{ "cells": [ { "cell_type": "markdown", "metadata": { "id": "view-in-github", "colab_type": "text" }, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "id": "uR3cXokfwmR6" }, "source": [ "# Kinematic chain in a plane (2D)\n", "\n", "> Marcos Duarte, Renato Naville Watanabe \n", "> [Laboratory of Biomechanics and Motor Control](https://bmclab.pesquisa.ufabc.edu.br/) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": { "toc": true, "id": "IdBezYFLwmR8" }, "source": [ "
Figure. The human body modeled as a kinematic chain (image from Wikipedia).
" ] }, { "cell_type": "markdown", "metadata": { "id": "5ppCqsknwmR-" }, "source": [ "## Properties of kinematic chains\n", "\n", "For a kinematic chain, the **base** is the extremity (origin) of a kinematic chain which is typically considered attached to the ground, body or fixed. The **endpoint** is the other extremity (end) of a kinematic chain and typically can move. In robotics, the term **end-effector** is used and usually refers to a last link (rigid body) in this chain.\n", "\n", "In topological terms, a kinematic chain is termed **open** when there is only one sequence of links connecting the two ends of the chain. Otherwise it's termed **closed** and in this case a sequence of links forms a loop. A kinematic chain can be classified as **serial** or **parallel** or a **mixed** of both. In a serial chain the links are connected in a serial order. A serial chain is an open chain, otherwise it is a parallel chain or a branched chain (e.g., hand and fingers). \n", "\n", "Although the definition above is clear and classic in mechanics, it is not the definition used by health professionals (clinicians and athletic trainers) when describing human movement. They refer to human joints and segments as a closed or open kinematic (or kinetic) chain simply if the distal segment (typically the foot or hand) is fixed (closed chain) or not (open chain). This difference in definition sometimes will result in different classifications. For example, a person standing on one leg is an open kinematic chain in mechanics, but closed according to the latter definition. In this text we will be consistent with mechanics, but keep in mind this difference when interacting with clinicians and athletic trainers.\n", "\n", "Another important term to characterize a kinematic chain is **degree of freedom (DOF)**. In mechanics, the degree of freedom of a mechanical system is the number of independent parameters that define its configuration or that determine the state of a physical system. A particle in the 3D space has three DOFs because we need three coordinates to specify its position. A rigid body in the 3D space has six DOFs because we need three coordinates of one point at the body to specify its position and three angles to to specify its orientation in order to completely define the configuration of the rigid body. For a link attached to a fixed body by a hinge joint in a plane, all we need to define the configuration of the link is one angle and then this link has only one DOF. A kinematic chain with two links in a plane has two DOFs, and so on.\n", "\n", "The **mobility** of a kinematic chain is its total number of degrees of freedom. The **redundancy** of a kinematic chain is its mobility minus the number of degrees of freedom of the endpoint." ] }, { "cell_type": "markdown", "metadata": { "id": "CWHi7Z2AwmR-" }, "source": [ "## The kinematics of one-link system\n", "\n", "First, let's study the case of a system composed by one planar hinge joint and one link, which technically it's not a chain but it will be useful to review (or introduce) key concepts. \n", "