Lie cheatsheet

A 4×4 homogeneous transformation matrix represents a rigid-body configuration. A rotation matrix $R \in SO(3)$ represents orientation. A six-dimensional twist describes spatial velocity. A six-dimensional wrench represents spatial force. The table uses the rigid-body motion notation of Lynch and Park (Kevin M. Lynch, 2017). $SO(n)$ and $SE(n)$ are Lie groups. They are the curved spaces of configurations. Their Lie algebras, $\mathfrak{so}(n)$ and $\mathfrak{se}(n)$, are flat tangent spaces at the identity that contain velocities. The matrix exponential maps a velocity to a configuration: $\exp: \mathfrak{so}(3) \to SO(3)$ and $\exp: \mathfrak{se}(3) \to SE(3)$. The matrix logarithm maps a configuration to a velocity that produces it in unit time: $\log: SO(3) \to \mathfrak{so}(3)$ and $\log: SE(3) \to \mathfrak{se}(3)$. The exponential is many-to-one, so a rotation has many logarithms. In the plane, the angles $\theta$ and $\theta + 2\pi$ give the same rotation. Choosing the principal angle in $(-\pi, \pi]$ makes the planar logarithm a function. On that branch, it inverts the exponential. In three dimensions, restricting the rotation angle to $[0, \pi]$ does not make the logarithm unique. A rotation by exactly $\pi$ has two logarithms because the axes $\hat{\omega}$ and $-\hat{\omega}$ give the same rotation. The logarithm is a single-valued inverse for angles below $\pi$. A rotation by $\pi$ requires a sign convention for the axis. The $[\cdot]$ operator maps a vector in $\mathbb{R}^3$ or $\mathbb{R}^6$ to its matrix form in $\mathfrak{so}(3)$ or $\mathfrak{se}(3)$.

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Rotations and rigid-body motions side by side

Lynch and Park summarize chapter 3 in four tables (Kevin M. Lynch, 2017). Each row pairs a fact about rotation with its rigid-body counterpart.

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References

• Kevin M. Lynch, F. (2017). Modern Robotics.

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