Dynamic Multi-Contact Motions in Robotics

• Dynamic multi-contact motions are advanced robotic movements involving simultaneous, sequential, or rapid contact switching for tasks like manipulation and locomotion.
• They leverage complementarity modeling and mode-free control synthesis to ensure stability, adaptability, and real-time performance in complex, underactuated environments.
• Experimental validations on humanoid, quadruped, and manipulation systems demonstrate effective hierarchical planning, model predictive control, and scalable simulation.

Dynamic multi-contact motions describe robot behaviors that involve simultaneous, sequential, or rapidly changing interactions with multiple contact points in their environment. Complex tasks in manipulation, locomotion, parkour, and dexterous control require physical consistency, stability, and adaptability as robots make and break contacts—often under high-speed, underactuated conditions and in unstructured settings. The modern literature develops mathematical models, control architectures, and optimization frameworks that fully capture the hybrid, combinatorial and complementarity structure of multi-contact dynamics, moving decisively beyond heuristic mode switching.

1. Mathematical Modeling of Multi-Contact Dynamics

Dynamic multi-contact motions fundamentally rely on physical models that encode the instantaneous interactions between the robot and environment. At the core is the manipulator equation: $M(q)\,\dot{v} + C(q,v) = B\,u + J(q)^\top\,\lambda$ where $q$ are generalized coordinates, $v$ velocity, $u$ control input, and $\lambda$ the vector of active contact forces (Aydinoglu et al., 2019). Contact events are represented via complementarity conditions: $0 \leq \lambda \perp \phi(q) \geq 0$ with $\phi(q)$ the signed contact gap and $\lambda_i\,\phi_i=0$ enforcing both non-interpenetration and unilateral (no-adhesion) contact.

Linearization about nominal trajectories yields a Linear Complementarity System (LCS): $\dot{x} = A x + B u + D \lambda$

$0 \leq \lambda \perp E x + F \lambda + c \geq 0$

Uniqueness and well-posedness are guaranteed if $F$ is a $P$-matrix. By substituting the contact force solution $\lambda = \psi(E x + c)$, the closed-loop dynamics become a piecewise-affine ODE with non-smooth but globally Lipschitz right-hand side.

Complementarity modeling generalizes naturally to multi-contact rigid-body and floating-base systems, underpinning advanced methods in trajectory optimization, value-function computation, and receding-horizon control (Aydinoglu et al., 2019, Kim et al., 2023, Aydinoglu et al., 2023).

2. Control Synthesis Strategies for Multi-Contact Motions

Mode-free control synthesis is critical for real-time feasibility and robustness. Instead of enumerating $2^m$ discrete contact modes, modern frameworks embed the complementarity structure directly in the control law. Specifically,

$u(x, \lambda) = K x + L \lambda$

with $K$ and $L$ as gain matrices. This yields a continuous, piecewise-affine policy in the state $x$ (Aydinoglu et al., 2019).

Stability is certified via piecewise-quadratic Lyapunov functions,

$$V(x, \lambda) = \left[\begin{array}{cc} x^\top & \lambda^\top \end{array}\right] M \left[\begin{array}{c} x \ \lambda \end{array}\right]$$

where $M$ is block positive-definite. The stabilization condition evaluates the derivative $V'(x;\dot{x})$ along closed-loop trajectories, encoding controller gains and system parameters as a bilinear matrix inequality (BMI). Off-the-shelf BMI solvers typically yield certificates in minutes for compact systems. Tactile and force sensors feed real-time measurement of $\lambda$ directly into the feedback channel, eliminating ad hoc mode scheduling.

Complementarity-based control design connects seamlessly to modern contact-implicit MPCs and DDP variants. In particular:

• Contact-implicit DDP merges hard contact simulation with smooth analytic gradients from relaxed complementarity constraints, permitting rapid exploration of new contact modes without pre-scheduled gait patterns (Kim et al., 2023).
• Consensus Complementarity Control (C³) applies ADMM with separable per-time-step contact projections, enabling parallelizable, hybrid MPC at rates up to 100 Hz (Aydinoglu et al., 2023).

3. Hierarchical and Sample-Efficient Planning for Multi-Contact Behaviors

Agile behaviors in legged and manipulation tasks necessitate co-optimization of contact sequencing and whole-body trajectories. Hierarchical optimization decomposes the discrete contact plan from the continuous motion, often with novel convexification techniques:

• MILPs select feasible contact schedules and approximate wrenches subject to tight convex relaxation of bilinear terms via binary encoding (Shirai et al., 11 Mar 2025).
• Downstream NLP refines robot and object trajectories under full second-order dynamics and complementarity contact laws.

Sampling-based planners (e.g., Monte-Carlo tree search or T-RRT) coupled with powerful local trajectory optimizers find acyclic, dynamically feasible contact sequences and patch assignments within tens of seconds—even for high-DOF robots navigating cluttered environments (Dhédin et al., 18 Aug 2025, Zhang et al., 2023). Infinite-programming and exchange algorithms dynamically instantiate only the necessary candidate contacts during local planning.

Kino-dynamic alternating optimization further enables consensus between centroidal momentum plans and body kinematics at modest computational cost (Ponton et al., 2020).

4. Model Predictive Control and Real-Time Multi-Contact Execution

Multi-contact MPC architectures vary in their approach to contact mode handling:

• Explicit scheduling: MPCs with pre-programmed contact activation functions (binary schedules $\sigma_{Ci}(t)$) encode foot, hand, and object contacts, maintaining a time-indexed but convex QP for efficient real-time solution (Li et al., 2022, Murooka et al., 29 May 2025).
• Contact-implicit: DDP variants (single/multiple shooting) and consensus ADMM solvers treat contacts as decision variables subject to complementarity constraints, discovering gaits or manipulation primitives online without pre-planned mode sequences (Kim et al., 2023, Aydinoglu et al., 2023).

Preview control (closed-form Riccati-based) is exploited to compute centroidal reference trajectories, drastically reducing computation versus full-horizon constrained QPs. Wrench projection and distribution QPs then enforce contact feasibility post hoc, with feedback policies ensuring robustness to disturbances (Murooka et al., 29 May 2025).

Receding-horizon planners achieve real-time operation by approximating long-horizon value functions via convex relaxations or learned oracles, dramatically shrinking NLP sizes and reliably producing dynamic multi-contact motion in dynamic and changing environments (Wang et al., 2023).

5. Complementarity, Stability Certificates, and Implementation Caveats

Rigorous stability and feasibility guarantees rely on Lyapunov theory for non-smooth, piecewise-affine closed-loop systems. Under positive-definiteness of matrix parameters and complementarity conditions, mode-free controllers are globally Lipschitz and Lyapunov stable (Aydinoglu et al., 2019). Time-stepping or measure-differential-inclusion frameworks are required for impulsive contacts.

Modeling and simulation often use PATH solvers for LCPs, while synthesis occurs in YALMIP/PENBMI or convex QP/SOCP backends. For multi-contact systems with hundreds to thousands of contacts (e.g., soft-body simulation), velocity-level fixed-point iteration with contact nodalization and diagonalization enables scalable, accurate, and highly parallelized simulation (Lee et al., 2022).

Current limitations include requirement for soft contact models (continuity of $\lambda(x)$), and restricted support for impulsive/hard impacts. Future work aims to generalize to instantaneous impact handling and pushing tasks.

6. Experimental Demonstrations and Benchmarking

Multiple frameworks have been validated across manipulation and locomotion tasks:

• Cart–pole with soft walls, three-cart juggling with partial state, acrobot swing-up with soft joint limits (Aydinoglu et al., 2019)
• Humanoid robots performing dynamic loco-manipulation: walking while picking, dropping, and throwing objects, with contact-aware policies outperforming naive LQR and stop-and-manipulate strategies (Li et al., 2022)
• Quadruped robots climbing boxes, crossing gaps, and transferring between complex acyclic environments in simulation and hardware (Dhédin et al., 18 Aug 2025)
• Manipulation: bimanual pivot-and-stow, sliding-and-regrasp, multi-finger contact-rich tasks (Shirai et al., 11 Mar 2025)
• Real-world hardware with SEIKO: interactive teleoperation with safety and automatic feasibility filtering, extreme reaching, active pushing, contact switching, and traversing uneven terrain (Rouxel et al., 2022)

Simulations consistently demonstrate sub-millisecond computation per control cycle, high robustness to disturbances, and reliable enforcement of friction, torque, and contact stability constraints.

7. Impact, State-of-the-Art, and Open Challenges

The development of provably stable, mode-free controllers for dynamic multi-contact motions marks profound progress toward general-purpose robotic manipulation and locomotion in real environments. The fusion of complementarity modeling, tight convex relaxations, hierarchy in planning, and scalable real-time control allows for versatile behaviors—climbing, running, parkour, dexterous manipulation—unachievable by prior heuristic, mode-switching, or statically stable approaches.

Ongoing challenges include universalization to impulsive and frictional contact laws, scalable combination of discrete and continuous plans, further reduction in computation for on-the-fly replanning, and development of robust learning-based surrogates for long-horizon value modeling.

The consensus in the current literature affirms that embedding the complementarity structure both in models and controllers achieves the dual goals of convexity and expressive power—enabling rigorous guarantees and high-performance multi-contact robots (Aydinoglu et al., 2019, Li et al., 2022, Kim et al., 2023, Shirai et al., 11 Mar 2025, Rouxel et al., 2022).