test_nvector.py

"""
This file contains solutions to the examples given at
www.navlab.net/nvector

The content of this file is based on the following publication:

Gade, K. (2010). A Nonsingular Horizontal Position Representation, The Journal
of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.
(www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf)

Copyright (c) 2015, Norwegian Defence Research Establishment (FFI)
All rights reserved.

Originated: 2015.03.26 Kenneth Gade, FFI

NOTES:
- All angles are by default assumed to be in radians, if an angle is
in degrees, the variable name has the following ending: _deg

- The dot product (inner product) of vectors x and y is written dot(x,y)
here to make the code more readable for those unfamiliar with
Matlab. In Matlab one would normally write x'*y (i.e. x transposed
times y)
"""
import numpy as np

import unittest
from nvector import (unit, deg, rad, lat_lon2n_E, n_E2lat_lon,
                     R2xyz, xyz2R, R2zyx, zyx2R,
                     n_EA_E_and_n_EB_E2p_AB_E, n_EA_E_and_p_AB_E2n_EB_E,
                     p_EB_E2n_EB_E, n_EB_E2p_EB_E,
                     mean_horizontal_position,
                     great_circle_distance, euclidean_distance,
                     n_EA_E_distance_and_azimuth2n_EB_E,
                     n_EA_E_and_n_EB_E2azimuth,
                     n_E_and_wa2R_EL, n_E2R_EN, R_EL2n_E, R_EN2n_E)
from numpy.testing import assert_array_almost_equal


class TestNvector(unittest.TestCase):
    @staticmethod
    def test_Ex1_A_and_B_to_delta_in_frame_N():

        # Positions A and B are given in (decimal) degrees and depths:
        lat_EA, lon_EA, z_EA = rad(1), rad(2), 3
        lat_EB, lon_EB, z_EB = rad(4), rad(5), 6

        # Find the exact vector between the two positions, given in meters
        # north, east, and down, i.e. find p_AB_N.

        # SOLUTION:
        # Step1: Convert to n-vectors (rad() converts to radians):
        n_EA_E = lat_lon2n_E(lat_EA, lon_EA)
        n_EB_E = lat_lon2n_E(lat_EB, lon_EB)

        # Step2: Find p_AB_E (delta decomposed in E).
        # WGS-84 ellipsoid is default:
        p_AB_E = n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA, z_EB)

        # Step3: Find R_EN for position A:
        R_EN = n_E2R_EN(n_EA_E)

        # Step4: Find p_AB_N
        p_AB_N = np.dot(R_EN.T, p_AB_E)
        # (Note the transpose of R_EN: The "closest-rule" says that when
        # decomposing, the frame in the subscript of the rotation matrix that
        # is closest to the vector, should equal the frame where the vector is
        # decomposed. Thus the calculation np.dot(R_NE, p_AB_E) is correct,
        # since the vector is decomposed in E, and E is closest to the vector.
        # In the example we only had R_EN, and thus we must transpose it:
        # R_EN'=R_NE)

        # Step5: Also find the direction (azimuth) to B, relative to north:
        azimuth = np.arctan2(p_AB_N[1], p_AB_N[0])
        # positive angle about down-axis

        print('Ex1, delta north, east, down = {0}, {1}, {2}'.format(p_AB_N[0],
                                                                    p_AB_N[1],
                                                                    p_AB_N[2]))
        print('Ex1, azimuth = {0} deg'.format(deg(azimuth)))

        assert_array_almost_equal(p_AB_N[0], 331730.23478089)
        assert_array_almost_equal(p_AB_N[1], 332997.87498927)
        assert_array_almost_equal(p_AB_N[2], 17404.27136194)
        assert_array_almost_equal(deg(azimuth), 45.10926324)

    @staticmethod
    def test_Ex2_B_and_delta_in_frame_B_to_C_in_frame_E():
        # delta vector from B to C, decomposed in B is given:
        p_BC_B = np.r_[3000, 2000, 100].reshape((-1, 1))

        # Position and orientation of B is given:
        n_EB_E = unit([[1], [2], [3]])  # unit to get unit length of vector
        z_EB = -400
        R_NB = zyx2R(rad(10), rad(20), rad(30))
        # the three angles are yaw, pitch, and roll

        # A custom reference ellipsoid is given (replacing WGS-84):
        a, f = 6378135, 1.0 / 298.26  # (WGS-72)

        # Find the position of C.
        # SOLUTION:
        # Step1: Find R_EN:
        R_EN = n_E2R_EN(n_EB_E)

        # Step2: Find R_EB, from R_EN and R_NB:
        R_EB = np.dot(R_EN, R_NB)  # Note: closest frames cancel

        # Step3: Decompose the delta vector in E:
        p_BC_E = np.dot(R_EB, p_BC_B)
        # no transpose of R_EB, since the vector is in B

        # Step4: Find the position of C, using the functions that goes from one
        # position and a delta, to a new position:
        n_EC_E, z_EC = n_EA_E_and_p_AB_E2n_EB_E(n_EB_E, p_BC_E, z_EB, a, f)

        # When displaying the resulting position for humans, it is more
        # convenient to see lat, long:
        lat_EC, long_EC = n_E2lat_lon(n_EC_E)
        # Here we also assume that the user wants output height (= - depth):
        msg = 'Ex2, Pos C: lat, long = {},{} deg,  height = {} m'
        print(msg.format(deg(lat_EC), deg(long_EC), -z_EC))

        assert_array_almost_equal(deg(lat_EC), 53.32637826)
        assert_array_almost_equal(deg(long_EC), 63.46812344)
        assert_array_almost_equal(z_EC, -406.00719607)

    @staticmethod
    def test_Ex3_ECEF_vector_to_geodetic_latitude():

        # Position B is given as p_EB_E ("ECEF-vector")

        p_EB_E = 6371e3 * np.vstack((0.9, -1, 1.1))  # m

        # Find position B as geodetic latitude, longitude and height

        # SOLUTION:
        # Find n-vector from the p-vector:
        n_EB_E, z_EB = p_EB_E2n_EB_E(p_EB_E)

        # Convert to lat, long and height:
        lat_EB, long_EB = n_E2lat_lon(n_EB_E)
        h_EB = -z_EB
        msg = 'Ex3, Pos B: lat, long = {} {} deg, height = {} m'
        print(msg.format(deg(lat_EB), deg(long_EB), h_EB))

        assert_array_almost_equal(deg(lat_EB), 39.37874867)
        assert_array_almost_equal(deg(long_EB), -48.0127875)
        assert_array_almost_equal(h_EB, 4702059.83429485)

    @staticmethod
    def test_Ex4_geodetic_latitude_to_ECEF_vector():

        # Position B is given with lat, long and height:
        lat_EB_deg = 1
        long_EB_deg = 2
        h_EB = 3

        # Find the vector p_EB_E ("ECEF-vector")

        # SOLUTION:
        # Step1: Convert to n-vector:
        n_EB_E = lat_lon2n_E(rad(lat_EB_deg), rad(long_EB_deg))

        # Step2: Find the ECEF-vector p_EB_E:
        p_EB_E = n_EB_E2p_EB_E(n_EB_E, -h_EB)

        print('Ex4: p_EB_E = {0} m'.format(p_EB_E.ravel()))

        assert_array_almost_equal(p_EB_E.ravel(),
                                  [6373290.27721828, 222560.20067474,
                                   110568.82718179])

    @staticmethod
    def test_Ex5_great_circle_distance():

        # Position A and B are given as n_EA_E and n_EB_E:
        # Enter elements as lat/long in deg:
        n_EA_E = lat_lon2n_E(rad(88), rad(0))
        n_EB_E = lat_lon2n_E(rad(89), rad(-170))

        r_Earth = 6371e3  # m, mean Earth radius

        # SOLUTION:
        s_AB = great_circle_distance(n_EA_E, n_EB_E, radius=r_Earth)
        d_AB = euclidean_distance(n_EA_E, n_EB_E, radius=r_Earth)

        msg = 'Ex5, Great circle distance = {} km, Euclidean distance = {} km'
        print(msg.format(s_AB / 1000, d_AB / 1000))

        assert_array_almost_equal(s_AB / 1000, 332.45644411)
        assert_array_almost_equal(d_AB / 1000, 332.41872486)

    @staticmethod
    def test_Ex6_interpolated_position():

        # Position B at time t0 and t2 is given as n_EB_E_t0 and n_EB_E_t1:
        # Enter elements as lat/long in deg:
        n_EB_E_t0 = lat_lon2n_E(rad(89), rad(0))
        n_EB_E_t1 = lat_lon2n_E(rad(89), rad(180))

        # The times are given as:
        t0 = 10
        t1 = 20
        ti = 16  # time of interpolation

        # Find the interpolated position at time ti, n_EB_E_ti

        # SOLUTION:
        # Using standard interpolation:
        n_EB_E_ti = unit(n_EB_E_t0 +
                         (ti - t0) * (n_EB_E_t1 - n_EB_E_t0) / (t1 - t0))

        # When displaying the resulting position for humans, it is more
        # convenient to see lat, long:
        lat_EB_ti, long_EB_ti = n_E2lat_lon(n_EB_E_ti)
        msg = 'Ex6, Interpolated position: lat, long = {} {} deg'
        print(msg.format(deg(lat_EB_ti), deg(long_EB_ti)))

        assert_array_almost_equal(deg(lat_EB_ti), 89.7999805)
        assert_array_almost_equal(deg(long_EB_ti), 180.)

    @staticmethod
    def test_Ex7_mean_position():

        # Three positions A, B and C are given:
        # Enter elements as lat/long in deg:
        n_EA_E = lat_lon2n_E(rad(90), rad(0))
        n_EB_E = lat_lon2n_E(rad(60), rad(10))
        n_EC_E = lat_lon2n_E(rad(50), rad(-20))

        # Find the horizontal mean position:
        n_EM_E = unit(n_EA_E + n_EB_E + n_EC_E)

        # The result is best viewed with a figure that shows the n-vectors
        # relative to an Earth-model:
        print('Ex7, See figure')
        # plot_Earth_figure(n_EA_E,n_EB_E,n_EC_E,n_EM_E)
        assert_array_almost_equal(n_EM_E.ravel(),
                                  [0.384117, -0.046602, 0.922107])
        # Alternatively:
        n_EM_E = mean_horizontal_position(np.hstack((n_EA_E, n_EB_E, n_EC_E)))
        assert_array_almost_equal(n_EM_E.ravel(),
                                  [0.384117, -0.046602, 0.922107])

    @staticmethod
    def test_Ex8_position_A_and_azimuth_and_distance_to_B():

        # Position A is given as n_EA_E:
        # Enter elements as lat/long in deg:
        lat, lon = rad(80), rad(-90)

        n_EA_E = lat_lon2n_E(lat, lon)

        # The initial azimuth and great circle distance (s_AB), and Earth
        # radius (r_Earth) are also given:
        azimuth = rad(200)
        s_AB = 1000  # m
        r_Earth = 6371e3  # m, mean Earth radius

        # Find the destination point B, as n_EB_E ("The direct/first geodetic
        # problem" for a sphere)

        # SOLUTION:
        # Step1: Convert distance in meter into distance in [rad]:
        distance_rad = s_AB / r_Earth
        # Step2: Find n_EB_E:
        n_EB_E = n_EA_E_distance_and_azimuth2n_EB_E(n_EA_E, distance_rad,
                                                    azimuth)

        # When displaying the resulting position for humans, it is more
        # convenient to see lat, long:
        lat_EB, long_EB = n_E2lat_lon(n_EB_E)
        print('Ex8, Destination: lat, long = {0} {1} deg'.format(deg(lat_EB),
                                                                 deg(long_EB)))

        assert_array_almost_equal(deg(lat_EB), 79.99154867)
        assert_array_almost_equal(deg(long_EB), -90.01769837)
        azimuth1 = n_EA_E_and_n_EB_E2azimuth(n_EA_E, n_EB_E, a=r_Earth, f=0)
        assert_array_almost_equal(azimuth, azimuth1+2*np.pi)

    @staticmethod
    def test_Ex9_intersect():

        # Two paths A and B are given by two pairs of positions:
        # Enter elements as lat/long in deg:
        n_EA1_E = lat_lon2n_E(rad(10), rad(20))
        n_EA2_E = lat_lon2n_E(rad(30), rad(40))
        n_EB1_E = lat_lon2n_E(rad(50), rad(60))
        n_EB2_E = lat_lon2n_E(rad(70), rad(80))

        # Find the intersection between the two paths, n_EC_E:
        n_EC_E_tmp = unit(np.cross(np.cross(n_EA1_E, n_EA2_E, axis=0),
                                   np.cross(n_EB1_E, n_EB2_E, axis=0), axis=0))

        # n_EC_E_tmp is one of two solutions, the other is -n_EC_E_tmp. Select
        # the one that is closet to n_EA1_E, by selecting sign from the dot
        # product between n_EC_E_tmp and n_EA1_E:
        n_EC_E = np.sign(np.dot(n_EC_E_tmp.T, n_EA1_E)) * n_EC_E_tmp

        # When displaying the resulting position for humans, it is more
        # convenient to see lat, long:
        lat_EC, long_EC = n_E2lat_lon(n_EC_E)
        msg = 'Ex9, Intersection: lat, long = {} {} deg'
        print(msg.format(deg(lat_EC), deg(long_EC)))
        assert_array_almost_equal(deg(lat_EC), 40.31864307)
        assert_array_almost_equal(deg(long_EC), 55.90186788)

    @staticmethod
    def test_Ex10_cross_track_distance():

        # Position A1 and A2 and B are given as n_EA1_E, n_EA2_E, and n_EB_E:
        # Enter elements as lat/long in deg:
        n_EA1_E = lat_lon2n_E(rad(0), rad(0))
        n_EA2_E = lat_lon2n_E(rad(10), rad(0))
        n_EB_E = lat_lon2n_E(rad(1), rad(0.1))

        r_Earth = 6371e3  # m, mean Earth radius

        # Find the cross track distance from path A to position B.

        # SOLUTION:
        # Find the unit normal to the great circle:
        c_E = unit(np.cross(n_EA1_E, n_EA2_E, axis=0))

        # Find the great circle cross track distance:
        s_xt = -np.arcsin(np.dot(c_E.T, n_EB_E)) * r_Earth

        # Find the Euclidean cross track distance:
        d_xt = -np.dot(c_E.T, n_EB_E) * r_Earth
        msg = 'Ex10, Cross track distance = {} m, Euclidean = {} m'
        print(msg.format(s_xt, d_xt))

        assert_array_almost_equal(s_xt, 11117.79911015)
        assert_array_almost_equal(d_xt, 11117.79346741)

    @staticmethod
    def test_R2xyz():
        x, y, z = rad((10, 20, 30))
        R_AB1 = xyz2R(x, y, z)
        R_AB = [[0.81379768, -0.46984631,  0.34202014],
                [0.54383814,  0.82317294, -0.16317591],
                [-0.20487413,  0.31879578,  0.92541658]]
        assert_array_almost_equal(R_AB, R_AB1)
        x1, y1, z1 = R2xyz(R_AB1)
        assert_array_almost_equal((x, y, z), (x1, y1, z1))

    @staticmethod
    def test_R2zxy():
        x, y, z = rad((10, 20, 30))
        R_AB1 = zyx2R(z, y, x)
        R_AB = [[0.813798, -0.44097, 0.378522],
                [0.469846, 0.882564, 0.018028],
                [-0.34202, 0.163176, 0.925417]]

        assert_array_almost_equal(R_AB, R_AB1)
        z1, y1, x1 = R2zyx(R_AB1)
        assert_array_almost_equal((x, y, z), (x1, y1, z1))

    @staticmethod
    def test_n_E_and_wa2R_EL():
        n_E = np.array([[0], [0], [1]])
        R_EL = n_E_and_wa2R_EL(n_E, wander_azimuth=np.pi/2)
        R_EL1 = [[0, 1.0, 0],
                 [1.0, 0, 0],
                 [0,  0, -1.0]]
        assert_array_almost_equal(R_EL, R_EL1)

        R_EN = n_E2R_EN(n_E)
        assert_array_almost_equal(R_EN, np.diag([-1, 1, -1]))

        n_E1 = R_EL2n_E(R_EN)
        n_E2 = R_EN2n_E(R_EN)
        assert_array_almost_equal(n_E, n_E1)
        assert_array_almost_equal(n_E, n_E2)

if __name__ == "__main__":
    # import syssys.argv = ['', 'Test.testName']
    unittest.main()