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Optimization and experimental verification for the antagonistic stiffness in redundantly actuated mechanisms: a five-bar example
Mechatronics 15 (2005) 213–238
Optimization and experimental verification for the antagonistic stiffness in redundantly actuated mechanisms: a five-bar example Sang Heon Lee a, Jae Hoon Lee b, Byung-Ju Yi Soo Hyun Kim a, Yoon Keun Kwak a
b,*
,
a
b
Department of Mechanical Engineering, KAIST, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, Republic of Korea School of Electrical Engineering and Computer Science, Hanyang University, 1271 Sa-1 dong, Ansan, Kyungki-do 475-791, Republic of Korea Received 19 February 2003; accepted 28 July 2004
Abstract A closed-chain mechanism having redundancy in force domain can produce an effective spring effect by proper internal load distribution. The so-called antagonistic stiffness is provided by redundant actuation in conjunction with nonlinear geometric constraints. The objective of this study is twofold. The first one is to propose a methodology for optimal kinematic design of antagonistic stiffness and the second one is to verify the phenomenon of the antagonistic stiffness through experimentation. In this work, a five-bar mechanism that can modulate antagonistic stiffness by using redundant actuators is employed as an exemplary system. The optimal structure of the five-bar mechanism that can maximize efficiency in generation of antagonistic stiffness is evaluated and analyzed. A measure of stiffness isotropy and the gradient property of the measure are employed as the performance indices in optimization. To deal with multi-criteria based design, a composite design index based on max–min principle of fuzzy theory is used as an objective function. Two optimization results are obtained. Furthermore, a five-bar mechanism has been developed for experimental verification. The
*
Corresponding author.
0957-4158/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.07.008
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phenomenon of antagonistic stiffness has been successfully demonstrated by free vibration and indirect force control experimentation. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Stiffness (or compliance) has been one of fundamental properties in the analysis and control of general dynamic systems. Specifically, compliance control capability is an important feature for robot interacting with environment or another robot [1– 17]. Salisbury [2] presented a compliance control scheme in the operational space. Once operational compliance matrix (or operational stiffness matrix [Kuu]) is designed, the joint space compliance matrix (or joint stiffness matrix [K//]) is obtained T as ½K // ¼ ½Gu/ ½K uu ½Gu/ , where ½Gu/ denotes the Jacobian relating the joint velocity to the operational velocity. Based on this principle, many compliance control algorithms have been reported in the field of redundant manipulators [12–15] and multiple arms [19] holding a common object. In Salisbury [2] and other related literatures [1–14], evaluations of the stiffness (or compliance) matrix of a robot have been limited to unloaded cases (i.e., zero externally applied loads), which always produce symmetric stiffness matrices. Yi et al. [10] made a geometric analysis on the additional stiffness term due to the external forces and pointed out that the physical stiffness matrix can be asymmetric for loaded cases, which yields difficulties in symmetric stiffness modulation [31,32]. The asymmetry obtained from externally applied load components is coincident to the analyses made by Griffis and Duffy [17], Chen and Kao [16]. Choi et al. [15] extended this idea to stiffness analysis of multi-fingered hands. As a useful application of the additional stiffness term, Hogan [3] initially addressed the concept of antagonistic stiffness and followed by Tong and Somerset [18]. When redundant actuators activate the mechanism antagonistically, an effective spring-like property can be produced by antagonistic internal load with conjunction with complex nonlinear geometry of the mechanism. They claimed that the antagonistic stiffness could be modulated arbitrarily by controlling actuation inputs without feedback control. Yi and Freeman [30–33] proposed a general methodology for analysis and application of the antagonistic stiffness created by antagonistic redundant actuation. This property has been also understood as an inherent property in the human-body that is usually driven by abundant inputs [31,39]. Even though a general mathematical formulation has been made on the concept of antagonistic stiffness, optimal design and experimental verification of the antagonistic stiffness has not been carried out yet. Thus, in this work, we investigate an optimal kinematic structure, specifically for a five-bar mechanism that efficiently generates antagonistic stiffness via redundant actuation. To do this, based on the antagonistic stiffness model, a stiffness modulation index is proposed, which represents the isotropy of the transformation matrix between the actuation efforts and
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the antagonistic stiffness. The transform matrix is the second derivative of the kinematic constraint [31,32]. Also in order to obtain the global isotropic characteristic, a global design index is employed which is defined as the average value of the isotropic indices calculated over the workspace. And a gradient design index is proposed to obtain uniform distribution of the isotropic index over the workspace. To cope with these design indices in optimization, composite design index based on max–min principle of fuzzy theory [36] is defined and used as objective function. The optimization is performed in two paths. One is optimizing X-directional stiffness modulation, and the other is corresponding to Y-directional stiffness. The result from optimization is suitable for generating stiffness more effectively as well as general force/velocity transmission. A five-bar mechanism was developed based on the kinematic parameters obtained by optimization. The characteristic of the antagonistic property will be verified through experimentation.
2. Concept of antagonistic stiffness When the number of actuation input is greater than the mobility, the system is called a redundantly actuated system. Structure of a system with redundant actuation usually has closed-chain mechanism in which many potential input locations exist. The redundantly actuated system can employ many optimal load distribution schemes [20–27] as well as beneficial internal load generation schemes [28–35]. Fig. 1 represents a four-bar mechanism that has mobility one. The four-bar requires at least one actuator to control the motion. This mechanism is redundantly actuated when it has more than two inputs. It is remarked that specially, when the mechanism is activated antagonistically by two joint actuators, an effective springlike property can be produced by antagonistic internal load with conjunction with complex nonlinear geometry of the mechanism. The antagonistic stiffness can be modulated arbitrarily by controlling actuation inputs without feedback control [3,32]. In a state of static equilibrium formed by antagonistic activation of the input load, given a disturbance to the system, the system oscillates about the equilibrium position like a physical spring.
K Fig. 1. Four-bar driven by redundant actuators.
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3. Kinematic modeling 3.1. Open-chain kinematics The five-bar mechanism shown in Fig. 2 has one closed-kinematic chain that consists of two open chains by cutting the middle joint. It is identical with cooperating two robots handling a common object. The end-effector positions u of the two open chains are identical, and they are described by l1 c1 þ l2 c12 x l5 þ l4 c5 þ l3 c45 ¼ ¼ : ð1Þ y l1 s1 þ l2 s12 l4 s5 þ l3 s45 And the orientation angle of link 2 is u ¼ h1 þ h2 ¼ h3 þ h4 þ h5 þ p:
ð2Þ
Adopting the standard Jacobian representation for the velocity of dependent output vector u 2 RN in terms of independent input coordinates r /_ 2 RM of rth open-chain, one has h i _ r ¼ 1; 2: u_ ¼ r G u/ r /: ð3Þ Here, h
u rG /
i
ou ¼ 2 RN M or /
ð4Þ
_ is the first-order kinematic influence coefficient (KIC) relating u_ to r /. Acceleration vector € u also can be represented in terms of input coordinates [37,32] h i h i € þ r /_ T r H u r /; _ € ð5Þ u ¼ r G u/ r / //
Fig. 2. Five-bar mechanism.
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where the element of the second-order KIC array h i o2 ui u ¼ : r H // ijk or /j or /k
h
u r H //
i
217
2 RN MM is defined as ð6Þ
3.2. Internal kinematics In redundantly actuated systems, the independent coordinates can be selected arbitrary. Since the mobility of the five-bar mechanism is 2, at least two actuation inputs are required to operate the system. In this work, the minimum independent (input) coordinates and dependent coordinates are selected as /a ¼ ðh1 ; h5 Þ
T
ð7Þ T
/p ¼ ðh2 ; h3 ; h4 Þ :
ð8Þ
From the equivalent velocity relationship of the two open-chains h i h i _ u_ ¼ 1 G u/ 1 /_ ¼ 2 G u/ 2 /;
ð9Þ
the velocity relation between independent and dependent coordinate set can be obtained by rearranging the above equation: ½A/_ p ¼ ½B/_ a ; where ½A ¼
hh
u 1 G/
i :2
ð10Þ
h i h i i ; 2 Gu/ ; 2 Gu/ 2 R33 ; :1
:2
h h i h i i ½B ¼ 1 Gu/ ; 2 Gu/ 2 R32 : :1
:3
ð11Þ ð12Þ
In Eq. (11), ½r Gu/ :i is ith column vector of the matrix. The velocity vector for the dependent coordinate set is obtained by premultiplying the inverse of [A] to Eq. (10): ð13Þ /_ ¼ ½A1 ½B/_ ¼ ½G p /_ ; p
½Gpa
a
a
a
32
2 R and the velocity vector of total systemÕs joints can be expressed in where terms of the velocity vector of the independent joint set by ½I ð14Þ /_ ¼ ½G /a /_ a ¼ /_ : ½G pa a € to /_ can be easily obThe second-order KIC array ½H paa 2 R322 relating / p a tained in a similar manner [32]. 3.3. Forward kinematics Since the joints of the rth open-chain is composed of some of the independent and dependent joints, r /_ can be expressed in terms of the total systemÕs independent joints by
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_ ¼
r/
r
G /a /_ a ;
ð15Þ 52
where ½r G /a is formed by rearranging rows of ½G /a 2 R . Thus the forward kinematics for the common object is obtained by embedding the first-order KIC into one of the rth pseudo open-chain kinematic expression as follows: h i h i u_ ¼ r G u/ r /_ ¼ r G u/ r G /a /_ a ¼ ½G ua /_ a ; ð16Þ where ½Gua 2 R22 . Using the same augmentation method employed in Eq. (16) evaluation of the second-order forward kinematic array ½H uaa 2 R222 is also straightforward. It is given by T
½H uaa ¼ ½r G u/ ½r H /aa þ ½r G /a ½r H u// ½r G /a
ð17Þ
where ÔÕ means a generalized scalar dot product [32,37]. 3.4. Statics of the five-bar mechanism According to the duality existing between the velocity and force vector, the force relation between the independent and dependent joints can be described. So the effective load referenced to the independent joints is given by T a ¼ T a þ ½G pa T T p ¼ ½G /a T T /
ð18Þ
where Ta, Tp, and T/ are force or torque at the independent joints, the dependent joints, and the total systemÕs joints, respectively. In static equilibrium, the torque at the independent coordinates can be expressed as T T a ¼ G Aa T A ¼ 0; ð19Þ where TA is the torque at the actuated joints /A. ½G Aa 2 RA2 , where A denotes the number of active joint, denoting the first-order KIC relating /_ A to /_ a is obtained by rearranging the rows of ½G /a .
4. Modeling of antagonistic stiffness Given a disturbance to the system under static equilibrium by redundant actuation, a spring-like behavior occurs to the system. Assuming that the magnitude of TA remains constant, the effective stiffness matrix ½K aa 2 R22 with respect to the independent coordinates is obtained as ½K aa ¼
oT a ¼ ðT A ÞT H Aaa o/a
ð20Þ
by differentiating Eq. (19) with respect to the independent coordinate set /a. It is observed from (20) that the so-called antagonistic stiffness is provided by redundant
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actuation in conjunction with nonlinear geometric constraints denoted by A H aa 2 RA22 . The stiffness relationship ½K uu 2 R22 between the output coordinates and the independent coordinates is given by [2] T
½K uu ¼ ½G au ½K aa ½G au : ½G au
22
ð21Þ ½G ua
where 2 R denotes the inverse of given in Eq. (16). Substituting Eq. (20) into Eq. (21) yields the following stiffness matrix expressed with respect to the output space. T ½K uu ¼ ðT A Þ H Auu ; ð22Þ A T where H uu ¼ ½G au ½H Aaa ½G au 2 RA22 is obtained by plane-by-plane multiplication [33,37]. An alternative form of Eq. (22) given in a matrix form is described by ð23Þ K u ¼ H Au T A ; where Ku is consists of the upper diagonal elements of [Kuu] and ½H Au 2 R3A is also obtained by collecting the upper diagonal columns of the three-dimensional array ½H Auu , which are defined as follows: K u ¼ ðK xx ; K xy ; K yy Þ
T
2 A 3 H uu ;xx 7 A 6 6 H A 7 H u ¼ 6 uu ;xy 7: 4 5 H Auu ;yy
ð24Þ
ð25Þ
Necessary (but not sufficient) conditions for full stiffness generation are derived in [32]: a closed chain is capable of full stiffness generation only if it satisfies NA P D þ N
ð26Þ
where N is DOF (system mobility), D is number of independent stiffness elements, and NA is number of actuated joints and NC ¼ ðIC LCÞ P D;
ð27Þ
where IC is number of independent constraint equations, NC is number of independent nonlinear constraint equations, and LC is number of independent linear equations. The five-bar mechanism has only one independent closed chain and thus it has two independent nonlinear constraint equations, so only two elements of Ku can be independently modulated. According to the condition of Eq. (26), at least four actuators are needed to modulate two stiffness elements. Four joints except the third joint are actuated in the given five-bar. There are five potential input locations in the five-bar mechanism. Thus, five sets of actuators exist. Selection of an actuator set can be based on analysis of the geometry associated with antagonistic stiffness. An
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optimal actuator set can be found from this process. However, in the following analysis, we fix the location of the actuator set and seek the optimal kinematic parameters that maximize the antagonistic stiffness generation.
5. Kinematic optimization for antagonistic stiffness 5.1. Stiffness modulation index 5.1.1. Local design index With consideration of the operational force vector and the independent joint vector given by T
T u ¼ ½G au T a : Eq. (19) can be also expressed as T T u ¼ G Au T A ¼ 0; where
G Au
T
ð28Þ
T ¼ ½G au G Aa 2 R2A
and T u is the effective load at the end-effector coordinates. Eq. (28) representing static equilibrium is the primary task for antagonistic stiffness generation. For Eq. (28), the general solution for TA is obtained as T þ T T A ¼ ½I G Au G Au ð29Þ e1 ; where e1 is an arbitrary vector and the superscript Ô+Õ means pseudo-inverse. Eq. (29) denotes the secondary subtask. Substituting Eq. (29) into Eq. (23) yields T þ T A K u ¼ H u ½I G Au G Au ð30Þ e1 and ½c H e1 ¼ K u ;
ð31Þ
where
T þ T A c ½ H ¼ H u ½I G Au G Au : 2 R2A :
ð32Þ
If the exact solution of e1 exists for Eq. (31), it implies that the second subtask associated with stiffness generation can be realized. The general solution of e1 is þ þ H K u þ ð½I ½c H ½c H Þe2 ; e1 ¼ ½ c
ð33Þ
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where e2 is an arbitrary vector. Then the solution of TA is represented as T þ T þ T A ¼ ½I G Au G Au ½c H Ku T þ T þ þ ½I G Au G Au H Þe2 : ð½I ½c H ½c
221
ð34Þ
The first term of right-hand side of the above equation is reduces to ½c H þ K u , by using condition of the symmetry and idempotency of T þ T AA A A ½I G u Gu 2 R . We can realize another subtask (the third-priority subtask) by e2 if the system has more redundancy. Assuming that there is no other subtask (e2 = 0), the solution becomes þ T A ¼ ½c H K u:
ð35Þ
So the relationship between Ku and TA which satisfies Eqs. (23) and (28) is T þ T K u ¼ H Au ½I G Au G Au ð36Þ TA: Now, the stiffness modulation index is defined by isotropic index of ½c H 2 RA3 , which shows isotropy characteristic in the stiffness modulation process: rmin ; ð37Þ j¼ rmax where rmin and rmax are the minimum and maximum singular values of the matrix ½c H , respectively. 5.1.2. Global design index The design index j defined in Eq. (37) is a local index that varies as the pose of the five-bar mechanism changes, so it cannot represent overall characteristics of the mechanism throughout the workspace. Therefore a design index that represents the global feature should be defined and used in optimization. The global design index that represents an average value of j over the workspace is defined as R j dW K ¼ RW ; ð38Þ dW W where W denotes workspace. Since the value of the local design index changes as the configuration of mechanism changes, gradient design index is also considered so that final design can have even distribution of the local design indices over the workspace. The gradient design index represents the change rate of the local design index [27]. The gradient design index is defined as the maximum of the differences among the local design indices calculated at adjacent points throughout the workspace. The smaller the global gradient design index, the more uniformly distributed the design index over the workspace, therefore the gradient design index should be minimized.
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5.1.3. Composite design index The design of a mechanism can be made based on any particular criterion. However, the single criterion-based design does not provide sufficient control on the range of the design parameters involved. Therefore, multi-criteria based design has been proposed. But various design indices are usually incommensurate concepts due to differences in unit and physical meanings, and therefore should not be combined with normalization and weighting functions unless they are transferred into a common domain. In other words, quantitative combination should be avoided. Instead, these design indices should be combined qualitatively. To consider these facts, we employ a composite design index. As an initial step to this process, preferential information should be given to each design parameter and design index. Then, each design index is transferred to a common preference design domain that ranges from zero to one. Here, the preference given to each design criterion is subjective to the designer. Preference can be given to each criterion by weighting. This provides flexibility in design. For K, the best preference is given the maximum value, and the least preference was given the minimum value of the criterion. Then the design index is transferred into common preference design domain as below e ¼ K K min ; K ð39Þ K max K min where ÔÕ implies the index which is transferred into common preference design domain. Reversely, if the best preference is given the minimum value, and the least preference is given the maximum value of the criterion, like the gradient design index Kd, the design index transferred into common preference design domain is given by d d e d ¼ K max K : K ð40Þ d d K max K min Note that each transferred design index is constructed such that a large value represents a better design. A set of optimal design parameters was obtained based on max–min principle of fuzzy theory [6]. Initially, minimum values among the design indices for all set of design parameters are obtained, and then a set of design parameters, which has the maximum value of the minimum values already obtained, is chosen as the optimal set of design parameters. Based on this principle, the composite design index (CDI) is given by a
d b
e ; ðK e Þ Þ: CDI ¼ minð K ð41Þ The upper Greek letter is the degree of weighting, and large value implies large weighting. The weightings are all set to ones in this work. 5.2. Optimization Optimization problem is formulated as follows: The objective is to evaluate optimal link lengths, which maximizes CDI. The design variables are the ratios of link lengths to the base link length l5, and they are defined by
S.H. Lee et al. / Mechatronics 15 (2005) 213–238
xi ¼ li =l5 ;
223
i ¼ 1; 2; 3; 4:
ð42Þ
Constraints to the design variables are 0:5 6 xi 6 1:5
ð43Þ
l5 ¼ 0:2 m:
ð44Þ
and
Two objective functions are defined and used in optimization. One noted as CDIx corresponds to the modulation of the X-directional stiffness. It contribute to enhance the isotropic characteristic for Kxx and Kxy. The other noted as CDIy corresponds to modulation of the Y-directional stiffness. It contribute to enhance the isotropic characteristic for Kxy and Kyy. Three numerical methods are used to deal with multi-variable nonlinear optimization problem with constraints. The exterior penalty function method is employed to transform the constrained optimization problem into an unconstrained problem. PowellÕs method is applied to obtain an optimal solution for the unconstrained multivariable problem, and quadratic interpolation method is used for unidirectional optimization [38].
Optimized Shape 0.4 0.35 0.3 0.25
Y
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.1
0
0.1
0.2
X Fig. 3. Optimized shape of Design 1.
0.3
0.4
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S.H. Lee et al. / Mechatronics 15 (2005) 213–238
Optimization result for modulation of the X-directional stiffness is given as ¼ 0:697434392; ¼ 1:250000000; ¼ 1:487984717; ¼ 0:729829688
l1 =l5 l2 =l5 l3 =l5 l4 =l5
and the shape of optimized mechanism (Design 1) is shown in Fig. 3. Optimization result for modulation of the Y-directional stiffness is given as ¼ 0:750000100; ¼ 0:750000050; ¼ 0:693700249; ¼ 1:499999810
l1 =l5 l2 =l5 l3 =l5 l4 =l5
and the shape of optimized mechanism (Design 2) is shown in Fig. 4. As shown in Fig. 3, the length of distal floating link is longer than that of the link attached to the base. This structure is suitable to translate the torque given at the two base joints to the forces at the end-effector. In Fig. 4, conversely, the length of right connecting link shows extreme value, and this represents bad force transmission from base actuator to end-effector. By intuition, the shape of the optimized fivebar should be symmetric when solely considering the isotropy in optimization. However, both optimization results show unsymmetrical configurations. It is remarked Optimized Shape 0.35
0.3
0.25
Y
0.2
0.15
0.1
0.05
0
-0.05 -0.15
-0.1
-0.05
0
0.05
0.1
0.15
X Fig. 4. Optimized shape of Design 2.
0.2
0.25
0.3
S.H. Lee et al. / Mechatronics 15 (2005) 213–238
225
that the asymmetry is caused by consideration of the gradient characteristic of the isotropic index in optimization. The contours of stiffness modulation index for Design 1 are shown in Fig. 5. As shown in Fig. 5, jx is high at the center of workspace. Isotropies for Y-directional
Fig. 5. Isotropic index of Design 1. (a) Distribution of jx. (b) Distribution of jy.
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S.H. Lee et al. / Mechatronics 15 (2005) 213–238
stiffness modulation are also high over most workspace. This factor is advantageous to operate the mechanism in both x and y directions. Fig. 6 shows the contours of stiffness modulation index for Design 2. The workspace is narrow and the peak is not on the center of workspace. Therefore this design can not be said to be useful. Contrastively the workspace of Design 1 has shape of a fan and is close to a diamond, and it is another advantage. As shown in Fig. 6(a), very low isotropy is detected on the middle of the workspace. Although Design 2
Fig. 6. Isotropic index of Design 2. (a) Distribution of jx. (b) Distribution of jy.
S.H. Lee et al. / Mechatronics 15 (2005) 213–238
227
is optimized to maximize Y-directional stiffness modulation effect only not to optimize X-directional stiffness modulation, but this design is unfit for general purpose. Shape of optimization result for maximizing isotropy of Jacobian ½G au and minimizing the gradient of its isotropy also shows like that of Design 1, though the result is not represented here. Therefore, Design 1 is suitable for both general operational purpose and antagonistic stiffness generation.
6. Experimental verification 6.1. Feed forward frequency modulation When the stiffness of a system can be modulated by redundant actuation, the frequency of the vibration motion of the system also can be modulated. To do this, the dynamics of the system, such as inertia matrix should be known, because the frequency is determined by the inertia and the stiffness of the system. Assuming that the system is in a state of static equilibrium, the velocity at the endeffector can be neglected. u_ ¼ 0:
ð45Þ
The dynamic equation of the system is simplified as ½I uu € u ¼ T u ;
ð46Þ
where ½I uu 2 R . When a small displacement from the neutral position un is occurred, the effective force T u at the end-effector frame will be changed, and Eq. (46) will be modified to 22
u dT u ¼ 0: ½I uu d€
ð47Þ
When the actuation input is determined to modulate the stiffness at the end-effector coordinate frame, then the change of the effective force is dT u ¼ ½K uu du; where the stiffness matrix at the end-effector frame is T ½K uu ¼ ðT A Þ H Auu :
ð48Þ
ð49Þ
So Eq. (47) is rearranged as 1
d€ u þ ½I uu ½K uu du ¼ 0:
ð50Þ
The multiplied matrices ½I uu 1 ½K uu is defined as the frequency matrix ½W uu 2 R22 whose elements are " 2 # xxx x2xy kxx kxy ¼ : ð51Þ kyx kyy x2yx x2yy The frequency matrix is also defined as a function of the actuation input because of the stiffness matrix is determined by the actuation inputs.
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n o 1 1 T ½W uu ¼ ½I uu ½K uu ¼ ½I uu ðT A Þ H Auu ¼ ðT A ÞT ½I uu 1 H Auu ¼ ðT A ÞT W Auu :
ð52Þ
In the above equation, the multiplication of the inverse of inertia matrix and the second-order KIC ½H Auu 2 R4A is operated in a plane-by-plane fashion. Eq. (52) can be rewritten in a matrix form as follows: 2 A 3 0 1 W uu ;11 kxx 6 A 7 7 B k C 6 W uu ;12 7 B xy C 6 W u ¼ B C ¼ 6 A 7T A ¼ W Au T A : ð53Þ 7 @ kyx A 6 W uu ;21 5 4 A kyy W uu ;22 The above equation can be obtained similar to (23). If it is desired that the motion along the x-direction vibrate with the angular velocity of xxd, then the following two equations should be satisfied. xxx ¼ xxd
and
xxy ¼ 0:
ð54Þ
Then the dynamic equation along the x-directional coordinate is d€ ux þ x2xd dux ¼ 0
ð55Þ
and a solution of dux is dux ¼ Ax sinðxxd tÞ:
ð56Þ
So the motion along the x-direction becomes sinusoidal curve with the frequency of fx ¼ xxd =2p:
ð57Þ
The procedure of determination of the actuation inputs is as follows: From Eq. (53), the actuating torques must satisfy A W ux T A ¼ W xd ; ð58Þ where ½W Aux 2 R2A that is composed of the first and second rows of is the matrix T ½W Au , and W xd ¼ x2xd 0 . The state of static equilibrium also must be satisfied. A T G u T A ¼ 0: ð59Þ The number of solution of the Eq. (59) is infinite. The general solution is obtained using pseudo-inverse as T þ T T A ¼ ½E4 G Au G Au ð60Þ e1 ; where e1 is an arbitrary vector. The general solution must satisfy Eq. (60). T þ T A W ux ½E 4 G Au G Au e1 ¼ W xd :
ð61Þ
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229
The solution of unknown vector e1 for above equation is obtained also using pseudoinverse. c þ W xd þ f½E 4 ½ W c þ ½ W c ge2 ; e1 ¼ ½ W
ð62Þ
where
T þ T A c ½ W ¼ W ux ½E 4 G Au G Au 2 RAA :
ð63Þ
e2 is arbitrary vector, but it is set to zero because additional actuation redundancies do not exist in the system. Then the solution of TA is T þ T c þ W xd T A ¼ ½E 4 G Au G Au ð64Þ t½ W and can be simplified into c þ W xd ; T A ¼ ½W ð65Þ T þ T because the matrix ½E 4 G Au G Au 2 RAA is symmetric and idempotent. When the y-directional motion is modulated with the angular velocity of xyd, Eq. (58) should be changed into ½W Auy T A ¼ W yd ;
ð66Þ
where ½W Auy 2 R2A that is composed of the third and forth rows of is the matrix T ½W Au , and W yd ¼ 0 x2yd . 6.2. Indirect force control algorithm When the stiffness of a system can be modulated by redundant actuation, a direct force control scheme can be also implemented. The previous force control scheme is based on high position feedback or a direct force feedback using a force sensor attached at the end effector of a robot manipulator. Due to the nature of feedback scheme, the inevitable delay combined with high feedback gain often exhibits unstable phenomenon such as chattering. The proposed scheme will be based on feed forward scheme in which the operational stiffness is generated by using redundant actuation. Thus, more precise force control can be possibly achieved. When a robot manipulator is driven by abundant actuators and is contacting with the environment, the effective load expressed in the independent coordinates is given as T T a ¼ G Aa T A þ ½G ua T T ex ð67Þ u ¼ 0: Taking derivative with respect to the independent coordinates results in an effective stiffness model, given by T T T u T ½K aa ¼ ðT A Þ H Aaa þ ðT ex ð68Þ u Þ ½H aa :
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S.H. Lee et al. / Mechatronics 15 (2005) 213–238 T
Now, using the effective stiffness relation given by ½K aa ¼ ½G ua ½K uu ½G ua , Eq. (68) can be rewritten as T T T T u T ðT A Þ H Aaa ¼ ½G ua ½K uu ½G ua þ ðT ex ð69Þ u Þ ½H aa : The last term of Eq. (69) denotes an additional stiffness term due to externally applied load T ex u . Salisbury [2] ignored this additional term in usual force control scheme, and analysis and consideration of this term have been conducted [9,10]. Rearranging Eq. (69) as a vertor form, we have ½H uA ðT A Þ ¼ K:
ð70Þ
where ½H uA 2 RBA and K 2 RB (B denotes the number of operational stiffness elements in control) Then, if we desire to modulate a stiffness characteristic so that a proper force control can be made in certain direction, we can specify a proper stiffness for each direction. In order to decouple the action, the coupling stiffness element is set as zero. Also, since the feed forward stiffness modulation is made under static equilibrium, the equilibrium equation given by A T G u T A ¼ 0; ð71Þ should be combined with Eq. (70). Assuming that only x-directional stiffness elements (i.e., kxx and kxy) is to be controlled, the final form is now given as 2 T 3 2 3 0 G Au 6 7 7 6 ½H u 7T A ¼ 6 ð72Þ 4 kxx 5: 4 A 1; 5 u kxy ½H A 2; Indirect force control is performed in such a way that when a robot contacts an environment and plans to control a force along the surface normal direction, an offset displacement between the actuator end point and the desired position of robot is multiplied by the stiffness element of the surface normal direction.
7. Experimental verification 7.1. Experimental setup A five-bar for experimental verification of the antagonistic property is developed as shown in Fig. 7. The system is constructed as three main parts. The first is the fivebar mechanism made of aluminum and steel, the second is AC servomotors and their drives, and the third is a PC and interface system for control. The links of the fivebar are made of aluminum, and the end-effector contacting with environment has the shape of a disk and can rotate freely to remove the sliding friction effect. Kinematic and dynamic parameters for each link are summarized in Table 1. The position of mass center and the mass moment of inertia are calculated from 3D CAD data.
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231
Fig. 7. Five-bar system.
Table 1 Kinematic and inertial parameters of links
Link 1 Link 2 Link 3 Link 4 End-effector
Length (m)
Mass center (m)
Mass (kg)
Moment of inertia (E 3 kg m2)
0.150 0.300 0.300 0.150 0.0
0.0891 0.1557 0.1513 0.0891 0.0
0.3642 0.6123 0.6012 0.3642 0.0632
0.8741 5.5851 5.5479 0.8741 0.0293
Actuator torque at motor is transmitted to the rotating joints by timing belt and pulley system with the reduction ratio of 2.6. We employed AC servomotor whose torque can be directly controlled. PC is used to control the system, and the control algorithm is programmed by using LabWindows on the MS Windows environment. 7.2. Experimental results 7.2.1. Free vibratory motion Experimentation has been performed to corroborate the algorithm proposed in the previous section. Four actuators activate the five-bar mechanism. If the motions in the X- and Y-directions are to be controlled such that the natural frequency for each motion is given xxx ¼ xxd xxy ¼ 0
or
xyy ¼ xyd xyx ¼ 0
;
respectively, the actuation input to satisfy this should be calculated. Once the calculated input load is given to the system, the system maintains a static equilibrium. At the moment that a displacement is given to the end-effector, the end-effector initiates
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a vibratory motion like a pendulum. The characteristic of this motion is measured by using encoders attached to the joints of the systems. A damped vibration is observed. This comes from the damping effect caused by frictions existing at the joints and timing belts. The dynamic behavior of the measured vibratory motion can be characterized by the natural frequency and the damping ratio. They can be estimated by comparing the experimental data to the analytic formulation of one degree-of-freedom vibratory motion, given by pffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ Ae1xn t sin xn 1 12 t þ / ;
ð73Þ
where xn and 1 denoting the natural frequency and the damping ration, respectively, are estimated by using the curve-fitting technique based on the experimental data. T T Initially, the equilibrium position is given as un ¼ ð xn y n Þ ¼ ð 0:1 0:3 Þ m. Fig. 8 denotes the vibratory motion when the frequency of the X-directional motion is given p rad/s. The dashed line and the solid line represent the experimental data and the model obtained by the curve fitting, respectively. The natural frequency obtained by the analytic model 3.41 rad/s, which is about 8.5% higher that that of the desired frequency. Fig. 9 shows the plot for 4.0 rad/s. The frequency calculated by
X-directional motion 0.35 0.3
Displacement (m)
0.25 0.2 0.15 0.1 0.05 0 -0.05 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 8. X-directional motion with the frequency of p rad/s.
4.5
5
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233
X-directional motion
0.3 0.25
Displacement (m)
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0
1
2
3
4
5
6
Time (s) Fig. 9. X-directional motion with the frequency of 4.0 rad/s.
Y-directional motion
0.45
Displacement (m)
0.4
0.35
0.3
0.25
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 10. Y-directional motion with the frequency of p rad/s.
4.5
5
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Y-directional motion
0.45
Displacement (m)
0.4
0.35
0.3
0.25
0.2 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 11. Y-directional motion with the frequency of 4.0 rad/s.
curve fitting is 4.23 rad/s, which is also 5.7% higher than that of the desired one. In general, the system shows increased stiffness and more damping. The Y-directional motion shows the same trend. Figs. 10 and 11 denote the plots for Y-directional motion with p rad/s and 4.0 rad/s. The frequencies for these cases are calculated as 3.48 and 4.24 rad/s, respectively.
Fig. 12. Experimental setup.
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235
Overall, the experimentation result exhibits the phenomenon of the antagonistic stiffness correctly. However, both the stiffness and damping of the prototype are a 1
0
Ns
-1
-2
-3
-4
-5 0
0.5
1
1.5
2
(a)
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
sec 1
0
Ns
-1
-2
-3
-4
-5 0
(b)
0.5
1
1.5
2
2.5
sec
Fig. 13. Experimental result of force control. (a) Without including the effect due to external force. (b) With including the effect of external force.
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little bit higher than the desired ones of the system. This is mainly due to the tension of the timing belts that connect the floating joints to the actuators placed on the base location. 7.2.2. Indirect force control Fig. 12 shows the experimental setup for force control. A force/torque sensor is attached on the wall. The trajectory of the five-bar mechanism is given to the x-direction. The offset distance denoted by Dx multiplied by the stiffness in the x-direction becomes the force exerted on the wall by the five-bar mechanism. The magnitude of the x-directional stiffness is given as Kxx = 50 N/m, and the offset distance is given as 0.05 m in 2 s. Therefore, at last 2.5 N should be exerted in the x-direction. The stiffness is created by antagonistic redundant actuation and the additional stiffness term (Eq. (69)) due to the external force is also taken into account. Fig. 13 shows the experimental result. Fig. 13(a) denotes the case that the additional stiffness due to the external force is not being considered, while Fig. 13(b) shows the case that the additional stiffness due to external force is not being considered. It is remarked that the additional term due to the external force should not be ignored for precise force control and that in Fig. 13(b) the measured force converges to 2.5 N, which is identical to desired force value. The special feature of antagonistic stiffness is that it is created in a feed forward fashion, but not feedback fashion. Thus, the inherent instability problem reported in many previous force control implementation, which is based on feedback scheme, can be avoided.
8. Conclusions The concept of antagonistic stiffness, resulting from antagonistic redundant actuation, has been studied extensively. However, previous works only have dealt with the theory and simulation study about this concept. The contribution of this study is twofold; methodology for optimal kinematic design of antagonistic stiffness and experimental verification of antagonistic stiffness. A five-bar mechanism that can modulate antagonistic stiffness by using redundant actuators is employed as an exemplary system. Optimization to obtain uniform stiffness characteristic has been carried out. And based on the optimal kinematic parameters, a five-bar mechanism has been developed for experimentation. The phenomenon of antagonistic stiffness has been successfully demonstrated by free vibration experimentation and the effective antagonistic stiffness generated by redundant actuation was also employed to indirect force control. Even though the proposed scheme for antagonistic stiffness generation was examined though a planar 2 degree of freedom device, the proposed scheme can be generalized to general redundantly actuated mechanisms. The fundamental requirement for antagonistic stiffness generation is given by the necessary conditions as noted in Eqs. (26) and (27). The mathematical tools for actual implementation are identical to those given in Sections 5 and 6. More in-depth analysis on the second-order kine-
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237
matic influence coefficient denoted as three dimensional array [H] will be needed for spatial type closed chain mechanisms [32,37].
Acknowledgement This study was supported by a grant of the Korea Health 21 R&D Project, Ministry of Health & Welfare, Republic of Korea. (02-PJ3-PG6-EV04-0003).
References [1] Mason MT. In: Brady M, editor. Compliant motion–Robot motion. M.I.T Press; 1983. [2] Salisbury JK. Active stiffness control of manipulator in Cartesian coordinates. In: Proc IEEE 19th Conf on Decision and Control. 1980, p. 95–100. [3] Hogan N. Mechanical impedance control: An approach to manipulation: Part II—implementation. ASME Trans Dyn Sys, Meas Contr 1985;107:8–16. [4] Fresonke DN, Hernandez E, Tesar D. Deflection prediction for serial manipulator. In: Proc IEEE Int Conf on Robotics and Automation, (Philadelphia, PA), vol. 1. 1988. p. 482–7. [5] Goswami A, Peshikin M, Colgate JE. Passive robotics an exploration of mechanical computation. In: Proc IEEE Int Conf on Robotics and Automation, (Cincinnati, OH). 1990. p. 279–84. [6] Loncaric J. Geometrical Analysis of Compliant Mechanisms in Robotics. PhD Dissertation, Harvard University, 1985. [7] Patterson T, Lipkin H. A classification of robot compliance. In: Proc 21st ASME Mechanism Conf, (Chicago, IL), 1990. [8] Kazerooni H, Guo J. Direct-drive, active compliant end-effector. In: Proc IEEE Int Conf on Robotics and Automation, 1987. p. 758–66. [9] Kao I, Cutkosky MR, Johansson RS. Robotic stiffness control and calibration as applied to human grasping tasks. IEEE Trans Robotics and Automation 1997;13(4):557–66. [10] Yi B-J et al. Geometric stability in force control. In: Proc IEEE Int Conf on Robotics and Automation. 1991. p. 281–6. [11] Kato A, Kondo N, Narita H, Ito K, Luo ZW. Compliance control of direct drive manipulator using ultrasonic motor. In: Proc of RoMan (Gdansk Poland) September 12–15, 1994. [12] Mussa-Ivaldi A et al. Integrable solutions of kinematic redundancy via impedance control. Int J Robot Res 1991;10(5):481–91. [13] Yokoi K, Maekawa H, Tanie K. A method of compliance control for a redundant manipulator. In: Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems, 1992. p. 1927–34. [14] Yokoi K, Kaneko H, Tanie K. A compliant control method suggested by muscle networks in human arms. In: Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems, 1988. p. 385–90. [15] Choi HR, Chung WK, Youm Y. Stiffness analysis and control of redundant manipulators. In: Proc IEEE Int Conf on Robotics and Automation, 1994. p. 689–95. [16] Chen S-F, Kao I. Geometrical method for modeling of asymmetric 6 · 6 Cartesian stiffness matrix. In: Proc 2000 IEEE/RSJ Int Conf on Intelligent Robots and Systems, 2000. p. 1217–22. [17] Griffis M, Duffy J. Comparing structures of stiffness matrices using invariants. In: Proc VIII CISMIFMoMM Symposium, Ro. Man. Sy Ô92, 1992. [18] Tong J, Somerset J. Control, performance, and application of antagonistic manipulator joints. In: Proc of American Control Conf, 1985. p. 63–4. [19] Tao JM, Luh JYS, Zheng YF. Compliant coordination control of two moving industrial robots. IEEE Trans Robot Autom 1990;6(3):322–30. [20] Uchiyama M et al. Adaptive load sharing for hybrid controlled two cooperative manipulators. In: Proc IEEE Int Conf on Robotics and Automation (Sacramento, CA), 1991. p. 986–91.
238
S.H. Lee et al. / Mechatronics 15 (2005) 213–238
[21] Pelletier M, Daneshmend LK. An adaptive compliance motion controller for robot manipulators based on damping control. In: Proc IEEE Int Conf on Robotics and Automation, 1987. [22] Yoshikawa T, Zheng XZ. Coordinated dynamic hybrid position/force control for multiple robot manipulators handling one constrained object. Int J Robot Res 1993;12(3):219–30. [23] Zheng YF, Luh JYS. Optimal load distribution for two industrial robots handling a single object. In: Proc IEEE Int Conf on Robotics and Automation (Philadelphia, PA), 1988. p. 344–9. [24] Nakamura Y, Ghodoussi M. Dynamic computation of closed-link robot mechanisms with nonredundant and redundant actuators. IEEE Trans Robot Automat 1989;5:294–302. [25] Nahon MA, Angeles J. Force optimization in redundantly-actuated closed kinematic chains. In: Proc IEEE Int Conf on Robotics and Automation (Scottsdale, AZ), 1989. p. 951–6. [26] Kumar VJ, Gardner J. Kinematics of redundantly actuated closed chain. IEEE Trans Robot Automat 1990;6:269–73. [27] Kurz R, Hayward W. Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy. IEEE Trans Robot Automat 1992;8:644–51. [28] Bonitz RG, Hsia TC. Internal force-based impedance control for cooperating manipulators. IEEE Trans Robot Automat 1996;12(1):78–89. [29] Uchiyama M, Dauchez P. A symmetric hybrid position/force control scheme for the coordination of two robots. In: Proc IEEE Int Conf on Robotics and Automation (Philadelphia, PA), 1988. p. 350–6. [30] Yi B-J, Oh S-R, Suh IH. A five-bar finger mechanism involving redundant actuators: analysis and its applications. IEEE Trans Robot Autom 1999;15(6):1001–10. [31] Yi B-J. Analysis of redundantly actuated mechanisms with applications to design and control of advanced robotic systems. Dissertation of the University of Texas at Austin, Dec. 1991. [32] Yi B-J, Freeman RA. Geometric analysis of antagonistic stiffness in redundantly actuated parallel mechanisms. Special Issues on Parallel Closed-Chain Mechanism. J Robot Syst 1993;10:581–603. [33] Yi B-J, Suh IH, Oh S-R. Analysis of a 5-bar finger mechanism having redundant actuators with applications to stiffness and frequency modulations. In: Proc IEEE Int Conf on Robotics and Automation (Albuquerque, NM), 1997. p. 759–65. [34] Lee JH, Yi B-J, Oh S-R, Suh IH. Optimal design of a five-bar finger with redundant actuation. In: Proc IEEE Int Conf on Robotics and Automation (Leuven, Belgium), 1998. p. 2068–74. [35] Lee SH, Yi B-J, Kwak YK. Optimal kinematic design of an anthropomorphic robot module with redundant actuators. Mechatronics 1997;7(5):443–64. [36] Klir GJ, Folger TA. Fuzzy sets, uncertainty and information. Prentice Hall; 1988. [37] Freeman RA, Tesar D. Dynamic modeling of serial and parallel mechanisms/robotic systems. Part I: Methodology. Part II: Applications. In: Proc ASME Biennial Mechanisms Conf DE-vol. 15-3, 1988. p. 7–27. [38] Luenberger DG. Linear and nonlinear programming. 2nd ed. Addison Wesley; 1984. [39] Yi B-J, Freeman RA. Feedforward spring-like impedance modulation in human arm models. In: Proc IEEE Int Conf on Robotics and Automation (Nagoya, Japan), 1995. p. 3121–8.
Optimization and experimental verification for the antagonistic stiffness in redundantly actuated mechanisms: a five-bar example Sang Heon Lee a, Jae Hoon Lee b, Byung-Ju Yi Soo Hyun Kim a, Yoon Keun Kwak a
b,*
,
a
b
Department of Mechanical Engineering, KAIST, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, Republic of Korea School of Electrical Engineering and Computer Science, Hanyang University, 1271 Sa-1 dong, Ansan, Kyungki-do 475-791, Republic of Korea Received 19 February 2003; accepted 28 July 2004
Abstract A closed-chain mechanism having redundancy in force domain can produce an effective spring effect by proper internal load distribution. The so-called antagonistic stiffness is provided by redundant actuation in conjunction with nonlinear geometric constraints. The objective of this study is twofold. The first one is to propose a methodology for optimal kinematic design of antagonistic stiffness and the second one is to verify the phenomenon of the antagonistic stiffness through experimentation. In this work, a five-bar mechanism that can modulate antagonistic stiffness by using redundant actuators is employed as an exemplary system. The optimal structure of the five-bar mechanism that can maximize efficiency in generation of antagonistic stiffness is evaluated and analyzed. A measure of stiffness isotropy and the gradient property of the measure are employed as the performance indices in optimization. To deal with multi-criteria based design, a composite design index based on max–min principle of fuzzy theory is used as an objective function. Two optimization results are obtained. Furthermore, a five-bar mechanism has been developed for experimental verification. The
*
Corresponding author.
0957-4158/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.07.008
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phenomenon of antagonistic stiffness has been successfully demonstrated by free vibration and indirect force control experimentation. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Stiffness (or compliance) has been one of fundamental properties in the analysis and control of general dynamic systems. Specifically, compliance control capability is an important feature for robot interacting with environment or another robot [1– 17]. Salisbury [2] presented a compliance control scheme in the operational space. Once operational compliance matrix (or operational stiffness matrix [Kuu]) is designed, the joint space compliance matrix (or joint stiffness matrix [K//]) is obtained T as ½K // ¼ ½Gu/ ½K uu ½Gu/ , where ½Gu/ denotes the Jacobian relating the joint velocity to the operational velocity. Based on this principle, many compliance control algorithms have been reported in the field of redundant manipulators [12–15] and multiple arms [19] holding a common object. In Salisbury [2] and other related literatures [1–14], evaluations of the stiffness (or compliance) matrix of a robot have been limited to unloaded cases (i.e., zero externally applied loads), which always produce symmetric stiffness matrices. Yi et al. [10] made a geometric analysis on the additional stiffness term due to the external forces and pointed out that the physical stiffness matrix can be asymmetric for loaded cases, which yields difficulties in symmetric stiffness modulation [31,32]. The asymmetry obtained from externally applied load components is coincident to the analyses made by Griffis and Duffy [17], Chen and Kao [16]. Choi et al. [15] extended this idea to stiffness analysis of multi-fingered hands. As a useful application of the additional stiffness term, Hogan [3] initially addressed the concept of antagonistic stiffness and followed by Tong and Somerset [18]. When redundant actuators activate the mechanism antagonistically, an effective spring-like property can be produced by antagonistic internal load with conjunction with complex nonlinear geometry of the mechanism. They claimed that the antagonistic stiffness could be modulated arbitrarily by controlling actuation inputs without feedback control. Yi and Freeman [30–33] proposed a general methodology for analysis and application of the antagonistic stiffness created by antagonistic redundant actuation. This property has been also understood as an inherent property in the human-body that is usually driven by abundant inputs [31,39]. Even though a general mathematical formulation has been made on the concept of antagonistic stiffness, optimal design and experimental verification of the antagonistic stiffness has not been carried out yet. Thus, in this work, we investigate an optimal kinematic structure, specifically for a five-bar mechanism that efficiently generates antagonistic stiffness via redundant actuation. To do this, based on the antagonistic stiffness model, a stiffness modulation index is proposed, which represents the isotropy of the transformation matrix between the actuation efforts and
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the antagonistic stiffness. The transform matrix is the second derivative of the kinematic constraint [31,32]. Also in order to obtain the global isotropic characteristic, a global design index is employed which is defined as the average value of the isotropic indices calculated over the workspace. And a gradient design index is proposed to obtain uniform distribution of the isotropic index over the workspace. To cope with these design indices in optimization, composite design index based on max–min principle of fuzzy theory [36] is defined and used as objective function. The optimization is performed in two paths. One is optimizing X-directional stiffness modulation, and the other is corresponding to Y-directional stiffness. The result from optimization is suitable for generating stiffness more effectively as well as general force/velocity transmission. A five-bar mechanism was developed based on the kinematic parameters obtained by optimization. The characteristic of the antagonistic property will be verified through experimentation.
2. Concept of antagonistic stiffness When the number of actuation input is greater than the mobility, the system is called a redundantly actuated system. Structure of a system with redundant actuation usually has closed-chain mechanism in which many potential input locations exist. The redundantly actuated system can employ many optimal load distribution schemes [20–27] as well as beneficial internal load generation schemes [28–35]. Fig. 1 represents a four-bar mechanism that has mobility one. The four-bar requires at least one actuator to control the motion. This mechanism is redundantly actuated when it has more than two inputs. It is remarked that specially, when the mechanism is activated antagonistically by two joint actuators, an effective springlike property can be produced by antagonistic internal load with conjunction with complex nonlinear geometry of the mechanism. The antagonistic stiffness can be modulated arbitrarily by controlling actuation inputs without feedback control [3,32]. In a state of static equilibrium formed by antagonistic activation of the input load, given a disturbance to the system, the system oscillates about the equilibrium position like a physical spring.
K Fig. 1. Four-bar driven by redundant actuators.
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3. Kinematic modeling 3.1. Open-chain kinematics The five-bar mechanism shown in Fig. 2 has one closed-kinematic chain that consists of two open chains by cutting the middle joint. It is identical with cooperating two robots handling a common object. The end-effector positions u of the two open chains are identical, and they are described by l1 c1 þ l2 c12 x l5 þ l4 c5 þ l3 c45 ¼ ¼ : ð1Þ y l1 s1 þ l2 s12 l4 s5 þ l3 s45 And the orientation angle of link 2 is u ¼ h1 þ h2 ¼ h3 þ h4 þ h5 þ p:
ð2Þ
Adopting the standard Jacobian representation for the velocity of dependent output vector u 2 RN in terms of independent input coordinates r /_ 2 RM of rth open-chain, one has h i _ r ¼ 1; 2: u_ ¼ r G u/ r /: ð3Þ Here, h
u rG /
i
ou ¼ 2 RN M or /
ð4Þ
_ is the first-order kinematic influence coefficient (KIC) relating u_ to r /. Acceleration vector € u also can be represented in terms of input coordinates [37,32] h i h i € þ r /_ T r H u r /; _ € ð5Þ u ¼ r G u/ r / //
Fig. 2. Five-bar mechanism.
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where the element of the second-order KIC array h i o2 ui u ¼ : r H // ijk or /j or /k
h
u r H //
i
217
2 RN MM is defined as ð6Þ
3.2. Internal kinematics In redundantly actuated systems, the independent coordinates can be selected arbitrary. Since the mobility of the five-bar mechanism is 2, at least two actuation inputs are required to operate the system. In this work, the minimum independent (input) coordinates and dependent coordinates are selected as /a ¼ ðh1 ; h5 Þ
T
ð7Þ T
/p ¼ ðh2 ; h3 ; h4 Þ :
ð8Þ
From the equivalent velocity relationship of the two open-chains h i h i _ u_ ¼ 1 G u/ 1 /_ ¼ 2 G u/ 2 /;
ð9Þ
the velocity relation between independent and dependent coordinate set can be obtained by rearranging the above equation: ½A/_ p ¼ ½B/_ a ; where ½A ¼
hh
u 1 G/
i :2
ð10Þ
h i h i i ; 2 Gu/ ; 2 Gu/ 2 R33 ; :1
:2
h h i h i i ½B ¼ 1 Gu/ ; 2 Gu/ 2 R32 : :1
:3
ð11Þ ð12Þ
In Eq. (11), ½r Gu/ :i is ith column vector of the matrix. The velocity vector for the dependent coordinate set is obtained by premultiplying the inverse of [A] to Eq. (10): ð13Þ /_ ¼ ½A1 ½B/_ ¼ ½G p /_ ; p
½Gpa
a
a
a
32
2 R and the velocity vector of total systemÕs joints can be expressed in where terms of the velocity vector of the independent joint set by ½I ð14Þ /_ ¼ ½G /a /_ a ¼ /_ : ½G pa a € to /_ can be easily obThe second-order KIC array ½H paa 2 R322 relating / p a tained in a similar manner [32]. 3.3. Forward kinematics Since the joints of the rth open-chain is composed of some of the independent and dependent joints, r /_ can be expressed in terms of the total systemÕs independent joints by
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_ ¼
r/
r
G /a /_ a ;
ð15Þ 52
where ½r G /a is formed by rearranging rows of ½G /a 2 R . Thus the forward kinematics for the common object is obtained by embedding the first-order KIC into one of the rth pseudo open-chain kinematic expression as follows: h i h i u_ ¼ r G u/ r /_ ¼ r G u/ r G /a /_ a ¼ ½G ua /_ a ; ð16Þ where ½Gua 2 R22 . Using the same augmentation method employed in Eq. (16) evaluation of the second-order forward kinematic array ½H uaa 2 R222 is also straightforward. It is given by T
½H uaa ¼ ½r G u/ ½r H /aa þ ½r G /a ½r H u// ½r G /a
ð17Þ
where ÔÕ means a generalized scalar dot product [32,37]. 3.4. Statics of the five-bar mechanism According to the duality existing between the velocity and force vector, the force relation between the independent and dependent joints can be described. So the effective load referenced to the independent joints is given by T a ¼ T a þ ½G pa T T p ¼ ½G /a T T /
ð18Þ
where Ta, Tp, and T/ are force or torque at the independent joints, the dependent joints, and the total systemÕs joints, respectively. In static equilibrium, the torque at the independent coordinates can be expressed as T T a ¼ G Aa T A ¼ 0; ð19Þ where TA is the torque at the actuated joints /A. ½G Aa 2 RA2 , where A denotes the number of active joint, denoting the first-order KIC relating /_ A to /_ a is obtained by rearranging the rows of ½G /a .
4. Modeling of antagonistic stiffness Given a disturbance to the system under static equilibrium by redundant actuation, a spring-like behavior occurs to the system. Assuming that the magnitude of TA remains constant, the effective stiffness matrix ½K aa 2 R22 with respect to the independent coordinates is obtained as ½K aa ¼
oT a ¼ ðT A ÞT H Aaa o/a
ð20Þ
by differentiating Eq. (19) with respect to the independent coordinate set /a. It is observed from (20) that the so-called antagonistic stiffness is provided by redundant
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219
actuation in conjunction with nonlinear geometric constraints denoted by A H aa 2 RA22 . The stiffness relationship ½K uu 2 R22 between the output coordinates and the independent coordinates is given by [2] T
½K uu ¼ ½G au ½K aa ½G au : ½G au
22
ð21Þ ½G ua
where 2 R denotes the inverse of given in Eq. (16). Substituting Eq. (20) into Eq. (21) yields the following stiffness matrix expressed with respect to the output space. T ½K uu ¼ ðT A Þ H Auu ; ð22Þ A T where H uu ¼ ½G au ½H Aaa ½G au 2 RA22 is obtained by plane-by-plane multiplication [33,37]. An alternative form of Eq. (22) given in a matrix form is described by ð23Þ K u ¼ H Au T A ; where Ku is consists of the upper diagonal elements of [Kuu] and ½H Au 2 R3A is also obtained by collecting the upper diagonal columns of the three-dimensional array ½H Auu , which are defined as follows: K u ¼ ðK xx ; K xy ; K yy Þ
T
2 A 3 H uu ;xx 7 A 6 6 H A 7 H u ¼ 6 uu ;xy 7: 4 5 H Auu ;yy
ð24Þ
ð25Þ
Necessary (but not sufficient) conditions for full stiffness generation are derived in [32]: a closed chain is capable of full stiffness generation only if it satisfies NA P D þ N
ð26Þ
where N is DOF (system mobility), D is number of independent stiffness elements, and NA is number of actuated joints and NC ¼ ðIC LCÞ P D;
ð27Þ
where IC is number of independent constraint equations, NC is number of independent nonlinear constraint equations, and LC is number of independent linear equations. The five-bar mechanism has only one independent closed chain and thus it has two independent nonlinear constraint equations, so only two elements of Ku can be independently modulated. According to the condition of Eq. (26), at least four actuators are needed to modulate two stiffness elements. Four joints except the third joint are actuated in the given five-bar. There are five potential input locations in the five-bar mechanism. Thus, five sets of actuators exist. Selection of an actuator set can be based on analysis of the geometry associated with antagonistic stiffness. An
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optimal actuator set can be found from this process. However, in the following analysis, we fix the location of the actuator set and seek the optimal kinematic parameters that maximize the antagonistic stiffness generation.
5. Kinematic optimization for antagonistic stiffness 5.1. Stiffness modulation index 5.1.1. Local design index With consideration of the operational force vector and the independent joint vector given by T
T u ¼ ½G au T a : Eq. (19) can be also expressed as T T u ¼ G Au T A ¼ 0; where
G Au
T
ð28Þ
T ¼ ½G au G Aa 2 R2A
and T u is the effective load at the end-effector coordinates. Eq. (28) representing static equilibrium is the primary task for antagonistic stiffness generation. For Eq. (28), the general solution for TA is obtained as T þ T T A ¼ ½I G Au G Au ð29Þ e1 ; where e1 is an arbitrary vector and the superscript Ô+Õ means pseudo-inverse. Eq. (29) denotes the secondary subtask. Substituting Eq. (29) into Eq. (23) yields T þ T A K u ¼ H u ½I G Au G Au ð30Þ e1 and ½c H e1 ¼ K u ;
ð31Þ
where
T þ T A c ½ H ¼ H u ½I G Au G Au : 2 R2A :
ð32Þ
If the exact solution of e1 exists for Eq. (31), it implies that the second subtask associated with stiffness generation can be realized. The general solution of e1 is þ þ H K u þ ð½I ½c H ½c H Þe2 ; e1 ¼ ½ c
ð33Þ
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where e2 is an arbitrary vector. Then the solution of TA is represented as T þ T þ T A ¼ ½I G Au G Au ½c H Ku T þ T þ þ ½I G Au G Au H Þe2 : ð½I ½c H ½c
221
ð34Þ
The first term of right-hand side of the above equation is reduces to ½c H þ K u , by using condition of the symmetry and idempotency of T þ T AA A A ½I G u Gu 2 R . We can realize another subtask (the third-priority subtask) by e2 if the system has more redundancy. Assuming that there is no other subtask (e2 = 0), the solution becomes þ T A ¼ ½c H K u:
ð35Þ
So the relationship between Ku and TA which satisfies Eqs. (23) and (28) is T þ T K u ¼ H Au ½I G Au G Au ð36Þ TA: Now, the stiffness modulation index is defined by isotropic index of ½c H 2 RA3 , which shows isotropy characteristic in the stiffness modulation process: rmin ; ð37Þ j¼ rmax where rmin and rmax are the minimum and maximum singular values of the matrix ½c H , respectively. 5.1.2. Global design index The design index j defined in Eq. (37) is a local index that varies as the pose of the five-bar mechanism changes, so it cannot represent overall characteristics of the mechanism throughout the workspace. Therefore a design index that represents the global feature should be defined and used in optimization. The global design index that represents an average value of j over the workspace is defined as R j dW K ¼ RW ; ð38Þ dW W where W denotes workspace. Since the value of the local design index changes as the configuration of mechanism changes, gradient design index is also considered so that final design can have even distribution of the local design indices over the workspace. The gradient design index represents the change rate of the local design index [27]. The gradient design index is defined as the maximum of the differences among the local design indices calculated at adjacent points throughout the workspace. The smaller the global gradient design index, the more uniformly distributed the design index over the workspace, therefore the gradient design index should be minimized.
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5.1.3. Composite design index The design of a mechanism can be made based on any particular criterion. However, the single criterion-based design does not provide sufficient control on the range of the design parameters involved. Therefore, multi-criteria based design has been proposed. But various design indices are usually incommensurate concepts due to differences in unit and physical meanings, and therefore should not be combined with normalization and weighting functions unless they are transferred into a common domain. In other words, quantitative combination should be avoided. Instead, these design indices should be combined qualitatively. To consider these facts, we employ a composite design index. As an initial step to this process, preferential information should be given to each design parameter and design index. Then, each design index is transferred to a common preference design domain that ranges from zero to one. Here, the preference given to each design criterion is subjective to the designer. Preference can be given to each criterion by weighting. This provides flexibility in design. For K, the best preference is given the maximum value, and the least preference was given the minimum value of the criterion. Then the design index is transferred into common preference design domain as below e ¼ K K min ; K ð39Þ K max K min where ÔÕ implies the index which is transferred into common preference design domain. Reversely, if the best preference is given the minimum value, and the least preference is given the maximum value of the criterion, like the gradient design index Kd, the design index transferred into common preference design domain is given by d d e d ¼ K max K : K ð40Þ d d K max K min Note that each transferred design index is constructed such that a large value represents a better design. A set of optimal design parameters was obtained based on max–min principle of fuzzy theory [6]. Initially, minimum values among the design indices for all set of design parameters are obtained, and then a set of design parameters, which has the maximum value of the minimum values already obtained, is chosen as the optimal set of design parameters. Based on this principle, the composite design index (CDI) is given by a
d b
e ; ðK e Þ Þ: CDI ¼ minð K ð41Þ The upper Greek letter is the degree of weighting, and large value implies large weighting. The weightings are all set to ones in this work. 5.2. Optimization Optimization problem is formulated as follows: The objective is to evaluate optimal link lengths, which maximizes CDI. The design variables are the ratios of link lengths to the base link length l5, and they are defined by
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xi ¼ li =l5 ;
223
i ¼ 1; 2; 3; 4:
ð42Þ
Constraints to the design variables are 0:5 6 xi 6 1:5
ð43Þ
l5 ¼ 0:2 m:
ð44Þ
and
Two objective functions are defined and used in optimization. One noted as CDIx corresponds to the modulation of the X-directional stiffness. It contribute to enhance the isotropic characteristic for Kxx and Kxy. The other noted as CDIy corresponds to modulation of the Y-directional stiffness. It contribute to enhance the isotropic characteristic for Kxy and Kyy. Three numerical methods are used to deal with multi-variable nonlinear optimization problem with constraints. The exterior penalty function method is employed to transform the constrained optimization problem into an unconstrained problem. PowellÕs method is applied to obtain an optimal solution for the unconstrained multivariable problem, and quadratic interpolation method is used for unidirectional optimization [38].
Optimized Shape 0.4 0.35 0.3 0.25
Y
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.1
0
0.1
0.2
X Fig. 3. Optimized shape of Design 1.
0.3
0.4
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Optimization result for modulation of the X-directional stiffness is given as ¼ 0:697434392; ¼ 1:250000000; ¼ 1:487984717; ¼ 0:729829688
l1 =l5 l2 =l5 l3 =l5 l4 =l5
and the shape of optimized mechanism (Design 1) is shown in Fig. 3. Optimization result for modulation of the Y-directional stiffness is given as ¼ 0:750000100; ¼ 0:750000050; ¼ 0:693700249; ¼ 1:499999810
l1 =l5 l2 =l5 l3 =l5 l4 =l5
and the shape of optimized mechanism (Design 2) is shown in Fig. 4. As shown in Fig. 3, the length of distal floating link is longer than that of the link attached to the base. This structure is suitable to translate the torque given at the two base joints to the forces at the end-effector. In Fig. 4, conversely, the length of right connecting link shows extreme value, and this represents bad force transmission from base actuator to end-effector. By intuition, the shape of the optimized fivebar should be symmetric when solely considering the isotropy in optimization. However, both optimization results show unsymmetrical configurations. It is remarked Optimized Shape 0.35
0.3
0.25
Y
0.2
0.15
0.1
0.05
0
-0.05 -0.15
-0.1
-0.05
0
0.05
0.1
0.15
X Fig. 4. Optimized shape of Design 2.
0.2
0.25
0.3
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225
that the asymmetry is caused by consideration of the gradient characteristic of the isotropic index in optimization. The contours of stiffness modulation index for Design 1 are shown in Fig. 5. As shown in Fig. 5, jx is high at the center of workspace. Isotropies for Y-directional
Fig. 5. Isotropic index of Design 1. (a) Distribution of jx. (b) Distribution of jy.
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stiffness modulation are also high over most workspace. This factor is advantageous to operate the mechanism in both x and y directions. Fig. 6 shows the contours of stiffness modulation index for Design 2. The workspace is narrow and the peak is not on the center of workspace. Therefore this design can not be said to be useful. Contrastively the workspace of Design 1 has shape of a fan and is close to a diamond, and it is another advantage. As shown in Fig. 6(a), very low isotropy is detected on the middle of the workspace. Although Design 2
Fig. 6. Isotropic index of Design 2. (a) Distribution of jx. (b) Distribution of jy.
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is optimized to maximize Y-directional stiffness modulation effect only not to optimize X-directional stiffness modulation, but this design is unfit for general purpose. Shape of optimization result for maximizing isotropy of Jacobian ½G au and minimizing the gradient of its isotropy also shows like that of Design 1, though the result is not represented here. Therefore, Design 1 is suitable for both general operational purpose and antagonistic stiffness generation.
6. Experimental verification 6.1. Feed forward frequency modulation When the stiffness of a system can be modulated by redundant actuation, the frequency of the vibration motion of the system also can be modulated. To do this, the dynamics of the system, such as inertia matrix should be known, because the frequency is determined by the inertia and the stiffness of the system. Assuming that the system is in a state of static equilibrium, the velocity at the endeffector can be neglected. u_ ¼ 0:
ð45Þ
The dynamic equation of the system is simplified as ½I uu € u ¼ T u ;
ð46Þ
where ½I uu 2 R . When a small displacement from the neutral position un is occurred, the effective force T u at the end-effector frame will be changed, and Eq. (46) will be modified to 22
u dT u ¼ 0: ½I uu d€
ð47Þ
When the actuation input is determined to modulate the stiffness at the end-effector coordinate frame, then the change of the effective force is dT u ¼ ½K uu du; where the stiffness matrix at the end-effector frame is T ½K uu ¼ ðT A Þ H Auu :
ð48Þ
ð49Þ
So Eq. (47) is rearranged as 1
d€ u þ ½I uu ½K uu du ¼ 0:
ð50Þ
The multiplied matrices ½I uu 1 ½K uu is defined as the frequency matrix ½W uu 2 R22 whose elements are " 2 # xxx x2xy kxx kxy ¼ : ð51Þ kyx kyy x2yx x2yy The frequency matrix is also defined as a function of the actuation input because of the stiffness matrix is determined by the actuation inputs.
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n o 1 1 T ½W uu ¼ ½I uu ½K uu ¼ ½I uu ðT A Þ H Auu ¼ ðT A ÞT ½I uu 1 H Auu ¼ ðT A ÞT W Auu :
ð52Þ
In the above equation, the multiplication of the inverse of inertia matrix and the second-order KIC ½H Auu 2 R4A is operated in a plane-by-plane fashion. Eq. (52) can be rewritten in a matrix form as follows: 2 A 3 0 1 W uu ;11 kxx 6 A 7 7 B k C 6 W uu ;12 7 B xy C 6 W u ¼ B C ¼ 6 A 7T A ¼ W Au T A : ð53Þ 7 @ kyx A 6 W uu ;21 5 4 A kyy W uu ;22 The above equation can be obtained similar to (23). If it is desired that the motion along the x-direction vibrate with the angular velocity of xxd, then the following two equations should be satisfied. xxx ¼ xxd
and
xxy ¼ 0:
ð54Þ
Then the dynamic equation along the x-directional coordinate is d€ ux þ x2xd dux ¼ 0
ð55Þ
and a solution of dux is dux ¼ Ax sinðxxd tÞ:
ð56Þ
So the motion along the x-direction becomes sinusoidal curve with the frequency of fx ¼ xxd =2p:
ð57Þ
The procedure of determination of the actuation inputs is as follows: From Eq. (53), the actuating torques must satisfy A W ux T A ¼ W xd ; ð58Þ where ½W Aux 2 R2A that is composed of the first and second rows of is the matrix T ½W Au , and W xd ¼ x2xd 0 . The state of static equilibrium also must be satisfied. A T G u T A ¼ 0: ð59Þ The number of solution of the Eq. (59) is infinite. The general solution is obtained using pseudo-inverse as T þ T T A ¼ ½E4 G Au G Au ð60Þ e1 ; where e1 is an arbitrary vector. The general solution must satisfy Eq. (60). T þ T A W ux ½E 4 G Au G Au e1 ¼ W xd :
ð61Þ
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229
The solution of unknown vector e1 for above equation is obtained also using pseudoinverse. c þ W xd þ f½E 4 ½ W c þ ½ W c ge2 ; e1 ¼ ½ W
ð62Þ
where
T þ T A c ½ W ¼ W ux ½E 4 G Au G Au 2 RAA :
ð63Þ
e2 is arbitrary vector, but it is set to zero because additional actuation redundancies do not exist in the system. Then the solution of TA is T þ T c þ W xd T A ¼ ½E 4 G Au G Au ð64Þ t½ W and can be simplified into c þ W xd ; T A ¼ ½W ð65Þ T þ T because the matrix ½E 4 G Au G Au 2 RAA is symmetric and idempotent. When the y-directional motion is modulated with the angular velocity of xyd, Eq. (58) should be changed into ½W Auy T A ¼ W yd ;
ð66Þ
where ½W Auy 2 R2A that is composed of the third and forth rows of is the matrix T ½W Au , and W yd ¼ 0 x2yd . 6.2. Indirect force control algorithm When the stiffness of a system can be modulated by redundant actuation, a direct force control scheme can be also implemented. The previous force control scheme is based on high position feedback or a direct force feedback using a force sensor attached at the end effector of a robot manipulator. Due to the nature of feedback scheme, the inevitable delay combined with high feedback gain often exhibits unstable phenomenon such as chattering. The proposed scheme will be based on feed forward scheme in which the operational stiffness is generated by using redundant actuation. Thus, more precise force control can be possibly achieved. When a robot manipulator is driven by abundant actuators and is contacting with the environment, the effective load expressed in the independent coordinates is given as T T a ¼ G Aa T A þ ½G ua T T ex ð67Þ u ¼ 0: Taking derivative with respect to the independent coordinates results in an effective stiffness model, given by T T T u T ½K aa ¼ ðT A Þ H Aaa þ ðT ex ð68Þ u Þ ½H aa :
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Now, using the effective stiffness relation given by ½K aa ¼ ½G ua ½K uu ½G ua , Eq. (68) can be rewritten as T T T T u T ðT A Þ H Aaa ¼ ½G ua ½K uu ½G ua þ ðT ex ð69Þ u Þ ½H aa : The last term of Eq. (69) denotes an additional stiffness term due to externally applied load T ex u . Salisbury [2] ignored this additional term in usual force control scheme, and analysis and consideration of this term have been conducted [9,10]. Rearranging Eq. (69) as a vertor form, we have ½H uA ðT A Þ ¼ K:
ð70Þ
where ½H uA 2 RBA and K 2 RB (B denotes the number of operational stiffness elements in control) Then, if we desire to modulate a stiffness characteristic so that a proper force control can be made in certain direction, we can specify a proper stiffness for each direction. In order to decouple the action, the coupling stiffness element is set as zero. Also, since the feed forward stiffness modulation is made under static equilibrium, the equilibrium equation given by A T G u T A ¼ 0; ð71Þ should be combined with Eq. (70). Assuming that only x-directional stiffness elements (i.e., kxx and kxy) is to be controlled, the final form is now given as 2 T 3 2 3 0 G Au 6 7 7 6 ½H u 7T A ¼ 6 ð72Þ 4 kxx 5: 4 A 1; 5 u kxy ½H A 2; Indirect force control is performed in such a way that when a robot contacts an environment and plans to control a force along the surface normal direction, an offset displacement between the actuator end point and the desired position of robot is multiplied by the stiffness element of the surface normal direction.
7. Experimental verification 7.1. Experimental setup A five-bar for experimental verification of the antagonistic property is developed as shown in Fig. 7. The system is constructed as three main parts. The first is the fivebar mechanism made of aluminum and steel, the second is AC servomotors and their drives, and the third is a PC and interface system for control. The links of the fivebar are made of aluminum, and the end-effector contacting with environment has the shape of a disk and can rotate freely to remove the sliding friction effect. Kinematic and dynamic parameters for each link are summarized in Table 1. The position of mass center and the mass moment of inertia are calculated from 3D CAD data.
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Fig. 7. Five-bar system.
Table 1 Kinematic and inertial parameters of links
Link 1 Link 2 Link 3 Link 4 End-effector
Length (m)
Mass center (m)
Mass (kg)
Moment of inertia (E 3 kg m2)
0.150 0.300 0.300 0.150 0.0
0.0891 0.1557 0.1513 0.0891 0.0
0.3642 0.6123 0.6012 0.3642 0.0632
0.8741 5.5851 5.5479 0.8741 0.0293
Actuator torque at motor is transmitted to the rotating joints by timing belt and pulley system with the reduction ratio of 2.6. We employed AC servomotor whose torque can be directly controlled. PC is used to control the system, and the control algorithm is programmed by using LabWindows on the MS Windows environment. 7.2. Experimental results 7.2.1. Free vibratory motion Experimentation has been performed to corroborate the algorithm proposed in the previous section. Four actuators activate the five-bar mechanism. If the motions in the X- and Y-directions are to be controlled such that the natural frequency for each motion is given xxx ¼ xxd xxy ¼ 0
or
xyy ¼ xyd xyx ¼ 0
;
respectively, the actuation input to satisfy this should be calculated. Once the calculated input load is given to the system, the system maintains a static equilibrium. At the moment that a displacement is given to the end-effector, the end-effector initiates
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a vibratory motion like a pendulum. The characteristic of this motion is measured by using encoders attached to the joints of the systems. A damped vibration is observed. This comes from the damping effect caused by frictions existing at the joints and timing belts. The dynamic behavior of the measured vibratory motion can be characterized by the natural frequency and the damping ratio. They can be estimated by comparing the experimental data to the analytic formulation of one degree-of-freedom vibratory motion, given by pffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ Ae1xn t sin xn 1 12 t þ / ;
ð73Þ
where xn and 1 denoting the natural frequency and the damping ration, respectively, are estimated by using the curve-fitting technique based on the experimental data. T T Initially, the equilibrium position is given as un ¼ ð xn y n Þ ¼ ð 0:1 0:3 Þ m. Fig. 8 denotes the vibratory motion when the frequency of the X-directional motion is given p rad/s. The dashed line and the solid line represent the experimental data and the model obtained by the curve fitting, respectively. The natural frequency obtained by the analytic model 3.41 rad/s, which is about 8.5% higher that that of the desired frequency. Fig. 9 shows the plot for 4.0 rad/s. The frequency calculated by
X-directional motion 0.35 0.3
Displacement (m)
0.25 0.2 0.15 0.1 0.05 0 -0.05 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 8. X-directional motion with the frequency of p rad/s.
4.5
5
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233
X-directional motion
0.3 0.25
Displacement (m)
0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0
1
2
3
4
5
6
Time (s) Fig. 9. X-directional motion with the frequency of 4.0 rad/s.
Y-directional motion
0.45
Displacement (m)
0.4
0.35
0.3
0.25
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 10. Y-directional motion with the frequency of p rad/s.
4.5
5
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Y-directional motion
0.45
Displacement (m)
0.4
0.35
0.3
0.25
0.2 0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s) Fig. 11. Y-directional motion with the frequency of 4.0 rad/s.
curve fitting is 4.23 rad/s, which is also 5.7% higher than that of the desired one. In general, the system shows increased stiffness and more damping. The Y-directional motion shows the same trend. Figs. 10 and 11 denote the plots for Y-directional motion with p rad/s and 4.0 rad/s. The frequencies for these cases are calculated as 3.48 and 4.24 rad/s, respectively.
Fig. 12. Experimental setup.
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Overall, the experimentation result exhibits the phenomenon of the antagonistic stiffness correctly. However, both the stiffness and damping of the prototype are a 1
0
Ns
-1
-2
-3
-4
-5 0
0.5
1
1.5
2
(a)
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
sec 1
0
Ns
-1
-2
-3
-4
-5 0
(b)
0.5
1
1.5
2
2.5
sec
Fig. 13. Experimental result of force control. (a) Without including the effect due to external force. (b) With including the effect of external force.
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little bit higher than the desired ones of the system. This is mainly due to the tension of the timing belts that connect the floating joints to the actuators placed on the base location. 7.2.2. Indirect force control Fig. 12 shows the experimental setup for force control. A force/torque sensor is attached on the wall. The trajectory of the five-bar mechanism is given to the x-direction. The offset distance denoted by Dx multiplied by the stiffness in the x-direction becomes the force exerted on the wall by the five-bar mechanism. The magnitude of the x-directional stiffness is given as Kxx = 50 N/m, and the offset distance is given as 0.05 m in 2 s. Therefore, at last 2.5 N should be exerted in the x-direction. The stiffness is created by antagonistic redundant actuation and the additional stiffness term (Eq. (69)) due to the external force is also taken into account. Fig. 13 shows the experimental result. Fig. 13(a) denotes the case that the additional stiffness due to the external force is not being considered, while Fig. 13(b) shows the case that the additional stiffness due to external force is not being considered. It is remarked that the additional term due to the external force should not be ignored for precise force control and that in Fig. 13(b) the measured force converges to 2.5 N, which is identical to desired force value. The special feature of antagonistic stiffness is that it is created in a feed forward fashion, but not feedback fashion. Thus, the inherent instability problem reported in many previous force control implementation, which is based on feedback scheme, can be avoided.
8. Conclusions The concept of antagonistic stiffness, resulting from antagonistic redundant actuation, has been studied extensively. However, previous works only have dealt with the theory and simulation study about this concept. The contribution of this study is twofold; methodology for optimal kinematic design of antagonistic stiffness and experimental verification of antagonistic stiffness. A five-bar mechanism that can modulate antagonistic stiffness by using redundant actuators is employed as an exemplary system. Optimization to obtain uniform stiffness characteristic has been carried out. And based on the optimal kinematic parameters, a five-bar mechanism has been developed for experimentation. The phenomenon of antagonistic stiffness has been successfully demonstrated by free vibration experimentation and the effective antagonistic stiffness generated by redundant actuation was also employed to indirect force control. Even though the proposed scheme for antagonistic stiffness generation was examined though a planar 2 degree of freedom device, the proposed scheme can be generalized to general redundantly actuated mechanisms. The fundamental requirement for antagonistic stiffness generation is given by the necessary conditions as noted in Eqs. (26) and (27). The mathematical tools for actual implementation are identical to those given in Sections 5 and 6. More in-depth analysis on the second-order kine-
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matic influence coefficient denoted as three dimensional array [H] will be needed for spatial type closed chain mechanisms [32,37].
Acknowledgement This study was supported by a grant of the Korea Health 21 R&D Project, Ministry of Health & Welfare, Republic of Korea. (02-PJ3-PG6-EV04-0003).
References [1] Mason MT. In: Brady M, editor. Compliant motion–Robot motion. M.I.T Press; 1983. [2] Salisbury JK. Active stiffness control of manipulator in Cartesian coordinates. In: Proc IEEE 19th Conf on Decision and Control. 1980, p. 95–100. [3] Hogan N. Mechanical impedance control: An approach to manipulation: Part II—implementation. ASME Trans Dyn Sys, Meas Contr 1985;107:8–16. [4] Fresonke DN, Hernandez E, Tesar D. Deflection prediction for serial manipulator. In: Proc IEEE Int Conf on Robotics and Automation, (Philadelphia, PA), vol. 1. 1988. p. 482–7. [5] Goswami A, Peshikin M, Colgate JE. Passive robotics an exploration of mechanical computation. In: Proc IEEE Int Conf on Robotics and Automation, (Cincinnati, OH). 1990. p. 279–84. [6] Loncaric J. Geometrical Analysis of Compliant Mechanisms in Robotics. PhD Dissertation, Harvard University, 1985. [7] Patterson T, Lipkin H. A classification of robot compliance. In: Proc 21st ASME Mechanism Conf, (Chicago, IL), 1990. [8] Kazerooni H, Guo J. Direct-drive, active compliant end-effector. In: Proc IEEE Int Conf on Robotics and Automation, 1987. p. 758–66. [9] Kao I, Cutkosky MR, Johansson RS. Robotic stiffness control and calibration as applied to human grasping tasks. IEEE Trans Robotics and Automation 1997;13(4):557–66. [10] Yi B-J et al. Geometric stability in force control. In: Proc IEEE Int Conf on Robotics and Automation. 1991. p. 281–6. [11] Kato A, Kondo N, Narita H, Ito K, Luo ZW. Compliance control of direct drive manipulator using ultrasonic motor. In: Proc of RoMan (Gdansk Poland) September 12–15, 1994. [12] Mussa-Ivaldi A et al. Integrable solutions of kinematic redundancy via impedance control. Int J Robot Res 1991;10(5):481–91. [13] Yokoi K, Maekawa H, Tanie K. A method of compliance control for a redundant manipulator. In: Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems, 1992. p. 1927–34. [14] Yokoi K, Kaneko H, Tanie K. A compliant control method suggested by muscle networks in human arms. In: Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems, 1988. p. 385–90. [15] Choi HR, Chung WK, Youm Y. Stiffness analysis and control of redundant manipulators. In: Proc IEEE Int Conf on Robotics and Automation, 1994. p. 689–95. [16] Chen S-F, Kao I. Geometrical method for modeling of asymmetric 6 · 6 Cartesian stiffness matrix. In: Proc 2000 IEEE/RSJ Int Conf on Intelligent Robots and Systems, 2000. p. 1217–22. [17] Griffis M, Duffy J. Comparing structures of stiffness matrices using invariants. In: Proc VIII CISMIFMoMM Symposium, Ro. Man. Sy Ô92, 1992. [18] Tong J, Somerset J. Control, performance, and application of antagonistic manipulator joints. In: Proc of American Control Conf, 1985. p. 63–4. [19] Tao JM, Luh JYS, Zheng YF. Compliant coordination control of two moving industrial robots. IEEE Trans Robot Autom 1990;6(3):322–30. [20] Uchiyama M et al. Adaptive load sharing for hybrid controlled two cooperative manipulators. In: Proc IEEE Int Conf on Robotics and Automation (Sacramento, CA), 1991. p. 986–91.
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[21] Pelletier M, Daneshmend LK. An adaptive compliance motion controller for robot manipulators based on damping control. In: Proc IEEE Int Conf on Robotics and Automation, 1987. [22] Yoshikawa T, Zheng XZ. Coordinated dynamic hybrid position/force control for multiple robot manipulators handling one constrained object. Int J Robot Res 1993;12(3):219–30. [23] Zheng YF, Luh JYS. Optimal load distribution for two industrial robots handling a single object. In: Proc IEEE Int Conf on Robotics and Automation (Philadelphia, PA), 1988. p. 344–9. [24] Nakamura Y, Ghodoussi M. Dynamic computation of closed-link robot mechanisms with nonredundant and redundant actuators. IEEE Trans Robot Automat 1989;5:294–302. [25] Nahon MA, Angeles J. Force optimization in redundantly-actuated closed kinematic chains. In: Proc IEEE Int Conf on Robotics and Automation (Scottsdale, AZ), 1989. p. 951–6. [26] Kumar VJ, Gardner J. Kinematics of redundantly actuated closed chain. IEEE Trans Robot Automat 1990;6:269–73. [27] Kurz R, Hayward W. Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy. IEEE Trans Robot Automat 1992;8:644–51. [28] Bonitz RG, Hsia TC. Internal force-based impedance control for cooperating manipulators. IEEE Trans Robot Automat 1996;12(1):78–89. [29] Uchiyama M, Dauchez P. A symmetric hybrid position/force control scheme for the coordination of two robots. In: Proc IEEE Int Conf on Robotics and Automation (Philadelphia, PA), 1988. p. 350–6. [30] Yi B-J, Oh S-R, Suh IH. A five-bar finger mechanism involving redundant actuators: analysis and its applications. IEEE Trans Robot Autom 1999;15(6):1001–10. [31] Yi B-J. Analysis of redundantly actuated mechanisms with applications to design and control of advanced robotic systems. Dissertation of the University of Texas at Austin, Dec. 1991. [32] Yi B-J, Freeman RA. Geometric analysis of antagonistic stiffness in redundantly actuated parallel mechanisms. Special Issues on Parallel Closed-Chain Mechanism. J Robot Syst 1993;10:581–603. [33] Yi B-J, Suh IH, Oh S-R. Analysis of a 5-bar finger mechanism having redundant actuators with applications to stiffness and frequency modulations. In: Proc IEEE Int Conf on Robotics and Automation (Albuquerque, NM), 1997. p. 759–65. [34] Lee JH, Yi B-J, Oh S-R, Suh IH. Optimal design of a five-bar finger with redundant actuation. In: Proc IEEE Int Conf on Robotics and Automation (Leuven, Belgium), 1998. p. 2068–74. [35] Lee SH, Yi B-J, Kwak YK. Optimal kinematic design of an anthropomorphic robot module with redundant actuators. Mechatronics 1997;7(5):443–64. [36] Klir GJ, Folger TA. Fuzzy sets, uncertainty and information. Prentice Hall; 1988. [37] Freeman RA, Tesar D. Dynamic modeling of serial and parallel mechanisms/robotic systems. Part I: Methodology. Part II: Applications. In: Proc ASME Biennial Mechanisms Conf DE-vol. 15-3, 1988. p. 7–27. [38] Luenberger DG. Linear and nonlinear programming. 2nd ed. Addison Wesley; 1984. [39] Yi B-J, Freeman RA. Feedforward spring-like impedance modulation in human arm models. In: Proc IEEE Int Conf on Robotics and Automation (Nagoya, Japan), 1995. p. 3121–8.