Complementarity-Free Contact Modeling

• Complementarity-Free Contact Modeling is a design principle that eliminates explicit complementarity constraints by using explicit maps, convex penalties, or smooth nonlinear programs.
• It employs methods such as closed-form dual updates, smoothing functions, and convex compliance to enable efficient time-stepping, trajectory optimization, and robust multi-contact simulation.
• Applications in dexterous manipulation, MPC, and finite-element analysis demonstrate high success rates while addressing challenges like approximation errors and boundary effects.

Searching arXiv for the cited papers and related work on complementarity-free contact modeling. Complementarity-free contact modeling denotes a family of contact formulations that avoid explicit complementarity conditions such as normal force–gap complementarity, frictional stick–slip complementarity, or discrete mode selection, while still representing unilateral contact and, in many cases, Coulomb friction. Across recent work, the term encompasses several distinct but related constructions: dual-cone closed-form time stepping for dexterous manipulation (Jin, 2024), unconstrained convex compliant contact in primal velocities (Castro et al., 2021), smooth and variable-smooth contact laws for trajectory optimization (Onol et al., 2018), robust guard-band formulations for uncertain contact timing (Zhao et al., 2024), overlap-volume energies for finite element contact (Wang et al., 10 Sep 2025), barrier-based thickness-aware codimensional contact processing (Gomez-Nogales et al., 11 Oct 2025), and GPU-parallel analytical dual-cone simulation (Borse et al., 12 Mar 2026). The common objective is to replace non-smooth complementarity systems and combinatorial mode switching with explicit maps, convex penalties, or smooth nonlinear programs that are more amenable to differentiation, optimization, scalability, and real-time control.

1. Conceptual basis and relation to classical contact models

Classical rigid contact is commonly expressed through unilateral gap constraints and Coulomb friction together with “no force at a distance” and stick–slip complementarity. In the Signorini form, these conditions are written as

$g \ge 0,\quad f_n \ge 0,\quad g\, f_n = 0,$

with Coulomb friction given by

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$

Discrete time-stepping schemes such as Stewart–Trinkle, Anitescu–Potra, and Moreau-type formulations lead to LCPs, NCPs, CCPs, or constrained conic programs. These formulations are non-smooth and exhibit combinatorial mode switching between separation, stick, and slip (Jin, 2024, Castro et al., 2021).

Complementarity-free formulations remove these constructs rather than solving them more efficiently. In the dexterous manipulation setting, this means expressing contact interactions without explicit complementarity constraints or mode selection and replacing them with a closed-form “spring-like” law in the dual friction cone (Jin, 2024). In convex compliant contact, it means analytically eliminating contact constraints to obtain an unconstrained convex objective in the primal variables (Castro et al., 2021). In trajectory optimization, it can mean using a smooth normal-force law

$\gamma(q) = k e^{-\alpha \phi(q)}$

or its variable-smooth extension

$\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$

thereby removing complementarity constraints from the nonlinear program (Onol et al., 2018). In robust hybrid optimal control, complementarity-free may refer specifically to avoiding complementarity constraints in the optimization by staying in pre-contact dynamics and enforcing guard-band traversal constraints instead (Zhao et al., 2024).

This suggests that “complementarity-free” is not a single model class but a design principle. The unifying feature is the replacement of KKT-style unilateral contact logic by explicit, convex, or smooth mechanisms that preserve tractability while retaining some or all of the physical structure of contact.

2. Dual-cone closed-form models for rigid multi-contact dynamics

A prominent rigid-body instance is the formulation in “Complementarity-Free Multi-Contact Modeling and Optimization for Dexterous Manipulation” (Jin, 2024). The starting point is a quasi-dynamic time-stepping scheme for a combined object–robot system with stacked velocity $v = [v_o; v_r]$, time step $h$, object mass regularization $\epsilon > 0$, and robot impedance $K_r$, with

$$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$

Using a polyhedral approximation of the dual friction cone, the primal QP is

$$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$

Its regularized dual yields the complementarity system

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$0

The complementarity-free step replaces $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$1 by a positive diagonal matrix $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$2, giving the elementwise dual update

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$3

and the explicit time-stepping map

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$4

No LCP, NCP, or CCP solve is required per step (Jin, 2024).

Within this construction, the net contact impulse at contact $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$5 is recovered as

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$6

so that

$$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$7

by the triangle inequality, meaning that Coulomb’s cone is automatically enforced without explicit cone constraints (Jin, 2024). The model is vectorized over all contacts and directions via stacked $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$8 and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$9. Stick–slip transitions and separation occur through the sign structure of

$\gamma(q) = k e^{-\alpha \phi(q)}$0

with $\gamma(q) = k e^{-\alpha \phi(q)}$1 corresponding to lift-off and negative enough components of $\gamma(q) = k e^{-\alpha \phi(q)}$2 producing active impulses (Jin, 2024).

A closely related but more simulation-oriented development appears in ComFree-Sim (Borse et al., 12 Mar 2026). There, contact impulses are computed analytically in the dual cone through a prediction–correction update. For each contact, mode $\gamma(q) = k e^{-\alpha \phi(q)}$3, and polyhedral facet $\gamma(q) = k e^{-\alpha \phi(q)}$4, the violation signal under the smooth prediction is

$\gamma(q) = k e^{-\alpha \phi(q)}$5

and the per-facet impulse is

$\gamma(q) = k e^{-\alpha \phi(q)}$6

The corrected velocity is then

$\gamma(q) = k e^{-\alpha \phi(q)}$7

This preserves the complementarity-free structure while extending the contact model to tangential, torsional, and rolling friction in a unified 6D wrench cone (Borse et al., 12 Mar 2026).

3. Convex, smooth, and energy-based alternatives

A distinct line of work eliminates contact constraints in the primal rather than the dual. In “An Unconstrained Convex Formulation of Compliant Contact” (Castro et al., 2021), the rigid normal condition is replaced by a one-sided convex compliance potential

$\gamma(q) = k e^{-\alpha \phi(q)}$8

and Coulomb friction is represented by the dissipation potential

$\gamma(q) = k e^{-\alpha \phi(q)}$9

The resulting midpoint step computes $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$0 as the minimizer of a strongly convex unconstrained objective

$\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$1

where $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$2 consists of the kinetic mismatch to free motion plus per-contact normal and tangential potentials (Castro et al., 2021). The corresponding tangential impulse is available analytically through a closed-form SOC projection: $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$3 This reproduces sticking when $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$4 and sliding otherwise, without explicit cone constraints (Castro et al., 2021). The paper emphasizes that all contact “constraints” are eliminated analytically and reappear as convex per-contact penalties with closed-form reactions, with the scheme implemented in Drake (Castro et al., 2021).

Smooth normal-force laws provide another complementarity-free route in trajectory optimization. In the comparison study of manipulation trajectory optimization (Onol et al., 2018), the smooth contact model uses

$\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$5

with fixed parameters $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$6, $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$7, and $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$8. The variable smooth contact model instead treats $\gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)},$9 as a decision variable and sets

$v = [v_o; v_r]$0

with $v = [v_o; v_r]$1 and $v = [v_o; v_r]$2 (Onol et al., 2018). These models are unconstrained at the level of the optimizer; friction is handled internally by MuJoCo during forward simulation rather than by optimizer-side stick–slip constraints (Onol et al., 2018).

In rigid dual-cone models, non-smoothness from the elementwise max can be smoothed directly. The dexterous manipulation work replaces $v = [v_o; v_r]$3 with

$v = [v_o; v_r]$4

yielding

$v = [v_o; v_r]$5

with $v = [v_o; v_r]$6 (Jin, 2024). With $v = [v_o; v_r]$7 and $v = [v_o; v_r]$8 frozen over an MPC horizon, the derivative with respect to the control is explicit, which is exploited by CasADi + IPOPT in contact-implicit MPC (Jin, 2024).

An energy-based finite-element variant is the overlap-volume method using Fiber Monte Carlo (Wang et al., 10 Sep 2025). There, the contact energy is

$v = [v_o; v_r]$9

with $h$0 adopted in the paper, so

$h$1

The contact force is obtained from

$h$2

FMC estimates the overlap

$h$3

and differentiates boundary intersection parameters by the implicit function theorem, producing a differentiable, complementarity-free frictionless normal contact force without master–slave pairing or local projection iterations (Wang et al., 10 Sep 2025).

4. Time stepping, differentiability, and optimization structure

Complementarity-free formulations are often motivated by time stepping and optimization rather than by constitutive novelty alone. The dexterous manipulation model yields a fully closed-form dual update and explicit time-stepping law; with $h$4 block diagonal and $h$5, computing $h$6 costs $h$7, $h$8 is elementwise, and $h$9 also costs $\epsilon > 0$0, so the per-step cost scales linearly in the number of active contact directions $\epsilon > 0$1 (Jin, 2024). Because no residual equation, QP, or complementarity problem is solved per step, the map is particularly suited to MPC (Jin, 2024).

The same paper formulates contact-implicit MPC as

$\epsilon > 0$2

subject to

$\epsilon > 0$3

with $\epsilon > 0$4 and $\epsilon > 0$5 frozen at $\epsilon > 0$6 for the horizon and only $\epsilon > 0$7 applied in receding horizon. The solver stack is CasADi + IPOPT, collision detection uses MuJoCo/libccd, and the paper reports that differentiability is exploited by IPOPT to converge in $\epsilon > 0$8 iterations per solve (Jin, 2024).

The SAP formulation in Drake (Castro et al., 2021) solves an unconstrained strongly convex problem each step by assembling analytic contact contributions, computing $\epsilon > 0$9 and an SPD approximation $K_r$0, solving

$K_r$1

by sparse Cholesky, and using exact line search or Armijo backtracking (Castro et al., 2021). Strong convexity of $K_r$2 gives a unique global minimizer and global convergence of descent/Newton-like methods (Castro et al., 2021). In this case, the contact model remains non-smooth at transition manifolds such as $K_r$3 and $K_r$4, but the optimization problem is still unconstrained and convex (Castro et al., 2021).

In uncertain-contact optimal control, complementarity-free structure appears at the transcription level. The robust-phase formulation introduces an uncertainty band around a guard and requires

$K_r$5

while retaining smooth pre-contact dynamics and guard inequalities elsewhere (Zhao et al., 2024). The result is a standard nonlinear programming problem solved with CasADi Opti stack and IPOPT, without complementarity constraints or integer variables (Zhao et al., 2024).

Codimensional IPC with barrier filtering also fits this pattern. The step minimizes

$K_r$6

where the barrier

$K_r$7

or its biphasic variant $K_r$8 replaces explicit non-penetration constraints by a smooth objective term with ACCD-based line search (Gomez-Nogales et al., 11 Oct 2025). This remains complementarity-free while preserving non-intersection guarantees (Gomez-Nogales et al., 11 Oct 2025).

5. Applications and empirical performance

The dexterous manipulation study evaluates fingertip 3D in-air manipulation, TriFinger in-hand manipulation, and Allegro hand on-palm reorientation over 17 diverse objects from ContactDB scaled to hand workspace, with diameters $K_r$9 mm (Jin, 2024). It reports a 96.5% average success rate across all objects and tasks, an average reorientation error of $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$0, an average position error of $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$1 mm, and contact-implicit MPC running at $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$2 Hz for all objects and tasks (Jin, 2024). The paper further reports 97.0% $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$3 4.5% overall success rate and 13 $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$4 3 ms average solve time for TriFinger, and 97.64% $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$5 5.18% overall success rate with 19 $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$6 3 ms average solve time for Allegro hand (Jin, 2024). The stated tuning burden is low: $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$7 or $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$8, $$Q := \operatorname{diag}( \epsilon M_o / h^2, K_r ),\qquad b(u) := [ \tau_o; K_r u + \tau_r ].$$9, $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$0, and robust performance over $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$1 (Jin, 2024).

In smooth trajectory optimization for pushing, the variable smooth contact model is the only method reported to have consistently satisfactory position error $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$2 across all initial distances $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$3 m (Onol et al., 2018). At $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$4 m, the reported final position error is 0.0836 m for VSCM compared with 0.1908 m for SCM and 0.1924 m for CCCM, and the final orientation error is 0.0415 rad for VSCM compared with 0.4547 rad for SCM and 0.3475 rad for CCCM (Onol et al., 2018). The study concludes that the proposed variable smooth contact model provides a good trade-off between physical fidelity and quality of motions at the cost of increased computation time (Onol et al., 2018).

The robust guard-band approach under contact timing uncertainty reports approximately 30–50% reduction in hopper impact compared to nominal optimal control over roughly 70% of the uncertain region when impact minimization is used (Zhao et al., 2024). In cart-pole stabilization via wall impact, the robust approach attains higher success rates than nominal in both early and late contact scenarios (Zhao et al., 2024). These results do not model contact through a smooth penalty; rather, they avoid complementarity by optimizing only over pre-contact trajectories and contact-relevant robustness criteria (Zhao et al., 2024).

In finite elements, the FMC overlap-energy method reproduces Hertzian pressure distributions, wedge and cone indentation responses, large-deformation hyperelastic contact, and dynamic collisions (Wang et al., 10 Sep 2025). The paper reports maximum relative errors of 0.036% for 2D circle area and 0.322% for its derivative, and 0.039% for 3D sphere volume and 0.225% for its derivative in geometric verification tests (Wang et al., 10 Sep 2025). The conical indentation benchmark reports a predicted contact force of 17.4 N, compared with the analytic 14.5 N and 19.5 N from Abaqus (Wang et al., 10 Sep 2025).

ComFree-Sim extends complementarity-free rigid contact to GPU-parallel dense-contact simulation (Borse et al., 12 Mar 2026). It reports near-linear runtime scaling with contact count, approximately linear step-time scaling in dense convex-primitive scenes, and $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$5 higher throughput than MJWarp in dense contact scenes (Borse et al., 12 Mar 2026). In a penetration-depth benchmark with $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$6 s, MJWarp reports mean $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$7 mm $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$8 mm, while ComFree-Sim reports $$\min_v \;\frac{1}{2} h^2 v^T Q v - h v^T b(u) \quad \text{subject to}\quad \tilde{J} v + \tilde{\phi}/h \ge 0.$$9 mm $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$00 mm for $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$01 and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$02 mm $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$03 mm for $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$04 (Borse et al., 12 Mar 2026). In real-time MPC on a real-world multi-fingered LEAP hand, MPPI with ComFree-Sim achieves 13.9–28.2 ms per MPC step, enabling $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$05 Hz control, and reports +27 percentage points on average over eight tasks relative to MJWarp (Borse et al., 12 Mar 2026).

For codimensional rods and shells, barrier-filtering integrated with IPC/C-IPC guarantees “contact-locking-free, non-intersecting” simulations while removing resolution restrictions that otherwise produce nonphysical self-contact forces between stencil-neighboring elements (Gomez-Nogales et al., 11 Oct 2025). The paper reports zero intersections, locking-free behavior, preserved patterns under dynamic exercises, and superior robustness to culling-based alternatives, which can generate intersections, tunneling, and pull-through (Gomez-Nogales et al., 11 Oct 2025).

6. Limitations, trade-offs, and open questions

Complementarity-free models do not remove approximation; they relocate it. In the dual-cone dexterous manipulation model, replacing $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$06 by diagonal $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$07 decouples directions and is not the exact dual KKT operator, so $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$08 may differ from the primal QP solution (Jin, 2024). The same work notes possible boundary effects, potential benefit from damping in highly dynamic cases, and degradation when $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$09 and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$10 are frozen over long horizons with large motions (Jin, 2024). Failure modes include self-collision near workspace boundaries, large objects relative to hand or palm, and poor choices of $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$11 that make the dual response too soft or too stiff (Jin, 2024).

The convex compliant SAP formulation replaces rigid complementarity by compliance, so penetration is limited by $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$12 and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$13 rather than prohibited exactly (Castro et al., 2021). It also exhibits gliding artifacts characteristic of compliant models, with micro-slip scaling with $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$14 and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$15, and may require user overrides of stiffness heuristics in deep contact stacks (Castro et al., 2021). Its strengths are global convexity and physical interpretability of $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$16, $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$17, and $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$18, but it is not a rigid exact-contact method (Castro et al., 2021).

Smooth exponential contact laws aid contact discovery from poor initial guesses, yet the pushing study documents virtual forces acting at a distance and arm repulsion when smoothing is too aggressive (Onol et al., 2018). In SCM, the object can begin moving before actual contact due to virtual forces; VSCM reduces this effect by letting $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$19 go to zero away from contact, but adds decision variables and can increase optimization time (Onol et al., 2018).

The robust timing-uncertainty method avoids complementarity only in the optimization, not necessarily in downstream impact evaluation. Its limitations are conservatism, dependence on approximated viability constraints such as convex hulls and stopping-distance fits, nonconvexity, and restriction to pre-contact robustness rather than full hybrid optimality across multiple uncertain switches (Zhao et al., 2024).

The FMC overlap-volume method is currently frictionless; tangential frictional contact is not modeled in the present formulation (Wang et al., 10 Sep 2025). It also lacks a consistent contact tangent $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$20 for monolithic Newton, and estimator variance depends on the number of fibers $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$21 and fiber length $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$22 (Wang et al., 10 Sep 2025). Local oscillations may appear near highly non-smooth features due to abrupt changes in the overlap region and sampling noise (Wang et al., 10 Sep 2025).

ComFree-Sim inherits the strengths and limitations of an analytical dual-cone surrogate. Very high friction coefficients or extremely stiff contacts may require smaller $$\|f_t\| \le \mu f_n,\quad s = 0 \Rightarrow f_t \in \{\|f_t\| \le \mu f_n\},\quad s \neq 0 \Rightarrow f_t = -\mu f_n\, \frac{s}{\|s\|}.$$23 or carefully tuned impedance to avoid chatter, and scenarios demanding exact complementarity enforcement or precise stick–slip transitions under tight tolerances may still favor optimization-based solvers (Borse et al., 12 Mar 2026). The paper also notes that compliant or deformable contacts and micro-slip phenomena are only approximated through torsional and rolling cones rather than modeled from first principles (Borse et al., 12 Mar 2026).

In codimensional simulation, barrier filtering preserves IPC robustness while altering pairwise activation distances for locking-prone internal stencil pairs (Gomez-Nogales et al., 11 Oct 2025). This is a practical contact-processing model rather than a first-principles constitutive derivation of thickness. A plausible implication is that future work may need tighter coupling between transverse compression, thickness-aware barriers, and constitutive rod or shell mechanics in order to reduce the gap between geometric robustness and material modeling (Gomez-Nogales et al., 11 Oct 2025).

Taken together, these developments show that complementarity-free contact modeling has become a broad methodological category rather than a single algorithmic doctrine. Some methods aim for closed-form rigid updates in the dual cone (Jin, 2024, Borse et al., 12 Mar 2026); some replace unilateral constraints by convex compliance and proximal maps (Castro et al., 2021); some use smooth or variable-smooth distance-based forces for trajectory optimization (Onol et al., 2018); some reformulate uncertain contact timing as a robust pre-contact NLP (Zhao et al., 2024); some derive forces from overlap-volume energies (Wang et al., 10 Sep 2025); and some preserve barrier-based non-intersection while repairing thickness-induced locking in codimensional geometries (Gomez-Nogales et al., 11 Oct 2025). The shared theme is the elimination of explicit complementarity from the computational core in favor of explicit maps, convex penalties, or smooth optimization landscapes that better support differentiation, real-time control, scalable simulation, and robust numerical behavior.