#### Summary of 2D geometry and pose

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We summarise the important points from this lecture.

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We summarise the important points from this lecture.

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The relationship between world coordinates, image coordinates and camera spatial velocity is elegantly summed up by a single matrix equation that involves what we call the image Jacobian.

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If we want to process images the first thing we need to do is to read an image into MATLAB as a variable in the workspace. What kind of variable is an image? How can we see the image inside a variable? How do we refer to to individual pixels within an image.

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For real robots such as those with 6 joints that move in 3D space the inverse kinematics is quite complex, but for many of these robots the solutions have been helpfully derived by others and published. Let’s explore the inverse kinematics of the classical Puma 560 robot.

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The simplest smooth trajectory is a polynomial with boundary conditions on position, velocity and acceleration.

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We learn how to describe the 3D pose of an object by a 4×4 homogeneous transformation matrix which has a special structure.

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The orientation of a body in 3D can also be described by a single rotation about a particular axis in space.

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A problem arises when using three-angle sequences and particular values of the middle angle leads to a condition called a singularity. This mathematical phenomena is related to a problem that occurs in the physical world with mechanical gimbal systems. Note that in Robotics, Vision & Control (second edition) and RTB10.x the default definition of roll-pitch-yaw […]

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We like robots but there’s also an element of fear, perhaps stoked by all those books and movies about our new robot overlords. I’m going to speculate a little about where the fear comes from.

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If your knowledge of dynamics is a bit rusty then let’s quickly revise the basics of second-order systems and the Laplace operator. Not rusty? Then go straight to the next section.