robotic task sequencing problem: a survey

Noname manuscript No.(will be inserted by the editor)

Robotic Task Sequencing Problem: A Survey

Sergey Alatartsev · Sebastian Stellmacher · Frank Ortmeier

Received: date / Accepted: date

Abstract Today, robotics is an important cornerstone

of modern industrial production. Robots are used for

numerous reasons including reliability and continuously

high quality of work. The main decision factor is the

overall efficiency of the robotic system in the produc-

tion line. One key issue for this is the optimality of the

whole set of robotic movements for industrial applica-

tions, which typically consist of multiple atomic tasks

such as welding seams, drilling holes, etc. Currently,

in many industrial scenarios such movements are opti-

mized manually. This is costly and error-prone. There-

fore, researchers have been working on algorithms for

automatic computation of optimal trajectories for sev-

eral years. This problem gets even more complicated

due to multiple additional factors like redundant kine-

matics, collision avoidance, possibilities of ambiguoustask performance, etc. This survey article summarizes

and categorizes the approaches for optimization of the

robotic task sequence. It provides an overview of exist-

ing combinatorial problems that are applied for robot

task sequencing. It also highlights the strengths and the

weaknesses of existing approaches as well as the chal-

lenges for future research in this domain. The article

S. Alatartsev,Otto-von-Guericke-Universitat,Universitatsplatz 2, D-39106, Magdeburg, GermanyTel.: +49 391-67-52662E-mail: [email protected]

S. StellmacherOtto-von-Guericke-Universitat,Universitatsplatz 2, D-39106, Magdeburg, GermanyE-mail: [email protected]

F. OrtmeierOtto-von-Guericke-Universitat,Universitatsplatz 2, D-39106, Magdeburg, GermanyTel.: +49 391-67-52804E-mail: [email protected]

is meant for both scientists and practitioners. For sci-

entists, it provides an overview on applied algorithmic

approaches. For practitioners, it presents existing solu-

tions, which are categorized according to the classes of

input and output parameters.

Keywords Robotic task optimization · Robotic task

scheduling · Multi-goal path planning · Traveling

Salesman Problem

1 Introduction

Improving the production efficiency is the key factor of

using robots in industry. The faster the robot moves,

the more it produces, the more efficient the produc-

tion process is. Therefore, a lot of research is concen-trated on making robots more productive: making effi-

cient control [72], optimizing trajectories by optimizing

velocity profiles [9] or end-effector path [4], planning

collision-free paths [60]. Mostly these approaches are

oriented on the movement from one point to another.

However, a vast majority of robot’s applications in in-

dustry consists of a set of tasks: taking a set of pictures

or deburring a set of edges. In that case, the sequence

of tasks and the way the robot is moving between them

greatly influence the overall performance. The easiest

way to solve the task sequencing problem is to model it

as the Traveling Salesman Problem (TSP) [6]. The goal

of the TSP is to find a minimal-cost cyclic tour through

a set of points such that every point is visited once. A

simple example of a task sequencing problem modeled

as TSP is shown in Fig. 1. In this scenario the robot

has to cut five holes and deburr the border of the detail.

Every cutting contour is approximated with a point and

a TSP tour is calculated. However, to meet industrial

requirements, multiple additional factors of the robotic

2 Sergey Alatartsev et al.

Fig. 1 Given a set of tasks {A,B,C,D,E,F} that robot has to visit with its end-effector. The goal is to find the optimal/near-optimal task sequence. Here, the optimal sequence is (A,C,E,D,B,F).

site have to be considered, e.g., obstacle avoidance, re-

dundant kinematics, partial order of the sequence, com-

plicated objective function (e.g., time or energy), a set

of possible robot base locations or complicated task ge-

ometry. All these factors make the optimization process

much more difficult and lead to a large search space.

In this survey, we observe papers that solve the

robotic task sequencing problem taking into account

constraints and freedom of the robot’s site. Researchers

referred to this problem differently. In general, there

are two common names: task sequencing/scheduling [5,

55, 56] and multi-goal path planning [49, 89]. Multi-

goal planning considers obstacles in the environment,

therefore, a collision-free path is the output. Task se-

quencing/scheduling may consider obstacles [95] or may

not [56, 94], therefore, the output could be either a se-

quence or a path. In this survey, we refer to the issue of

optimizing robot site in presence of multiple goals/tasks

as the task sequencing problem. This problem is rele-

vant to both service and industrial robotics. However, it

has greater impact for industrial robotics, as in produc-

tion environments the same tasks have to be repeated

multiple time.

The scope of this survey paper is limited to the task

sequencing problem, so that:

– Algorithms for offline sequencing with known envi-

ronment are considered.

– Tasks have no complex logic relations between each

other, e.g., such relations that are based on the task

geometry and task states.

– Robotic site with a single robot is observed.

– Sequencing is done for robot articulated arm. The

approaches for mobile robotics are included only if

they can be easily applied for robotic-arm scenarios.

– The core part of the survey is concentrated on se-

quencing approaches. Collision-free path planning is

considered to be an extra feature.

There are many other sequencing and planning do-

mains which are outside of the survey scope. We outline

them briefly in Section 2.

There are several recent surveys that cover the re-

lated domains, e.g., multi-robot patrolling algorithms

[74], assembly process planning [88], methods to solve

Traveling Salesman Problem [76] and coverage path

planning for robotics [33]. To the best of our knowl-

edge, there is no survey that covers the task sequenc-

ing problem for articulated robots. This survey provides

an overview of related combinatorial TSP-like problems

that are applied in robot task sequencing. We cover the

major milestones in the task sequence optimization for

the last 25 years. The article is useful for both scien-

tist and practitioners. For practitioners, the problems

are categorized by their input and output parameters.

It would allow them to choose the appropriate method

for a particular industrial task with a desired output:

sequence or path. For scientists, the survey allows to un-

derstand what problems exist, which of them are solved

and what perspective research directions exist.

The remainder of the paper is organized as follows.

Section 2 covers related task planning problems. Section

3 outlines the background in robotics and the TSP-like

problems applied in sequencing. The ways of catego-

rization are covered in Section 4. Section 5 outlines the

general ideas of the reviewed approaches. The conclu-

Robotic Task Sequencing Problem: A Survey 3

sion and outlook to the future work is presented in Sec-

tion 6. The acronyms used in this survey are listed in

the Appendix A.

2 Related Planning Problems in Robotics

The survey is concentrated on task sequencing algo-

rithms. In this chapter, we briefly outline other related

planning problems domains.

2.1 Collision-free path planning

In recent years a significant amount of research has been

focused on collision-free planning. The goal is to calcu-

late an optimal path between two robot configurations.

Optimality is expressed with multiple criteria: minimal

traveled distance, time or energy.

Graph-based: One of the most popular collision-free ap-

proaches is Probabilistic Roadmap (PRM) [54]. PRM

allows to construct a graph by random sampling of

collision-free vertices in space. After the graph is ob-

tained, it is used to navigate without collisions from one

configuration to another. PRM is an example of a multi-

request algorithm, as after the roadmap is obtained it

could be reused multiple times. Graph can also be con-

structed as a grid or with Voronoi diagrams [35]. The

navigation in the graph can be made with known graph-

search techniques, e.g., Dijkstra algorithm or A* [93].

Tree-based: Rapidly-growing Random Trees (RRT) [57]

is a single-request approach that builds a tree in a goal

direction. RRT checks for goal reachability from any

branch of the tree within the tree construction process.

Bidirectional RRT (BiRRT) is a variant of RRT, when

two trees are constructed: one is from the starting po-

sition to the goal and the second tree is from the goal

to the starting position.

Artificial potential fields The general idea of the arti-

ficial potential field is to apply two fields: one that at-

tracts to the goal and another that repels from the ob-

stacles [65]. One major drawback of the potential field

approach, is that it can easily stack in the local opti-

mum. Some methods were proposed to avoid this, e.g.,

application of deterministic annealing [26].

Sample based techniques, e.g., RRT/BiRRT or PRM

work well with high-dimensional spaces. Artificial po-

tential fields are more popular for mobile applications.

After the collision-free path is obtained, it is smoothed

and the desired velocity profile is applied in order to ob-

tain the robot trajectory [60]. There exist many path

planning libraries, e.g., MPK [40], MSL [59] or OMPL

[23]. Further information about collision-free path plan-

ners can be found in the surveys [47] and [52]. For

more details and comparison of the collision-free plan-

ners see [32].

The approaches stated above allow to calculate

collision-free paths between two robot configurations

automatically. However, they do not consider the

optimization of a task sequence. Collision-free path

planning is considered as one of the features of robotic

task sequencing problem.

2.2 Production scheduling

Note, that the problem we discuss here is different to

the scheduling cell optimization problem that is often

mentioned in the economics domain. This problem ad-

dresses optimization of a production facility, i.e., how

many robots should be used, how many parts a pro-

duced detail consist of and how to spread the work

evenly between the robots in order to increase the over-

all cell throughput. Due to the high computational com-

plexity of the problem, collision factors and kinematics

are ignored. As a consequence, solutions are obtained

based on Cartesian distance cost, but not cost of the

robot joints. This kind of research is mostly concen-

trated on how to organize production processes, rather

than how to construct optimal movements for a single

machine or robot [24].

This survey paper concentrates on the motion of a

robot, without involving the information on production

process.

2.3 Multi-robot task planning

In the scenarios with multiple robots, the task have to

be distributed among robots. One of the first who inves-

tigate this problem was Maimon [63]. He proposed the

clustering graph-based method based on cell layout and

robot specification. Later, Zhang et al. [97] proposed

the approach for task allocation for a team of robots.

The approach uses Stochastic Clustering Auction tech-

nique along with Simulated Annealing. This allows to

make optimization not only based on the greedy prin-

cipal, but also make worse steps for purpose, to avoid

local minima. The general principal is to recluster the

tasks until termination condition is satisfied. If the clus-

ter is better than the previous, than it is accepted as

a current by following Simulated Annealing strategy.

Further, this approach was improved to be able to solve


heterogeneous scenarios. The clusters are calculated ac-

cording to different capabilities of different robot mod-

els. Further the efficiency of the approach was improved

by using a Swendsen-Wang method [96]. More details

on multi-robot planning can be found in the survey [74].

In the following of this survey paper task sequencing

approaches for a single robot are presented.

2.4 Task-level planning

In task-level planning, the robot actions are specified

by their interaction with objects. Often the final goal is

known, e.g, “put the object from the box to the table”.

The goal of the task level planning is to find a sequence

of actions that a robot has to perform to modify the en-

vironment from the initial state to the goal state [63].

For example, the solution for this example will consist

of the following actions: 1) “open the box”, 2) “grasp

the object”, 3) “move the object to the table”, 4) “re-

lease the object”. Every single action is then planned

with domain dependent planner. Cao et al. [17] pro-

posed a net called AND/OR used for reasoning about

geometrical task constraints. The general idea is to map

proposed net to Petri net. Then, the solution search

is performed by building a reachability tree from the

Petri net. Later, Chien et al. [18] [19] proposed the ef-

ficient way to incorporate the domain information into

the planner for the indoor robot scenario. The data is

represented with object-oriented model. This model in-

cludes relation between objects, categories and physical

laws. The case study that the solver is capable to solve

was to put all metal parts to the bench in the assembly

room. This type of planning ignores the kinematics and

collision planning. Task-level planning was also applied

to mobile robotics [34], where the hierarchical planning

technique was proposed to reduce the computational

expenses.

In this survey paper, the tasks have no complex logic

relations based on physical laws. In task sequencing,

the tasks are areas that have to be visited, but not

the objects with its handling features as in task-level

planning. However, task sequencing problem may allow

partial order of the sequence.

2.5 Combination of Task-level planning and path

planning

Often, researchers observe task-level and path planning

as a separate problems. However, in real life, the plan

created by the task-level planner often cannot be ex-

ecuted due to the fail of the path-planner. For exam-

ple, the task plan forces the robot to put the cup into

the dish washer, as there is an empty space for it, but

the path-planner cannot construct such a path that al-

lows to perform this motion without collision. In that

case, the task-level planner has to construct a new plan.

Therefore, it is critical to design a task-level planner

in combination with a collision-free planner. Bhatia et

al. [13] proposed the architecture that allows to combine

task-level planning with low-level motion planning by

introducing new synergy level. Gaschler et al. [36] pro-

posed the knowledge volume approach. The main idea

was to make an intermediate stage between continuous

robot motions and symbolic planning.

2.6 Online control-based planning

Sometimes, the constraints for the complex task execu-

tion might be changed online. Sensor might detect these

changes and the control-based approach should reorga-

nize the task execution. Mansard et al. [64] proposed

the approach to sequence tasks to reach the general

goal. The main idea is that a complex task is sepa-

rated into a set of subtasks. The sequence of these sub-

tasks is represented as a stack. The three level control-

architecture is proposed, which maintain the stack of

subtasks to satisfy the constraints. Due to environment

or robot restrictions, controller removes or add a new

task to the stack.

In this survey, the offline approaches for the known

environment are observed.

2.7 Manipulation planning

Another well-known robot related planning problem is

the manipulation planning problem. It occurs when a

robot has to move an object in an environment to the

specific location. The goal of manipulation planning is

to find a sequence of actions the robot has to make to

accomplish this task [60]. Often manipulation planning

is observed in combination with extra degrees of free-

dom or constraints, e.g., manipulation planning with

caging grasp [25] or manipulation of deformable ob-

jects [11]. The further details on manipulation plan-

ning of deformable objects can be found in the follow-

ing survey [50] and for dual-arm manipulation in the

survey [83].

3 Preliminaries

This section describes background on robotics and

related combinatorial problems for modeling task

sequencing.


3.1 Robotic background

Before starting with the optimization, it is required to

specify what the robot pose is and how it is defined. In

general, two spaces are involved in robotics to describe

the robot pose.

T-space – task space SE(3), is used to designate

the position R3 and the orientation of the end-effector

SO(3), i.e., SE(3) = R3 × SO(3). Normally, it is spec-

ified with homogeneous coordinates [60]:

M =

[R3×3 T3×1

01×3 1

]M is the T-space point that denotes the end-effector

pose, where R is a 3×3 rotation matrix that stands for

the end-effector orientation and T is a 3×1 translation

vector that stands for the end-effector position. As de-

scription the end-effector orientation with a matrix is

not intuitive for a human, often a shorter representa-

tion with Euler angles is used [22]. Euler angles depict a

sequence of rotations about the axes of coordinate sys-

tem. For example, to describe any rotation with Euler

angles in 3D space, three angles are required, i.e., one

angle denotes rotation about one axis.

C-space – configuration space C, is the set of

possible robot joint angles. Standard industrial

robots have 6 degrees of freedom (DOF), therefore,

C = R6. Sometimes it is also called joint or axis

space. The relation between T-space and C-space is

described with two mappings. Forward Kinematics

(FK) takes the robot joint angles and calculates the

corresponding end-effector position and orientation:

FK : C → SE(3). Inverse Kinematics (IK) takes the

end-effector position and orientation and calculates the

set of possible robot configurations: IK : SE(3)→ C.

Often, up to 8 solutions exist for a standard 6-DOF

industrial robot. An example of reaching one task

point with two robot configurations is shown in the

Fig. 2.

3.2 Task sequencing in industrial robotics

Today, there exist two different ways to program robots

in an industrial setting: either it is done online or offline.

In online programming, the robot is directly taught a

movement, which it has to replicate later in produc-

tion mode, e.g., with the lead-through method [82]. Ac-

cordingly, the complete movement (e.g., movement on

welding seams or movement between seams) must be

taught. This is a lengthy process, which requires a lot

of intuition and experience if high quality movements

are desired. The quality of the path depends only on

Fig. 2 The T-space point is (x0, y0, z0, α, β, γ). It is reachedwith two robot configurations: (Θ1, Θ2, Θ3, Θ4, Θ5, Θ6) and(Φ1, Φ2, Φ3, Φ4, Φ5, Φ6).

the experience of the programmer and almost no opti-

mization approaches are involved [14,71].

In contrast, an offline approach is typically based

on simulation. The trajectory that has to be executed

by the real robot later in production is calculated or

taught in a simulated environment. This approach has

some major benefits, e.g., the possibility of testing the

calculated path for potential collisions before real exe-

cution – and, therefore, preventing the expensive hard-

ware from being damaged.

Task sequence optimization is either fully ommited

in offline simulation environments (e.g., RobotWorks1)

or presented with limited functionality (e.g., customiza-

tion for DELMIA2 adapted for drilling applications3).

Customization for DELMIA allows automatic sequenc-

ing of drilling tasks, but applies a simple greedy algo-

rithm, i.e., the next drilling task is always the nearest.

This trivial algorithm does not provide good results, as

it often converges to a local minimum. In addition, the

tasks are specified with one entry robot configuration.

In summary, the task sequencing in industry is of-

ten done manually. In the following, we focus on the

sequencing approaches that are applied in an offline

mode.

3.3 Sequencing problems

Tour-searching combinatorial problems build the foun-

dation for task sequencing in robotics. In this section

the related combinatorial problems are described and

1 Compucraft Ltd, RobotWorks www.compucraftltd.com2 Dassault Systemes http://www.3ds.com3 DELMIA V5 Robotic Drilling Application

http://www.delfoi.com/web/products/delfoi products/en GB/drilling-app/


Inp

ut:

pri

mit

ive t

asks

GTSPTSP SSP++SSPSOP

"A before B"

ATSP

A

B

cost(j, i)

cost(i, j)

cost(j, i)

cost(i, j)

Output: sequence of the tasks

Output: path

Inp

ut:

com

ple

x t

asks

CETSP

Output: sequence of the task entry points

TPP

1

2

3

4

5

Traveling Salesman

ProblemAsymmetric TSP Sequential Ordering

Problem

Generalized TSPShortest Sequence

Problem

Shortest Sequence

Problem++

Output: path

MTP

Multi-Goal Path

Planning Problem

Touring-a-sequence-of-Polygons

ProblemClose-Enough TSP

TSPN

TSP with Neighborhoods

SRP

Safari Route Problem

ZRP

Zookeeper Route Problem

MTPGR

MTP for Goal Regions

GTSPN

Generalized TSPN

Fig. 3 Sequencing problems are categorized according to their input and output parameters. The tour/path is denoted withblue and obstacles are denoted with red. The dashed circles denotes the clusters.

categorized by their input and output values. These

problems are illustrated in Fig. 3.

The tasks can be represented as points or finite sets

of points. We refer to such type of tasks as primitive

tasks. The tasks can be also modeled as areas, in that

case we refer to them as complex tasks. For primitive

tasks the output can be a sequence or a path. The path

includes information about the sequence. For complex

tasks, knowledge of the sequence is not enough, as it

is also required to know from which entry point each

task has to be started. Therefore, the output is either

a sequence of entry points or a path.

3.3.1 Input: primitive tasks

Output: sequence of the tasks The most well known

problem of searching for the optimal sequence is the

Traveling Salesman Problem (TSP) [6]. The goal of the

TSP is to find a minimal-cost cyclic tour through a

set of points such that every point is visited once. It

is a NP-hard combinatorial optimization problem and

has been well known in industrial robotics for many

years. The cost between two points in TSP is the same

for both directions. If the distance between two points

depends on the direction, then such problem is called

Asymmetric TSP (ATSP) [42]. Important extension to

ATSP is the constraints on order, e.g., when a point A

has to be visited before B in the tour. This problem is

referred to as Sequential Ordering Problem (SOP) [68].

Shortest Sequence Problem (SSP) is similar to TSP,

but with a difference that the salesman does not have

to return to a starting point [90]. If the points from SSP

are substituted with sets of points, then this problem is

called SSP++. The goal of SSP++ is to find a minimal-


cost tour that contains a point from every set with no

need to return to the starting point. A generalization

similar to the SSP++ exist for the TSP. This problem is

called Generalized TSP (GTSP) [85]. The goal of GTSP

is to find the minimal-cost cyclic tour that contains a

point from every set. GTSP is also known as TSP++,

set TSP or One-of-a-Set TSP.

Output: path Multi-Goal Path Planning Problem

(MTP), a variant of GTSP, was introduced by Wurll et

al. [90] especially for robotics. Each set here is a set of

the inverse kinematics solutions for a T-space goal. The

objective of the MTP is to calculate a minimal-cost

cyclic collision-free path that visit one point from each

set.

3.3.2 Input: complex tasks

Output: sequence of the task entry points In case the

goals are not points but polygons and the sequence

is given, then this problem is known as a Touring-a-

sequence-of-Polygons Problem (TPP) [27]. The goal of

the TPP is to find the minimal-cost cyclic tour that vis-

its a predefined sequence of regions. If the points should

be visited within a certain radius, then this TSP exten-

sion is known as Close-Enough TSP (CETSP) [66]. In

other words, every point in 2D space is represented as

a disk. One of the most general problems is the TSP

with Neighborhoods (TSPN) [7]. The goal of the TSPN

is to find the minimal-cost cyclic tour through a set of

regions such that every region is visited once. If the

neighborhoods are clustered then this problem is called

Generalized Traveling Salesman Problem with Neigh-

borhoods (GTSPN) [87]. The goal of the GTSPN is to

visit at least one neighborhood from each cluster.

Output: path If the TSPN tour must lay inside a bound-

ary then this problem is referred to the Safari Route

Problem (SRP) [69]. If the tour is not allowed to be

within given convex areas but must only visit their

borders, then this problem is referred to as Zookeeper

Route Problem (ZRP) [20]. In case obstacles are inte-

grated into TSPN and areas are convex polygons, then

this problem is referred to as MTP for Goal Regions

(MTPGR) [49]. In particular, this problem is the most

general and computationally difficult to be solved.

3.3.3 Solving methods

One possible way to solve these problems is to for-

mulate them as the Mixed-Integer NonLinear Program

(MINLP) (e.g., solution for TSPN [38]). However, the

exact methods are efficient only for a small number of

tasks. To be able to solve problems with large num-

ber of tasks, heuristic approaches are applied. In con-

trast to exact methods, heuristics do not guarantee

the optimum but often find a near-optimal solution

in a feasible time. Johnson et al. [51] classified TSP

heuristics into two groups: tour-construction (e.g., In-

sertion Heuristic [45]) and tour-improvement heuristics

(e.g., 2-Opt, 3-Opt [15, 77], Tabu Search [41]). While

tour-construction heuristics calculate a feasible solu-

tion, tour-improvement heuristics start with an initial

tour and reduce the cost.

The problems stated above with no appliance

to robotics are well researched and several efficient

solving-methods exist. To name a few: particle

swarm optimization-based algorithms for TSP and

GTSP [81], Lin–Kernighan heuristic for GTSP [53],

tour-construction and tour improvements heuristics for

CETSP [66], Rubber-band algorithm for TPP [27, 70],

approximation algorithms for TSPN [8,30,67].

The presented TSP-like problems and stated

approaches are the foundation for the optimization

of the robotics task sequence. However, applying

only the stated solving methods is not enough for

task sequencing in robotics, as multiple additional

robot specific features have to be considered: inverse

kinematics, multiple degrees of the task specification

freedom, collision-free planners, etc.

4 Categorization

We split the observed approaches for the robotic task

sequencing problem into the following four groups by

their input and output parameters:

Group 1: Input: primitive tasks, Output: sequence

Group 2: Input: primitive tasks, Output: path

Group 3: Input: complex tasks, Output: sequence

Group 4: Input: complex tasks, Output: path

The approaches from the Groups 1 and 3 might be

useful for environments with none or few obstacles that

have small impact on the robot motion. As Group 1 in-

volves less extra parameters to optimize, its approaches

could be more efficient for a large number of tasks. The

approaches from the Groups 2 and 4 are important for

highly-cluttered environments, however the calculation

of the collision-free paths needs more computational ef-

forts.

The summary of the approaches considered in this

study is provided in Table 1. The details of the par-

ticular realization of the observed approaches can be

found in Section 5. The overall optimization process

can involve large number of factors that influence the

sequence. Further, these factors are described in detail.


Table 1 Overview of the observed approaches. The “T-point” and “C-point” stand for the point from T-space and C-spacerespectively. “Cart. path” and “C-path” is the Cartesian and C-space path distance respectively.

Inp

ut

Ou

tpu

t

Ap

pro

ach

Collis

ion

free

Mu

ltip

leIK

Base

layou

t

Part

ial

ord

er

Taskspecification

ObjectiveActual

problemSolved

asApplied

algorithms

Sim

ple

task

s

Seq

uen

ce

[28] − − − + T-point cycle time TSP, SOP ATSP, SOP BB[1] − +a + − T-point cycle time GTSP TSP TSA[29] − + − − T-point cycle time TSP TSP NN[2] − +a − − T-point cycle time GTSP TSP TSA[94] − + − − T-point cycle time GTSP TSP GA[78] − − − − T-point Cart. path TSP TSP Random swaps[10] − + + − T-point cycle time GTSP TSP GA[62] − + − − T-point C-path GTSP TSP GA, SA[92] − − − − T-point complexb TSP TSP ENM, GA[55] − + − − T-point complexc GTSP GTSP Constraint solver

Path

[90] + + − − T-point C-path MTP GTSP, SSP++ GA[89] + + − − T-point C-path MTP GTSP, SSP++ GA, A*[73] + + − − T-point cycle time MTP TSP ENM[84] + − − − C-point C-path MTP TSP IH[80] + − − − C-point cycle time MTP TSP PA[79] + + − − T-point cycle time MTP GTSP PA[44] +d + − − T-point cycle time MTP TSP 2-Opt, LK[16] + + + + T-point cycle time MTP severale ACA[91] + + − − T-point cycle time MTP TSP GA[95] + + − − T-point cycle time MTP TSP GA[48] + + + − T-point complexf MTP TSP MOPSO, PSO, NNA[58] +g + − − T-point cycle time MTP TSP LK

Com

ple

xta

sks

Seq

uen

ce [37] − − − − 2D/3D areas Cart. path TSPN TSPN MINLP[3] − − − − 2D areas Cart. path TSPN TSP, TPP IH, 3-Opt[87] − − − − Rn areas Cart. path GTSPN GTSPN HRKGA[56] −h −h − − 3D volumes complexi TSPN TSP FIH, TS, 2-Opt

Path

[49] + − − − Polygons Cart. path MTPGR MTPGR SOM[43] + + − − T-point+2Dj cycle time MTPGR TSP SA[39] + − − − 7D curves cycle time MTPGR TSPN HRKGA[31] +k −h − − 3D volumes complexi MTPGR TSP FIH, TS, 2-Opt

a two IK solutions are observedb cycle time and welding distortionc cycle time and painting quality metricd using the redundancy of the systeme fixed, unfixed(i.e., TSP) or mixed sequence (i.e., SOP)f cycle time and cost of the candidate manipulatorg only for the end-effectorh applied afterwardsi Lexicographical order of three criteria: minimal cycle time / minimal path length (scanner head) / minimal path length

(laser end point)j certain freedom along z-axis and rotation around z-axisk only for sensor-workpiece, for the whole robot it is applied afterwards.

Collision-free planning It is computationally cheap

to calculate the cost of the point-to-point movement

(PTP) for the robot. However, when the environment

is cluttered with obstacles, often the PTP movement

is not possible to perform and, therefore, a collision-

free path has to be calculated. Integration of the

collision-free planner into the sequencing algorithm is

a challenging problem. The column “Collision-free” in

the Table 1 indicates the sequencing approaches that

consider obstacle avoidance.

Multiple inverse kinematics solutions Often the ar-

ticulated robot could reach a T-space point (e.g.,

drilling point) with several C-space points (e.g., several

robot configurations). Although this greatly increases

the search space and makes the sequencing problem

more difficult to solve, it brings a large potential


for optimization. The freedom of choosing a robot

configuration is used to avoid collisions and reduce the

duration of the movement. The column “Multiple IK”

in the Table 1 depicts the presence of this feature in

the considered approaches.

Robot base layout The location of the robot in the envi-

ronment in relation to the task locations greatly influ-

ences the sequence and vice versa. The main challenge

is that in the worse case the change of the robot loca-

tion requires the recalculation of the whole optimiza-

tion problem. This feature is presented in the column

“Base-layout” in the Table 1.

Task specification Robot tasks vary by their complex-

ity. Often they are modeled in a simple way as T-space

or C-space points. At a first glance, a simple task such

as a hole drilling could be represented with the ap-

propriate robot configuration, i.e., C-space point. How-

ever, this definition is too explicit, as the task could

be reached with multiple configurations. In that case

a T-space point might be a sufficient representation.

Nevertheless, the rotation along the drilling axis is not

important for the manufacturing process, thus it could

be also used for optimization, as it greatly influences the

robot motion behavior. Therefore, even simple robotics

tasks usually bring multiple possibilities for optimiza-

tion. Thus, there is also a group of approaches that con-

sider the task geometry and represent them as 2D-3D

areas, closed-contours, etc.

Partial order Sometimes, the certain task cannot be

performed before another task is accomplished. It is

strongly motivated by the industrial applications, e.g.,

when the robot has to pick up the detail from the con-

veyor and then drill it [28]. The possibility to make the

constraints on the sequence provides industrial robot

programmer with a control on optimization process.

The column “Partial order” in the Table 1 depicts ei-

ther the approach involves partial order feature or not.

Objective The optimization objective in robotic task

sequencing can be minimization of the time, length of

the Cartesian path of the end-effector or the C-space

path as well as specific industrial objectives. The mo-

tion time in the PTP movement is dictated by the slow-

est joint velocity and acceleration. In the approaches,

where the kinematics is not considered, often the Carte-

sian path length has to be minimized. The objective

could be also dictated by the specificity of the man-

ufacturing process, e.g., minimization of the material

distortion caused by welding [92] or minimization of

the laser path length [56].

Problem models When taking several features into ac-

count the researchers might obtain a too complicated

problem that is difficult or impossible to solve within

a feasible amount of time even using the heuristic ap-

proaches. Therefore, often complicated problems are re-

duced to simpler problem models. That brings errors

but makes the optimization feasible. For example, when

solving MTPGR, first TSPN can be solved and then

collision-free paths can be calculated [56]. When solv-

ing TSPN, at first the sequence could be found (i.e.,

TSP) and after that the position of the points (i.e.,

TPP) [66]. Therefore, the actual problem obtained by

combining several features is often different to the se-

quencing problem that is solved at the end. See column

“Actual Problem” and “Solved as” in the Table 1 re-

spectively.

5 Description of the approaches

5.1 Input: primitive tasks, Output: sequence

One of the first research works that discovered the ap-

plication of TSP-like problems in industrial robotics for

PTP movements was done by Dubowsky et al. [28].

ATSP was applied instead of TSP, as in robotics the

cost between two points depends on the direction of

the movement due to gravity and kinematics. The gen-

eral idea of the proposed method is to manipulate the

weights in the cost matrix in order to satisfy the de-

sired constraints on the sequence. Scenarios with no

constraints or with partial constraints on the sequence

are considered. Application of the sequence constraints

was motivated by the industrial scenario, when at first

the robot has to pick up the detail from the conveyor

and after that drill it. The cases when the robot has to

change its tool during the sequence execution were inte-

grated into the planning. The branch-and-bound tech-

nique [61] was applied to obtain the exact ATSP solu-

tion. Nevertheless, the method proposed by Dubowsky

et al. is independent from the particular ATSP solv-

ing algorithm. The main limitation is that no multiple

inverse kinematics solutions were taken into account.

Therefore, an obtained sequence is far from being opti-

mal in real-life applications.

Abdel-Malek et al. [1] improved a manufacturing

cell, in which a 3-DOF robot has to solve a TSP problem

while considering the multiple solutions of the inverse

kinematics. TSP was solved with Traveling Salesman

Algorithm (TSA) [46]. Optimization of the robot base

location was performed to minimize the cycle time. For

that a grid search scheme was applied. It limits the

problem to the X and Y coordinates, which means that

the robot location can not move vertically in the cell.


The robot is placed on every point of the grid and the

cycle time is calculated. The position with the shortest

cycle time is considered to be the optimal location for

that grid. However, there is a trade-off between accu-

racy and computation time when choosing the granu-

larity of the grid. Later Abdel-Malek et al. [2] extended

the approach with support of several types of robots:

cartesian, cylindrical, spherical, articulated.

Edan et al. [29] proposed the approach for robot

task sequencing. It was motivated by the fruit picking

scenario. The main idea of the approach is straight-

forward: calculate the cost matrix and then apply NN

algorithm to solve TSP. The goal of the approach is to

minimize the cycle time while considering both robot

kinematics and dynamics. It was suggested than the

approach can be also applied for the task-based robot

choice. As far as the algorithm requires the full cost

matrix, it makes the integration of collision-free plan-

ning unfeasible for real life scenarios, due to the need

of extremely large computational time.

Zacharia et al. [94] presented a method based on the

Genetic Algorithm (GA) and involved multiple inverse

kinematic solutions in the optimization process. The

method is capable of searching the sequence for a 6-

DOF robot in a feasible time. In order to take multiple

inverse kinematics solutions into account, an innovative

encoding for the GA was introduced. The GA chromo-

some consists of two parts. The first part encodes the

sequence and the second part encodes the robot config-

urations for the correspondent sequence.

Later Baizid et al. [10] extended the approach by

Zacharia et al. [94] with a robot base layout optimiza-

tion. It was done by adding a third component into the

chromosome that denotes the position of the base.

Although applied heuristic approaches in [94] [10]

allows to efficient search space exploration it does no

guarantee optimal solution to be found. Such iterative

search heuristics can be stopped by the exceeding the

maximum number of iterations, therefore, it does not

reflect how far is the obtained solution from optimum.

Reinhart et al. [78] proposed an algorithm for

solving the sequencing problem for open-end curves

in remote-laser-welding applications. At first, every

curve is represented with one point and a random tour

is constructed. Then random swaps are performed to

improve the tour. Later the direction on the curves are

randomly chosen. The algorithm randomly changes the

direction to obtain a better tour. Although, a certain

improvement of the solution can be obtained, random

swaps efficiency should drop down with computation

time.

Loredo-Flores et al. [62] developed a GUI that al-

lows users to manually create the robot’s path. Later,

the constructed sequence is optimized by GA and Sim-

ulated Annealing (SA) approaches. The output is the

sequence of robot configurations with the objective to

minimize the robot joint displacement, i.e., C-space dis-

tance.

Yang et al. [92] considered the task sequencing prob-

lem for welding applications with the objective to mini-

mize the time and the distortion of the welding process.

The general idea is to split the tasks into two categories

by their influence on the product quality, i.e., welding

deformation. The sequence of tasks from the category

with minimal influence is optimized by an Elastic Net

Method (ENM) [73] to minimize the cycle time. The

category with high influence on deformation is opti-

mized with a GA to minimize the product distortion.

The main idea of ENM is to apply two forces on the

net: one is trying to keep the points closer to each other

and another force attracts them to the goal positions.

When two optimized sequences are obtained, they are

merged with the nearest neighbor criterion. The pro-

posed approach was evaluated in a 2D environment, as

the calculation of the distortion in 3D is much more

complicated problem. However, the general ideas are

domain-independent and, therefore, can be reused in

other domains.

Kolakowska et al. [55] motivated robot task

sequencing with painting scenario. The proposed

approach is strictly aim at solving task scheduling

problem and completely independent from the task

and motion planning stages. The main contribution

of the paper is the mathematical constraint model,

that can be further used in any constraint solver.

Proposed approach is efficient for scenarios with up

to 10 goals. The downside of solver application is the

large computational time for larger case studies. Due

to the large search space, the solver was not able to

finish calculation of the solution in some scenarios

within 40 hours.

5.2 Input: primitive tasks, Output: path

Wurll et al. [90] introduced Multi-Goal Path Planning

(MTP). The cost matrix for MTP is almost impossible

to pre-calculate because distances should be obtained

by taking obstacles into account and collision-free plan-

ners are computationally expensive algorithms. There-

fore, the costs are unknown and have to be obtained

on the solving stage. The solution process was split

into three stages: controlling, path planning and short-

est sequence planning. Controlling is the stage where

the decision is made, which pair of start and goal con-

figurations should be sent to the path planning stage.


They proposed four different strategies for the control-

ling stage. The collision-free path planning between two

configurations is calculated in a C-space grid by using

the best first search method. In the third stage, the

approach uses a GA to solve the TSP. The proposed

approach loops in these three stages and guarantees

that every newly obtained tour will be not worse than

the previous one. Later Wurll et al. [89] extended the

approach by parallelization of a bidirectional search to

reduce computation time and used the A* search for

collision-avoidance method.

Petiot et al. [73] solved the task scheduling problem

for industrial robots with an ENM. The collision avoid-

ance feature was integrated as another repulsive force

into the ENM, i.e., a penalty term in the ENM function.

The limitation of this approach is that obstacles must

be convex and easily representable in the C-space. The

algorithm performs well on robots with small number of

DOFs (e.g., with 2 or 3). But it is difficult to generalize

the approach for robots with high number of DOFs, as

it becomes computationally expensive to map obstacles

to the robot configuration space.

Spitz et al. [84] proposed a general framework suit-

able for a variety of robots. Domain specific knowledge

could be integrated into the planner, as a local plan-

ner or an extension of the roadmap created with PRM.

Based on this roadmap, another graph is created, which

consists of only target points and all superfluous points

are deleted. It is then used for tour optimization with

known TSP solvers. The method assumes that task is

reachable with one robot configuration, that limits po-

tential of optimization due to committing the kinemat-

ics redundancy.

Saha et al. [80] observed the problem of multi-goal

planning among the robot configurations. The use of

C-space points instead of T-space points limited the

potential freedom for optimization. Therefore, the ap-

proach was improved by specifying the input points in

T-space [79]. Lower-bound approximation is used for

computing the optimal path (normally the shortest dis-

tance) and the call of the collision-free path planner is

delayed. A bidirectional tree-expansion PRM was used

for collision-free planning. For TSP solving the Prim’s

algorithm was applied [21]. The main assumption of this

approach is that calculation of a sequence is computa-

tionally less expensive than calculation of the collision-

free movements. Therefore, collision-free paths are cal-

culated after the sequence is obtained. If the collision-

free path cost is significantly higher than the estimated

cost, the sequence should be recalculated taking the

newly obtained path cost into account.

Gueta et al. [44] proposed a method for solving

the TSP problem for goals located on a revolvable ta-

ble. The table and the robot are seen as one redun-

dant system. The goals are put into clusters and a se-

quence of these clusters is calculated with the 2-Opt

algorithm. Then for each cluster two goals are deter-

mined as connectors for the previous and following clus-

ters in the sequence. After that the TSP is solved with

Lin–Kernighan (LK) heuristic in each cluster. The dis-

tance between the goals is considered as a straight line

in C-space. The system redundancy is used to select

such configurations of the robot that allow to obtain a

collision-free path.

Bu et al. [16] presented a method for the optimiza-

tion of the robot base layout and robotic task sequence.

The objective is the minimization of the cycle time. The

input sequence could be either fixed (linear topology),

unfixed (complete directed graph topology) or fixed and

unfixed sequence (mixed topology). The first step of

the solution process is to calculate the 5D space of the

possible base locations, where 3D stands for position,

1D stands for orientation around the base axis and 1D

shows if the base axis is upward or downward. Then

this 5D space is divided into discrete grid cells. Opti-

mization of the sequence is done with an Ant Colony

Algorithm (ACA) for each cell. The next step is to find

the local optimum for the robot base position and ori-

entation with the pattern search for each cell in rela-

tion to the costs of the obtained sequences. Finally, the

global optimum is the result of comparing all local op-

tima. The approach does not provide automatic smart

technique for collision avoidance. Obstacle-avoidance is

organized in a way that when the collision occurs be-

tween the robot and a workpiece, an intermediate frame

should be added between the two target frames.

Xidias et al. [91] investigated the problem of de-

termining the optimal sequence of task points for an

articulated robot with the obstacle avoidance feature

in a 2D environment. The problem is divided into the

optimal sequencing problem – TSP and the path plan-

ning problem. The objective is to obtain the shortest

cycle time. The proposed concept is an adaptation of

the GA, proposed by Zacharia et al. [94]. Furthermore,

they introduce the Bump Surface concept for collision

checking. The general idea of the Bump-surface is to

represent the 2D working environment in a unified way

by mapping it into a 3D space. It is done by discretizing

the 2D map into a uniform grid of points and assigning

a value from [0, 1] to each point. If the point is out-

side the obstacle then the value is 0 and if it is inside

then the value is from the range (0, 1]. The search space

is represented as one mathematical entity, a B-spline.

Therefore, the complexity of the problem does not de-

pend on the complexity of the environment (shape and


location of obstacles). The limitation of this approach

is that the working environment is considered to be 2D.

Further, Zacharia et al. [95] extended two ap-

proaches: [94] by adding obstacle avoidance and [91]

by adapting it to the real-world 3D environment.

The optimization problem was solved with the GA.

The chromosome was extended with the information

about the intermediate configurations required for

collision-free movement. The search space is repre-

sented with the Bump-surface as a single mathematical

entity. The innovative characteristic of the proposed

approach is that sequencing, motion planning and

obstacle avoidance are incorporated into the objective

function. It leads to the situation, when all paths

(both collision and collision-free) are considered to be

feasible but collision-free paths are preferred by the

search algorithm.

Huang et al. [48] observed the task of selecting a

specific manipulator for the particular multi-goal task

planning in a system that consists of a robot arm and

a position table. The input parameters for the algo-

rithm are the list of points and the candidate manip-

ulator specification (Denavit–Hartenberg parameters).

As it is not possible to explore the whole search space

within feasible time, the authors decomposed the prob-

lem into the three nested stages: 1) manipulator se-

lection (solved by Multiple-Objective Particle Swarm

Optimization (MOPSO)), 2) layout design (solved by

Particle Swarm Optimization (PSO)) and 3) motion

planning (solved by Nearest Neighborhood Algorithm

(NNA)). Less computationally expensive algorithms are

chosen for the inner loops (i.e., stage 2 and 3) due to the

fact that stages are nested in each other and, therefore,

the inner loops have to be repeated more often than

outer loops. The output is the manipulator structural

configuration as well as positions of the manipulator

and rotational table.

Lattanzi et al. [58] proposed the extension to the

LK heuristic by integrating collision-free planning into

the cost function. As it is computationally expensive

to obtain a collision-free path for the whole robot, the

collision-free path was calculated only for the robot’s

end-effector. The robot end-effector path between two

Cartesian points is the shortest (in terms of robot mo-

tion time) when the robot performs PTP movement. It

was necessary to make sure that this PTP movement

does not cause collision of the end-effector. PTP joint

path is transferred into the end-effector path with FK

and represented as a poly-line. In case of a collision, an

intermediate path-point is inserted and PTP movement

is recalculated. As LK is a tour improvement heuristic,

it was evaluated with different tour construction heuris-

tics: the shortest/farthest insertion and the greedy ap-

proach. The work was motivated by a visual inspection

task in an industrial setup.

5.3 Input: complex tasks, Output: sequence

Within the last five years, the use of task shape poten-

tial for optimization attracted a lot of attention. Fur-

ther these approaches are discussed in detail.

Gentilini et al. [37] used TSPN for modeling the se-

quencing problem of camera inspection tasks. The robot

has to take a picture with a camera mounted on its

end-effector. The problem does not require a precise

position but rather an area from which the pictures

have to be taken. TSPN problem was formulated as

Mixed-Integer Non-Linear Program (MINLP). It was

shown that searching for an exact solution requires un-

reasonable amount of time. Therefore, a heuristic was

introduced to a MINLP solver to speed up the calcula-

tion time. The method was evaluated on test instances

with up to 16 areas. Although the obtained solutions

are very close to the optimum (the average error is less

than 0.6% on tests with up to 16 areas), the approach

omits collision-free path planning. Furthermore, the op-

timization process is based on the Euclidean distance

metrics, thus the robot kinematics flexibility is not con-

sidered.

Alatartsev et al. [3] used TSPN for modeling the

problem of sequencing closed-contour tasks. The work

was illustrated with the industrial use case when the

robot has to cut out a set of holes from a toboggan.

A tour construction method – Constricting Insertion

Heuristic was proposed to solve TSPN. The main idea

was to slit TSPN into TSP and TPP and solve them si-multaneously. The evaluation showed that the solution

could be obtained within a short period of time for a

large number of contours (e.g., 30s. for 150 contours).

However, the approach did not make use of the robot

kinematics, and applied only Cartesian distance as a

metric. Later, the quality of the solution was improved

by developing a tour improvement heuristic based on

3-Opt [5].

Sequencing of tasks with extra freedom was investi-

gated by Kovacs [56]. This problem was motivated by a

specific industrial application – robot remote laser weld-

ing. An efficient combination of the TSP solver (Far-

thest Insertion Heuristic, Tabu search and 2-Opt) and

the path planning method was proposed. Collision-free

path planning is not considered during the sequencing,

but rather done after the sequence is calculated. The

planning is done in Cartesian space and after that the

solution is converted into the robot configuration space

by using inverse kinematics transformation. The min-

imization objective is the lexicographical order of the


three criteria: minimal cycle time / minimal path length

(scanner head) / minimal path length (laser end point).

Vicencio et al [87] proposed GTSPN problem. It is

the extension of the well-known TSPN problem, where

the neighborhoods can be clustered and each cluster

should be visited one. The A Hybrid Random-Key Ge-

netic Algorithm (HRKGA) was proposed to solve the

GTSPN. It was shown that HRKGA is capable of effec-

tive and efficient solving of scenarios up to 300 neigh-

borhoods. The algorithm was tested on 3D and 7D in-

stances. The author stressed the importance of the dy-

namics integration into sequencing. Due to the problem

complexity, the collision-free planning was omitted.

5.4 Input: complex tasks, Output: path

Faigl et al. [49] presented the multi-goal path planning

problem for goal regions (MTPGR) in the polygonal

domain. The areas could be arbitrary shapes and are

allowed to overlap. The proposed approach is based on

self-organizing map (SOM) algorithm and was evalu-

ated in a 2D environment. The presented method ap-

pears to be very efficient and is capable of solving test

cases with up to 106 areas in less than 6 seconds.

Later Gueta et al. [43] extended the previously

described approach [44] by representing the goals

as T-space points with two parameters that provide

additional freedom: 1) distance of the camera from

the goal point along the approach vector and 2)

rotation about the approach vector. In addition, the

tool attachment was optimized using a Simulated

Annealing (SA) method.

Gentilini et al. [39] used the TSPN to model the task

sequencing problem of the camera inspection tasks for

the case when the inspected object is located on the

rotated table. The neighborhoods (sets of robot con-

figurations for each goal placement) are represented as

curves in the seven-dimensional configuration space (6D

is the robot configuration space and 1D is added by the

table rotation). The paper concentrates on exploring

the redundancy of the system, rather than kinematics,

therefore, only one configuration was chosen among the

set of possible inverse kinematics solutions. Due to the

search space explosion, the exact methods are limited to

a small amount of goals. Therefore, the heuristic Hybrid

Random-key Genetic Algorithm was applied. A single

query BiRRT was used to construct the roadmap that

is further used for collision-free planning.

Erdos et al. [31] proposed the improvement of the

previous work of Kovacs [56]. The objective was the

same – minimization of the cycle time of the remote

laser welding operations. During the planning stage

only the scanner-workpiece was checked for collision,

but not the whole robot. Due to the search space fast

growth, collision-free path planners for the whole robot

have to be applied only after the sequence is obtained.

6 Conclusion

6.1 Summary

This paper surveys the approaches for the task sequenc-

ing problem in robotics. The sequencing of tasks with

primitive geometry attracted a lot of research atten-

tion, in contrast to the sequencing of tasks with com-

plex geometry. However, complex task are more com-

mon in industry. There is still no approach that is able

to efficiently optimize task sequence and consider all

degrees of freedom and constraints, e.g., collision-free

planning, multiple-IK, robot base layout optimization,

making use of the task specification fleixibily (see Sec-

tion 4). The fast growth of the search space makes the

optimization a very challenging problem. Thus, appli-

cation of the exact methods is problematic. Therefore,

the use of heuristics is preferred.

Mainly, two general concepts were applied: de-

composition and metaheuristics. In the first case,

researchers decomposed the complicated problems into

simple ones that can be solved efficiently: consecutive

solving of subproblems (e.g., [56, 79, 89]), solving of

nested subproblems (e.g., [3, 48]) or parallel solving of

subproblems and merging afterwards (e.g., [92].) Meta-

heuristics are problem-independent algorithms that are

capable of efficient search space exploration and finding

near-optimal solution. Metaheuristics are proved to

be efficient for tour-searching problems and as a

consequence suit well for task sequencing in robotics.

Researchers preferred to use GA (e.g., [39, 94]), ENM

(e.g., [73, 92]) and SA (e.g., [43, 62]). General trends

to obtain collision-free paths are: grid decomposi-

tion [90,91], sample based planning (e.g., PRM or/and

RRT) [39,79,84], redundancy of the system [44].

MTPGR (see Fig.3) is the most general problem

that is able to describe the majority of tasks. However,

it is too complicated to be solved for articulated robots.

In some industrial scenarios its use can be excessive. For

example, for many welding scenarios the probability to

collide with obstacles is small because tasks are located

roughly in one 2D plane. Therefore, there is no need

to make excessive collision-free planning, as the move-

ments between seams can be performed as “up - move

horizontally - down”. When there is no need to return

to the current position SSP or SSP++ might provide

good results. These problems in particular suit better

to service robotics, where jobs are rapidly changing and


after performing one task sequence the robot has to im-

mediately start with the next one without returning to

the starting configuration.

6.2 Future Challenges

Further research is clearly needed in the task sequenc-

ing domain. In this section we highlight the possible

directions.

High DOF robots It was already demonstrated that re-

dundancy, i.e., high DOF systems, could be used for

collision avoidance [43] [44] and in general leads to more

effective solutions. One future direction is to observe the

case when robots have a large or even infinite number

of inverse kinematics solutions. It greatly increases the

complexity of the problem, but brings a huge potential

for optimization.

Moving goals and obstacles Mainly, stated approaches

observe static goals and obstacles. In presence of a con-

veyor or other moving manufacturing devices the task

sequencing becomes more challenging. Wurll et al. [89]

stated importance of the moving goals scenario and sug-

gested it as a future research direction. This requires to

include the time dimension into the solution process.

Additional research is needed in this direction as it is

not clear how the moving laws will influence the com-

plexity of the problem.

Kinodynamic task sequencing Dubowsky et al. [28] in

1989 proposed ATSP to be the most suitable modeling

problem for robotic task sequencing, because the costs

of the movements “A to B” and “B to A” are differ-

ent due to the robot dynamics and gravity. However,

after that, the researchers considered robot kinemat-

ics but ignored robot dynamics. Till now TSP is the

most popular combinatorial problem for modeling the

robotic task sequencing problem. The exception is Edan

et al. [29] who included robot dynamics into the se-

quencing. The importance of the dynamics integration

was stated in [16] and [87]. Although, integration of the

robot dynamics increases the search space and compu-

tational time, it allows to utilize robots in production

much more efficiently. Involving the dynamics into the

task sequencing problem and, therefore, making use of

ATSP is still an open research question.

Constraints on sequence order The majority of the ob-

served approaches assume that tasks are independent

from each other within a sequence. Nevertheless, many

industrial robotic applications impose constraints on

order. There are cases, when a certain task has to be

performed before or after another one in the sequence.

Among the observed approaches, only two methods

(i.e., [28] [16]) considered the possibility to set sequence

constraints. This issue was also referred in [79] as a

future problem to be solved. The research in this

direction is needed as it will allow to broaden the

application domains to the scenarios that require the

constraints on order. It will also allow industrial robot

programmers to partially control the optimization

process.

Evaluation and benchmarks There are test instances

for benchmarking TSP-like problems, e.g., TSP4,

TSPN5,6, CETSP7. There is some research on evalua-

tion of collision-free planners [32]. However, there are

still no benchmarks for the task sequence optimization

problem, though it is known for many years. It is

difficult to compare existing approaches, as researchers

use different industrial case studies, robots, included

features and constraints. Nowadays, such platforms

as ROS [75] allow high re-usability of the code and,

therefore, might be used as a base for creating bench-

marks. Nevertheless, it is difficult to create a general

benchmark, therefore, domain dependent test instances

can be the first step, e.g., instances for spot-welding,

palletizing or deburring domains. Developing of the

benchmark for the task sequencing in robotics is still

an open research question.

Considering more freedom in task description Within

the last 5 years, sequencing of tasks with complicated

shapes attracted a lot of research attention. However,

there is still a lot of potential for improvement. The

observed approaches provide the freedom for a particu-

lar industrial tasks, e.g., closed-contours cutting [3] or

laser-welding [56]. But still, there is no general frame-

work that allows a general task description and opti-

mization techniques that are independent from the par-

ticular industrial task. At the same time, approaches

for collision-free “goal-to-goal” planning allow specifi-

cation of goals with a certain freedom, e.g., [12], [86].

This experience might be reused for creating a general

framework for the robot task sequencing problem.

“Super” TSP problem It is clear that TSP problem is

not enough for effective modeling real-life applications.

Therefore, researchers formalized TSP-like generaliza-

tions. We outlined TSP-like problems for robotic task

4 http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/5 https://cse.cs.ovgu.de/cse/robotics/tspn/6 http://wpweb2.tepper.cmu.edu/fmargot/ampl.html7 http://drum.lib.umd.edu/handle/1903/9822


sequencing in 3. The most complex and general problem

is MTPGR. However, even such a wide range problem is

not enough to describe real-world problems in robotics.

This problem clearly misses three features introduced in

other TSP-like problems: 1) sequence constraints from

SOP, 2) asymmetrical costs from ATSP and 3) areas

clustering from GTSPN. The generalization that would

incorporate the MTPGR+SOP+ATSP+GTSPN in one

“Super” TSP problem is a very challenging issue. How-

ever, it would allow to model the robot task sequencing

problem much more precise.

Acknowledgements The final publication is available atSpringer via http://dx.doi.org/10.1007/s10846-015-0190-6.Expected to be published in: Journal of Intelligent & RoboticSystems. DOI: 10.1007/s10846-015-0190-6

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Appendix A

The list of acronyms used in this survey is stated in

Table A.1.


Table 1 List of acronyms

ACA Ant Colony AlgorithmATSP Asymmetric TSPBB Branch and BoundBiRRT Bidirectional RRTCETSP Close-Enough TSPDH DenavitHartenberg parametersDOF Degrees of FreedomENM Elastic Net MethodFIH Farthest Insertion HeuristicsGA Genetic AlgorithmGTSP Generalized TSPGTSPN Generalized TSPNHRKGA Hybrid Random-Key Genetic AlgorithmIH Insertion HeuristicLK Lin-Kernighan heuristicMINLP Mixed-Integer Non-Linear ProgramMOPSO Multiple-Objective PSOMTP Multi-Goal Path PlanningMTPGR MTP for Goal RegionsNNA Nearest Neighborhood AlgorithmPA Prims AlgorithmPRM Probabilistic RoadMapsPS Pattern SearchPSO Particle Swarm OptimizationPTP Point-to-pointRLW Remote Laser WeldingRRT Rapidly-growing Random TreesSA Simulated AnnealingSOM Self-organizing MapSOP Sequential Ordering ProblemSRP Safari Route ProblemSSP Shortest Sequence ProblemTPP Touring-a-sequence-of-Polygons ProblemTS Tabu SearchTSA Traveling Salesman AlgorithmTSP Traveling Salesman ProblemTSPN TSP with NeighborhoodsWRP Watchman Route ProblemZRP Zookeeper Route Problem

robotic task sequencing problem: a survey

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