SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 4(1); June (2023), pp. 46–54
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Stability Analysis of Frictionless Planar Enveloping Grasps andGrasp Parameter Effects
Takayoshi Yamada1, Hidehiko Yamamoto2

Department of Mechanical Engineering, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan
Gifu University, Yanagido 1-1, Gifu 501-1193, Japan

1
2

ARTICLE INFO
Article History
Received 26 November 2022
Accepted 06 September 2023
Keywords
Robot
Hand
Enveloping grasp
Grasp stability
Wrench vector
Grasp stiffness matrix
Curvature effect
Stiffness effect

ABSTRACTThis paper discusses quasi-static grasp stability of frictionless enveloping grasps in two
dimensions. The stability is investigated from the viewpoint of potential energy stored in thegrasps. The system of the grasps is replaced with elastic property, in which joint position and link
surface properties are represented by linear stiffness. The contact constraints between a grasped
object and finger links are formulated. The potential energy of the grasp system is obtained fromthe stiffness and the joint and link surface displacements. A wrench (i.e., force and moment)
vector and a stiffness matrix of the grasp system are derived from partial differentiations by thepose (i.e., translation and rotation) displacement of the object. The grasps are stable if both thewrench vector is zero and the matrix is positive definite. The stability is evaluated by the
eigenvalues of the matrix. Since, in this paper, the wrench vector and the stiffness matrix arederived in an analytical way, the matrix is given as a function of grasp positions, grasp forces,
local curvatures, joint and surface stiffnesses, and so on at contact points, explicitly. We
investigate curvature and stiffness effects on the grasp system by partially differentiating thematrix by the curvatures and stiffnesses. The positive definiteness of the matrix differentiations
is analyzed. Validity of our analysis is confirmed through numerical examples.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

squeezing, slipping out, breaking the parts, appropriate
grasp forms (i.e., grasp types, grasp positions, grasp
forces, etc.) are determined from the various types ofgrasps.
In order to obtain suitable grasps, machine learning
methods are tackled in recent years. Siddiqui et al. [1], Liet al. [2], and Dong et al. [3] discussed grasp stability
based on large amount of experimental data acquired
from vision and/or force sensor. In general, the learning
methods require strong CPU power, large amount oftraining data, and so on.
On the other hands, in traditional methodologies, grasp
and manipulation of objects by multi-fingered hands aretackled from the viewpoint of mechanics, kinematics,
statics, and dynamics in details. As one of the methods,
Hanafusa et al. [4] and Nguyen [5] proposed a concept of

1. Introduction
Dexterous and flexible hands are inherent in humanbeings. The hands can grasp and manipulate objects ofvarious shapes skillfully. In various fields like as
production lines including assembly tasks, picking tasks,
handling tasks, and so on, manual handworks by skillful
workers are remained. Human hands are utilized for not
only parts handling but also drilling and screw tightening
with some tools. However, the handworks will be able tobecome bottlenecks in the production lines. Therefore,thehuman-likeadvanced
technologies
andmethodologies are required for robots in mechanicalfunctions.
Various types of grasp forms can be imagined like as
pinching, enveloping, hooking, and so on. To avoid

Corresponding author’s E-mail: yamada.takayoshi.d9@f.gifu-u.ac.jp, yamamoto.hidehiko.u4@f.gifu-u.ac.jp

46

stiffness effects on the grasp system by partiallydifferentiating the matrix by the curvatures andstiffnesses. In Section 6, we show a numerical example
to confirm validity of our analysis. In Section 7, we show
our conclusion.

grasp stability. The stability was investigated from theviewpoint of potential energy stored in the grasp system.
In the system, every finger displacement was replacedwith linear elastic property. The stability expresses theindex whether or not the grasp system returns to its initialpose (i.e., position and orientation) after external
disturbance disappears. Watanabe [6] and Dong et al. [7]attacked effects of softness at contact points. In Ref. [8],we explored grasp stability of two spatial objects.
Contact surface geometry (i.e., curvature, torsion, andmetric tensor) was considered. Ref. [9] explored grasp
stability of pinching grasps in which each finger isconstructed of three revolute or prismatic joints. Effectsof local curvatures on the stability were also tackled. Ref.
[10] investigated grasp position effects on the grasp
stability and proposed an algorithm for automatic
generation of optimal grasp. In grasp forms, there arepinching grasps by fingertips, enveloping grasps by
finger link surfaces, and so on. Refs. [4], [10] exploredpinching grasps.

2. Problem Formulation,
An object is enveloped by an m-fingered hand of which
each finger consists of n-joints and links. We supposethat every link of the fingers is in contact with the object
and the number of contact points of each link is one. Bothevery link and object surfaces are curved at the contact
point but not overlapped each other.

Figure 2: Object and finger coordinate frames
Figure 1: An enveloping grasp in two dimensions

2.1. Notations
Finger number is denoted as 𝑖𝑖 (𝑖𝑖 = 1, 2, ⋯ , 𝑚𝑚) , andjoint and link number is denoted as 𝑗𝑗 (𝑗𝑗 = 1, 2, ⋯ , 𝑛𝑛). As
shown in Figure 2, some coordinates are defined asfollows:
Σ𝑏𝑏 : Base coordinate frame of the grasp system.
Σ𝑜𝑜 : Object coordinate frame fixed on the object.
Σ𝑏𝑏𝑏𝑏 : Initial pose of Σ𝑜𝑜 .
Σ𝑖𝑖0 : Base coordinate frame of the i-th finger.
Σ𝑖𝑖𝑖𝑖 : Joint and link coordinate frame of the j-th joint of
the i-th finger.
Σ𝑏𝑏𝑏𝑏𝑏𝑏 : Initial pose of Σ𝑖𝑖𝑖𝑖 .
Σ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 , Σ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 : Contact coordinate frames on the object
and finger surfaces, respectively.
Σ𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 , Σ𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 : Initial poses of Σ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 and Σ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 ,respectively.
Σ𝑝𝑝𝑝𝑝𝑝𝑝 : Deformed coordinate frame of Σ𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 .

In this paper, we explore quasi-static grasp stability offrictionless enveloping grasps in two dimensions (Figure1). The grasp systems are analyzed by replacing joint
position displacement and finger surface displacementwith elastic model. In Section 2, displacement parameters,
potential energies, and contact constraints are formulated.
In Section 3, partial derivatives of the potential energy
are derived. In Section 4, frictionless conditions at
contact points are formulated and the wrench vector andstiffness matrix of the grasp system are derived. By using
the eigenvalues of the matrix, the grasp stability isevaluated. In our formulation, the wrench and matrix are
analytically derived, it is explicitly shown that the matrix
is given as a function of grasp positions, grasp forces,
local curvatures, joint and surface stiffnesses, and so on
at contact points. In Section 5, we analyze curvature and47

In Appendix A, we show vectors and matrices used in
the following sections.
2.2. Displacements of joint position and finger
surface
In this subsection, we explain position of each finger
joint and pose of each finger link surface.
Configurations of the finger links are represented bypositions of the finger joints. We define position of the jth joint as symbol 𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 ∈ ℝ, which is represented as thefollowing components:
𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑞𝑞𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 + 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 , (𝑗𝑗 = 1, 2, ⋯ , 𝑛𝑛) (1)
𝑞𝑞𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
Natural component
Compressed component by the initial force� 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Displaced component by the object displacement
The relative pose of Σ𝑖𝑖𝑖𝑖 with respect to Σ𝑖𝑖(𝑗𝑗−1) isrepresentedasthefollowinghomogeneoustransformation matrix:𝑖𝑖(𝑗𝑗−1)
𝐴𝐴𝑖𝑖𝑖𝑖 �𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 �𝑖𝑖(𝑗𝑗−1)

𝐴𝐴𝑏𝑏𝑏𝑏𝑏𝑏 �𝑞𝑞𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 , 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �𝐴𝐴𝑟𝑟 �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �for revolute joint= � 𝑖𝑖(𝑗𝑗−1)
𝐴𝐴𝑏𝑏𝑏𝑏𝑏𝑏 �𝑞𝑞𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 , 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �𝐴𝐴𝑡𝑡 �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝒖𝒖1 � for prismatic joint(𝒋𝒋 = 𝟏𝟏, 𝟐𝟐, ⋯ , 𝒏𝒏)(𝟐𝟐)

The link surface is in contact on the object. We supposethat deformation of the link surface is approximated as
the pose displacement of the surface coordinate. We
define the deformation as symbol 𝒒𝒒𝑝𝑝𝑝𝑝𝑝𝑝 ∈ ℝ3 , which isrepresented as the following components:
𝒒𝒒𝑝𝑝𝑝𝑝𝑝𝑝 = 𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∈ ℝ3 , (𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛)(3)Compressed component by the initial force
𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
�
𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Displaced component by the object displacement
The relative pose of Σ𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 with respect to Σ𝑖𝑖𝑖𝑖 isrepresentedbythefollowinghomogeneoustransformation matrix:
𝑖𝑖𝑖𝑖
𝑖𝑖𝑖𝑖
(4)
𝐴𝐴𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 �𝒒𝒒𝑝𝑝𝑝𝑝𝑝𝑝 � ≔ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝 �𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �𝐴𝐴𝑡𝑡𝑡𝑡 �𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �As a result, displacement parameters on the j-th link aresummarized as the following vector:
𝑇𝑇(5)
𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑 ≔ �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 , 𝒒𝒒𝑇𝑇𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � ∈ ℝ4
2.3. Potential energy of the grasp

We define an elastic coefficient of the joint position
displacement as 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 ∈ ℝ and the potential energy of thej-th joint as
1
2(6)
𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � ≔ 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 �𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �2
Initial joint torque 𝜏𝜏𝑎𝑎𝑎𝑎𝑎𝑎 ∈ ℝ is formulated as(7)
𝜏𝜏𝑎𝑎𝑎𝑎𝑎𝑎 ≔ 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
48

The coefficient 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 is generated by some joint stiffness
control and then is changeable. The torque 𝜏𝜏𝑎𝑎𝑎𝑎𝑎𝑎 is alsochangeable by adjusting 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 and 𝑞𝑞𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 .
We define a stiffness coefficient of the elasticity of thelink surface displacement as 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 ∈ ℝ3×3 and thepotential energy of the j-th link of the i-th finger as
1
𝑇𝑇
𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝 �𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � ≔ �𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 �𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �2(8)
Initial contact force 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 is formulated as𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 ≔ 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 𝒒𝒒𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(9)
The coefficient matrix 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 is inherent in the mechanicalsurface property.
We obtain the potential energy of the i-th finger as𝑛𝑛

𝑈𝑈𝑖𝑖 (𝒒𝒒𝑑𝑑𝑑𝑑 ) ≔ �

𝑗𝑗=1

�𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � + 𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝 �𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ��(10)where the symbol 𝒒𝒒𝑑𝑑𝑑𝑑 is defined as the following vector.
(11)
𝒒𝒒𝑑𝑑𝑑𝑑 ≔ [𝒒𝒒𝑇𝑇𝑑𝑑𝑑𝑑1 , ⋯ , 𝒒𝒒𝑇𝑇𝑑𝑑𝑑𝑑𝑑𝑑 ]𝑇𝑇 ∈ ℝ4𝑛𝑛
Since the number of fingers is denoted as 𝑚𝑚, we obtain
the potential energy of the grasp system as
𝑚𝑚

𝑈𝑈(𝒒𝒒𝑑𝑑 ) ≔ �

𝑖𝑖=1

𝑈𝑈𝑖𝑖 (𝒒𝒒𝑑𝑑𝑑𝑑 ) ,𝒒𝒒𝑑𝑑 ≔ [𝒒𝒒𝑇𝑇𝑑𝑑1 , ⋯ , 𝒒𝒒𝑇𝑇𝑑𝑑𝑑𝑑 ]𝑇𝑇

2.4. Contact constraints of the object and finger
link surfaces(12)

Since the j-th link surface is in contact on the object,
the contact constraint is given by the following formula.
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑏𝑏
𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 �
𝐴𝐴𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝐴𝐴𝑜𝑜 (𝜺𝜺𝑜𝑜 ) 𝑜𝑜𝐴𝐴𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑖𝑖(𝑗𝑗−1)
𝑏𝑏
𝑖𝑖0
𝐴𝐴𝑖𝑖𝑖𝑖 �𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 �= 𝐴𝐴𝑖𝑖0 𝐴𝐴𝑖𝑖1 (𝑞𝑞𝑎𝑎𝑎𝑎1 ) × ⋯ ×𝑖𝑖𝑖𝑖

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

× 𝐴𝐴𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 �𝒒𝒒𝑝𝑝𝑝𝑝𝑝𝑝 � 𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 � 𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛)(13)
The parameter 𝜺𝜺𝑜𝑜 ∈ ℝ3 denotes pose (i.e., position andorientation) displacement of the object. The parameters
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 ∈ ℝ and 𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 ∈ ℝ denote the contact point
displacements on the object and link surfaces,
respectively.
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 𝐴𝐴𝑟𝑟 �𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 � 𝐴𝐴−1
𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 � ≔𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 � ≔𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 𝐴𝐴𝑟𝑟 �𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 �
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 −1
𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅

(14)𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 are defined aswhere the matrices𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 andthe following forms:
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
−1
−1
𝒖𝒖1 �
𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 ≔ 𝐴𝐴𝑡𝑡 �−𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝒖𝒖1 �,
𝐴𝐴𝜅𝜅𝜅𝜅𝜅𝜅𝜅𝜅 ≔ 𝐴𝐴𝑡𝑡 �−𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓(15)
Local curvature at contact point is denoted as 𝜅𝜅. If thecurvature 𝜅𝜅 is positive, zero, or negative, then the surface
is convex, flat, or concave, respectively. Because of no
overlapping between the object and finger link surfaces,
we have 𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 + 𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 > 0. Since the link surface contacts

on the object surface, the relation between Σ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 andΣ𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 is given as
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(16)
𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 𝐴𝐴𝑟𝑟 (𝜋𝜋)From the contact constraint (13), we obtain𝑖𝑖(𝑗𝑗−1) −1
𝑝𝑝𝑝𝑝𝑝𝑝𝐴𝐴𝑖𝑖𝑖𝑖 �𝑞𝑞𝑎𝑎𝑎𝑎𝑎𝑎 � × ⋯𝐴𝐴𝑡𝑡𝑡𝑡 �𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 � = 𝐴𝐴𝑖𝑖𝑖𝑖
𝑖𝑖0
𝑏𝑏𝑏𝑏
𝑜𝑜
(𝑞𝑞
)
𝐴𝐴
𝐴𝐴
× 𝑖𝑖0𝐴𝐴−1
𝑎𝑎𝑎𝑎1
𝑏𝑏𝑏𝑏
𝑜𝑜 (𝜺𝜺𝑜𝑜 ) 𝐴𝐴𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑖𝑖1
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 −1
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 �×
𝐴𝐴𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜 � 𝐴𝐴−1
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(17)
The surface displacement 𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 depends on the object
displacement 𝜺𝜺𝑜𝑜 , joint position displacement 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (𝑘𝑘 =
1,2, ⋯ , 𝑗𝑗), and contact position displacement 𝜶𝜶𝑖𝑖𝑖𝑖 .
𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖𝑖𝑖 , 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑1 , ⋯ , 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �,(𝑗𝑗 = 1,2, ⋯ , 𝑛𝑛)(18)where the symbol 𝜶𝜶𝑖𝑖𝑖𝑖 is defined as the following vector.
𝛼𝛼𝑜𝑜𝑜𝑜𝑜𝑜(19)𝜶𝜶𝑖𝑖𝑖𝑖 ≔ �𝛼𝛼 � ∈ ℝ2
𝑓𝑓𝑓𝑓𝑓𝑓Consequently, the energy (10) is transformed to
𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ) ≔
𝑛𝑛

=�

𝑗𝑗=1

𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑘𝑘 ≔𝑛𝑛= 𝜏𝜏𝑎𝑎𝑎𝑎𝑎𝑎 − 𝒗𝒗𝑇𝑇𝜁𝜁 �

𝑗𝑗=𝑘𝑘

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝑇𝑇
𝐵𝐵𝑖𝑖𝑖𝑖𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 ∈ ℝ

The detailed derivations are omitted for page space.

(23)

3.2. The second partial derivatives
In this subsection, we show the second partial
derivatives of the potential energy by 𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , and 𝜷𝜷𝑖𝑖 .
Considering the initial condition, we have the following
matrices:𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 ≔𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜺𝜺𝑇𝑇𝑜𝑜
0
𝑛𝑛

=�

𝑗𝑗=1

�

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐵𝐵𝑜𝑜𝑇𝑇 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐵𝐵𝑜𝑜 + 𝒗𝒗𝜁𝜁 �

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇
𝒑𝒑𝑜𝑜 𝐼𝐼23 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 �𝒗𝒗𝑇𝑇𝜁𝜁 �∈ ℝ3×3
𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑇𝑇= 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 ≔�
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
0= [𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝛼𝛼1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝛼𝛼𝑛𝑛 ] ∈ ℝ3×2𝑛𝑛
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑇𝑇= 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 ≔�
𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
0= [𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝑞𝑞1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝑞𝑞𝑛𝑛 ] ∈ ℝ3×𝑛𝑛
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼 ≔𝜕𝜕𝜶𝜶𝑖𝑖 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
0
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼1𝛼𝛼1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼1𝛼𝛼𝑛𝑛⋮⋱⋮
� ∈ ℝ2𝑛𝑛×2𝑛𝑛=�𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑛𝑛𝛼𝛼1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑛𝑛 𝛼𝛼𝑛𝑛

�𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 �𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �+ 𝑈𝑈𝑝𝑝𝑝𝑝𝑝𝑝 �𝒒𝒒𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 �𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖𝑖𝑖 , 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑1 , ⋯ , 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ���(20)
where the symbols 𝜶𝜶𝑖𝑖 and 𝜷𝜷𝑖𝑖 are defined as thefollowing vectors:
𝜶𝜶𝑖𝑖1
𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑1(21)𝜶𝜶𝑖𝑖 ≔ � ⋮ � ∈ ℝ2𝑛𝑛 , 𝜷𝜷𝑖𝑖 ≔ � ⋮ � ∈ ℝ𝑛𝑛
𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝜶𝜶𝑖𝑖𝑖𝑖

3. Partial derivatives of the potential energy

2𝜕𝜕2 𝑈𝑈𝑖𝑖𝑖𝑖 �𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ��
𝜕𝜕𝜷𝜷𝑖𝑖 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
0
𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞1 𝛼𝛼1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞1𝛼𝛼𝑛𝑛

𝑇𝑇
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼= 𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽 ≔3.1. The first partial derivatives
In this subsection, we show the first partial derivativesof the potential energy by 𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , and 𝜷𝜷𝑖𝑖 . Considering theinitial condition, we have the following vectors:𝑛𝑛𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇
� =�
𝐵𝐵𝑜𝑜 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 ∈ ℝ3
𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀 ≔𝜕𝜕𝜺𝜺𝑜𝑜𝑗𝑗=1
0
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼1
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )� = � ⋮ � ∈ ℝ2𝑛𝑛
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼 ≔𝜕𝜕𝜶𝜶𝑖𝑖
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑛𝑛
0𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞1
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽 ≔� = � ⋮ � ∈ ℝ𝑛𝑛
𝜕𝜕𝜷𝜷𝑖𝑖
𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑛𝑛0where the elements 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑘𝑘 and 𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑘𝑘 are obtained as
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑇𝑇
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑘𝑘 ≔� = 𝐾𝐾𝑖𝑖𝑖𝑖
𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 ∈ ℝ2
𝜕𝜕𝜶𝜶𝑖𝑖𝑖𝑖
0−𝒖𝒖2 −𝒖𝒖2
3×2
𝐾𝐾𝑖𝑖𝑖𝑖 ≔ � 𝜅𝜅−𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 � ∈ ℝ𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝜕𝜕𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
0

=�

⋮

⋱
⋯

⋮

� ∈ ℝ𝑛𝑛×2𝑛𝑛

𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑛𝑛𝛼𝛼1
𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑛𝑛 𝛼𝛼𝑛𝑛
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽 ≔�
𝜕𝜕𝜷𝜷𝑖𝑖 𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
0
𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞1𝑞𝑞1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞1𝑞𝑞𝑛𝑛⋱⋮
� ∈ ℝ𝑛𝑛×𝑛𝑛=� ⋮𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑛𝑛 𝑞𝑞1 ⋯ 𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑛𝑛 𝑞𝑞𝑛𝑛

(24)
The details of the elements are shown in Appendix B.
4. Enveloping grasps with frictionless sliding
contact(22)

4.1. Frictionless constraints at contacts

49

Since the finger link surface is in contact on the object
with frictionless sliding, the contact point displacement
𝜶𝜶𝑖𝑖 and the joint position displacement 𝜷𝜷𝑖𝑖 have to locallyminimize the energy. Consequently, the following
constraints are handled:
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )(25.1)= 02𝑛𝑛𝜕𝜕𝜶𝜶𝑖𝑖
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )(25.2)= 0𝑛𝑛
𝜕𝜕𝜷𝜷𝑖𝑖
From (25.1) at the initial condition, we have 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑘𝑘 = 02 .
Using (23), we have the following contact force:𝑇𝑇𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 = 𝐼𝐼23

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝒇𝒇,

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝑓𝑓𝑥𝑥

𝒇𝒇 = � 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 � ∈ ℝ2 ,𝑓𝑓𝑦𝑦
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑓𝑓𝑦𝑦 = 0𝑓𝑓𝑥𝑥 < 0,

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

If 𝐺𝐺 𝑓𝑓𝑓𝑓 = 03 and 𝐻𝐻 𝑓𝑓𝑓𝑓 ≻ 03×3 at the initial condition, thenthe grasp is stable. The condition 𝐺𝐺 𝑓𝑓𝑓𝑓 = 0 means that thegrasp is in wrench equilibrium. The condition 𝐻𝐻 𝑓𝑓𝑓𝑓 ≻03×3 means that the matrix is positive definite, and thegrasp returns to the initial pose after external disturbances
disappear. Using the eigenvalues and eigenvectors of thematrix 𝐻𝐻 𝑓𝑓𝑓𝑓 , we can evaluate the grasp stability.
5. Effect of grasp parameters

The matrix 𝐻𝐻 𝑓𝑓𝑓𝑓 depends on grasp parameters which
include contact positions, contact directions, contact
forces, local curvatures, stiffness properties at the contact
points. In our formulation, the matrix is analyticallyderived. We can differentiate the matrix by the grasp
parameters.

(26)

The symbol
𝑓𝑓𝑥𝑥 and𝑓𝑓𝑦𝑦 are normal and tangential
components of the contact force, respectively. Because
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
the contact force
𝒇𝒇 is represented in the frame Σ𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 ,𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑓𝑓𝑥𝑥 is negative. From (25.2) at the initial condition, we
have 𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑘𝑘 = 0. Using (23), we have the following joint
torque:𝑛𝑛𝜏𝜏𝑎𝑎𝑎𝑎𝑎𝑎 = 𝒗𝒗𝑇𝑇𝜁𝜁 �

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

0

𝑗𝑗=1

𝑗𝑗=𝑘𝑘

𝑇𝑇
𝐵𝐵𝑖𝑖𝑖𝑖𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝

5.1. Partial derivative of the local curvatures
In this subsection, we derive effects of the local
curvatures at contact points. Partially differentiation of𝑓𝑓𝑓𝑓
the stiffness matrix 𝐻𝐻𝑖𝑖 by the local curvature 𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 yields
the following matrix:𝑇𝑇
𝑓𝑓𝑓𝑓
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀𝜕𝜕𝐻𝐻𝑖𝑖𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀
𝑓𝑓𝑓𝑓 𝑇𝑇=+���� �𝑄𝑄𝑖𝑖 �𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀
𝑓𝑓𝑓𝑓
�
��
+𝑄𝑄𝑖𝑖 �𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼 𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽
𝑓𝑓𝑓𝑓
𝑓𝑓𝑓𝑓 𝑇𝑇��� �𝑄𝑄𝑖𝑖 �+𝑄𝑄𝑖𝑖 �𝑈𝑈
𝑈𝑈
𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿=
𝑓𝑓𝑥𝑥 𝒃𝒃𝑜𝑜𝑜𝑜𝑜𝑜 𝒃𝒃𝑇𝑇𝑜𝑜𝑜𝑜𝑜𝑜 ≼ 03×3
(34)

(27)

Since (25) has 3𝑛𝑛 constraints, the number of independent
parameters of the grasp system is reduced to three.
Finally, the displacements 𝜶𝜶𝑖𝑖 and 𝜷𝜷𝑖𝑖 are given byfunctions of the parameter 𝜺𝜺𝑜𝑜 .
𝑓𝑓𝑓𝑓
(28)
𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 ) ≔ 𝑈𝑈𝑖𝑖𝑖𝑖 �𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 (𝜺𝜺𝑜𝑜 ), 𝜷𝜷𝑖𝑖 (𝜺𝜺𝑜𝑜 )�
Considering (25), the following gradient is obtained.
𝑛𝑛
𝑓𝑓𝑓𝑓
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 )𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑓𝑓𝑓𝑓
𝐺𝐺𝑖𝑖 ≔
� = � 𝑜𝑜𝑊𝑊𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝒇𝒇(29)
𝜕𝜕𝜺𝜺𝑜𝑜The derivatives

Deriving partial derivative of (25) by the object
displacement 𝜺𝜺𝑜𝑜 , we have the following formula:𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
𝑓𝑓𝑓𝑓�� �
𝑄𝑄𝑖𝑖 ≔ �𝜕𝜕𝜺𝜺𝑜𝑜 0 𝜕𝜕𝜺𝜺𝑜𝑜 0(30)
𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼 −1
3×3𝑛𝑛
𝑈𝑈
𝑈𝑈
� ∈ℝ
= −[ 𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 ] �
𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽 𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽Appendix C. In a similar manner, we have the following
matrix:𝑓𝑓𝑓𝑓
𝜕𝜕𝐻𝐻𝑖𝑖
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑇𝑇(35)=
𝑓𝑓𝑥𝑥 𝒃𝒃𝑓𝑓𝑓𝑓𝑓𝑓 𝒃𝒃𝑓𝑓𝑓𝑓𝑓𝑓≼ 03×3𝜕𝜕𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓
where the vectors 𝒃𝒃𝑜𝑜𝑜𝑜𝑜𝑜 and 𝒃𝒃𝑓𝑓𝑓𝑓𝑓𝑓 ∈ ℝ3 are given by
02(𝑗𝑗−1)×1
02(𝑗𝑗−1)×1
𝒖𝒖1
𝒖𝒖2
𝑓𝑓𝑓𝑓
𝑓𝑓𝑓𝑓
𝒃𝒃𝑜𝑜𝑜𝑜𝑜𝑜 ≔ 𝑄𝑄𝑖𝑖 �0� , 𝒃𝒃𝑓𝑓𝑓𝑓𝑓𝑓 ≔ 𝑄𝑄𝑖𝑖 �0� (36)
2(𝑛𝑛−𝑗𝑗)×1
2(𝑛𝑛−𝑗𝑗)×1
0𝑛𝑛×1
0𝑛𝑛×1
𝑓𝑓𝑓𝑓
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
Because we have𝑓𝑓𝑥𝑥 < 0, the derivatives 𝜕𝜕𝐻𝐻𝑖𝑖 �𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝑓𝑓𝑓𝑓
and 𝜕𝜕𝐻𝐻𝑖𝑖 �𝜕𝜕𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 are negative semi-definite. It is shownthat the values of the local curvatures are made small, thegrasp stability is enhanced. Its directions are given by thevectors 𝒃𝒃𝑜𝑜𝑜𝑜𝑜𝑜 and 𝒃𝒃𝑓𝑓𝑓𝑓𝑓𝑓 , respectively.

𝑓𝑓𝑓𝑓

The second partial derivative of 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 ) is obtained as
𝑓𝑓𝑓𝑓
𝐻𝐻𝑖𝑖 ≔𝑓𝑓𝑓𝑓

𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 )
𝑓𝑓𝑓𝑓 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼� = 𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 + 𝑄𝑄𝑖𝑖 ��𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜺𝜺𝑇𝑇𝑜𝑜0(31)

We have the potential energy of the grasp system as
𝑚𝑚

𝑈𝑈𝑓𝑓𝑓𝑓 (𝜺𝜺𝑜𝑜 ) ≔ �

𝑖𝑖=1

𝑓𝑓𝑓𝑓

𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 )(32)

Total grasp wrench 𝐺𝐺 𝑓𝑓𝑓𝑓 and grasp stiffness matrix 𝐻𝐻 𝑓𝑓𝑓𝑓
are given by𝑚𝑚𝜕𝜕𝑈𝑈𝑓𝑓𝑓𝑓 (𝜺𝜺𝑜𝑜 )𝑓𝑓𝑓𝑓
𝐺𝐺 𝑓𝑓𝑓𝑓 ≔� = � 𝐺𝐺𝑖𝑖 ∈ ℝ3
𝜕𝜕𝜺𝜺𝑜𝑜
𝑖𝑖=1
0
(33)
𝑚𝑚
𝜕𝜕 2 𝑈𝑈𝑓𝑓𝑓𝑓 (𝜺𝜺𝑜𝑜 )
𝑓𝑓𝑓𝑓
𝑓𝑓𝑓𝑓
3×3
�
=
�
𝐻𝐻
∈
ℝ
𝐻𝐻 ≔
𝑖𝑖
𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜺𝜺𝑇𝑇𝑜𝑜 0𝑖𝑖=1

𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀,, ⋯ are briefly shown in𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜

50

5.2. Partial derivative of the spring stiffnesses

𝐿𝐿𝑏𝑏 = 0.010 m, 𝐿𝐿 = 0.030 m, 𝐿𝐿𝑐𝑐 = 0.015 m,𝜅𝜅𝑜𝑜 = 100 m−1 , 𝜅𝜅𝑓𝑓 = 200 m−1 , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 = 10 Nm/rad,𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 = diag[500 N/m 500 N/m 10 Nm/rad](40)
For simplicity of discussion, the lengths of all finger links
are same, the contact positions on the links areintermediate on the links, and the surface curvatures areconvex. The center of the local curvature at contact pointon each link surface are located on the link axis. The
centers of the local curvatures at contact points on theobject with the i-th finger are located at one point.

In this subsection, we derive effects of the spring
𝑓𝑓𝑓𝑓
stiffnesses. Partially differentiation of the matrix 𝐻𝐻𝑖𝑖 by
the joint stiffness 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 yields the following formula:𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖(37)= 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝒃𝒃𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≽ 03×3
𝜕𝜕𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎
02𝑛𝑛×1⎡0⎤(𝑗𝑗−1)×1
𝑓𝑓𝑓𝑓⎥ ∈ ℝ3
𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≔ 𝑄𝑄𝑖𝑖 ⎢⎢ 1 ⎥⎣0(𝑛𝑛−𝑗𝑗)×1 ⎦In the case that the surface stiffness is given by 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 =𝑓𝑓𝑓𝑓
diag�𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 , 𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 , 𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �, the partial derivatives of 𝐻𝐻𝑖𝑖 bythese elements are obtained by𝑓𝑓𝑓𝑓
𝜕𝜕𝐻𝐻𝑖𝑖= 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝒃𝒃𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≽ 03×3
𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖= 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝒃𝒃𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≽ 03×3
𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖(38)= 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝒃𝒃𝑇𝑇𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≽ 03×3
𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
where the vectors 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∈ ℝ3 and thematrix 𝐶𝐶𝑖𝑖𝑖𝑖 ∈ ℝ3×3 are given by
𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≔ 𝐶𝐶𝑖𝑖𝑖𝑖 𝒗𝒗𝑥𝑥 , 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≔ 𝐶𝐶𝑖𝑖𝑖𝑖 𝒗𝒗𝑦𝑦 ,𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≔ 𝐶𝐶𝑖𝑖𝑖𝑖 𝒗𝒗𝜁𝜁02(𝑗𝑗−1)×3
⎡
⎤⎫
⎧
𝑇𝑇
⎢ � 𝐾𝐾𝑖𝑖𝑖𝑖 � ⎥⎪⎪⎪⎢ 02(𝑛𝑛−𝑗𝑗)×3 ⎥⎪
⎪
⎪
(39)
⎥
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇
𝑓𝑓𝑓𝑓 ⎢𝑇𝑇
𝐶𝐶𝑖𝑖𝑖𝑖 ≔𝐵𝐵𝑜𝑜 + 𝑄𝑄𝑖𝑖 ⎢ 𝒗𝒗𝑇𝑇𝜁𝜁 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵𝑖𝑖1
⎡
⎤⎥⎬
⎨
⎢ ⎢
⋮
⎥⎥
⎪
⎢− ⎢ 𝑇𝑇 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇 ⎥⎥⎪
⎪
⎪
𝒗𝒗
𝐵𝐵
𝑖𝑖𝑖𝑖 ⎥⎥⎪⎪⎢ ⎢ 𝜁𝜁
⎩⎣ ⎣ 0(𝑛𝑛−𝑗𝑗)×3 ⎦⎦⎭The derivatives

𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖,𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖,𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖𝜕𝜕𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

, and

𝑓𝑓𝑓𝑓

𝜕𝜕𝐻𝐻𝑖𝑖𝜕𝜕𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝(a) Link coordinate frames, link lengths, and joint angles

are positive

semi-definite. It is shown that the parameter 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎 , 𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ,𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 and 𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 are larger, positive definiteness of thestiffness matrix are larger. Its corresponding directions
are also obtained as 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and 𝒃𝒃𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 .

(b) Contact coordinate frames and local curvatures
Figure 3: Numerical example of an enveloping grasp

6. Numerical examples

As shown in (2), in this paper, we treated not only
revolute but also prismatic joint type. Due to page space,in this section, we only consider the revolute type.
As shown in Figure 3, we investigate the grasp that an
object enveloped by a 2-finger 4-joint hand. In thisexample, the grasp is symmetric. The link lengths, local
curvatures, and spring stiffnesses are set as follows:

In this example, the grasp will be stable. We show thegrasp stiffness matrix, and eigenvalues and eigenvectorsof the matrix. Moreover, we show the directions of thecurvature and stiffness effects.
The grasp stiffness matrix is calculated as51

7. Conclusions

620.7
0−0.092(41)
𝐻𝐻 𝑓𝑓𝑓𝑓 = � 0
630.1
0 �
−0.0920
0.01
As shown in (41), not only diagonal elements but alsointerference elements between x-translation and rotation
appear. The eigenvalues and eigenvectors of 𝐻𝐻 𝑓𝑓𝑓𝑓 are
shown in Table 1. In this grasp, all eigenvalues are
positive, then the grasp is stable. In the first mode, theobject displacement is obtained in y translation. In thesecond mode, the object displacement is mainly obtainedin x translation. In the third mode, the object
displacement is mainly obtained in rotation.
Table 2 shows the direction vectors of the parameter
effects. The vectors 𝒃𝒃𝑜𝑜11 , 𝒃𝒃𝑓𝑓11 , 𝒃𝒃𝑜𝑜12 , and 𝒃𝒃𝑓𝑓12 express
curvature effects, then these appear in tangential
directions at contact points. The vectors 𝒃𝒃𝑠𝑠𝑠𝑠11𝑥𝑥 , 𝒃𝒃𝑠𝑠𝑠𝑠11𝑦𝑦 ,𝒃𝒃𝑠𝑠𝑠𝑠12𝑥𝑥 , 𝒃𝒃𝑠𝑠𝑠𝑠12𝑦𝑦 express stiffness effects at thecorresponding contact points, then these appear invertical direction at contact surface.

In order to obtain appropriate grasps, we treated grasp
stability from the viewpoint of potential energy.
Frictionless enveloping grasps in two dimensions wasinvestigated. Not only joint position displacements but
also link surface displacements are replaced with elastic
properties. The potential energy of the grasp system
including the displacements was derived. The wrench
vector and the grasp stiffness matrix were obtained by the
first and the second partial derivatives, respectively. The
derivatives are formulated in an analytical way. It isshown that the wrench and the matrix are given byfunctions of the grasp parameters. Using partial
derivatives of the stiffness matrix by local curvatures atcontact points, the local curvature effects on the grasp
stability were clarified. Stiffness effects were also
derived. To confirm validity of our analysis, we showed
a numerical example.
Using our analysis, a grasp system can be evaluatedwhen contact positions, contact directions, local
curvatures, spring stiffnesses, and so on are inputted.
Consequently, this method can be used for searching an
optimum grasp and/or generating training data formachine learning.
In [9], we considered masses of the object and finger
links in the analysis. We can also include the masses inthis paper but omitted due to page space.
The case of rolling contact at contact points was omitted
due to page space. We will discuss the case in our future
publication. In our future work, we will discuss three
dimensional grasps.

Table 1: Eigenvalues and eigenvectors of the grasp
Eigenvectors
mode Eigenvalues
𝜆𝜆𝑝𝑝 (𝐻𝐻 𝑓𝑓𝑓𝑓 )𝒗𝒗𝑝𝑝 (𝐻𝐻 𝑓𝑓𝑓𝑓 )𝑝𝑝[0, 1, 0]𝑇𝑇
1630.1
[1.00, 0, 0.00]𝑇𝑇
2620.7[0.00, 0, 1.00]𝑇𝑇
30.0064

Table 2: Direction vectors of the parameter effects for𝑖𝑖 = 1. (The case of 𝑖𝑖 = 2 is omitted for page space.)𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓
𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎
𝑗𝑗
1

𝒃𝒃𝑜𝑜11−0.64= �−0.64�
−0.02
𝒃𝒃𝑜𝑜122
0.63= �−0.65�
−0.02
Table 2 (continue)
𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑗𝑗
1

2

𝒃𝒃𝑠𝑠𝑠𝑠11𝑥𝑥−0.71= � 0.71 �
0.01
𝒃𝒃𝑠𝑠𝑠𝑠12𝑥𝑥−0.71= �−0.71�
−0.01

𝒃𝒃𝑓𝑓11−0.32= �−0.32�
−0.00
𝒃𝒃𝑓𝑓120.32= �−0.32�
−0.00

𝒃𝒃𝑠𝑠𝑠𝑠11−0.66= �−0.27�
−0.00
𝒃𝒃𝑠𝑠𝑠𝑠12−0.57= �−0.47�
−0.00

𝒃𝒃𝑠𝑠𝑠𝑠11𝑦𝑦
0.25= �0.25�
0.00
𝒃𝒃𝑠𝑠𝑠𝑠12𝑦𝑦−0.24= � 0.26 �
0.00

𝒃𝒃𝑠𝑠𝑠𝑠11𝜁𝜁
−0.06= �−0.06�
−0.00
𝒃𝒃𝑠𝑠𝑠𝑠12𝜁𝜁
0.06= �−0.06�
0.00

𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

References[1] M. S. Siddiqui, C. Coppola, G. Solak L. Jamone,
Grasp Stability Prediction for a Dexterous Robotic Hand
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𝑠𝑠𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

52

Appendix B: Partial derivatives of 𝑼𝑼𝒊𝒊𝒊𝒊[5] V. D. Nguyen, Constructing Stable Grasps, Int. J.
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1-15, 2017.

The second partial derivatives of 𝑈𝑈𝑖𝑖𝑖𝑖 are obtained asfollows:
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝛼𝛼𝑘𝑘 ≔�𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖𝑖𝑖
0
1 𝑇𝑇
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇=
𝐵𝐵𝑜𝑜 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 𝐾𝐾𝑖𝑖𝑖𝑖 + 𝒗𝒗𝜁𝜁 �𝒗𝒗𝑇𝑇𝑥𝑥 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 � � �1
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀 𝑞𝑞𝑘𝑘 ≔𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
0
𝑛𝑛

= −�
𝑗𝑗=𝑘𝑘
2

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐵𝐵𝑜𝑜𝑇𝑇 �𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝑇𝑇
𝐵𝐵𝑖𝑖𝑖𝑖 𝒗𝒗𝜁𝜁 + 𝐼𝐼23
Ω𝐼𝐼23 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 �𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝜕𝜕𝜶𝜶𝑖𝑖𝑖𝑖 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖𝑖𝑖
0𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝑇𝑇𝑇𝑇
𝐾𝐾 𝑆𝑆 𝐾𝐾 + �𝒗𝒗𝑥𝑥 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 � �𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜= � 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖
02×2
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑙𝑙 𝛼𝛼𝑘𝑘 ≔𝜕𝜕𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖𝑖𝑖
0

𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑙𝑙 𝛼𝛼𝑘𝑘 ≔𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜−𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 �

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐵𝐵𝑖𝑖𝑖𝑖𝑇𝑇 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝 𝐾𝐾𝑖𝑖𝑖𝑖 − �𝒗𝒗𝑇𝑇𝑥𝑥 𝝉𝝉𝑝𝑝𝑝𝑝𝑝𝑝 �[101×2𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�𝑈𝑈𝑖𝑖𝑖𝑖,𝑞𝑞𝑙𝑙 𝑞𝑞𝑘𝑘 ≔𝜕𝜕𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝜕𝜕𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 0=�−𝒗𝒗𝑇𝑇𝜁𝜁

1](𝑙𝑙 = 𝑘𝑘)(otherwise)(𝑙𝑙 ≤ 𝑘𝑘)(otherwise)𝑛𝑛

Appendix A: Vectors and matrices
In planar case, a homogeneous transformation matrix offrame Σ𝑏𝑏 with respect to frame Σ𝑎𝑎 is denoted as
𝑎𝑎
𝑎𝑎
𝑅𝑅
𝒑𝒑𝑏𝑏
𝑎𝑎
(42)
� ∈ ℝ3×3
𝐴𝐴𝑏𝑏 ≔ � 𝑏𝑏
01×2
1
where 𝑎𝑎𝒑𝒑𝑏𝑏 ∈ ℝ2 is a position vector and 𝑎𝑎𝑅𝑅𝑏𝑏 ∈ ℝ2×2 isa rotation matrix. We use the following vectors andmatrices:101
𝒖𝒖1 ≔ � � , 𝒖𝒖2 ≔ � � , 𝒛𝒛 ≔ � �
0
1
−1
𝒖𝒖
𝒖𝒖
0
𝒗𝒗𝑥𝑥 ≔ � 1 � , 𝒗𝒗𝑦𝑦 ≔ � 2 � , 𝒗𝒗𝜁𝜁 ≔ � 2×1 �
0
0
1
𝑥𝑥
𝒙𝒙
𝜺𝜺 ≔ � 𝜁𝜁 � , 𝒙𝒙 ≔ �𝑦𝑦�
cos 𝜁𝜁 − sin 𝜁𝜁
Rot(𝜁𝜁) ≔ ��,sin 𝜁𝜁
cos 𝜁𝜁
𝜋𝜋0 −1Ω≔�� = Rot � �
1 0
2
𝐼𝐼2
𝒙𝒙
Rot(𝜁𝜁) 02×1
� , 𝐴𝐴𝑟𝑟 (𝜁𝜁) ≔ �
𝐴𝐴𝑡𝑡 (𝒙𝒙) ≔ ��
01×2 101×2
1
𝐴𝐴𝑡𝑡𝑡𝑡 (𝜺𝜺) ≔ 𝐴𝐴𝑡𝑡 (𝒙𝒙)𝐴𝐴𝑟𝑟 (𝜁𝜁),𝑎𝑎
𝑅𝑅−Ω 𝑎𝑎𝒑𝒑𝑏𝑏
𝑎𝑎
� , 𝐼𝐼23 = [𝐼𝐼2 02×1 ] ∈ ℝ2×3
𝐵𝐵𝑏𝑏 ≔ � 𝑏𝑏
01×2
1
𝑎𝑎
𝑅𝑅𝑏𝑏
𝑇𝑇
𝑎𝑎
(43)
𝑊𝑊𝑏𝑏 ≔ � 𝑎𝑎
� = �𝐼𝐼23 𝑏𝑏𝐵𝐵𝑎𝑎 �
𝒑𝒑𝑏𝑏 × 𝑎𝑎𝑅𝑅𝑏𝑏=⎧𝑠𝑠 + ��𝒗𝒗𝑇𝑇 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵𝑇𝑇 𝑆𝑆 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵 𝒗𝒗 + 𝝉𝝉𝑇𝑇 𝐼𝐼 𝑇𝑇 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝒑𝒑 �
𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 𝜁𝜁
𝑖𝑖𝑖𝑖
𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝
𝑝𝑝𝑝𝑝𝑝𝑝 23
𝜁𝜁
⎪
𝑗𝑗=𝑘𝑘
⎪
(𝑙𝑙 = 𝑘𝑘)

⎨
⎪
⎪
⎩

𝑛𝑛

��𝒗𝒗𝑇𝑇𝜁𝜁

𝑗𝑗=𝑘𝑘

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝐵𝐵𝑖𝑖𝑖𝑖𝑇𝑇 𝑆𝑆𝑝𝑝𝑝𝑝𝑝𝑝

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝑇𝑇
𝐵𝐵𝑖𝑖𝑖𝑖 𝒗𝒗𝜁𝜁 + 𝝉𝝉𝑇𝑇𝑝𝑝𝑝𝑝𝑝𝑝 𝐼𝐼23

(𝑘𝑘 ≤ 𝑙𝑙 ≤ 𝑗𝑗)

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

𝒑𝒑𝑖𝑖𝑖𝑖 �(44)𝜕𝜕 𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ) 𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )��𝜕𝜕𝜺𝜺𝑜𝑜𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ) 𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )�=�𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ) 𝜕𝜕 2 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )
⎡
⎤
𝑇𝑇
𝑇𝑇
𝜕𝜕𝜶𝜶𝑖𝑖 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
𝜕𝜕𝜶𝜶𝑖𝑖 𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖
𝜕𝜕𝜶𝜶𝑖𝑖 𝜕𝜕𝜷𝜷𝑖𝑖 ⎢
⎥
�⎢ 2
+�
2
𝜕𝜕𝜺𝜺𝑜𝑜 𝜕𝜕𝜺𝜺𝑜𝑜 ⎢𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 ) 𝜕𝜕 𝑈𝑈𝑖𝑖𝑖𝑖 (𝜺𝜺𝑜𝑜 , 𝜶𝜶𝑖𝑖 , 𝜷𝜷𝑖𝑖 )⎥⎥𝜕𝜕𝜷𝜷𝑖𝑖 𝜕𝜕𝜶𝜶𝑇𝑇𝑖𝑖
𝜕𝜕𝜷𝜷𝑖𝑖 𝜕𝜕𝜷𝜷𝑇𝑇𝑖𝑖⎦⎣= [03×2𝑛𝑛 03×𝑛𝑛 ](45)
Appendix C: Curvature effects

The second partial derivatives of 𝑈𝑈𝑖𝑖𝑖𝑖 are obtained asfollows:

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𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀= 03×3 ,= 0𝑛𝑛×𝑛𝑛 ,𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝑇𝑇𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀 𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽== 03×𝑛𝑛 ,𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝑇𝑇
𝑇𝑇
𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝛼𝛼𝑘𝑘 𝒛𝒛 02(𝑘𝑘−1)×1
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝜀𝜀𝜀𝜀==
� 𝒖𝒖1 � ∈ ℝ3×2𝑛𝑛 ,𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 + 𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 02(𝑛𝑛−𝑘𝑘)×1
02(𝑘𝑘−1)×1 𝑇𝑇⎤
02(𝑘𝑘−1)×1 ⎡ 𝑈𝑈
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼𝑖𝑖𝑖𝑖,𝛼𝛼𝑘𝑘 𝛼𝛼𝑘𝑘 𝒛𝒛 ⎥⎢= � 𝒖𝒖1 � ⎢⎥𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
02(𝑛𝑛−𝑘𝑘)×1 ⎢𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 + 𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 ⎥⎣ 02(𝑛𝑛−𝑘𝑘)×1 ⎦02(𝑘𝑘−1)×1⎡⎤ 02(𝑘𝑘−1)×1 𝑇𝑇⎢ 𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝑘𝑘𝛼𝛼𝑘𝑘 𝒛𝒛 ⎥+⎢
� 𝒖𝒖1 �𝜅𝜅 + 𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 ⎥ 0⎢ 𝑜𝑜𝑜𝑜𝑜𝑜⎥ 2(𝑛𝑛−𝑘𝑘)×1⎣ 02(𝑛𝑛−𝑘𝑘)×1 ⎦02(𝑘𝑘−1)×1 02(𝑘𝑘−1)×1 𝑇𝑇
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑇𝑇+�
𝒇𝒇 𝒖𝒖1 � � 𝒖𝒖1 � � 𝒖𝒖1 � ,02(𝑛𝑛−𝑘𝑘)×1 02(𝑛𝑛−𝑘𝑘)×1
𝑇𝑇
𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛼𝛼𝛼𝛼 𝜕𝜕𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛽𝛽=𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜𝜕𝜕𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜
𝑈𝑈𝑖𝑖𝑖𝑖,𝛽𝛽𝛼𝛼𝑘𝑘 𝒛𝒛 𝑇𝑇
𝒖𝒖 0𝑛𝑛×2(𝑛𝑛−𝑘𝑘) �.
= �0𝑛𝑛×2(𝑘𝑘−1)𝜅𝜅𝑜𝑜𝑜𝑜𝑜𝑜 + 𝜅𝜅𝑓𝑓𝑓𝑓𝑓𝑓 1(44)
Authors Introduction

Dr. Takayoshi Yamada
He is a Professor with the Departmentof Mechanical Engineering, Gifu
University. He received the B.E.,
M.E, Ph.D. degrees in MechanicalEngineering from the Nagoya
Institute of Technology, Japan, in
1991, 1993, 1995, respectively. He isa member of the Japan Society of
Mechanical Engineers (JSME), theRobotics Society of Japan (RSJ), the Society of Instrument
and Control Engineers (SICE) , IEEE, and the Japan
Society of Precision Engineering (JSPE). His researchinterests include grasping, manipulation, sensing, andautomation systems.
Dr. Hidehiko Yamamoto
He is currently a specially appointed
professor, Gifu University. He
received the M.E. and Ph. D. degrees
in Mech. Eng. from Nagoya Instituteof Technology, in 1980, 1991. He wasa professor with the Department. of
MechanicalEngineering,
Gifu
University.

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