A Screw-Theoretic Framework for Musculoskeletal Modeling and Analysis

A Screw-Theoretic Framework for Musculoskeletal Modeling and

Analysis

Michael J. Del Signore ([email protected])

Advisor: Dr. Venkat KroviMechanical and Aerospace Engineering State University of New York at Buffalo

December 16th 2005

Michael J. Del SignoreDecember 16th 2005Slide 2 of 57

Automation, Robotics, and Mechatronics LAB – SUNY Buffalo

Agenda

• Introduction• Background• Case Scenario

• System Modeling• GUI Implementation

• Simulation Framework• Mechanical Prototype Design • Future Work• Conclusion



Motivation

• 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.

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



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.

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

Musculoskeletal System Analysis

Powerful Tool

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




Challenges

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

• Complex Asymmetric Geometric Complex Asymmetric Geometric Shapes (i.e. muscle, bone).Shapes (i.e. muscle, bone).

• Each specimen is unique.Each specimen is unique.

• Dealing with (trying to simulate) Dealing with (trying to simulate) living tissue.living tissue.

• Multiple Muscles: More actuators Multiple Muscles: More actuators than degrees of freedom.than degrees of freedom.

• Infinite set of actuator (muscle) Infinite set of actuator (muscle) forces can produce the same end-forces can produce the same end-effector force.effector force.

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




• 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

Existing Tools

Musculoskeletal System Analysis




• While being successful at handling complex musculoskeletal systems these programs require:

• In depth physiological knowledge. • Extensive application specific programming and coding.

Existing Tools• Musculoskeletal System Analysis Tools

• In resent years tools have been developed to specifically model and analyze musculoskeletal systems.

• Examples: SIMM, AnyBody, LifeMod …

High Degree of Modeling and

Simulation Detail

Rapid Real-Time Simulation and Analysis Relatively

Impossible




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).




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]




Mathematical Preliminaries• Screw Coordinates

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.

Unit Screw

u - Unit vector pointing along the direction of the screw axis.

0ˆ ˆ ˆu r u ul´ += v- Moment of the screw axis about the origin.

- Location of a point on the screw axis. rv

- Pitch, the ratio of translation to rotation.




Mathematical Preliminaries• Screw Coordinates

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.

Twists (Velocity)

Wrenches (Force)

Linear Velocity

Angular Velocity

Applied Force

Moment caused by Fo




Musculoskeletal Analysis of the Jaw Closure of the Smilodon

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




Preliminary Simulations• Undertaken using traditional articulated mechanical system

tools.


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

Discovery Channel Model Virtual Recreation



Simulation of Mechanical Smilodon• Implemented using a prescribed motion analysis within

VisualNastran

• Simulation was successful but more complexity was desired.




Virtual Prototyping of Smilodon from Fossil Records

• Virtual representation created from actual fossil records

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




Smilodon Virtual Prototype – VisualNastran


• Constraints were placed on the system to represent:

• 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.

• 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.



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



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.




Model Set UpIntroduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion

• 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 • (XE, YE) End Effector Frame:

- Created with the application point of the external/ desired or bite force.




• 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.

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.





• Unit screws

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

Upper Revolute Joint Lower Revolute Joint Jaw Revolute Joint


• Prismatic Joints• Unit Screw with a pitch of infinity ( =∞)

Prismatic Joint




• Unit screws


Unit Direction Vectors

Distance Vectors




• 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




Screw Theoretic ModelingIntroduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion

• 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 Prismatic Joint Formulation

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

iJ%- Modified Jacobian, in-active

joints only..

Jaw Joint Formulation Jaw Joint Formulation



Screw Theoretic ModelingIntroduction Background Case Scenario Simulation Mechanical Prototype Future Work Conclusion

• 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

• Used to ensure that all muscle forces are acting in the same direction.

• Redundancy Resolution • Pseudo-Inverse Solution

Pseudo-Inverse of W



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




• Find the full rank null space component of the system.• Singular-Value-Decomposition of H

• – Number of columns of containing non-zero singular values.

Muscle Force Optimization


Rank deficient

1 2

| | |

| | |

S U U U #

Wf W $ νS= + ¢v v

Design Variables



22 1 2 1

1

1 1m m m

oo Po

mm n Pm n n

Sf fν

Sf f

• Pseudo-Inverse Solution

• Separate Solution Components

Muscle Force Optimization

Jaw Joint Reaction Forces

Actuator (Muscle) Forces

• Force Optimization




Muscle Activity Optimization• Normalized Muscle Activity


Muscle Force

Maximum Muscle Force

• Muscle/ reaction forces in terms of activity.

• System Equilibrium Equation (Activity)

• Pseudo-Inverse Solution



Muscle Activity Optimization


• Pseudo-Inverse (Activity) Solution

• Separate Solution Components

Jaw Joint Reaction Activities

Actuator (Muscle) Activities

• Activity Optimization

oo Po

mPmm

Sf fν

Sff

• Forces



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.



MATLAB Analysis GUI


• Mode Selection• Applied Force Definition• Muscle Location Definition • Muscle Range Definition• Jaw Gape Definition• Optimization and Plot Options• Mode1 Results - Single Static• Mode2 Results - Stepped Static



GUI Solution Validation


• Analytic Solution

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

• Solved Analytically



GUI Solution Validation




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.




Data / Information Flow




User Inputs• Desired Jaw Gape Angle Curve

• Jaw gape angle over time• Simulation Time




User Inputs• Desired Bite Force Curve

• Bite Force with respect to upper jaw over time




User Inputs• Initial Muscle Locations at (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



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

VisualNastran.




VisualNastran Model• Two-Dimensional representation of the skull/ mandible

musculoskeletal system.




VisualNastran Model• Measure Bite Force

• Check for compatibility with applied bite force




Framework Simulations• Four simulations

• Identical Simulation Parameters – tmax, t, … etc

• Varying/ Constant Jaw Gape• Varying/ Constant Bite Force




Simulation 1 – Constant Angle/ Constant Force


• Angle - 30° Force - 1000N



Simulation 2 – Constant Angle/ Varying Force


• Angle - 30° Force - 1000N to 500N



Simulation 3 – Varying Angle/ Constant Force


• Angle - 30° to 0° Force - 1000N



Simulation 4 – Varying Angle/ Varying Force


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



Simulation Summary


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



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.




Dentition Castings• CAD models developed from fossil records.




Mechanism Adjustability


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

Rotation Point LocationSkull/Mandible Location



Mechanical Prototype – Force/ Torque Analysis GUI




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).




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.




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.




Thank You

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A Screw-Theoretic Framework for Musculoskeletal Modeling and Analysis