Mathematics of joint movements
The "Mathematics of Joint Movements" explores the relationship between human joint mechanics and mathematical modeling. Joints, where bones connect, typically allow for rotational movements that are fundamental to everyday activities. Understanding these rotations involves the application of trigonometric functions and matrix algebra, specifically through the use of the Jacobian matrix, which relates joint angles to hand positions in three-dimensional space. The complexity increases when multiple joints, like the shoulder and elbow, are involved, leading to scenarios such as kinematic redundancy, where multiple configurations can achieve the same end position.
Additionally, the study of joint movements highlights the noncommutativity of rotations in three-dimensional space, meaning the order of joint movements affects the final configuration. Techniques like Euler angles and quaternions provide standardized methods for describing these rotations. Furthermore, joint movements are influenced by muscle forces and external factors, with the resulting forces at the joints, known as joint loading, being significantly higher than external forces alone. Understanding these dynamics is crucial for fields such as biomechanics and robotics, where precise joint configuration and load estimation play a vital role in movement and function.
Mathematics of joint movements
- SUMMARY: Joints allow bones to move—a movement that is modeled and analyzed using mathematics.
A jointwhere bones joingenerally allows motion of those bones relative to each other. The motion, typically, is a rotation about the joint. Such rotations underlie almost all the movements humans perform in everyday life. Mathematics plays a crucial role in understanding the causes and consequences of the joint rotations, singly or in combination, and also in estimating the forces to which the joints are subjected.
Simple Joint Movement
Assume, for simplicity, that rotation is confined to the elbow joint. The forearm would move in a plane, and the position of the hand would be represented by extrinsic (x, y) coordinates that involve trigonometric—sine and cosine—functions of the elbow angle. When many joints participate, such as the shoulder, elbow, and wrist, the description of a hand movement, like reaching for a cup, involves combinations of trigonometric functions of the joint angles. The relationship between changes in the joint angles and the resulting changes in the extrinsic coordinates is expressed in the form of a matrixcalled the “Jacobian matrix”consisting of rows and columns of trigonometric functions. The methods of matrix algebra can be used for understanding the consequences of a sequence of changes in joint angles.
The inverse problem of finding the joint angles when the extrinsic coordinates are given can have an infinite number of solutions, called “kinematic redundancy.” For example, there are many ways of configuring an arm so as to get a finger to touch one’s nose. Why a person chooses a certain configuration is not known, though various hypotheses have been proposed. This is a crucial issue also in robotics, where “joint” angles have to be computed in order to reach a prescribed position in space. Various mathematical methods have been utilized for picking an “optimal” solution to this problem.
Three-Dimensional Joint Movement
The importance of mathematics in understanding and describing joint function is further emphasized when considering motions in three-dimensional space because certain phenomena arise that are far from intuitive. As an example, assume the shoulder to be a ball-and-socket joint and imagine the following two sequences of 90-degree rotations about the right shoulder, starting each time with the arm horizontal and stretched out to point to the right:
- rotation about the vertical axis (bringing the arm to point to the front), followed by rotation about the left-right axis (raising the arm up, above the head)
- the opposite sequence of rotations, first about the left-right axis (twisting the arm about its long axis), followed by rotation about the vertical axis (bringing the arm pointing to the front)
The two sequences lead to different configurations. The dependence of the final outcome on the sequence of the rotations is expressed by mathematicians as the “noncommutativity” of rotations in three-dimensional space. It means that rotations can not be described simply by three numbers, unless the sequence is also specified. Certain ways of specifying the sequence have been standardized, such as a rotation being described by three “Euler angles”yaw, pitch, and roll. There are also several other mathematical techniques, involving matrices, for dealing with rotations in 3-dimensional space, as matrices too have the property of noncommutativity (A × B ≠ B × A). Another technique, which uses four rather than three numbers to represent a rotation, is the method of “quaternions.” These abstract entities were proposed originally as extensions of complex numbers. Incidentally, the designers of computer visualizations, like video games, utilize quaternions for programming the rotational motions of the objects.
Forces
Motions about joints result from muscle and external forces. It is the moments of these forces that matter for rotation. In multijoint movements, a muscle moment about one joint can cause motions about several joints; specifically, even a fully relaxed joint would flop when there is motion about nearby joints. This phenomenon is described by rather complicated differential equations, which the neural control system takes into account in its planning. But the force with which the bones at a joint push against each other cannot be determined simply from the moments of forces. This force (called “joint loading”) depends upon both external and muscle forces, and is typically many times greater than any external forces. The wear and tear of the joint—natural or artificial—depends upon the loading. Also, joints being nearly frictionless, slippage occurs if the load has a substantial component parallel to the surface of contact. Noninvasive techniques for estimating the joint loading force are highly computational. Given the external forces and observed motions, one determines the needed muscle torques at each joint, and then, knowing the anatomical layout of the muscles and their strengths, one estimates the distribution of forces among the muscles. With all other forces thus known or estimated, one can derive the joint loading.
Bibliography
Alexander, R. McNeill. Principles of Animal Locomotion. Princeton University Press, 2006.
Burstein, Albert H., and Timothy M. Wright. Fundamentals of Orthopaedic Biomechanics. Williams & Wilkins, 1994.
Hanson, Andrew. Visualizing Quaternions. Morgan Kaufmann, 2006.
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