Trajectory-Independent Linearizations

• Trajectory-independent linearizations are methods that yield local or globally valid linear approximations without relying on a specific state trajectory.
• They leverage inherent system structures, geometric properties, and algebraic constraints to produce coordinate-free representations for analysis in robotics and control.
• By circumventing trajectory-based limitations, these techniques enable robust control design, invariant analysis, and data-driven system modeling in nonlinear dynamics.

A trajectory-independent linearization is a mathematical or computational procedure that generates a linear—or locally linear—representation of a nonlinear system’s dynamics, properties, or local structures without explicit reference to a fixed trajectory through state-space. Unlike classical linearization techniques that expand the dynamics around a specific reference solution, trajectory-independent linearizations provide structural, coordinate-free, or local representations valid over entire regions, manifolds, or families of system behaviors. Such approaches are central to modern systems theory, geometric mechanics, nonlinear control, robotics, and phase space analysis.

1. Fundamental Concepts

Traditional linearization of a nonlinear dynamical system, such as $\,\dot{x}=f(x)\,$, generally involves Taylor expanding $f$ about a particular state or trajectory, producing a local linear approximation. In contrast, trajectory-independent linearizations aim to provide:

• Local approximations that remain valid over sets or classes of trajectories, not just one.
• Linear representations derived from system structure (e.g., symmetries, geometric properties) or from algebraic constraints, rather than from trajectory correspondence.
• Tools to infer invariant or dominant features (e.g., phase space structures, reachability sets) across a family of possible behaviors.

Examples include local quadratic forms (e.g., the trajectory divergence rate), geometric or group-based local linearizations, and structural reductions for control-affine systems under special algebraic conditions.

2. Local and Geometric Linearizations on Manifolds

In robotics and geometric mechanics, trajectory-independent linearizations address the challenges of global system structure, especially for moving-base (floating-base) robots whose configuration space includes nontrivial manifolds such as $SE(3)$. A trajectory-independent geometric linearization is constructed as follows (Bos et al., 2022):

• Coordinate-Free State Representation: The system state is represented as $q = (H,s) \in SE(3) \times \mathbb{R}^{n_J}$ with velocities in the corresponding Lie algebra, avoiding local coordinate charts and their singularities.
• Geometric Linearization Operator: For systems on a Lie group $G$, the dynamics $\dot{g} = f(g,u)$ are rewritten in left-trivialized form $\dot{g} = g\ \lambda(g,u)$. Perturbations $z \in \mathfrak{g}$ obey the linearized system:

$\dot{z} = A(t)\,z + B(t)\,w$

with $$A = \mathrm{D}_1\lambda\circ \mathrm{D}L_g(e) - \mathrm{ad}_{\lambda}$$ and $B = \mathrm{D}_2\lambda$.

• Recursive Algorithms: The key Jacobians and Hessians required for $A$ and $B$ are computed using $\mathcal{O}(n)$ recursions analogous to standard inverse-dynamics procedures, but without requiring local orientation parameterizations.
• Singularity-Free Nature: Because orientation perturbations are handled in the Lie algebra ($\mathfrak{se}(3)$), the method is globally valid and free from gimbal-lock or coordinate singularities.
• Validation: Comparison against geometric finite differences shows that these linearizations achieve relative errors around $10^{-6}$, confirming precision and structural independence (Bos et al., 2022).

This approach establishes a mathematically exact, trajectory-independent, singularity-free local linearization for highly nonlinear, manifold-valued dynamical systems.

3. Structural Linearization via System Decomposition

For nonlinear affine control systems, a trajectory-independent linearization structure emerges when the system satisfies the linearizing assumption (Löber, 2016):

• Affine Control System:

$$\dot{x}(t) = R(x(t)) + B(x(t))\,u(t),\quad y(t) = h(x(t))$$

• Linearizing Assumption: The Moore–Penrose projectors $P(x)=B(x)B^+(x)$ and $Q(x)=I-P(x)$ are constant; the unactuated drift $Q\,R(x)$ is affine-linear; and the output map $y=h(x)$ is linear.
• Constraint Dynamics: The unforced dynamics for the unactuated subspace $Qx$ (dimension $n-p$) are governed by a constant-coefficient LTI operator, driven by the “free” variables $Px$:

$Q\dot{x}_d(t) = Q A Q x_d(t) + Q A P x_d(t) + Q b$

The solution for $Q x_d(t)$ is the convolution of the LTI system’s resolvent with $P x_d$.

• Input-Output Equivalence: The mapping $P x_d(\cdot) \to Q x_d(\cdot)$ is trajectory-independent and identical to that of a constant LTI system. As a result, controllability conditions and trajectory planning can be reduced to those of classical linear systems by checking the rank of the Kalman-like controllability matrix built from the constant blocks $F = Q A Q$, $E = Q A P$ (Löber, 2016).

This structure allows for linear system concepts and tools, including exact trajectory realization and controllability verification, to be applied to a broad, nontrivial class of nonlinear affine control systems.

4. Trajectory-Independent Phase Space Structures

Instantaneous phase space features, such as invariant manifolds and coherent sets, can be revealed via purely local, trajectory-independent linearizations. The trajectory divergence rate provides such a tool (Jr. et al., 2017):

• Definition ($\rho(x)$): Given $\dot{x}=f(x)$ and $S(x)=\frac{1}{2}[Df(x) + Df(x)^{\top}]$ (the symmetric part of the Jacobian), the trajectory divergence rate at $x$ is

$\rho(x) = n(x)^{\top} S(x) n(x)$

where $n(x)$ is a unit vector normal to the flow direction at $x$.

• Physical Interpretation: $\rho(x)$ gives the first-order (in time) rate of normal separation between infinitesimally close trajectories at $x$, allowing the identification of repelling and attracting structures (such as slow manifolds or hyperbolic LCSs) without trajectory integration.
• Extension to Higher Dimensions: For $x \in \mathbb{R}^n$, with the tangent $e$ and the normal bundle projector $\Pi = I - e e^{\top}$, the principal normal stretching rates are the eigenvalues of the projected strain tensor $\Pi S \Pi$.
• Algorithmic Implications: The computation of $\rho(x)$ requires only $f(x)$ and its Jacobian $Df(x)$, enabling rapid, mesh-based scanning of phase space for dominant structures without solving for system trajectories (Jr. et al., 2017).

This method realizes a local, trajectory-independent linearization concept useful for dynamic systems analysis, especially in complex or high-dimensional phase spaces.

5. Data-Driven and Parameter-Varying Linear Embeddings

Recent developments extend trajectory-independent linearization concepts via data-driven and parameter-varying embeddings (Morato et al., 2023):

• Trajectory Feature Spanning: For linear time-invariant (LTI) systems, admissible trajectories can be spanned by input-output dictionaries under persistent excitation—central to the behavioral systems framework.
• Extension to Nonlinear Systems: Nonlinear dynamics can be embedded in a quasi-Linear Parameter Varying (qLPV) framework, by associating outputs to scheduling variables via a known analytic function. Differential inclusion techniques permit the adaptation of linear analysis tools to nonlinear systems within this embedding, generalizing the trajectory-independent behavioral structure.
• Analysis Tools: Provides simulation, prediction, and parameter-dependent dissipativity analysis that remains valid across entire bounded operation regions, not tied to any specific nominal trajectory.
• Practical Validation: High-fidelity simulations and experimental test-benches (e.g., a rotational pendulum and an electromechanical platform) demonstrate that—even with misspecified scheduling—the representation structure captures the system’s output dynamics reliably (Morato et al., 2023).

A plausible implication is that, by exploiting analytic scheduling functions, trajectory-independent linear representations for nonlinear systems can be realized in a data-driven manner across broad operating envelopes.

6. Local Optimal Control under Linearization Validity

Minimum-energy control laws for linearized systems are only locally valid in regions where the linearization accurately represents the true nonlinear dynamics. A trajectory-independent characterization of validity is given by control reachability ellipsoids (Klickstein et al., 2017):

• Reachability Ellipsoid $\Omega(t,E)$: For a linearized system $\dot{x} = A x + f + B u$, the set of all states reachable at time $t$ with control energy bounded by $E$ is

$$\Omega(t, E) = \{ x \mid (x - g(t))^{\top} W^{-1}(t) (x - g(t)) \leq E \}$$

where $g(t)$ is the zero-input trajectory and $W(t)$ the controllability Gramian.

• Local Validity Constraint: The ellipsoid $\Omega(t, E)$ must be contained within the region $\mathcal{N}$ where the linearization error $\|f(x) - f^{(0)} - A x\|$ remains below a prescribed threshold.
• Piecewise Linearization (LOCS Algorithm): For large excursions, the state space is traversed by concatenating locally valid minimum-energy linear control segments. At each knot point, the system is re-linearized, and the new reachability ellipsoid is constructed, ensuring that the piecewise trajectory never exits its corresponding validity region.
• Theoretical Guarantee: As long as each segment remains within its local validity neighborhood, the entire control synthesis remains locally accurate, and the sequence of control laws provides an effective, trajectory-independent roadmap for state transition and control (Klickstein et al., 2017).

This approach yields an explicit, quantifiable trajectory-independent domain of validity for linearized control laws and enables systematic, robust composition of complex control actions across nonlinear state spaces.

7. Significance and Methodological Connections

Trajectory-independent linearizations enable a range of advanced methodologies:

• Global and local stabilization without pre-computing or tracking explicit trajectories.
• Scalable, singularity-free dynamics computations in robotic and multi-body systems.
• Rapid extraction of phase space structures without computationally expensive trajectory integration.
• Formal reduction of nonlinear control and analysis problems to LTI system techniques within well-characterized domains.
• Support for data-driven and parameter-varying systems analysis methods.

The unifying thread is the provision of locally or structurally linear descriptions or tools that remain valid, interpretable, and effective across entire regions, manifolds, or under algebraic invariants—sidestepping the limitations imposed by trajectory-centric linearizations. These frameworks are crucial in modern control theory, robotic system design, nonlinear dynamics, and systems identification (Morato et al., 2023, Bos et al., 2022, Klickstein et al., 2017, Jr. et al., 2017, Löber, 2016).