> ## Documentation Index > Fetch the complete documentation index at: https://aegean.ai/llms.txt > Use this file to discover all available pages before exploring further. # Representing rigid motion > Configuration spaces, rotations, angular velocity, poses and twists, and the Lie theory that connects them. Motion algebras provide consistent representations of position, orientation and velocity. Lie theory connects these representations. The chapter starts from the configuration space of a robot and treats it as a Lie group. It then builds rotations, angular velocity and exponential coordinates, extends them to poses, twists and screw motions, and ends with calculus on Lie groups. The [kinematics](/book/kinematics/index) and [dynamics](/book/dynamics/index) chapters build on these representations. Describe robot configurations using degrees of freedom, joint spaces, and coordinates. Treat rigid-body configuration spaces as Lie groups, then define tangent spaces, Exp and Log. Describe frames and rotation matrices in the plane and in 3-D. Use a rotation matrix to represent, re-express and move a frame. Derive angular velocity in the plane and in space, and write it as a skew-symmetric matrix in so(3). Turn a constant angular velocity into a finite rotation with the matrix exponential, Rodrigues' formula and the matrix logarithm. Examine the simplest Lie group through its tangent space and exponential map. Describe poses, twists and screw motions in 3-D and in the plane. Use the exponential map to turn a twist into motion. Represent points and transformations in projective space. Build unit quaternions from complex numbers and relate them to the double cover of SO(3). Apply plus and minus operations, the adjoint and Jacobians. Connect them to uncertainty and the formulas in Modern Robotics. Look up the elements, dimension and meaning of each Lie group and Lie algebra used for rigid-body motion. ## References * [Motion in a plane](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane) (Khan Academy). If your vectors or circular motion are rusty, watch these short videos before the rotation and angular velocity pages. The unit's projectile motion videos are not needed here. * Vectors: [adding and subtracting vectors](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/x7183bffa9768c609:scalars-and-vectors/v/visually-adding-and-subtracting-vectors) and [unit vector notation](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/in-in-phy-review-unit-vectors/v/unit-vector-notation). * Products of vectors: [the dot product](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/x7183bffa9768c609:scalar-and-vector-product-of-vectors/v/the-dot-product), [cross product 1](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/x7183bffa9768c609:scalar-and-vector-product-of-vectors/v/cross-product-1) and [cross product 2](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/x7183bffa9768c609:scalar-and-vector-product-of-vectors/v/cross-product-2). * Circular motion: [angular motion variables](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/uniform-circular-motion-introduction/v/angular-motion-variables), [arc length from angular displacement](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/uniform-circular-motion-introduction/v/distance-or-arc-length-from-angular-displacement), [angular velocity and speed](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/uniform-circular-motion-introduction/v/angular-velocity-and-speed) and [relating angular and regular motion variables](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/in-in-class11-centripetal-acceleration/v/relating-angular-and-regular-motion-variables). * Centripetal acceleration: [deriving it from angular velocity](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/in-in-class11-centripetal-acceleration/v/deriving-formula-for-centripetal-acceleration-from-angular-velocity-in-accent), [a visual explanation](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/in-in-class11-centripetal-acceleration/v/visual-understanding-of-centripetal-acceleration-formula) and [a calculus proof](https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/in-in-class11-centripetal-acceleration/v/calculus-proof-of-centripetal-acceleration-formula-in-accent). *** Found an error on this page? 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