A Screw-Theoretic Framework for Musculoskeletal Modeling and Analysis

1
A Screw-Theoretic Framework for Musculoskeletal
Modeling and Analysis
Michael J. Del Signore
(mjd24_at_eng.buffalo.edu)
December 16th 2005
Advisor Dr. Venkat Krovi Mechanical and
Aerospace Engineering State University of New
York at Buffalo
2
Agenda

• Introduction
• Background
• Case Scenario
• System Modeling
• GUI Implementation
• Simulation Framework
• Mechanical Prototype Design
• Future Work
• Conclusion

3
Motivation
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Computational advances in the past decade have
  revolutionized engineering!!
• Improved Infrastructure
• Advanced Algorithms and Methodologies

• Such advancements have been seen far lesser in
  other professional arenas e.g. Biological
  Sciences
• Applications developed within this area could
  bring about similar advances and benefits.

4
Research Issues
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Significant gap halting the integration of
  engineering tools into the Biological Sciences
  fields.
• Need for specialized (problem specific) tools.
• Users need to be familiar with use and supporting
  theory.

• Three Critical Steps
• Model creation with adequate fidelity.
• Analysis of various actions/ behaviors.
• Iterative testing for refining hypotheses.

Powerful Tool

• Integration and application of certain
  engineering principles and techniques into one of
  the candidate biological sciences fields

Musculoskeletal System Analysis
5
Challenges
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Unlike traditional engineering systems,
  musculoskeletal systems inherently possess
  considerable irregularities and redundancies.

Irregularities
Redundancies

• Multiple Muscles More actuators than degrees
  of freedom.
• Infinite set of actuator (muscle) forces can
  produce the same end-effector force.

• Complex Asymmetric Geometric Shapes (i.e.
  muscle, bone).
• Each specimen is unique.
• Dealing with (trying to simulate) living tissue.

• Musculoskeletal analysis tools need to take these
  characteristics into account.

6
Existing Tools
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Traditional Articulated Mechanical System
  Analysis Tools
• Virtual Prototyping Virtual product simulation
  testing
• Examples VisualNastran, ADAMS, Pro-Mechanica
• Physics, Dynamics, FEA, Contact, Friction
  Implementation into real-time control frameworks

• The limitations of these tools can be seen when
  dealing with more complex phenomena and systems.
• Complex Geometries
• Redundant Actuation
• High Number of Contacts

Musculoskeletal System Analysis
7
Existing Tools
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Musculoskeletal System Analysis Tools
• In resent years tools have been developed to
  specifically model and analyze musculoskeletal
  systems.
• Examples SIMM, AnyBody, LifeMod

• While being successful at handling complex
  musculoskeletal systems these programs require
• In depth physiological knowledge.
• Extensive application specific programming and
  coding.

High Degree of Modeling and Simulation Detail
Rapid Real-Time Simulation and Analysis
Relatively Impossible
8
Research Goal
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• The development of computational tools that can
  analyze a redundant musculoskeletal system,
  incorporating
• An adequate degree of speed
• Accurate redundancy resolution
• Application in a real-time model based control
  framework

• Undertaken using screw-theoretic modeling
  methods
• Typically seen with the context of parallel
  manipulators.
• Convenient basis for redundancy resolution and
  optimization.

• Critical aspects addressed within a specific case
  scenario
• Musculoskeletal Analysis of the Jaw Closure of a
  Saber-Tooth Cat (Smilodon-Fatalis).

9
Related Works
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Musculoskeletal Modeling
• Multi-body Dynamics Approach
• Forster, 2003
• Detailed Muscle Modeling (Hill Model)
• Wolkotte, 2003
• Muscle Modeling and Software Development
  (Anybody)
• Rasmussen, Damsgaard, Surma, Christensen,
  de Zee, and Vondrack, 2003
• Konakanchi, 2005

• Screw-Theoretic Modeling
• Redundancy Resolution
• Firmani and Podhorodeski, 2004
• Parallel Manipulation
• Tsi, 1999
• Wrench Based Modeling and Analysis
• Ebert-Uphoff and Voglewede, 2004
• Kumar and Waldron, 1988

10
Mathematical Preliminaries
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Screw Coordinates

Unit Screw
The displacement of a rigid body can be defined
as a screw displacement, such that its motion can
be broken down into a rotation about a unique
axis (line) and a translation about the same
unique axis called the screw axis.
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Mathematical Preliminaries
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Screw Coordinates

Twists (Velocity)
Linear Velocity
Angular Velocity
Wrenches (Force)
Applied Force
Moment caused by Fo
The displacement of a rigid body can be defined
as a screw displacement, such that its motion can
be broken down into a rotation about a unique
axis (line) and a translation about the same
unique axis called the screw axis.
12
Musculoskeletal Analysis of the Jaw Closure of
the Smilodon
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Accurately model and simulate the skull/ mandible
  musculoskeletal structure of the Smilodon

13
Preliminary Simulations
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Undertaken using traditional articulated
  mechanical system tools.

• Virtual Simulation of Mechanical Saber-Tooth Cat
• Discovery Channel Model

Discovery Channel Model
Virtual Recreation
14
Simulation of Mechanical Smilodon
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Implemented using a prescribed motion analysis
  within VisualNastran

• Simulation was successful but more complexity was
  desired.

15
Virtual Prototyping of Smilodon from Fossil
Records
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• VisualNastran simulation created to calculate
  muscle forces necessary to produce a desired bite
  force.

• Virtual representation created from actual fossil
  records

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Smilodon Virtual Prototype VisualNastran
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Constraints were placed on the system to
  represent

• The simulation was met with limitations
• Due to the software's inability to handle
  redundancy in terms of resolving the multiple
  muscle forces in an inverse dynamics setting.
• These shortcomings provided the motivation for
  the development of our own low-order
  computationally tractable model based on
  screw-theoretic methods.

• Muscles ? Linear Actuators

• Skull/ Mandible Interaction ? Revolute Joint

• External forces (or alternately a prescribed
  motion) was applied to the skull as
  user-specified input to the system.

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Our Model
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Representation
• The underlying articulated structure and
  superimposed musculature is modeled as a
  redundantly actuated parallel mechanism.
• Goal Development of a Screw-Theoretic Framework
• Accurately calculate the muscle forces needed to
  produce a specific desired applied bite-force.
• Serve as a mathematical basis for
• Redundancy resolution and optimization
  implementation.
• Implementation into and analysis GUI and
  simulation framework

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Model Set Up
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Assumptions
• Planar
• Skull and mandible are rigid bodies.
• The skull is attached to the mandible via a
  revolute joint.
• Muscle act along the line of action joining the
  origin and insertion points.

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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Model Set Up

• Coordinate Frames

• (Xo, Yo) Inertial Frame
• Fixed in Space
• Main Calculation Frame

• (XU, YU) Upper Jaw Frame
• Attached to Skull (Upper Jaw)
• Related to Inertial Frame through jaw gape angle
  q.

• (XE, YE) End Effector Frame
• Created with the application point of the
  external/ desired or bite force.

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Screw Theoretic Modeling
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Each muscle is modeled as a Revolute-Prismatic-Rev
  olute (RPR) serial chain manipulator with an
  actuated prismatic joint.
• An external (desired bite) force is applied to
  the system.
• Need to calculate the actuator (muscle) forces
  needed to produce the external bite force.

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Screw Theoretic Modeling
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Calculate end-effector twist generated by every
  serial chain present in the system.

RPR Chains (Muscles)
Revolute Jaw Joint Serial Chain
Jacobian matrix whose column vectors represent
the unit screws associated with each joint in the
ith RPR serial chain.
Unit screw created by the jaw joint.
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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Screw Theoretic Modeling

• Unit screws

• Revolute Joints
• Unit Screw with a pitch of zero (l 0)

• Prismatic Joints
• Unit Screw with a pitch of infinity (l 8)

Upper Revolute Joint
Lower Revolute Joint
Jaw Revolute Joint
Prismatic Joint
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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Screw Theoretic Modeling

• Unit screws

Unit Direction Vectors
Distance Vectors
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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Screw Theoretic Modeling

• Combine and generate the Jacobian matrices
  corresponding to every serial chain in the system
  and simplify to 2-dimensions.

RPR Serial Chains (Muscles)
Jaw Joint
25
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Screw Theoretic Modeling

• Reciprocal Wrench Formulation
• Calculate the Selectively-Non-Reciprocal-Screws
  (SNRS) associated with the active joints
  (prismatic) in every serial chain.
• SNRS a screw which is reciprocal to all screws
  except the given screw.

Prismatic Joint Formulation
Jaw Joint Formulation

• WP,i is the SNRS to the unit screw corresponding
  to the Pi joint that satisfies

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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Screw Theoretic Modeling

• fP Particular Solution
• Equilibrating force field
• Least-squares solution
• fH Homogeneous Solution
• Interaction force field
• Used to ensure that all muscle forces are acting
  in the same direction.

• System Equilibrium Equation
• Collect all SNRSs Prismatic Joints and Jaw
  Joint.

• Redundancy Resolution
• Pseudo-Inverse Solution

Pseudo-Inverse of W
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Muscle Optimization
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Muscles produce force in only one direction
  (contraction).
• Implemented optimization routines minimize muscle
  forces while constraining them to remain positive
  (unidirectional)
• Two optimization routines are developed and
  implemented.
• Muscle Force Optimization
• Muscle Activity Optimization

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Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
Muscle Force Optimization
Rank deficient

• Find the full rank null space component of the
  system.
• Singular-Value-Decomposition of H
• r Number of columns of S containing non-zero
  singular values.

Design Variables
29
Muscle Force Optimization
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Pseudo-Inverse Solution
• Separate Solution Components

• Force Optimization

Jaw Joint Reaction Forces
Actuator (Muscle) Forces
30
Muscle Activity Optimization
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Normalized Muscle Activity

• System Equilibrium Equation (Activity)

Muscle Force
Maximum Muscle Force

• Muscle/ reaction forces in terms of activity.

• Pseudo-Inverse Solution

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Muscle Activity Optimization
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Pseudo-Inverse (Activity) Solution
• Separate Solution Components

• Activity Optimization

Jaw Joint Reaction Activities

• Forces

Actuator (Muscle) Activities
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Implementation into a MATLAB Graphical-User-Interf
ace (GUI)
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Analysis GUI - Computational Simulation Tool
• Uses the screw-theoretic model as a basis.
• Parametrically analyze the muscles forces
  associated with an applied desired bite force.
• User specifies the magnitude and location of the
  applied desired bite force and the location or
  location range of four separate muscles.
• GUI calculates the muscle forces needed to
  produce the applied bite force.

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MATLAB Analysis GUI
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Mode Selection

• Applied Force Definition

• Muscle Location Definition

• Muscle Range Definition

• Optimization and Plot Options

• Mode1 Results - Single Static

• Mode2 Results - Stepped Static

• Jaw Gape Definition

34
GUI Solution Validation
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• System Set Up
• One Active Muscle
• D.O.F nm
• Solved Analytically

• Analytic Solution

35
GUI Solution Validation
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
36
Virtual Model Simulation and Analysis Framework
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Simulation of the simplified (2D) representation
  of the Smilodon musculoskeletal system.
• Implemented within Simulink and VisualNastran.
• Screw-Theoretic Model main solution engine.
• Basis for real-time control/ hardware-in-the-loop
  (HIL) simulation of a mechanical model of the
  system.

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Data / Information Flow
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
38
User Inputs
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Desired Jaw Gape Angle Curve
• Jaw gape angle over time
• Simulation Time

39
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
User Inputs

• Desired Bite Force Curve
• Bite Force with respect to upper jaw over time

40
User Inputs
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Initial Muscle Locations at q(0) Maximum Forces
• Block also serves as the link to the
  screw-theoretic model/ optimization (activity)
  routine.
• Optimization feasibility check
• Provides muscle (actuator) forces to
  VisualNastran model.

Screw-Theoretic Model/ Activity Optimization
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VisualNastran Simulink Block
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Dynamic in-the-loop link between Simulink and
  VisualNastran.

42
VisualNastran Model
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Two-Dimensional representation of the skull/
  mandible musculoskeletal system.

43
VisualNastran Model
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Measure Bite Force
• Check for compatibility with applied bite force

44
Framework Simulations
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Four simulations
• Identical Simulation Parameters tmax, Dt,
  etc
• Varying/ Constant Jaw Gape
• Varying/ Constant Bite Force

45
Simulation 1 Constant Angle/ Constant Force
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Angle - 30 Force - 1000N

46
Simulation 2 Constant Angle/ Varying Force
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Angle - 30 Force - 1000N to 500N

47
Simulation 3 Varying Angle/ Constant Force
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Angle - 30 to 0 Force - 1000N

48
Simulation 4 Varying Angle/ Varying Force
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Angle - 30 to 0 Force - 1000N to 500N

49
Simulation Summary
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Error peaks occur at same time.
• Simulation Settling.
• Rotation of arbitrary material.

50
Design of a Mechanical Bite-Testing Prototype
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Designed to simulate biting actions of various
  large felines
• Accepts various dentition castings adjustable.
• Initial design developed for manual operation
  with eventual implementation of computer control
  (HIL simulations)
• Currently in preliminary manufacturing stages.

51
Dentition Castings
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• CAD models developed from fossil records.

52
Mechanism Adjustability
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Ensure proper dentition location.
• Locks in place during use.

Rotation Point Location
Skull/Mandible Location
53
Mechanical Prototype Force/ Torque Analysis GUI
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion
54
Future Work
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Completion of Mechanical Prototype
• Implementation of cable-actuation strategy
  simulating muscles.
• Implementation into real-time HIL control
  analysis framework.
• Extension Screw-Theoretic Model to
  Three-Dimensions
• Higher degree of complexity and realism.
• Additional analysis GUI.
• Provide modeling and solution basis for HIL
  simulations.
• Implementation of Muscle Physiological Properties
• Max muscle force currently only property
  considered.
• Insight into what types of muscles are needed to
  produce desired bite force.
• Preliminary inclusion of physiological muscle
  properties explored using Virtual Muscle
  (Simulink muscle model).

55
Conclusions
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Application of existing tools to musculoskeletal
  system analysis was explored.
• Traditional engineering tools found inadequate at
  handling inherent system redundancies.
• Specific musculoskeletal modeling tools require a
  high amount modeling detail and application
  specific programming rapid real-time simulation
  and analysis relatively impossible.
• Developed a screw-theoretic framework for
  modeling and analyzing the skull/mandible
  musculoskeletal system of a saber-tooth cat.
• Modeled as a redundantly actuated parallel
  manipulator.
• Framework resolves muscle forces needed to
  produce a desired bite force.
• Redundancy resolution scheme implemented a
  typical pseudo-inverse solution methodology.
• Muscle force and activity optimizations were
  explored and implemented.

56
Conclusions
Introduction Background Case Scenario
Simulation Mechanical Prototype Future Work
Conclusion

• Screw-Theoretic Framework provided the basis for
  the development of a MATLAB analysis GUI
• Parametrically analyses the muscle forces or
  activities (four muscle) needed to produce a
  desired bite force.
• Virtual simulation framework developed.
• Simulated a virtual representation of the
  saber-tooth cat.
• Implemented within Simulink and VisualNastran.
• Measured bite force compared to the applied
  bite-force.
• Overall the simulation was successful.
• Introduced a mechanical bite-testing prototype.
• Perform bite testing simulations on various large
  felines.
• Basis for implementation into real-time HIL
  analyses.
• Overall the developed screw-theoretic modeling
  and analysis framework shows significant promise
  at speeding up the musculoskeletal system
  analysis processes.

57
Thank You
Questions?