Article
Design of Actuators for a Humanoid Robot with Anthropomorphic
Characteristics and Running Capability
Chathura Semasinghe
, Drake Taylor and Siavash Rezazadeh *
Department of Mechanical and Materials Engineering, University of Denver, 2155 East Wesley Avenue,
Denver, CO 80210, USA; chathura.semasinghe@du.edu (C.S.); drake.taylor@du.edu (D.T.)
* Correspondence: siavash.rezazadeh@du.edu
Abstract: In this paper, we present the details of the actuator design for our humanoid
robot, Mithra. Mithra has been designed to match an average adult human in terms of
kinematic and kinetic characteristics. This poses various challenges in actuator design that
we have addressed in this work. First, we discuss how the high-level design can help in
achieving anthropomorphic traits. Next, the detailed design is verified and finalized using
stress and fatigue analyses. Further, we conduct experiments to validate the actuator’s
bandwidth and backdrivability, and discuss the outcomes in comparison with human
characteristics. The results show that Mithra’s actuators have sufficient structural strength
to withstand high running forces, and at the same time, provide human-like traits and
capabilities to accommodate human-inspired control paradigms.
Keywords: humanoid robots; multi-DoF actuator design; finite element analysis; fatigue
analysis; backdrivability
1. Introduction
Academic Editors: Chien-Sheng Liu,
Yu-Jen Wang and Chien-Yu Chen
Received: 17 April 2025
Revised: 11 May 2025
Accepted: 12 May 2025
Published: 13 May 2025
Citation: Semasinghe, C.; Taylor, D.;
Rezazadeh, S. Design of Actuators for
a Humanoid Robot with
Anthropomorphic Characteristics and
Running Capability. Actuators 2025,
14, 243. https://doi.org/10.3390/
act14050243
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Actuators 2025, 14, 243
Leg design for humanoid robots with the goal of achieving human-like locomotion
poses major difficulties in structural and actuator design [1,2]. Many state-of-the-art humanoid robots such as Atlas [3], ASIMO [4], LOLA [5], TALOS [6], and Digit [7] have
achieved remarkable advancements in performing versatile and stable bipedal locomotion.
However, these robots still fall short of their biological counterparts due to distinct limitations across various aspects, such as energy consumption, locomotion agility, and physical
interaction with the environment during highly dynamic locomotion [8–11]. Researchers
continue to explore strategies to bridge the gap between humanoid robots and humans in
achieving human-like locomotion, while balancing energy efficiency, agility, and stability.
The functionality and overall performance of a humanoid robot heavily depend on
the design of its robotic legs, as these are the primary components for locomotion. Several
technical aspects must be considered during leg development, including mechanical structure, communication architecture, actuators, and sensors. Among these, actuator design
plays a crucial role, as actuators are the primary source of motion and force required to
perform tasks with the desired efficiency and precision [12,13]. Due to the widespread
nature of this challenge, extensive literature exists on this topic. For instance, Atlas from
Boston Dynamics [3] and the Festo Bionic Kangaroo [14] use hydraulic and pneumatic actuators, respectively, to perform highly dynamic maneuvers with outstanding environmental
compliance. However, these actuation systems require significantly more energy than their
biological counterparts and come with several other drawbacks. In particular, hydraulic
actuators are prone to energy inefficiency and maintenance issues such as oil leaks [15],
https://doi.org/10.3390/act14050243
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while pneumatic actuators suffer from low precision, noisy operation, and susceptibility to
ambient temperature and pressure changes [16]. Moreover, both of these actuation types
limit the robot’s operational time due to energy inefficiencies and often require tethering
the robot to a power source [16]. Given these limitations, electromechanical actuators have
become a popular choice due to their potential for lightweight, energy-efficient, and compact design, and the ability to deliver relatively high peak torque when needed. Notably,
the Atlas robot has recently transitioned from hydraulic to electric actuation with the goal
of enhancing energy efficiency, strength, dexterity, and agility [17].
A well-known method for the design of electric actuators (i.e., combination of motor
and gearbox) was proposed in [18], selecting the gear ratio by matching the effective inertia
of the motor with the external load. This essentially minimizes the power demand of the
chosen motor. However, high energy efficiency typically requires high gear ratios, which
hinder fast joint movements and make the leg system undesirable for force/impedance
control [12]. For high-fidelity force control and better compliance, series elastic actuators
(SEAs) are used as robotic legs, which integrates an elastic element between the actuator
and joint load in series connection [19,20]. SEAs enhance stability and adaptability in
uneven terrain, improve energy efficiency, absorb shocks, and reduce damage to mechanical
components [19,21,22]. However, the implementation of SEAs at robotic joints demands
larger actuator setup with higher gear ratios [23], which reduces the robot’s ability to
perform high-speed, agile movements. To address these issues, Parallel Elastic Actuators
(PEAs) [24,25] and Variable Stiffness Actuators (VSAs) [26,27] have been introduced. These
actuators also reduce motor torque demands and improve performance, precision, and
energy efficiency [20]. However, they often sacrifice control bandwidth and introduce
mechanical complexity, especially for a compact leg design. The addition of components
to modulate stiffness introduces nonlinearities to the system that then complicate control
strategies as well [28].
Recent advancements in brushless direct-current (BLDC) motors have enabled the
development of quasi-direct-drive actuators, which combine high-torque-density BLDC
motors with low-reduction-ratio transmissions. These systems enable accurate force control
via motor current with high transparency [29,30]. MIT Cheetah [31] and SpaceBok [32]
are two popular robots equipped with high-torque-density motor actuators with large
gap radii. However, for biped robots requiring multiple degrees of freedom and dynamic
performance, such large motors are impractical [29]. Other proposed solutions, such as
continuously variable transmissions and infinitely variable transmissions [33,34], offer
promising capabilities; however, they introduce considerable mechanical complexity, making them difficult to integrate into compact actuators, and often result in increased system
weight and component count [29]. Nonetheless, the optimal selection of the gear reduction
ratio enables the achievement of highly instantaneous joint power across a wide range
of movements without significantly increasing the actuator’s size, weight, or mechanical
impedance of the system [29].
In our previous work [8], we proposed a unified optimization framework for the
optimal design of actuators for an anthropomorphic robotic leg that operates only in
the sagittal plane. The framework focused on the optimal selection of electric motors
and transmission mechanisms to enable energy-efficient, precisely controllable, and agile
bipedal locomotion. Next, we extended this framework to account for motion in all three
planes and used it in the design and development of our new humanoid platform named
Mithra [35]. Mithra was specifically developed with the goal of matching an average
human in overall size, mobility, kinematics, kinetics, and actuation performance. While the
previous paper focused on the optimization framework and matching anthropomorphic
traits, in this paper, we present the details of Mithra’s leg actuators’ design and development.
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In particular, we present the results of both theoretical and experimental evaluations to
assess the actuators’ functionality and overall joint performance, including comparisons
with the leg designs of other bipedal robots, wherever applicable.
This paper is organized as follows: Section 2 outlines the high-level design considerations of the humanoid robot Mithra. Section 3 presents a comprehensive description of
each leg actuator design, along with the relevant engineering analyses. Section 4 discusses
the results obtained from both experimental and theoretical evaluations on the developed
joint actuators and compare with other legged robots. Finally, Section 5 concludes this
paper with a summary and future research directions.
2. High-Level Design Consideration of Mithra
With the goal of achieving human-like locomotion through an energy-efficient and
agile bipedal robot, we developed Mithra [35]. Figure 1 presents both the 3D computeraided design (CAD) model and the final manufactured version of Mithra. The robot
features a single rigid-body torso and anthropomorphic robotic legs, each with six degrees
of freedom (DoF), enabling joint movements in all three anatomical planes of the human
body. The total height, leg segment masses, and inertia distribution closely resemble those
of an average human, and the joint range of motion is also comparable to that of human
legs. Beyond stable walking, Mithra was designed and developed to perform dynamic and
demanding tasks such as stair navigation, running, and squatting [35].
(a)
(b)
(c)
Figure 1. Mithra humanoid robot platform. (a) A 3D CAD model of mechanical structure; (b) a 3D
CAD model with covers; and (c) the final manufactured version.
2.1. Design Consideration for the Mechanical Structure
To accurately achieve human-like movements, it is essential to consider several key
factors, one of the most important being the degrees of freedom. While the human leg
possesses numerous DoFs, its primary kinematic characteristics can be effectively captured
using six DoFs with sufficient accuracy: three at the hip, one at the knee, and two at the
ankle [36]. Mithra adopts this fundamental kinematic structure, which is also employed by
many well-known humanoid robots such as ASIMO [4], HUBO [37], and TALOS [6], as
illustrated in Figure 2a. This six-DoF configuration not only captures essential leg dynamics,
but also enables a closed-form solution for inverse kinematics, simplifying control and
motion planning [38].
Since the hip joint forms the base of the leg’s kinematic chain, three on-axis actuators
were selected for joint motions in all three anatomical planes. To replicate human hip
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motion and simplify joint control, each actuator’s rotational axis was directly aligned
to intersect at a common point, effectively simulating a spherical joint through on-axis
actuation. Another key design consideration was the hip configuration that determines
the order of placement of actuators in the three-DoFs’ hip joints. To minimize leg inertia
during hip flexion/extension while maximizing leg swing acceleration and considering
the range of motion (RoM) of each hip joint movement, a yaw–roll–pitch configuration
was selected for the hip joint. A comparative analysis of leg inertia across different hip
configurations in existing humanoid robots further supported this hip configuration [39].
Figure 2b illustrates Mithra’s proposed hip joint design, where all three rotational axes
intersect at a single point, thereby facilitating simple and accurate decoupled joint control.
While the hip joint is well suited for on-axis actuation, the knee and ankle joints present
different design challenges. Positioning joint actuators farther down the leg significantly
increases the effective inertia, particularly around the hip, due to the added actuator
mass. This increased inertia negatively impacts dynamic performance and brings greater
demands on control effort while imposing higher actuator requirements in terms of power
output, size, and weight. To mitigate these drawbacks, a favorable approach involves
placing actuators off-axis and closer to the proximal end of the limb. This configuration
reduces distal mass and inertia, but requires lightweight power transmission mechanisms,
such as belt drives or linkage mechanisms, to transmit power to the joints. For Mithra’s leg
structure, anthropometric parameters of the lower limb were derived based on an average
adult with a height of 1.75 m and a weight of 75 kg, as reported in [36]. While the leg
structure was designed to match human-like link lengths—0.45 m between the ankle and
knee joints, and 0.45 m between the knee and hip joints—achieving equivalent mass and
inertia distributions for each link posed significant challenges. These difficulties arose
from constraints related to geometric space, actuator integration, structural strength, and
mechanical assembly requirements for joint actuation. To balance leg inertia, mechanical
simplicity, structural integrity, and anthropomorphic form, a design compromise was
adopted: the knee actuator was positioned at mid-thigh, and the ankle actuator was placed
at the knee joint. Power transmission to both joints was achieved using four-bar linkage
mechanisms, as illustrated in Figure 2b.
(a)
3-DoF Hip
Joint
(b)
Hip actuator for
internal/external rotation
(c)
Hip joint
Hip actuator for
abduction/adduction
Hip actuator for
flexion/extension
1-DoF Knee
Joint
Ankle actuators for
ankle joint motions
2-DoF Ankle
Joint
Thigh link
Knee actuator for
flexion/extension
Parallelogramshaped four-bar
linkage
Parallel linkage
mechanism
Universal
Joint
Shank link
Robot foot
Figure 2. Mithra’s leg structure. (a) Kinematic structure; (b) schematic of the joint actuation; and (c) a
3D model of the leg design.
Given that Mithra’s knee joint has a single degree of freedom (DoF) and considering
that the mid-thigh region offers limited space, an open-configuration four-bar linkage
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mechanism was employed for off-axis actuation. For the two-DoF ankle joint, a parallel
mechanism with coupled actuation was implemented [35]. This coupled actuation strategy
combines the output of both actuators at each joint to reduce the torque demand on
individual motors. As a result, smaller and more efficient geared electric motors could be
used, contributing to improved system compactness and overall energy efficiency [40]. By
positioning the geared electric motor for knee actuation at the mid-thigh and locating the
two geared electric motors for ankle actuation at the knee joint, Mithra’s lower limb achieves
reduced leg inertia and a mass distribution that closely resembles that of a human leg.
2.2. Design Consideration for the Leg Actuation
Among various types of electric motors, brushless DC (BLDC) motors were selected
for Mithra due to their advantages in durability, low acoustic noise, and high efficiency. For
power transmission, aside from four-bar linkages, the planetary gearboxes were chosen
due to their significantly higher efficiency compared to harmonic drives [41], as well
as their simpler manufacturing process relative to cycloidal drives [42]. To minimize
additional power losses and inertial effects, multi-stage planetary gear systems were
excluded from consideration. As a result, two gearbox configurations were considered.
The first configuration (Figure 3a) is a standard planetary gearbox comprising a sun gear, a
set of planet gears, and a ring gear. This configuration typically provides gear ratios up
to 10:1. The second option is a compound planetary gearbox (Figure 3b), which enables
higher gear ratios without any significant drop in efficiency [43,44].
Based on the mechanical design considerations, each of the three rotational axes in
the hip joint was actuated using an electric motor and gearbox assembly, as illustrated
in Figure 3c. Since the knee joint was designed as a 1-DoF active joint with an off-axis
actuation, its actuation system comprised a motor, gearbox, and a parallelogram four-bar
linkage mechanism for power transmission (Figure 3c). The 2-DoF ankle joint employed
a dual-actuation scheme, in which each actuator included a motor, gearbox, and linkage
system. These actuators were coupled to generate coordinated motion through a parallel
linkage mechanism, enabling joint motion at the ankle (Figure 3c). A universal joint was
incorporated in the foot to couple these degrees of freedom and facilitate human-like
ankle motions.
(a)
Ring Gear
(c)
Hip Actuation System
Motor
Planet Gear
(b)
Motor
Ring Gear
Gearbox
Linkage
Load
Ankle Actuation System
Planetary
Carrier
Sun Gear
Load
Knee Actuation System
Sun Gear
Ring-Planet
Gear
Gearbox
Motor
Gearbox
Linkage
Sun-Planet
Gear
Load
Motor
Gearbox
Linkage
Figure 3. Joint actuation system of Mithra. (a) Standard planetary gearbox; (b) compound planetary
gearbox; and (c) actuation system configuration for each joint.
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Motors for Mithra’s leg joint actuators were selected from torque-dense models, specifically the Allied Motion Megaflux and T Motor P series, using an optimization process
detailed in our previous works [8,35]. This optimization framework also identified the most
suitable power transmission configurations, for both gearboxes and linkage mechanisms,
to support energy-efficient, agile, and stable locomotion. See [35] for further details.
3. Six-DoF Anthropomorphic Legs of Mithra
With the objective of achieving human-like motion and dynamics in walking and
running locomotion patterns, Mithra was designed with high-performance and powerful
joints. While Mithra has been developed with the capability of conducting various human
tasks, its most demanding case is characterized by running at 3 m/s. Thus, each joint was
designed and developed to provide sufficient torque and rotational speed for running at
this speed. To ensure a robust yet lightweight mechanical design, engineering analyses such
as finite element and fatigue analyses were conducted, with a minimum factor of safety
of 2 considered for all mechanical components. Based on the outcomes of these analyses,
the structural components of Mithra were refined and finalized to enhance its mechanical
performance while aligning with anthropomorphic kinematic and kinetic attributes.
3.1. Desired Joint Kinematics and Kinetics
To identify the required range of motion and joint moments for each leg joint, averaged
and normalized normative human trajectory data were utilized from [45,46]. The mass
normalized joint moment data were then denormalized based on the target mass of Mithra
(75 Kg). Additionally, transmission efficiency was factored in, along with the direction of
power flow, to ensure robust performance and a long operational life for the robot.
To support the structural design while ensuring safety and lightweight characteristics,
a quasi-static force analysis was performed in the design of each structural component. This
analysis used ground reaction forces and joint moments during running at 3 m/s to compute
the peak forces and moments applied to each leg structural component. These forces were
used in the finite element analysis (FEA) of the respective structural components.
3.2. Hip Actuator Design and Analysis
The 3D CAD model of Mithra’s hip joints is shown in Figure 4. Each actuator assembly
consists of a brushless direct-current (BLDC) motor paired with a compound planetary
gearbox. In the exploded view of the abduction/adduction hip actuator, one end of the
motor’s rotor is connected to an incremental encoder, while the other end is coupled to the
sun gear of the gearbox. The gearbox contains three planetary gear sets, each comprising a
sun–planet gear and a ring–planet gear. The gear wheels for each part of these gearboxes
were selected using the optimization framework detailed in [8,35]), ensuring that they
withstand the worst-case loading conditions. This was accomplished through constraining
the nominal tangential load on the pitch circle of each gear to be below the allowable
tangential load calculated using the AGMA bending stress equation [47]. Considering the
uncertain nature of impact loads in dynamic robotic locomotion and the critical importance
of maintaining a lightweight design, a factor of safety (FoS) of 2 was applied to all gear
wheels during the optimization process. All three hip gearboxes utilize a transmission ratio
of 15:1. The gear wheels were purchased off-the-shelf to minimize manufacturing time and
cost. However, modifications were made to reduce the weight and moment of inertia of
the gears, maintaining the minimum FoS at 2. The cylindrical shaft, which is connected to
the planetary carrier, serves as the joint’s output shaft and is supported by a roller bearing
embedded in a bearing holder (Figure 4).
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Hip actuator for
internal/external rotation
Hip internal/external
rotation joint back mount
Joint encoder
mount
Hip internal/external
rotation joint mounting
frame
Joint encoder
(absolute)
Z
Y
X
Hip actuator for abduction/adduction
Hip-abd actuator mount
Motor encoder
(incremental) Sun gear
Planetary carrier
Joint shaft
bearing
BLDC motor
Ring gear
Ring-planet gear
Gear shaft
Sun-planet gear bearings
Joint
shaft
Bearing
retainer
Hip abduction/adduction
joint mounts
Hip actuator for
flexion/extension
Hip flexion/extension
joint mounts
Joint bearing holder
Figure 4. The design of Mithra’s 3-DoF hip joint.
In Mithra’s hip joint model, the kinematic chain of the robotic leg begins with the
actuator designated for internal and external rotation. This actuator includes connection
plates that ensure a solid and secure interface with the robot torso, as shown in Figure 4.
At the actuator output, a combined radial and thrust bearing was used to accommodate
multidirectional loads, particularly high axial loads in the z-direction encountered during
both the stance and swing phases. This bearing also enables smooth, low-friction rotation
around the vertical axis, which is essential for the accurate and rapid turning of the entire
leg during locomotion. The actuator’s output shaft is connected to the hip internal/external
rotation joint mount, which is a U-shaped frame that supports the remainder of the leg
assembly. This mount is an assembled structure composed of three components and serves
as one of critical components of Mithra’s leg. It is subjected to the highest forces in all
directions during the stance phase and supports the entire weight of the leg (excluding
the internal/external rotation actuator). It is also subjected to the significant inertial forces
during leg swing. The actuator for hip abduction and adduction is mounted directly onto
this joint mount. Two hip abduction and adduction joint mounts, connected via cylindrical
shafts that rotate within radial bearings, are used to attach the hip flexion/extension actuator to this U-shaped frame, as illustrated in Figure 4. The output of the flexion/extension
actuator is linked to the hip flexion/extension joint mount frame, which connects the rest
of Mithra’s leg. Each actuator in Mithra is equipped with both a high-resolution incremental encoder for motor angular position and an absolute encoder for direct joint position
measurement. This absolute encoders facilitate accurate joint position identification during
system startup while the incremental encoders can provide high-resolution control on the
motor side.
After careful evaluation of material options for Mithra’s mechanical structure, aluminum 7075-T6 was selected due to its optimal combination of yield strength, modulus
of elasticity, density, and cost. To prevent excessive weight in the leg structure, all structural components were analyzed and optimized through extensive FEA. All leg structural
components were designed to ensure structural integrity, resistance to deformation, and
protection against buckling under the maximum forces expected during dynamic locomotion. While all structural components of Mithra underwent FEA, this paper focuses on
the most critical structures in the hip actuation system: the hip abduction/adduction joint
mount, the mount and assembled mounting frame for hip flexion/extension joint, and the
back mount and assembled mounting frame for hip internal/external rotation joint. These
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components and assemblies were identified as critical because they endure the highest
loads during the worst-case scenario, which is running at 3m/s, and any failure or lack of
rigidity in these parts would significantly impair the function of the entire leg mechanism.
For the FEA of these mechanical components and assemblies, quasi-static structural
analysis was performed under the loads corresponding to the maximum forces and torques
obtained from the force analysis. To ensure the reliability and accuracy of the results, a mesh
convergence study was conducted. The mesh was refined incrementally until changes in
maximum stress and displacement were less than 0.1 percent, confirming that the solution
was not further sensitive to mesh density and independent of mesh refinement. All final
design versions of the structural components were validated to meet the minimum FoS of
2 under the highest loading conditions, while also aiming to minimize weight and maintain
better appearance and functional leg design.
Figure 5 presents the FEA results of the final versions of the selected components,
following an iterative design process aimed at removing as much material as possible to
produce lightweight parts with improved aesthetics and structural integrity. As shown in
Figure 5a,b, the hip abduction/adduction joint mount and the hip flexion/extension joint
mount were designed to achieve a minimum FoS of 2 when fabricated with aluminum 7075-T6.
In contrast, the hip internal/external rotation back mount (see Figure 5c) demonstrated a
significantly higher minimum FoS, approximately 4. This was due to design constraints
associated with embedding a radial bearing within the component, which required increased
wall thickness to accommodate the press fit of bearing. To reflect real-world conditions
more accurately, components were analyzed both individually and within their respective
assemblies. For the assembly-level FEA, fastener and bearing connections were defined to
capture more realistic contact interactions, and additional boundary conditions and load
distributions were considered. This approach allows for a more accurate representation of
real-world effects, including the combined influence of multiple loads and the presence of
contact stresses between mating parts. Figure 5d,e show the results of the analysis for the
assembled components. In both cases, the minimum factor of safety remained close to 2,
confirming that Mithra’s hip structure is capable of withstanding impact loads and high stress
conditions encountered during running at 3 m/s.
3.3. Knee Actuator Design and Analysis
The parallelogram four-bar linkage configuration of Mithra’s knee actuation system
enables the gearbox output torque to be transmitted to the knee joint with a 1:1 transmission
ratio. The knee actuator shares the same motor and gearbox configuration as the actuators
used for the hip abduction and flexion, with one important difference: the knee motor,
an Allied Motion MF0127020 BLDC motor, features a customized winding to increase the
torque constant k t from the default 0.16 Nm/A to 0.26 Nm/A. As a result, the actuator
design was based on a similar concept. The 3D CAD model of Mithra’s knee actuator is
shown in Figure 6. Because the gearbox output in the knee joint connects directly to the
parallelogram linkage, the planetary carrier was redesigned to attach the knee input link
instead of using the planetary carrier and the joint output shaft found in the hip actuators.
To accommodate the larger diameter but relatively thin profile of the planetary carrier disk,
and to ensure high load tolerance with smooth rotation, needle-roller bearings were used
between the carrier disk and its holder.
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(b)
(a)
(c)
(e)
(d)
Figure 5. FEA results of structural components in the hip actuation system. (a) Hip abduction/adduction joint mount; (b) hip flexion/extension joint mount; (c) hip internal/external rotation
joint back mount; (d) hip flexion/extension joint mounting frame; and (e) hip internal/external
rotation joint mounting frame.
Knee actuator for
flexion/extension
Motor encoder
(incremental)
Knee input link
Knee
transmission link
Thigh side
bracket top
Knee actuator
mount
Knee side
bracket top
Knee side
bracket bottom
Joint encoder
(absolute)
Ankle side
bracket right
Ankle actuator
mount
Knee connecting
column
Hardstops
Z
X
Y
Motor driver
holder
Figure 6. The design of Mithra’s knee actuator.
To preserve the desired thigh length, two knee side brackets (top and bottom) were
designed as shown in Figure 6. The axis of rotation for the knee joint is defined by the
connection axis between the bottom knee bracket and the ankle side bracket. Mechanical
hard stops were integrated between the left and right knee bottom brackets and the ankle
bracket to prevent hyperextension of the knee joint during motion (Figure 6). These stops
also help to avoid singular configurations in the linkage, where the input and transmission
links become colinear, potentially causing control instability and a loss of mechanical
advantage. At the contact surfaces of the hard stops, polymer strips made from D3O,
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an impact-absorbing material, were used to mitigate impact forces, reduce post-impact
vibration, and protect structural components from damage under high-shock conditions.
The knee joint gearbox utilizes a custom-designed ring gear, as it is required to deliver
the highest torque among all joints in Mithra’s leg. Based on the AGMA bending stress
equation [47], the ring gear was designed with a width of 25 mm, which is significantly
wider than the 15 mm off-the-shelf ring gears used in the hip gearboxes, to withstand the
elevated stresses and forces acting on the gear teeth. Due to these high loads, the ring gear
was assembled onto its holder using heavy-duty set screws, rather than the press-fit method
employed in the hip gearboxes. The complete knee actuation assembly was designed to be
mounted to the base plate of the hip flexion/extension mounting frame.
As with the hip actuation setup, aluminum 7075-T6 was selected for constructing all
mechanical structures in Mithra’s knee actuation system, except for the knee transmission
link rod, which was fabricated using carbon fiber. All mechanical components were analyzed and optimized through extensive FEA, following the same methodology described
earlier for the hip assembly. This process ensured structural integrity, resistance to deformation, and prevention of buckling under the maximum forces expected during dynamic
locomotion. Although every mechanical component of the knee setup was subjected to
FEA, here, we only present the simulation results for the most critical structures: knee
side brackets, knee planetary carrier, knee input link, knee connecting column, and knee
transmission link assembly. Figure 7 shows the FEA results of the final design version of
these structural components after an iterative weight reduction process aimed at producing
lightweight parts with better appearance.
(a)
(c)
(b)
(d)
(e)
(f)
Figure 7. FEA results of structural components in the knee actuation system. (a) Knee side bracket
top; (b) knee side bracket bottom; (c) knee planetary carrier; (d) knee input link; (e) knee connecting
column; and (f) knee transmission link assembly.
Based on the quasi-static force analysis, approximately half of the maximum forces
acting on the knee joint are distributed to each of the side brackets, which include the
top and bottom brackets that support the thigh structure. Under these loading conditions, FEA results showed that all final design versions of knee side brackets achieved a
minimum factor of safety of 2, as illustrated in Figure 7a,b, confirming their suitability
for the worst-case loading condition during running. Further, our analyses showed that
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the knee planetary carrier achieved a minimum factor of safety of 2.3 under maximum
actuator torque output, as shown in Figure 7c. The knee input link, the transmission link
assembly, and the connecting column are critical components of the knee linkage system.
These components were designed with precise dimensions to maintain proper clearance
for link movement near the gearbox output and internal joints. This ensured a feasible and
collision-free linkage configuration that could achieve the full knee joint range of motion
from 00 to 1450 . Under the maximum loading conditions, all three components exhibited a
minimum FoS greater than 2, as presented in Figure 7d–f, with small deflections.
3.4. Ankle Actuation Design and Analysis
3.4.1. Ankle Actuation Mechanism Design
Based on the optimization results presented in [35], the ankle joint actuation setup was
designed using a parallel four-bar linkage mechanism with coupled actuation driven by two
geared brushless DC (BLDC) motors, each equipped with a single-stage planetary gearbox
(Figure 2b,c). Figure 8 illustrates the design of the ankle actuation system for Mithra.
Motor shaft
Top universal
coupler
Top universal joint cross
joint fork
BLDC
Motor encoder
motor right
(incremental)
Ankle side
bracket right
BLDC
motor left
Motor shaft
Ankle side
bracket left
Motor encoder holder
Ankle actuator mount
Shank link
connector top
Ankle transmission
link left
Ankle actuator for parallel actuation
Planetary carrier
Gear shaft bearing
Sun gear
Ankle
input link
Back
left
universal
joint fork
Ankle transmission
link right
Shank link
(linkage fixed link)
Transmission link
mount
Shank link
connector bottom
Main
Planetary
universal
joint cross carrier bearing
DP Joint encoder
(absolute)
Main universal
joint fork
Z
X
Y
foot
IE Joint encoder
(absolute)
Planet gear
Ring gear
Planetary carrier
holder
Figure 8. The 2-DoF actuation for Mithra’s ankles.
Each gearbox output connects to the foot through a transmission link and two universal joints, i.e., top and back (see Figure 8), enabling the ankle joint to achieve two
degrees of freedom (2-DoF) motion at the main universal joint located on the foot. The
dorsi/plantarflexion motions are achieved by rotating both planetary carriers of the ankle
gearboxes in the same direction at the same speed. In contrast, inversion/eversion motions
are realized by driving the planetary carriers in opposite directions. With this coupled
actuation scheme, the ankle joint can deliver twice the torque of a single actuation setup
(comprising a motor, gearbox, and single 4-bar linkage) while retaining its output speed. In
other words, this configuration allows the ankle to perform high-torque operations at high
joint speeds. The ankle actuator differs from the hip and knee actuators, since it employs
an outrunner-type BLDC motor paired with a single-stage planetary gearbox. As shown
in Figure 8, the ankle motor is mounted on the ankle side bracket, and its motor shaft
is attached via a shaft coupler. The shaft passes through a high-resolution incremental
encoder for motor position and velocity measurements, and a sun gear is mounted at the
end of the shaft.
The exploded view of the ankle actuator in Figure 8 shows the gearbox assembly,
which uses a planetary carrier similar in style to the knee gearbox. However, in this case,
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the ankle input link shaft, which serves as the gearbox output, is mounted at a radial offset
from the center of the planetary carrier disk. To ensure smooth and reliable operation under
load, a universal joint (the top universal joint) is used at the distal end of the ankle input
link shaft. The ankle actuator mount is a rigid component that secures the actuator and
encoders, and serves as the structural connection between the shank and the knee joint.
The shank connects to the foot through the main universal joint, while the transmission
link connects to the foot through the back universal joint, as shown in Figure 8. The ankle’s
four-bar parallel linkage mechanism is illustrated in 2D in Figure 9a, along with the optimal
link lengths derived from our actuator optimization process.
Integrating parallel mechanisms into the leg posed several design challenges, particularly in maintaining a desired joint range of motion comparable to those of a human’s
while achieving compactness. The ankle, having two degrees of freedom, required the
inclusion of two gearboxes and motors within a compact space at the knee joint. This led to
a wider knee joint in Mithra’s leg. Despite this, the ankle actuators are positioned higher
on Mithra’s leg than in most existing robots, so it reduces the distal mass for enhanced leg
dynamic movements. In contrast, robots such as ASIMO [4] and COMAN [48] position
their ankle actuators below the knee joint. Although the positional difference is only a few
centimeters, the associated reduction in distal mass significantly influences the dynamic
behavior of the leg.
(a)
(c)
(b)
(d)
θm
Li
θ1x
θ1y
L1
V1
Ls
Shank - Ls
(450mm)
Lc
θjx
θjy
L4
Output link
(60mm)
L5
Phase angle
(100)
Z
Z
X
Foot
Y
X
L2
L3
θ2y
Z
θ2x
X
V2
Y
Figure 9. Kinematic chain of the ankle setup. (a) The 2D four-bar linkage; (b) the kinematic loops
shown on the spatial actuation mechanism; (c) the kinematic loops shown schematically; and (d) vectors V1 and V2 used for solving the inverse kinematics.
3.4.2. Kinematic Analysis of the Ankle Actuation Mechanism
The parallel four-bar linkage mechanism of the ankle actuation requires a closed-form
inverse kinematics solution for several reasons, including accurate ankle control, force
analysis for structural component design, and determining motor positions at robot startup
using feedback from joint encoders. In the inverse kinematics calculation of the ankle,
two distinct kinematic chains were defined between the two universal joints located at
the end of the shank and the ankle input link. One chain passes through the shank link
(highlighted in blue in Figure 9b), while the other passes through the ankle transmission
link (highlighted in red). These two kinematic chains are further illustrated on a simplified
3D schematic shown in Figure 9c.
The blue kinematic chain begins at the main universal joint located at the end of the
shank. From this point, the transformation proceeds as follows: a translation along the
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z-axis by a distance of Ls to the center of the ankle gearbox, followed by a rotation about the
gearbox output axis by an angle θm representing the geared motor angle; next, a translation
along the x-axis from the center of the gearbox to the rotational axis indexed by 1y; and
finally, a translation along the y-axis to reach the universal joint at the end of the ankle
input link. The final homogeneous transformation matrix [49] for this kinematic chain is
given by

cos(θm )
 0

Hjb = 
 sin(θm )
0
− sin(θm )
0
cos(θm )
0
0
1
0
0

− Li cos(θm )

− L1


Ls − Li sin(θm )
1
(1)
Likewise, the final homogeneous transformation matrix for the red kinematic chain
can be obtained by considering rotations at ankle joint (i.e., θ jx , θ jy ), rotations at linkage
joints (i.e., θ1x , θ1y , θ2x , θ2y ), and translations in all x, y, and z directions:
Hjr = Hr1 Hr2 Hr3 Hr4 Hr5 Hr6 Hr7

where

1

0
Hr4 = 
0
0

1

0
Hr5 = 
0
0
1

0
Hr1 = 
0
0
0
cos(θ2x )
sin(θ2x )
0
0
1
0
0
0
0
1
0

0
0
− sin(θ jx ) 0

,
cos(θ jx )
0
0
1


0
0
1

− sin(θ2x ) 0

0
, Hr5 = 
cos(θ2x )
0
0
0
1
0

0
cos(θ jx )
sin(θ jx )
0

cos(θ jy )

 0
Hr2 = 
 sin(θ jy )
0

0 0
0

1 0 − L2 
,
0 1
0 
0 0
1
− sin(θ jy )
0
cos(θ jy )
0

cos(θ2y )

0

Hr6 = 
 sin(θ2y )
0
0
1
0
0
(2)

0

0
,
0
1

1

0
Hr3 = 
0
0
0
1
0
0
0
1
0
0
− sin(θ2y )
0
cos(θ2y )
0

0

0
,
0
1
0
0
1
0

− L4
− L3 

,
− L5 
1
0

0
.
Lc 
1
Since Lc and L2 are fixed link lengths in the linkage mechanism, the total distance
from the top universal joint (at the ankle input link) to the back universal joint (at the heel)
can be computed using the Pythagorean theorem, as follows:
∥V1 − V2 ∥ =
q
L2c + L22
(3)
where V1 and V2 are the position vectors of the center of top and back universal joints,
respectively, relative to the ankle joint (Figure 9d). Note that V1 can be obtained from Hjb .
A new homogeneous transformation matrix, Hj2 , which is constructed from the matrix
product Hr1 Hr2 Hr3 , can be used for computing V2 . After expanding Equation (3) and
simplifying, the following relation can be derived:
sin θm + a cos θm = b
(4)
where a and b depend only on the (fixed) linkage lengths and the ankle angles, and not on
θm . Solving Equation (4) we obtain
θm = sin−1
√
b ± a a2 − b2 − 1
1 + a2
(5)
where plus and minus correspond to the left and right kinematic chains, respectively,
in the coupled actuation of parallel linkage mechanism. Using the previously derived
relationship for the ankle gearbox output position, θm , the corresponding output velocity,
θ̇m , and acceleration, θ̈m , can be obtained by considering the first (Jacobian) and second
time derivatives of θm , respectively. Through further analysis of the kinematic chains, the
additional joint angles θ1x , θ1y , θ2x , and θ2y can also be determined. These intermediate
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angles are essential for estimating quasi-static forces acting on the linkage components
during both walking and running, which are to be used in the engineering analysis of
structural components.
3.4.3. Structural Analysis
As with the hip and knee actuation setups, aluminum 7075-T6 was selected for the
construction of all major structural components in the ankle actuation system, except for the
shank link rod and transmission link rods, which were designed using carbon fiber tubes.
Additionally, stainless steel 410 was chosen for the planetary gearbox output shaft due
to its higher strength and wear resistance requirements. All structural components were
designed to be lightweight while ensuring structural integrity, resistance to deformation,
and the prevention of buckling under the maximum forces anticipated during dynamic
activities. All designed components of Mithra’s ankle actuation system were further
analyzed and optimized using extensive FEA, following the same methodology applied to
the hip and knee actuation setups. Figure 10 presents the FEA results of the final design
version of the most critical structures, including the ankle side bracket, ankle actuator
mount, ankle planetary carrier assembly with output shaft, transmission link assembly,
shank link assembly, main universal joint fork, and main universal joint cross.
(a)
(d)
(b)
(e)
(c)
(f)
(g)
Figure 10. FEA results of the structural components in the ankle actuation system. (a) Ankle side
bracket right; (b) ankle actuator mount; (c) ankle planetary carrier with shaft assembly; (d) ankle
transmission link assembly; (e) shank assembly; (f) main universal joint cross; and (g) main universal
joint fork.
The FEA on the ankle side brackets was conducted by applying fixtures at the array
of clearance holes that is used to fasten the bracket to the ankle actuator mount and
applying the calculated forces to the extruded shaft in all three directions. Figure 10a
shows the outcomes of this analysis. The results indicate that the final design version of
the bracket has a minimum FoS greater than 2, confirming its structural integrity under
the maximum loading conditions. The ankle actuator mount is another critical component,
as it houses both ankle geared motors. It experiences high stress during foot–ground
impacts due to its direct connection to the shank. Further, during the swing phase, the
knee actuator force is applied to and transmitted through this mount. The FEA results for
the final design version of this component in Figure 10b shows a minimum FoS of 2 at the
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worst-case conditions, validating its readiness for integration into Mithra. For the ankle
planetary carrier, the FEA was performed similar to the analysis conducted for the knee
planetary carrier. Considering that the output shaft is fabricated from stainless steel 410
and is positioned near the boundary of the carrier disk, the analysis was conducted on the
assembly of the planetary carrier and the ankle input link. As shown in Figure 10c, the final
design version demonstrated a minimum FoS of 2.1, which is sufficient for safe operation
under maximum load.
Figure 10d presents the FEA results for the ankle transmission link assembly under
the maximum loading conditions. The final design achieved a minimum FoS of 2.3. Given
the importance of link length to joint position and power transmission, deflection analysis
was also performed. The link was found to deflect only 0.0018 mm at maximum load,
which is negligible and does not compromise actuator performance. A higher safety factor
was required for components closer to the foot due to the increased impacts experienced
at the distal end of the leg. To address this, a minimum FoS of 2.5 was targeted for all
foot-level structural components. Figure 10e–g present the FEA results for the final design
versions of the shank assembly, main universal joint cross, and the joint fork located below
the knee, respectively. All components met or exceeded the FoS threshold of 2.5, ensuring
structural integrity, resistance to deformation, and protection against buckling under the
maximum dynamic loads. During both piecewise and assembly-level FEA, we aimed to
replicate real-world conditions as accurately as possible. For example, in the case of the
main universal joint, fixtures were defined to simulate proper support at the roller bearing
connections, along with directional fixtures to represent the placement of snap rings that
fix the component both radially and axially.
3.5. Fatigue Analysis
All the structural components of Mithra have been optimized with respect to weight,
size, and structural integrity to enhance both overall performance and appearance. However, to ensure long-term reliability and durability, fatigue analysis had also been performed
on critical components that are subjected to repetitive loading. The critical components for
fatigue analysis include those that regularly experience high stress under cyclic loading,
parts with complex geometries that are more expensive and time-consuming to manufacture, and components that are difficult to replace, especially if replacement involves
disassembling press-fit connections or undertaking labor-intensive procedures. Given that
walking is the most common locomotion pattern in humans [50] and align with our study
of human-inspired robotic locomotion, fatigue analysis was conducted based on the forces
derived from quasi-static analysis scaled to Mithra’s total weight. The analysis considered data from fast walking at a speed of 2.0 m/s, which was chosen as the worst-case
scenario for fatigue loading due to the relatively high force repetition involved in this
locomotion mode.
Although the S–N curve for Al7075-T6 is available in the published literature (e.g., [51,52]),
directly applying these experimentally derived values to custom structural designs is often
unrealistic. This is primarily due to variations in factors such as surface finish, loading type,
stress concentrations, and operating temperature, all of which significantly influence fatigue
performance in real-world applications. To address these discrepancies, a method for adjusting
the endurance limit by incorporating six modifying factors has been proposed [53]: surface
condition, ksur f ; size, ksize ; loading type, kload ; temperature, ktemp ; reliability factor, kreli ; and
miscellaneous items, kmsc . These factors are used to find the material’s endurance limit more
accurately to reflect the realistic conditions of the customized structural components during
operation. The adjusted endurance limit calculation using Marin’s equation is formulated as
′
Se = k sur f k size k load k temp kreli k msc Se
(6)
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Based on the guidelines provided in [54], Mithra’s structural components were assumed to
be polished, machined parts. Therefore, the surface condition factor was chosen as ksur f = 0.9.
Since the average sizes of components operate at room temperature, the size and temperature
factors were both set to ksize =1 and ktemp = 1, respectively. The loading factor was assigned
as kload = 0.85, reflecting the axial nature of the loading condition. For the reliability factor,
kreli = 0.868 was selected, corresponding to a 95% reliability level, as discussed in [54]. This
selection considers the high-performance nature of the robotic applications and the expectation
that structural components should endure for the full operational lifespan of the robot with no
planned replacements. Since no additional effects such as case hardening or residual stresses
were considered, the miscellaneous factor was set to kmsc = 1. After incorporating all these factors,
an adjusted endurance limit for Al7075-T6 was found with the consideration of maximum
fatigue strength found in the literature [51,52].
From the published data for the S–N curve of Al7075-T6 [51,52], it is observed that
the high-cycle fatigue domain extends from 103 to 108 cycles, with the maximum fatigue
strength reaching approximately 150 MPa. To construct an adjusted S–N curve suitable for
our customized structural components, we considered the ultimate tensile strength of the
material as the reference point at the onset of the high-cycle fatigue domain (103 cycles) and
adjusted endurance limit at the 108 cycles. The adjusted S–N curve for Al7075-T6, used in
our fatigue analysis, is presented in Figure 11. This curve was derived using the following
empirical equation provided in [54]:
σa = σ′f (2N ) β
(7)
where σa is the adjusted stress amplitude, σ′f is the ultimate tensile strength of the material,
β is the fatigue exponent determined using the endurance strength and its corresponding
number of cycles, and N is number of cycles to failure. Using the adjusted S–N curve for
Al7075-T6, fatigue analysis was performed on critical structural components of Mithra
based on the quasi-static force data derived for a fast walking gait. Among those critical
components, the results of the fatigue analysis for the hip internal/external rotation joint
back mount, knee input link, ankle side bracket, ankle planetary carrier assembly, and main
universal joint cross are presented in Figure 12.
Figure 11. Adjusted S-N curve for Al7075-T6 used for fatigue analyses.
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(a)
Total Life (cycle)
Min : 4.000 x107
4.004 x107
(b)
Total Life (cycle)
(c)
Total Life (cycle)
1.000 x108
4.004 x107
4.004 x107
8.888 x107
4.004 x107
4.003 x107
7.776 x107
4.003 x107
6.664 x107
4.003 x107
4.002 x107
5.552 x107
4.002 x107
4.002 x107
4.440 x107
4.002 x107
4.001 x107
3.328 x107
4.001 x107
4.001 x107
2.216 x107
4.001 x107
4.000 x107
1.104 x107
4.000 x107
Min : 1.104 x107
(d)
Total Life (cycle)
(e)
Total Life (cycle)
1.000 x1011
4.004 x107
8.751 x1010
4.004 x107
7.501 x1010
4.003 x107
6.251 x1010
4.003 x107
5.002 x1010
4.002 x107
3.753 x1010
4.002 x107
4.001 x107
2.503 x1010
Min : 4.000 x107
7 7
Min 4.003
: 4.000x10
x10
1.253 x1010
4.000 x107
Min : 4.000 x107
4.001 x107
4.000 x107
Figure 12. Fatigue analysis results of the structural components. (a) Hip internal/external rotation
joint back mount, (b) knee input link, (c) ankle side bracket, (d) ankle planetary carrier assembly, and
(e) universal joint cross.
Assuming that a fast-walking scenario with each trial lasts for about 20 min and an
average gait cycle duration of approximately 0.8 s, the robot completes around 1500 gait
cycles per trial. Over a planned operational lifespan of 5000 trials, the total number of the
cycles amounts to approximately 7,500,000. Based on fatigue analysis of critical structural
components, the minimum fatigue life is observed at the knee input link, with a simulated
total cycles of 11,000,000 for fast-walking conditions. Therefore, it is theoretically evident
that Mithra can undergo more than 5000 sessions of 20-minute fast-walking tests without
experiencing structural failure or any maintenance requirement due to fatigue.
4. Results and Discussion
Mithra was developed with the goal of providing a reliable and accessible humanoid
platform for the advancement of control strategies for human-like locomotion. Mithra’s
key anthropometric parameters are available in [35]. Each leg is equipped with six active
joints, three at the hip, one at the knee, and two at the ankle, and all actuation systems were
carefully engineered and constructed to optimize performance for human-like locomotion.
This section presents the design validation and performance assessment of Mithra’s actuators in generating human-like dynamic motions and adapting to external environmental
disturbances. In our previous work [35], we reported preliminary actuator performance
results, including peak joint torque, joint speed, and range of motion, with comparisons
to other well-known humanoid robots. In the present paper, we extend those findings by
presenting results from both theoretical and experimental evaluations of Mithra’s actuators. These results are intended to demonstrate the effectiveness of Mithra’s actuators in
supporting improved control and overall performance.
4.1. Anthropomorphic Design
As previously mentioned, one of the main objectives in the design of Mithra was
achieving kinematic and kinetic attributes similar to those of humans. The outcomes of this
anthropomorphic design, as detailed in [35], closely match the target human parameters
not only in total mass and height, but in segmental attributes such as mass distribution,
link lengths, link inertias, and joint range of motion. We also compared these characteristics
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with some humanoid robots designed with the same goal (including WALK-MAN [39],
TALOS [6], and JAXON3-P [55]), and showed that Mithra outperforms them in almost all
metrics. Please refer to [35] for details.
4.2. Hip Joint Configuration
The yaw–roll–pitch configuration is the most commonly implemented hip joint arrangement in humanoid robots [39]. In our design process, we investigated several hip
joint configurations to identify the most suitable option that aligns with the design considerations and high-level objectives outlined in Section 2. Table 1 presents the inertia values
corresponding to different hip configurations evaluated for Mithra, assuming that the leg
structure below the hip remains the same in each case.
Table 1. Inertia comparison for different hip actuator configurations.
Effective Inertia at
Hip Actuator
Configuration
Roll–Yaw–Pitch
Roll–Pitch–Yaw
Pitch–Roll–Yaw
Yaw–Roll–Pitch
Hip Yaw
Hip Roll
Hip Pitch
(kgm2 )
(kgm2 )
(kgm2 )
0.09
0.06
0.06
0.16
3.89
3.22
3.56
2.03
2.15
3.21
3.61
2.10
The yaw–roll–pitch configuration was confirmed to be the most favorable in terms of
effective inertia, as anticipated during the design phase. This configuration is also commonly used in humanoid robots such as Hubo, HRP-2, and Valkyrie. It provides minimal
effective inertia at the hip pitch and roll joints, indicating that the actuation setup for these
joints contributes to a leg design with lower effective inertia compared to alternative configurations. This reduction in inertia enhances hip joint control by enabling more precise
movements that would be difficult to achieve with higher-inertia joints. Moreover, the low
inertia at the hip is advantageous for fast stepping motions, as it allows for higher hip swing
acceleration. This characteristic supports stable locomotion, particularly in dynamic environments. However, adopting a spherical joint-like hip configuration in Mithra introduced
some trade-offs. While it ensures human-comparable ranges of motion, it contributes to a
bulkier appearance and limits joint motion compared to robots such as WALK-MAN [39],
which employs a non-spherical hip joint with a roll–yaw–pitch configuration that enables
larger joint ranges.
4.3. Actuator Bandwidth
In general, a real-world physical system behaves like a low-pass filter, attenuating
high-frequency inputs. This characteristic is especially relevant for electric actuators in
legged robots, where the cutoff frequency of the actuator system is a critical parameter for
characterizing joint responsiveness and agility. Since closed-loop position control is typically employed during the swing phase of bipedal locomotion, we conducted closed-loop
position control bandwidth experiments on all joint actuators using benchtop test setups.
Figure 13a shows the benchtop setups used for actuators of hip internal/external rotation
and hip abduction/adduction. To facilitate continuous testing, the mechanical output of
the joint actuators was disconnected, eliminating restrictions imposed by mechanical hard
stops. The closed-loop bandwidth test was conducted using a proportional-derivative
(PD) controller, with the desired velocity set to zero. Sinusoidal position commands with
three different amplitudes, 5, 10, and 15 degrees, were applied as the inputs for controllers
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whose gains were set to 76.4 Nm/rad for the proportional gain (k p ) and 0.64 Nm/rad/s
for the derivative gain (k d ). Testing began at a low frequency of 0.1 Hz and gradually
increased until the experiment was terminated. The stopping criteria were either the motor
current exceeding the 100 A limit imposed by the motor controller or excessive vibration in
the fixture.
(a)
Actuator for hip internal/external
rotation
Actuator for hip abduction/adduction
(b)
0
−1
−2
−3
100
102
101
frequency (rad/s)
0
−20
−40
−60
100
102
101
frequency (rad/s)
Figure 13. Actuator bandwidth experiment: (a) benchtop setup for testing the actuators and (b) bode
plot results from the test on the hip flexion/extension actuator.
To compare the experimental results with the theoretical ones, bode plots from the
transfer function of the system was also included in Figure 13:
G (s) =
Js2 + (k
kp
d + B)s + k p
(8)
where k p is 76.4 Nm/rad, k d is 0.64 Nm.s/rad, and J and B are the inertia and damping of
the chosen actuator with the values of 0.0007958 kg.m2 and 0.001 Nm.s/rad, respectively.
Figure 13b depicts the bode plot containing both theoretical and experimental results. The
plots of both experimental and theoretical results indicate that the actuator system exhibits
the characteristics of a first-order linear system. In all three cases of sinusoidal position
commands, the actuator did not reach the cutoff frequency (3 dB below the low-frequency
magnitude), as the experiment had to be stopped before reaching this frequency for the
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aforementioned reasons. This was due to the fact that the dynamic response of the actuators
is so fast that its cutoff frequency is above the practical operation range.
To assess the frequency spectrum relevant to human-like motion, we applied a Fast
Fourier Transform (FFT) to the time-domain joint angle data of hip and knee flexion during
Mithra’s maximum considered speed, i.e., running at 3 m/s. The resulting frequency
spectrum for both hip and knee joint motions are shown in Figure 14, highlighting the
dominant frequencies that the actuators must be capable of tracking for the accurate
reproduction of human-like running behavior.
(1.4, 11.7)
(2.8, 10.6)
Figure 14. Frequency spectrum of joint trajectories for hip and knee during a human running scenario
at 3 m/s.
In the results derived from the FFT analysis, for the hip and knee joint angles, the
representative frequency (after which the amplitudes were negligible) was about 2.8 Hz.
The amplitude for the knee at this frequency was 10.6 degrees. For the hip angle, since
the amplitude at this frequency was rather small, we focused on the previous significant
frequency content point, i.e., 1.4 Hz, at which the amplitude is 11.7 degrees. Based on
these results and compared with the frequency response results of Figure 13, it is evident
that Mithra’s hip and knee actuators can easily follow these amplitudes with frequencies
multiple times higher than the FFT frequency contents. As such, both hip and knee actuators
are expected to track human-like joint trajectories for running at 3 m/s with minimal error
and sufficient responsiveness.
4.4. Joint Backdrivability
All leg joints in Mithra are designed to be backdrivable, offering several advantages,
including improved impact mitigation, enhanced compliance to external forces, and increased battery life through power regeneration. Comparable humanoid robots including
WALK-MAN [39], TALOS [6], and JAXON3-P [55], due to their use of transmission systems such as harmonic drives or ball-screws, have mostly non-backdrivable joints. In
contrast, through detailed optimization and use of planetary gearing, Mithra’s design
led to highly backdrivable joints, which is essential for human-like locomotion. In our
previous work [35], we provided a comprehensive analysis of Mithra’s joint backdrivability,
including the required backdriving torque for each joint and demonstrations of free-swing
behavior in the hip and knee joints. In this paper, we extend that work further by quantitatively assessing and comparing the backdrivability of leg joints and demonstrate the
backdrivability of the ankle joint.
4.4.1. Knee Joint
We quantified the backdrivability of Mithra’s joints using the impact mitigation factor
(IMF) metric proposed in [31]. This provides a meaningful basis for comparing backdrivability across robotic leg systems by directly accounting for electric actuator design. Also, it
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emphasizes the importance of actuator design in impact mitigation, particularly considering the inertia-dependent effects on backdrivability. The actuator’s reflected inertia is a
crucial parameter, as it governs the peak loads experienced by structural components, gearboxes, and transmission systems during impact events. With this consideration, the IMF
metric quantifies how effectively the free dynamics of the actuation system reduce impact
impulses on the robot’s floating body by comparing it to a worst-case scenario in which
a system with all active joints are rigidly locked, thus reflecting the maximum possible
inertia. As noted in [31], the IMF is configuration-dependent and a dimensionless quantity
that ranges between 0 and 1. An IMF value of 1 indicates perfect inertial backdrivability,
where all impact impulses are fully mitigated by the system’s free dynamics. In contrast, an
IMF value of 0 corresponds to a fully non-backdrivable system. In this paper, we present
IMF values with respect to the knee joint to assess the backdrivability of Mithra’s leg design.
This joint was chosen because the IMF values are strongly correlated with the position of
the knee joint, which undergoes a wide range of motion and represents the most powerful
joint in the leg system. The calculation of IMF requires access to the detailed design of
robots, which are in general not available. However, in [31], the IMF for the humanoid
HUBO has been calculated and reported, which we use here as a baseline for comparison
with Mithra’s IMF. Figure 15 shows the computed IMF as a function of knee angle, along
with a comparison to the IMF values of the HUBO robot [37] and a hypothetical version of
HUBO equipped with series elastic actuators (SEAs), as reported in [31].
Figure 15. Impact mitigation factor in terms of the knee angle for Mithra in comparison with Hubo
(with and without SEAs). The vertical green line corresponds to Mithra’s maximum knee flexion angle.
The low IMF values observed at smaller knee angles in both Mithra and HUBO,
as shown in Figure 15, are primarily due to reduced backdrivability in the z-direction,
which is less dynamically active during locomotion. In comparison to HUBO, Mithra
begins with an IMF value (ε) of 0 at a knee angle of zero. This may be attributed to the
fact that HUBO defines zero knee flexion in a bent reference position, as it allows for
knee hyperextension up to −10 degrees [37], whereas Mithra is limited to 0 degrees at
its maximum knee extension, corresponding to a fully straightened leg. However, aside
from the low-angle region, Mithra demonstrates significantly higher IMF values across
the rest of the knee’s range of motion compared to HUBO. This is due to Mithra’s highly
backdrivable knee actuator, in contrast to HUBO’s actuator, which uses a harmonic drive
that is known for poor backdrivability at high reduction ratios. Figure 15 also illustrates that
while SEAs aim to improve backdrivability in otherwise non-backdrivable systems, Mithra
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achieves high backdrivability even without incorporating elastic elements. This analysis
proves that Mithra’s leg structure is indeed well suited to mitigate ground impacts. Its low
mechanical impedance and high sensitivity to external forces contribute to improved force
and impedance control, enabling stable and adaptive locomotion.
4.4.2. Ankle Joint
Most robots lack human-like ankle capability, both for flexion and inversion/eversion
degrees of freedom, due to the difficulty of placing actuators of considerable size and
weight near the foot. As discussed in detail in [35], a main contribution of our novel
design framework was an ankle actuator that can accommodate human-like running
at 3 m/s without increasing the shank mass and inertia beyond the anthropomorphic
range. Further, our multi-objective optimization framework results in other desirable
characteristics, including joint backdrivability [35]. In particular, the inversion/eversion
motion has been designed to be highly backdrivable in order to naturally adapt to rough
terrains and obstacles with minimal control. This is inspired by the findings on how humans
adopt passive or minimal proprioceptive control for walking on irregular terrains [56]. To
test this property, we conducted a simple experiment where the leg was brought down
onto an uneven surface, allowing the ankle to respond passively. Figure 16a shows the
ankle joint angle measurements and corresponding snapshots of the inversion/eversion
motion (rotation around the x-axis) captured during one of the trials. Note that during these
experiments, the dorsiflexion/plantarflexion axis of the ankle was actively controlled to
remain at zero by applying appropriate gains in a closed-loop position controller specifically
only for that joint movement.
(a)
0
−5
−10
−15
−20
Ankle Flexion/Extension
Ankle Inversion/Eversion
−25
0
(b)
0.5
1
1.5
Time (s)
2
2.5
0
−5
−10
−15
−20
−25
0
Ankle Flexion/Extension
Ankle Inversion/Eversion
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s)
Figure 16. Ankle backdrivability tests. (a) Passive response and (b) active control of ankle inversion/eversion with low impedance gains.
It is evident from the experiment that Mithra’s ankle joint utilizes its backdrivability to
adapt to uneven surfaces upon ground contact. Since the closed-loop position controller for
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the ankle’s x-axis (inversion/eversion) was configured with zero gains, the ankle motion
remained unchanged in that direction, even when the leg was subsequently lifted. This
behavior demonstrates the natural mechanical compliance of the ankle enabled by its
backdrivability, which allows the joint to adapt to irregular terrains without active control
and its associated complications and delays.
A less extreme case in interaction with irregular terrains occurs when the actuator is
operating using low-gain impedance control. In this case, the actuator behaves similarly to
a soft spring-damper, which allows it to adapt to the environment with some resistance.
To investigate this behavior, a second experiment was conducted in which non-zero gains
were applied to the closed-loop position controller for the inversion/eversion axis as well.
As shown in Figure 16b, the joint encoder data and corresponding snapshots illustrate that
the ankle adapted to the uneven surface when the foot made contact with the ground, and
then returned to its commanded joint angle (zero degrees) when the leg was lifted. Together,
these two experiments highlight the backdrivability of Mithra’s ankle joint in enabling the
foot to conform to uneven terrain and respond compliantly to external disturbances, rather
than rigidly resisting them.
5. Conclusions
To our knowledge and as detailed in our previous paper [35], among the state-of-theart humanoid robots, Mithra’s kinematics and kinetics are closest to those of an average
human. Although the optimization framework proposed in our previous work provided
results for actuator selection for our robot, several additional steps were necessary to arrive
at a design ready for manufacturing. In particular, this paper addresses key questions
such as how the actuators should be arranged, which mechanisms are most appropriate,
and how the actuator structure can withstand high running forces. We conducted detailed
FEA and fatigue analyses to ensure sufficient structural strength and lifespan, all while
preserving human-like inertial and backdrivability characteristics. Further, the performance
of the actuators were assessed by benchtop experiments to quantify closed-loop control
bandwidth, which proved that the actuators’ agility is indeed higher than what is needed
for human-like running. This was achieved through the careful design of the actuators’
outputs, and specifically, their low inertia and friction. These design outcomes also resulted
in high backdrivability of the actuators, which was tested through human-like adaptation
of the ankle joint to irregular terrains. Future works will include testing the robot for
anthropomorphic walking and running, and assessing the performance of the actuators
during these tasks.
Author Contributions: Conceptualization, S.R., D.T., and C.S.; methodology, S.R., D.T., and C.S.;
software, C.S. and D.T.; validation, C.S. and D.T.; formal analysis, C.S. and D.T.; investigation,
C.S. and D.T.; resources, S.R.; data curation, C.S.; writing—original draft preparation, C.S., D.T.,
and S.R.; writing—review and editing, C.S. and S.R.; visualization, C.S.; supervision, S.R.; project
administration, S.R.; funding acquisition, S.R. All authors have read and agreed to the published
version of the manuscript.
Funding: This research was in part funded by National Science Foundation, grant number 1953908,
subawarded through the University of Michigan with subaward number SUBK00012024.
Data Availability Statement: The raw data supporting the conclusions of this article will be made
available by the authors on request.
Acknowledgments: The authors express their thanks for the support of Justin Huff and Harsha
Karunanayake for their assistance with development of Mithra and conducting the experiments.
Conflicts of Interest: The authors declare no conflicts of interest.
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