{
"cells": [
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"import sympy as sp\n",
"sp.init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [],
"source": [
"# Source kinematics\n",
"px, py, pz = sp.symbols('px, py, pz')\n",
"vx, vy, vz = sp.symbols('vx, vy, vz')\n",
"ax, ay, az = sp.symbols('ax, ay, az')\n",
"\n",
"# speed of sound\n",
"sos = sp.symbols('sos')\n",
"\n",
"# Source time, used for the source side of the estimation (before convergence)\n",
"st = sp.symbols('st')\n",
"\n",
"# Apply source kinematics\n",
"def source_pos(t):\n",
" sx = px + vx * t + 0.5 * ax * t**2\n",
" sy = py + vy * t + 0.5 * ay * t**2\n",
" sz = pz + vz * t + 0.5 * az * t**2\n",
" return sx, sy, sz\n",
"\n",
"# Receiver location\n",
"#rx, ry, rz = sp.symbols('rx, ry, rz')\n",
"\n",
"# Receiver time, at which we're actually sampling\n",
"rt = sp.symbols('rt')"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dt*sos - sqrt((0.5*ax*(dt - rt)**2 + px - vx*(dt - rt))**2 + (0.5*ay*(dt - rt)**2 + py - vy*(dt - rt))**2 + (0.5*az*(dt - rt)**2 + pz - vz*(dt - rt))**2)\n",
"dt - (dt*sos - sqrt((0.5*ax*(dt - rt)**2 + px - vx*(dt - rt))**2 + (0.5*ay*(dt - rt)**2 + py - vy*(dt - rt))**2 + (0.5*az*(dt - rt)**2 + pz - vz*(dt - rt))**2))/(sos - ((ax*(2*dt - 2*rt) - 2*vx)*(0.5*ax*(dt - rt)**2 + px - vx*(dt - rt))/2 + (ay*(2*dt - 2*rt) - 2*vy)*(0.5*ay*(dt - rt)**2 + py - vy*(dt - rt))/2 + (az*(2*dt - 2*rt) - 2*vz)*(0.5*az*(dt - rt)**2 + pz - vz*(dt - rt))/2)/sqrt((0.5*ax*(dt - rt)**2 + px - vx*(dt - rt))**2 + (0.5*ay*(dt - rt)**2 + py - vy*(dt - rt))**2 + (0.5*az*(dt - rt)**2 + pz - vz*(dt - rt))**2))\n"
]
}
],
"source": [
"# Single Newton-Raphson iteration\n",
"\n",
"# Initial dt estimate\n",
"dt = sp.symbols('dt')\n",
"# Estimate position with the initial time estimate\n",
"(sx, sy, sz) = source_pos(rt - dt)\n",
"# Distance between estimated source and receiver\n",
"d = sp.sqrt((sx)**2 + (sy)**2 + (sz)**2)\n",
"\n",
"# This is the distance error\n",
"err = (dt * sos) - d\n",
"\n",
"# The derivative of the distance error with respect to dt\n",
"err_dt = sp.diff(err, dt)\n",
"\n",
"# Newton-Raphson iteration\n",
"dt2 = dt - err / err_dt\n",
"\n",
"print(sp.simplify(err))\n",
"print(sp.simplify(dt2))\n",
"\n",
"\n",
"nr_lambda = sp.lambdify((dt, px, py, pz, vx, vy, vz, ax, ay, az, rt, sos), dt2, 'numpy')\n",
"err_lambda = sp.lambdify((dt, px, py, pz, vx, vy, vz, ax, ay, az, rt, sos), err, 'numpy')\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'dt - (dt*sos - ((0.5*ax*(-dt + rt).powi(2) + px + vx*(-dt + rt)).powi(2) + (0.5*ay*(-dt + rt).powi(2) + py + vy*(-dt + rt)).powi(2) + (0.5*az*(-dt + rt).powi(2) + pz + vz*(-dt + rt)).powi(2)).sqrt())/(sos - ((1_f64/2.0)*(ax*(2*dt - 2*rt) - 2*vx)*(0.5*ax*(-dt + rt).powi(2) + px + vx*(-dt + rt)) + (1_f64/2.0)*(ay*(2*dt - 2*rt) - 2*vy)*(0.5*ay*(-dt + rt).powi(2) + py + vy*(-dt + rt)) + (1_f64/2.0)*(az*(2*dt - 2*rt) - 2*vz)*(0.5*az*(-dt + rt).powi(2) + pz + vz*(-dt + rt)))/((0.5*ax*(-dt + rt).powi(2) + px + vx*(-dt + rt)).powi(2) + (0.5*ay*(-dt + rt).powi(2) + py + vy*(-dt + rt)).powi(2) + (0.5*az*(-dt + rt).powi(2) + pz + vz*(-dt + rt)).powi(2)).sqrt())'"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.rust_code(dt2)"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'dt - (dt*sos - math.sqrt((0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt))**2 + (0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt))**2 + (0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt))**2))/(sos - ((1/2)*(ax*(2*dt - 2*rt) - 2*vx)*(0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt)) + (1/2)*(ay*(2*dt - 2*rt) - 2*vy)*(0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt)) + (1/2)*(az*(2*dt - 2*rt) - 2*vz)*(0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt)))/math.sqrt((0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt))**2 + (0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt))**2 + (0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt))**2))'"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.pycode(dt2)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"-2574.64085072153, 0.07911502216656353\n",
"-0.30138234576462253, 0.07912428511186843\n",
"-2.364280200189179e-09, 0.0791242851119411\n",
"-4.567774729886359e-13, 0.07912428511194111\n",
"-4.567774729886359e-13, 0.07912428511194113\n",
"0.0, 0.07912428511194113\n",
"0.0, 0.07912428511194113\n",
"0.0, 0.07912428511194113\n",
"0.0, 0.07912428511194113\n",
"0.0, 0.07912428511194113\n"
]
}
],
"source": [
"# Initial estimation\n",
"\n",
"px_ = -20\n",
"py_ = 1\n",
"pz_ = 0\n",
"vx_ = 90\n",
"vy_ = 0\n",
"vz_ = 0\n",
"ax_ = 0\n",
"ay_ = 0\n",
"az_ = 9\n",
"rt_ = 0\n",
"sos_ = 343\n",
"\n",
"import math\n",
"def nr_py(dt, px, py, pz, vx, vy, vz, ax, ay, az, rt, sos):\n",
" return dt - (dt*sos - math.sqrt((0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt))**2 + (0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt))**2 + (0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt))**2))/(sos - ((1/2)*(ax*(2*dt - 2*rt) - 2*vx)*(0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt)) + (1/2)*(ay*(2*dt - 2*rt) - 2*vy)*(0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt)) + (1/2)*(az*(2*dt - 2*rt) - 2*vz)*(0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt)))/math.sqrt((0.5*ax*(-dt + rt)**2 + px + vx*(-dt + rt))**2 + (0.5*ay*(-dt + rt)**2 + py + vy*(-dt + rt))**2 + (0.5*az*(-dt + rt)**2 + pz + vz*(-dt + rt))**2))\n",
"\n",
"\n",
"last_dt = 0\n",
"errs = []\n",
"dts = []\n",
"for i in range(10):\n",
" errs.append(err_lambda(last_dt, px_, py_, pz_, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_))\n",
" new_dt = nr_py(last_dt, px_, py_, pz_, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_)\n",
" dts.append(new_dt)\n",
" last_dt = new_dt\n",
"\n",
"\n",
"for err_, dt_ in zip(errs, dts):\n",
" print(f'{err_ * 44100 / sos_}, {dt_}')\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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]
}
],
"source": [
"# Incremental update\n",
"px_ = 0\n",
"py_ = 1\n",
"pz_ = 0\n",
"vx_ = 0\n",
"vy_ = 0\n",
"vz_ = 0\n",
"ax_ = 0\n",
"ay_ = 0\n",
"az_ = 0\n",
"rt_ = 0\n",
"sos_ = 343\n",
"\n",
"rvx = 900\n",
"rvy = 0\n",
"rvz = 0\n",
"rax = 0\n",
"ray = 0\n",
"raz = 0\n",
"\n",
"# Initial estimate\n",
"ldt = 0\n",
"\n",
"for i in range(10):\n",
" ldt = nr_lambda(ldt, px_, py_, pz_, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_)\n",
"\n",
"for sample in range(1024):\n",
" tadv = sample / 44100\n",
" pxn = px_ - (rvx * tadv + 0.5 * rax * tadv**2)\n",
" pyn = py_ - (rvy * tadv + 0.5 * ray * tadv**2)\n",
" pzn = pz_ - (rvz * tadv + 0.5 * raz * tadv**2)\n",
"\n",
"\n",
" sdt = ldt\n",
" serr = err_lambda(sdt, pxn, pyn, pzn, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_)\n",
" ldt = nr_lambda(sdt, pxn, pyn, pzn, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_)\n",
" nerr = err_lambda(ldt, pxn, pyn, pzn, vx_, vy_, vz_, ax_, ay_, az_, rt_, sos_)\n",
"\n",
" print(f'{serr * 44100 / sos_}, {ldt}, {nerr * 44100 / sos_}')\n",
" "
]
}
],
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