COMS W4160— Computer Graphics Spring 2017
Programming Assignment 5
out: April 22, Saturday
due: May. 12, Friday, 10:00AM. Note: the due time is in the morning!
This assignment is about forward and inverse kinematics for computer animation. It is an assignment
that we have decided to merge with the final project. Thus, it has a larger scope than the previous
programming assignments. You are expected to code more, and of course this assignment will be weighted
more in terms of your final grade.
1 Skeleton Structure
A skeleton has a tree structure, as illustrated in Figure 1. The skeleton consists of links connected by
revolute joints. There is always a root link. One end of the root link might be fixed by a revolute joint
(refer to Joint2D.java). Each link may connect with one or more child links. Note that a joint always
connects one parent link with one child link (1:1). If a link is connected with multiple child links, then
there are multiple joints, one connection for each child link (refer to LinkConnection2D.java). If one end of
a link is free, that end is referred as an end effector (see red circles in Figure 1).
In an inverse kinematic system, the user should be able to select an end effector and drag it around, and
your program needs to keep updating the angles of revolute joints to move the end effector to the desired
position.
2 Starter Code
Our starter code is built on the code of programming assignment 1. The files that you need to finish with
your code are under the c2g2/kinematics folder.
Starter code Structure. We provide a starter code framework for 2D skeletons. The skeleton is main-
tained in the class of Skeleton2D. A skeleton consists of a set of RigidLink2D instances organized in a tree
structure. Each link might be connected with multiple child links. Each pair of parent/child links are
connected through a revolute joint (RevoluteJoint2D), and such a connection is maintained in the class of
LinkConnection2D.
Skeleton specification. We provide a simple XML parser to specify a skeleton structure (see Skele-
ton2D.java). An example of the skeleton XML file is shown here:
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COMS W4160— Computer Graphics Spring 2017
Revolute Joint
End Effector
Root Link
Figure 1: Skeleton structure.
Here the root link is specified by its two end positions. The second end position is connected with child
links. All the child links are specified by the positions of joints. The other end of each child link is connected
to its parent link. You can visualize an example skeleton by running the starter code directly (see the class
c2g2.kinematics.engine.Main).
3 Programming Requirements
Your implementation consists of the following components: (i) forward kinematics on a 2D skeleton, (ii)
inverse kinematics on a 2D skeleton, (iii) forward kinematics in 3D, and (iv) setting keyframes using inverse
kinematics and creating animations using Cubic Bézier interpolation.
3.1 Forward Kinematics in 2D
You have two sub-tasks in this step. (i) You need to create a user interface (e.g., by pressing certain
keys or adjusting slide bars) to set the rotational angles of all joints. You can bind any keys or design
your user interface for updating the joint angles. Please refer to the starter code of previous assignments
(e.g., PA-4) for the codes of binding specific keys. (ii) Implement the forward kinematics algorithm (in
c2g2/kinematics/ForwardKinematics.java) that we discussed in class to update the positions of the joints.
In general, you need to travel through the tree structure of the skeleton, and concatenate transformation
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COMS W4160— Computer Graphics Spring 2017
matrices brought on by each revolute joint. You need to repeat the tree traversal and update the position
of each joint.
Every time you update the parameters of the joints, make sure that the OpenGL visualization re-renders
the current skeleton structure. Please refer to the starter code for the OpenGL visualization, and feel free
to improve it.
3.2 Inverse Kinematics in 2D
You have two sub-tasks in this step. (i) You need to follow the starter code and allow the user to choose
an end effector of the skeleton through mouse clicking on the end effector. Then, the user should be able
to drag the mouse and thereby specify the desired end effector position in a time continuous fashion. To
this end, you need to travel through the skeleton structure, find all the end effectors, and identify the
end effector over which the mouse is currently dragging. (ii) Implement the inverse kinematics algorithm
(in c2g2/kinematics/InverseKinematics.java) that we discussed in class to find joint angles that realize the
desired end effector position. Then, you need to update the skeleton structure with the resulting joint
angles, and render it in OpenGL visualization. You should perform the inverse kinematic solve when the
user is dragging the end effector, so the structure is updated continuously.
Solving a linear system. In the inverse kinematic algorithm, you need to solve linear systems (recall the
derivation in class and slides of lecture-20 on CourseWorks) during Newton’s iterations. There exist many
Java libraries for solving a linear system. For example, we recommend the Efficient Java Matrix Library
(EJML). You are allowed to choose other libraries, but make sure that all the external libraries are included
in your submission. It is your responsibility to make sure that your code can be compiled successfully when
the TAs grade it.
3.3 Forward Kinematics in 3D
Next, you need to implement a forward kinematics algorithm for 3D. We don’t provide starter code for a
3D skeleton. You will need to write your own code by augmenting the 2D code. In 3D, you can use revolute
joints to connect 3D links. Unlike the joints in 2D skeletons, those 3D revolute joints have revolution axis
which might point along an arbitrary direction. You need to take into account the revolution axis of each
joint when constructing the transformation matrices, as they might point in different directions.
Similar to the 2D forward kinematics, you need to implement a user interface to set the rotational angle
of each joint. You also need to implement a simple 3D visualization to display the 3D skeleton on the
screen.
3.4 Creating Character Animation by Interpolating Keyframes
Your last task is to create a skeleton animation by interpolating keyframes. You can set up key frames via
inverse kinematics. For example, when setting up a key frame, the user can drag the end effectors to pose
the skeleton. You need to implement a user interface to select a time frame and use inverse kinematics to
specify the pose at that time frame. Then, you need to use the Cubic Bézier curve interpolation to construct
an animation. You can implement this simple character animation system in 2D (using 2D forward and
inverse kinematics), although we will give (10%) bonus points if you implement this in 3D.
Each animation should include at least five key frames. The total time length should be at least 4
seconds (assuming the animation has 30 frames per second). In your report, for each animation you submit
(see below), please include a screen capture of each of the key frame poses and clearly label them.
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https://courseworks2.columbia.edu/courses/26680/files/folder/slides?preview=1188853
http://ejml.org
COMS W4160— Computer Graphics Spring 2017
4 Submission
Submission Checklist: Submit your assignment as a zip file via CourseWorks. Your submission must
consist of the following parts:
1. Documented code: Your code should be reasonably documented and be readable/understandable
by the TAs. If the TAs are not able to run and use your program, they will be unable to grade it. To
ensure the TAs can grade your assignment, please make sure your code can be compiled successfully.
It is your responsibility to make sure the code can be correctly compiled. Otherwise, you’ll lose some
points.
2. Skeleton file: You need to create your own skeletons. Please include the XML files of your skeletons
in your submission.
3. Video demos: You need to include five videos, each has a length less than 15 seconds: (1) a video of
screen capture demonstrating the 2D forward kinematics; (2) a video of screen capture demonstrating
the 2D inverse kinematics; (3) a video demonstrating 3D forward kinematics; (4) a video demon-
strating the interpolated skeleton animation. In this demo, you should use only a chain of links (with
at least 5 links, no branches) to demonstrate the inverse kinematics (to set up key frames) and the
interpolated animation; and (5) another video demonstrating the interpolated skeleton animation. In
this demo, you should use more complex skeletons with branches (i.g., a character-like skeleton) to
demonstrate the inverse kinematics (to set up key frames) and the interpolated animation.
4. Brief report: Include a description of what youve attempt, special features you have implemented,
and any instructions on how to run/use your program. Please also include the skeleton key
frames used in your video (4) and (5). In compliance with Columbia’s Code of Academic
Integrity, please include references to any external sources or discussions that were used to achieve
your results.
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https://courseworks.columbia.edu/welcome/
Skeleton Structure
Starter Code
Programming Requirements
Forward Kinematics in 2D
Inverse Kinematics in 2D
Forward Kinematics in 3D
Creating Character Animation by Interpolating Keyframes
Submission