1. Introduction
The rapid advancement of new technologies has sparked the development and significant commercial impact of wearable smart devices that can be attached to the body to improve quality of life through safety, assistance, and entertainment [
1,
2,
3]. The concept of wearable robotics as person-oriented systems was coined over a decade ago to primarily refer to exoskeletons, orthotic robots, and prosthetic robots [
4,
5]. However, the growing interest in lightweight and compact devices has broadened the notion of robot wearability to include miniature robots that can either be affixed to the body [
6,
7,
8,
9] or move along its surface. In this sense, dynamic wearable technology [
10], where small robots can move on or around the human body, can extend the capabilities of wearable smart devices in applications such as healthcare measurements [
11,
12], haptic interfaces [
13,
14], fashion expression [
15], and support and assistance with activities of daily living (ADL) [
16,
17].
Nevertheless, on-body mobile robotics is a challenging topic that requires specific research in key technological aspects such as adhesion, locomotion design, and control.
Generally, climbing robots for industrial and maintenance applications have adopted various adhesion mechanisms such as vacuum suction cups, grippers, magnetic systems, or bioinspired methods mimicking geckos or insects [
18,
19,
20]. Even if the requirements for wearable robots differ significantly from industrial solutions [
21], some studies have proposed locomotion designs that adapt some of these mechanisms to on-body robots. For instance, SkinBot [
11,
21] has an inch-worm locomotion system that employs two legs with suction cups to navigate on human skin and collect biosignals, yet vacuum pumps pose a challenge for achieving untethered devices. Furthermore, some works have investigated cloth climbing mechanisms that could be used for on-body robots. Thus, gripper-based locomotion can be employed to grasp fabric, taking advantage of the soft and deformable characteristics of loose clothing. Clothbot [
12] employs a gripper consisting of a passive spring and two tangential differential drive wheels, allowing it to roll along the fold within the gripper. Given that movement direction is restricted by the fold, an articulated tail can use gravity to assist steering [
22]. In Rubbot [
23], a passive folder frame holds two grippers for differential motion. Moreover, Rovables [
24] are differential drive miniature robots that use magnetic attraction between wheels and a rod that moves on the fabric’s inner side. Additionally, gecko-inspired adhesion was used for the feet of the CLASH lightweight micro-hexapod to climb on loose clothing [
25]. In general, cloth-climbing robots allow for navigation on any part of the body where clothing is smooth. However, they are not suitable for applications requiring direct contact with the body, such as healthcare measurements. Moreover, their design primarily focuses on managing folds and other interactions with flexible fabric, which limits maneuverability. Alternative locomotion designs have avoided adhesion issues by employing smooth rails or tracks affixed to the clothing worn by the user [
26]. The rail locomotion design of the Calico robot [
14] relies on wheels, a rail attachment system, and a track-switching mechanism for bifurcations.
In contrast, on-limb mobile robots designed to move along the wearer’s arm [
13,
16,
17,
27] are based on locomotion systems that make direct contact with the limb, either on bare skin or through tight clothing. These on-limb mobile robot designs prioritize a snug fit by exerting gentle pressure against the limb, relying less on adhesion, and using surface materials and textures that favor both grip and comfort. Additionally, on-limb robots require adaptability to the varying diameters of the limb (e.g., from the wrist to the upper arm). On-limb mobile robots are related to out-of-pipe climbing robots in industrial applications [
28,
29]. Some pipe climbing robots [
30,
31] have addressed diameter variation with adaptable clamping mechanisms that can be mechanically adjusted to different pipe sizes before use, but are not designed for pipes with diameter variations along their length. In general, the designs and construction of industrial solutions are mostly unsuitable for human limbs due to their heavy and complex structures [
31,
32], even if current research in soft robots [
33,
34] aims at lighter and compliant pipe crawling mechanisms that could be suitable for human–robot interaction [
35]. In this sense, Knitskin [
27] is a textile sleeve-like robot designed for moving along the user’s forearm that needs a compressor for pressurizing pneumatic actuators. Nevertheless, most methods for on-limb motion rely on rigid locomotion with self-actuated electric servo-motors. Thus, the Movelet robot for positional haptic feedback [
13] consists of a closed bracelet with four evenly distributed wheels connected by pairs of spring suspensions, allowing for small diameter adjustments along the forearm. The wearable bracelet robot proposed by Kimura et al. [
16] is based on rolling spherical capsules with internal motors, linked by alternating rigid links and a flexible rubber band with an actuated reel for diameter adaptation. Alternatively, the bathing assistance robot proposed by Liu et al. [
17] is a four-wheel mechanism designed for single-track locomotion along the user’s arm, pulling a cleaning fabric sleeve that also serves to hold the robot in place. All of these mechanisms depend on closed bracelets or sleeves that must be slid over the wearer’s hand, which limits applicability in situations where active user involvement or human assistance is challenging or unexpected, such as remote health monitoring for patients or victims.
In this work, we propose an open mechanism for on-limb locomotion that can be put on or removed at any point of the limb without slipping the robot over the hand or foot. This novel locomotion design has a minimum number of wheels: two actuated spherical rollers for differential drive and another actuated spherical roller for grasping and stabilization. A triangular configuration provides stability (i.e., three points of contact provide a stable support base for an object) and can accommodate uneven surfaces. Some studies [
36,
37] have demonstrated that using three fingers on robotic grippers improves the capability to grasp more complex irregular objects. In particular, spherical rollers [
38] have been employed as fingertips for in-hand manipulation [
36], facilitating the manipulation and reorientation of objects with different shapes and enabling continuous rotation and adaptability to changes in the contact state.
In particular, the major contributions of this article are as follows:
A new robot designed to move on human limbs featuring an open grasping mechanism with a spring linkage. One side has a pivoting differential drive base (PDDB) with two actuated spherical rollers, and the other side has an actuated roller for grasping and stabilization.
The kinematic analysis of the proposed robot configuration estimates the circumcenter coordinate, the current limb radius, and the actual roller contact points from the joint angle measurements. These values are necessary for estimating the actual differential drive wheel distance for motion control.
A cascade control system, combined with the passive spring linkage, allows for adaptation to varying limb diameters. The outer loop ensures stable grasping, while the inner loop adjusts the trajectory using PDDB roller velocities. A Lyapunov stability analysis is also provided.
Additionally, the article presents experimental results from both simulations and a real on-arm prototype. To our knowledge, this is the first analysis of a locomotion system based on spherical rollers with proprioceptive estimation of limb diameter changes.
The rest of this paper is organized as follows.
Section 2 presents the proposed design.
Section 3 offers a kinematic analysis locomotion system. The cascade control system is described in
Section 4.
Section 5 discusses the experimental results. Finally,
Section 6 summarizes the key findings and offers conclusions.
2. On-Limb Mobile Robot Design
This section offers an overview of the proposed design and describes the grasping model, highlighting the concept of central alignment for ensuring stability while moving on the limb.
2.1. Design Overview
The concept and main elements of the proposed robot design are illustrated in
Figure 1. The system consists of an open symmetrical two-link mechanism with a passive spring joint, where the first link holds a differential drive base with a passive pivot joint. The pivoting differential drive base (PDDB) has two active spherical rollers that move along the limb. The spring joint presses the PDDB over the limb, ensuring the rollers are always in contact with it. The centerline of the PDDB is connected to Link 1 using a free pivot joint aligned with the motion axis corresponding to the measured articulation variable
. Moreover, Link 1 and Link 2 are connected by a passive spring joint (measured articulation variable
) to facilitate grasping and adaptation to changes in limb diameter. The synergy between the pivot and spring joints enables stable and adaptable locomotion along the limb. The second link has a fixed (i.e., non-articulated) platform holding a 1 DoF spherical roller that provides the grasping point. Furthermore, this wheel is active to maintain an aligned robot pose relative to the limb.
The rotation axes of the three rollers (i.e., two spherical wheels in a differential drive configuration plus an opposing grasping wheel) are in the same plane. Spherical rollers allow for stable grasps independent of its orientation [
36] and limb radii. Also, the spherical design of the rollers allows them to have individual DC motors inside, with encoders for velocity control, reducing hardware size and utilizing internal volume [
36]. The PDDB rollers are used to change the heading. At the same time, the actuator of the third one provides a grasping effect, holding the device on the limb with a velocity equivalent to the PDDB’s linear velocity. These rollers are suitable for curved surfaces, providing a single contact point regardless of surface curvature. Moreover, the combination of the spring joint, the free-pivoting mechanism, and the spherical rollers allows continuous 3-wheel contact with the limb surface. PDDB wheels can be used to correct the orientation of the trajectory, preventing the deterioration of the stable grasp during locomotion.
All in all, the proposed design enables adaptability to variations in the limb section: the passive spring provides continuous adaptation to size changes, the PDDB ensures contact and allows for differential drive heading corrections against shape variations, and the continuous curvature of spherical rollers provides smooth motion independent of the actual contact points. Additionally, the mechanism can be easily attached or removed at any point on the limb without requiring the robot to be slipped over the hand or foot.
2.2. Grasp Model
Figure 2 shows a cross-sectional diagram of the proposed robot configuration on a cylindrical limb. The robot can be considered planar, with all roller axles in the same plane, and the axles of the spring and pivot joints are perpendicular to this plane. In this work, point coordinates in the local robot frame
are denoted by subindexes
x,
y, and
z (e.g.,
). In particular,
is placed at the center of the PDDB, where the
Z-axis coincides with the pivot axis, and the
X-axis coincides with the rotation axle of the differential drive wheels. Hence, the pivot joint point is
. For the kinematic chain, we have followed a Denavit–Hartenberg convention. The origin of the
frame for Link 1, which holds the PDDB, coincides with that of the robot’s frame, with the
-axis aligned with
Z. Link 1 has a length
and rotates around the
Z-axis with a pivot joint angle
measured relative to the
X-axis.
The local frame for Link 2, which has a length , originates at the spring joint point and rotates around the -axis, with the -axis aligned with the link segment . The spring joint angle is defined between and . The joint vector is defined as .
The frames for the left, right, and grasping spherical rollers (i.e., , , and ) are placed at the respective centers L, R, and G, with rotation around the corresponding x-axis. The Cartesian coordinates for the grasp roller center G are computed from and the spring-joint angle . The contact points relevant for grasp stability and motion control are denoted as , , and for the left, right, and grasp rollers, respectively.
The locomotion of the PDDB along the limb’s longitudinal axis is illustrated in
Figure 3, which shows a top view of the robot. This locomotion is characterized by independently controlled wheels (left and right) working in tandem to maintain a straight-line trajectory. The vehicle heading is parallel to the longitudinal limb axis in the centered orientation, meaning the pivot point is in the center. An off-centered state occurs when the robot deviates from the longitudinal axis. Moreover, orientation error
affects the cylindrical section shape, which becomes elliptical. The PDDB’s linear velocity is denoted as
V, while the right and left wheels’ linear velocities are
and
, respectively.
2.3. Stability of Grasp and Alignment Error
When considering grasp with
d fingers under a planar friction-less grasp, a stable equilibrium grasp
is achieved when the total wrench
due to the finger forces
is zero. In the case of the proposed circular geometry, with no frictions, the forces are pure normal forces with respect to the center of the grasped body (i.e.,
). The condition for the equilibrium grasp [
39] for a three finger (i.e.,
) friction-less grasping is that
with
and
(i.e., positive
coefficients can be found).
In this case, the planar grasping configurations that make possible a null wrench
are illustrated in
Figure 4. In
Figure 4a, a null wrench is possible with non-negative
.
Figure 4b shows the limit case, where a positive value of
makes a positive Wrench under the friction-less assumption. Finally,
Figure 4c shows an unstable grasping configuration. Cases (b) and (c) occur also, respectively, for negative values of
. Additionally, when considering surface frictions, the Coulomb friction replaces the normal forces with a friction cone [
40], making the grasping area wider and providing a bigger, more practical, and stable grasp.
Graphically, in the desired robot grasping configuration (see
Figure 4a), point
G lies on the
Y axis (i.e.,
) so the contact points
,
, and
form an isosceles triangle whose circumcenter coincides with
C. Additionally, the angle
of
with respect to the
Y axis is null, indicating that the contact point triangle is centrally aligned with the limb section. Static grasping stability is maintained as long as
C remains within the contact point triangle. However, as the absolute value of
increases, the degree of central alignment decreases, potentially causing the robot to detach from the limb when
. Specifically,
is reached when
C lies on the segment
.
Therefore, the central alignment of the contact triangle and the limb section corresponds to maximum grasping stability, and
can be considered as the alignment error for controlling grasping stability. Alignment error
can be computed from the joint variable measurements
and
, as well as the
and
lengths. First, the angle
of
with respect to
is obtained using the law of sines:
Then,
is given by:
4. Control Strategy
In this section, the control strategy is formulated based on several assumptions. The pivot mechanism is assumed to be rigid and symmetric with respect to the longitudinal axis of the limb. Furthermore, the model considers a perpendicular contact point of the roller with the skin, so traction and adherence are guaranteed. Additionally, the analysis presupposes the absence of slippage. Moreover, we assume the robot’s path is continuous and does not constitute a control objective. Also, the forward velocity is constant. Dynamics analysis and the influence of gravity are excluded.
The proposed cascade control scheme is shown in
Figure 5, with the alignment control represented externally and the heading control internally (within the shaded box). The variables to control are
and
. The aim of the control system is to minimize the
error estimation and to ensure that the robot motion remains parallel to the limb (i.e.,
is null).
The subindexes a, d, and e (e.g., ) are used to represent their respective relationships with the actual a, the desired d, and the error e value.
4.1. Central Alignment Control
The external control follows a proportional control law , outputting a desired heading that is limited to maintain the grasp stability. The controller input is derived from the central alignment and radius error estimation block, which receives information from the joint sensors and .
Additionally, note that the alignment and radius error estimation block computes the effective differential wheel distance
, radii
, and
based on Equations (
8), (
9), and (
10), respectively. These data are important for calculating inverse kinematics, which determines
,
, and
as shown in Equations (
12) and (
13). The heading limits saturate the central alignment control to ensure the robot stays centered within
.
4.2. Heading Control
The second loop controls the heading error
with a proportional control law with gain
. The estimated orientation error
is computed from roller odometry. The robot follows the trajectory varying the velocity of the PDDB’s rollers; these are deduced by the differential drive inverse kinematics with
and
as input references. The output is the desired velocity of each roller, given as
=
. As mentioned in
Section 3.3, the drift in the odometric estimation of the heading has an effect on the overall control system. As the external control loop has a proportional control, the system error would grow proportional to the
drift. To avoid this, an additional term can be added to the heading estimation (
15) that compensates for the estimation error based on the integral of the error
with a gain
:
4.3. Velocity Control
This is an additional velocity control that provides independent control of each roller of the PDDB. A PID is implemented to control the velocity of the wheels. The feedback for this control is obtained from the encoders. A first-order system has been implemented to create a model that simulates a simplified representation of the real system. The first-order transfer function simulates the loop response of the control system, with a time constant of
ms. Also, the actual velocities can be obtained as
. The robot vehicle block determines the global position of the robot, integrating roller velocities from Equation (
11).
4.4. Stability Analysis
This study aims to ensure the robot is stable while moving along a straight path along the cylinder’s longitudinal axis. We propose the control’s input
for the grasp stability analysis. The time derivative of
and
are established at Equations (
16) and (
17):
involving the temporal variation of
and
, where
is the angle between
and
and is derived in Equation (
1). Distributing the
term, we obtain
. Solving
in terms of
and
, the expression is:
To guarantee the convergence and stability, a Lyapunov function candidate
V is chosen as
, which is continuous, and positive definite
. Computing the time derivative
we have:
If we substitute
from model Equation (
18), then:
This shows that the term is semidefinite negative. This guarantees the stable behavior in the system. Additionally, achieving global asymptotic stabilization of , indicating that the system converges to a stable equilibrium point.
6. Conclusions
We have presented a novel on-limb mobile robot mechanism adaptable to human limb sizes. Concerning the mechanical design, the combination of the spring joint, the free pivoting mechanism, and the spherical wheels provides a solution that guarantees continuous three-wheel contact with the limb. Also, the open mechanism allows for the robot to be easily put on or removed at any point on the limb without slipping the robot over the hand or foot. While alternative solutions, such as a closed mechanisms or bracelet, offer intrinsic grasping stability, achieving mechanical stability poses a challenge in open mechanisms. Implementing a double cascade control system enables the robot to achieve grasping stability while enhancing maneuverability through differential drive steering. Moreover, the controller includes drift compensation in odometric heading estimation.
The simulations have demonstrated the effectiveness of the controller to maintain heading and central alignment. Furthermore, an on-limb robot prototype has been built to test its applicability to human arms.
Nevertheless, further work is required for practical application, including clinical experiments to evaluate user experience and comfort during extended use. Additionally, there is potential to optimize materials and robot design to reduce weight and facilitate the integration of electronics for untethered operation. Moreover, the estimation of heading error has been performed using odometry. A more accurate estimation of this value is related to the complex problem of on-body robot localization, which requires additional proprioceptive and/or exteroceptive sensors. Moreover, analyzing gravity and wheel–skin tissue interactions with terramechanics principles, involving wheel surface materials and human tissue modeling, is an open research area for on-limb robots.