Rhombot: Modular Reconfigurable Robot

• Rhombot is a planar lattice modular self-reconfigurable robot that leverages a unique morphpivoting motion primitive to maintain global connectivity during continuous configuration changes.
• The system uses a single degree-of-freedom actuation via a centrally mounted servo-driven cable and integrates fixed docking electromagnets to achieve sub-centimeter precision.
• Experimental results demonstrate robust, repeatable reconfiguration cycles with RMSE around 4.8 mm in x and 15 mm in y, confirming the system’s high positional and docking accuracy.

Rhombot is a planar lattice modular self-reconfigurable robot (MSRR) constructed from rhombus-shaped modules designed for robust, medium-independent reconfiguration. Each module is a deformable parallelogram actuated through a single centrally mounted servo-driven cable. The system achieves morphing, docking, and locomotion with low control complexity. Rhombot's central contribution is the morphpivoting motion primitive, enabling continuous reconfiguration that maintains global structural connectivity. Experimental results demonstrate sub-centimeter positional and docking accuracy, as well as reliable operation under repeated reconfiguration cycles (Gu et al., 27 Jan 2026).

1. Rhombus Module Design

Each Rhombot module consists of a planar rhombus skeleton of rigid carbon-fiber links with side length $2a$, where $a = 140$ mm. Opposite edges remain parallel through actuation, while a central parallelogram linkage enforces a single folding degree of freedom ($\theta$) defined as the interior angle between adjacent edges. The skeleton actively constrains its two diagonals such that $45^\circ \leq \theta \leq 135^\circ$, preventing undesired planar rotation.

A single micro-servo motor drives a cable winch (radius $r = 5$ mm), using a bevel-gear reduction ($Z_1=12$, $Z_2=24$ teeth) to increase delivered torque. The cable, looped beneath the skeleton and routed through guide pulleys, symmetrically pulls a pair of opposite vertices together or allows them apart, folding the rhombus along its long diagonal. This actuation both morphs the shape and realizes the morphpivoting motion.

Key module mechanical parameters include:

Parameter	Value	Description
$a$	140 mm	Half edge length
$b$	70 mm	Distance: module center to electromagnet
$\theta$	$45^\circ-135^\circ$	Folding angle range
$T$ (servo stall torque)	4.5 kg·cm	Torque capability
$r$ (winch radius)	5 mm
$F_e$ (electromagnet)	$\approx$25 N	Docking holding force

The actuation moment between two adjacent edges for folding angle $\theta$ is:

$$M_d = \frac{T\,Z_1}{r\,Z_2}\;2a\,\sin\Bigl(\frac{\theta}{2}\Bigr)$$

This compact mechanical package minimizes vertical profile, supports a single-degree-of-freedom (DoF) actuation, and embeds four docking electromagnets at fixed intervals for module connectivity.

2. Kinematic and Dynamic Modeling

Individual modules are modeled as quadrilaterals $M_i$ with edges $E_{i0}$–$E_{i3}$, where $E_{i0}$ is typically oriented "downward" (grounded or parental), and the others serve as inter-module interfaces. Each module's local coordinate frame $O_i$ is defined at the $E_{i0}$ midpoint, with $y_i$ directed toward the center and $z_i$ perpendicular to the module plane.

The single DoF is parameterized by $\sigma_i$ ($\sigma_i \equiv \theta_i$ in the undeformed configuration). Homogeneous transforms from $E_{i0}$ to midpoints of adjacent edges are:

• $T^0_1(\sigma)$:

$$\begin{pmatrix} R_z(\sigma+\pi) & [\,a + a\cos\sigma,\;a\sin\sigma,\;0\,]^T \ 0\;0\;0 & 1 \end{pmatrix}$$

• $T^0_2(\sigma)$:

$$\begin{pmatrix} I_3 & [\,2a\cos\sigma,\;2a\sin\sigma,\;0\,]^T \ 0\;0\;0 & 1 \end{pmatrix}$$

• $T^0_3(\sigma)$:

$$\begin{pmatrix} R_z(\sigma) & [\,a - a\cos\sigma,\;a\sin\sigma,\;0\,]^T \ 0\;0\;0 & 1 \end{pmatrix}$$

Where $R_z(\cdot)$ is a $3 \times 3$ rotation about $z$.

The forward kinematics for a chain of connected modules, forming a tree or loop (triangulated by constraint), is:

${}^bT_e = \prod_{j=0}^e T^{(k_j)}(\sigma_{k_j})$

Actuator requirements for connection/disconnection are set by the frictional resisting torque of the electromagnetic connector:

$M_f = F_e\,(2a - b) + \varepsilon$

with $\varepsilon \approx 0.84$ kg·cm. To guarantee single-sided release during docking, $M_d > M_f$ is always enforced.

3. Morphpivoting Motion Primitive

Morphpivoting constitutes the atomic reconfiguration operation, defined as pivoting a single module about a still-connected edge by exploiting the module’s ability to fold and unfold. Execution occurs over four sequential phases:

1. Pre-pivot morphing: Both pivoting and adjacent modules adjust $\theta$ to an intermediate $\sigma_{\rm pre}$ to create mechanical clearance.
2. Connection: The electromagnet on a previously unconnected adjacent edge-pair is activated.
3. Disconnection: The former connector is deactivated, so only the new axis forms the pivot.
4. Post-pivot morphing: Both modules restore angles to $\sigma_{\rm post}$, finalizing the new configuration.

The transformation for the connection shift, considering module $M_a$ with neighbors $M_b$ (old) and $M_c$ (new), involves computing the net transformation between relevant edge frames before and after:

$${}^{F_{\rm new}}X_{M_a} = \Bigl[T^{F_{\rm new}}_{\rm pre}(\sigma_{\rm pre})\Bigr]\, \Bigl[T^{F_{\rm old}}_{\rm pre}(\sigma_{\rm pre})\Bigr]^{-1}\,{}^{F_{\rm old}}X_{M_a}$$

This formalism supports issuing four servo-angle commands per module to perform each morphpivot.

4. Reconfiguration Planning and Control

Reconfiguration proceeds via a planning algorithm that takes as input a list of desired (Connect, Disconnect) edge-pairs. The current kinematic structure is maintained as a tree (KTree), which is updated after each primitive by re-establishing parent/child relationships.

For each requested edge-pair exchange, the system:

• Identifies all modules and edges implicated
• Invokes the Morphpivoting primitive with appropriate inputs
• Updates the KTree and recomputes all $\sigma_i$ mappings

Control is decentralized and minimal: each module contains a single actuator, a single angular sensor (AS5600 magnetic encoder), and independent on/off control of its four docking electromagnets. No external vision or global localization is required; all docking frames are defined from the local kinematic tree.

Throughout all steps, the morphpivoting approach ensures modules never fully detach, maintaining at least one connection and retaining global topological stability at all times.

5. Structural Stability and Precision Docking

During every morphpivoting action, the pivoting module keeps at least one edge connection, enabling uninterrupted global connectivity of the MSRR. The robot maintains static stability by keeping each module's center-of-mass within its support polygon, leveraging omnidirectional ball casters for low-friction ground support.

The docking system utilizes hermaphroditic connectors comprised of a normally‐on electromagnet and an iron plate. The nominal state is locked (powered off); power is applied only for intentional disconnection. The relationship $M_d > M_f$ ensures controlled, single-sided release.

Empirical positional accuracy is evidenced by kinematic-chain tests (4 modules), with end-effector root mean squared error (RMSE) $\approx 4.8$ mm in $x$ and $15$ mm in $y$ (the increased $y$ error is due to friction). During seven sequential morphpivotings, all connector offsets remained within the mechanical tolerance range of the electromagnet (a few millimeters), validating reliable and repeatable sub-centimeter docking precision.

6. Experimental Results, Limitations, and Future Work

The hardware deployment comprises multiple Rhombot modules on a ball-caster chassis atop a printed grid, powered by a wired 12 V Li-ion battery and coordinated through wireless nRF24L01 modules. Each module is controlled via Arduino Nano microcontrollers, servo drivers, and AS5600 angle sensors.

Demonstrated reconfigurations include three-module morphing (e.g., forming a 3-loop structure, reconnecting, and restoring shape) and simulated seven-module transitions (e.g., “μ” → “F”). Single servos in each module are responsible for all morphing and docking; robust operation was observed over many morphpivot cycles with no uncommanded disconnections.

Quantitative findings are as follows:

Metric	Value
Kinematic-chain RMSE	$4.77$ mm ($x$), $14.96$ mm ($y$)
Docking accuracy	Within sub-centimeter mechanical tolerance
Robustness	No spontaneous disconnections (dozens of cycles)

Limitations include cable slack-induced hysteresis of approximately 1500 encoder counts near end stops ($\Delta\theta \approx 131^\circ$), mitigated by hardware tensioners. Anticipated improvements cite the use of an irregular-profile winch and development of an autonomous high-level sequencer for flexible, automatic (Connect, Disconnect) pair selection (Gu et al., 27 Jan 2026).

Rhombot demonstrates that single-DoF, rhombus-shaped modules with centrally mounted actuation yield stable, medium-independent self-reconfiguration. The morphpivoting primitive, in conjunction with robust local kinematic trees and electromagnetic docking, achieves low control complexity alongside high positional and docking accuracy.