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Robotica (2010) volume 28, pp. 929–936. © Cambridge University Press 2010
doi:10.1017/S0263574709990804
Parameter self-adaptation in biped navigation employing
nonuniform randomized footstep planner
Zeyang Xia†,‡,∗, Jing Xiong†and Ken Chen†,§
†Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, China.
‡Mechanical and Aerospace Engineering, Nanyang Technological University, 639798 Singapore.
§State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China.
(Received in Final Form: December 3, 2009. First published online: January 15, 2010)
SUMMARY
In our previous work, a random-sampling-based footstep
planner has been proposed for global biped navigation.
Goal-probability threshold (GPT) is the key parameter that
controls the convergence rate of the goal-biased nonuniform
sampling in the planner. In this paper, an approach to
optimized GPT adaptation is explained by a benchmarking
planning problem. We first construct a benchmarking model,
in which the biped navigation problem is described in
selected parameters, to study the relationship between
these parameters and the optimized GPT. Then, a back-
propagation (BP) neural network is employed to fit this
relationship. With a trained BP neural network modular, the
optimized GPT can be automatically generated according
to the specifications of a planning problem. Compared with
previous methods of manual and empirical tuning of GPT for
individual planning problems, the proposed approach is self-
adaptive. Numerical experiments verified the performance
of the proposed approach and furthermore showed that
planning with BP-generated GPTs is more stable. Besides
the implementation in specific parameterized environments
studied in this paper, we attempt to provide the frame of the
proposed approach as a reference for footstep planning in
other environments.
KEYWORDS: Biped navigation; Randomized sampling;
Parameter self-adaptation; Rapidly exploring random trees;
Neural network.
1. Introduction
Humanoid robotics research has been one of the most exciting
topics in the field of robotics. With its rapid progress in recent
years, people concentrate more and more on their application
in human services and other task executions besides their
laboratory implementations. To realize the biped locomotion
in existing unstructured human-living environments, the
global navigation is an unavoidable and a new issue to be
resolved.
Global path planning and navigation strategies for mobile
robots require constructing boundary representation of
obstacles in the configuration space, and they always make
the robots circumvent obstacles.1However, biped robots, as
* Corresponding author. E-mail: zeyang.xia@ieee.org
well as other legged robots, have the ability of stepping over
or upon certain kinds of obstacles, which makes it impossible
to give exact obstacle boundary representations. In contrast,
the sampling-based motion-planning approach, which only
uses obstacle information by retral collision checking,2,3is a
practical resolution for the global biped navigation. Kuffner
and Chestnutt4–8realized the sampling-based footstep
planning for global biped navigation using forward dynamic
programming to compute footstep placement sequences.
Ayaz9,10 improved the smoothness of trajectories for posture
transitions; Michel11 applied this approach in some dynamic
environments; Chestnutt12,13 applied dynamic adjustment of
the footstep transition model and furthermore implemented
this approach to navigation for multilegged robots. Xia and
Chen14 implemented a compound footstep transition model
that decreases the planning complexity.
All the above-mentioned methods are realized by using
deterministic-sampling strategies, in which the expansion
of the search trees is directed by deterministic functions.
These functions are designed with the aim of generating
relatively optimal sequences of footstep placements to reach
the goal, such as smooth sequences with least numbers of
footsteps and least unnecessary footsteps to step over or upon
obstacles. The deterministic-sampling-based approaches are
very practical in environment with open areas. However, a
critical problem with the existing approaches is that they rely
on the design of the sampling set in the footstep transition
model and sampling-directing functions, which is usually
referred to as resolution complete. This character may result
in planning failures: the search tree may be not able to
converge, i.e., may get trapped, in the required planning
duration or sampling number of times. This situation is
especially visible during the implementation in areas with
local minima or/and narrow passages, which exist in human-
living environments, such as homes or office buildings (see
refs. [7, 15] for case studies). Besides the improvement on
the design of footstep transition model or the sampling-
directing function, another option regarding this problem is
to improve the ability to randomly avoid obstacles of the
footstep planner by employing random-sampling strategies
instead of deterministic-sampling strategies.
In our previous works, a random-sampling-based footstep
planner has been proposed.15 In this planner, a goal-biased
nonuniform sampling strategy is employed to improve the
rate at which the sampling tree converges to the goal.
930 Parameter self-adaptation in biped navigation
Goal-probability threshold (GPT) is the key parameter that
controls the convergence rate. In previous motion-planning
implementation employing nonuniform sampling strategies,
some parameters have to be manually and empirically preset
according to individual planning problems. Considering (a)
that it is difficult to preset an optimized GPT manually and
empirically and (b) that it is impossible to preset a fixed GPT
that is an optimized one for a planning problem with changing
specifications, a self-adaptive approach to GPT optimization
for our proposed footstep planner is quite necessary.
The current paper proposes a self-adaptive approach to
GPT optimization. We first construct a benchmarking model,
in which the biped navigation problem is described in
selected parameters, to study the relationship between these
parameters and the optimized GPT. Then, a back-propagation
(BP) neural network is employed to fit this relationship. With
a trained BP neural network modular, the optimized GPT can
be automatically generated according to the specifications of
a planning problem.
The rest of the paper is organized as follows: Section 2
introduces the randomized-footstep-planning approach and
the employed goal-biased sampling strategy; Section 3
studies the characters of GPT basing on a benchmarking
problem of planning in environments with local-minima
areas; Section 4 employs a BP neural network to
realize the GPT self-adaptation; Section 5 implements the
proposed parameter self-adaptive approach and verifies
its performance with comparison experiments; Section 6
discusses the further implementation of the proposed
approach and concludes the present paper.
2. Randomized Footstep Planning
2.1. Sampling-based footstep planning
The sampling-based footstep planner is a biped navigation
algorithm.15,16 It builds a search tree originated from the
initial footstep placement of the biped robot. The search
tree is expanded by footstep placement sampling in the
planning space. Footstep placements resulting in collision
are pruned from the tree by collision checking based on
the robot state and environment information. The planning
continues until some footstep placement in the search tree
reaches the goal region. Figure 1 gives the block diagram
of the footstep planner. The footstep placement sampling
is directed according to the footstep transition model that
predefines a discrete set of feasible footstep locations for the
swing foot (see Fig. 2).
2.2. Multi-RRT-GoalBias footstep planner
To resolve the problems of existing approaches using
deterministic-sampling strategy stated in the previous
section, we propose a randomized-sampling strategy based
on rapidly exploring random trees (RRTs).15–17
The key issue of the RRT-based footstep planner is
to randomly produce a temporary goal region, in place
of the actual goal region, at each step of footstep
placement sampling. Directed by these temporary goal
regions randomly distributed in the planning environment,
the search tree is provided with a better ability of avoiding
Fig. 1. Sampling-based footstep planner for humanoid robots.
Fig. 2. Footstep transition model. When the robot is supported by
one foot, we can obtain a region which can be reached by the
swing foot (left); if we consider the height offsets, the 2-D region is
expanded to a 3-D space. A set of footstep placements are selected
from the reachable region, which configures the footstep transition
model (left). Different footstep placements are used to perform
different locomotion functions, such as walking forward/backward
and turning left/right.
Fig. 3. The single-step expansion of the search tree of the RRT-
based footstep planner: Finit denotes the initial footstep placement;
Fgoal and ˜
Fgoal denote the goal region and the temporarily produced
goal region respectively; Fnew denotes the footstep placement newly
added to the search tree and is added to the search tree because it
is the nearest one to the temporarily produced goal region ˜
Fgoal
in the existing search tree. Note that the distance refers not to the
Euclidean distance but to a distance in the footstep placement space.
obstacles as well as traversing areas in which the sampling
tree may get trapped. Figure 3 explains the single-step
extension of the randomized footstep placement sampling.
We modified the RRT-based footstep planner in two
aspects. The first one was to add all footstep placements
of the footstep transition model to the search tree during
Parameter self-adaptation in biped navigation 931
Fig. 4. A single-step expansion of the search tree of the Multi-RRT
footstep planner. We use points to denote footstep placements in
order to make understanding the figure easier.
Fig. 5. Algorithm of footstep planner using the Multi-RRT: ρ2
denotes the measure function from a footstep placement to a region,
which considers not only the Euclidean distance but also the height
offset and orientation, among others.
the single-step expansion, since our previous studies verified
that the character that the RRT only adds one footstep
placement to the search tree may result in ill-conditioned
footstep placement sequences (see refs. [15, 16] for details).
Figure 4 shows the single-step expansion of the search tree
that adds multiple footstep placements. Figure 5 gives the
algorithm of the Multi-RRT footstep planner. Numerical
experiments showed that the Multi-RRT inherits the ability
of quick expansion in the planning space as well as the
characteristic of probabilistic completeness from the basic
RRT algorithm.18
Another modification was to apply a nonuniform sampling
strategy: the sampling distribution is biased to the goal
region controlled by a probability parameter, termed as
GPT (denoted as Pgoal ∈[0,1]). A goal-biased strategy can
improve the rate at which the random tree converges to
the goal region, compared with the way in which randomly
sampled footstep placements are uniformly distributed in the
planning space. While producing the temporary goal region
(see step 3 of the Multi-RRT algorithm), the planner returns
the actual goal region, instead of a randomized goal region,
at a probability of Pgoal (see Fig. 6). We termed the footstep
Fig. 6. Goal-biased random sampling controlled by Pgoal:P=rand()
is a computer-generated probability number uniformly distributed
in [0,1].
Fig. 7. A benchmarking model for a typical footstep planning
problem in an environment with local minima. The planning is
processedina4×4m
2area. The bottom and top circles denote
the original and goal regions of the biped robot separately, and the
distance between them is d. The obstacle is w×bin dimension,
and its center is in the line linking the original and goal regions of
the robot. The distance between the obstacle center and the robot
original region is λd. The field angle of the obstacle to the original
region is φ.
planner with the above-mentioned improvements as “Multi-
RRT-GoalBias footstep planner.”
3. Goal-Probability Threshold: Case Study of a
Benchmarking Problem
Now, Pgoal is a parameter that controls the expansion of the
random tree and dominantly affects rate of its convergence
of random tree. Our previous sampling experiments provided
some empirical guidelines for setting Pgoal values.15,16
However, a manually defined value cannot be assured to
be an optimized one. In addition, in order to realize the
planning in dynamic environments, it is necessary to have
a solution for Pgoal self-adaptation. Toward this objective,
we investigate the characters of Pgoal and self-adaptation of
optimized Pgoal (denoted as OPgoal), using the benchmarking
footstep planning problems in an environment with local
minima.
3.1. Benchmarking planning problem
A typical footstep planning problem in an environment
with local minima is defined as showed in Fig. 7. To
932 Parameter self-adaptation in biped navigation
obviate unnecessary coupling effects, the original and goal
regions of the robot are defined as constants. The field
angle and the distance from the obstacle to the robot, φ
and λd (see Fig. 7 for parameter descriptions), are two key
parameters considered in robot motion decision for obstacle
avoidance. The parameters in the benchmarking problem
have a geometry constraint as
φ=2 tan−1ω
2λd −b.(1)
So, instead of φand λd, two independent parameters λ
and ware selected to describe the planning problem1.
A benchmarking planning problem, denoted as P, can be
parameterized as
P=(λ, w)∈R2.(2)
To ensure the universality, we have geometric constraints for
benchmarking problem as follows: (a) A resolution exists;
i.e., the original and goal regions of the robot are legal.
(b) The robot cannot walk straightly to the goal; i.e., there
must be obstacles between the original and goal regions.
(c) No narrow-passage situation exists; i.e., there should be
a passage wide enough between the obstacle edges and the
margin of the planning area2. Considering these constraints,
we have
λ∈[0.2,0.8],w∈[0.4m,3m].(3)
In the implementation of sampling-based approaches, the
sampling number of times is related to the time complexity of
the planning algorithms and is independent of other planning
modules and the computing hardware as well; so it is a
logical index to indicate the convergence rate of the footstep
planner. The analysis on randomized-sampling approaches
is also based on statistical data of a number of experimental
trials. So the average sampling number of times of a number
of trials is used as the reference index of the benchmarking
studies.
Let P|Pgoal be a planning object of planning Pwith Pgoal.
For each P|Pgoal,wehave2×102trials. The sampling upper
limit for each trial is 1 ×103. We compute the average
sampling number of times and its standard deviation of k
1Independent parameters are preferred to describe benchmarking-
planning problems because of the following two considerations:
(a) The BP neural network will be employed that normally
requires independent input parameters, and the parameter constraint
relationship may lead to coupled effects. (b) The numerical planning
experiments that study the characters of GPT and its relationship
with the parameters of the benchmarking-planning problems also
demand independent parameters to describe the benchmarking-
planning problems.
2The third geometric constraint is given to avoid coupled effects
due to both local minima and narrow passages. The narrow passage
would occur should wapproach 4 m.
Table I. Benchmarking planning problems.
P(λ, w)¯
Nmin Pgoal(¯
Nmin)a
PN1(0.25, 1.60) 235 0.05
PN2(0.50, 1.60) 132 0.15
PN3(0.75, 1.60) 135 0.15
PW1(0.25, 2.40) 373 0.1
PW2(0.50, 2.40) 200 0.1
PW3(0.75, 2.40) 236 0.2
aPgoal(¯
Nmin) denotes the Pgoal value of planning P|Pgoal
with minimal average sampling number of times.
Fig. 8. The average sampling number of times continuously changes
with Pgoal.
times of successful trials as
¯
N=1
kk
i=1Ni,
σ=1
k−1k
i=1(Nk−¯
N)2,
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(4)
where Ni(1 ≤i≤k, k ≤200) is the sampling number of
times of ith trial. Those trails with a deviation of over 3σ, i.e.,
|Ni−¯
N|>3σ, are eliminated. The final average sampling
number of times for reference, denoted as ¯
N(P|Pgoal), is
computed with the rest of the trials by (4).
3.2. Characters of Goal-Probability Threshold
3.2.1. ¯
N(P|Pgoal)∼Pgoal relationship. We experiment
with six benchmarking problems in Table I, {PN1,PN2,
PN3,PW1,PW2,PW3}, with continuously varying GPTs,
Pgoal ∈{0,0.05,0.1,0.15,...,0.95}. Figure 8 shows how
the average sampling number of times changes with Pgoal,
from which we can see the following: (a) ¯
N(P|Pgoal) changes
smoothly and continuously with Pgoal; (b) there is only
one minimum of ¯
N(P|Pgoal) for each planning problem P;
(c) ¯
N(P|Pgoal) changes obviously with Pgoal . The above
phenomenon brings us to the conclusion that there is a Pgoal
value that makes the planner complete with a low average
Parameter self-adaptation in biped navigation 933
Fig. 9. OPgoal changes with wand fitting by conic and cubic curves.
The fitting errors by conic and cubic curves reach 0.174 and 0.161
separately.
Fig. 10. OPgoal changes with λand fitting by conic and cubic curves.
The fitting errors by conic and cubic curves reach 1.101 and 0.079
separately.
sampling number of times, i.e., optimized Pgoal, denoted as
OPgoal.
3.2.2. OPgoal ∼P(λ, w)relationship. Based on the above-
given conclusion, a critical issue is to obtain OPgoal for
individual planning problem P(λ, w). We first study the
relationship between OPgoal and (λ,w), the parameters
describing the benchmarking planning problem. Note that
actually it is impossible to know the exact value of OPgoal,
since there is no analytical solution for how it should be
computed. However, we can numerically compute a Pgoal
value with a limited error, which is termed as fitted OPgoal.
In the following, we do not specifically discriminate fitted
OPgoal and OPgoal. See ref. [16] for how to compute fitted
OPgoal for individual problem P(λ, w).
Figures 9 and 10 show how OPgoal changes with (λ, w) and
fitted curves of two sets of benchmarking planning problems,
from which we can see that for a given planning problem
P(λ, w), OPgoal changes continuously with the problem
specifications.
Fig. 11. The employed “2-20-10-1” BP neural network.
4. OP
goal Fitting by BP Neural Network
On the basis of the observations in Section 3.2, this section
discusses the approach to self-adaptation of OPgoal for a
planning problem ρ2.
4.1. Problem description
To obtain OPgoal for each P(λ,w), a mapping is defined as
:P(λ, w)→OPgoal.(5)
Since the analytical description of is not available, an
alternative solution is to employ an approximating function.
Let ˜
be the mapping using an approximating function
˜
:P(λ, w)→
OPgoal,(6)
where
OPgoal is the approximate OPgoal. For a planning
problem P(λ, W ), if the approximate error OP =
|
OPgoal −OPgoal|is within an acceptable error, ˜
can be
considered a successful approximating mapping of . With
a˜
modular employed in the footstep planner,
OPgoal can be
automatically generated regarding each individual planning
problem. Note that for each type of planning problems, only
one benchmarking model is needed for self-adaptation of the
optimized GPT.
4.2. Fitting by BP neural network
However, the fitting errors by analytical functions are not
acceptable (see Figs. 9 and 10), considering a fitting error of
over 0.1 will result large fluctuations of average sampling
number of times (see Fig. 8). A BP neural network is
employed to approximate mapping ˜
(see Fig. 11 and Table II
for specifications).
The BP neural network is trained by the Powell–Beale
conjugate gradient method with a number of learning samples
in . The samples are obtained by orthogonally designed
planning problems, such as
S(P(λ(i),w
(j)),OP(i,j )
goal ),(7)
where λ(i)=0.2+0.02i∈{0.2,0.25,...,0.8}(i=0∼
30) and w(j)=0.4+0.1j∈{0.4,0.5,...,3}(j=0∼26).
934 Parameter self-adaptation in biped navigation
Table II. BP neural network specificationsa.
Item Descriptions
Input (x1,x
2)=(λ, w),λ ∈[0.2,0.8],w ∈[0.4,3]
Layer I s(1)
i=
2
j=1
w(1)
ij xj+b(1)
i,y(1)
i=tanh s(1)
i,
Layer II s(2)
i=
20
j=1
w(2)
ij y(1)
j+b(2)
i,y(2)
i=tanh s(2)
i,
Output s(3)
i=
10
j=1
w(3)
ij y(2)
j+b(3)
i,y(3) =1+e−s(3) −1
OPgoal =y(3)
atanh(x)=ex−e−x
ex+e−x;w(k)
ij (k=1,2,3) denotes the weight of
the jth input of ith neuron in layer k;s(k)
idenotes the linear
summary of inputs of ith neuron in layer k;y(k)
idenotes the
output of ith neuron in layer k.
Fig. 12. The BP neural network converges after 531 trainings. The
learning rate is 0.01, and the error limit is 0.001.
Fig. 13. Comparison between OPgoal from the samples and
OPgoal
generated by BP neural network.
With total 31 ×27 =837 samples, the BP neural network
converges. The training specifications are given in Fig. 12.
The trained BP neural network is used to generate
OPgoal
for 30 samples randomly extracted out of the 837 ones.
Figure 13 compares OPgoal of the samples and
OPgoal
generated by BP neural network. The maximal error is
0.047 and the error deviation is 0.031, which are quite
satisfying, considering that the average sampling number
of times changes flatly while Pgoal is in the neighborhood of
OPgoal (see Fig. 8).
So far, for each planning problem P(λ, W), the BP neural
network employed in the Multi-RRT-GoalBias footstep
planner can generate the corresponding
OPgoal.And
Fig. 14. Average sampling number of times with different values of
Pgoal.
Fig. 15. Standard deviation of average sampling number of times
with different values of Pgoal.
with planning in dynamic environments, the BP module
can generate real-time
OPgoal according to the planning
problem specifications. Note that for different types of
planning problem, alternative benchmark models and the
corresponding BP neural networks should be constructed.
5. Implementation
5.1. Verification of self-adaptation approach
In order to compare the planner with BP neural network
and with manually given Pgoal, we planned with Multi-RRT-
GoalBias footstep planner for a set of randomly generated
benchmarking problems by the computer. Considering the
empirical observation that OPgoal is normally small except
in the cases of planning in clear environments, four manual
Pgoal are set as {0.05, 0.25, 0.45, 0.65}. Figures 14 and 15
compare the average sampling numbers of times and their
Parameter self-adaptation in biped navigation 935
Fig. 16. Footstep sequences of 10 trials by the Multi-RRT-GoalBias
footstep planner with the BP-generated Pgoal.
Fig. 17. A footstep sequence by the Multi-RRT-GoalBias footstep
planner with the benchmark model discussed in this paper.
standard deviations of planning with BP-generated Pgoal and
manual Pgoal. Comparison of average sampling numbers of
times verified that planners with self-adapted Pgoal by BP
neural network can complete with less complexities.
Figure 14 shows that the distribution of average sampling
numbers of times of planners with parameter self-adaptation
is more concentrative than planners with manual Pgoal, which
shows that goal-biased randomized planning with optimized
Pgoal is much stable. Figure 16 shows the footstep placement
sequences of 10 trials with the BP-generated Pgoal.
5.2. Planning by Multi-RRT-GoalBias footstep planner
We implemented the Multi-RRT-GoalBias footstep planner
for a planning problem reported in ref. [7]. Compared
with the results by deterministic planners,7the Multi-
RRT-GoalBias footstep planner can generate desirable
footstep sequences within quite acceptable planning duration
and sampling number of times (see Fig. 17 for one
footstep placement sequence). The
OPgoal generated by
BP neural network is 0.173. Statistical data of 200 trials
include average sampling number of times of 338, average
searched nodes of 3387, and average planning duration of
0.332 s. Implementation of the Multi-RRT-GoalBias footstep
planner in different planning problems relies on alternative
parameterized benchmark models with the corresponding BP
Fig. 18. A footstep sequence by the Multi-RRT-GoalBias footstep
planner with a benchmark model describing planning problems in
a typical environment with narrow passages.
neural networks. Figure 18 shows one footstep sequence
generated with a benchmark model describing a planning
problem in a typical environment with narrow passages (see
ref. [16] for more discussion).
6. Discussion and Conclusion
GPT is a key parameter controlling the convergence rate
and affecting the feasibility of the goal-biased nonuniform
sampling in the randomized footstep planner. Self-adaptation
of GPT regarding the planning problem specifications is an
important issue. In this paper, we proposed an approach to
GPT self-adaptation by parameterizing of planning problems
and presented how it was realized in the cases of footstep
planning in environments with local minima. Three key
steps of implementation of the proposed method include (a)
construction of parameterized model of planning problems,
(b) construction and training of BP neural network, and (c)
integration of BP module in the planner. The comparison
experiments verified the feasibility and performance of the
proposed approach.
Besides the implementation in specific parameterized
environments, such as environments with local minima stated
in the current paper, we attempt to provide this method
as a reference for implementation of planning in other
environments. The critical issue of further implementations
is to construct the parameterized model. We are in the
process of using the proposed approach for biped navigation
for humanoid robot soccer competition, and the related
publication will follow.
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