math

Sufficient axial rotation range of motion is critical for performing such simple tasks as reaching into your back pocket (internal axial rotation), or combing your hair (external axial rotation). I previously described how axial rotation can be measured in a physically meaningful manner. In this project, I explore how scapular motion can generate axial rotation. This mechanism has not been previously described in biomechanics literature and provides a more comprehensive view of shoulder motion.

For long bones, axial rotation is defined as a rotation around the axis running through the shaft of the bone. Sufficient axial rotation range of motion is critical for performing such simple tasks as reaching into your back pocket (internal axial rotation), or combing your hair (external axial rotation). Restoration of axial rotation range of motion is a critical consideration in shoulder replacement surgery and upper extremity prosthetic implant design. Yet, there is no consensus in the biomechanics community on how to measure axial rotation in a physically meaningful manner. This project 1) measures axial rotation in a physically meaningful manner for a cohort of 20 subjects performing arm elevation and rotation, 2) describes why commonly utilized techniques incorrectly measure axial rotation, 3) provides a JavaScript webapp for visualizing and quantifying physiologic arm motion (shoulder kinematics).

Although extensively utilized to estimate bone kinematics, skin-marker motion capture is plagued by errors arising from soft-tissue artefact (STA). The error caused by STA is substantial and “puts at risk the validity of a significant body of research in the basic, clinical, and applied sciences”. This project quantifies and visualizes STA for the humerus and scapula in 20 healthy subjects. The generated dataset and visualizations will serve as a guide for designing and validating STA suppression algorithms.

Unlike a traditional socket prosthesis, an osseointegrated (OI) prosthesis attaches directly to the bone of the residual limb. OI prostheses provide upper-extremity amputees increased range of motion, more natural movement patterns, and enhanced proprioception. However, the direct skeletal attachment of the prosthesis elevates the risk of bone fracture. To minimize the risk of fracture, it's important to mechanically characterize the bone-prosthesis interface under the same conditions that it would experience in vivo. In this project, I robotically replicate the motion of the humerus as recorded via motion capture while subjects performed activities typical of an active amputee. The robotically replicated motions will be utilized in future investigations to mechanically characterize the bone-prosthesis interface of an OI prosthesis.

In my first Ph.D. project, I robotically replicated the motion of the humerus as recorded via motion capture while subjects performed activities typical of an active amputee. The first task of this project was to program the position and orientation of the humerus onto the robot. In this post, I describe my method for accomplishing this task. To me, this is an interesting topic because it uses the same concepts as my previous post on establishing the position and orientation of a rigid body; but, the pen and paper are replaced by a robot and motion-tracking system.

The Singular Value Decomposition (SVD) is an incredibly useful tool with a staggering number of applications in seemingly unrelated fields. In this post I want to write about how the SVD is utilized to determine the orientation of a body segment from the skin markers attached to it. It's an interesting application of the SVD because it has straightforward geometrical interpretation.

Before building more complex logic into my codebase, I wanted to compare the performance of different methods of computing common biomechanics linear algebra operations in Numpy. As is common in biomechanics and robotics, I use a 4x4 homogeneous matrix to represent a coordinate system or pose, and correspondingly 3D vectors are upgraded to homogeneous coordinates. I knew that numpy.einsum could accommodate all linear algebra operations I am interested in performing, but I was curious to compare its performance against numpy.matmul for operations that could be performed just by matrix multiplication. Thanks to Numpy's broadcasting algorithm a considerable number of operation can be performed using numpy.matmul.

In this post I want to explore a simple insight about rotating between two orientations. Rotations are notoriously confusing and unintuitive, as evidenced by the fact that Wikipedia lists six different formalisms for representing rotations. Rotations are important to understand, however, because they play an integral role in physics, robotics, biomechanics, 3D animations, and a multitude of other disciplines.

When I first began studying continuum mechanics, tensors were a conundrum to me. I understood how to manipulate them algebraically but obtaining an intuitive understanding of them was a more difficult path. The purpose of this post is to simplify tensors for students who may have also run into the same roadblocks as I did. This post assumes familiarity with orthonormal bases, coordinate systems (frames), and measuring the position and orientation of one frame with respect to another.

In scientific/engineering endeavors, measuring the position and orientation of an object is important because it allows us to quantify movement characteristics. This post aims to provide a simple explanation of the process of measuring the position and orientation of an object in space. The goal is to not use any mathematical notation, although I presume that we all share a common intuitive understanding of the concept of a point and axis in the 3D space we inhabit. My failure to provide even a decent description of how to measure an object's position and orientation highlights the need for the precision introduced via mathematics (or my inability to explain a simple concepts succinctly - I pick the former). For simplicity, I will only consider measuring the position and orientation of rigid bodies, i.e. objects that do not deform.