An alternative for smooth motion between poses is Cartesian interpolated motion which leads to straight line motion in 3D space.
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We will introduce resolved-rate motion control which is a classical Jacobian-based scheme for moving the end-effector at a specified velocity without having to compute inverse kinematics.
We combine what we’ve learnt about smoothly varying position and orientation to create smoothly varying pose, often called Cartesian interpolation.
We revisit the fundamentals of geometry that you would have learned at school: Euclidean geometry, Cartesian or analytic geometry, coordinate frames, points and vectors.
We revisit the fundamentals of 3D geometry that you would have learned at school: coordinate frames, points and vectors.
Let’s recap the important points from the topics we have covered about homogeneous coordinates, image formation, camera modeling and planar homographies.
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.
Let’s recap the basics of homogeneous coordinates to represent points on a plane.
The Jacobian matrix provides powerful diagnostics about how well the robot’s configuration is suited to the task. Wrist singularities can be easily detected and the concept of a velocity ellipse is extended to a 3-dimensional velocity ellipsoid.
The end-effector is not able to move equally fast in all directions, and that in fact depends on the pose of the robot. We will introduce the velocity ellipse to illustrate this.