Herman J. Woltring

06-02-1991, 01:46 AM

Dear Biomch-L readers,

The two recent postings from Ian Stokes and Ton van den Bogert on the Instan-

taneous Centre of Rotation (ICR) contain many points worthy of further consi-

deration, but I propose to address only some of them at this time -- slightly

patterned after the original November 1990 discussion on this list.

Let me begin with some definitions and formulae. A joint is defined as the

interface of two adjacent body segments such as tibia and femur in the case

of the knee, or femur and pelvis in the case of the hip, and joint movement

is defined as movement of the distal segment w.r.t. the proximal one, where

the moving, distal segment is viewed under a `free body' paradigm. Thus, we

are *not* concerned with contact points and planes, or with the curvature

centres of segment surfaces, but purely with the kinematics and *net* kine-

tics at the joint. My rationale for this limitation is that, at the present

state of the art in (routine) *in-vivo* analysis of human motion, it is feas-

ible to assess kinematics and net kinetics for the segments and joints, but

not the (highly important) joint contact, ligament, and muscular/tendon forces

that together sum up to the joint's net forces and moments at any selected

reference point. Various investigators believe that net forces and moments

at, and net power flows through the joints may provide meaningful information

not sufficiently available from other movement descriptors; research during

the past decade has been concerned with providing the methodological tools

for doing this properly.

Whether 2-D/3-D centres of rotation are meaningful in more comprehensive

analyses is still waiting for proper experimentation that will hopefully

take place in not too distant a future. I agree with many of Ian's and Ton's

concerns *once* we go beyond net joint kinetics, in an attempt to address

the forces in (and the moments caused by) individual muscles and tendons,

ligaments, and interbone contact points. For those of you who have German,

Alfred Menschik's "Biometrie -- Das Konstruktions-prinzip des Kniegelenks,

des Hueftgelenks, der Beinlaenge und der Koerpergroesze" (i.e., Biometrics

-- the construction principle of the knee joint, hip joint, leg length, and

body size), Springer, Berlin etc. 1987 contains many good ideas about 2-D

centres of rotation, centres of curvature, and surface shape. On the 3-D

level, no such work is available at this time, and it seems realistic to

surmise that data on these matters will not soon be routinely available,

since they require anatomical information from CT and/or MRI plus much

additional, 3-D post-processing.

Given external measurements of the kinematics and prior estimated mass

distribution parameters of the moving, distal segment, one can assess the

position, attitude, velocities, accelerations, net force and net moment in

3-D at the segment's centre of mass (c.o.m.):

Position vector Pc Attitude matrix Rc

Translation velocity vector Vc Rotation velocity vector Wc

Translation acceleration vector Ac Rotation acceleration vector Zc

Net force Fc Net moment Mc

Linear power term Qc = Fc'Vc Rotational power term Uc = Mc'Wc

Total segment power Tc = Qc + Uc

For the time being, these data are asumed to be exact, without any `noise'.

Because of the equivalence of forces and moments at different points for a

freely moving body, these entities can also be assessed with respect to an

other reference point Pa than the c.o.m. Pc, with the following relations

between these entities (* denotes the vector product, and ' the dot product):

Pa = Pc + Pac Ra = Rc

Va = Vc + Wc * Pac Wa = Wc

Fa = Fc Ma = Mc + Fc * Pac

Qa = Fa'Va Ua = Ma'Wc

Total segment power Ta = Qa + Ua

with Pac = Pa - Pc the position vector from the c.o.m. Pc to the new reference

point Pa (i.c., the `free body' equivalent of the physical joint). Substitu-

tion and some algebraic manipulation shows that Qa = Qc, as one should expect:

irrespective of what reference point Pa is used, the power generated or dissi-

pated in the moving segment should remain the same as when Pc is used. If the

external, net force and moment working at the distal end are know (zero in the

case of a swinging foot, hand, or head; measured via a force plate at the foot

during stance; obtained via Newton's third law if there is another, more dis-

tal segment at the segment under consideration), these should be accounted for

in the values of Fc and Mc: the remainder is the equivalent force, moment and

power at the proximal joint.

There ar two important uses of the net joint moment Ma in today's Functional

Movement Analysis:

( I) as a potentially useful entity for clinical and/or ambulatory care in

itself;

(II) as an intermediate entity to assess total joint power Ta from the rota-

tional power Ua, while the linear power term Qa is ignored.

The above formulae show what problems are incurred here:

( I) the value of the net joint moment Ma depends on the position Pac, while

the net joint force Fa is invariant with Pac;

(II) the total joint power Ta at Pac is equal to the rotational power Ua if

and only if the linear power term Qa = Fa'Va vanishes.

For ( I), it becomes necessary to agree upon what point Pa should be taken

in order to obtain valid comparisons between institutions and protocols;

for (II), it becomes necessary to assess under what circumstances the linear

power term vanishes, AND/OR to persuade the biomechanical community that the

assumption of a vanishing, linear power term does not always hold.

In classical, planar analysis with the assumption of fixed joint centres at

the hip, knee and ankle, Va is forced to be zero at the joint, so the linear

term does indeed vanish. Recent reports from unpublished sources and in the

`grey literature' suggest that this approximation entails significant power

balance errors, and this is one motivation to investigate better free-body

models.

While the asssumption of a fixed, 3-D centre of rotation in the hip seems

reasonable for healthy, normal movement, it is more realistic to allow full

6 d.o.f. movement in (the pathological hip and in) the knee as apparent from

research in The Netherlands and Belgium during the past decade. For the knee

joint, a variable axis has been observed about and along which so-called heli-

cal (or screwing) motions occur: in-vivo knee motion is certainly not pure

rotation about a fixed or variable axis, but significant *shift* (translation)

velocities may simultaneously take place along such an axis. If the net joint

force is not perpendicular to this instantaneous shift velocity, the linear

power term Qc will not vanish. How strong this component is during realistic

movement has not been analysed yet, and I would propose that it is about time

to stop pure speculation and to do some proper experiments. May-be, that's

what Newton meant when he said "hypotheses non fungor" -- I don't entertain

hypotheses?

In my mind, the most straightforward generalisation from a joint with a fixed

pivot or axis to a more complicated system is to find those points Pa that

maximally approximate the `ideal' situation of a joint centre with zero (2-D)

or minimal (3-D) velocity. In the 2-D case, Va is zero at the 2-D ICR with

position

Picr = Pc + R(90) Vc / Wc

(in the 2-D case, Wc is a scalar, while R(90) is an attitude matrix corres-

ponding with a clockwise rotation through 90 degrees).

In the 3-D case, the Instantaneous Helical Axis or IHA about and along which

the segment is instantaneously moving is the locus of all points with the

smallest (shift) velocity of all points on the moving segment. The formulae

for the IHA have been stated before on this list, during the November 1990

exchanges with Fabio Catani and Ed Chao. Both the 2-D ICR and the 3-D IHA

become undefined if Wc goes through zero, and they are *ill-determined* from

noisy data if Wc becomes `small' -- however, one can assess them via analy-

tical continuation in the higher derivatives if Wc is only momentarily equal

to zero, i.e., if the rotation acceleration Zc is non-zero at that time. If

also Zc is too small, it becomes arguable that the movement is insufficient-

ly rotational to warrant the quest for some optimal centre or axis of rota-

tion -- this is a point for further research.

So, these are my "in defense of the 2-D ICR and 3-D IHA" for problem (II)

above: any point on the IHA will do, since they all have the same (shift)

velocity, and since Fa is invariant with Pa. For the 2-D ICR, this analysis

provides also a unique choice for problem (I), but *not* for the 3-D IHA since

Ma is *not* invariant with the position of Pa on the IHA. By straightforward

continuation of the `minimal, instantaneous movement requirement', I have pro-

posed to choose the 3-D ICR, i.e., that point on the IHA about which the IHA

itself performs a helical movement, for which it has been shown that it has

the smallest acceleration of all points on the IHA. If this point becomes

ill-defined because of vanishing direction change velocity of the IHA, the

quest for this unique point may not be warranted because of insufficient

rotatory effects (perhaps, one might consider that point on the IHA for which

the moment becomes as small as possible, i.e., coplanar with Fc and with the

IHA -- a pure hypothesis to be tried out by experiment). Whether either the

3-D ICR or the co-planar moment point can be located far away from the physi-

cal joint is an interesting research question.

The above was concerned with `what' and `why' (theory, science); the question

`how' (practice, engineering) remains. Here I hope to to tackle some of Ian

Stokes' thought-provoking statements.

In an inverse dynamics situation, *all data* needed for assessing the 2-D ICR,

the 3-D IHA and the central pivot or 3-D ICR are also needed for other aspects

of the modelling and calculation procedure. Contact points etc. require addi-

tional measurements, which seems to counter Ian Stokes' argument

"I question why he [i.e., Fabio Catani -- HJW] should use the center

of rotation rather than a more easily located reference".

There is one exception: if Wc is zero only instantaneously, i.e., the rotation

acceleration Zc is significantly non-zero, the IHA may be calculated by ana-

lytical continuation requiring knowledge of the third rotation derivative

(`jerk'); however, is is currently not known how strong its contribution is.

Furthermore, nowhere in the above analysis reference has been made to `small

motion steps' which Ian claims cause tremendous difficulties in estimating

joint centres of rotation,

"Instantaneous center of rotation is impossible to measure in

practice, since the errors tend to infinity as the increment

of motion tends to zero."

However, we typically do not *measure* these centres (and axes) of rotation,

but we try to *estimate* them, to the best of our abilities, from inexact

data, by combining other knowledge with our noisy measurements. If I were

to estimate instantaneous velocity from a very small step, with each meas-

urement afflicted by additive, uncorrelated, zero-mean noise with standard

deviation sigma, and if the time step between the measurements is T, then

the standard deviation of the finite step velocity estimate becomes

sigma * sqrt(2) / T, going to infinity if T becomes vanishingly small.

Does this mean that we cannot estimate velocities from noisy position data?

Clearly not, *if* we have reason to assume that the signal underlying the

noisy data has a finite bandwidth, and that the noise contains high frequen-

cies -- which are amplified in (approximate) differentiation. In that case,

there are many good algorithms around for reducing the effect of noise by

low-pass smoothing/filtering. After such filtering, the noise in adjacent

samples in the data record is no longer uncorrelated, and the above formula

for the finite difference estimate's standard deviation no longer holds.

Under an ideal low-pass filtering paradigm, the above s.d. formula is repla-

ced by sigma * Wo * SQRT{(Wo T)/( 3 PI)}, where Wo is the low-pass filter's

cut-off freqency. This formula shows that it may be advantageous to have a

very small step size T, as long as the model assumptions of white, uncorre-

lated noise with standard deviation sigma in each sample are met. Note

that the smoothing operation should be applied to the raw data or to such

transformation of them which ensures the low-pass nature of the underlying

signal and the wide-band, additive nature of the noise lest low-pass fil-

tering becomes an invalid procedure.

Furthermore, if the movement is rigid, the condition that interlandmark

distances (e.g., in a photogrammetric measurement setup) are constant should

be imposed; in the opposite case, one should entertain models that allow

separating pure movement from pure deformation. For the latter, continuum

mechanics offers some interesting models.

Herman J. Woltring, Eindhoven/NL

The two recent postings from Ian Stokes and Ton van den Bogert on the Instan-

taneous Centre of Rotation (ICR) contain many points worthy of further consi-

deration, but I propose to address only some of them at this time -- slightly

patterned after the original November 1990 discussion on this list.

Let me begin with some definitions and formulae. A joint is defined as the

interface of two adjacent body segments such as tibia and femur in the case

of the knee, or femur and pelvis in the case of the hip, and joint movement

is defined as movement of the distal segment w.r.t. the proximal one, where

the moving, distal segment is viewed under a `free body' paradigm. Thus, we

are *not* concerned with contact points and planes, or with the curvature

centres of segment surfaces, but purely with the kinematics and *net* kine-

tics at the joint. My rationale for this limitation is that, at the present

state of the art in (routine) *in-vivo* analysis of human motion, it is feas-

ible to assess kinematics and net kinetics for the segments and joints, but

not the (highly important) joint contact, ligament, and muscular/tendon forces

that together sum up to the joint's net forces and moments at any selected

reference point. Various investigators believe that net forces and moments

at, and net power flows through the joints may provide meaningful information

not sufficiently available from other movement descriptors; research during

the past decade has been concerned with providing the methodological tools

for doing this properly.

Whether 2-D/3-D centres of rotation are meaningful in more comprehensive

analyses is still waiting for proper experimentation that will hopefully

take place in not too distant a future. I agree with many of Ian's and Ton's

concerns *once* we go beyond net joint kinetics, in an attempt to address

the forces in (and the moments caused by) individual muscles and tendons,

ligaments, and interbone contact points. For those of you who have German,

Alfred Menschik's "Biometrie -- Das Konstruktions-prinzip des Kniegelenks,

des Hueftgelenks, der Beinlaenge und der Koerpergroesze" (i.e., Biometrics

-- the construction principle of the knee joint, hip joint, leg length, and

body size), Springer, Berlin etc. 1987 contains many good ideas about 2-D

centres of rotation, centres of curvature, and surface shape. On the 3-D

level, no such work is available at this time, and it seems realistic to

surmise that data on these matters will not soon be routinely available,

since they require anatomical information from CT and/or MRI plus much

additional, 3-D post-processing.

Given external measurements of the kinematics and prior estimated mass

distribution parameters of the moving, distal segment, one can assess the

position, attitude, velocities, accelerations, net force and net moment in

3-D at the segment's centre of mass (c.o.m.):

Position vector Pc Attitude matrix Rc

Translation velocity vector Vc Rotation velocity vector Wc

Translation acceleration vector Ac Rotation acceleration vector Zc

Net force Fc Net moment Mc

Linear power term Qc = Fc'Vc Rotational power term Uc = Mc'Wc

Total segment power Tc = Qc + Uc

For the time being, these data are asumed to be exact, without any `noise'.

Because of the equivalence of forces and moments at different points for a

freely moving body, these entities can also be assessed with respect to an

other reference point Pa than the c.o.m. Pc, with the following relations

between these entities (* denotes the vector product, and ' the dot product):

Pa = Pc + Pac Ra = Rc

Va = Vc + Wc * Pac Wa = Wc

Fa = Fc Ma = Mc + Fc * Pac

Qa = Fa'Va Ua = Ma'Wc

Total segment power Ta = Qa + Ua

with Pac = Pa - Pc the position vector from the c.o.m. Pc to the new reference

point Pa (i.c., the `free body' equivalent of the physical joint). Substitu-

tion and some algebraic manipulation shows that Qa = Qc, as one should expect:

irrespective of what reference point Pa is used, the power generated or dissi-

pated in the moving segment should remain the same as when Pc is used. If the

external, net force and moment working at the distal end are know (zero in the

case of a swinging foot, hand, or head; measured via a force plate at the foot

during stance; obtained via Newton's third law if there is another, more dis-

tal segment at the segment under consideration), these should be accounted for

in the values of Fc and Mc: the remainder is the equivalent force, moment and

power at the proximal joint.

There ar two important uses of the net joint moment Ma in today's Functional

Movement Analysis:

( I) as a potentially useful entity for clinical and/or ambulatory care in

itself;

(II) as an intermediate entity to assess total joint power Ta from the rota-

tional power Ua, while the linear power term Qa is ignored.

The above formulae show what problems are incurred here:

( I) the value of the net joint moment Ma depends on the position Pac, while

the net joint force Fa is invariant with Pac;

(II) the total joint power Ta at Pac is equal to the rotational power Ua if

and only if the linear power term Qa = Fa'Va vanishes.

For ( I), it becomes necessary to agree upon what point Pa should be taken

in order to obtain valid comparisons between institutions and protocols;

for (II), it becomes necessary to assess under what circumstances the linear

power term vanishes, AND/OR to persuade the biomechanical community that the

assumption of a vanishing, linear power term does not always hold.

In classical, planar analysis with the assumption of fixed joint centres at

the hip, knee and ankle, Va is forced to be zero at the joint, so the linear

term does indeed vanish. Recent reports from unpublished sources and in the

`grey literature' suggest that this approximation entails significant power

balance errors, and this is one motivation to investigate better free-body

models.

While the asssumption of a fixed, 3-D centre of rotation in the hip seems

reasonable for healthy, normal movement, it is more realistic to allow full

6 d.o.f. movement in (the pathological hip and in) the knee as apparent from

research in The Netherlands and Belgium during the past decade. For the knee

joint, a variable axis has been observed about and along which so-called heli-

cal (or screwing) motions occur: in-vivo knee motion is certainly not pure

rotation about a fixed or variable axis, but significant *shift* (translation)

velocities may simultaneously take place along such an axis. If the net joint

force is not perpendicular to this instantaneous shift velocity, the linear

power term Qc will not vanish. How strong this component is during realistic

movement has not been analysed yet, and I would propose that it is about time

to stop pure speculation and to do some proper experiments. May-be, that's

what Newton meant when he said "hypotheses non fungor" -- I don't entertain

hypotheses?

In my mind, the most straightforward generalisation from a joint with a fixed

pivot or axis to a more complicated system is to find those points Pa that

maximally approximate the `ideal' situation of a joint centre with zero (2-D)

or minimal (3-D) velocity. In the 2-D case, Va is zero at the 2-D ICR with

position

Picr = Pc + R(90) Vc / Wc

(in the 2-D case, Wc is a scalar, while R(90) is an attitude matrix corres-

ponding with a clockwise rotation through 90 degrees).

In the 3-D case, the Instantaneous Helical Axis or IHA about and along which

the segment is instantaneously moving is the locus of all points with the

smallest (shift) velocity of all points on the moving segment. The formulae

for the IHA have been stated before on this list, during the November 1990

exchanges with Fabio Catani and Ed Chao. Both the 2-D ICR and the 3-D IHA

become undefined if Wc goes through zero, and they are *ill-determined* from

noisy data if Wc becomes `small' -- however, one can assess them via analy-

tical continuation in the higher derivatives if Wc is only momentarily equal

to zero, i.e., if the rotation acceleration Zc is non-zero at that time. If

also Zc is too small, it becomes arguable that the movement is insufficient-

ly rotational to warrant the quest for some optimal centre or axis of rota-

tion -- this is a point for further research.

So, these are my "in defense of the 2-D ICR and 3-D IHA" for problem (II)

above: any point on the IHA will do, since they all have the same (shift)

velocity, and since Fa is invariant with Pa. For the 2-D ICR, this analysis

provides also a unique choice for problem (I), but *not* for the 3-D IHA since

Ma is *not* invariant with the position of Pa on the IHA. By straightforward

continuation of the `minimal, instantaneous movement requirement', I have pro-

posed to choose the 3-D ICR, i.e., that point on the IHA about which the IHA

itself performs a helical movement, for which it has been shown that it has

the smallest acceleration of all points on the IHA. If this point becomes

ill-defined because of vanishing direction change velocity of the IHA, the

quest for this unique point may not be warranted because of insufficient

rotatory effects (perhaps, one might consider that point on the IHA for which

the moment becomes as small as possible, i.e., coplanar with Fc and with the

IHA -- a pure hypothesis to be tried out by experiment). Whether either the

3-D ICR or the co-planar moment point can be located far away from the physi-

cal joint is an interesting research question.

The above was concerned with `what' and `why' (theory, science); the question

`how' (practice, engineering) remains. Here I hope to to tackle some of Ian

Stokes' thought-provoking statements.

In an inverse dynamics situation, *all data* needed for assessing the 2-D ICR,

the 3-D IHA and the central pivot or 3-D ICR are also needed for other aspects

of the modelling and calculation procedure. Contact points etc. require addi-

tional measurements, which seems to counter Ian Stokes' argument

"I question why he [i.e., Fabio Catani -- HJW] should use the center

of rotation rather than a more easily located reference".

There is one exception: if Wc is zero only instantaneously, i.e., the rotation

acceleration Zc is significantly non-zero, the IHA may be calculated by ana-

lytical continuation requiring knowledge of the third rotation derivative

(`jerk'); however, is is currently not known how strong its contribution is.

Furthermore, nowhere in the above analysis reference has been made to `small

motion steps' which Ian claims cause tremendous difficulties in estimating

joint centres of rotation,

"Instantaneous center of rotation is impossible to measure in

practice, since the errors tend to infinity as the increment

of motion tends to zero."

However, we typically do not *measure* these centres (and axes) of rotation,

but we try to *estimate* them, to the best of our abilities, from inexact

data, by combining other knowledge with our noisy measurements. If I were

to estimate instantaneous velocity from a very small step, with each meas-

urement afflicted by additive, uncorrelated, zero-mean noise with standard

deviation sigma, and if the time step between the measurements is T, then

the standard deviation of the finite step velocity estimate becomes

sigma * sqrt(2) / T, going to infinity if T becomes vanishingly small.

Does this mean that we cannot estimate velocities from noisy position data?

Clearly not, *if* we have reason to assume that the signal underlying the

noisy data has a finite bandwidth, and that the noise contains high frequen-

cies -- which are amplified in (approximate) differentiation. In that case,

there are many good algorithms around for reducing the effect of noise by

low-pass smoothing/filtering. After such filtering, the noise in adjacent

samples in the data record is no longer uncorrelated, and the above formula

for the finite difference estimate's standard deviation no longer holds.

Under an ideal low-pass filtering paradigm, the above s.d. formula is repla-

ced by sigma * Wo * SQRT{(Wo T)/( 3 PI)}, where Wo is the low-pass filter's

cut-off freqency. This formula shows that it may be advantageous to have a

very small step size T, as long as the model assumptions of white, uncorre-

lated noise with standard deviation sigma in each sample are met. Note

that the smoothing operation should be applied to the raw data or to such

transformation of them which ensures the low-pass nature of the underlying

signal and the wide-band, additive nature of the noise lest low-pass fil-

tering becomes an invalid procedure.

Furthermore, if the movement is rigid, the condition that interlandmark

distances (e.g., in a photogrammetric measurement setup) are constant should

be imposed; in the opposite case, one should entertain models that allow

separating pure movement from pure deformation. For the latter, continuum

mechanics offers some interesting models.

Herman J. Woltring, Eindhoven/NL