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A Dual-Cam Parallel Elastic Actuator with Shared Gas-Spring Compensation for Humanoid Ankles
Jingcheng Jiang, Yifang Zhang, Nikos G. Tsagarakis
TL;DR
Humanoid ankles need high torque capacity and energy efficiency despite constrained space and the limitations of conventional actuation. This paper proposes a dual-cam, shared-gas-spring 2-DoF PEA, develops coupled modeling and optimized cam synthesis, and validates the design through integration and simulation. The reported results support compact, customizable two-axis torque compensation and demonstrate mechanical feasibility and compensation effectiveness.
Problem
Humanoid ankle actuation must provide high torque capacity and energy efficiency within constrained space, while dedicated elastic elements increase multi-DoF system weight and volume.
Method
The paper combines dual cams with a shared gas spring, formulates a coupled 2-DoF model, and optimizes cam profiles from prescribed torque references.
Results
The design achieves two-axis torque compensation, with CAD integration, simulations, and static FEA demonstrating mechanical feasibility and compensation effectiveness.
Takeaways & Limitations
The proposed architecture provides a compact, customizable solution for 2-DoF humanoid ankle torque compensation.
Abstract
from arXiv · showhide
To improve torque capacity and energy efficiency of humanoid ankles, this paper proposes a 2-DoF parallel elastic actuator (PEA). The main novelty of the proposed design lies in its dual-cam, single-gas-spring architecture, which enables torque compensation in both pitch and roll using a shared elastic element, thereby improving structural compactness compared with conventional multi-element compensation schemes. By leveraging parallel gas springs and customized cam modules, the proposed architecture provides dual-axis torque assistance tailored to specific task requirements. The second key contribution is the formulation of a coupled 2-DoF mathematical model that explicitly captures the interdependence between the two compensation units through the shared spring. Based on this model, an optimization-based design framework is developed to synthesize customized cam profiles from prescribed torque references, establishing a systematic link from task requirements to hardware realization. The complete lower-leg CAD integration is presented in detail. Static FEA and kinematic simulations confirm the design's feasibility and torque-relief effectiveness. The results highlight the proposed design as a compact, customizable solution for 2-DoF humanoid ankle torque compensation.
I. INTRODUCTION
Humanoid ankles demand actuators that combine high load capacity, dynamic performance, and energy efficiency within constrained space. The paper addresses this challenge with a compact 2-DoF PEA using dual cams and a shared gas spring, supported by coupled modeling and optimization-based cam design.
- Humanoid robots require high-performance actuation to achieve load capacity, dynamic capability, and energy efficiency in unstructured environments.
- High-ratio gearheads increase output torque but introduce transmission losses, added mass, reflected inertia, and reduced energy efficiency.
- Lower-ratio gearboxes improve dynamic performance and impact resilience but can increase electrical losses because current-related losses scale with the square of current.
- Elastic elements support torque compensation and energy storage, with PEAs typically used to improve actuator energy efficiency through parallel elastic assistance.
- Dedicated elastic elements for each actuator increase weight and volume in multi-DoF systems, creating a challenge for compact ankle designs.
- The proposed 2-DoF PEA uses dual cams and a shared gas spring for pitch-and-roll compensation, with coupled modeling, customizable cam-profile optimization, and CAD, simulation, and FEA validation.
II. METHODOLOGY
The methodology develops a torque compensation unit from cam–roller geometry and extends it to two coupled units sharing one gas spring. The resulting model relates cam geometry and spring compression to joint-angle-dependent compensation torque across multiple configurations.
- 2-DoF Torque Compensation Mechanism: The proposed 2-DoF mechanism consists of two torque compensation units sharing the same gas spring.
- Torque Compensation Unit: Cam–roller contact generates compensation torque because the spring-balanced contact force acts with an offset moment arm from the pole.
- Torque Compensation Unit: Aligning the cam pole with the robotic joint rotation center converts eccentric-cam radial variation into gas-spring compression and required spring force.
- Torque Compensation Unit: For a polar cam curve R(θ), the model derives contact force and moment arm from the curve geometry, roller radius, normal direction, and related trigonometric relationships.
- Torque Compensation Unit: Gas-spring compression is determined by the change in pole-to-roller-center distance as the joint rotates from its zero position.
- Torque Compensation Unit: The model incorporates initial spring force, preload displacement, and stiffness to obtain spring force and compensation torque as functions of joint angle.
- Torque Compensation Unit: Four TCU configurations vary roller placement relative to cam concavity and whether the spring acts on the cam or roller assembly.
B. 2-DoF Torque Compensation Mechanism
The proposed ankle mechanism uses two torque compensation units with concave-side rollers, distal cams, and a shared gas spring to compensate pitch and roll torques in constrained space.
- The concave-side roller configuration reduces material volume and produces a lighter assembly than the convex configuration for the same cam profile radius.The compact arrangement is better suited to the ankle’s constrained spatial environment.
- Two detachable cams are placed distally while two proximal rollers share one gas spring, supporting compact and customizable ankle integration.The layout keeps cams away from the structurally complex 2-DoF rotational axes.
- The orthogonal ankle DoFs use separate compensation units, with one TCU handling pitch and the other handling roll.The schematic combines upper and lower TCUs from orthographic views rotated 90° for representation.
- The shared gas spring couples the two TCUs because total compression equals the sum of both units’ displacements.The resulting spring force acts simultaneously on both rollers, making each output torque depend on both contact-point angles.
C. Optimization-based Torque Compensation
The paper parameterizes each cam profile and optimizes its unknown parameters to minimize reference-torque mismatch while enforcing stroke and curvature constraints; the 2-DoF formulation uses coupled bivariate torques.
- The cam radius R(θ) is represented by an m-th order polynomial with unknown parameters to balance approximation flexibility and computational tractability.R0 denotes the initial radius at the zero position.
- The optimization minimizes the sum of squared torque errors between prescribed reference torques and compensation torques.Sampled positions are evaluated through the TCU model to construct the reference and compensation torque vectors.
- The design constrains spring compression below its maximum stroke and requires the cam’s minimum curvature radius to exceed the roller radius.These constraints limit feasible profiles and prevent geometric undercut.
- Optimized parameters are substituted into R(θ) to obtain cam profiles for subsequent mechanical design.The formulation extends the single-DoF approach while retaining an analogous optimization structure.
- The 2-DoF extension makes both reference and compensation torques bivariate functions of pitch and roll because the TCUs share one gas spring.Stroke and curvature constraints are extended from the 1-DoF formulation.
III. SYSTEM DESIGN
The optimized torque-compensation results are translated into physical cam, gas-spring, roller, motor, transmission, and lower-leg CAD designs.
- The system design converts optimized cam profiles into a complete 2-DoF ankle implementation with selected commercial components and integrated lower-leg CAD.The design also specifies motors and transmission systems before assembly in CREO.
A. Cam Design workflow
The cam-design workflow links task-specific ankle torque requirements and available component specifications to cam-profile synthesis and physical component selection.
- The workflow begins with reference torque trajectories extracted from robotic ankle task scenarios.These trajectories are combined with specifications for available gas springs and roller bearings.
- The workflow uses a structured parts library containing gas-spring force and stroke data and roller-bearing radii as design inputs.The optimization-based process establishes a path from functional requirements to hardware implementation.
B. Actuator and Transmission Design
The lower-leg design uses parallel ball-screw actuation and a dual-cam ankle compensation mechanism to support compact 2-DoF pitch and roll motion.
- Parallel actuation reduces each motor’s peak torque requirement, enabling lighter and more compact actuators.
- Ball-screw transmissions provide torque multiplication without auxiliary planetary or harmonic reducers.
- Two parallel ball screws drive crank-slider mechanisms for 2-DoF ankle motion in pitch and roll.
- The lower-leg assembly, including the foot, has a total height of 495 mm and mass of 3.56 kg.
- Two customizable cams flank rollers connected to a shared gas spring in the ankle compensation mechanism.
IV. SIMULATION VALIDATION
The study evaluates cam-based torque compensation and uses static FEA to assess the strength of the resulting mechanism.
- The designed cams are evaluated for dual-output compensation torque and analyzed with static FEA for strength validation.
A. Torque Compensation Validation
Torque compensation is validated with decoupled and coupled sinusoidal references, payload-based equilibrium conditions, optimized cam profiles, and motor-torque comparisons.
- Three sinusoidal torque-reference sets are selected for preliminary validation using a 10% compensation-ratio peak and a robot nominal mass of approximately 65 kg.
- The decoupled and coupled references represent separate target pitch and roll torque scenarios, while a 4 kg payload defines additional single-leg equilibrium conditions.
- A fourth-order polynomial approximates the sinusoidal references, and optimization is performed for three reference sets with two springs.
- Torque fitting is almost perfect for decoupled references but more constrained for fully coupled scenarios because pitch and roll compensation torques are mathematically interdependent.
- Figure 8 reports average and peak motor-torque drops associated with the performance summarized in Table I.
- The optimized profiles for the 4 kg payload reference are used to formulate the corresponding pitch and roll cam designs.
B. Finite Element Analysis and Validation
Static FEA assesses the pitch and roll cam modules under the 4 kg peak-loading reference using 7075-T6 aluminum alloy.
- The cam modules are evaluated under peak loading conditions corresponding to the 4 kg load reference.
- Approximately 500 MPa maximum subsurface von Mises stress approaches the tensile yield strength of 7075-T6 aluminum alloy.
- The analysis treats the loading as a single-occurrence static condition, excluding contact-fatigue failure induced by alternating loads.
V. CONCLUSIONS AND FUTURE WORK
The paper presents a customizable 2-DoF parallel elastic actuator whose coupled model and optimization-based cam design support task-specific ankle torque compensation. CAD integration and simulations demonstrate mechanical feasibility and compensation effectiveness, while physical prototype experiments remain future work.
- The proposed architecture provides torque compensation across two ankle degrees of freedom using parallel gas springs and customized cam modules.The design targets improved torque capacity and energy efficiency for humanoid robotic ankles.
- A coupled 2-DoF mathematical model captures the interconnection between the compensation units through the shared spring force.The analysis extends a single torque compensation unit to a coupled two-axis mechanism.
- Constrained optimization derives cam profiles from task-specific torque references, establishing the design workflow for customized compensation behavior.The optimization results are translated into the physical torque compensation mechanism.
- CAD integration and simulation studies under various torque references demonstrate the mechanism’s mechanical feasibility and the optimization-based framework’s effectiveness.The implementation includes component selection, transmission systems, and a complete lower-leg CAD assembly with the 2-DoF ankle joint.
- Physical prototype fabrication and experiments to quantify torque compensation and energy-efficiency improvements across locomotion tasks are identified as future work.The current future-work plan begins with fabrication of a single-leg prototype based on the established CAD assembly.