Frequently we want a trajectory that moves smoothly through a series of points without stopping.
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A more efficient trajectory has a trapezoidal velocity profile.
The simplest smooth trajectory is a polynomial with boundary conditions on position, velocity and acceleration.
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.