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The document discusses the inference of computational models for tendon networks using experimental data to improve the understanding of musculoskeletal systems, specifically focusing on human finger mechanics and leg models. It highlights various methodologies for measuring tendon tensions and determining the resulting fingertip forces, as well as comparing symbolic regression to polynomial regression for accuracy in modeling. The significance of the research is emphasized through experimental validation and the potential applications it may have in robotics and prosthetics.

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Inference of computational models of tendon networks via sparse experimentation Manish Umesh Kurse Apr 11, 2012 1 Brain-Body Dynamics Laboratory Ph.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 femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), Muscles cooperate to exert force Solutions in muscle activation s task-specific activation se 2 12 3 LIMB 3 1 Fy Fx Muscle 2 Muscle 1 Target x-for Target y-f Task-specific activation Fig. 1. Three muscle ‘‘schematic model’’ conceptually illustrates the nece region of force space, the feasible force set, is achievable given this mu y-force. (c) The valid coordination patterns for the x and y targets can a Kutch andValero-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 We examined 5 different postures in 3 specimens, and 3 different posture the final specimen. Each posture was neutral in add-abduction. The exami postures were chosen to cover the workspace and simulate those found everyday tasks. After positioning the finger in a specific posture, we determi the action matrix for the finger: we applied 128 combinations of tendon tensi representing all possible combinations of 0 and 10 N across the seven tendons, held each combination for 3 s. The fingertip forces resulting from each coordi tion pattern was determined by averaging the fingertip load cell readings acr the hold period. Linear regression was performed on each fingertip fo component using the tendon tensions as factors. In this way, the fingertip fo vector generated by 1 N of tendon tension was determined for all muscles. force vector generated by each muscle was scaled by an estimate for maxim muscle force (Valero-Cuevas et al., 2000) to generate the columns of the act matrix for each specimen and posture examined. 2.2. Action matrix for human leg model We also studied the necessity of muscles for mechanical output fo simplified, but plausible, sagittal plane model of the human leg (hip, knee, ankle joints). The model contained 14 muscles/muscle groups (Kuo and Za 1993) (muscle/muscle group abbreviation in parentheses): medial and lat gastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peron brevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendeno biceps femoris long head (hamstring), biceps femoris short head (bfsh), rec femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adduc Muscles cooperate to exert force 2 12 3 LIMB 3 1 Fy Fx Kutch andValero-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 = ✓2 1 + 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 & IEEETBME ’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 motors Positon encoders 1 2 3 4 5 6 1
Finger tapping 11 1 2 3 4 5 6
Slow finger movements 12 All intact Radial nerve palsy Median nerve palsy Ulnar nerve palsy fm Iteratively, 1 2 3 4 5 6
Finger equilibrium 13 ⌧ = RFm 0 Neutral equilibrium Stable equilibrium 1 2 3 4 5 6
14 NeutralStable FDP FDS EI EDC LUM FDI FPI 1 2 0 1 2 3 4 Distancefromnullspace(N) 1 2 0 2 4 6 8 Coordination pattern Tendontensions i Distance 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 = ✓2 1 + 9e✓2 1 2 3 4 5 6
Inference of analytical functions 17 Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 1 2 3 4 5 6 2 • Analytical functions for tendon excursions s = f(✓) Deshpande et al. 2009 • State of the art : Polynomial regression s s ‘Controller’ ‘Plant’ • Why? R(✓) = @s @✓ ⌧(✓) = R(✓)Fm R ✓ R1 R2 (✓)
• 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/2 tan(θ/2) ) s = 1.8θ s = f(✓1, ✓2, ✓3)
21 Schmidt and Lipson, 2009 Polynomial regression Koza 1992 Symbolic regression vs. Linear Quadratic Cubic Quartic 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 2 n 4 n 8 n 16 n 32 n 64 n 128 n 256 X X X XX n/256 n/16 n/64 2 5 10 20 n n 2 n 4 n 8 n 16 n 32 n 64 n 128 n 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 Linear Quadratic Cubic Dataset size (n =1688) Dataset size (n =1688) X X X Cross-validation Extrapolation RMSerror(%) RMSerror(%) 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 20 Tendon 3 RMSerror(%) Extrapolation by volume (%) 0 25 15075 10050 125 Symbolic Quartic Linear Quadratic Cubic 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 2 n 4 n 8 n 16 n 32 n 64 n 128 Training set size (n =1688) Extrapolationbyvolume(%) Symbolic Quartic Linear Quadratic 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/2 tan(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 extrap for some combinations of Landsmeer’s models I, II, III 1 2 3 4 5 6
Error vs. number of parameters 25 RMSerror(%) Cross-validationExtrapolation Symbolic Quartic Linear Quadratic Cubic Experimental data With no noise Number of parameters Simulated data With noise added 1 2 3 0 20 40 1 2 3 5 .0001 .01 1 0 20 40 .0001 .01 1 0 20 40 1 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/2 tan(θ/2) ) s = 1.8θ Landsmeer, 1961, Brook 1995 3 Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-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 encoders Servo 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. Linear Quadratic Cubic Quartic Landsmeer 1 2 3 4 5 6
Across trials 32 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 Symbolic Landsmeer Quartic Linear Quadratic Cubic 2 10 50 2 5 10 20 50 FDP 2 10 50 2 5 10 20 50 FDS 2 10 50 2 5 10 20 50 EIP 2 10 50 2 5 10 20 50 EDC 2 10 50 2 5 10 20 50 LUM 2 10 50 2 5 10 20 50 FDI 2 10 50 2 5 10 20 50 FPI Tendon Number of parameters NormalizedRMSerror(%) 1 2 3 4 5 6
Across hands 33 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 2 5 10 20 50 Tendon NormalizedRMSerror(%) Symbolic Landsmeer Quartic Linear Quadratic Cubic 1 2 3 4 5 6 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 Symbolic Landsmeer Quartic Linear Quadratic Cubic 10 20 50 Tendon NormalizedRMSerror(%)
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 = ✓2 1 + 9e✓2 1 2 3 4 5 6
Anatomy-based modeling 36 Co-author: Dr. FranciscoValero-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+UB RB=0.133 RI+0.167 EDC+0.667 LU UB=0.313 UI+0.167 EDC ES=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 FPI 0 20 40 60 Direrror(degrees) FDP FDS EIP EDC LUM FDI FPI 0 200 400 600 800 1000 Magerror% Magnitude errors Direction errors Flex Tap Extend
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) 44 Differential 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 5 0 1 2 3 4 5 RF Magnitude (Model) in N RFMagnitude(Data)inN RF Node 1 RF Node 2 x=y line
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 postures 1 2 3 4 5 6
Results: Sensitivity analysis 49 0 5 10 15 20 25 30 35 40 Directiondeviation(deg) −100 −50 0 50 100 150 200 Magnitudedeviation(%) 1 2 3 4 1 2 3 4 Network parameters Topology Network parameters Topology Fully flexed Tap Fully 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 models Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 6 R ✓ R1 R2 (✓)
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 models Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 6 1 2 3 4 5 6
Data 530 5000 10000 15000 0.1 0.5 2 10 50 Fitness error vs iterations TotalRFerroras% 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 5N 3N 1N1.5N 3N Start3 Random tests Measured data Evolve models No End Two best tests selected Estimation Exploration 1N 3N3N Identify most `intelligent’ tests (posture + tendon tensions) 1 2 3 4 5 6
Inference results 58 0 100 200 300 400 500 600 700 RMSMagerror(%) An Chao 1979 Optimized (best 5) Full flex Tap Full Ext 0 5 10 15 20 25 30 35 40 45 50 RMSDirerrordeg 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 = ✓2 1 + 9e✓2
Conclusions and future work 62 •Applies to other systems. •Step towards subject-specific models inferred from data. R System ✓ s R 2 Infer structure and parameters from input- output data (✓) tension in each cord, which was fed back to the motor so that a desired amount of tension could be maintained on each tendon. The fingertip was rigidly attached to 6 DOF load cell (JR3, Woodland, CA). We examined 5 different postures in 3 specimens, and 3 different postures in the final specimen. Each posture was neutral in add-abduction. The examined postures were chosen to cover the workspace and simulate those found in everyday tasks. After positioning the finger in a specific posture, we determined the action matrix for the finger: we applied 128 combinations of tendon tensions representing all possible combinations of 0 and 10 N across the seven tendons, and held each combination for 3 s. The fingertip forces resulting from each coordina- tion pattern was determined by averaging the fingertip load cell readings across the hold period. Linear regression was performed on each fingertip force component using the tendon tensions as factors. In this way, the fingertip force vector generated by 1 N of tendon tension was determined for all muscles. The force vector generated by each muscle was scaled by an estimate for maximum muscle force (Valero-Cuevas et al., 2000) to generate the columns of the action matrix 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 a simplified, but plausible, sagittal plane model of the human leg (hip, knee, and ankle joints). The model contained 14 muscles/muscle groups (Kuo and Zajac, 1993) (muscle/muscle group abbreviation in parentheses): medial and lateral gastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peroneus brevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendenosis/ biceps femoris long head (hamstring), biceps femoris short head (bfsh), rectus femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adductor longus (addlong), iliacus (iliacus), tensor fac (glmax). Moment arms for hip flexion, knee fle of these muscles were obtained from a compute et al., 2010). When necessary, multiple muscles muscle groups. We derived a 3 Â 3 square Jaco knee, and ankle angle to the foot position in t orientation of the foot in space. This Jacobian m combined with the moment arms and maxima matrix mapping muscle activation to forces Cuevas, 2005b), although our analysis of muscl with respect to the endpoint forces. 2.3. Analyzing the action matrix to determine m We used the action matrix to determine for a given desired output force using standard The muscle redundancy problem can be expre (Chao and An, 1978; Spoor, 1983). These ineq activation for each muscle lie between 0 and 1, equal to the desired force. The inequality con activation space called the task-specific activatio produce the desired output force (Kuo and Z Valero-Cuevas et al., 2000, 1998). We comput specific activation set using a vertex enumera 1992). We then found the task-specific activat output force for each muscle by projecting a coordinate axes to determine the minimum and While previous studies have used similar experi Muscles cooperate to exert force Feasible force set, one target force vector Fy Fx Target x-force Target y-f Feasible force set Target force vector 2 12 3 LIMB 3 1 Fy Fx Muscle 2 J.J. Kutch, F.J. Valero-Cuevas / Journal of Biomechanics 44 (2011) 1264–1270 Kutch andValero-Cuevas, 2011
Acknowledgements 63 Dr. Francisco Valero-Cuevas Dr. Hod Lipson Dr. Gerald Loeb Dr. Eva Kanso Dr. Jason Kutch Josh 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 Health NIAMS/NICHD R01-AR050520; R01-AR052345
Thank you! 65

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Manish Kurse PhD research slides

  • 1. Inference of computational models of tendon networks via sparse experimentation Manish Umesh Kurse Apr 11, 2012 1 Brain-Body Dynamics Laboratory Ph.D. committee: Dr. Francisco J.Valero-Cuevas, Dr. Hod Lipson, Dr. Gerald E. Loeb, Dr. Eva Kanso
  • 2. 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 femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), Muscles cooperate to exert force Solutions in muscle activation s task-specific activation se 2 12 3 LIMB 3 1 Fy Fx Muscle 2 Muscle 1 Target x-for Target y-f Task-specific activation Fig. 1. Three muscle ‘‘schematic model’’ conceptually illustrates the nece region of force space, the feasible force set, is achievable given this mu y-force. (c) The valid coordination patterns for the x and y targets can a Kutch andValero-Cuevas, 2011 Computational modeling of musculoskeletal systems
  • 3. 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. 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 We examined 5 different postures in 3 specimens, and 3 different posture the final specimen. Each posture was neutral in add-abduction. The exami postures were chosen to cover the workspace and simulate those found everyday tasks. After positioning the finger in a specific posture, we determi the action matrix for the finger: we applied 128 combinations of tendon tensi representing all possible combinations of 0 and 10 N across the seven tendons, held each combination for 3 s. The fingertip forces resulting from each coordi tion pattern was determined by averaging the fingertip load cell readings acr the hold period. Linear regression was performed on each fingertip fo component using the tendon tensions as factors. In this way, the fingertip fo vector generated by 1 N of tendon tension was determined for all muscles. force vector generated by each muscle was scaled by an estimate for maxim muscle force (Valero-Cuevas et al., 2000) to generate the columns of the act matrix for each specimen and posture examined. 2.2. Action matrix for human leg model We also studied the necessity of muscles for mechanical output fo simplified, but plausible, sagittal plane model of the human leg (hip, knee, ankle joints). The model contained 14 muscles/muscle groups (Kuo and Za 1993) (muscle/muscle group abbreviation in parentheses): medial and lat gastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peron brevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendeno biceps femoris long head (hamstring), biceps femoris short head (bfsh), rec femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adduc Muscles cooperate to exert force 2 12 3 LIMB 3 1 Fy Fx Kutch andValero-Cuevas, 2011
  • 5. 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
  • 6. Computational models 6 Analytical models Anatomy-based models s2 = ✓2 1 + 9e✓2
  • 9. 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 & IEEETBME ’12 ASB ’11 CSB ’12 ASME SBC ’09 9
  • 10. Cadaver finger control 10 Load cells Strings to the tendons Motion capture markers 6 DOF Load cell DC motors Positon encoders 1 2 3 4 5 6 1
  • 12. Slow finger movements 12 All intact Radial nerve palsy Median nerve palsy Ulnar nerve palsy fm Iteratively, 1 2 3 4 5 6
  • 13. Finger equilibrium 13 ⌧ = RFm 0 Neutral equilibrium Stable equilibrium 1 2 3 4 5 6
  • 15. 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
  • 16. Computational models 16 Analytical models Anatomy-based models s2 = ✓2 1 + 9e✓2 1 2 3 4 5 6
  • 17. Inference of analytical functions 17 Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 1 2 3 4 5 6 2 • Analytical functions for tendon excursions s = f(✓) Deshpande et al. 2009 • State of the art : Polynomial regression s s ‘Controller’ ‘Plant’ • Why? R(✓) = @s @✓ ⌧(✓) = R(✓)Fm R ✓ R1 R2 (✓)
  • 18. • 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. 19 Schmidt and Lipson, 2009 Koza 1992 Symbolic regression 1 2 3 4 5 6
  • 20. 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/2 tan(θ/2) ) s = 1.8θ s = f(✓1, ✓2, ✓3)
  • 21. 21 Schmidt and Lipson, 2009 Polynomial regression Koza 1992 Symbolic regression vs. Linear Quadratic Cubic Quartic 1 2 3 4 5 6
  • 22. 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 2 n 4 n 8 n 16 n 32 n 64 n 128 n 256 X X X XX n/256 n/16 n/64 2 5 10 20 n n 2 n 4 n 8 n 16 n 32 n 64 n 128 n 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 Linear Quadratic Cubic Dataset size (n =1688) Dataset size (n =1688) X X X Cross-validation Extrapolation RMSerror(%) RMSerror(%) 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 20 Tendon 3 RMSerror(%) Extrapolation by volume (%) 0 25 15075 10050 125 Symbolic Quartic Linear Quadratic Cubic X X X All extrapolation errors > 5% Achievable extrapolation > 150% Fewer training data points required More extrapolatable 2
  • 23. 23 Extrapolation by volume (%) 0 25 50 75 100 125 150 >150 n n 2 n 4 n 8 n 16 n 32 n 64 n 128 Training set size (n =1688) Extrapolationbyvolume(%) Symbolic Quartic Linear Quadratic 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. 24 Landsmeer model I Landsmeer model II Landsmeer model III s = 3.6sin(0.5θ) s = 0.6θ + 3.2(1 − θ/2 tan(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 extrap for some combinations of Landsmeer’s models I, II, III 1 2 3 4 5 6
  • 25. Error vs. number of parameters 25 RMSerror(%) Cross-validationExtrapolation Symbolic Quartic Linear Quadratic Cubic Experimental data With no noise Number of parameters Simulated data With noise added 1 2 3 0 20 40 1 2 3 5 .0001 .01 1 0 20 40 .0001 .01 1 0 20 40 1 2 5 10 1 2 5 10 1 2 3 4 5 6
  • 26. 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. 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
  • 28. 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/2 tan(θ/2) ) s = 1.8θ Landsmeer, 1961, Brook 1995 3 Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-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
  • 29. 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
  • 30. Cadaver experimental setup 30 MoEon   capture Load cells Position encoders Servo motors Markers 1 2 3 4 5 6 s1 = f(✓1, ✓2, ✓3, ✓4) s7 = f(✓1, ✓2, ✓3, ✓4) . . .
  • 31. 31 Schmidt and Lipson, 2009 Polynomial and LandsmeerKoza 1992 Symbolic regression vs. Linear Quadratic Cubic Quartic Landsmeer 1 2 3 4 5 6
  • 32. Across trials 32 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 Symbolic Landsmeer Quartic Linear Quadratic Cubic 2 10 50 2 5 10 20 50 FDP 2 10 50 2 5 10 20 50 FDS 2 10 50 2 5 10 20 50 EIP 2 10 50 2 5 10 20 50 EDC 2 10 50 2 5 10 20 50 LUM 2 10 50 2 5 10 20 50 FDI 2 10 50 2 5 10 20 50 FPI Tendon Number of parameters NormalizedRMSerror(%) 1 2 3 4 5 6
  • 33. Across hands 33 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 2 5 10 20 50 Tendon NormalizedRMSerror(%) Symbolic Landsmeer Quartic Linear Quadratic Cubic 1 2 3 4 5 6 FDP FDS EIP EDC LUM FDI FPI 2 5 10 20 Symbolic Landsmeer Quartic Linear Quadratic Cubic 10 20 50 Tendon NormalizedRMSerror(%)
  • 34. 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
  • 35. Computational models 35 Analytical models Anatomy-based models s2 = ✓2 1 + 9e✓2 1 2 3 4 5 6
  • 36. Anatomy-based modeling 36 Co-author: Dr. FranciscoValero-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. 37 • Widely used representation: An-Chao normative model (1978, 79) TE=RB+UB RB=0.133 RI+0.167 EDC+0.667 LU UB=0.313 UI+0.167 EDC ES=0.133 RI+0.313 UI+0.167 EDC+0.333 LU Chao et al. 1978,79 1 2 3 4 5 6
  • 38. 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
  • 39. 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 FPI 0 20 40 60 Direrror(degrees) FDP FDS EIP EDC LUM FDI FPI 0 200 400 600 800 1000 Magerror% Magnitude errors Direction errors Flex Tap Extend
  • 40. 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
  • 41. 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
  • 42. Import MRI scan of bones. Define tendon network. Tendon network simulator Solve the nonlinear finite element problem. 1 2 3 4 5 6 42
  • 43. 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
  • 44. Tension (N) 44 Differential loading as observed by Sarrafian, 1970, Micks 1981
  • 45. 45
  • 46. Validation 46 Motors Load cells Hemisphere Reflective markers Fixed nodes Hemisphere 1 2 3 4 5 6 0 1 2 3 4 5 0 1 2 3 4 5 RF Magnitude (Model) in N RFMagnitude(Data)inN RF Node 1 RF Node 2 x=y line
  • 47. 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. 48 Fully flexed Tap Fully extended Fully flexed Tap Fully extended Fixed nodes Sensitivity of fingertip force output in three postures 1 2 3 4 5 6
  • 49. Results: Sensitivity analysis 49 0 5 10 15 20 25 30 35 40 Directiondeviation(deg) −100 −50 0 50 100 150 200 Magnitudedeviation(%) 1 2 3 4 1 2 3 4 Network parameters Topology Network parameters Topology Fully flexed Tap Fully 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
  • 50. 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. 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 models Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 6 R ✓ R1 R2 (✓)
  • 52. 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 models Co-authors: Dr. Hod Lipson, Dr. FranciscoValero-Cuevas 6 1 2 3 4 5 6
  • 53. Data 530 5000 10000 15000 0.1 0.5 2 10 50 Fitness error vs iterations TotalRFerroras% Num evaluations CPU 1 CPU 2 CPU N ... ? Topology and parameter inference of 3D models 1 2 3 4 5 6
  • 54. 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
  • 55. 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
  • 56. Estimation-exploration algorithm Models Tests Bongard et al., Science, 2006 1 2 3 4 5 6 56
  • 57. Inference using EEA 57 Test suite Converged? 5N 5N 3N 1N1.5N 3N Start3 Random tests Measured data Evolve models No End Two best tests selected Estimation Exploration 1N 3N3N Identify most `intelligent’ tests (posture + tendon tensions) 1 2 3 4 5 6
  • 58. Inference results 58 0 100 200 300 400 500 600 700 RMSMagerror(%) An Chao 1979 Optimized (best 5) Full flex Tap Full Ext 0 5 10 15 20 25 30 35 40 45 50 RMSDirerrordeg Full flex Tap Full Ext 1 2 3 4 5 6 5 10 15 20 25 N 0
  • 59. 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
  • 61. 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 = ✓2 1 + 9e✓2
  • 62. Conclusions and future work 62 •Applies to other systems. •Step towards subject-specific models inferred from data. R System ✓ s R 2 Infer structure and parameters from input- output data (✓) tension in each cord, which was fed back to the motor so that a desired amount of tension could be maintained on each tendon. The fingertip was rigidly attached to 6 DOF load cell (JR3, Woodland, CA). We examined 5 different postures in 3 specimens, and 3 different postures in the final specimen. Each posture was neutral in add-abduction. The examined postures were chosen to cover the workspace and simulate those found in everyday tasks. After positioning the finger in a specific posture, we determined the action matrix for the finger: we applied 128 combinations of tendon tensions representing all possible combinations of 0 and 10 N across the seven tendons, and held each combination for 3 s. The fingertip forces resulting from each coordina- tion pattern was determined by averaging the fingertip load cell readings across the hold period. Linear regression was performed on each fingertip force component using the tendon tensions as factors. In this way, the fingertip force vector generated by 1 N of tendon tension was determined for all muscles. The force vector generated by each muscle was scaled by an estimate for maximum muscle force (Valero-Cuevas et al., 2000) to generate the columns of the action matrix 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 a simplified, but plausible, sagittal plane model of the human leg (hip, knee, and ankle joints). The model contained 14 muscles/muscle groups (Kuo and Zajac, 1993) (muscle/muscle group abbreviation in parentheses): medial and lateral gastrocnemius (gastroc), soleus (soleus), tibialis posterior (tibpost), peroneus brevis (perbrev), tibialis anterior (tibant), semimembranoseus/semitendenosis/ biceps femoris long head (hamstring), biceps femoris short head (bfsh), rectus femoris (rectfem), gluteus medialis/glueteus minimus (glmed/min), adductor longus (addlong), iliacus (iliacus), tensor fac (glmax). Moment arms for hip flexion, knee fle of these muscles were obtained from a compute et al., 2010). When necessary, multiple muscles muscle groups. We derived a 3 Â 3 square Jaco knee, and ankle angle to the foot position in t orientation of the foot in space. This Jacobian m combined with the moment arms and maxima matrix mapping muscle activation to forces Cuevas, 2005b), although our analysis of muscl with respect to the endpoint forces. 2.3. Analyzing the action matrix to determine m We used the action matrix to determine for a given desired output force using standard The muscle redundancy problem can be expre (Chao and An, 1978; Spoor, 1983). These ineq activation for each muscle lie between 0 and 1, equal to the desired force. The inequality con activation space called the task-specific activatio produce the desired output force (Kuo and Z Valero-Cuevas et al., 2000, 1998). We comput specific activation set using a vertex enumera 1992). We then found the task-specific activat output force for each muscle by projecting a coordinate axes to determine the minimum and While previous studies have used similar experi Muscles cooperate to exert force Feasible force set, one target force vector Fy Fx Target x-force Target y-f Feasible force set Target force vector 2 12 3 LIMB 3 1 Fy Fx Muscle 2 J.J. Kutch, F.J. Valero-Cuevas / Journal of Biomechanics 44 (2011) 1264–1270 Kutch andValero-Cuevas, 2011
  • 63. Acknowledgements 63 Dr. Francisco Valero-Cuevas Dr. Hod Lipson Dr. Gerald Loeb Dr. Eva Kanso Dr. Jason Kutch Josh 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
  • 64. Acknowledgements 64 The  NaEonal  Science  FoundaEon:   CAREER award, EFRI - COPN to FVC The National Institutes of Health NIAMS/NICHD R01-AR050520; R01-AR052345