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Continuous Jumping for Legged Robots on Stepping Stones via Trajectory Optimization and Model Predictive Control
Chuong Nguyen, Lingfan Bao, Quan Nguyen
TL;DR
Dynamic jumping across stepping stones remains challenging and comparatively underexplored. The paper combines full-body trajectory optimization with an MPC-based jumping controller, and reports robust consecutive jumping in simulation and robust single-jump hardware experiments.
Problem
Dynamic jumping on stepping stones remains less explored than walking, despite challenges from short contacts, hard impacts, and long flights.
Source: §I · p. 1Method
Full-body trajectory optimization generates periodic jumping references for different distances, which an MPC and joint-PD controller track through jumping transitions.
Source: §I · p. 2Results
The framework is validated for consecutive jumps across varied stepping-stone conditions in simulation and for a single jump on uneven terrain with unknown height perturbation in hardware.
Source: §I · p. 2, §VI · p. 7Takeaways & Limitations
The framework achieves robust, continuous jumping on uneven stepping stones in simulation, while hardware validation demonstrates robustness for a single jump.
Source: §I · p. 2, §VI · p. 7Hardware experiments validate robustness for a single jump; realizing continuous stepping-stone jumps on robot hardware remains future work.
Source: §I · p. 2, §VI · p. 7
Abstract
from arXiv · showhide
Performing highly agile dynamic motions, such as jumping or running on uneven stepping stones has remained a challenging problem in legged robot locomotion. This paper presents a framework that combines trajectory optimization and model predictive control to perform robust and consecutive jumping on stepping stones. In our approach, we first utilize trajectory optimization based on full-nonlinear dynamics of the robot to generate periodic jumping trajectories for various jumping distances. A jumping controller based on a model predictive control is then designed for realizing smooth jumping transitions, enabling the robot to achieve continuous jumps on stepping stones. Thanks to the incorporation of MPC as a real-time feedback controller, the proposed framework is also validated to be robust to uneven platforms with unknown height perturbations and model uncertainty on the robot dynamics.
I. INTRODUCTION
The paper addresses the limited exploration of dynamic jumping across stepping stones, where short contacts, hard impacts, and long flights make consecutive jumps challenging. It combines full-body trajectory optimization with MPC-based feedback and reports validation in simulation and hardware.
I. INTRODUCTION
- Dynamic jumping on stepping stones remains less explored than walking, and earlier jumping work largely focused on single jumps.The paper also notes that some prior approaches lacked body-motion feedback or used simplified dynamics without leg dynamics and actuator constraints.Source: §I · p. 1
- Short contact windows, hard impacts, and long flights make transitions difficult; small body-motion errors in flight can carry into landing and subsequent jumps.The paper identifies real-time ground-reaction-force feedback as important for mitigating impact perturbations to whole-body motion.Source: §I · p. 1
- Trajectory optimization uses the robot’s full-body dynamics to coordinate motion while satisfying physical constraints and supporting jumps across significant gaps.The resulting periodic gait references are intended for continuous jumps.Source: §I · p. 1
- Full-body trajectory optimization generates periodic references, while MPC and joint PD track them and support smooth transitions between consecutive jumps.The framework uses a gait library for different jumping distances and reports robustness to unknown platform-height perturbations and model uncertainty, such as an unknown load.Source: §I · pp. 1–2
- Hardware experiments validate controller robustness for a single jump on uneven terrain with unknown height perturbation, while A1-model simulations cover varied terrain and unknown loads.The simulation cases include randomly placed stones and unknown load, as well as combinations of terrain conditions.Source: §I · p. 2
II. OVERVIEW OF THE FRAMEWORK
The framework generates references for different jump distances from offline periodic trajectories and a gait library. An MPC and joint-PD feedback controller tracks those references and supports transitions between jumps.
II. OVERVIEW OF THE FRAMEWORK
- Offline trajectory optimization supplies periodic references; a gait library and interpolation policy generate online references for the next jump from the measured stepping-stone distance.The references are sent to the MPC and joint-PD controllers, which track the motion and enable smooth transitions.Source: §II · p. 2
A. Periodic Jumping Gait Optimization
- Periodic gait optimization uses full-body dynamics to coordinate the robot’s motion while respecting physical constraints and actuator limits.The optimized periodic gaits are extended to consecutive jumps with different distances.Source: A. Periodic Jumping Gait Optimization · p. 2
1) Dynamical Model for Jumping:
The jumping model focuses on sagittal-plane motion and represents the robot as a five-link rigid-body system. Periodic gaits connect four contact and flight phases, with optimization constraints governing dynamics, landing, friction, and geometry.
1) Dynamical Model for Jumping:
- The model reduces jumping to 2D sagittal-plane motion and represents the robot as a five-link rigid-body system with contact forces and actuator torques.The generalized coordinates include CoM position, body pitch, and joint angles.Source: 1) Dynamical Model for Jumping · pp. 2–3
2) Contact Schedule:
- Each periodic jump comprises front-leg contact, double contact, rear-leg contact, and flight, with the initial pose matching the final pose.All phases are solved through offline optimization.Source: 2) Contact Schedule · p. 3, Figure 3 · p. 3
3) Cost function and Constraints:
- The optimization enforces periodicity, full-body dynamics, pre-landing configuration, friction limits, and geometric constraints while weighting joint positions more heavily than torque.The stated example weights are wτ = 0.005 and wq = 1; geometric constraints provide terrain clearance and prevent self-collision.Source: 3) Cost function and Constraints · p. 3
B. Jumping Gait Library
Because solving full-body trajectory optimization in real time is impractical, the paper uses a gait library and interpolation to update jumping references quickly. The library covers four nominal jump distances and interpolates trajectories for intermediate distances.
B. Jumping Gait Library
- The gait library and interpolation policy update full-body motion references quickly because solving full-body trajectory optimization in real time is impractical.The reference is updated at the beginning of each jump.Source: B. Jumping Gait Library · p. 3
- The library contains four periodic jumping gaits with nominal step lengths of 0.6, 0.7, 0.8, and 0.9 m.Source: B. Jumping Gait Library · p. 3
- Linear interpolation produces a desired trajectory for arbitrary step lengths between neighboring nominal distances.The interpolated trajectory includes joint positions, velocities, torques, and forces.Source: B. Jumping Gait Library · p. 3
IV. JUMPING CONTROLLER
The jumping controller combines MPC with joint PD control to track an optimization-generated reference. The paper reports smooth transitions and accurate trajectories on uneven stepping stones.
IV. JUMPING CONTROLLER
- The controller combines MPC and joint PD control to track the trajectory-optimization reference.Source: §IV · p. 3
- This controller combination is intended to produce smooth jumping transitions.Source: §IV · p. 3
- The controller achieves accurate jumping trajectories on uneven stepping stones.Source: §IV · p. 3
1) Jumping Controller:
The controller uses moving-horizon MPC and joint PD control to track optimized jumping references, with a formulation designed for long flight phases and a wide range of body orientations. It also emphasizes pre-takeoff tracking and converts MPC force corrections into actuator torque.
1) Jumping Controller:
- The controller is designed for a wide range of body orientations and long flight phases, unlike prior MPC formulations focused on short-flight locomotion and small orientation variations.Source: 1) Jumping Controller · p. 4
- A moving-horizon quadratic program solves for ground reaction forces while minimizing weighted errors in body trajectory and force references.The references come from trajectory optimization.Source: 1) Jumping Controller · pp. 4–5
- The MPC imposes friction-cone and force-limit constraints alongside the shifting contact schedule.The cost assigns greater weight to the last horizon to reduce errors before takeoff.Source: 1) Jumping Controller · p. 4
- The MPC force solution is used to compensate for differences between actual and reference jumping, then combined with joint PD control and trajectory torque references.The resulting feed-forward torque is applied to the robot actuators.Source: 1) Jumping Controller · p. 5
- The MPC and joint PD combination is reported to provide accurate tracking and robustness to uncertainty.Source: 1) Jumping Controller · p. 5
2) Jumping Transitions:
For jumping transitions, the controller uses Cartesian PD control to place swing feet on the next stones and reapplies MPC using observed contact states. Recomputing MPC at double contact addresses contact-schedule mismatch that can accumulate errors across jumps.
2) Jumping Transitions:
- Swing-leg Cartesian PD control places feet on the next stepping stones, with target foot position set at the start of a pre-landing configuration.Source: 2) Jumping Transitions · p. 5
- The controller reapplies MPC for stance legs at impact and uses actual contact states because touchdown timing can differ from the predicted schedule.Source: 2) Jumping Transitions · p. 5
- At all-leg impact, MPC is recomputed from double contact to reduce errors that can accumulate from contact-schedule mismatch during limited contact time.Source: 2) Jumping Transitions · p. 5
A. Numerical Simulation
Simulations on the A1 robot model evaluate consecutive jumps across random stone gaps, unknown height perturbations, and an unknown carried load. The controller tracks references across these conditions while respecting actuator and friction limits.
A. Numerical Simulation
- The simulations use the A1 robot model, with trajectory optimization solved using Casadi and platforms modeled in Matlab-Simscape.Three cases vary stone distance, stone height, and carried load.Source: A. Numerical Simulation · p. 5
- The controller tracks references during consecutive jumps across random gaps, unknown stone-height perturbations, and an unknown carried load.Case II also reports successful jumps with orientation-reference tracking; Case III uses the Case I controller parameters.Source: A. Numerical Simulation · pp. 5–6, Figure 5 · p. 6, Figure 6 · p. 6
- In Case II, actuator torques remain within limits and MPC force commands satisfy the friction-cone constraint during consecutive jumps.The reported force-command condition is |Fx/Fz| ≤0.6.Source: A. Numerical Simulation · p. 6
B. Experimental Verification
Hardware experiments test a single jump from an uneven initial configuration. Under an unknown platform-height disturbance, the joint PD baseline fails, while the proposed controller maintains tracking and compensates for the disturbance.
B. Experimental Verification
- The comparison tests whether the rear legs clear a 5 cm box and reach the target, against a 60 cm baseline jump from a predefined initial configuration.The box height is about 33% of the robot’s initial height.Source: B. Experimental Verification · p. 6
- With an unknown disturbance, the joint PD baseline fails, while the proposed controller maintains high tracking performance and compensates for the disturbance.The comparison places a 5 cm box under the front feet; the controller combines MPC body-position and orientation feedback with joint-level PD tracking.Source: B. Experimental Verification · pp. 6–7, Figure 10 · p. 7
VI. CONCLUSION AND FUTURE WORK
The framework combines full-body trajectory optimization and MPC for robust, continuous jumping on uneven stepping stones. Validation covers random stone distances, unknown height perturbations, and unknown carried mass, while hardware experiments demonstrate controller robustness for a single jump; continuous hardware jumping remains future work.
VI. CONCLUSION AND FUTURE WORK
- The framework combines full-body trajectory optimization and model predictive control for robust, continuous jumping on uneven stepping stones.Validation covers random stone distances, unknown height perturbations, and unknown carried mass; hardware experiments demonstrate robustness for a single jump, with continuous hardware jumping left to future work.Source: §VI · p. 7