Jumping Control for a Quadrupedal Wheeled-Legged Robot via NMPC and DE Optimization

Xuanqi Zeng; Lingwei Zhang; Linzhu Yue; Zhitao Song; Hongbo Zhang; Tianlin Zhang; Yun-Hui Liu

cs.ROFeb 25, 2026

Jumping Control for a Quadrupedal Wheeled-Legged Robot via NMPC and DE Optimization

Authors: Xuanqi Zeng, Lingwei Zhang, Linzhu Yue, Zhitao Song, Hongbo Zhang, Tianlin Zhang, Yun-Hui Liu

Organizations: Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong

Abstract

Quadrupedal wheeled-legged robots combine the advantages of legged and wheeled locomotion to achieve superior mobility, but executing dynamic jumps remains a significant challenge due to the additional degrees of freedom introduced by wheeled legs. This paper develops a mini-sized wheeled-legged robot for agile motion and presents a novel motion control framework that integrates the Nonlinear Model Predictive Control (NMPC) for locomotion and the Differential Evolution (DE) based trajectory optimization for jumping in quadrupedal wheeled-legged robots. The proposed controller utilizes wheel motion and locomotion to enhance jumping performance, achieving versatile maneuvers such as vertical jumping, forward jumping, and backflips. Extensive simulations and real-world experiments validate the effectiveness of the framework, demonstrating a forward jump over a 0.12 m obstacle and a vertical jump reaching 0.5 m.

Figures & tables

Panel of Figure 1 — Fig. 1: The jumping motion on our quadrupedal wheeled-legged robot: (a) The robot jumping over a 0.12 m obstacle; (b) The robot achieves a vertical jump with a maximum height of 0.5 m.

Fig. 2: The hardware design of our mini-size quadrupedal wheeled-legged robot (unit: mm).

Fig. 3: Overview of the control framework. The locomotion of our wheeled-legged robot is implemented by Nonlinear Model Predictive Control (NMPC), which generates the required forward velocity before the jump. The Differential Evolution (DE) algorithm is activated only during the take-off phase to solve the jumping trajectory optimization problem. The $d$ subscript indicates the desired value. The term $x_{c,\textnormal{roll}}$ represents the displacement of the Center of Mass (CoM) in the $x$ -direction before the flight phase. The desired jumping state in the $x$ - $z$ plane, denoted as $[x_{c},z_{c},\theta_{c}]$ , corresponds to the desired jumping distance, height, and base rotation angle, respectively.

Fig. 4: The model of centroidal dynamic on our robot. $f_{ee1,x}$ , $f_{ee1,y}$ , $f_{ee1,z}$ indicates the GRF $\bm{f}_{ee1}$ in the $x$ , $y$ , $z$ direction w.r.t Body Frame.

Fig. 5: The foot’s constraints when the wheel contacts the ground. It includes the velocity constraints of the wheel in the three directions ( $\bm{t}$ , $\bm{n}$ , $\bm{b}$ ) and the friction cone constraints of foot force $\bm{f}_{eei}$ . $\omega_{i}$ denotes the rotation speed of the wheel.

Fig. 6: Three phases of jumping motion

Fig. 7: The locomotion performance of the robot in the simulation. The one-step height of the stair is 0.1 m, and the slope is 20 degrees.

Fig. 8: The simulation of the forward jumping on the robot. The obstacle is 0.25 m high, and the jumping height of the robot is 0.58 m.

Panel of Figure 9 — Fig. 9: The joint data of the left front leg during jumping. LF: left front leg; HAA: hip abduction/adduction joint; HFE: hip flexion/extension joint; KFE: knee joint.

Panel of Figure 10 — Fig. 10: The backflip simulation. Although the robot achieves the backflip motion successfully by the jumping trajectory optimization, the data indicate that such aggressive motion has already exceeded the hardware limit.

Fig. 11: The snapshot of the forward jumping experiment.

Panel of Figure 12 — Fig. 12: The joint data of the robot’s left front leg during the forward jumping and vertical jumping experiments.

Robot Locomotion

Robot locomotion studies how robots move efficiently and reliably across varied terrains, environments, and movement modes.

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May 5, 2026J/K move · Enter open · S save

Height Control and Optimal Torque Planning for Jumping With Wheeled-Bipedal Robots

Yulun Zhuang, Yuan Xu, Binxin Huang, Mandan Chao+4

Department of Mechanical and Energy Engineering, Southern University of Science and Technology, Shenzhen, 518055, China · Shenzhen Key Laboratory of Biomimetic Robotics and Intelligent Systems and Guangdong Provincial Key Laboratory of Human-Augmentation and Rehabilitation Robotics in Universities, Department of Mechanical and Energy Engineering, Southern University of Science and Technology, Shenzhen, 518055, China · Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, V6T1Z4, Canada

Robotic Control Legged Robotics Robotics Robot Locomotion

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This paper mainly studies the accurate height jumping control of wheeled-bipedal robots based on torque planning and energy consumption optimization. Due to the characteristics of underactuated, nonlinear estimation, and instantaneous impact in the jumping process, accurate control of the wheeled-bipedal robot's jumping height is complicated. In reality, robots often jump at excessive height to ensure safety, causing additional motor loss, greater ground reaction force and more energy consumption. To solve this problem, a novel wheeled-bipedal jumping dynamical model(W-JBD) is proposed to achieve accurate height control. It performs well but not suitable for the real robot because the torque has a striking step. Therefore, the Bayesian optimization for torque planning method(BOTP) is proposed, which can obtain the optimal torque planning without accurate dynamic model and within few iterations. BOTP method can reduce 82.3% height error, 26.9% energy cost with continuous torque curve. This result is validated in the Webots simulation platform. Based on the torque curve obtained in the W-JBD model to narrow the searching space, BOTP can quickly converge (40 times on average). Cooperating W-JBD model and BOTP method, it is possible to achieve the height control of real robots with reasonable times of experiments.

Abstract

Robot Locomotion

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