We will learn about the relationship, in 2D, between the velocity of the joints and the velocity of the end-effector — the velocity kinematics. This relationship is described by a Jacobian matrix which also provides information about how easily the end-effector can move in different Cartesian directions.
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We learn how to describe the 2D pose of an object by a 3×3 homogeneous transformation matrix which has a special structure.
We summarise the important points from this masterclass.
We can also derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame.
We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties.
The pose of an object can be considered in two parts, the position of the object and the orientation of the object.
We consider multiple objects each with its own coordinate frame. Now we can describe the relationships between the frames and find a vector describing a point with respect to any of these frames. We extend our algebraic notation to ease the manipulation of relative poses.
We introduce the idea of attaching a coordinate frame to an object. We can describe points on the object by constant vectors with respect to the object’s coordinate frame, and then relate those to the points described with respect to a world coordinate frame. We introduce a simple algebraic notation to describe this.
To fully describe an object on the plane we need to not only describe its position, but also which direction it is pointing. This combination is referred to as pose.
We revisit the fundamentals of geometry that you would have learned at school: Euclidean geometry, Cartesian or analytic geometry, coordinate frames, points and vectors.