Dynamical Feasibility & Contact Alignment

• Dynamical Feasibility and Contact Alignment are defined as compliance with physical constraints and proper contact force orientation using complementarity and friction laws.
• Hierarchical and contact-implicit optimization approaches enable efficient refinement from coarse, centroidal models to full-order dynamic simulations.
• Integration of trust regions and contact-aware feedback control supports robust performance in agile locomotion and dexterous manipulation applications.

Dynamical feasibility and contact alignment are foundational concepts in the planning, simulation, and control of robotic systems interacting with environments through multiple, possibly changing contacts. These form the mathematical and algorithmic backbone of trajectory optimization, manipulation, legged locomotion, hybrid control, and dynamic simulation across applications from dexterous manipulation to agile locomotion. This entry presents the formal definitions, algorithmic mechanisms, constraint structures, and practical methodologies underpinning the computation and enforcement of dynamical feasibility and contact alignment in contact-rich systems, drawing from the leading literature in optimization, numerical simulation, and planning.

1. Foundational Definitions and Principles

Dynamical feasibility denotes compliance with all physical constraints—including equations of motion, actuator limits, and, critically, hybrid unilateral contact and friction laws—during robot motion in contact with the environment. Any candidate trajectory (discrete sequence or continuous path) is dynamically feasible if, at every instant, the joint configuration, velocities, accelerations, control inputs, and all contact forces together satisfy both the (possibly nonlinear) rigid-body dynamics and all complementarity/inequality constraints due to contact, non-penetration, friction, and actuation.

Contact alignment refers to the correct placement and orientation of contact forces and locations so that interaction is physically admissible: all contact forces are supported by actual geometric contact (no tension at a distance), friction forces remain within the friction cone, and transitions between contact modes (onset, break, stick, slip) honor complementarity, resulting in smooth, artifact-free physical evolution and transferability to real systems.

Crucially, these properties must be enforced globally (across all possible contact modes and transitions), locally (within trust regions/horizons of optimization), and in real-time (in control loops), often without explicit enumeration of contact sequences.

2. Mathematical Formulations of Contact-Rich Feasibility

The core mathematical structures combine rigid-body mechanics with hybrid nonsmooth constraints:

• Rigid-Body Dynamics with Contact:

$$M(q)\ddot{q} + C(q, \dot{q}) + G(q) + J(q)^T\lambda = B u$$

with generalized coordinates $q$, velocities $\dot{q}$, contact Jacobian $J(q)$, Lagrange multipliers $\lambda$, and actuation $u$.

• Complementarity / Unilateral Constraints:

For each potential contact $i$:

$0 \leq \phi_i(q) \perp \lambda_{n,i} \geq 0$

where $\phi_i(q)$ is the signed distance between the bodies, $\lambda_{n,i}$ is the normal contact force.

• Friction Cone or Pyramid:

$\| \lambda_{t,i} \| \leq \mu_i \lambda_{n,i}$

where $\lambda_{t,i}$ is the tangential force, $\mu_i$ is the friction coefficient.

• Hybrid Mode Switching via Complementarity:

Contact onset and breakage, as well as stick-slip transitions, are modeled using complementarity conditions:

$$0 \leq \gamma_i \perp F(\lambda_{t,i}, v_{t,i}) \geq 0$$

with $v_{t,i}$ the relative tangential velocity.

• Contact Alignment Constraints:

Force direction aligns with local normal:

$n^T \mathbf{u}_{\text{contact}} \geq 0$

Frictional direction is constrained within the local tangent, and position is constrained to lie on or within the contact patch.

These constraints define the feasible set for trajectory optimization, planning, or simulation.

3. Hierarchical and Complementarity-Based Planning

A dominant paradigm is hierarchical two-stage or contact-implicit optimization:

• Centroidal/Approximate Stage: Low-dimensional approximation (e.g., centroidal model) is first solved, applying linear or convex relaxations and complementarity constraints for coarse CoM trajectories and contact forces. This is computationally tractable and prunes infeasible gross motions.
• Full-Order Refinement: Using the centroidal solution for warm start, a full-order (floating-base and joints) Mathematical Program with Complementarity Constraints (MPCC) incorporates complete rigid-body dynamics and all contact complementarity, often over a short horizon (Mastalli et al., 2019, Shirai et al., 11 Mar 2025).
• Contact Discovery and Alignment: Instead of scheduling touches, complementarity constraints cause the solver to "discover" the timing and sequence where force activates, aligning interactions with physical contacts emergently. No explicit combinatorial search is required—valid transitions (touchdown, liftoff, stick, slip) are generated by the MPCC solution that minimizes cost and satisfies all feasibility conditions.
• Cutting-Plane and Relaxation Feedback: When convex-relaxed, mixed-integer, or approximate models produce infeasible contact patterns during nonlinear refinement, the pattern is eliminated from the search (e.g., by adding a cut to the MILP), iterating until a feasible, contact-aligned solution emerges (Shirai et al., 11 Mar 2025).

4. Local and Global Feasibility via Trust Regions and Contact Cones

Guaranteeing local dynamical feasibility in trajectory optimization—especially with contact—is problematic when naive trust regions (simple ellipsoids) permit physically invalid steps (such as small steps that inadvertently leave feasible contact modes). To remedy this:

• Contact Trust Region (CTR):

A set of perturbations $(δq, δu)$ is defined to not only stay within an ellipsoid but also satisfy linearized (primal and/or dual) feasibility for non-penetration and friction cone at the next step (Suh et al., 4 May 2025):

$$\mathcal{S}_{\Sigma,\kappa}(\bar q, \bar u) = \{ (δq, δu) \mid \cdots, \hat{\lambda}_{+,i} \in \mathcal{K}_i^\star \ \forall i \}$$

where $\mathcal{K}_i^\star$ is the dual cone (friction cone in force space), and $\hat{\lambda}_{+,i}$ is the linearized contact force (including sensitivities). Only perturbations guaranteed to preserve physical admissibility are permitted—eliminating unphysical "force at a distance" artifacts.

• Acceleration Feasibility Cone (Wrench Cone):

For multi-contact setups, the force closure is characterized mathematically: the intersection of dual cones is empty if and only if any net force can be achieved at the CoM. Otherwise, CoM acceleration is bounded to a polyhedral subset defined by the wrench constraint matrix (Nikolić et al., 2016):

$$W \cdot \begin{pmatrix} m(a-g) \ \dot{L} \end{pmatrix} \geq 0$$

This characterization allows for efficient computation and rapid re-evaluation as contact configurations change.

• Mixed Nonlinear Complementarity for Non-Point Contacts:

Convex contact patches (planar, line, or surface) are treated via KKT conditions on the closest-point problem, embedded within the complementarity-augmented discrete-time dynamics. This ensures a unique equivalent contact point and globally consistent alignment of force and location for each step (Xie et al., 2020).

5. Integration in Multi-Modal Planning and Real-Time Control

• Monte Carlo Tree Search and Sampling: Dynamical feasibility is enforced only at the terminal node of candidate contact sequences, via full nonlinear MPC simulation and constraint-checking (not via a learned classifier) (Akizhanov et al., 16 Jul 2024). Kinematic checks handle preliminary pruning, but only NMPC verdicts define true feasibility.
• Contact-Aware Feedback Control: Controllers using the complementarity structure incorporate contact force feedback into the control law, yielding stabilization guarantees and ensuring that commanded actions always reside within the feasible hybrid state-contact-force space. Non-smooth Lyapunov functions are constructed over the (state, contact-force) domain to synthesize stabilizing gains that provably maintain feasibility and contact alignment (Aydinoglu et al., 2020).
• Sequential MPC and Global Roadmaps: For dexterous manipulation, contact-rich plans are synthesized locally using MPC under CTR constraints. Global repositioning is achieved by stitching these contact-consistent local plans into a graph, yielding globally feasible, contact-aligned, and dynamically realizable sequences (Suh et al., 4 May 2025).

6. Quantitative Metrics and Empirical Validation

Approach / Metric	Feasibility Guarantee	Contact Alignment Guarantee	Sample Results (where available)
Hierarchical MPCC (Mastalli et al., 2019)	Full dynamic model + complement.	Contact discovery by complementarity	Up to 48% time reduction vs. single full MPCC
Wrench Cone (Nikolić et al., 2016)	Linear program + convex hull	Bounded acceleration cone (polyhedral)	Constraint update in ~0.04 ms / config
CTR-based MPC (Suh et al., 4 May 2025)	Local contact trust region	Linearized dual/primal cone	AllegroHand: 2.2 mm / 9.8 mrad mean error (R-CTR)
MILP/NLP Hierarchy (Shirai et al., 11 Mar 2025)	MILP approx. + NLP refinement	Normals constrained, <5° misalignment	71% success vs. 8–22% for alternative relaxations
Contact-aware control (Aydinoglu et al., 2020)	BMI synthesis over LCS	CLF ensures alignment	87%–100% task completion vs. 49% for LQR (acrobot exp.)

Empirical studies across legged, manipulation, and simulated/hardware platforms characterize the dynamics- and contact-consistent nature of the trajectories, the speed and success rate of the solvers, and the degree of contact misalignment (e.g., angle between actual and planned normals), affirming that enforcing these constraints is both computationally tractable and necessary for physical validity and transferability.

7. Summary and Outlook

Contemporary frameworks for contact-rich robotics generally eschew explicit classification in favor of direct feasibility enforcement using complementarity constraints, trust-region geometry, and hierarchical optimization or control. Complementarity-based modeling automatically discovers feasible transitions and contact alignments, robustly connecting discrete contact-mode combinatorics with continuous-time dynamics. Recent advances in computational relaxation, trust region design, and controller synthesis explicitly encode dynamical feasibility and contact alignment within optimization- and control-loop iterations. Ongoing efforts focus on further reducing computational overhead, extending these principles to stochastic and learning-augmented planners, and achieving formally certified safety and robustness in more general multi-body, multi-contact scenarios (Akizhanov et al., 16 Jul 2024, Suh et al., 4 May 2025, Shirai et al., 11 Mar 2025, Aydinoglu et al., 2020, Xie et al., 2020, Sleiman et al., 2021, Mastalli et al., 2019, Nikolić et al., 2016).