Discrete Optimization-Based Kinematic Modeling

• Discrete optimization-based kinematic modeling is a method that converts continuous kinematic systems into finite, tractable states using numerical optimization.
• It employs techniques like sequential least squares programming and Levenberg–Marquardt to estimate configurations in tendon-driven robots and multibody dynamics.
• This approach ensures high-fidelity models with significantly reduced errors by rigorously enforcing holonomic and nonholonomic constraints using drift-free methods.

Discrete optimization-based kinematic modeling refers to a class of methodologies that utilize numerical optimization to predict or estimate the kinematic behavior of complex mechanical systems—often under constraints that may be holonomic, nonholonomic, or frictional—by discretizing the configuration space or trajectory and solving finite-dimensional nonlinear programs. These techniques underpin state-of-the-art models for tendon-driven continuum robots, trajectory optimization of multi-body systems, and constrained robotic manipulators, and are distinguished by their capacity to tightly enforce constraints, encode manufacturing realities, and faithfully replicate kinematic and geometric structure.

1. Mechanical Discretization of Kinematic Structures

A central theme in discrete optimization-based kinematic modeling is the mechanical discretization of the underlying continuous system. For tendon-driven continuum robots (TDCRs), mechanical discretization replaces the Cosserat rod paradigm with a chain of universal joints and rigid links. Each synovial universal joint module implements 2-DOF (pitch and yaw), with open channels accommodating tendon routing and tool passage. The backbone is thus reduced from an infinite-dimensional curve to a finite set of joint angles $\theta \in \mathbb{R}^{2n}$, with $n$ the number of universal joints, and fixed inter-joint link lengths $L \in \mathbb{R}^n$. General tendon routing is encoded by way-points $\bar p_i \in \mathbb{R}^{3 \times (2n+1)}$ defined in local link-joint coordinates (Shentu et al., 2024).

This discretization strategy yields a parametrically concise state (typically $2n$ DOF), facilitating tractable optimization and direct ties to hardware constraints. In multibody dynamics, a maximal-coordinate representation—comprising positions in $\mathbb{R}^3$ and orientations in unit quaternions $\mathbb{H}_0$ for $n_b$ bodies—is adopted, along with regularized holonomic and nonholonomic constraints (Marklund et al., 10 Feb 2025).

2. Optimization Problem Formulation

Discrete optimization-based kinematic modeling casts the prediction of system configurations, tendon tensions, and physical parameters as a nonlinear programming (NLP) problem subject to equality and inequality constraints. For TDCRs, the decision variables $x=(\theta, T)$ are estimated by minimizing the least-squares discrepancy between static-equilibrium bending moments and normalized joint angles: $$J(\theta, T) = \sum_{k=1}^{2n} \Bigl(\frac{m_k(\theta,T)}{\|m(\theta,T)\|_2} - \frac{\theta_k}{\|\theta\|_2}\Bigr)^2$$ subject to:

• Tendon displacement constraints: $\, \ell_j(\theta) = \ell^0_j + \Delta\ell_j$
• Joint limits: $$\, \theta_i^{\min} \le \theta_i \le \theta_i^{\max}$$
• Tendon tension nonnegativity: $\, T_j \ge 0$ (Shentu et al., 2024)

In multibody dynamics, joint parameter and state estimation proceeds via weighted nonlinear least squares over the inverse-dynamics and observation residuals: $$J(\theta,X) = \sum_{k=0}^{N-1} \|\,\Delta p_k(\theta,X)\|^2_{\Omega_k^{-1}} + \sum_{k=0}^{N-1}\|\,r_{\rm obs}(x_k,y_k;\theta)\|^2_{R_k^{-1}}$$ where $\theta$ includes inertial, frictional, and motor parameters, and $(q_k,v_k)$ are the discrete state and velocity (Marklund et al., 10 Feb 2025).

For general trajectory optimization, direct collocation methods formulate the problem over polynomial splines parameterized by knot values and internal states, enforcing dynamics and constraints at collocation nodes (Bordalba et al., 2023). Projection and local-coordinates approaches enforce constraint satisfaction at knots and—if desired—continuously along the trajectory.

3. Numerical Solution Strategies

Solving discrete optimization-based kinematic models necessitates nonlinear solvers capable of handling equality and inequality constraints, large state vectors, and nonconvexities arising from geometric or hardware restrictions.

• For TDCRs, Sequential Least Squares Programming (SLSQP), a SQP-family solver, is employed. Initialization uses straight configuration and unit tendon tensions; iterative updates proceed until both variable and objective convergence tolerances are met (e.g., $10^{-6}$ for $x$, $10^{-8}$ for $J$) (Shentu et al., 2024).
• In multibody estimation under frictional constraints, the Levenberg–Marquardt algorithm is used. Differentiation of the residuals leverages forward-mode automatic differentiation, with custom JVP rules for nonsmooth problems (e.g., complementarity for Coulomb friction, quaternion algebra). Fast convergence is reported: 4–6 iterations and total runtime of 5–10 seconds for joint state and parameter estimation (Marklund et al., 10 Feb 2025).
• Polynomial mechanics with pseudo-spectral collocation employs SQP solvers (SNOPT), with analytic Jacobians and spectral sparsity exploited for efficiency. The spectral convergence of approximation entails few discretization nodes for smooth problems, but with denser Jacobians (Srinivasan et al., 2014).

The following table summarizes solvers and performance diagnostics: | System type | Solver | Typical runtime | |----------------------|----------------------|------------------| | TDCR | SLSQP | 0.1–0.5 s/config | | Multibody dynamics | Levenberg–Marquardt | 5–10 s (13D parm)| | Trajectory opt. | SNOPT (SQP) | $\sim$100–500 it.|

4. Constraint Enforcement and Drift Elimination

A critical aspect of optimization-based kinematic modeling is the enforcement of holonomic and nonholonomic constraints. In direct collocation, standard transcription may lead to drift—accumulated violations of the constraint manifold—due to discretization error.

Classical stabilization remedies, such as Baumgarte, introduce artificial compliance and energy dissipation, requiring tuning of additional stabilization parameters. The Hermite–Simpson (PKT) approach injects loop-closure impulses, but distorts dynamics at collocation midpoints and restricts the polynomial degree (Bordalba et al., 2023).

Drift-free methods (projection and local-coordinates):

• Projection: Projects end-of-interval states onto the constraint manifold using auxiliary multipliers, enforcing $F(x_{k+1})=0$ at every knot.
• Local-coordinates: Transcribes the entire trajectory in a tangent-space chart defined by the nullspace of the constraint Jacobian, ensuring that reconstructed states lie exactly on the manifold throughout.

Numerical studies demonstrate that drift-free local-coordinates methods can reduce kinematic error by more than $10^{10}$-fold over unconstrained or stabilized schemes, e.g., $E_K\approx10^{-13}$ for a five-bar manipulator, without artificially modifying system dynamics (Bordalba et al., 2023).

5. Model Validation, Performance, and Design Implications

Validation of discrete optimization-based kinematic models relies on cross-comparison with physical prototypes and careful assessment of tip-position, kinematic, and dynamic errors.

• For TDCRs, comparison to hardware (170 mm robot with various tendon routings) shows average tip-position errors of $8.12\,\text{mm}\pm4.55\,\text{mm}$ ($4.8\%\pm2.7\%$ backbone length) for two-tendon actuation. Friction-aware models outperform frictionless baselines by 30–50% (Shentu et al., 2024).
• Joint parameter estimation on the Furuta pendulum identifies a full 13D parameter vector (inertial, viscous, dry friction, tilt, motor gain) with cross-validation MSE $2\times10^{-5}$, dependent on regularization weight $\kappa$ and constraint compliance $\epsilon$ (Marklund et al., 10 Feb 2025).
• For robotic manipulators and wheeled platforms, local-coordinates collocation achieves time-averaged kinematic errors orders-of-magnitude below those of Baumgarte or PKT, eliminating physically inconsistent drift (Bordalba et al., 2023).

Design implications:

• Discrete universal-joint models enable direct encoding of manufacturing constraints, arbitrary tendon routing, and frictional effects, while maintaining a compact finite-dimensional state space.
• Spectral collocation delivers exponential convergence for smooth mechanical systems, allowing reduced discretization size at the cost of denser Jacobians (Srinivasan et al., 2014).
• Model-based pipelines built on these optimization frameworks facilitate rapid design, hardware integration, and high-fidelity trajectory tracking in physically constrained environments.

6. Extensions and Generalization

Discrete optimization-based kinematic modeling extends across mechanism design, state/parameter estimation, and kinodynamic planning for systems with complex constraints and hardware realities.

Projection and local-coordinates collocation schemes generalize to:

• Closed-chain robot mechanism synthesis
• Wheeled vehicle modeling with rolling-contact constraints
• Multi-body dynamics on $SO(3)$
• Discrete-time system identification under frictional/nonsmooth constraints

These approaches support arbitrary-degree splines and hp-adaptive mesh refinement, yielding high-accuracy models for advanced robotics and mechanical systems. The embedding of physically regularized steppers within nonlinear solvers allows for robust joint estimation, even with limited observability or stiff constraints, provided regularization and weighting parameters are appropriately tuned (Marklund et al., 10 Feb 2025, Bordalba et al., 2023).

A plausible implication is that integration of discrete optimization with drift-free constraint management will become foundational for the next generation of model-based design and control pipelines in soft robotics, parallel mechanisms, and articulated robotic platforms.