Contact-Implicit MPC for Robotic Control

• Contact-Implicit MPC is a control paradigm that integrates contact dynamics directly into trajectory optimization, enabling robots to autonomously plan contact events in tasks like legged locomotion and dexterous manipulation.
• It employs complementarity constraints and differentiable contact models to seamlessly handle hybrid dynamics and switch between contact modes without explicit scheduling.
• Recent implementations utilize bi-level planning, consensus solvers, and adaptive estimation techniques to maintain real-time performance and robustness under disturbances and model mismatches.

Contact-Implicit Model Predictive Control (CI-MPC) is a trajectory optimization and control paradigm in robotics that integrates contact modeling directly into the predictive control problem. Unlike conventional MPC, which typically handles contact forces as external inputs or relies on explicit scheduling of contact modes, CI-MPC treats contacts as endogenous phenomena—allowing robots to autonomously discover when, where, and how contacts should occur during task execution. This integration is achieved through the use of complementarity constraints or differentiable contact models, enabling real-time planning and control in legged locomotion, dexterous manipulation, and interactive tasks subject to mode-switching dynamics.

1. Mathematical Foundations and Contact Modeling

CI-MPC is typically formulated as a finite-horizon optimal control problem with embedded contact dynamics. The state $x_k$, control $u_k$, and contact force $\lambda_k$ evolve according to both the robot's equations of motion and contact laws. Common choices for contact modeling include:

• Linear Complementarity Problems (LCPs)/Linear Complementarity Systems (LCS):

$x_{k+1} = A x_k + B u_k + D \lambda_k + d$

subject to

$$0 \le \lambda_k \perp E x_k + F \lambda_k + H u_k + c \ge 0$$

These complementarity relations encode unilateral constraints (e.g., non-penetration and friction).

• Nonlinear and Smooth Contact Models:

Models such as compliant spring-damper, regularized friction, and convex-cone complementarity are used for better numerical conditioning; e.g.

$\lambda_{normal} = f(D(p_{ee}))$

with $f$ typically a tanh- or barrier-based function acting on signed distance $D(p_{ee})$.

• Hard Contact via Differential Dynamic Programming (DDP):

Contact forces are solved using optimization problems (e.g., quadratic programs subject to Signorini–Coulomb constraints) during trajectory rollout:

$\lambda^{contact} = -A^{-1}b$

with gradients adapted for smooth exploration of contact modes.

Embedding contact physics directly allows CI-MPC to plan through contact transitions—impact, stick-slip, rolling—without the need for explicit mode schedules.

2. Algorithmic Architectures and Solver Design

Recent CI-MPC implementations employ bi-level or consensus architectures to decouple trajectory optimization from contact scheduling:

• Bi-level Planning (cf. (Cleac'h et al., 2021, Liu et al., 3 Nov 2024)):

Upper-level optimizes the motion trajectory, while the lower-level solves, on each prediction step, for the contact dynamics via LCP or other models. State is often lifted to consecutive configurations to accommodate contact events.

• Consensus Complementarity Control (C3, C3+; cf. (Aydinoglu et al., 2023, Bui et al., 22 Oct 2025)):

Reformulate the MPC problem such that each time step’s contact projection is solved independently, leveraging ADMM to alternate between quadratic trajectory optimization and parallel contact projection. This enables high-frequency MPC and rapid mode reasoning.

• Sampling-Based Global Augmentation (Venkatesh et al., 19 May 2025):

Augment local CI-MPC by sampling global candidate end-effector locations; each sample is evaluated under local contact-rich MPC, providing a route to overcome local minima.

• Inverse Dynamics Trajectory Optimization (Kurtz et al., 2023):

Formulation in terms of positions only, with compliant contact models and custom trust-region solvers, further reduces dimensionality and improves real-time performance.

A central algorithmic challenge is to maintain computational efficiency—solving nonconvex problems with hybrid constraints—while robustly tracking trajectories and handling disturbances in real time. Innovations such as structure-exploiting interior-point solvers (Cleac'h et al., 2021), consensus projections (Bui et al., 22 Oct 2025), and parallelization of line searches or projection steps (Kong et al., 2022, Aydinoglu et al., 2023) are key to reported performance.

3. Adaptive Estimation and Online System Identification

CI-MPC often integrates adaptive estimation for improved robustness in unknown or changing environments:

• Online System Identification (MIAC, (Minniti et al., 2021)):

Model parameters (mass, stiffness, damping, friction) of the environment are estimated in real time using Kalman filtering or gradient-based updating, then injected into the MPC model for online adaptation.

• Model Reference Adaptive Control (MRAC, (Minniti et al., 2021)):

Augments MPC with adaptive terms that minimize trajectory error, using Lyapunov-based adaptation laws:

$$\dot{\hat{\theta}} = \Gamma\, (\text{regressor})^T \sigma$$

yielding robust tracking even with imprecise force measurement.

• Online Residual Learning (Huang et al., 2023):

Introduces a residual term to correct the hybrid contact model, updated at rates up to 20 Hz via mini-batch gradient descent, reducing the gap between prior models and real system behavior.

• Differentiable Parameter Estimation (Haninger et al., 2023):

Uses differentiable, compliant contact primitives and extended Kalman filters to jointly estimate robot state and contact parameters, allowing the MPC to adapt in force-rich manipulation tasks.

These adaptive mechanisms enable CI-MPC to generalize across objects, morphologies, and contact conditions without manual re-tuning or offline model fitting.

4. Applications and Experimental Validation

CI-MPC has demonstrated efficacy across a range of robotic domains:

Application Domain	Key Experiments/Outcomes	Reference
Legged Locomotion	Quadruped robots dynamically switching gaits, robust to payload variation	(Cleac'h et al., 2021, Kim et al., 2023)
Dexterous Manipulation	In-hand rotation, multi-object planar pushing, non-prehensile manipulation	(Jiang et al., 29 Feb 2024, Bui et al., 22 Oct 2025)
Mobile Manipulation	Ballbot door opening, robust object lifting	(Minniti et al., 2021)
Humanoid Loco-Manipulation	Package transfer, object throwing while walking	(Li et al., 2022, Esteban et al., 21 Feb 2025)
Obstacle Avoidance	Ballbot uses arms to push off walls, achieves rapid maneuvers	(Liu et al., 3 Nov 2024)

Metrics reported include tracking error (RMSE, final error), control frequency (10 Hz to 6.8 kHz depending on task and algorithm), and success rates (up to 98–99.9% in planar pushing tasks (Bui et al., 22 Oct 2025)). CI-MPC is robust to large disturbances, model mismatch, actuator noise, and abrupt task changes.

5. Technical Features and Design Variants

Distinctive technical features and design decisions in CI-MPC literature include:

• Relaxed Complementarity and Smooth Gradients (Kim et al., 2023):

Analytical gradients computed through relaxed complementarity ($v^n \lambda^n = \rho$) enable reliable exploration of contact mode sequences, facilitating discovery of non-periodic and acyclic contact events (e.g., jumping, rearing, pushing).

• Multi-Contact Feedback Loops (Han et al., 13 Mar 2024):

Integration with high-frequency particle filter algorithms (MCP-EP) allows for real-time estimation of unknown contact points and forces.

• Hybrid Model Integration (Kong et al., 2022, Liu et al., 3 Nov 2024):

Use of hybrid (piecewise) dynamics and event-driven simulation, together with cost function extension via saltation matrices, captures discontinuities in contact transitions and improves stabilization after impacts.

• Reduced-Order Model Guidance (Esteban et al., 21 Feb 2025):

Reference motion from simplified models such as HLIP or LIP provides nominal gait parameters, whereas CI-MPC refines whole-body motion and contact schedule for robust adaptation to terrain and obstacles.

• Differentiable Compliant Contact Models (Haninger et al., 2023, Jiang et al., 29 Feb 2024):

Smoothing contact dynamics enables gradient-based optimization, aiding both real-time MPC and adaptive estimation for variable contact geometry and stiffness.

• Sampling for Global Exploration (Venkatesh et al., 19 May 2025):

Sampling end-effector positions enables CI-MPC to transcend local minima in non-convex tasks such as multi-object manipulation.

6. Computational and Practical Considerations

Reported computational advances permit the use of CI-MPC in real time across large systems (e.g., 24-DOF humanoids at 50 Hz (Esteban et al., 21 Feb 2025), bi-manual arms at 100 Hz (Kurtz et al., 2023), planar pushing at sub-second per loop (Bui et al., 22 Oct 2025)). Analytical 1D projections (C3+), parallelized MIQP solves, consensus formulations, and tailored trust-region/dogleg solvers contribute to these achievements.

Practical recommendations include:

• Prefer soft contact modeling and slack/decomposition when possible to mitigate ill-conditioning.
• Incorporate multi-stage planning (contact-free then contact-rich) and sampling for complex environments.
• Maintain adaptive mechanisms for unknown or time-varying contact properties.
• Use high-frequency feedback and real-time estimation, especially when multi-contact and unstructured environments are present.

7. Future Directions and Implications

Current challenges for CI-MPC include further improving global optimality, broadening to more dynamic tasks (fast locomotion, agile manipulation), scaling to higher-dimensional contact scenarios (multi-fingered hands, multi-limb robots), and integration with vision-based perception (robust tracking under occlusion and clutter). Promising directions are:

• Automated trajectory generation and learning-based contact modeling for better generality.
• Integration of additional safety guarantees via control barrier functions and formal hybrid system analysis.
• Extension to 3D multi-contact manipulation, leveraging advances in mesh-based sampling and high-throughput pose estimation.
• Combined hierarchical architectures mixing reduced-order reference generation with whole-body dynamic adaptation.

The reviewed advances collectively point to CI-MPC as a foundation for next-generation robotic control—capable of adaptive, robust, and general-purpose interaction in both locomotion and manipulation, even under severe uncertainty in contact dynamics and environmental properties.