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A Screw-Theoretic Framework for
Musculoskeletal Modeling and
Analysis
Michael J. Del Signore
(mjd24@eng.buffalo.edu)
December 16th 2005
Advisor: Dr. Venkat Krovi
Mechanical and Aerospace Engineering
State University of New York at Buffalo
Agenda
• Introduction
• Background
• Case Scenario
• System Modeling
• GUI Implementation
•
•
•
•
Simulation Framework
Mechanical Prototype Design
Future Work
Conclusion
Michael J. Del Signore
December 16th 2005
Slide 2 of 57
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Motivation
• 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.
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Research Issues
• 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.
Powerful
Tool and
• Integration
and
application
of
certain
engineering
principles
• Analysis of various actions/ behaviors.
techniques into one of the candidate biological sciences fields:
• Iterative testing for refining hypotheses.
Musculoskeletal System Analysis
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Challenges
• Unlike traditional engineering systems, musculoskeletal systems
inherently possess considerable irregularities
redundancies.
Irregularities andRedundancies
• Complex Asymmetric Geometric
Shapes (i.e. muscle, bone).
• Multiple Muscles: More actuators
than degrees of freedom.
• Dealing with (trying to simulate)
living tissue.
effector force.
•• Each
Musculoskeletal
analysis tools need• to
take these characteristics
Infinite set of actuator (muscle)
specimen is unique.
into account.
forces can produce the same end-
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Existing Tools
• Traditional Articulated Mechanical System Analysis Tools
• Virtual Prototyping – Virtual product simulation & testing
• Examples: VisualNastran, ADAMS, Pro-Mechanica …
• 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
• Physics, Dynamics, FEA, Contact, Friction – Implementation
into real-time control frameworks
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Existing Tools
• 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
Michael J. Del Signore
December 16th 2005
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Rapid Real-Time Simulation
and Analysis Relatively
Impossible
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Research Goal
• 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).
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Related Works
• 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]
Michael J. Del Signore
December 16th 2005
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Mathematical Preliminaries
• Screw Coordinates
Unit Screw
û - Unit vector pointing along the direction of the screw axis.
v
uˆ 0 = r ´ uˆ + l uˆ - Moment of the screw axis about the origin.
v
r - Location of a point on the screw axis.
l - Pitch, the ratio of translation to rotation.
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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Mathematical Preliminaries
• 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.
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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Musculoskeletal Analysis of the Jaw Closure of the
Smilodon
• Accurately model and simulate the skull/ mandible
musculoskeletal structure of the Smilodon
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Preliminary Simulations
• Undertaken using traditional articulated mechanical system
tools.
• Virtual Simulation of Mechanical Saber-Tooth Cat
• Discovery Channel Model
Discovery Channel Model
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December 16th 2005
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Virtual Recreation
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Simulation of Mechanical Smilodon
• Implemented using a prescribed motion analysis within
VisualNastran
• Simulation was successful but more complexity was desired.
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December 16th 2005
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Virtual Prototyping of Smilodon from Fossil Records
• 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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Smilodon Virtual Prototype – VisualNastran
• Constraints were placed on the system
to represent:
• The simulation was met with limitations:
• Muscles
 Linearin
Actuators
• Due to the software's inability to handle
redundancy
terms of
resolving the multiple muscle forces
an inverse
• in
Skull/
Mandibledynamics
Interaction 
setting.
Revolute Joint
• These shortcomings provided the motivation for the
development of our own low-order computationally tractable
model based on screw-theoretic methods.
• External forces (or alternately a prescribed
motion) was applied to the skull as userspecified input to the system.
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December 16th 2005
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Our Model
• 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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December 16th 2005
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Model Set Up
• 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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December 16th 2005
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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
•
(XE, YE) End Effector Frame:
•
(XU, YU) Upper Jaw Frame:
- Attached to Skull (Upper Jaw)
- Related to Inertial Frame
through jaw gape angle q.
- Created with the application point of the external/ desired or bite force.
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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Screw Theoretic Modeling
• Each muscle is modeled as a Revolute-Prismatic-Revolute (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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Screw Theoretic Modeling
• 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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Screw Theoretic Modeling
• Unit screws
• Prismatic
Revolute Joints
Joints
• Unit Screw with a pitch of infinity
zero (l =(l0)=∞)
Upper
Revolute
Prismatic
Joint Joint
Michael J. Del Signore
December 16th 2005
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Lower Revolute Joint
Jaw Revolute 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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December 16th 2005
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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)
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December 16th 2005
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Jaw Joint
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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:
J%i -
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December 16th 2005
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Modified Jacobian, in-active
joints only.
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Screw Theoretic Modeling
• System Equilibrium Equation
• Collect all SNRS’s – Prismatic
Joints and Jaw Joint.
fP – Particular Solution
• Equilibrating force field
• Least-squares solution
fH – Homogeneous Solution
• Interaction force field
• Redundancy Resolution
• Pseudo-Inverse Solution
• Used to ensure that all
muscle forces are acting in
the same direction.
Pseudo-Inverse of W
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Muscle Optimization
• 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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December 16th 2005
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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.
|
| 
 |
v
v
#


¢
S  U1 U 2
Ur
f = W $W + S ν


|
| 
 |
Design Variables
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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Muscle Force Optimization
• Pseudo-Inverse Solution
• Force Optimization
• Separate Solution Components
Jaw Joint Reaction Forces
 fo 21   fPo 21   So  2r  
 ν  r 1



 fm nm 1   fPm nm 1  Sm  nm r  
Actuator (Muscle) Forces
Michael J. Del Signore
December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Muscle Activity Optimization
• Normalized Muscle Activity
Muscle Force
• System Equilibrium
Equation (Activity)
Maximum Muscle Force
• Muscle/ reaction forces in
terms of activity.
Michael J. Del Signore
December 16th 2005
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• Pseudo-Inverse Solution
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Muscle Activity Optimization
• Pseudo-Inverse (Activity)
Solution
• Activity Optimization
• Separate Solution Components
Jaw Joint Reaction Activities
 fo   f   So  
    Po   
ν
 fm   fPm   Sm  
• Forces
Actuator (Muscle) Activities
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December 16th 2005
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Implementation into a MATLAB Graphical-User-Interface (GUI)
• 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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
MATLAB Analysis GUI
Mode2
Stepped
Static
Mode
Selection
Force
Definition
• Applied
Muscle
Location
Range
Definition
Definition
Mode1
Optimization
Results
and
- Single
Plot Options
Static
Jaw
Gape
Definition
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December 16th 2005
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GUI Solution Validation
• System Set Up
• One Active Muscle
• D.O.F = nm
• Solved Analytically
• Analytic Solution
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December 16th 2005
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GUI Solution Validation
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December 16th 2005
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Virtual Model Simulation and Analysis Framework
• 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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December 16th 2005
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Data / Information Flow
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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
User Inputs
• Desired Jaw Gape Angle Curve
• Jaw gape angle over time
• Simulation Time
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December 16th 2005
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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
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December 16th 2005
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User Inputs
• 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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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
VisualNastran Simulink Block
• Dynamic in-the-loop link between Simulink and
VisualNastran.
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December 16th 2005
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VisualNastran Model
• Two-Dimensional representation of the skull/ mandible
musculoskeletal system.
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December 16th 2005
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VisualNastran Model
• Measure Bite Force
• Check for compatibility with applied bite force
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December 16th 2005
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Framework Simulations
• Four simulations
• Identical Simulation Parameters – tmax, Dt, … etc
• Varying/ Constant Jaw Gape
• Varying/ Constant Bite Force
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December 16th 2005
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Simulation 1 – Constant Angle/ Constant Force
• Angle - 30°
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December 16th 2005
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Force - 1000N
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Simulation 2 – Constant Angle/ Varying Force
• Angle - 30°
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December 16th 2005
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Force - 1000N to 500N
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Simulation 3 – Varying Angle/ Constant Force
• Angle - 30° to 0°
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December 16th 2005
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Force - 1000N
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Simulation 4 – Varying Angle/ Varying Force
• Angle - 30° to 0°
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December 16th 2005
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Force - 1000N to 500N
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Simulation Summary
• Error peaks occur at same time.
• Simulation Settling.
• Rotation of arbitrary material.
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December 16th 2005
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Design of a Mechanical Bite-Testing Prototype
• 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.
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December 16th 2005
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Dentition Castings
• CAD models developed from fossil records.
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December 16th 2005
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Mechanism Adjustability
• Ensure proper dentition location.
• Locks in place during use.
Skull/Mandible
Location
Rotation Point Location
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December 16th 2005
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Mechanical Prototype – Force/ Torque Analysis GUI
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December 16th 2005
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Future Work
• 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).
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Conclusions
• 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.
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December 16th 2005
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Introduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion
Conclusions
• 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.
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December 16th 2005
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Thank You
Questions?
Michael J. Del Signore
December 16th 2005
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