--- title: Complementarity-Free Contact Modeling url: https://www.emergentmind.com/topics/complementarity-free-contact-modeling type: topic --- # Complementarity-Free Contact Modeling 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 [2408.07855], unconstrained convex compliant contact in primal velocities [2110.10107], smooth and variable-smooth contact laws for trajectory optimization [1806.01425], robust guard-band formulations for uncertain contact timing [2407.11478], overlap-volume energies for finite element contact [2509.08609], barrier-based thickness-aware codimensional contact processing [2510.10256], and GPU-parallel analytical dual-cone simulation [2603.12185]. 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 [2408.07855; 2110.10107]. 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 [2408.07855]. In convex compliant contact, it means analytically eliminating contact constraints to obtain an unconstrained convex objective in the primal variables [2110.10107]. 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 [1806.01425]. 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 [2407.11478]. 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” [2408.07855]. 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 \[ 0 \le \beta \perp \frac{1}{h}\Big[( \tilde{J} Q^{-1} \tilde{J}^T + R ) \beta + ( \tilde{J} Q^{-1} b + \tilde{\phi} )\Big] \ge 0. \] The complementarity-free step replaces \(( \tilde{J} Q^{-1} \tilde{J}^T + R )^{-1}\) by a positive diagonal matrix \(K(q) \succ 0\), giving the elementwise dual update \[ \beta^+ = \max\!\big( -h K(q) [ \tilde{J} Q^{-1} b + \tilde{\phi} ], 0 \big), \] and the explicit time-stepping map \[ v^+ = \frac{1}{h} Q^{-1} b + \frac{1}{h} Q^{-1} \tilde{J}^T \max\!\big( -K(q) [ \tilde{J} Q^{-1} b + \tilde{\phi} ], 0 \big). \] No LCP, NCP, or CCP solve is required per step [2408.07855]. Within this construction, the net contact impulse at contact \(i\) is recovered as \[ h \lambda_i^n = \Big(\sum_{j=1}^{n_d} \beta_{i,j}^+\Big)n_i,\qquad h \lambda_i^t = \mu_i \sum_{j=1}^{n_d} \beta_{i,j}^+ d_{i,j}, \] so that \[ \|\lambda_i^t\| \le \mu_i \lambda_i^n \] by the triangle inequality, meaning that Coulomb’s cone is automatically enforced without explicit cone constraints [2408.07855]. The model is vectorized over all contacts and directions via stacked \(\tilde{J}\) and \(\tilde{\phi}\). Stick–slip transitions and separation occur through the sign structure of \[ z := \tilde{J} Q^{-1} b + \tilde{\phi}, \] with \(\beta^+ = 0\) corresponding to lift-off and negative enough components of \(z\) producing active impulses [2408.07855]. A closely related but more simulation-oriented development appears in ComFree-Sim [2603.12185]. There, contact impulses are computed analytically in the dual cone through a prediction–correction update. For each contact, mode \(s \in \{\mathrm{t},\mathrm{tor},\mathrm{rol}\}\), and polyhedral facet \(j\), the violation signal under the smooth prediction is \[ s^{(j)}_{\mathrm{s}} := \tilde{\mathbf{J}}^{(j)}_{\mathrm{s}} \mathbf{v}^{+}_{\mathrm{smooth}}, \] and the per-facet impulse is \[ \lambda^{(j)}_{\mathrm{s}} = \Big( K^{(j)}_{\mathrm{s}}\big( s^{(j)}_{\mathrm{s}} dt + \tilde{\phi}^{(j)}_{\mathrm{s}} \big) + D^{(j)}_{\mathrm{s}} s^{(j)}_{\mathrm{s}} \Big)_+. \] The corrected velocity is then \[ \mathbf{v}^{+} = \mathbf{v}^{+}_{\mathrm{smooth}} + \mathbf{M}^{-1} \left( \sum_{\mathrm{s}\in\{\mathrm{t},\mathrm{tor},\mathrm{rol}\}} \sum_{j=1}^{n_{\mathrm{s}}} (\tilde{\mathbf{J}}^{(j)}_{\mathrm{s}})^{\!\top}\lambda^{(j)}_{\mathrm{s}}\, dt \right). \] This preserves the complementarity-free structure while extending the contact model to tangential, torsional, and rolling friction in a unified 6D wrench cone [2603.12185]. ## 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” [2110.10107], the rigid normal condition is replaced by a one-sided convex compliance potential \[ \Psi_n(g) := \frac{k_n}{2}[-g]_+^2, \qquad f_n(g) = k_n [-g]_+, \] and Coulomb friction is represented by the dissipation potential \[ \Phi_t(s; p_n) = \mu p_n \|s\| + \frac{c_t}{2}\|s\|^2. \] The resulting midpoint step computes \(v_{k+1}\) as the minimizer of a strongly convex unconstrained objective \[ \min_{v_{k+1}} J(v_{k+1}), \] where \(J\) consists of the kinetic mismatch to free motion plus per-contact normal and tangential potentials [2110.10107]. The corresponding tangential impulse is available analytically through a closed-form SOC projection: \[ p_t = \operatorname{proj}_{\|x\|\le \mu p_n}(c_t s). \] This reproduces sticking when \(\|c_t s\| \le \mu p_n\) and sliding otherwise, without explicit cone constraints [2110.10107]. 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 [2110.10107]. Smooth normal-force laws provide another complementarity-free route in trajectory optimization. In the comparison study of manipulation trajectory optimization [1806.01425], the smooth contact model uses \[ \gamma(q) = k e^{-\alpha \phi(q)}, \] with fixed parameters \(k = 100\), \(c = 5\times 10^3\), and \(\alpha = c/k = 50\). The variable smooth contact model instead treats \(k_l\) as a decision variable and sets \[ \alpha_l = c/k_l,\qquad \gamma_l(q_l) = k_l e^{-(c/k_l)\phi_l(q_l)}, \] with \(k_l \in [0,100]\) and \(c = 5\times 10^3\) [1806.01425]. 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 [1806.01425]. In rigid dual-cone models, non-smoothness from the elementwise max can be smoothed directly. The dexterous manipulation work replaces \(\max(x,0)\) with \[ \operatorname{softplus}(x) = \ln(1 + e^{\gamma x})/\gamma, \] yielding \[ v^+ = \frac{1}{h} Q^{-1} b + \frac{1}{h} Q^{-1} \tilde{J}^T \operatorname{softplus}\!\big( -K(q) [ \tilde{J} Q^{-1} b + \tilde{\phi} ] \big), \] with \(\gamma = 100\) [2408.07855]. With \(\tilde{J}\) and \(\tilde{\phi}\) frozen over an MPC horizon, the derivative with respect to the control is explicit, which is exploited by CasADi + IPOPT in contact-implicit MPC [2408.07855]. An energy-based finite-element variant is the overlap-volume method using Fiber Monte Carlo [2509.08609]. There, the contact energy is \[ E_c(V) = k V^m, \] with \(m = 1\) adopted in the paper, so \[ E_c(V) = kV. \] The contact force is obtained from \[ f_c(q) = -\frac{\partial E_c(V(q))}{\partial q} = -k m V(q)^{m-1}\frac{\partial V(q)}{\partial q}. \] FMC estimates the overlap \[ \hat{V} = \frac{1}{n}\sum_{i=1}^n \int_0^1 \chi_{\Omega_1}(r_i(t)) \chi_{\Omega_2}(r_i(t))\,dt, \] 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 [2509.08609]. ## 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 \(Q\) block diagonal and \(m = n_c n_d\), computing \(z\) costs \(O(mn)\), \(\beta^+\) is elementwise, and \(\tilde{J}^T \beta^+\) also costs \(O(mn)\), so the per-step cost scales linearly in the number of active contact directions \(m\) [2408.07855]. Because no residual equation, QP, or complementarity problem is solved per step, the map is particularly suited to MPC [2408.07855]. The same paper formulates contact-implicit MPC as \[ \min_{u_{0:T-1}\in[u_{lb},u_{ub}]} \sum_{t=0}^{T-1} c(q_t,u_t) + V(q_T) \] subject to \[ q_{t+1} = q_t “+” h v_t^+,\qquad v_t^+ = \mathrm{CF\!-\!smooth}(q_t,u_t), \] with \(\tilde{J}\) and \(\tilde{\phi}\) frozen at \(q_0\) for the horizon and only \(u_0^*\) 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 \(\sim 20\) iterations per solve [2408.07855]. The SAP formulation in Drake [2110.10107] solves an unconstrained strongly convex problem each step by assembling analytic contact contributions, computing \(\nabla J\) and an SPD approximation \(H\), solving \[ H \Delta v = -\nabla J \] by sparse Cholesky, and using exact line search or Armijo backtracking [2110.10107]. Strong convexity of \(J\) gives a unique global minimizer and global convergence of descent/Newton-like methods [2110.10107]. In this case, the contact model remains non-smooth at transition manifolds such as \(g=0\) and \(s=0\), but the optimization problem is still unconstrained and convex [2110.10107]. 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 \[ g_i(\mathbf{x}_i)=d \quad (i=\mathcal{K}_0),\qquad g_i(\mathbf{x}_i)=-d \quad (i=\mathcal{K}_{-1}),\qquad \dot{g}_i(\mathbf{x}_i)\le 0 \quad \forall i\in\mathcal{K}, \] while retaining smooth pre-contact dynamics and guard inequalities elsewhere [2407.11478]. The result is a standard nonlinear programming problem solved with CasADi Opti stack and IPOPT, without complementarity constraints or integer variables [2407.11478]. Codimensional IPC with barrier filtering also fits this pattern. The step minimizes \[ \frac{1}{2h^2}(q^{t+1}-\tilde{q})^T M (q^{t+1}-\tilde{q}) + E_{\mathrm{elastic}}(q^{t+1}) + \sum_{(i,j)\in C} b(d_{ij}(q^{t+1}), a_{ij}, \eta_{ij}) + E_{\mathrm{friction}}(q^{t+1}), \] where the barrier \[ b(d,a)= \begin{cases} -(d-a)^2 \ln(d/a), & 0