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Unformatted text preview: Lecture 9 Forward Dynamic Simulations EML 5595 Mechanics of the Human Locomotor System Outline Numerical Integration Simulation Problems Journal Article Reviews Gilchrist and Winter (1997) Outline Numerical Integration Forward Dynamics Process Measure kinematics and ground reactions Perform inverse dynamics analysis to estimate joint torques Perform optimization analysis to estimate muscle m scle forces Define initial positions and velocities and j q p estimated joint torque or muscle force inputs Numerically integrate equations of motion to predict new motion Forward Dynamics Equations
M() = N()T + G() + V(, ) q = , u = M(q )u = N(q )T + G(q ) + V(q , u) u M(q ) -1 (N(q )T + G(q ) + V(q, u) ) q = u u u y = , y = q q y = f (y ) Types of Integrators
Let y = f (t , y ) t = h ), Explicit Euler Integrator
y i +1 = y i + f (ti , y i )h Implicit Euler Integrator
y i +1 = y i + f (ti +1 , y i +1 )h Explicit equation for y i +1 Implicit equation for y i +1
ReRe it R -write as nonlinear li rootroot-finding problem y i +1 - y i + f (ti +1 , y i +1 )h = 0
Iterate guess of y i +1 until the equation = 0 to within some tol When to Use Each Type Try an explicit integrator first classic y p g example is 4th Order Runge-Kutta integrator RungeTry an implicit integrator only if the explicit integrator is slow Integration speed is related to number of equations solved and the numerical "stiffness" of the equations " tiff " f th ti Stiff Systems Stiff systems typically exhibit motions on different time scales (e.g., a rigid body system possessing stiff springs) s stem Numerical stiffness is an issue of accuracy and stability A stiff system is one that requires an extremely small step size to meet accuracy requirements and remain stable Can you think of some examples of stiff systems? Stiff System Example Explicit integrator Implicit integrator Integration Accuracy
Integration Step Size Integrator Order Order 1st 2nd 3rd 4th 5th Outline Numerical Integration Simulation Problems Problems and Solutions
Problem Solution Slow integration Motion drift Motion instability M ti i t bilit Joint hyperextension Implicit integrator Smaller time step, higher order integrator, or spline-fit controls splineFeedback control F db k t l Ligament torques Outline Numerical Integration Simulation Problems Journal Article Reviews Gilchrist and Winter (1997) Simulation Problems What was the goal of the Gilchrist and Winter paper? What problems did they encounter in trying to achieve it? What l ti Wh t solutions did th attempt t resolve th they tt t to l these problems? Can you think of additional modifications to get their forward dynamics simulation to reproduce the experimental motion and ground reactions better? Critique Wh t aspects of the authors'' simulation approach What t f th th i l ti h
did you like? General modeling philosophy to keep things as simple as possible. i l ibl Three-dimensional model. Three No prescribed trajectories of segments. Control of trunk orientation (though achieved using feedback torques on the hips). Limitations on joint ranges of motion via torsional spring/damper systems. Visco-elastic (i.e., spring-damper) two-segment Viscospringtwofoot f t-ground contact model. footd t t d l Critique What information (if any) did you feel was missing
from the article? How was the inverse dynamics analysis performed? What was the "root" segment for the model? What were the magnitudes of the inverse dynamics residuals loads on the root segment? Why was the decision made to lock the back joint? Critique Whi h modeling assumptions did you thi k were Which d li ti think
appropriate, and which assumptions did you think were questionable? No arms? N ? Locked back joint? No off-axis springs in the foot model? off- For Next Time Download and read Fregly and Zajac (1996) Journal of Biomechanics 29: 81-90. 81Download Do nload and read Higginson et al (2006) al. Journal of Biomechanics 39: 1769-1777. 1769- ...
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This note was uploaded on 02/15/2012 for the course EML 5595 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08