Fatigue Model Validation

The myoFati* environments and the mjlab fatigue twins use the three-compartment controller model with a rest-recovery multiplier (3CC-r): CumulativeFatigue on CPU and TorchFatigueState on mjlab. This page checks its sustained-contraction endurance times against published data. The numbers are pinned by myosuite/tests/test_fatigue_validation.py.

Model

Each muscle’s motor units are split into active (MA), resting (MR) and fatigued (MF) fractions (Xia & Frey-Law 2008). With the commanded excitation as target load TL:

\[\dot{MA} = C(t) - F\,MA, \qquad \dot{MR} = -C(t) + R_r\,MF, \qquad \dot{MF} = F\,MA - R_r\,MF\]

The controller C(t) moves MA towards TL (limited by MR when recruiting) at the rates of MuJoCo’s muscle activation dynamics (tauact / taudeact), integrated exactly over the control step.

The rest-recovery multiplier r (Looft et al. 2018) follows Rakshit et al. (2021, Eq. 7, where it is called k): r(k, TL) = k if TL = 0 and 1 if TL > 0, so R_r = r R at rest and R under any load. Rakshit et al. define rest as a zero command. A command through the muscle sigmoid never reaches zero, so MyoSuite counts TL <= 0.01 (FATIGUE_REST_THRESHOLD) as rest. Negative commands, which occur with the [-1, 1] muscle control range of myoFatiChallengeChaseTagFBP2-v0 and which MuJoCo’s muscle dynamics clamp to zero excitation, also count as rest. Earlier versions, including the legacy v2.x model, applied the multiplier whenever MA >= TL. That boosted recovery whenever the command dropped below the active fraction, even under load.

The rule changes nothing for sustained contractions, so the validation below is unaffected. It matters for how a task commands its muscles:

  • Envs that map actions through the muscle sigmoid 1 / (1 + exp(-5 (a - 0.5))) reach rest at a <= -0.42 (excitation 0.01, the threshold); the smallest excitation is 0.00055 at a = -1 and 0.076 at a = 0. This covers 56 of the 63 registered myoFati* envs: every pose, reach, key-turn, object-hold, pen, reorient and torso env, myoFatiLegStandRandom and every challenge env except the full-body ChaseTag.

  • Envs that pass the action to the muscles directly reach rest at the lower end of their action space and recover at r R there. These are myoFatiLegWalk-v0 and myoFatiLeg{Rough,Hilly,Stair}TerrainWalk-v0 (actions in [0, 1], also on their mjlab twins), myoFatiElbowPoseTask{Fixed,Random}-v0 ([0, 1]) and myoFatiChallengeChaseTagFBP2-v0 ([-1, 1], at or below 0).

Under random excitations drawn uniformly from [0, 1] every 20 ms, of which 1% rest, the mean fatigued fraction after 120 s rises from 0.40 to 0.55 (Default row), 0.46 to 0.56 (Elbow), 0.37 to 0.48 (Knee) and 0.53 to 0.77 (Shoulder) compared with the old rule.

Parameters and sources

MUSCLE_FATIGUE_PARAMS maps functional muscle groups (myosuite/core/muscle_groups.py) to F, R and r:

  • Default: the “general” fit of Frey-Law et al. (2012, Table 1), F = 0.00970, R = 0.00091, with r = 15, which Looft et al. (2018) found optimal for the ankle, knee and elbow.

  • Default_v2_4 (use_uniform_params and the JAX models): the elbow fit of Frey-Law et al. (2012), F = 0.00912, with R reduced tenfold (0.000094) and r raised tenfold (150), as in MyoSuite <= 2.x.

  • Joint-level rows (Ankle, Elbow, Hand, Knee) and the muscle-group and sex-specific rows: Rakshit et al. (2021, Table 2), fitted to torque decline in sustained and intermittent isometric contractions. Every value matches the published table. Toe copies Ankle; Wrist and Finger copy Hand; Wrist-Flexor is the general handgrip group.

  • Shoulder: the shoulder fit of Frey-Law et al. (2012, Table 1), F = 0.01820, R = 0.00168, with r = 15, which Looft & Frey-Law (2020) found somewhat better than r = 30 for intermittent shoulder flexion.

  • Muscles without a group, namely hip and hamstring muscles of the leg models and every torso muscle, use Default.

Method

A constant target load TL (10-90% MVC) is held with the repo’s CumulativeFatigue.compute_act at a 0.02 s step, the control step of the myoFati* envs. Task failure is defined as by Frey-Law et al. (2012): the time at which “the sum of the resting and active states fall below target levels, MR + MA < TL”. For loads at least 5 points above the asymptote, the integrated endurance time (ET) agrees with the closed-form sustained-load solution ET = -ln(1 - R (1 - TL) / (F TL)) / R to within 1.2% (0.9% at a 0.01 s step, 1.9% at 0.05 s; the test allows 2%). Below the asymptote TL = R / (F + R) the load is held indefinitely (∞ in the tables).

The empirical reference is the joint-specific power model of Frey-Law & Avin (2010, Table 2), ET = b0 TL^b1 (ET in s, TL as a fraction), fitted in log-log space to 369 endurance times from 194 studies. Errors are (model - empirical) / empirical.

Results

Empirical ET (s), Frey-Law & Avin (2010), Table 2

Curve

b0

b1

10%

20%

30%

40%

50%

60%

70%

80%

90%

General

21.92

-1.98

2093

531

238

135

86.5

60.3

44.4

34.1

27.0

Ankle

34.71

-2.06

3985

956

415

229

145

99.4

72.4

55.0

43.1

Trunk

22.69

-2.27

4225

876

349

182

109

72.3

51.0

37.7

28.8

Elbow

17.98

-2.21

2916

630

257

136

83.2

55.6

39.5

29.4

22.7

Grip

33.55

-1.61

1367

448

233

147

102

76.4

59.6

48.1

39.8

Knee

19.38

-1.88

1470

399

186

109

71.3

50.6

37.9

29.5

23.6

Shoulder

14.86

-1.83

1005

283

135

79.5

52.8

37.8

28.5

22.4

18.0
Published 3CC fits (Frey-Law et al. 2012, Table 1): model ET (s) and error

Parameters

F / R (1/s)

Curve

10%

20%

30%

40%

50%

60%

70%

80%

90%

Frey-Law 2012 Ankle

0.00589 / 0.00058

Ankle

3749 (-6%)

863 (-10%)

450 (+9%)

276 (+20%)

179 (+24%)

117 (+18%)

74.4 (+3%)

43.0 (-22%)

19.0 (-56%)

Frey-Law 2012 Knee

0.01500 / 0.00149

Knee

1508 (+3%)

340 (-15%)

177 (-5%)

108 (-0%)

70.3 (-1%)

46.0 (-9%)

29.2 (-23%)

16.9 (-43%)

7.5 (-68%)

Frey-Law 2012 Trunk

0.00755 / 0.00075

Trunk

2995 (-29%)

675 (-23%)

352 (+1%)

215 (+18%)

140 (+28%)

91.4 (+26%)

58.1 (+14%)

33.6 (-11%)

14.8 (-49%)

Frey-Law 2012 Shoulder

0.01820 / 0.00168

Shoulder

1059 (+5%)

274 (-3%)

144 (+7%)

88.8 (+12%)

57.7 (+9%)

37.9 (+0%)

24.1 (-16%)

13.9 (-38%)

6.2 (-66%)

Frey-Law 2012 Elbow

0.00912 / 0.00094

Elbow

2796 (-4%)

566 (-10%)

293 (+14%)

179 (+31%)

116 (+39%)

75.8 (+36%)

48.1 (+22%)

27.8 (-6%)

12.3 (-46%)

Frey-Law 2012 Grip

0.00980 / 0.00064

Grip

1385 (+1%)

473 (+6%)

258 (+11%)

161 (+10%)

106 (+3%)

69.6 (-9%)

44.4 (-25%)

25.8 (-46%)

11.4 (-71%)

Frey-Law 2012 General

0.00970 / 0.00091

General

2045 (-2%)

517 (-3%)

272 (+14%)

167 (+24%)

108 (+25%)

71.0 (+18%)

45.1 (+2%)

26.1 (-23%)

11.5 (-57%)
MyoSuite joint-level rows: model ET (s) and error

Parameters

F / R (1/s)

Curve

10%

20%

30%

40%

50%

60%

70%

80%

90%

Default

0.00970 / 0.00091

General

2045 (-2%)

517 (-3%)

272 (+14%)

167 (+24%)

108 (+25%)

71.0 (+18%)

45.1 (+2%)

26.1 (-23%)

11.5 (-57%)

Default_v2_4

0.00912 / 0.00009

General

1036 (-51%)

448 (-16%)

259 (+9%)

166 (+23%)

110 (+28%)

73.4 (+22%)

47.1 (+6%)

27.5 (-19%)

12.2 (-55%)

Elbow

0.01086 / 0.00225

Elbow

∞

785 (+25%)

294 (+14%)

166 (+21%)

103 (+24%)

66.1 (+19%)

41.4 (+5%)

23.7 (-20%)

10.4 (-54%)

Hand

0.01227 / 0.00134

Grip

3047 (+123%)

429 (-4%)

220 (-6%)

134 (-9%)

86.4 (-16%)

56.5 (-26%)

35.8 (-40%)

20.7 (-57%)

9.1 (-77%)

Wrist

0.01227 / 0.00134

Grip

3047 (+123%)

429 (-4%)

220 (-6%)

134 (-9%)

86.4 (-16%)

56.5 (-26%)

35.8 (-40%)

20.7 (-57%)

9.1 (-77%)

Finger

0.01227 / 0.00134

Grip

3047 (+123%)

429 (-4%)

220 (-6%)

134 (-9%)

86.4 (-16%)

56.5 (-26%)

35.8 (-40%)

20.7 (-57%)

9.1 (-77%)

Wrist-Flexor

0.01235 / 0.00135

Grip

3066 (+124%)

426 (-5%)

218 (-6%)

133 (-10%)

85.8 (-16%)

56.1 (-27%)

35.6 (-40%)

20.6 (-57%)

9.1 (-77%)

Knee

0.00825 / 0.00076

Knee

2326 (+58%)

605 (+51%)

319 (+71%)

196 (+80%)

127 (+78%)

83.4 (+65%)

53.0 (+40%)

30.7 (+4%)

13.6 (-43%)

Knee-Extensor

0.00825 / 0.00076

Knee

2326 (+58%)

605 (+51%)

319 (+71%)

196 (+80%)

127 (+78%)

83.4 (+65%)

53.0 (+40%)

30.7 (+4%)

13.6 (-43%)

Ankle

0.01485 / 0.00333

Ankle

∞

683 (-28%)

223 (-46%)

123 (-46%)

76.3 (-47%)

48.7 (-51%)

30.4 (-58%)

17.4 (-68%)

7.6 (-82%)

Toe

0.01485 / 0.00333

Ankle

∞

683 (-28%)

223 (-46%)

123 (-46%)

76.3 (-47%)

48.7 (-51%)

30.4 (-58%)

17.4 (-68%)

7.6 (-82%)

Shoulder

0.01820 / 0.00168

Shoulder

1059 (+5%)

274 (-3%)

144 (+7%)

88.8 (+12%)

57.7 (+9%)

37.9 (+0%)

24.1 (-16%)

13.9 (-38%)

6.2 (-66%)

Default (torso muscles)

0.00970 / 0.00091

Trunk

2045 (-52%)

517 (-41%)

272 (-22%)

167 (-8%)

108 (-1%)

71.0 (-2%)

45.1 (-11%)

26.1 (-31%)

11.5 (-60%)
MyoSuite muscle-group and sex-specific rows (Rakshit et al. 2021) against the joint curve

Parameters

F / R (1/s)

Curve

10%

20%

30%

40%

50%

60%

70%

80%

90%

Ankle-Dorsiflexor-F

0.00746 / 0.00081

Ankle

4677 (+17%)

704 (-26%)

361 (-13%)

220 (-4%)

142 (-2%)

92.8 (-7%)

58.9 (-19%)

34.0 (-38%)

15.0 (-65%)

Ankle-Dorsiflexor-M

0.00725 / 0.00096

Ankle

∞

786 (-18%)

385 (-7%)

231 (+1%)

148 (+2%)

96.3 (-3%)

60.9 (-16%)

35.1 (-36%)

15.5 (-64%)

Ankle-Dorsiflexor

0.00828 / 0.00204

Ankle

∞

2082 (+118%)

419 (+1%)

226 (-1%)

139 (-4%)

88.0 (-11%)

54.7 (-24%)

31.2 (-43%)

13.6 (-68%)

Ankle-Plantarflexor-F

0.00702 / 0.00098

Ankle

∞

834 (-13%)

402 (-3%)

240 (+5%)

153 (+6%)

99.7 (+0%)

63.0 (-13%)

36.3 (-34%)

16.0 (-63%)

Ankle-Plantarflexor-M

0.00683 / 0.00093

Ankle

∞

846 (-11%)

411 (-1%)

246 (+7%)

157 (+9%)

102 (+3%)

64.7 (-11%)

37.3 (-32%)

16.4 (-62%)

Ankle-Plantarflexor

0.00695 / 0.00096

Ankle

∞

838 (-12%)

405 (-2%)

242 (+6%)

155 (+7%)

101 (+1%)

63.6 (-12%)

36.6 (-33%)

16.1 (-63%)

Elbow-Extensor-F

0.01874 / 0.00206

Elbow

2222 (-24%)

281 (-55%)

144 (-44%)

87.5 (-36%)

56.6 (-32%)

37.0 (-33%)

23.5 (-41%)

13.6 (-54%)

6.0 (-74%)

Elbow-Extensor-M

0.01269 / 0.00085

Elbow

1087 (-63%)

367 (-42%)

200 (-22%)

125 (-9%)

81.6 (-2%)

53.8 (-3%)

34.3 (-13%)

19.9 (-32%)

8.8 (-61%)

Elbow-Extensor

0.01559 / 0.00125

Elbow

1024 (-65%)

310 (-51%)

166 (-36%)

103 (-25%)

66.9 (-20%)

44.0 (-21%)

28.0 (-29%)

16.2 (-45%)

7.2 (-68%)

Elbow-Flexor-F

0.00965 / 0.00197

Elbow

∞

861 (+37%)

328 (+28%)

186 (+36%)

116 (+39%)

74.3 (+34%)

46.5 (+18%)

26.6 (-10%)

11.7 (-49%)

Elbow-Flexor-M

0.01302 / 0.00188

Elbow

∞

459 (-27%)

219 (-15%)

130 (-5%)

83.0 (-0%)

53.9 (-3%)

34.0 (-14%)

19.6 (-33%)

8.6 (-62%)

Elbow-Flexor

0.01703 / 0.00494

Elbow

∞

∞

229 (-11%)

116 (-15%)

69.4 (-17%)

43.6 (-22%)

26.9 (-32%)

15.3 (-48%)

6.7 (-71%)

Hand-Adductor-Pollicis-F

0.00476 / 0.00093

Grip

∞

1636 (+265%)

655 (+181%)

373 (+154%)

234 (+128%)

150 (+97%)

94.1 (+58%)

53.9 (+12%)

23.6 (-41%)

Hand-Adductor-Pollicis-M

0.00586 / 0.00202

Grip

∞

∞

808 (+247%)

360 (+146%)

209 (+104%)

129 (+69%)

79.2 (+33%)

44.7 (-7%)

19.4 (-51%)

Hand-Adductor-Pollicis

0.00558 / 0.00283

Grip

∞

∞

∞

506 (+245%)

250 (+144%)

146 (+91%)

86.6 (+45%)

47.9 (-0%)

20.5 (-48%)

Hand-First-Dorsal-Interossei-F

0.03999 / 0.03983

Grip

∞

∞

∞

∞

146 (+42%)

27.5 (-64%)

14.0 (-76%)

7.2 (-85%)

3.0 (-93%)

Hand-First-Dorsal-Interossei-M

0.01637 / 0.00360

Grip

∞

589 (+32%)

200 (-14%)

111 (-24%)

69.0 (-33%)

44.1 (-42%)

27.5 (-54%)

15.7 (-67%)

6.9 (-83%)

Hand-First-Dorsal-Interossei

0.02686 / 0.00656

Grip

∞

578 (+29%)

129 (-45%)

69.6 (-53%)

42.7 (-58%)

27.1 (-64%)

16.9 (-72%)

9.6 (-80%)

4.2 (-89%)

Knee-Extensor-F

0.01407 / 0.00185

Knee

∞

404 (+1%)

198 (+6%)

119 (+9%)

76.3 (+7%)

49.6 (-2%)

31.4 (-17%)

18.1 (-39%)

8.0 (-66%)

Knee-Extensor-M

0.01420 / 0.00153

Knee

2292 (+56%)

369 (-8%)

189 (+2%)

115 (+6%)

74.6 (+5%)

48.8 (-4%)

30.9 (-18%)

17.9 (-39%)

7.9 (-67%)

Knee-Extensor

0.00825 / 0.00076

Knee

2326 (+58%)

605 (+51%)

319 (+71%)

196 (+80%)

127 (+78%)

83.4 (+65%)

53.0 (+40%)

30.7 (+4%)

13.6 (-43%)

Findings

  • The implementation reproduces the published model. With the Frey-Law et al. (2012) parameters, the endurance times lie within 0.54-1.39 times the empirical curves at 20-80% MVC for every joint. Like the published model, 3CC with a constant F underestimates endurance near maximal loads (46-71% short at 90% MVC, where ET ~ (1 - TL) / (F TL)).

  • Tolerance. The regression test requires 20-80% MVC endurance times within a factor of 2 of the joint curve. This is the band the published fits meet, with their worst case being grip at 80% MVC (0.54x). The 90% load is reported here but not pinned.

  • Default, Default_v2_4, Elbow, Knee / Knee-Extensor and Shoulder are inside the band. Default_v2_4 falls 51% short at 10% MVC, because its tenfold smaller R lowers the asymptote to 1%. Knee holds 40-80% longer than the knee curve at 20-70% MVC.

  • Shoulder. With the Frey-Law et al. (2012) shoulder fit, the Shoulder row is within 0.62-1.12 times the shoulder curve at 20-80% MVC. The deltoid, rotator-cuff, pectoral and latissimus muscles of the arm, bimanual, relocate, table-tennis and full-body models therefore fatigue as the data rank the shoulder: the most fatigable joint.

  • Ankle and Toe (Rakshit et al. 2021 ankle joint, F = 0.01485, 2.5 times the Frey-Law et al. 2012 ankle F) fall 46-58% short of the ankle curve at 30-70% MVC and 68% short at 80%. That curve is the most fatigue-resistant one in Frey-Law & Avin (2010).

  • Hand, Wrist, Finger and Wrist-Flexor are within 16% of the grip curve at 20-50% MVC and fall 57% short at 80% MVC, where the published grip fit falls 46% short.

  • Torso muscles use Default. Against the trunk curve it is within the band but 41% short at 20% MVC. The Frey-Law et al. (2012) trunk fit would hold longer. Hip and hamstring muscles also use Default. Frey-Law & Avin (2010) have no hip curve, and this is what the general fit is for.

  • The muscle-group and sex-specific rows have no matching empirical curves, so they are compared with their joint curve for information only. Single muscles deviate most: the adductor pollicis rows hold 2.0-3.5 times longer than handgrip at 30-50% MVC (some loads indefinitely), and the female first dorsal interosseous row (F ≈ R) holds any load below 50% MVC indefinitely but fails within 28 s at 60% MVC.

These deviations are reported, not retuned: the rows match their source. The regression test marks the joint-level ones (Ankle, Toe, Hand, Wrist, Finger, Wrist-Flexor) as strict expected failures, so any change to these rows shows up.

References