#### Inverting the Jacobian Matrix

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By inverting the Jacobian matrix we can find the joint velocities required to achieve a particular end-effector velocity, so long as the Jacobian is not singular.

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By inverting the Jacobian matrix we can find the joint velocities required to achieve a particular end-effector velocity, so long as the Jacobian is not singular.

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We learn the mathematical relationship between angular velocity of a body and the time derivative of the rotation matrix describing the orientation of that body.

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As we did for the simple planar robots we can invert the Jacobian and perform resolved-rate motion control.

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We resume our analysis of the 6-link robot Jacobian and focus on the rotational velocity part.

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For a simple 2-link planar robot we introduce and derive its Jacobian matrix, and also introduce the concept of spatial velocity.

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We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties. Try your hand at some online MATLAB problems. You’ll need to watch all the 2D “Spatial Maths” lessons to complete the problem set.

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Since an image in MATLAB is just a matrix of numbers, we could write code to fill in the elements of the matrix. Let’s look at some simple examples such as squares, circles and lines and more complex images formed by pasting these shapes together.

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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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We can describe the relationship between a 3D world point and a 2D image plane point, both expressed in homogeneous coordinates, using a linear transformation – a 3×4 matrix. Then we can extend this to account for an image plane which is a regular grid of discrete pixels.

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Images contain many pixels and the normal way to process them is with nested for loops that index each pixel in turn. This is slow and somewhat cumbersome to write. MATLAB has a facility called vectorization that allows us to perform complex matrix operations without any loops.