 Research Article
 Open Access
Cagingbased grasping of deformable objects for geometrybased robotic manipulation
 Dabae Kim^{1},
 Yusuke Maeda^{2}Email authorView ORCID ID profile and
 Shun Komiyama^{3}
 Received: 31 December 2018
 Accepted: 18 March 2019
 Published: 27 March 2019
Abstract
In this paper, we present a novel method for cagingbased grasping of deformable objects. This method enables manipulators to grasp objects simply with geometric constraints by using position control of robotic hands, and not through force controls or mechanical analysis. Therefore, this method has cost benefits and algorithmic simplicity. In our previous studies, we mainly focused on cagingbased grasping of rigid objects such as 2D/3D primitiveshaped objects. However, considering realistic objects, manipulation of deformable objects is also required frequently. Hence, this study is motivated to manipulate deformable objects, adopting a cagingbased grasping approach. We formulate cagingbased grasping of deformable objects, and target three types of deformable objects: a rigid object covered with a soft part, a closedloop structure, and two rigid bodies connected with a string, which can be regarded as primitive shapes. We then derive concrete conditions for grasp synthesis and conduct experimental verification of our proposed method with an industrial manipulator.
Keywords
 Robotic manipulation
 Caging
 Grasping
 Cagingbased grasping
 Deformable objects
Background
On the other hand, the caging approach [3, 4] is also introduced as an object restraint method, which cages objects using position controls as shown in Fig. 1b (we call this approach “Caging”, simply). In general, rigid objects are caged by multiple agents in [5, 6], whereas [7, 8] are recently conducted by robotic hands. When caging is realized, rigid objects cannot escape from the robotic hands physically, even if the objects take any possible pose. Caging enables robotic hands to restrain objects without using mechanical analysis or force sensing, in contrast to grasping. However, it cannot determine object poses uniquely, so it has to be performed when manipulating objects roughly. In other words, although caging is not a more accurate approach than grasping, it is an effective approach from the viewpoint of enhancing the versatility of manipulation with a simple algorithm. Comparing grasping and caging, it can be said that there are advantages and disadvantages, and it is necessary to choose the proper approach in accordance with manipulating situations.
In our previous studies [9, 10], we formulated cagingbased grasping, derived concrete conditions with geometric constraints, and confirmed that 2D/3D cagingbased grasping can be realized by experimental verification. However, the studies were restricted to rigid objects, which cannot be deformed by external force. Most caging studies also consider rigid objects or their equivalents only (e.g., [11–14]). However, caging deformable objects is conducted in [15, 16]. Gopalakrishnan and Goldberg [15] defines “Dspace” by modeling deformable objects as linear and elastic polygons, and [16] computes topological features such as necks or double forks. For grasping deformable objects, [17, 18] worked with deformable viscid objects with FEM analysis based on linear/nonlinear theory considering a friction cone. As another grasping approach, visionbased grasping of deformable objects in real time is also conducted in [19–21], which requires mechanical analysis or FEM analysis. However, the above studies require difficult modeling for deformable objects. In this paper, we focus on cagingbased grasping of deformable objects with simple geometrybased algorithms, and aim to extend the versatility of cagingbased grasping.
Problem statement
The rigid object covered with a soft part is a rigid body, which has a soft part around it. In this paper, we consider 2D primitive shapes including crosses, circles, and rectangles. The soft part indicates elastic bodies, which generate reaction forces when deformed by external forces. Rigidfingered hands are used as robotic hands for 2D cagingbased grasping, which is conducted via deformations of the soft parts of objects.
The closedloop structure has one closed loop, composed of rigid links and rotation joints. Rigidfingered hands with soft parts are used as robotic hands for 2D cagingbased grasping, which is conducted via deformations of the soft parts of robotic hands.
The two rigid bodies connected with a string consist of two 3D rigid bodies such as spheres, cuboids, and cylinders that are connected with a string. Parallel grippers with soft parts are used as robotic hands for 3D cagingbased grasping, which is made possible because of the deformations of the soft parts of robotic hands.
Cagingbased grasping conditions of deformable objects are formed by only geometric constraints, whereas there are also mechanical constraints which are complicate to deal with and require high computational cost. For these reasons, we assume: (1) the position control of the hand is perfect even if a reaction force is given by the object, and (2) the reaction force by the deformation of the object/hand soft part is not large enough to destroy the object or hand.
Cagingbased grasping

Objects are deformable.

Soft parts may be located on objects as well as hands.
Definition

Rigidpart caging condition: The object is caged in a closed region formed by the rigid parts of a robot hand. This condition consists of the following three subconditions:
 (a)
Closed region formation: A closed region through which the object cannot pass is formed by the rigid parts of the robot hand.
 (b)
Object inside: The object is within the closed region formed by the rigid parts of the robot hand.
 (c)
No interference: The rigid parts of the robot hand do not overlap with the rigid parts of the object.
 (a)

Softpart deformation condition: Assuming that the soft parts of the robot hand and/or the object become rigid, the closed region for caging in the configuration space of the object becomes empty.
Formulation

n: Number of rigid robot bodies.

C: Configuration space of the rigid parts of the object for their movements.

\({\mathcal {A}}_{\mathrm {obj}}\): Occupied region in real space of rigid parts of the object.

\({\mathcal {A}}'_{\mathrm {obj}}\): Occupied region of the entire object including rigid and soft parts in real space without deformations of soft parts.

\({\mathcal {A}}_{i}\): Occupied region in real space of the ith rigid part of the robot, \((i = 1,\ldots,n)\).

\({\mathcal {A}}'_{i}\): Occupied region of the ith rigid part and its attached soft parts in real spaces without deformations of soft parts.

\(\varvec{q}_{\mathrm {obj}}\): Current configuration of the object.

Rigidpart caging condition

Closed region formation: The rigid parts of the robot form a closed region, through which the object cannot pass no matter how the object is moved or deformed:$$\begin{aligned} \exists {\mathcal {C}}_{\mathrm {closed}} \text { such that } {\mathcal {C}}_{\mathrm {closed}} \cap {\mathcal {C}}_{\mathrm {free \_ ECS}} = \emptyset . \end{aligned}$$(7)

Object inside: The rigid parts of the object exist inside the closed region formed by the rigid parts of the robot:$$\begin{aligned} \varvec{q}_{\mathrm {obj}} \in {\mathcal {C}}_{\mathrm {closed}}. \end{aligned}$$(8)

No interference: The rigid parts of the robots do not interfere with the rigid parts of the object:$$\begin{aligned} \varvec{q}_{\mathrm {obj}} \in {\mathcal {C}}_{\mathrm {free}}. \end{aligned}$$(9)


Softpart deformation condition: The object cannot exist in closed regions formed by the rigidpart caging condition, when regarding the soft parts of the robot and the object as rigid bodies:$$\begin{aligned} {\mathcal {C}}'_{\mathrm {free \_ICS}} = \emptyset . \end{aligned}$$(10)
Rigid object covered with a soft part
In this section, we introduce a class of the cagingbased grasping of the rigid object covered with a soft part. Concretely, we set targets of cagingbased grasping as cross, H, and Ushaped objects with twofingered rigid hands, and circular, elliptic, triangular, rectangular, Lshaped, and Tshaped objects with threefingered rigid hands. In this paper, we representatively introduce concrete conditions of the crossshaped object with the twofingered rigid hand for 2D cagingbased grasping.
Hand and object model
Rigidpart caging condition
Closed region formation
 (i)
\((W_1t_1)/2 \ge r\), and \((W_2t_2)/2 \ge r\)
 (ii)
\((W_1t_1)/2 < r\), and \((W_2t_2)/2 \ge r\)
 (iii)
\((W_1t_1)/2 < r\), and \((W_2t_2)/2 < r\)
Softpart deformation condition
 (i)
\((W_3t_3)/2 \ge r\), and \((W_4t_4)/2 \ge r\)
 (ii)
\((W_3t_3)/2 < r\), and \((W_4t_4)/2 \ge r\)
 (iii)
\((W_3t_3)/2 \ge r\), and \((W_4t_4)/2 < r\)
 (iv)
\((W_3t_3)/2 < r\), and \((W_4t_4)/2 < r\)
Experimental verification

Hand measurement [mm]: \(r = 7\).

Object measurements [mm]: \(W_{1} = 79.2\), \(W_{2} = 79.2\), \(W_{3} = 99.7\), \(W_{4} = 100.4\), \(t_{1} = 15.8\), \(t_{2} = 15.8\), \(t_{3} = 35.6\), \(t_{4} = 38.2\).
Closedloop structure
In this section, we introduce a class of cagingbased grasping of the closedloop structure. Concretely, we set targets of cagingbased grasping as four, five, and sixbar closedloop structures, and an infinitebar closedloop structure (strap). In this paper, we representatively introduce concrete conditions of the fourbar closedloop structure with a twofingered hand with the soft part, for 2D cagingbased grasping.
Hand and object model

\(\mathrm {J}_{i}\): joint centers (\(i = 14\)).

\(q_{j}\): finger centers (\(j =\) m, n).

\(r_{\mathrm {rigid}}\): radius of the rigid part.

\(r_{\mathrm {soft}}\): radius of the soft part.

b: distance between the center of the hand and that of the finger.

\(d_{mn}\): finger center distance.

a: link length.

2t: link thickness.

\(\alpha\): angle between xaxis and link.

\(x', y'\): intersection of perpendicular line dropped from \(q_{n}\) to \(\overline{\mathrm {J}_{1}\mathrm {J}_{2}}\).
Rigidpart caging condition
Closed region formation
In this paper, it is obvious that this condition is satisfied automatically by the existence of the rigid part of the finger, and the closed loop of the object.
Softpart deformation condition
Experimental verification

Hand measurements [mm]: \(r_{\mathrm {rigid}} = 7, r_{\mathrm {soft}} = 10.5\).

Object measurements [mm]: \(a = 79.9, t = 3.7\).
Two rigid bodies connected with a string
In this section, we introduce a class of cagingbased grasping of two rigid bodies connected with a string. Concretely, we set the targets of cagingbased grasping as two spheres, cuboids, and cylinders connected with a string. In this paper, we representatively introduce concrete conditions of the two cuboids connected with a string with the parallel gripper, which consists of rigid parts and soft parts, for 3D cagingbased grasping.
Hand and object model

\(r_{\mathrm {rigid}}\): rigid link part radius.

\(r_{\mathrm {string}}\): string radius.

T: jaw tip length.

L: jaw side length.

\(d_{\mathrm {tip}}\): jaw tip distance.

\(d_{\mathrm {soft}}\): soft part distance in open/close direction of jaw.
Rigidpart caging condition
Closed region formation
Softpart deformation condition
Experimental verification

Hand measurements [mm]: \(L = 30.7\), \(T = 20.0\), \(r_{\mathrm {rigid}} = 7\).

Object measurements [mm]: \(a = 25.1\), \(b = 45.1\), \(c = 55.2\), \(r_{\mathrm {string}} = 2.5\), \(l = 100\).
Discussion and conclusion
In this study, we proposed a method of cagingbased grasping to manipulate deformable objects with only geometric constraints. We firstly formulated the cagingbased grasping approach of deformable objects. This was realized by caging of the objects by rigid parts of hands with deformations of soft parts. This formulation was defined using only shape information, which consists of rigidpart caging condition and a softpart deformation condition, even for deformable objects. Next, three types of deformable objects: a rigid object covered with a soft part, a closedloop structure, and two rigid bodies connected with a string are defined as deformable objects. These can be regarded as the primitive shapes of deformable objects. In these types, soft parts, joints, and strings were regarded as the deformable components, respectively. Also, we derived concrete conditions for grasp synthesis as sufficient conditions respectively. Through pickandplace experiments, we confirmed that it was possible to manipulate deformable objects with cagingbased grasping, simply by knowing the shape information of the objects.
There are also limitations of our proposed method as a tradeoff. When closing the robotic hands for satisfying the cagingbased grasping condition, we set \(d_{mn}\) (or \(d_{\mathrm {tip}}\)) properly in the range of calculated results. However, object weights or insufficient friction causes objects to slide and escape from the hands in the case of 2D cagingbased grasping. For solving this problem, calculating the optimized \(d_{mn}\), which maximizes the margin spaces of closed regions, also remains as future works.
Declarations
Authors' contributions
YM proposed the basic formulation of the proposed method, checked the manuscript, and carried out the conceptual supervising. DK revised the formulation of the proposed method, derived the concrete conditions, designed and carried out the experiments, analyzed the experimental results, and wrote the manuscript. SK checked the derived conditions, and corrected errors found in them. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Funding
This work was supported by Yokohama National University.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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