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

Binxin Huang; Chenglong Fu; Guowei Shi; Kuangen Zhang; Mandan Chao; Xin Yang; Yuan Xu; Yulun Zhuang

arxiv: 2605.03302 · v1 · submitted 2026-05-05 · 💻 cs.RO

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

Yulun Zhuang , Yuan Xu , Binxin Huang , Mandan Chao , Guowei Shi , Xin Yang , Kuangen Zhang , Chenglong Fu This is my paper

Pith reviewed 2026-05-07 15:59 UTC · model grok-4.3

classification 💻 cs.RO

keywords wheeled-bipedal robotsjumping controltorque planningBayesian optimizationheight controlenergy optimizationrobot dynamics

0 comments

The pith

Wheeled-bipedal robots achieve precise jump heights through Bayesian-optimized continuous torque profiles that also cut energy use.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a wheeled-bipedal jumping dynamical model to predict and control exact jump heights despite underactuation and impact effects. Direct use of the model yields stepped torques that are impractical on hardware, so the authors add a Bayesian optimization routine that searches for smooth torque curves by narrowing the search space with the initial model. In simulation this combination reduces height error by 82.3 percent and energy consumption by 26.9 percent while converging in roughly 40 iterations on average. A reader would care because robots currently overshoot heights for safety, wasting motor energy and increasing structural loads. The work therefore offers a practical route to accurate, efficient jumping without requiring a perfect analytic model from the start.

Core claim

The authors claim that the W-JBD dynamical model supplies usable initial torque curves, which the BOTP method then refines into continuous optimal profiles; the resulting plans deliver accurate height control together with lower energy cost, as shown by an 82.3 percent drop in height error and 26.9 percent drop in energy use when tested on the Webots platform.

What carries the argument

The BOTP (Bayesian optimization for torque planning) method, which uses the W-JBD model to shrink the torque search space and then optimizes smooth profiles without needing an exact dynamic model.

Load-bearing premise

Simulation results from Webots will transfer to physical robots and the optimized torque profiles will stay near-optimal when friction, motor delays, and sensor noise are present.

What would settle it

Apply the same continuous torque curves on a physical wheeled-bipedal robot and check whether height error remains reduced by at least 70 percent relative to the unoptimized baseline.

Figures

Figures reproduced from arXiv: 2605.03302 by Binxin Huang, Chenglong Fu, Guowei Shi, Kuangen Zhang, Mandan Chao, Xin Yang, Yuan Xu, Yulun Zhuang.

**Figure 1.** Figure 1: Motion states A. Stance Model In order to control the balance of the wheeled-bipedal robot, two knee joints’ angles maintain unchanged. A single inverted pendulum model abstracts the essential dynamics of wheeled-bipedal robot. Equations of the force and torque of wheels in steady state are given below.    Fx = f − mαr¨ τ = 1 2mr2α¨ + fr 2Fx = M d dt2 (x + Lsin θ) 2Fy − Mg = M d dt2 (Lco… view at source ↗

**Figure 3.** Figure 3: The motion planning curve for squatting C. W-JBD Model To jump to a specified height, we model the robot’s jumping process. Some researches have used the SLIP model for robot jumping. However, it considers the robot a mass point at the top, ignoring other parts’ mass. During the flying phase, the robot will contract legs to overcome obstacles, which is unconsidered in the SLIP model. Therefore, this paper … view at source ↗

**Figure 4.** Figure 4: The W-JBD Model of the Wheeled-Bipedal Robots view at source ↗

**Figure 5.** Figure 5: The W-JBD Model of the Wheeled-Bipedal Robots    mbL˙ 2 = (mω + mb)VCoM V 2 CoM = 2ghc (mω + mb)∆h = 2mbr[sin αi 2 − sin αf 2 ] hw = hc + ∆h (6) finally, deriving the height of wheel hω = 1 2g ( mω + mb mb ) 2 + 2r[sin αi 2 −sin αf 2 ] mb mω + mb (7) Substitute Eqn(7) into Eqn(5), we can derive the relationship between designed spring displacement ∆Le, spring stiffness Ks and desired height e… view at source ↗

**Figure 6.** Figure 6: The simulation-based joint optimization framework A. Bayesian Optimization Firstly, we build the optimization model. The optimization goal is the difference between the actual jumping height of the robot’s wheels hw and the target height ehw. The actual jumping height of the robot’s wheels is calculated by the simulation physics engine. In the model and controller view at source ↗

**Figure 8.** Figure 8: The afterimage of jumping view at source ↗

**Figure 9.** Figure 9: The jumping trajectory of W-JBD and BOTP(desired height is 0.3m) B. Optimization Performance BOA can optimize the objective values of the given jumping task within 100 iterations and be close to convergence within 40 iterations (Fig.10). By using the joint optimization framework, we optimize different jumping heights to obtain the control parameter sequence corresponding to the certain jumping height, and… view at source ↗

**Figure 7.** Figure 7: Simplified model for simulation C. Joint Optimization Framework The overall simulation-based optimization framework is set up by jointly combining Advisor optimizer and Webots simulator as shown in Fig.6 Initially, we determine the optimizer setting (i.e., number of iterations, parameters specifications) in a configuration file, and then run the Advisor optimizer. Advisor optimizer invokes BOA after a warm… view at source ↗

**Figure 10.** Figure 10: Typical decreasing objective values during optimization In the experiment, we analyze the optimization parameters of three different desired jumping heights (hw = 0.2, 0.3, 0.4) shown in Fig.11, the error bars represent the standard deviation of top ten sets of optimal parameters. This shows that the robot’s best optimal take-off speed has a clear positive correlation with desired jumping height view at source ↗

**Figure 13.** Figure 13: The toque trajectory of W-JBD model in stance and take-off phase BOTP is employed to find a proper torque curve τ (t), which is continuous and will not damage the motors. With W-JBD providing the torque curve that can determine a reasonable searching space for Bayesian optimization, significantly reducing the number of iterations of parameter optimization, the W-JBD model helps the BOTP method to work be… view at source ↗

**Figure 11.** Figure 11: Optimal taking off speed with different desired jumping height C. The Precision of Jumping Height The first indicator of evaluating the performance is the accuracy of jumping height shown as Fig.12. It plots that the W-JBD model and BOTP method both show good performance in this indicator, as the error is limited within 4% for W-JBD model(corresponding absolute error is limited at most 0.015m with maximu… view at source ↗

**Figure 14.** Figure 14: The toque trajectory of BOTP method in stance and take-off phase E. Energy Cost Energy Cost is the work energy of the knee motors, which is transformed to gravitational potential energy and damping energy loss. Normally, the energy cost increases along with the height increment(Fig.15). Analyzing the data in Fig.15, BOTP takes less energy, though it needs the larger torque but its take-off phase traveled … view at source ↗

**Figure 12.** Figure 12: The height error of W-JBD model and BOTP method with different desired height D. The Torque Trajectory Our experiments focus on the take-off phase optimization, i.e., the curve shown below between the blue points. Fig.13 illustrates the torque curve of the knee motors in the case of the W-JBD model with the desired height of 0.3m. Notice that there is a striking step from -4.96N·m to -35N·m at the first b… view at source ↗

**Figure 15.** Figure 15: The energy cost using W-JBD model and BOTP method with different desired height view at source ↗

**Figure 16.** Figure 16: The encoder value of knee motors via time, the curve between the points(red or blue respectfully) stands for the take-off phase V. CONCLUSION In this paper, a novel wheeled-bipedal jumping dynamical (W-JBD) model is proposed for achieving accurate height control for jumping. The W-JBD model performs well on height control, but it does not consider the joint’s torque limit so that the improper torque plann… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a simulation-only recipe for smoother torque profiles on wheeled-bipedal jumps that cuts height error and energy use, but offers no hardware data to support its real-robot claims.

read the letter

The main thing to know is that the authors built a wheeled-bipedal jumping model (W-JBD) and then used Bayesian optimization to find continuous torque curves instead of the stepped ones the model produces. In Webots they report an 82% drop in height error and 27% lower energy cost, with the search converging in about 40 iterations on average because the model narrows the space. That combination is the concrete new piece here, and the simulation numbers are presented clearly enough to show the method works inside that environment. The torque profiles look smoother and more practical than the raw model output, which is a useful engineering step for this robot class. The paper is honest that the model alone is not ready for hardware and that the optimization step is meant to fix it. What is missing is any physical robot test or even a basic sim-to-real calibration. There are no trials on the actual machine, no sensitivity checks for motor lag, friction variation, or ground compliance, and no comparison against standard baselines with reported statistics. The final sentence that the approach will let real robots hit target heights with reasonable experiments is therefore an extrapolation rather than a demonstrated result. The work is aimed at people already building or controlling wheeled bipeds who need a practical way to plan jump torques in simulation. A reader looking for new theory or broad insights will not find much, but someone facing the same torque-step problem might borrow the Bayesian search idea. It is worth sending to referees because the simulation evidence is specific and the method is reproducible from the description, even though the authors will almost certainly be asked to add hardware validation or stronger caveats before publication.

Referee Report

2 major / 1 minor

Summary. This paper claims to solve the problem of accurate height control for jumping wheeled-bipedal robots by proposing a wheeled-bipedal jumping dynamical model (W-JBD) that enables precise control but produces discontinuous torques unsuitable for hardware. To address this, it introduces the Bayesian optimization for torque planning (BOTP) method, which is largely model-free and yields continuous optimal torque curves. In Webots simulations, BOTP reduces height error by 82.3% and energy cost by 26.9%, converging in about 40 iterations on average when the search space is narrowed using the W-JBD model. The authors assert that this combination makes real-robot height control feasible with a reasonable number of experiments.

Significance. The results, if they hold under hardware conditions, represent a meaningful advance in control strategies for underactuated robotic jumping systems by combining model-based insights with model-free optimization to produce practical torque plans. The reported quantitative gains in accuracy and efficiency, along with the rapid convergence of BOTP, highlight the potential of Bayesian optimization in robotics applications where accurate models are hard to obtain. Credit is due for the simulation-based validation showing clear improvements.

major comments (2)

[Abstract] The reported performance improvements of 82.3% in height error and 26.9% in energy cost are presented without any information on the baseline methods, the number of trials, statistical significance, or the methods used to measure jumping height and energy consumption. This omission undermines the ability to fully assess the validity and reproducibility of the central quantitative claims.
[Abstract] The final claim that the W-JBD model and BOTP method together 'achieve the height control of real robots with reasonable times of experiments' lacks any supporting evidence from physical robot experiments. The validation is confined to the Webots simulation platform, with no analysis of sim-to-real transfer, robustness to uncertainties (e.g., friction variations, motor delays, sensor noise), or calibration data, which is a load-bearing gap for the paper's broader applicability assertion.

minor comments (1)

[Abstract] The description of BOTP as operating 'without accurate dynamic model' is slightly inconsistent with the use of the W-JBD model to narrow the searching space; a brief clarification on the degree of model dependency would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive evaluation of the work's significance. We address each major comment below and will revise the manuscript to improve clarity and accuracy.

read point-by-point responses

Referee: [Abstract] The reported performance improvements of 82.3% in height error and 26.9% in energy cost are presented without any information on the baseline methods, the number of trials, statistical significance, or the methods used to measure jumping height and energy consumption. This omission undermines the ability to fully assess the validity and reproducibility of the central quantitative claims.

Authors: We agree that the abstract should include more context on these elements to support the claims. The baseline is the direct torque planning from the W-JBD model. The improvements are obtained from repeated Webots simulations with results averaged over multiple runs, height measured from the simulated robot's kinematic state, and energy computed as the time integral of electrical power. We will revise the abstract to briefly note the baseline, the simulation-based nature of the evaluation, and the measurement methods, while referring readers to the results section for trial counts and any statistical details. revision: yes
Referee: [Abstract] The final claim that the W-JBD model and BOTP method together 'achieve the height control of real robots with reasonable times of experiments' lacks any supporting evidence from physical robot experiments. The validation is confined to the Webots simulation platform, with no analysis of sim-to-real transfer, robustness to uncertainties (e.g., friction variations, motor delays, sensor noise), or calibration data, which is a load-bearing gap for the paper's broader applicability assertion.

Authors: We acknowledge that the manuscript contains only simulation results and no hardware experiments. The statement is an extrapolation from the rapid convergence of BOTP (approximately 40 iterations) when the search space is narrowed by the W-JBD model. We agree this claim requires qualification. We will revise the abstract to state that the approach shows promise for real-robot height control with a reasonable number of experiments, subject to future hardware validation. We will also add discussion of sim-to-real considerations, including sensitivity to friction, delays, and noise, based on our simulation analyses. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on independent simulation experiments

full rationale

The paper introduces the W-JBD model and BOTP method as original contributions. Height-error and energy reductions (82.3 % and 26.9 %) are measured outcomes of running BOTP inside the external Webots simulator, not quantities derived by algebraic substitution from the model equations themselves. Narrowing the BOTP search space with W-JBD torque curves is a heuristic that still leaves the optimizer free to produce new continuous profiles; the reported convergence count (40 iterations) and performance deltas are therefore empirical results, not tautological re-statements of the inputs. No self-citations appear in the provided text, and no fitted parameter is relabeled as a prediction. The forward claim about real-robot feasibility is an untested extrapolation rather than a circular derivation step.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract provides no explicit free parameters, mathematical axioms, or external benchmarks; the primary new element is the W-JBD model itself.

invented entities (1)

W-JBD model no independent evidence
purpose: Dynamical model for wheeled-bipedal jumping height control
New model introduced to address underactuation and impact issues; no independent evidence outside the paper.

pith-pipeline@v0.9.0 · 5550 in / 1230 out tokens · 61958 ms · 2026-05-07T15:59:05.824897+00:00 · methodology

discussion (0)

Reference graph

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