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Kinematics
Jehee LeeSeoul National University
Kinematics
• How to animate skeletons (articulated figures)
• Kinematics is the study of motion without regard to the forces that caused it
Hierarchical Models
• Tree structure of joints and links– The root link can be chosen arbitrarily
• Joints– Revolute (hinge) joint allows rotation about a fixed axis– Prismatic joint allows translation along a line– Ball-and-socket joint allows rotation about an arbitrary axis
Human Joints• Human joints are actually much more complicated
Forward and Inverse Kinematics
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)F(),( iq=qp ),(F 1 qp-=iqForward Kinematics Inverse Kinematics
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Forward Kinematics: A Simple Example
• A simple robot arm in 2-dimensional space– 2 revolute joints– Joint angles are known– Compute the position of the end-effector
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3D Position and Orientation
• The position and orientation of an object is represented as a rigid transformation
• Vector & 3x3 Matrix
• 4x4 Matrix
• Vector & Quaternion
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Forward Kinematics: A Simple Example
• Forward kinematics map as a coordinate transformation– The body local coordinate system of the end-effector was initially
coincide with the global coordinate system– Forward kinematics map transforms the position and orientation
of the end-effector according to joint angles
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Thinking of Transformations
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Thinking of Transformations• In a view of body-attached coordinate system
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Thinking of Transformations
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Thinking of Transformations
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How to Handle Ball-and-Socket Joints ?
• Three revolute joints whose axes intersect at a point (equivalent to Euler angles), or
• 3D rotation about an arbitrary axis
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Floating Base
• The position and orientation of the root segment are added
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Joint & Link Transformations• Each segment has its own coordinate frame• Forward kinematics map is an alternating multiple of
– Joint transformations : represents joint movement– Link transformations : defines a frame relative to its parent
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The position and orientation of the root segment
1st link transformation
1st joint transformation
Joint & Link Transformations
• Both are rigid transformations in general– Joint transformations may include translation
• Human joints are not ideal hinges– Link transformations may include rotation
• Some links are twisted
Denavit-Hartenberg Notation
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about measured and between angle the along measured to from distance the
about measured and between angle the along measured to from distance the
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Link Transformations• How do you compute the link transform for the ith joint if
you know the position and orientation of the ith joint as well as its parent’s position and orientation at the neutral pose ?
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Representing Hierarchical Models• A tree structure
– A node contains a joint transformation– A arc contains a link transformation
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Depth-First Tree Traversal• Draw graphics needs to compute the position and
orientation of all links• OpenGL’s matrix stack is useful
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Summary• Kinematics is the study of motion of articulated figures
– Kinematics does not consider physics (forces, mass, …)
• Forward kinematics is straightforward– Forward kinematics map can be considered as a coordinate
transformation
• The next lecture: How to solve the inverse problem
Programming Assignment #1• Create a tree-structured hierarchical model of human
figures– You have to compute the forward kinematics map of your model– You may use a matrix stack to build the forward kinematics map
incrementally
• The model may be displayed using three-dimensional primitives such as polygons, boxes, cylinders, and spheres
• The joints are either revolute or ball-and-sockets
Programming Assignment #1• Animate the model to demonstrate the hierarchical
structure– For example, use sin/cos functions to make it walk and run
• Implement a scene 3D viewer– You should be able to change the viewpoint (scene
rotation/translation)– The viewer will be used in all the other programming assignment
• Submit a report of at most three A4 pages– Describe the tree-structure, joints, and geometric primitives– Describe how the joints are animated
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