Improving Back-drivability of Robot Joint by Reducing Reflected Inertia Using Multi-motor System

Zexin Shan, Mitsuru Endo, Yukio Tsutsui, and Shimpei Tanaka

2025 IEEE/SICE International Symposium on System Integrations SII 2025

WedP2T2.6

• T2R2

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Background

Enhancing back-drivability in robot joints is crucial for safe and effective physical human-robot interaction (pHRI). Traditional high-ratio gearboxes increase reflected inertia, reducing efficiency and back-drivability. This study proposes a multi-motor system (MMS) to distribute torque across multiple motors, allowing for lower gear ratios and reduced reflected inertia. An optimization model is developed to minimize reflected inertia while meeting load and geometric constraints.

Back-drivability is essential in applications such as exoskeletons, service robots, and collaborative robots. While powered back-drivability relies on advanced control algorithms [1], unpowered back-drivability depends on mechanical design improvements [2]. This study focuses on the latter, specifically reducing reflected inertia to improve passive back-drivability by optimizing the design of a proposed multi-stage MMS (as shown in Figure 1).

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Method

1. Reflected Inertia in Robot Joints

Back-drivability is affected by reflected inertia, which is scaled by the square of the gear ratio. Using high-ratio gearboxes amplifies motor inertia, making back-driving more difficult.

The torque required to back-drive the joint is given by:

\[T_{BD} = \left[J_{w} + G_t^2(J_p+J_m)\right] \ddot{\theta}_L + G_tT_{f,r}\]

where $J_w$ is wheel inertia, $J_p$ is pinion inertia, $J_m$ is motor inertia, and $G_t$ is the total gear ratio.

The total reflected inertia for an MMS with $a$ stages is:

\[J_{r,t} = nJ_{m}G_{t}^2 + n\sum_{i=1}^{a-1} \left[(J_{pi}G_{i}^2+J_{wi})\prod_{j=i+1}^{a}G_{j}^2\right] + nJ_{pa}G_{a}^2 + J_{ring}\]

where $J_{pi}$ and $J_{wi}$ are the inertia of the pinions and wheels of each stage, and $J_{ring}$ is the inertia of the ring gear.

2. Optimization Constraints

The optimization process includes constraints ensuring feasibility and functionality:

• Load Constraint: Ensures that the system generates the required torque:

  \[h(\overline{G}, T_m, n) = T_{load} - n T_m \prod_{i=1}^{a}G_i = 0\]

  where $T_{load}$ is the required torque, $T_m$ is the rated torque per motor, and $G_i$ are the stage-wise gear ratios.

• Geometric Constraints: Ensure gear and motor placements do not interfere within the joint space. These include:

  • Interference avoidance: Motors and gears must be arranged without overlapping. The clearance between any two motors, or between motors and gears, must be at least a predefined safe distance.
  • Maximum system diameter: The total system diameter must fit within the given mechanical space constraints. The combined size of the gear train and motors must not exceed the housing limits.
  • Stage-wise alignment: Each gear stage must have sufficient space for placement, ensuring that no two stages interfere in physical positioning. The final stage must accommodate all required transmission components without causing mechanical strain.
  • Axial alignment: The axes of the motors must be distributed symmetrically around the ring gear to balance torque distribution, ensuring uniform force application and preventing asymmetric loads.
  • Component clearance: The pinion gears and wheels must have sufficient spacing to prevent meshing interference, which can lead to increased friction and wear over time.

3. Optimization Problem

An optimization framework using the nested DIRECT method is proposed, minimizing reflected inertia while ensuring sufficient load capacity and geometric feasibility. The objective function is:

\[\underset{a, n, J_m, \overline{G}}{\text{minimize}} \quad J_{r,t}\]

subject to:

• Load constraints ensuring required torque
• Geometric constraints preventing interference
• Stress constraints ensuring gear durability

The optimization considers motor catalogs, allowable system dimensions, and material properties to find the best MMS configuration.

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Result

1. Case Study: Industrial Robot Shoulder Joint

The study applies MMS optimization to the UR10e shoulder joint. The reference single-motor system uses a Maxon EC 90 flat motor with a 160:1 harmonic drive, resulting in a reflected inertia of 12.19 kgm².

2. Optimized MMS Design

The optimized MMS consists of 5 motors and a 3-stage gear train with the following gear ratios: [2.79, 2.79, 5.79]. The total reflected inertia is reduced to 1.37 kgm², an 88.7% reduction compared to the reference system. The gear system design also considers:

• Material properties (Aluminum 7075-T6)
• Modified gears’ face widths and teeth numbers to prevent from failures caused by bending and surface contact stresses
• Uniform torque distribution among motors

3. Simulation Results

A MATLAB/Simulink simulation comparing back-driving acceleration shows that the MMS achieves a significantly higher acceleration (0.7304 m/s² vs. 0.08206 m/s²), confirming improved back-drivability.

4. Comparison with Non-Optimized MMS

A non-optimized MMS design with similar torque capacity achieved only a 70.4% reduction in reflected inertia (3.61 kgm²). This comparison highlights the effectiveness of the optimization framework in further reducing reflected inertia.

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Conclusion

This study demonstrates that a multi-motor system (MMS) significantly improves back-drivability by reducing reflected inertia. The proposed optimization framework effectively minimizes inertia while maintaining mechanical feasibility.

Future work includes:

• Considering weight constraints to prevent excessive mass increase
• Incorporating friction and efficiency analysis to refine gear system performance
• Extending optimization to stress analysis to ensure long-term reliability
• Developing control strategies for motor synchronization in MMS
• Investigating alternative MMS layouts for better spatial utilization

These advancements will further enhance robot safety and performance in physical human-robot interaction (pHRI).

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Reference

[1] S. Yamada and H. Fujimoto,“Position-based high backdrivable control using load-side encoder and backlash, ” IEEJ J. Ind. Appl., vol. 10, no. 2, pp. 142–152, 2021.

[2] H. Matsuki, K. Nagano, and Y. Fujimoto, “ Bilateral drive gear―A highly backdrivable reduction gearbox for robotic actuators, ” IEEE ASME Trans. Mechatron., vol. 24, no. 6, pp. 2661–2673, 2019.

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