Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Fall 2024 semester.

Spatial Algebra

Throughout these notes we've introduced the basic rules of spatial algebra. I find myself looking back at them often! For ease of reference, I've compiled them again here. These make heavy use of our "monogram notation".

The Drake documentation also has a very nice summary of the multibody quantities and how they map to the monogram notation.

Position, Rotation, and Pose

As we introduced here, we use $^{B} p_{C}^{A}$ to denote a position of point or frame $A$ relative to point or frame $B$ expressed in frame $C$ . We use $^{B} R^{A}$ to denote the orientation of frame $A$ measured from frame $B$ ; unlike vectors, pure rotations do not have an additional "expressed in" frame. Similarly, we use $^{B} X^{A}$ to denote the pose/transform of frame $A$ measured from frame $B$ . We do not use the "expressed in" frame subscript for pose; we always want the pose expressed in the reference frame.

The basic rules of spatial algebra are as follows:

Positions expressed in the same frame can be added when their reference and target symbols match: $\begin{matrix} (1) & ^{A} p_{F}^{B} +^{B} p_{F}^{C} =^{A} p_{F}^{C} . \end{matrix}$ Addition is commutative, and the additive inverse is well defined: $\begin{matrix} (2) & ^{A} p_{F}^{B} = -^{B} p_{F}^{A} . \end{matrix}$ Those should be pretty intuitive; make sure you confirm them for yourself.
Multiplication by a rotation can be used to change the "expressed in" frame: $\begin{matrix} (3) & ^{A} p_{G}^{B} =^{G} R^{F}^{A} p_{F}^{B} . \end{matrix}$ You might be surprised that a rotation alone is enough to change the expressed-in frame, but it's true. The position of the expressed-in frame does not affect the relative position between two points.
Rotations can be multiplied when their reference and target symbols match: $\begin{matrix} (4) & ^{A} R^{B}^{B} R^{C} =^{A} R^{C} . \end{matrix}$ The inverse operation is also simply defined: $\begin{matrix} (5) & {[^{A} R^{B}]}^{- 1} =^{B} R^{A} . \end{matrix}$ When the rotation is represented as a rotation matrix, this is literally the matrix inverse, and since rotation matrices are orthonormal, we also have $R^{- 1} = R^{T} .$
Transforms bundle this up into a single, convenient notation when positions are measured from a frame (and the same frame they are expressed in): $\begin{matrix} (6) & ^{G} p^{A} =^{G} X^{F}^{F} p^{A} =^{G} p^{F} +^{F} p_{G}^{A} =^{G} p^{F} +^{G} R^{F}^{F} p^{A} . \end{matrix}$
Transforms compose: $\begin{matrix} (7) & ^{A} X^{B}^{B} X^{C} =^{A} X^{C}, \end{matrix}$ and have an inverse $\begin{matrix} (8) & {[^{A} X^{B}]}^{- 1} =^{B} X^{A} . \end{matrix}$ Please note that for transforms, we generally do not have that $X^{- 1}$ is $X^{T},$ though it still has a simple form.

Spatial velocity

Add acceleration

As we introduced here, we represent the rate of change in pose using a six-component vector for spatial velocity: $\begin{matrix} (9) & ^{A} V_{C}^{B} = [\begin{matrix} ^{A} ω_{C}^{B} \\ ^{A} v_{C}^{B} \end{matrix}] . \end{matrix}$ $^{A} V_{C}^{B}$ is the spatial velocity (also known as a "twist") of frame $B$ measured in frame $A$ expressed in frame $C$ , $^{A} ω_{C}^{B} \in R^{3}$ is the angular velocity (of frame $B$ measured in $A$ expressed in frame $C$ ), and $^{A} v_{C}^{B} \in R^{3}$ is the translational velocity (along with the same shorthands as for positions). Spatial velocities fit nicely into our spatial algebra:

Velocities add when they are expressed in the same frame: $\begin{matrix} (10) & ^{A} v_{F}^{B} +^{B} v_{F}^{C} =^{A} v_{F}^{C},^{A} ω_{F}^{B} +^{B} ω_{F}^{C} =^{A} ω_{F}^{C}, \end{matrix}$ and have the additive inverse, $^{A} V_{F}^{C} = -^{C} V_{F}^{A},$ .
Rotations can be used to change between the "expressed-in" frames: $\begin{matrix} (11) & ^{A} v_{G}^{B} =^{G} R^{F}^{A} v_{F}^{B},^{A} ω_{G}^{B} =^{G} R^{F}^{A} ω_{F}^{B} . \end{matrix}$
Translational velocities compose across frames with: $\begin{matrix} (12) & ^{A} v_{F}^{C} =^{A} v_{F}^{B} +^{B} v_{F}^{C} +^{A} ω_{F}^{B} \times^{B} p_{F}^{C} . \end{matrix}$
This reveals that additive inverse for translational velocities is not obtained by switching the reference and measured-in frames; it is slightly more complicated: $\begin{matrix} (13) & -^{A} v_{F}^{B} =^{B} v_{F}^{A} +^{A} ω_{F}^{B} \times^{B} p_{F}^{A} . \end{matrix}$ .

Spatial force

As we introduced here, we define a six-component vector for spatial force, using the monogram notation: $\begin{matrix} (14) & F_{name, C}^{B_{p}} = [\begin{matrix} τ_{name, C}^{B_{p}} \\ f_{name, C}^{B_{p}} \end{matrix}] or,if you prefer {[F_{name}^{B_{p}}]}_{C} = [\begin{matrix} {[τ_{name}^{B_{p}}]}_{C} \\ {[f_{name}^{B_{p}}]}_{C} \end{matrix}] . \end{matrix}$ $F_{name, C}^{B_{p}}$ is the named spatial force applied to a point, or frame origin, $B_{p}$ , expressed in frame $C$ . The name is optional, and the expressed in frame, if unspecified, is the world frame. For forces in particular, it is recommended that we include the body, $B$ , that the force is being applied to in the symbol for the point $B_{p}$ , especially since we will often have equal and opposite forces.

Spatial forces fit neatly into our spatial algebra:

Spatial forces add when they are applied to the same body in the same frame, e.g.: $\begin{matrix} (15) & F_{total, C}^{B_{p}} = \sum_{i} F_{i, C}^{B_{p}} . \end{matrix}$
Shifting a spatial force from one application point, $B_{p}$ , to another point, $B_{q}$ , uses the cross product: $\begin{matrix} (16) & f_{C}^{B_{q}} = f_{C}^{B_{p}}, τ_{C}^{B_{q}} = τ_{C}^{B_{p}} +^{B_{q}} p_{C}^{B_{p}} \times f_{C}^{B_{p}} . \end{matrix}$
As with all spatial vectors, rotations can be used to change between the "expressed-in" frames: $\begin{matrix} (17) & f_{D}^{B_{p}} =^{D} R^{C} f_{C}^{B_{p}}, τ_{D}^{B_{p}} =^{D} R^{C} τ_{C}^{B_{p}} . \end{matrix}$