Linear and Angular (Spacial) Velocities

Given a transformation matrix that represents the configuration $\{b\}$ as seen from $\{s\}$

the spacial velocity can be calculated analogous to the angular velocity.

with

• $R^T\dot{R} = [\omega_b]$: skew-symmetric representation of angular velocity (expressed in $\{b\}$ coordinates)
• $\dot{p}$: linear velocity of origin of $\{b\}$ expressed in $\{s\}$ coordinates
• $R^T\dot{p} = v_b$: linear velocity (expressed in $\{b\}$ cooridnates)

A twist is the velocity of a screw motion.

Matrix Representation of a Twist (Spatial Velocity)

Body Twist

The spacial velocity in the body frame, or body twist is (in matrix representation):

where:

• $[\omega_b] \in so(3)$: Angular velocity (skew-symmetric matrix representation) expressed in $\{b\}$
• $v_b \in \mathbb{R}^3$: linear velocity of a point at the origin of $\{b\}$ expressed in $\{b\}$

The set of all $4 \times 4$ matrices of the form of $[V_b]$ is called $se(3)$ (Lie algebra for the Lie group $SE(3)$).

$se(3)$ is the set of the matrix representations of the twists associated with the rigid-body configuration $SE(3)$.

$se(3)$ consists of all possible $\dot{T}$ (when $T= I$).

Spacial Twist

The spacial twist (spacial velocity in the space frame) is analogous to the body twist

Where:

• $[\omega_s] \in so(3)$: Angular velocity (skew-symmetric matrix representation) expressed in $\{s\}$
• $v_s \in \mathbb{R}^3$: linear velocity of a point at the origin of $\{s\}$ expressed in $\{s\}$

Conversion between Body-Frame and Space-Frame (adjoint map)

From body frame $\{b\}$ to space frame $\{s\}$:

From space frame $\{s\}$ to body frame $\{b\}$:

Where $[Ad_T] \in \mathbb{R}^{6 \times 6}$ is the adjoint representation (adjoint map) of $T = (R,p) \in SE(3)$.

The fixed-frame representation of a twist $V_s$ does not depend on the choice of the body frame $\{b\}$.

And the body-frame representation $V_b$ of the same twist does not depend on the choice of the fixed frame $\{s\}$.

Properties of the Adjoint Map

• Composition:
  • $Ad_{T_1}(Ad_{T_2}(V)) = Ad_{T_1T_2}(V)$
  • or: $[Ad_{T_1}][Ad_{T_2}]V = [Ad_{T_1T_2}]V$
  • with:
    • $T_1, T_2 \in SE(3)$
    • $V = (\omega, v)$
• Inverse:
  • $[Ad_T]^{-1} = [Ad_{T^{-1}}]$ for any $T \in SE(3)$:

Screw interpretation of a Twist

A twist $V$ can be viewed as a screw axis $S$ and a velocity $\dot{\theta}$ (just like an angular velocity $\omega$ can be combined as $\hat{\omega}\dot{\theta}$).

The screw axis $S$ is defined using a normalized version of any twist $V = (\omega, v)$ corresponting to motion along the screw:

1. if $\omega \neq 0$:
   • Screw axis $S$ is just $V$ normalized by length of angular velocity vector: $S = V / \left\| \omega \right\| = (\omega/\left\| \omega \right\|, v/\left\| \omega \right\|)$
   • Angular velocity about screw axis is $\dot{\theta} = \left\| \omega \right\|$ such that $S\dot{\theta} = V$
2. if $\omega = 0$:
   • Screw axis $S$ is just $V$ normalized by length of linear velocity vector: $S = V / \left\| v \right\| = (0, v/\left\| v \right\|)$
   • Linear velocity along screw axis is $\dot{\theta} = \left\| v \right\|$ such that $S\dot{\theta} = V$

Normalized Screw Axis

A unit screw axis is a noramlized twist defined as:

Where:

• either pitch $h$ is finite
  • $\left\| S_{\omega} \right\| = 1$ (equivalent: $\left\| \omega \right\| = 1$)
  • $\dot{\theta}$: rotational speed
  • then: $v = -\omega \times q + h\omega$
    • $q$: a point on the screw axis
    • $h$: pitch of the screw ($h=0$ for pure rotation)
• or pitch $h$ is infinite ($h \to \infty$)
  • $S_{\omega} = 0$ (equivalent: $\omega = 0$)
  • $\left\| S_{v} \right\| = 1$ (equivalent: $\left\| v \right\| = 1$)
  • $\dot{\theta}$: linear speed
  • pure translation along the axis defined by $v$

A screw axis $S$ is just a normalized twist. It can be represented as a $4 \times 4$ matrix $[S]$ of $S = (\omega, v)$:

with:

• $[\omega] \in so(3)$: the screw-symmetric representation of $\omega$

Conversion of Screw Axes

A screw axis represented in any frame $\{a\}$ can be represented in another frame $\{b\}$ (with a modified version of the subscript cancellation rule):

With:

• $S_a$: Screw axis represented in $\{a\}$ coordinates
• $S_b$: Same screw axis represented in $\{b\}$ coordinates
• $[Ad_{T_{ab}}]$: Adjoint map (from $b$ to $a$)

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017