subspaces with respect to the connectivity of the object. The relationship we ... for medical and industrial applications is so important that considerable effort has ...
Classification of Grasps by Multifingered Robot Hands Yuru Zhang* and William A. Gruver Intelligent Robotics and Manufacturing Systems Laboratory School of Engineering Science Simon Fraser University Burnaby, BC V5A 1S6 Canada finger tips, further research is needed on grasping and manipulation with multiple contacts between one finger and the grasped object. It has been recognized by several researchers that using a whole finger will enhance the performance of the robot hand and make it possible to grasp objects with a wider range of shapes and sues. Different names have been proposed to describe grasps using fingertipsand other links ofthe fingers and the palm, such as enveloping grasp, power grasp, and whole finger grasp. Research on power grasps involves several issues. The representation and optimization of contact forces has been investigated [2, 81. Dynamic simulations were performed [3], and quality measures for stability were proposed [ 2 , 5 ] .Trinkle, et al., [7] developed a system forplanning and simulating the enveloping grasp of convex polygons in the absence of frictional and dynamic effects. A neural network was applied to obtain force control of power grasps ~41. From an examination of human grasping, fingertip grasps achieve dexterity by holding the object by the tips of the fingers and thumb. Power grasps are distinguished by large areas of contact between the object and the fingers and palm and by little or no ability to impart motions with the fingers [l]. When similar strategies of human grasping are used by robot hands, it is necessary to develop quantitative models to characterize grasps. Bang and Nakamura, et al., [5] defined a power grasp as that which passively resists extemal forces without the control of joint torques. Bicchi [6] pointed out that uncontrollable intemal forces may exist in some enveloping grasps. For identifying the uncontrollable intemal forces, Melchiorri [9] decomposed the space of contact forces into four subspaces. This idea is further developed in this paper by defining four subspaces using the nullspaces of the grasp and Jacobian matrices and their orthogonal complements. Our method establishes relationships between the dimensions of the subspaces and the connectivity of the object. This
Abstract This research characterizesgrasping by multifingered robot hands through an investigation of the space of contactforces betweenfingers and the grasped object. By decomposing the space of contactforces intofour subspaces, we develop a method to determine the dimensions of the subspaces with respect to the connectivity of the object. The relationship we obtain reveals the diflkrences between three qpes of grasps classified in the paper and indicates how the contactforce can be decomposed corresponding to each type of grasp. Therefore, it provides a guideline for formulating the distribution of contactforces as a basisfor optimization. The subspaces and the determination of their dimensions are illustrated by examples.
1. Introduction Robot end-effectors range from the simplest design of parallel jaw gripper to complex configurations of dextrous mechanical hands. Most parallel jaw grippers are designed to grasp objects by fingertips. Grasping performance is, therefore, more limited compared with human grasping. The ability to imitate functions of the human hand for medical and industrial applicationsis so important that considerable effort has been devoted to the development and design of dextrous hands with three, four and five fingers [121. Because of the technical difficulties in actuation, sensingand control, however, dextroushands are still far from being able to perform the fwnctions of human grasping. An extension of the fingertip grasp is a multifingered gripper that can use its whole finger and palm to grasp the object. Although a dextrous mechanical hand can also do more than merely grasp an object using its
*
On leave from Beijing University of Aeronautics and Astronautics, Beijing, China
Proc. IROS 96 0-7803-3213-X/96/ $5.00
01996 IEEE 1052
brings insights on differentgrasps, which results in a classification of grasps into three types. The classification indicates how the contact forces can be decomposed corresponding to each type of grasp, and in which grasps the uncontrollable intemal forces may exist. Consequently, it provides a basis to formulate the distribution of contact forces and optimize the contact force by controlling joint torques of the fingers. In reference [ 111, we presented a new deftnition of power grasps and a classificationof grasps. In this paper, that investigation is generalized in two aspects. Firstly, new definitions of the subspaces are provided. Secondly, singular configurationsof grasps are considered. In addition, new examples are provided to demonstrate the decomposition of contact forces and the physical meaning of each component of a decomposed contact force. The dimensions of the subspace are determined with respect to the connectivity of the object. The dimensional equations obtained in this research are valid for both nonsingular configurations and singular configurations.
fp is a particular solution and fh is a homogeneous solution in the null space N(W) . Physically, $ is the component that generates a wrench balancing the external wrench and fh ,the component that produces zero resultant wrench, is called the intemal force. Since the contact force f is generated by the joint torque 2, i f f lies in the null space N(JT) it is impossible to generate a contact force by controlling the joint torque. Taking this into consideration, either $ or fh may contain two parts, one of which lies in the null space N(JT ) . 2.1 Definitions of subspaces
We decompose the contact force into four parts = $1 + 6 1 + $ 2
The first parts of the particular and homogeneous solutions resulting fiom the joint torque are called the active force and the controllableinternal force fhl . Because the second parts cannot be produced by the joint torques, they are called the passive force, G2 and the uncontrol-
2. Contact force decomposition
G1
In this section we characterizegrasps by investigating the space of contact forces. We shall classify grasps into three types and provide some examples for each type of grasp. Contact forces and moments between fmgers and the object are determined by the equilibrium equations of the object formulated as
Wf=w
fh2
lable internal force, fh2. Then, the space of contact forces can be decomposed into the following subspaces:
(1)
where f E R" is a vector consisting of all contact forces and moments, n is the number of constraints, w E R6 is the external wrench consisting of the external force and the moment applied to the object. W E R6xn is called the grasp matrix. To obtain Eq. (l), we assume f is the contact force applied to the fingers, while the reaction force - f is applied to the object. For the resulting contact force, the corresponding joint torques of the fingers are determined by
JTf = Z
NI (W) and N I (JT ) are the orthogonal complements of the N(W) and N( JT ) .According to the above where
definition, the four components of the contact force satisfy
(2)
where z E R m is the vector consisting of all joint torques, m is the number of joints and J E Rnm is the Jacobian matrix of the hand. The general solution to Eq. (1) is f = $ + fh ,where
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The dimension N, is called the indeterminacy because
2.2 Dimensions of the subspaces
U E N(WT)represents the motion of the object that rema& fiee by contact constraints [ 101. Using Eq. (6), we have
As shown in Fig. 1, r G R" represents the space of contact forces containing the null spaces N(W) and N(JT ),The intersection of these two sets is the subspace r,,.The remaining regions of the two null spaces are the
dimR(W)= 6- Ni
subspaces r h , and rp2, respectively.The subspace rpl is shown by the white region. The range spaces R ( W and
(7)
The range space R(J T ) is the space ofjoint torques whose orthogonal complement R I (J T ) contains all joint veloci-
R(JT)are subspaces of R6 and Rm.
ties that satisfy ~~q = 0,i.e. JQ= 0, so that q E N(J). Therefore, RI UT) = N(J). DefLning
The dimension N, is called the redundancy, because q E N(J) represents the motions of the joints that can be produced without affecting the motion of the object. Using Eq. (S), we obtain
dimR(JT)=m-N,
(9)
From Eq. (4), we have Figure 1. Subspaces of contact forces Fig. 1 denotes the dimensional relationships between the subspaces:
where
dim rpl+ dim rp2 = dim R(W) dimrpl + dimr,, = dimR(JT) dim rh2 + dim rhl= dim N(W) dimrh2+dimrp2= dimNUT)
Therefore r h , = N(A), and
The range space R(W) is the space of external wrenches whose orthogonal complement R*(W) contains all ob-
Substituting Eq. (7), (9) and (11) into ( 5 ) , we obtain
ject velocities that satisfy wTU= 0, where U E R6 is the vector consisting of the linear and angular velocities of the
dimr,, =m+6-rank(A)-NP -Ni dimrp2 = -m + rank(A) + N, dimr,, = -6 + rank(A) + N,
object. From Eq. (l), we know that wTU= 0 implies
WTU= 0,
so
that
U E N(WT).
There-
Eqs. (11- 12) are valid in the most general cases in which Ni > 0 , N, > 0 and a decreasedrank(A) may occur. Note that the sum of the dimensions of all subspaces must equal n because f E R".
fore, R'(W) = N(WT).Suppose
Ni = di"(WT)
(12)
(6)
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In order to see how the contact force c m be decomposed corresponding to different value of the connectivity of the object, in the following, we develop a relationship between the dimensions of the subspaces and the connectivity. The connectivity between two particular bodies in a kinematic system is the number of independent parameters necessary to specify the relative positions of the two bodies. We use N, to denote the connectivity of the object relative to the palm. N, can be determined by using the relationship
M=N,+N,
tive movements between the object and the fingers along the direction of contact normal cannot be obtained. If firiction is used as a contact constraint, it is possible that the tkiction is sufficient to prevent sliding at the contact points. Therefore, considering a rigid-body model for the handobject system and assuming no slippage between the object and the fingers, the object and the fmgers must have the same velocities in the directions of contact constraints, i.e., xo = x,. Combining this equality with Eq. (16), we obtain
[WT -J[:]=OER”
(13)
where M is the mobility of the hand-object system which can be computed using the following [ 131:
This means the velocity space of
qT], belongs to the null
[UT
[W’ -I]. Therefore, the mobility of the system
equals the dimension of the null space, where 4 is the number of degrees of freedom @OF) of the ith joint, gj is the number of DOF of motion at the jth contact, and L is the number of independent loops in the system. The inequality in the latter relation results from the fact that constraints in the system may not be independent. Suppose each joint of fingers has one DOF. Because the number of contacts p and the number of independent loops are related by, p = L + 1, the mobility equation can be modified into
M=m+6-n
M = m+ 6- rank(A)
(17)
Note that, when A has full rank and M 2 0, Eq. (17) is equivalent to Eq. (15). In this context, Eq. (17) is more general than Eq. (15). The mobility obtained from Eq. (15) can be negative, which means it describes an over constrained system, whereas the mobility obtained from Eq. (17) is always nonnegative. Usually a stable grasp is required. Therefore, we consider grasps that possess form closure which requires Ni = 0, because if Ni > 0, the grasps cannot resist arbitrary external wrenches [ 131. We also assume N, = 0 if M I0. This assumptionis reasonablebecause redundancy has no practical advantage when motion of the object is not required. To see how the connectivity of the object directly relates to the dimensions of the subspaces, we suppose A has full rank. From Eqs. (10) and (15), it is clear that
(15)
where m is the number of joints and n is the number of contact constraints. Eq. (15) is valid only when constraints in the system are independent. To obtain a mobility equation which is valid when constraints in the system are not independent, we consider the following kinematic equations of the system [ 101,
I
where x, and xf E R” are velocities of the object and the fingers whose elements correspond to the elements of the contact force f. For instance, in the case of point contact without friction, each element of x, is the normal velocity of a contact point on the object. For point contact with friction, each element of 3, is one of the three components of the velocity of a contact point on the object, and itf is the correspondingvelocity of the fingers. For a rigid body, rela-
N c
dimr,, dimr,,
dimr,,, dimr,
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1
6 6
I
5 -1 N
C
l o
IC0
0
0 6
0
6 -N,
6
n-6
n-6
m
o
0
0
1
m -Nc
Combining the above equation with Eq. (15 ) and Eqs. (1112), we obtain the results shown in Table 1.
3. Classification of grasps Table 1 indicates that the dimensions of the subspaces vary with respect to the connectivity. The following analysis shows how the decomposition characterizes different &rasps. Case (i) N, = 6 In this case, as indicated by dimTpl = 6, the grasp can actively resist arbitrary extemal wrenches applied to the object by controlling the joint torques and, therefore, can generate arbitrary motions of the object. In other words, all the external wrenches are balanced by the joint torques. Moreover, all internal forces can be generated by controlling the joint torques, because dimrh2= 0. Fig. 2 shows a three-fingered hand using fingertips to grasp the object. Let us assume point contacts with friction and Ni = N,= 0. Each contact point results in three constraints so that n=9. Then using Eq. (15), N, = 6 , Therefore, arbitrary motion of the object can be obtained by controlling the joint variables.
trolling the joint torques. Fig. 3 is an example of this case. Two fingers have line contacts with the object at their last links from the palm and the third fmger has a point contact. Considering friction, each line contact results in five constraints, thus n=13. Then we obtain N, = 2 which means the object has two independent motions. In this example, only the rotation and the translation about and along the axis of the cylinder may be generated without violating the contact constraints, the other four motions of the object are constrained. This may happen as a transit state between the above type of grasps and the following during the operation of grasping and manipulation.
Figure 3. Grasp with 0 < N, < 6 Case (iii) N,= 0 In this case, the grasp can only passively resist arbitrary extemal wrenches applied to the object. In other words, no motion of the object is allowed by the grasp as indicated by dim rpl= 0. Moreover, all the intemal forces lie
in an m-dimensional subspace that are controllable. Fig. 4 is a grasp with three point contact and one line contact between the object and the fingers and palm. Considering friction, we have m=8, n=14 and N,= 0. Therefore, no motion of the object can be generated by the grasp.
Figure 2. Grasp with N, = 6 Case (ii) 0 < N,
