2D and 3D Caging-Based Grasping of Objects in Various Shapes* Tomohiro EGAWA and Yusuke MAEDA
Abstract— We proposed a novel method to grasp objects: caging-based grasping. This method is caging an object by rigid parts of a robot hand and simultaneously grasping by soft parts attached to the rigid parts. By this method, even with a robot hand of simple structure, we can grasp objects in various shapes with only geometrical information. We derive concrete conditions for 2D and 3D caging-based grasping of objects in various shapes. We then simulate cagingbased grasping on a PC and conduct experiments with mobile robots and robot hands. In this paper, we show the results of experiments and simulations.
I. INTRODUCTION In the robot manipulation, grasping is the most common method to constrain objects. Grasping is useful to uniquely determine the position and posture of objects. However, it generally needs complicated force control and mechanical analysis. Caging is another method to constrain objects. It makes an object inescapable from the cage composed of robots [1]. Therefore, caging can not uniquely determine the position and posture of the object in the cage. However, it can constrain objects by position-controlled robots based on only geometrical information. Many researchers have used caging as a substitute for or a complement to conventional grasping in robotic manipulation [2]∼[3]. We proposed a novel approach to grasping based on caging with a robot hand covered with soft parts: cagingbased grasping [4]. An object is caged by the rigid parts of the hand, and a complete grasp is achieved by the soft parts (Fig. 1). This method does not require complicated force control and mechanical analysis, and only requires geometrical information. Furthermore, it can allow sensing error and control error to some extent. In our conventional research, we formulated caging-based grasping and realized for only circles and spheres, but it is insufficient for cagingbased grasping of various objects. In this paper, we realize caging-based grasping of objects in various shapes. There are some conventional researches on grasping by position-controlled soft fingers (for example, [5] and [6]). However, these researches require stability analysis of grasps based on mechanical information to guarantee successful grasping. On the other hand, our caging-based grasping can be achieved only by geometrical information, and does not require explicit mechanical analysis of grasping that depends on contact friction and elasticity. *This work was partly supported by JSPS KAKENHI 22700200. 1 Tomohiro EGAWA is with Maeda Lab, Dept. of Mechanical Engineering, Div. of Systems Integration, Graduate School of Engineering, Yokohama National University. 2 Yusuke MAEDA is with Div. of Systems Research, Faculty of Engineering, Yokohama National University. 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501 JAPAN
[email protected]
Robot
(rigid part)
Finger
(soft part)
(rigid part) (soft part)
Object
Object
(a) 2D example Fig. 1.
(b) 3D example
Caging-based grasping
II. C AGING -BASED G RASPING [4] A. Definition Caging-based grasping holds when the following two geometrical conditions are both established: • Rigid-part caging condition: The rigid parts of the robot hand is caging the object. • Soft-part deformation condition: Assuming that the soft parts of the robot hand are rigid, the object can not exist in the closed region for caging. Furthermore, rigid-part caging condition consists of the following three conditions: 1) Closed-region formation: The closed region through which the object can not pass is formed by the rigid parts of the robot hand. 2) Object inside: The object is in the closed region which is formed by the rigid parts of the robot hand. 3) No interference: The rigid parts of the robot do not collide with the object. Namely, the rigid-part caging condition holds when the object is in the closed area formed by the rigid parts of the robot hand without collision. And the soft-part deformation condition holds when the soft parts of the robot hand cannot help deforming in the closed region. Consequently, the object is caged by the rigid parts of the robot hand and grasped by reaction forces of the deformed soft parts. Note that both of the conditions can be tested geometrically and explicit mechanical analysis is not necessary. B. Formulation Here we introduce the formulation of caging-based grasping presented in [4]. Let us define the following symbols: • n: The number of robot bodies. • C: The configuration space of the object. • Aobj (q): The object in the workspace when its configuration is q ∈ C. • Ai : The rigid part of the i-th robot body in the workspace (i = 1, . . . , n). • Ai : The rigid and soft parts of the i-th robot body (i = 1, . . . , n) without deformation of the soft parts. • qobj : The configuration of the object.
The free configuration space of the object in which the object is free from interference with the rigid robot bodies A1 , . . . , An can be written as follows: n Cfree := q ∈ C Aobj (q) ∩ Ai = ∅ . (1)
Robot
(rigid part) (soft part)
2
1
Fig. 2.
i=1
3
4
Caging-based grasping of a triangle
Similarly, the free configuration space of the object in which the object is free from interference with the robot bodies A1 , . . . , An can be written as follows: n Ai = ∅ . (2) Cfree := q ∈ C Aobj (q) ∩ i=1
(a) Ellipse
Cfree
is the free configuration space when assuming soft parts ⊆ Cfree . of the robots rigid. Naturally we obtain Cfree When the object is caged by the rigid parts of the robots, Cfree can be separated into two parts: an inescapable configuration space (ICS) and an escapable configuration space (ECS). In the latter space, the object can escape far away. Here we define the latter space Cfree ECS as follows: Cfree ECS := Cfree max (q). (3) q∈Qdist
where Qdist (⊂ C) is a set of object configurations that can be regarded as “distant”; Cfree max (q) is the maximal connected subset of Cfree that includes q. Now the rigid-part caging condition can be written as follows: 1) Closed-region formation: ∃Cclosed such that Cclosed ∩ Cfree ECS = ∅. (4) 2) Object inside: 3) No interference:
qobj ∈ Cclosed .
(5)
qobj ∈ Cfree .
(6)
When only the rigid-part caging condition holds, the object can freely move in Cfree ICS . On the other hand, if Cfree ICS is occupied by the soft parts of the robots, the object cannot exist without the deformation of the soft parts. In this case, the object is grasped by the soft parts and localized somewhere in Cfree ICS . The soft-part deformation condition can be written as follows: Cfree ICS = ∅,
(7)
where Cfree ICS := Cfree \Cfree ECS Cfree Cfree ECS := max (q),
(8) (9)
q∈Qdist where Cfree max (q) is the maximal connected subset of Cfree that includes q. Thus the caging-based grasping can be formulated as the combination of (4)–(6) and (7) as follows: ⎧ ∃Cclosed such that Cclosed ∩ Cfree ECS = ∅, ⎪ ⎪ ⎪ ⎨q ∈ C obj closed , (10) ⎪ qobj ∈ Cfree , ⎪ ⎪ ⎩ Cfree ICS = ∅.
Fig. 3.
(b) Rectangle
Caging-based grasping of an ellipse and a rectangle
III. 2D C AGING -BASED G RASPING A. By mobile robots We consider 2D caging-based grasping by three circular robots (iRobot Create), and use triangles, ellipses and rectangles for objects. Urethane foams are attached to all the robots as soft parts. We derived concrete conditions for caging-based grasping of them to satisfy eq.(10), and conducted manipulation experiments of caging-based grasping for validation of the derived conditions. Here, we omit concrete conditions and show only experimental results. In the experiment shown in Fig. 2, we located the robots and the object at the positions where the conditions we derived was satisfied. Then, we moved the three robots at the same time and they translated the object keeping their formation by open-loop control. As a result, the object was successfully grasped and manipulated. In the same way, we conducted experiments of cagingbased grasping for an ellipse and a rectangle shown in Fig. 3. As a result, the objects were successfully grasped and manipulated, too. B. By a parallel gripper We consider 2D caging-based grasping by a parallel gripper, and use triangles, rectangles and H-, L-, T-, Uand cross-shaped objects. We omit concrete conditions for caging-based grasping of them, and show only simulation results for validation of the derived conditions. We simulated caging-based grasping with MATLAB and ODE (Open Dynamics Engine) [7]. First we calculated the values to satisfy the derived conditions with MATLAB, and simulated caging-based grasping of objects to satisfy the values on ODE. In the simulation shown in Fig. 4, we gave the external force to the object and validated that it did not pass through the gap between the tips of the gripper on ODE. Therefore, we could validate that caging-based grasping is successful on the simulation. In the same way, we conducted experiments of cagingbased grasping for other objects shown in Fig. 5. As a result, the objects were successfully grasped and manipulated, too.
①～④：Actuator
Gripper (rigid part)
③
①
(soft part)
④
②
1 Fig. 4.
3
2
Caging-based grasping of a cross-shaped object
Finger
(a) Rigid parts Fig. 7.
(b) With soft parts 2-fingered hand
⑤
(a) Triangle
(b) Rectangle
① ③ ⑥ ② ④
(c) H-shaped
①～⑥：Actuator
Finger
(a) Rigid parts Fig. 8. (d) L-shaped Fig. 5.
(e) T-shaped
(f) U-shaped
Caging-based grasping of objects in various shapes
IV. 3D C AGING -BASED G RASPING A. By multi-fingered hands We consider 3D caging-based grasping by a two- or threefingered hand with the following features (Fig. 6): • The hand consists of a flat palm and two or three fingers. • All the fingers are identical in their structure; each of them has three cuboid links as rigid parts, covered with cylindrical soft parts and connected by two rotary joints. • All the fingers are attached on the palm with circular symmetry. • All the angles of the first and the second joints of the fingers are identical, respectively. As objects, we use spheres, cuboids, ellipsoids, tori, hollow and solid cylinders and dumbbell- and bulb-shaped objects. In this paper, we omit concrete conditions for caging-based grasping, and show only experimental results for validation of the derived conditions. We fabricated a two-fingered robot hand as shown in Fig. 7(a) and a three-fingered robot hand as shown in Fig. 8(a). Each finger had two RC servo motors (Futaba RS405CB). It was attached to an industrial manipulator, Fanuc LR-Mate 200iA. For caging-based grasping, urethane foam was attached to each of the finger links of the hand (Fig. 7(b), 8(b)) to form a cylindrical soft part. Palm
Palm
Joint
1
2
4
5 Fig. 9.
1
3
6
Caging-based grasping of a torus
2
3
Finger (rigid part)
(rigid part)
(soft part)
(soft part)
Fig. 6.
In the experiment shown in Fig. 9, 10, we controlled the two- or three-fingered hand to satisfy the derived conditions for tori and cuboids. Then, we picked up a torus and a cuboid by caging-based grasping, carried it to a destination, and placed it successfully. In the same way, we conducted experiments of cagingbased grasping of other objects shown in Fig. 11. As a result, the relative position among the object and the robots was maintained throughout the manipulation, which is difficult in conventional caging.
Joint
Finger
(a) two-fingered hand
(b) With soft parts 3-fingered hand
(b) three-fingered hand Multi-fingered hand
4
Fig. 10.
5
6
Caging-based grasping of a cuboid
1
(a) Hollow cylinder
(b) Dumbbell-shaped object
(c) Solid cylinder
(d) Ellipsoid (egg-shaped object)
Fig. 13.
2
1
Fig. 11.
3
Caging-based grasping of a dumbbell-shaped object on ODE
Fig. 14.
2
3
Caging-based grasping of a solid cylinder on ODE
Caging-based grasping of objects in various shapes
Palm
Palm
Gripper
Gripper
(a) two-fingered gripper Fig. 12.
(a) Torus
(b) Hollow cylinder (c) Cup-shaped object
(rigid part) (soft part)
(rigid part) (soft part)
(b) four-fingered gripper
Multi-fingered gripper
B. By multi-fingered grippers We consider caging-based grasping by a two- or fourfingered gripper with the following features (Fig. 12): • The gripper consists of two or four fingers. • Each of fingers has cylindrical links as rigid parts, covered with cylindrical soft parts and connected by fixing joints. • All the fingers are attached on the palm. • Two-fingered gripper consists of U-shaped fingers. • Four-fingered gripper consists of U-shaped fingers and two cylindrical fingers. As objects, we use spheres, cuboids, tori, hollow and solid cylinders and dumbbell-, bulb- and cup-shaped objects. In this paper, we omit concrete conditions for caging-based grasping, and show only simulation results on PC for validation of the derived conditions. We simulated caging-based grasping with MATLAB and ODE. In the simulation shown in Fig. 13, 14, we moved the two- or four-fingered gripper at random and validated that the dumbbell-shaped object and solid cylinder did not fall from the gripper on ODE. Therefore, we could validate that caging-based grasping is successful on the simulation. In the same way, we conducted experiments of cagingbased grasping of other objects shown in Fig. 15. As a result, we could validate that caging-based grasping of objects in various shapes is successful on the simulations.
(d) Sphere Fig. 15.
(e) Cuboid
(f) Bulb-shaped object
Caging-based grasping of objects in various shapes
V. C ONCLUSION In this research, we derived concrete conditions for 2D and 3D caging-based grasping of objects in various shapes. Then, we show the experimental results of caging-based grasping with multi-fingered hands and, simulations of caging-based grasping on MATLAB and ODE. There are many issues on caging-based grasping to be addressed, which include how to grasp and release objects, how to select appropriate softness for soft parts, how to derive concrete conditions for caging-based grasping for various robot hands, and its application to various tasks. R EFERENCES [1] E. Rimon and A. Blake, “Caging planar bodies by one-parameter twofingered gripping systems,” Int. J. of Robotics Research, vol. 18, no. 3, pp. 299–318, 1999. [2] Z. Wang and V. Kumar, “Object closure and manipulation by multiple cooperating mobile robots,” in Proc. of IEEE Int. Conf. on Robotics and Automation, 2002, pp. 394–399. [3] S. Makita and Y. Maeda, “3D multifingered caging: Basic formulation and planning,” in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 2008, pp. 2697–2702. [4] Y. Maeda, N. Kodera and T. Egawa, “Caging-based grasping by a robot hand with rigid and soft parts,” in Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 5150–5155, 2012. [5] M. R. Cutkosky and I. Kao, “Computing and controlling the compliance of a robotic hand,” IEEE Trans. on Robotics and Automation, vol. 5, no. 2, pp. 151–165, 1989. [6] T. Inoue and S. Hirai, “Experimental investigation of mechanics in soft-fingered grasping and manipulation,” in Experimental Robotics, ser Springer Tracts in Advanced Robotics, O. Khatib, V. Kumar and D. Rus, Eds. Springer, 2008, vol. 39, pp. 13–22. [7] “Open Dynamics Engine,” http://www.ode.org/.
