* File: ANGLES3D TOPIC * Date: Jan 6, 1993 * Editor: Ton van den Bogert This file contains the full text of discussions on Biomch-L between November 1989 and March 1992 on the various methods for quantification of 3-D joint rotations. ========================================================================= Date: Mon, 13 Nov 89 17:27:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Standardization in Kinematics Dear Biomch-l readers, Within the CAMARC project (see below), the question has been posed on how to standardize body segment co-ordinate systems. There seem to be a variety of these, e.g., Z is the longitudinal axis in an oblong segment like thigh, shank or foot, X is the medio-lateral axis, and Y the anterior-posterior axis. I would be grateful for any pointers to relevant standardization proposals in the litterature or elsewhere, and information on how (and why!) you define your own coordinate systems. Related to this is the notorious question on how to define joint angles via Cardan ("Euler") angles `a la Grood & Suntay (they defined a mechanization of a particular Cardanic convention, with the flexion/extension axis imbedded as the 1st rotation axis in the proximal segment, the endo-/exo-rotation axis imbedded as the 3rd axis in the distal segment, and ab-/ad-duction the `floating' axis perpendicular to the two other axes but imbedded in neither segment. See their paper in the Journal of Biomechanical Engineering 105(1983), 136-144 for further details, and the Letter to the Editor by Andrews in the Journal of Biomechanics 17(1984)2, 155-158. Alternatively, I have proposed to use the Finite Axis of Rotation which is de- fined by a unit direction vector N (N'N = 1), and a rotation angle theta ran- ging between 0 and 180 degrees. Taking the product theta * N we obtain a `vec- tor' which can be used as an attitude descriptor with respect to the reference attitude. This representation does not exhibit `gimbal lock' as does the Car- danic convention `a la Grood & Suntay; furthermore, the three components are completely symmetrical with respect to each other, unlike Cardanic representa- tion. I'll be glad to send a copy of a recent paper to anyone interested. In clinical practise, joint angulation is merely defined in the simple, `planar' case, where two of the three components in any convention are zero. Thus, we have the liberty to choose any definition for compound rotations (where at least two components are non-zero) in such a fashion as to minimize disadvantageous properties like gimbal-lock; with various manufacturers now proposing their own standards, there is a chance that de-facto standards are created with less than optimal properties. Standardization is becoming an important issue now an increasing number of (cli- nical) laboratories are beginning to pool data in order to acquire suitable databases on normal and pathological movenent. Apart from naming joint axes and defining rotation parametrizations, there is the perhaps even more serious problem of how to define segment coordinate sys- tems from anatomical landmarks. For example, the flexion-extension axis in the femur is often said to pass through the peaks of the condyles. However, these peaks are rather flat, so ambiguity may be the result. Any comments, either on BIOMCH-L or directly to ELERCAMA@HEITUE5.BITNET would be most welcome. Herman J. Woltring (CAMARC/Netherlands) CAMARC ("Computer Aided Movement Analysis in a Rehabilitation Context") is a project under the Advanced Informatics in Medicine action of the Commission of the European Communities (AIM/DG XIII-F/CEC), with academic, industrial, public-health, and independent partners from Italy, France, U.K. and The Netherlands. Its scope is pre-competitive. ========================================================================= Date: Fri, 17 Nov 89 18:51:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Standardization (II) Dear Biomch-l readers, Among the replies received until now on my standardization query of a few days ago, there was one with a review paper currently in print: Goeran Selvik, Roentgen Stereophotogrammetric Analysis - A Review Article, Acta Radiologica 31(1990), Fasc. 2. In this paper, X is defined as the transverse axis (from the right, -, to the left, +, of the body in the `anatomical' position), Y is the vertical axis (from the feet, -, to the head, +), and Z is the sagittal axis (from the posterior, -, to the anterior, +, direction). This applies equally to left, central, and right body segments; by consequence, anatomically oriented terms such as endo-/ exorotation, ab-/adduction and medio-lateral translation in the limbs would have opposite signs between the ipsi- and contralateral sides. As a motivation for this choice, reference was made to M.M. Panjabi, A.A. White & R.A. Brand in the Journal of Biomechanics 7(1974), 385 and to M. Williams & H.R. Lissner, Biome- chanics of Human Motion, W.B. Saunders Co., Philadelphia 1962. Furthermore, segment rotations are defined as a Cardanic convention about body- fixed axes, in the sequence X, Y and Z from the neutral or reference attitude. Assuming (as is the usual situation) that joint motion is defined as distal w.r.t. proximal (in the case of the limbs, not necessarily for other parts of the body), this results in `body-fixed' axes denoting flexion-extension in the proximal segment and ab-/adduction in the distal segment, while endo-/exorota- tion of the joint is defined about a `floating' axis, in the sense of Grood & Suntay (1983). This is different from these authors' definition who chose endo-/exorotation about a distally fixed axis, and ab-/adduction about the `floating' axis. As indicated in a 1988 paper by Blankevoort et al. in the Journal of Biomechanics, the numerical difference is small for typical knee angulations. However, if one desires to standardize rotational parametriza- tions, it might be useful to define one which can also be used for other joints and for segment motion definitions. Since Roentgenstereophotogrammetry is a highly advanced field of quantitative biokinematics, its results should be carefully considered for their transfer- ability to whole-body kinematics in gait and other forms of functional move- ment. In the former, much interest is directed to translations as in, e.g., joint loosening, and less to rotations. In whole-body movement studies, it is just the opposite, thus special attention to rotational parametrizations might deserve even more attention than given in Selvik's work. I would be grateful for further reactions -- Herman J. Woltring (CAMARC/NL). ========================================================================= Date: Sun, 19 Nov 89 16:44:00 N Reply-To: Ton van den Bogert Sender: Biomechanics and Movement Science listserver From: Ton van den Bogert Subject: RE: Body segment axes Dear BIOMCH-L readers, In last week's posting, Herman Woltring raised the question of how to define body segment axes and joint rotations. This is indeed a fundamental problem in basic as well as applied biomechanics, which can only be solved by some sort of standardization. My own experience on this subject comes from 2-dimensional multibody modelling, and (to a lesser extent) from clinically oriented 3D kinematic analysis on horses. 1. Multibody models are commonly used for (inverse) dynamic analysis of movement, using measured kinematics as input. Usually, these models assume hinge or ball joints between the body segments. I think that this is the reason why in these models the 'joint centers' are often used to define the long axis (being one of the coordinate axes) of the segment. This definition also slightly simplifies the equations of motion. In 3D models a second axis must be defined somehow, usually the lateromedial axis. Kinematic data, obtained from landmarks placed at arbitrary points on the segments, can be used for dynamic analysis if the positions of these points with respect to the segmental axes are known. For my 2D work, I obtained this information from radiographs showing the 'joint centers', as well as the kinematical landmarks (marked by steel rings). I think this method has a few serious drawbacks: - It relies heavily on the assumption that joints have a fixed center of rotation, and that these points can somehow be identified. - For 3D applications, the definitions are not sufficiently strict, and transformation of marker data to rigid body kinematic variables depends on marker coordinates with respect to the segmental reference frame. It is difficult to obtain this information. Possibly it would be wiser to define standard segment axes with help of three well-defined points on the outside of each bone. If these axes do not coincide with the traditional anatomical axes, this will have to be accepted. 2. For clinical applications, it is practically inevitable to use segment axes based on external markers. For this reason, I tend to prefer methods of kinematic analysis that give results that are (within certain limits) independent of marker placement. This is the best way to ensure that data obtained from different recording sessions or different individuals can be compared safely. In fact, this seems to be the way most software supplied with kinematic analysis systems works. A typical analysis method is to put two markers (proximal and distal) on each body segment, and make sure that during the recording the walking direction is along one of the coordinate axes (e.g. the Y-axis) of the laboratory reference frame. Joint angles are obtained from projections of the 'stick diagrams' on the sagittal (YZ) and frontal (XZ) planes. Incorrect marker placement will only produce a constant error in the angles, and it is easy to extract parameters from the signals that are insensitive to this. Of course, the example given above does not provide data for all six degrees of freedom (DOF) of the body segments. In my opinion, this loss of information is not too serious for clinical applications because the set of kinematic variables describing human movement is by no means independent. A more practical reason for this 2x2D approach is, that a complete 3D analysis of, for example, the femur requires a simultaneous view of markers on the medial and lateral condyles. The system we are currently using (CODA-3) does not allow this, you would need a video- based system with many cameras. Summarizing: - For basic research (requiring full 3D kinematic data, 6 DOF per body segment) standard segmental axes should be adopted, based on the same palpable bone landmarks where the markers are attached. - In clinical biomechanics, we should only measure those variables that are reliable and reproducible. If a 3D analysis produces less reliable results or requires too much effort, stick to a 2D (or 2x2D) approach. These are my own personal opinions, and I would welcome further discussion on BIOMCH-L about this subject. Ton van den Bogert Dept. of Veterinary Anatomy University of Utrecht, The Netherlands. ========================================================================= Date: Wed, 14 Feb 90 21:43:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Joint attitudes: a debate Dear Biomch-L readers, The abstract below has been accepted for presentation at the previously announ- ced, 4th International Biomechanics Seminar in Gothenburgh/Sweden on 26 and 27 April, 1990. In view of various comments about this proposal, both from Dr Grood in Cincinnati/Ohio and elsewhere, Ed Grood has agreed to a debate on the pro's and con's of various 3-D joint attitude parametrizations. For the sake of clarity, this text deviates slightly from the accepted text. Following earlier items on Biomch-L last year, I hope that this debate may be one of many to follow. It is for this kind of activities that an email discus- sion list can be much more efficient than, say, Letters to the Editors in formal journals. This is not to say that such letters are not useful; rather, they might become the result of more interactive debates, whether at conferences, during laboratory visits, or from behind a terminal. Ed, it's up to you, now! Herman J. Woltring Eindhoven, The Netherlands. 3-D ATTITUDE REPRESENTATION : A NEW STANDARDIZATION PROPOSAL (1) Herman J. Woltring, Eindhoven, The Netherlands Grood & Suntay (Journal of Biomechanical Engineering 1983, 136) have proposed a `sequence independent, oblique co-ordinate system' in which the current orienta- tion (i.e., position and attitude) is thought to be reached from a predefined reference orientation via an ordered sequence of rotations ijk of three elemen- tary, helical displacements a b o u t (PHI.) and a l o n g (D.) the axes i, j, and k of an electrogoniometric linkage system. The terminal axes i and k are imbedded in the body segments comprising the joint, and they are identical to prior selected, Cartesian co-ordinate axes defined in these segments; the intermediate or `floating' axis j is normal to the two imbedded axes, and iden- tical to the `line of nodes' in classical handbook descriptions of Euler/Cardan angles. Although t e m p o r a l sequence dependency of Cardan/Eulerian rotation conventions is avoided in such predefined electrogoniometric systems, a similar effect is now imposed by the g e o m e t r i c a l choice of imbedded and floating axes. Thus, different numerical results may be obtained for current joint attitudes (given identical segment co-ordinate systems), and adverse effects such as g i m b a l - l o c k (i.e., for some PHIj, either the sum or the difference of PHIi and PHIk is undefined) and C o d m a n ' s P a r a - d o x (i.e., both {PHIi,PHIj,PHIk} and {PHIi+PI,PI-PHIj,PHIk+PI} (N.B.: -PI works also) are valid solutions; cf. A.E. Codman, The Shoulder, Boston 1934) continue to occur. Instead of defining joint orientation or movement in terms of an ordered sequence of three helical displacements, it seems more appropriate to view the current orientation in terms of a s i n g l e helical displacement, to be decomposed into orthogonal components in either body segment's co-ordi- nate system which, apart from a sign inversion, appear to be identical. For attitude representation (position representation is more complicated), one can define an attitude `vector' THETA = theta * N, where N is the unit direction vector about which the helical, scalar rotation theta occurs. This vector THETA, while not a true vector as rotations are not additive, is symmetrical in its three components and not affected by gimbal-lock or Codman's Paradox. Unlike N, THETA is well-determined from noisy measurements even for small theta. A further advantage is that the helical representation corresponds approximately with the mean value of all valid, Cardanic representations once Codman's Paradox is accounted for. (1) This is a paper under the CAMARC project. CAMARC, for "Computer Aided Movement Analysis in a Rehabilitation Context", is a project under the Advanced Informatics in Medicine action of the Commission of the European Communities, XIII-F/CEC), with academic, public-health, industrial, and independent partners from Italy, France, U.K. and The Netherlands. Its scope is pre-competitive. ========================================================================= Date: Sat, 17 Feb 90 12:27:00 EST Reply-To: GROOD@UCBEH Sender: Biomechanics and Movement Science listserver From: GROOD@UCBEH Subject: Joint Attitude Debate: First Response Dear Biomch-L Readers, I appreciate Dr. Herman Woltring's invitation to debate his proposal for standardization of 3-D joint attitude representation. In this first response I have three goals. These are: a) to outline my current position on Herman's proposal; b) to address the common misconceptions about sequence dependency of finite rotations, and c) to stimulate thought and discussion about the description of particle displacements and how the principles used are applied to the description of rigid body displacements. My Current Position Let me start with the area of agreement. I completely accept Dr. Woltring's proposal for the use of the helical axis to describe joint attitude and rotational displacement. The helical axis has the important characteristic that the magnitude of the rotation and translation are independent of the coordinate system chosen. The analogy to particle displacement is clear, the magnitude of the total displacement vector is also independent of the coordinate system chosen. Now the area of disagreement. Herman has proposed the helical displacement be decomposed into orthogonal components in either body segment's coordinate system. This produces six components, three for translation and three for rotation. I believe this is not fully satisfactory for describing joint translation and incorrect for the joint rotation components. It is my position that the proper joint rotation components are those describe by Fred Suntay and I. My reasons for this will become apparent in the course of the debate. Misconceptions Herman and many others refer to the rotations Fred and I described as being an "ordered sequence of rotations". I would agree they are an ordered triple, just like the orthogonal components of particle displacement are an ordered triple. I disagree with the terminology "ordered sequence" because the final displacement is not dependent upon the sequence the rotations are performed. Am I missing some other meaning of this phrase? There is a general, and incorrect, belief that finite three dimensional rotations are sequence dependent. This is not surprising as almost every text on the subject gives the example of a book rotated using two different sequences resulting in two different final positions. This example is passed along without any careful analysis of what is actually happening. The basic problem with the book example is the three axes used are those of an orthogonal coordinate system located in one of the body segments. While such axes do define independent translational degrees-of-freedom (dof), they do not define independent rotational dof. The independent rotational degrees-of-freedom are those Herman referred to: a fixed axis in each body segment and their common perpendicular. The orientation of the fixed axes are chosen for convenience. This is similar to selecting an appropriate orthogonal system for describing particle displacements. To better understand the origin of the sequence dependency I will give a similar example for particle displacement. It starts by first specifying the displacements (x,y,z) without specifying the three independent dof. Next, perform the displacements along the axes of any orthogonal coordinate system and note its final location. Third change the orientation of the orthogonal coordinate system. We still have three independent dof but the directions have a different physical significance. Now perform the three component displacements in any sequence. The final position is clearly not the same as before. The problem is not that particle displacements are sequence dependent, it's that we changed the independent dof used. Independent Rotational Degrees-of-Freedom At the risk of being unnecessarily redundant I will again describe an appropriate set of independent rotation axes. First locate two body fixed axes, one in each body segment. These axes are selected so that rotation about them is a motion of interest. The third rotation axis is the common perpendicular to the two body fixed axes. The three angles which define the orientation were described in the paper with Fred Suntay. Briefly, they are: 1. The rotation about the common perpendicular axis is given by the angle between the two body fixed axes. 2. The rotation about each body fixed axis is given by the angle between the common perpendicular and a reference line located in the same body as the fixed axis. It is convenient to select the reference line so it is also perpendicular to the body fixed axis. In this way the body fixed axes are normal to the plane which contains both the reference line and the common perpendicular axis. In closing this first round of the debate I will state the primary reasons for using the system we proposed as the components of the helical rotation. 1. They are independent components. 2. They add (in a screw sense) to the total helical rotation. 3. They correspond to common clinical descriptions of joint rotation. 4. They are easy to compute from the rotation matrix and have a well defined mathematical relationship with the total helical rotation. Edward S. Grood Cincinnati, Ohio, USA ========================================================================= Date: Mon, 19 Feb 90 22:01:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: About vectors and matrices: round #2 THE CURIOUS INCIDENT OF THE VECTORIAL TRIBE It is rumoured that there once was a tribe of Indians who believed that arrows are vectors. To shoot a deer due northeast, they did not aim an arrow in the northeasterly direction; they sent two arrows simultaneously, one due north and one due east, relying on the powerful resultant of the two arrows to kill the deer. Skeptical scientists have doubted the truth of this rumour, pointing out that not the slightest trace of the tribe has ever been found. But the complete disappearance of the tribe through starvation is precisely what one would ex- pect under the circumstances; and since the theory that the tribe existed con- firms two such diverse things as the NONVECTORIAL BEHAVIOR OF ARROWS and the DARWINIAN PRINCIPLE OF NATURAL SELECTION, it is surely not a theory to be dismissed lightly. A. Banesh Hoffmann, About Vectors. Prentice-Hall, New Jersey 1966; Dover, New York 1975, pp. 11-12. Dear Biomch-L readers, Ed Grood's reply of last Saturday is to be commended, and I'll address his points more-or-less in the (time, first-last, top-bottom) sequence provided by him. 1. `Current position': While the (finite) helical axis has the usuful property of defining total dis- placement (translation and rotation) between two orientations, where one can be a reference orientation, and the other a current or actual orientation, this does not preclude its utility for other purposes. The beguiling analogy between rigid-body and point displacements will be discussed below. I don't think that I proposed to decompose helical movement into s i x compo- nents, with three orthogonal translation/position, and three orthogonal transla- tion/attitude components. In the context of the current debate, my main focus is on attitude/rotation parametrization as apparent from the original title. I agree that position description is somewhat awkward under the helical approach (as defined by the position of the helical axis, i.e., of some point on it re- quiring 2 d.o.f.'s, and of the amount of translation along the helical axis when moving from the reference to the current orientation, requiring 1 more d.o.f.), but I disagree that the helical decomposition is incorrect for joint (or seg- ment) attitude/rotation. While the G&S convention is useful, I would disagree that it is the best (or, for truely 3-D joint rotation as in hip and shoulder, even a good convention). Actually, position/translation description is also awkward under the G&S approach since gimbal-lock affects both the rotational a n d the translational parametrizations in that model... As said in the 14 Feb posting, gimbal-lock does not exist under the helical convention, and Ed has not yet commented that point. 2. `Misconceptions and verbal ambiguities': It seems that the confusion (including the wording in my 14 Feb posting) exists between two different kinds of `sequence effects': (a) Sequences in a `geometrical, top-down, or left-right' sense, as in the matrix product Ri(PHIi)*Rj(PHIj)*Rk(PHIk) or in the vector sum Xi( Di )+Xj( Dj )+Xk( Dk ), (b) Sequences in a temporal sense, where the variables PHI. and D. in the above expressions are changed from their reference values (zero) to the final values that describe the current or actual orientation. Different sequences under (a) are Ri(PHIi)*Rj(PHIj)*Rk(PHIk) and Rk(PHIk)*Ri(PHIi)*Rj(PHIj), or Xi( Di )+Xj( Dj )+Xk( Dk ) and Xk( Dk )+Xi( Di )+Xj( Dj ). Now, the important point is that matrix multiplication is non-commutative, that is, the resulting matrix products are generally n o t identical, while vector addition i s commutative, that is, the vector sums are always identical. One particular t i m e - sequence in the sense of (b) is: Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0) 1st displacement: Ri(PHIi)*Rj( 0 )*Rk( 0 ), Xi(Di)+Xj( 0)+Xk( 0) 2nd displacement: Ri(PHIi)*Rj(PHIj)*Rk( 0 ), Xi(Di)+Xj(Dj)+Xk( 0) 3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk) and a different t i m e - sequence is: Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0) 1st displacement: Ri( 0 )*Rj( 0 )*Rk(PHIk), Xi( 0)+Xj( 0)+Xk(Dk) 2nd displacement: Ri( 0 )*Rj(PHIj)*Rk(PHIk), Xi( 0)+Xj(Dj)+Xk(Dk) 3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk) Here, initial and final orientations are identical, buth the displacement occurs (or is thought to occur) along a different p a t h. The G&S approach imposes one particular sequence in the sense of (a), and then proceeds by declaring it sequence-independent in the sense of (b). The standard handbooks, however, interpret the term s e q u e n c e in the sense of (a)... [The many Dutch readers on the list will undoubtedly recall the recent parliamentory debate on `Social Innovation', where a highly ranking civil servant gave four completely different explanations of the meaning of that term.] I believe that the major motivation underlying the G&S approach -- or, for that matter, of any approach in terms of Cardanic/Eulerian rotations -- is to try and mimick the p a t h properties of vectorial movement. If I decompose a position or translation in terms of components X, Y, and Z, I may reach the position (X,Y,Z)' from the reference position (0,0,0)' by sequence-independent steps in the (b) sense above, but a l s o by sequence-independent steps in the (a) sense above. For rotations, this is n o t the case, as I can only have a sequence-independence in the (b) sense above. In the linkage-terminology of the G&S approach: if the linkage is made to merely allow translations along the linkage's axes, but no rotations about them, the choice of which axes are to be the imbedded ones and which are to be the floating ones is irrelevant, and has no influence on the amount of translation along each (permutated) axis. For rotations, this sequence-independence in the (a) sense does not apply. While it may be attractive to have an attitude parametrization that has the physical property of path description, this property is in no way necessary for the attitude parametrization goal. In fact, the imposition of this desire entails a number of awkward side-effects such as gimbal-lock and Codman's para- dox. Using a simple, while not exact analogon: while I may elect to travel from Eindhoven to Cincinnati by first moving south (or north, via the North Pole) until I reach the equator, thence west (or east) until I am due south of Cincinnati, and finally north (or south, via the South Pole) until I can join Ed for a drink, this does not imply that this is an elegant way to describe Ed's position with respect to me. The helical decomposition results in three orthogonal components that generally do n o t have path description properties. If two of the three components are zero, however, the single remaining non-zero term has a path describing nature. While this may be regrettable, I am really not interested in thinking how to get from some neutral hip or shoulder attitude to a complex, possibly pathological one, in terms of elementary rotations about some anatomically or technically defined co-ordinate axes, but only in a unique, well-behaved, minimally singular parametrization; I think that the helical decomposition meets these requirements quite well. Current clinical practise is more-or-less in agreement on how to define pure flexion-extension, pure ab-adduction, and pure endo-exorotation; such agreement does, at this time, not exist when these elementary (`planar') rotations occur simultaneously. Thus, we have the freedom to choose that particular convention which corresponds with the already established, special cases (the G&S approach, all other Cardanic conventions, and the helical approach meet that condition), a n d which is best behaved in the complex, 3-D case. Summarizing: Both the G&S and the helical approach exhibit three `independent' components except for the singular cases of gimbal-lock; in the helical approach, the com- ponents are mutually orthogonal but do not have trajectory properties, and the three axes are symmmetrical with respect to each other, without preferred `body- fixed' and `floating' axes; in the G&S approach, the opposite is the case. I believe that the balance is in favour of the proposed, helical standard, but it is up to the relevant community to accept or reject such a proposal. Herman J. Woltring , Eindhoven, The Netherlands. ========================================================================= Date: Thu, 22 Feb 90 12:40:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: joint attitude math & s/w Dear Biomch-L readers, Some readers on the list have shown interest in the mathematics and programming details underlying the current joint attitude parametrization debate. For this purpose, I'd like to refer them to a test package PRP FORTRAN which is available to the readership by sending the request SEND PRP FORTRAN BIOMCH-L to LISTSERV@HEARN.EARN (or .BITNET); the request should be sent from the address at which the Biomch-L listserver knows the requester. A FORTRAN-77 package is then returned which accomodates all possible Cardanic conventions and the proposed helical convention. While the PRP package was written for use under a VAX/VMS system, only a few I/O statements in the package's testprogramme may have to be modified for use in different environments. Basically, the relation between a 3*3 orthonormal matrix Rijk (with |Rijk| = +1) and the selected Cardanic angles can be described as follows: Let Rijk = Ri(PHIi) * Rj(PHIj) * Rk(PHIk), where PHIi, PHIj, and PHIk denote `planar' rotations, in the right-handed sense, about selected axes of a Carte- sian co-ordinate system (1: x-axis, 2: y-axis, 3: z-axis). Defining ijk as cyclic (i.e., 123, 231, or 312), the elements of Ri(PHIi) are defined as rii = 1; rij = rji = rik = rki = 0 rjj = rkk = cos(PHIi); rkj = -rjk = sin(PHIi) whence the elements r.. of Rijk can be expressed as: rkj + rji = {1 + sin(PHIj)} sin(PHIi + PHIk) rjj - rki = {1 + sin(PHIj)} cos(PHIi + PHIk) rkj - rji = {1 - sin(PHIj)} sin(PHIi - PHIk) rjj + rki = {1 - sin(PHIj)} cos(PHIi - PHIk) This yields two unique pairs -- as per Codman's Paradox -- for PHIi and PHIk unless |PHIj| = PI/2 -- gimbal-lock; in the latter case, two of the four rela- tions above are zero. Note that all angles are modulo 2*PI. Given the choosen PHIi and PHIk, PHIj can be derived from rik = sin(PHIj) {rii cos(PHIk) + rkk cos(PHIi)} - {rij sin(PHIk) + rjk sin(PHIi)} = cos(PHIj) If ijk is anti-cyclic, i.e., 321, 213 or 132, the cyclic case can be obtained via the intermediate transformation Rijk(PHIi,PHIj,PHIk) = Rkji(-PHIk,-PHIj,-PHIi)' Note that for other ijk, i.e., i=k, the n e u t r a l attitude corresponds to gimbal-lock, with either PHIi + PHIk or PHIi - PHIk undefined. Interestingly, Leonhard Euler's original publication `De Immutatione Coordinatarum, Caput IV, Appendix de Superficiebus' in his `Introductio in Analysin Infinitorum' (Lausan- ne, France 1748) provides such a case. For the helical convention, the following formulae apply: (Rijk - Rijk')/2 = sin(theta) A(N) (Rijk + Rijk')/2 = cos(theta) I + {1-cos(theta)} N N' THETA = theta N, N = (n1,n2,n3)', |N| = 1. where A(N) is a skew-symmetric matrix with elements aji = -aij = nk and akk = 0, for cyclic ijk. This system has a unique solution for 0 < theta < PI. If theta = 0, no solution exists for N, but THETA is zero; for theta = PI, there are two solutions, one THETA, the other -THETA, as one would expect for any periodic solution modulo 2*PI. Those who are interested in written material may get in touch with me directly, and I'll be glad to send a mathematically oriented paper presented during a 1989 CAMARC Workshop on Clinical Protocols in Ancona/Italy. Regards -- Herman J. Woltring, tel & fax +31.40.41 37 44 ========================================================================= Date: Sun, 25 Feb 90 16:09:00 EST Reply-To: GROOD@UCBEH Sender: Biomechanics and Movement Science listserver From: GROOD@UCBEH Subject: Debate: Are rotation components screws or vectors? Excerpt From "A Dynamic Parable" "Mr. Cartesian was very unhappy. ... He had an invincible attachment to the x,y,z, which he regarded as the "ne plus ultra" of dynamics. `Why will you burden the science,' he sighs, `with all these additional names? Can you not express what you want without talking about ... twists, and wrenches, ... instantaneous screws, and all the rest of it?' `No,' said Mr. One-to-One, `there can be no simpler way of stating the results than the natural method we have followed. ... "We are dealing with questions of perfect generality, and it would involve a sacrifice of generality were we to speak of the movement of a body except as a twist, or a system of forces except as a wrench.'" Ball, R.S., 1900, "A Treatise on the Theory of Screws" Cambridge University Press I suppose in a debate one good parable deserves another. Hoffman's parable on arrows is quite appropriate and its point (pun intended) supports my position. Rigid body rotations, like arrows, are not vectors. Rotations, but not arrows, are, as Ball states, twists (ie. screws or helices). My objection to your proposal, Herman, is that you want to decompose a screw as if it were a vector. The resulting components might be independent in the vector sense, but they are not independent rotations. You can not obtain the total rotation by performing the three component rotations obtained this way. Is it possible I'm a stronger supporter of the screw axis than you? I believe the component rotations must also be screws and follow the rules of screw addition to produce the total screw. Here, again, I make an analogy to the simpler case of particle displacements. The total displacement is a vector, the components are vectors and follow the rules of vector addition to produce the total displacement. You referred to my analogies as beguiling, which Webster defines as "to lead by deception". Am I deceptive when I teach my students they must specify the independent coordinates (ie. establish the coordinate system) they plan to use prior to solving a problem in particle kinematics? I think not. Nor is it deceptive for me to argue the same practice for the description of rigid body motions. For both particle displacements and rigid body rotations the independent degrees of freedom are specified using lines (or axes). In both cases one degree-of-freedom is obtained by the cross-product of unit vectors along the other two degrees-of-freedom. For particle displacements the three axes so formed are mutually orthogonal and are either fixed to a point on the path (path coordinates) or to a stationary system the motion is referred to. For rigid body rotations one axis is fixed to each of the bodies whose relative rotation are to be described and the third axis is there mutual perpendicular. It still surprises me that so little attention is paid to the physical axes that correspond to the rotational degrees-of-freedom while so much attention is paid to the axis used for particle displacements. Your program PRP is a nice way to examine the differences between the various Euler/Cardanic systems. I recommend the following question be asked whenever it is used, "What physical axes correspond to the independent degrees-of- freedom for each of the systems?" The limitations of electronic mail prevent me showing figures. However, the figure in Goldstein's book "Classical Mechanics" which is used to show the angles is excellent for this purposes. All you need to do is also focus on the axes the rotations are performed about. These axes are clearly shown in the figure and are perpendicular to the plane of the rotation I will try to describe the figure in words for those who do not have a copy of the figure readily available. I will also mail a copy to any who send an Email request to "Grood@UCBEH.SAN.UC.EDU". This figure shows two circular planes which intersect along a common diameter. The normals to each plane and the line formed by the intersection of the planes (line of nodes) are the three axes which define the three independent degrees-of-freedom. In this figure, one plane and its normal are in the fixed system while the other plane and normal are in the moving system. The magnitude of the rotations preformed about the body fixed axes are the angle between the line of nodes and reference lines located in the plane of rotation for each body. The reference lines are attached to the body and move with the rotation. The rotation about the line of nodes is given by the angle between the two body fixed axes. The different Euler/Cardanic systems correspond to different orientations of the planes in the fixed and moving bodies (ie. to different sets of three independent degrees-of freedom). All of the different systems have the path descriptive properties that you discussed in you last response. Analogy time again. The different orientations of the planes produced by the various Euler\Cardanic systems is equivalent to specifying different Cartesian coordinate systems to characterize particle displacements. The particle displacement produced by an x,y,z triple is invariant to changes in sequence only when the same set of axes are always employed. It is not invariant when both the sequence of the component displacements and the orientation of the Cartesian system change. The fact that each Euler/Cardanic system produce a different set of independent coordinates is reflected by the fact that each set is acceptable for use as generalized coordinates in writing the Lagrangian (or Hamiltonian) for 3-D dynamical problems. Such generalized coordinates must have the property of being independent. The solution of a single problem with different conventions will produce different, but equivalent, descriptions of the same motion. This is exactly what happens when a problem in particle kinematics is solved using Cartesian systems with different orientations. Having spent a lot of time on the important fundamentals of our disagreement, I will only briefly address the issues of gimbal lock and Codman's paradox. Gimbal lock is not a serious problem and only happens when the two body fixed axes are both parallel and co-linear. This is a rare occurrence for biological joints and can be easily avoided when analyzing experimental data. If the presence of a singularity in an equation were reason to reject an approach we would have to throw out a lot of good physics and mathematics. Use of the total helical axis alone solves the problem by refusing to look at the component rotations. Using the vector components of the helical axis to describe the motion does not provide a set of rotations that yield the total helical axis when the rotations are performed. Codman's paradox, which you cite as an argument against my point a view I cite as an argument for my point of view. The paradox exists independent of how the motion is described. I like to express the paradox as follows. Why does an abduction of the shoulder by 180 degrees followed by an extension of 180 degrees result in no net abduction or extension, but an external rotation of 180 degrees instead? I propose the value of a system be judged by its ability to explain the paradox. If the system proposed by Fred Suntay and I for the knee is applied to the shoulder the following explanation can be deduced. First I specify the degrees-of-freedom for a right arm. The flexion axis is embedded in the glenoid and points away from the body (unit vector e1). The internal rotation axis is in the humerus and points proximally when the arm is by your side (unit vector e3). The orientation of the abduction axis is obtained by the cross-product of unit vectors along the flexion and internal rotation axes, e2 = e1 x e3 / magnitude(e1 x e3) . The sense of the abduction axis is obtained from the right hand rule and by taking the cross-product from the flexion to the internal rotation axis. With the arm at your side it points posteriorly. The sense of the abduction axis is also given by the sine of the angle alpha between the flexion and internal rotation axes through the standard formula for the cross-product of two vectors. At the starting position alpha is 90 degrees and the sin(90)=+1. The first part of Codman's path is an abduction of 180 degrees. The arm goes from being at your side (palm toward the body) to straight overhead (palm away from the body). As the shoulder is abducted the angle alpha increases from its initial value of 90 degrees. As the arm passes through the 90 degree abduction position (arm straight out with palm down) alpha becomes greater than 180 degrees, the sin(alpha) becomes negative, and there is a reversal in the sense of the abduction axis which now points anteriorly. Since the sense of the abduction axis is reversed, motion of the arm from the 90 degree abducted position to the straight overhead position is actually an adduction of 90 degrees that takes the net adduction back to zero. The sense reversal of the abduction axis also affects both flexion and internal rotation. Since the abduction axis is a reference line for both motions a change in its sense changes both motions by 180 degrees. (This can be seen by changing the direction of either of a pair of lines that forms any angle.) No net displacement of the arm occurs when this happens because the flexion and internal rotation axes are parallel and the corresponding physcial rotations are in opposite directions. Because the axis have opposite sense, internal rotation points toward the body and flexion away, the signs of the rotations are the same. Both decrease and the arm becomes extended and externally rotated by 180 degrees in addition to the 90 degrees abduction. As the arm is raised to the overhead position, abduction returns to 0 leaving the arm in a position of 180 degrees extension and 180 degrees external rotation. The 180 degrees of flexion which return the arm to the side of the body reduce the net flexion to zero leaving only the 180 degrees of external rotation. Thus the angles of the system I support predict the net result of motion along Codman's path and therefore explain the paradox. I will leave it as a mental exercise to show that at the end of the motion the abduction axis points posteriorly as it did at the start of the motion. Hint: The abduction axis is always perpendicular to both the flexion and internal rotation axes because it is formed by cross-product of unit vectors along these axes. What always amazes me about the above explanation is the apparent mathematical quirk which occurs when the abduction axis changes sense is actually expressed by the physical motion of the arm. Summarizing: 1. I support the use of the helical (screw) axis to describe the rotational motions between body segments. 2. I believe the proper component rotations are also screws and can not be expressed as vector like the arrows in Hoffman's tale. To quote Ball "We are dealing with questions of perfect generality, and it would involve a sacrifice of generality were we to speak of the movement of a body except as a twist, or a system of forces except as a wrench." 3. I do not believe the existence of gimbal lock is a serious problem. 4. Codman's paradox can be easily explained by the use of the component screw approach I support. Herman, I also agree with you it is now time to hear from the community. Edward S. Grood Grood@UCBEH.SAN.UC.EDU Noyes-Giannestras Biomechanics Laboratories University of Cincinnati 2900 Reading Rd. Cincinnati, OH USA 45221-0048 ========================================================================= Date: Mon, 26 Feb 90 18:01:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Joint attitudes: (final?) round 3. "I don't know what you mean by `glory'," Alice said. Humpty Dumpty smiled con- temptuously. "Of course you don't know - till I tell you. I meant `there's a nice knock-down argument for you!'." "But `glory' doesn't mean `a nice knock- down argument'," Alice objected. "When I use a word," Humpty Dumpty said in a rather scornful tone, "it means just what I choose it to mean - neither more nor less." "The question is," said Alice, "whether you can make words mean so many different things." "The question is," said Humpty Dumpty, "which is to be master - that's all." Lewis Carrol (1872), Through the Looking Glass, Chapter VI. Dear Biomch-L readers, If we substitute `independent' or `commutative' for `glory,' there's not much new under the sun. (a) Commutation. In mathematics, c o m m u t a t i v i t y stands for the property (a o b) = (b o a) of a binary operation (a o b), for all valid a and b, where a and b are the entities upon which the binary operation o performs its function (e.g., +, -, *, /). If a and b are scalars or vectors, + and - are commutative operators, whereas - (and, for scalars, /) are not. If a and b are matrices, + is commutative while - and * are not; furthermore, / is generally undefined, although one may define certain classes of operations that more-or-less correspond to scalar division. (b) If a and b are functions of certain scalar parameters, say a1 and b1, respectively, the t e m p o r a l `commutation' as originally envisaged by Ed Grood and others [e.g., Bernard Roth, Finite Position Theory Applied to Mechanism Design, Journal of Applied Mechanics, Sept. 1967, p. 600, left column], is nothing else but the temporal order in which these parameters are changed from their reference values (zero or one, usually) to their final or current settings. This has nothing to do with mathematical com- mutativity, as proposed in one of my previous postings in this debate, nor with `similarity of matrices,' as Roth would have it. (c) Independence. I don't know what Ed means with `independent'. From a sta- tistical point of view, dependence and correlation of error sources are important when we are close to or at gimbal-lock. Using the formulae in Woltring et al, J. of Biomechanics 1985 on the Finite Helical Axis and Finite Centroid, in combination with formulae (2.32) in J. Wittenburg, Dynamics of Rigid Bodies (B.G. Teubner, Stuttgart/FRG, 1977), the `radial s.d.' or root-sum-of-squares of the s.d.'s for the three Cardanic angles is a function of PHIj due to the PHIj-dependent correlation between the SIGMA. errors, and can be derived as SIGMA(PHIj) = sqrt[SIGMAi**2 + SIGMAj**2 + SIGMAk**2] := SIGMA(PHIj=0) * sqrt[{1 + 2/cos(PHIj)**2}/3] As before, PHIj denotes rotation anbout the floating axis, with gimbal-lock when |PHIj| = PI/2. If PHIj is close to gimbal-lock, SIGMA is much larger than when PHIj = 0 degrees. The closer we approach the gimbal-lock situa- tion, more erratic the calculated angles become. If a physician wishes to interpret joint angle graphs for hip or shoulder, e.g., in complex sportive movement, these adverse effects should preferably be avoided. (d) The utility of `orthogonal attitude components' is even more apparent once we become interested in both kinematics and kinetics. Positions and translational and rotational velocities, accelerations, forces, and moments are all vectors, only attitudes are not. Should we really decompose all these vectorial entities in terms of the non-vectorial behaviour of rota- tions/attitudes? I think that it is much better to try and find a rota- tional representation which maximally approaches the vectorial properties of these other, biomechanically highly relevant entities. Again, I see no reason why we should impose or require trajectorial properties for our attitude parametrizations. Given the fact that there are different, Cardanic commutations, and the con- comitant asymmetries between the `floating' and `imbedded' axes (i.e., the floating axis follows from the vector product of the two imbedded ones, while neither imbedded axes is generally derivable as the vector product of the two other axes), I think that a symmetric representation like the helical decompo- sition continues to be a better candidate. After Ed's reply to this 3rd round, I believe that the debate should be suspended --- unless others on the list wish to enlighten (or to confuse) the readership with their views. At any rate, I have been enjoying the exchange of views, and it will certainly help me in preparing a somewhat more formal presentation on this issue in April at, I hope, both sides of the Atlantic. If Ed should finally agree with me, I would be delighted to ask him as a co- author ... Herman J. Woltring Research Associate , Eindhoven University of Technology, The Netherlands. ========================================================================= Date: Wed, 28 Feb 90 11:49:00 EDT Reply-To: MEGLAN%GAIT1@ENG.ENG.OHIO-STATE.EDU Sender: Biomechanics and Movement Science listserver From: MEGLAN%GAIT1@ENG.ENG.OHIO-STATE.EDU Subject: Joint Orientation, the saga continues... or won't die at least I guess it's about time that someone besides Herman and Ed commented on the method of describing the 3D orientation of the body segments relative to one another. In the recent complete rewrite of the analysis and display software for the gait lab here at Ohio State University, I included both the Grood & Suntay description, which I refer to as the Joint Coordinate System (JCS), of the joint rotations as well as the standard Pitch-Roll-Yaw (PRY) euler angles plus calculation of the finite screw across the joint (the position and orientation of the segments is known in 3D using the marker position data) using several different algorithms. In addition, the rotation about the screw axis can be described using Herman's concept of attitude angles (basically multiply the direction vector of the screw axis by the magnitude of the rotation about the axis to form a scaled vector) in terms of the proximal segment local coordinate system (LCS), distal segment LCS, or the JCS. All these different methods are included because I wanted to compare them on actual experimentally acquired data plus since there is no standardization at the moment I decided to use the shotgun approach and try to cover the majority of the possible orientation description methods. I did a poster on some of the results for the recent Orthopaedic Research Society meeting in New Orleans. Interestingly, Murali Kadaba had a poster right across the aisle which dealt with finite screws also. Basically, the results using multiple trials of normal subject gait data (we use a marker set consisting of 21 markers total for the upper and lower body with 3 markers on most segments) show that for sagittal plane joint angles there really is no statistically siginificant difference between any of the two rotation angle or three attitude 1angle schemes used. This was true for all the joints of the lower body as well as the pelvis, whose angle is not a relative angle but instead is calculated relative to the global lab coordinate system (GCS). In the out of plane angles (ab/adduction, in/external, pro/supination, in/eversion) there were significant differences primarily for the ab/aduction and in/eversion angles. These differences increase with the magnitude of the flexion angle which not surprising. The interesting point however is that the attitude angles are significantly different so the coordinate system in which they are expressed is critical. Also, the attitude angles expressed in terms of the JCS were different from the angles calculated with the JCS rotation angle method. Based upon these results, plus various other experiments which I have tried out, it would seem that while the calculation method, i.e. rotation angles or attitude angles, is important, the coordinate system used to express the attitude angles is just as important. Which leads to the obvious point that the positioning of these segment LCS's will have a critical effect upon the calculated joint angles. I did some perturbation experiments to look at the sensitivity of the calculated joint angles to slight changes in the orientation of either or both of the segment LCS's used as input data for the angle calculation. Each perturbation had some effect upon one or more of the calculated angles- not surprising. The effects are widely varying however and it is a relevant concern to identify those effects which are going to most readily occur with a given marker set and motion system configuration. Even rigid bodies with more than 3 markers attached to the body segment are sensitive to this problem since it is difficult to control the position/ orientation of the marker carrier relative to the underlying bones with accuracy and consistency. This is getting a little of track of the discussion focusing on rotation angles and finite screw attitude angles... Oh well, I guess we all have our soap box on one issue or another :-) At this point, I am inclined to go with the finite screw attitude angle technique expressed relative to the JCS axes except for the shoulder whose angles I feel are more properly represented by a spherical angle system. This was used by Andy An from the Mayo Clinic in a paper at the ORS. I personally think that the singularity issue ('gimbal lock' if you insist) for the JCS is theoretically relevant but not really in practice. True, as Herman pointed out, as the joint approaches this orientation the angles can become inconsistant due to the fact that both the numerator and denominator of the inverse tangent approach zero and thus noise in the input data will have a more significant effect. But this situation can be detected and corrected numerically (I did just such a thing until I decided to switch to the spherical system- basically you just monitor the magnitudes of both numerator and denominator of the inverse tangent and when they both go below a predetermined threshold, you skip the angle calculations for that frame of data and then go back later and interpolate the missing angle back in. It's a kludge but it worked nicely for our situation...) The finite screw is more proper representation of the actual motion process occuring in the joint. I.e., the joint is actually rotating about some line in space not some arbitrarily selected coordinate axes. Unfortunately, this line of reasoning falls apart when you try to express the screw in more clincally oriented terms since it must be expressed relative to just such a set of axes as in the case of the attitude angles concept. I don't think that using either the proximal or distal LCS as the base for expressing the attitude angles is a good idea because even though the system axes will orthogonal and hence will never be singular, the angles are physically meaningful to the clinician. They are valid descriptors of the joint orientation but since eventually the data is to be used by a phsycian to make a clinical decision, at least here that is the case, the data should be in a form with which they are familiar. The JCS axes pretty much align with the way that the orthopaedist describes/visualizes joint orientation. There is the singularity problem but that cannot physically occur in any major extremity joint except the shoulder unless there is a very severe pathology present. Even though the screw does represent full 6 DOF motion, if only the attitude angle representation is desired then the rotation matrix relating the segment orientations is all that is required since this calculation is a subpart of the full finite screw calculation. This calculation, I might add, is no more computational difficult than the standard technique of extracting rotaion angles from the rotation matrix. As an aside, the numerical method by which the screw is calculated is significant as to its sensitivity to noise in the input data. The input data consists of the position and orientation of one rigid body relative to another in 3D. This data has noise within it as result of the resolution of the system used to measure the position/orientation and its method of measuring. We have used both VICON based marker measuring as well as a 6DOF spatial linkage designed and built here to collect this information. I have done some experiments using data from these using the algorithm given in Herman's papers, which I believe is more or less the same as that of Spoor-Veldpaus (sp?), as well as using one based upon derivations done by Ken Waldron, who is a prominent kinematics/robotics researcher at the Mechanical Engineering Department here at Ohio State. I found that Waldron's method was less sensitive to noise. In addition, there was a paper recently given at the ASME conference on Advances in Design Automation-1989 (Univ. of California, Davis) intitled 'Comparison of Methods for Determining Screw Parameters of Finite Rigid Body Motion from Initial and Final Position Data' by R.G. Fenton and X. Shi. This compared 5 different methods, including Spoor-Veldpaus, and concluded that the best, based upon sensitivity to noise, is one by Bottema and Roth (of the textbook Theoretical Kinematics fame) while Spoor-Veldpaus was noise sensitive. I know that there are other versions of the Spoor- Veldpaus method for use with more than 3 markers to decrease the noise sensitivity, but based upon its use with straight position/orientation information of the rigid bodies as input (as I use it) it does not appear to be the best choice. I will be doing some more work with this area in the near future, say the next month or two, and if anyone is interested I can post a follow up. In summary, I'm trying to stay out of the 'one is better than the other because...' argument but instead am trying to look at the several methods as they are used in practice. The only place where a firm decision as to a single calculation method is critical is for exchange of data between institutions. That is how all of this got started here at Ohio State. We have been involved in a Multi-Institutional study of cerebral palsy child gait for a number of years now that has used sagittal joint angles exclusively. We would like to expand our focus but before this can happen this matter of joint angle calculation methods and coordinate system positioning/orienting must be resolved, at least between the sites involved in the study. For an individual research group this is not such a critical issue just so long as they establish a method and stick to it. This issue is more of academic interest in this case. It is also possible to convert back and forth between the various angle systems with relative ease as demonstrated by Herman's program, but the matter of the definition of the coordinate systems is a definite problem... By the way, that paper on body segment mass/inerita parameters that mentioned in the fall from Wright-Patterson Air Force Medical Research Laboratory had an interesting approach to the coordinate system definition problem. The CS's were defined relative to several easily identifiable bony landmarks for each body segment. The down side of this method is the need to measure the 3D location of a lot of points. Enough said for now. I hope some others might join in on this discussion... Dwight Meglan Research Engineer and Phd-in-training (finished product coming to a journal near you sometime in 1990/91 ;-) meglan%gait1@eng.ohio-state.edu In the current spirit of disclaimers, the above text was not written by anyone, to anyone, about anyone, or for anyone no matter what the text actually says:-) ========================================================================= Date: Sat, 3 Mar 90 14:29:00 EST Reply-To: GROOD@UCBEH Sender: Biomechanics and Movement Science listserver From: GROOD@UCBEH Subject: Round 3 Response Dear Biomch-L readers, Karl Popper, in his famous book, "Logic of Scientific Discovery" proposed a solution to the problem of how to distinguish scientific from non-scientific theories. Popper noted that a theory can never be proved, only disproved. This is because no matter how may observations are made that support the theory there are an infinite number of future observations that might disprove it. This led Popper to the belief that a scientific theory must be falsifiable. In this way he was able to distinguish scientific theories from theories that are defended regardless of the weight of the evidence. While I would not classify the issue before us as scientific theory in the same sense Popper used, his logic can still be applied to the present argument. This can be done by Herman and I stating what points, if proved or disproved would cause us to change our mind. If no such points exist, this debate can have no scientific resolution. We might just as well be discussing political ideology or religion. In this spirit I have identified in my arguments below, the key points which, if disproved, would cause me to change my mind. These points go to the heart of the technical issues which underlie my position. It is possible, of course, for others to identify arguments I overlooked which would also disprove my position. MEANING OF "INDEPENDENT" This term is used in the common non-statistical mathematical sense. A set of variables are independent if each can change its value without any change occurring in any of the other variables. INDEPENDENT GENERALIZED COORDINATES To describe the position of a mechanical system it is necessary to employ a set of k independent variables, where k is equal to the number of degrees-of freedom (dof) in the system. A particle has 3 dof in 3-D space. A collection of N independent particles has 3N dof. If the particles are constrained so the distance Lij between any pair of particles i and j is constant, the particles comprise a rigid body. The constant distance between particles can be expressed by constraint equations of the form: Lij = const i,j = 1...N, i not equal to j. For two particles, N=2, we have one constraint condition, L12 = const which reduces the dof from 6 to 5. When a third particle is added (not co-linear with the first two) there are two additional constraints L13 = const and L23 = const so only one dof is added for a total of 6 dof. Each additional particle has 3 or more constraints of which only 3 are independent. As a result the addition of more particles does not further change the number of dof. This explains why 6 independent variables are required to describe the position and orientation of a rigid body, we need one for each of the 6 degrees-of-freedom. Such independent variables are also called generalized coordinates. Generalized coordinates (often written qi) have the trajectory property Herman referred to. This property is simply a manifestation of their independence from each other. The trajectory (or path) is obtained by expressing each coordinate as a function of time, qi = qi(t) i=1,..,6. Thus, at any time, t, there are six coordinates qi(t) i=1,..,6 which completely define the position and orientation of a rigid body. There is no unique set of 6 generalized coordinates. Numerous sets are possible and often used. For example, the location of a particle can be described with Cartesian, cylindrical, and spherical coordinates or by the length and direction cosines of its position vector. In a similar manner, the orientation of a rigid body can be described by any of the Eulerian/Cardanic sets of coordinates, or by the magnitude of the rotation and the direction cosines of the helical axis. Point 1. THE THREE ANGLES FOR EACH EULERIAN/CARDANIC SYSTEM ARE INDEPENDENT COORDINATES WHICH DESCRIBE THE ORIENTATION OF RIGID BODIES SEQUENCE DEPENDENCE Herman, following arguments made by Jim Andrews, differentiates two types of sequence dependence, geometrical and temporal. The geometric sequence refers to the order rotations are performed about the axes of a Cartesian system fixed to either body segment. Point 2. EACH SEQUENCE OR ORDER OF ROTATION PRODUCES A DIFFERENT SET OF INDEPENDENT COORDINATES. Herman asserts that rotations are sequence dependent but particle displacements are not. I disagree. When properly defined neither are sequence dependent. When poorly defined it is possible to produce sequence dependency in particle displacements by following the same recipe used for the sequence dependency of rotations. Point 3. PARTICLE DISPLACEMENTS ARE SEQUENCE DEPENDENT WHEN THE INDEPENDENT COORDINATES ARE CHANGED ALONG WITH THE ORDER OF THE DISPLACEMENTS COMMUTATIVE PROPERTY Once the generalized coordinates which describe a system have been selected we can check them for the commutative property under some specific operation. Let us restrict the discussion to addition. Herman's definition is correct, but incomplete. The entities "a", "b", and the result of the binary operation (a o b) must all be valid elements of the same set. Let the entities, designated by the triple (x,y,z), be valid generalized rotational coordinates. Let rk represent a triple at position k. The reference position is designated as r0 = (0,0,0). Thus, the triple (x,y,z) not only represents the orientation of a rigid body, but also a displacement from the reference position. The operation of addition is taken to represent the physical process of combining two displacements and is defined in the conventional manner used for vectors: r1 + r2 = r3 r1 = x1, y1, z1; r2 = x2, y2, z2; r3 = x1+x2, y1+y2, z1+z2 In order for the addition to be commutative, not only must r1 + r2 = r2 + r1 = r3 r1, r2, and r3 must all be elements of the same set (ie; they must all be valid coordinates or displacements). All of the Eulerian/Cardanic systems satisfy these requirements as do the Cartesian, cylindrical, and spherical coordinates used to describe particle locations and displacements. Herman refers to this property as temporal commutativity to distinguish it from the geometric sequence used to define the generalized coordinates. The distinction between geometric and temporal commutativity was, to my knowledged, first introduced by Jim Andrews in his letter to the editor (J. Biomech.) discussing my paper with Fred Suntay. I find this new terminology to be confusing. It is not employed when discussing particle dis- placements, and is unnecessary when discussing rotational displacements. We only need to talk about specifying the set of coordinates to be used for a particular problem and whether such coordinates are commutative in the ordinary sense. Interestingly, neither the total particle displacement vector or the helical axis are commutative under the operation of addition (as defined above). Both descriptions employ direction cosines which are limited in value to the range from -1 to +1. Adding two direction cosines of value greater than 1/2 produces a result greater than 1. Clearly the result is no longer a member of the same set of coordinates. Herman makes an interesting proposal which, at first glance, seems to avoid this problem. He proposes to first create a vector parallel to the helical axis whose magnitude is equal to the total helical rotation. This vector can then be described by its vector components with respect to the Cartesian system in either body segment. I will call these components Woltring angles to differentiate them from traditional helical axis coordinates which employ direction cosines. A Woltring rotation (or displacement) will be a change in body attitude produced by a chance in Woltring angles. If I subject Woltring angles to the arguments above I conclude they are a valid set of generalized attitude coordinates. They certainly define the helical axis and therefore the orientation or rotational displacement of a rigid body. Further, the components are independent as they can be changed one at a time. Finally, they have path descriptive properties as you can reconstruct the trajectory followed if you know the time history of the Woltring angles. Why then should I object to their use? There are several reasons. First, Point 4. WOLTRING ANGLES DO NOT CORRESPOND TO ANY REAL ANGLES AS DEFINED BY TWO LINES IN SPACE. Second, while Woltring angles and displacements satisfy the mathematical requirements for commutativity, the operation of addition can not be interpreted as combining two Woltring displacements. That is, if a body is subjected to two finite helical dispalcements, the Woltring angles of the combined displacement are not the sum of the Woltring angles of the individual displacements. A corrolary to this is, Point 5. THE ORIENTATION OF A BODY DESCRIBED BY A SET OF WOLTRING ANGLES CAN NOT BE OBTAINED BY PERFORMING WOLTRING ROTATIONS ABOUT A SET OF THREE AXES. Finally, I am fearful that these properties will be ignored and that Woltring angles will be improperly interpreted as rotations about each of the coordinate axes. The three axis approach I recommend has the following advantages: 1. The coordinates are real angles and displacements are real rotations about well defined axes in space. 2. The axes can be selected so the angles correspond to common clinical descriptions of joint position. 3. The angles are valid generalized coordinates that can be used in dynamical analysis of joint motions. 4. The angles are commutative and rotational displacements can be combined by adding and subtracting the angles. 5. The system provides insight in to the nature of joint motions by explaining such phenomena as Codman's paradox. At this point I will withhold further comment until others in the community have an opportunity to share their thoughts. I will, however, answer specific questions directed to me. Herman, I appreciate your invitation to co-author a paper. In order to reach some agreement, I have identified 5 points which are the foundation of my arguments. Proving any of them wrong would cause me to carefully re-evaluate my position. The problem of gimbal lock and the associated computational complexities seem minor to me. As noted above, I see Codman's paradox as supportive of my position. If we can not find a way to agree there is the possibility of putting this debate in print. Edward S. Grood Grood@UCBEH.SAN.UC.EDU Noyes-Giannestras Biomechanics Laboratories Department of Aerospace Engineering and Engineering Mechanics University of Cincinnati, OH 45221-0048 2900 Reading Rd. Cincinnati, OH 45221-0048 ========================================================================= Date: Tue, 6 Mar 90 11:22:00 N Reply-To: "Giovanni LEGNANI. University of Brescia" Sender: Biomechanics and Movement Science listserver From: "Giovanni LEGNANI. University of Brescia" Subject: angles, screws and matrices......... Dear biomch-l subscriber, Returning from a few days holiday I found other papers on angles, screw axes and so on... It looks as if everybody wants to show how good he is in math.... And although my English and my Math are very bad, I also want to do the same... Like many people, I have studied many different ways to express rotations. A chapter of my doctoral dissertation (1986) was devoted to this study. I do believe that there is NOT a BEST way to represent the angular position of rigid bodies. Every method has good points and bad points. Probably each method has been established because it was useful in a practical application. For example, I guess, that the Euler's angles have an interesting meaning in Astronomic study about planets. Any way I devoted a little time in studying the following sets of "angular parameters": 1) Euler's angles 2) Cardan's angles (also known as Tait-Brians' angles) 3) Euler's parameters and "quaternions" 4) Rodriguez-Hamilton's parameters 5) Euler's axis and angle (also known as finite rotational axis) 6) a system based on Latitude and longitude 7) director cosines and rotational matrices 8) screw theory Each of the methods, except 8), falls in one of the two following groups: a) a three independent parameters system 1) 2) 4) 5) b) a more (>3) NOT-independent parameters system 3) 6) 7) the screw theory describes either rotations and displacements and from the rotational point of view is "similar" to method 5. Every system which consists of 3 parameters has mathematical singularities for a few particular values of its parameters. (i.e. the Rodriguez parameters for TETA==90 degrees or the Euler angles if the node axis is not defined, ....) and there are high numerical errors when the parameters are closed to these singularity points. Any way this is not a problem if one is sure that in its problem he will never reach or approach these points. Every system consisting in more than 3 parameters removes the singularity but forces to use a set of NOT independent "coordinates". The meaning of a set of coordinates of group 2 is also generally less expressive or clear for Humans. I do think that the choice of an angular parameter system can be done taking into account many aspects as (for instance): 1) the meaning of each parameter in each practical application 2) the risk to fall into a singularity 3) the math simplicity ...... .... and last but not the least 999) the researcher's experience and beliefs... I like to use a 4*4 matrix method based on the well known Transformation matrices approach. I extended this method to full kinematics (speed and acceleration) and dynamics (wrenches (forces+torques), linear and angular momentum, inertial terms). I think that, at least for computer applications, this method is very convenient for its programming simplicity and because it has not any math singularity. A 3*3 sub-matrix of a 4*4 matrix is the well known rotational matrix that can be easily built from whatever of the above coordinate systems. However, the inverse transformation (i.e. from matrix to parameters) is, of course, possible and easy only if we are not close to one of the math sin- gularities of that parameters set. Using this approach, my colleagues and I have been developing SPACELIB, a computer library for the kinematic and dynamic analy- sis of systems of rigid bodies. Using this library we have reali- zed many computer programs for the study of robots and for human body simulation (direct and inverse dynamics and direct and inver- se kinematics). Summarizing: 1) each method has good and bad points 2) I like matrices 3) I am interested in the exchange of papers, SHORT mails and computer libraries. If someone is interested in my topics, he can have a look at the proceedings of the last Int. Congress of Biomechanics (LA 1989) or its satellite meeting on computer simulation (DAVIS, CA 1989) or write or mail to me directly. I am looking forward to hearing from someone about angles, matrices or grammar mistakes. Yours Faithfully Giovanni LEGNANI Giovanni LEGNANI University of Brescia Mech. Eng. Dep. Via Valotti 9 25060 MOMPIANO BS ITALY tel +39 30 3996.446 fax +39 30 303681 ========================================================================= Date: Fri, 9 Mar 90 11:18:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Joint attitude debate: final reply & summary Dear Biomch-L readers, The number of (lengthy) responses on the current joint `angles' debate has been rather small, both posted and emailed to me, so I agree with Ed Grood that it is time to put things to an end with this reply. Of course, other subscribers are free to continue, but the major philibusters should, perhaps, exercise some con- straint. * * * 1. Dwight Meglan's posting (Wed, 28 Feb 90 11:49 EDT) * * * I am delighted that someone took the trouble of processing real joint data and to show the different graphical results; time permitting, this is precisely what I have been planning to do. Regretfully, Dwight's abstract as published in the ORS '90 Proceedings on p. 558 (D.A. Meglan, J. Pisciotta, N. Berme and S.R. Simon, Effective Use of Non-Sagittal Plane Joint Angles in Clinical Gait Analysis) merely refers to what were called `different Euler angle systems: 1) The fixed xyz axis system [Inman, V. et al., Human Walking, Williams & Wilkins, 1981], and 2) the floating xyz axis system (or Joint Coordinate System) [Chao, E.Y., J. Biomech. 13:989-1006, 1980; Grood, E.S. & Suntay, W.J., J. Biomech. Eng., 104:126-144, 1983]'. Thus, helical `angles' were not anticipated at the time of abstract submission. On the other hand, co-ordinate system changes were discussed, and Dwight's point that these should also be taken into account is quite appropriate. However, this does not mean that the current joint `angle' debate is superfluous since the differences in calculated angles under different conventions can be quite dramatic, other things being equal. Actually, Dwight's fixed xyz Euler angles are, in my mind, not at all what are usually seen as Euler angles, i.e., some (well defined) sequence in the Carda- nic/Eulerian sense that we have been debating on this list, but angles between projections in fixed laboratory coordinate planes (XY, YZ, ZX) of spatial lines (e.g., longitudinal limb axes). Since these angles are even less well-behaved than Cardanic angles, I decided not to discuss them in the current debate. Of course, the problem with abstracts submitted to large conferences is that they may be obsolete or incomplete (because of new results) once the conference takes place, and this may have been the case with the OSU study. Fortunately, most conference organizers have the flexibility to accept such changes. That I make this seemingly unfriendly remark has a real purpose: when I quoted the debate between Alice and Humpty Dumpty, this was not merely a (poor?) joke, but an indication of what seems to happen perpetually in the present debate. Words are used in slightly different meanings, with all the concomitant, Babylo- nian confusion. Before the Iron Curtain became torn, there were big signs once one entered Eastern from Western Germany saying "You are now entering the Ger- man Democratic Republic" -- which I, because of a western bias, just thought to have left. Jim Andrew's Letter to the Editor [J. Biomech. 1984, 155-158], referred to by Ed Grood in his last posting, claims that Ed's use of the term `co-ordinate system' is unconventional, and that Ed, in fact, merely described a fictitious linkage system. In a similar fashion, I think that Ed has reinter- preted the term `sequence independence' in an unusual and unnecessarily confu- sing way. While Dwight's `shotgun' approach is to be commended, his explanations and con- clusions worry me to some extent. When he says `It is important to note, how- ever, that the finite screws describe the joint motion as a single rotation about an axis in space which is exactly what the joint motion is, not a sequence of ordered rotations', I believe that he is quite mistaken. Again, the helical convention merely proposes to describe a current or actual joint attitude AS IF it is attained from the reference attitude via a single, helical motion about some directed line in space, it does NOT claim that this, in fact, occurs. What one can do, though, is to view the movement, at each moment in time, as an i n s t a n t a n e o u s rotation about plus translation along some directed line in space. Now we talk about the Instantaneous Helical Axis which is some- thing quite different. At each time instant, the `amount' of movement is defi- ned by the instantaneous rotation velocity about and translation velocity along this IHA, while the `mode' of the movement is defined by the position of (some point on) the IHA and the unit direction vector of the IHA. Dwight suggests that one should decompose the helical angle `vector' into com- ponents along the generally oblique axes of a Cardanic linkage system like the one preferred by Ed Grood. It may be that many orthopaedists are now familiar with this Cardanic convention (but I believe that the non-Eulerian, `projected angles' in San Diego are well understood there), but that is not a valid reason to stick to them, especially if there are serious disadvantages. Good surgeons are keen on learning new things once they believe that it will help them in their work. Jim Andrews mentions three key arguments for any joint angle definition: they should explain the movement (or orientation) easily, they should not exhibit singularities, and they should be easy to calculate. The first argument ap- plies, in my mind, both to helical and Cardanic angles, with a preference for the former since I think that sequence effects are more difficult to explain than the notion of orthogonally decomposing the helical `vector'. The second argument is in favour of the helical approach, where it might be useful to note that close to gimbal-lock, certain differential displacements of a joint or body will hardly be reflected by the chosen joint angle `co-ordinates', whereas other differential displacements of the same magnitude will result in very strong changes in these angles; this makes visual interpretation of angular graphs rather difficult. The third argument is obsolete with current computa- tional facilities. Instead, the advantages of maximizing orthogonality should be clear to anyone who wishes to describe joint angulation and to relate it to forces and moments which are true vectors, commonly decomposed in (truely) independent, orthogonal components. Under various Cardanic conventions, strong flexion, abduction, and endorotation under one convention become about the same flexion, but adduction and/or exo- rotation under another one, while no strong differences are observable when all angles are small. Thus, different p a t t e r n s of joint angles are to be expected necessitating agreement between investigators (and their institutions?) in order to allow valid comparisons. Since the `helical convention' more-or-less provides the mean values of all possible Cardanic conventions, this might be an additional argument in its favour, despite the generally non-physical nature of its component angles. * * * 2. Ed Grood's posting of Sat 3 Mar 90 14:29 EST * * * I appreciate Ed's attempt to clearly define his key considerations, which should make it easy to follow the debate. a. "Independent" in the non-statistical, mathematical sense. Agreed, but in- dependence is better if it applies also in the statistical, mathematical sense with orthogonal (uncorrelated) components. Besides, some of Ed's examples for his 6 generalized co-ordinates do not reflect independence in his use of the term: the direction cosines of a position or direction `vector' are not, since their squares add up to unity. In an earlier email note or posting (off my head, there's too much paper on this debate already), Ed suggested using the length and two of the three direction cosines as independent variables; this, in my mind, is extremely unelegant and at variance with Jim Andrew's first two conditions. b. I agree that both Eulerian/Cardanic and helical angles are independent in the above sense, but close to gimbal lock, they are quite differently behaved. This may not be a problem in level, straight gait analysis (the current paradigm), but it certainly is a problem in complex, sportive movements. When Ed claims that generalized co-ordinates have trajectory properties, he reinterprets a word from its intended meaning (for which I am the guilty one if I have been insufficiently clear in my words): certainly, I thought in terms of a physical path about and along the axes of Ed's linkage system, not in terms of some abstract, mathematical space whose parameters have the dimension of angles, behave as angles in certain special cases, but which are, in general, not real, physically identifiable angles. ["It looks like a duck, walks like a duck, quacks like a duck, so it must be a ..."]. Similarly, I did not think in terms of the continuous time-dependent movement that our joints exhibit, but about how to `optimally' describe a given attitude at some specific time only. c. Ed's argument on `different sets of independent co-ordinates' comes back on what I stated above with Dwight Meglan. Ed defines `sequence' in an unusual, and, in my mind, unnecessarily confusing way. If I first (i.e., proximally) translate x along the X-axis, then (distally) y along the (displaced) Y-axis of a given, Cartesian co-ordinate system, I wind up in the same position as when I had first (proximally) translated y along the Y-axis and then (distally) x along the (displaced) X-axis. For rotations, however, different attitudes are attained. Furthermore, the distinction between temporal and geometrical se- quences is not used with particle displacements because it is not needed there. When Ed says that this terminology `... is unnecessary when discussing rotatio- nal displacements. We only need to talk about specifying the set of co-ordina- tes to be used for a particular problem and whether such co-ordinates are com- mutative in the ordinary sense', I do not know whether he refers to Cartesian co-ordinates or linkage co-ordinates (when are they the same?) and to what kind of `ordinary sequence' he refers to. d. `Woltring angles'. While I feel honoured to see my name attached to gene- rally non-existing angles (at least in real, physical space), I would prefer to stick to the name `helical angles'. I maintain my position that, for attitude description purposes, there is no need that these angles be generally identifi- able with some specific physical angles; their orthogonality is the more impor- tant property. Jim Andrews made some related remarks on this point. e. At the present time, there is no clinically well-accepted convention for 3-D joint attitude parametrization. Various orthopaedic surgeons accept what their engineers tell them, but if these engineers cannot agree amongst themselves ... f. Cardanic/Eulerian angles fail close to and at gimbal-lock, in both direct and inverse dynamics. g. Ed's angles are `commutative' and `additive' in his definition of the terms, at the expense of a generally non-Cardanic, oblique `co-ordinate system'. h. There is no need to advocate the use of any proposed joint convention for joint ATTITUDE quantification because of its properties to explain certain peculiarities of particular conventions. Ed, let's wait and see what the (electronic) community has to say; may-be, its contribution will eventually make a published debate useful. * * * 3. Dr Legnani's posting of Tue 6 Mar 90 11:22 N * * * Dr Legnani has given a nice summary of various attitude parametrization methods; in the present debate (human interpretation of joint angle graphs), his `3 para- meter models' are the relevant ones. His claim that `(e)very system which consists of three parameters has mathema- tical singularities for a few particular values of its parameters' does (unless I am mistaken) not always hold true. If they were, the covariance matrix for these parameters should become unbounded when the singular points are approach- ed. This is not the case when calculating the covariance matrix for the `vec- tor' THETA (with 0 .le. theta .le. pi) using the relation THETA = theta N and formulae (20) and (21) in a paper on finite helical axes and centres of rotation in the Journal of Biomechanics 1985, p. 382. Working out the various partial derivatives and matrix products yields COV(THETA) = k [ {theta^2/(1 - cos(theta))} (I - NN') + 2 cos(theta)^2 NN' ] which, for theta --> 0, reduces to 2 k I, where I is the identity matrix, and k the variance of incremental disturbances on the attitude matrix in arbitrary directions; see the quoted paper for further details where k is a function of isotropic measurement noise per co-ordinate axis and of an isotropic landmark distribution in photogrammetric rigid-body reconstruction. If theta approaches 2 k pi, with k integer and non-zero, the covariance matrix becomes unbounded, but this is irrelevant in the present debate where arbitrary attitudes can be represented for 0 .le. theta .le. pi; cf. the planar case where the unit direction vector N is replaced by a + or - sign, with theta periodic in 2 pi both in the 2-D and 3-D cases. While I have no proof, I believe that the `helical vector' is well behaved also for other situations than the isotropic case referred to above, as long as the landmark distribution and the noise are non-pathological. Proving this might be a nice challenge for a mathematically oriented MSc or PhD thesis ? Herman J. Woltring Eindhoven, The Netherlands ========================================================================= Date: Fri, 9 Mar 90 11:46:36 MET Reply-To: Leendert Blankevoort Sender: Biomechanics and Movement Science listserver From: Leendert Blankevoort Subject: EMOTION (E-mail Motion) Dear Biomch-L readers, Following the discussions on the description of joint motions, I would like to contribute the following points, which are based on the experiences we have in our Biomechanics Lab. at the University of Nijmegen. My intention is not to be in favor of any proposed convention, but to discuss some items which were not directly addressed as yet and to clarify some points. 1) JCS vs. Euler/Cardan angles According to our analyses, the rotations in the JCS system and the Euler/ Cardan angles are equivalent, provided that the rotation sequence is applied which is implied in the geometric definition in the JCS system. This can be derived from the equations given in the paper of Grood and Suntay (1983). For instance, if for the knee flexion, ad-abduction (or varus-valgus) and internal-external rotation are specified of the tibia relative to the femur, then in the JCS system the flexion axis is fixed to the femur, the ad-abduction angle is the floating axis and the internal- external rotation axis is fixed to the tibia. Using the Euler/Cardan convention, the rotations are then specified as rotations around the body- fixed axes of the tibia moving relative to a space fixed femur for which the rotation sequence flexion, ad-abduction, internal-external rotation is to be used in order to obtain the same values for the rotations as in the JCS system. I believe that despite the controversies between Grood and Woltring, both "chief" discussers can agree on this. 2) Translations So far much of the attention was paid to the rotations and only little was said on the translations, probably because the rotations are considered to be clinically meaningful. However, the anteroposterior translations as measured in instrumented AP-laxity tests, are important for quantifying AP laxity and AP stiffness. The value of the translations of one bone relative to another depend on the choice of the so-called base points, which generally are identical with the origins of the coordinate systems in the bones. Thus, special care is to be taken when comparing translational data between different research groups and between different measurement devices. When using the finite helical axis description, the translation along the axis is independent of the choice of the coordinate systems and represents the "true" translation of one bone relative to another for a finite motion step. For pure translations however, the helical axis is not defined, but then the value of the translation does not depend of the choice of the base point. 3) Variations of the coordinate systems in experiments As pointed out by Meglan, variations of the orientation of the coordinate systems relative to the joint anatomy and the variations of the anatomic reference position account, at least partly, for the differences in the resulting rotations between joint specimens or between subjects. This was discussed and illustrated in a paper from our group in the Journal of Biomechanics in 1988 (Blankevoort et al., 1988). This does not mean that as long as we are not able to uniquely define coordinate systems in different joint specimens or different individuals, we should not bother about the choice of the kinematic convention for the reason that the variations between the convention are smaller than the observed variations between joint specimens or individuals. Different kinematic conventions will introduce systematic differences which have no relation whatsoever with biologic variations or statistical standard deviations. 4) Mathematics and representation The present discussion at the list was mainly focussed on how the rotations are to be represented. Of course, if one is presenting data on joint angles, then the mathematical background is to be specified. The reader can then always reconstruct the motion patterns and express them with his (or hers) own convention. However, most of the reports fail to give all numbers, e.g. three rotations and three translations, because only those data is reported which is relevant to the subject of the paper. Some standardization may be necessary for commercially available measurement systems to be used on a routine basis in the clinic. However, during the process of data acquisition and data processing or in the formulations of kinematic models, one is free to use any of the conventions as long as it meets the criteria for the data processing and the mathematics of the model. In the final stage, the kinematics data can be transformed to any desired system. For instance in gait analysis, one can process the marker data by some rotation convention which is found to suit best the requirements for the subsequent dynamic analyses and can represent the kinematics by use of the JCS or the Woltring system. 5) Future steps Following the discussions on the list, it became clear to me that because of the complexity of the description of 3-D joint motions, there remains a lot of "teaching" to do in the field of Biomehcanics to make clinicians, anatomists as well as biomechanicians aware of the difficulaties and pitfalls of kinematics. Basic understanding seems more important than the normalisation of motion description by adopting some standard convention, since the consequences of any choice should be properly understood by the users of such standard system. I am certainly in favor of producing an extensive paper (or maybe a book?) on the ins and outs of biokinematics, which should be aimed at a broad audience in the field of biomechanics, to people with a poor mathematical background as well as people with poor knowledge of (human) anatomy. I encourage Grood and Woltring to write something (after they have cleared their confusions). References Grood, E.S. and Suntay, W.J. (1983) A joint coordinate system for the clinical description of three-dimensional motions: applications to the knee. J. Biomech. Engng 105, 136-144. Blankevoort, L., Huiskes, R., Lange, A. de (1988) The Envelope of passive knee joint motion. J. Biomechanics 21, 705-720. Leendert Blankevoort Biomechanics Section Institute of Orthopedics University of Nijmegen P.O.box 9101 NL-6500 HB NIJMEGEN The Netherlands U462005@HNYKUN11.EARN ========================================================================= Date: Fri, 9 Mar 90 20:38:00 EST Reply-To: GROOD@UCBEH Sender: Biomechanics and Movement Science listserver From: GROOD@UCBEH Subject: Joint Attitude Debate Dear Biomch-L Subscribers I will be in Switzerland for the next week and therefor unable to respond to any questions or comments for a while. While this debate has certainly been worthwhile and interesting, I do note the ever widening circle of points to argue and debate about. There is a clear need to bring some focus and resolve fundamental issues. If anyone has a suggestion on how to accomplish this I would sure like to hear it. Edward S. Grood Cincinnati, OH USA ========================================================================= Date: Mon, 12 Mar 90 10:40:00 EST Reply-To: MEGLAN%GAIT1@PHEM1.IRCC.OHIO-STATE.EDU Sender: Biomechanics and Movement Science listserver From: MEGLAN%GAIT1@PHEM1.IRCC.OHIO-STATE.EDU Subject: Let me fix that... Greetings once again, In Herman's recent message which summarized the postings by several of us, he mentioned that he was concerned with the statement I made about finite screw axis defining the axis about which the actual joint rotation is occuring at a given instant in time. Well, he is right. As written in the posting, my statement is incorrect. I was thinking one thing and writing another. For our purposes here, we use two different types of finite and instantaneous screws. We call them absolute and relative screws. The absolute is the screw motion which defines the position of a body relative to a base coordinate system. Essentially, it is the motion the body would go through if it started out coincident with the base coordinate system and ended up in its final position. This is the screw used by ourselves to calculate the screw attitude angles and I believe this is the same definition used by Herman. The relative screw is the axis about which the body would move when it goes from a position/ orientation at time t to a position/orientation at time t+1. This screw axis can be defined in terms of any coordinate system. In our case we usually look at it in terms of the proximal segment local coordinate system (LCS), although the distal segment LCS and lab global coordinate system (GCS) versions are also calculated. This is the screw axis that I was refering to in my posting. This I believe is the axis about which the joint rotates at a given instant in time. Of course, the relative instantaneous screw axis should be a more correct description and in the limit as the sampling rate becomes infinitesimally small the finite and instantaneous screw axes should coincide. Note: I'm refering to the direction of the screw axes not the entire screw (i.e., the rotation and translation along and about the axis and well the screw's position). Both the absolute and relative screws are useful as can be seen in Herman's application of the absolute screw and in Leendert Blankevoort's works on joint kinematics where he has used both absolute and relative screws I believe. The relative screw description is what has been used to define the pierce point of the axis of rotation of the knee on to a sagittal plane as following a C shaped curve located in the midst of the femoral condyles. Sorry for the confusion. I hope this clarifies what I meant rather than what I said :-) One last comment. I fully agree with Leendert Blankevoort's comments about translation definitions. We've tried a number of different definitions usually based upon the locus of the instantaneous screws of a given motion and it is definitely more difficult to define translations than rotations are far as I'm concerned. With translations not only is the placement of the embedded coordinate systems critical, but also the orientation. This is an even more difficult task for trying to come up with a standardized definition. Dwight Meglan The Ohio State University Gait Analysis Lab meglan%gait1@eng.ohio-state.edu ========================================================================= Date: Thu, 29 Mar 90 10:41:00 N Reply-To: "Giovanni LEGNANI. University of Brescia - Italy - European Economic Community" Sender: Biomechanics and Movement Science listserver From: "Giovanni LEGNANI. University of Brescia - Italy - European Economic Community" Subject: joint attitude debate...... Dear biomch-l readers, I was asked by a number of you to point out better my opinion on the joint attitude and angles debate. I was also asked to give information about SPACELIB. I will send a personal answer to everybody but I'd also like to summarize my ideas here. I begin with SPACELIB. This is a software library I realized with the help of a few colleagues in order to study spatial systems of rigid bodies. We apply it in robotics as well as in biomechanics. SPACELIB is based on a matrix approach involving 4*4 matrices. We have six different kind of matrices to describe three kinematic and three dynamic entities. They are POSITION, VELOCITY AND ACCELERATION of bodies (or points) and ACTIONS (forces and torques), MASS DISTRIBUTION (inertia moments) and MOMENTUM (both angular and "linear"). Using this library we have written a program for the whole body motion analysis we presented at the last symposium on computer simulation in Biomechanics (DAVIS CA 1989). This matrix approach, that can be considered being the extension of the homogeneous transformation approach to the whole kinematics and dynamics, is documented in many papers. The library written in C-language consists of about 40 modules which perform all the elementary steps necessary in developing of the analysis of any spatial system of rigid bodies. We are planing to realize in the future an academic free sharable version of SPACELIB. We will be happy to send further documentation or papers to anyone who is interested in. **** Angles debate: In a previous e-mail I summarized the most famous sets of angular coordinates. I outlined also my opinion on the fact that I don't believe that any of these sets are THE BEST in EVERY situation. Before answering to whom asked me a deep discussion I want to state a few points: 1) I am interested in (three dimensional) kinematic and dynamic aspects of biomechanics. 2) I am NOT an expert in smoothing raw data. 3) I am not trying to give THE FINAL answer to the debate but I am just trying to point out a few relevant aspects. 4) This mail isn't a well-organized paper but just a "hand- written" note edited in order to give a general idea of my opinion. I believe that the central idea is "if we want to study something (e.g. the elbow, the knee, ... the whole body ...), initially we create a model of that thing (e.g. we decide if we will consider or not the knee being a perfect revolute pair) and at last WE CHOOSE A SET OF COORDINATES WHICH EMPHASIZE THE MAIN IMPORTANT CHARACTERISTICS OF THE MOVEMENT" I find that the relative angular position between two bodies A and B generally falls in one of the following situations: a) The two bodies are not linked by any angular constrains. In this situation body B can assume any angular position with res- pect to A (e.g. the trunk of a man during a jump can rotate freely with respect to the Earth). b) The two bodies are connected by a non-spherical joint. Many situations can occur -- the most relevant are: b.1) The two bodies are connected by a revolute pair or an approximate rev.pair (i.e. arm and forearm are connected by the elbow (an approximate revolute pair)). b.2) The two bodies are connected by a "joint" which permits two or more degrees of freedom (e.g. the head is connected to the body by means of the neck (an approximately spheri- cal joint)). c) The two bodies are connected by a revolute pair or by a joint which can be considered the sum of two or three revolute pairs. (although this situation does not happen in Biomechanics, some joints (e.g., the elbow) can be considered as approximately be- longing to this category.) In situation a) there is no preferred direction, axis or angle, so it appears very obvious to choose a set of angular coordinates which is "neutral" or "symmetric" with respect to the body and to the reference frame (e.g. screw angle system (also called the Euler angle and axis)). On the contrary, in situation b.1) The movement is usually composed of one large rotation (around the joint axis) and possibly two smaller rotations. Anyway all these three rotation angles are always contained in a well-known range. I think that in this situation a system of coordinates which gives a different importance to the first rotation (e.g. flexion-extension of the elbow) with respect to the others (ab-adduction and endo-exorota- tion) is more significant than other angular parameter sets. At last in situation b.2) all the three angles can vary of about the same maximum range but from a "physical" point of view we can identify two different kinds of angular displacements. Let us consider, for instance, the movement of the head with respect to the trunk. It can be decomposed into two "flexions" and one "twist" of the neck. Also in this case it looks to me that the flexions and the twist can be considered having a different "priority". A further consideration is that, while in case a) the angles can vary from 0 to +/- 180 degrees (or 0 -/+ 360),in the cases b) their values are generally lower than 90 degrees. Situations b.1 and b.2 can be possibly handled considering the "joint" composed by two different joints connected in succession. The first joint being a two degrees of freedom joint while the second is a one degree of freedom joint. The first joint is a "semi-spherical" joint which allows the orientation of an axis "a" but do not permits a twist movement around the axis itself. The second joint is a revolute pairs which allow a rotation about axis "a" displaced by the previous joint. (I hope that anyone will invent a easy way to send pictures on e-mail). The three angles giving the body orientation can be: a sort of Latitude and Longitude angles of axis "a" plus the twist angle around the axis itself. These latitude and longitude angles can be defined giving to both of them the same importance and making these angles true measurable angles. Situation c) "requires" the adoption of a "planar" convention (one angle) or (sometimes) the adoption of an Euler/Cardanic convention if the joint is considered being constituted of a series of revolute pairs. Any of these systems of angles can originate a rotation matrix very useful in performing calculations. Summarizing: I suggest to identify a (little) number of different situations and to choose for each of them an appropriate "standard" set of angular coordinates. ************** At last a one million dollars question ******** How can H. Woltring be so frequently present on e-mail, how can he produce so fast new papers (by the way: thanks for the acknowledgement) and how can he continue working at his lab at the same time? Do the days in The Nederland have more than 24 hours? Or are his hours longer than everywhere else? I'd like to have any suggestion from Herman in order to increase my productivity. have a good work. Giovanni LEGNANI P.S. Please generally excuse me for my bad English. ========================================================================= Date: Wed, 9 May 90 17:01:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: J. of Orthopaedic Research 8(3) Dear Biomch-L readers, The last issue of the Journal of Orthopaedic Research (vol. 8, nr. 3, 1990) contains a number of interesting articles: William G. Negendank, Felix R. Fernandez-Madrid, Lance K. Heilbrun, and Robert A. Teige, Magnetic Resonance Imaging of Meniscal Degeneration in Asymptomatic Knees (pp. 311 - 320) A. van Kampen and R. Huiskes, The Three-Dimensional Tracking Pattern of the Human Patella (pp. 372 - 382) M.P. Kadaba, H.K. Ramakrishnan and M.E. Wootten, Measurement of Lower Extremity Kinematics During Level Walking (pp. 383 - 392) In the context of the recent debate on joint angles, the latter two papers seem to implement two different joint angle conventions: Albert van Kampen en Rik Huiskes (Nijmegen/NL) implement the Selvik convention (flexion/extension imbed- ded proximally, endo/exorotation floating, and ab/adduction imbedded distally), while Murali Kadaba and his colleagues (West Haverstraw, NY/USA) have adopted the Grood & Suntay definition with proximally imbedded flexion/extension axis, floating ab/adduction axis, and distally imbedded endo/exorotation axis. They call the corresponding joint angles even `orthopaedic angles'... As Leendert Blankevoort (Nijmegen/NL) claimed in a recent paper in the Journal of Biomechanics, the difference between these conventions is not very large. However, flexion in his study did not go beyond about 90 degrees, if I recall correctly, while knee flexions up to 150 degrees are reported in the present Nijmegen study. It would be interesting to see how different the various curves look under the two conventions. In recent research with Sandro Fioretti in Ancona/Italy, we found that the heli- cal angles and the Grood & Suntay angles are rather different for ab/adduction and endo/exorotation, but quite similar for flexion/extension during regular level walking, if the neutral attitudes are defined with the subject in the standard, anatomical position. In fact, flexion/extension was similar for all possible, Cardanic permutations, while the two other angles could be quite dif- ferent. In one case where flexion/extension was the floating angle, the two other angles were very large, presumably, because of a positive correlation between the terminal angles when the floating angle is not small. I'd like to pose the following question to the readership: it seems that there are different interpretations of the terms `sagittal, frontal, and transverse axes'. While Kapandji defines the sagittal and frontal axes to be horizontal within the planes of the same names (again, for the subject in the standard, anatomical position), others view these axes as normal to these planes. I would be grateful for any comments and pointers to literature defining these alterna- tives. Herman J. Woltring, Eindhoven/NL ========================================================================= Date: Fri, 10 Aug 90 11:30:00 N Reply-To: FIORETTI@ANVAX2.CINECA.IT Sender: Biomechanics and Movement Science listserver Comments: INFN.IT domain is equivalent to BITNET domain: INFNET; INFNET has been disestablished Dec 31, 1988 From: FIORETTI@ANVAX2.CINECA.IT Subject: INFORMAL WORKSHOP AARHUS, 8 JULY 90 Minutes and comments on the informal workshop on 3-D Joint and Segment Kinematics, held in Aarhus, Denmark, 8 July 90. Dear Biomch-l reader, Being a participant in the workshop, these are my comments about the discussion held at the informal workshop organized by Dr. H.J. Woltring in the margin of the VII ESB-meeting, in Aarhus. As a CAMARC partner, I think that it is useful to give information about this meeting to the largest scientific community and to obtain feedback from other participants in the meeting and from all other interested readers. Sandro Fioretti Department of Electronics and Automatica University of Ancona via Brecce Bianche 60131 Ancona - Italy email: FIORETTI@ANVAX2.CINECA.IT Report and Comments An informal workshop under CAMARC has been held in Aarhus during the VII Congress of the European Society of Biomechanics. CAMARC (Computer Aided Movement Analysis in a Rehabilitation Context) is a European project financed by the Advanced Informatics in Medicine (AIM) Programme of the European Communities. The workshop was organized by Dr.ir. H.J. Woltring (CAMARC partner) and took place in a room of the Orthopaedic Hospital in Aarhus/DK. About thirty persons were present coming from Europe and USA. In particular Prof. Grood was the main participant after the long debate occurred with dr. Woltring via the Biomch-L electronic mail discussion list. Almost one half of the participants were physicians coming from public health institutions. Aim of the workshop was the discussion of some practical problems in Movement Analysis. In particular: - Joint angle definition (projected, cardanic, helical angles); - Definition of Body-Segment Co-ordinate systems; - Landmark cluster definitions per body segment; - Anything else thought to be relevant. From the discussion it emerged that: - It was debated, without final agreement, that a standard for the definition of joint angle parameterisation can be valid for all joints. For some joints (such as the shoulder complex) the helical representation could be more efficient than the cardanic ones, while for other joints the opposite could be true. - It is recommended to pose the problem of the superiority of one angle description with respect to another in the sense of its higher level of regularity, globality and simmetry. - There is the problem of the anatomical meaning of the angles obtained with a particular parameterisation. This problem emerged very clearly during the workshop because all non-technical participants ignored the problems connected with the 3-D angle representations while they expected to obtain from the various parameterisations the usual anatomical definition of the joint angles. - The above two points suggest that it could be useful to develop a software tool mapping the results obtained with the most appropriate angle parameterisation for a particular joint into an anatomical conventional, easily understandable, representation of the same variables. - These points suggest that it might be useful to think about how to properly teach technically superior parameterisations with less physical interpretability. - The correct definition of the anatomical co-ordinate systems by external marker positioning is unanimously recognized as a real problem for Movement Analysis. - The procudure proposed by dr. Woltring for a better alignment of the technical axes with the anatomical ones was discussed with data obtained at Ancona Movement Analysis Laboratory and relative to the analysis of knee joint movement during locomotion. The correction of the cross-talks among flexion with ab/adduction and with endo/exorotation constraining to zero the latter two angles (obtained under the helical convention) in correspondence of the maximum of knee flexion was not approved. It was affirmed that the same procedure could be useful if correction is performed in correspondence of 45 degrees of knee flexion during the swing phase of gait. [NB: The above procedure was merely proposed for demonstration; its main purpose was to show how one m i g h t perform such corrections -- HJW] ========================================================================= Date: Tue, 8 Oct 91 10:18:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Volkmann (19th century) Dear Biomch-L readers, Last year February and March, there was an extensive debate between, a.o., Ed Grood and myself on the pro's and con's of Cardanic angles as mechanized in the Grood & Suntay `joint coordinate system', and the so-called `helical angles' as proposed by myself. During last week's second Mathematics in Orthopaedics symposium at the Oskar-Helene-Heim at the Free University in Berlin, I was told that a mr(s) Volkmann in Germany described this convention about one century ago. Unfortunately, I have up to now been unable to locate a reference, and I would be grateful for any help from the readership. Thanks in advance -- Herman J. Woltring, Eindhoven/NL ========================================================================= Date: Wed, 18 Dec 91 01:38:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: 3-D joint angles and standardisation Dear list readers, During ISB-13 in Perth last week, an announcement was made that a draft proposal on 3-D kinematics will be published shortly in the ISB Newsletter, with a request for comments. Following the joint angle debate on Biomch-L during February and March 1990, I am pleased to be able to say that two vendors of 3-D biokinematics equip- ment have promised (during the recent Clinical Gait Analysis meeting in Richmond, Virginia/USA) to implement the so-called "helical" convention next to the other convention(s) already available in their systems. This shall certainly facilitate the community to provide the comments that will be asked for by the ISB standardisation committee. In addition, I might point out to our recent subscribers that a FORTRAN test programme implementing these various conventions can be obtained from the Biomch-L fileserver, by sending the following command (Subject: line is irrelevant) to LISTSERV@HEARN.BITNET or to LISTSERV@NIC.SURFNET.NL, get prp fortran With kind regards -- Herman J. Woltring (via TELNET from UoNSW, Sydney) ========================================================================= Date: Tue, 14 Jan 92 21:15:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Angle(s) et Axe d'Euler (1980/1984) Dear Biomch-L readers, In my posting of 8 October, I referred to Paul Volkmann in Germany who, about one century ago, allegedly introduced the helical angles for the first time. While this rumour has not yet been confirmed (see the ATTJOB TEX manuscript mentioned earlier today on this list), a few hours ago I received a highly interesting fax from one of our readers, Giovanni Legnani of the University of Brescia in Italy. It is a copy from section 1.4.4 "Angle et Axe d'Euler [6]" (note the singular form "angle") in Alain Liegeois, LES ROBOTS, Vol. 7, Analyses des Performances et C.A.O., Hermes Publishing (France) 1984. My translation of the most rele- vant parts is as follows: Another non-redundant and representative way for defining the rotation is of the form N f(theta) which provides a vector in R3. The simplest form is f(theta) = theta, resulting in the rotation vector [Vrx] [nx theta] Vr = [Vry] = [ny theta] (1-42) [Vrz] [nz theta] Except for the trivial case theta = 0 where N is indetermined, on can derive from this 2 2 2 theta = +/- SQRT(Vrx + Vry + Vrz), (1-43) nx = Vrx/theta, ny = Vry/theta, nz = Vrz/theta which define Euler's angle and axis. The vector Vr can be calculated from the rotation matrix R since ... ( ... formulae omitted, similar to those in Spoor & Veldpaus, Journal of Biomechanics 13(1980)4, 391-393 ... ) As regards the instantaneous rotation velocity of the rigid object, this is simply o omega = N theta (1-47) The reference [6] in the paragraph's title is: [6] J.-C. Latombe, J. Mazer, D'efinition d'un langage de programmation pour la robotique (L.M.). Rapport de Recherche No. RR 197, Labora- toire de Math'ematiques Appliqu'ees et Informatique, Grenoble, March 1980. At the present time, this is the earliest source for what (lacking a better term) I have called `helical angles'. Note that the last equation presupposes that N is constant; the ATTJOB TEX manuscript contains a more general formula describing the case that both N and theta in the product THETA = N theta are time-varying. The book's next section, 1.4.5 "Angles d'Euler" (plural!) seems to discuss the classical Cardanic/Eulerian angles -- I merely have its first three lines. Anyone in the readership who knows of an older reference in the published or gray literature? With kind regards, Herman J. Woltring, Eindhoven/NL, FAX +31.40.413 744 ========================================================================= Date: Mon, 20 Jan 92 17:17:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: Euler angles Dear Biomch-L readers, Further to my recent posting on Euler's angle and axis, I have obtained the English translation of Liegeois' book: Alain Liegeois, Performance and Computer-Aided Design Vol. 7, Robot Technology Series Hermes Publishing, London - Paris - Lausanne 1985 Following Section 1.4.4 on Euler's Angle and Axis, the next two sections are concerned with Euler's Angles and Tate/Bryant (i.e., Cardanic) Angles, i.e., those that describe three successive rotations about the axes of either a fixed or moving, Cartesian co-ordinate system from a reference attitude (I) to a current one (R). Interestingly, the author states about *both* types of angles: "In general, (these) angles are not easy to measure and it is better to limit their use to the cases where the three rotations are actually carried out: for example when the end effector of a Cartesian manipulator (ie an arm with three transla- tions) is constructed such that the three successive rotations correspond to linear functions of (these) angles." He does not motivate this preference, though. While the calculus of Eulerian and Cardanic angles is relatively simple once we have an attitude matrix via photogrammetric, ultrasonic, or electromagne- tic means, human movement is fortunately not confined to actually carrying out these successive rotations when getting from I to R, or worse, from R1 to R2 ... Herman J. Woltring, Eindhoven/NL ========================================================================= Date: Mon, 27 Jan 92 20:13:00 N Reply-To: "Herman J. Woltring" Sender: Biomechanics and Movement Science listserver From: "Herman J. Woltring" Subject: 3-D vectors and inertial navigation systems Dear Biomch-L readers, In another posting today, reference was made to RoboTech@uscvm.bitnet which, it did appear, is not a list on robotics. Another query, onto the NA-net list on Numerical Analysis last week was more succesful. This morning, I received the latest issue with the response quoted below. Again, it appears that similar things are (re)discovered in rather different fields, and that, instead of reinventing the wheel and the gunpowder, it makes sense to look around in other but potentially related disciplines ... I have looked up the Bortz and Miller papers which provide some interesting and complementary ideas. In particular, their interest is to assess attitude angles from a measured rotation velocity vector; in addition to a concise angular representation from a given attitude matrix, my interest is just the opposite: to assess the rotation velocity and acceleration vectors from the attitude matrix and/or angles. Herman J. Woltring, Eindhoven/NL - - - - - - - - - - - - - - - - - - - - - - - - From: Daniel Johnson Date: Mon, 20 Jan 1992 08:52:15 -0600 Subject: Re: 3-D Attitude "Vectors" Sender: NA-net Numerical Analysis Digest, Vol. 92, Nr. 4 Herman J. Woltring asks about references to representing rotations as a three-dimensional vector where the direction is the axis of rotation, and the length is the amount of rotation. As a navigation house, we tend to collect different ways of parametrizing rotations. We refer to his sug- gestion as the "Bortz Rotation Vector", based on a paper by John Bortz in 1971: A New Mathematical Formulation for Strapdown Inertial Navigation John E. Bortz IEEE Transactions on Aerospace and Electronic Systems January 1971, Vol. AES-7, No. 1, pp 61-66 He refers to a report by J. Laning in 1949 which I have not