Summary.
The chapter begins with rigid-body kinematics under a strict notational convention: displacement vectors and rotation/transformation matrices are expressed in the reference frame, while body linear and angular velocities are expressed in the body frame. Differentiating a rotation matrix naively does not resemble angular velocity, but enforcing orthogonality reveals that R^T\dot R is skew-symmetric, yielding the body angular velocity \omega=\lfloor R^T\dot R\rfloor. Because two displacement vectors compose only when expressed in a common frame, rigid-body displacements form a semidirect product SE(3) = SO(3) \ltimes \mathbb{R}^3, captured exactly by the 4\times4 homogeneous transformation matrix and its ordinary matrix product.
Seeking a unified six-dimensional velocity, the chapter shows that T^{-1}\dot T=\lceil V\rceil\in\mathfrak{se}(3) collects the body linear and angular velocities into the body twist V=(v^T,\omega^T)^T, whereas \dot T T^{-1} produces the spatial twist. Comparing the matrix form with the adjoint action of Chapter~\ref{Chap:WhyLieGroupRobotics} yields the six-dimensional adjoint transformation \mathrm{Ad}_T and the clean composition rule for twists across frames. Differentiating the adjoint transformation introduces the adjoint operator \mathrm{ad}_V, which behaves like a six-dimensional cross product and reproduces the Lie bracket.
The chapter then turns to dynamics. Body wrenches F=(f^T,n^T)^T pair forces and moments expressed in the body frame; by the principle of virtual work, wrenches transform by the inverse-transpose adjoint, F_a= \mathrm{Ad}_{^aT_b}^{-T}F_b. Starting from the Newton–Euler equations in the inertial frame and converting to body quantities, the single equation of motion F=A\dot V+BV is obtained, with the constant block-diagonal body inertia A=\mathrm{diag}(mI_3,I_b). The bias admits infinitely many factorizations, but the physically correct one must satisfy the skew-symmetry condition \dot A=B+B^T. Introducing the co-adjoint operator, the chapter derives the unique Christoffel-consistent factorization of B, which stems from the metric-compatible, torsion-free Levi-Civita connection induced by treating the body inertia as a left-invariant metric on SE(3).
The remainder develops the transformation and composition of dynamics. The equation of motion is recast in an arbitrary body frame, recovering the Huygens–Steiner (parallel-axis) theorem. Body momentum \mathcal{L}=AV transforms like a wrench, consistent with coordinate-free kinetic energy. Finally, the composition of body inertia across the center-of-mass(CoM), modeling, and body frames is given, allowing the inertias of multiple components to be assembled into a single link inertia.
Connection to subsequent chapters.
This chapter is the dynamical kernel replicated and assembled in Part III. The body-frame equation F=A\dot V+BV and the Christoffel-consistent bias matrix scale directly to the closed-form articulated equation of motion \tau=M(\theta)\ddot\theta+C(\theta,\dot\theta)\dot\theta in Chapter 6, where the single-body skew-symmetry \dot A=B+B^T becomes the system passivity property \dot M=C+C^T. The adjoint-based transformations of twists, wrenches, and inertias underpin the recursive Newton–Euler and articulated-body-inertia algorithms of Chapter 7. The relative-transformation error definitions previewed here are used for the kinematic control of Chapter 5 and the SO(3)/SE(3) tracking and impedance controllers of Chapter 10, while the passivity structure is exploited by the joint-space controllers of Chapter 9.