Source-linked AI summary
A Bi-level Co-design Framework for Lunar Monoped with 2D Jump and Landing Control
Sanjay Ramkumar, Suraj Kumar
TL;DR
The paper examines whether controller choice changes the optimal morphology of a lunar hopper, a design question shaped by long flight phases and limited attitude control through ground forces. It uses bi-level co-design for vertical jumps and extends to 2D jumping and landing. Trajectory optimization outperforms QP-TSC across the compared morphologies, and both in-place and forward jumps are demonstrated in simulation.
Problem
The study asks whether the optimal lunar-hopper morphology depends on the controller used to evaluate it, in a setting where long flight phases make attitude regulation harder.
Source: §I · p. 1Method
A bi-level framework searches link lengths using differential evolution and evaluates candidates with either QP-TSC or trajectory optimization, then extends control to 2D jumping and landing.
Source: §I · p. 1Results
Trajectory optimization achieves lower COT than QP-TSC on both optimized morphologies, with COTs of 1.280 versus 1.461 on the trajectory-optimized morphology and 1.605 versus 1.550 on the QP-TSC morphology.
Source: §IX · p. 5Takeaways & Limitations
Controller choice influences the selected absolute link lengths, while trajectory optimization retains an advantage across the compared morphologies.
Source: §IX · p. 5All results are from simulation, design variables are limited to link lengths with ideal actuators, and the 2D jump is demonstrated on the baseline morphology.
Source: C. Limitations · p. 5
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper asks whether controller choice changes the optimal lunar-hopper morphology, then extends the study to planar jumping and landing. It situates this work in low-gravity hopping and co-design research and describes the monoped model used.
A. Low Gravity Legged and Hopping Robots
- Low-gravity hopping can be energy efficient, but long flight phases make attitude regulation harder because ground forces cannot alone control attitude and take-off angular momentum must be managed.Prior examples include platforms using reaction wheels, learned in-flight reorientation, and simulated lunar hopping with a torso reaction wheel and PID flight control, [5],, [7].Source: §I · p. 1, A. Low Gravity Legged and Hopping Robots · p. 1
C. Landing Stability and Regions of Attraction
- The 2D controller regulates centroidal angular momentum for in-place and forward jumps, then uses LQR for landing; its region of attraction is characterized empirically across touchdown conditions.Formal estimation is difficult for hybrid legged systems because contact causes state resets [4].Source: §I · p. 1, C. Landing Stability and Regions of… · p. 1
B. Co-design of Legged Robots
- Prior co-design approaches combine morphology with motions or policies, while this study varies the inner controller to isolate its effect on optimal morphology.Related methods include sensitivity analysis, policy learning [9], differentiation through a motion planner [10], joint feedback and morphology optimization, trajectory optimization, actuator and gearbox modeling, and Mars-gravity grid search [14].Source: C. Landing Stability and Regions of… · p. 1, B. Co-design of Legged Robots · p. 1
III. SYSTEM MODEL
- The monoped’s state uses base height and joint angles in 1D, adding horizontal position and base pitch in 2D.The coordinates are z, q1, and q2 for 1D, and z, x, θ, q1, and q2 for 2D.Source: §III · p. 2
A. Robot Model
- The planar two-link leg uses thigh and shank lengths as co-design variables, with constant-density links and a fixed-mass cuboid base.The links have density 1989.4 kg/m3 and radius 0.02 m; the base is 3 kg, and the direct-drive ideal actuators have 25 Nm torque limits.Source: A. Robot Model · p. 2
B. Robot Dynamics
- The model describes the monoped with generalized coordinates and an actuation selection matrix.The supplied passage identifies these components but does not provide their equations.Source: B. Robot Dynamics · p. 2
IV. BI-LEVEL CO-DESIGN FRAMEWORK
A bi-level search varies thigh and shank lengths, evaluating each candidate with an inner controller. The framework compares controller-dependent jump performance using cost of transport.
IV. BI-LEVEL CO-DESIGN FRAMEWORK
- The outer loop searches thigh and shank lengths, while the inner controller attempts a 0.2 m jump and returns cost of transport as the design fitness.The initial hip angle is fixed at 45°, and the initial knee angle is set so the center of mass lies above the foot in static equilibrium.Source: §IV · p. 2
A. Differential Evolution Outer Loop
- Differential evolution searches link lengths without gradients, retaining lower-COT trials and penalizing failed tasks, failed controllers, or statically unstable designs.Each jump evaluation runs a MuJoCo simulation that could fail, making COT gradients difficult to obtain.Source: A. Differential Evolution Outer Loop · p. 2
B. Performance Metric
- Cost of Transport is used to compare morphology efficiency fairly, since raw actuator energy would favor shorter links as mass and inertia decrease.The task is a 0.2 m jump in lunar gravity; COT uses total positive stance-phase actuator work, robot mass, gravity, and jump height.Source: B. Performance Metric · p. 2
C. Inner Controller 1: QP based Task Space Controller
- The QP task-space controller tracks phase-dependent references under constraints, targeting liftoff COM velocity during stance and landing preparation during flight.During landing, it targets impact absorption and a nominal pose; stance also enforces a fixed-foot constraint.Source: C. Inner Controller 1: QP based Task… · p. 3
D. Inner Controller 2: Trajectory Optimization
The trajectory-optimization controller formulates stance as a constrained nonlinear optimization that minimizes positive actuator work. Flight and landing are handled separately with joint-space PD control.
D. Inner Controller 2: Trajectory Optimization
- Stance jumping is formulated as a constrained nonlinear trajectory optimization, while joint-space PD control handles flight and landing.CasADi and IPOPT solve the optimal control problem.Source: D. Inner Controller 2: Trajectory… · p. 3
- The optimization minimizes positive mechanical work performed by the actuators.A smooth softplus approximation is used for positive-work calculation to improve convergence.Source: D. Inner Controller 2: Trajectory… · p. 3
- The stance optimization enforces robot dynamics, jump height, initial configuration, trapezoidal collocation, fixed-foot contact, friction and normal-force constraints, and actuator torque limits.Source: D. Inner Controller 2: Trajectory… · p. 3
A. Landscapes and Optima
The co-design comparison finds similar optimal link-length ratios but different absolute designs and COT; trajectory optimization performs better across most morphologies. The QP-TSC optimum also raises a physical-realizability concern, while the 2D extension uses angular-momentum regulation to return the base upright at touchdown.
A. Landscapes and Optima
- The COT landscapes compare controller performance over link lengths and show their difference; the optimal designs are summarized from five independent differential-evolution runs.The supplied passages give no landscape values or table entries.Source: A. Landscapes and Optima · p. 3, Figure 3 · p. 4, Table I · p. 4
B. Cross-Evaluation
- Trajectory optimization performs better than QP-TSC across most morphologies, although both controllers converge to similar link-length ratios (1.71 and 1.59) with different absolute designs.The optima are reported as means of five independent differential-evolution runs.Source: A. Landscapes and Optima · p. 3, B. Cross-Evaluation · p. 4
C. Physical Realizability
- The QP-TSC optimum uses links of 0.54 m and 0.34 m, with leg mass totaling 2.2 kg of the 5.2 kg robot, an implausible fraction once actuator mass is included.A physical-design pipeline should add leg-mass-fraction constraints.Source: C. Physical Realizability · p. 4
A. From 1D to 2D
- In the 2D extension, trajectory optimization regulates centroidal angular momentum and redistributes leg velocities in flight so the base returns upright at touchdown.This addresses the competing demands of damping leg motion and controlling base pitch.Source: A. From 1D to 2D · p. 4
B. Trajectory Optimization with Angular Momentum Regulation
For the 2D jump, the method constrains angular momentum and horizontal COM velocity, then uses an LQR controller for landing impact and stabilization. The landing region of attraction is assessed empirically, while the 1D co-design comparison shows asymmetric cross-controller performance and notes scope and tuning caveats.
B. Trajectory Optimization with Angular Momentum Regulation
- In-place hopping constrains total angular momentum about the COM to zero at lift-off and horizontal COM velocity to zero; the latter can be changed for forward or backward hopping.Other constraints are similar to the 1D case.Source: B. Trajectory Optimization with… · p. 4
C. Landing: Impact Absorption and LQR Stabilization
- The LQR landing controller is linearized around a nominal configuration with a fixed foot; touchdown foot position is selected to limit peak body pitch during impact.The trajectory optimization uses an extended-leg touchdown configuration for impact-absorbing stroke.Source: C. Landing: Impact Absorption and LQR… · pp. 4–5
D. Region-of-Attraction Analysis
- The landing region of attraction is estimated by varying touchdown conditions two parameters at a time, with convergence requiring pose, rate, and peak-pitch thresholds.The reported criteria are within 2° of equilibrium, within 0.3 rad/s of zero rate, and peak body pitch below 90°.Source: D. Region-of-Attraction Analysis · p. 5
A. Jump Trajectories
- TABLE III is identified as reporting 2D jump metrics, but the supplied passage does not specify its metrics or values.No specific comparison can be recovered from the caption alone.Source: Table III · p. 5
A. 1D Co-Design
- The trajectory-optimized design loses 3.5% under QP-TSC, while the QP-TSC-optimized design loses 14% under trajectory optimization.The QP-TSC landscape is shallower; both landscapes also contain a triangular region where static stability is violated.Source: A. 1D Co-Design · p. 5
B. 2D jump and Landing Controller
The paper connects morphology and controller choice to performance, then demonstrates 2D jumping with momentum regulation and LQR landing control. Its conclusions are bounded by simulation-only evaluation and identify more realistic models and terrain as future work.
B. 2D jump and Landing Controller
- The LQR stabilizes a good touchdown pose but does not recover from a poor one, with the COM-to-foot position offset dominating landing outcomes.The landing controller's empirical region of attraction is characterized over touchdown conditions; the supplied figure caption identifies Fig. 5 as that region.Source: B. 2D jump and Landing Controller · p. 5, Figure 5 · p. 6
- The empirical position- and rate-perturbation ROA regions differ, but their different units preclude a direct sensitivity comparison.Position perturbations form a narrow diagonal band, while rate perturbations cover a broad region.Source: B. 2D jump and Landing Controller · p. 5
C. Limitations
- The study is simulation-only, varies link lengths with ideal actuators, and demonstrates the 2D jump on the baseline morphology.No hardware testing was done.Source: C. Limitations · p. 5
D. Future Work
- Future work proposes checking whether the co-designed morphology holds in 2D, adding actuator models and torque-speed curves, and exploring more robust landing controllers.The paper also proposes extending both controllers to lunar terrain.Source: D. Future Work · p. 5
IX. CONCLUSION
- The trajectory-optimized morphology achieved COT 1.280 with its co-designed controller and 1.605 with QP-TSC; the QP-TSC-optimized morphology achieved 1.461 with trajectory optimization and 1.550 with QP-TSC.The conclusion also reports that trajectory optimization is stronger in absolute terms across morphologies, while morphology and controller jointly determine performance.Source: §IX · pp. 5–6
- Both in-place and forward jumps were demonstrated in simulation, using stance-phase angular-momentum regulation and LQR control during landing.The planar results identify angular momentum during liftoff and foot placement at touchdown as the two points where pitch must be handled.Source: B. 2D jump and Landing Controller · p. 5, §IX · p. 6
APPENDIX A IMPLEMENTATION DETAILS
The appendix lists implementation settings for the QP-TSC, trajectory optimization, differential evolution, and LQR design.
A. QP-TSC parameters •
- The stance phase uses Kp = [0,0,0] and Kd = [300,0,0], while the flight phase uses Kp = [0,50,50] and Kd = [0,10,10]; switching is contact based with debounce.Source: A. QP-TSC parameters • · p. 6
B. Trajectory optimization
- Trajectory optimization discretizes the 1D case into 75 nodes and the 2D case into 75 stance nodes and 60 flight nodes.Stance duration is a free variable.Source: B. Trajectory optimization · p. 6
C. Differential Evolution Parameters
- Differential evolution searches link lengths from 0.15 m to 0.6 m using random selection with exponential crossover.Source: C. Differential Evolution Parameters · p. 6
APPENDIX B LQR DERIVATION AND GAINS
- The LQR derivation obtains equations of motion using a Lagrangian formulation after introducing generalized coordinates for the foot-pinned monoped.The hip and knee are actuated, while body pitch is unactuated.Source: Appendix B · p. 6
A. Equilibrium Configuration
The appendix specifies the static equilibrium used for linearization and reports that the resulting linearized system is fully controllable.
A. Equilibrium Configuration
- The equilibrium subsection specifies the touchdown configuration, solves body pitch to satisfy G1(q0) = 0, and gives the corresponding equilibrium torques.The linearization is performed about a static equilibrium.Source: A. Equilibrium Configuration · pp. 6–7
B. Linearized Model
- The linearized system's controllability matrix has full rank, indicating that the system is fully controllable.Source: B. Linearized Model · p. 7
C. LQR Controller
The LQR section reports its feedback law and closed-loop eigenvalues, while the DE results show that the selected morphology depends on the controller and that raw energy favors smaller links.
C. LQR Controller
- The LQR control law adjusts the nominal input according to the state’s deviation from its nominal value.The displayed law is u = u0 − KLQR(x − x0).Source: C. LQR Controller · p. 7
- The reported closed-loop eigenvalues are {−1.0055, −1.6688, −1.8803, −2.0672, −2.1098, −23.2090}.Source: C. LQR Controller · p. 7
A. DE results for both conservative and extended bounds
- In preliminary 0.15–0.3 m bounds, mean optima were l2 = 0.221 m (COT 1.61) for QP-TSC and l2 = 0.225 m (COT 1.35) for trajectory optimization; l1 hit the upper bound, prompting an expanded search.The search range was extended to 0.15–0.6 m to find the true minima.Source: A. DE results for both conservative… · pp. 7–8
- Raw actuator energy biases the optimizer toward smaller link lengths, so Cost of Transport is used to compare morphologies fairly.The stated reason is that mass and inertia scale down with link lengths.Source: A. DE results for both conservative… · p. 8