Domain A.4: Rigid-Body Dynamics
Domain A.3 answers: given $\boldsymbol{\omega}(t)$, how does attitude evolve? Domain A.4 answers the harder question: given inertia and torques, what angular acceleration occurs, how do angular momentum and kinetic energy evolve, and when is the motion stable?
Key shift: rotational motion is governed by moment balance $\,\mathbf{M}=\left.\dfrac{d\mathbf{H}}{dt}\right|_{N}\,$, not by $\,\mathbf{F}=m\mathbf{a}\,$ directly.
Six-DOF picture and the “rotational-only” reduction
A rigid body has 6 degrees of freedom: translation of the center of mass (COM) and rotation about the COM. In spacecraft attitude dynamics we often “peel off” translation by choosing the reference point at the COM (or by using a convenient inertial reference point when justified).
Moment balance drives rotation: $\mathbf{M}_O=\left.\dfrac{d\mathbf{H}_O}{dt}\right|_{N}$. Forces govern translation; moments govern how $\mathbf{H}$ evolves.
Angular momentum: orbital + spin decomposition
For mass elements $dm$, the angular momentum about a point $O$ is
\[ \mathbf{H}_O=\int \mathbf{r}\times \dot{\mathbf{R}}\,dm, \]
where $\mathbf{r}$ is measured from $O$ and $\mathbf{R}$ is the inertial position. Decomposing motion into COM translation plus motion relative to the COM produces:
- Orbital contribution (associated with COM translation),
- Spin contribution (pure rotation about the COM).
Physical meaning: torque does not “create” angular momentum from nothing; it changes the direction and/or magnitude of an existing $\mathbf{H}$ vector.
Body frame, transport theorem, and why dynamics become nonlinear
Introduce a body-fixed frame $B=\{\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\}$ embedded in the rigid body. Any vector $\mathbf{v}$ satisfies the transport theorem:
\[ \left(\frac{d\mathbf{v}}{dt}\right)_{N}= \left(\frac{d\mathbf{v}}{dt}\right)_{B}+\boldsymbol{\omega}\times \mathbf{v}. \]
This identity explains the coupling in rigid-body equations: the basis vectors rotate, so “component derivatives” are not the same as inertial derivatives.
Inertia matrix / dyadic and the inertia geometry
About a chosen reference point (typically the COM), inertia maps angular velocity to angular momentum:
\[ \mathbf{H}=\mathbf{J}\boldsymbol{\omega}. \]
The inertia matrix $\mathbf{J}$ is symmetric positive definite. A coordinate-free dyadic form is
\[ \widehat{\mathbf{J}}=\int \left(\rho^2\widehat{\mathbf{I}}-\boldsymbol{\rho}\boldsymbol{\rho}^{T}\right)\,dm. \]
Key idea: $\mathbf{J}$ is the “shape-of-mass” operator. Products of inertia create cross-axis coupling and strongly influence stability.
Parallel-axis theorem (offset reference points)
If inertia about the COM is known, inertia about another point $O$ offset by $\mathbf{r}_c$ is
\[ \widehat{\mathbf{J}}_{O}=\widehat{\mathbf{J}}_{C}+ m\left(r_c^2\widehat{\mathbf{I}}-\mathbf{r}_c\mathbf{r}_c^{T}\right). \]
Where this shows up immediately:
- actuator lines not passing through the COM,
- booms/appendages shifting effective inertia,
- internal moving masses and simplified slosh models.
Principal axes and diagonalisation
There exists an orthonormal body frame (principal axes) in which
\[ \mathbf{J}=\mathrm{diag}(J_1,J_2,J_3), \qquad J_1\ge J_2\ge J_3\ \text{(common ordering)}. \]
Principal axes are obtained from the eigenvalue problem \[ (\lambda\mathbf{I}-\mathbf{J})\mathbf{e}=0. \]
Euler’s rotational equations
Angular momentum balance in inertial form:
\[ \mathbf{M}=\left.\frac{d\mathbf{H}}{dt}\right|_{N}. \]
Convert to body derivatives:
\[ \mathbf{M}=\left.\frac{d\mathbf{H}}{dt}\right|_{B}+\boldsymbol{\omega}\times \mathbf{H}. \]
With $\mathbf{H}=\mathbf{J}\boldsymbol{\omega}$ (rigid body, $\mathbf{J}$ constant in $B$):
\[ \boxed{\ \mathbf{M}=\mathbf{J}\dot{\boldsymbol{\omega}}+\boldsymbol{\omega}\times(\mathbf{J}\boldsymbol{\omega})\ }. \]
In principal axes:
\[ \begin{aligned} J_1\dot{\omega}_1-(J_2-J_3)\omega_2\omega_3 &= M_1,\\ J_2\dot{\omega}_2-(J_3-J_1)\omega_3\omega_1 &= M_2,\\ J_3\dot{\omega}_3-(J_1-J_2)\omega_1\omega_2 &= M_3. \end{aligned} \]
Interpretation: the nonlinear cross term $\boldsymbol{\omega}\times(\mathbf{J}\boldsymbol{\omega})$ is why commanding a torque about one axis can excite motion in the other two.
Torque-free motion: invariants and geometric picture
For $\mathbf{M}=0$, two quantities are conserved:
- angular momentum magnitude $\|\mathbf{H}\|$,
- rotational kinetic energy $T$.
\[ T=\tfrac12 \boldsymbol{\omega}^{T}\mathbf{J}\boldsymbol{\omega}=\text{const}, \qquad \mathbf{H}=\mathbf{J}\boldsymbol{\omega},\ \ \|\mathbf{H}\|=\text{const}. \]
The intersection of the constant-energy ellipsoid with the constant-$\|\mathbf{H}\|$ surface confines the motion. The body-frame curve is the polhode; the inertial counterpart is the herpolhode.
Stability of principal-axis spin (intermediate-axis instability)
For $J_1>J_2>J_3$, pure spin about a principal axis is an equilibrium of the torque-free system. Linear stability results:
- spin about $J_1$ (maximum inertia axis) → stable,
- spin about $J_3$ (minimum inertia axis) → stable,
- spin about $J_2$ (intermediate axis) → unstable.
Nondimensional torque-free Euler system (structured integrals)
A useful scaled form (with respect to $\tau$) is:
\[ \dot{x}_1-x_2x_3=0,\quad \dot{x}_2+x_3x_1=0,\quad \dot{x}_3-x_1x_2=0. \]
Two conserved quantities
\[ x_2^2+x_3^2=A,\qquad x_1^2+x_2^2=B. \]
Useful identities: $x_1^2-x_3^2=B-A,\ \ x_1^2+2x_2^2+x_3^2=A+B$.
Second-order nonlinear oscillator forms
\[ \ddot{x}_1+(A-2B)x_1+2x_1^3=0, \]
\[ \ddot{x}_2+(A+B)x_2-2x_2^3=0,\qquad \ddot{x}_3+(B-2A)x_3+2x_3^3=0. \]
These forms make separatrices and phase portraits “pop out” once invariants are used correctly.
Constant body-fixed torque: equilibria, manifolds, separatrices
With a constant torque applied along a body axis, Euler’s equations become forced. In reduced phase portraits, separatrices divide qualitatively different motion families: bounded (libration-like) behavior versus transitions into new rotation regimes.
Engineering meaning: thrust misalignment drives coning/precession; spin-up maneuvers can cross separatrix boundaries; near-boundary behavior is sensitive to small errors.
Axisymmetric spinner with constant transverse torque (misalignment model)
For $J_1=J_2=J$ and $\omega_3\approx n$ approximately constant, a reduced transverse model often takes the form:
\[ \dot{\omega}_1=\lambda\omega_2+\mu,\qquad \dot{\omega}_2=-\lambda\omega_1. \]
The solutions are sinusoidal with an offset, producing combined precession + nutation.
Orbit–attitude coupling: gravity-gradient torque
In circular orbit, the gravitational field varies across the spacecraft. The net force passes through the COM (no translation torque), but the distributed mass creates a gravity-gradient torque.
\[ \boxed{\ \mathbf{M}_{gg}=3n^2\,\mathbf{a}_3\times(\mathbf{J}\mathbf{a}_3)\ }. \]
- $n$ is the orbital mean motion,
- $\mathbf{a}_3$ is the local vertical (radial) direction, expressed in the body frame for the cross product.
Linearised attitude equations in LVLH: stability conditions
For small attitude errors relative to LVLH, the equations linearize into coupled second-order ODEs. Pitch often decouples; roll/yaw typically remain coupled and yield characteristic polynomials whose stability depends on inertia ratios.
\[ k_1=\frac{J_2-J_3}{J_1},\qquad k_3=\frac{J_2-J_1}{J_3}. \]
Engineering meaning: gravity-gradient stabilization is configuration-dependent; inertia ordering can stabilize one axis while destabilizing another.
Gyrostats and internal angular momentum (reaction wheels / momentum bias)
A gyrostat is a rigid body with internal spinning rotors. Total angular momentum is
\[ \mathbf{H}=\mathbf{J}\boldsymbol{\omega}+\mathbf{h}. \]
The body-frame dynamics become
\[ \mathbf{M}=\left.\frac{d}{dt}\right|_{B}\left(\mathbf{J}\boldsymbol{\omega}+\mathbf{h}\right)+ \boldsymbol{\omega}\times\left(\mathbf{J}\boldsymbol{\omega}+\mathbf{h}\right). \]
Dual-spin spacecraft + platform damper: moving mass and dissipation
Dual-spin designs introduce multiple rotating sub-bodies, relative rates, and damping devices that remove nutation energy. A compact model typically includes composite momentum balance, rotor spin dynamics, and damper motion with stiffness $k$ and damping $c$, plus coupling through body rates.
Big picture: dissipation effectively shrinks accessible energy, guiding the system toward stable spin states.
Engineering insight: what to remember for GNC
- Torque ≠ angular acceleration direction unless inertia is spherical.
- Cross-axis coupling is normal via $\boldsymbol{\omega}\times(\mathbf{J}\boldsymbol{\omega})$.
- Stability has structure: max/min inertia spins are stable; intermediate-axis spin is unstable.
- Orbit matters: LVLH rotation and gravity-gradient terms reshape the dynamics.
Kinematics: $\boldsymbol{\omega}\Rightarrow$ attitude evolution. Dynamics: $(\mathbf{J},\mathbf{M})\Rightarrow \dot{\boldsymbol{\omega}}\Rightarrow$ stability + response.