The Geometry of Motion — Volume I: Tangent-Space Methods for Nonlinear Control

The Geometry of Motion — Volume I: Tangent-Space Methods for Nonlinear Control

Tangent-space foundations, variational dynamics, contraction metrics, local optimal control, and counterfactual decompositions for nonlinear mechanical systems.

Volume I: Tangent-Space Methods for Nonlinear Control develops local geometric tools for analyzing motion along reference trajectories. Equilibrium linearization, feedback linearization and trajectory-based methods address different questions; the relevant choice depends on the model and control objective.

For a differentiable flow, derivatives of trajectories with respect to initial conditions or inputs satisfy exact linear variational equations along the nominal trajectory. These are infinitesimal variations. Finite trajectory differences generally contain higher-order terms, and a moving frame also contributes a basis-transport term.

The chapters connect that distinction to local optimal control, contraction certificates and counterfactual calculations. Linear superposition requires a common linear operator; same-state acceleration additivity in a control-affine model does not imply additivity of finite nonlinear interventions. Drift denotes the model’s declared zero-input dynamics, which may include gravity, elastic forces, dissipation and retained baseline actuation. It is not synonymous with passive momentum or absence of muscle activity.

This volume is presented as an addressable, chapter-by-chapter reading path where each derivation retains its own testable assertions, mathematical proofs, and canonical URL.

Geometry and Variational Dynamics

The first four entries develop the geometry and coordinate representations of motion:

Foundations — Tangent spaces, moving frames, coordinate invariance, and exactness of first variations.
Variational Dynamics — State-transition matrices, variational differential equations, and transport of perturbations.
Superposition — Same-state acceleration decomposition and the limits of trajectory superposition.
Induced Acceleration in Biomechanics — Model-conditioned attribution of joint accelerations to declared forces, velocity effects and gravity.

Control, Contraction, and Applications

The remaining six chapters develop contraction metrics, optimal control algorithms, and constrained mechanical applications:

Contraction — Differential contraction analysis, Riemannian metric certificates, and exponential convergence tubes.
Optimal Control — Finite-horizon trajectory optimization, iterative LQR (iLQR), differential dynamic programming, and Riccati backward sweeps.
Duality — Riccati value and stability identities under their stated linear-quadratic assumptions.
Counterfactuals — Forward-dynamics counterfactual simulations, drift-control ratios, and causal attribution limits.
Applications — Practical engineering applications in robotic manipulation, gait, and biomechanical athletic swings.
Parallel Mechanisms, Constrained Dynamics, and Geometric Control — Kinematic closed loops, algebraic constraints, and projection methods.

Prerequisites and Next Steps

Readers should complete Volume 0 or possess equivalent fluency in Lie groups, rigid-body kinematics, and state-space mechanics. Continue directly to Volume II: Control Is Motion for trajectory architecture, transverse coordinates, and orbital funnel synthesis.

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@online{untitled,
  author = {},
  title = {The {Geometry} of {Motion} — {Volume} {I:} {Tangent-Space}
    {Methods} for {Nonlinear} {Control}},
  url = {https://affinedrift.com/articles/The_Geometry_of_Motion/quarto/volume1.html},
  langid = {en}
}

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--- title: "The Geometry of Motion — Volume I: Tangent-Space Methods for Nonlinear Control" description: "Tangent-space foundations, variational dynamics, contraction metrics, local optimal control, and counterfactual decompositions for nonlinear mechanical systems." --- **Volume I: Tangent-Space Methods for Nonlinear Control** develops local geometric tools for analyzing motion along reference trajectories. Equilibrium linearization, feedback linearization and trajectory-based methods address different questions; the relevant choice depends on the model and control objective. For a differentiable flow, derivatives of trajectories with respect to initial conditions or inputs satisfy exact linear variational equations along the nominal trajectory. These are infinitesimal variations. Finite trajectory differences generally contain higher-order terms, and a moving frame also contributes a basis-transport term. The chapters connect that distinction to local optimal control, contraction certificates and counterfactual calculations. Linear superposition requires a common linear operator; same-state acceleration additivity in a control-affine model does not imply additivity of finite nonlinear interventions. Drift denotes the model's declared zero-input dynamics, which may include gravity, elastic forces, dissipation and retained baseline actuation. It is not synonymous with passive momentum or absence of muscle activity. This volume is presented as an addressable, chapter-by-chapter reading path where each derivation retains its own testable assertions, mathematical proofs, and canonical URL. ## Geometry and Variational Dynamics The first four entries develop the geometry and coordinate representations of motion: 1. [Foundations](ch01_foundations.html) — Tangent spaces, moving frames, coordinate invariance, and exactness of first variations. 2. [Variational Dynamics](ch02_variational.html) — State-transition matrices, variational differential equations, and transport of perturbations. 3. [Superposition](ch03_superposition.html) — Same-state acceleration decomposition and the limits of trajectory superposition. 4. [Induced Acceleration in Biomechanics](ch03b_induced_acceleration_biomechanics.html) — Model-conditioned attribution of joint accelerations to declared forces, velocity effects and gravity. ## Control, Contraction, and Applications The remaining six chapters develop contraction metrics, optimal control algorithms, and constrained mechanical applications: 5. [Contraction](ch04_contraction.html) — Differential contraction analysis, Riemannian metric certificates, and exponential convergence tubes. 6. [Optimal Control](ch05_optimal_control.html) — Finite-horizon trajectory optimization, iterative LQR (iLQR), differential dynamic programming, and Riccati backward sweeps. 7. [Duality](ch06_duality.html) — Riccati value and stability identities under their stated linear-quadratic assumptions. 8. [Counterfactuals](ch07_counterfactuals.html) — Forward-dynamics counterfactual simulations, drift-control ratios, and causal attribution limits. 9. [Applications](ch08_applications.html) — Practical engineering applications in robotic manipulation, gait, and biomechanical athletic swings. 10. [Parallel Mechanisms, Constrained Dynamics, and Geometric Control](ch09_parallel_mechanisms_constrained_dynamics.html) — Kinematic closed loops, algebraic constraints, and projection methods. ## Prerequisites and Next Steps Readers should complete [Volume 0](volume0.html) or possess equivalent fluency in Lie groups, rigid-body kinematics, and state-space mechanics. Continue directly to [Volume II: Control Is Motion](volume2.html) for trajectory architecture, transverse coordinates, and orbital funnel synthesis.