updating notebook (#364)

book/trajectories/exercises/door_opening.ipynb

@@ -265,7 +265,7 @@
     "\n",
     "In the lecture, we've seen an example of a robot manipulating a cylindrical object in the DARPA Robotics Challenge. By using a similar optimization-based inverse kinematics, We will solve a series of IK problems that can open a cupboard door.\n",
     "\n",
-    "If you open up Meshcat at the link you got by running the first cell, you should be able to see that we provided you with setpoints ${}^W X^G(t)$ in the end-effector space, which gives us a nominal trajectory to reach and open the cupboard, which was computed by dividing the motion into following segments:\n",
+    "If you open up Meshcat at the link you got by running the first cell, you should be able to see that we provided you with setpoints ${}^W X^G(t)$ in the end-effector space, which gives us a nominal trajectory to reach and open the cupboard. \"Nominal\" refers to an ideal or baseline quantity. This trajectory was computed by dividing the motion into following segments:\n",
     "\n",
     "- $0 \\leq t \\leq 5$: The end-effector linearly interpolates between the initial pose at $t=0$, and the nominal grasp pose required to grip the cylinder (${}^W\\mathbf{X}^H$ at $t=5$), while having the gripper open.\n",
     "- $5 \\leq t \\leq 6$: The end-effector stays still at ${}^W\\mathbf{X}^H$, and the gripper is closed.\n",
@@ -305,7 +305,7 @@
    "source": [
     "**Part a** [3pts]: Let us parametrize ${}^H X ^G$ with six degrees of freedom: xyz positions and roll-pitch-yaw rotation (i.e. rotations around the xyz axii, respectively) of frame $G$ with respect to frame $H$. Which three would you constrain to be exactly equal, and which three are you willing to provide some slack? Explain your reasoning.\n",
     "\n",
-    "(HINT: you should constrain two positions and one orientation).\n",
+    "(HINT: you should constrain two positions and one orientation. Also remember the x-y-z axes are represented in red-green-blue colours respectively.)\n",
     "\n",
     "**Part b** [2pts]: Write down an optimization problem to find $q$ with a joint centering cost on $q_{nom}$, and the constraints you wrote down in Part a. You should write an equality constraint for the constrained degrees of freedom, and inequality constraints for the unconstrained one. Writing a constraint on some principal axis of orientation is fairly involved due to rotation formalisms, so in this problem you may full constrain the orientation degrees of freedom (i.e. using equality constraints).\n",
     "\n",