Notation For Relative Reference Frames In Robotics

Expressing one reference frame relative to another is a universal problem in robotics because relative reference frames form the basis of rigid body transformations. For example, if you have a robotic hand at the end of an arm you will often want to express the reference frame of the hand relative to the base frame of the arm.

There are at least five notations with which I am familiar for expressing some quantity in one frame relative to another frame. Define a vector \(p\) in frame 1. To express \(p\) in frame 0, we can write:

1. \(p_1^0\): current frame in the subscript and base frame in the superscript, or \(p_\text{curr}^\text{base}\).
2. \(p_0^1\): current frame in the superscript and base frame in the subscript, or \(p_\text{base}^\text{curr}\).
3. \(^0p_1\): base frame in the pre-superscript and current frame in the subscript, or \(^\text{base}p_\text{curr}\).
4. \(^0_1p\): base frame in the pre-superscript and current frame in the pre-subscript, or \(^\text{base}p_\text{curr}\).
5. \(p_{10}\): current and base frames in the subscripts.

The best notation for handwriting is (1), \(p_\text{curr}^\text{base}\).

Notation (2) is the worst because it places \(p\)’s identifier, the current frame, in the superscript. This grossly violates the universal scientific convention of putting a variable’s identifier in the subscript, e.g. when you want to note a vector’s Cartesian components you write \(v_x, v_y, v_z\), not \(v^x, v^y, v^z\).

Notation (3) has the benefit of standing out as a relative reference frame due to the prefixed superscript. Some might consider the non-standard superscript a downside because it is not standard. Regardless, notation (3) is worse than (1) when handwriting because prefixing a superscript is awkward and potentially messy. Notation (3) can also lead to confusion with you pre-multiply by another variable. Consider \(a\) times \(^2p_1\), or \(a^2p_1\). This is ambiguous and could easily be interpreted as \(a^2\) times \(p_1\), particularly when handwriting, which probably wouldn’t even make sense in an equation with relative reference frames (because \(p_1\) would appear to be a free vector). To avoid this ambiguity, one must take care to put extra space or add a multiplication symbol before \(^2p_1\). These steps take time and are downsides to (3).

Some may say Notation (3) has the added benefit of being syntactically compatible with superscripts such as powers, e.g. you can insert a power of two like \(^0p_1^2\). They may say that (1) (and (2)) are incompatible with superscripts because you’d end up with something confusing like \({p_1^0}^2\). To avoid this you’d have to use a parenthetical like \(({p_1^0})^2\). Now the question is whether the confusion or parenthetical make (1) worse than (3).

I think (3)’s \(^0p_1^2\) is almost as confusing as (1)’s \({p_1^0}^2\) because (3) just has too many numbers surrounding the \(p\). If you add a parenthetical, like \((^0p_1)^2\), you fall back to (1)’s parenthetical case.

If you are familiar with other notations let me know.