Mike Schwartz wrote:

> Kinematic fitting modifies some/all joint parameters and

> joint angles, often in a nested manner, to minimize a cost

> function (e.g. cost = sum of squared distances between

> measured and predicted marker trajectories). Further

> explanation can be found in Lu T-W and O'Connor JJ, J

> Biomech, 32:129-134, Charleton et al, Gait Posture,

> 20:213-221, or Rienbolt et al, J Biomech, 38:621-626.

>

> As a personal note -- I have had **no end of grief** trying

> to use optimization approaches. I believe that this is due to

This is my personal experience also. It is easy to get carried away

with this concept because mathematically it is perfectly valid, but in

my experience you need to be very careful not have too many unknown

model parameters.

Let's say you have N unknown joint parameters p1...pN. And you have a

model with M degrees of freedom q1...qM, and you capture Nf frames of

motion data. Then the number of unknowns is N + M*Nf, since the model

parameters do not change during movement. The number of forward

kinematic equations is Nf times the number of markers times 3 (for 3D).

For large enough number of frames, and when using more than M/3 markers,

this will always exceed the number of unknowns. Then we can solve this

with a least squares method and a cost function which is the sum of

squared residuals as defined by Mike above.

The nested optimization (guess p1...pN in the outer loop, and guess

q1...qM in each frame in an inner loop) is just a way to partition the

problem but it still should find the same solution.

If you test this with simulated motion data, it always works. But in

the real world there are errors in model and data which can easily

produce false minima in the cost function. We got this optimization to

work in our two-axis ankle model (Smith et al, J Biomech 1994) but

discovered the following limitations:

(1) Motion data must span sufficient range of motion in all joints.

(2) Optimization must start from many initial guesses to ensure finding

the global optimum.

(3) Orientation of the subtalar joint in the horizontal plane was

sensitive to measuring error.

(4) With data collected during weightbearing, the method failed,

probably because there was too much foot deformation which violated the

ideal two-axis model.

I expect that these problems get worse when trying to do this for

multiple joints simultaneously. It may be possible if you carefully

select a low number of joint parameters to estimate, leaving others

fixed. For instance, the axial rotation axis of the knee would not be

estimated from kinematics, but be anatomically defined along the line

from knee center to ankle center. I would be interested in hearing

Richard Baker's observations and opinions on this.

Some other related comments.

The kinematic fitting method is a "global optimization" approach (using

the term coined by Lu & O'Connor), which assumes certain kinematic

connections between body segments (hinge, ball, or even coupled

rotation/translation as in the SIMM knee model). This contrasts with

the 6-DOF method mentioned by Frank Buczek, which makes no such

assumptions. The 6-DOF method does not suffer from potential modeling

errors (since there are no joint models) but the global optimization

approach has some nice advantages:

(1) Fewer than 3 markers per segment are needed, so you can sometimes

avoid using markers on "wobbly" sites.

(2) Less sensitive to skin motion artifacts.

(3) Degrees of freedom can be made compatible with whole body dynamics

models

(4) Degrees of freedom can be made compatible with graphics (animation)

models

Depending on the scientific question, subject population, and the

movement being studied, we must carefully balance the effects of model

error (in global optimization methods) against the effects of data error

(greater in 6-DOF methods). In our lab we use global optimization

methods, but in one of our projects we model each joint as three slider

joints and three hinge joints, i.e. six degrees of freedom. This made

the results compatible with historical data which used 6-DOF analysis.

It also allowed the joints to "absorb" marker wobble at impact which

would otherwise lead to (brief) overestimation of knee valgus and

flexion. This was a problem because the movement was jump landing, and

the subject population included some with the potential for substantial

marker wobble.

Incidentically, this example shows that the software tools for global

optimization can be used for 6-DOF analysis. You can even model some

joints with 6 DOF and other joints with 1 DOF in the same model. The

6-DOF software tools probably do not have this versatility.

The philosophical problem with 6-DOF analysis is that we have large skin

marker errors, causing errors in translational motion in the joints

which are usually larger than the translational motion itself.

Therefore assuming zero translation is a logical approach, leading to

global optimization with many practical advantages. But as the above

example illustrates, modeling the translational motion can sometimes

make the rotational motions more accurate. Even if the translational

motion itself is poorly measured and of no interest.

I agree with Dan Benoit that we must be very careful when presenting

data on non-sagittal knee rotations. Apart from the skin movement

problem, it is well known that these results are sensitive to the

orientation of the joint coordinate system (Ramakrishnan & Kadaba,

1991). This is the main motivation for using coordinate systems with

functional axes, which are hopefully more reproducible than axes based

on anatomy.

I would like to propose that for joints or degrees of freedom that are

very "stiff" with limited range of motion, the joint moment is

potentially much more reliable than the joint motion. The joint moment

can be quite large even when there is almost zero motion in the joint.

Obviously the joint angle results would be totally overwhelmed by errors

in that case.

Generally I think there is no single correct model or marker set. It

always depends on purpose of the study, movements, subjects, etc. It's

best not to be dogmatic about these things. That is easy enough to say

in basic research, but the clinical labs need standardized methods. I

think it was Mike Schwarts who mentioned that he routinely uses standard

and non-standard models and markersets simultaneously. To me this to be

a very sensible way to gain insight into how much the results are

affected by these choices. If not much, we gain some confidence in the

results of our analysis.

This was much more than I intended to write. This is a useful

discussion and it is great to have these contributions in the Biomch-L

archives.

--

Ton van den Bogert

Biomch-L co-moderator

===================================

Cleveland Clinic is ranked one of the top 3 hospitals in

America by U.S.News & World Report. Visit us online at

http://www.clevelandclinic.org for a complete listing of

our services, staff and locations.

Confidentiality Note: This message is intended for use

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and may contain information that is privileged,

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> Kinematic fitting modifies some/all joint parameters and

> joint angles, often in a nested manner, to minimize a cost

> function (e.g. cost = sum of squared distances between

> measured and predicted marker trajectories). Further

> explanation can be found in Lu T-W and O'Connor JJ, J

> Biomech, 32:129-134, Charleton et al, Gait Posture,

> 20:213-221, or Rienbolt et al, J Biomech, 38:621-626.

>

> As a personal note -- I have had **no end of grief** trying

> to use optimization approaches. I believe that this is due to

This is my personal experience also. It is easy to get carried away

with this concept because mathematically it is perfectly valid, but in

my experience you need to be very careful not have too many unknown

model parameters.

Let's say you have N unknown joint parameters p1...pN. And you have a

model with M degrees of freedom q1...qM, and you capture Nf frames of

motion data. Then the number of unknowns is N + M*Nf, since the model

parameters do not change during movement. The number of forward

kinematic equations is Nf times the number of markers times 3 (for 3D).

For large enough number of frames, and when using more than M/3 markers,

this will always exceed the number of unknowns. Then we can solve this

with a least squares method and a cost function which is the sum of

squared residuals as defined by Mike above.

The nested optimization (guess p1...pN in the outer loop, and guess

q1...qM in each frame in an inner loop) is just a way to partition the

problem but it still should find the same solution.

If you test this with simulated motion data, it always works. But in

the real world there are errors in model and data which can easily

produce false minima in the cost function. We got this optimization to

work in our two-axis ankle model (Smith et al, J Biomech 1994) but

discovered the following limitations:

(1) Motion data must span sufficient range of motion in all joints.

(2) Optimization must start from many initial guesses to ensure finding

the global optimum.

(3) Orientation of the subtalar joint in the horizontal plane was

sensitive to measuring error.

(4) With data collected during weightbearing, the method failed,

probably because there was too much foot deformation which violated the

ideal two-axis model.

I expect that these problems get worse when trying to do this for

multiple joints simultaneously. It may be possible if you carefully

select a low number of joint parameters to estimate, leaving others

fixed. For instance, the axial rotation axis of the knee would not be

estimated from kinematics, but be anatomically defined along the line

from knee center to ankle center. I would be interested in hearing

Richard Baker's observations and opinions on this.

Some other related comments.

The kinematic fitting method is a "global optimization" approach (using

the term coined by Lu & O'Connor), which assumes certain kinematic

connections between body segments (hinge, ball, or even coupled

rotation/translation as in the SIMM knee model). This contrasts with

the 6-DOF method mentioned by Frank Buczek, which makes no such

assumptions. The 6-DOF method does not suffer from potential modeling

errors (since there are no joint models) but the global optimization

approach has some nice advantages:

(1) Fewer than 3 markers per segment are needed, so you can sometimes

avoid using markers on "wobbly" sites.

(2) Less sensitive to skin motion artifacts.

(3) Degrees of freedom can be made compatible with whole body dynamics

models

(4) Degrees of freedom can be made compatible with graphics (animation)

models

Depending on the scientific question, subject population, and the

movement being studied, we must carefully balance the effects of model

error (in global optimization methods) against the effects of data error

(greater in 6-DOF methods). In our lab we use global optimization

methods, but in one of our projects we model each joint as three slider

joints and three hinge joints, i.e. six degrees of freedom. This made

the results compatible with historical data which used 6-DOF analysis.

It also allowed the joints to "absorb" marker wobble at impact which

would otherwise lead to (brief) overestimation of knee valgus and

flexion. This was a problem because the movement was jump landing, and

the subject population included some with the potential for substantial

marker wobble.

Incidentically, this example shows that the software tools for global

optimization can be used for 6-DOF analysis. You can even model some

joints with 6 DOF and other joints with 1 DOF in the same model. The

6-DOF software tools probably do not have this versatility.

The philosophical problem with 6-DOF analysis is that we have large skin

marker errors, causing errors in translational motion in the joints

which are usually larger than the translational motion itself.

Therefore assuming zero translation is a logical approach, leading to

global optimization with many practical advantages. But as the above

example illustrates, modeling the translational motion can sometimes

make the rotational motions more accurate. Even if the translational

motion itself is poorly measured and of no interest.

I agree with Dan Benoit that we must be very careful when presenting

data on non-sagittal knee rotations. Apart from the skin movement

problem, it is well known that these results are sensitive to the

orientation of the joint coordinate system (Ramakrishnan & Kadaba,

1991). This is the main motivation for using coordinate systems with

functional axes, which are hopefully more reproducible than axes based

on anatomy.

I would like to propose that for joints or degrees of freedom that are

very "stiff" with limited range of motion, the joint moment is

potentially much more reliable than the joint motion. The joint moment

can be quite large even when there is almost zero motion in the joint.

Obviously the joint angle results would be totally overwhelmed by errors

in that case.

Generally I think there is no single correct model or marker set. It

always depends on purpose of the study, movements, subjects, etc. It's

best not to be dogmatic about these things. That is easy enough to say

in basic research, but the clinical labs need standardized methods. I

think it was Mike Schwarts who mentioned that he routinely uses standard

and non-standard models and markersets simultaneously. To me this to be

a very sensible way to gain insight into how much the results are

affected by these choices. If not much, we gain some confidence in the

results of our analysis.

This was much more than I intended to write. This is a useful

discussion and it is great to have these contributions in the Biomch-L

archives.

--

Ton van den Bogert

Biomch-L co-moderator

===================================

Cleveland Clinic is ranked one of the top 3 hospitals in

America by U.S.News & World Report. Visit us online at

http://www.clevelandclinic.org for a complete listing of

our services, staff and locations.

Confidentiality Note: This message is intended for use

only by the individual or entity to which it is addressed

and may contain information that is privileged,

confidential, and exempt from disclosure under applicable

law. If the reader of this message is not the intended

recipient or the employee or agent responsible for

delivering the message to the intended recipient, you are

hereby notified that any dissemination, distribution or

copying of this communication is strictly prohibited. If

you have received this communication in error, please

contact the sender immediately and destroy the material in

its entirety, whether electronic or hard copy. Thank you.