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Technical Briefs

# On Representations for Joint Moments Using a Joint Coordinate System

[+] Author and Article Information
Oliver M. O'Reilly

Professor
Mem. ASME
Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
e-mail: oreilly@berkeley.edu

Mark P. Sena

UCB-UCSF Joint Program in Bioengineering,
Department of Orthopaedic Surgery,
University of California,
San Francisco, CA 94143

Brian T. Feeley

Assistant Professor in Residence
Department of Orthopaedic Surgery,
University of California,
San Francisco, CA 94143

Jeffrey C. Lotz

Professor
Department of Orthopaedic Surgery,
University of California,
San Francisco, CA 94143

The dual Euler basis is also related to the dual basis used in the screw motion descriptions of rigid body motions in [7,8] and contravariant basis vectors in tensor calculus [9].

The corresponding results for a 1-2-3 or 2-3-1 set of Euler angles can be obtained by relabeling the basis vectors for the distal and proximal frames.

By way of notation, $M·ei$ in Ref. [1] corresponds to our $M·gi$. However, “$Mj$” in Ref. [1] corresponds to our $Mj$ because we have chosen to employ the conventions from differential geometry for covariant and contravariant components of vectors.

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received January 28, 2013; final manuscript received August 16, 2013; accepted manuscript posted September 6, 2013; published online September 26, 2013. Assoc. Editor: Richard E. Debski.

J Biomech Eng 135(11), 114504 (Sep 26, 2013) (4 pages) Paper No: BIO-13-1046; doi: 10.1115/1.4025327 History: Received January 28, 2013; Revised August 16, 2013; Accepted September 06, 2013

## Abstract

In studies of the biomechanics of joints, the representation of moments using the joint coordinate system has been discussed by several authors. The primary purpose of this technical brief is to emphasize that there are two distinct, albeit related, representations for moment vectors using the joint coordinate system. These distinct representations are illuminated by exploring connections between the Euler and dual Euler bases, the “nonorthogonal projections” presented in a recent paper by Desroches et al. (2010, “Expression of Joint Moment in the Joint Coordinate System,” ASME J. Biomech. Eng., 132(11), p. 11450) and seminal works by Grood and Suntay (Grood and Suntay, 1983, “A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee,” ASME J. Biomech. Eng., 105(2), pp. 136–144) and Fujie et al. (1996, “Forces and Moment in Six-DOF at the Human Knee Joint: Mathematical Description for Control,” Journal of Biomechanics, 29(12), pp. 1577–1585) on the knee joint. It is also shown how the representation using the dual Euler basis leads to straightforward definition of joint stiffnesses.

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Topics: Knee , Rotation

## References

Desroches, G., Chèze, L., and Dumas, R., 2010, “Expression of Joint Moment in the Joint Coordinate System,” ASME J. Biomech. Eng., 132(11), p. 114503.
Morrow, M. M. B., Hurd, W. J., Kaufman, K. R., and An, K.-A., 2009, “Upper-Limb Joint Kinetics Expression During Wheelchair Propulsion,” J. Rehabil. Res. Dev., 46(7), pp. 939–944. [PubMed]
Schache, A. G., and Baker, R., 2007, “On the Expression of Joint Moments During Gait,” Gait and Posture, 25(3), pp. 440–452. [PubMed]
O'Reilly, O. M., 2007, “The Dual Euler Basis: Constraints, Potentials, and Lagrange's Equations in Rigid-Body Dynamics,” ASME J. Appl. Mech., 74(2), pp. 256–258.
O'Reilly, O. M., 2008, Intermediate Dynamics for Engineers: A Unified Treatment of Newton-Euler and Lagrangian Mechanics, Cambridge University Press, Cambridge, England.
O'Reilly, O. M., Metzger, M. F., Buckley, J. M., Moody, D. A., and Lotz, J. C., 2009, “On the Stiffness Matrix of the Intervertebral Joint: Application to Total Disk Replacement,” ASME J. Biomech.Eng., 131(8), p. 081007.
Howard, S., Žefran, M., and Kumar, V., 1998, “On the 6 × 6 Cartesian Stiffness Matrix for Three-Dimensional Motions,” Mech. Mach. Theory, 33(4), pp. 389–408.
Žefran, M., and Kumar, V., 2002, “A Geometrical Approach to the Study of the Cartesian Stiffness Matrix,” ASME J. Mech. Des., 124(1), pp. 30–38.
Simmonds, J. G., 1994, A Brief on Tensor Analysis, second ed., Springer-Verlag, New York.
Christophy, M., Curtin, M., Faruk Senan, N. A., Lotz, J. C., and O'Reilly, O. M., 2013, “On the Modeling of the Intervertebral Joint in Multibody Models for the Spine,” Multibody Syst. Dyn. (in press).
Grood, E. S., and Suntay, W. J., 1983, “A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee,” ASME J. Biomech. Eng., 105(2), pp. 136–144.
Fujie, H., Livesay, G. A., Fujita, M., and Woo, S. L.-Y., 1996, “Forces and Moment in Six-DOF at the Human Knee Joint: Mathematical Description for Control,” J. Biomech., 29(12), pp. 1577–1585. [PubMed]

## Figures

Fig. 1

Comparison of the components of a vector A = A1g1 + A3g3 = A1g1 + A3g3. In the interest of clarity, the A2 = A2 components of this vector are assumed to be zero. (a) The components A1,3, (b) the components A1,3, and (c) two representations of the vector A. The presence of g in the distances shown in (b) arises because g1 and g3 have magnitudes of g-1. For the 3-2-1 set of Euler angles g = cos(θ).

Fig. 2

Graphical representation of the dual Euler basis vector and Euler basis vectors and their relationships with the proximal P and distal D frames for a set of 3-1-2 Euler angles. Explicit expressions for these vectors can be found in Eqs. (5) and (6). In this figure θ>0.

Fig. 3

Schematic of the right knee joint showing the proximal and distal frames of the femur and tibia, respectively, and the Euler basis vectors associated with rotational motions of this joint. The axes are drawn to intersect in the interest of clarity and the condyles C1 and C2 are also shown. For the illustrated case θ<0.

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