HW04.ipynb

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    "# Homework 04\n",
    "\n",
    "1.  Implement the simplest walking model and find a periodic gait. The\n",
    "    file `simplestwalkingstep` provides equations of motion, event\n",
    "    detection, the step-to-step transition, and even sample animation\n",
    "    code.\n",
    "  1.  Produce a simulation similar to the rimless wheel. Use parameter\n",
    "    values alpha = 0.3, gamma = 0.03 (3% downward slope), and try\n",
    "    initial guess $x = [0.3 -0.3 -0.3 -0.25]$.\n",
    "  2. Add a test for energy conservation, similar to the rimless wheel.\n",
    "  3.  Using the initial guess, try to find a periodic motion (limit cycle)\n",
    "    that includes stance phase ending at heelstrike, and the\n",
    "    step-to-step transition to yield the new state.<br/><br/>\n",
    "    \n",
    "2.  <img src=\"simplestwalkingimage1.png\" align=\"right\" width=\"20%\"/>Use the embedded constraint method to\n",
    "    produce a drop-in replacement `fsimpwalk2Jp` for the equations of motion for the\n",
    "    simplest walking model, which are provided in symbolic form in the\n",
    "    sample code `testsimpwalk2eom`. The foot O is attached to the ground origin with a\n",
    "    hinge joint, and the swing leg also rotates about the pelvis P with\n",
    "    a hinge joint. Both legs are of length L, and have orientations\n",
    "    measured counter-clockwise with respect to vertical. The\n",
    "    corresponding pose Jacobian looks like this:\n",
    "$$J_p = \\begin{bmatrix}\n",
    "-L\\cos q_{1} & 0 \\\\\n",
    "\\ldots & 0 \\\\\n",
    "1 & 0 \\\\\n",
    "\\ldots & \\ldots \\\\\n",
    "-L\\sin q_{1} & L\\sin q_{2} \\\\\n",
    "0 & 1 \\\\\n",
    "\\end{bmatrix}$$\n",
    "The only masses are point masses M and m; the legs have no mass or\n",
    "inertia.\n",
    "  1.  Produce a state-derivative function `fsimpwalk2Jp`, which should have identical\n",
    "    inputs and outputs to the sample `fsimpwalk2m`. Demonstrate it by\n",
    "    comparing one-step simulations against the symbolic function. Look in `testsimpwalk2eom` for places where you should enter your own code.\n",
    "  2.  Test for your function against the provided `fsimpwalk2m`, as performed in `testsimpwalk2eom`.<br/><br/>\n",
    "\n",
    "3.  <img src=\"simplestwalkingimage2.png\" align=\"right\" width=\"20%\"/>Use the explicit constraint method to produce a drop-in replacement `fsimpwalk2Jc`\n",
    "    for the equations of motion for the simplest walking model. Here the\n",
    "    segments should be treated as separate, but then require a\n",
    "    constraint Jacobian $J_c$ (also called $C$ in lecture) to enforce the joints. The matrix should look\n",
    "    like this:\n",
    "$$J_c = \\begin{bmatrix}\n",
    "1 & 0 & L\\cos q_{1} & 0 & 0 & 0 \\\\\n",
    "0 & 1 & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n",
    "-1 & 0 & 0 & 1 & - L\\cos q_{2} & 0 \\\\\n",
    "\\ldots & \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n",
    "\\end{bmatrix}$$\n",
    "(Note that the centers of mass are located at the ends of the segments,\n",
    "and the constraints describe how the segment ends are connected\n",
    "together.)\n",
    "  1.  Produce a state-derivative function `fsimpwalk2Jc`, which should have identical\n",
    "    inputs and outputs to the sample `fsimpwalk2m`. Demonstrate it by\n",
    "    comparing one-step simulations against the symbolic function.\n",
    "  2.  When using the constraint Jacobian $J_c$ method, it is helpful to also compute the pose Jacobian $J_p$. If these two quantities are consistent with each other, the product $J_c\\cdot$J_p$ should be zero. Compute the product and verify that this is true. Also explain briefly why this should be the case.\n",
    "  3.  Test for your function against the provided `fsimpwalk2m`, as performed in `testsimpwalk2eom`.<br/><br/>\n",
    "  \n",
    "4. The problems above are challenging, and deserve additional clues or clarifications. Please file GitHub issues to ask for hints or corrections."
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