Gary Tad Yamaguchi

A new technique for determining 3-D joint angles: the tilt/twist method

OBJECTIVE To develop a new method of representing 3-D joint angles that is both physically meaningful and mathematically stable. DESIGN The two halves of a joint are modeled as overlapping cylinders. This simple physical model is easily understood and yields mathematically stable angle equations. BACKGROUND Two currently-used methods are the Euler/Cardan (joint coordinate system) method and the projection angle method. Both of these methods approach a singularity at 90 degrees that limits their use. The helical angle (attitude vector) method is mathematically stable but has limited physical meaning and is difficult to communicate. METHODS Calculation of the tilt/twist angles is described. Tilt/twist angles are compared to Euler/Cardan, projection, and helical angles in terms of behavior and stability. RESULTS Through a small range of angulation, tilt/twist angles match the specific projection and Euler/Cardan angles previously found to be appropriate for describing spinal motion. Through larger ranges, tilt/twist angles do not match the other angles studied. Although not as stable as helical angles, tilt/twist angles are twice as stable as Euler/Cardan and projection angles, reaching a singularity only at 180 degrees. CONCLUSIONS Because of their mathematical stability and simple physical interpretation, tilt/twist angles are recommended as a standard in describing angular joint motion.

Methods for determining spinal flexion/extension, lateral bending, and axial rotation from marker coordinate data: Analysis and refinement

Abstract Angular coupling patterns in the spine are often described by quantifying flexion/extension, lateral bending, and axial rotation angles as functions of one another. The most common methods for calculating these angles from marker coordinate data are the Euler method and the projection method. Both methods have the problem that they may be applied to spinal motion in a variety of ways, depending on the sequence chosen for Euler rotations or the vectors chosen for projection. The spinal angles calculated by each permutation of both methods vary significantly, leading to difficulties in reporting and comparing results between studies. The ambiguities of the Euler and projection techniques may be resolved and the two techniques standardized for application to the spine by considering vertebral symmetry. Using symmetry considerations, unique vectors may be chosen for determining the planar projection angles that best describe coupling in the spine. Because of the close relationship, presented herein, between projection angles and Euler angles, the same considerations allow one Euler rotation sequence to be chosen over the five alternate sequences. To validate the need for standardization of these techniques and to demonstrate the utility of the method presented, the results from a published study describing angular coupling patterns in the upper cervical spine are reexpressed in terms of the newly chosen Euler sequence and projection angle set. The reevaluated angles are consistent in both methods and lead to a conclusion different from the published conclusion with regard to the pattern of lateral bending coupling at C1C2 during axial rotation.

Overview of Dynamic Musculoskeletal Modeling

It is now the new millenium, and 3-D is where it’s at! The past few decades have seen an endless parade of analyses and modeling efforts confined to a two dimensional plane. In the past, limitations in computational power required such models to be simplified as far as one dared. Small and simple was good in many ways. Results could be easily interpreted, and generalities of the system were easily illuminated by elegant models, analyses, and interpretations. However, times have changed ! With the advent of the inexpensive and powerful desktop computer, the new focus is upon creating models that are ever more realistic and accurate in terms of their predictive power.

Chapter 6 Dynamic Equations of Motion

To introduce the reader to Kane’s Method as a means of deriving the dynamic equations of motion for a musculoskeletal system.

Vector Based Kinematics

To find the positions and velocities of every point at which a force acts, the angular velocities of every body on which a torque acts, and the accelerations of every mass center.

Models of the Skeletal System

To define inertial properties of the body segments and ways in which they are joined together to form kinematic linkage models of the musculoskeletal system.

An Introduction to Modeling Muscle and Tendon

To apply mathematically predictable driving forces to the segments of the body using lumped-parameter models of the muscles and tendons.

Rigid Bodies and Reference Frames

The ability to simultaneously work in multiple reference frames greatly simplifies kinematic and dynamic analyses in three dimensions. In this chapter, the reader is introduced to direction cosines, which provide the key to analyzing models with multiple degrees of freedom.

Optimal control model of arm configuration in a reaching task

It was hypothesized that the configuration of the upper limb during a hand static positioning task could be predicted using a dynamic musculoskeletal model and an optimal control routine. Both rhesus monkey and human upper extremity models were formulated, and had seven degrees of freedom (7-DOF) and 39 musculotendon pathways. A variety of configurations were generated about a physiologically measured configuration using the dynamic models and perturbations. The pseudoinverse optimal control method was applied to compute the minimum cost C at each of the generated configurations. Cost function C is described by the Crowninshield-Brand (1981) criterion which relates C (the sum of muscle stresses squared) to the endurance time of a physiological task. The configuration with the minimum cost was compared to the configurations chosen by one monkey (four trials) and by eight human subjects (eight trials each). Results are generally good, but not for all joint angles, suggesting that muscular effort is likely to be one major factor in choosing a preferred static arm posture.