{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Iterative Linearization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The goal is to have a more accurate linearization of the diff equations. For every time step the optimization is iterativelly repeated using he previous optimization results **u_bar** to approximate the vehicle dynamics, instead of a random starting guess and/or the rsult at time t-1.\n",
    "\n",
    "In previous case the results at t-1 wer used to approimate the dynamics art time t!\n",
    "\n",
    "This maks the results less correlated but makes the controller slower!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "from scipy.interpolate import interp1d\n",
    "import cvxpy as cp\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.style.use(\"ggplot\")\n",
    "\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Control problem statement.\n",
    "\"\"\"\n",
    "\n",
    "N = 4  # number of state variables\n",
    "M = 2  # number of control variables\n",
    "T = 20  # Prediction Horizon\n",
    "DT = 0.2  # discretization step\n",
    "\n",
    "\n",
    "def get_linear_model(x_bar, u_bar):\n",
    "    \"\"\"\n",
    "    Computes the LTI approximated state space model x' = Ax + Bu + C\n",
    "    \"\"\"\n",
    "\n",
    "    L = 0.3  # vehicle wheelbase\n",
    "\n",
    "    x = x_bar[0]\n",
    "    y = x_bar[1]\n",
    "    v = x_bar[2]\n",
    "    theta = x_bar[3]\n",
    "\n",
    "    a = u_bar[0]\n",
    "    delta = u_bar[1]\n",
    "\n",
    "    A = np.zeros((N, N))\n",
    "    A[0, 2] = np.cos(theta)\n",
    "    A[0, 3] = -v * np.sin(theta)\n",
    "    A[1, 2] = np.sin(theta)\n",
    "    A[1, 3] = v * np.cos(theta)\n",
    "    A[3, 2] = v * np.tan(delta) / L\n",
    "    A_lin = np.eye(N) + DT * A\n",
    "\n",
    "    B = np.zeros((N, M))\n",
    "    B[2, 0] = 1\n",
    "    B[3, 1] = v / (L * np.cos(delta) ** 2)\n",
    "    B_lin = DT * B\n",
    "\n",
    "    f_xu = np.array(\n",
    "        [v * np.cos(theta), v * np.sin(theta), a, v * np.tan(delta) / L]\n",
    "    ).reshape(N, 1)\n",
    "    C_lin = DT * (\n",
    "        f_xu - np.dot(A, x_bar.reshape(N, 1)) - np.dot(B, u_bar.reshape(M, 1))\n",
    "    )\n",
    "\n",
    "    return np.round(A_lin, 4), np.round(B_lin, 4), np.round(C_lin, 4)\n",
    "\n",
    "\n",
    "\"\"\"\n",
    "the ODE is used to update the simulation given the mpc results\n",
    "I use this insted of using the LTI twice\n",
    "\"\"\"\n",
    "\n",
    "\n",
    "def kinematics_model(x, t, u):\n",
    "    \"\"\"\n",
    "    Returns the set of ODE of the vehicle model.\n",
    "    \"\"\"\n",
    "\n",
    "    L = 0.3  # vehicle wheelbase\n",
    "    dxdt = x[2] * np.cos(x[3])\n",
    "    dydt = x[2] * np.sin(x[3])\n",
    "    dvdt = u[0]\n",
    "    dthetadt = x[2] * np.tan(u[1]) / L\n",
    "\n",
    "    dqdt = [dxdt, dydt, dvdt, dthetadt]\n",
    "\n",
    "    return dqdt\n",
    "\n",
    "\n",
    "def predict(x0, u):\n",
    "    \"\"\" \"\"\"\n",
    "\n",
    "    x_ = np.zeros((N, T + 1))\n",
    "\n",
    "    x_[:, 0] = x0\n",
    "\n",
    "    # solve ODE\n",
    "    for t in range(1, T + 1):\n",
    "\n",
    "        tspan = [0, DT]\n",
    "        x_next = odeint(kinematics_model, x0, tspan, args=(u[:, t - 1],))\n",
    "\n",
    "        x0 = x_next[1]\n",
    "        x_[:, t] = x_next[1]\n",
    "\n",
    "    return x_\n",
    "\n",
    "\n",
    "def compute_path_from_wp(start_xp, start_yp, step=0.1):\n",
    "    \"\"\"\n",
    "    Computes a reference path given a set of waypoints\n",
    "    \"\"\"\n",
    "\n",
    "    final_xp = []\n",
    "    final_yp = []\n",
    "    delta = step  # [m]\n",
    "\n",
    "    for idx in range(len(start_xp) - 1):\n",
    "        section_len = np.sum(\n",
    "            np.sqrt(\n",
    "                np.power(np.diff(start_xp[idx : idx + 2]), 2)\n",
    "                + np.power(np.diff(start_yp[idx : idx + 2]), 2)\n",
    "            )\n",
    "        )\n",
    "\n",
    "        interp_range = np.linspace(0, 1, np.floor(section_len / delta).astype(int))\n",
    "\n",
    "        fx = interp1d(np.linspace(0, 1, 2), start_xp[idx : idx + 2], kind=1)\n",
    "        fy = interp1d(np.linspace(0, 1, 2), start_yp[idx : idx + 2], kind=1)\n",
    "\n",
    "        final_xp = np.append(final_xp, fx(interp_range))\n",
    "        final_yp = np.append(final_yp, fy(interp_range))\n",
    "\n",
    "    dx = np.append(0, np.diff(final_xp))\n",
    "    dy = np.append(0, np.diff(final_yp))\n",
    "    theta = np.arctan2(dy, dx)\n",
    "\n",
    "    return np.vstack((final_xp, final_yp, theta))\n",
    "\n",
    "\n",
    "def get_nn_idx(state, path):\n",
    "    \"\"\"\n",
    "    Computes the index of the waypoint closest to vehicle\n",
    "    \"\"\"\n",
    "\n",
    "    dx = state[0] - path[0, :]\n",
    "    dy = state[1] - path[1, :]\n",
    "    dist = np.hypot(dx, dy)\n",
    "    nn_idx = np.argmin(dist)\n",
    "\n",
    "    try:\n",
    "        v = [\n",
    "            path[0, nn_idx + 1] - path[0, nn_idx],\n",
    "            path[1, nn_idx + 1] - path[1, nn_idx],\n",
    "        ]\n",
    "        v /= np.linalg.norm(v)\n",
    "\n",
    "        d = [path[0, nn_idx] - state[0], path[1, nn_idx] - state[1]]\n",
    "\n",
    "        if np.dot(d, v) > 0:\n",
    "            target_idx = nn_idx\n",
    "        else:\n",
    "            target_idx = nn_idx + 1\n",
    "\n",
    "    except IndexError as e:\n",
    "        target_idx = nn_idx\n",
    "\n",
    "    return target_idx\n",
    "\n",
    "\n",
    "def get_ref_trajectory(state, path, target_v):\n",
    "    \"\"\"\n",
    "    Adapted from pythonrobotics\n",
    "    \"\"\"\n",
    "    xref = np.zeros((N, T + 1))\n",
    "    dref = np.zeros((1, T + 1))\n",
    "\n",
    "    # sp = np.ones((1,T +1))*target_v #speed profile\n",
    "\n",
    "    ncourse = path.shape[1]\n",
    "\n",
    "    ind = get_nn_idx(state, path)\n",
    "\n",
    "    xref[0, 0] = path[0, ind]  # X\n",
    "    xref[1, 0] = path[1, ind]  # Y\n",
    "    xref[2, 0] = target_v  # sp[ind] #V\n",
    "    xref[3, 0] = path[2, ind]  # Theta\n",
    "    dref[0, 0] = 0.0  # steer operational point should be 0\n",
    "\n",
    "    dl = 0.05  # Waypoints spacing [m]\n",
    "    travel = 0.0\n",
    "\n",
    "    for i in range(T + 1):\n",
    "        travel += abs(target_v) * DT  # current V or target V?\n",
    "        dind = int(round(travel / dl))\n",
    "\n",
    "        if (ind + dind) < ncourse:\n",
    "            xref[0, i] = path[0, ind + dind]\n",
    "            xref[1, i] = path[1, ind + dind]\n",
    "            xref[2, i] = target_v  # sp[ind + dind]\n",
    "            xref[3, i] = path[2, ind + dind]\n",
    "            dref[0, i] = 0.0\n",
    "        else:\n",
    "            xref[0, i] = path[0, ncourse - 1]\n",
    "            xref[1, i] = path[1, ncourse - 1]\n",
    "            xref[2, i] = 0.0  # stop? #sp[ncourse - 1]\n",
    "            xref[3, i] = path[2, ncourse - 1]\n",
    "            dref[0, i] = 0.0\n",
    "\n",
    "    return xref, dref"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-2-e22409964dd8>:127: RuntimeWarning: invalid value encountered in true_divide\n",
      "  v /= np.linalg.norm(v)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CVXPY Optimization Time: Avrg: 0.5979s Max: 0.8275s Min: 0.2939s\n"
     ]
    }
   ],
   "source": [
    "track = compute_path_from_wp(\n",
    "    [0, 3, 4, 6, 10, 12, 14, 6, 1, 0], [0, 0, 2, 4, 3, 3, -2, -6, -2, -2], 0.05\n",
    ")\n",
    "\n",
    "# track = compute_path_from_wp([0,10,10,0],\n",
    "#                              [0,0,1,1],0.05)\n",
    "\n",
    "sim_duration = 200  # time steps\n",
    "opt_time = []\n",
    "\n",
    "x_sim = np.zeros((N, sim_duration))\n",
    "u_sim = np.zeros((M, sim_duration - 1))\n",
    "\n",
    "MAX_SPEED = 1.5  # m/s\n",
    "MAX_ACC = 1.0  # m/ss\n",
    "MAX_D_ACC = 1.0  # m/sss\n",
    "MAX_STEER = np.radians(30)  # rad\n",
    "MAX_D_STEER = np.radians(30)  # rad/s\n",
    "\n",
    "REF_VEL = 1.0  # m/s\n",
    "\n",
    "# Starting Condition\n",
    "x0 = np.zeros(N)\n",
    "x0[0] = 0  # x\n",
    "x0[1] = -0.25  # y\n",
    "x0[2] = 0.0  # v\n",
    "x0[3] = np.radians(-0)  # yaw\n",
    "\n",
    "for sim_time in range(sim_duration - 1):\n",
    "\n",
    "    iter_start = time.time()\n",
    "\n",
    "    # starting guess for ctrl\n",
    "    u_bar = np.zeros((M, T))\n",
    "    u_bar[0, :] = MAX_ACC / 2  # a\n",
    "    u_bar[1, :] = 0.0  # delta\n",
    "\n",
    "    for _ in range(5):\n",
    "        u_prev = u_bar\n",
    "\n",
    "        # dynamics starting state\n",
    "        x_bar = np.zeros((N, T + 1))\n",
    "        x_bar[:, 0] = x_sim[:, sim_time]\n",
    "\n",
    "        # prediction for linearization of costrains\n",
    "        for t in range(1, T + 1):\n",
    "            xt = x_bar[:, t - 1].reshape(N, 1)\n",
    "            ut = u_bar[:, t - 1].reshape(M, 1)\n",
    "            A, B, C = get_linear_model(xt, ut)\n",
    "            xt_plus_one = np.squeeze(np.dot(A, xt) + np.dot(B, ut) + C)\n",
    "            x_bar[:, t] = xt_plus_one\n",
    "\n",
    "        # CVXPY Linear MPC problem statement\n",
    "        x = cp.Variable((N, T + 1))\n",
    "        u = cp.Variable((M, T))\n",
    "        cost = 0\n",
    "        constr = []\n",
    "\n",
    "        # Cost Matrices\n",
    "        Q = np.diag([20, 20, 10, 0])  # state error cost\n",
    "        Qf = np.diag([30, 30, 30, 0])  # state final error cost\n",
    "        R = np.diag([10, 10])  # input cost\n",
    "        R_ = np.diag([10, 10])  # input rate of change cost\n",
    "\n",
    "        # Get Reference_traj\n",
    "        x_ref, d_ref = get_ref_trajectory(x_bar[:, 0], track, REF_VEL)\n",
    "\n",
    "        # Prediction Horizon\n",
    "        for t in range(T):\n",
    "\n",
    "            # Tracking Error\n",
    "            cost += cp.quad_form(x[:, t] - x_ref[:, t], Q)\n",
    "\n",
    "            # Actuation effort\n",
    "            cost += cp.quad_form(u[:, t], R)\n",
    "\n",
    "            # Actuation rate of change\n",
    "            if t < (T - 1):\n",
    "                cost += cp.quad_form(u[:, t + 1] - u[:, t], R_)\n",
    "                constr += [\n",
    "                    cp.abs(u[0, t + 1] - u[0, t]) / DT <= MAX_D_ACC\n",
    "                ]  # max acc rate of change\n",
    "                constr += [\n",
    "                    cp.abs(u[1, t + 1] - u[1, t]) / DT <= MAX_D_STEER\n",
    "                ]  # max steer rate of change\n",
    "\n",
    "            # Kinrmatics Constrains (Linearized model)\n",
    "            A, B, C = get_linear_model(x_bar[:, t], u_bar[:, t])\n",
    "            constr += [x[:, t + 1] == A @ x[:, t] + B @ u[:, t] + C.flatten()]\n",
    "\n",
    "        # Final Point tracking\n",
    "        cost += cp.quad_form(x[:, T] - x_ref[:, T], Qf)\n",
    "\n",
    "        # sums problem objectives and concatenates constraints.\n",
    "        constr += [x[:, 0] == x_bar[:, 0]]  # starting condition\n",
    "        constr += [x[2, :] <= MAX_SPEED]  # max speed\n",
    "        constr += [x[2, :] >= 0.0]  # min_speed (not really needed)\n",
    "        constr += [cp.abs(u[0, :]) <= MAX_ACC]  # max acc\n",
    "        constr += [cp.abs(u[1, :]) <= MAX_STEER]  # max steer\n",
    "\n",
    "        # Solve\n",
    "        prob = cp.Problem(cp.Minimize(cost), constr)\n",
    "        solution = prob.solve(solver=cp.OSQP, verbose=False)\n",
    "\n",
    "        # retrieved optimized U and assign to u_bar to linearize in next step\n",
    "        u_bar = np.vstack(\n",
    "            (np.array(u.value[0, :]).flatten(), (np.array(u.value[1, :]).flatten()))\n",
    "        )\n",
    "\n",
    "        # check how this solution differs from previous\n",
    "        # if the solutions are very\n",
    "        delta_u = np.sum(np.sum(np.abs(u_bar - u_prev), axis=0), axis=0)\n",
    "        if delta_u < 0.05:\n",
    "            break\n",
    "\n",
    "    # select u from best iteration\n",
    "    u_sim[:, sim_time] = u_bar[:, 0]\n",
    "\n",
    "    # Measure elpased time to get results from cvxpy\n",
    "    opt_time.append(time.time() - iter_start)\n",
    "\n",
    "    # move simulation to t+1\n",
    "    tspan = [0, DT]\n",
    "    x_sim[:, sim_time + 1] = odeint(\n",
    "        kinematics_model, x_sim[:, sim_time], tspan, args=(u_bar[:, 0],)\n",
    "    )[1]\n",
    "\n",
    "    # reset u_bar? -> this simulates that we don use previous solution!\n",
    "    u_bar[0, :] = MAX_ACC / 2  # a\n",
    "    u_bar[1, :] = 0.0  # delta\n",
    "\n",
    "\n",
    "print(\n",
    "    \"CVXPY Optimization Time: Avrg: {:.4f}s Max: {:.4f}s Min: {:.4f}s\".format(\n",
    "        np.mean(opt_time), np.max(opt_time), np.min(opt_time)\n",
    "    )\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "[encoded data, 102652 characters]\n",
      "text/plain": [
       "<Figure size 1080x720 with 5 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot trajectory\n",
    "grid = plt.GridSpec(4, 5)\n",
    "\n",
    "plt.figure(figsize=(15, 10))\n",
    "\n",
    "plt.subplot(grid[0:4, 0:4])\n",
    "plt.plot(track[0, :], track[1, :], \"b+\")\n",
    "plt.plot(x_sim[0, :], x_sim[1, :])\n",
    "plt.axis(\"equal\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.xlabel(\"x\")\n",
    "\n",
    "plt.subplot(grid[0, 4])\n",
    "plt.plot(u_sim[0, :])\n",
    "plt.ylabel(\"a(t) [m/ss]\")\n",
    "\n",
    "plt.subplot(grid[1, 4])\n",
    "plt.plot(x_sim[2, :])\n",
    "plt.ylabel(\"v(t) [m/s]\")\n",
    "\n",
    "plt.subplot(grid[2, 4])\n",
    "plt.plot(np.degrees(u_sim[1, :]))\n",
    "plt.ylabel(\"delta(t) [rad]\")\n",
    "\n",
    "plt.subplot(grid[3, 4])\n",
    "plt.plot(x_sim[3, :])\n",
    "plt.ylabel(\"theta(t) [rad]\")\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  }
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