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Momentum Control with Hierarchical Inverse Dynamics on a Torque-Controlled Humanoid
Alexander Herzog, Nicholas Rotella, Sean Mason, Felix Grimminger, Stefan Schaal, Ludovic Righetti
TL;DR
The paper addresses the difficulty of implementing hierarchical inverse dynamics with momentum feedback on torque-controlled humanoids under model inaccuracies, sensor noise, and real-time constraints. It reformulates cascaded QPs for real-time control, integrates LQR-designed momentum regulation, and evaluates the approach experimentally, finding robust feedback performance across balancing and tracking tasks, including one-foot support.
Problem
Hierarchical inverse dynamics had not been demonstrated as task-space feedback control on a torque-controlled humanoid under model inaccuracies, sensor noise, and real-time computation requirements.
Method
The paper combines a reformulated hierarchical QP cascade with a dynamics decomposition, LQR-based momentum control, and extensive experiments on a torque-controlled humanoid.
Results
Experiments show robust balancing and tracking performance with recovery from pushes and disturbances in single and double support, including balancing on one foot.
Takeaways & Limitations
Hierarchical inverse dynamics with momentum control is a feasible approach for feedback control under real robot conditions despite model inaccuracies and computation constraints.
Takeaways & Limitations
During contact transitions, interpolated LQR gains do not guarantee stable behavior, and sufficiently strong pushes can require stepping because stationary-feet constraints become too restrictive.
Abstract
from arXiv · showhide
Hierarchical inverse dynamics based on cascades of quadratic programs have been proposed for the control of legged robots. They have important benefits but to the best of our knowledge have never been implemented on a torque controlled humanoid where model inaccuracies, sensor noise and real-time computation requirements can be problematic. Using a reformulation of existing algorithms, we propose a simplification of the problem that allows to achieve real-time control. Momentum-based control is integrated in the task hierarchy and a LQR design approach is used to compute the desired associated closed-loop behavior and improve performance. Extensive experiments on various balancing and tracking tasks show very robust performance in the face of unknown disturbances, even when the humanoid is standing on one foot. Our results demonstrate that hierarchical inverse dynamics together with momentum control can be efficiently used for feedback control under real robot conditions.
1 Introduction
The paper evaluates hierarchical inverse dynamics and momentum-based feedback on torque-controlled humanoids, where model accuracy, sensor noise, torque bandwidth, and real-time computation complicate implementation. Extensive experiments assess balancing, tracking, and contact switching, including one-foot support.
- Torque-control algorithms can provide high-performance, compliant motion and direct control of contact interactions in dynamic, uncertain environments.
- Full-dynamics inverse-dynamics controllers suit dynamic motions but are challenging because of sensor noise, limited torque bandwidth, precise-model requirements, and fast-loop computation demands.
- Hierarchical inverse dynamics expresses complex behaviors at task level while enforcing strict priorities, such as preserving balance over lower-priority goals.
- Replay of hierarchical-inverse-dynamics trajectories with joint-position control demonstrates real-system usability but does not provide task-space feedback or direct multi-contact force control.
- Momentum regulation can benefit humanoid balancing and walking by controlling both linear momentum associated with the CoG and angular momentum.
- The paper extends preliminary work with real-time algorithm modifications, LQR-based momentum gains, and experiments showing robust balancing, tracking, and contact switching, including one-foot balancing.
2 Hierarchical Inverse Dynamics
The controller formulates robot dynamics, contact, actuation, and task objectives as prioritized quadratic programs, then simplifies the dynamics formulation to reduce optimization size. The resulting cascade supports real-time hierarchical feedback control.
- Problem formulation: Rigid-body equations model inertia, non-contact forces, underactuation, commanded torques, contact Jacobians, and generalized contact forces.
- Physical constraints: Contact constraints enforce stationary feet or hands, while CoP and friction inequalities maintain support and prevent slipping.
- Physical constraints: Torque and joint limits are included explicitly so optimized commands remain valid for the robot's actuation and configuration bounds.
- Problem formulation: The controller optimizes joint accelerations, contact forces, and torques subject to dynamics, physical constraints, and prioritized control objectives.
- Hierarchical solver: Conflicting objectives are traded off with quadratic-program slack variables, while a sequence of QPs preserves optimality across task priorities.
- Hierarchical solver: Nullspace mappings eliminate locked degrees of freedom between hierarchy levels, reducing variables and potentially reducing inequalities as constraints become fixed.
- Hierarchical solver: Combining inequality prioritization with variable elimination yields the benefits of both hierarchical QP formulations.
- Dynamics decomposition: Replacing torques using the decomposed dynamics reduces optimization variables from 2n + 6 + 6c to n + 6 + 6c and enables a 1 kHz feedback loop.
3 Linear and angular momentum regulation
The paper formulates momentum regulation as a force-control task and uses LQR to compute feedback gains for hierarchical inverse dynamics. This addresses tuning difficulties and momentum coupling while simplifying real-robot application.
- Momentum formulation: Momentum regulation can be expressed either as a kinematic task based on joint accelerations or as a force task based on external forces.The corresponding matrix is treated as the task Jacobian.
- Momentum formulation: The force formulation avoids computing the numerically differentiated centroidal momentum matrix derivative, which can amplify noise.External forces can also be interpreted as the system’s control inputs for controller design.
- PD limitations: PD momentum control requires contact-dependent gains and does not exploit coupling between linear and angular momentum rate of change.These issues create many open parameters and complicate tuning on the real robot.
- LQR design: The proposed method linearizes the force-based dynamics and computes an LQR controller from a selected performance cost.The resulting law includes feedback and feedforward terms and supplies the desired closed-loop momentum behavior.
- LQR design: The controller projects forces into momentum space to exploit multi-contact redundancy for further internal-force optimization.This projection is unavailable when using the direct force-control expression alone.
- LQR design: The approach accounts for momentum coupling and computes optimal gains for contact and pose configurations at low computational cost.The authors report that this drastically simplified application on the real robot.
- Implementation boundary: During contact transitions, gains are interpolated between key-pose controllers, but this interpolation does not guarantee stable behavior theoretically.The authors report that it worked well in practice because transitions were fast and trajectories were planned in advance.
4 Experimental Setup
Experiments used the lower Sarcos Humanoid with torque feedback, state estimation, and controlled disturbances. The setup achieved good torque tracking despite an imperfect dynamic model.
- Sarcos Humanoid Robot: The experiments used the lower Sarcos Humanoid, consisting of two legs and a torso, with the torso degrees of freedom frozen.Each leg has 7 DoFs, while the torso has 3 DoFs and negligible supported mass in this setup.
- Sarcos Humanoid Robot: Each joint uses hydraulic actuation with piston-force load cells and position sensors, while foot force sensors and a pelvis IMU provide measurements.The sensing supports torque control and floating-base state estimation.
- Low-level torque control: The low-level controller computes valve commands from desired and measured force using PID feedback, piston-velocity feedback, and a constant bias.The piston velocity is computed from joint velocity and the kinematic model.
- Low-level torque control: Good torque tracking was necessary for good hierarchical inverse-dynamics performance during balancing experiments.Figure 2 compares desired and actual torques for selected left-leg joints.
- State estimation: Floating-base pose and twist are estimated with an extended Kalman filter that fuses IMU and leg-kinematics information.The filter handles contact switching without assuming the gait or world contact location.
- Model limitations: The CAD-based dynamic model omits hydraulic hoses, electronics, and friction, yet the experiments obtained good results without a perfect model.The authors expect better performance after dynamic identification.
- Disturbance experiments: Disturbances were generated with a force-sensing push stick and rolling or tilting platforms instrumented with an IMU.These scenarios were used to evaluate behavior under applied and platform-induced disturbances.
5 Experiments
Experiments evaluated hierarchical inverse dynamics with momentum control in balancing and tracking tasks, including disturbances, rolling and balancing platforms, squatting, and single-support pushes. The controller maintained balance and recovered CoG tracking under substantial disturbances, while the decomposition enabled real-time operation and hierarchy-based constraint handling.
- Experimental setup: The experiments evaluated the controller as task-space feedback on balancing and tracking tasks in single and double support.Desired torques were sent directly to the robot without joint PD stabilization.
- 5.1 Processing Time: 40% lower worst-case computation time from the proposed decomposition enabled a 1 kHz control loop on the 14 DoF robot.The full 25 DoF simulation task increased peak computation time from 1 ms to 3 ms, and computation varied with constrained end effectors.
- 5.2 Balance Control Experiments: Peak forces up to 290 N and impulses of 9.5 Ns were absorbed without falling, while momentum was damped quickly and CoG tracking resumed after disturbances.These values summarize the strongest pushes reported for the momentum-control experiments.
- 5.2.2 Comparison of momentum controllers: LQR gains produced similar maximum angular-momentum deviations but improved CoG recovery and faster damping than diagonal gains in push experiments.The LQR response remained better behaved despite a higher peak force in the representative comparison.
- Balance and tracking experiments: The controller maintained balance with compliant posture adaptation under constant pushes and when the feet were not co-planar, including one foot placed on a block.The experiments also tested rapid rolling-platform and balancing-board disturbances while monitoring CoG, momentum, platform motion, and CoPs.
- Limitations: At sufficiently large pushes, the fixed-foot constraint became too restrictive, the optimization failed to find balancing solutions, and the biped fell.The reported boundary motivates a higher-level controller that can account for stepping.
- Tracking and single-support experiments: Hierarchical constraints kept CoPs inside the support polygon during squatting and supported CoG velocity tracking, while single-support pushes up to 150 N were recovered with bounded CoPs.The hierarchy prevented foot tilting when CoP constraints became active, and swing-leg tracking was barely affected during a strong hip push.
6 Discussion
The discussion finds that hierarchical inverse dynamics can preserve physical consistency and prioritize balancing while operating in real time, with LQR momentum control improving adaptability and performance. Practical limitations include restricted degrees of freedom, task-specific tuning, and the need for faster solvers on larger robots.
- Task design and hierarchies: Hierarchical separation keeps solutions physically consistent when reaction-force constraints conflict with lower-priority tasks, trading off task performance to preserve balance.The authors state that this guarantee is unavailable with a standard QP formulation.
- Task design and hierarchies: The 14-DoF robot limits available hierarchy levels, motivating equal-priority posture and momentum tasks with weight-based importance adjustment.The authors expect a full humanoid with arms to provide more flexibility, including higher-priority manipulation tasks.
- Task design and hierarchies: Experiments used only a small set of task weights, while more task-specific parameter and hierarchy adjustments could further improve performance without joint PD stabilization.The reported balancing behavior is attributed to the hierarchical inverse dynamics controller.
- Relation to other balancing approaches: The controller combines momentum control with a single optimization problem that enforces joint, acceleration, torque, CoP, and contact-force constraints consistently.This contrasts with approaches that simplify the optimization by solving for contact forces before accelerations and torques.
- Relation to other balancing approaches: LQR momentum control accounts for linear–angular momentum coupling and automatically recomputes gains across poses and support conditions without manual retuning.This was particularly useful for contact switching and single-support experiments.
- Relations to other hierarchical inverse dynamics solvers: The QP cascade was sufficiently fast for the evaluated lower-body robot, but a more efficient algorithm is required for feedback control on the full 25 DoF robot.Prioritized inequality constraints and variable elimination provide complementary benefits, but speed remains a scalability concern.
7 Conclusion
The paper presents a modified hierarchical inverse-dynamics QP cascade with LQR momentum control in a 1 kHz feedback loop on a torque-controlled humanoid. Experiments evaluate its balancing and tracking capabilities under real-robot conditions.
- 7 Conclusion: A modified QP cascade was implemented in a 1 kHz feedback-control loop for hierarchical inverse dynamics on a torque-controlled humanoid.The paper uses LQR to formulate momentum controllers for balancing and tracking tasks.
- 7 Conclusion: The experiments systematically evaluated the robot’s balancing and tracking capabilities using the proposed control framework.The evaluation focused on experimental control performance on the torque-controlled humanoid.