SmoothLookahead.ipynb

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   "id": "ecc8c4e9-9a1d-46de-b3d4-69dbe149c7de",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b48bf4dd-15a5-4e26-bad0-c3ee837b5d62",
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   "outputs": [],
   "source": [
    "v0, v1, v2 = sp.symbols('v_0, v_1, v_2', positive=True) # Entry, cruising, and exit velocities\n",
    "t, ta, tc, td = sp.symbols('t t_a, t_c t_d', positive=True)\n",
    "a, k, d = sp.symbols('a k d', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6278725f-bf42-4fa0-ae87-617654fbb970",
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   "outputs": [],
   "source": [
    "# No cruising, models (accel and decel durations) -> (distance travelend, end speed)\n",
    "va = v0 + sp.integrate(a*t, (t, 0, t))\n",
    "vd = va.subs(t, ta) + sp.integrate(-a*t, (t, 0, t))\n",
    "\n",
    "d_tot = sp.simplify(sp.integrate(va*t, (t, 0, ta)) + sp.integrate(vd*t, (t, 0, td)))\n",
    "v_end = sp.simplify(vd.subs(t, td))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8a7e1bbf-8dc5-443a-b8a0-103c4f85725d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{a} = \\frac{\\sqrt{- 2 v_{0} + \\sqrt{4 a d + 2 v_{0}^{2} + 2 v_{2}^{2}}}}{\\sqrt{a}}$"
      ],
      "text/plain": [
       "Eq(t_a, sqrt(-2*v_0 + sqrt(4*a*d + 2*v_0**2 + 2*v_2**2))/sqrt(a))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{d} = \\frac{\\sqrt{- 2 v_{2} + \\sqrt{2} \\sqrt{2 a d + v_{0}^{2} + v_{2}^{2}}}}{\\sqrt{a}}$"
      ],
      "text/plain": [
       "Eq(t_d, sqrt(-2*v_2 + sqrt(2)*sqrt(2*a*d + v_0**2 + v_2**2))/sqrt(a))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle v_{1} = \\frac{\\sqrt{4 a d + 2 v_{0}^{2} + 2 v_{2}^{2}}}{2}$"
      ],
      "text/plain": [
       "Eq(v_1, sqrt(4*a*d + 2*v_0**2 + 2*v_2**2)/2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solve for accel and decel durations, then peak velocity (v1)\n",
    "tas, tds = sp.solve([d_tot - d, v_end - v2], [ta, td])[0]\n",
    "v1s = sp.simplify(va.subs(t, tas))\n",
    "\n",
    "display(sp.Eq(ta, tas))\n",
    "display(sp.Eq(td, tds))\n",
    "display(sp.Eq(v1, v1s))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "fae9eddd-7666-48c2-9918-a82cf4686b19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle d_{c} = d \\left(1 - k\\right)$"
      ],
      "text/plain": [
       "Eq(d_c, d*(1 - k))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{a} = \\frac{\\sqrt{- 2 v_{0} + \\sqrt{4 a d k + 2 v_{0}^{2} + 2 v_{2}^{2}}}}{\\sqrt{a}}$"
      ],
      "text/plain": [
       "Eq(t_a, sqrt(-2*v_0 + sqrt(4*a*d*k + 2*v_0**2 + 2*v_2**2))/sqrt(a))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{d} = \\frac{\\sqrt{- 2 v_{2} + \\sqrt{4 a d k + 2 v_{0}^{2} + 2 v_{2}^{2}}}}{\\sqrt{a}}$"
      ],
      "text/plain": [
       "Eq(t_d, sqrt(-2*v_2 + sqrt(4*a*d*k + 2*v_0**2 + 2*v_2**2))/sqrt(a))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle t_{c} = \\frac{\\sqrt{2} d \\left(1 - k\\right)}{\\sqrt{2 a d k + v_{0}^{2} + v_{2}^{2}}}$"
      ],
      "text/plain": [
       "Eq(t_c, sqrt(2)*d*(1 - k)/sqrt(2*a*d*k + v_0**2 + v_2**2))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Solves for accel and decel durations when the peak velocity is reduced by using a virtual acceleration = k * a (0 < k <= 1)\n",
    "v1sk = v1s.subs(a, k * a)\n",
    "tas, tds = sp.solve([va.subs(t, ta) - v1sk, vd.subs(t, td) - v2], [ta, td])[0]\n",
    "# Cruised distance and time:\n",
    "d_cruise = sp.simplify(d - d_tot.subs({ta: tas, td: tds}))\n",
    "t_cruise = sp.simplify(d_cruise / v1sk)\n",
    "\n",
    "display(sp.Eq(sp.Symbol('d_c'), d_cruise))\n",
    "display(sp.Eq(ta, tas))\n",
    "display(sp.Eq(td, tds))\n",
    "display(sp.Eq(tc, t_cruise))"
   ]
  }
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