Source-linked AI summary
Highly Dynamic Quadruped Locomotion via Whole-Body Impulse Control and Model Predictive Control
Donghyun Kim, Jared Di Carlo, Benjamin Katz, Gerardo Bledt, Sangbae Kim
TL;DR
Dynamic quadruped locomotion requires control of short stance periods, aerial phases, and high-speed leg swings. The paper combines MPC-based reaction-force planning with WBC that executes those forces while stabilizing posture and controlling swing legs. On Mini-Cheetah, the controller supported varied gaits and reached 3.7 m/s.
Problem
Dynamic legged locomotion lacks established control schemes that handle aerial phases, short stance times, and high-speed leg swings.
Method
The controller combines MPC for longer-horizon reaction-force planning with WBIC for reaction-force execution, posture stabilization, and swing-foot control.
Results
3.7 m/s running was achieved on the Mini-Cheetah quadruped robot, alongside demonstrations of various gaits in outdoor and other environments.
Takeaways & Limitations
The integrated controller supports highly dynamic locomotion with aerial phases and multiple gaits on real quadruped hardware.
Abstract
from arXiv · showhide
Dynamic legged locomotion is a challenging topic because of the lack of established control schemes which can handle aerial phases, short stance times, and high-speed leg swings. In this paper, we propose a controller combining whole-body control (WBC) and model predictive control (MPC). In our framework, MPC finds an optimal reaction force profile over a longer time horizon with a simple model, and WBC computes joint torque, position, and velocity commands based on the reaction forces computed from MPC. Unlike existing WBCs, which attempt to track commanded body trajectories, our controller is focused more on the reaction force command, which allows it to accomplish high speed dynamic locomotion with aerial phases. The newly devised WBC is integrated with MPC and tested on the Mini-Cheetah quadruped robot. To demonstrate the robustness and versatility, the controller is tested on six different gaits in a number of different environments, including outdoors and on a treadmill, reaching a top speed of 3.7 m/s.
I. INTRODUCTION
The paper addresses dynamic quadruped locomotion with a controller that prioritizes reaction-force execution over body-trajectory tracking. Combining MPC and WBC supports aerial phases, multiple gaits, and high-speed running on hardware.
- Motivation: Existing WBCs make frequent non-contact phases such as high-speed running difficult because they focus on trajectory following through contact-force manipulation.The motivating challenges include short stance periods, aerial phases, and high-speed swing-leg motion.
- Control architecture: The controller uses MPC to compute reaction forces and WBC to modify them for body stabilization and swing-leg control.The resulting WBC commands are sent to the robot for dynamic locomotion.
- Whole-body impulse control: WBIC primarily tracks the ground-reaction-force profile while relaxing floating-base trajectory tracking, allowing body posture and swing-foot control during dynamic gaits.This formulation incorporates pre-computed MPC reaction forces while retaining posture stabilization and reaction-force execution.
- Versatility: Changing the contact sequence enables gait switching, while hopping, pronking, and bounding can be handled without manually planning around flight or underactuation.The controller is designed to support varied behaviors with minimal reconfiguration of desired motions.
- Implementation and contributions: The implementation runs MPC at 30 Hz and WBIC at 500 Hz on Mini-Cheetah’s low-power computer using rotor-aware dynamics and linear-algebra optimizations.The paper identifies versatile robust control and real-hardware demonstrations as its two main contributions.
II. HYBRID CONTROL ARCHITECTURE
The hybrid architecture separates long-horizon reaction-force planning from high-bandwidth full-body control. MPC uses a simple lumped-mass model, while WBIC uses full-body dynamics to generate accurate joint commands.
- Architecture: The method reduces complexity by separating locomotion control into two simpler controllers.The first controller plans reaction-force profiles over one complete gait cycle.
- Model predictive control: MPC finds optimal reaction-force profiles over a full gait cycle using a simple lumped-mass model.The model represents body position, reaction forces, gravity, mass, contacts, rotational inertia, angular velocity, and contact-point moment arms.
- Whole-body impulse control: WBIC uses full-body dynamics and high-frequency feedback to compute more accurate torque commands than the lumped-mass model.Its dynamics include generalized mass, Coriolis and gravitational forces, joint torque, augmented reaction force, and contact Jacobian.
- Supporting components: A Kalman Filter-based state estimator supplies global body position and velocity from kinematics and acceleration data.A custom dynamics engine includes rotor effects in the mass matrix and Coriolis vector, although these components are outside the paper’s detailed scope.
III. MODEL PREDICTIVE CONTROL
The MPC plans reaction forces over a horizon using a convex formulation built from simplified lumped-mass dynamics. Contact scheduling, linearization assumptions, force-variable elimination, and friction constraints make the optimization computationally tractable.
- The MPC objective finds reaction forces that make the lumped mass follow a commanded trajectory.A predefined contact sequence from the gait scheduler and step planner keeps the optimization convex, enabling a unique global minimum.
- Small roll and pitch angles simplify the coordinate transformation used to obtain a convex MPC formulation.The model also assumes states remain near the commanded trajectory and pitch, roll velocities and inertia off-diagonal terms are small.
- The simplified rotational dynamics approximate d/dt(Iω) as Iω̇ by neglecting the cross-product term under the stated assumptions.
- The resulting discrete dynamics are used to construct a quadratic program minimizing trajectory-tracking and reaction-force costs under dynamics and initial-condition constraints.Friction cones are approximated through ground reaction force constraints.
- Removing force variables for feet that are not in contact reduces both cost and constraint matrices and provides a speedup of over 10 times.Because contacts are modeled as points, the formulation omits the non-flip condition.
B. Gait Scheduler
The gait scheduler defines periodic gaits using phase offsets and stance periods for each foot, while cycle duration controls gait frequency.
- Two parameters per foot—phase offset and stance period—specify the periodic gait type.Different gaits arise from the spacing of swing and stance periods within one cycle.
- Changing the cycle duration changes gait frequency while preserving the gait type.
C. Foot Step Planner
The foot-step planner selects upcoming landing locations from body position, leg geometry, and a Raibert symmetry heuristic tied to commanded velocity.
- The planner computes each upcoming footstep from the body position and the corresponding leg shoulder location in the global frame.
- The Raibert heuristic enforces identical landing and leaving angles when the robot travels at commanded velocity.The setup uses a feedback gain of 0.03.
IV. WHOLE-BODY IMPULSE CONTROL
Whole-body impulse control converts MPC reaction-force commands into joint commands while combining prioritized task execution, posture stabilization, and torque optimization. Joint-level position feedback complements torque control for high-frequency motion stabilization.
- Whole-Body Impulse Control: WBIC uses MPC reaction forces to compute joint position, velocity, and torque commands.Inverse kinematics maintains strict task priority, while quadratic programming balances acceleration and reaction-force tracking under inequality constraints.
- Whole-Body Impulse Control: Joint position and velocity commands stabilize posture through joint-level position controllers alongside the torque command.For Mini-Cheetah, joint PD control runs at 40kHz, 80 times faster than high-level full-body control.
- Whole-Body Impulse Control: Operational space control replaces configuration impedance control because Mini-Cheetah’s proprioceptive actuators are highly backdrivable and support open-loop joint torque control.
- Prioritized Task Execution: A null-space projection technique executes prioritized tasks by enforcing a strict task hierarchy efficiently.The formulation uses full configuration-space quantities and iteratively computes task commands and joint updates.
- Prioritized Task Execution: Task position errors and acceleration commands use position and velocity feedback gains Kp and Kd, while dynamically consistent pseudo-inverses are used for acceleration computation.The projected Jacobian used for acceleration differs from those used in kinematics computations.
- Prioritized Task Execution: Computed joint commands are sent to the joint-level PD controller, and acceleration commands are delivered to the quadratic program for torque computation.
B. Quadratic Programming
The WBIC quadratic program combines MPC reaction-force commands with acceleration and contact constraints, while relaxing floating-base acceleration to accommodate flight phases.
- QP formulation: The QP computes final reaction forces using the previous acceleration command and MPC reaction forces.It uses an open-source QP solver designed for small problems.
- QP formulation: The formulation includes reaction forces, a floating-base selection matrix, contact Jacobians, contact constraints, and relaxation variables.The relaxation variables apply to floating-base acceleration and reaction forces.
- Flight-phase handling: Floating-base acceleration relaxation allows the base to remain uncontrolled during flight phases but can introduce tracking errors in other tasks.The authors accept this trade-off to avoid responding to unpredictable floating-base motion with more complex prioritized control.
- Torque computation: The final WBIC step solves for joint torque commands from the reaction forces and configuration-space acceleration.Substituting these quantities into the dynamics equation yields the joint torque command.
V. EXPERIMENTAL RESULTS
Experiments use the Mini-Cheetah with one fixed task and gain setup across the tests, while prioritizing tasks in the QP.
- Experimental setup: The Mini-Cheetah experiments use the same task setup and feedback gains across all tests.Task priorities are summarized in Table I, and all joint controllers share common gains except the abduction joints.
- Experimental setup: The QP assigns reaction-force tracking weight 1 and floating-base control weight 0.1.These weights define the relative emphasis of the two listed control objectives.
- Experimental setup: Joint feedback gains are kp = 3 N m/rad and kd = 0.3 N m s/rad, with a higher derivative gain for abduction joints.The common gains apply to every joint except the abduction joints.
A. High Speed Running
The controller achieves stable high-speed Mini-Cheetah running by combining reaction-force and trajectory tracking, including aerial phases and hardware-intensive joint operation.
- High-speed running: 3.7 m/s is the highest stable forward running speed achieved by the new MPC-WBIC controller.The robot reached 4 m/s briefly but lost balance immediately afterward.
- High-speed running: At higher speed, front-foot step width narrows and hind-foot step width widens to prevent leg collisions.This adjustment permits crossed front and hind leg trajectories without collision.
- High-speed running: The test reaches 34 rad/s hip velocity, 25.5 N m hip torque, and 280 W power output.The commanded torque can exceed actuator limits because the actuator truncates output at its maximum torque.
- High-speed running: WBIC follows MPC reaction-force commands while regulating posture and swing feet rather than enforcing the constant body trajectory exactly.The resulting height and pitch vary around their commands during jumps and landings.
B. Outdoor Test
Outdoor experiments show that the controller switches among dynamic gaits, follows commanded motion, and operates across wet, slippery, and rough terrain.
- Outdoor gait tests: The controller performs various gait types with stable and robust locomotion outdoors.The same gains and weights are used across the gait tests.
- Outdoor gait tests: The measured velocity and direction follow the commanded velocity and direction well during outdoor locomotion.Gait changes are accompanied by corresponding commanded reaction-force profiles.
- Terrain robustness: The robot maintains posture and controls swing legs on wet, slippery grass and runs over rough gravel terrain.The gravel includes rolling elements that make body and step control difficult.
VI. CONCLUSION AND DISCUSSION
The controller achieves varied, high-speed quadruped locomotion with minimal user intervention, including outdoor gait execution and push-recovery on rough terrain. The authors identify portability across robot systems and future locomotion-manipulation extensions as next steps.
- Conclusion: 3.7 m/s is the reported top speed for Mini-Cheetah running using the proposed control scheme.The experiments verify the MPC-WBC combination on the Mini-Cheetah robot.
- Conclusion: Outdoor tests demonstrate multiple gaits above 1 m/s and push-recovery on rough terrain.These tests support the controller’s reported robustness across outdoor conditions.
- Conclusion: The control scheme is designed to accomplish varied quadruped locomotion with minimal user intervention.The paper presents this as a central outcome of the proposed controller.
- Future work: The authors expect the same framework to transfer to Cheetah 3 by switching system dynamics, because the formulation is independent of system parameters.They describe this as an immediate next implementation step.
- Future work: A planned extension adds manipulation tasks to cheetah robots equipped with an additional manipulator.The authors state that only a small addition to the current scheme would be needed.
- Future work: The authors plan to extend the controller to biped walking and running with additions such as new contact sequences and a step-location planner.They state that the proposed scheme is not limited to the target system’s topology.