Underactuated Metamorphic Loading Manipulators

• UMLM is defined as a loading manipulator combining underactuation and metamorphic behavior, enabling topology reconfiguration with minimal actuation.
• The design utilizes a single motor to drive both the metamorphic arm and passive gripper through slider-actuated mode switching, achieving adaptive grasping and lifting.
• Detailed kinematic and stiffness analyses, along with optimization studies, demonstrate practical load redistribution and energy-efficient performance in vehicle-mounted applications.

Searching arXiv for the cited UMLM-related papers to ground the article in current literature. Underactuated metamorphic loading manipulators (UMLMs) are loading manipulators that combine underactuation with metamorphic behavior: they have fewer actuators than degrees of freedom, and their topology and effective mobility can change during operation. In the formulation explicitly proposed for vehicle-mounted systems, a UMLM integrates a metamorphic arm with a passively adaptive gripper, uses geometric constraints to realize topology reconfiguration and flexible motion trajectories without additional actuators, and completes the sequence of descending, pre-grasping, adaptive grasping, and lifting with only one actuator (Mao et al., 11 Sep 2025). Across adjacent literature, the term also usefully covers a broader family of systems in which passive joints, variable morphology, passive or compliant grasping, and load-triggered or mode-triggered transmission changes shape the feasible motion and load-bearing behavior of the manipulator (Chen et al., 2019, Jang et al., 11 Jan 2026).

1. Definition and architectural scope

In the most direct UMLM formulation, the manipulator is defined as an underactuated metamorphic loading manipulator because it combines two properties: underactuation and metamorphic behavior. Underactuation means that the system has fewer actuators than degrees of freedom; in the cited vehicle-mounted design, the entire system is driven by a single motor. Metamorphic behavior means that the mechanism can change topology and effective mobility during operation, through geometric constraints and release or engagement of internal constraints (Mao et al., 11 Sep 2025).

The architecture proposed for vehicle-mounted UMLMs has two major subsystems. The first is a metamorphic arm that performs lowering toward the object, lifting after grasping, and driving the gripper through an internal slider. The equivalent arm mechanism is described as composed of six rods and one slider, with revolute joints at points $A, B, C, D, F, G$, slider segment $DE$ moving along upper arm $GE$, and torque input applied at point $A$. The second is a passive or adaptive gripper that is underactuated, passively adaptive, and driven mechanically by motion transmitted from the arm via a slider and linkage system (Mao et al., 11 Sep 2025).

The practical operation is organized into two metamorphic configurations and a four-stage operational sequence: initial state, descending or lowering state, pre-grasping state, grasping state, and lifting state. When the slider is restrained by a limit block, the motor drives the arm as a lifting or lowering mechanism; when the slider is released, the same motor input causes lateral slider motion, which closes or opens the gripper. The gripper is therefore not separately actuated; the arm mechanically transmits motion to the gripper through the slider-linkage chain (Mao et al., 11 Sep 2025).

A broader mechanistic interpretation is supported by adjacent underactuated manipulation literature. The SPINE gripper, for example, is a single-input underactuated manipulator whose function changes passively when an internal load or torque threshold is crossed. It exhibits grasping mode and rotation mode, with a mode change occurring without control switching, sensors, or additional actuators. That system is not the same as the vehicle-mounted UMLM, but it is a closely related example of loading-governed reconfiguration in an underactuated manipulator (Jang et al., 11 Jan 2026). This suggests that UMLM is best understood not only as a particular mechanism class, but also as a design principle: embed task sequencing, load sensitivity, and topology-dependent functionality directly into the mechanism.

2. Metamorphic arm and passive gripper principles

The metamorphic characteristic of the arm lies in a change of constraint relations rather than replacement of physical members. In one configuration, the slider is constrained and the mechanism behaves as a lifting or lowering arm; in the other, the slider is free and motion is redistributed to drive gripper closure. The practical metamorphic condition is therefore slider constrained $\Rightarrow$ arm lifting or lowering topology, and slider released $\Rightarrow$ gripper actuation topology (Mao et al., 11 Sep 2025).

This mode sequencing implements a mechanically embodied control strategy. The cited UMLM begins in an elevated posture with the gripper open. During approach, the slider is restrained and the actuator rotates counterclockwise so that the arm lowers toward the object. In pre-grasping, the end effector reaches the object vicinity. During adaptive grasping, the actuator rotates clockwise, the slider is released, and the gripper closes around the object. As clockwise rotation continues and the slider reaches its limit, the topology effectively switches again and the arm lifts the grasped object (Mao et al., 11 Sep 2025). The central engineering claim is that task sequencing is embodied in mechanism design, not in multi-actuator control software.

The passive adaptive gripper conforms to different object sizes and shapes because finger segments continue rotating until they encounter object contact, after which contact redistributes torque and causes subsequent links to reposition. The gripper is driven entirely by the arm-side actuation. The coupling chain is

$$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$

No dedicated gripper motor is required (Mao et al., 11 Sep 2025).

The same general principle appears in the SPINE gripper’s passive mode-transition logic. There, input torque twists a Twisted Underactuated Mechanism, axial contraction closes the fingers, grasp force builds while a friction generator keeps the gripper rotationally locked to the frame, and when input torque exceeds the maximum static friction torque of the friction generator, the lock breaks and the mechanism passively transitions to rotation mode (Jang et al., 11 Jan 2026). The paper identifies three operational phases—approaching phase, force buildup phase, and rotating phase—and shows that identical contraction occurs regardless of rotation direction. A plausible implication is that UMLM mechanisms need not rely exclusively on discrete slider release: they can also realize metamorphic function through friction-threshold-induced transmission path change.

Related variable-stiffness work on malleable robots broadens the architectural picture further. Malleable robots are reduced-DOF serial robot arms with changeable geometry, enabled by a malleable link whose offset is variable in all six relative pose components. The robot can switch between flexible mode for reshaping and rigid mode for task execution, mainly via layer jamming. This is not a canonical UMLM, but it is a strong example of metamorphic morphology achieved through variable stiffness rather than serial extension of actuated joints (Clark et al., 6 Feb 2025).

3. Kinematic, kinetostatic, and stiffness formulations

The vehicle-mounted UMLM literature develops the model in stages: arm kinematics, slider-to-vertical-input transmission, vertical-input-to-gripper transmission, and gripper force analysis (Mao et al., 11 Sep 2025). For the arm, with Cartesian frame $Oxy$, the loop-closure equation is

$$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$

Projected on the axes, it yields nonlinear relations among $\theta_0,\theta_1,\theta_2,\theta_4$: $DE$0

$DE$1

Velocity and acceleration analyses then produce Jacobian-like linear systems for passive joint rates and accelerations during lifting and grasping (Mao et al., 11 Sep 2025).

A key transmission variable is the slider displacement $DE$2, related to $DE$3 through a geometry-based expression, with corresponding velocity and acceleration relations: $DE$4 and

$DE$5

These relations mediate the shift from arm posture change to gripper actuation (Mao et al., 11 Sep 2025).

The gripper itself is analyzed by contact geometry and virtual work. Contact point positions $DE$6, contact force vectors $DE$7, and contact point velocities are written explicitly, and the virtual-work balance is

$DE$8

From this, the contact-force expressions are derived: $DE$9

$GE$0

$GE$1

The paper emphasizes that as $GE$2 increases, $GE$3 decreases quickly while $GE$4 increases sharply, a distribution interpreted as desirable for enveloping grasp (Mao et al., 11 Sep 2025).

For UMLMs with passive joints and load-dependent internal posture, the stiffness problem is not captured by unloaded kinematics alone. A transferable formulation is given by the loaded equilibrium equations for manipulators with passive joints: $GE$5 with $GE$6 denoting perfect passive joint coordinates and $GE$7 denoting virtual spring and preloaded passive joint coordinates (Pashkevich et al., 2011). The resulting Cartesian stiffness matrix can be rank-deficient, and the framework explicitly captures load-induced changes in Jacobians and Hessians, as well as buckling and bifurcation phenomena. This is especially pertinent for UMLMs because the same mechanism may be locally stiff in one metamorphic phase and nearly singular in another (Pashkevich et al., 2011).

A fixed-structure control-affine perspective is also relevant in planar phases with one passive revolute joint. For an $GE$8-link horizontal planar manipulator with one unactuated joint, partial feedback linearization yields

$GE$9

and for $A$0, if the first joint is actuated, the manipulator is accessible from almost any state and is STLC from a subset of equilibrium points; otherwise it is neither accessible nor STLC from any state (Chen et al., 2019). This does not model metamorphosis, but it gives a fixed-structure phase theory for UMLMs during a given morphology and no-contact phase.

4. Underactuation, controllability, and reduced-order behavior

Underactuation in UMLMs is not limited to “one motor drives many links.” It also appears as restricted accessible subsets of configuration space, passive self-motion, and task-dependent manifolds. Tendon-driven underactuated chains make this explicit. In the general tendon-driven formulation, actuator coordinates $A$1 and joint coordinates $A$2 satisfy

$A$3

with constrained equilibrium defined by

$A$4

subject to

$A$5

and

$A$6

The paper emphasizes that the reachable set in joint space is a hardware-dependent manifold that can be low-dimensional and or discontinuous (Islam et al., 2024). For UMLMs, this suggests that metamorphic transmission design can be treated as manifold shaping, not only as geometry selection.

A different reduced-order mechanism appears in mixed groups of planar fully actuated and passive-active manipulators. For a passive-active manipulator starting from zero initial joint velocity,

$A$7

which integrates, under stated angular-domain conditions, to the holonomic relation

$A$8

This produces reduced differential kinematics

$A$9

with a $\Rightarrow$0 reduced Jacobian $\Rightarrow$1 (Peng et al., 2023). The direct setting is distributed formation control, not loading manipulation, but the core idea is transferable: when passive coordinates become explicit functions of active coordinates under a mode-specific constraint, task-space control should be built around the reduced map rather than a fictitious fully actuated model.

Underactuated behavior can also arise from passive or low-DoF support branches rather than from tendon transmission alone. In modular task-driven co-design, an assist branch connected via a shared module can reduce torque at the base or common elbow joint, and an optimized morphology may contain an assist branch with 0 DoF that still reduces torque through static mass distribution (Lei et al., 18 Dec 2025). A plausible implication is that some UMLM behaviors commonly attributed to “extra actuation” may instead be realized by appropriately placed passive structure.

Supervisory planning work on cooperative underactuated manipulation highlights the algorithmic side of this issue. In a two-manipulator task where object yaw is not directly actuated, a policy over the augmented state

$\Rightarrow$2

outputs the next subtask, such as robot identity and next drop-off point, within a Task and Motion Planning decomposition (Witte et al., 2023). This is not a direct UMLM model, but it demonstrates that underactuated variables can be controlled indirectly through sequencing of cooperative or structural modes. That planning logic is directly suggestive for UMLM mode scheduling.

5. Optimization, planning, and design co-synthesis

The direct UMLM design paper uses Particle-Swarm Optimization to refine gripper dimensional parameters with the explicit aim of equalizing contact forces. The force extrema are

$\Rightarrow$3

and the objective is

$\Rightarrow$4

with

$\Rightarrow$5

Search bounds are

$\Rightarrow$6

$\Rightarrow$7

The paper reports worst and best parameter groups

$\Rightarrow$8

$\Rightarrow$9

with objective values

$\Rightarrow$0

The interpretation given is that the best design nearly equalizes the three contact forces (Mao et al., 11 Sep 2025).

Task-driven co-design in adjacent underactuated literature is more explicit about planner-in-the-loop optimization. For tendon-driven underactuated chains, the hardware parameter set

$\Rightarrow$1

is co-optimized with a control policy so that the hardware-induced equilibrium manifold becomes smoother and more task-aligned (Islam et al., 2024). The key computational device is a neural proxy model $\Rightarrow$2 combined with periodic hardware extraction via CMA-ES: $\Rightarrow$3 The paper shows that with a fixed optimization budget and 100 random actions, SLSQP on the optimized manifold gives lower error than on the unoptimized manifold: unoptimized hardware error $\Rightarrow$4, optimized hardware error $\Rightarrow$5 (Islam et al., 2024). This suggests that UMLM design optimization should target not only endpoint performance but also favorable geometry of the feasible manifold.

For modular reconfigurable manipulators, planner-in-the-loop design uses a hybrid search over morphology, branch arrangement, and mounted pose. A continuous latent vector $\Rightarrow$6 is mapped to morphology $\Rightarrow$7, branch mounting order $\Rightarrow$8, and mounted pose $\Rightarrow$9, while HMPC computes feasible motion trajectories under kinematic limits and inverse-dynamics torque checks (Lei et al., 18 Dec 2025). The objective becomes feasibility-first: $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$0 Although the architecture is not explicitly underactuated, the framework is highly relevant because it treats morphology, load distribution, and trajectory feasibility as a single design problem (Lei et al., 18 Dec 2025).

A control-oriented soft-robot counterpart proceeds from a mechanics-preserving Discrete Elastic Rod model and rewrites the external force as

$$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$1

This yields a control-affine model and an actuation-consistent inverse-dynamics trajectory generation law

$$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$2

This is a methodological template for UMLMs with distributed compliance or large deformation, even though the paper does not include metamorphic topology changes or payload variation (Liu et al., 23 Mar 2026).

6. Applications, performance, and engineering implications

The direct UMLM simulations are performed in a robot simulation platform and Adams multibody simulation software using a 12 W DC motor. Two representative object classes are tested. The spherical object has diameter $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$3 and mass $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$4; the cylindrical object has base diameter $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$5, length $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$6, and mass $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$7 (Mao et al., 11 Sep 2025). For the sphere, grasping action completes around the 7th second. The paper states that the manipulator can apply a force in excess of $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$8 when grasping objects, and that the optimized parameter group $$\text{motor torque at arm} \rightarrow \text{arm motion} \rightarrow \text{slider motion} \rightarrow \text{vertical input } h \rightarrow \text{gripper linkage motion} \rightarrow \text{finger closure}.$$9 achieves stable grasping with smaller force than group $Oxy$0, reducing power consumption (Mao et al., 11 Sep 2025).

The same simulations support four operational claims: one motor is used for descending, grasping, lifting, and releasing; the passive adaptive gripper conforms to both spherical and cylindrical objects; the same mechanism handles geometrically distinct objects without controller redesign or finger-level actuation; and topology change with passive adaptation makes the manipulator suitable for changing task stages and variable object conditions (Mao et al., 11 Sep 2025). These are the strongest application-level statements currently available for a system explicitly named UMLM.

Related task-driven modular design work demonstrates the importance of morphology for load handling. In a drilling task, the workspace is enlarged from

$Oxy$1

and optimized bi-branch morphologies attain maximum torques $Oxy$2, $Oxy$3, $Oxy$4, and $Oxy$5 in simulation, with one design measured at $Oxy$6 in experiment, all near but below the $Oxy$7 rating (Lei et al., 18 Dec 2025). The paper states that the single-branch version of the same morphology exceeds the threshold at several points, while the bi-branch version remains below the limit. This gives direct evidence that morphology can be used as a load-redistribution mechanism rather than merely as a reach-extending device (Lei et al., 18 Dec 2025).

Soft underactuated trajectory generation gives a complementary performance picture. In a 2-segment pneumatic soft robot with four chambers, DER-based trajectory generation yields mean total tip-tracking errors of $Oxy$8 cm, $Oxy$9 cm, and $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$0 cm across three scenarios, compared with PCC baseline errors of $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$1 cm, $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$2 cm, and $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$3 cm. DER mean error stays below 1 cm in all scenarios, while PCC exceeds 1 cm in all scenarios (Liu et al., 23 Mar 2026). This is not a UMLM validation, but it shows the practical benefit of actuation-consistent planning for high-dimensional underactuated structures.

Constraint-aware whole-body control of an underactuated aerial manipulator shows a similar lesson in another domain. A quadratic program computes dynamically consistent generalized accelerations under underactuation equality constraints, rotor thrust bounds, and joint-position-derived acceleration limits, while a passivity-based integral action improves disturbance rejection without compromising feasibility (Laribi et al., 13 Jan 2026). With passivity-based integral action, base position RMSE is reduced to $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$4 m and end-effector position RMSE to $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$5 m under uncertainty (Laribi et al., 13 Jan 2026). A plausible implication is that UMLMs operating under variable load, topology-dependent inertia, or model mismatch may require comparable separation between hard feasibility constraints and robustness augmentation.

7. Limitations, adjacent concepts, and research directions

The current UMLM literature is specific and still narrow. The direct vehicle-mounted formulation neglects friction between fingers and object in grasp-force modeling, neglects finger gravity, validates primarily in simulation, and focuses optimization mainly on force uniformity rather than full multiobjective trade-offs such as speed, fatigue life, or collision robustness (Mao et al., 11 Sep 2025). These are not minor omissions: they bound the present scope of UMLM theory.

Fixed-structure controllability theory also has clear limits. The $$\vec{L}_{AB} + \vec{L}_{BC} + \vec{L}_{CF} = \vec{L}_{AO} + \vec{L}_{OG} + \vec{L}_{GF}. \tag{1}$$6-link horizontal planar results do not model morphology changes, changing payload or load distribution, contact constraints, nonplanar motion, gravity-dominated operation, variable inertia due to telescoping or reconfiguration, friction, backlash, compliance, flexible links, or hybrid transitions between metamorphic modes (Chen et al., 2019). Consequently, those results are best treated as mode-wise local controllability tools rather than a complete theory of UMLM operation.

Similarly, mixed-network formation control of passive-active manipulators relies on a special integrability property valid only under zero initial joint velocity and under stated angular-domain conditions (Peng et al., 2023). Supervisory learning for cooperative manipulation is predominantly kinematic and treats underactuation at the manipulation level rather than at the mechanism level (Witte et al., 2023). Modular planner-in-the-loop design is not explicitly underactuated, while malleable robots are only partially underactuated and only weakly loading manipulators in the heavy-payload sense (Lei et al., 18 Dec 2025, Clark et al., 6 Feb 2025). These mismatches matter and should not be obscured.

At the same time, the adjacent literature suggests a coherent research agenda. One direction is mode-by-mode loaded-stiffness analysis: define passive, compliant, and actuated coordinates in each morphology; solve the exact loaded equilibrium; compute Jacobians and Hessians at that equilibrium; and derive the local Cartesian stiffness while allowing rank deficiency (Pashkevich et al., 2011). Another is task-based co-design of reachable manifolds, where transmission and morphology parameters are optimized jointly with policy so that the feasible state manifold is connected, smooth, and aligned with the task (Islam et al., 2024). A third is planner-in-the-loop structural synthesis, where topology, mounting, and feasible motion are optimized together under torque, collision, and tracking constraints (Lei et al., 18 Dec 2025).

Taken together, the literature supports a precise interpretation of UMLM research. A UMLM is not merely a manipulator with fewer motors. It is a mechanism in which topology reconfiguration, passive adaptation, load-dependent transmission paths, and actuator scarcity are co-designed so that mechanical intelligence substitutes for control complexity. The direct vehicle-mounted UMLM work shows that descending, pre-grasping, adaptive grasping, and lifting can be realized with one actuator (Mao et al., 11 Sep 2025). The broader arXiv record suggests that the next step is a unified theory spanning controllability in fixed phases, stiffness and instability under passive joints, task-driven morphology optimization, and hybrid planning across metamorphic modes (Chen et al., 2019, Pashkevich et al., 2011, Islam et al., 2024, Lei et al., 18 Dec 2025).