Geometric and Force-Based Strategies for Dual-Mode Planar Manipulation of Deformable Linear Objects

Abstract

This paper investigates the fundamental challenges in planar manipulation of deformable linear objects (DLOs), where conventional rigid-body pushing and rotation strategies are often inadequate due to complex deformation dynamics. While the robotic manipulation of rigid objects has been extensively explored, the inherent conflict between the infinite degree of freedom in DLOs and the limited control points available in a robotic system presents significant obstacles to effective shape maintenance and force regulation. To address these limitations, we proposed a unified systematic framework for two-dimensional DLO manipulation that integrates object shape modeling with constraint force derivation. By leveraging the principles of system energy minimization and Lagrangian mechanics, our method generates gripper trajectories that simultaneously satisfy the requirement of object shape deformation and force constraints. The efficacy of the framework is validated via a dual-mode manipulation of DLOs, comprising (1) pushing with a static contact point, followed by (2) rotation-based surface alignment through continuous changing contact points. Results demonstrate that our approach achieves integrated shape and force regulation within a single computational framework.

1. Introduction

The manipulation of rigid bodies through pushing and rotating operations constitutes two elementary primitives for their spatial reconfiguration. While extensive research has been devoted to modeling and controlling these operations within the realm of rigid-body dynamics, practical applications frequently necessitate the manipulation of deformable objects, which requires more delicate and sophisticated control strategies to achieve desired deformable object configurations through pushing or rotational forces. For instance, deformable linear object (DLO) manipulation is prevalent in various real-world applications, including floor sweeping with brooms, wall brushing using paintbrushes, and poster placement on vertical walls. These tasks illustrate the remarkable capability of humans to manipulate such tools with ease. Humans intuitively bend the compliant heads of these tools to slide or rotate against contact surfaces, often doing so without relying on explicit mathematical models. However, these seemingly simple tasks present significant challenges for robotic systems in terms of deformation prediction, contact mechanics, and control policies that fundamentally differ from rigid objects.

The concept of rigid object has brought great convenience in object modeling for developing useful manipulation techniques. The properties of rigid transformations, e.g., translation and rotation, which preserve the distances and angles among the particles that make up the rigid body [1], facilitate a comprehensive and predictable understanding of the configuration of the object under external forces and render these primitives particularly useful. For instance, common manipulation concepts and procedures for pushing through or rotating rigid bodies are well-established and understood, as illustrated in Figure 1. Nevertheless, directly applying similar control strategies for rigid bodies to the manipulation of soft objects can lead to compromised results. As demonstrated in Figure 2, the deformation induced by significant internal forces within the soft object may result in different kinds of unpredictable displacements. In practical scenarios, humans typically rely on extensive life experience to manage the uncertainties associated with softness, as depicted in Figure 3. These intuitive methods, however, are not systematic and exhibit significant drawbacks. First, while human intuition may be plausible, it does not necessarily represent the optimal approach. Second, soft objects are vulnerable to damage from overstretching, and excessive contact can also harm the soft material. Lastly, maintaining a consistent force that aligns with the surface curvature necessitates a complex feedback control system.

To address these challenges, this paper proposes a novel strategy for manipulating flexible linear objects in a two-dimensional plane. Instead of treating the object shape as a mere consequence of external forces, where the magnitude and contact position of external forces dictate the final deformation, our strategy aims to determine desired external forces based on the manipulation goals through active shaping of the object. The primary contributions of this work are twofold:

• From the perspective of the object, we propose a framework that establishes the relationship between the shape of the object and reaction forces, thereby delineating a region of desired object configurations that facilitate manipulation tasks.
• From the perspective of the robot, we develop and define a dual-mode manipulation primitive that provides explicit guidance to the robot for manipulating deformable, linear objects (DLOs).

Ultimately, a sequence of gripper waypoints is generated to implement the proposed manipulation primitive while preserving the desired object configuration. This enables the execution of practical tasks, including effective pushing, and facilitates continuous, stable rotation of the object.

The remainder of this paper is organized as follows: Section 2 reviews the mechanism of pushing and rotation for rigid objects as well as methods for soft object modeling. Section 3 depicts the specific scenario addressed in this paper. Section 4 elaborates on the derivation of the novel method in detail. Then, Section 5 evaluates the proposed method through experiments conducted with a real robotic system. Finally, Section 6 summarizes the key findings of the paper and outlines future expectations for the research area.

2. Related Work

Robotic manipulation encompasses a variety of skills with a general goal of changing object configuration to the desired one. While pick-n-place operation forms a firm grasp and focuses on selecting appropriate grasping points on objects to enable their free movement within a given space, the most common manipulation primitives, pushing and rotation for non-prehensile grasping, pertain to scenarios where the object must interact with an external environment, such as a rigid surface. The complexity of these tasks increases significantly when the target object is deformable. In this section, we will first review existing solutions for two-dimensional pushing and rotation of rigid objects, followed by a brief introduction to soft object modeling and an overview of the prevalent techniques developed for the manipulation of deformable objects.

2.1. Planar Pushing and Rotation for Rigid Objects

Initially, we review the mechanisms underlying the pushing and rotation of rigid objects. For instance, [2] presents a voting theorem that determines the direction of rotation relative to the center of mass of a pushed object, based on the consensus derived from three vectors—the boundaries of two friction cones and the direction of the applied force. [3,4,5] develop the concept of limit surface, which provides geometric descriptions that relate object motion to external frictional forces. Data-driven approaches are also prevalent; for instance, [6] employs multiple regression models in conjunction with density estimation to predict the motion of pushed rigid objects. Furthermore, [7] integrates both analytical and data-driven methods, incorporating position and orientation changes modeled with Gaussian processes into a model predictive control framework. The reorientation of an object can be specified as rotating around a point (pivoting) [8] or along a line (tumbling) [9].

2.2. Soft Object Modeling

Deformable objects can be classified into three categories: (1) linear objects, (2) thin objects, and (3) lump objects that are composed of limp materials [10]. Much of the research in soft robotics focuses on predicting the shape or deformation of soft objects in relation to gripper motion [11,12]. In [13], an object is modeled as a collection of N particles, where the displacement of each particle under external forces is governed by the laws of physics. Mass–spring (MS) systems are also widely used as a physics-based modeling approach [14]. In addition to force-based models, position-based methods are also effective in giving geometrical constraints directly [15,16]. To simulate the entire mesh, the Finite Element Method (FEM) [17,18] is another well-established technique, yielding more accurate results, albeit at a higher computational cost. Furthermore, learning-based methods have been utilized to address the challenges associated with the numerous parameters required by model-based approaches [19,20].

2.3. Manipulation Techniques for Deformable Objects

Diverse manipulation skills have been developed to manipulate deformable objects, especially for linear elastic objects. This subsection discusses relevant works pertaining to the manipulation of deformable objects that can be modeled linearly. For non-elastic ropes, shape control is generally achieved by controlling several selected waypoints. For example, Refs. [21,22] presents a sequence of manipulation steps that guarantee the feasibility of shape control and then deform the object through manipulating multiple feature points sequentially. In another study, [23] utilizes a dual-arm robotic system, modeling the DLO as a kinematic multi-body system through a keypoint detection network. Researchers have also explored methods for solving problems associated with elastic rods. For instance, [24] proposes parameterized regression features, which are used to construct a compact feature vector, e.g., Bézier and NURBS, to quantify objects’ shapes. Besides model-based methods, as we mentioned the modeling complexity of deformable objects earlier, learning-based methods have also been developed. Refs. [25,26] established a control framework based on a deep reinforcement learning approach to make the shape of a deformable object reach a set of desired points. Notably, advancements in 2D deformable object manipulation have been observed. [27] explores the relationship between static deformation of sheet metal and the bending moments exerted on it using Lagrange’s equations based on a finite element model. Moreover, [26] successfully aligns the end edge of a flexible sheet object through position-based and image-based visual serving.

Despite existing literature demonstrating the successful modeling of shape and deformation, as well as a wide range of methodologies that have been developed in the reconfiguration of soft objects, continuous shaping considering the entire mobility of soft objects remains underexplored. This oversight may arise from the inherent deformability of such objects, complicating instantaneous shape prediction and making it challenging to derive consistent motion plans. This study concentrates on linear and flexible deformable objects, developing manipulation techniques that empower these soft objects to interact with their environment in a manner comparable to that of rigid bodies.

3. Problem Description

The primary challenge in the manipulation of deformable linear objects (DLOs) [28] stems from the intricate, nonlinear coupling between geometric configuration, internal stress distribution, and external constraint forces. This complex interdependency makes it particularly difficult to determine the appropriate external forces required to achieve a desired object state, as small variations in applied forces can lead to significant changes in both the internal force distribution and final configuration. By modeling the coupled dynamics of deformable object mechanics, contact constraints, and manipulator kinematics, the research seeks to formulate a unified framework for motion planning that accounts for both shape deformation and environmental force distribution. The objective is to develop an effective strategy for synthesizing optimal gripper motions that simultaneously accomplish the dual aims of active shape control and desired contact force maintenance.

3.1. Target Object and Environment Configuration Definition

The interested objects in this work are conceptualized as a deformable linear object. We posit that these objects possess elastic properties while remaining inextensible. This model is applicable to a diverse range of materials, including flexible wires, strips of paper, and polymers. Initially, one end of the object is assumed to be securely grasped by an end-effector, while the opposing end makes contact with the target surface, as illustrated in Figure 4a. Under the assumption of isotropic properties, any deformation occurring in the cross-section of the x-z plane is uniform across all y-coordinates. Specifically, the configuration of the target object in a 2D plane, presented in Figure 4b, is characterized by the following parameters:

$${\mathbb{C}}_{DLO}=\{\mathbf{L},\hspace{0.17em}\mathbf{P},\hspace{0.17em}\theta ,\hspace{0.17em}\mathbf{F},\hspace{0.17em}n\}$$

where:

• $\mathbf{L}\in {\mathbb{R}}^{+}$ represents the total length $l_{obj}$ of the object.
• $\mathbf{P}=\{(p_{x_0},\hspace{0.17em}{p}_{z_0}),\hspace{0.17em}(p_{x_n},\hspace{0.17em}{p}_{z_n})\hspace{0.17em}|\hspace{0.17em}\sqrt{(p_{x_n}-p_{x_0}{)}^2+(p_{z_n}-p_{z_0}{)}^2}<l_{obj}\}$ denotes positions of two fixed ends of the object within a certain manipulation scenario.
• $\mathit{\theta }=\left\{\right\{{\theta }_i\in (-\pi ,\pi ]{)}_{i=0}^n,\hspace{0.17em}\forall i\in \{0, 1,\dots ,n\})$ describes the curvature profile with each ${\theta }_i$ representing the local bending angle with respect to the x-axis at the $i$th discretized segment of the entire object.
• $\mathbf{F}=(F_n\hspace{0.17em},\hspace{0.17em}{F}_s)$ is a force vector that represents the normal and shear forces applied at one extremity of the object for shape maintenance.
• $n\in {\mathbb{R}}^{+}$ is the number of discretizations of the target object.

The geometrical information of the target surface is represented as follows:

$$\mathbb{S}=\left\{\right(s_{x_0},\hspace{0.17em}{s}_{z_0}),\hspace{0.17em}\dots ,\hspace{0.17em}(s_{x_i},\hspace{0.17em}{s}_{z_i}),\hspace{0.17em}\dots \hspace{0.17em},(s_{x_m},\hspace{0.17em}{s}_{z_m})\hspace{0.17em}|\hspace{0.17em}m\in {\mathbb{R}}^{+}\}$$

where $m$ refers to the number of segments of the surface, and $(s_{x_i},\hspace{0.17em}{s}_{z_i})$ denotes the starting position of the $i$th segment.

Figure 4. Analysis of 2D modeling for thin, deformable objects in manipulation scenarios. (a) Abstracting the process by observing variation across cross-sections under the assumption of isotropy. (b) Approximating curvature using discrete line segments, where angular variations measure local deformation.

Figure 4. Analysis of 2D modeling for thin, deformable objects in manipulation scenarios. (a) Abstracting the process by observing variation across cross-sections under the assumption of isotropy. (b) Approximating curvature using discrete line segments, where angular variations measure local deformation.

3.2. Task Description and Key Considerations

The manipulation process is modeled as a two-dimensional planar motion within the x-z coordinate plane, as depicted in Figure 4. To ensure computational feasibility, the deformable linear object is divided spatially into finite segments, possessing both a proximal and a distal end. The proximal end, firmly held by the end-effector, acts as the actuation input, whereas the distal end interacts with the environment.

In practical applications, the manipulation tasks involving deformable linear objects (DLOs) can be categorized into two modes: static and dynamic contact points, based on the conditions of object–surface interaction. To illustrate the effectiveness of the proposed framework, we introduce a dual-mode manipulation strategy that explicitly depends on these object–surface contact conditions.

• Mode 1—Pushing with Static Contact Point

Scenarios such as floor sweeping and wall brushing necessitate the deformation of the tool head to exert pressure against and slide along the target surface while maintaining a fixed contact point on the object. This mode examines the optimization of arrangements aimed at enhancing the generation of normal force at the distal end to improve pushing efficiency while ensuring a stable contact point. To highlight the significance of our approach in achieving improved manipulation results through systematic configuration optimization, the validation of results entails a comparison between an empirically derived object shape and one optimized using the proposed method.

• Mode 2—Rotating with Dynamic Shifting Contact Points

Building upon Mode 1, this task broadens the scope to include rotational manipulation with contact points that vary over the task while being able to maintain static at each discrete time step. It is applicable to materials such as paper sheets, cellphone screen protectors, and other thin-film materials, facilitating alignment with target surfaces. In this context, the object’s configuration is continuously optimized in relation to the remaining length of the object, the current pivot center, and the instant surface normal, beginning with the distal end aligning with the specified starting point on the target surface. With the appropriately generated motion trajectory of the proximal end, the object is able to synchronize with the target surface through a continuous rotation primitive. Two primary requirements are pursued: (1) adapting the first segment of the unaligned part to conform to the local surface curvature at each step and (2) reducing interruptions between aligned segments and those that remain unaligned, thereby ensuring the preservation of existing alignment states.

The key consideration to the success of prescribed tasks hinges on achieving a fully controllable contact condition at the distal end during the manipulation. Conversely, unexpected changes can be caused by unstable object shapes arising from improperly positioned coordinates of the proximity end, as illustrated in Figure 5. In the subsequent section, we will derive the relationship between object shape and constraint force, thereby exploring the region of desired coordinates for the proximity end of the object.

4. Hybrid Geometric-Force Planning Formulation

In this section, we describe the mechanism of the employed strategy for deriving a feasible trajectory of gripper motion for executing target planar motion. The derivation pipeline, illustrated in Figure 6, comprises two primary modules. The Pre-Planning module is designed to explore a feasible region of object configuration through theoretical derivation of the relationship between object geometry and the constraint forces at both extremities of the object. Initially, by specifying the positions of the two endpoints, we model the deformed shape of the linear flexible object of unit length through the minimization of total potential energy. Subsequently, we employ Lagrangian mechanics to calculate the reaction forces at the two extremities of the object. These computed forces are refined in accordance with the key considerations outlined in the previous section, which facilitate the maintenance of a stable state at the distal end. The corresponding coordinates of the object’s proximity end thus formulate a feasible operational region for the successful execution of the proposed manipulation tasks.

This geometric and reaction force estimation process is applied to each combination of relative positions of the two extremities throughout the entire reachable workspace. The Experiment Validation module utilizes an applicable coordinate of the object proximity end, selected from the feasible operational region derived from the previous module, integrated with the physical parameters of real-world objects to generate a sequence of waypoints, which collectively define the actual trajectory of the robotic gripper for implementation within the robot execution module.

4.1. Pre-Planning

The module will execute iteratively across the entire reachable workspace, concentrating on validating the feasibility of object configuration based on the coordinate of the object’s proximity end $p_n$ at each iteration. For $\mathrm{i}$th iteration, the module takes the specified coordinate $\hspace{0.17em}(p_{0i},\hspace{0.17em}{p}_{ni})$, representing a pair of endpoints’ positions for a unit length object, as input. The derived optimal energy is further incorporated into Lagrangian mechanics to estimate horizontal and vertical constraint forces $({\mathrm{F}}_{\mathrm{n}},\hspace{0.17em}{\mathrm{F}}_{\mathrm{s}})$. Upon completion of the iterative process, this module records a set of feasible operating points for the robot end-effector, ensuring that these points are associated with appropriate constraint forces necessary to maintain a stable contact state at the distal end.

4.1.1. Geometry Modeling

We model the object as a two-dimensional curved line composed of multiple line segments, which display isotropic physical properties along the y-axis. The total object length $\mathbf{L}=l_{obj}$ is discretized into $n$ segments with equal length $\delta l={\displaystyle \frac{l_{obj}}{n}}$. The position of the $(i+1)$th segment $p_{i+1}$ relative to the $i$th segment $p_i$ with a rotation angle ${\theta }_i$ can be calculated cumulatively as follows:

$$x_{i+1}=x_i+\delta l\ast \cos {\theta }_i=\sum _{j=0}^i\delta l\ast \cos {\theta }_j$$

$$z_{i+1}=z_i+\delta l\ast \sin {\theta }_i=\sum _{j=0}^i\delta l\ast \sin {\theta }_j$$

In general, the potential energy associated with a soft object can be categorized into several distinct components, each of which contributes to the overall energy state of the system as follows:

$$\mathbb{U}={\mathbf{U}}_{flex}+{\mathbf{U}}_{tor}+{\mathbf{U}}_{ext}+{\mathbf{U}}_{grav}$$

where ${\mathbf{U}}_{flex}, {\mathbf{U}}_{tor}, {\mathbf{U}}_{ext}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{U}}_{grav}$ represent the flexural, torsional, extensional, and gravitational energy of the object, respectively.

Given that the object’s motion is constrained to a two-dimensional plane, while neglecting stretching or rotation about its centerline, and recognizing that its slender shape renders gravitational energy negligible, we formulate the optimization problem by focusing exclusively on the flexural energy ${\mathbf{U}}_{flex}$ as the primary determinant of the final object shape, as depicted below:

$$\underset{{\theta }_i}{\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}}\hspace{1em}{\mathbf{U}}_{flex}={\displaystyle \frac{1}{2}}{\sum }_{i=0}^{n-1}{R}_f\Delta {\theta }_i^2\delta l$$

(1)

$$\mathrm{c}1: s.t.\hspace{1em}{p}_0=\left[0, 0\right],\hspace{1em}{p}_n=\left[x_n, z_n\right]$$

$$\mathrm{c}2: {\theta }_0=0$$

$$c3: z_i\ge 0$$

with the subsequent definitions of symbols and constraints:

• $R_f$: the flexural rigidity of the object.
• $\Delta {\theta }_i$: the curvature angle between two adjacents, $(\mathrm{i}$ − 1) and $\mathrm{i}$th segments.
• $\delta l$: the discretized length of each line segment of the object.
• $\mathrm{c}1$: represents the position constraint of two extremities.
• $\mathrm{c}$2: the first segment of the object is parallel to the surface for proper alignment.
• $\mathrm{c}3$: restricts the entire object to remain above the surface.

Remark 1. 

Notice in our formulation, the flexure coefficient $R_f$ is treated as a constant parameter. This treatment ensures that the optimization outcome of a set of curvature angles $\mathit{\theta }=\{{\theta }_0,\hspace{0.17em}{\theta }_1,\hspace{0.17em}\dots ,\hspace{0.17em}{\theta }_n\}$ remains invariant to variations in $R_f$, excluding the case $R_f=0$, thereby demonstrating that our proposed strategy is independent of the object’s flexural property. Consequently, the derived results maintain their validity and can be effectively generalized to deformable linear objects with other flexural properties with the same set of $\mathit{\theta }$. Furthermore, objects comprising multiple flexural rigidities can also utilize our results by discretizing into piecewise constant segments.

The established minimization problem Equation (1), which incorporates both inequality and equality constraints, can be mathematically solved using nonlinear programming techniques, such as the multiplier method [29]. This approach transforms the objective function with constraints into an unconstrained minimization problem as Equation (2), providing a guaranteed lower bound for the optimal value of the original problem.

$$\mathcal{L}({\mathbf{U}}_{flex},{\lambda }_{\mathcal{H}},{\lambda }_{\mathcal{G}})={\mathbf{U}}_{flex}(\mathit{\theta })+{\lambda }_{\mathcal{H}}\cdot \mathcal{H}+{\lambda }_{\mathcal{G}}\cdot \mathcal{G}$$

(2)

where ${\lambda }_{\mathcal{H}}$ = [${\lambda }_{h_1},\hspace{0.17em}{\lambda }_{h_2}$] and ${\lambda }_{\mathcal{G}}$ = [${\lambda }_{g_1}$] are the Lagrangian multipliers associated with the holonomic constraints $\mathcal{H}=[h_1:$c1,$h_2:$c2] and non-holonomic constraint $\mathcal{G}=[g_1:$c3], respectively. The constraints $\mathrm{c}1~\mathrm{c}3$ can be further converted to their analytic format as follows:

$c1: x_0+\sum _{j=1}^i\delta l\ast \cos {\theta }_j=x_n$, where $x_0=0$

c2: $z_0+\sum _{j=1}^i\delta l\ast \sin {\theta }_j=z_n$, where $z_0=0$

${\theta }_0=$ 0

$c3:z_i=z_0+\sum _{j=1}^i\delta l\ast \sin {\theta }_j\ge 0$, where $\mathrm{i}\in [1,\mathrm{n}]$

The simulation results are selectively presented in Figure 7, utilizing an object of unit length. These results offer insights into the overall object shape in relation to various positions of the proximity end, thereby enhancing our understanding of the system’s behavior under different conditions.

It is important to emphasize that the specific geometric configuration or curvature of each segment of the simulated object is not the primary focus of this investigation. Instead, our methodology emphasizes the acquisition of the proximal endpoint position through the entire shape modeling, which serves as a critical waypoint in the gripper’s motion trajectory. Furthermore, flexural energy formulation is employed to derive the constraint forces, as detailed in the subsequent Section 4.1.2.

4.1.2. Constraint Force Estimation

Estimating reaction forces at the endpoints of deformable linear objects (DLOs) is critical for precise robotic manipulation of them, yet it poses significant challenges. Unlike rigid bodies, DLOs exhibit complex nonlinear dynamics and distributed internal forces, making full-state force modeling computationally expensive or even intractable with current sensing technologies. However, for practical applications—such as assembly, medical robotics, or compliant grasping—only the endpoint forces directly interacting with the environment are often required. Focusing on these extremities simplifies the problem while preserving functional relevance.

This subsection develops methods for the strategic estimation and maintenance of the requisite constraint forces at the two endpoints of the object, enabling reliable physical interaction without exhaustive internal force analysis. By bridging the gap between theoretical complexity and practical necessity, we implement the approach of Lagrangian mechanics, which is computationally superior for complex systems, such as deformable objects, because it avoids vectorial force/torque balances and instead leverages scalar energy functions, prioritizes computational efficiency, and has real-world applicability.

• Mechanics vs. Math: Discussion and Derivation

Theoretically, Lagrangian mechanics provides a powerful framework for deriving external forces acting on mechanical systems by reformulating Newtonian dynamics through energy principles. Given the system energy, such as the flexural energy possessed by a linear flexible object, the Euler–Lagrange equation [30] is expressed as follows:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{q}}}\right)-{\displaystyle \frac{\partial L}{\partial q}}=Q_{\mathrm{e}\mathrm{x}\mathrm{t}}$$

(3)

where $L(q_1,\dots ,q_n,\hspace{0.17em}{\dot{q}}_1,\dots ,{\dot{q}}_n)=T-V$ is the Lagrangian (kinetic minus potential energy); $\mathit{q}=(q_1,\dots ,q_n)\in {\mathbb{R}}^n$ represents the generalized coordinates; and $Q_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is generalized external forces.

In the presence of holonomic constraints (constraints that depend solely on the generalized coordinates $\mathit{q}$), which is denoted as

$$\mathcal{H}=\left\{h_i\right(q_1,\dots ,q_n),\hspace{1em}i\in (0,m)\hspace{0.17em}\hspace{0.17em}|\hspace{0.17em}\hspace{0.17em}{h}_i=0\}$$

the Lagrangian can be modified to form the augmented Lagrangian $\overline{L}$:

$$\overline{L}=L+{\lambda }_1{h}_1+\cdots +{\lambda }_j{h}_j+\cdots +{\lambda }_m{h}_m$$

(4)

Applying Hamilton’s principle to the above modified Lagrangian, and noting that all reaction forces arise from environmental constraints, we can ignore the generalized forces from external forces $Q_{\mathrm{e}\mathrm{x}\mathrm{t}}$:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{\partial L}{\partial {\dot{q}}_i}}-{\displaystyle \frac{\partial L}{\partial q_i}}={\sum }_j{\lambda }_j{\displaystyle \frac{\partial h_j}{\partial q_i}}$$

(5)

The right-hand term in Equation (5) represents the generalized force associated with the constraint $h_j$. The ${\lambda }_j$ is the Lagrange multiplier corresponding to the $j$th holonomic constraint, which encapsulates information about the constraint forces corresponding to $h_j$. Substituting $L(q_1,\dots ,q_n,{\dot{q}}_1,\dots ,{\dot{q}}_n)=T-V$ into Equation (5), where $T$ and $\mathrm{V}$ denote the kinetic and potential energy of the system, respectively, and with the assumption of quasi-static manipulation, we obtain the following:

$${\displaystyle \frac{\partial V}{\partial q_i}}+{\sum }_{j=1}^m{\lambda }_j{\displaystyle \frac{\partial h_j}{\partial q_i}}=0$$

(6)

Equation (6) represents the relationship between the system’s energy and external forces.

On the other hand, the nominal optimal value of the system energy is already attained through differentiating Equation (2) and letting the result be ${\mathcal{L}}^{\prime }({\mathbf{U}}_{flex}, {\lambda }_{\mathcal{H}}, {\lambda }_{\mathcal{G}})=0$, where

$${\mathbf{U}}_{flex}(\theta {)}^{\prime }+{\sum }_{j=1}^m{\lambda }_{h_j}{\displaystyle \frac{\partial h_j}{\partial q_i}}+{\sum }_{k=1}^n{\lambda }_{g_k}{\displaystyle \frac{\partial g_k}{\partial q_i}}=0$$

(7)

To render the system energy calculated from the optimization framework (Equations (2) and (7)) applicable for the derivation of constraint forces using Lagrangian mechanics (Equation (6)), we conduct a comparative analysis between the Lagrangian mechanics formulation (Equation (6)) and the optimization approach (Equation (7)). This analysis reveals structural parallels between the two frameworks; however, two critical points of potential divergence are also notable:

■

From the perspective of optimization theory [31], when solving an optimization problem using Lagrangian duality, the relation between the dual solution $d*$ and the primal (original) solution $p*$ is governed by the duality theory:

$$d*\hspace{0.17em}\le \hspace{0.17em}p*$$

This indicates that the optimization result ${\mathbf{U}}_{flex}(\mathsf{\theta })$ may not necessarily reflect the actual system energy.

■

From the perspective of Lagrangian mechanics, the Euler–Lagrange equation is theoretically inapplicable to nonholonomic constraints, such as the inequality constraint exemplified by the partial derivative term $(\sum _{k=1}^n{\lambda }_{g_k}{\displaystyle \frac{\partial g_k}{\partial q_i}})$ in Equation (7).

To leverage the results derived from Equation (2), the following claims should be verified:

Claim 1. 

${\mathit{U}}_{flex}(\mathit{\theta })=V$, where ${\mathit{U}}_{flex}(\mathit{\theta })$ is the optimization result of Equation (5), and $V$ is the actual system energy.

Proof. 

In the context of nonlinear optimization, if the primal problem is convex, and the Slater’s condition is satisfied, indicating that the inequality constraints hold with strict inequalities, such as

$$g_k\left(i\right)<0,\forall i=1,\cdots ,m.$$

(8)

Based on the above, the strong duality holds. In our scenario, the objective function ${\mathbf{U}}_{flex}$ is quadratic and, therefore, convex. The minimization result is achieved without violating the inequality constraint c3 strictly, with the exception of the initial segments. However, the final contribution of the segment energy to the overall object is negligible as the discretization increases. Thereby satisfying Slater’s condition. Consequently, the optimization result aligns with the potential energy in reality. Thus, the first claim has been validated. □

Claim 2. 

The presence of terms related to the inequality constraints ${\displaystyle \frac{\partial g_k}{\partial q_i}}$ does not influence the optimization results concerning the magnitudes of the multipliers ${\lambda }_{h_j}$, which are associated with equality constraints.

Proof. 

Since the Karush–Kuhn–Tucker (KKT) optimality conditions are satisfied for this optimization problem, one of the conditions is referred to as complementary slackness, which states the following:

$${\lambda }_i^{\ast }{g}_i\left(x^{\ast }\right)=0,\hspace{0.17em}\forall i=1,\cdots ,m.$$

(9)

This implies that the $i$th optimal Lagrange multiplier will be zero unless the $i$th constraint is active, meaning the equality condition is satisfied at the optimum. Consequently, the multipliers associated with constraint c3 are all zero and do not affect the magnitudes of the other multipliers. Therefore, the resulting solution is optimal and unique from both an optimization perspective and within the framework of Lagrangian mechanics. In conclusion, the multipliers obtained from the optimization process can be directly applied in Lagrangian mechanics. □

In summary, the preceding analysis clarifies the ambiguity between the abstract mathematical framework and the classical mechanical principles by establishing a unified model that connects geometrical modeling with force estimation for DLOs. Through this established relationship, the constraint forces at both end points of a deformable object across various relative positions can be successfully derived.

• Feasible Configuration Identification

The derived constraint forces among the workspace are further distilled to identify operating points that are feasible to complete the proposed manipulation tasks. The feasibility of a robotic configuration is characterized by the capacity of the distal end to maintain persistent contact with the environment under specified force constraints. The feasible configuration ${\mathcal{C}}_f$ denotes a region that contains the admissible set of coordinates of the proximal end in the two-dimensional workspace. Feasible configuration identification utilizes established geometry–force relationships to explore stable object configurations while preventing slippage or curling at the object tip.

Compared with rigid objects, soft objects exhibit less predictability in their behavior as they respond to external forces. Once undesirable deformation manifests, the system may become challenging to manage. Consequently, drawing inspiration from rigid object manipulation, the fundamental concept is to achieve immobilization of the contact point and implement quasi-static operation, where inertial effects are neglected. Hence, two criteria should be guaranteed during the manipulation process:

• Normal force dominance ($F_n\gg F_s$) to minimize required ${\mu }_{\mathrm{e}\mathrm{n}\mathrm{v}}$.
• Positive normal force ($F_n>0$) at the distal end to prevent loss of contact.

Thus, the feasible configuration ${\mathcal{C}}_f\subset {\mathbb{R}}^2$ is the set of proximal end positions $(x_p,z_p)$ that can be mathematically formalized as follows:

$${\mathcal{C}}_{\mathrm{f}}≔\{({\mathrm{x}}_{\mathrm{p}},{\mathrm{z}}_{\mathrm{p}})\in {\mathbb{R}}^2{\displaystyle \genfrac{}{}{0pt}{}{\left|{\mathrm{F}}_{\mathrm{s}}\right|\le {\mathsf{\mu }}_{\mathrm{e}\mathrm{n}\mathrm{v}}{\mathrm{F}}_{\mathrm{n}}\hspace{1em}\left(\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{a}\mathrm{v}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\right)}{{\mathrm{F}}_{\mathrm{n}}\ge {\mathrm{F}}_{\mathrm{n},\mathrm{m}\mathrm{i}\mathrm{n}} \left(\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{g}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{e}\right)}},\forall \mathrm{t}\in [0,\mathrm{T}]\}$$

(10)

where:

• $\mathbf{F}=[F_n, F_s{]}^T$ denotes the contact wrench vector, where $F_n$ and $F_s$ represent the normal and shear force components, respectively.
• ${\mu }_{\mathrm{e}\mathrm{n}\mathrm{v}}$ denotes the static friction coefficient characterizing the environmental contact interface.
• $F_{n,\mathrm{m}\mathrm{i}\mathrm{n}}$ specifies the lower bound for the normal contact force required to maintain a stable interaction.
• The discrete temporal domain is parameterized by $t\in \{\mathrm{1,2},\dots ,T\}$, where $t$ indexes the $t$th time step and $T$ defines the complete manipulation horizon.

To identify feasible configurations, we apply the preceding definitions to the calculated constraint forces at the corresponding coordinates. We first present the qualified normal and shear constraint forces ($F_n>0$ and ${|F}_s|<F_{threshold}$) in the top row of Figure 8. Then, utilizing the distilled constraint forces, a map of coefficient of friction, bounded within the interval (0,1], is depicted in the bottom left of Figure 8. Consequently, the overall criterion defined as {High $F_n$, Low CoF} delineates the admissible configuration space, represented by the red contour boundary in the bottom right panel of Figure 8.

Remark 2. 

The region of feasible configurations ${\mathcal{C}}_{f,unit}$ is derived under normalized length conditions, exhibiting scale invariance. Consequently, the feasible configuration for a practical physical system can be obtained through appropriate dimensional scaling:

$${\mathcal{C}}_{f,\hspace{0.17em}scaled}=\alpha {\mathcal{C}}_{f,unit },\hspace{1em}\alpha =l_{actual }/l_{unit}$$

where ${\mathcal{C}}_{f,\hspace{0.17em}scaled}$ is the actual region of feasible configurations for the target system, and $\alpha$ represents the scaling factor by the actual object length $l_{actual}$ over the unit object length $l_{unit}$.

4.2. Gripper Trajectory Generation

Following the derivation of feasible configurations, we designate the representative operational point ${\mathbf{P}}_{\mathbf{d}}=(-0.2, 0.4)$ for experimental validation. The selected point is approximated to be the geometric center of the area for ease of maintenance within the region during gripper motion, as illustrated by the blue line in Figure 9. The two recalled demonstration scenarios are described in Section 3. In the context of the static contact point task, the objective is to maintain this optimized shape to generate greater normal forces. Similarly, but with increased complexity, the dynamic shifting contact points task, which aligns the deformable linear object (DLO) through continuous rotation, necessitates the preservation of the desired shape throughout the manipulation process, even as the object length changes. Therefore, a dedicated gripper motion trajectory will be developed utilizing the algorithms outlined in Algorithm 1.

Initially, we define the base coordinate system using the x-z axes, as illustrated in Figure 4, with the origin positioned at the initial starting point of the alignment process, which corresponds to the first contact point between the target surface and the proximity end of the object. The representative operational point ${\mathbf{P}}_{\mathbf{d}}$ is identified as the coordinates (−0.2, 0.4) within this base coordinate system.

To compute the waypoints, during the $i$th iteration of the for-loop, the trajectory generator calculates the desired waypoint $p_i$, which facilitates the transformation of the initial segment of the object to conform to the local surface. This process is informed by combining the reference operational point ${\mathbf{P}}_{\mathbf{d}}$ with the local coordinate system, along with the actual remaining object length $l_i$. The local coordinate system is defined by the instantaneous geometric information of the surface segment and is transformed back to the base coordinate system using the transformation matrix $T$. The loop concludes once all segments of the target surface have been traversed. By connecting the generated sequence of waypoints at each step, the complete trajectory of the gripper is formulated.

Algorithm 1: Gripper Trajectory Generation
	Input: Target Configuration: ${\mathit{p}}_{\mathit{d}}=({\mathit{x}}_{\mathit{d}},{\mathit{z}}_{\mathit{d}})$
		Object Length: $l_{obj}$
		Surface Geometry: $f(x,z)$
		Step Number: n
	for $i$in $n$steps do
		// update the remaining object length
		${ l}_i=l_{obj}-{\int }_0^{x_i}\sqrt{1+{\left(f^{\prime }\right(x_i,z_i\left)\right)}^2}dx$
		// get the rotation matrix
		${ R}_i\left({\theta }_i\right)\leftarrow {\theta }_i=arctan\left(f^{\prime }\right(x_i,z_i\left)\right)$
		// get the translation matrix
		$t_i=(x_i,z_i)$
		// get the transformation matrix
		$T=\left(R_i,T_i\right) \in SE\left(2\right)$
		// calculate the instant gripper position for step $i$
		$p_i\leftarrow T·p_d·l_i$
		// add $p_i$ to the trajectory
		$M\leftarrow add p_i$
	end
	Output: Gripper Trajectory $M(p_1,p_2,\dots ,p_n)$

The Figure 10 below shows two sample cases to visualize the planned gripper trajectory on curved surfaces for the task of rotation for alignment of the deformable linear objects. The primitives proposed within the dual-mode framework are clearly demonstrated, whereby each individual alignment step can be considered as achieving Mode 1. This mode is focused on maintaining a stable state at the contact point. Moreover, Mode 2 is achieved through the successful completion of multiple phases of Mode 1.

5. Experiments and Analysis

5.1. Hardware Setup

The proposed manipulation strategy is systematically evaluated through a series of experiments conducted using the hardware configuration depicted in Figure 11, consisting of a robotic manipulation system and various test specimens.

Robotic manipulation system: A Universal Robots UR10 industrial robotic arm, featuring six degrees of freedom (DOF), is implemented. The end-effector is a 3D-printed fixture specifically designed for this research, fabricated using polylactic acid (PLA) material. This fixture serves to securely clamp one terminal of the test specimens while enabling precise control of its position in conjunction with the end-effector.

The test specimens: There are three classes of linear flexible objects with different flexural rigidities: a standard A4 paper sheet and two steel sheets. The specimens are prepared to ensure dimensional consistency, with each sample cut to $400 \mathrm{m}\mathrm{m}\times 50 \mathrm{m}\mathrm{m}$.

5.2. Effective Pushing with a Static Contact Point

Two linear test specimens with identical flexural rigidity are bent into distinct geometric configurations following the shapes shown in Figure 9, respectively. As illustrated in Figure 12, the left panel demonstrates the initial deformation profiles at the onset of the experiment, under additional loads applied to establish a stable state:

Red Contour: The red contour represents the conventional deformation pattern, which emulates the characteristic bending observed in domestic cleaning tools (e.g., brooms and brushes) during typical wall-cleaning or floor-sweeping operations by humans. This configuration serves as our baseline reference case.

Blue Contour: The blue contour follows the optimized object geometry derived from the feasible region analysis result presented in Figure 8.

The right panel presents results following the removal of the externally applied stabilizing force, where the displacement vector indicates leftward motion of the junction point of two testing specimens. This evidence qualitatively validates the superior performance of the optimized configuration compared to empirical ones, confirming our theoretical predictions regarding force transmission efficiency.

5.3. Rotation: Precise Alignment Through Continuous Pivoting

The experimental evaluation demonstrates robust performance in object placement tasks across both planar and curved surfaces. Through systematic testing with 20 repeated trials per surface type, the method achieved perfect success rates (100%) in all cases. However, quantitative analysis revealed the following two key limitations: (1) When the grasped segment length approaches parity with the unconstrained portion, the actual bending profile deviates significantly from simulated predictions, reducing model effectiveness. (2) For thin objects, the required end-effector proximity to the target surface introduces practical challenges in final placement execution.

5.3.1. Placement on a Flat Surface

In this experiment, illustrated in Figure 13a–c, although the unconstrained placement configuration of the distal end permitted full translational freedom of test objects, allowing the possibility of sliding freely on the surface for all objects, the high normal pressing force acting at the contact point facilitated a firm contact during the whole placement process. Post-experiment observations indicated that the displacement between the initial and final positions of the distal end was negligible.

5.3.2. Placement on Convex/Concave Surfaces

The effectiveness of our approach is further validated through its application to non-planar geometries. We extend our method to encompass more complex yet prevalent curved surfaces. As demonstrated in Figure 13d, the distal end is initially adhered to a convex surface to mitigate the effects of gravity. Similarly, Figure 13e illustrates the alignment process on a concave surface, achieved without any fixation on the distal end. Successful placement on those curved surfaces confirms the method’s efficacy and effectiveness in suppressing internal force disturbances.

6. Conclusions and Future Development

This study systematically investigates the fundamental challenges in executing canonical pushing and rotation actions on soft objects, where the primary difficulty stems from the nonlinear and unpredictable deformation characteristics of such objects. The proposed manipulation framework establishes a novel theoretical connection between geometric shaping requirements and external constraint forces through a synthesis of first principles and Lagrange mechanics. By explicitly modeling deformation dynamics, we derive an optimized configuration region that enables effective primitive execution for deformable linear objects. The methodology demonstrates significant practical advantages, particularly in its low hardware requirements, as it does not involve complex sensing systems, and inherent scalability across different physical systems through dimensionless parameterization in the optimization process.

While experimental validation on physical robotic systems has confirmed the approach’s efficacy across diverse test cases, several inherent limitations warrant discussion. The current framework only mitigates but does not fully eliminate the influence of internal forces. Consequently, the method can be less effective in more geometrically complex scenarios, such as handling composite surfaces with mixed convex–concave geometries, where accumulated strain energy from unmodeled higher-order deformation modes exceeds the system’s self-stabilization threshold. These limitations stem fundamentally from the trade-off between model simplicity and physical completeness—while our reduced-order approach enables real-time computation, it necessarily approximates certain higher-order deformation modes. Future model refinements could incorporate nonlinear strain-energy potentials to better capture these effects.

The current implementation employs a simplified planning algorithm that selects operation points within the feasible configuration region. However, this approach encounters significant limitations when environmental constraints (e.g., obstacles) prevent direct access to optimal manipulation points. Future research directions should address the following two key aspects: (1) development of sampling-based motion planners that incorporate real-time collision checking to dynamically identify feasible manipulation waypoints in constrained environments, and (2) theoretical generalization of the framework to handle material with heterogeneous elasticity, as the current formulation’s independence from absolute elasticity values suggests potential for broader material applicability. These enhancements would substantially improve the method’s robustness and practical applicability in real-world manipulation scenarios.