Robot Trajectory Optimization for the Relaxed End-effector Path

Sergey Alatartsev, Anton Belov, Mykhaylo Nykolaychuk and Frank Ortmeier

Chair of Software Engineering, Otto-von-Guericke University, Universit¨atsplatz 2, Magdeburg, Germany

Keywords:

Trajectory Optimization, Minimum-jerk Trajectory, Path Relaxing, Pattern Search, Rubber-band Algorithm.

Abstract:

In this paper we consider the trajectory optimization problem for the effective tasks performed by industrial

robots, e.g., welding, cutting or camera inspection. The distinctive feature of such tasks is that a robot has to

follow a certain end-effector path with its motion law. For example, welding a line with a certain velocity has

an even inﬂuence on the surface. The end-effector path and its motion law depend on the industrial process

requirements. They are calculated without considering robot kinematics, hence, are often “awkward” for the

robot execution, e.g., cause high jerks in the robot’s joints. In this paper we present the trajectory optimization

problem where the end-effector path is allowed to have a certain deviation. Such path is referred to as relaxed

path. The goal of the paper is to make use of this freedom and construct the minimal-cost robot trajectory. To

demonstrate the potential of the problem, jerk of the robot joint trajectory was minimized.

1 INTRODUCTION

All robot movements can be divided into two cate-

gories: effective and supporting movements (Alatart-

sev et al., 2013). Effective movements are required

to perform a certain task, e.g., welding, deburring

or glue dispensing. Supporting movements are re-

quired to move from one effective task to another,

e.g., motion between two welding seams. Effective

movements are task-depended, therefore, robot end-

effector path and its motion law are often deﬁned in

a strict way to meet the requirements of the industrial

process. As a consequence, the obtained robot trajec-

tory is “awkward” for the robot execution,e.g., causes

high jerks in robot joints.

Effective tasks often allow a freedom of execu-

tion. For example, laser-welding can be performed

with a set of possible tool orientations (Kov´acs, 2013)

or cutting can be performed with a set of possible tool

positions and orientations (Alatartsev and Ortmeier,

2014). As a consequence, the end-effector path is not

unique. There is a continuous set of admissible paths,

that are equal in terms of quality of task performing.

However, such paths are not equal in terms of robot

kinematics and lead to different robot trajectories. We

call the set of admissible paths a relaxed path.

We present the general problem of ﬁnding such

an end-effector path from the relaxed path that would

lead to a minimum cost robot trajectory. It is assumed

that motion law is given, e.g., imposed by an indus-

trial application or already optimized. The heuristic

approach proposed in this paper is independent from

a way of path relaxation and a cost function. In this

paper we are interested in the continuous planning

rather than in point-to-point trajectories, e.g., motion

law should be maintained throughout the whole path

and not only in its via-points. The trajectory cost is a

domain-dependent parameter and can be, for exam-

ple, time, energy or material inﬂuence metric. We

show on a robot application from the medical-domain

that by exploiting the freedom of path relaxation the

trajectory cost can be signiﬁcantly reduced.

2 BACKGROUND

Robot trajectory is speciﬁed with a geometrical path

and a motion law by which the path must be tracked

(Biagiotti and Melchiorri, 2008). The path can be

speciﬁed either in the task space (T-Space) as a se-

quence of robot end-effector positions and orienta-

tions (Path

) or in the conﬁguration space (C-space)

as a sequence of robot joints angles (Path

), see Fig.

1. By the end-effector path, we imply the path for the

tool center point (TCP) or end of arm (EOA) point. To

obtain the robot trajectory, the motion law has to be

applied either to the end-effector path or to the robot

joint path and then it is denoted as ML

and ML

re-

spectively. Converting between T-space and C-space

is done by means of Inverse Kinematics (IK) and For-

385

Alatartsev S., Belov A., Nykolaychuk M. and Ortmeier F..

Robot Trajectory Optimization for the Relaxed End-effector Path.

DOI: 10.5220/0005093103850390

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 385-390

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

T-space

C-space

Path

Forward

Kinematics

Inverse

Kinematics

Path

(ML

(t))

Path

(ML

(t))

Trajectory

Figure 1: Overview of the robot trajectory calculation.

ward Kinematics (FK) transformations (Craig, 2005).

FK obtains a unique end-effector pose for the given

robot conﬁguration. IK obtains a set of possible robot

conﬁgurations for the given end-effector pose.

The decision on what path and motion law are re-

quired to obtain the robot trajectory is based on the

type of movement the robot has to make. For support-

ing movements, the motion law ML

is speciﬁed for

the C-space path Path

, as it is not important how ex-

actly the robot end-effector should move. On the con-

trary, effective movements are task-depended, there-

fore, robot end-effector path Path

and its motion

law ML

are often deﬁned in the T-space to meet

the requirements of the industrial process. In this pa-

per trajectory optimization problem is observed for

the effective tasks. Therefore, formal deﬁnitions for

the Path

and ML

are given below.

End-effector path is deﬁned as follows:

Path

= (P

,..., P

), where n is a number of

via-points. The via-points belong to 6D T-space,

∈ R

, where three dimensions stand for position

and the remaining three dimensions stand for orien-

tation (when using the Euler angle convention). The

path is normally represented by a smooth interpola-

tion function in the domain [0,1]. Motion law is a

function that maps the time value from [0,T] to the

value from [0,1], i.e., ML

: [0, T] → [0,1], where T

is the desired motion duration.

The output of the trajectory planning is the

C-space robot trajectory as it uniquely describes

the robot motion. It is a tuple of trajectories for

every robot joint. C-space trajectory is obtained by

applying the motion law ML

to the path Path

, i.e.,

IK(Path

(ML

(t))) = Path

(ML

(t)) = Tra j

(t).

3 PROBLEM DESCRIPTION

3.1 Path Relaxation

It is important to emphasize that relaxing the path is a

domain-dependent process. However, general meth-

ods can be inherited from the path planning domain,

Figure 2: Relaxed path RelPath

= (V

,...,V

) that con-

sists of spheres and path Path

= (P

,..., P

) that belongs

to it. Smooth function of the end-effector movement is de-

picted with red curve.

e.g., to apply intervals to each coordinate instead of

a single value (Berenson et al., 2011). Path freedom

was also applied in the task sequencing domain, when

for the laser welding application, the truncated cones

are used instead of the T-space points (Kov´acs, 2013).

We deﬁne relaxed end-effector path as follows:

RelPath

= (V

,...,V

), where n is a number of via-

volumes. Via-volume is a subset of the 6D T-space,

i.e., V

⊂ R

The path Path

belongs to the relaxed path

RelPath

, when all its points belong to the corre-

sponding volumes, i.e., P

∈ V

, where i ∈ 1,...,n, see

Fig. 2. It should be clear that an inﬁnite number of

possible Path

belongs to the RelPath

. This fact

is used to search for such a Path

that leads to a

minimal-cost robot trajectory.

3.2 Problem Statement

The problem is formulated as follows:

Given a relaxed robot end-effector path

RelPath

, end-effector motion law ML

and a trajectory duration T, ﬁnd such

a path Path

belonging to RelPath

that leads to a minimal-cost robot C-space

trajectory Traj

(t) = IK(Path

(ML

(t))),

where t ∈ [0,T].

Note that the problem does not require constraints or

cost function to be convex. The optimization cost

could be: energy, jerk or any domain-speciﬁc param-

eter. During optimization, it is important to verify that

a C-space trajectory is feasible, i.e., maximum joints

velocities, accelerations bounds and joint limits are

not violated.

3.3 Problem Discussion

Modeling the Problem: The problem stated in this

paper bears a resemblance to the well-known geo-

metrical problem - Touring a sequence of Polygons

Problem (TPP) (Dror et al., 2003). The goal of the

TPP is to construct a minimal-cost tour through the

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

386

sequence of polygons. Solution of the TPP is a list

of points through which the tour visits each poly-

gon. This problem can be rephrased to ﬁt the prob-

lem stated in this paper. Find a minimal-cost C-space

trajectory such that its T-space path visits every given

multi-dimensional volume. This way techniques for

TPP can be applied for trajectory optimization. In

this paper we applied Rubber-band Algorithm (RBA)

(Pan et al., 2010) that is used for solving TPP.

State of the Art in Trajectory Optimization: Tra-

jectory optimization problem received large attention

for the last 30 years and numerous variations of this

problem exist. The comprehensive overview on state

of the art in robot trajectory optimization can be found

in the survey (Ata, 2007). In general, smooth func-

tions are used for the T-space path interpolation and

it is often assumed that via-points should be visited

strictly without any deviation (Liu et al., 2013). The

problem proposed in this paper is different from the

trajectory tracking problem (Ata and Myo, 2005), as

we are concerned in making the end-effector trajec-

tory more suitable for a robot, rather than in a precise

following of the given end-effector trajectory. There

are approaches mainly oriented on supporting tasks as

constraints are set in the C-space, e.g., (Chettibi et al.,

2004) or (Gasparetto and Zanotto, 2010). Continuous

end-effector path planning problem for the effective

tasks instead of a point-to-point movements was ob-

served by (Olabi et al., 2010). A robot has to strictly

follow the end-effector path and its motion-law was

optimized. In this paper, we do exactly the opposite –

the path geometry is optimized, however, the motion-

law is followed strictly.

Path Relaxation in Trajectory Optimization: In

(Aspragathos, 1998) path is relaxed by extending the

given end-effector path with possible deviation. The

generation of the C-space trajectory is computation-

ally expensive for a large number of end-effector path

via-points. Therefore, Aspragathos proposed an algo-

rithm to minimize the number of via-points and as a

consequence to reduce the number of IK calls but still

guarantee that the end-effector is within a certain de-

viation from the given end-effector path. In contrast,

our problem has a ﬁxed number of via-points but it

allows them to vary within the allowed freedom.

Another similar problem was proposed by (Kolter

and Ng, 2009) who used cubic splines to construct

T-space smooth trajectories. All constraints are con-

vex and are set for the T-space path via-points. Then

the problem is solved with a general purpose convex

solver. The presented approach is powerful and can

incorporate numerous objective functions from the T-

space, except minimization of the trajectory duration

time. Cost functions from the C-space can also be

used but in that case Jacobian approximation should

be performed along the trajectory. As a consequence,

only one of many IK solutions is considered. The

main limitation is that this technique is not suitable

for the cases when the path must go through the non-

convex narrow corridors in the robot C-space. That is

often the case in industrial robotics during handling

the effective tasks.

The freedom for the paint gun orientation was de-

scribed by (From et al., 2011). They proposed an

approach for the real-time calculation of the optimal

paint gun orientation for each time step for the given

constant velocity value. The minimal-cost here means

that the displacements of the paint gun are minimized.

The freedom is given as convex constraints for orien-

tation. The objective function must be convex as well.

The problem presented in this paper is a generaliza-

tion of their problem, as the freedom is provided both

for the end-effector orientation and position. We do

not require the constraints or objective function to be

convex. There is also no requirement that velocity has

to be a constant value, it can be an arbitrary function.

Summarizing, the previously described ap-

proaches often ignore freedom of the path. Those ap-

proaches that exploit path freedom are either domain-

speciﬁc or not applicable for the industrial applica-

tions. The idea proposed in this paper does not com-

pete with the related approaches but rather attempts to

enhance their scope. The proposed problem deﬁnition

is domain-independent and formulated with no de-

pendence on the solving approach, as a consequence,

there are no special requirements for the constraints

or the cost function.

4 CASE STUDY

We illustrate the usefulness of the proposed idea with

the medical application, 3D-angiographythat is based

on the C-Arm and provides computed tomography3D

volumes. Such 3D volume is obtained by stitching

multiple picture-scans taken from different positions.

The C-arm is a horseshoe-shaped device mounted on

the robot and it consists of two components: the X-

ray source and the detector, see Fig. 3 and Fig. 4. We

consider only degrees of freedom of the robot. For

simplicity, degrees of freedom in the C-arm device

are ignored.

It is critical to know the exact position and a

time when each picture was taken. Imprecise trajec-

tory following inﬂuences the ﬁnal 3D volume quality,

i.e., makes it blurry. The path of the X-ray source

RobotTrajectoryOptimizationfortheRelaxedEnd-effectorPath

387

Figure 3: Layout of the robot equipped with C-arm.

is speciﬁed for a certain task without considering

robot kinematics. One possible way to obtain high-

quality source trajectory is to make the robot trajec-

tory smooth by minimizing joint jerks. Jerk mini-

mization reduces the error of the path tracker. In ad-

dition, trajectories with small jerk reduce wear of the

robot and, as a consequence, increase its life span (Si-

mon, 1993). Thus, the objective to minimize is the

maximum joints jerks throughout trajectory duration:

max

i∈[1,...,ndo f]

( max

t∈[0,...,T ]

(

∂

Tra j

(t)

∂t

)) → min (1)

where ndof is the number of robot degrees of free-

dom.

This application scenario allows a certain freedom

for the path. For example, the source might have the

deviation of being closer or further to the point of in-

terest (in our scenario this deviation is 0.02 m.). In

addition, the approaching vector might have deviation

of 6

◦

. This freedom results in the truncated-cone via-

volumes, which the C-arm source path has to visit.

The path and the freedom are shown in the Fig. 4.

In any point of the via-volume, approaching vector of

the source is directed to the point of interest – isocen-

ter. Similar freedom description was used for a laser-

welding application (Kov´acs, 2013).

4.1 Solution Approach

Exhaustive search strategies are impractical due to the

large search space of the presented problem. The con-

vex solvers cannot be applied, as we do not restrict the

problem constraints and cost function to be convex.

The way to solve the problem is to apply a heuris-

tic approach. Heuristics do not guarantee ﬁnding the

optimum, however, they can provide near-optimal so-

lution to the real-life scenarios in a reasonable time.

We propose heuristic search that is based on the RBA

(Pan et al., 2010) and on the Pattern Search (PS)

(Hooke and Jeeves, 1961).

The general steps of the optimization process are

presented in Algorithm 1. The algorithm takes re-

laxed robot end-effector path RelPath

, end-effector

motion law ML

and a motion duration T. Its output

X-ray source

Detector

Isocenter

Via-volume

Figure 4: The path of the C-arm X-ray source is designated

with blue, the path of the detector is red. The point of inter-

est is the middle of the sphere. The relaxed path consist of

light-blue volumes.

is an optimized C-space trajectory. Note, that T is a

domain-speciﬁc value and if it is too small, then al-

gorithm will not ﬁnd the solution, as the robot joint

velocity limits will be violated. Initially, the algo-

rithm constructs a feasible path Path

that belongs

to the given relaxed path in Algorithm 1 line 1. In this

application, initial path Path

consists of the central

points of the via-volumes. Then a C-space robot tra-

jectory is calculated with the algorithm GetTraj that

is discussed further and its cost is obtained with the

function GetCost calculated with Formula (1).

The general idea of RBA is to iterate while stop-

ping condition is not satisﬁed (lines 4 – 17 in Al-

gorithm 1) and in each iteration run through the all

points from the Path

and optimize position and ori-

entation of each point one by one (lines 5 – 16 in Al-

gorithm 1). The stopping condition can be a number

of iterations, an elapsed calculation time, etc.

Optimization of a single point can be done in a

number of ways. In the current implementation PS is

applied (lines 6 – 15 in Algorithm 1). At ﬁrst, PS

modiﬁes the point P

(line 7). The modiﬁcation is

done as a change of one of the point’s coordinates by

a certain small value. PS loop breaks when no mod-

iﬁcation is possible (line 6), i.e., all coordinates have

already been modiﬁed. For every new modiﬁcation,

a new path Path

′

is obtained and a trajectory is re-

calculated with the further described method GetTraj

(line 8) and its cost is obtained (line 9). If the modi-

ﬁcation leads to the cost decrease (line 10), then save

the Path

′

, cost

′

, Traj

′

(lines 11 –13). The algo-

rithm guarantees that the path worse than the initial

one will not be returned. The algorithm only varies

the path of the end-effector but keeps the motion law

unchanged.

The C-space trajectory Tra j

is calculated with

the method GetTraj. The straightforward way to

obtain a C-space trajectory is to map every point of

the end-effector trajectory to the robot conﬁguration

with IK. However, it requires a large number of IK

calls that are normally computationally expensive. In

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Algorithm 1: Heuristic search.

Input: RelPath

, ML

, T

Output: Traj

1 Get feasible initial path Path

∈ RelPath

;

2 Tra j

← GetTraj(Path

,ML

,T);

3 cost ← GetCost(Traj

) ;

4 while stopping condition is not satisﬁed do

5 foreach P

∈ Path

6 while Modiﬁcations are possible do

7 Path

′

← Modi f y(Path

);

8 Traj

′

← GetTraj(Path

′

,ML

,T);

9 cost

′

← GetCost(Traj

′

) ;

10 if cost

′

< cost then

11 Path

← Path

′

;

12 cost ← cost

′

;

13 Traj

← Traj

′

;

14 end

15 end

16 end

17 end

18 return Traj

;

this paper, at ﬁrst, Path

is obtained by applying IK

only to the Path

via-points. Then Path

is inter-

polated with a smooth function with evenly spread

parameter from [0,1]. Later the interpolation param-

eter is rescaled in a way that the end-effector tra-

jectory follows the given ML

. It is done by iter-

ating through the spline domain with a step size of

1\ ( frequency× T). Then, save the obtained spline

values into the array Value

new

and save distances be-

tween end-effector positions for two sequential steps

into the array Parameter

new

. The sum of all values

from Parameter

new

equals to the T-space path length.

Then parametrize these distances to the interval [0,1].

Finally, construct a new spline on the domain param-

eters Parameter

new

and codomain values Value

new

This reduces the number of IK calls. In case if more

control on precision is desired, the number of via-

points can be increased.

In this paper cubic splines were applied for inter-

polation of the path and motion law, as they are twice

continuous differentiable and provide constant jerk.

Higher order splines generally suffer from unwanted

high osculation and might lead to a retrograde motion

(Macfarlane and Croft, 2003).

4.2 Evaluation

Two cases are considered: the motion law that re-

sults in a trapezoidal velocity proﬁle (case “A”) and

minimum-jerk optimized velocity proﬁle for the ini-

tial path (case “B”), see Fig. 5. In the case “A”, the

desired trapezoidal velocity allows to obtain picture

points with the constant velocity in the center of the

0.00

0.05

0.10

0.15

0.20

0.25

Velocity (m/s)

0.0

0.2

0.4

0.6

0.8

1.0

Time (s)

Path paramer

0 2

6 8 10 12

0 2

6 8 10 12

Path paramer

0 2

6 8 10 12

Velocity (m/s)

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Case "A"

Case "B"

Motion law

Velocity

Time (s)

Figure 5: Given motion laws and computed end-effector ve-

locities.

100

120

140

1 2 3 4 5 6 7 8 9 10

Number of iterations

1 2 3 4 5 6 7 8 9 10

Maximal jerk (Rad/s

)

Number of iterations

Case "A"

Case "B"

Maximal jerk (Rad/s

)

Figure 6: Convergence rates of optimization for the given

motion laws.

path but that leads to an “awkward” robot C-space tra-

jectory with high jerks, i.e., the cost of the initial tra-

jectory is 123 Rad/s

. In the case “B”, the motion law

was optimized to obtain the minimum-jerk C-space

trajectory for the initial path. For motion law opti-

mization, an idea similar to the algorithm proposed

by (Chettibi et al., 2004) was applied. Uniformly dis-

tributed nodes were taken on the motion law curve

and then positions were optimized with the Pattern

Search. The obtained velocity proﬁle in depicted in

Fig. 5. The trajectory cost after motion law minimiza-

tion for the case “B” is 30.2 Rad/s

. Due to multiple

calls of inverse kinematics the computational time is

63 min. As the C-arm robot movements are typical

and predeﬁned, it is possible to calculate them ofﬂine.

After applying the proposed heuristic for 10 iter-

ations, the cost was decreased to 26 Rad/s

for the

case “A” and to the 17.5 Rad/s

for the case “B”. The

rate of convergence is depicted in Fig. 6. This study

shows that the end-effector path relaxation leads to

the decrease of the robot trajectory cost regardless of

whether the motion law was optimized or not. How-

ever, it is more effective to relax the path in conjunc-

tion with the motion law optimization.

RobotTrajectoryOptimizationfortheRelaxedEnd-effectorPath

389

5 CONCLUSION

The problem of the minimal-cost trajectory planning

for a given end-effector motion law and a relaxed

path is proposed in this paper. We deﬁne this prob-

lem as domain-independent and formulate it with no

regard to the solution method. As a consequence,

we do not impose any requirement on the constraints

or the cost function. It was shown that relaxing the

path can lead to a signiﬁcant trajectory cost reduc-

tion. This improvement is achieved with no depen-

dency on whether the motion law was deﬁned by an

industrial process or was optimized. The limitation of

the approach follows from its generality. It cannot be

applied in real time, as the used heuristic is computa-

tionally slower than the convex optimization solvers.

One way to achieve better results is to generalize

the problem further by relaxing the motion law, as in

the current problem formulation it is considered to be

given and ﬁxed. Currently, we considered only one

IK solution, e.g., “elbow-up”. However, making use

of the multiplicity of IK solutions might provide bet-

ter results. Robot base location in the environment

greatly inﬂuences the cost of the C-space trajectory

obtained for the end-effector path. In many applica-

tions robot base location is not important, or at least

can vary within a certain area. In this paper a greedy

local search method was presented. However, the po-

tential of the problem can be utilized evenmore by ap-

plying more sophisticated search techniques that have

mechanisms to avoid local optimum, e.g., Genetic Al-

gorithm or Variable Neighborhood Search.

ACKNOWLEDGEMENTS

The work in this paper is partly funded by the German

Ministry of Education and Research (BMBF) within

the Forschungscampus STIMULATE (grant number

03FO16101A).

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