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Balancing experiments on a torque-controlled humanoid with hierarchical inverse dynamics
Alexander Herzog, Ludovic Righetti, Felix Grimminger, Peter Pastor, Stefan Schaal
TL;DR
Hierarchical inverse-dynamics controllers had theoretical advantages but lacked torque-controlled humanoid demonstrations under model and computational uncertainties. The paper simplifies their optimization, implements them in a fast torque loop, and evaluates balance and tracking, finding feasible feedback control and robust single-support balance.
Problem
Hierarchical inverse-dynamics controllers had not been implemented on torque-controlled robots where model inaccuracies and real-time computation requirements are problematic.
Method
The paper experimentally evaluates cascaded-QP hierarchical inverse dynamics, simplifies the dynamic constraints, and implements momentum-based balance and tracking controllers.
Results
The controllers ran in a fast torque loop, and momentum-based balance remained robust against unknown disturbances, including single-foot standing, while tracking accuracy was demonstrated.
Takeaways & Limitations
Hierarchical inverse-dynamics controllers are feasible for humanoid feedback control despite model inaccuracies and computational complexity.
Takeaways & Limitations
The experiments used an inaccurate CAD-based dynamic model and simple floating-base state estimation based on integrated accelerations with decay damping.
Abstract
from arXiv · showhide
Recently several hierarchical inverse dynamics controllers based on cascades of quadratic programs have been proposed for application on torque controlled robots. They have important theoretical benefits but have never been implemented on a torque controlled robot where model inaccuracies and real-time computation requirements can be problematic. In this contribution we present an experimental evaluation of these algorithms in the context of balance control for a humanoid robot. The presented experiments demonstrate the applicability of the approach under real robot conditions (i.e. model uncertainty, estimation errors, etc). We propose a simplification of the optimization problem that allows us to decrease computation time enough to implement it in a fast torque control loop. We implement a momentum-based balance controller which shows robust performance in face of unknown disturbances, even when the robot is standing on only one foot. In a second experiment, a tracking task is evaluated to demonstrate the performance of the controller with more complicated hierarchies. Our results show that hierarchical inverse dynamics controllers can be used for feedback control of humanoid robots and that momentum-based balance control can be efficiently implemented on a real robot.
I. INTRODUCTION
The paper evaluates hierarchical inverse-dynamics control for humanoid balance under real-robot uncertainties and proposes computational simplifications for fast torque control. Experiments target momentum-based balance and operational-space tracking.
- Torque-controlled robots offer compliant motion and direct contact-interaction control, but require controllers suited to uncertain, changing environments.
- Full-dynamics controllers support dynamic motions but are difficult to implement because of model requirements, sensor noise, torque bandwidth, and fast-loop timing.
- Hierarchical inverse-dynamics methods formulate general task hierarchies, including inequalities, as cascades of quadratic programs.
- The paper evaluates these controllers on a humanoid under model uncertainty, sensor noise, estimation errors, and limited control bandwidth.
- A dynamic-constraint simplification reduces inverse-dynamics computational complexity and enables implementation in a 1 kHz real-time control loop.
A. Modelling Assumptions and Problem Formulation
The controller models robot dynamics, contact conditions, actuation limits, and task objectives as affine constraints over accelerations, contact forces, and torques. Objectives are ordered by priority, with physical consistency above balance, tracking, and redundancy tasks.
- Rigid-body dynamics relate inertia, generalized forces, joint torques, and contact forces through the robot’s equations of motion.
- Contact modeling enforces stationary end-effectors, support-polygon CoP constraints, and friction-cone restrictions on ground reaction forces.
- Actuation and joint limits constrain commanded torques and joint accelerations within admissible bounds.
- Motion and force objectives are expressed as linear combinations of joint accelerations, contact forces, and torques.
- At each control cycle, dynamics, physical constraints, and control objectives are assembled as affine equations over the optimization variables.
- The hierarchy prioritizes physical consistency, then balancing and motion tracking, followed by redundancy resolution.
B. Hierarchical Inverse Dynamics Solver
The hierarchical solver resolves competing objectives through prioritized quadratic programs and preserves higher-priority optima while optimizing lower-priority tasks. Nullspace parameterization enables recursive optimization across the task hierarchy.
- Slack variables allow conflicting objectives and constraints to be traded off within quadratic programs.
- Weighted tasks alone cannot guarantee strict prioritization, so lower-priority objectives must not sacrifice higher-priority physical consistency.
- Each priority level parameterizes remaining solutions in the nullspace of preceding equality constraints.
- The recursive formulation constrains lower-priority optimization to preserve the optimality of higher-priority tasks.
C. Decomposition of Equations of Motion
The authors decompose the equations of motion so torque variables can be eliminated from the optimization, leaving momentum evolution as the dynamics constraint. This reduces problem size and enables faster hierarchical inverse-dynamics control.
- Hierarchical inverse-dynamics formulations enforce physical consistency through equations of motion expressed as optimization constraints.
- The decomposition separates joint-space dynamics from floating-base Newton–Euler equations governing whole-system momentum.Torques occur only in the joint-space equation and are determined by accelerations and contact forces.
- Because torques are linearly determined by accelerations and contact forces, optimization needs only the floating-base momentum equation as a dynamics constraint.
- The substitution reduces optimization variables from 2n + 6 + 6c to n + 6 + 6c, saving as many variables as the robot has degrees of freedom.The reduction is reported as crucial for reducing optimizer time and implementing a 1 kHz feedback loop.
- The authors note that this decomposition had generally been ignored in related work despite the need for computationally fast algorithms.
III. EXPERIMENTAL SETUP
Experiments used the lower body of the Sarcos Humanoid, with torque commands sent at 1 kHz from an offboard computer. The setup supported balancing experiments and torque-control evaluation on a compact biped platform.
- The experiments used the lower part of the Sarcos Humanoid, consisting of two seven-DoF legs and a three-DoF torso.The torso degrees of freedom were frozen because the torso supported negligible mass and had little effect on dynamics.
- The robot’s legs are 0.82 m high, while each foot is 0.09 m wide and 0.25 m long.The paper characterizes the feet as rather small for a biped.
- Figure 2 evaluates torque tracking during balancing for selected hip, knee, and ankle joints by comparing desired and actual torques.Desired torques are shown in blue and actual torques in green.
- An offboard computer sends control commands to the robot at a 1 kHz rate.Commands specify the desired current applied to each valve.
B. Low-level torque control
A low-level actuator controller converts desired joint torques into valve commands using force feedback and piston-velocity feedback. The design achieved good torque tracking, while experiments retained model and state-estimation limitations.
- Each actuator uses a torque-feedback controller to produce the desired force generated by the balance controller.The controller computes desired valve commands, which represent desired hydraulic flow.
- The valve command combines PID force feedback, constant-gain piston-velocity feedback, and a constant bias.
- The controller design achieved good torque tracking performance, which the paper identifies as necessary for hierarchical inverse-dynamics control.Figure 2 illustrates tracking during a balancing experiment.
- The dynamic model omits hydraulic hoses, electronics, and friction, while base velocity and position are estimated by integrating acceleration readings with decay damping.
- The authors report that these limitations were not too problematic in the experiments and could be addressed using system identification and filtering methods.
IV. EXPERIMENTS
Experiments on the Sarcos Humanoid evaluated hierarchical inverse dynamics in real-time stepping, balance, disturbance recovery, and single-support scenarios. A proposed decomposition reduced computation time, while momentum-based control maintained balance under varied disturbances and exposed limits under sufficiently large pushes.
- Experimental setup: The controller ran as a feedback law that directly computed desired torque commands without joint PD stabilization.Feedback control was performed only in task space.
- Processing time: The proposed decomposition removed 25 equality constraints and 25 optimization variables, reducing worst-case computation time by 40%.The speedup enabled a 1 kHz control loop for the 14-DoF robot; computation varied with the number of constrained effectors.
- Balance control: The momentum-based balance controller regulated linear and angular momentum, with physical constraints prioritized above balancing, force regularization, and postural control.The task hierarchy is summarized in Table II.
- Balance control: After impulsive pushes, the robot kept standing while momentum was damped, center-of-mass tracking recovered, and centers of pressure remained inside support polygons.In the planar posture, model-predicted centers of pressure were approximately correct within 2 cm error.
- Balance control: On a rapidly moved rolling platform and a balancing board with lateral slopes from [−9.5°; 9.5°], the robot remained balanced with feet flat and centers of pressure inside the foot soles.The balancing-board experiment also revealed discrepancies between predicted and real contact forces, yet the robot continued balancing.
- Balance control: Under sufficiently large pushes, the momentum could not be damped quickly enough, the optimization became infeasible, and the biped fell because stationary-foot constraints were too restrictive.A higher-level controller incorporating stepping was identified as necessary to increase the stability margin.
C. Tracking Experiments
The tracking experiment evaluated center-of-mass motion under a hierarchical controller with additional posture and contact-force tasks. The robot tracked commanded trajectories closely while retaining limited disturbance-handling ability.
- The experiment tracked desired center-of-mass motion while regularizing joint posture and tangential contact forces.
- The commanded trajectories included a 0.3 Hz sine wave with 0.02 m forward and 0.03 m vertical amplitudes, plus a 0.06 m vertical sine wave.
- The robot handled some disturbances during tracking, but less effectively than when angular momentum was also regulated.
- The controller computed commands in an average of 0.9 ms with a 0.045 ms standard deviation for the more deeply prioritized hierarchy.The authors report that this was near the limit of the 1 kHz control cycle.
- The desired and actual vertical center-of-mass positions and velocities matched closely in both tracking tasks.The reported position RMSE was 0.0094 m for the upper plot and 0.0058 for the lower plot.
D. Balance in Single Support
The controller transitioned the humanoid to single support and maintained balance during impulsive pushes. It recovered center-of-mass tracking while preserving continuous control and respecting contact constraints.
- The experiment tested contact switching by lifting the swing foot 10 cm while balancing on the stance foot.
- The transition used three phases: moving the center of mass over the stance foot, unloading the swing-foot contact, and controlling the subsequent single-support motion.
- Figure 7 compares center-of-mass tracking with the measured push force during single-foot balancing.
- The contact-switching strategy maintained continuous control commands, with computation generally below 1 ms and occasional peaks of 1.05 ms during unloading.
- 150 N peak pushes with 4.5 Ns to 5.8 Ns impulses produced fast, damped center-of-mass recovery above the support polygon.
V. DISCUSSION
The discussion relates the presented results to other approaches.
- The discussion compares the presented results with other balancing approaches.
- The comparison concerns the results presented in the preceding experiments.
- The discussion addresses how the results relate to prior approaches.
A. Relation to other balancing approaches
The paper positions its balance controller as a constrained, single-optimization reformulation that differs from sequential or kinematics-only alternatives. The experiments support hierarchical inverse dynamics as feasible on a torque-controlled humanoid despite model inaccuracies and computational demands.
- A. Relation to other balancing approaches: The balance controller solves one optimization problem while enforcing joint, acceleration, torque, CoP, and contact-force constraints consistently.
- A. Relation to other balancing approaches: Unlike sequential optimization approaches, the formulation searches the full solution set and incorporates contact forces explicitly rather than using purely kinematic control.
- A. Relation to other balancing approaches: The controller differs from another full-dynamics approach that first determines contact forces and then computes accelerations and torques through least squares.
- A. Relation to other balancing approaches: The hierarchical solver computes one priority’s null-space mapping in parallel with solving the QP at that priority, while alternative inequality handling may trade construction time against solve time.
- A. Relation to other balancing approaches: The experiments suggest that complete dynamic models and task hierarchies are feasible for humanoid feedback control despite model inaccuracies and computational complexity.