How can I add angular noise to my vector?

For example, there is an unit vector X

X = [p q r]; % norm(X) = 1

And what I wanna do is to make some vectors which have the contained angle with X less than certain degree. Additionally, I want them to be distributed normally. How can I do this?

Answer 1

[KSSV]

Read about rand.

X = X+rand(size(X)) ; 

Answer 2

[Tanner Jones]

If I understand the question, you want to perturb a unit vector, , by some random angle, θ, sampled from a normal distribution many times?

Start by creating an orthogonal vector, . This can be done by generating a random unit vector, , and calculating the cross product of the two vectors: .

Then rotate about an axis defined as by θ. To do this first create a Direction Cosine Matrix (DCM) from and θ, by the following formula: , where I is the identity matrix, and is a function which produces a cross product or skew matrix (see Cross product section). Once you have calculated the DCM you will multiply it by (the DCM must be on the left to perform this multiplication) to get the desired vector.

I have implemented a solution below:

N = 100;  % number of output vectors
results = zeros(3,N);  % allocate memory for the resulting vectors
angleErrors = zeros(N);  % allocate memory for resulting angle errors
X = sqrt(3)/3*ones(3,1);  % given unit normal vector X
thetaSTD = deg2rad(1.0);  % error has a standard deviation of 1 degree

for i = 1:N
    result = perturbVector(X, thetaSTD);
    angleErrors(i) = acos(dot(X, result));  % calculate error
    results(:,i) = result;  % record result
end

plot(rad2deg(angleErrors))



function x_hat = perturbVector(x, sigma)
theta = sigma*randn;  % noise sample
v = rand(3,1);  % random vector
v = v/norm(v);  % normalize to a unit vector
y = cross(v,x);  % calculate cross product
DCM = eye(3) - sin(theta) * skew(y) + (1 - cos(theta)) * skew(y)^2;
x_hat = DCM * x;
end

function vSkew = skew(v)
vSkew = zeros(3,3);
vSkew(1,2,:) =-v(3,:);
vSkew(1,3,:) = v(2,:);
vSkew(2,1,:) = v(3,:);
vSkew(2,3,:) =-v(1,:);
vSkew(3,1,:) =-v(2,:);
vSkew(3,2,:) = v(1,:);
end