Bio-inspired Thrust-Assisted Trotting on a legged-Aerial Robot via Real-Time QP-MPC
Authors: Shreyansh Pitroda, Eric Sihite, Alireza Ramezani, Morteza Gharib
Organizations: The authors are with SiliconeSynapse Labs, the Department of Electrical Engineering, Northeastern University, USA. · The authors are with the Department of Aerospace Engineering, California Institute of Technology, USA.
This paper presents the hardware implementation of a quadratic program-based controller on Harpy: thruster-assisted bipedal robot to perform thruster-assisted trotting. This is inspired by wing-assisted locomotion observed in birds such as the chukar partridge, which employ thrust vectoring to manipulate the ground reaction force. The proposed controller formulates the thruster-assisted spring-loaded inverted pendulum dynamics as a condensed QP-based model predictive control problem, enabling simultaneous regulation of ground contact force and active thrust vectoring at 500 Hz. A decoupled control architecture is used, where the robot's sagittal dynamics are controlled using QP-MPC while the frontal dynamics is regulated using a cascade orientation stabilization controller. Two experiments were conducted with different forward reference velocities of 0 m/s and 0.1 m/s. While the instantaneous velocity oscillates, the mean velocities can track their respective reference. Additionally, joint tracking error remains within 1 deg while tracking the desired leg length given by the leg trajectory controller and desired swing trajectory given by Raibert's heuristic. The QP-MPC problem is consistently solved with 0.18ms, which is well within the 2ms computation budget while satisfying the GRF friction cone constraint.
Figures & tables
Fig. 1: Illustrates biological inspiration for thruster-assisted bipedal robot. Birds employ wing-assisted locomotion to manipulate GRF [ 2 ] . Harpy replicates this strategy by replacing biological wings with electric ducted fans, enabling active thrust vectoring.
Fig. 2: (Left) Full-order kinematic model of our robot highlighting joint, and thruster position. (Right) Planar 2D Spring-loaded Inverted Pendulum reduced-order model used for QP-MPC formulation.
PWM
Side thruster
Upper thruster
(%)
(kgf)
(kgf)
10
0.092
0.0875
20
0.37
0.175
30
0.809
0.2625
40
1.346
0.35
50
1.97
0.4375
TABLE I: Mapping PWM to thruster forces
Fig. 3: Control architecture of the proposed QP-TSLIP controller.
Fig. 4: Sequential snapshots from hardware Experiment B. The robot performs thruster-assisted trotting locomotion using the proposed QP-TSLIP controller.
Fig. 5: Illustrates QP solver performance on hardware. (Left) Histogram of QP total computation time for Experiment A (blue) and Experiment B (red). (Right) Distribution of solver iteration, displaying majority convergence within 6-7 steps.
Fig. 6: Illustrates GRF distribution within the friction cone for Experiment A (Left) and Experiment B (Right). Each point represents the horizontal and vertical force commanded by the QP solver, colored by time (cyan = start and magenta = end). The red line denotes the friction cone boundary with coefficient μ=0.5 .
Fig. 7: Thruster force commands during hardware experiments. (Top) QP-commanded forward thrust ut,x for both the experiments. The thrust frequently remains near the saturation limit of ±8.7N . (Bottom-left) Side thruster forces for Experiment A and (Bottom-right) for Experiment B, showing left and right thruster output combining PID orientation controller and QP controller. The dashed line at 5N denotes the baseline produced by the QP controller.
Fig. 8: Body state trajectory during hardware experiments. (Top-left) COM position in x-direction for Experiment A (blue) and Experiment B (red) with dotted position reference line (black). (Top-right) body velocity in the x-direction, along with the mean values and velocity reference. (Bottom-left) Pitch angle remains bounded within ±9∘ throughout both experiments. (Bottom-right) Roll angle stays within ±4∘ , validating the effectiveness of the cascade control strategy. The filled black circle (∙) denotes the end of Experiment B.
Fig. 9: Box plot for joint trajectory tracking error for Experiment A and Experiment B.
Fig. 10: MPC prediction rollout validated on hardware for Experiment A. Forward (top-left), and vertical (Bottom-left) COM velocity trajectory comparing the QP-MPC model prediction against the hardware measurement (blue). The zoomed section illustrates the receding horizon strategy where at each state feedback point (red dot), the linearized SLIP model rollout (black) are calculated for 5 horizon.
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.
Kazuki Sugihara, Kei Okada
Department of Mechano-Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
This paper presents a multi-phase whole-body model predictive control (MPC) approach for bipedal walking, combining a detailed whole-body model in the near horizon with a simplified single-rigid-body model in the later prediction steps. This reduces computational complexity while retaining prediction capabilities. The resulting nonlinear optimal control problem is solved entirely within the general-purpose, off-the-shelf nonlinear MPC framework acados, using sequential quadratic programming (SQP). Given a contact schedule and a target walking speed, the controller optimizes joint torques without depending on preselected footstep locations. The controller is validated in MuJoCo simulation on the 18-DoF bipedal robot HyPer-2.
Franek Stark, Felix Wiebe, Shubham Vyas +2
Robotics Innovation Center at the German Research Center for Artificial Intelligence (DFKI), Bremen, Germany · University Bremen, Germany
This paper presents dynamics-relaxed model predictive control (DR-MPC), a novel MPC formulation for legged locomotion, and a tailored interior-point method (IPM) solver. The formulation combines online optimization feasibility by construction with a contact-aware input parameterization. DR-MPC moves the dynamics equality and affine input constraints into quadratic penalties and retains only nonempty box constraints. The resulting box-constrained quadratic program (QP) has a block-arrow Hessian that enables the state and affine-output directions to be eliminated through a Schur complement. The solver factors only the reduced control system after swing-force elimination and contact-aligned move blocking. For the evaluated implementations using the same DR-MPC formulation, our method achieves median end-to-end MPC speedups of 16.0× over HPIPM and 4.4× over OSQP, with comparable locomotion performance in simulation. DR-MPC achieves a median onboard MPC end-to-end time of 4.4 ms and is validated on a Unitree Go1 quadruped. Open-source code will be made available after publication.
Run Wang, Alapati Tuerxun, Shuo Liu +4
Department of Automation, Tsinghua University, Beijing, China · Department of Mechanical Engineering, Boston University, Brookline, MA, USA · School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore +1