The investigation of the maneuverability deterioration based on acceleration radius theory

(1)

The investigation of the maneuverability deterioration based on acceleration

radius theory

Kuei-Jen Cheng, Pi-Ying Cheng

*

Department of Mechanical Engineering, National Chiao-Tung University, 1001 University Road, Hsinchu City, Taiwan 300, ROC

a r t i c l e

i n f o

Article history: Received 23 September 2007 Accepted 27 March 2009 Keywords: Maneuverability Acceleration radius Deterioration rate

a b s t r a c t

Maneuverability is the measure of the dynamic performance of a manipulator in a speciﬁc posture or con-ﬁguration, and acceleration radius is one of the most utilized indices of it. Acceleration radius can be uti-lized as the reference to judge whether further dynamic analysis should be performed when evaluating the controllability and feasibility of the manipulator following the prescribed path with assigned kine-matic and kinetic requirements in the planning phase. When utilizing acceleration radius as the dynamic reference in the planning phase, it can prevent wasting the calculation cost due to these non-necessary dynamic analyses, and it can also be utilized as the benchmark in the on-line control.

However, the existence of the configuration errors is inevitable in reality, and it deteriorates the dynamic performance of a manipulator with the ideal configuration parameters and leads to the potential risk of failing to achieve an assigned dynamic task. To investigate the adverse behavior caused by the con-figuration errors and to provide some clues to avoid or reduce their influence, this article proposes a novel and systematic method which can be used to evaluate the maneuverability deterioration of a non-redun-dant serial manipulator system due to the influence of configuration errors, and it also provides an index, deterioration rate, to quantitate this kind of deterioration. Deterioration rate can be utilized to quantitate the maneuverability deterioration due to the influence of configuration errors in a prescribed workspace or region and can also be treated as the safety or derating margin when proceeding with the control or path planning.

1. Introduction

Acceleration radius is utilized to describe the ability of a manip-ulator to accelerate its end-effecter and defined as the maximum achievable acceleration in all directions of the end-effecter in a specific posture or configuration by the known output limits of joint actuators. For easily being understood, the definition of accel-eration radius is briefly demonstrated inFig. 1with a two degrees of freedom example. Acceleration radius is one of the common indices of maneuverability which usually is utilized as the refer-ence to judge whether further dynamic analysis should be performed in the planning phase to reduce the calculation cost and relieve the burden of a control or path designer when planning the control strategies or trajectories.

However, when manufacturing and assembling a manipulator, the existence of the configuration errors due to these processes is inevitable. Since acceleration radius is defined as the maximum achievable acceleration in all directions, it shall include the adverse influence due to the configuration errors in its mathematical model to prevent planning an unattainable or infeasible dynamic task.

In the 80s and the early 90s, the literature studied in this field discusses the best expression to present the maneuverability or how to calculate it with the fastest speed or the maximum effi-ciency[1–3]. In the 90s, the direction of the studies changed to investigate the maneuverability in the redundant robot systems and its application[4–9]. In the recent years, the study focused on what the influence of velocity has on the maneuverability

[10–13]. Although lots of literature has been published in this field, none of them mentions the influence of configuration errors in their studies. This leads the conclusions to not fully match the reality. To redeem this insufficiency, this article conducts the mathematical model of the acceleration radius, which includes the influence of configuration errors, and proposes a systematic method to evaluate the maneuverability deterioration of a non-redundant serial manipulator due to the influence of these errors. Besides, this article also provides a new index, deterioration rate, to quantitatively express the maneuverability deterioration due to the influence of these configuration errors.

For effectively expressing the proposed method and index, this article is arranged as follows. The kinematics of a manipulator based on Denavit–Hartenberg transformation matrix (D–H trans-formation matrix) with conﬁguration errors is conducted in Section2. In Section3, it interprets how to derive the acceleration 0957-4158/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechatronics.2009.03.011

* Corresponding author. Tel.: +886 3 5712121 55131; fax: +886 3 5720634. E-mail address:pycheng@cc.nctu.edu.tw(P.-Y. Cheng).

Contents lists available atScienceDirect

Mechatronics

(2)

ellipsoid and acceleration radius with the influence of configura-tion errors. In this secconfigura-tion, the meaning and expression of deterio-ration rate is also interpreted. In Section 4, two examples are utilized to demonstrate the maneuverability deterioration due to the influence of configuration errors in the prescribed workspace. Some conclusions are presented in Section5.

2. Kinematics with conﬁguration errors

2.1. D–H Transformation matrix with conﬁguration errors between two consecutive links

D–H transformation matrix is wildly used in kinematic analysis of the serial type manipulator. Between two consecutive links, link

i 1 and i, D–H transformation matrix can be determined by four D–H link parameters, hi, di, ai, and

a

i, as shown inFig. 2 [14].

From the deﬁnitions of D–H link parameters demonstrated in

Fig. 2and[14], the standard form of D–H matrix,i1_ASDH

i , can be

ex-pressed as(1).

i1_ASDH i ¼

Chi ShiC

a

i ShiS

a

i aiChi Shi ChiC

a

i ChiS

a

i aiShi 0 S

a

i C

a

i di 0 0 0 1 2 6 6 6 4 3 7 7 7 5 ð1Þ

However, when conﬁguration errors exist in the manipulator, the standard D–H transformation matrix can not fully describe the whole system due to the angle error of the rotation about the yiaxis which can not be compensated or covered with

rea-sonable values by other standard D–H link parameters when the orientations of the two consecutive joint axes are parallel or near parallel[15–18]. For this reason, another link parameter, bi, which is the rotation angle about the yiaxis as shown inFig. 2

must be introduced into the standard D–H transformation matrix to redeem this insufﬁciency. It must be emphasized that the necessity of biis only held when the orientations of two

consec-utive joint axes are parallel or near parallel. Except the condition stated above, the standard D–H transformation matrix can fully describe the whole system with conﬁguration errors. The reason for putting biinto the discussion is to get the general form of a

manipulator system with conﬁguration errors. In any case, bi is

always set to be zero, and only its error, Dbi will be used and

discussed.

The modiﬁed D–H transformation matrix,i 1_A

i, aims to redeem

the insufﬁciency stated above and is conducted by post-multiply-ing the standard D–H transformation matrix,i1_ASDH

i , with the

rota-tion homogeneous matrix of bias shown in(2).

i

Link i

Joint i

Joint i+1

Z

X

H

d

a

i i i-1 i-1 i-2 i-2 i i-1 i i i i-1

O

Y

i-2 i-1

Y

i

Y

i

H

i

Fig. 2. Deﬁnitions of D–H link parameters.

(X-X )

Ac

cel

era

tion R

adi

us

(r

i

r)

e

Acceleration Ellipsoid

off Y X off (X-X ) or

(3)

i1_Ai_¼i1_ASDH i Aðyi;biÞ

¼

Chi ShiC

a

i ShiS

a

i aiChi Shi ChiC

a

i ChiS

a

i aiShi 0 S

a

i C

a

i di 0 0 0 1 2 6 6 4 3 7 7 5 Cbi 0 Sbi 0 0 1 0 0 Sbi 0 Cbi 0 0 0 0 1 2 6 6 4 3 7 7 5 ¼

ChiCbi ShiS

a

iSbi ShiC

a

i ChiSbiþ ShiS

a

iCbi aiChi ShiCbiþ ChiS

a

iSbi ChiC

a

i ShiSbi ChiS

a

iCbi aiShi C

a

iSbi S

a

i C

a

iCbi di 0 0 0 1 2 6 6 4 3 7 7 5 ð2Þ where A(yi, bi) is the rotation homogeneous matrix of bi. When biis

equal to zero,(2)is fully equivalent to the standard D–H transfor-mation matrix. When configuration errors exist, the corrective modified D–H transformation matrix that includes configuration er-rors can be expressed as the sum of the original modified D–H transformation matrix and the differential change matrix which is due to the influence of these errors[16]and shown in(3). i1_AC

i ¼i1Aiþ dAi ð3Þ

wherei1_AC

i is the corrective modiﬁed D–H transformation matrix, i 1

Ai is the modiﬁed D–H transformation matrix with nominal

D–H link parameters, and dAiis the differential change matrix due

to the inﬂuence of conﬁguration errors. Because biis always set to

be zero, then Cbi= 1 and Sbi= 0. SettingDhi,Ddi,Dai,D

a

i, andDbi

are the errors of hi, di, ai,

a

i, and bi, respectively. Because these errors

are always much smaller than the nominal design link parameters, dAican be presented as the linear combination of these errors, as

shown in(4). dAi¼@Ai @hi

D

hiþ @Ai @di

D

diþ @Ai @ai

D

aiþ @Ai @

a

i

D

a

iþ @Ai @bi

D

bi ð4Þ Set @Ai @hi¼ Dh i1_A i;@A@dii¼ Dd i1_A i;@A@aii¼ Da i1_A i;@A@aii¼ Da i1_A i, and @A@bii¼ Dbi1Ai,where Dh¼ 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5; Dd¼ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5; Da¼ 0 0 Shi diShi 0 0 Chi diChi Shi Chi 0 0 0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5; Da¼ 0 0 0 Chi 0 0 0 Shi 0 0 0 0 0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5; Db¼

0 S

a

i ChiC

a

i aiShiS

a

i diChiC

a

i S

a

i 0 ShiC

a

i aiChiS

a

i diShiC

a

i ChiC

a

i ShiC

a

i 0 aiC

a

i

0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5 From(4), dAi¼ Dh

D

hiþ Dd

D

diþ Da

D

aiþ Da

D

a

iþ Db

D

bii1Ai ð5Þ Set di1_A

i= DhDhi+DdDdi+DaDai+DaD

a

i+DbDbi, then the

correc-tive modiﬁed D–H transformation matrix can be expressed as(6). i1_AC

i ¼i1Aiþ di1Ai¼i1Aiþ ðdi1AiÞi1Ai ð6Þ where di1_A i¼ 0 Dhi ShiDai ChiD

a

i diShiDai Dhi 0 ChiDai ShiD

a

iþ diChiDai ShiDai ChiDai 0 Ddi 0 0 0 0 2 6 6 4 3 7 7 5 þ 0 S

a

iDbi ChiC

a

iDbi ðaiShiS

a

i diChiC

a

iÞDbi S

a

iDbi 0 ShiC

a

iDbi ðaiChiS

a

i diShiC

a

iÞDbi ChiC

a

iDbi ShiC

a

iDbi 0 aiC

a

iDbi 0 0 0 0 2 6 6 4 3 7 7 5

2.2. Total transformation matrix with conﬁguration errors

From(6), the corrective modiﬁed D–H transformation matrix of two consecutive links can also be presented as(7).

i1_AC

i ¼ ðI þ di1AiÞi1Ai ð7Þ

where I is the 4 4 identical matrix. From(7), the total corrective modiﬁed D–H transformation matrix, TCn, can be presented as(8).

TCn¼ Yn i¼1 i1_AC i ¼ Yn i¼1

ðI þ di1_AiÞi1_Ai _ð8Þ where n is the number of the consisting links.

Because the kinematic deviation due to the error items, di 1_A i,

is relatively small, the influence of the second and higher order terms can be omitted without any significant influence on the re-sult. In the following, only the first order approximation of(8)will be utilized, and it can be presented as(9).

2.3. Jacobian matrix with conﬁguration errors

Jacobian matrix is utilized to map the joint velocities in the joint space into the end-effecter velocity in the world space. In the gen-eral form, it can be shown as(10).

_xn¼

v

n wn

¼ Jnn_qn ð10Þ

where n is the dimension of joint and end-effecter velocities in their corresponding spaces,

_v

nis the linear velocity vector and wnis the

angular velocity vector of end-effecter in the world space, Jnn is

the n n Jacobian matrix, and _qnis the joint velocity vector in the

joint space. In(11) and (12), they show

_v

nand wn, respectively [19].

v

n¼ Xn i¼1 _hiðzC i1i1p C nÞ þ z C i1_di h i ð11Þ TCn¼ Tnþ Xn i¼1 Ti1di1_Aii_Tn_¼ _{I þ}X n i¼1

Ti1di1_AiT1 i1 ! Tn ¼ R C n P C n 0 1 " # ¼ Rnþ Pn i¼1 Ri1driR1 i1 Rn Pnþ P n i¼1 ðRi1driR1 i1Þ PnP n i¼1 Ri1driR1

i1Pi1 Ri1dpi 0 1 2 4 3 5 ¼ I þ Pn i¼1

Ri1driR1_i1

Rn I þP n i¼1

Ri1driR1_i1

PnP n i¼1

Ri1driR1_i1Pi1 Ri1dpi

0 1 2 4 3 5 ð9Þ

(4)

wn¼X n i¼1 _hizC i1 ð12Þ where zC

i1 is the z axis direction of the i 1 frame which is

de-scribed in the reference frame and equivalent to the 3rd column vector of RC

i1;i1pCnis the corrective position vector from

end-effect-er to the origin of the i 1 frame and is also described in the refend-effect-er- refer-ence frame, _hiis the angular velocity value of the ith revolute joint,

and _diis the linear velocity value of the ith prismatic joint.

From the conduction in(9), zC

i1andi1pCncan be presented as

(13) and (14), respectively. zC i1¼ zi1þ Xi1 j¼1 Rj1drjR1 j1 ! Ri1 " # z ð13Þ i1_PC

n¼ Ri1½i1TnPþ Ri1½di1TnPþ dRi1½i1TnPþ dRi1½di1TnP ¼i1_Pn_{þ Ri1} X n j¼i i1_Tj1dj1_Aji1_T1 j1 ! i1_Tn " # P þ X i1 j¼1 Rj1drjR1_j1 ! Ri1 " # ½i1_Tn P þ X i1 j¼1 Rj1drjR1_j1 ! Ri1 " # Xn j¼i i1_Tj1dj1_Aji1_T1 j1 ! i1_Tn " # P ð14Þ where zi 1is the z axis direction of the i 1 frame presented in the

reference frame which is equivalent to the 3rd column vector of Ri 1,i 1pnis the position vector from end-effecter to the origin

of the i 1 frame and also is described in the reference frame, and drjis the rotational part of dj 1Aj. The subscript ‘‘P” means

the translation part of the bracketed transformation matrix and the subscript ‘‘Z” means the z axis direction of the bracketed rota-tion matrix equivalent to the 3rd column vector.

From(10)–(12), the corrective Jacobian matrix can be expressed as(15). JC ¼ JC1;J C 2; . . . ;J C n h i ð15Þ where JCi ¼ zC i1i1pCn zC i1

is for a revolution joint, and JCi ¼

zC i1

0

is for a prismatic joint.

Substitute(13) and (14)into(15)and eliminate the second or-der term, JC

i can be presented as(16).

JC

i ¼ Jiþ dJi ð16Þ

where Jiis the ith column of the nominal Jacobian matrix without

the inﬂuence of conﬁguration errors, and dJi is the differential

change Jacobian matrix due to the inﬂuence of these errors. In(17) and (18), they show Jiand the ﬁrst order dJiof a revolute

joint, respectively. Ji¼ zi1 i1_p n zi1 " # ð17Þ dJi¼

zi1 Peþ zdei1_Pn zde

" #

ð18Þ Similarly, in(19) and (20), they represent Jiand dJiof a prismatic

joint, respectively. Ji¼ zi1 0 ð19Þ dJi¼ zde 0 ð20Þ where zde¼ Pi1j¼1Rj1drjR1j1 Ri1 h i

zis the direction error of the z axis

of the i 1 frame, and Pe¼ Ri1 Pnj¼ii1Tj1dj1Aji1T1j1

i1_T n h i Pþ Pi1 j¼1Rj1drjR1j1 Ri1 h i i1_T n Pþ ð Pi1 j¼1Rj1drjR1j1ÞRi1 h i ðPnj¼ii1Tj1 h dj1_A

ji1T1j1Þi1TnPis the position error due to the inﬂuence of

conﬁg-uration errors from the end-effecter to the i 1 link. 3. Acceleration radius with conﬁguration errors

The ﬁrst and second order differential kinematic equations of the end-effecter of a non-redundant serial manipulator can be ex-pressed as(21) and (22) [2,3]. _xn¼ JCnnðqÞ _qn ð21Þ €xn¼ _JC nnðqÞ _qnþ J C nnðqÞ€qn ð22Þ

where q is the joint variable, x is the position variable of the end-effecter, and JC

nnis the n n corrective Jacobian matrix.

The dynamic equation of a manipulator can be presented as

(23).

s

¼ MðqÞ€q þ cðq; _qÞ þ gðqÞ ð23Þ

where

s

e

Rn _{is the vector of the joint forces, torques, or both,}

M(q)

e

Rnn _{is the symmetric, positive deﬁnite inertia matrix,}

cðq; _qÞ 2 Rnis the vector of the centrifugal and Coriolis forces, tor-ques, or both, and g(q)

e

Rn_{is the vector of the external forces,}

tor-ques, or both.Rearrange(23), the €q can be presented as(24). €

q ¼ M1

ð

s

c gÞ ð24Þ

Substitute(24)into(22), €x can be presented as(25). €x ¼ JC_M1

ð

s

c gÞ þ _JC_{_q} ¼ JCM1

_s

þ ðJCM1_{c þ _J}C_{_qÞ þ ðJ}C_M1_{gÞ ¼ J}C_M1

_s

þ €xoff ð25Þ where €x is the acceleration vector of the end-effecter, JC_M1

_s

_{is the}

acceleration vector which is contributed by the actuation of each consisting joint actuator, and €xoff¼ ðJCM1c þ _JC_qÞ þ ðJCM1gÞ is

the acceleration vector due to the linear velocity, angular velocity, external force and torque exists in each consisting link.

Usually, the output torque or force of each joint actuator has symmetric upper and lower limits and can be expressed as(26).

s

lim it

i 6

s

i6

s

lim iti ; i ¼ 1 n ð26Þ After normalizing, the normalized output vector, ^

s

, can be pre-sented as(27).

^

s

¼ L1

s

ð27Þ

where L is the diagonal matrix which the value of each diagonal ele-ment is equal to the output limit of the corresponding joint actuator and is expressed as(28). L ¼

s

lim it i 0 0 0 .. . 0 0 0

s

lim it n 2 6 6 4 3 7 7 5 ð28Þ

From(26) and (27), ^

s

has the characteristics as shown in(29) and (30).

j^

s

j 6 1 ð29Þ

^

s

T_^

_s

61 _ð30Þ

Substitute(27)into (25)and then rearrange, €x and €

s

can be pre-sented as(31) and (32), respectively.

€x ¼ JM1_L^

_s

þ €xoff ð31Þ

^

s

¼ L1MJ1ð€x €xoffÞ ð32Þ

Substitute(32)into(30), the equation of the acceleration ellipsoid can be conducted and expressed as(33).

(5)

ð€x €xoffÞTJT_MT

LT_L1_MJ1_{ð€x €xoff}_{Þ 6 1} _ð33Þ Substitute Q = JT_MT_LT_L1_MJ1_into₍₃₃₎_{, a simpler form can be}

pre-sented in(34).

ð€x €xoffÞTQð€x €xoffÞ 6 1 ð34Þ The value of acceleration radius is equal to the value of the ra-dius of the smallest inner tangent sphere of the acceleration ellip-soid which is centered in the origin of the reference frame. When €

xoff¼ 0, the acceleration radius is equal to the reciprocal of the

square root of the largest eigenvalue of Q. If the value of the accel-eration radius is less than the prescribed accelaccel-eration at some point of the path in planning or operation, this manipulator may not perform its assigned dynamic task due to the insufficient accel-eration ability at this posture or configuration. This means further dynamic analysis should be performed to judge the assigned dy-namic task at this posture or configuration is not toward these directions without sufficient acceleration ability to assure the as-signed dynamic requirements are achievable.

For reasonably and effectively quantitating the adverse maneu-verability deviation due to the influence of configuration errors and being as the derating margin of the manipulation, a new index, deterioration rate, is proposed and defined in(35).

DRP¼ri re

ri ð35Þ

where DRPis the deterioration rate at a speciﬁc posture or

conﬁgu-ration, riand reare the acceleration radius without and with

conﬁg-uration errors, respectively.

For a small workspace or region, a representative index of der-ating margin of manipulation over this workspace or region is use-ful, and this index can be expressed as(36) [3].

DRW¼ I

W ð36Þ

where DRWis the deterioration rate in a prescribed workspace or

re-gion, I =Rw(dr)dw is the integral of deterioration rate over this

workspace or region, W =Rwdw presents the workspace or region

in discussion, dr is the differential function of deterioration rate, and dw presents the differential of the workspace or region.When in application, the exact derating margin of manipulation at a spe-ciﬁc posture or conﬁguration must be represented by DRP, but when

roughly estimating the derating margin for a region, DRWwould be

a better choice to reduce the cost and effort of judgment. When the prescribed workspace or region is a speciﬁc posture or conﬁgura-tion, DRPand DRWwill be equivalent.

4. Examples

In this section, a two-link planar manipulator and a PUMA 560 robotic arm will be taken as the examples to demonstrate the pro-posed method and the maneuverability deterioration due to the prescribed conﬁguration errors which usually can be found as the uncertainty or tolerance in the product speciﬁcation. From

(34), it is easy to find the influences caused by the centrifugal force, Coriolis force, gravity force, external force and torque just simply shift the center of the acceleration ellipsoid and can be easily rein-troduced into the conducted results by the basic definition of accel-eration radius. To simplify, both examples discussed in this section are assumed in standing, _q ¼ 0 and omitted the influences of exter-nal forces and torques.

4.1. Two-link planar manipulator

The ﬁrst example is to evaluate the maneuverability deteriora-tion of a two-link planar manipulator which usually can be used to simulate the motion of arms or legs of a human being and shown in

Fig. 3. Because the inverse Jacobian matrix does not exist when

Li

nk

1 L

i

n

k2

1

2 X0

Y0

X1

Y1

X2

Y2

Fig. 3. Link parameter deﬁnitions of the two-link planar manipulator.

Table 1

Prescribed inertia properties, errors, and D–H parameters of the two links planar manipulator.

Link i Link parameters Errors

d h a a b Weight Torque limits Dd Dh Da Da Db

1 0 h1 0.7 0 0 3.5 ±5 0 ±0.1° ±0.0007 0 0

(6)

near or in the singular posture or configuration, in this example, it evaluates the maneuverability deterioration over a region which is slightly smaller than the real achievable workspace in the first quadrant. InTable 1, the setting of the inertia properties, configu-ration errors, and link parameters are presented. After taking all the settings presented inTable 1into the proposed method, the deterioration rate of this region is derived, and its value is 0.16% which can be used as the rough estimation of derating margin of this region. The distribution of the deterioration rate is demon-strated inFigs. 4a and b and this distribution can be used as the ex-act reference of the derating margin of any posture or configuration in this region. In Figs. 4a and b, two phenomena can be observed easily. One is when the manipulator gets closer to the singular posture or configuration (the boundaries), no mat-ter full stretched or folded, the demat-terioration rate becomes greamat-ter with the same configuration errors. The other is with the same dis-tance between the end-effecter and base joint, the deterioration rate will also be the same, and this phenomenon implicitly means the value of h1has no influence on the deterioration rate.

To further investigate these observations, the analytical forms of the acceleration radius and its sensitivity with regard to h2of this

example are conducted and shown in(37) and (38), respectively.

Acceleration radius ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP1 P2 P3þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P2 3 P4 0:1225P3 q r ð37Þ

Sensitivity with regard to h2 ¼ @h2 P1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P3þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 P2 3 P4 0:1225P3 q r ¼ P7 P5 2P8P9P6 P3 5 þ P14_P51P15 P2 6 P3 5 þP6 P5 P9þP161 P2 5 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P12 P2 5 þP28P9 P4 5 r 0 B B @ 1 C C A ffiffiffi 2 p P8P9 P2 5 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P12 P2 5þ P2 8P9 P4 5 r 3 2 ð38Þ where P1¼ ffiffiffi 2 p ; P2= 0.35430 0.00286 cos h2, P3= 0.1225 sin2h2, P4= 0.00085, P5= sin h2, P6= cos h2, P7= 0.02332, P8= 8.16327, P9= 0.35430 0.00286 cos h2, P10= 0.11334, P11= 0.19038, and P12= 0.05667.

From(37), it can easily be observed that no h1item exists in the

analytic form, and this means h1has no inﬂuence on the

accelera-tion radius. This matches the prior observaaccelera-tion in this example. In

Fig. 5, it shows the sensitivity of the acceleration radius with re-gard to h2, and the range of h2is from 0 to

p

. FromFig. 5, it also

can easily be observed that when h2 closes to 0 or

p

, it has the

greatest sensitivity. This means when h2closes to 0 or

p

, the same

variation in h2will cause greater deviation in the acceleration

ra-dius. In other words, when this manipulator is in full stretch or fold, conﬁguration errors will have the greatest inﬂuence on maneuverability deterioration, and this matches the prior observation.

Fig. 4b. Deterioration rate distribution of Example 1 (b) Contour plot.

0.5

1

1.5

2

2.5

3 θ

2

Rad

−0.4

−0.2

0.2

0.4 Sensitivity

Fig. 5. Sensitivity of acceleration radius with regard to h2.

Fig. 6. Zero position with attached coordinate frames of PUMA 560.

Table 2

DH link parameters of PUMA 560.

Frame i di(m) hi(°) ai(m) ai(°) 1 0 h1 0 90 2 0.2435 h2 0.4318 0 3 0.0934 h3 0 90 4 0.4331 h4 0.0203 90 5 0 h5 0 90 6 0 h6 0 0

(7)

4.2. PUMA 560

In the second example, it discusses the maneuverability deteri-oration due to the prescribed conﬁguration errors of PUMA 560. The zero position with the attached frames of PUMA 560 is shown inFig. 6, and its link parameters are shown inTable 2. InTable 3, the inertia and joint torque parameters of each link and its at-tached actuator are presented, respectively[20,21]. Because the design purpose of the wrist is dedicated to change the orientation of the end-effecter and not to be the kinetic functions provider, the conﬁguration errors of the wrist will be omitted in the following discussion. Since h1only changes the orientation of the

accelera-tion ellipsoid without changing its shape, the value of h1is

inde-pendent of the acceleration radius [8]. Based on the reasons stated above, in the following discussion, only h2and h3will be

ta-ken as the control variables, and their ranges are speciﬁed in accor-dance with the one which covers most pick and place operations. Except h2and h3, others will be taken as the constants, and their

values will be explained as follows. For h4, h5, and h6, their values

are 41°, 37°, and 16°, respectively for preventing the occurrence of the singular conﬁgurations in the discussed region. Also, in

accordance with the capability of current precise machining tech-nology, the values of the length and the angular errors are basically assigned as the 0.1% of the nominal dimensions and 0.1°, respectively.

For investigating the necessity ofDbi, and the effect of the

mag-nitude of the conﬁguration errors to the deterioration rate, three cases are used to discuss these two issues in the following simula-tions. The link parameters and their errors of each case are pre-sented inTables 4–6, respectively, and the workspace under the discussed joint regions is shown inFig. 7. FromFigs. 8a and 9a, it can be observed that cases 1 and 2 have similar distributions of the deterioration rate, but case 2 has a much lower deterioration rate than the one in case 1. This phenomenon explicitly points out without including the effect ofDbiwill lead to underestimate

the deterioration rate, and this also shows the necessity of Dbi

Table 3

Inertial parameters of each link of PUMA 560.

Link i M (kg) rx(m) ry(m) rz(m) Ixx(kg m2) Iyy(kg m2) Izz(kg m2) Ixy= Iyz= Izx;(kg m2) Torque limit (N m)

1 0 0 0 0 0 0 0.35 0 ±97.6 2 17.4 0.068 0.006 0.2275 0.13 0.524 0.539 0 ±186.4 3 4.8 0 0.070 0.0794 0.066 0.0125 0.086 0 ±89.4 4 0.82 0 0.0203 0.4141 1.8 103 _{1.8 10}3 _{1.3 10}3 ₀ _±24.2 5 0.34 0 0 0.032 0.3 103 _{0.3 10}3 _{0.4 10}3 ₀ _±20.1 6 0.09 0 0 0.064 0.15 103 0.15 103 0.04 103 0 ±21.3 Table 4

The prescribed conﬁguration and its errors of case 1. Link i Parameters Errors

h Dd Dh Da Da Db 1 0° 0 ±0.1° 0 ±0.1° ±0.1° 2 45° to 45° ±0.00024 ±0.1° ±0.00043 ±0.1° ±0.1° 3 95°–135° ±0.00009 ±0.1° 0 ±0.1° ±0.1° 4 41° ±0.00043 0° ±0.00002 0° 0° 5 37° 0 0° 0 0° 0° 6 16° 0 0° 0 0° 0° Table 5

h Dd Dh Da Da Db 1 0° 0 ±0.1° 0 ±0.1° 0° 2 45° to 45° ±0.00024 ±0.1° ±0.00043 ±0.1° 0° 3 95–135° ±0.00009 ±0.1° 0 ±0.1° 0° 4 41° ±0.00043 0° ±0.00002 0° 0° 5 37° 0 0° 0 0° 0° 6 16° 0 0° 0 0° 0° Table 6

h Dd Dh Da Da Db 1 0° 0 ±0.5° 0 ±0.5° ±0.5° 2 45° to 45° ±0.00120 ±0.5° ±0.00215 ±0.5° ±0.5° 3 95–135° ±0.00045 ±0.5° 0 ±0.5° ±0.5° 4 41° ±0.00215 0° ±0.00010 0° 0° 5 37° 0 0° 0 0° 0° 6 16° 0 0° 0 0° 0°

(8)

when evaluating the conﬁguration errors causing maneuverability deterioration of PUMA 560.

When observingFigs. 8a and 10a, these two diagrams also have similar distributions but different amplitudes. This phenomenon also means the magnitude of the conﬁguration errors does inﬂu-ence the deterioration rate, and when the errors are greater, the greater deterioration rate will be.

From the above discussions, no matter the effect ofDbiis

omit-ted, or the configuration errors are magnified, the distributions of the deterioration rates have similar trends but different ampli-tudes. This phenomenon can be deduced that the distribution of the maneuverability deterioration rate due to the influence of

con-figuration errors mainly depends on the concon-figuration of the dis-cussed manipulator, but the actual amplitude is controlled by the magnitude of these errors and the configuration.

Besides, inFigs. 8a, 9a and 10a, one common phenomenon can be observed, and that is when h3gets closer to 95° which is the

out-er boundary of the working region, the detout-erioration rate will be greater. InFig. 11, it shows the two extreme conﬁgurations of Fig. 10b. Deterioration rate distribution in the X–Z coordinates of case 3 (b) In h2

and h3joints space.

Fig. 9b. Deterioration rate distribution in the X–Z coordinates of case 2 (b) In h2and

h3joints space.

Fig. 10a. Deterioration rate distribution in the X–Z coordinates of case 3 (a) In the X–Z coordinates.

Fig. 8b. Deterioration rate distribution in the X–Z coordinates of case 1 (b) In h2and

h3joints space.

Fig. 8a. Deterioration rate distribution in the X–Z coordinates of case 1 (a) In the X– Z coordinates.

Fig. 9a. Deterioration rate distribution in the X–Z coordinates of case 2 (a) In the X– Z coordinates.

(9)

the discussed region which are located on the outer and inner boundaries. From Fig. 11a, it is easily observed that when h3= 90, this robotic arm is in a singular posture due to the full

stretch between link 2 and link 3. Also fromFig. 11b, it can be found that when h3= 135, it is the posture or conﬁguration that

is far from the singular ones, no matter the one in full stretch or fold is and has smaller deterioration rate. From this observation, one deduction can be conducted, and that is even with the same conﬁguration errors, when getting closer to the singular posture or conﬁguration, the deterioration rate will be greater.

5. Conclusions

The existence of configuration errors is inevitable when manu-facturing and assembling a manipulator. When evaluating the maneuverability of a manipulator system without including the influence of these errors, the system maneuverability will be over-estimated and will cause system control failure or other unpredict-able adverse outcomes especially in the high speed or relative heavy loading applications with kinetic or dynamic requirements, e.g. the limbs of a walking robot or a robot arm simulating a ball throw of a human arm. To redeem this insufficiency, this article proposes a systematic method to include the influence of configu-ration errors into the analytical model and proposes an index, dete-rioration rate, to quantitate the maneuverability detedete-rioration due to the influence of these errors. By utilizing this index, it is easy for a control or path designer to assign a reasonable derating margin of manipulation to plan control strategies or trajectories to attain an assigned dynamic task without any potential failure risk caused by the maneuverability deterioration due to the influence of configu-ration errors.

From the observations found in Section4, three conclusions can be conducted. The first one is when getting closer to the singular postures or configurations, the deterioration rate will be greater, even with the same configuration errors. The second is the distri-bution of the deterioration rate mainly depends on the configura-tion, but the actual amplitude is controlled by the magnitude of these errors and the configuration. The last one is thatDbihas its

existence necessity when evaluating maneuverability deteriora-tion due to the influence of configuradeteriora-tion errors. When omitting the influence ofDbi, it will lead to underestimate the deterioration

and may cause some unpredictable adverse outcomes.

Based on these observations, for a control or path designer, it is better to choose the path or working region far from the singular postures or configurations when numerous ones are available to reduce the influence of configuration errors on maneuverability, and if choosing the one close to the singular postures or configura-tions is inevitable, greater derating margin must be kept when the control strategies or trajectories are implemented. From the obser-vations in Section4.1, when the implemented path or working re-gion is short or small and far from the singular postures or configurations, DRWis a good estimation of the derating margin

for the entire path or region. However, when close to the singular postures or conﬁgurations is inevitable, DRpmust be investigated

for all the segments or sections which are close to these singular postures or conﬁgurations to get the information of setting the der-ating margin to them.

In this article, it proposed a systematic method to evaluate the maneuverability deterioration due to the influence of configuration errors, and it also provides an index, deterioration rate, to quanti-tate the maneuverability deterioration due to the influence of these errors. The proposed method and index are useful for a Fig. 11. Two extreme configurations of the discussed region (a) h2= 45and h3= 95(b) h2= 45and h3= 135.

(10)

control or path designer to decide the derating margin of manipu-lation or choose the path or region with less influence of configu-ration errors to ensure the achievement of the assigned dynamic task in a prescribed workspace or region. The examples not only show the relations between the maneuverability deterioration and the configuration errors but also find the guideline to choose the best path or region when numerous ones are available, and all these make a great benefit to the control or path designers when they plan the control strategies or trajectories to achieve an as-signed dynamic task.

References

[1] Tsuneo Yoshikawa. Manipulability of robotic mechanisms. Int J Robot Res 1985;4(2):3–9.

[2] Graettinger Timothy J. The acceleration radius-a global performance measure for robotic manipulators. IEEE J Robot Automat 1988;4(1):60–9.

[3] Gosselin C, Angeles J. A global performance index for the kinematic optimization of robotic manipulator. ASME J Mech Des 1991;113:220–6. [4] Kim Kim-Kap, Yoon Yong-San. Trajectory planning of redundant robots by

maximizing the moving acceleration radius. Robotica 1992;10:195–203. [5] Choi Byoung Wook, Won Jon Hwa, Chung Myung Jin. Evaluation of dexterity

measures for a 3-link planar redundant manipulator using constraint locus. IEEE Trans Robot Automat 1995;11(2):282–5.

[6] Doty Keith L, Claudio Melchiorri, Schwartz Eric M, Claudio Bonivento. Robot manipulability. IEEE Trans Robot Automat 1995;11(3):462–8.

[7] Pasquale Chiacchio. A new dynamic manipulability ellipsoid for redundant manipulators. Robotica 2000;18:381–97.

[8] Kovecses Jozsef, Fenton Robert G, Cleghorn William L. Effects of joint dynamics on the dynamic manipulability of geared robot manipulator, vol. 11. Pergamon Mechatronics; 2001. p. 43–58.

[9] Gravagne Ian A, Walker Ian D. Manipulability and force ellipsoids for continuum robot manipulators. In: Proceedings of the 2001 IEEE international conference on intelligent robots and systems; 2001. p. 304–11. [10] Rosenstein Michael T, Grupen Roderic A. Velocity-dependent dynamic

manipulability. In: Proceedings of the 2002 IEEE international conference on robotics and automation; 2002. p. 2424–9.

[11] Bowling Alan P, Khatib Oussama. The actuation efﬁciency – a measure of acceleration capability for non-redundant robotic manipulators. In: Proceedings of the 2003 IEEE international conference on intelligent robots and systems; 2003. p. 3325–30.

[12] Bowling Alan P, Oussama Khatib. The dynamic capability equations-a new tool for analyzing robotic manipulator performance. IEEE Trans Robot Automat 2005;21(1):115–23.

[13] Bowling Alan P, ChangHwan Kim. Velocity effects on robotic manipulator dynamic performance. ASME Trans Mech Des 2006;128:1236–45.

[14] Denavit J, Hartenberg RS. A kinematic notation for lower-pair mechanisms based on matrices. ASME J Appl Mech 1955;22:215–21.

[15] Hayati Samad A. Robot arm geometric link parameter estimation. In: Proceedings of the 22nd IEEE international conference on decision and control; 1983. p. 1477–83.

[16] Veitchegger WK, Wu Chi-Haur. Robot accuracy analysis based on kinematics. IEEE J Robot Automat 1986;2(3):171–9.

[17] Veitchegger WK, Wu Chi-Haur. Robot calibration and compensation. IEEE J Robot Automat 1988;4(6):643–56.

[18] Caenen JL, Angue JC. Identiﬁcation of geometric and non geometric parameters of robots. In: Proceedings of the 1990 IEEE international conference on robotics and automation; 1990. p. 1032–7.

[19] Tsai Lung-Wen. Robot analysis, 1st ed., vol. 1. New York: John Wiley and Sons; 1999.

[20] Armstrong Brian, Khatib Qussama, Burdick Joe. The explicit dynamic model and inertial parameters of the PUMA 560 arm. In: Proceedings of the 1986 IEEE international conference on robotics and automation; 1986. p. 510–8. [21] Mozaryn J, Kurek JE. Design of decoupled sliding mode control for the PUMA

560 robot manipulator, In: Proceedings of the third international workshop on robot motion and control RoMoCo 2002, Bukowy Dworek, Poland; 2002. p. 45– 50.