nyx_space/

utils.rs

/*
    Nyx, blazing fast astrodynamics
    Copyright (C) 2018-onwards Christopher Rabotin <christopher.rabotin@gmail.com>

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU Affero General Public License as published
    by the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU Affero General Public License for more details.

    You should have received a copy of the GNU Affero General Public License
    along with this program.  If not, see <https://www.gnu.org/licenses/>.
*/

use crate::cosmic::Orbit;
use crate::linalg::{
    DefaultAllocator, DimName, Matrix3, OVector, Vector3, Vector6, allocator::Allocator,
};
use nalgebra::Complex;

/// Returns the skew-symmetric matrix (also known as the tilde matrix)
/// corresponding to the provided 3D vector.
///
/// The skew-symmetric matrix of a vector `v` is defined as:
///
/// ```plaintext
///  0   -v.z  v.y
///  v.z  0   -v.x
/// -v.y  v.x  0
/// ```
///
/// This matrix has the property that for any vector `w`, the cross product `v x w`
/// can be computed as the matrix product of the skew-symmetric matrix of `v` and `w`.
pub fn tilde_matrix(v: &Vector3<f64>) -> Matrix3<f64> {
    Matrix3::new(0.0, -v.z, v.y, v.z, 0.0, -v.x, -v.y, v.x, 0.0)
}

/// Checks if the provided 3x3 matrix is diagonal.
///
/// A square matrix is considered diagonal if all its off-diagonal elements are zero.
/// This function verifies this property for a given 3x3 matrix.
/// It checks each off-diagonal element of the matrix and returns `false` if any of them
/// is not approximately zero, considering a tolerance defined by `f64::EPSILON`.
///
/// # Arguments
///
/// * `m` - A 3x3 matrix of `f64` elements to be checked.
///
/// # Returns
///
/// * `bool` - Returns `true` if the matrix is diagonal, `false` otherwise.
///
/// # Example
///
/// ```
/// use nyx_space::utils::is_diagonal;
/// use nyx_space::linalg::Matrix3;
///
/// let m = Matrix3::new(1.0, 0.0, 0.0,
///                      0.0, 2.0, 0.0,
///                      0.0, 0.0, 3.0);
/// assert_eq!(is_diagonal(&m), true);
/// ```
///
/// # Note
///
/// This function uses `f64::EPSILON` as the tolerance for checking if an element is approximately zero.
/// This means that elements with absolute value less than `f64::EPSILON` are considered zero.
pub fn is_diagonal(m: &Matrix3<f64>) -> bool {
    for i in 0..3 {
        for j in 0..3 {
            if i != j && m[(i, j)].abs() > f64::EPSILON {
                return false;
            }
        }
    }
    true
}

/// Checks if the given matrix represents a stable linear system by examining its eigenvalues.
///
/// Stability of a linear system is determined by the properties of its eigenvalues:
/// - If any eigenvalue has a positive real part, the system is unstable.
/// - If the real part of an eigenvalue is zero and the imaginary part is non-zero, the system is oscillatory.
/// - If the real part of an eigenvalue is negative, the system tends towards stability.
/// - If both the real and imaginary parts of an eigenvalue are zero, the system is invariant.
///
/// # Arguments
///
/// `eigenvalues` - A vector of complex numbers representing the eigenvalues of the system.
///
/// # Returns
///
/// `bool` - Returns `true` if the system is stable, `false` otherwise.
///
/// # Example
///
/// ```
/// use nyx_space::utils::are_eigenvalues_stable;
/// use nyx_space::linalg::Vector2;
/// use nalgebra::Complex;
///
/// let eigenvalues = Vector2::new(Complex::new(-1.0, 0.0), Complex::new(0.0, 1.0));
/// assert_eq!(are_eigenvalues_stable(&eigenvalues), true);
/// ```
/// # Source
///
/// [Chemical Process Dynamics and Controls (Woolf)](https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Chemical_Process_Dynamics_and_Controls_(Woolf)/10%3A_Dynamical_Systems_Analysis/10.04%3A_Using_eigenvalues_and_eigenvectors_to_find_stability_and_solve_ODEs#Summary_of_Eigenvalue_Graphs)
pub fn are_eigenvalues_stable<N: DimName>(eigenvalues: &OVector<Complex<f64>, N>) -> bool
where
    DefaultAllocator: Allocator<N>,
{
    eigenvalues.iter().all(|ev| ev.re <= 0.0)
}

/// Returns the provided angle bounded between 0.0 and 360.0.
///
/// This function takes an angle (in degrees) and normalizes it to the range [0, 360).
/// If the angle is negative, it will be converted to a positive angle in the equivalent position.
/// For example, an angle of -90 degrees will be converted to 270 degrees.
///
/// # Arguments
///
/// * `angle` - An angle in degrees.
///
pub fn between_0_360(angle: f64) -> f64 {
    let mut bounded = angle % 360.0;
    if bounded < 0.0 {
        bounded += 360.0;
    }
    bounded
}

/// Returns the provided angle bounded between -180.0 and +180.0
pub fn between_pm_180(angle: f64) -> f64 {
    between_pm_x(angle, 180.0)
}

/// Returns the provided angle bounded between -x and +x.
///
/// This function takes an angle (in degrees) and normalizes it to the range [-x, x).
/// If the angle is outside this range, it will be converted to an equivalent angle within this range.
/// For example, if x is 180, an angle of 270 degrees will be converted to -90 degrees.
///
/// # Arguments
///
/// * `angle` - An angle in degrees.
/// * `x` - The boundary for the angle normalization.
pub fn between_pm_x(angle: f64, x: f64) -> f64 {
    let mut bounded = angle % (2.0 * x);
    if bounded > x {
        bounded -= 2.0 * x;
    }
    if bounded < -x {
        bounded += 2.0 * x;
    }
    bounded
}

/// The Kronecker delta function
pub fn kronecker(a: f64, b: f64) -> f64 {
    if (a - b).abs() <= f64::EPSILON {
        1.0
    } else {
        0.0
    }
}

/// Returns a rotation matrix for a rotation about the X axis.
///
/// # Arguments
///
/// * `angle_rad` - The angle of rotation in radians.
///
/// # Warning
///
/// This function returns a matrix for a COORDINATE SYSTEM rotation by `angle_rad` radians.
/// When this matrix is applied to a vector, it rotates the vector by `-angle_rad` radians, not `angle_rad` radians.
/// Applying the matrix to a vector yields the vector's representation relative to the rotated coordinate system.
///
/// # Example
///
/// ```
/// use nyx_space::utils::r1;
///
/// let angle_rad = std::f64::consts::PI / 2.0;
/// let rotation_matrix = r1(angle_rad);
/// ```
///
/// # Source
///
/// [NAIF SPICE Toolkit](https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/eul2xf_c.html)
pub fn r1(angle_rad: f64) -> Matrix3<f64> {
    let (s, c) = angle_rad.sin_cos();
    Matrix3::new(1.0, 0.0, 0.0, 0.0, c, s, 0.0, -s, c)
}

/// Returns a rotation matrix for a rotation about the Y axis.
///
/// # Arguments
///
/// * `angle` - The angle of rotation in radians.
///
/// # Warning
///
/// This function returns a matrix for a COORDINATE SYSTEM rotation by `angle_rad` radians.
/// When this matrix is applied to a vector, it rotates the vector by `-angle_rad` radians, not `angle_rad` radians.
/// Applying the matrix to a vector yields the vector's representation relative to the rotated coordinate system.
///
/// # Example
///
/// ```
/// use nyx_space::utils::r2;
///
/// let angle_rad = std::f64::consts::PI / 2.0;
/// let rotation_matrix = r2(angle_rad);
/// ```
///
/// # Source
///
/// [NAIF SPICE Toolkit](https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/eul2xf_c.html)
pub fn r2(angle_rad: f64) -> Matrix3<f64> {
    let (s, c) = angle_rad.sin_cos();
    Matrix3::new(c, 0.0, -s, 0.0, 1.0, 0.0, s, 0.0, c)
}

/// Returns a rotation matrix for a rotation about the Z axis.
///
/// # Arguments
///
/// * `angle_rad` - The angle of rotation in radians.
///
/// # Warning
///
/// This function returns a matrix for a COORDINATE SYSTEM rotation by `angle_rad` radians.
/// When this matrix is applied to a vector, it rotates the vector by `-angle_rad` radians, not `angle_rad` radians.
/// Applying the matrix to a vector yields the vector's representation relative to the rotated coordinate system.
///
/// # Example
///
/// ```
/// use nyx_space::utils::r3;
///
/// let angle_rad = std::f64::consts::PI / 2.0;
/// let rotation_matrix = r3(angle_rad);
/// ```
///
/// # Source
///
/// [NAIF SPICE Toolkit](https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/eul2xf_c.html)
pub fn r3(angle_rad: f64) -> Matrix3<f64> {
    let (s, c) = angle_rad.sin_cos();
    Matrix3::new(c, s, 0.0, -s, c, 0.0, 0.0, 0.0, 1.0)
}

/// Rotate a vector about a given axis
///
/// # Arguments
///
/// * `v` - A vector to be rotated.
/// * `axis` - The axis around which to rotate the vector.
/// * `theta` - The angle by which to rotate the vector.
///
/// # Returns
///
/// A new vector that is the result of rotating `v` around `axis` by `theta` radians.
pub fn rotv(v: &Vector3<f64>, axis: &Vector3<f64>, theta: f64) -> Vector3<f64> {
    let k_hat = axis.normalize();
    v.scale(theta.cos())
        + k_hat.cross(v).scale(theta.sin())
        + k_hat.scale(k_hat.dot(v) * (1.0 - theta.cos()))
}

/// Returns the components of vector a orthogonal to b
///
/// # Arguments
///
/// * `a` - The vector whose orthogonal components are to be calculated.
/// * `b` - The vector to which `a` is to be made orthogonal.
///
/// # Returns
///
/// A new vector that is the orthogonal projection of `a` onto `b`.
pub fn perpv(a: &Vector3<f64>, b: &Vector3<f64>) -> Vector3<f64> {
    let big_a = a.amax();
    let big_b = b.amax();
    if big_a < f64::EPSILON {
        Vector3::zeros()
    } else if big_b < f64::EPSILON {
        *a
    } else {
        let a_scl = a / big_a;
        let b_scl = b / big_b;
        let v = projv(&a_scl, &b_scl);
        (a_scl - v) * big_a
    }
}

/// Returns the projection of a onto b
///
/// # Arguments
///
/// * `a` - The vector to be projected.
/// * `b` - The vector onto which `a` is to be projected.
///
/// # Returns
///
/// * A new vector that is the projection of `a` onto `b`.
pub fn projv(a: &Vector3<f64>, b: &Vector3<f64>) -> Vector3<f64> {
    let b_norm_squared = b.norm_squared();
    if b_norm_squared.abs() < f64::EPSILON {
        Vector3::zeros()
    } else {
        b.scale(a.dot(b) / b_norm_squared)
    }
}

/// Computes the Root Sum Squared (RSS) state errors between two provided vectors.
///
/// # Arguments
///
/// * `prop_err` - A vector representing the propagated error.
/// * `cur_state` - A vector representing the current state.
///
/// # Returns
///
/// A f64 value representing the RSS state error.
pub fn rss_errors<N: DimName>(prop_err: &OVector<f64, N>, cur_state: &OVector<f64, N>) -> f64
where
    DefaultAllocator: Allocator<N>,
{
    prop_err
        .iter()
        .zip(cur_state.iter())
        .map(|(&x, &y)| (x - y).powi(2))
        .sum::<f64>()
        .sqrt()
}

/// Computes the Root Sum Squared (RSS) orbit errors in kilometers and kilometers per second.
///
/// # Arguments
///
/// * `prop_err` - An Orbit instance representing the propagated error.
/// * `cur_state` - An Orbit instance representing the current state.
///
/// # Returns
///
/// A tuple of f64 values representing the RSS orbit errors in radius and velocity.
pub fn rss_orbit_errors(prop_err: &Orbit, cur_state: &Orbit) -> (f64, f64) {
    (
        rss_errors(&prop_err.radius_km, &cur_state.radius_km),
        rss_errors(&prop_err.velocity_km_s, &cur_state.velocity_km_s),
    )
}

/// Computes the Root Sum Squared (RSS) state errors in position and in velocity of two orbit vectors [P V].
///
/// # Arguments
///
/// * `prop_err` - A Vector6 instance representing the propagated error.
/// * `cur_state` - A Vector6 instance representing the current state.
///
/// # Returns
///
/// A tuple of f64 values representing the RSS orbit vector errors in radius and velocity.
pub fn rss_orbit_vec_errors(prop_err: &Vector6<f64>, cur_state: &Vector6<f64>) -> (f64, f64) {
    let err_radius = (prop_err.fixed_rows::<3>(0) - cur_state.fixed_rows::<3>(0)).norm();
    let err_velocity = (prop_err.fixed_rows::<3>(3) - cur_state.fixed_rows::<3>(3)).norm();
    (err_radius, err_velocity)
}

/// Normalize a value between -1.0 and 1.0
///
/// # Arguments
///
/// * `x` - The value to be normalized.
/// * `min_x` - The minimum value in the range of `x`.
/// * `max_x` - The maximum value in the range of `x`.
///
/// # Returns
///
/// A normalized value between -1.0 and 1.0.
pub fn normalize(x: f64, min_x: f64, max_x: f64) -> f64 {
    2.0 * (x - min_x) / (max_x - min_x) - 1.0
}

/// Denormalize a value between -1.0 and 1.0
///
/// # Arguments
///
/// * `xp` - The value to be denormalized.
/// * `min_x` - The minimum value in the original range.
/// * `max_x` - The maximum value in the original range.
///
/// # Returns
///
/// A denormalized value between `min_x` and `max_x`.
pub fn denormalize(xp: f64, min_x: f64, max_x: f64) -> f64 {
    (max_x - min_x) * (xp + 1.0) / 2.0 + min_x
}

/// Capitalize the first letter of a string
///
/// # Arguments
///
/// `s` - The string to be capitalized.
///
/// # Returns
///
/// A new string with the first letter capitalized.
///
/// # Source
///
/// <https://stackoverflow.com/questions/38406793/why-is-capitalizing-the-first-letter-of-a-string-so-convoluted-in-rust>
pub fn capitalize(s: &str) -> String {
    let mut c = s.chars();
    match c.next() {
        None => String::new(),
        Some(f) => f.to_uppercase().collect::<String>() + c.as_str(),
    }
}

#[macro_export]
macro_rules! pseudo_inverse {
    ($mat:expr) => {{
        use $crate::md::TargetingError;
        let (rows, cols) = $mat.shape();
        if rows < cols {
            match ($mat * $mat.transpose()).try_inverse() {
                Some(m1_inv) => Ok($mat.transpose() * m1_inv),
                None => Err(TargetingError::SingularJacobian),
            }
        } else {
            match ($mat.transpose() * $mat).try_inverse() {
                Some(m2_inv) => Ok(m2_inv * $mat.transpose()),
                None => Err(TargetingError::SingularJacobian),
            }
        }
    }};
}

/// Returns the order of mangitude of the provided value
/// ```
/// use nyx_space::utils::mag_order;
/// assert_eq!(mag_order(1000.0), 3);
/// assert_eq!(mag_order(-5000.0), 3);
/// assert_eq!(mag_order(-0.0005), -4);
/// ```
pub fn mag_order(value: f64) -> i32 {
    value.abs().log10().floor() as i32
}

/// Returns the unit vector of the moved input vector
pub fn unitize(v: Vector3<f64>) -> Vector3<f64> {
    if v.norm() < f64::EPSILON {
        v
    } else {
        v / v.norm()
    }
}

/// Converts the input vector V from Cartesian coordinates to spherical coordinates
/// Returns ρ, θ, φ where the range ρ is in the units of the input vector and the angles are in radians
pub fn cartesian_to_spherical(v: &Vector3<f64>) -> (f64, f64, f64) {
    if v.norm() < f64::EPSILON {
        (0.0, 0.0, 0.0)
    } else {
        let range_ρ = v.norm();
        let θ = v.y.atan2(v.x);
        let φ = (v.z / range_ρ).acos();
        (range_ρ, θ, φ)
    }
}

/// Converts the input vector V from Cartesian coordinates to spherical coordinates
/// Returns ρ, θ, φ where the range ρ is in the units of the input vector and the angles are in radians
#[allow(clippy::many_single_char_names)]
pub fn spherical_to_cartesian(range_ρ: f64, θ: f64, φ: f64) -> Vector3<f64> {
    if range_ρ < f64::EPSILON {
        // Treat a negative range as a zero vector
        Vector3::zeros()
    } else {
        let x = range_ρ * φ.sin() * θ.cos();
        let y = range_ρ * φ.sin() * θ.sin();
        let z = range_ρ * φ.cos();
        Vector3::new(x, y, z)
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use approx::assert_abs_diff_eq;
    use nalgebra::{Complex, OVector};

    #[test]
    fn test_stable_eigenvalues() {
        let eigenvalues = OVector::<Complex<f64>, nalgebra::U2>::from_column_slice(&[
            Complex::new(-1.0, 0.0),
            Complex::new(0.0, 0.0),
        ]);
        assert!(are_eigenvalues_stable(&eigenvalues));
    }

    #[test]
    fn test_unstable_eigenvalues() {
        let eigenvalues = OVector::<Complex<f64>, nalgebra::U2>::from_column_slice(&[
            Complex::new(1.0, 0.0),
            Complex::new(0.0, 0.0),
        ]);
        assert!(!are_eigenvalues_stable(&eigenvalues));
    }

    #[test]
    fn test_oscillatory_eigenvalues() {
        let eigenvalues = OVector::<Complex<f64>, nalgebra::U2>::from_column_slice(&[
            Complex::new(0.0, 1.0),
            Complex::new(0.0, -1.0),
        ]);
        assert!(are_eigenvalues_stable(&eigenvalues));
    }

    #[test]
    fn test_invariant_eigenvalues() {
        let eigenvalues =
            OVector::<Complex<f64>, nalgebra::U1>::from_column_slice(&[Complex::new(0.0, 0.0)]);
        assert!(are_eigenvalues_stable(&eigenvalues));
    }

    #[test]
    fn test_angle_bounds() {
        approx::assert_relative_eq!(between_pm_180(181.0), -179.0);
        approx::assert_relative_eq!(between_0_360(-179.0), 181.0);
    }

    #[test]
    fn test_positive_angle() {
        approx::assert_relative_eq!(between_0_360(450.0), 90.0);
        approx::assert_relative_eq!(between_pm_x(270.0, 180.0), -90.0);
    }

    #[test]
    fn test_negative_angle() {
        approx::assert_relative_eq!(between_0_360(-90.0), 270.0);
        approx::assert_relative_eq!(between_pm_x(-270.0, 180.0), 90.0);
    }

    #[test]
    fn test_angle_in_range() {
        approx::assert_relative_eq!(between_0_360(180.0), 180.0);
        approx::assert_relative_eq!(between_pm_x(90.0, 180.0), 90.0);
    }

    #[test]
    fn test_zero_angle() {
        approx::assert_relative_eq!(between_0_360(0.0), 0.0);
        approx::assert_relative_eq!(between_pm_x(0.0, 180.0), 0.0);
    }

    #[test]
    fn test_full_circle_angle() {
        approx::assert_relative_eq!(between_0_360(360.0), 0.0);
        approx::assert_relative_eq!(between_pm_x(360.0, 180.0), 0.0);
    }

    #[test]
    fn test_pseudo_inv() {
        use crate::linalg::{DMatrix, SMatrix};
        let mut mat = DMatrix::from_element(1, 3, 0.0);
        mat[(0, 0)] = -1407.273208782421;
        mat[(0, 1)] = -2146.3100013104886;
        mat[(0, 2)] = 84.05022886527551;

        println!("{}", pseudo_inverse!(&mat).unwrap());

        let mut mat = SMatrix::<f64, 1, 3>::zeros();
        mat[(0, 0)] = -1407.273208782421;
        mat[(0, 1)] = -2146.3100013104886;
        mat[(0, 2)] = 84.05022886527551;

        println!("{}", pseudo_inverse!(&mat).unwrap());

        let mut mat = SMatrix::<f64, 3, 1>::zeros();
        mat[(0, 0)] = -1407.273208782421;
        mat[(1, 0)] = -2146.3100013104886;
        mat[(2, 0)] = 84.05022886527551;

        println!("{}", pseudo_inverse!(&mat).unwrap());

        // Compare a pseudo inverse with a true inverse
        let mat = SMatrix::<f64, 2, 2>::new(3.0, 4.0, -2.0, 1.0);
        println!("{}", mat.try_inverse().unwrap());

        println!("{}", pseudo_inverse!(&mat).unwrap());
    }

    #[test]
    fn spherical() {
        for v in &[
            Vector3::<f64>::x(),
            Vector3::<f64>::y(),
            Vector3::<f64>::z(),
            Vector3::<f64>::zeros(),
            Vector3::<f64>::new(159.1, 561.2, 756.3),
        ] {
            let (range_ρ, θ, φ) = cartesian_to_spherical(v);
            let v_prime = spherical_to_cartesian(range_ρ, θ, φ);

            assert!(rss_errors(v, &v_prime) < 1e-12, "{} != {}", v, &v_prime);
        }
    }

    #[rustfmt::skip]
    #[test]
    fn test_diagonality() {
        assert!(!is_diagonal(&Matrix3::new(10.0, 0.0, 0.0,
                                           1.0, 5.0, 0.0,
                                           0.0, 0.0, 2.0)),
            "lower triangular"
        );
        assert!(!is_diagonal(&Matrix3::new(10.0, 1.0, 0.0,
                                           1.0, 5.0, 0.0,
                                           0.0, 0.0, 2.0)),
            "symmetric but not diag"
        );
        assert!(!is_diagonal(&Matrix3::new(10.0, 1.0, 0.0,
                                           0.0, 5.0, 0.0,
                                           0.0, 0.0, 2.0)),
            "upper triangular"
        );
        assert!(is_diagonal(&Matrix3::new(10.0, 0.0, 0.0,
                                           0.0, 0.0, 0.0,
                                           0.0, 0.0, 2.0)),
            "diagonal with zero diagonal element"
        );
        assert!(is_diagonal(&Matrix3::new(10.0, 0.0, 0.0,
                                          0.0, 5.0, 0.0,
                                          0.0, 0.0, 2.0)),
            "diagonal"
        );
    }

    #[test]
    fn test_projv() {
        assert_eq!(
            projv(&Vector3::new(6.0, 6.0, 6.0), &Vector3::new(2.0, 0.0, 0.0)),
            Vector3::new(6.0, 0.0, 0.0)
        );
        assert_eq!(
            projv(&Vector3::new(6.0, 6.0, 6.0), &Vector3::new(-3.0, 0.0, 0.0)),
            Vector3::new(6.0, 0.0, 0.0)
        );
        assert_eq!(
            projv(&Vector3::new(6.0, 6.0, 0.0), &Vector3::new(0.0, 7.0, 0.0)),
            Vector3::new(0.0, 6.0, 0.0)
        );
        assert_eq!(
            projv(&Vector3::new(6.0, 0.0, 0.0), &Vector3::new(0.0, 0.0, 9.0)),
            Vector3::new(0.0, 0.0, 0.0)
        );

        let a = Vector3::new(1.0, 1.0, 0.0);
        let b = Vector3::new(1.0, 0.0, 0.0);
        let result = projv(&a, &b);
        assert_abs_diff_eq!(result, Vector3::new(1.0, 0.0, 0.0), epsilon = 1e-7);
    }

    #[test]
    fn test_rss_errors() {
        use nalgebra::U3;
        let prop_err = OVector::<f64, U3>::from_iterator([1.0, 2.0, 3.0]);
        let cur_state = OVector::<f64, U3>::from_iterator([1.0, 2.0, 3.0]);
        approx::assert_relative_eq!(rss_errors(&prop_err, &cur_state), 0.0);

        let prop_err = OVector::<f64, U3>::from_iterator([1.0, 2.0, 3.0]);
        let cur_state = OVector::<f64, U3>::from_iterator([4.0, 5.0, 6.0]);
        approx::assert_relative_eq!(rss_errors(&prop_err, &cur_state), 5.196152422706632);
    }

    #[test]
    fn test_normalize() {
        let x = 5.0;
        let min_x = 0.0;
        let max_x = 10.0;
        let result = normalize(x, min_x, max_x);
        approx::assert_relative_eq!(result, 0.0);
    }

    #[test]
    fn test_denormalize() {
        let xp = 0.0;
        let min_x = 0.0;
        let max_x = 10.0;
        let result = denormalize(xp, min_x, max_x);
        approx::assert_relative_eq!(result, 5.0);
    }

    #[test]
    fn test_capitalize() {
        let s = "hello";
        let result = capitalize(s);
        assert_eq!(result, "Hello");
    }

    #[test]
    fn test_r2() {
        let angle = 0.0;
        let rotation_matrix = r2(angle);
        assert!((rotation_matrix - Matrix3::identity()).abs().max() <= f64::EPSILON);

        let angle = std::f64::consts::PI / 2.0;
        let rotation_matrix = r2(angle);
        let expected_matrix = Matrix3::new(0.0, 0.0, -1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0);
        assert!((rotation_matrix - expected_matrix).abs().max() <= f64::EPSILON);

        let v = Vector3::new(0.0, 1.0, 0.0);
        let rotated_v = rotation_matrix * v;
        assert!((rotated_v - v).norm() <= f64::EPSILON);
    }

    #[test]
    fn test_r1() {
        let angle = 0.0;
        let rotation_matrix = r1(angle);
        assert!((rotation_matrix - Matrix3::identity()).abs().max() <= f64::EPSILON);

        let angle = std::f64::consts::PI / 2.0;
        let rotation_matrix = r1(angle);
        let expected_matrix = Matrix3::new(1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, -1.0, 0.0);
        assert!((rotation_matrix - expected_matrix).abs().max() <= f64::EPSILON);

        let v = Vector3::new(1.0, 0.0, 0.0);
        let rotated_v = rotation_matrix * v;
        assert!((rotated_v - v).norm() <= f64::EPSILON);
    }

    #[test]
    fn test_tilde_matrix() {
        let vec = Vector3::new(1.0, 2.0, 3.0);
        let rslt = Matrix3::new(0.0, -3.0, 2.0, 3.0, 0.0, -1.0, -2.0, 1.0, 0.0);
        assert_eq!(tilde_matrix(&vec), rslt);

        let v = Vector3::new(1.0, 2.0, 3.0);
        let m = tilde_matrix(&v);

        approx::assert_relative_eq!(m[(0, 0)], 0.0);
        approx::assert_relative_eq!(m[(0, 1)], -v.z);
        approx::assert_relative_eq!(m[(0, 2)], v.y);
        approx::assert_relative_eq!(m[(1, 0)], v.z);
        approx::assert_relative_eq!(m[(1, 1)], 0.0);
        approx::assert_relative_eq!(m[(1, 2)], -v.x);
        approx::assert_relative_eq!(m[(2, 0)], -v.y);
        approx::assert_relative_eq!(m[(2, 1)], v.x);
        approx::assert_relative_eq!(m[(2, 2)], 0.0);
    }

    #[test]
    fn test_perpv() {
        assert_eq!(
            perpv(&Vector3::new(6.0, 6.0, 6.0), &Vector3::new(2.0, 0.0, 0.0)),
            Vector3::new(0.0, 6.0, 6.0)
        );
        assert_eq!(
            perpv(&Vector3::new(6.0, 6.0, 6.0), &Vector3::new(-3.0, 0.0, 0.0)),
            Vector3::new(0.0, 6.0, 6.0)
        );
        assert_eq!(
            perpv(&Vector3::new(6.0, 6.0, 0.0), &Vector3::new(0.0, 7.0, 0.0)),
            Vector3::new(6.0, 0.0, 0.0)
        );
        assert_eq!(
            perpv(&Vector3::new(6.0, 0.0, 0.0), &Vector3::new(0.0, 0.0, 9.0)),
            Vector3::new(6.0, 0.0, 0.0)
        );
        let a = Vector3::new(1.0, 1.0, 0.0);
        let b = Vector3::new(1.0, 0.0, 0.0);
        let result = perpv(&a, &b);
        assert_abs_diff_eq!(result, Vector3::new(0.0, 1.0, 0.0), epsilon = 1e-7);
    }

    #[test]
    fn test_r3() {
        let angle = 0.0;
        let rotation_matrix = r3(angle);
        assert!((rotation_matrix - Matrix3::identity()).abs().max() <= f64::EPSILON);

        let angle = std::f64::consts::PI / 2.0;
        let rotation_matrix = r3(angle);
        let expected_matrix = Matrix3::new(0.0, 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 1.0);
        assert!((rotation_matrix - expected_matrix).abs().max() <= f64::EPSILON);

        let v = Vector3::new(0.0, 0.0, 1.0);
        let rotated_v = rotation_matrix * v;
        assert!((rotated_v - v).norm() <= f64::EPSILON);
    }

    #[test]
    fn test_rotv() {
        use approx::assert_abs_diff_eq;
        let v = Vector3::new(1.0, 0.0, 0.0);
        let axis = Vector3::new(0.0, 0.0, 1.0);
        let theta = std::f64::consts::PI / 2.0;
        let result = rotv(&v, &axis, theta);
        assert_abs_diff_eq!(result, Vector3::new(0.0, 1.0, 0.0), epsilon = 1e-7);
    }
}