Spring-Mass Trajectory Libraries

• Spring-Mass Trajectory Libraries are comprehensive, precomputed sets defining periodic or transient solutions of spring-mass systems for vibrational analysis and legged locomotion.
• They use Lie-algebraic methods and normal mode decomposition to derive eigenfrequencies, mode shapes, and closed-form trajectory functions for varied system parameters.
• The libraries enable real-time control through deadbeat gain selection and efficient trajectory retrieval, ensuring robust performance under uncertainties in robotics.

Spring-mass trajectory libraries are comprehensive, precomputed sets of solutions to the equations of motion governing spring-mass systems, structured for efficient retrieval and application in both classical vibrational analysis and biomimetic locomotion control. These libraries encode parameterized families of periodic or transient state trajectories—typically center-of-mass or mass node positions—supporting analyses and control approaches across disciplines including physics, applied mathematics, and robotics (Urzúa et al., 2019, Sovukluk et al., 15 Dec 2025).

1. Foundations: Dynamical Systems and Model Classes

Spring-mass models fall into two main categories: multi-degree-of-freedom lattices (chains or networks of masses and springs), and reduced-order hybrid models for legged locomotion (e.g., the spring-loaded inverted pendulum, or SLIP, template). The canonical Hamiltonian for a chain of $N$ masses with nearest-neighbor springs is

$$H = \frac{1}{2}\sum_{i=1}^N \frac{p_i^2}{m_i} + \frac{1}{2}\sum_{i=1}^{N} k_i\bigl(x_{i+1}-x_i\bigr)^2,$$

with appropriate boundary conditions ($x_{N+1} \equiv x_1$ for circular; fixed or free for linear chains). The resulting equations of motion underlie all linear vibrational analyses and form the basis for constructing trajectory libraries for arbitrary mass and stiffness profiles (Urzúa et al., 2019).

For biomimetic locomotion, the 3D SLIP template is defined by the center-of-mass (CoM) position $\mathbf{p}_c$, fixed foot point $\mathbf{p}_f$, leg vector $\mathbf{r}=\mathbf{p}_c-\mathbf{p}_f$, mass $m$, stiffness $k$, rest length $r_0$, and touchdown angles $(\theta_1,\theta_2)$, with hybrid stance and flight phases dictated by norm thresholds $\|\mathbf{r}\| < r_0$ and apex-to-apex periodic search (Sovukluk et al., 15 Dec 2025).

2. Lie-Algebraic Methods and Normal Mode Decomposition

For finite-dimensional spring-mass networks, Lie-algebraic structure facilitates diagonalization of the dynamical matrix $\mathbf{M}$.

• Circular chains leverage shift operators $V$, $V^\dagger$ generating a discrete $U(1)$ algebra, diagonalized via the discrete Fourier transform $F$; normal frequencies are $\omega_j = 2\sqrt{\tfrac{k}{m}|\sin(\pi j/N)|}$ and mode shapes are $N$-periodic complex exponentials:

$\phi_j(i) = \frac{1}{\sqrt{N}}e^{2\pi i j i / N}.$

• Linear chains with special (binomial) mass/spring profiles employ $su(2)$ generators $K_0$, $K_+$, $K_-$, with orthonormal Chebyshev or Kravchuk polynomial-based transforms yielding explicit mode spectra and analytical diagonalizations for all $N$ (Urzúa et al., 2019).

In both cases, the exact time-evolution of each $x_i(t)$ is reconstructed as a superposition of mode shapes scaled by initial conditions and oscillatory terms $\cos(\omega_j t)$, $\sin(\omega_j t)$.

3. Library Generation and Trajectory Parameterization

Linear Systems (Chains and Lattices)

Library generation consists in tabulating, for a specified class (linear/circular, fixed/periodic boundaries, uniform/graded mass and stiffness):

1. Dynamical matrix $\mathbf{M}$ and associated parameters ($\{m_i\}$, $\{k_i\}$).
2. Diagonalizing transform $S$ (e.g., DFT, Chebyshev, $su(2)$ rotations).
3. Eigenfrequencies $\omega_j$ and mode shapes $\phi_j(i)$.
4. Explicit trajectory function:

$$x_i(t) = \sum_{j=1}^N \left[ \phi_j(i)x_j(0) \cos(\omega_j t) + \frac{\phi_j(i)\dot{x}_j(0)}{\omega_j} \sin(\omega_j t) \right].$$

Each entry combines the dynamical specification, initial conditions, and closed-form evolution, enabling efficient lookup and reuse for system simulation, control, or vibration analysis (Urzúa et al., 2019).

SLIP-based Apex-to-Apex Trajectories

For spring-mass template models relevant to locomotion:

1. Discretized grids over apex height $h$, forward speed $v_x$, stiffness $k$, and lateral touchdown angle $\theta_2$.
2. Nonlinear least-squares solved (e.g., via Ceres) for periodic solutions in variables $\zeta=[\theta_1,v_y]$ given apex state $x^*$, control input $u$, and parameterization $\left(h,k,v_x,\theta_2\right)$.
3. Numerical integration (Runge-Kutta) progresses flight and stance phases, checking periodicity and feasibility.
4. Resultant library entries store $h$, $k$, $v_x$, $\theta_2$, $\theta_1^*$, $v_y^*$ plus derived foot placements $(\delta_{i,x},\delta_{i,y})$ (Sovukluk et al., 15 Dec 2025).

Total generation time for 315 such periodic apex-to-apex trajectories is approximately $4.5\text{ s}$ offline.

4. Controller Synthesis and Real-Time Trajectory Selection

For real-time adaptive locomotion using spring-mass trajectory libraries, control is realized through:

• Deadbeat control gain libraries: At each periodic trajectory, local linearization yields Jacobians $J_x$, $J_u$ of the return map. The deadbeat gain matrix $K = -J_u^{-1}J_x$ enables feedback correction:

$\Delta u = -J_u^{-1}J_x\,\Delta x,$

guaranteeing aperiodic state error elimination after one step in the linearized regime.

• Selection policy: At runtime, given the sensed touchdown state $x_\text{TD}$ and leg state $u_\text{TD}$, all library entries are scored via the cost

$$\|u_\text{TD} - (u^*_i + K_i(x_\text{TD} - x^*_i))\|_2,$$

promoting high-probability convergence (Sovukluk et al., 15 Dec 2025). Obstacle and stepping-stone constraints are implemented by filtering entries according to apex clearance and target region reachability.

The entire selection process, including local filtering and evaluation, is $O(n)$ with measured $20\;\mu$s per step for $n=315$ library entries.

5. Whole-body Mapping, Adaptive Behaviors, and Practical Implementation

Mapping spring-mass trajectories to high-dimensional humanoid models is conducted via whole-body control (WBC) frameworks:

• Inverse dynamics constraints: Floating-base equations $M(q)\,\dot{y} + C(q,y)\,y + \tau_g$ are solved in a quadratic program, enforcing foot constraints, friction cones, actuator limits, and closed-kinematic-chain requirements.
• Collision avoidance: Task velocities are projected onto admissible tangent spaces if imminent collision is detected.
• Reactive limb swing: Nullspace projection of posture tasks and momentum regulation dampens disturbances.

C++-style data structures encapsulate each trajectory and associated deadbeat gain:

struct Trajectory { double θ2, h, k, vx, θ1, vy, δx, δy; Matrix3d K; }; std::vector<Trajectory> library;

Real-time retrieval filters the library by task-specific constraints before minimizing the cost. All agility behaviors—random stepping, slalom, direction changes, disturbance rejection—share a single trajectory/gain library and WBC parameterization with no per-trajectory tuning (Sovukluk et al., 15 Dec 2025).

6. Robustness and Performance under Uncertainty

Injected signal noise (velocity $\sigma_v\approx0.02$ m/s, angular $\sigma_\omega\approx0.1$ rad/s), actuator delays (1–2 ms), and $5\%$ mass/inertia disturbances are handled robustly. The combination of deadbeat-leg re-planning and two-step look-ahead mitigates the effect of errors, achieving steady-running root mean square errors of

$$v_x\text{ RMSE} \approx 0.085\text{ m/s},\quad v_y\text{ RMSE} \approx 0.021\text{ m/s}$$

during unstructured locomotion in simulation (Sovukluk et al., 15 Dec 2025).

7. Comparison of Spring-Mass Trajectory Libraries

Application Domain	Core Model	Library Components
Vibrational analysis (chains)	Hamiltonian chain	$(\mathbf{M}, S, \omega, \phi, x_i(t))$
Biomimetic locomotion (SLIP)	3D SLIP template	$$(h, k, v_x, \theta_2, \theta_1, v_y, \delta_x, \delta_y, K)$$
Whole-body robot control	SLIP + WBC mapping	Trajectory structs, deadbeat gains

Both approaches leverage the precomputation of parameter-rich trajectory sets to enable fast retrieval, analytical insight into system dynamics, and robust real-time performance in simulation and robotics.

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References: (Urzúa et al., 2019): Dynamical analysis of mass-spring models using Lie algebraic methods (Sovukluk et al., 15 Dec 2025): Humanoid Robot Running Through Random Stepping Stones and Jumping Over Obstacles: Step Adaptation Using Spring-Mass Trajectories