That adds to 11 dimensions of control. And their flight computer nailed it, without even zeroing one at a time! Only roll rate around the axis of the rocket was zeroed well before touchdown. Impressive!
The number of degrees you have to take car of is 6: X, Y, Z, yaw, pitch, and roll.
Here's a way to start thinking about this. Consider the 1D case, stopping a car with the goal of being at zero speed at a specified point. 2 goals, one controlled input. This is an under actuated problem. Setting a constant deceleration is not enough to do this. However, if you have a deceleration start time and a deceleration value after that point, you now have two variables, and can solve for zero speed at the desired point. This is essentially what Space-X is doing in the vertical direction.
They have limited ability to throttle the main engines (off, or 70% to 100%), and I suspect that in the final landing phase, they use that range to keep the vertical component constant while doing any horizontal positioning. The main engines gimbal; they don't have to change attitude for minor horizontal adjustments. So the simple approach is to get the attitude stabilized on approach, and the landing vertical. Their successful landings look like that. Their unsuccessful ones show non-vertical attitudes as the control system tries to make big horizontal adjustments.
Sure you do. Landing too fast in any of the three directions won't do.
gimbal is very intuitive, you just write down three angles: pitch, roll, and yaw. to imagine the objects rotation when I give you the three angles, you just apply them one after the other.
however, there is an issue. When the object's rotation is large enough that say, roll becomes 90, suddenly pitch and yaw correspond to the same 'thing '. which means one axis of rotation can no longer be realized.
in practice this can be seen with physical gimbals (three concentric rings, one per axis) or in numerical gimbal representation, where values might end up being unworkable.
quaternions eliminate this problem by using four values instead of three (quater, quattro, quad bike - the prefix stands for 4) 3 values are the components of a 3d vector, and the 4th is how much rotation to apply around it.
The above post argue that this 4th value seems like an extra degree of freedom, it isnt. it simply is a matter of representation if coded correctly.
The ocean landing is much more technically difficult, but it gives them the ability to use those extra percentage points of fuel on payload.