Motion algebras
Motion algebras provide consistent representations of position, orientation and velocity. Lie theory connects these representations. The motion algebras overview introduces the chapter.Configuration space
Describe robot configurations using degrees of freedom, joint spaces, and coordinates.
Lie I: groups
Treat rigid-body configuration spaces as Lie groups, then define tangent spaces, Exp and Log.
Rotation representations
Describe frames and rotation matrices in the plane and in 3-D. Use a rotation matrix to represent, re-express and move a frame.
Angular velocity
Derive angular velocity in the plane and in space, and write it as a skew-symmetric matrix in so(3).
Exponential coordinates
Turn a constant angular velocity into a finite rotation with the matrix exponential, Rodrigues’ formula and the matrix logarithm.
SO(2) as a manifold
Examine the simplest Lie group through its tangent space and exponential map.
Motion representations
Describe poses, twists and screw motions in 3-D and in the plane. Use the exponential map to turn a twist into motion.
Homogeneous coordinates
Represent points and transformations in projective space.
Quaternions
Build unit quaternions from complex numbers and relate them to the double cover of SO(3).
Lie II: calculus
Apply plus and minus operations, the adjoint and Jacobians. Connect them to uncertainty and the formulas in Modern Robotics.
Lie cheatsheet
Look up the elements, dimension and meaning of each Lie group and Lie algebra used for rigid-body motion.
Kinematics
Kinematics describes motion without considering its causes. It uses the motion algebras to relate a wheeled robot’s velocities to its pose, and an arm’s joint angles to the pose of its end effector. The kinematics overview introduces the chapter.Lie III: applications
Connect Lie theory to software libraries and estimation on manifolds, then compare the product of exponentials with Denavit-Hartenberg.
Wheeled robots
Develop kinematic models for wheeled mobile robots using chapter 13 of Lynch and Park.
AR4 arm kinematics
Follow a six-axis example from Denavit-Hartenberg parameters through forward and inverse kinematics to motor steps.
AR4 kinematics lab
Build FK and IK from the as-built URDF, then reconcile the results with the DH model.
Dynamics
Dynamics relates motion to its causes, including masses, forces and torques. The dynamics overview introduces the chapter.Rigid-body dynamics
Form equations of motion with twists and wrenches. Content is planned for December 2026.
Inverse dynamics
Compute the joint torques for a desired motion with the recursive Newton-Euler algorithm. Content is planned for December 2026.
Forward dynamics
Determine the motion produced by given joint torques and simulate the result. Content is planned for December 2026.
Constrained dynamics
Form equations of motion under Pfaffian constraints. Content is planned for December 2026.
Systems integration
A simulator lets you test a robot’s software against a modeled world before it runs on hardware. These sections build Gazebo worlds and connect them to perception and mapping through ROS.Simulation
Build and extend Gazebo simulation worlds for embodied AI.
Gazebo worlds
Read a world file, the SDF and Xacro formats, and a residential environment with 68 models.
Authoring worlds with LLMs
Generate SDF and Xacro worlds and 3-D models from natural language, then validate them.
Object detection
Run a pose-gated YOLOv8 and CLIP detection pipeline over Zenoh and store the results in pgvector.
Semantic graphs
Build a semantic property graph from detections, cluster places, and re-localize from landmarks.
References
- Modern Robotics Videos (Northwestern University), companion video lectures to Kevin Lynch and Frank Park’s Modern Robotics: Mechanics, Planning, and Control.
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