UCRL-JC- 126078 PREPRINT Displacements and Rotations of a Body Moving About an Arbitrary Axis in a Global Reference Frame K. Hollerbach Lawrence Llvermore National Laboratory Livermcm, California and A. Hollister, Rancho Los Amigos Medical Center Downey, California This paper was prepared fw submittal to the ASME Winter Annual Meeting, San Francisco, California November 12-17,1995.
DISCLAIMER This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.
presentedat the 1995 MME Conferenq November 1995, San Francisco DISPLACEMENTS AND ROTATIONS OF A BODY MOVING ABOUT AN ARBITRARY AXIS IN A GLOBAL REFERENCE FRAME K. Hollerbach, Institute for Scientific Computing Research, Lawrence Livermore National Laboratory, Livermore, CA A. Hollister, Rancho Los Amigos Medical Center, Downey, CA INTRODUCTION Measurementof humanjoint motion frequently involves the use of markersto describejoint motion in a global referenceframe. Results may be quite arbitraryif the reference fimne is not propexlychosen with respect to thejoint s rotationalaxis(es). In na~ jornt axes can exist at any cxientationand location relative to an arbitmrily chosen global reference frame. An arbitraryaxis is any axis thatis not coincident with a reference coadinate. We calculate the errorsthat result when joint motion occurs about an arbitraryaxis in a global reference frame. REVIEW AND THEORY Kinematic and anatomic analyses of a numberof humanjoints have suggested thatthey move about fixed revolutes thatare not parallel to the anatomic reference planes (Hollister, et al. 1991, Hollister, et al. 1993, Inman, 1976, London, 1981). Analysis of kinematic datahas traditionallybeendone usingjoint comdinate systems aligned with the anatomic referenceplanes (Go@ et al., 1983, Kinzeu et al., 1983, Lundborg, 1988) These analyses use Eulerrotationsabout the coordinatereference frame with or without calculated displacements. This method of describing motion is subject to error(de Lange, et al., 1990) and intra-obsemervariability. Furthermore, these methods for describing motion do not relatedmtly to the kinematic mechanism of thejoint itself. PROCEDURES AND RESULTS Rotations of a body moving about an arbitraryaxis in a reference frame are determined by the axis ct and 13 angles of offset fro-mthe reference fmme and the 9k angle of rotation about the arbitrary axis, k (Fig. 1). Displacements of a body moving about an arbhmry axis in a referenee ftame are determined by the axis a and B angles, the@ angle of rotation, r, the distance of the body fmm the axis of rotation, and dj the distance from the axis to the reference frame @lg. 2). When the arbhmy axis, k is coincident with the refenmcez-axis (d, a,~=o)andtithr. l,thexandy positions of the point&ace out cosine and sine waves, respectively, and the z position remains at zem for varying Ok. For perfect alignment but with r#l, the amplitude of the cosbe and sine curves is scaled accordingly. When the arbitrary axis is parallel to the reference z-axk but is transitedby non-zero xk, y k,aud/orzk,thecoirspoiid@ measuredl y, and z trajectoriesare shifted by x~ W, and zk, respectively (Fig. 3). kx ve +Ce kk~ve - be ldwe + kyse- Rke = krkyve+ kw k~% + C6 kykzve )&Se [ khe kyse kjwe + be kz ve + co where ve= 1 ce Figwe 1: Rotation about an arbitrary axis. Withthearbitraryaxisoffsetftomthemfenmce frame saxes, but still-passing throughthe origin (d= O),
presented at the 1995 ASME Conference, November 1995, San Francisco the xyz trajectories vary significantly with the offset angles, a and ~ @lg. 4). The trajectories can be made to vary qualitatively as well as quantitatively depending on the choice of ct and b. arbitrary axis with 10 deg offse~ this relationship breaks down: cxand ~ offset angles produce rotations about all three Euler coordinates, and the relationships are nonlin~ (Fig. 5). Different results are obtained if the a and ~ offsets are 80-, even though the axis lies within 1(Y of a reference coordinate. (0,0,0) ZIG Figure Z Rotation, ~, about ~ arbitrary axis, ~ with translation (by amounts xk, yk, zk) from the reference frame, x, y, z. he distance of the arbitrary axis is given by: d = (x# + yk2 + zk2)l~. The point moved about the arbitrary frame is a distance, r, from the arbitrary axis. c o 6 4 i!=-== theta k (deg) l%gure3: xyz positions with a shift of xk = 5, for Ok varying from zero to 90 deg. The z trajectory is mm; they tmjectcny traces out the cosine fimctiow the x tmjectory traces out an offset sine, centered about xk=5. 1 0.5 c o = o ; %-0.5-1 theta k (deg) Figure 4: xyz positions for axes of rotation with varying offset angles, a and ~ (d=o, r=l): shown are x trajectories (thick lines; et, ~ = O,solid C@= 10 and 20 deg, dashed; tx=80 deg. ~ = O,dashed), y trajectories (thin dashed), and z trajectories (thin, solid). Euler angle rotations are also affected by the et and ~ offsets. With zero offsets, the Euler angles correspond with the @ rotations (with variation in only one of the Euler angles) Whh rotation about a single -20 a 20 48 00 O 100 120 140 lea 1s0 200 Figure 5: Euler angle rotations calculated from rotations about an iubitrary axis with ct and ~ = 10 deg. DISCUSSION Slight offsets of the global reference frame from the axis of rotation produce signifkant errors in remrding displacements and rotations for motion about the joint axis. We conclude that the relationship of the reference frame to the axis of rotation must be known in order to perform accumte kinematic measurements.,. REFERENCES [1] GmocLEs., and Sunday, WJ. A joint coordinate system for the clinical description of threedimensional motion~ Application to the knee. J Biomech Engin 105,136-144,1983. [2] Hollister ~ et al. Ihe Axes of Rotation of the lhunb Carpometacarpal Join4 JO~ 104454.60,1991. [3] Hollister AM, et al. The axes of rotation of the knee, Clin OrthOp, 290:259-268,1993. [4] Inman, V.T. The Joints of the Ankle. pp. 45-112 Williams & Wilkins, 1976. [5] KinzeH G~ Gutkowski LJ. Joint models, degrees of freedornj and anatomical measurement. J Biomech Eng 105,55-62,1983. [6] de Lange, A., et al. Effects of data smoothing on the reconstruction of helical axis parameters in human joint kinematics.1mwnech.%g 115:108-13,1990. [7] London, J.T. Kinematics of the Elbow..l Bone.loirzr Surg, 63-A, 529-35, 1981. [8] Lundborg, A. Patterns of Motion of the Ankle/Foot Con@eX Stockhohn, Kongl Carolinska Medico Chirurgiska Institute~ 1988. This research was support in part by the United States Department of Energy, contract No. W-7405-ENG48.
Technical Information Department Lawrence Livermore National Laboratory University of California Livermore, California 94551