Helical trajectory generation using frenet frame

How to generate a helical trajectory using frenet frame? Position vector given as [x y z]^T=[[500sin(0.01t) 500cos(0.01t) −2t−20000]^T initial position is ζ (0) = [−300 0 −19800]^T initial line velocity is υ(0) =[5 0 0]^T, the initial attitude is γ (0) = [0, 0, 0]^T How to convert this data to frenet frame and find attitude coordinates and then plot a helical trajectory

Answer 1

[Vaibhav]

Hi Dhishya

To generate the helical trajector using frenet frame. We first define the helical trajectory using the given position vector. Then, we calculate the velocity and acceleration to derive the Frenet frame vectors (Tangent, Normal, Binormal). Finally, we can plot the trajectory and the Frenet frame vectors for visualization.

Here is the code for your reference:

% Define the trajectory
t = linspace(0, 20000, 10000);
x = 500*sin(0.01*t);
y = 500*cos(0.01*t);
z = -2*t - 20000;

% Calculate derivatives for velocity and acceleration
dx = gradient(x, t);
dy = gradient(y, t);
dz = gradient(z, t);
ddx = gradient(dx, t);
ddy = gradient(dy, t);
ddz = gradient(dz, t);

% Calculate Frenet frame vectors
T = [dx; dy; dz];
T_norm = sqrt(dx.^2 + dy.^2 + dz.^2);
T_unit = T ./ T_norm;
dTx = gradient(T(1,:), t);
dTy = gradient(T(2,:), t);
dTz = gradient(T(3,:), t);
N = [dTx; dTy; dTz];
N_norm = sqrt(dTx.^2 + dTy.^2 + dTz.^2);
N_unit = N ./ N_norm;
B = cross(T, N);
B_norm = sqrt(B(1,:).^2 + B(2,:).^2 + B(3,:).^2);
B_unit = B ./ B_norm;

% Plot the trajectory and Frenet frame vectors
figure;
plot3(x, y, z, 'LineWidth', 2);
hold on;
quiver3(x(1:100:end), y(1:100:end), z(1:100:end), T_unit(1,1:100:end), T_unit(2,1:100:end), T_unit(3,1:100:end), 'r');
quiver3(x(1:100:end), y(1:100:end), z(1:100:end), N_unit(1,1:100:end), N_unit(2,1:100:end), N_unit(3,1:100:end), 'g');
quiver3(x(1:100:end), y(1:100:end), z(1:100:end), B_unit(1,1:100:end), B_unit(2,1:100:end), B_unit(3,1:100:end), 'b');
xlabel('X');
ylabel('Y');
zlabel('Z');
title('Helical Trajectory with Frenet Frame');
legend('Trajectory', 'Tangent', 'Normal', 'Binormal');
grid on;
axis equal;

Hope it helps!