Overview: The goal of this assignment is to derive the set of kinematic equations for the joint hierarchy (i.e. skeleton) shown below that will be animated in subsequent assignments using a variety of techniques (motion capture, keyframing, IK and dynamic simulation).
<aside> 💡 (10 points) In Homework $\#4$ you will build an animation engine capable of computing the forward kinematics for the skeleton shown above.
In order to do this a data structure will need to be created called AJoint that stores the following information for each joint:
Assume the pelvis (joint 1) is the root joint of the skeleton shown above
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💡 2. list the AJoint information for each joint ($i=6$ to $9$) with the pelvis as the root.
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<aside> 💡 (25 points) Let $\mathbf{F}_j^i$ **be the frame of reference at **each joint **$j$ with respect to joint $i$, where $\mathbf{F}_j^i$ is a $4 \times 4$ homogeneous transformation matrix of the form:
$$ \mathbf{F}_j^i=\left[\begin{array}{cc} \mathbf{R}j^i\left(\boldsymbol{\theta}j\right) & \mathbf{d}{i j}^i \\ \mathbf{0} & 1 \end{array}\right] $$
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(a) Assume each joint is oriented by first rotating about the local z-axis, then about the local $y$-axis, then about the local $x$-axis (i.e. rotation order of $z->y->x$ ),
<aside> 💡 1. What is the general form of the corresponding rotation matrix $\mathbf{R}j^i\left(\boldsymbol{\theta}j\right)$ **for joint $j$ in terms of $*\theta{\mathrm{jx}}, \theta{\mathrm{jy}}, \theta_{\mathrm{jz}}*$, where $i$ is the parent joint's frame of reference?
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<aside> 💡 2. What are the joint displacement vectors $\mathbf{d}_{i j}^i$ (bones) for $j=6$ to $9, i=1,6$ to $8$ .
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<aside> 💡 3. What is the form of the corresponding Euler Angle rate to angular velocity matrix $\mathbf{L}_6\left(\boldsymbol{\theta}_6\right)$ for joint 6 .
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<aside> 💡 4. If the order of rotation for joint 6 was changed to $\mathrm{x}->\mathrm{y}->\mathrm{z}$, what would the Euler Angle rate to angular velocity matrix $\mathbf{L}_6\left(\boldsymbol{\theta}_6\right)$ look like then?
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