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Abstract— Minimizing the task completion time of

manipulator systems is essential in order to achieve high

productivity. In this paper, this problem is dealt with by

utilizing the redundant degrees of freedom (DOF) of a given

task and the tool attachment optimization. For example, in a

vision-based inspection where a camera is held by a

manipulator, the extra DOF can be brought about by allowing

the camera to be translated along its approach axis or rotated

about this axis when capturing images. Furthermore, the

manipulator end-effector position and orientation is optimized

by designing an additional linkage at the manipulator

end-effector which is called a tool attachment. A 7-DOF

manipulator system is used in the simulations to verify the

proposed approach. Results showed that this approach can

minimize the task completion time by about 17% compared to

conducting only motion coordination.

I. INTRODUCTION

are ANIPULATORS

manufacturing due to their accuracy and flexibility.

Typical tasks of manipulators are inspection and spot

welding wherein several goals have to be reached. In

realizing these tasks, there are two factors that are essential

in order to minimize the task completion time: (1) the

redundancy inherent to a given task and (2) the suitability of

the manipulator to the task.

The factor stated in (1) can be related to how goals are

defined as input to the problem of multiple goal task

realization. In [1], for instance, a goal corresponds to a

distinct solution in the manipulator inverse kinematics (IK).

In other works [2]-[3], goals are defined by positions with

corresponding orientations in the Cartesian spaces. In some

research [4], each goal is mapped to a goal group, which is a

set of manipulator configurations. In this study, however,

(1) is defined based on the given task without reference to

the manipulator system used. To expound, consider a

manipulator inspecting an object by capturing images on

some parts of it. In this task, a camera, which is attached at

commonly used in

L. B. Gueta, J. Cheng and J. Ota are with Research into Artifacts, Center for

Engineering, at the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,

Chiba, Japan, 277-8568.(email:{gueta, ota}@race.u-tokyo.ac.jp).

T. Arai is with Department of Precision Engineering, School of

Engineering, at the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo,

Japan, 113-8656. (email: arai-tamio@robot.t.u-tokyo.ac.jp).

R. Chiba is with Faculty of System Design, Tokyo Metropolitan

University, 6-6, Asahigaoka, Hino-shi, Tokyo, Japan, 191-0065.

(email: rchiba@sd.tmu.ac.jp).

T. Ueyama is with DENSO WAVE INCORPORATED, 1, Yoshiike,

Kusaki, Agui-cho, Chita-gun, Aichi, Japan, 470-2297

the manipulator end-effector, can capture images at a

number of candidate positions and orientations along the

approach axis. Such deviation or extra DOF of task can be

made as long as the captured image is usable in inspection.

As for (2), it is possible that a manipulator is mismatched

to a give task since normally the manipulator is assumed to

be given prior to defining a task. A 6-DOF manipulator with

a spherical wrist, for instance, is considered as a

general-purpose manipulator; however, it may have

limitations due to its structure such as link lengths. In

dealing with this limitation, the task-based manipulator

optimization proposes creating a manipulator based on a

given task specification [5]-[8]. The algorithms either

directly calculates the Denavit-Hartenberg parameters of a

manipulator [5], [6] or creates a manipulator based on some

pre-defined modules [7], [8]. Although it is effective in

minimizing task completion time, this optimization is not

practical in manufacturing where variability of tasks is

unavoidable. It could entail large fabrication costs and incur

substantial delay from design to actual system setup.

In this paper, we aim to minimize the task completion time

of a manipulator system by addressing the two

aforementioned factors. One method we employed, which is

referred to as the goal pose optimization, is to utilize the

redundancy inherent to a given task as in the case of

vision-based inspection. Moreover, we propose the addition

of tool attachment, which is a linkage added between the

manipulator end-effector and tool. It is more cost-effective

(i.e., lower fabrication cost) than designing an entire

manipulator. For simplicity and practical reasons, it is

designed as a fixed linkage without moving parts in this

paper. The tool attachment optimization was introduced in

our previous works [9], [10]. Since the manipulator system

(Figure 1) is kinematically redundant, we employ motion

Multiple-Goal Task Realization Utilizing Redundant Degrees of

Freedom of Task and Tool Attachment Optimization

Lounell B. Gueta, Jia Cheng, Ryosuke Chiba, Tamio Arai, Tsuyoshi Ueyama and Jun Ota

M

goals

Object

Rotating table

Tool

attachment

Tool

Approach

axis

E

T

(a) (b)

Fig.1 A manipulator system with a tool attachment which is a linkage

between the manipulator end-effector and tool.

2011 IEEE International Conference on Robotics and Automation

Shanghai International Conference Center

May 9-13, 2011, Shanghai, China

978-1-61284-380-3/11/$26.00 ©2011 IEEE 1714

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coordination as a part of the proposed approach.

II. PROBLEM FORMULATION

As shown in Fig. 1(a) the manipulator system is

composed of a 6-DOF manipulator and a 1-DOF rotating

table. The manipulator has to reach all goals in a specific

order. Concurrently, the table rotates and positions an object

placed on it. The task consists of n number of goals gi,

i=1...n which are located on the object. In executing the task,

the tool pose (e.g. camera position and orientation) has to be

positioned and oriented depending on gi.

The configuration of the manipulator and rotating table at

goal i is denoted as qi = ( i 0... i 6) where i 0 is the table

joint angle and i 1... i 6 are the manipulator joint angles. In

Fig. 1(b), E and T denote the reference coordinate frames of

the manipulator end-effector and tool tip, respectively. In

this paper, the transformation matrix between these two

frames is optimized through the tool attachment

optimization.

A. Goal Pose Definition

Figure 2 illustrates the definition of goal pose. Commonly,

a goal is defined by a position (x, y, z) and an orientation (roll,

pitch and yaw). In this paper, a goal is defined by a position

(x, y, z) with an associated tool approach axis (z-axis). This

definition is distinct in two ways:

1) The goal rotation about the z-axis, a, can be varied for

a certain range of values:

min

where min

maximum values of a.

2) The z value of the goal coordinate along the tool

approach axis can be varied and is bounded by:

z min

where zmin

of z a measured along the approach axis of a given goal. As

shown in Fig. 2, the possible goal poses or locations can be

a a max

a are user-defined minimum and

a (1)

a and max

a z a z max

a are the minimum and maximum values

a (2)

a and zmax

represented as a line along the approach axis with an

associated orientation about that axis.

B. Manipulator End-Effector Pose

With the addition of a tool attachment, the manipulator

end-effector E is adjusted relative to T depending on the tool

attachment design. The tool attachment is defined by the

following parameters: (ExT, EyT, EzT) is the displacement of

the tool tip T relative to the manipulator end-effector E. The

(ET, ET, ET) are the roll (rotation about the z-axis), pitch

(rotation about the y-axis), and yaw rotation about the

x-axis) describing the relative orientation of T and E.

C. Performance Index and Constraints

The total point-to-point motion of the manipulator and

table in reaching all goals has to be minimized and is

calculated using (3).

1

cggcf

ii

where f is the task completion time, c(•) is the motion time

from gi to gi+1 and g0 is the home position of the manipulator

and table before and after task execution.

Minimization of f is subject to the joint and velocity limits

of the manipulator and table. Further, a calculation time limit

is imposed so that solutions are derived within a reasonable

amount of time.

),(),(min

0

0

1

gg

n

n

i

, (3)

D. Design Variables

In summary, the following variables are designed:

A = (zi

B = (ExT, EyT, EzT, ET, ET, ET ), and

qi, i=1…n.

The subscript i corresponds to goal i. The parameter A

corresponds to the optimized goal pose parameters such as

the translation zi

approach axis. The variable zi

axis; thus if it is varied the xi, yi, zi values of gi will change.

The parameter B corresponds to the tool attachment

parameters wherein ExT, EyT,

the tool tip from the manipulator end-effector and ET, ET,

ET are the roll, pitch, and yaw describing the relative

orientation of the tool tip with respect to the end-effector.

The variable qi, i=1…n denotes the manipulator

configurations at gi i=1…n.

a, i

a), i=1...n,

a along and rotation i

a is defined along the approach

a about the tool

EzT describe the displacement of

III. PROPOSED APPROACH

In this section, the problem is first analyzed and the

proposed approach is then discussed.

A. Problem Analysis

Figure 3 illustrates the case when a tool attachment is

added between the manipulator end-effector and tool and is

z

zmina

zmaxa

a

z-axis

a, min

a, max

Tool attachment

Tool

Fig. 2 Goal pose definition. A goal shown as a red dot is defined by a range

of positions from zmin a to zmax a along the tool approach axis (the z-axis) and

a range of rotational values from a,min to a,max about the z-axis.

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rotated about the tool approach axis (i.e., when

The manipulator end-effector position and orientation E

relative to that of the tool tip T can be thought of as a single

vector for a goal defined by a single pose (i.e., position and

orientation). With the goal definition in this paper, the

possible positions of a tool attachment can be described by a

cylindrical surface as a result of the translation along and

rotation about the approach axis. These manipulator

end-effector positions are valid provided that the

manipulator can achieve its corresponding configuration and

no collision occurs. A valid location of E on this surface

depends primarily on the manipulator link lengths and joint

limits. The search space for selecting the optimized goal

pose and the tool attachment design can therefore be thought

of as surfaces of concentric cylindrical shape. In addition,

since the manipulator system is kinematically redundant, the

goal locations as the table is rotated into several angles are

numerous which means that the number of inverse kinematic

solutions is infinite.

The problem is highly-dimensional given the number of

design variables. For n number of goals, the number of

design variables in the goal pose optimization is 2n (i.e., zi

and i

number of unknowns increases with n. In the tool attachment

optimization, the number of design variables is 6 which are

the translation and rotation parameters of the transformation

matrix of T relative to E. Considering the manipulator

system having 7 DOFs, the total number of variables is at

least 9n+6.

In optimization problems, the mathematical expression

for the objective function is necessary to derive optimal or

near-to-optimal solutions. In this study, deriving this

expression is complex due to the nonlinearity in the

manipulator kinematics, the existence of several goals, the

redundancy of the system, and potential collision among the

manipulator, tool, and object. A straightforward solution to

this problem is to treat all the design variables as a single set

and use a monolithic solver or optimizer to find the best

a is varied).

a

a for every goal i). Note that in this optimization the

solution. Such a solution, however, may not consider the

uniqueness of sub-domain problems constituting the entire

problem that may have existing practical solutions.

Furthermore, a monolithic solver may require substantial

calculation time as the number of goals increases.

B. Overview of Proposed Approach

In this paper, we propose to divide the problem into

sub-problems for the purpose of finding a practical solution

within a reasonable amount of time. Figure 4 shows the

proposed approach composed of two stages. Stage 1 is the

goal pose and tool attachment optimizations while Stage 2 is

the motion coordination between the manipulator and table.

The tool attachment and goal pose are first derived; then the

motion coordination is dealt with. Afterwards, the task

completion time is calculated. The optimization is an

iterative process and is terminated if the design time limit

tdesign is reached.

The first stage precedes the second stage because the

former defines the manipulator system kinematics that is

necessary in the second stage to calculate manipulator

configurations, perform collision detection, and compute

task completion time. The motion coordination is important

so that the object can be oriented at a position reachable by

the manipulator. The goals can be located around the object

and may not reachable without motion coordination.

Stage 1 is discussed in Section III-C while Stage 2 is

discussed in Section III-D. The detail of task completion

time calculation is provided in Section III-E.

C. Goal Pose and Tool Attachment Optimizations

In the goal pose optimization, the objective is to derive the

A value (i.e., zi

tool attachment optimization it is to derive the B value (i.e.,

ExT,

mathematical function expressing f is difficult to derive;

therefore a direct method is used in this study. Exploring all

the solutions is not feasible given the size of the search space

(i.e., all the possible solutions). Moreover, due to collision

a and i

a), of every goal i i=1…n while in the

EyT,

EzT,

ET,

ET,

ET). As aforementioned, the

Possible pose of

manipulator end-effector E

T

a

Fig. 3 End-effector pose. The figure is the result of superimposing images

of the manipulator when the tool is rotated about its approach axis.

Fig. 4 Proposed approach

Input parameters Output parameters

yes

Task completion time calculation

-Tool positions and orientations

- Manipulator and table

kinematic properties.

- Design time limit, tdesign .

no

Is calculation

time< tdesign?

Motion coordination

- Tool attachment length

and orientation

- Goal position and

orientation

-Manipulator and table

configurations

Goal pose and tool attachment

optimizations

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occurrences, the search space can have several local minima

which prevent from deriving a good-quality solution. To

avoid such case, stochastic direct methods are applicable. In

this paper, we used the simulated annealing (SA) algorithm.

SA is based on stochastic local neighbor search method. It

is robust despite the search space complexity and can avoid

local minima. It is unique from other algorithms on how it

chooses its neighborhood and selects a worse solution over

the current solution. The probability of accepting a worse

solution is too high at the start and is subsequently reduced

for each phase. The decreasing probability of accepting

solution is analogous to the temperature reduction in the

actual annealing of a metal, which is slowly cooled from a

high to a low temperature to allow the formation of a

high-quality metal. See [11] and [12] for details of SA.

Figure 5 shows the steps in SA. Suppose we denote X as

the set of all design variables of the goal pose and tool

attachment optimizations (i.e., A and B). Moreover, current,

best and new denote the current, best, and new solutions.

The values of f from these solutions are denoted as f current

f new, and f best, respectively. The variable C controls the

degree of accepting a worse solution. Initially, C=Cmax,

which is pre-specified with a high value. Gradually, the

value of C is reduced by a factor Cfactor, a pre-specified value

between 0 and 1. SA iterates (i.e. one execution of Step 3) as

long as the calculation time is less than tdesign. One iteration

consists of several phases; the phasemax value sets the

maximum number of phase per iteration. In Step 3.1, the

neighbor( ) is a function that returns a randomly generated

value adjacent to which is calculated by incrementing or

decrementing the values of the design variables. In Step 3.4,

the function U returns a value between 0 and 1 from a

uniformly distributed random number.

,

D. Motion Coordination

In the motion coordination, the extra DOF (the table

rotation) is utilized to avoid collision while imposing a

straight-line motion in the manipulator configuration space.

A graph search method is employed wherein a stage is

defined as the motion of the manipulator and table from a

previous goal position to the next goal position. The stages

i = 0 and i = n+1 correspond to the position of the

manipulator and table at g0. Every vertex in each stage

represents the goal position rotated by the table at various

angles. The table angle i

variable l defined in (4).

i

d(r) = ( 0,max - 0,min )/r, r1 (5)

where r is a user-defined resolution of the table rotation, d is

the step, and 0,min, and 0,max

minimum search limit values for i

assigned with values ranging from 0,min to 0,max

assigned value, i

manipulator inverse kinematics (IK). If an IK solution

exists, the configuration is said to be valid. For every valid

configuration, the motion time is calculated using (6), which

assumes that a joint achieves its maximum velocity in a very

short time.

c (gi, gi+1)=

6...0

j

where v j,max is the maximum velocity of joint j.

A different motion profile can be considered that takes

into account acceleration; a related work is conducted in

[13] employing a trapezoidal joint motion profile.

In dealing with the multiple IK solutions, the next

configuration of a manipulator when reaching a goal is

selected based on the previous configuration. This method

does not provide a global minimum solution but reduces the

complexity of the configuration search space.

The size of the search space and the method employed in

finding the solution determine the required amount of design

time. With respect to the number of goals and resolution, the

size of the search space is n·r. An exhaustive search method

that evaluates all the possible i

of rn combinations which is exponential with n and can

require a large amount of calculation time.

0 is parameterized into a single

0(l) = 0,min + r·l·d(r) , l{0,1/r, 2/r…1} (4)

are the maximum and

0. The parameter i

0

is

. For every

1… i

6

are derived by solving the

max

(| i+1 j - i j |/v j,max) (6)

0 values can require a total

E. Task completion time calculation

Based on the above-mentioned search space, the joint

angle value of i

neighbor algorithm because it is fast and noting that the

motion coordination is invoked several times (i.e., it is

invoked every time an X value is evaluated). For the nearest

neighbor algorithm, the number of i

n·r, which is linear with n.

In calculating the task completion time based on the

nearest neighbor algorithm, the cl (gi, gi+1) is the motion time

from stage i to stage i+1 at a valid goal location defined by l

and C i is the least motion time at stage i.

)},({ min

10

1 ...0l

1

21

1 ...0

l

i

iil

l

1 ...0

0

for every gi is selected using the nearest

0 values evaluated is

1

ggcC

l

, (7)

2

)},({ minCggcC

l

, (8)

i

CggcC

)},({min

1

1

, (9)

Fig. 5 SA-based algorithm

1. Initialize design variable current.

2. Set best current and C = Cmax.

3. while calculation time < tdesign

3.1. new neighbor( current).

3.2. If fnew< fbest, then best

3.3. =fnew- fcurrent.

3.4. If ≤0, then current new. Else, generate uU.

3.4.1 If u e(-/C), then current new.

3.4.2 Else, do not change current.

3.5. phase=phase+1.

3.6. If phase<phasemax, go back to 3.1.

3.7. C= Cfactor*C.

4. Exit.

new.

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n

n

Cggcf

),(

0

. (10)

The f value is the calculated task completion time as a

result of evaluating the goal pose and tool attachment design

and deriving the manipulator and table configurations in the

motion coordination.

IV. SIMULATION, RESULTS AND DISCUSSION

This section describes the simulations conducted, the

results and discussion. The setting of the important

parameters is provided in Table I. In the values of zmin

zmax

given positions along their corresponding approach axes.

The rotation about the approach axis has limits of min

-180 and max

attachment parameters (ExT, EyT, EzT) is varied from 0 to 300

[mm], which is set based on the initial setting. Figure 6

shows the initial simulation setting wherein the tool is

directly held by the manipulator. The number of goals is

varied from 20 to 40. The positions are located around the

object such that the table has to be rotated so that the

manipulator can be able to reach goals without collision. The

goals are not lying on the object but are instead at a distance

from the object as in the case of an inspection task. The

initial manipulator base placement, although is not

optimized, is derived based on an empirical method, e.g.

70% of manipulator reach [13]. The calculation time limit is

1 hour which is, based on our experience, enough to derive a

good quality solution for the given number of goals. Similar

to typical SA implementation, C is initialized to a large value

C0 (e.g. 100) and is gradually reduced by a factor of 0.99.

The simulation is conducted using an Intel Core Duo

3.0GHz processor with 4GB memory.

a

and

a, gi are varied from -10[mm] to 10[mm] relative to the

a =

a= 180 to cover one full rotation. The tool

A. Evaluation of Proposed Approach and Other Methods

The following methods are compared:

MC: motion coordination only,

TA+MC: tool attachment (TA) with MC,

GO+MC: goal pose optimization (GO) with MC,

TA+GO+MC: TA with GO and MC (the proposed

approach)

MC is the reference method since it is the most basic; the

other methods are combinations of one or two optimizations.

Each method is run 5 times and the best solution of each

method is selected.

Figure 7 shows the performance of compared methods.

For all n settings, the same trend is observed in which

TA+GO+MC (the proposed approach) has the best

performance; the improvement is significant which is on the

average 16.65 % relative to MC. On the other hand, the

performance of TA+MC is 12.00% relative to MC. With

GO+MC, it can be seen that minimal task completion time

can be achieved by properly choosing the goal pose, which

is better than MC by 10.52%. Based on the result of

TA+MC and GO+MC, the tool attachment optimization

provides more flexibility than the goal pose optimization.

For example, the tool orientation can be varied by 3 design

variables using the tool attachment optimization whereas in

the goal pose optimization, only the rotation in the approach

axis is varied. Furthermore, a tool attachment adds

reachability (i.e., ExT,

performing task. The combined effect of TA and GO is

evident in the proposed approach, which achieves the least

task completion time among the compared methods.

Figure 8 shows some snapshot of the animation for

TA+GO+MC. These figures shows the manipulator and

table configurations when reaching goals. We observed that

when the manipulator is moving from one goal to another

goal especially in those goals at the corner of the object, the

tool is being rotated about its approach axis that results to

minimal motion time. This observation is not evident in MC

and TA+MC.

EyT,

EzT) to the manipulator in

E: (0, 0, 600)

T: (0, 0, 500)

B: (-600, 0, 0)

(0, 0, 0)

z

z

x

y

x

y

200

Values are in [mm].

100

Fig. 6 Initial simulation setting

TABLE I

SIMULATION SETTING

Parameters Values

zmin

min

(ExT, EyT,EzT)

(ET, ET,ET)

a

, zmax

a, max

a z a -/+ 10 [mm]

-/+ 90

(0,10..290, 300) [mm]

-180, -170…,170, 180)

a

n 20, 30, 40

0

0.5

1

1.5

2

2.5

3

3.5

4

30 4050

MC

Fig. 7 Performance of compared methods

TA+MCGO+MC TA+GO+MC

Task completion time (s)

20 30 40

Number of goals

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We also performed a simulation to know if integrating the

motion coordination is justified in our proposed approach.

This evaluation is necessary since the calculation time

requirement would be lessened if the motion coordination is

not included in the iteration. We found that the motion

coordination has to be integrated with the tool attachment

and goal pose optimizations in order to derive good-quality

solution. Without integrating the motion coordination, the

table rotation angles can be derived only at the start of the

optimization of the goal pose and tool attachment; the

improvement of this method is minimal (i.e., at most 3%

relative to MC).

The proposed approach can be evaluated to other

manipulator systems. Since the goals in this paper are at a

distance to the object inspected, it will be interesting to

consider more restrictive condition. In addition, instead of

employing simple motion coordination, sampling based

motion planning can be utilized to evaluate the proposed

approach.

V. CONCLUSION

In this paper, the motion of a 6-DOF manipulator and a

1-axis rotating table is minimized in a multiple-goal task.

We propose optimizing the goal and manipulator

end-effector poses. The proposed approach is different from

a conventional method wherein the manipulator end-effector

pose is directly derived based on given goal poses in a way

that the achievable manipulator configurations are

constrained by a given goal pose. As a solution, the goal

position along the tool approach axis and the rotation about

this axis are optimized. In addition, the tool attachment

optimization is employed to optimize the manipulator

end-effector pose. The result showed that the proposed

approach has the best performance compared to other

methods. The reduction in the task completion time is 17%

compared to employing only motion coordination.

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Fig. 8 Snapshots of the result in using the proposed approach with 20 goals. From left to right, the manipulator and table configurations at g1, g3, g4 and g5.

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