dist

Angular distance in radians

Syntax

distance = dist(quatA,quatB)

Description

distance = dist(quatA,quatB) returns the angular distance in radians between two quaternions, quatA and quatB.

Examples

Calculate Quaternion Distance

Calculate the quaternion distance between a single quaternion and each element of a vector of quaternions. Define the quaternions using Euler angles.

q = quaternion([0,0,0],"eulerd","ZYX","frame")

q = quaternion
     1 + 0i + 0j + 0k

qArray = quaternion([0,45,0;0,90,0;0,180,0;0,-90,0;0,-45,0],"eulerd","ZYX","frame")

qArray = 5x1 quaternion array
       0.92388 +         0i +   0.38268j +         0k
       0.70711 +         0i +   0.70711j +         0k
    6.1232e-17 +         0i +         1j +         0k
       0.70711 +         0i -   0.70711j +         0k
       0.92388 +         0i -   0.38268j +         0k

quaternionDistance = rad2deg(dist(q,qArray))

quaternionDistance = 5×1

   45.0000
   90.0000
  180.0000
   90.0000
   45.0000

If both arguments to dist are vectors, the quaternion distance is calculated between corresponding elements. Calculate the quaternion distance between two quaternion vectors.

angles1 = [30,0,15; ...
           30,5,15; ...
           30,10,15; ...
           30,15,15];
angles2 = [30,6,15; ...
           31,11,15; ...
           30,16,14; ...
           30.5,21,15.5];

qVector1 = quaternion(angles1,"eulerd","ZYX","frame");
qVector2 = quaternion(angles2,"eulerd","ZYX","frame");

rad2deg(dist(qVector1,qVector2))

ans = 4×1

    6.0000
    6.0827
    6.0827
    6.0287

Note that a quaternion represents the same rotation as its negative. Calculate a quaternion and its negative.

qPositive = quaternion([30,45,-60],"eulerd","ZYX","frame")

qPositive = quaternion
     0.72332 - 0.53198i + 0.20056j +  0.3919k

qNegative = -qPositive

qNegative = quaternion
    -0.72332 + 0.53198i - 0.20056j -  0.3919k

Find the distance between the quaternion and its negative.

dist(qPositive,qNegative)

ans = 
0

The components of a quaternion may look different from the components of its negative, but both expressions represent the same rotation.

Input Arguments

quatA,quatB — Quaternions to calculate distance between

quaternion objects | arrays of quaternion objects

Quaternions to calculate distance between, specified as comma-separated quaternion objects or arrays of quaternion objects of any dimensionality. quatA and quatB must have compatible sizes:

• size(quatA) == size(quatB), or

• numel(quatA) == 1, or

• numel(quatB) == 1, or

• if [Adim1,…,AdimN] = size(quatA) and [Bdim1,…,BdimN] = size(quatB), then for i = 1:N, either Adimi==Bdimi or Adim==1 or Bdim==1.

  If one of the quaternion arguments contains only one quaternion, then this function returns the distances between that quaternion and every quaternion in the other argument.

Output Arguments

distance — Angular distance (radians)

numeric scalar | numeric array

Angular distance in radians, returned as a numeric scalar or numeric array. The dimensions are the maximum of the union of size(quatA) and size(quatB).

Data Types: single | double

Algorithms

The dist function returns the angular distance between two quaternions.

A quaternion may be defined by an axis (u_b,u_c,u_d) and angle of rotation θ_q: $$q=\cos \left(\raisebox{1ex}{${\theta }_q$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)+\sin \left(\raisebox{1ex}{${\theta }_q$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\left(u_b\text{i}+u_c\text{j}+u_d\text{k}\right)$$.

Given a quaternion in the form, $$q=a+b\text{i}+c\text{j}+d\text{k}$$, where a is the real part, you can solve for the angle of q as $${\theta }_q=2{\cos }^{-1}(a)$$.

Consider two quaternions, p and q, and the product $$z=p*\text{conjugate}(q)$$. As p approaches q, the angle of z goes to 0, and z approaches the unit quaternion.

The angular distance between two quaternions can be expressed as $${\theta }_{\text{z}}=2{\cos }^{-1}\left(\text{real}\left(z\right)\right)$$.

Using the quaternion data type syntax, the angular distance is calculated as:

angularDistance = 2*acos(abs(parts(p*conj(q))));

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2019b