Lie groups and tangent spaces

Configuration space described the shapes of robot configuration spaces. A revolute joint lies on a circle $S^1$, a two-link arm on a torus $T^2$, and a wheeled chassis on $\mathbb{R}^2 \times S^1$. These spaces are curved, so any coordinates you put on them have a singularity somewhere. You met one at the poles of the sphere, where a tiny step can change the longitude by a large angle. Rigid-body configuration space has additional structure because its elements are motions that you can compose and invert. Rotations form the manifold of 3x3 rotation matrices, called the special orthogonal group $SO(3)$. Rigid motions form the special Euclidean manifold $SE(3)$. Lie theory defines derivatives, integrals and probability distributions on these curved spaces. You can differentiate a pose, integrate a velocity and represent uncertainty in an orientation without approximations that fail for large motions. A space that is both a group and a smooth manifold is called a Lie group.

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The group

Following Solà et al., let $\mathcal{X}$ denote an element of a Lie group $\mathcal{M}$, $\circ$ denote composition and $\mathcal{E}$ denote the identity. For rotations and poses, $\circ$ is matrix multiplication. In that setting, $\mathcal{E}$ is the identity matrix and $\mathcal{X}$ is a matrix such as $R$ or $T$. A group is a set with a composition operation that obeys four axioms (Solà eqs. 1 to 4): $$\begin{aligned} &\text{closure:} && \mathcal{X} \circ \mathcal{Y} \in \mathcal{M} \\ &\text{identity:} && \mathcal{E} \circ \mathcal{X} = \mathcal{X} \circ \mathcal{E} = \mathcal{X} \\ &\text{inverse:} && \mathcal{X}^{-1} \circ \mathcal{X} = \mathcal{X} \circ \mathcal{X}^{-1} = \mathcal{E} \\ &\text{associativity:} && (\mathcal{X} \circ \mathcal{Y}) \circ \mathcal{Z} = \mathcal{X} \circ (\mathcal{Y} \circ \mathcal{Z}) \end{aligned}$$ The identity represents no transformation. It is the group’s neutral element, analogous to $0$ under addition or $1$ under multiplication. Once you choose a world frame, the identity pose coincides with it. That choice does not make the identity physically special. A group can also act on other objects (Solà eqs. 5 to 7). For example, a rotation acts on a vector through $v \mapsto R\,v$, and a pose acts on a point. The identity action leaves an object unchanged. Applying $\mathcal{X}$ followed by $\mathcal{Y}$ has the same effect as applying $\mathcal{Y} \circ \mathcal{X}$ once.

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Three viewpoints of a Lie group

You can view the same Lie group in three ways.

• Topology describes the manifold’s shape. The manifold has a flat tangent space at every point, and the exponential map wraps tangent vectors onto it. SO(2) as a manifold develops this picture for planar rotations. Configuration space explains why no single coordinate system can cover such a space without singularities.
• Algebra describes the group operations, including composition $\mathcal{X} \circ \mathcal{Y}$ and inverse $\mathcal{X}^{-1}$, together with their axioms. These rules give closed-form expressions for chaining transforms and taking the exponential of a twist.
• Geometry connects group elements to model variables. A rotation matrix represents an orientation, a homogeneous transform represents a pose, and a tangent vector represents a velocity.

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A basis turns an object into coordinates

In linear regression, you express a function as a combination of basis functions: $$f(x) = \sum_i \theta_i \, \phi_i(x) ,$$ The coefficients $\theta_i$ are the coordinates of $f$ in that basis. The same construction applies to vectors, $v = \sum_i v_i e_i$, and matrices, $A = \sum_i a_i E_i$. In particular, the $3 \times 3$ matrices form a nine-dimensional vector space. Rotation matrices do not have this linear structure. They satisfy $R^\top R = I$ and $\det R = 1$, but the sum $R_1 + R_2$ of two rotations is not a rotation. Neither is $2R$. Rotations do not form a vector space, so you cannot construct them as linear combinations of basis elements. A linear description works near the identity. Write a rotation close to $I$ as $R \approx I + \epsilon\,\Omega$ for a small $\epsilon$. Substitution into $R^\top R = I$ gives $$(I + \epsilon\,\Omega)^\top (I + \epsilon\,\Omega) = I + \epsilon\,(\Omega^\top + \Omega) + \epsilon^2\,\Omega^\top\Omega = I .$$ The first-order term must be zero, which requires $\Omega^\top = -\Omega$. The possible directions away from the identity are therefore skew-symmetric matrices. These matrices form a vector space called the Lie algebra. Its basis elements are the generators $G_i$. Any element can be written as $\sum_i \tau_i G_i$, and the exponential maps that element to a rotation: $$R = \exp\Big(\sum_i \tau_i G_i\Big) .$$ The complex-numbers prerequisite gives the planar version of this construction. The angle $\theta$ is the coordinate, $i$ is the generator and $e^{i\theta}$ is the rotation. Coordinates enter the Lie algebra through an ordinary linear map. The exponential maps the result nonlinearly to the group.

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The tangent space and the Lie algebra

At each configuration $\mathcal{X}$, the manifold has a tangent space $T_{\mathcal{X}}\mathcal{M}$. It contains every possible instantaneous motion from that configuration. A moving body traces a curve $\mathcal{X}(t)$, and its velocity $\dot{\mathcal{X}}$ belongs to the tangent space at the current configuration. Using a separate tangent space at every configuration would be cumbersome. The group structure provides a common space. The tangent space at the identity is the Lie algebra, $\mathfrak{m} = T_{\mathcal{E}}\mathcal{M}$ (Solà eq. 8). Multiplication by $\mathcal{X}^{-1}$ brings any velocity into this space. Apply the product rule to $\mathcal{X}^{-1}\mathcal{X} = \mathcal{E}$. Because the identity is constant, $$\frac{d}{dt}\big(\mathcal{X}^{-1}\big)\,\mathcal{X} + \mathcal{X}^{-1}\dot{\mathcal{X}} = 0 ,$$ The velocity expressed at the identity is $$v^\wedge = \mathcal{X}^{-1}\dot{\mathcal{X}} \in \mathfrak{m}$$ (Solà eq. 9). You can represent every velocity in one linear space, regardless of the body’s configuration. For a planar rotation, the calculation is direct. If $R(\theta)$ is the rotation matrix, then $R^\top \dot R = \dot\theta\, G$, where $G$ is the quarter-turn matrix $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Although the Lie algebra is a matrix space, each element needs only a small coordinate vector. The hat operator constructs the matrix from its coordinates. The vee operator recovers those coordinates (Solà eqs. 10 and 11): $$\tau^\wedge = \sum_i \tau_i\, G_i \in \mathfrak{m}, \qquad (\tau^\wedge)^\vee = \tau \in \mathbb{R}^m .$$

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Exp and Log

The exponential map takes an element of the Lie algebra back to the group. In the planar case, its generator $G$ satisfies $G^2 = -I$, so the powers repeat in the order $G, -I, -G, I$. Separating the exponential series into even and odd terms gives $$e^{\theta G} = I\Big(1 - \frac{\theta^2}{2!} + \cdots\Big) + G\Big(\theta - \frac{\theta^3}{3!} + \cdots\Big) = I\cos\theta + G\sin\theta = R(\theta) .$$ This expression is the matrix counterpart of $e^{i\theta}$, with $G$ in place of $i$. More generally, a body with constant velocity $v^\wedge$ satisfies $\dot{\mathcal{X}} = \mathcal{X}\,v^\wedge$. Integrating this equation gives $$\mathcal{X}(t) = \mathcal{X}(0)\,\exp(v^\wedge t)$$ (Solà eqs. 12 and 13). Solà eqs. 14 and 15 define the exponential $\exp: \mathfrak{m} \to \mathcal{M}$ and its inverse, the logarithm $\log: \mathcal{M} \to \mathfrak{m}$. In calculations, you usually work with coordinate vectors. Solà defines capitalized forms for them (eqs. 23 and 24): $$\mathcal{X} = \mathrm{Exp}(\tau) = \exp(\tau^\wedge), \qquad \tau = \mathrm{Log}(\mathcal{X}) = \log(\mathcal{X})^\vee .$$

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References

• Solà, J., Deray, J., Atchuthan, D. (2018). A micro Lie theory for state estimation in robotics. arXiv [cs.RO].

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