Transcript MEGN 536 – Computational Biomechanics

MEGN 536 – Computational Biomechanics
Prof. Anthony J. Petrella
Inverse Kinematics in Musculoskeletal Analysis
 Need to know joint angles that
define model configuration at each
increment in the motion
 Markers used to track motion and
match model to individual
 We can then do dynamics calcs
to find joint reactions / torques
 Muscle forces and joint contact
forces can also be estimated
Ali, et al., CMBBE, 2013
Inverse Kinematics in Musculoskeletal Analysis
Problems…
 Model markers don’t perfectly
match subject
 Multiple sources of exp. error
 Segments may have multiple
markers  over-determined
 Cannot solve for model config
exactly (closed form)
 Have to use alternate method,
such as optimization
Ali, et al., CMBBE, 2013
Optimization in Inverse Kinematics
 We want to configure / orient the model so that the
model markers are as close as possible to the exp
marker measurements
 We can use RMS error as an indication of the overall
error in all marker pairs (model-experiment)…
 𝑒𝑟𝑟𝑜𝑟 =
𝑁
𝑖=1
2
𝑝𝑖𝑥 −𝑞𝑖𝑥 2 + 𝑝𝑖𝑦 −𝑞𝑖𝑦
𝑁
 Where 𝒑𝑖 are exp coords and 𝒒𝑖 are model coords
 The goal is to find the model configuration that
minimizes RMS error at each increment in the motion
Optimization Problem Statement
 Find x that minimizes f(x), subject to A*x ≤ b
and Aeq*x = beq
x1
x= :
xn
f(x)
A*x ≤ b
Aeq*x = beq
design vector
objective function
inequality constraints  e.g. -Fquad ≤ 0
equality constraints
 e.g. SFx = 0
Optimization Problem Statement
 Note that the goal is to minimize the objective
function in an absolute sense – not to make it small,
but to make it as negative as possible
 If you are solving a problem in which you wish to
maximize a value (e.g., find the maximum volume of
a box using a fixed amount of material for the sides of
the box), then you simply minimize the negative of
the objective function…
f(x) = -fvol(x) = -(L * W * H)
Optimization Example
 Given: the plane x1 + 2x2 + 4x3 = 7
 Problem: find point on the plane closest to the origin
design vector
x1
x = x2
x3
obj. function f(x)= dist = sqrt(x12 + x22 +x32)
ineq. constraints
eq. constraints
A*x ≤ b
- none x1 + 2x2 + 4x3 = 7
(Aeq = [1 2 4]; beq = [7])
Optimization Example - Solution
10
8
6
Z-Axis
4
2
0
-2
-4
10
-6
-10
0
-5
0
5
10
-10
Y-Axis
X-Axis
Optimization in MATLAB
 Type ‘doc fmincon’ at the MATLAB command line
 All of the inputs and outputs of fmincon are
explained in the MATLAB documentation
 To use fmincon you will need to write a MATLAB
function and use function handles