Multi-goal Trajectory Planning with Motion Primitives for Hexapod

Walking Robot

Petr Van

ek, Jan Faigl and Diar Masri

Department of Computer Science, Czech Technical University in Prague, Technick

a 2, 166 27 Prague, Czech Republic

Keywords:

Motion Planning, Traveling Salesman Problem.

Abstract:

This paper presents our early results on multi-goal trajectory planning with motion primitives for a hexapod

walking robot. We propose to use an on-line unsupervised learning method to simultaneously ﬁnd a solution of

the underlying traveling salesman problem together with particular trajectories between the goals. Using this

technique, we avoid pre-computation of all possible trajectories between the goals for a graph based heuristic

solvers for the traveling salesman problem. The proposed approach utilizes principles of self-organizing map

to steer the randomized sampling of conﬁguration space in promising areas regarding the multi-goal trajectory.

The presented results indicate the proposed steering mechanism provides a feasible multi-goal trajectory in a

less number of samples than an approach based on a priori known sequence of the goals visits.

1 INTRODUCTION

Walking robots provides a great ﬂexibility to oper-

ate in complex unstructured environments. On the

other hand, motion planning for such a vehicle is chal-

lenging because of its high-dimensional conﬁguration

space C. In this paper, we address a more challeng-

ing problem, where several optimal trajectories have

to be determined for a hexapod walking robot to re-

peatedly visit a given set of locations with a speciﬁed

precision. The problem is called the multi-goal mo-

tion planning (MGMP) and it combines challenges of

the motion planning together with challenges of the

combinatorial optimization as it is necessary to deter-

mine an optimal sequence of the goals visits together

with the cheapest trajectory connecting the goals.

The optimal sequence can be found as a solution

of the traveling salesman problem (TSP), which is

known to be NP-hard and efﬁcient heuristics have

been proposed in operational research (Applegate

et al., 2007). However, the main computational dif-

ﬁculty lies in the determination of the particular tra-

jectories between the goals, which is a challenging

problem itself, e.g., PSPACE-hard (Reif, 1979) for a

polyhedral problem representation. For n goals, up

to n

goal–goal trajectories can be needed for a stan-

dard graph-based TSP solver. Moreover, this bound

does not hold for kinodymamic or non-holonomic

constraints, where it can be desirable to ﬁnd a cost

efﬁcient trajectory for a sequence of the goals.

(a) (b)

Figure 1: Hexapod performing found multi-goal trajectory.

A trajectory between two goals can be found

by randomized sampling based motion planner, e.g.,

Rapidly-exploring Random Tree (RRT) and Rapidly-

exploring Random Graphs (RRG) (Karaman and

Frazzoli, 2011). However, a complex robotic platform

has a large set of possible control inputs, which does

not allow to efﬁciently use randomized planners with

all possible control inputs. Therefore, a set of motion

planning primitives can be considered for walking and

complex platforms (Bevly et al., 2000; Pivtoraiko and

Kelly, 2005). Motion primitives efﬁciently provide

only local motion, but they can be used in random-

ized planners, such as the RRT (Von

asek et al., 2013),

to ﬁnd a global trajectory between two conﬁgurations.

599

Van

ek P., Faigl J. and Masri D..

Multi-goal Trajectory Planning with Motion Primitives for Hexapod Walking Robot.

DOI: 10.5220/0005118405990604

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 599-604

ISBN: 978-989-758-040-6

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

We propose to leverage on motion primitives and

steer the randomized sampling in the RRG to directly

ﬁnd a solution of the multi-goal trajectory problem

for a hexapod walking robot without prior determina-

tion of the (optimal) sequence of the goals visits. The

idea is based on an on-line unsupervised learning on

top of the growing motion planning roadmap to con-

sider promising areas of C regarding the solution of

the sequencing part of the MGMP. Our early results

indicate the steering reduces the required number of

roadmap expansions to ﬁnd an admissible multi-goal

trajectory and thus demonstrate feasibility of the pro-

posed approach for multi-goal trajectory planning.

The paper is organized as follows. The problem

is introduced in Section 2 together with description

of the considered robotic platform. A reference algo-

rithm for the MGMP problem for a given sequence

of the goals visits is presented in Section 3. The pro-

posed unsupervised learning based MGMP algorithm

is described in Section 4 and its validation in selected

case studies is presented in Section 5. The concluding

remarks and future work are summarized in Section 6.

2 PROBLEM STATEMENT

The MGMP problem is studied in a 3D environment

W ⊂ R

represented by a set of triangles forming a

set of obstacles O. The trajectory planning can be for-

mulated using the notion of the conﬁguration space C.

Following (Karaman and Frazzoli, 2011), the simple

start–goal trajectory from a conﬁguration q

init

to the

desired goal conﬁguration q

goal

can be found as a mo-

tion query asking for a feasible path κ : [0,1] → C

f ree

such that κ(0) = q

init

and d(κ(1),q

goal

) < ε, where

f ree

denotes the obstacle free part of C as C

f ree

cl(C \ C

obs

), ε is an admissible distance of the trajec-

tory end-point to the desired goal, cl() is the closure

of a set, and d(,) is a distance between two conﬁgu-

rations.

Our motivational scenario is a visual inspection

planning problem to cover n given areas of interest

from particular locations G = {g

,... ,g

}. We

assume each location is described by a 6D vector

× SO(3) consisting of a position (x,y,z) in W

accompanied by a desired orientation of the camera

pointing to the requested goal area denoted as (α,β,γ)

for roll, pitch, and yaw, respectively. We assume it

is sufﬁcient the trajectory is within ε distance to the

goal location and provides a coverage of the area, i.e.,

a camera orientation is within particular angular lim-

its. For simplicity, we assume the camera is attached

at the center of the robot’s coordinate system and the

robot position is a 6D pose in W .

Similarly to a simple trajectory a multi-goal tra-

jectory visiting a set of goals G can be deﬁned as

follows. Let the sequence of the goals visits be

,... ,v

) for which v

∈ G and

1<i≤n

= G.

Then, we can deﬁne the feasible and admissible multi-

goal trajectory as a closed trajectory τ : [0,1] → C

f ree

such that τ(0) = τ(1) = q

start

and for which there ex-

ists n points on τ such that 0 ≤ t

≤ t

≤ ... ≤ t

and

d(τ(t

),v

) < ε. In other words, there exists n points

from which the robot will cover the objects of interest.

Having these preliminaries, the MGMP problem

can be formulated as follows: for the given set of

goals G, conﬁguration space C, admissible distance

ε, and monotonic, bounded, and strictly positive cost

function c: ﬁnd a feasible and admissible (accord-

ing to ε) trajectory τ

∗

such that c(τ

∗

) = min{c(τ) |

τ is feasible admissible multi-goal trajectory}.

The proposed approach is based on the RRG al-

gorithm used for an incremental construction of the

graph G

RRG

=(V

RRG

), which represents the mo-

tion planning roadmap constructed by the RRG algo-

rithm. The set of vertices V

RRG

represents particu-

lar conﬁgurations of the robot q ∈ C

f ree

and an edge

e ∈ E

RRG

describes a feasible motion between two

conﬁgurations v

∈ V

RRG

,i 6= j. The graph is a re-

sult of the graph expansion from the nearest vertex

of the graph towards a random sample by applying a

particular control command from a considered set of

possible commands / motion primitives. The assump-

tions and speciﬁcations of the RRG for the considered

hexapod walking robot are described in the following

section.

2.1 RRG for Hexapod Walking Robot

The robotic platform considered in this study is 18

degrees of freedom (DoF) hexapod walking robot

(PhantomX), see Fig. 1. The body of the robot has

dimensions about 26 cm × 20 cm and its dimensions

with fully deployed legs are about 75 cm × 70 cm.

Motion planning for an 18 DoF platform can be com-

putationally demanding, and therefore, we consider

a set of motion primitives that signiﬁcantly reduces

computational burden in randomized sampling based

motion planners for complex robots. The proposed

online-learning and steering of the randomized sam-

pling is general and does not assume a particular mo-

tion; however, we consider simple motion primitives

in this initial study because they allow more conve-

nient deployment and veriﬁcation.

The used motion primitives allow crawling in for-

ward/backward, left/right, and diagonal direction and

crawling along a part of the circle with the deﬁned ra-

dius. The primitives are designed in such a way that

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they can be interchangeably shifted after ﬁnishing one

step, which allows a spread variety of combination

of motion patterns. Each primitive is parametrized

by a number of steps to be performed. The diagonal

crawling is parametrized by a direction angle from the

current state to the desired position, and curvature of

the circle for circular crawling. Thus, these primi-

tives allow the robot to perform 14 possible actions

that means 28 different motions.

For construction of the roadmap G

RRG

by the

RRG only a single parameter for each primitive is

used and the number of steps is set to 1. The length

of the particular motion step slightly differs and it is

around 10 ± 1 cm. Notice, the motion primitives re-

strict ability to reach an arbitrary conﬁguration in few

RRG expansions. However, it provides a smooth tra-

jectory to reach vicinity of the desired goal locations.

In this study, we further simplify the problem by

considering robot motion on a plane. Besides, the

collision checking is computed using a bounding box

of the platform. These simpliﬁcations do not affect

the principle of the proposed steering but reduce the

computational burden. An additional speed improve-

ment is gained using goal bias and goal zooming tech-

niques (Lavalle and Kuffner, 2001) in sampling C

f ree

during expansion of the RRG.

3 REFERENCE MGMP

ALGORITHM

A simple multi-goal trajectory planner for a given se-

quence of goals visits is considered for a compari-

son with the proposed direct solution of the MGMP.

The planner represents a straightforward deployment

of the RRG algorithm for ﬁnding a multi-goal trajec-

tory. The required sequence of the goals visits can

be determined as a solution of the TSP, e.g., using the

CONCORDE solver (Applegate et al., 2003), with ap-

proximation of the goal–goal distances. The reference

planner follows the approach proposed in (Saha et al.,

2006) and Euclidean distances between the locations

are used for ﬁnding the sequence.

The reference algorithm is directly based on an in-

cremental construction of the roadmap G

RRG

by the

RRG. A robot starts at the position of the ﬁrst goal g

which also initializes the roadmap. The goals from

the given sequence of locations (g

,... ,g

) are iter-

atively alternating to be the temporal goal q

of the

goal bias. After k

expansions with the same temporal

goal, the new one is altered. This is repeated until k

total expansions of the RRG. Then, for each goal g

∈

G the closest conﬁguration q

= argmin

v∈V

RRG

d(v,g

)

is found. If the distance d(q

) > ε the algorithm

fails; otherwise, the trajectory between q

i−1

and q

found and connected to the ﬁnal trajectory τ. Finally,

the trajectory is enclosed connecting q

to q

. This

algorithm can provide a ﬁrst admissible trajectory in

RRG expansions and the trajectory can be then

further improved during the total of k expansions.

4 PROPOSED MGMP

ALGORITHM

The proposed multi-goal trajectory planner is based

on adaptation rules of the self-organizing map (SOM)

for the TSP (Faigl et al., 2011), which is a two lay-

ered neural network accompanied by an unsupervised

learning procedure. SOM provides a non-linear ap-

proximation of a high-dimensional input space into a

one dimensional output space that represents the re-

quested tour as a sequence of the output units. The

ﬁrst layer is the input layer for presenting goals to

be visited and towards which the network is adapted.

The output layer consists of m units N = {ν

,... ,ν

}

representing neurons weights.

The unsupervised learning of SOM for the TSP

is performed in a series of learning epochs. At each

epoch, all goals to be visited are presented to the net-

work in a random order and for each goal the units

compete to be a winner using its distance to the pre-

sented goal. The winner node is then adapted to the

goal together with its neighbouring nodes. This learn-

ing procedure is repeated until each goal has a distinct

winner sufﬁciently close (e.g., in a distance less than

a given threshold δ).

The adaptation of neuron weights can be imagined

as a movement of the node closer towards the goal,

which may provide an intuitive insight to the learning

process. In our case, the neuron weights are particular

conﬁgurations in C and to ensure they are in C

f ree

, the

evolution of SOM is considered on a motion roadmap

RRG

constructed by the RRG. Thus, the weights are

restricted to be at the graph edges or vertices and the

adaptation can be imagined as movements of neurons

along the graph edges (Yamakawa et al., 2006). Once

the network is stabilized, the sequence of the goals

visits can be retrieved by traversing the output layer

and selecting associated goals to the winning units,

i.e., the closest unit to a graph vertex representing the

desired goal location v

∈ V

RRG

The fundamental issue of using SOM in MGMP

is that the selection of the winner node to a presented

goal g is based on computing a distance between node

ν and g. Such a distance corresponds to the length

of the trajectory from ν to g, which is obviously not

known due to a sparse coverage of C by G

RRG

. This

Multi-goalTrajectoryPlanningwithMotionPrimitivesforHexapodWalkingRobot

601

issue is addressed by new proposed approximation of

the distance, which together with the proposed adap-

tation turns out to a steering strategy for randomized

sampling in the RRG planner.

The approximation combines the Euclidean dis-

tance (e.g., as in (Englot and Hover, 2011)) with

the current knowledge about C stored in the incre-

mentally built roadmap G

RRG

. The current roadmap

should be used as much as possible, because it pro-

vides a realistic estimation about the expected dis-

tance d(ν,g). Therefore, a part of d(ν, g) is based

on a trajectory in G

RRG

from ν towards the vertex v

ν,g

that is found as

ν,g

= argmin

v∈V

RRG

(c(κ

ν,v

) + |(v,g)|), (1)

where c(κ

ν,v

) is the cost of the trajectory κ from ν to

v. A path P(ν,g) from ν to g consists of a trajectory

ν,v

ν,g

in G

RRG

and a straight line segment (v

ν,g

,g)

from v

ν,g

to g as

P(ν,g) = κ

ν,v

ν,g

⊕ (v

ν,g

,g). (2)

On the other hand, the Euclidean distance between

two conﬁgurations would be always shorter or equal

to the length of the trajectory in G

RRG

, and therefore,

d(ν,v) has to respect the current knowledge of C and

the inﬂuence of the usually shorter Euclidean distance

should be suppressed. Therefore, we propose to com-

pute d(ν,g) for winner selection as

d(ν,g) = |P(ν,g)| = c(κ

ν,v

ν,g

) + |(v

ν,g

,g)|

. (3)

The adaptation modiﬁes the neuron weights to get

a neuron closer to the goal. Thus, new weights ν

of the neuron after its adaptation represents a point

on the path P(ν,g) at the distance from ν deﬁned by

the learning rules. The power of adaptation of winner

and its neighbors is controlled by the fractional learn-

ing rate µ and neighbouring function that has form

exp(−l

/σ

), where l is a distance of the node from

the winner and σ is called learning gain, which is de-

creased after each learning epoch. Therefore, the ex-

pected “traveled” distance of ν towards g is

|P(ν,g)| − |P(ν

,g)| = |P(ν,g)|µe

−l

/σ

. (4)

Such a new location can be out of the current roadmap

RRG

; however, we can consider the expected loca-

tion ν

as a temporal goal for expanding G

RRG

in k

iterations. Hence, ν

is used to steer the randomized

sampling in the RRG. After the expectation of G

RRG

is restricted to be on an edge or vertex of G

RRG

guarantee the neurons weights always represent feasi-

ble and reachable conﬁgurations.

An intuitive insight to the learning procedure and

the ﬁnal trajectory determination may arise from the

analyzing the situation at the ﬁnal learning epochs.

The winners are closed to the goals, and therefore, a

trajectory from a neighbouring node to the goal will

likely go over the edge (or vertex) where the win-

ner node is located after the adaptation. Hence, the

roadmap graph G

RRG

is expanded to support determi-

nation of the ﬁnal multi-goal trajectory by preferring

direction for expansion steered by the SOM evolution.

Completeness – Notice, the proposed procedure does

not affect the probabilistic completeness of the RRG.

It can be considered as a meta-heuristic strategy to

steer the randomized sampling towards the goals. The

unsupervised learning can be terminated at any time

and a regular RRG can be used to grow the roadmap

and ﬁnd optimal trajectories between each consecu-

tive goals in the found sequence. Notice, the learn-

ing gain is decreased after each learning epoch, which

means the effective adaptation is terminated in a ﬁnite

number of learning epochs. Thus, the main purpose

of the unsupervised learning is to force sampling to

quickly ﬁnd a feasible solution and also to determine

the sequence of the goals visits. These expectations

are veriﬁed for the robot motion model introduced in

Section 2.1 and our early results are presented in the

following section.

5 CASE STUDY

A validation of the proposed SOM based multi-goal

trajectory planner has been performed in two scenar-

ios to verify if the proposed steering strategy provides

a feasible multi-goal trajectory in less RRG expansion

steps than the reference method (denoted as MGMP

RRG) for the given sequence of goals. In the both

scenarios, the studied indicators of the planner per-

formance are the total number of RRG expansions

RRG

, the number of vertices and edges of the build

roadmap G

RRG

, and the required computational time

for the ﬁrst feasible admissible multi-goal trajectory

that is closer than the selected ε to all goals. All the

presented results have been computed using a single

core of the iCore7 processor running at 3.4 GHz.

The used number of expansions k

in the SOM

RRG is 15 iterations for the winner and 10 iterations

are used for neighbouring nodes.

5.1 Inﬂuence of the Admissible Distance

The ﬁrst evaluation is performed for a simple envi-

ronment, where the optimal sequence to visit 5 goals

can be determined using Euclidean distance. The en-

vironment is depicted in Fig. 4 where selected found

ﬁrst feasible paths are visualized. Both algorithms

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●

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 10000 30000 50000

ε [m]

Total no. of RRG Iterations

●

MGMP RRG

SOM RRG

(a) no.of RRG expansions

●

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 5000 10000 15000 20000

ε [m]

No. of Roadmap Vertices

●

MGMP RRG

SOM RRG

(b) no. of G

RGG

vertices

●

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 10 20 30 40

ε [m]

Path length [m]

●

MGMP RRG

SOM RRG

●

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 1 2 3 4 5

ε [m]

CPU Time [s]

●

MGMP RRG

SOM RRG

(d) computational time

Figure 2: Average values of the studied indicators in simple MGMP problem with 5 goals and admissible distance ε.

●

5 7 10 13 17 20

0 2000 6000 10000

Number of Goals

Total no. of RRG Iterations

●

MGMP RRG

SOM RRG

(a) no. of RRG expansions

●

5 7 10 13 17 20

0 5000 10000 15000 20000

Number of Goals

No. of Roadmap Vertices

●

MGMP RRG

SOM RRG

(b) no. of G

RGG

vertices

●

5 7 10 13 17 20

0 5000 10000 15000 20000

Number of Goals

No. of Roadmap Edges

●

MGMP RRG

SOM RRG

RGG

edges

●

5 7 10 13 17 20

0 1 2 3 4 5

Number of Goals

CPU Time [s]

●

MGMP RRG

SOM RRG

(d) computational time

Figure 3: Average values of the studied indicators for the ﬁrst feasible trajectory in potholes environment and number of goals.

RRG and SOM are randomized, and therefore, 50 tri-

als have been performed for each planner and the ad-

missible distance ε from the range 0.02 ≤ ε ≤ 0.1.

Notice, the reference planner uses the given se-

quence of the goals visits, while the SOM RRG ap-

proach also determines the sequence. Because of this,

it may happened that not the same sequence of the

goals visits as for the reference algorithms is deter-

mined. Therefore, only SOM MGMP solutions with

the identical sequence to the reference are considered.

The reference algorithms utilize k

= 200 RRG ex-

pansions for particular sampling towards the temporal

goal. In all cases, a feasible trajectory is found within

the given maximal number of RRG expansions, which

is set to 20 000.

(a) RRG, n

RRG

=2000 (b) SOM, n

RRG

=715

Figure 4: Found feasible solutions of the MGMP with 5

goals projected to a ground plan of the environment.

The average values of the studied indicators are

depicted in Fig. 2. Regarding the required n

RRG

ﬁnd the ﬁrst feasible multi-goal trajectory, the pro-

posed SOM based steering of the randomized sam-

pling requires the lowest number of the RRG expan-

sions. The main differences are for small ε, which

makes the problem more difﬁcult for the randomized

planners with motion primitives.

A feasibility of the found multi-goal trajectories

has been experimentally veriﬁed for a real traversabil-

ity by the hexapod walking robot using the considered

motion primitives. Although the robot was controlled

in the open loop it was able to follow the planned tra-

jectory and reach all the desired goal locations. Snap-

shots form the deployment are depicted in Fig. 1.

5.2 Inﬂuence of the Number of Goals

In the second scenario, we study performance indica-

tors for increasing complexity of the underlying TSP

with the number of goals n ∈ {5,7,10,13,17, 20}. For

each n, 20 random instances have been created in the

environment called potholes that gives 120 problems

in total. A single value of ε = 0.1 m has been se-

lected and the number of RRG expansions was in-

creased to k

=500 and the total number of expansions

to k =50 000 for the MGMP RRG algorithm. Each

problem is solved 20 times, which gives 2400 trials

for both MGMP RRG and SOM RRG algorithms.

Here, SOM may provide different sequence of goals

visits more frequently, and therefore, we include all

sequences in the computation of the average values

of the studied performance indicators. The average

(a) n

RRG

=1 045 (b) n

RRG

=10 570 (c) n

RRG

=41 500

Figure 5: Found feasible solutions for 5 goals in potholes

environment: (a) the ﬁrst feasible solution found by SOM

RRG; (b) evolved feasible SOM RRG solution; (c) reﬁned

solution by the MGMP RRG.

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603

values over all trials for the ﬁrst found admissible tra-

jectories are shown in Fig. 3.

Also in this case, SOM RRG requires less num-

ber of RRG expansions to ﬁnd a feasible solution.

Such a solution can be then improved by the RRG ex-

pansions in a similar way as the reference algorithm

works, which is demonstrated in Fig. 5.

5.3 Discussion

Despite of the relatively simple problems considered,

the presented early results validate a feasibility of the

proposed approach for steering the randomized sam-

pling in the RRG. The results indicate SOM for the

TSP can be applied directly in C and thus a construc-

tion of the motion planning roadmap prior sequenc-

ing part of the MGMP can be avoided. The SOM

RRG provides ﬁrst feasible solutions of the MGMP

in fewer expansions steps albeit the quality of such a

solution is worse than for reference algorithm utiliz-

ing the known sequence of the goals.

The SOM based approach seems to be faster than

the reference algorithm in ﬁnding the ﬁrst feasible so-

lution; however, we found out that the selection of the

winner node is computationally more demanding with

increasing learning epoch and the number of goals,

see Fig. 3(d). It is because the graph G

RRG

becomes

denser, which increases computational burden of the

determination of the vertex v

ν,g

by (1). This issue may

be addressed by removing non-perspective roadmap’s

vertices or by building an additional structure to sup-

port path queries in the roadmap.

6 CONCLUSION

In this paper, we address multi-goal trajectory plan-

ning with motion primitives and we introduce a new

steering strategy for randomized sampling in RRG to

improve its performance for a hexapod walking robot.

The proposed approach is based on principles of un-

supervised learning of self-organizing maps that al-

low to simultaneously solve the sequencing part of the

MGMP together with determination of the particular

trajectories. The presented early results of the pro-

posed idea in simple case studies provide a ground

work for a further research. Despite of the simple

problems considered, a straightforward MGMP plan-

ner based on a priori known sequence of goals visits

and the standard RRG algorithm requires more expan-

sion steps to ﬁnd a feasible solution than the proposed

approach. This indicates that the proposed steering of

the randomized sampling in the RRG can improve the

performance of the planner signiﬁcantly.

The encouraging results motives us to evaluate the

proposed idea in more complex problems and to com-

pare the performance regarding the solution quality

of the ﬁnal multi-goal trajectory provided by other

approaches based on postponed distance evaluation

techniques.

ACKNOWLEDGEMENTS

The presented work is supported by the Czech Sci-

ence Foundation (GA

CR) under research project No.

13-18316P. The work of Petr Van

ek was supported by

the Grant Agency of the Czech Technical University

in Prague, grant No. SGS14/203/OHK3/3T/13.

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

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