Load (N)
Figure 10. Tangential forces of the thumb and VF in the three-digit grasps as a function of
the load and friction, high (H) or low (L). The eight friction conditions were HHH, HLL,
HHL, HLH, LLL LHH, LHL, and LLH, where the letters correspond to the friction condition
462
Parallel Manipulators, Towards New Applications
for the thumb, index and middle fingers, respectively. The friction sets with the thumb at a
low friction contact (LLL, LHH, LHL and LLH) are printed with dotted lines. The solid lines
represent the tasks with the high friction contact at the thumb. In the left panel, LFE is the
local friction effect, i.e. the difference induced by the high or low friction contact at the thumb.
Two other smaller figure brackets show the synergic effects, i.e. the effect of friction at other
digits on the thumb force. The numbers in the bottom right insets are the regression
coefficients and intercepts (the regression model t
)
1
(
f = a + k
.
L
i
i
i
was used for
computations). Note the small values of the intercepts. Cf. the right and the left panels: the
thumb friction, H or L, induced opposite changes of the thumb and VF forces. (The figure is
from X. Niu, M. Latash, V.M. Zatsiorsky (2007) Prehension synergies in minimally
redundant grasps & the triple-product model of digit force control, Experimental Brain
Research, 98 (1): 16-28.)
Local and synergic reactions are not limited to adjustments to local friction. For instance,
similar reactions were observed when performers were holding a motorized handle while
the handle width was forcibly either increased or decreased (Zatsiorsky et al. 2006). Handle
expansion/contraction did not perturb the handle equilibrium; both the resultant force and
moment acting on the handle remained the same. However, when the handle width
increased each digit was perturbed (the length of the flexor muscle increased), and a
restoring force tending to return the digit to its previous position arose (the local digit force
adjustment). The local mechanisms, e.g. stretch reflexes, were directed to resist the imposed
digit displacement. These mechanisms violated the object equilibrium, whilst the synergic
force adjustments restored the equilibrium.
9. Principle of superposition in human prehension
The principle of superposition refers to decomposition of complex skilled actions into several
elemental actions, which can be controlled independently by several controllers. The
principle was first suggested in robotics (Arimoto et al. 2001; Arimoto & Nguyen 2001) and
was verified for the dexterous manipulation of an object by two soft-tip robot fingers. Such a
control can be realized via a linear superposition of two commands, one command for the
stable grasping and the second one for regulating the orientation of the object. In robotics,
such a decoupled control decreases the computation time.
When applied to multi-finger grasps in humans, the principle claims that at the VF level the
forces and moments during prehension are defined by two independent commands: “Grasp
the object stronger/weaker to prevent slipping” and “Maintain the rotational equilibrium of
the object”. The commands correspond to the two internal forces discussed previously, the
grip force and the internal moment. The effects of the two commands are summed up. The
validity of the principle was confirmed in a set of diverse experiments (for the review see
Zatsiorsky et al. 2004; see also Shim et al. 2005 and Shim & Park, 2007)). The principle allows
explaining the digit force adjustments to different factors such as (a) the load force and its
modulation associated with the handle acceleration; (b) the external torque and its
modulation; (c) the object orientation in the gravity field; (c) friction at the digit tips; and
some other variables. The variations of the above factors may require similar or opposite
adjustments. For instance, an increase of the object weight and a decrease in friction both
require a larger gripping force while, a decrease of the load and a decrease in friction require
opposite grasp force changes, a force decrease and increase, respectively. It has been
Human Hand as a Parallel Manipulator
463
suggested that the CNS responds to a mixture of similar or opposite requirements follows a
rule: Adjustment to the sum equals the sum of the adjustments (reviewed in Zatsiorsky& Latash
2008).
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!"#
$%&'(')*#%+#,-.)'.(#/.0.((1(#$.2'-3(.)%04#
Jing-Shan *hao, Fulei Chu and *hi-Jing Feng
Department of Precision Instruments and Mechanology,
Tsinghua University, Beijing 100084,
P. R. China.
"5#62)0%738)'%2#
This chapter focuses on the mobility analysis of spatial parallel manipulators. It first
develops an analytical methodology to investigate the instantaneous degree of freedom
(DOF) of the end-effector of a parallel manipulator. And then, the instantaneous
controllability of the end-effector is discussed from the viewpoint of the possible actuation
schemes which will be especially useful for the designers of the parallel manipulators. Via
comparing the differences and essential mobility of a set of underactuated, over actuated
and equally actuated manipulators, this chapter demonstrates that the underactuated, over
actuated and equally actuated manipulators are all substantially fully actuated mechanisms.
This work is significantly important for a designer to contrive his or her manipulators with
underactuated or over actuated structures.
Based on the analytical model of the DOF of a spatia ! $" "
!
#$ # 0
(16)
)! 3 I3&
where $ and !
$ are column vectors, # # '
$ , and I3 and ! 3 are 3* 3 identity and
(I3 ! 3%
zero matrices, respectively.
Similarly, if one gets a set of terminal constraints exerted to a rigid body, its free motion(s)
can also be solved through equation (16). Next, one can investigate the instantaneous
mobility of the end-effector of a parallel manipulator with equation (16).
!5"5#9:1#71;011#%+#+0117%<#%+#):1#127=1++18)%0#%+#.#-.0.((1(#<.2'-3(.)%0#
The free motions of the end-effector can be instantaneously expressed in a set of PlPcker
homogeneous coordinates in one Cartesian coordinate system. The main steps are:
!"#$%&'()*+,)'#)-'#.'/0*%,1#23%()/,*%)(#34#)-'#5*%'0,)*6#2-,*%(#
In general, any parallel manipulator can be decomposed into $! $ + "
1 kinematic chains
connecting the end effector with the base. In order to instantaneously analyze the mobility
properties of the end-effector, this section only establishes one absolute coordinate system.
After establishing the coordinate system, the Pl$cker homogeneous coordinates of all
kinematic pairs in a chain can be obtained. Group all of the kinematic screws of the same
chain to be %
$ ! % # ,
1 2 , ! , $" and solve the terminal constraint(s) r
$% with equation (16).
468
Parallel Manipulators, Towards New Applications
In fact, if all of the terminal constraints of the kinematic chains are gained, the constraints
exerted to the end-effector, denoted by '
$# , should also be obtained. The dimension of
constraint spaces spanned by the terminal constraints of kinematic chains can be simplified
as
1
) #
'
R'$( $# ,.
/
.
0
-
7"#831&'#)-'#9/''#:3)*3%;(<=# (
$# =#34#)-'#>%?@>44'6)3/#A*)-#>BC,)*3%#;!D<#
Naturally, the mobility properties of the end-effector is fully expressed by (
$# . Its number of
DOF can be expressed as:
(
*
R'$( 1
#
$ ,.
/
# 6 )
#
2
(17)
0
-
Now, the DOF of the end-effector of the parallel manipulator shown in Fig. 1 can be
instantaneously investigated with the above two steps. In this manipulator, the end-effector
1
+ + 2 3
+ has three identical PPRR kinematic chains connected with the fixed base. For the
sake of modelling, one can establish any Cartesian coordinate system for the manipulator.
Assume that the direction vector of the prismatic joint , ! % # 1, 2,3 is denoted by
%
"
/, # ! '% 1
%
.
-
1
" "
%
, the direction vector of the prismatic joint 0 ! % # 1, 2,3 is denoted by
%
"
%
1
/0 # ! '% 2
%
.
-
2
" "
%
, the rotational vector of the revolute joint
%
2
%
0 is denoted by
/ #
*
#
2
2
2
, the rotational vector of the revolute joint
!
/
/
. -
. -
' -
' -
' .
' .
0
,
0
! % 1 % 2 % 2 % 1 % 2 % 1 % 1 % 2 % 1 % 2