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Anti-Gravity Walking by a Flying Humanoid Robot via Thrust-Rate Input Whole-Body Model Predictive Control

Kazuki Sugihara, Kei Okada

arXiv:2609.07544v1cs.RO

TL;DR

Flying humanoid locomotion is largely limited to flight and near-flat ground, while ceiling walking requires sustained contact despite contact-switching dynamics that can produce thrust spikes. The paper uses thrust-rate whole-body MPC with smoothly transferred foot-force bounds, demonstrating anti-gravity walking in simulation and on hardware. The framework suppresses spikes in thrust, contact forces, and joint torques, and supports stable real-time walking.

  • Problem

    Flying humanoid locomotion has largely been limited to aerial flight and near-flat ground, leaving anti-gravity walking with whole-body dynamics and contact switching unaddressed.

    Source: §I · p. 1

  • Method

    The framework treats thrust as a state and thrust-rate as the control input, with smoothly transferred lower bounds on foot-normal contact force during double support.

    Source: §I · p. 2

  • Results

    The framework demonstrated anti-gravity walking in simulation and on hardware, suppressing thrust, contact-force, and joint-torque spikes; average MPC solving time was 6.69 ms, with 91.5% of solves within 10 ms.

    Source: 2) Results · pp. 5–6, 2) Result · p. 6, §VII · p. 8

  • Takeaways & Limitations

    The demonstrations show that a flying humanoid can walk in an anti-gravity environment using a real-time whole-body MPC framework.

    Source: §VII · p. 8

Abstract

from arXiv · show

Flying humanoids are expected to perform tasks in diverse environments, while their existing locomotion is mainly limited to aerial flight and ground walking. The capability to move in complex three-dimensional space can greatly expand their application range. For such walking motion on ceilings and similar anti-gravity environments, whole-body MPC is effective. However, the discontinuous changes in dynamic structure accompanying contact switching during walking can induce thrust spikes, resulting in control instability. Therefore, in this work, we propose and implement a real-time whole-body MPC framework for anti-gravity bipedal walking. First, we formulate whole-body MPC using the time derivative of thrust, namely thrust-rate, as the control input. This formulation guarantees continuity of the thrust trajectory during contact switching while preserving the sparse structure of the optimal control problem for fast computation. Second, we address the lack of natural support forces in anti-gravity environments. We introduce lower bounds on the foot-normal component of the contact force, and smoothly transfer them during the doublesupport phase. Finally, we implement the proposed framework and demonstrate anti-gravity walking by a flying humanoid through simulation and a hardware experiment. To the best of our knowledge, this is the first demonstration of multi-contact whole-body MPC for a transformable aerial robot and walking by a flying humanoid beyond the ground.

I. INTRODUCTION

The paper targets ceiling walking by flying humanoids, where contact switching can provoke thrust spikes and anti-gravity support requires continuous pressing force. It proposes thrust-rate whole-body MPC and smoothly transferred foot-force lower bounds, then demonstrates walking in simulation and hardware.

I. INTRODUCTION

  • Treating thrust as a state and thrust-rate as input guarantees thrust continuity through contact switching while preserving the OCP’s sparse structure.This avoids adding difference constraints between consecutive inputs, which would make real-time computation difficult.Source: §I · pp. 1–2
  • Lower bounds on foot-normal contact force, smoothly transferred during double support, address the lack of natural support forces in anti-gravity walking.The bounds are incorporated into the contact wrench cone condition.Source: §I · pp. 1–2
  • The framework was integrated into a real-time control system and used to demonstrate anti-gravity walking in simulation and on hardware.The stated demonstrations are by a flying humanoid.Source: §I · p. 2

A. Thruster-Equipped Legged Robots

  • Prior thruster-assisted walking relied on gravity to press feet onto floors or slopes; ceiling and wall walking requires thrust-generated normal force and had not been addressed.Earlier platforms used thrust to reduce contact force, support the body, or shape forces within friction constraints.Source: A. Thruster-Equipped Legged Robots · p. 2

B. Environmental Contact with Multirotor Platform

The paper couples joint motion and thrust allocation through whole-body dynamics in a single OCP, addressing the lack of unified treatment of multi-contact constraints. Its flying humanoid uses vectorable thrusters, while the thrust-rate formulation maintains thrust continuity without sacrificing OCP sparsity.

B. Environmental Contact with Multirotor Platform

  • Aerial manipulation approaches generate environmental interaction forces or use contact for stability, but typically do not jointly treat joint torques, thrusts, whole-body dynamics, and multi-contact constraints.The proposed approach couples joint motion and thrust allocation within a single OCP.Source: B. Environmental Contact with… · p. 2

C. Thrust-Aware Whole-Body MPC

  • Treating thrust as a state and its time derivative as input guarantees thrust continuity through state transitions without constraints coupling consecutive inputs.This preserves the sparse structure of the OCP; continuity is guaranteed by the state transition rather than encouraged by a penalty.Source: C. Thrust-Aware Whole-Body MPC · p. 2

1) Robot Dynamics:

  • The robot has two six-DoF legs and two independently vectorable thrusters that can direct thrust toward a contact-surface normal.The vectoring mechanism lets each rotor rotate around the torso link’s y-axis, supporting moment generation and pressing force for anti-gravity walking.Source: 1) Robot Dynamics · pp. 2–3, Figure 2 · p. 3

B. Whole-body MPC Formulation

The whole-body MPC is posed as a discrete-time optimal control problem with stage and terminal costs, dynamics, constraints, and an initial-state condition. Its formulation is solved numerically, including with sparse-structure solvers suited to real-time MPC.

B. Whole-body MPC Formulation

  • The discrete-time OCP minimizes a cost over state and control trajectories subject to dynamics, inequality and equality constraints, and an initial-state condition.The cost comprises stage and terminal terms; the equality constraints include contact consistency.Source: B. Whole-body MPC Formulation · p. 3
  • The state transition represents discretized whole-body dynamics, while inequality constraints express bounds and equality constraints express conditions such as contact consistency.The OCP also includes an initial-state constraint.Source: B. Whole-body MPC Formulation · p. 3
  • The OCP can be solved with numerical optimization such as SQP or fast nonlinear optimal-control solvers that exploit its sparse structure for real-time MPC.Source: B. Whole-body MPC Formulation · p. 3

IV. THRUST-RATE INPUT WHOLE-BODY MPC

The MPC uses thrust-rate as its control input so thrust remains continuous across contact switches, without sacrificing the sparse OCP structure needed for fast computation.

IV. THRUST-RATE INPUT WHOLE-BODY MPC

  • The formulation augments the state with thrust and uses its time derivative as the control input, which changes thrust through state integration.The thrust-rate input does not directly generate generalized force; the integrated thrust does.Source: §IV · p. 3
  • The augmented state transition is discretized with Euler integration, using the SE(3) integration operation for configuration evolution.The passages identify the discretization time step as Δt.Source: §IV · p. 3
  • Thrust-rate input makes thrust change continuously across MPC nodes, preserving OCP sparsity for real-time solvers.With a sufficiently small discretization step, the resulting thrust trajectory is smooth enough for real rotors to track through contact switching.Source: §IV · p. 3

V. ANTI-GRAVITY WALKING MPC FORMULATION

The walking MPC balances tracking, contact-wrench feasibility, thrust management, and regularization. It treats the contact-wrench-cone condition as a soft penalty to favor real-time computation.

V. ANTI-GRAVITY WALKING MPC FORMULATION

  • The contact-wrench-cone condition is a soft cost penalty rather than a hard inequality constraint to reduce solver computation time.The formulation prioritizes real-time performance and relies on swing-foot and CoM references to make the soft penalty sufficient.Source: §V · pp. 3–4

A. Cost and Constraints for Anti-Gravity Walking

  • The stage cost combines swing-foot and CoM tracking, contact-wrench-cone penalty, thrust regularization, and state/input regularization.Each term addresses a distinct tracking, contact, thrust, or regularization objective.Source: A. Cost and Constraints for… · p. 4
  • The terminal cost contains the stage-cost terms except for input regularization.The terminal cost therefore omits the input-regularization term included at intermediate stages.Source: A. Cost and Constraints for… · p. 4
  • The cost uses weighted squared norms and a deviation measure that is zero within bounds and signed outside them.The max and min operations defining the deviation are applied elementwise.Source: A. Cost and Constraints for… · p. 4

1) Swing Foot and CoM Tracking:

The formulation tracks swing-foot and CoM references to generate walking along the contact surface, while penalizing contact-wrench violations and enforcing a minimum foot-normal force.

1) Swing Foot and CoM Tracking:

  • Swing-foot references interpolate linearly in the walking direction and use a sinusoidal lift perpendicular to the contact surface.The target velocity comes from the motion planner, and the cost tracks the swing-foot pose error.Source: 1) Swing Foot and CoM Tracking · p. 4
  • The CoM reference maintains height from the contact surface, staying above the support foot and transitioning toward the next support foot during double support.The transition is by linear interpolation, and the cost tracks CoM position error.Source: 1) Swing Foot and CoM Tracking · p. 4

2) Contact Wrench Cone Penalty:

  • The contact-wrench-cone constraint bounds each foot’s contact wrench, while a lower bound on foot-normal force counters detachment risk when thrust minimization yields weak contact.The lower bound applies to the foot-normal component; the CWC bounds are determined by friction coefficient and foot size.Source: 2) Contact Wrench Cone Penalty · p. 4

3) Thrust Regularization and Saturation Penalty:

The formulation penalizes thrust magnitude and saturation, regularizes state and input deviations, and smoothly transfers minimum normal-force references between feet during double support.

3) Thrust Regularization and Saturation Penalty:

  • The thrust cost minimizes generated thrust and penalizes violations of rotor-thrust lower and upper limits.The formulation distinguishes thrust regularization from the saturation penalty.Source: 3) Thrust Regularization and… · p. 4

4) State and Input Regularization:

  • State and input regularization penalizes deviation from a standing reference pose with slightly bent knees and zero velocity.The state error excludes thrust; regularization suppresses state deviation, thrust change, and joint-torque magnitude.Source: 4) State and Input Regularization · p. 4

B. Load Transfer Strategy

  • The figure defines the contact-frame normal force and depicts its minimum-force reference across double-support and right- and left-swing phases.Fmin denotes the minimum normal force during double support.Source: Figure 3 · p. 5
  • Minimum foot-normal-force bounds transfer linearly between the feet during double support and apply Fmin only to the support foot in single support.The bounds change continuously at contact switching; Fmin was chosen experimentally for this robot and its friction conditions.Source: B. Load Transfer Strategy · pp. 4–5

C. Real-Time MPC Implementation

The implementation adapts Crocoddyl for thrust-rate whole-body MPC and keeps the MPC horizon aligned with real time. It also refreshes foot and CoM references at double support to limit accumulated slip and model error.

C. Real-Time MPC Implementation

  • The OCP is solved with warm-started BoxFDDP, one iteration per update, and parallel computation; the implementation supports articulated robots with arbitrary thruster distributions.The latest solution is initialized with the current state, and the latest command is sent to the robot.Source: C. Real-Time MPC Implementation · p. 5
  • The horizon advances only after elapsed wall-clock time exceeds the discretization step, preventing skipped nodes when a solve exceeds the control cycle but stays below that step.The latest solution is applied within each node interval, decoupling node shifts from per-cycle computation time.Source: C. Real-Time MPC Implementation · p. 5
  • At double support, the controller projects the actual foot position onto the contact surface and regenerates foot and CoM references to suppress accumulated slip and model error.Source: C. Real-Time MPC Implementation · p. 5

VI. EVALUATION

  • The evaluation verifies the methods through trajectory optimization, dynamics simulation, and a hardware experiment.Source: §VI · p. 5

A. Trajectory Optimization

Trajectory optimization compares thrust-rate input with conventional thrust input for a four-step ceiling-walking OCP. The thrust-rate formulation yields smoother thrust through contact switching and converges in fewer iterations; the section also specifies the subsequent MPC simulation setup.

1) Problem setup:

  • The four-step ceiling-walking motion was formulated as one OCP with 0 N minimum contact force and 0.2 s double support, using a static-equilibrium QP to initialize thrust and joint torques.The offline optimization used four parallel threads on an Intel Core i7-10850H laptop.Source: 1) Problem setup · p. 5

2) Results:

  • 16 iterations versus 26 for thrust-input optimization; thrust-rate input produced smooth thrust through contact switching instead of about 3 N step-like changes.The comparison also showed similar smoothing in foot contact force and leg joint torques.Source: 2) Results · pp. 5–6, Figure 4 · p. 6

1) Problem Setup:

  • The MPC simulation used a 100 Hz control frequency, 0.75 s double support, noisy torso-state estimates fused with IMU data, and a 5 N minimum contact force.Source: 1) Problem Setup · p. 6

2) Result:

With a 5 N minimum contact force, thrust-rate MPC achieved anti-gravity walking in simulation, with smooth thrust and stable support. Ablations without the foot-normal bound or with thrust as input destabilized contact and led to detachment.

2) Result:

  • With thrust-rate-input MPC and a 5 N minimum contact force, generated thrust periodically and smoothly changed between about 8 N and 12 N; foot-landing deviation stayed below approximately 0.02 m, while swing-phase support-foot normal force was at least 5 N.Source: 2) Result · p. 6
  • 6.69 ms was the average MPC solve time; 91.5% of solves finished within 10 ms, and the 18.9 ms worst case remained below the 25 ms discretization step.The system remained stable despite occasional constraint violations; the left-foot CoP and yaw-torque bounds were violated for 5.00%, 7.76%, and 3.01% of contact samples, respectively.Source: 2) Result · p. 6

3) Ablation study:

  • With thrust input and a 5 N minimum contact force, thrust oscillations destabilized ceiling contact and the robot eventually detached.The target thrust reached its 0 N lower bound, indicating that physically plausible trajectories were not generated.Source: 3) Ablation study · pp. 6–7
  • With the foot-normal contact-force lower bound set to 0 N, the robot became unstable and detached from the ceiling, particularly as yaw contact became unstable.The initial thrust was approximately 10 N for both thrusters.Source: 3) Ablation study · p. 6, Figure 6 · p. 7
  • The authors conclude that the framework is effective for contact-switching motions where gravity provides no natural support.Source: 3) Ablation study · p. 7

C. Hardware validation

The hardware experiment demonstrated ceiling walking with the proposed MPC, while the conclusion summarizes its thrust-rate and contact-force design and identifies broader environments as future work.

C. Hardware validation

  • The hardware setup used position-controlled, high-reduction servos and had an effective mass of 2.0 kg including wiring; foot and CoM references were corrected at each double-support phase.Joint positions followed MPC trajectories, with thrust added for roll and pitch stabilization.Source: C. Hardware validation · p. 7
  • 0.15 m in four steps over about 8.5 s, with thrust spikes and excessive vibrations suppressed, orientation error within 0.1 rad, and 99.7% of solves under 10 ms.Average MPC solve time was 5.47 ms.Source: C. Hardware validation · p. 7, Figure 9 · p. 8, Figure 8 · p. 8
  • High tracking weights led to joint-angle deviations, including the left knee shifting from 1.0 rad to about 0.3 rad; the hardware experiment also lacked terminal-state equality constraints and joint-torque force control.The authors identify inverse-kinematics references, a general-constraints solver, and low-reduction-ratio servos as expected ways to address these limitations.Source: C. Hardware validation · p. 7

VII. CONCLUSION

  • Thrust-rate MPC preserved thrust continuity at contact switches, suppressed thrust, contact-force, and joint-torque spikes, and sped trajectory optimization; foot-normal force lower bounds were smoothly transferred during double support.The lower bounds were incorporated into the contact-wrench-cone penalty.Source: §VII · p. 8
  • The authors propose extending the framework to wall-to-ceiling transitions, curved surfaces, and integration with aerial flight.Source: §VII · p. 8
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