* File: ICR TOPIC * Date: Oct 10, 1995 * Editor: Ton van den Bogert This file contains the full text of discussions on Biomch-L between November 1990 and June 1991 on the use of joint centers and joint axes in biomechanical analysis of movement. ========================================================================= Date: Mon, 5 Nov 1990 17:53:50 MET From: "FABIO CATANI, M.D." Subject: KINEMATICs AND KINETICs Sender: Biomechanics and Movement Science listserver To: Multiple recipients of I'm an orthopaedic surgeon at the University of Bologna, Istituto Ortopedico Rizzoli, Dept. of Orthopaedic Surgery. Since 1987 I have been working on orthopaedic patients using a 3-D gait analysis system - Elite. The purpose of my study is both to verify the repeatability and the accuracy of the system in clinical applications and how gait analysis assessment is helpfull in adding information regarding surgery and rehabilitation. During the past 2 years I have always used arrays of passive markers, at least 4 per array identifying pelvis, thigh, shank and foot limb segments: I used "TRACK" software (developped at M.I.T. in the Biomotion Lab) to determine 6 d.o.f. for each limb segment and to determine joint rotation. I'm trying now to calculate joint moments to study more in detail tibial osteotomy and total hip arthroplasty patients. What is really difficult to have is some routines calculating the centre of rotation having the 6 d.o.f. for each limb segment. I know that this is a crucial point first of all when the patient has a deteriorated joint (arthrosis or R.A.) or when he/she has a total joint replacement. I think in fact that is not possible to calculate moments with markers put on anatomical landmarks. Probably the best way could be to do some dynamic test and calculate the centre of rotation having the 6 d.o.f. for each limb segment. I look forward to any suggestions and help from an M.D. or Ph.D. involved with clinical gait assessment. Fabio Catani. ========================================================================= Date: Thu, 8 Nov 90 23:57:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: RE: Kinematics & Kinetics Dear Biomch-L readers, In reply to Fabio Catani's Biomch-L posting last Monday on how to assess 3-D ICR's, I have the following comments. Some of this is (to my knowledge) new information, and it seems interesting to announce this in a non-refereed, electronic bulletin rather than to wait for a comprehensive review process. (a) Utility: An instantaneous, 3-D centre of rotation for a moving rigid body (e.g., a distal segment w.r.t. a proximal segment in the case of 3-D joint motion) may be useful in a dynamics context for the following reasons: - If you are looking at net joint moments without regarding individual force and moment contributions in ligaments, muscles, and contact points, the numerical values and the graphical results will strongly depend on the chosen reference point at which the net moment M is assessed. Cappozzo in Rome/Italy has amply demonstrated this (Bristol ESB Meeting, 1987). - It is sometimes proposed that it is better to use as a reference point one which is stationary w.r.t. the anatomy. While this is possible within one segment, such a point will generally not be stationary w.r.t. the other segment. Thus, which of the two segments comprising a joint should be chosen? - The net power transmitted through a joint can be assessed from the rotatio- nal and translational contributions as M'omega + F'v, where F is the net force, and omega and v the rotation and translation velocity vectors of the joint (i.e., distal w.r.t. proximal or vice-versa). Of these entities, M and v are position-dependent, and v is minimal at the Instantaneous Helical Axis. The common practise to neglect the linear term F'v is appropriate if v == 0 (planar movement at the IHA) which includes the case of a fixed rotation axis or centre. While this may be the case for the healthy hip joint, it is certainly debatable for other joints. Note: while the net power is invariant with the position along the IHA, the net moment varies. - If interest is oriented to assessing the efficacy of individual muscles, the muscle's moment arm can be suitably assessed with respect to the IHA or ICR as defined below. (b) Definiton and Calculus/Estimation: A 3-D ICR can be defined by looking for the point with minimal, instantaneous displacement. Given the general rigid-body equation y(t) = R(t).x + p(t), R(t) orthonormal, x arbitrary but fixed one could try and find the point x for which the absolute velocity |dy/dt| and acceleration |d(dy/dt)/dt| are minimal in some weighted sense. For example, one might choose from all points with minimal velocity that particular one which has minimal acceleration. The class of points with minimal velocity is defined by the Instantaneous Helical Axis about and along which the body is instantaneously translating and rotating; it is defined by the projection s = p + omega * dp/dt / omega^2 of p onto it, and by its direction vector omega, where * denotes the external vector product, and where omega = (omx,omy,omz)' follows from Poisson's equa- tion, [ 0 -omz omy] A(omega) == [ omz 0 -omx] = dR/dt . R' (' denotes transposition). [-omy omx 0 ] The point q on the IHA with minimal acceleration then follows by minimizing |d(dy/dt)/dt|^2 as a function of the parameter r in the equation y = s+r.omega for an arbitrary point y(r) on the IHA. [N.B.: when working this out, be sure not to reverse the sequence of differentiating and substitution; like the Car- danic rotations of this spring's joint angle debate, these operations are not commutative.] The measurement system and protocol must be sufficiently accurate to allow reliable assessment of both first and second derivatives. However, these data are required anyhow if one plans to assess inverse dynamics including inertial effects. Thus, skin motion artefacts should be minimized if external markers are utilized, and the marker distribution should be sufficiently non-collinear. Note also that the 3-D ICR is ill-defined unless the rotational velocity OMEGA == omega * d(omega)/dt / |omega|^2 with which the IHA changes its own direction is sufficiently large. Thus, the 3-D ICR is ill-defined if the movement is nearly planar. Presumably, though, the net moment will in that case have a negligible tendency to rotate the IHA into an other direction (this might be an interesting research question). It turns out that the ICR as defined above coincides with the so-called central point c about which the IHA itself executes an instantaneous, helical movement with rotation velocity vector OMEGA (cf. Suh & Radcliffe, Kinematics and Mecha- nisms Design, Wiley 1978, Ch. 10). Fischer (1907) and Chao & An (ESB Nijmegen/ NL, 1982) have proposed that the central point may be a useful descriptor of spatial movement. The formal proof of the equivalence between ICR and c is quite tedious and error-prone; fortunately, the recent computer-algebra posting on this list has resulted in access to the Maple package (Maple Software, Waterloo, Canada) at the Computer Algebra Netherlands Expertise Centre in Amsterdam by means of which this relation could be verified [N.B.: SURFnet users in The Netherlands can obtain access to this system; for details, send a note to can@can.nl]. I hope that this answers Dr Catani's question to some extent (a paper is in preparation, which hopefully will contain some experimental results). Sincerely -- Herman J. Woltring, Eindhoven/NL. ========================================================================= Date: Mon, 12 Nov 90 16:32:29 -0600 Reply-To: cahalan@MAYO.EDU Sender: Biomechanics and Movement Science listserver From: cahalan@MAYO.EDU Subject: ISSUES RELATED TO JOINT MOMENT CALCULATION From: Edmund Chao, Rochester, MN To: BIOMCH-L Mail Readers, Fabio Catani and Herman Woltring Date: November 12, 1990 Subject: Issues related to joint moment calculation Dear BIOMCH-L Mail Readers: Tom Cahalan has shown me the letters sent by Fabio Catani and the response letter by Herman Woltring and I read them with great interest. As a noncomputer user, I seem to have missed many of the interesting communications through the computer BITNET system and, luckily, I have Tom Cahalan to keep me posted on some of the subjects that I have always had an interest in. I am deeply sympathetic towards Dr. Catani's problems, and I wish to address the issue of joint moment calculation in general and acknowledge that the suggested solution and calculation guidelines provided by Herman are the right way to go. First of all, we should all realize that the so-called "joint moments" currently used in the field are calculated mainly from the externally measured forces and the inertial effect of the musculoskeletal system. Hence, these are the quantities to be supposedly more reliable without the concerns of the Inverse Dynamic Problem and the Redundancy in muscle and joint constraint force determination. In order to calculate such moment, accurate determination of joint center (2-D) or joint axis (3-D) of rotation is extremely important, as stated by both Catani and Woltring. Unfortunately, the limited papers I have read in the literature about joint moment analysis have not gone into details concerning how these moments were calculated. In fact, when this subject was presented at some society meetings, I occasionally asked questions related to this concern. On this issue, I wish to raise the following fundamental questions. 1. The calculation of joint moment has many sources of error in addition to the relative motion of the markers due to soft tissue movement. The more critical issue here is that all the potential measurement errors, including ground reaction force magnitudes, the location of such forces and their directions, plus the determination of the center of mass as well as the inertial properties of the limb segments are all additive in accumulating the potential error of such calculation. These do not include the problem of locating joint axis of rotation yet. I realize that one may only con- cern relative changes instead of the absolute value. But, if the calcula- tion will be subjected to so many sources of error, one should question the accuracy and reliability of such quantities (i.e., joint moments in different planes). 2. In all seemingly exciting clinical applications, one should provide an error bound of all the moment data based on the experimental method and the theoretical calculation utilized. Otherwise, only relying on statistical analysis, the results and concluding remarks can be grossly misleading. The concerns on data normalization with respect to walking velocity or cadence, on the temporal occurrence of critical joint moment of concern and on the consideration of the entire pattern of moment as a function of time rather than looking at discrete instances must be addressed critically. We are all waiting to have the most exciting and reliable technique for clinical application of gait analysis, but we must also be extremely careful not to fall flat on what we have tried for so long to reach such goal. 3. Finally, the determination of instantaneous axis of rotation in 3-D rigid body kinematics or the hypothetical 3-D instantaneous center of rotation should be utilized to calculate joint moments. The joint articulating surface contact point (even if one could determine it reliably) does not necessarily represent the center of rotation, since the type of articula- ting surface motion can vary from sliding to spinning type of motion rather than rolling without slipping. Therefore, I wish to propose that the biomechanicians in the field who are interested in rigid body kinematics and kinetcs be more critical of the currently reported joint moment papers. There is no doubt in my mind that the present instrumentation and technology are able to determine approximately the instantaneous helical axes and refer them back to each connecting skeletal segment for the purpose of calculating joint resultant moment resulting from externally applied forces and the inertial effects. The method outlined by Herman appears to be very elegant and practical. The concept of three- dimensional instantaneous center of rotation proposed by me about ten years ago was only meant to represent a spatial trajectory illustrating the path of instantaneous helical axis movement of two loosely connected rigid bodies. Strictly speaking, such concept has rather limited utilization, and for the purpose of calculating instantaneous joint moment, one must utilize the instantaneous helical axis concept described by Herman. Please keep up with your excellent work, Herman! For those who may be interested in knowing the potential application of joint moment to clinical orthopedics, the following excellent references are highly recommended: 1. Podromos, et al., JBJS, 67-A:1188, 1985 2. Wang, et al., JBJS, 72-A:905, 1990 3. Berchuck, et al., JBJS, 72-A:71, 1990 In addition, Kadaba has performed an excellent experiment dealing with the issue of repeatability of gait kinematic, kinetic, and EMG data in normal adults (see J. of Orthopaedic Research, 7:849, 1989). In Kadaba's article, the repeatability issue has been addressed, but unfortunately, the issues of accuracy and reliability were not questioned. Such issues should be carefully considered, especially when one wishes to calculate the joint moments in the frontal plane and for more distal joints (such as the ankle and knee) because of the potential error magnification and the inherent technical limitation of the motion analysis system due to visual field resolution. Sorry I don't have Cappozzo's paper discussing joint moment determination as presented in the Bristol ESB meeting, 1987. I would very much like to read that to see if Cappozzo's opinion would in any way coincide with my concerns. We are dealing with a rather important subject, because our orthopedic clinical colleagues tend to take the surface value of a biomechanical article, and we must ensure that the data presented do indeed reflect the actual physiological phenomena quantitated as accurately as current instrumentation allows. If the method used to determine the joint moment is not theoretically correct or is technically inaccurate, one must be extremely carefull before making strong prognostic or therapeutic statements. I hope that this discussion and my personal opinion will stimulate a lot of debate as well as clarification. If joint moment can be well established for clinical application, everyone in the field should utilize it to its maximum extent. I will be eternally happy to see theoretical/objective gait analysis make a giant step forward in its practical utilization in orthopedics. Sincerely, Edmund Chao Rochester, MN P.S. Dr. Chao may be contacted through Tom Cahalan ========================================================================= Date: Fri, 23 Nov 90 10:50:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Biomechanics in Florida Dear Biomch-L readers, Following Professor Chao's highly interesting and kind response of November 12 to Fabio Catani's posting of November 5 and my reply of November 8, I had the pleasure to attend the recent ASB meeting in Miami. `New ideas' seldom occur in isolation, and I am pleased to refer to the paper "An intrinsic parameter for the study of complex three-dimensional human motion" by Raymond R. Brodeur & Robert W. Soutas-Little from the Department of Biomechanics at Michigan State University in East Lansing, MI 48824, U.S.A. In this paper, the central point of differential kinematics is mentioned in the context of the so-called distribution parameter which is the ratio of the first-order IHA's translation speed along and rotation speed about the second- order IHA. [Note: the second-order IHA is the IHA about which the (first-order) IHA itself executes an instantaneous, helical movement.] In other words (and further to a question by Prof. K.-N. An from the Mayo Clinic after Mr Brodeur's presentation), it is the so-called pitch of the second-order IHA, similar to the pitch of the first-order IHA which equals the ratio of a rigid-body's translation speed along and rotation speed about the first-order IHA. While the authors place all emphasis of their paper on the distribution parameter, it seems clear from their presentation that they have the central point in mind for joint centre representation, too. At the CAMARC meeting in Rome next week, there is another paper on joint centre estimation from gait data; the authors are unknown to me at this time. Thus, the time might be ripe for a paradigm change in whole-body movement analysis. Another, highly interesting paper at the Miami ASB meeting was from Prof. Jack M. Winters' group (Chemical, Bio & Materials Engineering) at Arizona State Uni- versity in Tempe, AZ 85060, U.S.A. [where, incidentally, the next ASB meeting will be held]. In "Head finite screw axis parameters during vertical, horizon- tal, and oblique tracking movements: normal and injured subjects" (cf. also the companion poster "Directional and spatial sensitivity of neck muscle acti- vity during comfortably-spaced 3-D head tracking movements"), the Abstract's initial part, co-authored with two members from the Fuer Chiropractic Clinic in Phoenix, AZ, runs as follows: This study documents how finite screw axis parameters (FSAP) for head rotation change during voluntary horizontal, vertical, and oblique point- to-point head tracking movements. Both "normal" (control) subjects (n=9) and subjects with "whiplash" conditions (n=10) are considered, with each subject evaluated twice, separated by six weeks. The information collec- ted here serves to document normal ranges, help validate 3-D computer models, and assist optimization studies by providing reference data for a performance criterion. Here we emphasize the first two of these. The foundation behind the approach rests with the concept that the head is the most distal link of an open kinematic chain that includes the neck. Con- sequently, changes in neck kinematics, due to changes in passive neck tis- sue biomechanics and/or in neuromuscular function, should be reflected in head kinematics (Chao et al., Abstract 319, XIIth ISB, UCLA 1989). Tradi- tional clinical measurements (e.g. ranges of motion; radiographs) appear limited in their information content. In this study changes in head orien- tation are essentially specified since the subject is asked to track targets with a laser pointer coupled to the head. By systematically varying target patterns, variation in helical screw axis parameters, and in particular the location and direction of the axis of rotation, can be documented for a wide variety of movements. Mr Brodeur, Prof. An and Prof. Winters are Biomch-L subscribers; their email addresses can be retrieved via the REVIEW BIOMCH-L (COUNTRIES command to LISTSERV @ { HEARN.BITNET | NIC.SURFNET.NL }. During the two days after the Miami ASB meeting, the Ninth Southern Biomedical Engineering Conference took place at the same location. Also this conference hosted a considerable number of Biomechanics presentations, and I'd like to mention in particular the paper by S.K. Mishra & D.B. Goldgof, Department of Computer Science and Engineering, University of South Florida in Tampa, FL/USA entitled "Nonrigid motion analysis and its biomedical applications". The paper "(...) describes a curvature-based approach to nonrigid motion analysis which uses three-dimensional data". I did not have the opportunity to hear its presentation, but this work might be extremely relevant in human motion analysis. Finally, I'd like to mention that the next East Coast Clinical Gait Analysis Meeting will take place at Prof. Soutas-Little's Department from 5-7 December 1990. I understand that the Proceedings will consist of 4-page papers instead of the 1-page abstracts of the preceeding meetings (1989: Helen Hayes Hospital, up-state New York; 1988: Penn State University). Herman J. Woltring, Eindhoven/NL. ========================================================================= Date: Tue, 28 May 91 23:47:00 EST Reply-To: Ian Stokes Sender: Biomechanics and Movement Science listserver From: Ian Stokes Subject: Moments of forces about C of R - Is it a good idea? To: Fabio Catani, Herman Woltring, Ed Chao and Biomch-l readers. During an email correspondence with our encyclopaedic moderator (HJW), he encouraged me to review the Biomch-l archive for November 1990 (prior to my membership in the list) in order to read the discussion initiated by Fabio Catani, and joined by Herman Woltring and Ed Chao concerning the use of center/axis of rotation as a reference point for forces and moments at joints. The following borrows heavily from a poster I presented at the Combined Canadian and American Societies of Biomechanics meeting, Montreal, Quebec, in August 1986. The abstracts were not published that year, other than in the Conference Proceedings. The poster was motivated by my desire to clarify my own mind about the assumptions we make about joint moment equilibrium and lines of actions of joint forces. Fabio Catani's original question was about how to find the center of rotation of the knee and hip joints in order to calculate the joint moments. He got excellent advise about how to do this, but I question why he should use the center of rotation, rather than a more easily located reference. There is moment equilibrium about any and all points, so why worry about where to place the reference point? Usually because we are interested in going from measurements of external moments and inertial forces to estimation of internal forces in anatomic structures - joints and muscles. Estimation of joint forces is usually solved by writing six equilibrium equations (for force, and torques in a 3 axis system). This method can be applied both to the complete kinematic analysis, and to the quasi-static case when it is assumed that acceleration terms are negligible. Usually there are more than six unknowns in the six equations, so assumptions have to be made so that possible solutions can be obtained. These assumptions can be divided into two groups: (a) assumptions about the line of action of the joint force and (b) assumptions about the distribution of muscle and ligament forces around the joint. The first assumption enables equations of moment equilibrium to be simplified. Since the unknown joint force has no moment about a point on its own line of action it can be eliminated from the equation of moments about such a point. If the direction of the joint force is also known, then a constraint equation can be written. People want to measure moments about the center of rotation, presumably, because they are looking for a point or axis about which the joint force has no moment. For my poster I reviewed 1483 papers published in the Journal of Biomechanics between Jan 1969 and Dec 1986. 56 were concerned with estimation or evaluation of articular forces and/or moments of muscles about joints. Among these 56 papers, the assumption about the line of action of the joint force was classified into one of 4 groups: (1) Contact Point (9 papers): Nine used observations of the area of contact between joint surfaces to define a point on the line of action of the joint force. (2) Axis of Rotation (17 papers): Fourteen assumed that the joint force passed through a point on the axis of rotation (or center of rotation for planar motion) and used this as a point about which to consider moment equilibrium. Two papers used this point and the further constraint of having the joint force pass through the joint contact area. One of these papers considered errors introduced by uncertainty about these points. One considered that both the center of curvature and the center of rotation are co-linear with the contact point, and therefore the joint force should pass through both of these points. (3) Center of Curvature (16 papers): Eleven papers assumed that the joint force passed through the center of curvature of an articular surface. Five papers made this assumption and also constrained the line of action of the joint force to pass through the contact area. (4) Miscellaneous (14 papers): One aligned the tibiofemoral forces in the knee with points half way across the condyles and perpendicular to the articular surfaces. Two papers used the center of curvature of the hip and the center of contact of the knee. One analyzed joint forces in the fingers and assumed that they acted along the long axes of phalanges, transecting the joint contact. One used previously published data for moment arms of the triceps surae. In three cases no assumption was stated. Two papers showed how the principle of virtual work could be used to find moment arms of muscles by measuring tendon length/joint angle relationships. One determined experimentally where there was a region of zero moment due to external forces and muscle forces, and related it to the anatomic features of the knee joint. One study of the shoulder investigated whether the joint center of rotation lay on the line of the joint force. Two papers addressed the question of what differences in the calculated joint forces would be produced by adopting either the joint center of curvature or the experimentally determined axis of rotation of the temporomandibular joint (TMJ) as a point about which to consider moment equilibrium. In the TMJ joint (which is relatively unconstrained by ligaments or by its articular surfaces) the axis of rotation could become quite remote from the joint itself. Thus, many different assumptions have been used both singly and in combination. The three major ones are that the joint force passes through the center of contact, the center of curvature, or the center of rotation. Clearly, an inter-articular force must pass through the point (or region) of contact between surfaces. In the absence of friction, it must also be aligned with the mutual perpendicular between the surfaces, so it would pass through the centers of curvature of the surfaces. But is it valid to assume that the joint force has zero moment about the center of rotation? On the face of it the center of rotation is a kinematic property of the joint. Why should it have any special significance for force and moment equilibrium? What are the constraints on the center of rotation? The relative motion at the cartilage surface can be considered to have three components: rolling, gliding, and spin. At any instant, there is a plane passing through the articular contact in which rolling and gliding take place. The center of rotation is the point where the axis of rotation cuts this plane. Spin is an rotation between the two components of the joint, about an axis which is the mutual perpendicular through the joint contact. It can be shown that the center of rotation for rolling and gliding lies on the perpendicular to the common tangent plane. Its position is at the articular contact for pure rolling, but increasingly distant from it as the amount of gliding is increased. Since the joint contact, the center of curvature and the center of rotation are co-linear under normal conditions, it is therefore correct to assume that the joint force has no torque about the center of rotation. This should be true under both static and dynamic conditions, providing the center of rotation is controlled by contact at joint surfaces with inelastic cartilage and there is zero friction. The axis or center of rotation is, however, notoriously difficult to measure experimentally. Joint laxity can make its position quite variable, so it is not certain that the center of rotation found in one experiment can be applied directly to estimating forces under different loading conditions. The same also applies to the contact area or center of curvature, but there are bounds on their positions, and they are not subject to large measurement error sensitivity. Instantaneous center of rotation is impossible to measure in practice, since the errors tend to infinity as the increment of motion tends to zero. All joint force calculations (especially those in which inertial forces are significant) would be simpler if the reference point were constant. Indeed, the three papers recommended by Ed Chao use a fixed 'joint center' based on anatomic reference points. For most real joints, especially those with minimal constraints on motion, neither the center of curvature, center of rotation nor the contact point can be expected to be fixed to either part of the joint. Therefore, as biomechanicians we must always be critical and imaginative in our analyses. There is no simple rule which can be applied in all cases. However, a rigid adherence to using the center of rotation of a joint is unwise. Neither biomechanical theory, nor practical considerations support it. Ian Stokes Burlington, VT, USA. ========================================================================= Date: Fri, 31 May 91 14:31:55 00100 Reply-To: DDIATVB@CC.RUU.NL Sender: Biomechanics and Movement Science listserver From: DDIATVB@CC.RUU.NL Subject: Re: Moments about C of R? Dear Biomch-L readers, The posting on joint centers of rotation by Ian Stokes is interesting; a (literal) shift of perspective from the center of rotation can be useful. I tend to disagree however, with his final conclusion that "...neither biomechanical theory, nor practical considerations support it (the use of joint centers as reference points)". If you do not make assumptions about the line of action of the joint force (JF), the JF must be described by 3 variables (2D) instead of 2: two components of force, and one for the line of action. This introduces one extra unknown variable into the moment equilibrium equation, and typically there are too many unknowns already. So you must make an assumption about the line of action of the JF. But there is only one thing a priori known about this: the JF goes through the instantaneous center of rotation (ICR). This can be proved using the principle of virtual work. A joint is defined as a 'kinematic' connection, i.e. the force associated with this connection generates or absorbs no power at any time. Picture one body (bone) as stationary while the other is moving. All points on the line of action of the joint force must have velocities perpendicular to this force (power is the dot product of force and velocity). In a moving rigid body, every line on which all velocities have the same direction *must* pass through the ICR. Please take a few seconds to verify this statement... So, the joint force also passes through the ICR. Incidentally, this also proves that the ICR of the knee joint during the swing phase coincides with the intersection of the cruciate ligaments. The joint force is in that case the resultant of ligament forces only. Note that this only applies to true kinematic connections, without frictional losses or energy storage in elastic cartilage or joint ligaments. Neglecting these small amounts of energy is probably allowed. Also note that in this definition, 'joint force' is taken to mean the total 'constraint reaction force' in mechanical terms, sometimes called 'net joint force'. If you only want the contact force, without ligaments, the ligament forces become additional unknowns in the equilibrium equations and that is not what you want. A joint, defined as a kinematic connection between two bodies, is more than just the bone-to-bone contact surfaces. It also includes the structures that guide the movement without exchanging energy with the system. I.e. ligaments that can be considered inextensible for the purpose of dynamic analysis. Remains the problem that the ICR has (in general) no fixed position on either bone, and that the ICR is not easily determined during actual movements. That is exactly why the ICR is taken as the reference point in the moment equation. That way you do not have to know it! Of course, this implies that all other moments must alse be calculated about the ICR. For muscular forces this is no problem: the moment arm with respect to the ICR is the partial derivative of origin-insertion length with respect to the joint angle (also to be proved by the principle of virtual work). Using this definition, moment arms of muscles are easily determined from cadaver measurements or a rigid-body model incorporating the line of action. Only for calculation of external (ground reaction force) moments is an estimated location of the ICR required. Hopefully, moment arms of ground reaction forces are large enough (or the moments small enough) to be insensitive to errors in the ICR. So, my opinion is that moments should be calculated about the ICR. I would like to hear Ian's reply, or other opinions. -- Ton van den Bogert University of Utrecht, Netherlands. ========================================================================= Date: Sun, 2 Jun 91 14:46:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: 2-D/3-D ICRs for net moments and powers 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 ========================================================================= Date: Wed, 5 Jun 91 00:59:30 00100 Reply-To: DDIATVB@CC.RUU.NL Sender: Biomechanics and Movement Science listserver From: DDIATVB@CC.RUU.NL Subject: Moments about ICR, continued Dear Biomch-L readers, Now that Herman Woltring has shown us the mathematical relationships between various definitions of joint moments/forces, as well as the 3-dimensional generalization, I want to point out a difference between two views on dynamic analysis. In my view this is important to clarify the discussion. Herman is deliberately limiting the discussion to *net* joint kinetics, i.e. the model consists of rigid links with one force and one moment transmitted by each joint. These variables are calculated, plus sometimes the joint powers (moment x angular velocity). The analysis essentially stops there, and individual muscles are not part of the model. This is probably a good method for clinical gait analysis, because no detailed information on muscle lines of action is needed, and no assumptions on load sharing of muscles have to be made. However, these joint forces and moments are not physical quantities but mathematical abstractions: they do not exist at all anywhere in the system. When I say 'not exist', I mean that there is no anatomical structure loaded by (= deformed as a function of) either the force or the moment. Is that not a good definition of 'physical existence' of a force: that it produces a deformation somewhere that has a one-to-one relationship to the force? Hmm, you could even say that force is then also a mathematical abstraction, and that only stresses 'exist'. But let's accept the concept of force (muscle force, ligament force, contact force...) for now. Another way to look at 'net kinetics' analysis is as a transformation of the original kinematic, kinetic and anthropomorphic measurements, intended to facilitate the (clinical) quantification of 'gait quality' or the recognition of certain abnormalities. A mechanical interpretation of the resulting 'net kinetics' variables is not the real purpose of the analysis (apart from the fact that, strictly speaking, it is not even allowed - see above). Looking at it this way, I must agree with Ian Stokes that it does not really matter which reference point is used to calculate the joint moment. Just as long as you use the same reference point when comparing results, and the variables obtained still contain useful information. In fact, using a fixed point is to preferred above the elusive ICR. The ICR can only be estimated when the kinematic data are of sufficient quality, and even then requires sophisticated filtering and analysis methods. For such a 'net kinetic' analysis it might be more reliable to use the lateral epicondyle as reference at the knee, rather than the ICR, because it can be marked directly and measured by the measuring system. The choice of reference point does require standardization however, to avoid problems when comparing published results. Many biomechanicians however, *are* interested in real muscle forces and real joint forces, and try to estimate them as well as possible. These forces are not mathematical but physical quantities. There are of course the well-known indeterminacy problems because the equilibrium equations for moment and force often have too many unknowns. For some situations however, such an analysis is the right tool for the job. In that case, the reasoning of my previous posting applies: the ICR is the only point about which the moment arms of muscle forces (dL/dA) and joint forces (zero) are easy to obtain. (For simplicity I limit the discussion to 2D). Note that the 'net joint force' resulting from this type of analysis is not the same as in the 'net kinetics' analysis, but is much larger (and more realistic). This force may also be only a resultant of several physical (contact & ligaments) forces, but the muscle forces that have been obtained are real physical quantities. So my revised opinion is: use the ICR as reference point when estimating muscle and joint forces. For a 'net kinetic' analysis, only standardization is required; there is no preferred reference point. Clinical usefulness seems to be more important than mechanical interpretation in that case. Finally, this is probably a very academic discussion without practical implications; the various definitions reviewed by Ian Stokes produce very similar results. But sometimes it is enlightening to think about why you do things one way, and not the other way. -- Ton van den Bogert University of Utrecht, Netherlands ========================================================================= Date: Thu, 6 Jun 91 11:47:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: ICR, joint moments, and stability Dear Biomch-L readers/posters, Following Ton van den Bogert's posting the other day, I'd like to add a few considerations, in the hope that this debate will not confine itself to one just between your two `moderators' ! (1) While net joint force and moment may be `abstract mathematical entities', i.e., the mathematical sum of all `physical' forces and moments (muscle force, ligament tension, interbone contact forces, etc. plus their moment effects), I would suggest that net joint *power* is quite physical, i.e., the effective power generated at / transmitted through a joint. The versatility and redun- dancy of the NMSK system entail that different muscles and muscle parts may be called upon to generate the *same* net kinetics and kinematics in order to counter fatigue and other detrimental factors in individual motor components. When considering (pathological) movement from a functional point of view, on the IDH (Impairment - Disability - Handicap) scale, Ian Stokes' and Ton van den Bogert's approach seems to lean more towards Impairment, while mine might, perhaps, be biased more towards Disability and perhaps even to Handicap? Furthermore, Ton wrote in a posting some time ago that high contact forces in the ankle joints of his horses (if I recall correctly) do not seem to have much detrimental effect, and I have suggested some years ago that high force *transients* (kinetic `jerk' -- 3rd derivative related) may be important, similar to metal fatigue effects. Of course, 3rd derivative estimation from noisy position/attitude data is even more difficult that conventional estima- tion of 1st and 2nd derivatives... Indeed, the skating biomechanics researchers at the Faculty of Human Movement Sciences at the Free University in Amsterdam typically assess joint and seg- ment power in their studies, as has been done by David Winter in Waterloo and his students; thus, I am not so sure about Ton's `sometimes' when he writes that I am "(...) deliberately limiting the discussion to *net* joint kinetics, i.e. the model consists of rigid links with one force and one moment transmitted by each joint. These variables are calculated, plus sometimes the joint powers (moment * angular velocity). The analysis essentially stops there, (...)" I should like to see comments from others on this list on their views on the utility of net forces, moments, *and* powers in FMA (Functional Movement Assessment). Any volunteers ? (2) While I agree that the net moment reference point does not *have* to be the (2-D or 3-D) ICR *if* linear power terms are not forgotten, it seems useful to consider it seriously in anticipation of the kind of applications for which both Ton and I deem the ICR meaningful. Thus, experience on its assessment and use can be obtained, and databases can be accumulated that can be used once we are able to routinely acquire the type of data that are necessary for more comprehensive analyses. In addition to Ton's arguments in favour of the (2-D) ICR, I might mention that another argument is the notion of knee joint stability in stance, where the normal from the knee joint's Centre of Rotation to the GRF (Ground Reaction Force) vector or its projection onto the sagittal plane is used as a measure for such stability considerations (with the assumption that inertial effects are negligible during stance). If the ICR can be modulated by different forms of muscular co-contraction (those forces cannot be gleaned from the external kinematics and GRF data only), we may have a useful gait assessment parameter in the ICR and its projection onto the (equally instantaneous) GRF vector. (3) When Ton writes about the `net joint force' in his more comprehensive analysis not being equal to the one in `net joint kinetics', I think that he is creating a Babylonian, linguistic confusion. I strongly suggest to reserve the term `net joint forces, moments, and powers' for the free-body analysis of my previous posting, with one total force and moment vector per joint, and to use different terms for other physical things or mathematical abstractions. (4) The moment arm as the partial derivative d(muscle length)/d(joint angle) applies in the planar case where the (straight) muscle is perpendicular to the IHA (Instantaneous Helical Axis, normal to the plane under consideration). In the general 3-D case, the issue is complicated because of the possibility that perpendicularity no longer holds, because of a translation component along the IHA, and because there is not a single `joint angle'. If a muscle and an IHA are parallel, for example, the muscle has no possibility to cause further rotation *about* the IHA, but only of *changing* the direction and position of the IHA, and these are different phenomena. In general, a muscle may show a mixture of both properties. Herman J. Woltring, Eindhoven/NL. ========================================================================= Date: Thu, 6 Jun 1991 11:22 EST Reply-to: HJS@PSUECL Sender: Biomechanics and Movement Science listserver From: HJS@PSUECL Subject: RE: ICR, joint moments, and stability I have been quite interested in the recent BIOMCH-L postings about joint kinematics and kinetics, in particular about the planar ICR (instant center of rotation) and the spatial ISA (instant screw axis) - also denoted as the spatial IHA (instant helical axis). I have two points which I would like to discuss in this regard. 1) We have developed linear least-squares methods to compute both two- and three-dimensional INSTANTANEOUS angular velocity and angular acceleration - and consequently the ICR/ISA and the AAA (angular acceleration axis) - from positon, velocity, and acceleration of multiple point landmarks on a rigid body. If using video/photo methods, the velocity and acceleration of the landmarks may be determined by smoothing and differentiating landmark position trajectories. These methods have been combined to also determine the instantaneous central point of the screw axode ruled surface (the point on the ISA with minimum acceleration about which the ISA instantaneously changes direction with time) denoted as the spatial ICR by Herman Woltring in recent BIOMCH-L postings. Mathematical development of these methods has been presented and published through ASME. Application of these methods to biomechanics will be presented in July at the Int. Symp. on 3D Analysis of Human Movement in Montreal. 2) Although Herman Woltring and I may be the only two people in the world interested, these methods may be easily extended to determine two- and three-dimensional angular jerk from third derivatives of landmark position trajectories. H.J Sommer, Professor of Mechanical Engineering The Pennsylvania State University University Park, PA 16802 USA phone (814)865-9214 FAX (814)863-4848 bitnet hjs@psuecl internet hjs@ecl.psu.edu ========================================================================= Date: Fri, 7 Jun 91 12:02:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: AAA, ICR, and central point Dear Professor Sommer and other Biomch-L readers/posters, It is with pleasure that I see others joining the ICR debate. In fact, Professor Sommers kindly sent me on 2 Jan 1991 a letter with some highly interesting (p)reprints of his most recent work, to be precise: (1) H.J. Sommer III, Determination of First and Second Order Instant Screw Parameters from Landmark Trajectories, Proc. 21st Mechanisms Conference, American Society of Mechanical Engineers, DE-25:429-437 (1990), also accepted for publication in the ASME Journal of Mechanisms, Trans- missions, and Automation in Design (scheduled to appear during Spring 1991); (2) H.J. Sommer III & F.L. Buckzek, Least Squares Estimation of the Instant Screw Axis and Angular Acceleration Axis, 1990 ASME Advances in Bio- engineering, BED-17:339-342 (1990), also to be presented at the Inter- national Symposium on 3-D Analysis of Human Movement whose programme was posted onto this list by Ian Stokes some weeks ago. My main problem with Professor Sommer's zero acceleration pivot (which can be calculated from the rotation velocity and acceleration vectors and the acceleration of some base point on the moving body) is the question what it can be used for: while it is the generally unique point with zero instanta- neous acceleration on (an extension of) a moving rigid body, it does not in general have the smallest, instantaneous velocity. Thus, it is -- in my mind -- less attractive a candidate for (straightforward) generalization from a fixed to an Instantaneous Centre of Rotation than the Instantaneous Helical Axis' central point or pivot; it is, however, the generally unique point which has instantaneous *stationary* movement by virtue of of its vanishing accele- ration, and this may have some special kine(ma)tic implications hopefully revealed in future research. From Professor Sommer's posting I understand that he claims ASME priority on what I have chosen to call the `3-D ICR', "These methods have been combined to also determine the instantaneous central point of the screw axode ruled surface (the point on the ISA with minimum acceleration about which the ISA instantaneously changes direction with time) ... Mathematical development of these methods has been presented and published through ASME. Application of these methods to biomechanics will be presented in July at the Int. Symp. on 3D Analysis of Human Movement in Montreal". I must confess not having been aware of prior ASME-published work in this area (but then, my Nov 1990 postings tried to make clear that I was not claiming any `inventors' primacy other than believing to have shown that the IHA's central pivot is that point on the IHA which has the smallest accele- ration; it is the point with the latter property that I choose to call the 3-D ICR). At any rate, the central point as such is an old notion, having been used in a finite displacement context by Otto Fischer in 1907, and proposed as an `instantaneous' centre of rotation by Ed Chao and Kai-Nan An at the Nijmegen ESB meeting about 10 years ago. Furthermore, the central point's instantaneous kinematics have been provided by Suh & Radcliffe in their 1978 book "Kinematics & Mechanisms Design", Chapter 10 (N.B.: Ian Stokes might think again about encyclopedias, but I must insist on declining that compliment: Professor Sommer does not only quote Suh & Radcliffe, but also Everett 1875 with work getting close to the above idea that the central point coincides with the 3-D ICR defined as the point of smallest accelera- tion of all points on the IHA). While the mathematics for assessing all these kinematic movement descriptors from rigid-body data and their derivatives is straightforward but tedious, assessing these intermediate rigid-body data from noisy landmark coordinates is not so easy. For example, optimally transforming noisy landmark data is a nonlinear least-squares problem under rather conventional noise conditions, and Professor Sommer has kindly quoted some recent litterature in this area. While there are certain linear procedures, they are not optimal from a mini- mum variance point of view; however, it is currently not known how suboptimal these linear methods are in practice. Last-but-not-least: obtaining reliable 1st and 2nd derivatives from noisy data -- especially if they contain genuine transients -- is far from easy; this is even more difficult for 3rd derivatives, and I look forward to the Montreal presentations about these and related signal processing challenges. Finally, I'd like to have some `democratic' feedback from the readership on whether this kine(ma)tics debate is thought interesting or too esoteric. Herman J. Woltring, Eindhoven/NL ========================================================================= Date: Mon, 10 Jun 91 17:31:00 EST Reply-to: Ian Stokes Sender: Biomechanics and Movement Science listserver From: Ian Stokes Subject: Center of Rotation - Joint moment debate Dear Biomch-l Readers, The Joint moment - center/axis of rotation (C of R) debate is, I believe, most important to the fields of Biomechanics and Human Motion science. Therefore, I am glad that the discussion initiated by Fabio Catani which I joined lately has provoked so much discussion and debate. My own contribution, in which I argued that the C of R is *not* an appropriate reference point for consideration of joint moments has stimulated a number of further contributions which have been most helpful. Some of these postings have invited me to respond, but as fast as I have tried to collect my thoughts, new ideas and opinions have been posted! I should add that other contributors have more experience of this kind of biomechanics in practice, but perhaps I have something to offer at least from a theoretical point of view. As of today I believe I understand the following: 1. Force and moment equilibrium about joints is a common tool in biomechanics. Many text books and many courses in biomechanics teach that the C of R is the reference point about which we consider moment equilibrium, because the joint force passes through it. The literature in the Journal of Biomechanics (and elsewhere) is not consistent about this reference point - centers of curva- ture, contact and rotation are used singly and in combination. Theoretical considerations support all of these (with certain conditions such as neg- ligible friction and surface compliance). We can prove this by analyzing geometry, statics and/or virtual work. 2. Practical considerations depend on the purpose of the study/analysis. Biomechanics studies can be divided into: - Quasi-static vs. dynamic analyses and - Studies of internal forces vs. studies of joints as actuators (actuators transmit torques and generate power). Considering dynamic analyses, use of the C of R is simpler, because relative motion has fewer degrees of freedom about the center/axis of motion. Therefore, the inertial terms are easier to deal with. However, the practical problems of finding the C of R are great, so in some joints and some situa- tions it would be better to look at the anatomy and constraints, and use other information (fixed center of rotation, or knowledge of joint contact or center of curvature). Considering 'joints as actuators' (net moments) vs. 'internal forces', the important question to ask is 'does it matter?' The objective in both cases is to have an expression for joint moment which includes the effects of muscles, in equilibrium with external and inertial forces. The joint force should be excluded by considering moments about a point on its line of action. (Ligament forces, and joints with two condyles complicate this.) It seems that the net moment on each side of this equilibrium is sensitive to the point about which moments are calculated, except that if the muscles forces are nearly parallel to the joint force, as probably is often true, the sensitivity could be small compared to other sources of errors. Certainly, in 'net moment' measurement and reporting for any particular joint, standardization in the biomechanics field would be very helpful. 3. As biomechanicians and teachers the most important thing we must remember is to be critical and to be sure of the assumptions on which we base our analyses. This is especially important in multi-disciplinary cases in which studies/analyses may be done by one person and interpreted or applied by another. This happens often in the clinical field where scientific findings from studies of a few people in a controlled research situation may be applied to a larger population. Also, sophisticated equipment, designed with known limitations can be adapted to turn-key operation for people not necessarily trained in all aspects of its interpretation. In anything as complex as human joint function there are no simple answers, but the basic principles must be clear to us before we get into the complexities. This debate certainly has helped me. Ian Stokes ========================================================================= Date: Tue, 11 Jun 91 12:22:44 00100 Reply-To: DDIATVB@CC.RUU.NL Sender: Biomechanics and Movement Science listserver From: DDIATVB@CC.RUU.NL Subject: ICR and IHA in 2D/3D dynamic analysis Dear fellow biomechanicians, Some of you will (I hope) answer Herman Woltring's call, and give comments on practical use of net forces/moments/powers in functional movement analysis. I want to reply to some technical statements made by Herman. Most differences of opinion seem to originate from different areas of application; my interest is in estimation of 'internal forces', which need not be part of 'functional movement analysis'. (1) Are joint powers physical quantities? In case of only monoarticular muscles, yes. In that case one joint power equals the sum of several muscle powers. When there are polyarticular muscles, the only equality is: sum of all joint powers = sum of all muscle powers. A joint power is then just another mathematical transformation of measured variables, without physical meaning. It may have practical value though, but others should comment on that. An example. If you say that ankle power is very large just before take- off in a vertical jump, you are restating what was observed: a large distance between GRF and joint center, simultaneous with a large extension velocity. The information was there already. But again, the information might be more useful (for pattern recognition?) in the transformed form; you get one variable instead of the original three: GRF vector and point of application, and joint angular velocity. Mechanical interpretation of joint power is dangerous: if you assume that all ankle power is generated by ankle muscles, you are wrong because the gastrocnemius muscle (even when it does not contract) can transmit power from the knee extensors to the ankle joint. Herman is very right that interpretation of physical joint forces is also not straightforward. Very little is known on how cartilage and bone react to specific loading patterns. But, you can not use differentiation of force to get the 'jerk' if you do not estimate the force in the first place. (2) I agree with Herman, that the ICR (or 3D IHA) should ideally be part of a biomechanical analysis. But using the ICR as moment reference point requires high-quality kinematics and processing. It would be a pity if a cheap and simple analysis would be considered below standard. Especially if there is no good reason to prefer this difficult transformation of measured variables over another. (3) I apologize for using confusing terminology. It is indeed logical to reserve the term 'net joint force' for the force obtained from 'net kinetic analysis', i.e. models with one force and one moment transmitted by each joint. In other types of analysis, which include estimation of muscle forces, the use of just 'joint force' seems more appropriate. When a joint is a complex (powerless) kinematic connection, this 'joint force' can be the resultant vector of several forces, e.g. contact and ligament forces. In that case, one might be tempted to add 'net'. To avoid confusion, the term 'constraint force' or 'constraint reaction force' from theoretical mechanics could be an alternative. (4) My discussion on muscle moment arms was, admittedly, not properly generalized to 3D. I had only joints in mind with one degree of freedom (DOF). In that case, the IHA (instantaneous helical axis) depends only on the joint angle, and muscles cannot change it. Within this limi- tation the moment arm is still d(length)/d(angle). When for example, the knee joint is part of a model, one must decide on the number of DOF. If it is simplified to one DOF, the above theory applies. If the laxity is an essential part of the analysis, more DOF are required. The removal of kinematic constraints means that the corresponding constraint forces are also lost (see below). This produces incorrect muscle forces, because only the muscles are assumed to be responsible for the observed movement, unless the actual physical constraints (the joint ligaments) are added to the model. This shows that it is best to reduce the degrees of freedom as much as possible in a dynamic analysis. In a truly general 3D-theory, the concept of generalized coordinates is convenient. If a joint has N degrees of freedom, you need N variables (generalized coordinates) to specify the position of body 2 relative to body 1. There are many ways to define such variables, as was shown by the '3D joint angles' debate some time ago on Biomch-L, but the theory always applies. Each of the N generalized coordinates is associated with a generalized force (the 'moments', if the coordinates are angles). The relationship between physical forces and generalized forces is linear, and the coefficients (the 'moment arms') can be found using the principle of virtual work: SUM(F.dr) = Q.dq (for all dq). Where dq is a small change in the N-vector q of generalized coordinates, dr is the resulting (3D-vector) change in position of the point of application of each force vector F. Q is the N-vector of generalized forces. Actually, the principle of virtual work defines Q. From this, we find for each component Qi of Q: Qi = SUM(F.(dr/dqi)) (the d's mean partial derivative here) For a muscle, the direction of the vector F is exactly opposite to the direction of lengthening, so F.(dr/dqi) = -|F|dL/dqi. For a ground reaction force, the full vector equation must be used. The conditions for static equilibrium are now simply: Q=0. Dynamic equations of motion can also be formulated in generalized coordinates: Q = M(q).q" I am not familiar with the method to find the inertia matrix M, which may depend in a complex way on q. My dynamics software (DADS) does not use generalized coordinates but 'cartesian' coordinates, which are more suitable for general-purpose software. Finally, there are 6-N (in 3D) constraint force variables (the 'joint force'). Examples: A ball-and-socket joint (the hip) has N=3, and the constraint force is a 3D force vector. A universal joint (in Dutch: "kruiskoppeling"), as used in machines, has N=2 because it does not allow internal/external rotation. The 4 constraint reaction forces are one 3D force vector, plus one torque. Note that joints can also have translational degrees of freedom ('slider' joints), where the corresponding qi is best measured in meters, not in radians. The generalized theory still applies. Enough of theoretical mechanics now, let's get back to more practical matters! -- Ton van den Bogert University of Utrecht, Netherlands. ========================================================================= Date: Wed, 19 Jun 91 08:41:31 -0500 Reply-To: cahalan@MAYO.EDU Sender: Biomechanics and Movement Science listserver From: cahalan@MAYO.EDU Subject: ICR debate Readers and debators: I have enjoyed the debate on the ICR issue thus far, but must admit that I have been confused and lost much of the time. I am a physical therapist working in the area of applied gait analysis, and admit that I do not know as much about kinematic and kinetics as the engineers. I take the advise of the engineers regarding how to apply the technology that is available. I would very much appreciate a brief overview and a basic (if possible) position stance from each of the debators. I believe that this would help me better understand the basic concepts and issues being discussed. Thank you Tom Cahalan ========================================================================= Date: Fri, 21 Jun 91 03:57:20 00100 Reply-To: DDIATVB@CC.RUU.NL Sender: Biomechanics and Movement Science listserver From: DDIATVB@CC.RUU.NL Subject: RE: ICR debate Dear biomecha(cli)nicians, In response to Tom Cahalan's request, here is the summary of my standpoint in the ICR debate. 1. When estimating internal forces from kinematic/kinetic data, use the ICR as the reference point for moment calculations. You do this automatically for the muscle and joint forces, when using the correct methods. So, you must also do it for the external (ground reaction) forces. The moment equilibrium equation requires that all moments are calculated with respect to the same point. 2. When using kinematic/kinetic data to calculate net joint moments, forces or powers, use a reference point that is close enough to the ICR so as not to hamper the (clinical) interpretation. In practice, you would use a point where you can reproducably attach a marker for the kinematic analysis. Here, standardization, reproducability and ease of use is more important than mechanical correctness. That's it; short and (I hope) clear. -- Ton van den Bogert, University of Utrecht [ Editor's note: After some more practical experience with 3-D inverse dynamics, I should add some comments to item 2 of my message of June 23, 1991. If moments are calculated with respect to a fixed point on a segment, it is important (for interpretation) that this point is close to the actual instantaneous center of rotation (ICR) at all times, and it may not be possible to attach a marker there. I can see two ways to determine a fixed 'joint center' which is close to the ICR: (1) Use three non-colinear markers per segment, attached at sites with minimal skin movement. During standing, attach temporary markers that indicate the joint center location (a suitable procedure for the hip joint is described by Bell et al., J. Biomech. 23:617-621, 1990). Measure 3-D coordinates of all markers, and remove the temporary markers before recording movement data, to avoid marker merging and tracking problems. Since the location of the joint center with respect to the three markers (or with respect to the segment coordi- nate system) is known, its location can be calculated at any time using a combination of 3-D rotation and translation (e.g. Challis, J. Biomech. 28:733-737, 1995). It is therefore possible to calculate joint moments with respect to this point. (2) Instead of the temporary markers, use a kinematic analysis to locate the 'average' center or axis of rotation for the entire range of motion of the joint. A procedure for the hip joint was described by Cappozzo (Hum. Movt. Sci. 3:27-50, 1984; also cited by Bell et al.). This is actually the procedure presented at ESB 1988 (Bristol), and referred to by Herman Woltring earlier in this discussion. Additional comments are welcome. Please post to Biomch-L@nic.surfnet.nl. -- Ton van den Bogert October 1995 ]