Walking Robot.ipynb

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  "metadata": {
    "colab": {
      "name": "CI_HW2_Walid_Shaker.ipynb",
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      "collapsed_sections": []
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      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
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  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "## CVXPY Installation "
      ],
      "metadata": {
        "id": "loUbTxPiqxOY"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
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        "outputId": "d9b0de2c-1428-4281-8c79-a0f387662c21"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Requirement already satisfied: CVXPY in /usr/local/lib/python3.7/dist-packages (1.0.31)\n",
            "Requirement already satisfied: numpy>=1.15 in /usr/local/lib/python3.7/dist-packages (from CVXPY) (1.21.6)\n",
            "Requirement already satisfied: multiprocess in /usr/local/lib/python3.7/dist-packages (from CVXPY) (0.70.12.2)\n",
            "Requirement already satisfied: scipy>=1.1.0 in /usr/local/lib/python3.7/dist-packages (from CVXPY) (1.4.1)\n",
            "Requirement already satisfied: ecos>=2 in /usr/local/lib/python3.7/dist-packages (from CVXPY) (2.0.10)\n",
            "Requirement already satisfied: scs>=1.1.3 in /usr/local/lib/python3.7/dist-packages (from CVXPY) (3.2.0)\n",
            "Requirement already satisfied: osqp>=0.4.1 in /usr/local/lib/python3.7/dist-packages (from CVXPY) (0.6.2.post0)\n",
            "Requirement already satisfied: qdldl in /usr/local/lib/python3.7/dist-packages (from osqp>=0.4.1->CVXPY) (0.1.5.post2)\n",
            "Requirement already satisfied: dill>=0.3.4 in /usr/local/lib/python3.7/dist-packages (from multiprocess->CVXPY) (0.3.4)\n"
          ]
        }
      ],
      "source": [
        "!pip install CVXPY"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Import Libraries"
      ],
      "metadata": {
        "id": "yx8W-p93q2Mk"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import cvxpy as cp\n",
        "import numpy as np\n",
        "import scipy\n",
        "import matplotlib.pyplot as plt\n",
        "from mpl_toolkits.mplot3d import Axes3D\n",
        "from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n",
        "%matplotlib inline"
      ],
      "metadata": {
        "id": "pphM9fvG-RK_"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Helper Functions \n"
      ],
      "metadata": {
        "id": "gDExp7e9vrrL"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def skew(vector):\n",
        "    \"\"\"\n",
        "    this function returns a numpy array with the skew symmetric cross product matrix for vector.\n",
        "    the skew symmetric cross product matrix is defined such that\n",
        "    np.cross(a, b) = np.dot(skew(a), b)\n",
        "\n",
        "    :param vector: An array like vector to create the skew symmetric cross product matrix for\n",
        "    :return: A numpy array of the skew symmetric cross product vector\n",
        "    \"\"\"\n",
        "    return np.array([[0, -vector[2], vector[1]], \n",
        "                     [vector[2], 0, -vector[0]], \n",
        "                     [-vector[1], vector[0], 0]])"
      ],
      "metadata": {
        "id": "R2kxGKyxvyVC"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## TASK1: Horizontal ground\n",
        "\n",
        "Stability equations:\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = -mg\\\\ \n",
        "\\end{equation}\n",
        "\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\\\\ \n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} =  0_{3\\times1}\\\\ \n",
        "\\end{equation}\n",
        "\n",
        "\n",
        "where $r_{c}$ is center of mass $r_{i}$ are position of the feet, $m$ is the mass of the robot, and $g$ = $\\begin{bmatrix}\n",
        "0\\\\\n",
        "0\\\\\n",
        "-9.8\n",
        "\\end{bmatrix}$ \n",
        "\n",
        "R is skew symmetric cross product matrix for vector such that \n",
        "\\begin{equation} \n",
        "(r_{i}-r_{c}){\\times}f_{i} = R f_{i}\n",
        "\\end{equation}\n",
        "\n",
        "\\begin{equation}\n",
        "R = \n",
        "\\begin{bmatrix}\n",
        "0 & -r_{z} & r_{y}\\\\\n",
        "r_{z} & 0 & -r_{x}\\\\\n",
        "-r_{y} & r_{x} & 0\n",
        "\\end{bmatrix}\n",
        "\\end{equation} \\\n",
        "\n",
        "Problem can be formulated as feasibility problem as follows:\\\n",
        "**Constraints**\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = \n",
        "\\begin{bmatrix} -mg\\\\ 0_{3\\times1}\n",
        "\\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "This can be written as \\begin{equation} Af = b \\tag{1}\\end{equation}\n",
        "\n",
        "and assuming that the reaction forces lie in the friction cone, with friction coefcient $\\mu$\n",
        "\n",
        "Let the normal to the plane is $n$ = $\\begin{bmatrix}\n",
        "0\\\\\n",
        "0\\\\\n",
        "1\n",
        "\\end{bmatrix}$ \n",
        "so, the tangential orthonormal basis that span the horizontal plane are $t$ = $\\begin{bmatrix}\n",
        "0 & -1\\\\\n",
        "1 & 0\\\\\n",
        "0 & 0\n",
        "\\end{bmatrix}$ \n",
        "\n",
        "\\begin{equation} \n",
        "\\lVert t^Tf_{i} \\rVert \\leq \\mu n^T f_{i} \\tag{2}\n",
        "\\end{equation} \n",
        "\n",
        "\n",
        "The optimazion problem: \n",
        "\\begin{aligned} \n",
        "\\min_{f} \\quad & 0\\\\ \n",
        "\\textrm{s.t.} \\quad & Af = b\\\\ \n",
        "  & \\lVert t^Tf_{i} \\rVert \\leq \\mu n^T f_{i}\n",
        "\\end{aligned}"
      ],
      "metadata": {
        "id": "Rk6vb-EfPB0N"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        ""
      ],
      "metadata": {
        "id": "z9Y59D16cF6u"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1.1 Problem assumptions "
      ],
      "metadata": {
        "id": "IaY__6Dlq-Cl"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "m = 2\n",
        "g = np.array([0,0,-9.8])\n",
        "mu = 0.25"
      ],
      "metadata": {
        "id": "fTAfTRq45MDW"
      },
      "execution_count": 4,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1.2 Define the horizontal plane  "
      ],
      "metadata": {
        "id": "hwKQEK28xAxH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.array([[0],[0],[1]])\n",
        "t = scipy.linalg.null_space(n.T)\n",
        "print(\"plane orthonormal basis:\\n\",t)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3cvBgj-0v8NH",
        "outputId": "e1f19a84-3082-4e9a-c1d7-9fc7b526b3f7"
      },
      "execution_count": 29,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "plane orthonormal basis:\n",
            " [[ 0. -1.]\n",
            " [ 1.  0.]\n",
            " [ 0.  0.]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1.3 Define feet positions and corresponding skew-symmeric matrices\n",
        "        \n"
      ],
      "metadata": {
        "id": "PUSXUrGmxb74"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "r1 = np.dot(t, np.random.randint(5,size=(2,1)))\n",
        "r2 = np.dot(t, np.random.randint(5,size=(2,1)))\n",
        "r3 = np.dot(t, np.random.randint(5,size=(2,1)))\n",
        "r4 = np.dot(t, np.random.randint(5,size=(2,1)))\n",
        "\n",
        "xc = (r1[0][0] + r2[0][0] + r3[0][0] + r4[0][0])/4\n",
        "yc = (r1[1][0] + r2[1][0] + r3[1][0] + r4[1][0])/4\n",
        "zc = (r1[2][0] + r2[2][0] + r3[2][0] + r4[2][0])/4\n",
        "\n",
        "rc = np.array([[xc],[yc],[zc]])\n",
        "\n",
        "R1 = skew((r1-rc)[:,0])\n",
        "R2 = skew((r2-rc)[:,0])\n",
        "R3 = skew((r3-rc)[:,0])\n",
        "R4 = skew((r4-rc)[:,0])\n",
        "\n",
        "# r1 = np.array([0,0,0])\n",
        "# r2 = np.array([2,0,0])\n",
        "# r3 = np.array([2,2,0])\n",
        "# r4 = np.array([0,2,0])\n",
        "# rc = np.array([1,1,0])\n",
        "\n",
        "# R1 = skew(r1-rc)\n",
        "# R2 = skew(r2-rc)\n",
        "# R3 = skew(r3-rc)\n",
        "# R4 = skew(r4-rc)"
      ],
      "metadata": {
        "id": "Z9lzxc8ewde_"
      },
      "execution_count": 39,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "fig = plt.figure(figsize=(5,5))\n",
        "ax = fig.add_subplot(projection='3d')\n",
        "\n",
        "r = np.vstack([r1.T, r2.T, r3.T, r4.T])\n",
        "# r = np.vstack([r1, r2, r3, r4])\n",
        "vertices = [list(zip(r[:,0],r[:,1],r[:,2]))]\n",
        "poly = Poly3DCollection(vertices, alpha=0.3)\n",
        "ax.add_collection3d(poly)\n",
        "\n",
        "ax.scatter(xs = r[:,0], ys = r[:,1], zs = r[:,2], alpha=1, label = \"robot legs\")\n",
        "ax.scatter(rc[0],rc[1],rc[2], alpha=1, label = \"center of mass\") \n",
        "\n",
        "ax.set_xlim(-5,5)\n",
        "ax.set_ylim(-5,5)\n",
        "ax.set_zlim(0,3)\n",
        "ax.set_xlabel(\"x-axis\")\n",
        "ax.set_ylabel(\"y-axis\")\n",
        "ax.set_zlabel(\"z-axis\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 303
        },
        "id": "7y7Se1PmrYWr",
        "outputId": "8ec57b26-0d0e-4f81-a308-b723cfbb218c"
      },
      "execution_count": 40,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 360x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1.4 Define A, b for the equality constraint"
      ],
      "metadata": {
        "id": "ItrTOnU4rc9B"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "row1 = np.hstack([np.eye(3), np.eye(3), np.eye(3), np.eye(3)])\n",
        "row2 = np.hstack([R1, R2, R3, R4])\n",
        "A = np.vstack([row1, row2])\n",
        "b = np.hstack([-m*g, 0, 0, 0])"
      ],
      "metadata": {
        "id": "LwMGYO5c4NH8"
      },
      "execution_count": 41,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 1.5 Problem solution"
      ],
      "metadata": {
        "id": "CwgcvZ22rk8Q"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "f = cp.Variable(12) #f1_x, f1_y, f1_z .... f4_z\n",
        "obj_function = 0\n",
        "constraint = [A@f==b]\n",
        "\n",
        "for i in range(0,12,3): \n",
        "  constraint += [cp.norm2(t.T@f[i:i+3]) <= mu*n.T@f[i:i+3]]\n",
        "\n",
        "obj = cp.Minimize(obj_function)\n",
        "proj = cp.Problem(obj, constraint)\n",
        "result = proj.solve()\n",
        "\n",
        "print(\"The problem status is\", proj.status)\n",
        "print(\"The optimal value is\", result) # The optimal objective value is returned by `proj.solve()`.\n",
        "print(f\"f1 = {f.value[0:3]}.T\") # The optimal value for f is stored in `f.value`.\n",
        "print(f\"f2 = {f.value[3:6]}.T\")\n",
        "print(f\"f3 = {f.value[6:9]}.T\")\n",
        "print(f\"f4 = {f.value[9:12]}.T\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AiCKXyNHDwrZ",
        "outputId": "345b6203-5aa5-4681-dc7e-499db593da0c"
      },
      "execution_count": 42,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The problem status is optimal\n",
            "The optimal value is 0.0\n",
            "f1 = [0.         0.         4.72265536].T\n",
            "f2 = [0.         0.         5.02667474].T\n",
            "f3 = [0.         0.         4.79866021].T\n",
            "f4 = [0.         0.         5.05200969].T\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## TASK2: Tilted ground\n",
        "\n",
        "To define the titled plane, let's introduce a random normal vector $n$ = $\\begin{bmatrix}\n",
        "0\\\\\n",
        "0\\\\\n",
        "1\n",
        "\\end{bmatrix} + random(3,1)$  \n",
        "\n",
        "After that get the normalized_n = $\\frac{n}{\\lVert n \\rVert}$\n",
        "\n",
        "Then get the tangential orthonormal basis that span the tilted plane $t=NullSpace(n^T)$  \\\n",
        "By taking the left null space of vector n, we will obtain matrix $t_{3\\times2} $ \n",
        "\n",
        "Generating random positions for the robot feet by making some random linear combinations of t basis as\n",
        "$r_{i} = t*random(2,1)$  \n",
        "\n",
        "For the skew symmetric matrix $R_{i} = $\\begin{bmatrix} (r_{i}-r_{c}) {\\times} \\end{bmatrix}\n",
        "\n",
        "The problem still can be formulated as feasibility problem as follows:\\\n",
        "**Constraints**\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = \n",
        "\\begin{bmatrix} -mg\\\\ 0_{3\\times1}\n",
        "\\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "This can be written as \\begin{equation} Af = b \\tag{1}\\end{equation}\n",
        "\n",
        "and assuming that the reaction forces lie in the friction cone, with friction coefcient $\\mu$\n",
        "\n",
        "\\begin{equation} \n",
        "\\lVert t^Tf_{i} \\rVert \\leq \\mu n^T f_{i} \\tag{2}\n",
        "\\end{equation} \n",
        "\n",
        "The optimazion problem: \n",
        "\\begin{aligned} \n",
        "\\min_{f} \\quad & 0\\\\ \n",
        "\\textrm{s.t.} \\quad & Af = b\\\\ \n",
        "  & \\lVert t^Tf_{i} \\rVert \\leq \\mu n^T f_{i}\n",
        "\\end{aligned}"
      ],
      "metadata": {
        "id": "h7mmS9Evn5Zt"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.1 Problem assumptions "
      ],
      "metadata": {
        "id": "zmJ-7M8DsDMP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "m = 2\n",
        "g = np.array([0,0,-9.8])\n",
        "mu = 2.5"
      ],
      "metadata": {
        "id": "7H8Td9CHp-pZ"
      },
      "execution_count": 71,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.2 Define the tilted plane  "
      ],
      "metadata": {
        "id": "n3fn6TlksIMK"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Normal vector to a tilted plane\n",
        "n = np.array([[0],[0],[1]]) + np.random.randn(3,1)\n",
        "n = n / np.sqrt(np.sum(n**2))\n",
        "print(\"normal vector: \\nn = \", n.T)\n",
        "\n",
        "# Orthonormal basis of the tilted plane \n",
        "t = scipy.linalg.null_space(n.T)\n",
        "print(\"\\nplane orthonormal basis:\\n\",t)\n",
        "\n",
        "# Check dot product = 0\n",
        "print(\"\\ndot product(t1.T, t2) = \", np.dot(t[:,0].T, t[:,1]))\n",
        "print(\"dot product(n.T, t1) = \", np.dot(n.T, t[:,0])[0])\n",
        "print(\"dot product(n.T, t2) = \", np.dot(n.T, t[:,1])[0])"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "slon1cebYFvd",
        "outputId": "13fb7c37-e786-41cd-cf50-7b1b07f837fe"
      },
      "execution_count": 72,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "normal vector: \n",
            "n =  [[-0.36107214 -0.7153266   0.59827649]]\n",
            "\n",
            "plane orthonormal basis:\n",
            " [[-0.7153266   0.59827649]\n",
            " [ 0.62405215  0.31443086]\n",
            " [ 0.31443086  0.73701999]]\n",
            "\n",
            "dot product(t1.T, t2) =  8.326672684688674e-17\n",
            "dot product(n.T, t1) =  -5.551115123125783e-17\n",
            "dot product(n.T, t2) =  5.551115123125783e-17\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.3 Define feet positions and corresponding skew-symmeric matrices"
      ],
      "metadata": {
        "id": "zEWv_DPTsyw7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "r1 = np.dot(t, np.random.randn(2,1))\n",
        "r2 = np.dot(t, np.random.randn(2,1))\n",
        "r3 = np.dot(t, np.random.randn(2,1))\n",
        "r4 = np.dot(t, np.random.randn(2,1))\n",
        "\n",
        "xc = (r1[0][0] + r2[0][0] + r3[0][0] + r4[0][0])/4\n",
        "yc = (r1[1][0] + r2[1][0] + r3[1][0] + r4[1][0])/4\n",
        "zc = (r1[2][0] + r2[2][0] + r3[2][0] + r4[2][0])/4\n",
        "\n",
        "rc = np.array([[xc],[yc],[zc]])\n",
        "\n",
        "R1 = skew((r1-rc)[:,0])\n",
        "R2 = skew((r2-rc)[:,0])\n",
        "R3 = skew((r3-rc)[:,0])\n",
        "R4 = skew((r4-rc)[:,0])"
      ],
      "metadata": {
        "id": "15npMN_sse8j"
      },
      "execution_count": 73,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "fig = plt.figure(figsize=(5,5))\n",
        "ax = fig.add_subplot(projection='3d')\n",
        "\n",
        "r = np.vstack([r1.T, r2.T, r3.T, r4.T])\n",
        "vertices = [list(zip(r[:,0],r[:,1],r[:,2]))]\n",
        "poly = Poly3DCollection(vertices, alpha=0.2)\n",
        "ax.add_collection3d(poly)\n",
        "\n",
        "ax.scatter(xs = r[:,0], ys = r[:,1], zs = r[:,2], alpha=1, label = \"robot legs\")\n",
        "ax.scatter(xc,yc,zc, alpha=1, label = \"center of mass\") \n",
        "\n",
        "ax.set_xlim(-5,5)\n",
        "ax.set_ylim(-5,5)\n",
        "ax.set_zlim(0,3)\n",
        "ax.set_xlabel(\"x-axis\")\n",
        "ax.set_ylabel(\"y-axis\")\n",
        "ax.set_zlabel(\"z-axis\")\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 303
        },
        "id": "c_JSTHNJzycp",
        "outputId": "1ff374c4-29fa-4e64-ee51-52557672de0f"
      },
      "execution_count": 74,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 360x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.4 Define A , b for the equality constraint"
      ],
      "metadata": {
        "id": "7D_HJzBntHQ1"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "row1 = np.hstack([np.eye(3), np.eye(3), np.eye(3), np.eye(3)])\n",
        "row2 = np.hstack([R1, R2, R3, R4])\n",
        "A = np.vstack([row1, row2])\n",
        "b = np.hstack([-m*g, 0, 0, 0])"
      ],
      "metadata": {
        "id": "Q2fQ-opopFow"
      },
      "execution_count": 75,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.5 Problem solution"
      ],
      "metadata": {
        "id": "H4gPwEXntXPJ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "f = cp.Variable(12) #f1_x, f1_y, f1_z .... f4_z\n",
        "obj_function = 0\n",
        "constraint = [A@f==b]\n",
        "\n",
        "for i in range(0,12,3): \n",
        "  constraint += [cp.norm2(t.T@f[i:i+3]) <= mu*n[:,0].T@f[i:i+3]]\n",
        "\n",
        "obj = cp.Minimize(obj_function)\n",
        "proj = cp.Problem(obj, constraint)\n",
        "result = proj.solve()\n",
        "\n",
        "print(\"The problem status is\", proj.status)\n",
        "print(\"The optimal value is\", result) # The optimal objective value is returned by `proj.solve()`.\n",
        "print(f\"f1 = {f.value[0:3]}.T\") # The optimal value for f is stored in `f.value`.\n",
        "print(f\"f2 = {f.value[3:6]}.T\")\n",
        "print(f\"f3 = {f.value[6:9]}.T\")\n",
        "print(f\"f4 = {f.value[9:12]}.T\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JO9gT-bmpOBc",
        "outputId": "2eacbf91-8124-4881-c17d-03f3eb6f67d4"
      },
      "execution_count": 77,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The problem status is optimal\n",
            "The optimal value is 0.0\n",
            "f1 = [0.03753015 0.01855974 4.95826524].T\n",
            "f2 = [0.00890092 0.00519051 4.93299467].T\n",
            "f3 = [0.04434614 0.0137798  4.91535817].T\n",
            "f4 = [-0.09077722 -0.03753005  4.79338192].T\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Note that the problem feasibility is not always guaranteed as it depends on the random generated feet positions on the tilted plane."
      ],
      "metadata": {
        "id": "ERIySPeN9A6b"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## TASK3: One foot on a vertical wall\n",
        "\n",
        "Assuming that 3 feet are standing on a horizontal plane $xy$ and the $4^{th}$ leg pushes against a vertical wall $yz$ plane. \\\n",
        "\n",
        "For the first 3 feet normal vector $n_{1-3}$ = $\\begin{bmatrix}\n",
        "0\\\\\n",
        "0\\\\\n",
        "1\n",
        "\\end{bmatrix}, t_{1-3} = \\begin{bmatrix}\n",
        "0 & -1\\\\\n",
        "1 & 0\\\\\n",
        "0 & 0\n",
        "\\end{bmatrix} $\n",
        "\n",
        "For the $4^{th}$ leg normal vector $n_{4}$ = $\\begin{bmatrix}\n",
        "-1\\\\\n",
        "0\\\\\n",
        "0\n",
        "\\end{bmatrix}, t_{4} = \\begin{bmatrix}\n",
        "0 & 0\\\\\n",
        "1 & 0\\\\\n",
        "0 & 1\n",
        "\\end{bmatrix} $\n",
        "\n",
        "\n",
        "The problem still can be formulated as feasibility problem as follows:\\\n",
        "**Constraints**\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = \n",
        "\\begin{bmatrix} -mg\\\\ 0_{3\\times1}\n",
        "\\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "This can be written as \\begin{equation} Af = b \\tag{1}\\end{equation}\n",
        "\n",
        "and assuming that the reaction forces lie in the friction cone, with friction coefcient $\\mu$ \\\n",
        "for $i = 1,2,3$\n",
        "\\begin{equation} \n",
        "\\lVert t^T_{1-3}f_{i} \\rVert \\leq \\mu n_{1-3}^T f_{i}\\tag{2}\n",
        "\\end{equation} \n",
        "\n",
        "for $i = 4$\n",
        "\\begin{equation} \n",
        "\\lVert t^T_{4}f_{4} \\rVert \\leq \\mu n_{4}^T f_{4}\\tag{3}\n",
        "\\end{equation} \n",
        "\n",
        "The optimazion problem: \n",
        "\\begin{aligned} \n",
        "\\min_{f} \\quad & 0\\\\ \n",
        "\\textrm{s.t.} \\quad & Af = b\\\\ \n",
        "  & \\lVert t^T_{1-3}f_{i} \\rVert \\leq \\mu n_{1-3}^T f_{i} \\\\\n",
        "  & \\lVert t^T_{4}f_{4} \\rVert \\leq \\mu n_{4}^T f_{4}\n",
        "\\end{aligned}"
      ],
      "metadata": {
        "id": "WDopI9GpB3TS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.1 Problem assumption"
      ],
      "metadata": {
        "id": "IYeD4RULD6eH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "m = 2\n",
        "g = np.array([0,0,-9.8])\n",
        "mu = 0.25"
      ],
      "metadata": {
        "id": "V9h-SmTe6U_S"
      },
      "execution_count": 78,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.2 Define the horizontal plane  "
      ],
      "metadata": {
        "id": "iMXwx_i06U_W"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.array([[0],[0],[1]])\n",
        "t = scipy.linalg.null_space(n.T)\n",
        "print(\"plane orthonormal basis:\\n\",t)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "b478a769-b42e-499b-bcca-94dcfee24816",
        "id": "wFC7MAGT6U_Z"
      },
      "execution_count": 79,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "plane orthonormal basis:\n",
            " [[ 0. -1.]\n",
            " [ 1.  0.]\n",
            " [ 0.  0.]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.3 Define feet positions and corresponding skew-symmeric matrices\n",
        "        \n"
      ],
      "metadata": {
        "id": "Up-zza3b6U_d"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "r1 = np.array([0,0,0])\n",
        "r2 = np.array([-4,0,0])\n",
        "r3 = np.array([-4,4,0])\n",
        "r4 = np.array([0,4,4])\n",
        "rc = np.array([-2,2,0])\n",
        "\n",
        "R1 = skew(r1-rc)\n",
        "R2 = skew(r2-rc)\n",
        "R3 = skew(r3-rc)\n",
        "R4 = skew(r4-rc)"
      ],
      "metadata": {
        "id": "G9_EoXgb6U_f"
      },
      "execution_count": 80,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "fig = plt.figure(figsize=(5,5))\n",
        "ax = fig.add_subplot(projection='3d')\n",
        "\n",
        "r = np.vstack([r1, r2, r3])\n",
        "vertices = [list(zip(r[:,0],r[:,1],r[:,2]))]\n",
        "poly = Poly3DCollection(vertices, alpha=0.3)\n",
        "ax.add_collection3d(poly)\n",
        "\n",
        "ax.scatter(xs = r[:,0], ys = r[:,1], zs = r[:,2], alpha=1, label = \"robot legs\")\n",
        "ax.scatter(rc[0],rc[1],rc[2], alpha=1, label = \"center of mass\") \n",
        "ax.scatter(r4[0],r4[1],r4[2], alpha=1, label = \"foot on vertical wall\")\n",
        "\n",
        "ax.set_xlim(-5,5)\n",
        "ax.set_ylim(-5,5)\n",
        "ax.set_zlim(0,5)\n",
        "ax.set_xlabel(\"x-axis\")\n",
        "ax.set_ylabel(\"y-axis\")\n",
        "ax.set_zlabel(\"z-axis\")\n",
        "ax.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 303
        },
        "id": "7VSuyvFI5EIV",
        "outputId": "901439ff-4044-4515-acb6-8826a639b1bc"
      },
      "execution_count": 81,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 360x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.4 Define A, b for the equality constraint"
      ],
      "metadata": {
        "id": "6kvl9oFX6U_h"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "row1 = np.hstack([np.eye(3), np.eye(3), np.eye(3), np.eye(3)])\n",
        "row2 = np.hstack([R1, R2, R3, R4])\n",
        "A = np.vstack([row1, row2])\n",
        "b = np.hstack([-m*g, 0, 0, 0])"
      ],
      "metadata": {
        "id": "B7ZNn-QZ6U_l"
      },
      "execution_count": 82,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 3.5 Problem solution"
      ],
      "metadata": {
        "id": "B90RUv3y6U_m"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "f = cp.Variable(12) #f1_x, f1_y, f1_z .... f4_z\n",
        "obj_function = 0\n",
        "constraint = [A@f==b]\n",
        "\n",
        "for i in range(0,9,3): \n",
        "  constraint += [cp.norm2(t.T@f[i:i+3]) <= mu*n.T@f[i:i+3]]\n",
        "\n",
        "G = np.array([[0,1,0],[0,0,1]])\n",
        "constraint += [cp.norm2(G@f[9:12]) <= mu*np.array([-1,0,0])@f[9:12]]\n",
        "\n",
        "obj = cp.Minimize(obj_function)\n",
        "proj = cp.Problem(obj, constraint)\n",
        "result = proj.solve()\n",
        "\n",
        "print(\"The problem status is\", proj.status)\n",
        "print(\"The optimal value is\", result) # The optimal objective value is returned by `proj.solve()`.\n",
        "print(f\"f1 = {f.value[0:3]}.T\") # The optimal value for f is stored in `f.value`.\n",
        "print(f\"f2 = {f.value[3:6]}.T\")\n",
        "print(f\"f3 = {f.value[6:9]}.T\")\n",
        "print(f\"f4 = {f.value[9:12]}.T\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "75368f04-0fea-4152-effc-3165cfee39bb",
        "id": "fhfIJrhA6U_o"
      },
      "execution_count": 83,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The problem status is optimal\n",
            "The optimal value is 0.0\n",
            "f1 = [ 0.5183811  -0.4807061   7.69696292].T\n",
            "f2 = [0.0282039  0.02531794 2.16891598].T\n",
            "f3 = [1.42528175 0.52126707 9.60295078].T\n",
            "f4 = [-1.97186676 -0.0658789   0.13117032].T\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## TASK4: One foot nailed to the floor\n",
        "\n",
        "Assuming that the feet are standing on a horizontal plane $xy$, then the\n",
        "normal vector $n$ = $\\begin{bmatrix}\n",
        "0\\\\\n",
        "0\\\\\n",
        "1\n",
        "\\end{bmatrix}, t = \\begin{bmatrix}\n",
        "0 & -1\\\\\n",
        "1 & 0\\\\\n",
        "0 & 0\n",
        "\\end{bmatrix} $\n",
        "\n",
        "But the $4^{th}$ leg is nailed to the ground, so no friction cone for this leg.\n",
        "\n",
        "\n",
        "The problem still can be formulated as feasibility problem as follows:\\\n",
        "**Constraints**\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = \n",
        "\\begin{bmatrix} -mg\\\\ 0_{3\\times1}\n",
        "\\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "This can be written as \\begin{equation} Af = b \\tag{1}\\end{equation}\n",
        "\n",
        "and assuming that the reaction forces lie in the friction cone, with friction coefcient $\\mu$ \\\n",
        "for $i = 1,2,3$\n",
        "\\begin{equation} \n",
        "\\lVert t^T_{1-3}f_{i} \\rVert \\leq \\mu n_{1-3}^T f_{i}\\tag{2}\n",
        "\\end{equation} \n",
        "\n",
        "The optimazion problem: \n",
        "\\begin{aligned} \n",
        "\\min_{f} \\quad & 0\\\\ \n",
        "\\textrm{s.t.} \\quad & Af = b\\\\ \n",
        "  & \\lVert t^T_{1-3}f_{i} \\rVert \\leq \\mu n_{1-3}^T f_{i} \n",
        "\\end{aligned}"
      ],
      "metadata": {
        "id": "7Pr9x1yyCfYq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4.1 Problem assumptions "
      ],
      "metadata": {
        "id": "bfDLdd4q7J7q"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "m = 2\n",
        "g = np.array([0,0,-9.8])\n",
        "mu = 0.25"
      ],
      "metadata": {
        "id": "PqjehGo67J7s"
      },
      "execution_count": 22,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4.2 Define the horizontal plane  "
      ],
      "metadata": {
        "id": "9JUw8OOh7J7t"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "n = np.array([[0],[0],[1]])\n",
        "t = scipy.linalg.null_space(n.T)\n",
        "print(\"plane orthonormal basis:\\n\",t)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "44b75da9-ea1a-47d9-e56b-71f3472652b0",
        "id": "DgdGpkLN7J7u"
      },
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "plane orthonormal basis:\n",
            " [[ 0. -1.]\n",
            " [ 1.  0.]\n",
            " [ 0.  0.]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4.3 Define feet positions and corresponding skew-symmeric matrices\n",
        "        \n"
      ],
      "metadata": {
        "id": "NDtnTk6e7J7w"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "r1 = np.array([0,0,0])\n",
        "r2 = np.array([-4,0,0])\n",
        "r3 = np.array([-4,4,0])\n",
        "r4 = np.array([0,4,-1])\n",
        "rc = np.array([-2,2,0])\n",
        "\n",
        "R1 = skew(r1-rc)\n",
        "R2 = skew(r2-rc)\n",
        "R3 = skew(r3-rc)\n",
        "R4 = skew(r4-rc)"
      ],
      "metadata": {
        "id": "MY_je4aI7J7y"
      },
      "execution_count": 24,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "fig = plt.figure(figsize=(5,5))\n",
        "ax = fig.add_subplot(projection='3d')\n",
        "\n",
        "r = np.vstack([r1, r2, r3])\n",
        "vertices = [list(zip(r[:,0],r[:,1],r[:,2]))]\n",
        "poly = Poly3DCollection(vertices, alpha=0.3)\n",
        "ax.add_collection3d(poly)\n",
        "\n",
        "ax.scatter(xs = r[:,0], ys = r[:,1], zs = r[:,2], alpha=1, label = \"robot legs\")\n",
        "ax.scatter(rc[0],rc[1],rc[2], alpha=1, label = \"center of mass\") \n",
        "ax.scatter(r4[0],r4[1],r4[2], alpha=1, label = \"foot nailed on floor\")\n",
        "\n",
        "\n",
        "ax.set_xlim(-5,5)\n",
        "ax.set_ylim(-5,5)\n",
        "ax.set_zlim(0,5)\n",
        "ax.set_xlabel(\"x-axis\")\n",
        "ax.set_ylabel(\"y-axis\")\n",
        "ax.set_zlabel(\"z-axis\")\n",
        "ax.legend()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 303
        },
        "id": "FweceGRK7nQ-",
        "outputId": "4047dca5-daed-470f-aefd-2c1456e9717e"
      },
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 360x360 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4.4 Define A, b for the equality constraint"
      ],
      "metadata": {
        "id": "PS7mXVSS7J71"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "row1 = np.hstack([np.eye(3), np.eye(3), np.eye(3), np.eye(3)])\n",
        "row2 = np.hstack([R1, R2, R3, R4])\n",
        "A = np.vstack([row1, row2])\n",
        "b = np.hstack([-m*g, 0, 0, 0])"
      ],
      "metadata": {
        "id": "Rk02w8Xt7J72"
      },
      "execution_count": 26,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 4.5 Problem solution"
      ],
      "metadata": {
        "id": "LhjQVVcM7J74"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "f = cp.Variable(12) #f1_x, f1_y, f1_z .... f4_z\n",
        "obj_function = 0\n",
        "constraint = [A@f==b]\n",
        "\n",
        "for i in range(0,9,3): \n",
        "  constraint += [cp.norm2(t.T@f[i:i+3]) <= mu*n.T@f[i:i+3]]\n",
        "\n",
        "obj = cp.Minimize(obj_function)\n",
        "proj = cp.Problem(obj, constraint)\n",
        "result = proj.solve()\n",
        "\n",
        "print(\"The problem status is\", proj.status)\n",
        "print(\"The optimal value is\", result) # The optimal objective value is returned by `proj.solve()`.\n",
        "print(f\"f1 = {f.value[0:3]}.T\") # The optimal value for f is stored in `f.value`.\n",
        "print(f\"f2 = {f.value[3:6]}.T\")\n",
        "print(f\"f3 = {f.value[6:9]}.T\")\n",
        "print(f\"f4 = {f.value[9:12]}.T\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "6feb4915-e4ea-4651-dd38-f19427f4d277",
        "id": "uNIiB4dX7J76"
      },
      "execution_count": 27,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The problem status is optimal\n",
            "The optimal value is 0.0\n",
            "f1 = [0.04105438 0.04105438 3.31257993].T\n",
            "f2 = [0.04049262 0.04049262 6.45676973].T\n",
            "f3 = [0.04105438 0.04105437 3.31257993].T\n",
            "f4 = [-0.12260137 -0.12260137  6.51807041].T\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## TASK5: Applied external force\n",
        "\n",
        "Assume the external force is given by $f_{e}$ applied at the center of mass\\\n",
        "Stability equations:\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = -mg+f_{e}\\\\ \n",
        "\\end{equation}\n",
        "\n",
        "There is no torque for this external force as it is at the center\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\\\\ \n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} =  0_{3\\times1}\\\\ \n",
        "\\end{equation}\n",
        "\n",
        "Problem can be formulated as feasibility problem as follows:\\\n",
        "**Constraints**\n",
        "\\begin{equation} \n",
        "\\begin{bmatrix} \n",
        "I_{3\\times3} & I_{3\\times3} & I_{3\\times3} & I_{3\\times3}\\\\ \n",
        "R1_{3\\times3} & R2_{3\\times3} & R3_{3\\times3} & R4_{3\\times3}\n",
        "\\end{bmatrix} \\begin{bmatrix} \n",
        "f1\\\\ f2\\\\ f3\\\\ f4\\\\  \n",
        "\\end{bmatrix} = \n",
        "\\begin{bmatrix} -mg+f_{e}\\\\ 0_{3\\times1}\n",
        "\\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "This can be written as \\begin{equation} Af = b \\tag{1}\\end{equation}\n",
        "This leads to \\begin{equation}f=A^{+}b + Nz \\tag{2}\\end{equation}\n",
        "where $N$ is the $NullSpace(A)$ and $z$ is the coordinates of $f$ in the basis $N$\n",
        "\n",
        "Let's introduce $Q$ matrix such that \\begin{equation}f=QA^{+}(f_{e}-mg) + QNz \\tag{3}\\end{equation}\n",
        "\n",
        "For the friction cone constraint, an approximation is made to represent the cone as a polytope as follows:\n",
        "\\begin{equation} \n",
        "\\lVert t^Tf_{i} \\rVert \\leq \\mu n^T f_{i} \\tag{4}\n",
        "\\end{equation} \n",
        "\n",
        "The approximation is $H-representation$ of a polytope\n",
        "\\begin{equation} Cf_{i} \\leq D \\tag{5} \\end{equation} \n",
        "\n",
        "thus \\begin{equation} c^{T}_{i}f \\leq d_{i} \\tag{6} \\end{equation} \n",
        "\n",
        "substitute eq[3] into eq[6], this leads to \\begin{equation}c^{T}_{i}(QA^{+}(f_{e}-mg) + QNz)  \\leq d_{i} \\tag{7} \\end{equation}\n",
        "\n",
        "\\begin{equation}c^{T}_{i}QA^{+}f_{e} + c^{T}_{i}QNz  \\leq d_{i} +c^{T}_{i}QA^{+}mg \\tag{8} \\end{equation} \\\n",
        "\n",
        "Let $h = d_{i} +c^{T}_{i}QA^{+}mg$, leads to \n",
        "\\begin{equation}c^{T}_{i}QA^{+}f_{e} + c^{T}_{i}QNz  \\leq h \\tag{9} \\end{equation} \\\n",
        "\n",
        "Given $\\lVert Mf_{e}+f_{e,0} \\rVert \\leq 1$ where M is positive definite matrix, and let $y=Mf_{e}+f_{e,0}$, so\n",
        "\n",
        "\\begin{equation}\\lVert y \\rVert \\leq 1 \\tag{10} \\end{equation} \\\n",
        "Then\n",
        "\\begin{equation}\\ f_{e} = M^{-1}(y-f_{e,0}) \\tag{11} \\end{equation} \\\n",
        "\n",
        "substitute eq[11] into eq[9], we get the following:\n",
        "\\begin{equation}\\ c^{T}_{i}QA^{+}M^{-1}y + c^{T}_{i}QNz  \\leq h + c^{T}_{i}QA^{+}M^{-1}f_{e,0} \\tag{12} \\end{equation} \\\n",
        "\n",
        "It is clear that worst-case scenario corresponds to the largest\n",
        "value of $c^{T}_{i}QA^{+}M^{-1}y$, meaning that $y$ should align with $c^{T}_{i}QA^{+}M^{-1}$ and have its maximum possible length $1$. From that we conclude that\n",
        "\n",
        "\\begin{equation}\\ y = \\frac{c^{T}_{i}QA^{+}M^{-1}}{\\lVert c^{T}_{i}QA^{+}M^{-1} \\rVert} \\tag{13} \\end{equation} \\\n",
        "\n",
        "substitute eq[13] into eq[12] leads to\n",
        "\\begin{equation}\\ \\lVert c^{T}_{i}QA^{+}M^{-1} \\rVert+ c^{T}_{i}QNz  \\leq h + c^{T}_{i}QA^{+}M^{-1}f_{e,0} \\tag{14} \\end{equation} \\\n",
        "\n",
        "Finally, the constraint can be written as linear constraint in the form of \n",
        "\\begin{equation}\\ c^{T}_{i}QNz  \\leq h + c^{T}_{i}QA^{+}M^{-1}f_{e,0} - \\lVert c^{T}_{i}QA^{+}M^{-1} \\rVert \\tag{15} \\end{equation} \\\n",
        "\n",
        "The optimazion problem: \n",
        "\\begin{aligned} \n",
        "\\min_{f} \\quad & 0\\\\ \n",
        "\\textrm{s.t.} \\quad & Af = b\\\\ \n",
        "  & c^{T}_{i}QNz  \\leq h + c^{T}_{i}QA^{+}M^{-1}f_{e,0} - \\lVert c^{T}_{i}QA^{+}M^{-1} \\rVert\n",
        "\\end{aligned}"
      ],
      "metadata": {
        "id": "5qUVng5vkw0C"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Appendix"
      ],
      "metadata": {
        "id": "Upq2qJtx8XA1"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Normal vector to a tilted plane\n",
        "n = np.array([[0],[0],[1]]) + np.random.randn(3,1)\n",
        "n = n / np.sqrt(np.sum(n**2))\n",
        "print(\"normal vector: \\nn = \", n.T)\n",
        "\n",
        "# Tangential vectors (vectors in the plane)\n",
        "x = y = 1\n",
        "z = (-n[0][0]*x - n[1][0]*y)/ n[2][0]\n",
        "t1 = np.array([[x],[y],[z]])\n",
        "t1 = t1 / np.sqrt(np.sum(t1**2))\n",
        "print(\"\\ntangential vector1: \\nt1 = \", t1.T)\n",
        "\n",
        "x, y = 1, 2\n",
        "z = (-n[0][0]*x - n[1][0]*y)/ n[2][0]\n",
        "t2 = np.array([[x],[y],[z]])\n",
        "t2 = t2 / np.sqrt(np.sum(t2**2))\n",
        "print(\"\\ntangential vector2: \\nt2 = \", t2.T)\n",
        "\n",
        "# Check dot product = 0\n",
        "print(\"\\ndot product(n.T, t1) = \", np.dot(n.T, t1)[0][0])\n",
        "print(\"dot product(n.T, t2) = \", np.dot(n.T, t2)[0][0])\n",
        "\n",
        "# Tilted plane orthonormal basis  \n",
        "t2 = t2 - t1*((np.dot(t1.T, t2) / np.dot(t1.T, t1))[0][0])\n",
        "print(\"dot product(t1.T, t2) = \", np.dot(t1.T, t2)[0][0])\n",
        "\n",
        "t = np.hstack([t1,t2])\n",
        "print(\"\\nplane orthonormal basis:\\n\",t)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "gQa8KcNs8Yuf",
        "outputId": "14938331-4700-46ed-ac33-21eea08a818e"
      },
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "normal vector: \n",
            "n =  [[-0.28628025 -0.67764315  0.67737979]]\n",
            "\n",
            "tangential vector1: \n",
            "t1 =  [[0.49844607 0.49844607 0.70929756]]\n",
            "\n",
            "tangential vector2: \n",
            "t2 =  [[0.30326852 0.60653703 0.73494288]]\n",
            "\n",
            "dot product(n.T, t1) =  -5.551115123125783e-17\n",
            "dot product(n.T, t2) =  5.551115123125783e-17\n",
            "dot product(t1.T, t2) =  -9.71445146547012e-17\n",
            "\n",
            "plane orthonormal basis:\n",
            " [[ 0.49844607 -0.18260784]\n",
            " [ 0.49844607  0.12066068]\n",
            " [ 0.70929756  0.04353225]]\n"
          ]
        }
      ]
    }
  ]
}