤ i + b (l)
Note that it forces to perform the auxiliary tasks during inference, instead of at training time only.

Experiments and discussion

Datasets

We experiment on the three widely used English datasets of SemEval2015-Task18 [START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF] (DM, PAS and PSD), which are acyclic graphs mainly representing predicate-argument relations. We also experiment on French deep syntactic graphs [START_REF] Ribeyre | Semi-Automatic Deep Syntactic Annotations of the French Treebank[END_REF] (Appendix D). These capture most of argument sharings (e.g. raising, obligatory and arbitrary control, subject sharing in VP coordination) but are closer to surface syntax in the sense that labels remain syntactic, even though syntactic alternations are neutralized (e.g. passive by-phrases are labeled as subjects). Cycles may appear e.g. in relative clauses.5

Experimental protocol

We chose to investigate the impact of the auxiliary tasks on a high baseline, using pretrained contextual representations. We use our own implementation6 of the biaffine parser, the BERT base-uncased model for English, and FlauBERT base-cased for French. 7 We used two settings (see Appendix A for details):

• BERT tuned : the first setting is intended to use the contextual representations as only source of pre-trained parameters, and defines the v i vectors as in equation ( 1) (no lemma nor POS embeddings), with the word embeddings being randomly initialized, and the BERT embeddings being fine-tuned for the SDP task.

• BERT froz +POS+lem: the second one is used to compare our results to previous work on the English SemEval2015-Task18 datasets: the BERT embeddings are frozen, additional POS and lemma embeddings are used (cf. v i definition as in equation ( 2)). The same pre-trained word and lemma embeddings as FG20 are used. Note this setting uses gold POS and lemmas and is not a realistic scenario.

For the BERT tuned mode, we tuned the hyperparameters on the French data, and applied the same configuration to the English datasets. After a few tests, we set a configuration (see Appendix A), and searched for the best combination of auxiliary tasks.

For each experiment, we report the labeled Fscore (LF), including root arcs, averaged over 9 runs.

Results on French deep syntactic graphs

Table 1 shows the results with and without various combinations of auxiliary tasks. We tested the impact of stack propagation using auxiliary tasks B+H. We observe a modest but statistically significant +.34 increment with weights c (B) =1 and c (H) =10. Considering that this makes the inference task more complex, we did not use it in later experiments on English. We then tested the B+H configuration on the English test sets. In Table 2, we observe that performance gains using B and H auxiliary tasks are systematic across datasets (DM, PAS, PSD) and across in-or out-of-domain test sets, which tends to show the robustness of our method. 10We can also measure the impact of the auxiliary tasks by evaluating how accurate the predicted graphs are, concerning the number of heads of tokens: on average on the English dev sets, the proportion of tokens receiving the right number of heads in the predicted graphs increases from 94.9 without auxiliary tasks to 95.5 with tasks B+H.

Results on English semantic graphs

Finally, we provide in Table 3 a comparison to FG20 results (thus using the BERT froz +POS+lem setting), which are the state-of-the-art for systems using a single source of contextual embeddings. While our results remain below, note that our auxiliary tasks can be used with their system, as well as with e.g. that of [START_REF] Wang | Automated concatenation of embeddings for structured prediction[END_REF], which achieve significant improvements with an automated concatenation of various contextual embedding models, reaching 91.7 for ID et 90.2 for OOD.

ID OOD FG20 BERT froz +POS+lem 90.7 88.8 Ours BERT froz +POS+lem, B+H 90.2 87.9 Ours BERT tuned , B+H 89.9 88.0

Table 3: Comparison to the state-of-the-art SDP parser using BERT, on English ID and OOD test sets, in BERT froz +POS+lem setting. FG20: [START_REF] Fernández | Transition-based semantic dependency parsing with pointer networks[END_REF].

Conclusion

When using a biaffine graph-based architecture for semantic dependency parsing (SDP), arcs are predicted independently from each other. Our contribution is a set of simple yet original auxiliary tasks that introduce some form of interdependence of arc decisions. We showed that training recurrent word-token representations both for the SDP task and for predicting the number of heads and the incoming labels of each word is systematically beneficial, when tested either on English or French, on semantic or on deep syntactic graphs, and on inor out-of-domain data.

A Training details

All experiments are run on a Nvidia GTX 1080 Ti GPU.

For all settings:

• lexical embedding size (e (word) i

) : 100

• lexical dropout : 0.4. At learning, the lexical embedding of a token is replaced with probability 0.4 by a special token *DROP*, whose embedding is learned.

• biLSTM : 3 layers of size 2 * 600, with 0.33 dropout

• MLPs for aux. tasks: 1 hidden layer (300), output layer (300), dropout 0.25

• Batch size = 8

• Optimizer = Adam, β 1 = β 2 = 0.9

• Learning is stopped when all labeled Fscores decrease on the dev set. Note that beside the main FL score, tasks H and B give rise to their labeled Fscore, computed using the number of heads as predicted by task H (resp. B).

BERT tuned setting

• BERT and FlauBERT models : fine-tuned

• MLPs for arc and label score : 1 hidden layer (600), output layer (600), dropout 0.33

• Learning rate = 2 × 10 5

• Loss combination : learnt weights (cf. section 3.2)

BERT froz +POS+lem setting

• BERT model : frozen

• word and lemma embeddings (100), initialized with embeddings by [START_REF] Ma | Stackpointer networks for dependency parsing[END_REF], fine-tuned

• POS embeddings (100), randomly initialized

• MLP for arc score : 1 hidden layer (500), output layer (500), dropout 0.33

• MLP for label score : 1 hidden layer (100), output layer (100), dropout 0.33 11

• Learning rate = 5 × 10 4

• loss combination : plain sum of losses for each task

B Unsuccessful tests

Various tests were abandoned as unsuccessful in our preliminary tests:

• Using pre-trained lexical embeddings with tuned contextual embeddings had no impact on performance on average.

• Freezing BERT's and FlauBERT's parameters without using word and lemma embeddings significantly decreased performance (by about 2 FL points).

• Increasing the level of parameter sharing between tasks was not successful: instead of applying the MLPs of the auxiliary tasks on the recurrent representations r i , we tested applying them on the outputs of the specialization MLPs (i.e. on h (arc-head) i

for task D, on h (arc-dep) i for task H, on h (lab-dep) i for tasks B and S). While this tends to increase the number of epochs, it does not improve the performance.

C Significance testing

We use a Fisher-Pitman exact permutation test to estimate the significance of the differences in performance between two configurations (as done for example by [START_REF] Bernard | Multiple tasks integration: Tagging, syntactic and semantic parsing as a single task[END_REF]). More precisely, suppose we consider two samples of Fscores, for nA runs corresponding to configuration A, and nB runs for configuration B, with on average configuration B better than A. The null hypothesis is that the two samples follow the same distribution. The p-value corresponds to the probability that separating the set of Fscores into two samples A' and B' of size nA and nB gives a difference in mean at 11 Sizes of the MLPs for arc and label scores are defined here to replicate [START_REF] He | Establishing strong baselines for the new decade: Sequence tagging, syntactic and semantic parsing with BERT[END_REF] settings. We observed very marginal differences when keeping the sizes used in BERTtuned setting (600 for both MLPs). least as large as the observed difference. With the exact test, the p-value is calculated exactly, on all possible splits into A' and B' samples. [START_REF] Abeillé | Enriching a French treebank[END_REF]).

D French data statistics and full results

Train

The complete results on dev set for the French data is provided in

  sollicité. This him has accustomed to be very solicited.

Figure 1 :

 1 Figure1: Top: English Semantic graph in the DM format, as part of the SemEval2015-Task18 dataset[START_REF] Oepen | SemEval 2015 task 18: Broad-coverage semantic dependency parsing[END_REF]. Bottom: French Deep syntactic graph as defined by[START_REF] Candito | Deep syntax annotation of the sequoia French treebank[END_REF].

Figure 2 :

 2 Figure 2: Example of competition for the sequence rule of thumb. Above arcs: correct MWE analysis (rule and of attached to the last MWE component thumb, and thumb being the head of the sequence). Below arcs: incorrect compositional analysis, in which rule is the head, e.g. attached wrongly as ARG1 of good (in red).

Table 1 :

 1 8 While no auxiliary task provide a significative increase 9 , the combinations B+H and B+D+H+S bring a statistically significant +0.53 and +0.56 increase on average (see Appendix C for significance test). Results on French dev set, for various tasks combinations, with and without stack propagation (H: nb of heads, D: nb of dependents, B: bag of labels,

	Auxiliary	Stack	On 9 runs
	tasks	propagation	meanLF stdev
	∅	NA	86.79	0.19
	B+H	no	87.32 * * * 0.18
	D+H	no	86.61	0.71
	H+S	no	87.04	0.17
	B+D+H+S no	87.35 * * * 0.26
	B+H	c (B) =1 c (H) =1	87.49	0.06
	B+H	c (B) =1 c (H) =10 87.66 +++ 0.18

S: label multi-set). Col3-4: average LF, and standard deviation. * * * : significative diff. wrt first line. +++ : significative diff. wrt second line (p <0.001).

Table 2 :

 2 Average LF (on 9 runs), in BERT tuned setting, on English in-domain (ID) and out-of-domain (OOD) test sets, using either no auxiliary task (∅) or tasks B and H (B+H), without stack propagation. B+H results are statistically higher than ∅ for DM ID, DM OOD, PAS ID, PAS OOD (p <0.001) and PSD ID (p <0.01).

		Tasks DM PAS PSD Avg
	ID	∅ B+H 94.2 94.3 81.2 89.9 93.7 93.9 80.7 89.4
	OOD	∅ B+H 91.0 92.8 80.2 88.0 90.3 92.0 79.8 87.4

Table 4 :

 4 Statistics of the deep French syntactic graphs, built on the French treebank

Table 5 :

 5 Table 5, of which Table 1 is a truncated version. =1 c (H) =10 87.66 +++ 0.18 Full results on French dev set, for various tasks combinations, with and without stack propagation (H: nb of heads, D: nb of dependents, B: bag of labels, S: label multi-set). Col3-4: average LF, and standard deviation. * * * : significative difference wrt first line (p <0.001). +++ : significative difference wrt B+H without stack propagation (p <0.001).

	Auxiliary	Stack	On 9 runs
	tasks	propagation	meanLF stdev
	∅	NA	86.79	0.19
	H	no	86.82	0.54
	D	no	86.83	0.40
	S	no	86.98	0.30
	B	no	87.05	0.49
	B+H	no	87.32 * * * 0.18
	D+H	no	86.61	0.71
	H+S	no	87.04	0.17
	B+D+H+S no	87.35 * * * 0.26
	B+H	c (B) =1 c (H) =1	87.49	0.06
	B+H	c (B)

[START_REF] He | Establishing strong baselines for the new decade: Sequence tagging, syntactic and semantic parsing with BERT[END_REF] report very slight improvements in OOD when using the average of all subwords, but an opposite trend in ID.

nbhj is interpreted as 1+ the log of the number of heads of wj, and same for nbdj, so as to penalize less errors in bigger numbers: e.g penalize more predicting one head instead of 0 than predicting 2 heads instead of 3.

The categories are label multi-sets because we neutralize the order of the heads when considering the incoming arcs. This limits the number of categories.

In these deep graphs, the root tokens (usually unique) are attached to a dummy root token in practise. Thus for this dataset, in all the above formulations, the sequence w1:n corresponds to a sentence of n -1 word-tokens, with a dummy root w1. For its contextual representation, we use the contextual vector of the beginning of sequence token.

6 https://github.com/mcandito/aux-task s-biaffine-graph-parser-

FindingsACL227 We used the HuggingFace library[START_REF] Wolf | Transformers: State-of-the-art natural language processing[END_REF].

Previous state-of-the art on this data is a non-neural system:[START_REF] Ribeyre | Accurate deep syntactic parsing of graphs: The case of French[END_REF] obtained LF=80.86, and went up to LF=84.91 thanks to features from constituency parses from the rich FrMG parser[START_REF] Villemonte | Convertir des dérivations TAG en dépendances[END_REF]. The biaffine architecture with contextual vectors, without auxiliary tasks, obtains a mean LF=86.79.

 9 See Appendix D for results with each auxiliary task.

The improvement tends to be higher for OOD, and for DM and PAS. One reason could be that the PSD graphs show less reentrancies, hence the number of heads is more predictable, and using a specific auxiliary task for it is less beneficial.


F Lassiaille
email: frederic.lassiaille@yahoo.com

D Μ Pd

Surrounding Matter Theory: first mathematical developments

Keywords: Relativity, gravitation, millennium, Yang-Mills, confinement, mass gap PACS numbers: 12.38.Aw General properties of QCD (dynamics, confinement, etc.)

Surrounding Matter Theory, an alternative theory to dark matter, suggests mathematical developments. Two of them are presented. Then they are used in the study of a different interpretation of General Relativity (GR) principles using a four-momentum in place of the stress-energy tensor. It is showned that the surrounding effect prevailing in Surrounding Matter Theory appears also as the inner part of such a model. A surrounding effect in the context of particle physics is tried.

Introduction

In Surrounding Matter Theory ? an alternative theory to dark matter was presented. Its development was suggested by another model more ambitious but quite complicated ? and constructed with assumptions. Therefore, it might be interesting to try to rewrite the latter in a more straightforward and argumentative way. In order to achieve that, some mathematical developments are required. The present document will present two of them, which deal with null four-vectors. It will be also revisited the famous issue of the absence of algebraic structure for the set of boosts in three dimensional space. It will be reminded a prediction of GR for the space-time deformation generated by a particle in motion. From those developments, some constraints will be applied on physics models trying to replace Einstein equation with a discrete equation using four-momentums in place of the stress-energy tensor. After the improvement of the coherence of one of those models, an insight in gravitation and in particle physics will be tried.

An algebraic structure for the set of loaded boosts

Let's write S tpa, B µ ν qu and let's call it the "set of loaded boosts". a is a strictly positive real number, B µ ν is a boost. Of course the speed which is associated with a boost is strictly below the speed of light. Therefore S is the set of pa, B µ ν q couples, which can be understood in physics the following way. a is the energy at rest of a non relativistic object, therefore a is a strictly positive real number. B µ ν is the boost associated with its motion in a given frame. So S is this set of such couples. Then the algebraic structure of S is inherited from the s isomorphism from the set of strictly positive four-vectors, to S, such as pa, B µ ν q spD µ q in the following way.

(

In ( 1) it has been written D µ in such a base that its two last components are null and its first component is positive. a and v are deduced from D µ by (1). It has been used γ 1{ 1 ¡ v 2 {c 2 . In (1) it has been supposed that the first component is positive, hence that the flow of time is positive. If this component is negative then identical calculations are possible, with an opposite direction of time. Then the "1" in ( 1) is replaced by "¡1". The mathematical structures which will be studied in the present document are valid with either positive or negative flow of time. But in the present document a positive flow of time will allways be supposed. Then the B µ ν boost of ( 2) is deduced from v.

B µ ν γ ¤ ¦ ¦ ¥ 1 ¡ v c 0 0 ¡ v c 1 0 0 0 0 1 0 0 0 0 1 BpD µ q (2)
The definition of ¦, the induced operator acting on S is the following.

pa 1 , B µ ν 1 q ¦ pa 2 , B µ ν 2 q sps ¡1 ppa 1 , B µ ν 1 qq s ¡1 ppa 2 , B µ ν 2 qqq.

(

) 3
Obvious notations have been used for this equation. It must be noticed that the image of S by s ¡1 is the set of strictly positive four-vectors. Let's call SP this set. Also let's call Q the set of speeds which are strictly below the speed of light. Therefore Q is a subset of IR 3 .

The result is that pS, ¦q is isomorphic to pSP, q. A more human sensitive result is that pS, ¦q is isomorphic to pIR ¦ Q, opq. op is the barycentric operator of IR ¦ Q, using IR ¦ for the barycentric loads, and addition in Q. The physics point of view is that this is isomorphic to the set of pa t , vq couples where a t is the total energy of an object, and v, being strictly weaker than c is the speed of the inertial center of the object in a given inertial frame, using the barycentric operator.

By the way here an answer is given to the famous GR question of the absence of algebraic structure for the set of boosts with the composition operator. This answer starts by associating each boost with a strictly positive real number.

Another GR famous issue which can be addressed now is the issue of the order of the boosts in their three dimensional composition. ?, ?, ? Let's suppose the Universe filled with a constant and uniform distribution of matter. Let's choose and write R a frame in which the universe is at rest. Let' study two A and B objects. If A and B are far enough from each other, the space-time deformation due to A is not noticeable around B and vice versa. In this case an inertial frame R B attached to B will appear in an inertial frame R A attached to A following the rule of the composition of the boosts and it will appear a Wigner rotation. "A frame attached

R E P R I N T May 21, 2022 12:33

Surrounding Matter Theory: first mathematical developments 3 to a particle" means that the particle is at rest in the frame. But the space-time structure will be determined without ambiguity. The answer to the question of the order will be conspicuous: R A to R and then R to R B . Now if the Universe becomes empty except A and B then no boost composition will be required, only one boost will be relevant, and no Wigner rotation will appear. Also if E A and E B are the respective total energies, in R, of A and B, if E A ¡¡ E B and if A and B are closed enough to each other so that the space-time deformation of the Universe is not noticeable around A, (that is, the deformations of A and B are the only one noticeable around B), then R B will appear in R with a Wigner rotation and the choice of the order will be obvious too: R to R A and then R A to R B . The final result is that the answer to the questions of which composition of boosts and which chosen composition order between the frames depends of the relative energies associated with those frames.

Null four-vectors

The two following paragraphs will depict the two following features of null fourvectors. The "projective loaded boost" term will be explained just below.

1) The set of projective loaded boosts is a barycentric generator of the set of loaded boosts. 2) There is a morphism between the set of null four-vectors with addition and the corresponding set of projective boosts with composition.

A barycentric generator for the set of loaded boosts

Let's study the extension of the s function to the case of null four-vectors. For those null four-vectors, (2) is replaced by its limit from v to c. Similarily S is extended into a set which contains S, and also S#, a subset of its projective space. Let's call "projective loaded boost" an element of S#. Such an element is of the form p0, H µ ν q, where 0 means that the corresponding energy at rest is null, and H µ ν is an element of the projective space of the EndpIR 4 q space, which gets the following shape in some given inertial frame.

H µ ν ¤ ¦ ¦ ¥ 1 ¡1 0 0 ¡1 1 0 0 0 0 1 0 0 0 0 1 (4)
Let's call "projective boost" such a H µ ν element. Let's write B o the set of boosts and B o # its projective set. Therefore B o # is the set of such H µ ν elements. This extension of s is such that there is spp1, 1, 0, 0qq p0, H µ ν q. Along with this extension of s there is an extension of the B function of (2) which allows to write Bpp1, 1, 0, 0qq H µ ν .

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4 Lassiaille
Composing two projective boosts yields a projective element of EndpIR 4 q. The composition operator of EndpIR 4 q can be transported to its projective space naturally. And the usual property is valid in which the result of composition in the projective space is the projective element which is the projection of the composition of any of their representations. That is, the definition of the transported operation does not depends of the choice of the elements of EndpIR 4 q.

Two projective loaded boosts paired with the ¦ operator yield a loaded boost.

There is the same result for a projective loaded boost paired with a loaded boost. Therefore, s is extended from SP , to P , the set of positive four-vectors. And this extension of s is still a morphism from pP, q to pS U S#, ¦q, but no longer an isomorphism. Indeed, such an extended s is not injective with the null four-vectors.

From now on in the present document, the s and B functions will mean their extended versions. The result of the present paragraph is that S#, the set of projective loaded boosts, is a generator of S, the set of loaded boosts, using the ¦ operator. And it can be said equivalently, using the s ¡1 function, that the set of null four-vectors is a generator of P with addition. Of course it is not a base of P because there are multiple ways to generate an element of P using addition of null four-vectors.

There is one and only one such way for any space direction. More precisely any strictly positive four-vector is the sum of two null four-vectors. In physics, it means that any non relativistic object can be imagined as being only composed of two particles moving at the speed of light. And once the space direction of one of them is knowned, then the motion of the other one is deduced.

A morphism between the set of null four-vectors with addition and the corresponding set of projective boosts with composition

The important remark of the present paragraph is given by ( 5), using the notation of ( 4).

H ρ ν H µ ρ 2 H µ ν . (5)
It must be noticed that 2 H µ ν is representing the same element as H µ ν in the projective space of EndpIR 4 q. This equation gets an interesting consequence in physics.

Let's write Dn one given half line of null four-vectors. Therefore one null fourvector is given, which is not equal to 0, let's call it n, and Dn is the set of αn null four-vectors such as α is a positive real number. Then let's write Br the set of the projected boosts which are associated with Dn. Therefore Br BpDnq and Br is composed of the projected boosts along the direction indicated by n. This is a trivial case because Br is in fact a singleton of B o #. In other words there exists an element of B o #, let's write it H µ ν , such as Br tH µ ν u.

The important point is that B is a morphism between each D n subset of IR

Consequences of those null four-vector features 4.1. New physics models

These features allow to construct any kind of physics model which is composed of the following assumption and equation.

i

The assumption is that matter is made of particles moving at the speed of light. ii The equation starts the calculation of space-time structure at any spacetime event, by addition of the four-momentums propagated along the gravitational waves which are received at the event.

The need of the assumption (i) will be explained further. Let's show the shape of Eq. (ii).

D µ pxq Σ V n0 1 w px, y n qfpx, y n qC µ py n q (6)
In ( 6), 1 w px, y n q is equal to 1 if x and y n events are connected by a null geodesic and if x is located after y n along this geodesic. It means that the gravitational wave generated by the particle located in y n is received in x. f px, y n q is the scalar positive function equal to 1 if y n is equal to x, and which expresses the attenuation of the gravitational wave energy generated by the particle located in y n .

C µ py n q is a four-vector. It represents the four-momentum of the particle which is located in the y n event. Under the assumption (i), this particle is moving at the speed of light and C µ py n q is a null four-vector. If f px, y n q 1, then f px, y n qC µ py n q represents in x the four-momentum of the gravitational wave which is generated by the particle located in y n . Of course in this equation it is supposed that such a gravitational wave energy is described this way. The validity of this supposition will be discussed further. Once received in the x event, all those gravitational waves add their four-momentums. The final sum is D µ pxq. Then, from D µ pxq will be calculated the space-time structure. Noticeably, this can be done by using the boost of (2), and then from this boost the evolution of the space-time structure can be derived in x.

There are many ways of constructing such a physics model. The choice of the f function, and even the particular way of constructing space-time structure from D µ pxq are degrees of freedom for this construction. Let's depict one of those ways of deriving space-time structure from (6).

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An important prerequisite

Before that, an important prerequisite must be showned. It is that in GR there is a privileged frame, and the boost which is associated to the motion of matter in this frame, describes the evolution of this privileged frame.

This can be showned using a thought experiment. This thought experiment is simply imagining the energy at rest of a particle increasing progressively, and at the same time the whole energy of the universe decreasing. The asymptotic and then final result is that the universe and the particle have their roles permuted. Now the particle contains the energy of the previous universe, and the universe contains the energy of the previous particle. The result is that the frame in which time elapses the most is no longer the frame attached to the universe. Now this frame in which time elapses the most is the frame attached to the particle. It means that the space-time structure is now the symmetrical result of a permutation of those two frames. It means also that during the experiment, the space-time structure has been modified progressively from the first state to the final one which is symmetric with respect to the first one. And this operation has allowed to revert the time dilatation. For example, if this was a twin paradox configuration, at the end of his brother's journey, the older twin would be the youngest after the thought experiment. Therefore this space-time modification is simply described by the boost transporting one frame into another. It can be noticed that this reasoning is using the well established supposition that GR is coherent. Now the need of naming the frames appears. Let's call Ru a frame attached to the universe. It can be supposed that the universe is filled with a constant, homogenous distribution of matter, therefore this matter is supposed to be at rest in Ru. Let's call Rp a frame attached to the particle. The result of the thought experiment is that the particle generates locally a space-time deformation which is described by the boost from Ru to Rp. Of course, this deformation is local to the particle but the more energy at rest of the particle, the more this deformation is valid around the particle. A "more valid deformation" means that the space-time deformation exists significatively over a larger space-time domain. Now in the particular case of a particle moving at the speed of light, this boost becomes the projective boost described by (4).

Of course it might happen that between the two opposite and extreme cases of the thought experiment, the space-time deformation would be completely different from the one occurring during those two cases. But this would be a strange situation. Indeed there would exists two different change of coordinates, one for the progressive determination of the frame in which time elapses the most, along the process from one extreme case to the other, and another which is the usual boost of Special Relativity (SR).

It is already knowned in relativity that the frame in which time elapses the most is a privileged frame and is required to be as such. Therefore the space-time deformation appearing in the experiment is described by a boost which allows to

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Surrounding Matter Theory: first mathematical developments 7 transform progressively this privileged frame from the first case to the second one. And it means that this frame remains privileged during the whole process, even though it is no longer the frame in which time elapses the most between the two extreme situations. This frame is simply a frame in which the particle is at rest. Its physics relevance is only local to the particle.

This ends the explanation of the prerequisite. The final result is that it exists potentially a privileged frame in any space-time event. And for any particle located in x, this privileged frame exists in x and is transformed by the particle, using the boost which is associated with the four-momentum of the particle. Roughly speaking for the understanding, let's write that this boost is calculated in the "old privileged frame", that is, the one "just before the particle", and that it transforms this old privileged frame into the new one, that is, the one "just after the particle".

From four-momentum to space-time structure

Now it is possible to depict briefly one of those ways of deriving space-time structure from (6). The first step is to calculate a boost from each D µ pxq four-momentum. It has been showned that a coherent way to do that is to use the B function of (2). The result is a B µ ν pxq boost. It is not a projective boost, because D µ pxq is not null. A null D µ pxq would not determine correctly a space-time structure because s is not injective for null four-vectors and energy information would be lost. It means that ( 6) is only relevant for two or more terms in the sum.

The second step is to derive the metric from B µ ν pxq. Let's call R 0 the privileged frame of the prerequisite, in x, before the reception of the gravitational waves of (6). Let's call R I 0 this privileged frame after the action of the boost which results from ( 6) and (2). Therefore, R 0 is the "old privileged frame", and R I 0 is the "new privileged frame". Let's write x I the first event in which this transformation takes place, along the time of R 0 . Now the second step can be done which is to derive the metric from this B µ ν pxq boost. R I 0 is obtained by transforming R 0 , in x, using the B µ ν pxq boost. It is required to rescale the lengths of the "boosted" time and space axis. The boosted time and space axis are the time and space axis which have been modified by the boost, in their states after the boost. Let's call "one step" this process in which R 0 is replaced by R I 0 and then by this rescaled frame. The rescaling is done in such a way that the resulting time line described successively by those successive steps is a geodesic. Equivalently this constraint is that the privileged frame must be inertial. This is detailed by the following equations, relating X Iν the coordinates after the boost, to X µ the coordinates in R 0 , and then relating X" ρ the final rescaled coordinates in

R I 0 to X Iν . X Iν px I q B ν µ pxqX µ pxq (7)
X" ρ px I q S ρ ν px I qX Iν pxq

S µ ρ px I q is a symetric transform which has the ability of being diagonalized in R I 0 . Its value is determined by the constraint above (the time line of the set of successive privileged frames must be a geodesic). Eq. ( 7), ( 8) and ( 9) show how g µν px I q the new metric is deduced from g αβ pxq the old one, due to the action of B ν µ pxq, which results from the D µ pxq added energy. This is a possible construction of a space-time structure which is done in order to be as much coherent as possible with relativity. This new model is discrete, as compared with GR which is continuous. These equations have been obtained by discretizing simply the spherically symmetric static case. In the Schwarzschild metric of this case, a free falling particle, being at rest when located infinitely far from the center of the symmetry, follows a time line which is transformed by those equations. ? Finally, from this discrete metric the continuous metric must be interpolated. For example, this can be done by using the sinc function. This yields a continuous metric which is valid at the microscopic scale. From it, for gravitation, an average value might be calculated, resulting in a metric more relevant for the macroscopic scale.

But whatever is that exact construction, without any of the features of the previous paragraph, this physics construction would be either not possible or incoherent. Let's detail this.

First feature

Without the algebraic structure of the set of loaded boosts, along with the first feature (projective loaded boosts being generators of loaded boosts using the ¦ operator), it would not be possible to calculate in a coherent way the boost from (6) and then (2). Let's explain this.

In the particular case where the particles are close to each other, this calculation of the boost which is generated by matter can be incoherent in the case of an object which is composed of several smaller objects. Indeed its calculation can be done first by calculating the boost of the big object, and it can be done also by calculating the boosts for the smaller objects, and then deducing from that the boost of the big object. Of course those two calculations must yield identical results. Those possible incoherences were exposed in Ref. ?. The same solution as the one chosen in Ref.

? is constructed in the present document, in a more detailed manner. This manner is to use assumption (i), (ii), and the ¦ operator. This was about the composition of different objects in space. For comparison the composition of different objects along the time axis will be studied in the next paragraph.

Another requirement, appearing in any cases for the calculation of the boosts with ( 6) and ( 2), is that associativity of this calculation is mandatory (without speaking about commutativity, or any neutral element). And associativity is given by the algebraic structure of the set of loaded boosts.

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Surrounding Matter Theory: first mathematical developments 9

It must be noticed that the reasoning of the present paragraph supports already the validity of assumption (i). Moreover, the next point is that this assumption might be simply required in GR.

Second feature

Without the second feature (morphism between null four-vectors and projected boosts), the new models which execute (6) and then (2) would not respect energy conservation. This is shown in the case of three particles, named for example P 1 , P 2 , P 3 . They are supposed to be located close to some given x event.

Their speeds are supposed to be colinear and sharing the same direction. Let's write D µ 1 pxqq, D µ 2 pxqq, and D µ 3 pxqq their respective four-momentums. It is supposed D 3 pxq D 1 pxq D 2 pxq. Now the time order of appearance of those particles in x must not lead to different spacetime structures in x. Otherwise, the conservation of energy principle would not be satisfied because the generated geodesics would be different. For example a fourth particle coming in the vicinity of x would follow different trajectories and therefore different evolution of its four-momentum in R 0 and R I 0 , the frames which are used by ( 7), ( 8) and [START_REF] Vitiuk | Triple nuclear collisions -a new method to explore the matter properties under new extreme conditions[END_REF]. It means that P 1 and P 2 without P 3 , or P 3 alone, must yield identical metrics. The result, for the evolution of the privileged frame calculated by (2), is [START_REF] Jaffe | Quantum Yang-Mills Theory (Clay Math Institute)[END_REF].

BpD µ 1 pxqq ¥ BpD µ 2 pxqq BpD µ 1 pxq D µ 2 pxqq (10)
And it has been showned that the domain of this morphism is the set of null four-vectors. More precisely, it has been showned that this morphism is valid only when restricted to any D n set. But this is exactly what is required in order to solve the physics incoherences which have been explained just above. Now let's consider once again the case of objects which are so close to each other that the gravitational waves do not take place and are replaced by immediate interaction. It results that the assumption (i) is required for having a valid Eq. ( 6) in this case.

Moreover what has been showned is that, in the context of GR, only from the prerequisite above and the second feature, conservation of energy requires assumption (i).

Assumption (i) is required in GR. Of course it does not imply any direct consequence in fundamental physics since the size of the particles of this assumption is not knowned. But the consequence in GR is that a discrete model arises. Indeed any particle which is not moving at the speed of light now must be modelled by a group of particles moving at the speed of light. Either the number of particles in such a group is finite, or infinite. If it is infinite then it is a countably infinite.

To say the least, this gives a very good reason for trying to develop such new models which are discrete and in which space-time structure is calculated under this assumption (i). From there, the previous reasonings imply (ii) (that is, Eq. ( 6)).

But ( 6) is assumed to sum null four-vectors in any cases, otherwise it might generate once again violations of the conservation of energy principle. Therefore,

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May 21, 2022 12:33 10 Lassiaille similarily the gravitational wave four-momentums must be assumed to be null fourvectors. This is coherent with the formulation of the stress-energy tensor of a gravitational wave in some literature, for example in Ref. ?.

The surrounding effect as an inner part of those physics models

Let's rewrite (6), shifting the energy at rest from left to right.

U µ pxq 1 1 ¡ v 2 c 2 ¡ 1, v x c , v y c , v z c © D µ pxq E{c (11) E E 2 t ¡ E 2 c ( 12
)
E t c Σ V n0 1 w px, y n qfpx, y n qC 0 py n q (13) E 2 c c 2 Σ 3 i1 pΣ V n0 1 w px, y n qfpx, y n qC i py n qq 2 (14)
U µ pxq is the four-velocity associated with D µ pxq. v x , v y , v z , are its x, y, z space components, in some given inertial frame. Practically speaking this frame will be the privileged frame described above. v is the space norm of this speed. E, E t , and E c are respectively the energy at rest, total energy, energy of motion of D µ pxq. The denominator of (11) has been detailed in (12), ( 13) and (14) using once again Eq. ( 6).

It has been illustrated in Ref. ?, that such an equation predicts a particular effect. This effect was called "surrounding" in Ref. ?. Stated in one sentence, this effect is an increase of the equivalent G constant (that is, an increase of each gravitational forces), in a way which is inversely proportional with the energy of matter surrounding the location where the force is exerted.

Let's try an illustrative calculation from Eq. ( 6). For this, let's suppose for simplification that only the particle which is located in y 0 pertains to an object generating a gravitational force. Let's suppose that the other particles, therefore those located in y n , with n ¡ 0, represent the contribution of the universe, that they are distributed symmetrically in space with respect to x. It can be easily supposed that the sum of their energies will be strong in front of the energy of the particle located in y 0 . Those latter suppositions are often valid in reality, at least for gravitation. One consequence is that the sum in x of the contributions of the particles which are located in y n , for n 1 to n V, will be a four-momentum having approximately no space components. The result is given by (15) and (16).

U µ pxq 1 w px, y 0 qfpx, y 0 qC µ py 0 q E{c Σ V n1 1 w px, y n qfpx, y n qC µ py n q E{c (15)

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Surrounding Matter Theory: first mathematical developments 11 U µ pxq 1 w px, y 0 qfpx, y 0 qC µ py 0 q E{c p1, 0, 0, 0q

Now let's suppose that Σ V n1 1 w px, y n qfpx, y n qC 0 py n q, the total energy surrounding the x event, increases. It is supposed that the object generating the gravitational force does not experience any modification during that increasing process. It follows that E increases, since from (12), ( 13) and ( 14) there is E Σ V n1 1 w px, y n qfpx, y n qC 0 py n q. Increasing E in (16) without modifying C µ py 0 q results in decreasing v. Therefore, v is decreasing in D µ pxq, and then in B ν µ pxq after the execution of (2). It results a weakening of the transformation, from g αβ pxq into g µν px I q, which is done by executing ( 7), [START_REF] Carroll | Space time and geometry, An introduction to General Relativity[END_REF], and (9). Then, any continuous g µν metric interpolated in a mathematical regular manner from this discretization, will experience a decrease of its curvature.

This calculation was done only on illustrative purposes. It was based on some suppositions. For example, the number of particles involved in (6) might happen to be much lower than what was supposed implicitely. In that case, another kind of calculation must be done, taking into account the size of the cells of the grid of the gravitational waves intersections. This kind of calculation is out of the scope of the present document.

v c Σ 3 i1 pΣ V n0 1 w px, y n qfpx, y n qC i py n qq 2 Σ V n0 1 w px, y n qfpx, y n qC 0 py n q (17)
But (17) derives directly from (6), whithout any supposition. And it shows that the space velocity of the resulting four-velocity is inversely proportional to the total surrounding energy. The denominator of the right-hand side of this equation is a sum of positive scalars, whether the three spatial sums of the numerator involve positive or negative scalars.

It was not showned how this effect takes place precisely in any cases, but the result of the present paragraph is that the surrounding effect is an inner part of those new physics models.

A development in particle physics

A physics picture

The figure 1 tries to show the architecture of physics in a very simple manner. The physics aspect of SR and GR are supposed to be embodied into the component untitled "gravitation" in the figure. In that particular envision, gravitation is using the mathematical features of SR and GR to create the gravitational predictions of today's physics. When presenting this picture the aim is to show that GR and SR are each of them used by gravitation, but only SR is used by particle physics. GR is not used by particle physics.

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Surrounding Matter Theory: first mathematical developments 13 the size of any knowned fermion. Now using (i), (ii), (that is, a new model for the mathematical GR as previously explained), and this (iii) physics assumption, then it results the two following consequences.

1) Gravitation is replaced by a new model in which the surrounding effect plays a central role. 2) Particle physics is modified: the surrounding effect arises.

In gravitation, such a modification has been described in Ref. ?. It shows that the surrounding effect suggests a solution to the most important gravitational issues of today.

In particle physics, the surrounding effect would modify noticeably the physics predictions in the case of triple nuclear collisions. ? Indeed those collisions would be predicted to behave in a completely different manner. This would be hardly noticeable for the electromagnetic and weak forces because they involve only two incoming particles in close interactions. But this would imply an important modification of the strong force because this one involves also three body interactions. Any group of three particles closed to each other would experience low values of the strong force between them, because the surrounding effect would be strong, due to their close proximity to one another. But any group of two particles or any group of three particles having one of them far enough from the two others, would experience stronger values of the strong force, because the surrounding effect would be weak.

It results a very simple scheme for a possible solution of the Millennium problem. ?, ?

Discussion

Mathematically, an algebraic structure for the set of loaded boosts was described, and two features of null four-vectors were showned.

Concerning physics, what was described by an assumption in Ref. ? was showned to be more than an assumption. It is a prerequisite. It is that a moving particle in a given frame generates locally the space-time deformation described by the boost which is associated with its speed in this frame. This implies the existence of a privileged frame, which in some cases is the frame in which time elapses the most.

From those mathematical developments, and from that prerequisite in physics, it has been showned that the assumption that matter is made of particle moving at the speed of light is also more than an assumption. It is a requirement.

Of course this last sentence can be discussed, because the reasoning leading to it is not pure mathematics but is based on a thought experiment. But to say the least, the developments of the new physics models of the present document have been showned to be relevant, if not required. What follows is the development of those new physics models in which it is tried to calculate space-time structure in a discrete manner. It is tried to replace Einstein continuous equation with a discrete equation

g

  αβ pxq B ρ α pxqB κ β pxqS µ ρ px I qS ν κ px I qg µν px I q

  May 21, 2022 12:33Surrounding Matter Theory: first mathematical developments 5 with addition operator, and its associated Br set with the composition operator. Moreover, B is such a morphism only when restricted to such Dn sets. In other words, if it is already knowned that B is a function from IR 4 ¦ to B o U B o #, it is a morphism from pIR 4 ¦ , q to pB o U B o #, oq only when restricted to each of those D n subsets.

4

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But under some particular assumptions, GR might have a role to play in particle physics.

"Awakening" GR in particle physics

A simple way to have particle physics affected by GR is to assume the following assumption.

iii "There are no four forces but only one which is gravitation. The three other forces are only macroscopic effects of gravitation in the particular context in which they operate".

Under this assumption, it is more than a guess that any increase or decrease of the underlying gravitation would be immediatly and proportionally, or almost proportionally, transfered to each of the three other forces.

Let's notice that under this assumption (iii), and assuming also assumption (i) (but this one has been showned to be a requirement), the conjecture that the gravitational waves used in ( 6) are transporting four-momentums being null is supported. Indeed, under the assumption (iii) except for its localization a particle is equivalent to the gravitational waves it generates. Also, the term "macroscopic" which is used in the formulation of assumption (iii) is coherent with assumption (i). Indeed, the size of such fundamental particles moving at the speed of light must be weaker than Fig. 1. This rough and simple picture of today's physics is presented in order to remind the role of relativity in gravitation and in particle physics. Only some components are presented. The "GR" and "SR" rectangles are representing only the mathematical models related to "GR" and "SR", respectively. The "GR" rectangle on the right is grayed because GR is not used in today's particle physics.

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May 21, 2022 12:33 14 Lassiaille using a four-momentum in place of the stress-energy tensor. Some constraints have been applied on such models.

Then it appears that the surrounding effect of Ref. ? is an inner part of those models. Therefore a surrounding effect arises in gravitation. A new gravitational model similar to the surrounding model ? might be derived.

The assumption that the four forces derive from the gravitational force enforces the coherence of those models. And it allows them to apply in particle physics. Then the surrounding effect arises also in particle physics. A confinement of the strong force appears immediatly.

The final result is that a modification of GR is relevant, if not required, in which the surrounding effect plays a central role. The first result is that this surrounding effect is added to gravitation. The second result is given under the assumption that the four forces derive from the gravitational force. It is that the surrounding effect arises in particle physics as well. Those new models suggest solutions to the most important gravitational problems of today as well as a solution to the Millennium Yang-Mills problem. Four-vector: an element of IR 4 . Null four-vector: a four-vector having a Minskowskian value of 0. Positive four-vector: a four-vector having a positive or null Minskowskian value. Strictly positive four-vector: a four-vector having a strictly positive Minskowskian value. SP : the set of strictly positive four-vectors. P : the set of positive four-vectors. Q: the set of space vectors which space norm is strictly below the speed of light.

EndpIR 4 q: the algebra of endomorphisms of IR 4 . SR: Special Relativity. GR: General Relativity. Boost: the particular Minkowskian isometry described by SR and used for describing the space-time coordinates transformation between two inertial frames, one of them being in constant velocity with respect to the other.


Olivier Sename
email: olivier.sename@gipsa-lab.inpg.fr

Daniel Simon
email: daniel.simon@inrialpes.fr

Mongi Ben Gaid
email: mongi.ben-gaid@ifp.fr

A LPV approach to control and real-time scheduling codesign: application to a robot-arm control

Keywords: control/computing co-design, robust control, LPV systems, resource management

This paper deals with real-time control under computational constraints. A robust control approach to control/real-time scheduling co-design is proposed using the H∞ framework for Linear Parameter Varying (LPV) polytopic systems. The originality consists in a new resource sharing between control tasks according to the controlled plant performances. Here the varying parameters are images of the control performance w.r.t. the sampling frequencies. Then a LPV based feedback scheduler is designed to adapt the control tasks periods according to the plant behavior and to the availability of computing resources. The approach is illustrated with a robot-arm controller design, whose feasibility is assessed in simulation.

I. INTRODUCTION

Optimisation of computing resources in computercontrolled systems is a challenging problem. Current solutions consist in on-line changing the algorithm or adapting the sampling period in order to increase the flexibility by adaptation of the processor utilisation.

This recent research field has received few attention in the past. In [START_REF] Cervin | Feedbackfeedforward scheduling of control tasks[END_REF] some sampling period dependent PID controller is developed and a feedback scheduler based on a LQ optimisation of the control tasks periods is proposed to change on-line the sampling periods of the controller according to the resource availability. In [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF] a processor load regulation is proposed and applied for real-time control of a robot arm. In [START_REF] Tan | Output-feedback control of LPV sampled-data systems[END_REF], some results are given using the lifting technique for output-feedback synthesis for LPV sampled-data systems, where the sampling period may also be parameter varying. Finally methods to design sampling period dependent controllers have been proposed in [START_REF] Robert | Sampling period dependent RST controller used in control/scheduling co-design[END_REF] for RST ones and in [START_REF]Synthesis of a sampling period dependent controller using LPV approach[END_REF], [START_REF]A reduced polytopic LPV synthesis for a sampling varying controller : experimentation with a T inverted pendulum[END_REF] using the H ∞ control approach LPV polytopic systems.

However in all these studies the variation of computing resources is linked to the real-time system performance only and not to the plant expected performances. In this paper we will provide a methodology to design a feedback scheduling controller which will make the resource utilization vary on line according to the resource availability and to the plant trajectory. The design of the real-time control scheme is done in the context of robust control for LPV systems.

Thanks to recent works on robust linear control [START_REF] Apkarian | A convex characterization of gain scheduled H∞ controllers[END_REF], [START_REF] Gahinet | Affine parameter-dependent lyapunov functions and real parametric uncertainty[END_REF]see and references therein, Linear Matrix Inequalities and optimization tools (see e.g. [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], [START_REF] Lofberg | Yalmip : A toolbox for modeling and optimization in matlab[END_REF]) and references therein, robust controller synthesis theory is now well established and widely used in control applications. This approach makes possible to tackle the non-linearities of a system by considering them as parameter uncertainties and to build robust controllers w.r.t. these uncertainties. LPV theory allows to model the non-linearities or to make the controller performances varying through the linear introduction of parameters. Hence, since a decade, LPV modelling is being increasingly used and allows to extend classical linear robust control methodology to a larger class of systems, keeping the usage of linear tools.

Following the preliminary results in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF] a LPV/H ∞ feedback scheduler is designed in this paper, which is new in the context of real-time control. The objective is to adapt on line the control task periods (i.e. the varying parameters) not only according to the availability of computing resources but also to the plant performances.

In section II the need for more flexible real-time scheduling is stated and some features on feedback scheduling are given. Section III presents the main result of the paper, i.e an original strategy for control-scheduling co-design which consists in a LPV feedback scheduler where the varying parameter is function of the process to be controlled. This new methodology is illustrated on a robot-arm control problem in section IV.

II. PROBLEM STATEMENT AND STATE OF THE ART

A. The need for more flexible real-time scheduling

Embedded digital control systems use a computer to periodically sample sensors, compute a control law and send control signals to the actuators of a continuous time physical process.

Implementation related issues in digital control are the control latencies and intervals, and also timing uncertainties such as deviations of the sampling period and computing delays, jitter and occasional data loss [START_REF] Åström | Computer-Controlled Systems[END_REF], [START_REF] Törngren | Fundamentals of implementing real-time control applications in distributed computer systems[END_REF], [START_REF] Simon | Design and analysis of synchronization for real-time closed-loop control in robotics[END_REF]. However, a shown in [START_REF] Cervin | Integrated control and real-time scheduling[END_REF], a robust closed-loop system is, to some extent, able to tolerate such problems with no loss of stability or integrity.

On the other hand the scheduling policy would give better results when chosen based on application's based requirements instead of traditional policies (such as Rate Monotonic for fixed priorities or Earliest Deadline First for dynamic ones), which are not control aware but only computing aware.

Finally off-line schedulability analysis rely on a right estimation of the tasks worst case execution time. However in embedded systems the processors use caches and pipelines to improve the average computing speed while decreasing the timing predictability. Another source of uncertainty may come from some pieces of the control algorithm. For example, the duration of a vision process highly depends on incoming data from a dynamic scene. In a dynamic environment, some control activities can be suspended or resumed and control algorithms with different costs can be scheduled according to various control modes leading to large variations in the computing load.

Thus real-time control design based on worst case execution time, maximum expected delay and strict deadlines inevitably leads to a low average usage of the computing resource and to a poor adaptivity w.r.t. a complex execution environment. All these drawbacks call for a better integration of control goals and computing capabilities through a codesign approach.

B. About feedback scheduling

Some preliminary works have been done in the last decade in view of control/scheduling co-design, for instance in [START_REF] Ryu | Streamlining real-time controller design: from performance specifications to end-to-end timing constraints[END_REF] for off-line iterative optimisation of scheduling parameters.

Concerning co-design for on-line implementation, recent results deal with varying sampling rates in control loops in the framework of linear systems: for example [START_REF] Schinkel | Optimal control for systems with varying sampling rate[END_REF] show that, while switching between two stable controllers, too frequent control period switches may lead to unstability. Moreover most real-life systems are non-linear and the extrapolation of timing assignment through linearisation often gives rough estimations of allowable periods and latencies or even can be meaningless. In fact, the knowledge of the plant's behaviour w.r.t varying sampling is necessary to get an efficient control/scheduling co-design: the on-line combination the control performance and implementation constraints lead to a feedback scheduling approach.

This approach has been initiated from both the realtime computing side [START_REF] Lu | Feedback control realtime scheduling: Framework, modeling, and algorithms[END_REF] and the control side [START_REF] Cervin | Feedbackfeedforward scheduling of control tasks[END_REF]. The idea consists in adding to the process controller an outer sampled feedback loop ("scheduling regulator") to control the scheduling parameters as a function of a QoC (Quality of Control) measure. Indeed the objective is to increase the control performance (via efficient resource sharing) and robustness w.r.t timing uncertainties.

Figure 1 gives an overview of a feed-back scheduler architecture where an outer loop (the scheduling controller) adapts in real-time the scheduling parameters from measurements taken on the computer's activity, e.g. the computing load. Besides this controller working periodically (at a rate larger than the sampling periods of the plant control tasks), the system's structure may evolve along a discrete time scale upon occurrence of events, e.g. for new tasks admission or exception handling. These decisional processes may be handled by another real-time task, the scheduling manager, which is not further detailed in this paper. Notice that such a manager may give a reference to the controller resource utilisation.

The design problem can be stated as control performance optimisation under constraint of available computing resources. Early results come from [START_REF] Eker | A feedback scheduler for realtime controller tasks[END_REF] where a problem of optimal control under computation load constraints is theoretically solved by a feedback scheduler, but leads to a solution too complex to be implemented in real-time. Then [START_REF] Cervin | Integrated control and real-time scheduling[END_REF] shows that this optimal control problem can be often simply implemented by computing the new tasks periods by the rescaling:

h k+1 i = h k i U U sp
where U sp is the utilisation set-point and U the estimated CPU load. The feedback scheduler then controls the processor utilisation by assigning task periods that optimise the overall control performance. This approach is well suited for a "quasi-continuous" variation of the sampling periods of real-time tasks under control of a preemptive real-time operating system.

Another approach has been used in the framework of the so-called (m,k)-firm schedulability policy, where the scheduling strategy ensures the successful execution of at least m instances of a given task (or message sending) for each time window of length k slots. Hence a selective data drop policy (as in [START_REF] Jia | Graceful degradation of the quality of control through data drop policy[END_REF]) or a computing power allocation to selected tasks (as in [START_REF] Ben Gaid | Optimal scheduling of control tasks with state feedback resource allocation[END_REF]) can be used to perform optimal control of a plant under constraint of computing or communication limitations. This latter approach is well suited for nonpreemptive scheduling of control tasks and for networked control systems subject to messages loss : the tasks or messages are scheduled to jointly perform congestion avoidance and optimal control.

Finally the authors have used in [START_REF] Sename | Feedback scheduling for realtime control of systems with communication delays[END_REF] a LQG approach to design the feedback scheduling controller while in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF] an H ∞ approach is proposed for a multi-task control systems, assuming an a priori distribution of the computing resources between the control tasks.

However the distribution is fixed and the feedback scheduling strategy only ensures the use of computing resources as a function of the resource constraints (availability). No plant information is used to make the resource distribution optimal w.r.t the closed-loop performances of the plant to be controlled by the real-time computer.

In the following section the approach in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF] is extended, allowing to make the distribution of the computing resources to vary according to the plant trajectory, which is new in the context of real-time feedback scheduling.

The next section concerns the main paper result, i.e. the design of a new methodology for real-time control. It consists in designing a LPV feedback scheduler where the varying parameter is function of the process to be controlled.

III. A LPV FEEDBACK SCHEDULER IN VIEW OF PLANT CLOSED-LOOP PERFORMANCES

Feedback scheduling is a dynamic approach allowing a better usage of the computing resources, in particular when the workload changes (e.g. due to the activation of an admitted new task). Here the CPU activity will be controlled according to the resource availability by adjusting scheduling parameters (i.e. period) of the plant control tasks. However the use of computing resources should also be linked to the dynamical behavior of the plant(s) to be controlled. Indeed while controlling different subsystems in a single computer it is natural to ensure the resource availability when large transient behaviors occur. The main result given in this section consists in deriving a new feedback scheduling controller which will depend on the plant trajectory in view of an "optimal" resource sharing. It is designed in the LPV/H ∞ framework for polytopic systems.

Following previous authors' results in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF], the feedback scheduling is illustrated in Fig 2 as a dynamical system between control task frequencies and processor utilisation. As far as the adaptation of the control tasks is concerned, the load of the other tasks is seen as an output disturbance. We assume that the CPU utilization is measured or estimated. Let us first recall that the scheduling is here limited to periodic tasks. In this case the processor load induced by a task is defined by U = c h where c and h are the execution time and period of the task. Hence processor load induced by a task is estimated, in a similar to way [START_REF] Cervin | Feedbackfeedforward scheduling of control tasks[END_REF], for each period h s of the scheduling controller, as:

Ûkhs = λ Û(k-1)hs + (1 -λ) c khs h (k-1)hs ( 1
)
where h is the sampling frequency currently assigned to the plant control task (i.e. at each sampling instant kh s ) and c is the mean of its measured job execution-time. λ is a forgetting factor used to smooth the measure (here λ = 0.3). Now for a n-multi-tasks control system, one should note that, as shown in [START_REF] Simon | Real-time and delay-dependent control co-design through feedback scheduling[END_REF], if the execution times are constant, then the relation, U = n i=1 C i f i (where f i = 1/h i is the frequency of the task) is a linear function (while it would not be the case if expressed as a function of the task periods). Therefore, using (1), the estimated CPU load is given as:

Û (kh S ) = (1 -λ) z -λ n i=1 c i (kh S )f i (kh S ) (2)
However in practice, the execution-time of the control tasks may vary according to the run-time environment (e.g. processor speed). As proposed in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF], a "normalized" linear model of the task i (i.e independent on the execution time), G ′ i , is used for the scheduling controller synthesis where c is omitted and will be compensated by on-line gain-scheduling (1/c) as shown below.

G ′ i (z) = Û (z) f i (z) = 1 -λ z -λ , i = 1, . . . , n (3)
Also, as explained above the use of computing resources is chosen to depend on the plant trajectory. Hence the control scheme of computing resource control is illustrated in figure 3 for a 2-tasks control systems for simplicity. In figure 3 the interval of frequencies is limited by the "saturation" block, α represents a set of real parameters {α 1 , α 2 , . . . , α n } dedicated to the set of control tasks {U 1 , U 2 , . . . , U n }. These parameters will be used to make the resource sharing vary according to the plant trajectory. For instance, in a 2 control-tasks system, where U = U 1 +U 2 , we will require that :

K(z, α)

1 c2 Task 2 6 - + Task 1 - ? -H(z) U others 6 - U r + + + f 1 f 2 1 c1 ? α
U 1 = αU (4) U 2 = (1 -α)U (5)
α being a varying parameter. This makes the control scheme flexible enough to distribute on-line the use of computing resources to the different control tasks. The choice of the value of the time-varying parameters {α 1 , α 2 , . . . , α n } can be realized by many different ways, from on-line computation of optimal cost functions, to a dependency on the control effort. It will be illustrated in details in section IV for the robot-arm control example.

Here the design of the controller K(α) is done using the H ∞ control approach for LPV systems. The H ∞ control scheme to synthesize the controller K(α) is given in figure 4.

K(α) - 6 - U r + - W e -e 1 j j Û + G ′ C ′ - H ? U others + M (α) W x -e 2
Ûtot G Fig. 4. A LPV Hinf controller for CPU resources

In figure 4, G ′ is the model of the scheduler, the output of which is the vector of all task loads. To get the sum of all task loads as in (3), we use

C ′ = [1 . . . 1]
. The H transfer function represents the sensor dynamical behaviour which measures the load of the other tasks. It may be a first order filter. The template W e specifies the performances on the load tracking error. It is chosen in the continuous-time domain as :

W e (s) = s/M s + ω b s + ω s ǫ (6)
with M s = 2, ω s = 10rad/s, ǫ = 0.01 to obtain a closedloop settling time of 300 ms, a static error less than 1 % and a good robustness margin.

The resource distribution is realized through the M (α) matrix defined below. Note that for a n-multi-tasks system:

U = U 1 + U 2 + . . . + U n (7) U = (α 1 + α 2 + . . . + α n )U (8)
where α 1 + α 2 + . . . + α n = 1. Then:

U 1 = α 1 U (9)
U 2 = α 2 U = α 2 α 1 U 1 ( 10
)
U 3 = α 3 U = α 3 α 2 U 2 (11)
. . . . . . ( 12)

U n = α n U = α n α n-1 U n-1 (13)
Then to ensure the on-line distribution of the computing resources M is chosen as follows:

M =      -α 2 α 1 0 . . . . . . 0 0 α 3 α 2 0 . . . 0 . . . • • • • • • • • • . . . 0 . . . • • • • • • • • • -α n α n-1      (14) = α 1 M 1 + α 2 M 2 + . . . + α n M n (15)
Using [START_REF] Apkarian | A convex characterization of gain scheduled H∞ controllers[END_REF] the LPV controller K(α) is obtained through the solution of the H ∞ control problem for polytopic systems, and consists in solving 2 LMIs. Then the design of K(α) can be done directly in the discrete-time domain or in the continuous-time one and then discretized. In this paper K(α) has been synthesized in the continuous-time domain using the H ∞ control approach for polytopic systems, as described in the Appendix.

By solving the H ∞ problem for the LPV system using the Yalmip interface and Sedumi solver [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], [START_REF] Lofberg | Yalmip : A toolbox for modeling and optimization in matlab[END_REF], one obtains γ opt = 1.8885, and a controller of order 7.

IV. APPLICATION TO A ROBOT-ARM CONTROL

We consider here a seven degrees of freedom Mitsubishi PA10 robot arm that has been previously modelled and calibrated [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF].

The problem under consideration is to track a desired trajectory for the position of the end-effector. Using the Lagrange formalism the following model can be obtained:

Γ = M (q)q + Gra(q) + C(q, q) ( 16
)
where q stands for the positions of the joints, M is the inertia matrix, Gra is the gravity forces vector and C gathers Coriolis, centrifugal and friction forces.

The structure of the (ideal) linearising controller includes a compensation of the Gravity, Coriolis/centrifugal effect and Inertia variations as well as a Proportional-Derivative (PD) controller for the tracking and stabilisation problem, of the form: [START_REF] Lu | Feedback control realtime scheduling: Framework, modeling, and algorithms[END_REF] leading to the linear closed-loop system M (q)q = K p (q dq) + K d ( qd -q), where q d and qd stand for the reference trajectory positions and velocities.

Γ = Gra(q) + C(q, q) + K p (q d -q) + K d ( qd -q),
The controller is split in five tasks, i.e. a specific task is considered for the PD control, the trajectory generation and for the Gravity, Inertia and Coriolis compensations, in order to use a multi-rate controller. In this feedback scheduling scheme, only the periods of the compensation tasks will be adapted, as they are time consuming compared with the PD task while being less critical for the stability.

A. Performance evaluation of the control tasks in view of optimal resource distribution

In order to associate the use of computing resources with the robot trajectory, the contribution of each of the 3 control tasks to the closed-loop system performances has been evaluated as a function of its execution period.

The methodology is the following. Assuming a nominal sampling period for each task of 1ms, the period of each compensation control task is changed, and new simulations are performed during which the following cost is computed:

J = t f ti (P hi,hg,hc (t) -P ref (t)) 2 - t f ti (P c (t) -P ref (t)) 2 (18)
where P ref is the desired position in the operational space of the end tip, computed from q d using the geometric model, P c is the position obtained when all the control tasks act with the minimal sampling period of 1 ms. Finally P hi,hg,hc is the position obtained when the sampling period of one of the compensation tasks is increased from 1 to 30 ms.

Simulations are performed for a particular robot trajectory, defined by the reference vector (q d , qd ) for all the robot joints. Here q d goes from π/2 to -π/2. We get the following results presented in figures 5, 6 and 7, where the evolution of the cost function J is shown for the three compensation control task. While it is difficult to infer the relations between the compensation tasks execution period and the trajectory tracking performance, a natural interpretation is as follows. First the Gravity compensation effect is very sensitive to the increase of the sampling period at the end of the trajectory, as the cost increases in the second part of the trajectory (first part of the graph as the trajectory goes from π/2 to -π/2). We will require to ensure the availability of CPU resources to this task, in a linear way with the trajectory position. Then the situation is almost reverse for the Inertia effect. Finally, even if some variations can be observed, we will ask for a constant use of CPU resources of the Coriolis compensation task, all along the trajectory.

We have then chosen that the distribution of control task periods should be:

U I = α I U, U G = α G U, U C = α C U (19)
where α C = 0.25, α I = 1 -α G , and α G is linked to the plan trajectory by: where [α min ; α M ax ] = [0.1; 0.65], q ini is the initial position and q end the final trajectory position.

α G = α min + (α M ax -α min ) × q d -q end q ini -q end (20)

B. Simulation with TrueTime

TrueTime (www.control.lth.se/truetime/) [14] is a MATLAB/Simulink-based simulator for real-time control systems that eases simulation of the temporal behaviour of a multi-tasking real-time system executing controller tasks. The tasks are controlling processes that are modelled as ordinary continuous-time Simulink blocks.

In this section, simulations have been performed using TrueTime. In this application, the period of the feedback scheduler has been fixed to 30ms to be larger than the robot control tasks periods, which limits have been set from 1ms to 30ms.

In the experiment depicted in figure 10 the desired CPU usage is initially set to 50% of the maximum usage. The upper plots show the tasks periods and CPU usage. The PD loop period is fixed at 1ms and the trajectory generator at 5ms.

As seen in figure 10, the load of the compensation tasks (Gravity, Coriolis and Inertia) vary on line as expected according to the parameter α I (see figure 12). The corresponding evolution of the task periods is shown in figure 11. Moreover, in figure 13, the adaptive LPV case (α varying) is compared with the constant case (α = 0.375). It can be seen that the LPV case leads to a smaller cost function which emphasizes the real interest of the provided approach.

V. CONCLUSIONS

In this paper a new approach for integrated control/realtime scheduling has been proposed in the framework of the robust theory for LPV systems. It consists in a LPV/H ∞ feedback control of computing resources, where the varying parameters can be linked to the controlled plant performances. The given control structure is open enough to allow for many plant performance criteria to be used for the on-line variation of the parameters. Here it is illustrated through the real-time control of a robot-arm, and the varying parameters are in this case linked to the robot reference trajectory, leading to a better use of computing resources Fig. 9. Control torques and an increase of closed-loop performances. The provided simulation results using TrueTime emphasize the interest and efficiency of the proposed methodology, which highlights the benefits of the robust control approach.

Note that, as explained in [START_REF] Simon | Robust control / scheduling codesign: application to robot control[END_REF] and depicted in Figure 14, the scheduling feedback loop can be easily implemented on top of an off-the-shelf real-time operating system (e.g. Posix) under the form of an additional real-time periodic task, i.e. a control module which function is specified and encoded by the control designer. The inputs are the measured execution times of the control tasks. The set point is a desired global computing load. Outputs are the sampling intervals of the Gravity, Coriolis and Inertia control tasks which are triggered by programmable timers provided by the operating system.

Thanks to the use of a hierarchical control structure, the given results may also be integrated with existing methods for the design of varying sampled controllers, as in [START_REF] Tan | Output-feedback control of LPV sampled-data systems[END_REF], [START_REF]A reduced polytopic LPV synthesis for a sampling varying controller : experimentation with a T inverted pendulum[END_REF]. This makes this integrated approach easier and generic.

Fig. 1 .

 1 Fig. 1. Hierarchical control structure

Fig. 2 .

 2 Fig. 2. Feedback scheduling bloc diagram

Fig. 3 .

 3 Fig. 3. Control scheme for CPU resources

Fig. 5 .Fig. 6 .

 56 Fig. 5. Cost variation due to varying sampling for gravity compensation task

Fig. 7 .

 7 Fig. 7. Cost variation due to varying sampling for Inertia compensation task

  Fig. 8. Positions

Fig. 10 .Fig. 11 .

 1011 Fig. 10. TrueTime real-time parameters: loads, periods

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Variation of α I according to the desired position

This work was supported by the French National Research Agency (ANR) project Safe-NeCS under grant No. ANR-ARA SSIA-NV-15

Scheduling Controller