HIERARCHICAL
PROBABILISTIC ESTIMATION OF ROBOT
REACHABLE WORKSPACE
Jing Yang, Patrick Dymond and Michael Jenkin
Department of Computer Science and Engineering, York University, 4700 Keele Street, Toronto, Canada
Keywords:
Reachable workspace, Probabilistic motion planning, Mobile robotics.
Abstract:
Estimating a robot’s reachable workspace is a fundamental problem in robotics. For simple kinematic chains
within an empty environment this computation can be relatively straightforward. For mobile kinematic struc-
tures and cluttered environments, the problem becomes more challenging. An efficient probabilistic method
for workspace estimation is developed by applying a hierarchical strategy and developing extensions to a prob-
abilistic motion planner. Rather than treating each of the degrees of freedom (DOFs) ‘equally’, a hierarchical
representation is used to maximize the volume of the robot’s workspace that is identified as reachable for each
probe of the environment. Experiments with a simulated mobile manipulator demonstrate that the hierarchi-
cal approach is an effective alternative to the use of an estimation process based on the use of a traditional
probabilistic planner.
1 INTRODUCTION
The reachable workspace W
reach
is defined as the
volume or space within which all the points can
be reached by a reference point of the mechanism,
for example, the centre of the end-effector (Kumar,
1980). Reachable workspace estimation is a fun-
damental problem in robotics as workspace proper-
ties can represent important criteria in the evalua-
tion and design of mechanical manipulators (Lenar-
cic and Umek, 1994; Badescu and Mavroidis, 2004),
robots (Zacharias et al., 2007) and environmental lay-
outs (Yang et al., 2008). The determination of W
reach
involves a considerable amount of numerical calcula-
tions, which increases with the number of degrees of
freedom of the mechanism and the complexity of the
environment.
For kinematic chains within an empty environ-
ment the computation of W
reach
can be relatively
straightforward. For mobile kinematic structures in
the presence of obstacles, the problem becomes more
challenging. Notably, estimating the W
reach
for a mo-
bile robot can be expressed in terms of the ability
of the device to plan motions within its environment.
Motion planning emerged as a crucial and productive
research area in robotics in the late 1960’s (Latombe,
1991) and its applications in real world problems con-
tinue to attract researchers from all over the world.
In basic motion planning (Latombe, 1991), given
a robot A and a static workspace W containing a set
of obstacles, the objective is to determine a collision-
free motion between the specified start and goal for
A. A configuration c of A is a specification of the
position and orientation of A in W . A configuration
c is said to be free if A positioned at c does not col-
lide with any obstacles in W . The free configuration
space C
f ree
is defined as the set of all free configura-
tions of A. The motion planning problem is therefore
formulated as computing a path in C
f ree
between two
given configurations.
A complete solution to the motion planning prob-
lem is known to be exponential to the robot’s degree
of freedom (DOF) (Canny, 1988). As a consequence
a number of heuristic approaches to path planning
have been developed. The main difference between
the probabilistic approaches and earlier complete ap-
proaches is that the probabilistic approaches do not
attempt to construct an exact representation of C
f ree
.
Rather they create a simplified graph that approxi-
mately “covers” C
f ree
and captures its connectivity in
reasonable time. The Probabilistic Roadmap Method
(PRM) (Horsch et al., 1994; Kavraki et al., 1996) is a
popular heuristic motion planner. The algorithm first
constructs a roadmap by connecting randomly sam-
pled collision free configurations and then answers
multiple queries by attempting to connect them to the
roadmap.
In estimating a robot’s workspace it is important to
60
Yang J., Dymond P. and Jenkin M. (2009).
HIERARCHICAL PROBABILISTIC ESTIMATION OF ROBOT REACHABLE WORKSPACE.
In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 60-66
DOI: 10.5220/0002205600600066
Copyright
c
 SciTePress

φ
1
φ
2
Mobile base
L
1
L
2
End-effector
φ
1
x
y
z
C
f
= (x, y, θ)
C
f
(a) (b)
Figure 1: (a) A consists of A
base
and A
arm
that has two links L
1
and L
2
connected by revolute joints. The configuration of A
is written as (x, y, θ, φ
1
, φ
2
). (b) Representations of the hierarchical occupancy of A are shown in different colors. The body
itself also has occupancy constraints but these are not shown.
observe that some DOFs are likely to be more impor-
tant than others in terms of their effect on how much
of the workspace is reachable. In addition, it may
be possible to construct subspaces of the workspace
defined by arbitrary configurations of certain com-
binations of joints. Motivated by these observations
we explore the use of a hierarchical structure of the
DOFs of the kinematic device to establish the en-
tire W
reach
. In this hierarchy we order the DOFs of
the robot in terms of their predefined “importance”.
Then we consider corresponding sub-versions of the
kinematic structure in which sub-versional joints are
considered over their range of motion. Each of these
sub-versions defines a reachability subspace that can
be established as being reachable with a single probe
into the robot’s workspace. This hierarchical search
mechanism can be used to enhance the probabilistic
algorithms for reachable workspace estimation.
This paper is structured as follows. Section 2
reviews existing algorithms for reachable workspace
estimation. Section 3 outlines our hierarchical ap-
proach. Section 4 includes comparison results from
applying the hierarchical approach and basic PRM to
a simulated mobile manipulator. Finally Section 5
summarizes the work and provides possible directions
for future research.
2 RELATED WORK
Although a range of techniques exist for reachable
workspace estimation (Badescu and Mavroidis, 2004;
Hsu and Kohli, 1987; Zacharias et al., 2007) most
existing approaches consider the problem for robotic
manipulators and do not consider arbitrary obstacles
in the environment. Robotic manipulators are fixed at
one end and this assumption provides certain efficien-
cies for reachable workspace estimation. For exam-
ple, one straightforward method to compute W
reach
is to take planer sections of the workspace defined
by the joint angles that make up the kinematic struc-
ture and determine the contour of the section in the
plane. Rotating and translating this plane based on
other joints in the chain yields the three-dimensional
workspace (Morecki and Knapczyk, 1999).
A numerical approach calculates the exact W
reach
by tracing boundary surfaces of a workspace (Kumar,
1980). In this approach, an imaginary force is applied
to the reference point at the end-effector in order to
achieve the maximum extension in the direction of the
applied force. The manipulator reaches its maximum
extension when the force’s line of action intersects all
joint axes of rotational joints and is perpendicular to
all joint axes of prismatic joints. The drawback of
this algorithm is its exponential time complexity and
that it only deals with manipulators that have ideal
joints (without limits). A more efficient system was
later developed for computing W
reach
for manipula-
tors with joint limits (Alameldin et al., 1990). The
system decomposed the problem into two subprob-
lems: workspace point generation by direct kinematic
based techniques and surface computation by extract-
ing the workspace contours utilizing a subset of the
points generated in the first module.
Much work has been done on capturing workspace
properties for interesting kinematic structures such as
human arms (Lenarcic and Umek, 1994; Zacharias
et al., 2007) and reconfigurable robotic arms (Bade-
HIERARCHICAL PROBABILISTIC ESTIMATION OF ROBOT REACHABLE WORKSPACE
61

Figure 2: Construction of R . Upper row shows the generation and classification of one node. The rank of a node c
r
is calcu-
lated by checking the hierarchical occupancy representation of A. The lower row shows the hierarchical PRM in operation.
The coverage and connectivity of R increases as more nodes are added.
scu and Mavroidis, 2004). Lenaric and Umek devel-
oped a simplified kinematic model to represent human
arm motion. W
reach
was determined by calculating
the reference point on the wrist for all combinations
of values of joint coordinates inside the given ranges.
Another approach presented by Zacharias et al. dis-
cretizes the workspace into equally sized small cubes.
Into each cube a sphere is inscribed and sample points
on the sphere are examined using inverse kinematics.
The percentage of the points on the sphere that are
reachable is used to represent its level of reachability.
The structure of W
reach
for a given device can be
very complex. Existing methods aim at capturing the
exact shape and volume of W
reach
and they involve
a considerable amount of numerical, time-consuming
calculations. It can be expensive to compute exact
W
reach
for a mobile robot in a cluttered environment.
In this paper we develop a probabilistic algorithm to
give a proper estimation for W
reach
for an arbitrary
mobile device operating in a known and cluttered en-
vironment.
3 HIERARCHICAL REACHABLE
WORKSPACE ESTIMATION
For a robot A moving in the workspace W , the
robot’s degrees of freedom (DOFs) are the mini-
mum set of independent displacements/orientations
that specify A’s complete position and orientation in
W . Thus a configuration c of A with n DOFs can be
specified as a set of n parameters, say j
1
, ..., j
n
, and
theoretically there can be O(n
2
) different orderings of
the joints. First we define an ordering of the DOFs
such that more important DOFs have a lower index.
Although there can be many definitions according to
the nature of the problem, here we define the impor-
tance of a DOF by its effect on the volume of A’s oc-
cupation in W . This importance weight is expressed
more formally below.
For a point a in A, let P
a
: C → W be the mapping
that calculates the position of point a in W when A is
placed at configuration c. Depending on the position
of a in A, P
a
is a function of c
0
⊆ c. We say that a is
determined by a DOF j
i
if j
i
∈ c
0
. Now we can define
a weight function of a DOF j
i
in terms of the volume
of A in W determined by it:
Defn 1: w( j
i
) =| {a ∈ A | a is determined by j
i
} |.
A DOF j
x
is more important than j
y
if w( j
x
) >
w( j
y
). Therefore we can write the configuration c as
a vector of length n in a decreasing order of their im-
portance, say c = ( j
1
, j
2
, ... j
n
), i.e. for i = 1, 2, ...n −
1, w( j
i
) ≥ w( j
i+1
).
Take the 5-DOF mobile manipulator shown in
Figure 1 as an example. x and y determines the en-
tire robot, and θ determines all the portions except
the rotation center of the mobile base (assume the mo-
bile robot can rotate around its center). φ
1
determines
links L
1
and L
2
. φ
2
determines link L
2
. Therefore,
w(x) = w(y) > w(θ) > w(φ
1
) > w(φ
2
), and the or-
dered configuration can be written as (x, y, θ, φ
1
, φ
2
)
(or (y, x, θ, φ
1
, φ
2
)).
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
62

Given an ordering of the DOFs, we seek a hier-
archical representation within which certain joint an-
gles are ‘free’ and can assume arbitrary values within
some previously defined domain. Let the domain of
j
i
be D
i
, c
r
= ( j
1
, j
2
, ... j
r
) is a subset of D
1
× D
2
×
...D
n
, given by {∀x
r+1
∈ D
r+1
, x
r+2
∈ D
r+2
, ...x
n
∈
D
n
| ( j
1
, j
2
, .. j
r
, x
r+1
, x
r+2
, ...x
n
)}. That is c
r
is the set
of possible configurations with joints 1...r having spe-
cific values but joints r +1...n being free. This hierar-
chical concept applies to general kinematic structures
in the domain of motion planning.
Algorithm 1: Node selection.
1: nodeFound ← f alse
2: while ¬nodeFound do
3: c ← a randomly chosen configuration in C
4: for k ← 1 to n do
5: if V (c
k
) then
6:
nodeFound
←
true
7: break
8: end if
9: end for
10: end while
11: N ← N ∪ {c
k
}
In order to integrate the above hierarchy into the
workspace estimation process, we first perform two
types of analysis to the hierarchical representations:
occupation analysis and reachability analysis. Define
the occupied area OA
c
(c
r
) of c
r
as the union of the
occupied volume in W of every element in c
r
, and
the reachable area RA
c
(c
r
) of c
r
as the union of the
reachable points of every element in c
r
. Under the
hierarchy nodes with lower r occupy and reach larger
workspace than those with higher r. To be precise, we
have these two lemmas:
Lemma 1. ∀i, j ∈ [0, n], i < j =⇒ OA
c
(c
i
) ⊇
OA
c
(c
j
). For some configuration of A, the occupied
workspace of the lower hierarchy is the superset of
that of the higher hierarchy.
Lemma 2. ∀i, j ∈ [0, n], i < j =⇒ RA
c
(c
i
) ⊇
RA
c
(c
j
). For some configuration of A, the reachable
workspace of the lower hierarchy is the superset of
that of the higher hierarchy.
In addition, let V (c) be the function that returns
true iff A is collision free when it is at configuration c.
Similarly, V (c
r
) returns true iff every element in c
r
is
valid, i.e. OA
c
(c
r
) does not collide with any obstacle
in the environment. Therefore, we have the following
lemma:
Lemma 3. ∀i, j ∈ [0, n], i < j ∧V (c
i
) =⇒ V (c
j
).
For some configuration of A, that its lower hierarchi-
cal representation is free implies the higher hierarchi-
cal representation is free, too.
Hierarchical representations of both the reachabil-
ity and occupancy can be very complex shapes de-
pending on the kinematics of the robot. In practice
the computation of the exact hierarchical represen-
tations is unnecessary. Conservative representations
of these complex shapes can provide significant com-
putational savings and this computation can be done
prior to the execution of the motion planner. This
needs to be done only once for each DOF of the robot,
independent of the robot’s configuration.
Algorithm 2: Connect(a
r
1
, b
r
2
).
1: τ ← the edge candidate returned by the local path
locator
2: Discretize τ into a list of configurations τ
0
=
(c
1
, c
2
, ..., c
m
)
3: r
current
← MAX(r
1
, r
2
)
4: for all c
i
∈ τ
0
do
5: for k ← r
current
to n do
6: if V (c
k
i
) then
7: r
current
← k
8: break
9: else
10: exit and report failure
11: end if
12: end for
13: end for
14: E ← E ∪ {(a, b)
r
current
}
To establish the general representation of OA(c
r
)
and RA(c
r
), we fix the first r DOFs and take all pos-
sible values of the remaining DOFs to construct the
hierarchical body. Note that RA(c
r
) defines a sub-
space in the robot’s workspace that is reachable by
some unique configuration within c
r
, and that for any
two configurations within c
r
there exists a continuous
path between them.
As an example, consider the mobile manipulator
A shown in Fig. 1(a), let D
1
= [x
min
, x
max
], D
2
=
[y
min
, y
max
], D
3
= [−π, π), D
4
= [−π, π), and D
5
=
[−π, π). Fig 1(b) shows the hierarchical representa-
tion of A from level 5 down to 2, indicated by color.
For simplicity we used ideal ranges [−π, π) for the
rotational joints of A.
The hierarchy can be integrated into workspace
estimation using probabilistic planners such as PRM.
We describe below the main steps involved in the con-
struction of a hierarchical roadmap R for efficient
reachable workspace estimation. Nodes with large
reachable areas are preferred (they establish more of
the environment as being reachable for each calcula-
tion). So for each configuration c, we look for the
minimum value r
min
such that V (c
r
min
) is true, and we
call r
min
the rank of c. The procedure described in
HIERARCHICAL PROBABILISTIC ESTIMATION OF ROBOT REACHABLE WORKSPACE
63

x
y
z
Top view 3: z=40
Top view 1: z=120
Front view 3: x=-100
Front view 2: x=0
Front view 1: x=100
Top view 2: z= 80
Top view 1 Top view 2 Top view 3
Front view 1 Front view 2
Front view 3
(a) 3D workspace view (b) Slices through the workspace
Figure 3: The estimation of the 3D reachable workspace W
reach
(green area) shown in different vertical and horizontal layers
of top and front views.
the pseudocode below finds a random configuration
and establishes its most general representation in the
hierarchy.
In the “for” loop from Line 4 to Line 9, the al-
gorithm computes the rank of the node by checking
collisions of the hierarchical representations of the
robot’s occupancy. Once the minimal valid hierarchi-
cal node is established the configuration together with
the computed rank is added to the set of nodes N (Line
11).
Whenever a new hierarchical node is found, we
select a number of candidate nodes from the current
set N and try to connect the new node to each of them.
In addition to the connection computation performed
by the traditional local planner, the rank of the edge
should be established. For an edge e we look for
the minimum hierarchy r
min
such that V (e
r
min
) is true
along the edge.
The hierarchical node interconnection is built
upon a local path locator and a hierarchy establisher.
The local path locator returns an edge candidate, i.e. a
local path that A can follow from one configuration to
another. Then the hierarchy establisher checks if the
edge candidate is collision free and meanwhile estab-
lishes the edge’s most general representation in the hi-
erarchy. The process of establishing the hierarchical
node interconnection is outlined in Algorithm 2.
In line 3, the hierarchy r is initialized to be the
maximum value of the ends of the edge. There is an
obvious lemma according to the definition of the hi-
erarchical edge connecting two nodes a
r
1
and b
r
2
:
Lemma 4: V (e
r
) = true =⇒ r ≥ r
1
∧ r ≥ r
2
, i.e.
the rank of an edge is not less than the rank of ei-
ther end node of the edge. Algorithm 2 searches over
the sequence of configurations on the edge for ver-
ification and hierarchy establishment. This general
approach is straightforward to implement. Note that
the hierarchy is established through the validation of
the configurations. Similarly, to apply this strategy
Figure 4: Experimental setup. (a) The mobile manipula-
tor is placed in a simple 3D environment; (b) The mobile
manipulator is placed in an apartment-like environment.
to other probabilistic motion planners it can be done
when samples are checked for collision.
After the hierarchical roadmap R is built, W
reach
can be computed from the connected component of
R that contains the initial configuration of A through
mapping function RA.
4 EXPERIMENTAL VALIDATION
We conducted experiments of our algorithm on the
simulated 5-DOF mobile manipulator A shown in
Fig. 1. First we provide an example that illustrates
the hierarchical strategy described in the previous sec-
tion. Fig. 2 provides details of the execution of the
hierarchical PRM on this example. Initially R con-
tains only one node that represents the initial config-
uration of A. The rank of each randomly generated
node is determined by looking for the most general
occupancy representation of A that does not collide
with any obstacles. Similarly the rank of each edge
is determined by looking for the most general occu-
pancy representation of A along the edge that does
not collide with any obstacles. The top row of Fig. 2
shows tests for a randomly generated node. c
2
and c
3
generate collisions while c
4
does not, so this specific
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
64

10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
250 750
1250 1750 2250
50000 70000
90000
110000
Size of Roadmap
Estimated reachable workspace
HPRM
PRM
Figure 5: Experimental result for the environment shown in
4(a). The graph shows a comparison of workspace volume
computed from hierarchical PRM and repetitive PRM sam-
pling. Each data point plots the average of ten experiments.
Standard deviations are plotted.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
250 750
1250 1750 2250 2750 3250 3750
110000
Size of Roadmap
Estimated reachable workspace
HPRM
PRM
Figure 6: Experimental result for the environment shown in
4(b). The graph shows a comparison of workspace volume
computed from hierarchical PRM and repetitive PRM sam-
pling. Each data point plots the average of ten experiments.
Standard deviations are plotted.
node is classified as c
4
. The construction of R is in-
cremental. The lower row shows incremental changes
in R . As more nodes are added both coverage and
connectivity of R increases.
The connected component of the constructed R
can be mapped to W such that W
reach
is obtained.
We use uniform cell decomposition to represent W .
Fig. 3 shows the estimated W
reach
in different layers
for the environment given in Fig. 4(a).
Fig. 5 and Fig. 6 show comparisons of the effec-
tiveness of the hierarchical PRM and random sam-
pling using repetitive PRM for reachable workspace
Figure 7: Comparison of the average running time of the
construction of traditional roadmaps and the hierarchical
ones. Averages are for 20 trials. Standard deviations are
shown.
estimation on the environments shown in Fig. 4 (a)
and (b) respectively. The x-axis represents the num-
ber of samples used in the estimation. The y-axis
represents the percentage of the estimated W
reach
(the true W
reach
of the robot was computed by brute
force search). Because randomness is involved, the
workspace volume for ten independent runs for each
case were averaged. Standard deviations are also plot-
ted. Both graphs show that the hierarchical approach
is more effective than repetitive PRM sampling for
reachable workspace estimation.
Finally we evaluate the time efficiency of our hi-
erarchical approach. Figure 7 shows the comparison
of the running time of the roadmap construction in
the basic PRM and the hierarchical approach for the
model given in 4(b). Because randomness is involved,
running times for 20 independent runs for each case
were averaged. The hierarchical PRM performed rea-
sonably well in these experiments. As can be seen
from the results, creating a hierarchical roadmap takes
less time than creating a traditional roadmap. This is
because the hierarchical PRM saves time in collision
checking in easy regions.
5 SUMMARY AND FUTURE
WORK
A hierarchical approach was presented for adapt-
ing probabilistic motion planners for reachable
workspace estimation. Unlike traditional probabilis-
tic motion planners that treat each DOF equally,
we order the DOF’s of the kinematic structure and
consider a hierarchical approach to the planning
HIERARCHICAL PROBABILISTIC ESTIMATION OF ROBOT REACHABLE WORKSPACE
65

task. Considering the characteristic of the reach-
able workspace estimation problem, this hierarchy
exploration improves the planning process through
two critical computations: occupational analysis and
reachability analysis. Validation of configurations be-
gins by doing fast tests on simple occupational rep-
resentations and only progresses to more accurate
(and more expensive) evaluations as necessary. Be-
cause randomness is involved it is hardly possible
to estimate the entire reachable workspace from the
probabilistic roadmap within reasonable time. How-
ever, by iteratively computing the maximal reachable
workspace from each node and edge our hierarchical
motion planner can be more effective in the computa-
tion process than the traditional ones.
The hierarchical workspace estimation algorithm
is especially useful for mobile robots in environments
with obstacles. Experiments were conducted on a
simulated 5-DOF mobile manipulator in two 3D en-
vironments. Experiments show that the hierarchical
approach can be an effective and efficient alternative
to the repetitive PRM for reachable workspace esti-
mation.
Our current hierarchical algorithm uses the
coarse-to-fine hierarchical nature in the process of es-
timating the workspace. The hierarchical character-
istic might also be employed in other aspects of mo-
tion planners. For example, one heuristic would be
to let the established hierarchy lead the sampling pro-
cess toward the boundaries of obstacles, i.e. to sample
more densely near nodes with higher hierarchy labels
than those with lower hierarchy labels.
We can also imagine a more sophisticated defini-
tion of reachable workspace which might involve es-
tablishing the number of configurations from which
the kinematic structure can reach a given location.
This might provide insights into different levels of
reachability. A space for where there exists many
reachable configurations should probably be consid-
ered more reachable than one with just a few.
ACKNOWLEDGEMENTS
The financial support of NSERC Canada is gratefully
acknowledged.
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