Struct Multidisc Optim (2012) 45:703–715
DOI 10.1007/s00158-011-0728-6
RESEARCH PAPER
Mobile robot path planning algorithm by equivalent conduction
heat flow topology optimization
Jae Chun Ryu · Frank Chongwoo Park ·
Yoon Young Kim
Received: 5 March 2010 / Revised: 6 March 2011 / Accepted: 10 March 2011 / Published online: 12 November 2011
c Springer-Verlag 2011
Abstract This paper addresses the path planning problem
for a point robot moving in a planar environment filled with
obstacles. Our approach is based on the principles of thermal conduction and structural topology optimization and
rests on the observation that, by identifying the starting and
ending configurations of a point robot as the heat source and
sink of a conducting plate, respectively, the path planning
problem can be formulated as a topology optimization problem that minimizes thermal compliance. Obstacles are modeled as regions of zero thermal conductivity; in fact, regions
can be assigned varying levels of non-uniform conductivity
depending on the application. We describe the details of our
path planning algorithm, including the use of artificial mass
constraints (particularly limits on the plate mass) to ensure
convergence, and the choice of penalty exponents. The feasibility and practicality of our approach is validated through
numerical experiments performed with several benchmarks,
with intriguing possibilities for extension to more complex
environments and real terrains. The benchmark problems
of this paper mainly consist of obstacle-free path planning
problems in two-dimensional space with maze-typed, symmetric, and spiral-type obstacles. We also address planning
problems involving user-specified checkpoints, and also
finding shortest paths on real three-dimensional terrain. To
The main idea of this work was first presented at WCSMO8 (June
1-5, 2009, Lisbon, Portugal) under the title of “Novel Mobile Robot
Path Planning Algorithm by Equivalent Conduction Heat Flow
Topology Optimization” by the authors.
J. C. Ryu · F. C. Park · Y. Y. Kim (B)
School of Mechanical and Aerospace Engineering,
Seoul National University, 599 Gwanangno, Gwanak-Gu,
Seoul 151-742, South Korea
e-mail: yykim@snu.ac.kr
the authors’ knowledge, the path planning going through a
stopover is considered for the first time.
Keywords Mobile robot · Path planning ·
Topology optimization · Conduction heat flow
1 Introduction
The path planning problem is one of the most fundamental
problems in robotics and has received considerable attention in the literature. In its simplest formulation, one seeks a
path in configuration space connecting a pair of given points
while avoiding any obstacles. Perhaps the most widely used
methods today are those classified as probabilistic roadmap
methods (PRM). An overview and survey can be seen in
the recent literatures of Overmars (2002) and Choset et al.
(2005). Typically the configuration space is high dimensional and often cluttered due to the presence of obstacles.
PRM methods take random samples from the feasible region
of the configuration space, and attempt to find a feasible
path connecting these points, typically using graph representations. The PRM methodology has been combined with
various optimal control and potential field-based methods
for improving performance (Choset et al. 2005; Rodriguez
et al. 2006; Perez 1987).
Another set of widely used methods are based on
artificial potential fields (Khatib 1986). In this method,
the goal point and obstacles are defined as attractive and
repulsive potentials, respectively. The direction of steepest decent, obtained by differentiating the total sum of
the potentials, is then chosen as the direction of motion.
Because of the simplicity of the algorithm, it is easy to
understand mathematically and the cost of calculation is
704
small. The main drawback is the occurrence of the so-called
deadlock, a kind of local minima. Various approaches have
been proposed to avoid deadlock and other phenomena.
In particular, Koditchek and Rimon (1990) proposed the
so-called navigation functions.
Motivated in part by a desire to more naturally overcome
the deadlock problem, a variety of potential field-based
methods inspired by physics and mechanics have recently
been suggested. Kim and Khosla (1992) suggested a method
using streams of particles within the potential flow, while
Masoud et al. (1994) proposed a biharmonic potential
approach using an analogy with a mechanical stress field.
Singh et al. (1996) used a magnetic analogy, while an
analogy with fluid dynamics was made by Keymeulen and
Decuyper (1994). Louste and Liegeois (2000) also suggested a method using a potential viscous fluid analogy; in
their approach, they provide both a minimum energy path as
well as a shortest path by varying the viscosity in the energy
function. The unsteady diffusion equations were also used
by Schmidt and Azarm (1992).
The work that is the closest in sprit to our approach is
that of Wang and Chirikjian (2000), who also proposed
to apply the steady state heat conduction equations to the
robot path planning problem. Obstacles and feasible regions
are distinguished by their thermal conductivity and starting and goal point are regarded as a heat source and heat
sink, respectively; a robot path is determined to be the path
minimizing the thermal resistance between a heat source
and heat sink. We elaborate on the differences below, but
the main difference with our approach is that they use an
artificial potential field-based method to solve the equations, with a different choice of an objective function and
optimization formulation. Also, in the present formulation,
obstacle regions cannot be passed through at any instance,
which always ensures feasible solutions.
In this paper, we suggest an alternative approach to navigate the shortest robot path. The main idea is to set up
Fig. 1 Analogy between (a)
robot path planning and (b)
topology optimization of a heat
path
J. C. Ryu et al.
the path planning problem as a topology optimization problem. Although topology optimization (Bendsøe and Kikuchi
1988) has been widely used in several disciplines, there has
been no attempt to use it for robot path planning problems.
The use of the topology optimization in path planning has
some advantages over existing methods, as shall be listed
below; the proposed approach possibly offers a different
perspective on the problem.
As mentioned earlier, our work bears a number of similarities to that of Wang and Chirikjian (2000), in that it
is also based on the steady state heat conduction problem.
However, the distinguishing feature of our work is that a
global thermal compliance criterion is minimized (Wang
and Chirikjian (2000) used a thermal resistance criterion)
because the problem is converted to a topology optimization
problem.
Compared to the work of Wang and Chirikjian (2000),
the present method has two distinct advantages. First, our
method can specify a unique starting point, whereas in
Wang and Chirikjian (2000) a collection of points near
the heat source is used as starting points; this is because
artificial potential field-based methods use potential gradients, and the heat source corresponds to a singular point.
Because the resulting paths depend on the choice of starting point, the costs associated with each starting point must
be compared in determining the globally optimal point.
A second advantage of our method is that it can precisely distinguish obstacles from feasible regions. Wang and
Chirikjian (2000) mentioned in their work that the lower
the thermal resistance is, the less the opportunities to meet
obstacles. This description implies that lower thermal conductivity does not guarantee obstacle avoidance; rather, it
only increases the probability of obstacle avoidance. Our
proposed method in contrast ensures that all feasible paths
avoid obstacles. While there are many articles applying the
topology optimization method to real heat transfer problems (Yoon and Kim 2005; Yin and Ananthasuresh 2002;
Mobile robot path planning algorithm
705
Table 1 Analogy between a heat path and a robot path
Topology optimization of steady state
Robot path navigation
conductive heat path
in the configuration space
Heat source
Initial configuration
Heat sink
Goal configuration
Conductive area (design domain)
Free configuration space
Insulated area (non-design domain)
Obstacle configuration space
Optimal heat path
where symbols K, θ and F denote the stiffness matrix, the
nodal temperature vector and the load vector. If is discretized by Ne finite elements, K, θ and F are assembled
by using element-level matrices (d Ke , v Ke ) and vectors
(θe , Fe ):
K=
Ne
d
Ke +
e=1
Final robot path
Bruns 2007; Hansen et al. 2006; Munoz et al. 2007),
no investigation using topology optimization has been
proposed yet.
Our method is inspired by natural similarities that exist
between these two physically distinct problems. For example, as depicted in Fig. 1, the regions where a robot can
move are regarded as the design domain over which conduction heat transfer occurs, obstacles are considered as thermal
insulators in the topology optimization problem, and the
starting and goal points are replaced by a heat source and
a heat sink, respectively. These intuitive analogies are organized in Table 1. The details of the proposed formulation
are given in Section 2 and several benchmarks robot path
planning problems are solved in Section 4.
F=
Ne
Ne
v
Ke
(5)
e=1
Fe
(6)
θe
(7)
e=1
θ=
Ne
e=1
The components of the element conduction matrix (d Ke )
and the element convection matrix (v Ke ) are given by
the following where Ni , N j (i, j = 1, 2, 3, 4) denote linear
interpolation functions:
d
∂Nj
∂ Ni ∂ N j
k
+ ke
∂x ∂x
∂y ∂y
=
K iej
e
e ∂ Ni
de
(8)
v
2 Heat transfer and topology optimization
K iej
=h
e
Ni N j de
(9)
e
We first review the basic element of heat transfer and
topology optimization. The governing equation of the twodimensional steady state heat transfer problem in a design
domain as depicted in Fig. 1(b) is expressed as (See, e.g.,
Incropera and DeWitt 2002).
∇ (−k∇θ) + hθ + Q = 0
in (1)
with boundary conditions:
θ = θ̄
on ∂1
(2)
q = q̄
on ∂2
(3)
The components of the element force vector Fe are
expressed as
f ie =
Ni Qde +
e
Ni qd∂e
∂e
In order to formulate a topology optimization problem to
find an optimal heat path, the thermal conductivity k e of
the e-th finite element needs to be interpolated as a function of the element density design variable γe (0 ≤ γe ≤ 1).
For instance, one may employ the following polynomial
interpolation (Yang and Chuang 1994):
p
In (1), k and h are the coefficients of thermal conductivity
and convective heat transfer while Q denotes heat generation. The symbol θ (= T − T∞ ) is an excess temperature
and θ in (2) is a pre-defined value on a boundary ∂1 . The
symbol q in (3) denotes the heat flux defined on ∂2 .
When (1)–(3) are implemented by the finite element
method, the following matrix equation can be obtained (See,
e.g., Zienkiewicz et al. 2005):
Kθ = F
(4)
(10)
k e = k 0 γe
(11)
where k0 and p denote the nominal thermal conductivity and
the penalty exponent, respectively. When γ e approaches its
upper and lower bounds, the element state is interpolated as
γe → 1 : e
γe → 0 : e
→ material element
→ void element
(12)
In the present formulation, the design objective is to minimize the thermal compliance subject to a mass constraint. A
706
J. C. Ryu et al.
typical formulation for thermal compliance minimization is
Bendsøe and Sigmund (2003):
minimize
(γ1 ,γ2 ,γ3 ,···,γ N e )
Ne
= FT θ = θT Kθ
γe m e − M 0 ≤ 0
(13)
(14)
of M0 could result in multiple robot paths or meaningless
paths. Therefore, formulations suitable for robot path planning will be examined in the next section. Nevertheless, the
problem formulation given by (13)–(15) is the underlying
formulation for robot path planning. Thus, the sensitivity
of the objective function in (13) is given here. By using
the direct differentiation method, one can easily find the
sensitivity ∂/∂γe as
e=1
0 < ε ≤ γe ≤ 1 (e = 1, 2, · · · , N e : ε = small value) (15)
∂ e T ∂Ke e θ
= θ
∂γe
∂γe
The mass constraint is stated as (14) where m e is the mass of
the e-th element and M0 , the allowed mass. It is noted that
the above formulation cannot be directly used in robot path
planning because it is neither possible to choose a specific
value of M0 nor useful because an improperly chosen value
The solution procedure to solving (13)–(15) is well-known,
but a simple test problem is considered to choose the interpolation strategy. Here, we use a continuation method,
which varies the penalty exponent value of p in the polynomial interpolation function in (11):
p=
1
for n ≤ n c
min (3, 1 + 0.1 × (n − 10)) for n > n c
(n : interation number)
where the parameter n c can be adjusted depending on the
problem (satisfactory results were obtained for 5 ≤ n c ≤
10). A specific test problem is depicted in the first figure
of Fig. 2, where there is a heat source S and a heat sink G
inside the square design domain discretized by 40×40 finite
elements. In this case, n c = 5 was used. Note that for the
robot path planning problem addressed in the next section,
S and G will be used as start and goal points, respectively.
In this model, the heat sink G is modeled by an element having a non-zero normal convection coefficient. All
other elements have zero normal convection coefficients.
The heat source S is modeled by an element having a
(16)
(17)
heat source yielding a no vanishing force components ( f ie ).
Also, the boundary ∂ of the design domain is assumed
to be thermally insulated. (Note that the modeling technique
described here will be used for robot path planning.)
Figure 2 also shows the layout history during optimization iterations. For this problem, the values M0 = 0.025 and
m e = 6.25 × 10−4 were used. As expected, the optimized
layout is straight, and converges to a distinct void-solid distribution. This test problem demonstrates the effectiveness
of the interpolation scheme suggested in (17) for this test
problem. Although the value of M0 is given in this problem, the topology optimization based robot path planning
should work without a specific M0 value, particularly since
M0 cannot be pre-determined by the proposed method.
3 Robot path planning
Fig. 2 A test heat path optimization problem and iteration histories
In this section, we will modify the topology optimization
formulation presented for thermal compliance minimization
in the previous section. Here, we make use of the analogies between thermal compliance minimization and robot
path planning as tabulated in Table 1. In making such analogies, some issues must be addressed: (i) how to convert the
robot configuration space C into the design domain of the
thermal compliance topology optimization problem, and (ii)
how to set up a minimization problem suitable for robot path
planning. These issues will be addressed in detail below.
Mobile robot path planning algorithm
707
Fig. 4 Starting point modeling when it is (a) inside and (b) lies
along ∂
Fig. 3 Mapping of (a) the configuration space for robot path planning
to (b) the design (analysis) space for topology optimization
3.1 Robot configuration space
In explaining the mapping of the robot configuration space
C into the domain for the topology optimization, C is
assumed to be discretized into a number of finite elements
as in Fig. 3(a). The configuration space C shown in Fig. 3(a)
may be divided as
1
2
C = CF + CO
+ CO
+ · · · + C S + CG
(18)
In (18), C F denotes a free space in which a robot can
i present the i-th space of obstacle
freely move and C O
through which a robot cannot go. Here, the starting point
and the goal point are assumed to each occupy one finite element, respectively. As illustrated in Fig. 3(b), the following
mapping of C into is suggested.
(19)
Thus, the whole analysis domain for the resulting topology optimization consists of four domains:
= F + 1O + 2O + · · · + S + G
f ie
Ni Qde (i = 1, 2, 3, 4)
=
(22)
e
When a starting point lies along the boundary as in Fig. 4(b),
the load vector is constructed by assuming incoming heat
flux along the boundary (the edge connecting nodes 2 and 3
in Fig. 4(b)). Hence the load vector is defined as
f ie =
C → , Cα → α (α = F, S, G) ,
i
CO
→ iO (i = 1, 2, · · ·)
From the viewpoint of heat transfer, the use of (21) implies
that obstacles and free regions are treated as thermal insulators and conductors, respectively.
Now let us consider the actual description of S and
G . As indicated in Fig. 3(b), only one finite element is
assigned to define S or G . Depending on the shape of
the configuration space C, the starting point may be located
inside F or along the boundary of F . Figure 4 depicts
the two cases. If a starting point is located inside the design
domain as in Fig. 4(a), a heat source Q is assumed to
be present at the element. In this case, the load vector
corresponding to S is defined as
Ni qd∂e (i = 2, 3)
(23)
∂e
The modeling of a goal point is depicted in Fig. 5, where
q n and q s denote the heat transfers due to surface normal
(20)
Here, F is the design domain in the topology optimization and the finite elements discretizing F have varying
thermal conductivity values based on the interpolation function given by (11). On the other hand, iO is the non-design
domain inside the analysis domain , so k0 = 0 is assigned
to all elements in iO . Thus,
k0 = 0 for e ⊂ iO (i = 1, 2, · · ·)
k0 = 0 (free-defined finite value) for e ⊂ F
(21)
Fig. 5 Goal point modeling when it is (a) inside and (b) lies
along ∂
708
J. C. Ryu et al.
convection and side convection, respectively. Note that no
convection is assumed to occur in all other elements in
the design domain . Depending on the location of the
goal point, different convection models, as described below,
are used:
nonzero surface normal convection q n if ∂G ⊂ ∂
nonzero side convection q s if ∂G ⊂ ∂
(24)
Based on (24) and the adopted finite element implementation scheme, the following non-zero element convection
stiffness matrices are used in the analysis.
v e
e
Ki j = h
Ni N j de if ∂G ⊂ ∂
(25)
e
v
K iej = h e
Ni N j d∂e
if ∂e ⊂ ∂
(26)
∂e
3.2 Optimization formulation for robot path planning
In the previous section, it was implied that if the suggested
analogy and procedure are employed, robot path planning
can be solved by an equivalent thermal compliance minimization problem. To solve it for robot path planning, we
may consider different formulations. Specifically, we may
consider the following three formulations (although we propose to use Formulation III for robot path planning). The
side condition on γ e , 0 ≤ γe ≤ 1 (e = 1, 2, · · · , Ne ) will
be omitted in equations below.
Formulation I:
minimize
(γ1 ,γ2 ,γ3 ,··· ,γ N e )
Ne
= θT Kθ =
Ne
θe
T
Ke θe
(27)
e=1
γe m e − M0 ≤ 0 (m e :the mass of the e-th element) (28)
e=1
Formulation II:
minimize
(γ1 ,γ2 ,γ3 ,··· ,γ N e )
= αTT KT +
Ne
γe m e
(29)
e=1
Formulation III:
minimize
(γ1 ,γ2 ,γ3 ,··· ,γ N e )
Ne
e=1
= θT Kθ =
γe m e − M0 (n) ≤ 0
Ne
θe
T
Ke θe
(30)
e=1
(31)
M0 (n) = max (ℵ (n) , Mmin )
Ne
ℵ (n) =
Mmin =
(32)
me
e=1
n2
|PG − P S | m e
Ne
me
(33)
(34)
e=1
(PG , PS : position vector of the goal and starting points)
For the above formulations, the element conductivity k e is
interpolated by (11) with the penalty exponent p governed
by (17). Formulation I, which was introduced in Section 2,
may be regarded as being the most typical. As long as the
allowed mass M0 is known, this formulation can be directly
used, but it is not possible to give a specific value of M0
in robot path planning. Alternatively, Formulation II may
be employed where α is a weighting factor. If α is properly
selected, we can find a robot path simultaneously minimizing the traveling distance and the average width of the path.
Although α may be adaptively adjustable, it is still difficult
to choose an appropriate value of α.
The formulation we are proposing is Formulation III,
which is a modified version of Formulation I. The main
modification is that the mass (equivalently interpreted as the
average width of the robot path) is continuously reduced as a
function of the optimization iteration number n. Examining
(32, 33) shows that the initial iteration (n = 1) starts with
the full mass (m e ×Ne). As n increases, the allowed mass
M0 (n) decreases as 1/n 2 but is ensured to be larger than
Mmin . The minimum mass defined in (34) is evaluated by
the Euclidian distance of a straight line connecting the goal
point and the starting point. For our path planning problem,
this constraint implies that no path connecting starting and
goal points is shorter than this straight line. It can be directly
applied to the topology optimization as an explicit mass constraint. In topology optimization, it is common that each
element constituting a converged structure has unit element
density. However, notice that we are just finding the path,
not a heat dissipation structure that must be compatible with
the thermal physics; there is no need for the density constituting a path to be unity in our numerical simulation. In fact,
in all of our simulations, each element density of the resulting structure is less than unity. For example, in CASE I-1 of
Fig. 6(a), the element density is 0.2717 because, although
the Euclidian distance equals the length of 25 finite elements, the finial structure is constructed using 92 finite
elements. This is a minor problem that can be addressed
easily by a simple post processing. By gradually reducing
the allowed mass M0 up to Mmin , this formulation can avoid
unnecessarily thick paths while yielding the shortest path.
Mobile robot path planning algorithm
709
Fig. 6 Obstacle-free robot path planning. (a) CASE I-1, (b) CASE I-2 and (c) CASE I-3 (d) CASE I-4
4 Path planning by proposed method
In this section, we consider several path planning problems
to test the effectiveness of the proposed method. Although
the problem is formulated as an equivalent topology optimization problem to find optimal heat paths, we do not need
to directly associate the results with heat transfer phenomena. Therefore, we can choose numerical data such as k0
and h quite arbitrarily as long as the analogies in Table 1.
can be achieved. The specific numerical data used for robot
path problems are given in Table 2. As explained earlier,
Formulation III will be employed for all problems to be considered. The penalty exponent p is continuously changed as
the optimization iterations proceed.
The following four typical cases are considered as
numerical examples:
CASE I:
CASE II:
CASE III:
CASE IV:
Robot path planning avoiding obstacles
CASE I with a stopover
CASE I with complex terrains
Robot path planning on a real terrain.
The finite element mesh used for each problem may be seen
in the corresponding figure. The optimality criteria method
(Bendsøe and Kikuchi 1988) is used as an optimization
algorithm with the following convergence criterion:
•
•
•
Stop if |n − n−1 | / |n | ≤ ε (n : the objective
function value at the n-th iteration)
Check the criterion for three consecutive iteration
numbers.
A typical value of ε is 0.01.
4.1 CASE I: Robot path planning avoiding obstacles
Figure 6 shows problem definitions, optimized paths and
iteration histories for three different obstacle-free path planning problems. In CASE I-1 shown in Fig. 6(a), obstacles
and starting and goal points are symmetrically located.
Without imposing any symmetry constraint in the optimization, the use of Formulation III successfully yielded
symmetric paths. Because M0 (n = 1) is the full mass, the
iteration starts with an initial guess of the design domain is
710
J. C. Ryu et al.
Fig. 6 (continued.)
fully filled (i.e., all γe = 1) and thus the objective function has the minimum value at the beginning. As M0 (n)
is decreased, all values of γe tend to be reduced, yielding
grey images where the total sum of densities of grey images
are around the mass constraint value. Then around n = 10,
the uniformly distributed mass through the free region starts
to be more effectively used by pushing some γe ’s towards
zero and, on the other hand, some γe ’s, which organize
the optimal path at the end, towards unity value. However,
because of the obstacle, the value of each element density
constituting the optimal path cannot be unity. In terms of
the objective function, as the total mass is decreased, the
value increases. Then, as the mass is redistributed more
Table 2 Numerical values used to solve the robot path planning
problem by an equivalent thermal compliance minimization problem
Nominal thermal conductivity of design domain k0
1 W/m2 K
Nominal thermal conductivity of non-design domain k0
0 W/m2 K
Convection coefficient h
1 W/m2 K
Square finite element edge length
le
Heat flux at initial configuration q
1m
1 W/m2
effectively, the objective function starts to decrease until a
convergence criterion is met.
For CASSE I-2 shown in Fig. 6(b), the free configuration
is spiral. The proposed method yielded the optimized path
without many iterations and the convergence behavior is
almost the same as CASE I-1. Although this problem looks
rather simple, this problem is not an easy one because of
the shape complexity of the given obstacle configuration.
CASE I-3 (see Fig. 6(c)) is the case where obstacles of
different shapes are scattered. As seen in the iteration history, several candidate paths are constructed at the initial
step. However, the shortest path is correctly found by the
proposed method.
CASE I-4 (see Fig. 6(d)) shows the results of a benchmark problem solved by Wang and Chirikjian (2000).
Figure 6(d) show the problem as defined, the obtained optimal path, and iteration history, respectively. The design
domain is constituted with 70 by 66 finite elements. For the
present manual result, it can be seen clearly that unlike the
result of Wang and Chirikjian (2000), our method can find
almost the same optimal path directly (to see the benchmark
problem and results, see Wang and Chirikjian 2000).
Mobile robot path planning algorithm
711
Fig. 7 Robot path planning with a stopover. (a) CASE II-1 and (b) CASE II-2
4.2 CASE II: Robot path planning with a stopover
Let us consider a problem to find an obstacle-free path with
a stopover. As case studies, two problems shown in Fig. 7
are solved. In Fig. 7, the symbol “I ” denotes the stopover.
The simplest way to solve this problem is to apply a path
planning algorithm twice; after finding two partial paths
from S to I and from I to G, one can connect the two paths
to form a complete path. Instead of solving two path planning problems, the present algorithm can treat the stopover
problem as a single path planning problem if the following
modeling is used:
S, G → heat sources ( S )
I → heat sink (G )
(35)
Here, both the starting point and goal point are treated as
heat sources while the stopover is treated as a heat sink.
Obviously, the modeling given by description (35) is easily implemented in the proposed algorithm and does not
cost extra computation time in comparison with path planning without a stopover. (If there is more than one stopover,
then one has to apply the proposed algorithm more than
once.) As the iteration histories in Fig. 7 show, the path
connecting S to I and I to G is found approximately in
the same number of iterations as for the cases without a
stopover. Because the stopover is located at the center of
the domain in CASE II-1, two symmetric paths crossing
the stopover are created (see Fig. 7(a)). In CASE II-2, an
optimal path passing through the stopover is successfully
found (see Fig. 7(b)). The solution convergence for CASE
II exhibits similar behavior to that for CASE I.
4.3 CASE III: Robot path planning for complex
terrain conditions
Here, we will consider a robot path planning problem for
which the terrain condition of a free space is complex. The
complexity of a terrain may result from different levels of
robot movability. In this problem, we show that the complexity of a free space terrain can be effectively modeled
by assigning different values of thermal conductivity k0
depending on terrain complexity. Figure 8(a) depicts the
712
J. C. Ryu et al.
Assume that the height of a given terrain is given by
g(x, y). To relate conductivity k to g(x, y), we propose
the following formula:
(k e )
x
(k e ) y
⎡
=⎣
2
1/ 1 + ∂g ∂ x
×
(k0 )x
(k0 ) y
0
⎤
0
2 ⎦
1/ 1 + ∂g ∂ y
(36)
In formula (36), the conductivities in the x and y directions
are assigned independently, but (k0 )x and (k0 ) y are set to
be k0 . (Since earlier analysis for k x = k y can be easily
extended to the case for k x = k y , the detailed analysis procedure will not be given here.) The rational to use formula
(36) is as follows.
To simplify the discussion, let us consider a onedimensional model shown in Fig. 9(a). In Fig. 9(a), the
actual
traveling infinitesimal distance ds is given by ds =
d x 1 + (dg/d x)2 on a real terrain, but all analyses should
be carried out on the x coordinate line. Therefore, one can
project A1 and A2 onto A
1 and A
2 on the x axis while
imposing T (A1 ) = T A
1 and T (A2 ) = T A
2 If the
Fig. 8 Robot path planning for complex terrain conditions. (a) Terrain
condition (The gray level corresponds to the difficulty of movability),
(b) optimized path, and (c) iteration history
problem in consideration. The feasible regions and obstacles
are located exactly the same as for CASE I-3 (and CASE II2). The gray regions belong to the feasible regions but the
gray level expresses different movability. Although different
levels of movability can be assigned, the same gray level
representing the same difficulty in movability is used here
for simpler modeling. Specifically, the value kreough = 0.2
is used for the gray regions. (The fully movable regions
have the value of k e = k0 = 1) The optimized path
obtained with the consideration of two levels of movability is shown in Fig. 8(b). It is quite different from the path
shown in Fig. 6(c) obtained for the single level movability.
The iteration history is shown in Fig. 8(c).
4.4 CASE IV: Robot path planning on a real terrain
Assigning different conductivity values to discretizing finite
elements simulates different movability levels. Therefore
the developed topology optimization based algorithm can be
also used if the real terrain information is properly reflected
in defining conductivity values.
Fig. 9 (a) One-dimensional model to relate k e and g(x, y), (b) the
three-dimensional view of a real terrain with the height given by (37)
Mobile robot path planning algorithm
713
Fig. 10 The result for the robot
path planning on a real terrain
shown in Fig. 9(b); (a)
optimized path obtained by
using (36), (b) nominal path
obtained without considering
the terrain height and (c) the
relation between the distance
traveled in the two-dimensional
map and the terrain height
amount of heat flux q is set to be equal in the original and
projected path, one can use the following equation:
q = − (k0 )x
dT
dT
= − ke x
ds
dx
(37)
Therefore, the projected problem onto the x axis becomes
equivalent to the heat problem defined on a curve g. Combining the ds-dx relation and (37) yields formula (36).
Formula (36) indicates that as ∂g/∂ x or ∂g/∂ y becomes
large, i.e., as the height varies rapidly, small conductivity
values are assigned to the corresponding elements.
As a specific case study, the following height function is
considered.
g(x, y) = 2 cos x 2 − 4 + 2 sin (y ∧ 2)
+ 5 (sin (x + 1) + cos (y + 2))2
+ 0.5 (sinh (x) − cosh (y)) + 30
x = [−4, 4] ; y = [−4, 4]
(38)
The three-dimensional view of the terrain having g(x, y)
in (38) is illustrated in Fig. 9(b). To find the shortest path
on the real terrain, the x-y plane is discretized into 50×50
finite elements where element conductivities are given by
formula (36).
The optimized path obtained by the present approach is
marked by a line in Fig. 10(a) where the color level indicates the height of the given terrain. To demonstrate that the
obtained path has a shorter traveling distance compared to
the nominal path ignoring the terrain height, the nominal
path is plotted on a real terrain in Fig. 10(b). Figure 10(c)
shows the distance traveled in the two-dimensional map versus the height of the terrain for the optimized and nominal
paths. In the two-dimensional plane map, the optimized
path is longer than the nominal path, but the actual traveling distance considering the height of the terrain is much
shorter than that of the nominal path; compare cumulative
distances listed in Fig. 10(a, b). This example demonstrates
the usefulness of the present approach in a real terrain
problem.
714
5 Conclusions
A point robot path algorithm utilizing an analogy between
the topology optimization of heat paths and robot path planning was newly developed. Although substantial advancements have been made in the field of topology optimization,
there has been apparently no attempt to use newly developed
topology optimization methods in robot path planning. In
this respect, this work suggests new potential applications
of topology optimization to non-traditional fields.
The analogy is based on the idea of viewing the start
and goal points as a heat source and sink, respectively. No
convection modeling is needed except for the goal points.
When convection takes place as in the real physical world,
numerical instability appears in the thermal topology optimization unless special treatment is given. In this respect,
the employed conduction-dominant formulation can help
yield stable numerical results. In employing the topology optimization method for path planning, the issue of
assigning a mass constraint must also be addressed: in conventional topology optimization, one must provide a mass
constraint. In robot path planning, however, it is not possible to select a specific upper bound on mass. We therefore
propose a modified formulation that continuously reduces
the mass upper bound as the iteration proceeds. Furthermore, the optimization is formulated so as to start with a
full mass because it is difficult to select other bound values.
This approach is shown to be effective through numerical examples. Due to the mass-constraint varying strategy,
the convergence behavior of the objective function in the
proposed planning problem is different from that typically
observed in conventional structural problems; in the present
problems, the objective function value always increases at
initial iteration steps and later reaches an optimal value.
The developed algorithm yields stable, rapid solution convergence in all test problems including path planning with
a stopover. It is also effective in problems involving complex terrain conditions, where different complexity levels
are modeled by different values of thermal conductivity. The
problem on a real terrain was also effectively handled by
relating terrain height derivatives to conductivity values.
From practical point of view, our benchmark problems
may be quite simple and idealistic. They are limited to
path planning of a point robot. Therefore, three-dimensional
problems that take into account the orientation of a robot
cannot be directly solved at the current stage of development. However, there are a few distinct advantages over
existing works: unique starting point specification, precise
distinction of obstacles from feasible regions, and easy handling of real terrains with varying heights. If the present
work is extended, in the future, to path planning problems
in a dynamic circumstance and the problems considering a
J. C. Ryu et al.
direction of non-circular polygon mobile robots, the present
approach will be practically more attractive.
Acknowledgments This research was supported by the National
Creative Research Initiatives Program (National Research Foundation
of Korea grant No. 2009-0083279) contracted through the Institute of
Advanced Machinery and Design at Seoul National University and
WCU (World Class University) program (Grant No. R31-2009-00010083-0) through the National Research Foundation of Korea funded
by the Ministry of Education, Science and Technology. F.C.Park was
also supported by the AIM Center.
References
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods
Appl Mech Eng 71(2):197–224
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, New York
Bruns TE (2007) Topology optimization of convection-dominated,
steady-state heat transfer problem. Int J Heat Mass Transfer
50:2859–2873
Choset H, Lymch KM, Hutchinson S, Kanor G, Burgard W, Kavraki
LE, Thrun S (2005) Principles of robot motion: theory, algorithm
and applications. MIT Press, Cambridge
Hansen AG, Bendsøe MP, Sigmund O (2006) Topology optimization of
heat conduction problems using the finite volume method. Struct
Multidisc Optim 31:251–259
Incropera FP, DeWitt DP (2002) Fundamental of heat and mass
transfer, 5th edn. Wiley, New York
Keymeulen D, Decuyper J (1994) The fluid dynamics applied to
mobile robot motion: the stream field method. In: Proc IEEE int
conf robotics and automation, San Diego, pp 378–385
Khatib O (1986) Real-time obstacle avoidance for manipulators and
mobile robots. Int J Rob Res 5(1):90–98
Kim JO, Khosla P (1992) Real-time obstacle avoidance using harmonic
potential functions. IEEE Trans Robot Autom 8(3):338–349
Koditchek ED, Rimon E (1990) Robot navigation functions on manifolds with boundary. Adv Appl Math 11(4):412–442
Louste C, Liegeois A (2000) Near optimal robust path planning for
mobile robots: the viscous fluid method with friction. J Intell
Robot Syst 27:99–112
Masoud A, Masoud S, Bayoumi M (1994) Robot navigation using a
pressure generated mechanical stress field: the biharmonic potential approach. In: Proc IEEE int conf robotics and automation, San
Diego, pp 124–129
Munoz E, Allaire G, Bendsøe MP (2007) On two formulations of an
optimal insulation problem. Struct Multidisc Optim 33:363–373
Overmars MH (2002) Recent developments in motion planning. In: Int
conf on computational science, Amsterdam, 21–24 April 2002
Perez TL (1987) A simple motion-planning algorithm for general robot
manipulators. J Robot Autom 3:224–238
Rodriguez S, Thomas S, Pearce R, Amato NM (2006) RESAMPL: a
region-sensitive adaptive motion planner. In: Proc int wkshp on
Alg found of rob, New York City, NY
Schmidt GK, Azarm K (1992) Mobile robot navigation in a dynamic
world using an unsteady diffusion equation strategy. In: Proc
IEEE int conf on intelligent robots and system, Raleigh,
pp 642–647
Mobile robot path planning algorithm
Singh L, Harry S, Wen J (1996) Real-time robot motion control with
circulatory fields. In: Proc IEEE int conf robotics and automation,
Minneapolis, pp 2737–2742
Wang Y, Chirikjian GS (2000) A new potential field method for robot
path planning. In: Proc IEEE int conf robotics and automation,
San Francisco, pp 977–982
Yang RJ, Chuang CH (1994) Optimal topology design using linear
programming. Comput Struct 52:265–275
715
Yin L, Ananthasuresh GK (2002) A novel topology design scheme for
the multi-physics problems of electro-thermally actuated compliant micromechanisms. Sens Actuators A 97–98:599–609
Yoon GH, Kim YY (2005) The element connectivity parameterization
formulation for the topology design optimization of multiphysics
system. Int J Numer Methods Eng 64(12):1649–1677
Zienkiewicz OC, Taylor RL, Zhu JG (2005) The finite element method:
its basis & fundamentals, 6th edn. Elsevier, Burlington