MD, PhD Ana Marques
email: ar_marques@chu-clermontferrand.fr

MD,PhD Christine Brefel-Courbon

CHRONIC PAIN IN PARKINSON'S DISEASE: CLINICAL AND PATHOPHYSIOLOGICAL ASPECTS

Keywords: Pain, Parkinson's disease, central, nociceptive parkinson's disease; PCP: Parkinsonian central pain Pain, Parkinson's disease, central, nociceptive

Pain is an increasingly recognized non-motor symptom of Parkinson's disease (PD), with significant prevalence and strong impact on quality of life of patients. Moreover, pain can occur with various features in PD and several subtypes may coexist in a same patient, leading to a complex presentation and difficult diagnosis and treatment. In this paper we review the clinical manifestations of painful phenomena in PD, with focus on classifications and algorithms allowing to standardize the diagnosis of pain and PD. We also discuss the pathophysiological mechanisms underlying pain in PD, particularly parkinsonian central pain, in regard to recent clinical, neurophysiological and imaging studies.

Introduction

Non-motor symptoms of Parkinson's disease (PD) are increasingly recognized, and among them pain is a frequent and debilitating manifestation of the disease. Painful experiences can occur with various features in PD and hereby are difficult to describe by patients and by clinicians. Moreover, several types of pain may coexist in one patient, leading to insufficient identification and treatment of pain. Efforts have been made to develop classifications and algorithms in order to standardize the diagnosis of pain in PD, and numerous studies have addressed the question of the pathophysiological mechanisms underlying pain in PD.

1.Clinical description of pain in PD 1.1. Prevalence of pain and pharmacoepidemiolgy

Painful phenomena are reported in 30 to 85% (mean 66%) of PD patients. [START_REF] Silverdale | A detailed clinical study of pain in 1957 participants with early/moderate Parkinson's disease[END_REF][START_REF] Beiske | Pain in Parkinson's disease: Prevalence and characteristics[END_REF][START_REF] Valkovic | Pain in Parkinson's Disease: A Cross-Sectional Study of Its Prevalence, Types, and Relationship to Depression and Quality of Life[END_REF][START_REF] Rana | Pain in Parkinson's disease: analysis and literature review[END_REF][START_REF] Defazio | Pain as a nonmotor symptom of Parkinson disease: evidence from a case-control study[END_REF] The discrepancies of prevalence among different studies reflect the absence of clear diagnostic criteria and a poor knowledge of this non-motor symptom by the medical community.

Moreover, this variability in the estimation of prevalence is also related to differences in sample sizes, different scales used for the diagnosis of pain depending on the studies, and different types of pain coexisting [START_REF] Defazio | Pain as a nonmotor symptom of Parkinson disease: evidence from a case-control study[END_REF][START_REF] Lee | A survey of pain in idiopathic Parkinson's disease[END_REF][START_REF] Nègre-Pagès | Chronic pain in Parkinson's disease: the cross-sectional French DoPaMiP survey[END_REF]. One study aimed to estimate the frequency of pain in PD patients using analgesic prescription histories and to compare it with the general population based on two samples of painful patients (diabetes and osteoarthritis) using a pharmacoepidemiological design [START_REF] Brefel-Courbon | Comparison of chronic analgesic drugs prevalence in Parkinson's disease, other chronic diseases and the general population[END_REF]. Pain estimated by analgesic drug prescription was more prevalent in PD patients than in the general population with an estimated 82% prevalence of pain in PD (versus 76.6% in the general population), and an estimated 33% prevalence of chronic pain in PD, (versus 20% in the general population). Prescriptions of antidepressants and antiepileptic drugs for their analgesic properties were substantially more frequent in PD patients in comparison with the three control populations, suggesting that the prescribing doctors may have suspected that their PD patients had neuropathic pain because these drugs are the most frequently used drugs for neuropathic pain [START_REF] Brefel-Courbon | Comparison of chronic analgesic drugs prevalence in Parkinson's disease, other chronic diseases and the general population[END_REF]. Thus pain observed in PD appears to be different than pain observed in osteoarthritis, and appears to be related to several subtypes of pain.

Pain impact on quality of life

Pain is one of the most invalidating non-motor symptoms in Parkinson's disease (PD) with a strong impact on patients' quality of life, sometimes even taking precedence over motor symptoms [START_REF] Silverdale | A detailed clinical study of pain in 1957 participants with early/moderate Parkinson's disease[END_REF][START_REF] Nègre-Pagès | Chronic pain in Parkinson's disease: the cross-sectional French DoPaMiP survey[END_REF]. In the general population, chronic pain has shown a clear correlation with severity of depression and reduced quality of life [START_REF] Fishbain | Chronic pain-associated depression: antecedent or consequence of chronic pain? A review[END_REF][START_REF] Bernfort | Severity of chronic pain in an elderly population in Sweden--impact on costs and quality of life[END_REF]. In PD as well, pain has been demonstrated to be a major detrimental factor affecting physical and mental health-related quality of life [START_REF] Quittenbaum | Quality of life and pain in Parkinson's disease: a controlled cross-sectional study[END_REF][START_REF] Roh | The relationship of pain and health-related quality of life in Korean patients with Parkinson's disease[END_REF] and also leading to reduced autonomy [START_REF] Santos-García | Pain in Parkinson's disease: prevalence, characteristics, associated factors, and relation with other non motor symptoms, quality of life, autonomy, and caregiver burden[END_REF]. A significant relation between pain and depression has been reported in PD, suggesting that pain issues should be taken into consideration when treating PD patients with depression, and reciprocally [START_REF] Ehrt | Pain and its relationship to depression in Parkinson disease[END_REF].

Yet, pain is an independent predictor of poorer quality of life, independently of its close relationship with depression. Pain has an effect on quality of life, which is greater than motor impairment and comparable to the effect of motor complications [START_REF] Kpps | Distribution and impact on quality of life of the pain modalities assessed by the King's Parkinson's disease pain scale[END_REF].

Risk factors for pain in PD

Female gender, dyskinesia, postural abnormalities, motor complications and depression are the main predictors for development of pain in PD [START_REF] Lee | A survey of pain in idiopathic Parkinson's disease[END_REF][START_REF] Nègre-Pagès | Chronic pain in Parkinson's disease: the cross-sectional French DoPaMiP survey[END_REF][START_REF] Tinazzi | Pain and motor complications in Parkinson's disease[END_REF]. Autonomic and affective symptoms, have also been associated with an increased risk of pain, while the severity of motor impairment and of cognitive impairment have not been found to have a statistically significant effect on the severity of pain [START_REF] Silverdale | A detailed clinical study of pain in 1957 participants with early/moderate Parkinson's disease[END_REF]. Younger age has been reported as associated with pain in some studies (1), while disease progression was reported as a risk factor for pain in PD in others [START_REF] Mylius | Pain sensitivity and clinical progression in Parkinson's disease[END_REF]. Finally, no relationship between pain and age at diagnosis, disease duration, motor examination, or PD stage was observed in several studies [START_REF] Beiske | Pain in Parkinson's disease: Prevalence and characteristics[END_REF][START_REF] Tinazzi | Pain and motor complications in Parkinson's disease[END_REF][START_REF] Mylius | Pain sensitivity and clinical progression in Parkinson's disease[END_REF][START_REF] Rana | A cross-sectional study investigating clinical predictors and physical experiences of pain in Parkinson's disease[END_REF]. Thus conflicting results are reported regarding the role of PD progression in the emergence of pain, and this could be related to different pain subtypes occurring at different moments in the course of PD.

Thus some studies have reported all pain types to be more prevalent in patients with advanced-stage PD than in early stages, whereas only nociceptive arthritic pain was more prevalent in early-stage PD [START_REF] Valkovic | Pain in Parkinson's Disease: A Cross-Sectional Study of Its Prevalence, Types, and Relationship to Depression and Quality of Life[END_REF]. Moreover, some studies suggest that pain could be present very early in the course of PD and could even precede motor symptoms [START_REF] Pont-Sunyer | The onset of nonmotor symptoms in Parkinson's disease (the ONSET PD study)[END_REF][START_REF] Gonera | Symptoms and duration of the prodromal phase in Parkinson's disease[END_REF]. Indeed, 16.4% of PD patients have been reported to have presented chronic pain before the diagnosis of PD [START_REF] Ozturk | Chronic pain in Parkinson's disease: Frequency, characteristics, independent factors, and relationship with health-related quality of life[END_REF]. It is then possible that pain observed at the premotor phase and pain observed late in the disease could have different characteristics leading to discrepant findings among studies. The same heterogeneity could explain the fact that pain in PD is reported to be worse in the OFF state by 17% of participants, whereas 2% describe it as worse in the ON state and 81% report there is no difference between the ON and OFF state with regards to pain severity (1).

Different pain subtypes

Pain in PD is difficult to describe for patients but also for neurologists, and various features are reported such as muscle cramps, painful dystonia, aching, numbness, and tingling, burning, vibrating, or lancinating pain, reflecting the heterogeneity and complexity of these painful experiences. Ford (1998) defined five subtypes in PD: musculoskeletal, radicular/peripheral neuropathy, dystonic, central pain and akathisia. Yet, different pain subtypes can coexist in a same patient [START_REF] Wasner | Pains in Parkinson disease--many syndromes under one umbrella[END_REF]. It has thus been found that 29% of PD patients with pain suffered from one type, 35% from two types, 10% from three types of pain, and 2% reported having four concurrent types of pain (3). In the same vein, it has been found that 74-82% of patients had three or less types of pain [START_REF] Beiske | Pain in Parkinson's disease: Prevalence and characteristics[END_REF][START_REF] Lee | A survey of pain in idiopathic Parkinson's disease[END_REF][START_REF] Kpps | Distribution and impact on quality of life of the pain modalities assessed by the King's Parkinson's disease pain scale[END_REF]. This is a very important finding, because each type of pain requires a different therapeutic approach.

The distinction between these different pain subtypes in Parkinson's patients is still poorly carried out in clinical practice, and their respective prevalence is therefore poorly known and varies from study to study (Table 1). Differences may have several explanations, such as the specific focus of the study, different severity stages in the studied PD populations, or methods of classification used. For example, Valkovic et [START_REF] Beiske | Pain in Parkinson's disease: Prevalence and characteristics[END_REF][START_REF] Defazio | Pain as a nonmotor symptom of Parkinson disease: evidence from a case-control study[END_REF][START_REF] Hanagasi | Pain is common in Parkinson's disease[END_REF]. Moreover, pain severity was highest in central parkinsonian pain followed by dystonic, musculoskeletal, and radicular/peripheral neuropathic pain [START_REF] Ozturk | Chronic pain in Parkinson's disease: Frequency, characteristics, independent factors, and relationship with health-related quality of life[END_REF]. Central parkinsonian pain is probably underdiagnosed due to its complexity and the lack of clear diagnostic criteria, leading to insufficient recognition and treatment [START_REF] Beiske | Pain in Parkinson's disease: Prevalence and characteristics[END_REF][START_REF] Ozturk | Chronic pain in Parkinson's disease: Frequency, characteristics, independent factors, and relationship with health-related quality of life[END_REF][START_REF] Marques | How to diagnose parkinsonian central pain?[END_REF]. Indeed, a successful approach to distinguish between these pain subtypes is required so that effective treatment strategies can be established.

Classifications for pain

The first recognized classification of pain in PD defined five subtypes: musculoskeletal, radicular/peripheral neuropathy, dystonic, central pain and akathisia (Table 2) [START_REF] Ford | Pain in Parkinson's disease[END_REF]. The main limitation of this classification is that it relies on numerous clinical features that may be present, meaning that they also may not be present. Before this classification, pain in PD was generally classified depending on its relationship to PD: pain "related to PD" or "not related to PD" (6,7). Pain was considered "related to PD" if there was a temporal and/or topographical relationship with PD (onset and location), if pain was influenced by motor complications, or by antiparkinsonian medication, and depending on the patient's opinion about the relationship between pain and PD. Increasing knowledge regarding the mechanisms of pain led to the emergence of more precise classifications that do not contradict previous approaches, but combine several aspects of previous classifications [START_REF] Wasner | Pains in Parkinson disease--many syndromes under one umbrella[END_REF]. Wasner [START_REF] Mylius | Pain sensitivity and clinical progression in Parkinson's disease[END_REF]. King's PD Pain Scale (KPPS) includes 14 items rating the severity and frequency of pain, covering seven main domains [START_REF] Chaudhuri | King's Parkinson's disease pain scale, the first scale for pain in PD: An international validation[END_REF]. However different domains are not specific, may be common to different subtypes of pain, thus KPPS is suggested but not recommended for pain classification in PD [START_REF] Perez-Lloret | Rating scales for pain in Parkinson's Disease: critique and recommandations[END_REF].

Among these different proposed classifications, the definition of parkinsonian central pain (PCP) varies and fails to be precise. Given the lack of reliable validated positive clinical criteria for PCP, and since clinical features of PCP are difficult to specify, it has been recently proposed to standardize PCP diagnosis. Indeed, from the new classification of chronic pain in general population [START_REF] Kosek | Do we need a third mechanistic descriptor for chronic pain states?[END_REF], it has been proposed to classify pain in PD into nociceptive, neuropathic and nociplastic pain, either specific or unspecific to PD (Table 3) [START_REF] Marques | How to diagnose parkinsonian central pain?[END_REF]. From this new proposed classification, an algorithm was suggested, aiming to disentangle PCP from other chronic pain subtypes in PD by specifically and sequentially ruling out what is not PCP [START_REF] Marques | How to diagnose parkinsonian central pain?[END_REF].

A better classification of pain in PD is crucial in order to homogenize the populations in studies aiming to improve our comprehension of pathophysiological mechanisms of pain in PD.

Pathophysiological aspects

Based on the physical pathological approach and according to the International Association for the Study of Pain (IASP), we can define different kinds of pain in PD according to their potential mechanisms [START_REF] Marques | How to diagnose parkinsonian central pain?[END_REF]. Nociceptive pain and neuropathic pain result from a lesion of the somatosensory system and nociplastic pain results from a dysfunction of the somatosensory central system without any evidence of lesion [START_REF] Kosek | Do we need a third mechanistic descriptor for chronic pain states?[END_REF]. These different kinds of pain can be specific or unspecific to PD. Unspecific pain is related to other painful conditions such as musculoskeletal pain, radicular pain or due to restless leg syndrome. They can be worsened by PD. Specific pain is defined by the presence of chronological and/or topographical links with no evidence of other painful condition. These specific pains can either be nociceptive (directly related to motor symptoms such as the dystonic pain) or nociplastic called central, PCP [START_REF] Marques | How to diagnose parkinsonian central pain?[END_REF]. Abnormal pain processing in PD has been suggested by several studies. For instance, reduced pain thresholds were shown in experimental animal models of PD [START_REF] Park | Pain perception in acute model mice of Parkinson's disease induced by 1-methyl-4-phenyl-1,2,3,6tetrahydropyridine (MPTP)[END_REF]. In addition PD patients also presented decreased subjective and objective pain thresholds to various stimuli in comparison to healthy volunteers [START_REF] Brefel-Courbon | Nociceptive brain activation in patients with neuropathic pain related to Parkinson's disease[END_REF][START_REF] Brefel-Courbon | Effect of levodopa on pain threshold in Parkinson's disease: a clinical and positron emission tomography study[END_REF][START_REF] Gerdelat-Mas | Levodopa raises objective pain threshold in Parkinson's disease: a RIII reflex study[END_REF]. Recently, two meta-analyses have confirmed the increase in pain perception in PD patients [START_REF] Sung | Pain sensitivity in Parkinson's disease: Systematic review and meta-analysis[END_REF][START_REF] Thompson | Pain perception in Parkinson's disease: A systematic review and meta-analysis of experimental studies[END_REF]. Beside pain threshold studies, other arguments in favor of this abnormal pain processing are the abnormalities of laser evoked potentials and the brain modifications induced by pain in PD patients [START_REF] Brefel-Courbon | Effect of levodopa on pain threshold in Parkinson's disease: a clinical and positron emission tomography study[END_REF][START_REF] Schestatsky | Neurophysiologic study of central pain in patients with Parkinson disease[END_REF][START_REF] Tinazzi | Muscular pain in Parkinson's disease and nociceptive processing assessed with CO2 laser-evoked potentials[END_REF].

In fact, functional imaging using H2 15 O-PET scan have shown that, in OFF levodopa condition, pain-free PD patients have overactivation in several nociceptive areas (ipsilateral prefrontal cortex, ipsilateral insula, and contralateral anterior cingulate) in comparison with normal subjects [START_REF] Brefel-Courbon | Effect of levodopa on pain threshold in Parkinson's disease: a clinical and positron emission tomography study[END_REF]. Thus, both sensory discriminative (insula/SII) as well as affective motivational processing of pain (anterior cingulate cortex, prefrontal cortex) are abnormal in PD. In line with these results, it has been reported that both pain threshold (evaluating the sensory-discriminative component of pain) and tolerance thresholds (referring to the psychological perception of pain, a complex balance between cognitive and affective functions) were increased in PD patients after subthalamic nucleus deep brain stimulation and after acute levodopa administration [START_REF] Marques | Central pain modulation after subthalamic nucleus stimulation: A crossover randomized trial[END_REF][START_REF] Dellapina | Effect of subthalamic deep brain stimulation on pain in Parkinson's disease[END_REF]. Thus, several findings -such as results of imaging studies and the abnormalities found in quantitative sensory assessments as well as laser evoked potentials and nociceptive flexion reflex -suggest that the abnormal pain processing could be located in the central nervous system. But peripheral deafferentation could also play a role in sensory dysfunction [START_REF] Nolano | Sensory deficit in Parkinson's disease: evidence of a cutaneous denervation[END_REF]. However, a recent study using laser evoked potentials, strongly suggests that abnormal nociceptive processing occurs mostly at a central level in PD patients [START_REF] Zambito-Marsala | Abnormal nociceptive processing occurs centrally and not peripherally in pain-free Parkinson disease patients: A study with laser-evoked potentials[END_REF]. All the previous results indicate that nociceptive signals are amplified somewhere along pain transmission pathways. This could be related to an increased facilitation through a central sensitization of pain pathways [START_REF] Lim | Do dyskinesia and pain share common pathophysiological mechanisms in Parkinson's disease?[END_REF] or to a decreased inhibition (reduced activity of the descending inhibitory control system) [START_REF] Mylius | Pain sensitivity and descending inhibition of pain in Parkinson's disease[END_REF][START_REF] Grashorn | Influence of Dopaminergic Medication on Conditioned Pain Modulation in Parkinson's Disease Patients[END_REF]. An interesting point is the potential existence of a pathophysiological difference between PD patients with persistent pain and pain-free PD patients. Conflicting results have been reported in regard to pain thresholds (lower or similar pain thresholds in painful and pain-free PD patients), so we cannot conclude that the more the patients are painful, the more their pain threshold is lower [START_REF] Schestatsky | Neurophysiologic study of central pain in patients with Parkinson disease[END_REF][START_REF] Tinazzi | Muscular pain in Parkinson's disease and nociceptive processing assessed with CO2 laser-evoked potentials[END_REF]. By contrast, functional imaging study comparing pain-induced activation in PD patients with central pain and without any pain showed a preferential recruitment of the medial affective system in painful patients. Thus the existence of painful sensation might result from an imbalance between the two systems (discriminative and affective) with an overactivation of the medial affective system [START_REF] Brefel-Courbon | Nociceptive brain activation in patients with neuropathic pain related to Parkinson's disease[END_REF].

Other imaging study confirmed these abnormalities using structural and functional magnetic resonance imaging [START_REF] Polli | Anatomical and functional correlates of persistent pain in Parkinson's disease[END_REF]. PD patients with persistent pain had significant thinning and functional changes in several nociceptive areas compared to PD patients without pain. All these imaging results suggest the existence of a network of brain regional alterations specifically associated with pain. These brain modifications are not related to chronic dopaminergic treatment because de novo PD patients never treated by dopaminergic drugs showed overactivations in several nociceptive areas at functional magnetic resonance imaging with experimental pain stimulation [START_REF] Tessitore | Central pain processing in "drug-naïve" pain-free patients with Parkinson's disease[END_REF]. Finally, the role of the dopaminergic system in the pathophysiology of pain remains an open question. Most studies have reported that L-Dopa reduces pain perception. For instance, L-Dopa raised subjective cold pain thresholds, decreased pain intensity and reduced cerebral overactivations induced by experimental pain stimulation [START_REF] Brefel-Courbon | Effect of levodopa on pain threshold in Parkinson's disease: a clinical and positron emission tomography study[END_REF][START_REF] Gerdelat-Mas | Levodopa raises objective pain threshold in Parkinson's disease: a RIII reflex study[END_REF][START_REF] Schestatsky | Neurophysiologic study of central pain in patients with Parkinson disease[END_REF][START_REF] Nebe | Pain intensity on and off levodopa in patients with Parkinson's disease[END_REF]. Nevertheless, a dopamine agonist (apomorphine) did not modify pain threshold or cerebral activations induced by pain stimulation [START_REF] Dellapina | Apomorphine effect on pain threshold in Parkinson's disease: a clinical and positron emission tomography study[END_REF]. Moreover, in an imaging correlation study, subjective pain threshold was not correlated with FP-CIT binding ratio in striatum and but there was a significant correlation with FP-CIT binding ratio in extrastriatal regions (posterior cingulate cortex and insula) suggesting that pain perception abnormalities may be related to other monoaminergic systems [START_REF] Dellapina | Dopaminergic denervation using [123I]-FPCIT and pain in Parkinson's disease: a correlation study[END_REF]. The same hypothesis was proposed in animal models. Norepinephrine and serotonin reuptake inhibitors decreased or prevented hypersensitivity in six OHDA rats [START_REF] Cao | Restoring Spinal Noradrenergic Inhibitory Tone Attenuates Pain Hypersensitivity in a Rat Model of Parkinson's Disease[END_REF][START_REF] Campos | Monoaminergic regulation of nociceptive circuitry in a Parkinson's disease rat model[END_REF]. In addition, these drugs normalized spinal encephalin immunoreactivity in the spinal cord in these rats. Thus, hyperalgesia in the rat model of PD is not mediated by nigral dopaminergic lesion alone but by other monoaminergic systems (norepinephrine and serotonin) which modulate the opioid system.

To conclude, several neurotransmitters such as dopamine, norepinephrine, serotonin and opioid could participate in the abnormal pain processing in PD.

Conclusion

Numerous clinical and preclinical studies show that PD should now be considered as a painful disease, at least, as much as osteoarthritis and diabetes. Yet, the coexistence of different pain subtypes in PD requires a successful approach to distinguish between these pains that effective treatment strategies can be established. In particular, the identification of parkinsonian central pain remains uneasy and clear positive diagnosis criteria still need to be validated. More, generally, a better classification of pain in PD and better homogenization of populations is of major importance for further studies aiming to improve our comprehension of pathophysiological mechanisms of pain in PD.

Abbreviations: PD: parkinson's disease; PCP: Parkinsonian central pain.

Funding sources: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors

  and Deuschl proposed a four-tier taxonomy to improve the characterization of the complexity of pain in PD. This taxonomy combines the categorization proposed earlier by Lee et al. or Nègre-Pages et al. depending on the relationship to PD, and assigns nociceptive, neuropathic and miscellaneous pains to distinct categories, as well as further characterization into subcategories. The limitation of this classification is the lack of clarity of miscellaneous subtypes (pain that cannot be clearly assigned as nociceptive or neuropathic). The Marburg-Sao-Paulo-Cre teil Questionnaire for Pain in Parkinson's Disease is also an original questionnaire for pain assessment in PD, based on practical clinical reasoning but does not include central pain

  al. reported that musculoskeletal pain (41%) and radicular back pain (27%) were the most prevalent types, and dystonic pain (17%) the least prevalent type (3). Beiske et al. reported 70% prevalence of musculoskeletal, 40% dystonic, and 20% radicular pain in their PD cohort (2). Rana et al. observed that musculoskeletal pain was prevalent only in 28% and dystonic pain, in 48% (4). In contrast, Broetz et al. found up to 36% occurrence of radicular pain (23). Valkovic et al. showed the highest prevalence (22%) for central neuropathic pain (3), in contrast to previous studies (4.5-

	12.7%)

Table 1 :

 1 TABLES Prevalence of pain subtypes according to different studies conducted in PD

		Muskulo-	Dystonic	Radicular-	Central	Others
		skeletal		Neuropathic
	Broetz et al 2007	74.3%		35.6%
	Defazio et al 2008	25.4%	11.4%	4.7%	4.5%
	Beiske et al 2009	70%	40%	20%	10%
	Rana et al 2013	48%	26%	13%	8%
	Valkovic et al 2015 Early stage	34%	11%	19%	19%	26%
	Advanced stage	47%	23%	34%	25%	23%
	Silverdale et al 2018 66%	16%	34%	27%

Table 2 :

 2 Clinical classification of painful or unpleasant sensations in PD according toFord et al.

	Category by description	Clinical features
	Musculoskeletal	• Aching, cramping, artralgic, myalgic
		sensations in joints, and muscles
		• Associated findings may include
		tenderness, arthritic changes,
		skeletal deformity, limited joint
		mobility, postural abnormalities, and
		arthralgic gait;
		• May be exacerbated by parkinonian
		rigidity, stiffness and immobility, nad
		relieved bu mobility;
		• May fluctuate with medications
		dosing and improve with levodopa
	Radicular/neuropathic	• Pain in a root or nerve territory,
		associated with motor or sensory
		signs of nerve or root entrapment
	Dystonia	• Associated with twisting movements
		and postures; muscular contractions
		often very forceful and painful;
		• Dystonia may involve any limb or
		extremity, as well as facila and
		pharyngeal musculature;
		• May fluctuate closely with
		medications dosing: early morning
		dystonia, beginning of dose and end
		of dose dystonia, peak dose dystonia
	Central primary pain	• Burning, tingling, formication,
		"neuropathic" sensations, often
		relentness and bizarre in quality, not
		confined to root or nerve territory;
		• Pain may have an autonomic
		character, with visceralsensations or
		dyspnea, and vary in parallel with
		the medication cycle as a non motor
		fluctuation;
		• Not explained by rigidity, dystonia
		musculoskeletal or internal lesion
	Akathisia	• Subjective sense of relentness, often
		accompanied by an urge to move;
		• May fluctuate with medicationeffect,
		and improve with levodopa

Table 3 :

 3 Clinical classification of painful or unpleasant sensations in PD according toMarques et al.

	Chronic pain	Nociceptive pain	Neuropathic pain	Nociplastic pain
	Unspecific to PD	Musculoskeletal	Radicular pain	RLS
		pain
	Specific in PD	Dystonic pain	-	Central Pain

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors


Daniel Ioan
email: daniel.ioan@l2s.centralesupelec.fr

Ionela Prodan
email: ionela.prodan@lcis.grenoble-inp.fr

Sorin Olaru
email: sorin.olaru@l2s.centralesupelec.fr

Florin Stoican
email: florin.stoican@acse.pub.ro

Silviu-Iulian Niculescu
email: silviu.niculescu@l2s.centralesupelec.fr

Navigation in cluttered environments with feasibility guarantees

Keywords: obstacle avoidance, constrained control, linear systems

This paper addresses the navigation problem in a multi-obstacle environment and makes use of convex lifting in trajectory planning problems with anti-collision constraints. The design problem is commonly stated in the literature in terms of a constrained optimization problem over a non-convex domain. In our framework, the convex lifting approach plays an instrumental role in the partitioning of the feasible space in accordance with the distribution of obstacles and in the subsequent generation of corridors in cluttered environments. We consider an adaptation of the generic MPC (Model Predictive Control) trajectory tracking problem, aiming to guarantee the feasibility and the convergence. Simulation results and proof of concepts illustrations prove the effectiveness of the proposed approach.

INTRODUCTION

Navigation through multi-obstacle environments has received significant attention from both control and robotics communities [START_REF] Janecek | OPTIPLAN: A Matlab Toolbox for Model Predictive Control with Obstacle Avoidance[END_REF] due to its many applications including, among others, monitoring or surveillance [START_REF] Puri | A survey of unmanned aerial vehicles uav for traffic surveillance[END_REF], autonomous overtaking [START_REF] Ballesteros-Tolosana | Collision-free trajectory planning for overtaking on highways[END_REF] or precision agriculture [START_REF] Jawad | Energy-efficient wireless sensor networks for precision agriculture: A review[END_REF]. The main challenge resides in the non-convexity of the feasible regions in the motion space.

To tackle the navigation problem, several approaches have been developed. To the best of authors knowledge, most of them divide the problem in two main sub-tasks: path/trajectory generation and trajectory tracking. Moreover, these (sub-)tasks are usually viewed as independent or decoupled problems [START_REF] Latombe | Robot motion planning[END_REF]. For instance, the classical sample-based 1 approaches, [START_REF] Karaman | Sampling-based algorithms for optimal motion planning[END_REF], [START_REF] Lavalle | Rapidly-exploring random trees: A new tool for path planning[END_REF], are prone to focus on the first task, neglecting the second one and, thus, simplifying the problem. Next, the optimization-based strategies, e.g. mixed-integer formulations [START_REF] Richards | Aircraft trajectory planning with collision avoidance using mixed integer linear programming[END_REF]; [START_REF] Prodan | Mixed-integer representations in control design: Mathematical foundations and applications[END_REF], convexification techniques [START_REF] Szmuk | Convexification and real-time on-board optimization for agile quad-rotor maneuvering and obstacle avoidance[END_REF], potential field methods [START_REF] Chen | UAV path planning using artificial potential 3 The number of edges that are incident to that node. field method updated by optimal control theory[END_REF] or settheoretic approaches [START_REF] Franzè | The obstacle avoidance motion planning problem for autonomous vehicles: A low-demanding receding horizon control scheme[END_REF] merge the planning and tracking tasks at the expense of a higher computational complexity, especially in the case of clut-

The author is also with Inria Saclay, "DISCO" Team, France. Institute of Engineering Univ. Grenoble Alpes 1 Sample-based approaches are generally related to the construction of a graph structure.

tered/congested multi-obstacle environments. In addition, there are a few works, e.g., [START_REF] Weiss | Motion planning with invariant set trees[END_REF], that have a more intrinsic approach, considering planning and tracking as distinct but interacting tasks [START_REF] Berntorp | Automated driving: safe motion planning using positively invariant sets[END_REF] and employing appropriate methods from the previously mentioned categories. For instance, in this last category, we may include [START_REF] Berntorp | Positive invariant sets for safe integrated vehicle motion planning and control[END_REF], where the authors proposed an extension of the RRT(rapidly-exploring random tree), using feedback control and positively invariant sets to guarantee collision-free tracking.

The main drawback of the methods involving samplebased techniques is their probabilistic completeness [START_REF] Barraquand | A random sampling scheme for path planning[END_REF]. More specifically, the probability that the algorithm returns a feasible solution tends to 1 if the number of sample points is sufficiently large (approaching ∞), as it was empirically shown in [START_REF] Hsu | On the probabilistic foundations of probabilistic roadmap planning[END_REF]. Moreover, these probabilistic completeness proofs do not provide any guarantee on the time in which the algorithm finds the optimal path (if there exists one). Therefore, the primary objective of our paper is the global feasibility and we aim to develop a method characterized by completeness, eliminating the risks of an heuristic/unpredictable behavior.

More precisely, we revisit the ideas proposed by [START_REF] Ioan | From obstacle-based space partitioning to corridors and path planning. a convex lifting approach[END_REF] and we proceed further in providing global feasibility guarantees and enhancing the effective control strategy. Basically, in a first stage we neglect the differential constraints and the physical limitations that can appear in the motion planning in order to generate a geometric path. This path ensures the avoidance of obstacles and has the potential to explicitly describe a feasible corridor similar to the ones in [START_REF] Liu | Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3-d complex environments[END_REF] or in [START_REF] Faulwasser | Nonlinear model predictive control for constrained output path following[END_REF]. At a second stage, using the geometric path and the corridor as starting points, we find some appropriate trajectory respecting the agent's dynamics and constraints using a MPC strategy.

The main contributions of this paper are twofold: i) providing a detailed analysis of the recursive feasibility of a corridor-constrained MPC; ii) providing a generic navigation strategy for cluttered environments with feasibility guarantees.

The remaining of the paper is organized as follows: Section 2 presents some basic set-theoretic tools, introduces the formulation of the problem and provides a mechanism able to generate feasible geometric paths through the multi-obstacle environment. Section 3 presents a prototype control strategy with recursive feasibility guarantees and the generic navigation strategy, pointing to the extensions to the problem of multiple agents. Finally, some concluding remarks end the paper Notation: The Minkowski sum of two sets is denoted as

A ⊕ B = {x : x = a + b, a ∈ A, b ∈ B}. Given a compact set S ∈ R n , C X (S)
denotes the complement of S over X ∈ R d , Com(S) the space of compact subsets of S, int(S) its interior and V(S) the set of its extreme points.

As well, proj [d1:d2] S stands for projection of S on the dimensions d 1 to d 2 . For a polyhedron P ∈ R d , V(P ) is the (finite) set of its vertices, and

F k i (P ) is the i-th face of the dimension k < d. For x ∈ R d we denote x 2 Q = x Qx. B p,r = {x ∈ R d : x -p ≤ r} is a ball of radius r ≥ 0 centered in p ∈ R d w.r.t. a given norm.

PREREQUISITIES AND PROBLEM FORMULATION

Consider a finite dimensional output space R d and a finite number of interdicted convex regions P j ⊂ Com(R d ), j∈ I = {1, . . . , N o } as obstacles:

P = No j=1 P j ; P i ∩ P j = ∅, ∀i =j, (1)
their union lies in a bounded2 cluttered environment X:

P ⊂ int(X) ⊂ R d (2)
while the obstacle-free (implicitly, non-convex) domain is C X (P) X \ P.

(3) Definition 1. (Corridors - [START_REF] Ioan | From obstacle-based space partitioning to corridors and path planning. a convex lifting approach[END_REF]). Given the obstacles P, a corridor between two points x 0 , x f ∈ int(C X (P)) is enabled by the existence of two continuous functions:

γ

: [0, 1] → C X (P), ρ : [0, 1] → R >0 (4) satisfying γ(0) = x 0 , γ(1) = x f and γ(θ) ⊕ B 0,ρ(θ) ⊂ C X (P), ∀θ ∈ [0, 1]. The corridor is defined as: Π = {x ∈ R d : ∃θ ∈ [0, 1] s.t. x ∈ γ(θ) ⊕ B 0,ρ(θ) }. (5) Definition 2. A family of sets {X i } i∈I verifying: i) X = No i=1 X i , ii) int(X i ) int(X j ) = ∅, ∀i = j ∈ I, iii) P i ⊂ int(X i ), ∀i ∈ {1 . . . , N o }
is called a partition of X induced by the obstacles P. Furthermore, if the sets X and X i , ∀i are polyhedral, then X = No i=1 X i is called a polyhedral partition. The problem of navigation in cluttered environments X can be divided in three sub-problems: P1) Starting from the set of forbidden regions P, determine a partition of the cluttered environment around them. P2) Considering any two points x 0 , x f ∈ int(C X (P)), construct a corridor linking them or provide a certificate of infeasibility. P3) Given a non-empty corridor, select/generate a continuous trajectory τ : [t 0 , t f ] → Π guaranteeing collision avoidance, i.e. τ (θ)

∩ P = ∅, ∀θ ∈ [0, 1].
Previous work [START_REF] Ioan | From obstacle-based space partitioning to corridors and path planning. a convex lifting approach[END_REF] addressed the first two problems P1) and P2). By developing a systematic solution, we aim hereinafter to treat in detail problem P3) and, additionally, to provide some appropriate feasibility guarantees. Prior to main developments, we recall the constructive method for the resolution of P1) and P2). Definition 3. [START_REF] Nguyen | Convex lifting: Theory and control applications[END_REF]). Given a collection of obstacles P = No j=1 P j with P i ∩ P j = ∅, ∀i = j, as defined in (1), and a partitioning of the cluttered environment

X ⊃ P, the function z : X → R is called a piecewise affine lifting if: z(x) = a i x + b i , x ∈ X i , (6) with X i satisfying int(X i ) ⊃ P i , ∀i , a i ∈ R d and b i ∈ R.
Taking P i as polyhedral sets (i.e, having a finite number of extreme points), the lifting can be constructed as the result of the following convex optimization problem:

min ai,bi No i=1 [a i b i ] 2 2 (7) s.t. a i v + b i ≥ a j v + b j + , ∀v ∈ V(P i ), ∀i = j, (8a) a i v + b i ≤ M, ∀v ∈ V(P i ), ∀i.
(8b) Based on the solutions of (7), we define the following "d+1"-dimensional polyhedron:

P = x z ∈ R d+1 : a i -1 x z ≤ -b i , i ∈ I .
Projecting the facets of P on X provides a polyhedral partition {X i } i=1:No . Corollary 4. The polyhedral partition {X i } i=1:No has the following properties:

i) P i ⊂ int(X i ), ∀i, ii) X i ∩ P j = ∅, ∀j = i.
Fig. 1 shows some partitioning of complicated 2D and 3D cluttered environments. Since in the motion planning context generally we do not go higher than 3D, we restricted the illustrations to these dimensions, but the construction from Corollary 4 holds also for arbitrary dimensions.

Once the partition {X i } i=1:No of the cluttered environment X is available, the goal is to construct a graph in order to generate feasible paths through X. Therefore, we have to select the nodes, the edges and the associated weights from the constructive parameters of the compact sets X i (vertices and faces). For illustration, we consider the case d = 2. The resulting graph is Γ 1 (N 1 , E 1 , f 1 ), where the nodes are the vertices of the polyhedral re-

gions X i : N 1 = No i=1
V(X i ); the edges are the facets of the partition regions X i (i.e., their support hyperplanes):

E 1 = F 1 (X i )
; and the function f 1 gives the Euclidean distance between the incident nodes of the edge.

Finding the shortest path through the graph between the start and final points x i , x f ∈ X \ P means adding them to the graph and finding the closest edge such that the connection to it does not intersect any obstacles:

(j ⊥ , x ⊥ )(x i ) = arg min j∈I f i min x∈facetj (Xi) x -x i (9a) s.t. αx + (1 -α)x i / ∈ P i , x i ∈ X i , (9b)
where I fi is the number of facets of X i . These projections of the points are also new nodes of the graph. Thus, the graph Γ(x i , x f ) preserves the properties of Γ. Finally, a graph search algorithm (e.g., Dijkstra's Algorithm Karaman and Frazzoli ( 2011)) is employed and the shortest path between x i and x f is derived.

Remark 5. For further use, we denote the shortest path through the graph between x i and x f as Path(x 0 , x f ) = (x 0 = x i , x1 , . . . , xn , xn+1 = x f ). This represents an ordered set of points where no segment defined by a pair of consecutive points cuts any of the obstacles. Altough it is not a path in the sense stated in problem P3), it represents a sufficient condition for the existence of a corridor (5).

Having the geometric path given by Path(x 0 , x f ), we can determine the values corresponding to γ(θ). Thus, the remaining part is to select the width of the corridor, namely ρ(θ). To ensure that Π ∩ P = ∅, ρ(θ) has to verify the inequality:

ρ(θ) ≤ min Pi∈P d H (P i , γ(θ)), ∀θ ∈ [0, 1]; ( 10
)
Remark 6. The corridor Π can be written as Π =

Nc i=1 Π i with Π i = {x ∈ R d : ∃ θ ∈ [0, 1] s.t. x ∈ γ i ( θ) ⊕ B 0,ρi( θ) } where γ i (0) = xi-1 and γ i (1) = xi .
For illustrating the construction of the corridor, we revisit the obstacle collection shown in Fig. 1a. Hence, we provide an approximation of the corridor width ρ (red area in Fig. 2 is the corridor defined in ( 5)). To compute the corridor width we sampled the continuous parameter θ and used the corresponding values in (10). Remark 7. Since we deal with a polytopic representation of the environment, for further use we consider the polytopic underapproximation of the corridors (blue area in Fig. 2). The construction of the corridor is based only on the topological characteristics, but the method proposed here is not unique. For example, instead of using Hausdorff distance, method in [START_REF] Liu | Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3-d complex environments[END_REF] relies on finding the maximal ellipsoid including a segment from a given path and not intersecting the obstacles.

CORRIDOR-CONSTRAINED MPC

This section introduces the formulation of a MPC controller and its proof of recursive feasibility. The strategy proposed in the sequel exploits the existence of feasible corridors resulted from the partitioning of the environment. First we recall some concepts and definitions and next we present the formulation associated to a single compact corridor (a segment from (5)). Then the extension to the entire corridor (5) is tackled.

Prototype MPC with recursive feasibility guarantees

In what follows, our aim is to obtain a collision-free trajectory inside a corridor Π i for an agent described by the following LTI dynamics:

x k+1 = Ax k + Bu k , ∀k (11)
with x k ∈ R d the state vector, u k ∈ R du the input vector and the matrices A, B of appropriate dimension. Also, the physical constraints lead to the compact sets X and U from R d and R du , respectively. The constraints corresponding to the corridor Π i ⊂ R d1 are usually only on a subset of the state variables X ⊂ R d (without loss of generality considered next the corridor to be the first d 1 subcomponents). We use the corridors in the original state to be described by the set:

Πi = {x ∈ X | I d1 0 d-d 1 +1 x ∈ Π i }. ( 12
)
For the MPC problem we consider a standard formulation with a quadratic cost:

J (Np, xi ) =   V f (x k ,u k ) x k+Np|k -xi 2 P + + Np-1 l=1 x k+l|k -xi 2 Q + Np-1 l=1 ∆u k+l|k 2 R V(x k ,u k )      (13)
where N p is the prediction horizon, the weight matrices P (terminal cost penalty), Q (output error penalty) and R (control action penalty) are positive semi-definite and of appropriate dimensions. For further implications, the value of P from the terminal cost V f (x k , u k ) is selected such that the function V (x) = x P x is a Lyapunov function for a pre-stabilizing state-feedback law u = Kx with K ∈ R d×du in a vicinity of xi [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. As well, the values of Q and R from the cost per stage V(x k , u k ) are chosen in order to enhance the tracking performances.

Therefore, the MPC problem to be solved at each time step throughout a corridor Πi can be formulated as:

P( Πi , N p ,X f , xi , X ) : min u J (N p , xi ) (14a) s.t. x k+l+1|k = Ax k+l|k + Bu k+l|k , (14b) u k+l|k ∈ U, ∀ = 1 : N p -1 (14c) x k+l|k ∈ Πi (14d) x k+Np|k ∈ X f , (14e)
Having the formulation ( 14), the question to be addressed is how to select the parameters Πi , N p , X f , xi , X in order to ensure the recursive feasibility. The recursive feasibility is one of the fundamental challenges in MPC literature. Basically, it represents the property that for all initial feasible states and for all optimal sequences of control inputs the MPC optimization problem remains feasible [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. Löfberg ( 2012) offers a broad overview on this topics, especially from the computational perspective.

Essentially, the selection of the parameters in ( 14) can be viewed as an additional analysis step, which builds on the, so-called, backward reachable set (BRS). Definition 8. (N-step BRS). The N-step (BRS) is the set of all states that can reach a final position xi in N stepsassociated to the system described by LTI dynamics (11):

R i N = A -N xi ⊕ N -1 j=0 -A j BU . ( 15
)
Since we have "hard" constraints on state (the ones given by the corridor), we have to compute the N -step BRSs taking into account these constraints: Ri N . In fact, the N -step BRSs remain the same as long as they check the corridor constraints. From that point, the computation can be done iteratively as in Algorithm 1.

Algorithm 1 Computing N-step BRS for (11) taking into account the state constraints Πi

1: Find first R i Nuc such that R i Nuc ⊂ Πi 2: For N < N uc : Ri N = R i N 3: Ri Nuc = R i Nuc ∩ Πi 4:
For N > N uc the computation is iterative and relies on: Ri

N = A -1 Ri N -1 ⊕ (-BU ) ∩ Πi .
The N-step BRSs have an instrumental role in providing the parameters of the MPC problem ( 14). For instance, the theoretical minimal value of N p is given by the condition

x 0 ∈ R i Np . Proposition 9. i , N p , X f , xi , X ) is feasible for all feasi- ble initial states if N p ≥ arg min N N s.t. x 0 ∈ Ri N . (16)
Proof. It is straightforward that x 0 ∈ Ri Np . That is, there exists at least one sequence of N p inputs so that the predicted final state is exactly the reference xi . As well, the existence of a terminal cost as in ( 13) ensures the convergence. Moreover, since xi ∈ X f , it is obvious that (14e) holds. Therefore, the feasibility of P(Π i , N p , X f , xi , X ) is guaranteed. As a consequence of Proposition 9, the recursive feasibility of a MPC controller based on P(•) follows straighforwardly.

Generic collision-free trajectory generation within cluttered environments

Since the recursive feasibility of an MPC strategy based on ( 14) is guaranteed for certain parameters, we aim to exploit these degrees of freedom for the resolution of the problem P3). To this end, an intuitive solution is to iteratively compute the parameters ensuring feasibility for each corridor segment. Specifically, for each initial position (γ i (0)) within the corridor segment, we compute the N-step BRS centered in x i f . The computational effort corresponding to this may be substantial relative to a consequent real-time implementation. To overcome this drawback, we can split the strategy since we are aware of the corridor characteristics.

(1) (OFF-LINE) For each Πi and x i f • compute the BRSs (as inAlgorithm 1) and N i pmin :

N i pmin = arg min N N s.t. Πi ⊆ Ri N , (17)
• X i f = ( Πi ∩ Πi+1 ) excepting the last segment, for which we have

X i f = Ri 1 (2) (ON-LINE) Apply Algorithm 2
As its name suggests, the idea behind the "Relay MPC" strategy is to ensure the transitions from a segment of corridor ( Πi ) to the next ( Πi+1 ). For that reason, we choose the terminal sets X fi as the intersections between two consecutive segments.

Algorithm 2 "Relay MPC"

Input: Π = Nc i=1 Π i , x 0 , N i pmin , X i f 1: for each Πi ⊂ Π do 2:
N p = N i pmin ;

3: repeat 4:

Apply MPC strategy solving P( Πi , N p , X i f , xi , X )

5: until x k+1|k ∈ X i f 6:
x 0 = x k+1|k 7:

Update the parameters of P 8: end for Proposition 10. If Path(x 0 , x f ) exists and the control law based on P(•) (14) is recursively feasible then the convergence x 0 → x f is guaranteed.

Proof. The existence of Path(x 0 , x f ) leads to the construction of the corridor Π = i Πi . The recursive feasibility and the selection of terminal sets as in (1) ensure the transition from Πi to Πi+1 . Moreover, for the last segment we have X i f = Ri 1 and the initial state in RNp which directly leads to x f being reachable, and, by consequence, it proves the convergence of the scheme.

For illustration purposes, consider an agent described by the dynamics (11) in R 4 with: As a side remark, for the same initial and final position an MIP based MPC strategy can be employed, considering the entire set of obstacles and imposing a large prediction horizon (N p = 40). Despite the large prediction horizon, the resulting trajectory does not converge to the final position, the agent remaining on the boundary of one of the obstacles (the pink trajectory in Fig. 3).

A = O 2 I 2 O 2 -µ M I 2 , B = O 2 M -1 I 2 , (18)
Regarding the offline part, as stated above, the computational effort is substantial, e.g., for the trajectory in Fig. 3 is around 3 minutes, but this can be improved by replacing the polyhedral representation of the sets with a zonotopic one [START_REF] Althoff | Cora 2015 manual[END_REF]. Intuitively, there is a connection between the length of a corridor segment (∆γ i = γ i (1)γ i (0) ) and the length of the minimal prediction horizon(N pmin ). Therefore, we depict in Table 1.

Remark 11. In practice, whenever the length of a segment of the corridor is greater than a user-defined value, that segment can be split. In this way, one can manage the trajectory tracking with similar length of the prediction horizons in concordance with the available computational constraints.

To emphasize the importance of the corridor and how it reflects in a real-world application, we can assume that the dynamics ( 18) are the nominal ones and any disturbance impacts, in fact, the dimension of the corridor. That is, the function ρ(•), as in ( 5), becomes ρ(•) = ρ(•) + ∆ρ.

In Table 2 we delineate: N goal -the number of steps to attain a neighborhood of the final point, t goal -the total time to compute the trajectory and t -the trajectory length. As it can be seen in Table 2, the computing time t goal has similar values (slight modifications), while N goal and t increase with the values of ∆ρ. This behavior is counter-intuitive, but it can be explained by the fact that the decreasing in terms of steps (or distance) gained along of the corridor are wasted with the maneuvers associated to changing the segments of the corridor. However, by moving all complex operations from on-line to off-line, we note that the computing time is kept reasonable, allowing a comparison with the classical sampled-based methods. Table 2.

Insights on the extensions to multi-agent system

Up to now we consider only one agent within the static multi-obstacle environment. There exist however applications which can be arguable better performed by a team of agents. Thus, a second strong requirement is the avoidance of collisions among the agents. As stated in Section 1, the method herein has two phases: planning and tracking.

Clearly, we can eliminate the risk of collision from the planning phase by selecting paths with disjoint sets of nodes (and, implicitly, of edges). Nevertheless, this may lead to trajectories too far (in terms of performance or consumed energy) from the optimal ones. Hence, in the planning phase we can only impose that the paths of the two agents share a minimal number of common nodes.

Since the graph is constructed relying on the partitioning from Corollary 4, it is straightforward that the degree (or valency) 3 of the nodes is 3, excepting the nodes from the boundaries of X. Thus, the minimal number of common nodes between the paths of the agents is two and, implicitly, the paths share a common edge. Consequently, in the tracking phase we have to treat the behavior at the proximity of these/this common nodes/edge. Concretely, we adjust Algorithm 2, specifically an adaptation of the parameters involved in Step 4. Since the corridor constraints affect only a subset of the state variables, we may affect the behavior of the agent in the proximity of conflict zones by changing the bounds on the remaining subset of variables. Thus, we modify the parameter X of the MPC strategy from Algorithm 2. A prospective solution is to tune the bounds X according to a reachability analysis. That is, by considering the reachable sets mapped on the velocity subspace, we can find the set of velocities that would lead to collision with a moving entity. Enforcing velocities apart from this set ensures that the agents are able to attain a slowdown before the potential conflict zones.

4. CONCLUSIONS The paper presents a constructive solution for the generation of collision-free trajectories between two points in an environment containing multiple obstacles in a ddimensional space. We use the geometry of the obstacles and the convex lifting procedure to describe a graph around the obstacles. This graph represents a key element in order to generate collision-free trajectories employing MPC controllers with recursive feasibility guarantees. As a perspective of this work we will extend the results for time-varying multi-obstacle environments, considering, as well, re-configuration and adjustments of the graph.

  Fig. 1. Space-partitioning {X i } i=1:No from (7).

  Fig. 2. Path(x i , x f and Π =

Fig. 3 .

 3 Fig.3. Algorithm 2 over the example in Fig.1a

Fig. 4 .

 4 Fig. 4. Input and velocity during Algorithm 2As a side remark, for the same initial and final position an MIP based MPC strategy can be employed, considering the entire set of obstacles and imposing a large prediction horizon (N p = 40). Despite the large prediction horizon, the resulting trajectory does not converge to the final position, the agent remaining on the boundary of one of the obstacles (the pink trajectory in Fig.3).

  Table 1 the values corresponding to (17).

	i	1	2	3	5	8	10	11
	∆γ i	0.77	4.02	0.09	1.27	4.50	2.39	0.53
	Np min	11	24	13	14	29	19	13

This assumption is always valid for some bounded X, due to the boundedness assumptions on P j , j ∈ I.

ACKNOWLEDGEMENTS

The research of Daniel Ioan is financially supported by the Ministry of the Armed Forces -Defence Procurement Agency (DGA) -no.2017352. The authors acknowledge Jacques Blanc-Talon, expert DGA, for the fruitful discussions.