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Torque Saturation in Bipedal Robotic Walking through Control Lyapunov Function Based Quadratic Programs
Kevin Galloway, Koushil Sreenath, Aaron D. Ames, J. W. Grizzle
TL;DR
The paper addresses the problem that user-imposed torque saturations can degrade CLF-based biped walking control and violate theoretical performance measures. It incorporates those bounds directly into a CLF controller through convex optimization, then demonstrates the approach in simulation and on MABEL at real-time control rates. The resulting method handles strict or soft saturation constraints and extends to broader user-defined online constraints.
Problem
User-imposed input saturations can degrade CLF controller performance, violate theoretical measures, and leave the system unable to respond to increasing tracking errors.
Method
The paper implements a pointwise min-norm CLF controller with user-defined saturation constraints through a convex optimization routine suitable for embedded hard real-time operation.
Results
Numerical simulations and experimental implementation on MABEL demonstrate that the saturation-aware control methods can be useful in systems requiring high real-time control update rates.
Takeaways & Limitations
The approach can handle strict or soft input saturations and incorporate a whole family of user-defined constraints into online controller-effort calculation.
Abstract
from arXiv · showhide
This paper presents a novel method for directly incorporating user-defined control input saturations into the calculation of a control Lyapunov function (CLF)-based walking controller for a biped robot. Previous work by the authors has demonstrated the effectiveness of CLF controllers for stabilizing periodic gaits for biped walkers, and the current work expands on those results by providing a more effective means for handling control saturations. The new approach, based on a convex optimization routine running at a 1 kHz control update rate, is useful not only for handling torque saturations but also for incorporating a whole family of user-defined constraints into the online computation of a CLF controller. The paper concludes with an experimental implementation of the main results on the bipedal robot MABEL.
I. INTRODUCTION
The paper builds on CLF-based stabilization of periodic biped gaits, where hybrid dynamics and underactuation make control challenging. It introduces a controller implementation that preserves CLF performance while respecting user-defined input saturations at a 1 kHz update rate.
- Motivation: Hybrid and underactuated biped dynamics make periodic walking control challenging.HZD addresses this by driving the robot to a lower-dimensional zero-dynamics manifold containing an exponentially stable walking orbit.
- Prior CLF control: RES-CLFs let users directly tune the exponential convergence rate through an additional parameter.Earlier CLF-based control established theoretical hybrid-system properties and successful MABEL experiments.
- Prior CLF control: Earlier MABEL experiments produced smoother, less noisy CLF torques than comparable PD control, but user-defined saturations remained active throughout much of the experiment.The saturation issue motivates a different controller implementation.
- Contribution: The paper introduces a constructive method that respects user-defined input saturation bounds while preserving desirable CLF performance characteristics.The method is implemented on an embedded hard real-time system operating at 1 kHz.
- Paper organization: The paper reviews hybrid biped dynamics and CLF control, develops saturation-aware quadratic programming, and evaluates it in simulation and experiments.The stated workflow culminates in experimental results and a discussion of future work.
II. CONTROL LYAPUNOV FUNCTIONS FOR HYBRID SYSTEMS REVISTED
The paper formulates biped walking around Hybrid Zero Dynamics and uses RES-CLFs to regulate transverse errors toward the zero-dynamics manifold. Its baseline controller selects minimum-norm inputs satisfying the CLF condition, but the accompanying MABEL torque data show that raw commands can substantially exceed implemented saturation limits.
- Biped model and HZD: The biped model uses configuration q and motor torques u, with outputs designed to impose virtual constraints.The robot dynamics are expressed in configuration variables and control torques before introducing output functions.
- Biped model and HZD: HZD drives output functions and their first derivatives to zero, defining a lower-dimensional zero-dynamics manifold for walking.The manifold is characterized by y(q)=0 and L_f y(q, q̇)=0.
- Transverse dynamics: A pre-control law decomposes the dynamics into transverse variables η and zero-dynamics states z, with feed-forward torque u* preserving the manifold when µ vanishes on Z.The resulting closed-loop dynamics are written in terms of (η,z).
- RES-CLF control: The RES-CLF condition defines admissible feedback inputs that enforce a Lyapunov decrease rate for the transverse dynamics.A feedback law µ_ε selected from the admissible set K_ε provides the stated convergence guarantee.
- RES-CLF control: The convergence rate toward the zero-dynamics manifold is directly controlled by ε, and the practical baseline chooses a pointwise minimum-norm feedback law.The minimum-norm formulation selects among feedback inputs satisfying the RES-CLF condition.
- Torque saturation: In the MABEL experiment, raw CLF torque commands for the leg-angle motor sometimes exceeded 400% of the implemented saturated torque over a substantial part of a step.The figure compares thinner unsaturated commands with thicker experimental saturated torques across four consecutive walking steps.
III. ADVERSE EFFECTS OF TORQUE SATURATION ON THE CLF CONTROLLER
User-imposed torque saturations can degrade CLF-controller performance by violating nominal bounds and weakening tracking-error correction. The section motivates saturation-aware CLF control for nonlinear systems.
- Observed effects: Active torque saturations during MABEL experiments significantly affected CLF control and frequently violated its nominal bounds.The saturations were necessary for safety but were not applied in a way that preserved the controller’s theoretical properties.
- Observed effects: When ideal control effort exceeds prescribed bounds, controller performance degrades and theoretical performance measures may be violated.Saturation bounds may be imposed for safety or physical necessity, including limiting current draw from a dying battery.
- Observed effects: Saturated inputs leave the system effectively open-loop, preventing responses to increasing tracking errors and often leading to eventual failure.
- Motivation: The paper’s objective is a CLF-based controller for general nonlinear systems that respects user-specified input saturations.This addresses the experimental importance of saturation-aware controller design beyond approaches for special system classes.
A CONVEX OPTIMIZATION
The CLF min-norm controller is recast as a convex optimization problem so input bounds can be enforced online. Slack variables and penalties trade CLF-rate violations against saturation violations, while hard bounds can be imposed exactly.
- Convex formulation: The pointwise min-norm CLF controller is equivalently formulated as a convex optimization problem.
- Convex formulation: The optimization enforces the CLF derivative bound ˙Vε(η) ≤−c3/ε Vε(η), yielding the original min-norm controller.
- Soft constraints: Input saturation constraints are added by relaxing the CLF derivative bound with slack d1 and penalizing constraint violations in the objective.The relaxed condition is ˙Vε(η) ≤−c3/ε Vε(η)+d1.
- Soft constraints: Penalty weights can prioritize saturation adherence over CLF-rate adherence, with p2 > p1 penalizing input-bound violations more severely.The input constraints apply to the total control input u* + v, not only the corrective term.
- Hard constraints: Hard torque bounds can instead be enforced perfectly by redefining the optimization problem when saturation limits cannot be relaxed.
- Hard constraints: Saturation limits may be specified as functions of time or system state because the convex program is solved at every instant.For periodic motions, state-dependent limits can vary along the walking orbit.
V. SIMULATION AND EXPERIMENTAL RESULTS
The evaluation validates the proposed control methods through numerical simulations and experiments using models that progress from a simple MABEL representation to a more detailed setup. The study uses MABEL’s compliant actuation and four torque-controlled outputs.
- Evaluation setup: Numerical simulations and experiments evaluate the control methods developed in Section IV.
- Evaluation setup: The evaluation progresses from a simple MABEL model to a complex model that includes compliant ground and cable stretch.
- Robot and controls: MABEL is a five-link biped with point feet and series-compliant actuation intended to improve agility and energy efficiency.
- Robot and controls: The four control inputs are leg-angle and leg-shape motor torques for the stance and swing legs.
A. Numerical simulation
Numerical studies apply hard input saturation across multiple cases and show that increasingly restrictive bounds gradually worsen tracking. One case destabilizes the gait, while complex-model phase portraits depart from the nominal limit cycle under stricter saturation.
- Numerical simulation: The simulations use the hard-input-saturation CLF controller across four cases with different control bounds.Case D specifies bounds dynamically as a function of the robot state.
- Numerical simulation: Case C leads to walking-gait instability, visible in both tracking errors and the Lyapunov-function plots.
- Numerical simulation: More restrictive saturation bounds increase tracking error, but the degradation is gradual in the representative walking-step simulations.
- Complex-model simulation: In the complex MABEL model, stricter saturations gradually deteriorate tracking as the torso-angle limit cycle deviates from the nominal orbit.The phase portrait covers 15 walking steps, and the saturation cases become more restrictive from Case I to Case IV.
B. Experimental results
The experiments implemented CLF controllers with soft and hard torque-saturation constraints on MABEL under a 1 kHz real-time requirement. Soft constraints enabled stable walking for 169 steps, while hard constraints respected specified bounds over 70 steps, aside from isolated optimization timeouts.
- Real-time implementation: The controller computations were completed within the 1 kHz real-time requirement using CVXGEN to solve the constrained quadratic program in a few hundred microseconds.The full computation had to finish in less than 1 ms.
- Soft saturation experiment: With soft torque bounds of 8 Nm for LA and 12 Nm for LS, MABEL completed 169 stable walking steps before operator termination.The optimization could relax the user-defined bounds when necessary.
- Experimental platform: The hard-constraint experiment used MABEL, a bipedal robot whose setup and coordinates are shown in Figure 5.The figure documents the experimental platform associated with the reported walking trials.
- Hard saturation experiment: Hard-bound violations occurred only at isolated times when the convex optimization failed to converge within the specified time constraints.The paper reports that these occurrences had no effect because a motor could not respond to them.
VI. CONCLUSION
The paper presents a quadratic-program implementation of the pointwise min-norm CLF controller that handles strict and soft input saturations. Simulations and MABEL experiments support its practical use at high real-time update rates and its extension to broader user-defined constraints.
- Contribution: The proposed method implements the pointwise min-norm CLF controller while handling user-defined input saturations as either strict or soft constraints.The conclusion identifies this as the paper’s alternative controller implementation.
- Validation: Numerical simulations and experimental implementation demonstrate practical use of the control methods in systems requiring a high real-time control update rate.The conclusion connects both evaluation modes to real-time applicability.
- Generalization: The approach can incorporate a whole family of user-defined constraints into the online calculation of controller effort.The paper frames dynamic torque saturation as one example of this broader constraint-handling capability.
- Potential application: Progressively lowering torque saturations as battery charge decreases could prolong remaining battery charge while allowing system performance to degrade gracefully.This is presented as a potential application for power-limited systems.