A Bi-level Co-design Framework for Lunar Monoped with 2D Jump and Landing Control

Sanjay Ramkumar; Suraj Kumar

cs.ROOct 27, 2026

A Bi-level Co-design Framework for Lunar Monoped with 2D Jump and Landing Control

Authors: Sanjay Ramkumar, Suraj Kumar

Organizations: Dept. of Aerospace Engineering, Indian Institute of Space Science and Technology. · Controls and Digital Area, U R Rao Satellite Center.

Abstract

Legged robots for lunar exploration must be designed for low gravity, where reduced contact forces and long flight phases make leg morphology and controller choice interdependent. We study this for a planar monoped in lunar gravity ( $1.625$ m/s $^2$ ) using a bi-level co-design framework: a differential evolution outer loop searches over link lengths, and each candidate is evaluated on a vertical jump by either a quadratic program based task-space controller (QP-TSC) or a direct-collocation based trajectory optimization. The two controllers yield different optimal link lengths, and trajectory optimization outperforms the QP-TSC across most of the design space, with 17% lower Cost of Transport (COT) at their respective optima. We then extend to a 2D jump, where trajectory optimization with centroidal angular momentum regulation produces stable in-place and forward jumps, and an LQR controller absorbs impact and stabilizes the robot after touchdown. This is demonstrated on a baseline morphology with both links 0.3 m long.

Figures & tables

Fig. 2 : Bi-Level framework

Panel of Figure 3 — Fig. 3 : COT landscapes over link lengths for the two controllers, and the difference between them.

Controller	$\mathbf{l_{1}}$ (m)	$\mathbf{l_{2}}$ (m)	Opt. COT	Base COT	Imp. (%)
QP-TSC	0.54	0.34	1.55	1.65	6.34
Traj-opt	0.36	0.21	1.28	1.57	18.96

TABLE I : Optimal designs from five independent DE runs

Design Optimized Using	Trajectory Optimization	QP-TSC
Trajectory Optimization	1.280	1.605
QP-TSC	1.461	1.550

TABLE II : Comparison of COT for Different Design-Controller pairs.

Fig. 4 : Pitch trajectory for various COM-foot position offset

Panel of Figure 7 — Fig. 5 : Empirical region of attraction of the LQR landing controller.

Quantity	In-place	Forward
Apex CoM height (m)	0.557	0.559
Liftoff CoM height (m)	0.353	0.333
Flight time (s)	0.70	0.76
Horizontal displacement (m)	$\approx 0$	0.20
Peak $\|$ pitch $\|$ before touchdown ( ∘ )	43	51
Peak $\|$ pitch $\|$ after touchdown ( ∘ )	17	79

TABLE III : 2D jump metrics.

Panel of Figure 10 — Fig. 7 : 2D jump trajectories.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

	Traj-opt			QP-TSC
Run	$l_{1}$	$l_{2}$	COT	$l_{1}$	$l_{2}$	COT
1	0.3562	0.2114	1.2824	0.5452	0.3387	1.5499
2	0.3564	0.2116	1.2825	0.5470	0.3401	1.5498
3	0.3571	0.2119	1.2824	0.5515	0.3438	1.5498
4	0.3568	0.2119	1.2830	0.5448	0.3383	1.5498
5	0.3575	0.2122	1.2825	0.5295	0.3319	1.5497

Appendix

TABLE IV : Five independent DE runs (extended bounds).

Panel of Figure 12 — Fig. 8 : Energy landscapes over link lengths.

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Legged robots can extend planetary exploration to rugged and discontinuous terrain that is inaccessible to conventional rovers. In lunar gravity, jumping further expands this mobility by enabling robots to cross obstacles and gaps that cannot be traversed by walking alone. We present a multistage reinforcement-learning approach that produces a single end-to-end control policy for commanded point-to-point jumping under lunar gravity. The resulting policy has a simple architecture and relies exclusively on mission-compatible onboard sensing, making its deployment on a real lunar mission plausible. In simulation, the learned policy performs accurate point-to-point jumps with landing errors smaller than the robot's body size. We analyze the torque and power demands of lunar jumping and characterize performance degradation under extreme outlier events, including control-update freezes of up to \SI{1}{s} during flight. Hardware transfer demonstrates that the policy can successfully execute a vertical jump under lunar-gravity offloading. Furthermore, the measured torque and power profiles are consistent with the simulation results and remain within the robot's hardware limits.

Abstract

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