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https://scholar.uwindsor.ca/cgi/viewcontent.cgi?article=5808&context=etd
Optimal Linkage Lengths
When looking at linkage lengths, a few things can be considered. The goal of a robotic arm is to execute a series of actions with as much precision and repeatability as possible. The speed at which it accomplishes these tasks also needs to be taken into account. Generally, having the joints closer to the end effector (and closer to each other) increases the precision of the positioning of the end effector. The further apart the joints are, the faster the arm will be (due to v = w x r). In many industrial robots, the wrist joints are as close as possible to the end effector to increase end effector positioning resolution, while the first two axes are as close to the base as possible to give velocity to the arm.
Manipulability
A common metric to compare robotic arms is called manipulability. This is a rough measure of how difficult it is for the arm to move in a given direction. It is generally given with the following equation:
Where J(q) is the Jacobian matrix for this manipulator. See the links and video below for more detail. Jacobian Matrices are explained in the arm theory.
https://engineerjau.wordpress.com/2013/05/04/advanced-robotics-manipulability-ellipsoids/
https://engineerjau.wordpress.com/2013/06/03/kinematic-manipulability-indicators-in-robotics/
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Velocity Ellipsoid
Manipulability can be illustrated through the use of a velocity ellipsoid. The previous measure is simply proportional to the volume of the ellipse. How to create the ellipsoid is shown in the image below (for 2D motion to keep it simple), taken from the following video
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