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Induced Acceleration Analysis for Everyone
Art Kuo University of Michigan, USA
GCMAS Tutorial, 14 May 2013
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Muscle induced acceleration analysis
Neptune et al. (2001) J. Biomech. Zajac (2002) J. Biomech.
Why does soleus produce negative vertical power?
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Some history
Zajac & Gordon (1989) Determining muscle's force and action in multi-articular movement. Exerc Sport Sci Rev. 17: 187 – 230.
Kepple, T.M., Siegel, K.L., Stanhope, S.J. (1997) Relative
contributions of the lower extremity joint moments to forward progression and support during stance. Gait & Posture. 6: 1 – 8.
Kuo, A. D. & Zajac, F. E. (1993) A biomechanical analysis of muscle
strength as a limiting factor in standing posture. J Biomech. 26 Suppl 1:137 – 150.
Kuo, A. D. & Zajac F. E. (1993) Human standing posture: Multi-joint movement strategies based on biomechanical constraints. Prog Brain Res. 97: 349 – 358.
Kuo, A. D. (1994) A mechanical analysis of force distribution between redundant, multiple degree-of-freedom actuators in the human: implications for central nervous system control. Hum. Movem. Sci. 13: 635 – 633.
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Controversies in induced accelerations Understanding muscle coordination… (Zajac, 2002 J Biomech.) Biomechanics & muscle coordination…I & II (Zajac et al. 2002, 2003 Gait Posture.)
15 – 30% gait: Emphasize proximal muscles accelerate trunk forward, as if to imply importance despite actual deceleration. 0 – 15% gait: Proximal hip & knee muscles perform forward acceleration “perplexing” as actually trunk decelerates Letter to editor (Chen, 2004 Gait Posture) Induced acceleration contributions to locomotion dynamics are not physically well defined (Chen, 2006 Gait Posture)
Propulsive adaptation... (Riley et al. 2001 J Biomech)
20 – 65% gait: Stance-leg moments impede forward propulsion because, hip flexor decelerate hip…inconsistent with well-known anterior ground reaction force (Neptune et al. 2006 J Biomech)
“The muscular component of pedaling force is not well defined.” (Ruina & van Soest, 2002 World Congr Biomech)
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Muscle contributions to accelerations with crouch gait
Steele et al. (2012) Gait & Posture
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Muscle contributions to accelerations in running
Hamner & Delp (2013) J Biomech.
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Goals of this tutorial
Gain intuitive understanding of induced acceleration analysis
Learn about controversies Use simple models to analyze Gain ability to read literature, form opinions
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Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion
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Equations of motion simplified
( ) ( ), TmM V G R Fθ θ θ θ+ + = ⋅
( ) ( )1 1 1, TmM V M G M R Fθ θ θ θ− − −= − − + ⋅
mass matrix Coriolis & centripetal
gravitational moment arms
muscle forces
Coriolis forces are usually small (gyroscopes, rockets, divers?) Centripetal forces are often small (leg swing, riding in car) Gravity is relatively easy Muscle moments can be guessed (in sign) Inertial coupling (from mass matrix) is main issue
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Equations of motion: Accelerations
( ) ( )1 1 1, TmM V M G M R Fθ θ θ θ− − −= − − + ⋅
[ A ]
1
21 2
m
mn
mn
ff
a a a
f
1 1 2 2gravity m m n mna f a f a fθ = + ⋅ + ⋅ + + ⋅
accelerations individual muscle forces
Accelerations are linear sum of effects from individual muscles plus gravity…
So we can examine each muscle individually, right?
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Forces induce rotation and translation
Torque acts here
Induces body rotation
And COM translation
Key: Torques Forces
Rotations Translations
T
F x
�̈�
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Exercise 1: Find the accelerations
Impulsive force here
(no gravity, floating in space)
2. COM translation?
3. This point?
1. Rotation?
Sketch arrows here
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Exercise 1: Find the accelerations
Impulsive force here
2. COM translation?
3. This point?
1. Rotation?
2. Forward
3. Backward!
1. Clockwise
(one point doesn’t move) (no gravity, floating in space)
Take-aways: Forces generally induce rotation and translation Not all points on a body accelerate in same direction
SOLUTION
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Exercise 1: Find the accelerations
Impulsive force here
DETAILS
length L
�̈�
cx
1x
2x
c
Fx
M=
M, ML2/12
2
12 62LF F
MLMLθ = ⋅ =
1
62
2 2c
L F L F Fx x
M ML Mθ= − = − = −
2
64
2 2c
L F L F Fx x
M ML Mθ= + = + =
Newton’s law
Euler’s eqn torquesIθ = ∑
forcescMx = ∑
Kinematics ( )1 1/ 1/c c ca a r rα ω ω= + × + × ×
F
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Induced accelerations respect constraints
Hinge joint (constraint)
Induces body rotation
And COM translation
gravity
Acceleration from gravity will not be very high. Why?
g
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Induced accelerations vs. length & mass
This point?
rα×Because Slower
due to mass
Heavier Translational accelerations scale with distance from pivot
Translational & rotational accelerations scale inversely with mass
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Active forces also induce constraint forces
Reaction force from hinge
(equal and opposite)
(forces must respect hinge)
Gravity & reaction forces largely cancel
Free body diagram
Reaction force on hinge
Resultant: Small force & moment
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Equations of motion: Forces
grav0
T Tm
qM C R FC Cqλ
+ += − −
Reaction forces are also linear sum of individual muscles plus gravity…
constraint reaction forces constraint matrix
Examine accelerations & reaction forces as linear sum e.g., ground reactions, pedal forces, COM accelerations
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Exercise 2: Find the accelerations & reactions
2. End point?
3. Reaction forces?
1. Rotations?
Take-aways: Multi-body systems induce multiple constraint forces Not all points on a body accelerate in same direction
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Exercise 2: Find the accelerations & reactions
2. End point?
3. Reaction forces?
1. Rotations?
Take-aways: Multi-body systems induce multiple constraint forces Not all points accelerate in same direction
SOLUTION
Counter-clockwise Clockwise
Accelerates downward
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What about additional segments?
To move endpoint up, shouldn’t proximal joint always rotate up?
Regardless of how many joints?
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General accelerations and forces
COM acceleration is weighted sum of accelerations (i.e., linear in accelerations)
ground
Same with reaction forces
Any acceleration or reaction force of interest linearly related to active forces
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Induced Accelerations for Everyone
Intuitive induced accelerations
Forces induce combined rotation & translation Induced accelerations respect constraints Accelerations scale with length & mass Forces also induce reaction forces
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Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion
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Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
Example: Gait
Discussion
Hip extension non-intuitive effects Ankle (2 ways) Multivariate perspective
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Exercise: How to pedal a bike
Goal: Perform positive work on pedal Which direction for hip torque?
pedal
Zajac (2002) J. Biomech.
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Exercise: How to pedal a bike
pedal
Hip extension performs positive work
Good!
Bad!
, flexes ankle but induces upward force on pedal
SOLUTION 1
Hip extension
Paradox: Hip extension obviously desirable, but doesn’t press on pedal??
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Exercise: How to pedal a bike
pedal
Hip flexion performs negative work , extends ankle and induces downward force on pedal
Bad!
Good!
SOLUTION 2
Hip flexion
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Exercise: How to pedal a bike
pedal
Now hip extension performs positive work and induces downward pedal force
Fuse joint
But when is a joint a joint??
SOLUTION 3
Ruina & van Soest (2002) The muscular component of pedaling force is not well defined. World Congr Biomech #5414.
Chen (2006) Gait Posture
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Another paradox
But when is a joint a joint?? Ruina & van Soest (2002) The muscular component of pedaling force is not well defined. World Congr Biomech #5414.
Move pedal slightly, hip extension has reverse effect
Hip extension
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Another component: Ankle extension
Find pedal force, thigh force from ankle extension
ankle extension
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Another component: Ankle extension
pedal
Ankle extension induces downward force on pedal, flexion of thigh
Good!
Bad!
Ruina & van Soest (2002) WCB
Zajac (2002) JB
“Don’t do this”
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Combine ankle and hip
Find pedal force, thigh force from both ankle & hip extension
pedal
“Extensor synergy” Zajac (2002) J. Biomech.
SOLUTION 4
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Multivariate perspective: Combine ankle and hip
pedal
pedal down
pedal up
ankle ext
ankle flx
hip torque
upward pedal force, flexes ankle
Kuo (1994) Hum Movem Sci
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Multivariate perspective: Combine ankle and hip
pedal
pedal down
pedal up
ankle ext
ankle flx
ankle torque
hip torque
downward pedal, extends ankle
Kuo (1994) Hum Movem Sci
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Multivariate perspective: Combine ankle and hip
pedal
Hip & ankle extension produces positive work (esp. at hip) and downward pedal force
pedal down
pedal up
ankle ext
ankle flx
ankle torque
hip torque
both
Kuo (1994) Hum Movem Sci
Good!
Good!
Multivariate approach may help
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Summary: How to pedal a bike
pedal
Hip ext: Positive hip work, upward pedal force
Ankle ext: Flex hip, downward force
Pedaling (and most other tasks) are multivariate
Avoid univariate interpretations
Watch out for model dependencies
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Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
Example: Gait
Discussion
Hip extension non-intuitive effects Ankle (2 ways) Multivariate perspective
![Page 39: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/39.jpg)
Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion
![Page 40: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/40.jpg)
How to throw a ball
velocity
velocity
Should torque be exerted to perform positive work, add energy?
ball Throw this way
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Exercise: Sketch induced accelerations
Should torque be exerted to perform positive work, add energy?
Throw this way
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Exercise: Sketch induced accelerations
negative acceleration!
Should torque be exerted to perform positive work, add energy?
SOLUTION
Throw this way
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Positive work doesn’t help ball
negative acceleration!
Should torque be exerted to perform positive work, add energy?
Shoulder flexion induces wrong acceleration for throwing
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Alternative strategy: Braking
velocity
velocity
Or opposite to accelerate ball?
ball
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Braking strategy analysis
Or opposite to accelerate ball?
extra acceleration!
despite negative work
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Braking strategy a moment later
Or opposite to accelerate ball?
much less acceleration
reversed
velocity
velocity
ball
Braking works, but fleetingly
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Summary: How to throw a ball
Flex shoulder: Add energy but decelerate ball
velocity
velocity
Reduce energy but accelerate ball for an instant
Flex elbow: Add energy and accelerate ball BEWARE snapshot analysis
when designing movement
1 2
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Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
Example: Gait
Discussion
Shoulder flexion, extension both viable Beware snapshot analysis
![Page 49: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/49.jpg)
Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion
![Page 50: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/50.jpg)
A quick study of gait
What is effect of knee extension moment?
How does joint moment cause forces on neighboring segments?
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More tips on induced reaction forces
Heavy Heaviest masses attract reaction forces 1. Draw line between joints on either
side of active torque. 2. Reaction forces fall on that line.
![Page 52: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/52.jpg)
More tips on induced reaction forces
Heaviest masses attract reaction forces
Heavy
Heavy
1. Draw line between joints on either side of active torque.
2. Reaction forces fall on that line.
![Page 53: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/53.jpg)
More tips on induced reaction forces
Heaviest masses attract reaction forces
Heavy
Heavy
1. Draw line between joints on either side of active torque.
2. Reaction forces fall on that line.
![Page 54: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/54.jpg)
More tips on induced reaction forces
Heaviest masses attract reaction forces
Heavy
Heavy
1. Draw line between joints on either side of active torque.
2. Reaction forces fall on that line. Heavy
![Page 55: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/55.jpg)
More tips on induced reaction forces
Heaviest masses attract reaction forces
Heavy
Heavy
1. Draw line between joints on either side of active torque.
2. Reaction forces fall on that line. Heavy
3. Aligned joints usually induce higher forces, lower in-line accelerations
4. Non-aligned usually induce lower reactions, higher in-line accelerations
Not always (Discuss)
![Page 56: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/56.jpg)
A quick study of gait: Trunk
Find effect on trunk, whole body COM
![Page 57: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/57.jpg)
A quick study of gait: Trunk
Find effect on trunk, whole body COM
Trunk pitches forward
![Page 58: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/58.jpg)
A quick study of gait: Trunk
Find effect on trunk, whole body COM
Trunk pitches forward
COM slowed, redirected upward
Trunk & COM need not experience same accelerations
![Page 59: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/59.jpg)
A quick study of gait: Swing leg
Leading leg can induce swing phase
![Page 60: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/60.jpg)
Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion Stance knee extension slows COM, speeds trunk Can induce swing phase
![Page 61: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/61.jpg)
Induced Accelerations for Everyone
Intuitive induced accelerations
Case study: Pedaling
Case study: Throwing a ball
A quick study of gait
Discussion
![Page 62: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/62.jpg)
The connotation problem
What is the goal of gait?
Propulsion Braking
gastrocnemius vasti soleus
Support
Falling tibialis anterior hamstrings
vasti gluteus medius
Promote propulsion, support? Reduce braking, falling?
Walking requires zero net work, zero net acceleration, zero net vertical displacement
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Some issues with induced accelerations
Issues Connotation problem Model dependencies Snapshot analysis
Interpretations often not predictive
Conclusions have not been well tested
![Page 64: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/64.jpg)
The future of induced accelerations
Recognize model dependencies
Avoid connotations Study movement over
time Perform more and
better experimental tests
![Page 65: Induced Acceleration Analysis for Everyone - etouches · PDF fileInduced Acceleration Analysis for Everyone Art Kuo . University of Michigan, USA . ... SOLUTION . Exercise 1: Find](https://reader033.vdocuments.mx/reader033/viewer/2022052319/5a7884517f8b9ab8768c2912/html5/thumbnails/65.jpg)
Selected Bibliography (chronological) Zajac, F.E. & Gordon, M.A. (1989) Determining muscle's force and action in multi-articular movement. Exerc Sport Sci
Rev. 17: 187 – 230. Kuo, A. D. & Zajac, F. E. (1993) A biomechanical analysis of muscle strength as a limiting factor in standing posture. J
Biomech. 26 Suppl 1:137 – 150. Kuo, A. D. & Zajac F. E. (1993) Human standing posture: Multi-joint movement strategies based on biomechanical
constraints. Prog Brain Res. 97: 349 – 358. Kuo, A. D. (1994) A mechanical analysis of force distribution between redundant, multiple degree-of-freedom actuators
in the human: implications for central nervous system control. Hum. Movem. Sci. 13: 635 – 633. Kepple et al. (1997) Relative contributions of the lower extremity joint moments to forward progression and support
during gait. Gait Posture 6: 1 – 8. Neptune RR, Kautz SA, Zajac FE. (2000) Muscle contributions to specific biomechanical functions do not change in
forward versus backward pedaling. J Biomech. 33(2): 155 – 164. Riley, P.O., Della Croce, U., Kerrigan, D.C. (2001) Propulsive adaptation to changing gait speed. J Biomech. 34(2):
197 – 202. Zajac, F.E. (2002) Understanding muscle coordination of the human leg with dynamical simulations. J Biomech. 35(8):
1011 – 1018. Zajac, F.E, Neptune, R.R., Kautz, S.A. (2002) Biomechanics and muscle coordination of human walking. Part I:
Introduction to concepts, power transfer, dynamics and simulations. Gait Posture. 16(3): 215 – 232. Ruina. A. & van Soest, A.J. (2002) The muscular component of pedaling force is not well defined. World Congr
Biomech #5414. Zajac, F.E., Neptune, R.R., Kautz, S.A. (2003) Biomechanics and muscle coordination of human walking. Part II:
Lessons from dynamical simulations and clinical implications. Gait Posture. 17(1): 1 – 17. Chen, G. (2004) Comments on biomechanics and muscle coordination of human walking: parts I and II. Gait Posture
19: 206 – 207 Neptune RR, Kautz SA, Zajac FE. (2006) Comments on “Propulsive adaptation to changing gait speed”. J Biomech.
34(12): 1667 – 1670. Chen, G. (2006) Induced acceleration contributions to locomotion dynamics are not physically well defined. Gait
Posture. 23(1): 37 – 44.