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Since f is a module homomorphism between free modules, it can be visualized as an (infinite) matrix with entries in R and M as its cokernel. A free presentation always exists: any module is a quotient of a free module: , but then the kernel of g is again a quotient of a free module: ′ . The combination of f and g is a free presentation of M.
In robotics, a manipulator is a device used to manipulate materials without direct physical contact by the operator. The applications were originally for dealing with radioactive or biohazardous materials, using robotic arms , or they were used in inaccessible places.
Serial architectures a.k.a. serial manipulators are very common industrial robots; they are designed as a series of links connected by motor-actuated joints that extend from a base to an end-effector. SCARA, Stanford manipulators are typical examples of this category.
A manipulator can move an object with up to 6 degrees of freedom (DoF), determined by 3 translation 3T and 3 rotation 3R coordinates for full 3T3R mobility. However, when a manipulation task requires less than 6 DoF, the use of lower mobility manipulators, with fewer than 6 DoF, may bring advantages in terms of simpler architecture, easier control, faster motion and lower cost. [2]
In space, the Canadarm and its successor Canadarm2 are examples of multi degree of freedom robotic arms. These robotic arms have been used to perform a variety of tasks such as inspection of the Space Shuttle using a specially deployed boom with cameras and sensors attached at the end effector, and also satellite deployment and retrieval manoeuvres from the cargo bay of the Space Shuttle.
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The Denavit and Hartenberg notation gives a standard (distal) methodology to write the kinematic equations of a manipulator. This is especially useful for serial manipulators where a matrix is used to represent the pose (position and orientation) of one body with respect to another.
Kinematic diagram of Cartesian (coordinate) robot A plotter is a type of Cartesian coordinate robot.. A Cartesian coordinate robot (also called linear robot) is an industrial robot whose three principal axes of control are linear (i.e. they move in a straight line rather than rotate) and are at right angles to each other. [1]