Download - Manish Kurse PhD research slides
Inference of computational models of tendon networks via
sparse experimentation
Manish Umesh KurseApr 11, 2012
1
Brain-Body Dynamics LaboratoryPh.D. committee: Dr. Francisco J. Valero-Cuevas, Dr. Hod Lipson,
Dr. Gerald E. Loeb, Dr. Eva Kanso
2
MSMS: Davoodi et al., 2007
Measurement of internal states Injury, deformity, surgery
h4p://www.ispub.com/
Ergonomics, prosthetic design, etc.
h4p://www.anybodytech.com
Inputs Outputs
tension in each cord, which was fed back to the motor so that a desired amount oftension could be maintained on each tendon. The fingertip was rigidly attached to6 DOF load cell (JR3, Woodland, CA).
We examined 5 different postures in 3 specimens, and 3 different postures inthe final specimen. Each posture was neutral in add-abduction. The examinedpostures were chosen to cover the workspace and simulate those found ineveryday tasks. After positioning the finger in a specific posture, we determinedthe action matrix for the finger: we applied 128 combinations of tendon tensionsrepresenting all possible combinations of 0 and 10 N across the seven tendons, andheld each combination for 3 s. The fingertip forces resulting from each coordina-tion pattern was determined by averaging the fingertip load cell readings acrossthe hold period. Linear regression was performed on each fingertip forcecomponent using the tendon tensions as factors. In this way, the fingertip forcevector generated by 1 N of tendon tension was determined for all muscles. Theforce vector generated by each muscle was scaled by an estimate for maximummuscle force (Valero-Cuevas et al., 2000) to generate the columns of the actionmatrix for each specimen and posture examined.
2.2. Action matrix for human leg model
We also studied the necessity of muscles for mechanical output for asimplified, but plausible, sagittal plane model of the human leg (hip, knee, andankle joints). The model contained 14 muscles/muscle groups (Kuo and Zajac,1993) (muscle/muscle group abbreviation in parentheses): medial and lateralgastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peroneusbrevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendenosis/biceps femoris long head (hamstring), biceps femoris short head (bfsh), rectusfemoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adductor
longus (addlong), iliacus (iliacus), tensor facia lata (tensfl), gluteus maximus(glmax). Moment arms for hip flexion, knee flexion and ankle dorsiflexion for allof these muscles were obtained from a computer model of the lower limb (Arnoldet al., 2010). When necessary, multiple muscles in were combined into the singlemuscle groups. We derived a 3!3 square Jacobian mapping changes in the hip,knee, and ankle angle to the foot position in the plane (2 components) and theorientation of the foot in space. This Jacobian matrix, inverted and transposed, wascombined with the moment arms and maximal muscle forces to form the actionmatrix mapping muscle activation to forces and torques at the foot (Valero-Cuevas, 2005b), although our analysis of muscle redundancy was only performedwith respect to the endpoint forces.
2.3. Analyzing the action matrix to determine muscle necessity
We used the action matrix to determine whether muscles are necessaryfor a given desired output force using standard tools in computational geometry.The muscle redundancy problem can be expressed as a set of linear inequalities(Chao and An, 1978; Spoor, 1983). These inequality constraints enforce that theactivation for each muscle lie between 0 and 1, and that the actual output force isequal to the desired force. The inequality constraints define a region in muscleactivation space called the task-specific activation set: any point inside that set willproduce the desired output force (Kuo and Zajac, 1993; Valero-Cuevas, 2005b;Valero-Cuevas et al., 2000, 1998). We computed the vertices defining the task-specific activation set using a vertex enumeration algorithm (Avis and Fukuda,1992). We then found the task-specific activation ranges to achieve the desiredoutput force for each muscle by projecting all vertices onto the seven musclecoordinate axes to determine the minimum and maximum task-specific activations.While previous studies have used similar experimental (Valero-Cuevas et al., 2000)
Muscles cooperateto exert force
Feasible force set,one target force vector
Solutions in muscle activation space,task-specific activation set
Fy
Fx
Target x-force
Target y-force
Feasible force set
Target force vector
2
12
3
LIMB
31
Fy
Fx
Task-specificactivation ranges
0
1
Muscle
1 2 3
Neces
sary
Neces
sary
Redun
dant
Muscle 3
Muscle 2
Muscle 1
Target x-force sol’nsTarget y-force sol’ns
Task-specific activation set
Fig. 1. Three muscle ‘‘schematic model’’ conceptually illustrates the necessity of muscles. (a) Muscles can be functionally visualized as force vectors at the endpoint. (b) Aregion of force space, the feasible force set, is achievable given this musculature. A particular target force vector can be decomposed into a target x-force and a targety-force. (c) The valid coordination patterns for the x and y targets can also be viewed in muscle activation space as planes; the portion of the intersection of these twoplanes that is inside the unit cube is the task-specific activation set. Any point on the task-specific activation set will generate the same target force vector. (d) The task-specific activation set can be projected onto the muscle coordinate axes, revealing the minimum and maximum activation in each muscle for the given applied force vector.The task-specific activation ranges can be constructed for each muscle, and reveal which muscles are necessary and which are redundant for a given target force vector.
J.J. Kutch, F.J. Valero-Cuevas / Journal of Biomechanics 44 (2011) 1264–1270 1265
Kutch and Valero-Cuevas, 2011
Computational modeling of musculoskeletal systems
3
Modeling : Structure based on observation + experimental measurement of some parameters.
Drawbacks: • Not possible to measure all parameters. • Not validated with experimental input-‐output data. • Structure assumed need not be funcEonally accurate representaEon.
R
System
✓s
R
Structure assumed,
parameters fit.
1 R2
Infer structure and parameters from input-
output data
(✓)
4
Develop computational methods to simultaneously infer structure and parameter values of functionally accurate
models of musculoskeletal systems directly from experimental input-output data.
Objective
Inputs Outputs
tension in each cord, which was fed back to the motor so that a desired amount oftension could be maintained on each tendon. The fingertip was rigidly attached to6 DOF load cell (JR3, Woodland, CA).
We examined 5 different postures in 3 specimens, and 3 different postures inthe final specimen. Each posture was neutral in add-abduction. The examinedpostures were chosen to cover the workspace and simulate those found ineveryday tasks. After positioning the finger in a specific posture, we determinedthe action matrix for the finger: we applied 128 combinations of tendon tensionsrepresenting all possible combinations of 0 and 10 N across the seven tendons, andheld each combination for 3 s. The fingertip forces resulting from each coordina-tion pattern was determined by averaging the fingertip load cell readings acrossthe hold period. Linear regression was performed on each fingertip forcecomponent using the tendon tensions as factors. In this way, the fingertip forcevector generated by 1 N of tendon tension was determined for all muscles. Theforce vector generated by each muscle was scaled by an estimate for maximummuscle force (Valero-Cuevas et al., 2000) to generate the columns of the actionmatrix for each specimen and posture examined.
2.2. Action matrix for human leg model
We also studied the necessity of muscles for mechanical output for asimplified, but plausible, sagittal plane model of the human leg (hip, knee, andankle joints). The model contained 14 muscles/muscle groups (Kuo and Zajac,1993) (muscle/muscle group abbreviation in parentheses): medial and lateralgastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peroneusbrevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendenosis/biceps femoris long head (hamstring), biceps femoris short head (bfsh), rectusfemoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adductor
longus (addlong), iliacus (iliacus), tensor facia lata (tensfl), gluteus maximus(glmax). Moment arms for hip flexion, knee flexion and ankle dorsiflexion for allof these muscles were obtained from a computer model of the lower limb (Arnoldet al., 2010). When necessary, multiple muscles in were combined into the singlemuscle groups. We derived a 3!3 square Jacobian mapping changes in the hip,knee, and ankle angle to the foot position in the plane (2 components) and theorientation of the foot in space. This Jacobian matrix, inverted and transposed, wascombined with the moment arms and maximal muscle forces to form the actionmatrix mapping muscle activation to forces and torques at the foot (Valero-Cuevas, 2005b), although our analysis of muscle redundancy was only performedwith respect to the endpoint forces.
2.3. Analyzing the action matrix to determine muscle necessity
We used the action matrix to determine whether muscles are necessaryfor a given desired output force using standard tools in computational geometry.The muscle redundancy problem can be expressed as a set of linear inequalities(Chao and An, 1978; Spoor, 1983). These inequality constraints enforce that theactivation for each muscle lie between 0 and 1, and that the actual output force isequal to the desired force. The inequality constraints define a region in muscleactivation space called the task-specific activation set: any point inside that set willproduce the desired output force (Kuo and Zajac, 1993; Valero-Cuevas, 2005b;Valero-Cuevas et al., 2000, 1998). We computed the vertices defining the task-specific activation set using a vertex enumeration algorithm (Avis and Fukuda,1992). We then found the task-specific activation ranges to achieve the desiredoutput force for each muscle by projecting all vertices onto the seven musclecoordinate axes to determine the minimum and maximum task-specific activations.While previous studies have used similar experimental (Valero-Cuevas et al., 2000)
Muscles cooperateto exert force
Feasible force set,one target force vector
Solutions in muscle activation space,task-specific activation set
Fy
Fx
Target x-force
Target y-force
Feasible force set
Target force vector
2
12
3
LIMB
31
Fy
Fx
Task-specificactivation ranges
0
1
Muscle
1 2 3
Neces
sary
Neces
sary
Redun
dant
Muscle 3
Muscle 2
Muscle 1
Target x-force sol’nsTarget y-force sol’ns
Task-specific activation set
Fig. 1. Three muscle ‘‘schematic model’’ conceptually illustrates the necessity of muscles. (a) Muscles can be functionally visualized as force vectors at the endpoint. (b) Aregion of force space, the feasible force set, is achievable given this musculature. A particular target force vector can be decomposed into a target x-force and a targety-force. (c) The valid coordination patterns for the x and y targets can also be viewed in muscle activation space as planes; the portion of the intersection of these twoplanes that is inside the unit cube is the task-specific activation set. Any point on the task-specific activation set will generate the same target force vector. (d) The task-specific activation set can be projected onto the muscle coordinate axes, revealing the minimum and maximum activation in each muscle for the given applied force vector.The task-specific activation ranges can be constructed for each muscle, and reveal which muscles are necessary and which are redundant for a given target force vector.
J.J. Kutch, F.J. Valero-Cuevas / Journal of Biomechanics 44 (2011) 1264–1270 1265
Kutch and Valero-Cuevas, 2011
Tendon networks of the fingers
5
Lateral bands
Central slip
Terminal slip
Retinacular ligament
Sagittal band
Transverse fibers
Clavero et al. (2003). “Extensor Mechanism of the Fingers: MR Imaging-Anatomic Correlation”, Radiographics
Netter, F. Atlas of Human Anatomy, 3rd edition, pp 447-453
Computational models
6
Analytical models Anatomy-based models
s2 = ✓21 + 9e✓2
Significance of research
7
http://www.myefficientassistant.com/
http://www.punchstock.com/
‘Plant’
‘Controller’
Motor control
http://www.ispub.com/
Clinical
Wilkinson et al. 2006
Robotics and prosthetics
?
Computational models
Experimentation Mathematical modeling
Inference algorithms
8
Dissertation outline
Experimental actuation of a cadaveric hand.
2
3
4
5
6
New inference approach to
learn functions of tendon
routing.
Application to the human
index finger.
Experimental validation of an existing
model.
Tendon network simulator and
sensitivity analysis.
Inference of anatomy-based models
from experimental data.
1
Analytical models Anatomy-based models
ASME SBC ’10 & IEEE TBME ’12
ASB ’11
CSB ’12
ASME SBC ’09
9
Cadaver finger control
10
Load cells
Strings to the tendons
Motion capture markers
6 DOF Load cell
DC motorsPositon encoders
1 2 3 4 5 6
1
Finger tapping
11
1 2 3 4 5 6
Slow finger movements
12
All intactRadial nerve palsyMedian nerve palsyUlnar nerve palsy
fmIteratively,
1 2 3 4 5 6
Finger equilibrium
13
⌧ = RFm
0
Neutral equilibrium
Stable equilibrium
1 2 3 4 5 6
14
NeutralStableFDPFDSEI
EDCLUMFDIFPI
1 20
1
2
3
4
Dis
tanc
e fro
m n
ull s
pace
(N)
1 20
2
4
6
8
Coordination pattern
Tend
on te
nsio
ns
iDistance from null space
1 2 3 4 5 6
Conclusions
• Spring-based (muscle-like) control effective to control movement.
• Simple tap requires a coordinated set of tendon excursions.
• Neutral equilibrium in specific postures and tendon tensions.
15
1 2 3 4 5 6
Computational models
16
Analytical models Anatomy-based models
s2 = ✓21 + 9e✓2
1 2 3 4 5 6
Inference of analytical functions
17
Co-authors: Dr. Hod Lipson, Dr. Francisco Valero-Cuevas
1 2 3 4 5 6
2
• Analytical functions for tendon excursionss = f(✓)
Deshpande et al. 2009
• State of the art : Polynomial regression
s s
‘Controller’
‘Plant’• Why? R(✓) =@s
@✓⌧ (✓) = R(✓)Fm
R
✓
R 1R
2(✓)
• Can we simultaneously learn form and parameter values from data?
• Compare accuracy with polynomial regression.
Specific Aims
18
1 2 3 4 5 6
s = f(✓)
19
Schmidt and Lipson, 2009
Koza 1992
Symbolic regression
1 2 3 4 5 6
Robotic tendon driven system
20
s1
s2
s3
2
1
3
Position encoders
Motors keeping tendons taut
Load cells
Motion capture markers
Motion capture camera
1 2 3 4 5 6
Landsmeer model I
Landsmeer model II
Landsmeer model III
s = 3.6sin(0.5θ)
s = 0.6θ + 3.2(1 − θ/2tan(θ/2)
)
s = 1.8θ
s = f(✓1, ✓2, ✓3)
21
Schmidt and Lipson, 2009
Polynomial regressionKoza 1992
Symbolic regression vs.
LinearQuadratic
CubicQuartic
1 2 3 4 5 6
Comparing symbolic and polynomial regressions
22
2
5
10
20
2
5
10
20
2
5
10
20
Tendon 1
2
5
10
20
Tendon 2
Tendon 3
2
5
10
20
n/256
n/16
n/64
n n
2n
4n
8n
16n
32n
64n
128n
256
X
X
X
XX
n/256
n/16
n/64
2
5
10
20
n n
2n
4n
8n
16n
32n
64n
128n
256
n/256
n/16
n/64
n/256
n/16
n/64
n/256
n/16
n/64
n/256
n/16
n/64
Symbolic
Quartic
LinearQuadraticCubic
Dataset size (n =1688) Dataset size (n =1688)
X
X
X
Cross-validation Extrapolation
RM
S er
ror (
%)
RM
S er
ror (
%)
X Error for all sizes > 5%Min training set size < n/256
2
5
10
20
Tendon 1
Tendon 2
2
5
10
20
25%
75%
125%
25%
75%
125%
25%
75%
125%
2
5
10
20Tendon 3
RM
S er
ror (
%)
Extrapolation by volume (%)
0 25 15075 10050 125 Symbolic
Quartic
LinearQuadraticCubic
X
X
X All extrapolation errors > 5%Achievable extrapolation > 150%
Fewer training data points required More extrapolatable
2
23
Extrapolation by volume (%)
0
25
50
75
100
125
150
>150
n n
2n
4n
8n
16n
32n
64n
128
Training set size (n =1688)
Extra
pola
tion
by v
olum
e (%
)
SymbolicQuartic
LinearQuadratic
Cubic
Comparing symbolic and polynomial regressions
Fewer training data points required
More extrapolatable
Kurse et al. 2012 (in press)
1 2 3 4 5 6
24
Landsmeer model I
Landsmeer model II
Landsmeer model III
s = 3.6sin(0.5θ)
s = 0.6θ + 3.2(1 − θ/2tan(θ/2)
)
s = 1.8θ
Simulated musculoskeletal systems
Landsmeer comb. Expressions
I, I, I
Target 1.8✓1 + 1.8✓2 + 1.8✓3
Evolved 1.8✓1 + 1.8✓2 + 1.8✓3
I, II, III
Target
1.8✓1 + 3.6sin(0.5✓2) + 0.6✓3 �(1.6✓3)/tan(0.5✓3) + 3.2
Evolved
1.8✓1 + 3.61sin(0.5✓2) + 1.54✓3 �0.778sin(✓3)
II, II, I
Target 3.6sin(0.5✓1)+3.6sin(0.5✓2)+1.8✓3
Evolved 3.6sin(0.5✓1)+3.6sin(0.5✓2)+1.8✓3
Table 1: Target and inferred expressions with training, cross-validation and extrapolation RMS errors(%)
for some combinations of Landsmeer’s models I, II, III
1
1 2 3 4 5 6
Error vs. number of parameters
25
RM
S er
ror (
%)
Cro
ss-v
alid
atio
nEx
trapo
latio
n
SymbolicQuartic
LinearQuadratic
Cubic
Experimental dataWith no noise
Number of parameters
Simulated dataWith noise added
1
2
3
0 20 401
2
3
5
.0001
.01
1
0 20 40
.0001
.01
1
0 20 401
2
5
10
1
2
5
10
1 2 3 4 5 6
Conclusions
• Symbolic regression outperforms polynomial regression
• Number of training data points
• Extrapolatability
• Robustness to noise
• Number of parameters
• Insight on physics
26
1 2 3 4 5 6
Kurse et al. 2012 (in press)
27
Novel method of inference of
analytical functions from data
Application to the human finger
Schmidt and Lipson, 2009
s1 = f(✓1, ✓2, ✓3, ✓4)
1 2 3 4 5 6
Analytical functions: Index finger
28
Constant moment arm
(Linear)
Polynomial regressions
Landsmeer based models
Landsmeer model I
Landsmeer model II
Landsmeer model III
s = 3.6sin(0.5θ)
s = 0.6θ + 3.2(1 − θ/2tan(θ/2)
)
s = 1.8θ
Landsmeer, 1961, Brook 1995
3
Co-authors: Dr. Hod Lipson, Dr. Francisco Valero-Cuevas
1 2 3 4 5 6
Eg. An et al. 1983, Valero-Cuevas et al. 1998
Eg. Franko et al. 2011
Eg. Brook et al. 1995
Specific aims
• Infer analytical functions for the seven tendons of the index finger.
• Compare against polynomial regression and Landsmeer based models.
29
1 2 3 4 5 6
Cadaver experimental setup
30
MoEon capture
Load cells
Position encodersServo motors
Markers
1 2 3 4 5 6
s1 = f(✓1, ✓2, ✓3, ✓4)
s7 = f(✓1, ✓2, ✓3, ✓4)
...
31
Schmidt and Lipson, 2009
Polynomial and LandsmeerKoza 1992
Symbolic regression vs.
LinearQuadratic
CubicQuartic
Landsmeer
1 2 3 4 5 6
Across trials
32
FDP FDS EIP EDC LUM FDI FPI2
5
10
20
SymbolicLandsmeer
Quartic
LinearQuadratic
Cubic
2 10 502
5
10
20
50FDP
2 10 502
5
10
20
50FDS
2 10 502
5
10
20
50EIP
2 10 502
5
10
20
50EDC
2 10 502
5
10
20
50LUM
2 10 502
5
10
20
50FDI
2 10 502
5
10
20
50FPI
Tendon Number of parameters
Nor
mal
ized
RM
S er
ror (
%)
1 2 3 4 5 6
Across hands
33
FDP FDS EIP EDC LUM FDI FPI2
5
10
20
2
5
10
20
50
Tendon
Nor
mal
ized
RM
S er
ror (
%)
SymbolicLandsmeer
Quartic
LinearQuadratic
Cubic
1 2 3 4 5 6
FDP FDS EIP EDC LUM FDI FPI2
5
10
20
SymbolicLandsmeer
Quartic
LinearQuadratic
Cubic
10
20
50
Tendon
Nor
mal
ized
RM
S er
ror (
%)
Conclusions
• For subject-specific models as well as generalizable models,
• Symbolic regression more accurate than other models.
• Error bounds on generalizability.
• Models insight on tendon routing.
34
1 2 3 4 5 6
Computational models
35
Analytical models Anatomy-based models
s2 = ✓21 + 9e✓2
1 2 3 4 5 6
Anatomy-based modeling
36
Co-author: Dr. Francisco Valero-Cuevas
Netter, F. Atlas of Human Anatomy, 3rd edition, pp 447-453
1 2 3 4 5 6
4
Clavero et al. 2003
Boutonniere deformity
http://www.ispub.com/
Swan-neck deformity
Mallet finger deformity
37
• Widely used representation: An-Chao normative model (1978, 79)
TE=RB+UBRB=0.133 RI+0.167 EDC+0.667 LUUB=0.313 UI+0.167 EDCES=0.133 RI+0.313 UI+0.167 EDC+0.333 LU
Chao et al. 1978,79
1 2 3 4 5 6
Validation of An-Chao model
38
6 DOF loadcell
Load cells measur-ing tendon tensions
Strings connecting tendons to motors
Fingertip force vector
1 2 3 4 5 6
Validation of An-Chao normative model
39
• Large magnitude and direction errors in fingertip force magnitude and direction.
1 2 3 4 5 6
(Sagittal plane)
FDP FDS EIP EDC LUM FDI FPI0
20
40
60
Dir
erro
r (de
gree
s)
FDP FDS EIP EDC LUM FDI FPI0
200
400
600
800
1000
Mag
erro
r %
Magnitude errors Direction errorsFlexTapExtend
40
• Let the physics and mechanics decide force distribution.
• Existing musculoskeletal modeling software do not model tendon networks.
• Environment to understand role of components in force transformation.
Valero-Cuevas and Lipson, 2004
1 2 3 4 5 6
Specific aims
• Develop a modeling environment to represent these tendon networks.
• Study sensitivity of fingertip force output to properties of the extensor mechanism.
41
1 2 3 4 5 6
Tendon network simulator and sensitivity analysis
5
Import MRI scan of bones.
Define tendon network.
Tendon network simulator
Solve the nonlinear finite element problem.
1 2 3 4 5 6
42
Iteratively,
• Node and element penetration testing.
• Apply input Forces in increments
• Solve by Newton-Raphson iteration method the displacements of nodes, U(i), for system equilibrium :
Finite Element Method
• Assemble the internal force vector and the tangent stiffness matrix in each element.
43
Tension (N)
44Differential loading as observed by Sarrafian, 1970, Micks 1981
45
Validation
46
Motors
Load cells
Hemisphere
Reflective markers
Fixed nodes
Hemisphere
1 2 3 4 5 6
0 1 2 3 4 50
1
2
3
4
5
RF Magnitude (Model) in N
RF
Mag
nitu
de (D
ata)
in N
0 1 2 3 4 50
1
2
3
4
5
RF Magnitude (Model) in N
RF
Mag
nitu
de (D
ata)
in N
RF Node 1RF Node 2x=y line
Node 1
Node 2
Node 1
Node 2
Sensitivity analysis of parameters and topology
47
Tessellated bones
i. Locations of nodes
ii. Cross-sectional areas
iii. Resting lengths
iv. Topology
1 2 3 4 5 6
48
Fully flexed
Tap
Fully extended
Fully flexed Tap Fully extended
Fixed nodes
Sensitivity of fingertip force output in three postures1 2 3 4 5 6
Results: Sensitivity analysis
49
0
5
10
15
20
25
30
35
40
Dire
ctio
n de
viat
ion
(deg
)
−100
−50
0
50
100
150
200
Mag
nitu
de d
evia
tion
(%)
1 2 3 4 1 2 3 4
Network parameters Topology Network parameters Topology
Fully flexedTapFully
extended
Fingertip force direction and magnitude
• Sensitive to:
- Posture
- Resting lengths
- Topology
• Less sensitive to
- Node positions.
- Cross-sectional areas.
1 2 3 4 5 6
Conclusions
50
• Developed a novel tendon network simulator to represent these tendon networks.
• Studied what properties the fingertip force output is most sensitive to.
51
1 2 3 4 5 6
Simultaneous inference of topology and parameter values
Valero-Cuevas et al. 2007 Saxena et al. (in review)
Inference of anatomy-based modelsCo-authors: Dr. Hod Lipson, Dr. Francisco Valero-Cuevas
6
R
✓
R 1R
2(✓)
52
Specific aims
•Simultaneous inference of 3D tendon networks from
input-output data in simulation.
•Inference of models of the finger’s extensor
mechanism directly from input-output data via sparse
experimentation.
Inference of anatomy-based modelsCo-authors: Dr. Hod Lipson, Dr. Francisco Valero-Cuevas
61 2 3 4 5 6
Data
530 5000 10000 150000.1
0.5
2
10
50
Fitness error vs iterations
Tota
l RF
erro
r as
%
Num evaluations
CPU 1
CPU 2
CPU N
...
?
Topology and parameter inference
of 3D models
1 2 3 4 5 6
Inference of tendon networks in simulation
54
6 DOF loadcell
Load cells measur-ing tendon tensions
Strings connecting tendons to motors
Fingertip force vector
3 Postures,3 sets of inputs
1 2 3 4 5 6
Inference parameters
55
Tessellated bones
i. Locations of 6 nodes
(4 variables)
ii. Resting lengths of 13 elements(7 variables)
iii. Topology : 8 elements
(4 variables)
1 2 3 4 5 6
Estimation-exploration algorithm
Models Tests
Bongard et al., Science, 2006
1 2 3 4 5 6
56
Inference using EEA
57
Test suite
Converged?
5N 5N3N
1N1.5N3N
Start3 Random testsMeasured data
Evolve models
No
End
Two best tests selected
Estimation Exploration
1N3N3N
Identify most `intelligent’
tests (posture +
tendon tensions)
1 2 3 4 5 6
Inference results
58
0
100
200
300
400
500
600
700
RM
S M
ag e
rror
(%)
An Chao 1979
Optimized (best 5)
Full flex Tap Full Ext0
5
10
15
20
25
30
35
40
45
50
RM
S D
ir e
rror
deg
Full flex Tap Full Ext
1 2 3 4 5 6
5
10
15
20
25 N
0
Conclusions
• Demonstrated for the first time the successful inference of model topology and parameters of a complex musculoskeletal system from experimental input-output data.
• Inferred models are more accurate than models in the literature.
59
1 2 3 4 5 6
60
?
Computational models
Experimentation Mathematical modeling
Inference algorithms
61
?
Computational models
Experimentation Mathematical modeling
Inference algorithms
http://www.myefficientassistant.com/
http://www.punchstock.com/
‘Plant’
‘Controller’
Motor control
Boutonniere deformity
http://www.ispub.com/
Swan-neck deformity
Mallet finger deformity
Clinical
Computational models
Analytical models Anatomy-based models
s2 = ✓21 + 9e✓2
Conclusions and future work
62
• Applies to other systems.
•Step towards subject-specific models inferred from
data.
R
System
✓s
R2
Infer structure and parameters from input-
output data
(✓)
tension in each cord, which was fed back to the motor so that a desired amount oftension could be maintained on each tendon. The fingertip was rigidly attached to6 DOF load cell (JR3, Woodland, CA).
We examined 5 different postures in 3 specimens, and 3 different postures inthe final specimen. Each posture was neutral in add-abduction. The examinedpostures were chosen to cover the workspace and simulate those found ineveryday tasks. After positioning the finger in a specific posture, we determinedthe action matrix for the finger: we applied 128 combinations of tendon tensionsrepresenting all possible combinations of 0 and 10 N across the seven tendons, andheld each combination for 3 s. The fingertip forces resulting from each coordina-tion pattern was determined by averaging the fingertip load cell readings acrossthe hold period. Linear regression was performed on each fingertip forcecomponent using the tendon tensions as factors. In this way, the fingertip forcevector generated by 1 N of tendon tension was determined for all muscles. Theforce vector generated by each muscle was scaled by an estimate for maximummuscle force (Valero-Cuevas et al., 2000) to generate the columns of the actionmatrix for each specimen and posture examined.
2.2. Action matrix for human leg model
We also studied the necessity of muscles for mechanical output for asimplified, but plausible, sagittal plane model of the human leg (hip, knee, andankle joints). The model contained 14 muscles/muscle groups (Kuo and Zajac,1993) (muscle/muscle group abbreviation in parentheses): medial and lateralgastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peroneusbrevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendenosis/biceps femoris long head (hamstring), biceps femoris short head (bfsh), rectusfemoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adductor
longus (addlong), iliacus (iliacus), tensor facia lata (tensfl), gluteus maximus(glmax). Moment arms for hip flexion, knee flexion and ankle dorsiflexion for allof these muscles were obtained from a computer model of the lower limb (Arnoldet al., 2010). When necessary, multiple muscles in were combined into the singlemuscle groups. We derived a 3!3 square Jacobian mapping changes in the hip,knee, and ankle angle to the foot position in the plane (2 components) and theorientation of the foot in space. This Jacobian matrix, inverted and transposed, wascombined with the moment arms and maximal muscle forces to form the actionmatrix mapping muscle activation to forces and torques at the foot (Valero-Cuevas, 2005b), although our analysis of muscle redundancy was only performedwith respect to the endpoint forces.
2.3. Analyzing the action matrix to determine muscle necessity
We used the action matrix to determine whether muscles are necessaryfor a given desired output force using standard tools in computational geometry.The muscle redundancy problem can be expressed as a set of linear inequalities(Chao and An, 1978; Spoor, 1983). These inequality constraints enforce that theactivation for each muscle lie between 0 and 1, and that the actual output force isequal to the desired force. The inequality constraints define a region in muscleactivation space called the task-specific activation set: any point inside that set willproduce the desired output force (Kuo and Zajac, 1993; Valero-Cuevas, 2005b;Valero-Cuevas et al., 2000, 1998). We computed the vertices defining the task-specific activation set using a vertex enumeration algorithm (Avis and Fukuda,1992). We then found the task-specific activation ranges to achieve the desiredoutput force for each muscle by projecting all vertices onto the seven musclecoordinate axes to determine the minimum and maximum task-specific activations.While previous studies have used similar experimental (Valero-Cuevas et al., 2000)
Muscles cooperateto exert force
Feasible force set,one target force vector
Solutions in muscle activation space,task-specific activation set
Fy
Fx
Target x-force
Target y-force
Feasible force set
Target force vector
2
12
3
LIMB
31
Fy
Fx
Task-specificactivation ranges
0
1
Muscle
1 2 3
Neces
sary
Neces
sary
Redun
dant
Muscle 3
Muscle 2
Muscle 1
Target x-force sol’nsTarget y-force sol’ns
Task-specific activation set
Fig. 1. Three muscle ‘‘schematic model’’ conceptually illustrates the necessity of muscles. (a) Muscles can be functionally visualized as force vectors at the endpoint. (b) Aregion of force space, the feasible force set, is achievable given this musculature. A particular target force vector can be decomposed into a target x-force and a targety-force. (c) The valid coordination patterns for the x and y targets can also be viewed in muscle activation space as planes; the portion of the intersection of these twoplanes that is inside the unit cube is the task-specific activation set. Any point on the task-specific activation set will generate the same target force vector. (d) The task-specific activation set can be projected onto the muscle coordinate axes, revealing the minimum and maximum activation in each muscle for the given applied force vector.The task-specific activation ranges can be constructed for each muscle, and reveal which muscles are necessary and which are redundant for a given target force vector.
J.J. Kutch, F.J. Valero-Cuevas / Journal of Biomechanics 44 (2011) 1264–1270 1265
Kutch and Valero-Cuevas, 2011
Acknowledgements
63
Dr. Francisco Valero-Cuevas
Dr. Hod Lipson
Dr. Gerald Loeb
Dr. Eva Kanso Dr. Jason
KutchJosh Inouye Sudarshan
Dayanidhi
Dr. Heiko Hoffmann
Dr. Anupam Saxena
Dr. Jae-Woong Yi
Kornelius Rácz
Brendan Holt
Alex Reyes Emily Lawrence
Dr. Srideep Musuvathy
John Rocamora
Dr. Marta Mora
Na-hyeon Ko
Alison HuDr. Evangelos Theodorou
Dr. Caroline LeClercq
Dr. Vincent Rod Hentz
Dr. Nina Lightdale
Dr. Isabella Fasolla
Kari Oki
Dr. Terrance Sanger
Acknowledgements
64
The NaEonal Science FoundaEon: CAREER award,
EFRI - COPN to FVC
The National Institutes of HealthNIAMS/NICHD R01-AR050520; R01-AR052345
Thank you!
65