Contact-Implicit Planning

Unlike traditional methods that require a pre-defined sequence of discrete contact events (e.g., foot strike, lift-off), contact-implicit planning treats contact as a continuous decision variable. The optimizer simultaneously solves for the robot's state trajectory, control inputs, and a contact force trajectory. It uses complementarity constraints or relaxed formulations to allow the solution to naturally select between sticking contact, sliding friction, and no contact at every point in time, based purely on the dynamics and cost function.

The core mathematical mechanism that governs contact logic without pre-specification. For each potential contact point, three conditions must be satisfied simultaneously:

• Non-penetration: The distance between the contact point and the surface must be non-negative.
• Non-adhesion: The normal contact force must be non-negative (pushing, not pulling).
• Complementarity: The product of the distance and the normal force must be zero. This enforces that either the distance is zero (in contact) and the force can be positive, or the distance is positive (no contact) and the force must be zero. Similar conditions model Coulomb friction, deciding between stick and slip.

Contact-implicit planning is framed as a single, large-scale Mathematical Program with Complementarity Constraints (MPCC) or a relaxed Nonlinear Program (NLP). A standard formulation includes:

• Decision Variables: System states (q, v), control inputs (u), and contact forces (λ) discretized over time.
• Cost Function: Minimizes effort, time, or deviation from a goal.
• Dynamic Constraints: Enforces the equations of motion: M(q)v̇ + C(q,v) = Sᵀu + J(q)ᵀλ.
• Complementarity/Contact Constraints: As described above.
• Path Constraints: Joint limits, torque bounds, etc. This is typically solved using direct transcription methods like direct collocation.

A key advantage is the ability to discover globally optimal or highly efficient motion strategies that a human engineer might not pre-script. For example, when planning for a monopod hopper on uneven terrain, the optimizer might discover that a brief, intentional foot slip reduces control effort compared to a strictly sticking gait. For a robotic arm manipulating an object, it might find an optimal sequence of rolling, sliding, and re-grasping that minimizes energy. This makes it powerful for novel mechanism design and exploring the full dynamic capabilities of a system.

Solving MPCCs is numerically challenging. The complementarity constraints are non-convex and non-smooth, leading to:

• Ill-conditioned Jacobians and Hessians near contact transitions.
• Multiple local minima (e.g., different contact sequences).
• Sensitivity to initial guesses for the solver. Consequently, solutions often require relaxation techniques (e.g., using a smooth Fischer-Burmeister function) or homotopy methods. Furthermore, small numerical inaccuracies in the planned contact forces can lead to significant sim-to-real transfer issues, as physical actuators cannot instantaneously realize the optimized impulse-like forces.

Contact-implicit planning occupies a distinct point in the design space:

• Vs. Model Predictive Control (MPC): Traditional locomotion MPC uses a pre-defined, fixed contact sequence over its prediction horizon. Contact-implicit planning is the optimization inside a theoretical MPC that also optimizes the contact sequence.
• Vs. Reduced-Order Models (ROMs): Methods based on the Linear Inverted Pendulum Model (LIPM) prescribe contact timing and use analytic solutions. Contact-implicit planning uses the full-order dynamics, making it more general but computationally heavier.
• Vs. Reinforcement Learning (RL): RL can also discover contact sequences through trial-and-error but is a sampling-based approach. Contact-implicit planning is a gradient-based, single-shot optimization for a specific scenario, offering different trade-offs in data efficiency and guarantee.

The overarching mathematical framework for planning motions. It involves finding a sequence of states and control inputs that minimize a cost function (e.g., energy, time) while satisfying system dynamics and constraints. Contact-implicit planning is a specific class of trajectory optimization where the contact sequence is not pre-defined but is part of the optimization variables. This contrasts with contact-explicit methods, which require the engineer to specify when and where contacts occur beforehand.

The mathematical formalism used to model contact mechanics without pre-specifying mode. For a foot and the ground, three conditions must hold simultaneously:

• Non-penetration: The foot cannot go below the ground.
• Non-adhesion: The normal force can only push, not pull.
• Complementary Slackness: Either the foot is in contact (distance=0, force>0) or it is not (distance>0, force=0). These non-linear, non-convex constraints are what make contact-implicit planning computationally challenging but physically accurate.

An alternative formulation for contact planning. Since a foot is either in contact (a binary 1) or not (a binary 0), the problem can be framed as a Mixed-Integer Program (MIP), where discrete variables represent contact modes and continuous variables represent states and forces. While conceptually clear, solving MIPs in real-time for complex robots is often prohibitively slow, leading to the preference for continuous optimization with complementarity constraints in modern contact-implicit approaches.

The real-time control layer that executes planned motions. While contact-implicit planning generates a dynamically feasible trajectory (including contact forces) over a future horizon, Whole-Body Control is the hierarchical, often QP-based controller that computes joint torques to track that plan at the millisecond level. The planned contact forces from the optimizer become reference inputs for the WBC's force control tasks, creating a planning-control pipeline.

The simplified dynamics of the robot's entire mass. Planning for a full humanoid robot is intractable. Contact-implicit planning often uses a centroidal dynamics model, which describes the motion of the center of mass and the angular momentum about it. The optimizer solves for contact forces and CoM motion that satisfy these aggregate dynamics, which are then distributed to individual limbs by the whole-body controller. This abstraction is key to computational tractability.

The critical deployment step. Contact-implicit plans are typically generated in a physics simulator (like MuJoCo, Drake, or Isaac Sim) that can accurately model the complementarity constraints. The success of the planned motion on physical hardware depends entirely on the fidelity of the simulation model. Discrepancies in friction, damping, or actuator dynamics can cause failures, making robust sim-to-real transfer techniques—like domain randomization or adaptive control—essential companions to this planning method.