Planar motion representations

This section borrows heavily from chapter 3, Rigid-Body Motions, of Kevin Lynch and Frank Park, Modern Robotics: Mechanics, Planning, and Control (Cambridge University Press, 2017). You can download the chapter as a PDF. This section covers rigid-body motion in the plane. A planar rigid body has three degrees of freedom: two for position and one for orientation. Every idea here has a direct counterpart in three dimensions, covered in spatial motion representations. Wheeled robots move in the plane, so this section is also the background for wheeled robots.

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Vectors and Reference Frames

A free vector $\mathbf{v} \in \mathbb{R}^n$ is a geometric quantity with magnitude and direction and is independent of any coordinate system. It acquires numerical values only when expressed in a coordinate frame. For example, a velocity vector may be $\mathbf{v} = [3, 1, 0]^T$ in frame $\{a\}$ and have different coordinates in frame $\{b\}$. Points are also represented as vectors. Their coordinates are defined relative to the origin of a reference frame, as shown below. A point and a free vector behave differently when the frame changes. A point moves with the origin, while a free vector ignores translation and only rotates. Homogeneous coordinates make this distinction explicit by appending a 1 to a point and a 0 to a free vector. All frames defined below here are inertial and right handed, despite that in reality they move / rotate as we are all on planet Earth. The following example is key to understand the convention adopted here with respect to inertial frames.

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Planar Rigid-Body Motions

We define a single fixed frame, called space frame $\{s\}$, with unit axes $x̂_s$ and $ŷ_s$.Sometimes we refer to this as the world frame. We also define a fixed at each instance body frame, denoted $\{b\}$, associated with the body with unit axes $x̂_b$ and $ŷ_b$. This frame is coincident with the body at any instant. In the plane, a rotation matrix $P \in SO(2)$ and a translation vector $\mathbf{p} \in \mathbb{R}^2$ describe a rigid-body configuration. The pair $(P, \mathbf{p})$ represents a body in the space frame. Consider a point $q$ at $[1, 1]^T$ in a frame that is rotated 60° counterclockwise and translated by $[2, 1]^T$. In the space frame, the transformation is given by: $$P = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, \quad q' = Pq + \mathbf{p}$$ Although P contains four numbers, three constraints apply. Each column of P must be a unit vector, and the columns must be orthogonal. The remaining degree of freedom is parametrized by θ. The pair (P, p) describes the orientation and position of $\{b\}$ relative to $\{s\}$. Any planar motion can also be interpreted as a rotation about a fixed point $\mathbf{s}$. The figure below shows such a motion. (a) The frame $\{d\}$ is fixed to an elliptical rigid body and initially coincides with $\{s\}$. A rotation by P followed by a translation by p displaces it to \{d0\}, which coincides with the stationary frame $\{b\}$. The pair (P, p) represents $\{b\}$ in $\{s\}$. The same transformation takes the frame $\{c\}$, also attached to the rigid body, to \{c0\}. Transformation 1 rotates $\{c\}$ about the origin of $\{s\}$. Transformation 2 then translates the frame by p expressed in $\{s\}$. (b) The rotation and translation can also occur simultaneously. In this view, the displacement is a rotation of β = 90° about a fixed point s. This is a planar example of a screw motion. The three screw coordinates $(β, s_x , s_y )$ parametrize the displacement. Here, $(s_x , s_y ) = (0, 2)$ gives the coordinates of point s, which is the screw axis out of the page, in the fixed frame $\{s\}$.

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Planar velocities and twists

Collect the rotation $P$ and the translation $\mathbf{p} = (x, y)$ into one $3 \times 3$ matrix: $$T = \begin{bmatrix} P & \mathbf{p} \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta & x \\ \sin\theta & \cos\theta & y \\ 0 & 0 & 1 \end{bmatrix} \in SE(2)$$ The matrix acts on the homogeneous coordinates of the plane. A point $(x, y, 1)$ is rotated and then translated, while a free vector $(v_x, v_y, 0)$ is only rotated. The configuration of a planar rigid body is the triple $q = (\theta, x, y)$, one number per degree of freedom. In the plane every rotation happens about the axis perpendicular to the plane, so the angular velocity is a single number, $\omega = \dot\theta$. Differentiating $P$ gives $\dot P = [\omega] P$, where $$[\omega] = \begin{bmatrix} 0 & -\omega \\ \omega & 0 \end{bmatrix} \in \mathfrak{so}(2)$$ is the same skew-symmetric matrix that appears as the tangent at the identity in SO(2) as a manifold. A twist packs the angular and linear velocity of the body into one object. As with frames, there are two ways to express it. The body twist comes from $T^{-1} \dot T$, and the space twist comes from $\dot T T^{-1}$: $$T^{-1} \dot T = \begin{bmatrix} [\omega_b] & \mathbf{v}_b \\ 0 & 0 \end{bmatrix}, \qquad \dot T T^{-1} = \begin{bmatrix} [\omega_s] & \mathbf{v}_s \\ 0 & 0 \end{bmatrix}, \qquad \omega_b = \omega_s = \dot\theta$$ Both matrices belong to $\mathfrak{se}(2)$. Each is described by three numbers, collected in a twist vector $\mathcal{V} = (\omega, v_x, v_y) \in \mathbb{R}^3$:

• $\mathbf{v}_b = P^T \dot{\mathbf{p}}$ is the velocity of the body-frame origin, written in the coordinates of $\{b\}$.
• $\mathbf{v}_s = \dot{\mathbf{p}} - [\omega]\mathbf{p} = (\dot x + \dot\theta\, y,\ \dot y - \dot\theta\, x)$ is the velocity of the point of the body that is currently at the origin of $\{s\}$, written in $\{s\}$. That point may lie outside the physical body; imagine the body extended far enough to contain it.

The body twist is related to the rate of change of the configuration by $$\mathcal{V}_b = \begin{bmatrix} \omega_b \\ v_{bx} \\ v_{by} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} \dot\theta \\ \dot x \\ \dot y \end{bmatrix}$$ This relation is the starting point for wheeled robots. A wheeled chassis is a planar rigid body with configuration $q = (\theta, x, y)$, and its wheels constrain its body twist. For example, a robot whose wheels cannot slide sideways must have $v_{by} = 0$. Its chassis can point anywhere and reach any position, yet at each instant it can only move forward, backward, or turn.

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References

Lynch and Park, Modern Robotics: Mechanics, Planning, and Control (2017) http://modernrobotics.org

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