A Framework for Optimal Ankle Design of Humanoid Robots

Abstract: The design of the humanoid ankle is critical for safe and efficient ground interaction. Key factors such as mechanical compliance and motor mass distribution have driven the adoption of parallel mechanism architectures. However, selecting the optimal configuration depends on both actuator availability and task requirements. We propose a unified methodology for the design and evaluation of parallel ankle mechanisms. A multi-objective optimization synthesizes the mechanism geometry, the resulting solutions are evaluated using a scalar cost function that aggregates key performance metrics for cross-architecture comparison. We focus on two representative architectures: the Spherical-Prismatic-Universal (SPU) and the Revolute-Spherical-Universal (RSU). For both, we resolve the kinematics, and for the RSU, introduce a parameterization that ensures workspace feasibility and accelerates optimization. We validate our approach by redesigning the ankle of an existing humanoid robot. The optimized RSU consistently outperforms both the original serial design and a conventionally engineered RSU, reducing the cost function by up to 41% and 14%, respectively.

• The paper introduces a unified framework combining closed-form kinematic analysis, RSU reparameterization, and multi-objective optimization to optimize humanoid ankle mechanisms.
• The framework validates both SPU and RSU architectures through rigorous evaluation, demonstrating RSU's superior performance with up to 41% cost reduction over serial designs.
• The study provides actionable insights on integrating commercial actuators and balancing performance metrics like speed, torque, and backdrivability for effective design selection.

A Unified Framework for Optimal Humanoid Ankle Mechanism Design

Introduction

The paper presents a comprehensive methodology for the optimal design and evaluation of parallel ankle mechanisms in humanoid robots. Recognizing the critical role of the ankle in ground interaction and overall locomotion agility, the authors focus on two representative parallel architectures: Spherical-Prismatic-Universal (SPU) and Revolute-Spherical-Universal (RSU). The framework integrates closed-form kinematic analysis, a novel RSU parameterization for guaranteed workspace feasibility, multi-objective optimization, and a scalar cost function for cross-architecture comparison. The approach is validated through the redesign of an existing humanoid robot's ankle, demonstrating significant improvements over both serial and conventionally engineered parallel designs.

Parallel Mechanism Architectures and Kinematic Analysis

The study targets two parallel mechanism topologies for the ankle joint, each with distinct actuation and kinematic properties:

• SPU Mechanism: Utilizes linear actuators (prismatic joints) in its legs, enabling direct control of foot orientation via joint elongations.
• RSU Mechanism: Employs rotary actuators (revolute joints), with a more complex kinematic chain involving crank and rod elements.

Both mechanisms are modeled as closed kinematic chains with two degrees of freedom, connecting the shin (fixed base) to the foot (end-effector) via three legs. The central leg is a universal joint, while the lateral legs differ in their joint sequence depending on the architecture.

Figure 1: Connectivity graphs of SPU and RSU mechanisms, illustrating closed-loop topology and independent kinematic loops.

Closed-form inverse kinematics (IK) solutions are derived for both mechanisms. For SPU, the IK reduces to solving for prismatic joint displacements given desired roll and pitch. For RSU, the IK involves solving for revolute joint angles, subject to geometric constraints that can render the problem unsolvable for certain configurations. The RSU analysis leads to an explicit existence condition for feasible workspace coverage.

Figure 2: SPU mechanism schematic, highlighting the arrangement of shin, foot, slider, and linear guide.

RSU Reparameterization for Guaranteed Workspace Feasibility

A key contribution is the RSU mechanism reparameterization, which introduces auxiliary parameters ($\gamma_i$, $\delta_i$) to guarantee that the desired operational region (in roll and pitch) is always contained within the feasible configuration space. This is achieved by deriving lower bounds for crank and rod lengths as functions of the operational region, ensuring that the IK problem admits solutions throughout the workspace.

Figure 3: RSU reparameterization, showing the operational region (red) fully contained within the feasible configuration space (gray), with singularity curves and infeasible regions explicitly visualized.

This parameterization enables robust optimization and design, eliminating the risk of unreachable configurations and facilitating automated mechanism synthesis.

Multi-Objective Optimization and Evaluation Framework

The design process is formalized as a multi-objective optimization problem, where the geometric parameters of each mechanism are tuned to minimize peak actuator force and velocity over a set of reference locomotion tasks. The optimization is performed independently for each actuator-architecture pair, accounting for actuator-specific constraints and physical integration requirements.

Figure 4: Example reference task—step-down maneuver—used for trajectory-based evaluation of candidate mechanisms.

Performance metrics are extracted from the optimized populations, focusing on high-level, architecture-agnostic criteria:

• Speed and torque transmission at the ankle joint
• Backdrivability (static friction at actuators)
• Manipulability ratio (isotropy of motion/force transmission)
• Mechanical compactness
• Actuation mass
• Center of mass (CoM) height

Region-based weighting is applied to metrics that vary across the workspace, prioritizing performance in task-critical regions via a raised cosine taper.

Figure 5: Raised cosine tapered weight map, with red dots marking foot configurations observed during simulated and experimental tasks.

All metrics are normalized and aggregated into a scalar cost function, enabling unified ranking and selection of optimal designs across architectures and actuators.

Experimental Validation and Comparative Results

The framework is implemented using Grasshopper and Python, with NSGA-II employed for multi-objective optimization. Commercial actuators are selected based on preliminary power demand analysis, and geometric constraints are enforced for collision-free integration.

Figure 6: Pareto front of multi-objective optimization for RSU mechanism with Synapticon actuators.

The scalar cost function is computed for all candidates, revealing that RSU mechanisms with Synapticon actuators consistently achieve lower cost and variance compared to other configurations.

Figure 7: Cost distribution $\xi$ for Pareto fronts of each actuator, with arrows indicating engineered RSU performance and dashed line for original serial architecture.

A detailed breakdown of the best SPU and RSU candidates shows that RSU outperforms SPU in most metrics, though SPU exhibits lower performance variability across the workspace.

Figure 8: Comparison of best SPU and RSU candidates across seven normalized performance metrics, with average and standard deviation visualized.

The automated optimization pipeline yields designs with strictly lower cost than conventionally engineered RSU mechanisms and the original serial architecture. The optimized RSU reduces the scalar cost by 41% over the serial design and by 14% over the engineered RSU, substantiating the efficacy of the proposed methodology.

Implications and Future Directions

The presented framework enables systematic, quantitative comparison and optimization of parallel ankle mechanisms for humanoid robots, bridging the gap between kinematic theory, actuator selection, and task-driven performance evaluation. The RSU reparameterization ensures robust workspace coverage, while the scalar cost function supports application-specific prioritization of design trade-offs.

Practically, this approach facilitates the integration of commercial actuators, supports rapid design iteration, and provides guarantees on workspace feasibility and task coverage. Theoretically, it advances the state-of-the-art in parallel mechanism synthesis and evaluation, offering a template for similar methodologies in other robot joints and closed-chain systems.

Future work will focus on embedding collision checking within the optimization loop and integrating the cost function directly into the search process for more efficient, performance-guided design.

Conclusion

This paper delivers a unified, task-informed framework for the optimal design and evaluation of parallel ankle mechanisms in humanoid robots. By combining closed-form kinematic analysis, robust RSU parameterization, multi-objective optimization, and scalar cost-based evaluation, the methodology enables informed selection and synthesis of ankle architectures tailored to specific locomotion requirements. The demonstrated improvements over serial and conventionally engineered designs underscore the practical value of the approach, with future extensions poised to further enhance its applicability and integration into advanced robotic systems.