Q4 {rotations}

'Q4' class for storing rotation data as quaternions

Description

Creates or tests for objects of class "Q4".

Usage

as.Q4(x, ...)

## Default S3 method:
as.Q4(x, theta = NULL, ...)

## S3 method for class 'SO3'
as.Q4(x, ...)

## S3 method for class 'Q4'
as.Q4(x, ...)

## S3 method for class 'data.frame'
as.Q4(x, ...)

is.Q4(x)

id.Q4

Arguments

x

object to be coerced or tested

...

additional arguments.

theta

vector or single rotation angle; if length(theta)==1, the same theta is used for all axes

Format

id.Q4 is the identity rotation given by the matrix [1,0,0,0]^\top.

An object of class Q4 with 1 rows and 4 columns.

Details

Construct a single or sample of rotations in 3-dimensions in quaternion form. Several possible inputs for x are possible and they are differentiated based on their class and dimension.

For x an n-by-3 matrix or a vector of length 3, the angle-axis representation of rotations is utilized. More specifically, each quaternion can be interpreted as a rotation of some reference frame about the axis U (of unit length) through the angle \theta. For each axis and angle the quaternion is formed through

q=[cos(\theta/2),sin(\theta/2)U]^\top.

The object x is treated as if it has rows U and theta is a vector or angles. If no angle is supplied then the length of each axis is taken to be the angle of rotation theta.

For x an n-by-9 matrix of rotation matrices or an object of class "SO3", this function will return the quaternion equivalent of x. See SO3 or the vignette "rotations-intro" for more details on rotation matrices.

For x an n-by-4 matrix, rows are treated as quaternions; rows that aren't of unit length are made unit length while the rest are returned untouched. A message is printed if any of the rows are not quaternions.

For x a "data.frame", it is translated into a matrix of the same dimension and the dimensionality of x is used to determine the data type: angle-axis, quaternion or rotation (see above). As demonstrated below, is.Q4 may return TRUE for a data frame, but the functions defined for objects of class 'Q4' will not be called until as.Q4 has been used.

Value

as.Q4

coerces its object into a Q4 type

is.Q4

returns TRUE or FALSE depending on whether its argument satisfies the conditions to be an quaternion; namely it must be four-dimensional and of unit length

Examples

# Pull off subject 1's wrist measurements
Subj1Wrist <- subset(drill, Subject == '1' & Joint == 'Wrist')

                               ## The measurements are in columns 5:8
all(is.Q4(Subj1Wrist[,5:8]))   #TRUE, even though Qs is a data.frame, the rows satisfy the
                               #conditions necessary to be quaternions BUT,
                               #S3 methods (e.g. 'mean' or 'plot') for objects of class
                               #'Q4' will not work until 'as.Q4' is used

Qs <- as.Q4(Subj1Wrist[,5:8])  #Coerce measurements into 'Q4' type using as.Q4.data.frame
all(is.Q4(Qs))                 #TRUE
mean(Qs)                       #Estimate central orientation for subject 1's wrist, see ?mean.Q4
Rs <- as.SO3(Qs)               #Coerce a 'Q4' object into rotation matrix format, see ?as.SO3

#Visualize the measurements, see ?plot.Q4 for more

  plot(Qs, col = c(1, 2, 3))