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Robots can move themselves, as cars and humanoids do (egomotion), or move objects around them (manipulation). Describing either kind of motion requires representations of position, orientation and velocity. The useful representation depends on the robot and its field. A self-driving car, a humanoid and an aircraft may describe the same motion differently. This chapter develops these representations and applies them to kinematics and dynamics.

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Motion algebras

Motion algebras provide consistent representations of position, orientation and velocity. Lie theory connects these representations.

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, rotation matrices, angular velocities and exponential coordinates in 3-D, with the plane as a special case.

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.

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Kinematics

Kinematics describes motion without considering its causes. From a video of a tennis match, for example, you can reconstruct the ball’s path without knowing its mass or the forces of gravity and drag. A wheeled robot’s kinematic model relates its linear and angular velocities to changes in its pose (x,y,θ)(x, y, \theta). For a robot arm, the model determines the end-effector pose from joint angles and their rates without considering the torques that cause the motion.

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.

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Dynamics

Dynamics relates motion to its causes, including masses, forces and torques.

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.

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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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