Robin Souron
email: robin.souron@univ-nantes.fr

Julian Colard

Juan D Ruiz-Cárdenas

Anicée Beltran

Pascale Duché

Mathieu Gruet

Development and assessment of test-retest reliability of a new field test to evaluate lower-limb muscle fatigability in young adults

Keywords: Performance fatigability, muscle testing, muscle power, muscle endurance, smartphone applications Fatigabilité, testing musculaire, puissance musculaire, endurance musculaire, applications mobiles

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Introduction

Muscle endurance is an important component of neuromuscular function and refers to the ability of a muscle or a muscle group to perform repeated work [START_REF] Caspersen | Physical activity, exercise, and physical fitness: definitions and distinctions for health-related research[END_REF]. The main outcomes informing on muscle endurance are either the total amount of work performed or the time to exhaustion in the case where the task requires to maintain or repeat a given level of voluntary contraction for as long as possible [START_REF] Evans | Is quadriceps endurance reduced in COPD?: a systematic review[END_REF]. While endurance time is a relevant indicator of muscle performance for such tests, it also greatly depends on psychological factors such as motivation and the ability to withstand pain, which exert an increasing influence as participants gets close to exhaustion. Therefore, it is useful to supplement this outcome with other indicators of muscle performance collected before symptom limitation. Within this framework, it is possible to incorporate performance fatigability indicators early in the test and at regular interval until exhaustion. Performance fatigability refers to the kinetics of change in a given performance indicator, e.g., muscle strength or power [START_REF] Enoka | Translating fatigue to human performance[END_REF]. For instance, many studies used changes in maximal muscle strength of a muscle group to track the kinetics of muscle fatigability development in various contexts and populations (Bachasson, Millet et al., 2013;[START_REF] Gruet | Skeletal muscle contractility and fatigability in adults with cystic fibrosis[END_REF][START_REF] Souron | Task failure during sustained low-intensity contraction is not associated with a critical amount of central fatigue[END_REF]. These tests generally assess maximal muscle strength at regular intervals throughout the exercise protocol, offering insights on muscle functioning across different effort levels and fatigue states, which may be pertinent to various activities of daily living. Overall, endurance time, fatigability assessments throughout the test and fatigability level at volitional exhaustion are complementary to provide a comprehensive assessment of neuromuscular performance during prolonged motor tasks. Bachasson, Millet et al. (2013) developed a hybrid protocol allowing to compute all these outcomes from a prolonged motor task consisting in intermittent isometric knee extensions performed at increasing force levels until exhaustion, interspaced with muscle fatigability assessments at regular intervals. Despite its numerous advantages, this test shares the common drawbacks of other fatiguing tests requiring specific ergometers, load cells/strain gauges and dedicated software, with large space requirements and elevated costs usually restricting their use to laboratory settings.

Here we sought to develop an innovative test allowing the assessment of both lower-limb muscle fatigability and endurance and gathering several advantages of more traditional fatigability protocol while remaining highly feasible and affordable. We considered the use of the isometric wall squat as the fatiguing exercise. This choice was guided by the fact that such exercise procedure i) is easy to standardize and replicate [START_REF] Lea | Validity and reliability of RPE as a measure of intensity during isometric wall squat exercise[END_REF], ii) does not require specific material as opposed to other tests (Bachasson, Millet et al., 2013), iii) targets various lower-limb muscle groups, i.e., quadriceps, hamstrings, gluteal [START_REF] Ayotte | Electromyographical analysis of selected lower extremity muscles during 5 unilateral weight-bearing exercises[END_REF], iv) isolates the neuromuscular function with low cardiorespiratory requirements [START_REF] Lea | Validity and reliability of RPE as a measure of intensity during isometric wall squat exercise[END_REF], v) can be implemented with a progressive difficulty by adjusting the duration of the exercise, vi) can be conducted in an intermittent fashion and interrupted without delay, allowing to incorporate regular fatigability assessments and vii) is feasible in various populations, including the general population, athletes and several clinical populations [START_REF] Baffour-Awuah | An evidence-based guide to the efficacy and safety of isometric resistance training in hypertension and clinical implications[END_REF][START_REF] Baffour-Awuah | An evidence-based guide to the efficacy and safety of isometric resistance training in hypertension and clinical implications[END_REF][START_REF] Goldring | The effects of isometric wall squat exercise on heart rate and blood pressure in a normotensive population[END_REF]Wiles et al., 2018[START_REF] Wiles | Development and assessment of test-retest reliability of a new field test to evaluate lower-limb muscle fatigability in young adults[END_REF]. We finally considered the use of simple indicators of muscle performance based on tests relevant to daily life (i.e., vertical jump and sit-to-stand) to track changes in muscle fatigability.

This work aimed to develop a new field test to evaluate lower-limb muscle fatigability in young adults, as detected by the kinetics of changes in squat-jump height (SJ H ) and sit-to-stand time (STS T ) computed using two smartphone applications (experiment A), and to assess its test-retest reliability in an independent sample of young adults (experiment B). We hypothesized that the test we developed in this work would induce a progressive development of muscle fatigability and that changes in the indicators computed to evaluate muscle fatigability would demonstrate a high test-retest reliability.

Materials and methods (experiment A)

Study design

Two independent experiments were built in a logical fashion. The first one (experiment A) aimed to determine if muscle fatigability could be detected from the kinetic of change in SJ H and STS T along the course of an incremental isometric exercise performed until exhaustion. The second one (experiment B) aimed to evaluate the test-retest reliability of the muscle fatigability test on an independent sample of participants. We planned to adapt the protocol for experiment B (e.g., duration of the stages, deletion of one measure if no change was detected for its associated outcome during the protocol) based on the results obtained during experiment A. For both experiments, none of the participants had known cardiovascular, musculoskeletal, auditory, or cognitive disorders that may impair their ability to perform the test. Participants were asked to refrain from strenuous and unaccustomed physical activity at least 24h prior testing. Participants were informed of the benefits and risks of the investigation prior to signing an institutionally approved informed consent to participate in the study. This study conformed to standards from the latest revision of the Declaration of Helsinki and was approved by a national research ethic committee (n°IRB00012476-2021-08-02-87).

Participants

We recruited twenty participants (eight females; mean ± SD age: 24 ± 2 years, height: 173 ± 8 cm, weight: 69 ± 7 kg) who visited the laboratory on two occasions, i.e., one familiarization and one experimental session. All participants were recreationally active, which reflected a commitment of 2-4 h of physical activity per week, i.e. level 2 of performance according to guidelines from De [START_REF] De Pauw | Guidelines to classify subject groups in sport-science research[END_REF]. During the familiarization session, participants were accustomed with all the procedures of the experimental session, including the procedures for the good execution of sit-to-stand and squat jump exercises, the fatigability protocol (i.e., completion of the first three stages of isometric wall squat, see below) and the report of rating of perceived exertion (RPE) scores.

Procedures

Squat jump performance

In line with our reasoning of developing a highly feasible test for optimal implementation, SJ H was evaluated with a valid and reliable mobile App called My Jump 2 installed on an iPad mini ® (Apple Inc., California, USA) with a 120-fps recording mode [START_REF] Balsalobre-Fernandez | The validity and reliability of an iPhone app for measuring vertical jump performance[END_REF][START_REF] Balsalobre-Fernandez | The validity and reliability of an iPhone app for measuring vertical jump performance[END_REF]. Participants started the jump with the feet a shoulder-width apart, the knee joint at approximately 90°and with their hands placed on their hips [START_REF] Balsalobre-Fernandez | The validity and reliability of an iPhone app for measuring vertical jump performance[END_REF]. The iPad was positioned in front of the participants, at 150 cm of his feet, enabling a clear view of the participant's lower limbs to ascertain take-off and post jump landing and any possible non-desirable counter movement [START_REF] Balsalobre-Fernandez | The validity and reliability of an iPhone app for measuring vertical jump performance[END_REF]. SJ H was calculated by manually selecting the take-off (i.e., the first frame in which both feet were off the ground) and the landing (i.e., the first frame in which at least one foot was touching the ground; see Figure 1B or http://carlosbalsalobre.com/#apps for a zoomed display) to estimate the flight time. Jump height was calculated using the following equation: SJ H = t 2 Â 1.22625, with SJ H being the jump height in cm and t representing the flight-time in s.

Sit-to-Stand performance

We calculated STS T using the Sit to Stand application (Ruiz-Cardenas et al., 2018Ruiz-Cardenas, Rodriguez-Juan, Smart, Jakobi, & Jones, 2018) installed on an Fig. 1. Panel A schematically represents the muscle fatigability protocol. The duration of the fatiguing exercise was progressively increased until exhaustion (Exh), that was defined as the moment when participants voluntarily stopped the exercise or the moment when participants were not able to maintain a correct wall squat position for more than three consecutive seconds. At the end of each stage, participants performed two squat jumps (SJ) and two sit-to-stand (STS) exercises. Panel B displays the experimental layout. The fatigability protocol began with the first 15-s stage. Ten second before the end of each stage, participants were asked to verbally report their perception of muscular and respiratory efforts (dotted grey arrow). At the end of each exercise stages, participants moved from the wall squat position to the SJ area [START_REF] Wiles | An alternative approach to isometric exercise training prescription for cardiovascular health[END_REF]. After performing two squat jumps, participants moved to the STS area and performed two STS exercises (2). Then, participants returned to the wall squat position to start a new fatiguing stage (3), and so on until exhaustion. Screenshot images for Sit to Stand and My Jump 2 applications were re-used with permission from Ruiz-Cárdenas and colleagues [START_REF] Ruiz-Cardenas | Validity and reliability of an iPhone App to assess time, velocity and leg power during a sit-to-stand functional performance test[END_REF] and from http://carlos-balsalobre.com, respectively).

iPhone X (240 fps; Apple Inc., California, USA). For the sit-to-stand test, we used a standard chair (46 cm height) with a flat seat and no armrest. Participants were asked to sit with their legs hip-width apart and with hip, knee and ankle joints at ∼90° [START_REF] Strassmann | Population-based reference values for the 1-min sit-to-stand test[END_REF]. Participants started in a sitting position and were asked to stand-up once as fast as possible (i.e., one transition from sitting to standing) while maintaining their arms crossed over the chest. The mobile device was positioned on a 0.7 m high tripod placed 3 m from the right side of the participants (Fig. 1B) [START_REF] Ruiz-Cardenas | Validity and reliability of an iPhone App to assess time, velocity and leg power during a sit-to-stand functional performance test[END_REF]. This mobile application was designed to analyse sit-to-stand test via high-speed video recording (i.e., 240 fps) to allow the calculation of the rising time (in s) between two frames manually selected by the experimenter (Fig. 1B).

A 90-s resting period separated the baseline measurements (i.e., the three squat jumps and sit-to-stand) from the beginning of the fatigability protocol.

Test protocol

The whole protocol is illustrated in Figure 1. The selected fatigability task consisted in maintaining an isometric wall squat position. First, posture was adjusted to obtain knee and hip angles at 100°using a clinical goniometer, with participants keeping their lower legs and trunk vertical (i.e., back against the wall). A mark was then placed on the midline of the femur and a laser system (CL2i STANLEY) was used to ensure the maintenance of the position. A wooden board was placed on the floor just in front of the feet of the participants. Both the laser system and the wooden board allowed for rapid and accurate replication of the position at the beginning of each new stage. Participants were instructed to wear sport shoes. A 50 Â 50 cm non-slippery surface was positioned under the feet of participants to ensure maximal adherence to the floor.

The duration of the isometric wall squat task was progressively incremented (i.e., 15-s incremental stages for the four first stages, 30-s incremental stages for the fifth to eighth stages and 60-s incremental from the ninth stages) until exhaustion (Fig. 1A). Exhaustion (Exh) was the moment where participants voluntarily stopped the exercise or the moment where they were not able to maintain the wall squat position for more than three consecutive seconds. Standardized sentences (e.g., "you're doing great, keep going") were pronounced by the experimenter with a neutral tone at the middle and 15 s before the end of each stage. In a 30-s window, participants performed two squat jumps and two sit-to-stand exercises immediately at the end of each stage, at exhaustion and 2 min after exhaustion (Post 2min ).

We used My Jump 2 (Balsalobre-Fernandez et al., 2015) and Sit to Stand (Ruiz-Cardenas et al., 2018) mobile Apps to assess SJ H and STS T . Internal validation showed an excellent intra-class correlation coefficient two-way mixed effect absolute agreement for the two squat jumps (ICC 2-k = 0.98) and the two sit-to-stand tests (ICC 2-k = 0.93) prior to the fatigability protocol. The kinetics of changes in these indicators were used as indices of muscle fatigability. The endurance time was calculated by summing the time of each completed stages and the time performed in the last stage, i.e., the one where the participants reached exhaustion. Participants were asked to verbally report their perception of muscular (RPE M ) and respiratory (RPE R ) efforts (Borg 6-20 scale) 10 s before the end of each stage. Similar to studies employing exercise modalities expected to predominantly activate specific systems (e.g. limb muscles during isolated isometric exercises), we opted for the use of differentiated RPE scores [START_REF] Bolgar | Effect of exercise intensity on differentiated and undifferentiated ratings of perceived exertion during cycle and treadmill exercise in recreationally active and trained women[END_REF][START_REF] Borg | Borg's perceived exertion and pain scales[END_REF][START_REF] Robertson | Perception of physical exertion: methods, mediators, and applications[END_REF], using the 6-20 RPE scale that has been used repeatedly for that purpose [e.g. [START_REF] Green | Overall and differentiated ratings of perceived exertion at the respiratory compensation threshold: effects of gender and mode[END_REF]; [START_REF] Gruet | Overall and differentiated sensory responses to cardiopulmonary exercise test in patients with cystic fibrosis: kinetics and ability to predict peak oxygen uptake[END_REF][START_REF] Gruet | Overall and differentiated sensory responses to cardiopulmonary exercise test in patients with cystic fibrosis: kinetics and ability to predict peak oxygen uptake[END_REF]; [START_REF] Pageaux | Prolonged mental exertion does not alter neuromuscular function of the knee extensors[END_REF][START_REF] Pageaux | Prolonged mental exertion does not alter neuromuscular function of the knee extensors[END_REF]]. The heart rate (HR, in bpm) was monitored throughout the fatigability test using a Polar Vantage M watch (Polar Electro, Kempele, Finland). The peak HR value (HR PEAK ) obtained during the fatigability test was expressed as a percentage of the theoretical maximal heart rate (HR MAX ). Theoretical HR MAX was calculated using the following equation [START_REF] Gellish | Longitudinal modeling of the relationship between age and maximal heart rate[END_REF]: HR MAX = 192-0.007 Â age 2

Statistical analysis

Sample size was estimated to detect differences between repeated measures with a moderate effect size (F test ≥ 0.27), an alpha level of 0.05, and a desired statistical power of 80% for a one-tailed test. A minimum of 16 participants was estimated. Normal distribution and homogeneity of variances analysis were confirmed using the Kolmogorov-Smirnov and Skewness test, respectively.

A group isotime analysis was performed [START_REF] Nicolò | A comparison of different methods to analyse data collected during time-to-exhaustion tests[END_REF]. This includes only the portion of the fatiguing test that was available for all participants, which is limited by participants with the shorted exercise time (i.e., the lowest number of completed stages). The lowest number of completed stages was three (S1, S2 and S3 for the first, second and third stages, respectively). In order to analyse changes in our fatigability indicators (i.e., SJ H , STS T , HR, RPE M , RPE R ) throughout the fatiguing protocol, a oneway repeated measures ANOVA with the fatigability indicators as dependent variable and time (i.e., Pre, S1, S2, S3, Exh, Post 2min ) as independent variable was used. If significant main effects were observed, these were followed up by post hoc Bonferroni-corrected pairwise comparisons. Partial eta square (h p 2 ) was reported as an estimate of effect size, with h p 2 ≥ 0.07 and h p 2 ≥ 0.14 used as moderate and large effects, respectively [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF]. Additionally, the minimal detectable change with 95% confidence (MDC 95 ) was calculated as SEM Â p 2 Â 1.96 and expressed as absolute and percentage of change [START_REF] Weir | Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM[END_REF]. The MDC 95 can be interpreted as the smallest change in a measure that can be considered real change beyond measurement error with 95% confidence. The alpha level for all statistical tests was set a p < 0.05.

3 Results (experiment A)

Endurance time, heart rate and rating of perceived exertion

The mean number of completed stages was 5.3 ± 2.0 (range 3-10 stages), with a mean exercise duration of 398 ± 339 s (range 110-1429 s). A significant time effect was found for HR (p < 0.001; ph 2 = 0.85) with a progressive increase throughout the fatiguing exercise (Tab. 1). The mean HR PEAK obtained during the test was 91 ± 7% of theoretical HR MAX (range 78-100%). Significant time effects were found for RPE M (p < 0.001; h p 2 = 0.93) and RPE R (p < 0.001; h p 2 = 0.78), with values recorded at Exh significantly different from values recorded at the end of the first stage (p < 0.001; Tab. 1).

Squat jump performance

A significant time effect was found for SJ H (p < 0.001; h p 2 = 0.62). SJ H progressively decreased throughout the fatiguing protocol, with mean decreases of 6.6 ± 5.4, 11.6 ± 5.7, 13.2 ± 8.2 and 22.3 ± 10.9% at S1, S2, S3 and Exh when compared to Pre, respectively (Fig. 2A). Mean of changes in SJ H was greater than the MDC 95 (2.7 cm; 9.71%) for S2, S3 and Exh. Two minutes of recovery were not sufficient for SJ H to fully recover, with a 11.5 ± 8.3% of mean decrease in performance at Post 2min being greater than the MDC 95 . However, this time frame was sufficient to induce a partial recovery, with SJ H measured at Post 2min being significantly different from Exh (p < 0.001) and nondifferent from S1 (p = 0.16), S2 (p = 1) and S3 (p = 1). Individual data analysis revealed that only 3 participants did not show a decrease in SJ H at Exh and more than half of the individuals (n = 11) did not recover their performance at Post 2min according to the MDC 95 (Fig. 2B).

Sit-to-stand performance

A significant time effect was found for STS T (p < 0.001; h p 2 = 0.73). STS T rose progressively along the fatiguing protocol, with mean increases of 2.2 ± 10.8, 7.0 ± 9.0, 12.6 ± 8.7 and 30.5 ± 13.3% at S1, S2, S3 and Exh when compared to Pre, respectively (Fig. 3A). This increase in STS T was non-significant at S1 but became significant from S2 and for all other times measurements (p < 0.01) when compared to baseline. Mean of changes in STS T was greater than the MDC 95 (0.053 s; 9.96%) only for S3 and Exh. Two minutes of recovery were not sufficient for STS T to fully recover (significant Pre vs. Post 2min differences; p < 0.001), however mean decrease in performance at Post 2min (8.4 ± 9.4%) was not greater than MDC 95 . Individual data analysis revealed that only one participant did not show an increase in STS T at Exh and less than half of the individuals (n = 8) did not recover their performance at Post 2min according to the MDC 95 (Fig. 3A). Raw data for SJ H and STS T recorded at each time measurements are displayed in Table 2.

4 Materials and methods (experiment B)

Participants

An independent sample of fifteen participants (four females; mean ± SD age: 22 ± 3 years, height: 173 ± 10 cm, weight: 66 ± 11 kg) participated in experiment B. They visited the laboratory on three occasions, i.e., one familiarization and two experimental sessions separated by 3-7 days. The familiarization and experimental sessions were exactly the same to the one of the Experiment A.

Procedures

Considering the above-presented results, no change in the fatigability protocol was necessary. The fatigability protocol used for experiment B was rigorously the same than in the experiment A (see Fig. 1). For squat jump performance assessment, we ran the My Jump 2 application on both an iPad mini ® (120 fps) and an iPad Pro ® (240 fps). This was done to compare the test-reliability of each recording mode to ensure that the fatigability outcomes could be reliably assessed with any commercial devices. Since no difference was found in SJ H between recording modes at any time point (i.e., Pre, S1, S2, S3, Exh, Post 2min ), and excellent reliability (ICC 2-k ≥ 0.99) was reported, only data obtained from the 240-fps mode are presented. For the evaluation of STS T , we used the same device as in the Experiment A (i.e., iPhone X, 240 fps; Apple Inc., California, USA) to run the Sit to Stand application.

Statistical analysis

Sample size estimation for Experiment B was based on an expected excellent reliability (rH 1 ≥ 0.9) based on the results of the Experiment A, an alpha error of 0.5, a statistical power of 80% for a two-tailed test, and a minimum acceptable reliability of moderate (rH 0 ≥ 0.5). The estimated sample size was 12 participants.

The same group isotime analysis was performed. The lowest number of completed stages was three (S1, S2 and S3). Differences in fatigability indicators (i.e., SJ H , STS T , HR, RPE M , RPE R ) between sessions were tested using two-way repeated measures ANOVAs [time (i.e., Pre, S1, S2, S3, Exh, Post 2min ) Â session (session 1, session 2)]. If significant main effects were observed, these were followed up by post hoc Bonferroni-corrected pairwise comparisons. Finally, for test-retest reliability analyses of fatigability indicators, the intra-class correlation coefficient two-way mixed effects consistency (ICC 3-1 ) was used as a measure of relative reliability and interpreted as poor (<0.5), moderate (0.50-0.74), good (0.75-0.89), and excellent (≥0.9). Absolute reliability was assessed using typical error expressed as a coefficient of variation (CV TE ) which was calculated using the online spreadsheet of Hopkins [START_REF] Hopkins | Reliability from consecutive pairs of trials. A new view of statistics[END_REF]. The alpha level for all statistical tests was set a p < 0.05.

Results (experiment B)

Endurance time, heart rate and rating of perceived exertion

The mean number of completed stages was 4.9 ± 1.4 (range 3-8 stages) and 5.0 ± 1.4 (range 3-8 stages) in sessions 1 and 2, respectively, with a mean exercise duration of 320 ± 169 (range 134-709 s) and 324 ± 175 s (range 111-704 s). For exercise duration, CV TE between both sessions was 19.4% and ICC 3-1 was 0.87 (95% CI: 0.65-0.96). A significant effect of time (p < 0.001; h p 2 = 0.85) but no time  session interaction (p = 0.08; h p 2 = 0.15) was found for HR. The mean HR PEAK obtained during the fatigability protocols was 86.3 ± 5.6 and 88.4 ± 5.6% of theoretical HR MAX for sessions 1 and 2, respectively. For this variable, CV TE was 2.0% and ICC 3-1 was 0.90 (95% CI: 0.73-0.97). While a main effect of time was found for both RPE R and RPE M (both p < 0.001, with values recorded at S2, S3 and Exh being significantly higher than S1), no time  session interaction was found for these two variables (p = 0.19 and 0.90, respectively). RPE M and RPE R recorded at Exh during the first and second sessions were 19.8 ± 0.6 and 13.8 ± 0.3, and 19.8 ± 0.4 and 13.7 ± 2.9, respectively (p > 0.05). ICCs 3-1 and CV TE were 0.79 (95% CI: 0.49-0.93) and 9.4% for RPE M and 0.5 (95% CI: 0.03-0.8) and 17.4% for RPE R , both being obtained at isotime (i.e., stage 3).

Squat jump and sit-to-stand performance

A significant main effect of time was found for SJ H (p < 0.001; h p 2 = 0.58), with each time measurements being significantly different when compared to Pre (Fig. 4A), and a significant time effect was found for STS T (p < 0.001; h p 2 = 0.72), with values recorded at S3 and Exh being significantly different from Pre (Fig. 4B). No time  session for SJ H and STS T interactions were found (p > 0.05). The reliability of SJ H and STS T at isotime (i.e., S3), exhaustion and during the recovery period is displayed in Table 3.

Discussion

Two distinct experiments were conducted to develop an easy-to-implement fatigability test and assess its testretest reliability. Progressive development of muscle fatigability was detected during the test performed in the first experiment. The second experiment reported that the test-retest reliability was excellent for both SJ H and STS T changes during and after the test.

Test ability to detect muscle fatigability

Our first objective was to evaluate if our test conception was effective to induce progressive muscle fatigability. Several studies reported a significant level of muscle fatigability when the indicator of muscle performance changed by ∼10% compared to baseline [START_REF] Bachasson | Neuromuscular fatigue and exercise capacity in fibromyalgia syndrome[END_REF][START_REF] Doyle-Baker | An Innovative Ergometer to Measure Neuromuscular Fatigue Immediately after Cycling[END_REF][START_REF] Souron | Task failure during sustained low-intensity contraction is not associated with a critical amount of central fatigue[END_REF]. When looking at individual data, SJ H was decreased by > 10% at Exh in 16/20 participants, and STS T was increased > 10% in 19/20 participants. This fatigability threshold of 10% is, however, arbitrary. Based on MDC 95 analysis, we further reported that only three and one participants did not show significant changes in SJ H and STS T at Exh, respectively. This confirms that our test induced important levels of muscle fatigability in almost all individuals. Importantly, muscle fatigability can be easily and affordably detected through two validated mobile applications highlighting its potential to be used in clinical settings where space and technology are often constrained. Monitoring the decrease in performance due to muscle fatigability during functional tasks such as rising from a chair or jumping could be more informative of its impact on daily living activities or athletic performance than monitoring muscle fatigability through isolated isometric contraction.

Test-retest reliability

Excellent reliability was found for SJ H and STS T (see Table 3). The reliability results of our test are nearly comparable with those of similar studies that developed more complex test using intermittent isometric (e.g., CV TE of 5.3% and ICC of 0.88 in Bachasson et al., 2013) and dynamic (e.g., ICC of 0.84-0.89 in [START_REF] Maffiuletti | Reliability of knee extension and flexion measurements using the Con-Trex isokinetic dynamometer[END_REF]) muscle contractions, but with the advantage to be portable and not to require any technical expertise. This is promising for the future dissemination of our test.

The total exercise duration exhibited a good relative (ICC 3-1 of 0.87) but only moderate absolute reliability (CV TE of 19.4%). These results are better than other timed wall squat tests [START_REF] Lubans | Test-retest reliability of a battery of field-based health-related fitness measures for adolescents[END_REF] but slightly lower when compared to submaximal isometric tests performed until exhaustion (Bachasson, Millet et al., 2013;[START_REF] Gruet | Long term reliability of EMG measurements in adults with cystic fibrosis[END_REF][START_REF] Gruet | Long term reliability of EMG measurements in adults with cystic fibrosis[END_REF]. Several arguments can explain this moderate absolute reliability. Endurance performance is influenced by external motivation [START_REF] Marcora | The limit to exercise tolerance in humans: mind over muscle[END_REF]. This is especially true for long-duration exercise modalities, but also for "open ended"-exhaustion tasks compared to timetrial where the endpoint is known [START_REF] Jeukendrup | A new validated endurance performance test[END_REF]. We decided not to provide strong encouragements during this test, as the frequency and intensity of the encouragements are not easy to standardize and replicate. Alternatively, we decided to use very simple sentences pronounced with a neutral tone at regular time intervals. We believe this procedure may reduce variability across testing locations, but it is still possible that the absence of uninterrupted strong encouragements close to exhaustion may have slightly increased intra-individual variability in the specific context of a reliability study, where participants performed the test several times in the absence of a highly motivational purpose. The lower reliability of endurance performance compared to indicators of fatigability further emphasizes the value of the latter. These indicators possess the advantage of being measured using valid, affordable and portable instruments as early as the first stages and throughout the exercise duration, with timepoints far to exhaustion being much less influenced by the potential motivation bias.

The test-retest reliability of perceptual measurements obtained at isotime was satisfactory for muscular (RPE M ) but not respiratory (RPE R ) efforts. Respiratory effort was only moderate at isotime (i.e., 9.9 ± 2.3). Participants likely experienced more difficulties in reporting RPE R as they were less accustomed to focus their attention on respiratory sensations which have a minor influence on the overall sensory integration process during activities of daily living. This may explain the lower test-retest reliability of RPE R compared to RPE M . We may anticipate that respiratory effort would be more important in specific populations (e.g., people with chronic respiratory disorders) that are more accustomed to these sensations in their everyday life, potentially leading to better reliability in RPE R . The reliability of RPE R can be important if our test is intended to be used in specific populations in the future, but we could argue that it is not useful to monitor this parameter when using this test in the general population, and that only RPE M should be collected and considered for further analyses.

Test feasibility

The feasibility of our test was excellent, with no adverse events reported in any participants. We did not evaluate the presence of muscle damage in the days that followed the test. No differences were found in baseline SJ H and STS T between the first and second sessions of the Experiment B, suggesting that the potential damages of the contractile apparatus have recovered before the second experimental session, i.e. at least 3 days. This has important implications as exercise-induced muscle damage may be disruptive for the engagement in subsequent physical activity, especially in frail populations, potentially decreasing the uptake of a test. The full protocol has an acceptable duration of < 30 min, and requires only little and unexpensive equipment, contrasting with classical time-consuming and less ecological muscle fatigability test (Bachasson, Millet et al., 2013;[START_REF] Maffiuletti | Reliability of knee extension and flexion measurements using the Con-Trex isokinetic dynamometer[END_REF][START_REF] Pageaux | Reliability of a novel high intensity one leg dynamic exercise protocol to measure muscle endurance[END_REF][START_REF] Pageaux | Reliability of a novel high intensity one leg dynamic exercise protocol to measure muscle endurance[END_REF]. The intermittent nature of our test together with its low exercise duration and submaximal cardiac stress further reinforce its feasibility for some specific populations who are sometimes unable to sustain prolonged periods of continuous exercise [START_REF] Beauchamp | Interval versus continuous training in individuals with chronic obstructive pulmonary disease-a systematic review[END_REF]. Sustained isometric contraction may lead to muscle ischemia due to increased intramuscular tissue pressure [START_REF] Sadamoto | Skeletal muscle tension, flow, pressure, and EMG during sustained isometric contractions in humans[END_REF], which could be viewed as a factor limiting the feasibility of isometric wall-squat test in some clinical populations. However, it should be noted that such isometric modalities have been employed in various populations (e.g. people with hypertension) for both testing and training purposes, with high level of feasibility and safety [START_REF] Baffour-Awuah | An evidence-based guide to the efficacy and safety of isometric resistance training in hypertension and clinical implications[END_REF][START_REF] Wiles | An alternative approach to isometric exercise training prescription for cardiovascular health[END_REF]. Altogether, these arguments support the potential of this test for being used in a large variety of populations (e.g. athletes, people with specific medical conditions), which remain to be confirmed by future studies investigating its reliability and feasibility in specific contexts.

Practical application

This test could be used in applied settings as a standard exercise test or a screening instrument, similar to other simple tests such as walking or sit-to-stand. These field tests have the drawback of being self-paced, with performance potentially influenced by pacing strategies. These tests may also suffer from a ceiling effect, limiting their use in the general population and even in people with chronic disease [START_REF] Radtke | The 1-min sit-to-stand test-A simple functional capacity test in cystic fibrosis?[END_REF]. Our test overcomes these limitations, and also has the advantage of cumulating two different indicators of muscle performance. This offers flexibility based on the population being studied. Additionally, this test can be used to assess the efficacy of a given acute and/or chronic intervention on muscle fatigability. The ease of our test favours multicentre studies, which is a real asset compared to most studies having muscle fatigability as a primary outcome which are often characterized by small sample size. Our test is not designed as most other fatigability tests used in laboratory settings that are generally based on a percentage of a maximal value (e.g., muscle strength). This experimental approach does not reflect ecologically valid conditions where the mobilized muscle mass and the needed muscle force are imposed by the task. Our test overcomes this limitation with the intensity of exercise reflecting an absolute quantity of muscle work, that is influenced by the participant' body mass.

Conclusion

We developed a new muscle fatigability test which demonstrates an excellent feasibility and test-retest reliability. The low resources requirements and easy-to-implement nature of this test make it a promising tool for evaluating muscle fatigability in both applied and laboratory settings. Future studies should i) confirm its feasibility and reliability in clinical populations and ii) focus on the establishment of reference values in a wide range of age groups, ensuring optimal interpretation in specific populations.

Fig. 2 .

 2 Fig. 2. Squat jump height (SJ H ) recorded before (Pre), during (S1, S2, S3), at exhaustion (Exh) and 2 min after the end of the fatiguing test (Post 2min ). Panel A displays the pooled data of the 20 participants. Panel B displays the individual data for values recorded before the protocol (Pre), at exhaustion (Exh) and after a 2-min recovery period (Post 2min ). Error bars on the panel A denote standard deviation. Horizontal dashed line represents the minimal detectable change (MDC 95 ). Data are expressed as percentage of Pre. Significantly different from Pre: ** p < 0.01, *** p < 0.001. Significantly different from Exh: $$$ p < 0.001

Fig. 3 .

 3 Fig. 3. Sit-to-stand time (STS T ) recorded before (Pre), during (S1, S2, S3), at exhaustion (Exh) and 2 min after the end of the fatiguing test (Post 2min ). Panel A displays the pooled data of the 20 participants. Panel B displays the individual data for values recorded before the protocol (Pre), at exhaustion (Exh) and after a 2-min recovery period (Post 2min ). Error bars on the panel A denote standard deviation. Horizontal dashed line represents the minimal detectable change (MDC 95 ). Data are expressed as percentage of Pre. Significantly different from Pre: * p < 0.05, *** p < 0.001. Significantly different from Exh: $$$ p < 0.001

Fig. 4 .

 4 Fig. 4. Squat jump height (SJ H ) recorded at 240 per second (panel A), and sit-to-stand time (STS T ; panel B) recorded before (Pre), during (S1, S2, S3), at exhaustion (Exh) and 2 min after (Post 2min ) the end of the fatiguing protocols performed in two separate experimental sessions. Error bars denote standard deviation. Data are expressed as percentage of Pre. Significantly different from Pre: * p < 0.05

Table 1 .

 1 Heart rate (HR), perceptions of muscular (RPE M ) and respiratory (RPE R ) efforts recorded during the first three stages of the test (S1, S2 and S3, respectively) and at exhaustion (Exh).

		S1	S2	S3	Exh
	HR (bpm)	124 ± 15	147 ± 12***	157 ± 15***	170 ± 15***
	RPE M (6-20 scale)	8.4 ± 1.7	10.8 ± 2.0***	13.4 ± 2.5***	19.1 ± 1.7***
	RPE R (6-20 scale)	6.8 ± 0.7	8.4 ± 1.6*	9.9 ± 2.3***	13.6 ± 2.8**

*

Perceptions of muscular and respiratory efforts were measured using the Borg' 6-20 scale. Significantly different from S1: * p <0.05, *** p < 0.001.

Table 2 .

 2 My Jump-and Sit to Stand-related outcomes recorded during the first three stages of the fatigability test (S1, S2 and S3), at exhaustion (Exh) and 2 min after the end of the fatigability test (Post 2min )

		Pre	S1	S2	S3	Exh	Post 2min
	SJ H (m)	0.278 ± 0.07	0.26 ± 0.07	0.248 ± 0.07	0.244 ± 0.07	0.217 ± 0.06	0.247 ± 0.07
	STS T (s)	0.536 ± 0.06	0.547 ± 0.86	0.575 ± 0.09	0.604 ± 0.09	0.699 ± 0.01	0.58 ± 0.08

Values are mean ± SD. SJ H : squat jump height; STS T : sit-to-stand time.

Table 3 .

 3 Between sessions reliability for squat jump height (SJ H ) and sit-to-stand time (STS T ) at isotime (i.e. stage 3), immediately after task failure (Exh) and after two min of recovery (Post 2min ).

		Session 1	Session 2	Change in mean (95% CI)	CV TE (95% CI)	ICC 3-1 (95% CI)
	Isotime
	SJ H (m)	0.238 ± 0.08	0.237 ± 0.09	0.0002 (-0.012 to 0.012)	6.6 (4.8 to 10.4)	0.97 (0.91 to 0.99)
	STS T (s)	0.569 ± 0.08	0.569 ± 0.08	-0.0006 (-0.007 to 0.006)	1.4 (1 to 2.2)	0.99 (0.97 to 0.99)
	Exh
	SJ H (m)	0.214 ± 0.08	0.217 ± 0.09	-0.0012 (-0.013 to 0.011)	7.4 (5.4 to 11.7)	0.96 (0.89 to 0.99)
	STS T (s)	0.65 ± 0.09	0.66 ± 0.08	-0.011 (-0.03 to 0.005)	3.2 (2.4 to 5.1)	0.93 (0.81 to 0.97)
	Post 2min
	SJ H (m)	0.253 ± 0.09	0.255 ± 0.09	-0.0019 (-0.015 to 0.012)	6.8 (5 to 10.7)	0.97 (0.91 to 0.99)
	STS T (s)	0.54 ± 0.06	0.55 ± 0.07	-0.005 (-0.01 to 0.005)	2.4 (1.7 to 3.8)	0.96 (0.88 to 0.98)
	Values are mean ± SD. 95% CI: 95% confidence interval.

R.Souron et al.: Mov Sport Sci/Sci Mot

R.Souron et al.: Mov Sport Sci/Sci Mot

Remerciements

Ce travail a bénéficié d'une subvention de la part de "Toulon, Provence, Méditerranée" (N°201445).

Le matériel de l'article a fait l'objet d'une présentation lors du congrès de l'Association des Chercheurs en Activités Physiques et Sportives (ACAPS) 2021, à Montpellier.

Funding

This work was supported by a grant from "Toulon, Provence, Méditerranée" (N°201445)

Data availability statement

Data generated or analysed during this study are provided in full within the published article.

Conflict of interests

The authors declare there are no competing interests.

Ethical approval

This study was performed in line with the principles of the Declaration of Helsinki. The CERSTAPS research ethic committee (n°IRB00012476-2021-08-02-87) approved all procedures. All participants provided written informed consent.


Tobias Marauli
email: tobias.marauli@jku.at

Haribhau Durgesh

Hubert Salunkhe

Andreas Gattringer

Damien Müller

Philippe Chablat

Wenger

Durgesh Haribhau Salunkhe
email: durgesh.salunkhe@ls2n.fr

Hubert Gattringer
email: hubert.gattringer@jku.at

Andreas Müller
email: a.mueller@jku.at

Damien Chablat
email: damien.chablat@ls2n.fr

Philippe Wenger
email: philippe.wenger@ls2n.fr

Time-Optimal Point-To-Point Motion Planning and Assembly Mode Change of Cuspidal Manipulators: Application to 3R and 6R Robots

Keywords: Optimization and Optimal Control, Kinematics, Mechanism Design

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

I. INTRODUCTION

Motivation: The ability to change the assembly mode (i.e. perform a motion that starts and ends with the same end-effector (EE) pose but with different inverse kinematic solution) without having to pass through a singularity is referred to as singularity-free assembly mode change. Robots having this ability are called cuspidal. Cuspidal robots thus possess at least one singularity free connected region in joint-space, called c-sheets [START_REF] Smith | Design of solvable 6r manipulators[END_REF] or aspects [START_REF] Borrel | A study of multiple manipulator inverse kinematic solutions with applications to trajectory planning and workspace determination[END_REF], with multiple inverse kinematics solutions (IKS). This was first reported in [START_REF] Burdick | On the Inverse Kinematics of Redundant Manipulators: Characterization of the Self-Motion Manifolds[END_REF], [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF]. Historically, the term cuspidal stems from the fact that a projection of the critical values (the forward kinematics map of the singularities) exhibits cusps, which is a sufficient condition for 3R manipulators to be cuspidal. This has triggered extensive research on cuspidal 3R serial manipulators [START_REF] Wenger | Comments on "A classification of 3R regional manipulator geometries and singularities[END_REF], [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Non-redundant Manipulators[END_REF], [START_REF] Wenger | Robots cuspidaux : théorie et applications[END_REF]. It was recently shown [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF] that for generic 3R robots the existence of cusps in workspace is sufficient and necessary to be cuspidal. Research on robot geometries satisfying the necessary cuspidality property was originally pursued for 3R regional robots [START_REF] Wenger | Comments on "A classification of 3R regional manipulator geometries and singularities[END_REF], [START_REF] Wenger | Classification of 3r positioning manipulators[END_REF]. A review to the research in cuspidal robots is given by [START_REF] Wenger | A review of cuspidal serial and parallel manipulators[END_REF], including also parallel mechanisms. Lately it was recognized that also 6R industrial robots are cuspidal, e.g. Fanuc CRX series and Kinova Jaco gen2, which increases the significance of cuspidality for practical applications. Consequently, to fully exploit this feature, cuspidality should be included in the motion planning process. Optimal trajectory planning based on a dynamic robot model has not yet been addressed for cuspidal robots. Moreover, cuspidality not only allows for singularity-free assembly mode change, it is also enables planning singularity-free point-to-point (PtP) motions. In the following, a example which further highlights the benefits of considering cuspidality in the motion planning process is discussed.

In the course of this paper n-DOF robotic manipulators with only revolute joints are considered. Denote with q(t) ∈ T n the joint coordinate vector, and with z ∈ W the end-effector pose, where W ⊆ SE (3) is the workspace. Further denote with z = f (q) the forward kinematics map, and with J the forward kinematics Jacobian. Kinematic singularities are defined by the hyper-surface det J(q) = 0, which divides the joint-space into aspects. For a conventional (non-cuspidal) wrist-partitioned robot, all IKS for a given EE pose z belong to one aspect, and are generally referred to as postures. For example, the elbow up and elbow down configurations are the two postures of a 2R robot. Similarly, wrist-partitioned robots possess 8 postures. The important point is that each IKS belongs exactly one aspect, and that the robot must pass through a singularity to change posture. Cuspidal robots, on the other hand, have multiple IKS in one aspect. Fig. 1 shows this for a 3R robot with DH parameters {a 1 , a 2 , a 3 } = {0.5, 1, 0.75}m, {d 1 , d 2 , d 3 } = {0, 0.5, 0}m, {α 1 , α 2 , α 3 } = {-π/2, π/2, 0}. The singularity loci, separating the aspects, are shown as solid lines in joint and task space. For the EE pose z a there are four IKS, denoted with q a,1 , . . . q a,4 , and for z b there are two IKS, denoted with q b,1 , q b,2 . Importantly, an aspect contains different IKS. This allows to transit from one IKS to another without leaving an aspect, i.e. not crossing a singualrity. The 3R robot can move from q a,1 to q b,1 corresponding to EE pose z a and z b , respectively. If the EE has to return to the inital z a (e.g. in a pick and place scenario), it could go to the IKS q a,2 . This leads to a nonsingular assembly mode change. Another motivation is that in this way collisions can be avoided with obstacles that were not present during the forward motion. Also from a control perspective, it may be more efficient or faster to move from q b,1 to q a,2 instead to q a,1 . Such a time optimal trajectory planning and assembly mode change is addressed in this paper.

Contribution: An algorithm for time-optimal point-topoint (PtP) trajectory planning in workspace is proposed in Sec. II, and for time-optimal assembly mode change in Sec. III, which is the special case where initial and terminal EEpose are identical. The approach solves the combinatorial problem of identifying that pair of IKS corresponding to the prescribed initial and terminal EE-pose which admits the fastest trajectory connecting them, while taking all technical limits of the robot into account. To this end, a set of pairs of candidate IKS is determined, and for each pair, a time-optimal trajectory planning problem is solved using a multiple shooting algorithm. The results when applying the method to a 3R robot (only EE positions are used) and a 6R industrial robot are shown in the course of this paper. To the best authors' knowledge, this is the first paper addressing singularity-free time-optimal trajectory planning and assembly mode change of cuspidal robots. In addition to the time-optimal motion, the algorithm enables checking whether singularity-free motions are actually feasible given the joint limits of real robots. The latter has never been taken into account when discussing cuspidal robots.

II. OPTIMAL NON-SINGULAR TRAJECTORY PLANNING

A. Problem Statement

A major benefit of serial cuspidal manipulators is the presence of regions in work space with more than two IKS that are connected by singularity-free curves in joint space. Denote with

I z = {q ∈ T n | z = f (q)} (1)
the set of IKS for given EE-pose z, which can be computed using e.g. the HUPF-algorithm [START_REF] Husty | A new and efficient algorithm for the inverse kinematics of a general serial 6r manipulator[END_REF], the approach presented in [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF] or roboter specific such as [START_REF] Gosselin | Polynomial inverse kinematic solution of the jaco robot[END_REF] for the Kinova Jaco robot. For a non-redundant robot, i.e. dim W = dim im f ≤ n, the IKS set consist of a finite n z number of IKS I z = {q 1 , . . . , q nz }. A necessary condition for two solutions to belong to the same aspect is that the determinant of the Jacobian has the same sign. This is not sufficient, however, since the sign may change multiple times along the trajectory. The set of candidate IKS for an initial solution q 0 and EEpose z is introduced as

R q0,z := {q ∈ I z | sign(det J(q 0 )) = sign(det J(q))}. (2)
Furthermore, the set of pairs of candidate IKS for a given initial EE-pose z 0 and EE-pose z is introduced as

R z0,z := {(q 0 , q) ∈ I z0 × I z | sign(det J(q 0 )) = sign(det J(q))} = {(q 0 , q) ∈ R q,z0 × I z } = {(q 0 , q) ∈ I z0 × R q0,z }. (3)
Problem 1 (Time-Optimal Non-Singular PTP Trajectory Planning in Task Space): Given initial and terminal EEpose, z 0 and z T . For all potentially valid IKS combinations (q 0 , q T ) ∈ R z0,z T , find the optimal (q * 0 , q * T ) that allows for the fastest trajectory q * (t), with q * (0) = q * 0 , q * (T ) = q * T , satisfying all kinematic and dynamic constraints of the robot.

B. Computational Algorithm

The time-optimal non-singular trajectory planning problem in workspace is to determine the optimal pair of candidate IKS that leads to the time-optimal non-singular trajectory between z 0 and z T . This corresponds to the problem of finding an optimal index i ∈ N = {1, 2, . . . , n(R z0,z )}, which lead to the the optimal pair (q 0 , q T ) * = (q 0 , q T ) i * with the minimal duration time T * between the two EEposes. This can be written as

i * = arg min i∈N {T 1 , T 2 , . . . } , (4)
where T i = T • (q 0 , q T ) i denotes the duration time of a time-optimal trajectory q * (t) connecting q 0,i and q T,i . To this end, a time-optimal PtP control problem (OCP) is solved ∀ (q 0 , q T ) ∈ R z0,z T , to provide to the input data to (4). 1) Time-Optimal Trajectory Planning: For each candidate pair (q 0 , q T ) ∈ R z0,z T , a time-OCP is solved that ensures singularity-free motion. The robot dynamics is represented by the equations of motion (EOM)

M(q)q + h(q, q) = τ (5)
where M(q) is the mass matrix and τ are the joint torques. The Coriolis, centrifugal and gravitational terms as well as the Coulomb and viscous friction are described by h(q, q). Planing non-singular trajectories requires a measure of distance to the singularity. To this end, the kinematic manipulability µ = √ det JWJ T (where W is a scaling matrix) is used [START_REF] Doty | Robot manipulability[END_REF]. While for 3R serial robots this is well-defined, appropriate scaling is necessary for spatial robots to account for disparate units of EE twists [START_REF] Angeles | Is there a characteristic length of a rigid-body displacement?[END_REF]. Denote with ε a threshold on the 'distance' to singularity. The condition on q(t) being a singularity-free trajectory is µ(q(t)) ≥ ε, sign(det J(q 0 )) = sign(det J(q(t))). [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Non-redundant Manipulators[END_REF] The OCP is to find a joint trajectory connecting the IKS q 0 and q T in minimum time, while satisfying the physical limits of the robot and ensuring smooth joint trajectories. In most applications it is sufficient to ensure continuous accelerations q. A common strategy to this end is to use the so-called flat system representation [START_REF] Khalil | Nonlinear systems third edition[END_REF], [START_REF] Markus | Flat control of industrial robotic manipulators[END_REF]. Then the joint jerk ... q is used as control input, and the non-linear system dynamics (5) serves as dynamic constraint. The integrator chain is described as ODE represented by ẋ = f (x, u), with state x T = q T , qT , qT and input u = ... q . The objective is a combination of terminal time T and a regularization term for smooth joint jerk [START_REF] Gasparetto | A technique for time-jerk optimal planning of robot trajectories[END_REF]. The OCP is then written as a non-linear optimization problem min T,x,u

T + k j T 0 u T u dt , s.t. ẋ = f (x, u), x(0) = x 0 , x(t T ) = x T , x ≤ x(t) ≤ x, u ≤ u(t) ≤ u, τ ≤ τ (t) ≤ τ , conditions (6) for t ∈ [0, t T ], (7)
with x T 0 = q T 0 , 0, 0 and x T T = q T T , 0, 0 as initial and terminal state. The limits on the joint coordinates, velocities, accelerations, jerks, and generalized torques (5) are expressed with the lower and upper bounds denoted with (), (). The parameter k j defines the trade-off between time-optimal and smooth joint jerk trajectories. Non-singular trajectories are ensured by the constraints [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Non-redundant Manipulators[END_REF]. The OCP is solved with a multiple shooting approach [START_REF] Bock | A multiple shooting algorithm for direct solution of optimal control problems*[END_REF] implemented in MATLAB using CasADi [START_REF] Andersson | CasADi -A software framework for nonlinear optimization and optimal control[END_REF] and Ipopt [START_REF] Wächter | On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming[END_REF] as solver.

2) Selection of the Optimal Pair of IKS: The OCP ( 7) is solved for all candidate pairs (q 0 , q T ) ∈ R z0,z T , and the fastest solution is selected. The time-optimal solution is assumed to be isotropic, i.e. R z0,z T is regarded as non-ordered set. That is, if (q 0 , q T ) has been treated, the pair (q T , q 0 ) obtained by swapping initial and terminal configuration is not considered. This leads to the Algorithm 1.

Algorithm 1 Time optimal non-singular trajectory planning Require: EE-pose z 0 and z T ⇒ IKS (q 0 , q T ) ∈ R z0,z T for all (q 0 , q T ) ∈ R z0,z T do Solve OCP (7) ⇒ q * i (t), T i end for With (4), i * ⇒ (q 0 , q T ) * and fastest trajectory q * (t)

Above algorithm takes care of numerical difficulties encountered by the fact that different revolute angles are equal modulo 2π. Since these joints can rotate freely within their technological limits, clockwise as well as counter clockwise rotations must be taken into account. The joint coordinates are defined by a n-torus T n . Therefore, adding ±2kπ with k ∈ N 0 does not change the IKS i.e. z = f (q) = f (q ± 2kπ). For practical applications only solutions within the interval q ∈ [-2π, 2π] have to be considered. As example consider planning singularity-free trajectories for a 3R serial manipulator (DH parameters in Sec. I) connecting the IKS in one aspect, as shown in Fig. 2a. A non-singular trajectory between q 1 and q 2 is readily found. On the other hand, planning a trajectory between q 3 and q 4 without crossing a singularity is not possible, since the OCP [START_REF] Wenger | Robots cuspidaux : théorie et applications[END_REF] does not consider the periodicity of the joint coordinates. Therefore, adding ±2π element-wise to the solution q 4 is extended to the interval [-2π, 2π] as shown in Fig. 2b. Thus, the IKS q 3 and q 4 can be connected by a non-singular trajectory, with a counter clockwise rotation of the third joint.

C. Examples

1) Non-orthogonal serial 3R: Consider a non-orthogonal 3R manipulator with DH parameters {a 1 , a 2 , a 3 } = {1.178, 0.339, 1}m, {d 1 , d 2 , d 3 } = {0, 0.32, 0.67}m, {α 1 , α 2 , α 3 } = {0, 1.55, -1.124}rad [START_REF] Paganelli | Topological analysis of singularity loci for serial and parallel manipulators[END_REF]. This synthetic robot model was chosen because the aspects are more complicated and consist of singularity encapsulated regions, as can be seen in Fig. 3a. Since no dynamic parameters and physical limits are available for this robot, the dynamic parameters and limits of the UR10 (first three links) are used to compute physically consistent results. The threshhold of the kinematic manipulability is chosen as ε = 10 -3 . Exemplary a singularity-free time optimal trajectory between an EE-pose z a in a 4-IKS region and z b in a 2-IKS region, as shown in Fig. 3b, is planned. The corresponding IKS q a ∈ I za and q b ∈ I z b are illustrated in Fig. 3a. Solving the time-optimal non-singular trajectory planning problem (4) with Algorithm 1 leads to the optimal pair (q 0 , q T ) * = (q a,1 , q b,1 ) and the fastest singularity-free time-optimal trajectory q * (t), dashed line in Fig. 3. It is worth noting, that all combinations including q a,2 lead to an infeasible OCP [START_REF] Wenger | Robots cuspidaux : théorie et applications[END_REF] because the constraint (6) can not be satisfied. This is due to the fact that, q a,2 is within an encapsulated region and therefore no singularity-free trajectory exists. Since, the joint jerk ... q is used as control input in the OCP [START_REF] Wenger | Robots cuspidaux : théorie et applications[END_REF] and the torques τ are the optimal feed-forward, their corresponding trajectories are shown in Fig. 4. Typical for time optimal-solutions at least one constraint is active. By viewing the trajectory ... q 3 the latter is true for nearly the whole trajectory. At the time steps, where ... q 3 is not limited, a joint velocity or acceleration is limited by the constraints. Since the singularities are independent of q 1 , the trajectory of the first joint is not restricted as much, which results in lower joint jerks in ... q = q (3) and motor torques τ of the 3R manipulator traversing from z a to z b . this case. Further, the resulting motor torques are smooth trajectories due to requiring smooth acceleration trajectories.

2) Serial 6R: For the serial 6R cuspidal manipulator the collaborative robot Fanuc CRX10iA/L shown in Fig. 5a is chosen, since it is known to be cuspidal. Visualizing the joint and workspace similar to the 3R cuspidal robots becomes quite challenging, since the singularities det J(q) = g(q 2 , q 3 , q 4 , q 5 ) depend on four coordinates and the workspace has 6-DOF. Therefore, a slice of the workspace with fixed orientation and y-coordinate is exemplary shown in Fig. 5b. Again, the optimal pair of IKS which lead to the time-optimal trajectory between an EE-pose z a , in a 12-IKS region, and z b , in a 8-IKS region has to be found. The joint limits of the Fanuc, are taken from the datasheet available on the Fanuc website. Since, the dynamic parameters of the robot are not known and the Fanuc roughly shares a similar structure to the UR10 the dynamic parameters of the UR10 are used again. The resulting fastest time-optimal non-singular trajectory connecting z a and z b is shown in Fig. 6 as sequence of configurations. The optimal joint jerk ... q and torque τ trajectories are shown in Fig. 11. In this case, different joint limits are limited by their corresponding constraints. Due to lack of space these plots can not be shown.

III. OPTIMAL SINGULARITY-FREE ASSEMBLY MODE CHANGE

A. Problem Statement

Changing the IKS solution without crossing singularity is a key feature of cuspidal robots. The problem of finding the corresponding path, connecting two IKS in one aspect without crossing a singularity is referred to as the connectivity problem. The latter problem can be defined in various ways. Since this paper focuses on planning time-optimal trajectories the latter problem is defined as time-OCP: Problem 2 (Time-Optimal Assembly Mode Change Trajectory Planning Problem): Given an IKS q 0 to the EEpose z. Find q * T of all q T ∈ R q0,z , which yields the fastest trajectory q * (t), with q * (0) = q 0 , q * (T ) = q * T , while satisfying all kinematic and dynamic constraints of the robot. This is a special case of the problem discussed in Sec. II. If the EE-pose z 0 and z T coincide i.e. z 0 = z T = z and the Fig. 7: Normalized joint jerk ... q = q (3) and motor torques τ of the Fanuc CRX10iA/L traversing from z a to z b . initial IKS q 0 is fixed, Problem 1 degenerated to Problem 2. Therefore, the same algorithm can be used for planning the time-optimal assembly mode change trajectory.

B. Application to Verifying of Cuspidality

Furthermore, solving the time-optimal non-singular assembly mode change problem gives raise to a different approach for checking cuspidality. It is worth noting that the presented approach can terminated without finding a connection between two IKS. Since only one EE-pose z is checked, this does not imply that the robot is not cuspidal. In order to decide if a robot is cuspidal or not, the connectivity problem has to be solved for the whole workspace until a connection of at least two IKS is found. In [START_REF] Chablat | Deciding cuspidality of manipulators through computer algebra and algorithms in real algebraic geometry[END_REF] a certificated algorithm is presented, which means that the algorithm always checks if a robot is cuspidal or not. The downside of this algorithm is, that it is hard to implement and comes with high computational costs. Therefore, the strategy discussed in this paper can be seen as practical counterpart for checking cuspidality (locally). Moreover, the connectivity problem is solved by considering joint limits in the OCP [START_REF] Wenger | Robots cuspidaux : théorie et applications[END_REF], which are normally not considered for checking cuspidality. Also, collision constraints can be incorporated easily in the verification process by extending the OCP (7) with the latter.

C. Examples

1) Non-orthogonal serial 3R: Consider again the 3R manipulator of Sec. II-C.1. In the following the singularityfree time-optimal assembly mode trajectory planning problem is solved for the EE-pose z a in a 4 IKS region, as shown in Fig. 8b. In order to show the effect of the threshold value ε on the trajectory planning, the problem is solved for two values ε 1 = 10 -3 and ε 2 = 30 -3 . Choosing q 0 = q a,1 as desired initial solution, results in the optimal terminal IKS q * T = q a,3 . The fastest non-singular timeoptimal solution is illustrated in Fig. 8 for either work-and joint-space. Comparing the optimal trajectories of the two ε 1 (a) joint-space (b) work-space Fig. 8: Singularity-free time-optimal assembly mode change of the non-orthogonal 3R manipulator from q a,1 to q a,3 .

Fig. 9: Normalized joint jerk ... q = q (3) and motor torques τ of the 3R manipulator performing the non-singular assembly mode change. and ε 2 a significant difference is already investigated, despite the small increase of ε. The minimal time increases from T 1 = 0.971 s to T 2 = 1.025 s, which is a loss of 5.6 % in time. It is worth noting that completely different trajectories can be planned, depending on the shape of the aspects and the value of ε. The corresponding control input ... q as well as the evolution of the feed-forward i.e. τ are shown in Fig. 9 for ε 1 . As can be seen, the joint jerk constraints are active most of the time.

2) Serial 6R: Finally, the connectivity problem is solved for the Fanuc CRX10iA/L. The non-singular time-optimal assembly mode change is computed for the same EE-pose z b as in Sec. II-C.2, which is in 8-IKS region. The threshhold for the singularity is chosen to be ε = 10 -3 . The resulting timeoptimal singularity-free assembly mode change is shown in Fig. 10 as sequence of configurations. These results, show that the Fanuc CRX10iA/L is even cuspidal, when joint limits are considered. For completeness, the optimal joint jerk ... q and torque τ trajectories are shown in Fig. 11.

IV. CONCLUSION

A method for time-optimal PtP trajectory planning and assembly mode change was proposed. The method selects the optimal combination among all pairs of admissible candiate ... q = q (3) and motor torques τ of the Fanuc CRX10iA/L performing the non-singular assembly mode change. IKS. To this end, time-optimal trajectories for all candidate pars are computed with a multiple shooting algorithm. An important issue that is automatically taken into account is to ensure that the non-singular trajectories (which may exist if no restrictions of the joint space is considered) are in fact executable by a real robot given the specific joint limits. Future work will incorporate collision avoidance, including time-dependent obstacles in the trajectory planning. This is when process, to further exploit the advantage of a nonsingular IKS change, is of high interest and will be addressed in future work. Moreover, the fact that IKS can vanish when following a path as addressed in [START_REF] Salunkhe | Trajectory planning issues in cuspidal commercial robots[END_REF] is a big issue, since the initial IKS can not be chosen freely. This also impacts the repeatability of closed EE-paths as required for e.g. welding processes, because the initial and terminal EE-pose are not necessarily equal. This briefly explained problems are of high research interest and will be covered also in future work.

Fig. 1 :

 1 Fig. 1: Joint space a) and spherical cut of the workspace b) of a cuspidal 3R robot.

Fig. 2 :

 2 Fig. 2: Example for considering clockwise and counter clockwise rotations in the IKS.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Optimal IKS combination and fastest singularityfree time optimal-trajectory between z a and z b for the nonorthogonal 3R manipulator.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Fanuc CRX 10iA/L and a slice of its workspace for y = 0.05 m and orientation represented as Tait-Bryan angles α o = 0.153 • , β o = 0.639 • , γ o = 0 • .

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Time-optimal singularity-free assembly mode change of the Fanuc CRX10iA/L at the EE-pose z b .


Giuseppe Coratella
email: giuseppe.coratella@unimi.it

Stefano Longo

Emiliano Cè

Fabio Esposito

Commentary on: "Viewpoint: Distinct modalities of eccentric exercise: different recipes, not the same dish"

Mail address: Via Giuseppe Colombo 71, 20133, Milano, Italy.

Phone number: 0039 3471948321 Fax number: 0039 0280315163

We read with interest the recent viewpoint on the different eccentric exercise modalities (4) and we commend the Authors for having raised this interesting topic. Since the eccentric-exercise family is variegated, characterizing each modality helps identifying their specific intrinsic stimuli. However, a between-modality comparison is not an easy matter.

Exercise volume is a key-factor in resistance training. A recent meta-analysis showed comparable exercise volumes led to similar strength gains [START_REF] Grgic | Effect of Resistance Training Frequency on Gains in Muscular Strength: A Systematic Review and Meta-Analysis[END_REF]. When performing an isoweight eccentric exercise, the volume can be manipulated a priori, given the total number of repetitions and for each repetition the range of motion, time under tension and intensity (i.e., the external load) [START_REF] Coratella | Unilateral eccentric resistance training: a direct comparison between isokinetic and dynamic constant external resistance modalities[END_REF]. When performing an isokinetic eccentric exercise, the intensity depends on the subject's ability to gradually or maximally perform each repetition. To possibly match the eccentric isoweight vs isokinetic exercise intensity, we previously calculated the maximal eccentric:concentric isokinetic ratio and then transferred such a ratio as percentage of the maximal concentric isoweight load (i.e., %1-RM) [START_REF] Coratella | Unilateral eccentric resistance training: a direct comparison between isokinetic and dynamic constant external resistance modalities[END_REF]. It resulted comparable isoweight vs isokinetic training intensity and volume led to overall similar muscle strength and architecture changes (2, 3).

To further entangle this picture, the eccentric phase of isoinertial exercise is preceded by a maximal explosive-concentric phase [START_REF] Berg | A gravity-independent ergometer to be used for resistance training in space. [Online][END_REF]. Thus, isonertial cannot be considered as a purely eccentric exercise because of the effects of the eccentric or concentric phase, or the combination of both.

Therefore, encouraging new studies to compare isoweight vs isokinetic vs isoinertial eccentric training, we recommend a thorough exercise volume calculation for appropriate matching.