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Hybrid Discrete-Continuous Path Planning for Lattice Traversal



Lattice structures allow robotic systems to operate in complex and hazardous environments, e.g. construction, mining and nuclear plants, reliably and effectively. However, current navigation systems for these structures are neither realistic, as they assume simplistic motion primitives and obstacle-free workspaces, nor efficient as they rely solely on global discrete search in an attempt to leverage the modular-ity of lattices. This paper tackles this gap and studies how robots can navigate lattice structures efficiently. We present a realistic application environment where robots have to avoid obstacles and the structure itself to reach target locations. Our solution couples discrete optimal search, using a domain-dependent heuristic, and sampling-based motion planning to find feasible trajectories in the discrete search space and in the continuous joint space at the same time. We provide two search graph formulations and a path planning approach. Simulation experiments, based on structures and robots created for the Innovate UK Connect-R project, examine scalability to large grid spaces while maintaining performances close to optimal.
Hybrid Discrete-Continuous Path Planning for Lattice Traversal
Santiago Franco*1, Julius Sustarevas*2, Sara Bernardini1,
Abstract Lattice structures allow robotic systems to operate
in complex and hazardous environments, e.g. construction,
mining and nuclear plants, reliably and effectively. However,
current navigation systems for these structures are neither
realistic, as they assume simplistic motion primitives and
obstacle-free workspaces, nor efficient as they rely solely on
global discrete search in an attempt to leverage the modular-
ity of lattices. This paper tackles this gap and studies how
robots can navigate lattice structures efficiently. We present a
realistic application environment where robots have to avoid
obstacles and the structure itself to reach target locations.
Our solution couples discrete optimal search, using a domain-
dependent heuristic, and sampling-based motion planning to
find feasible trajectories in the discrete search space and in the
continuous joint space at the same time. We provide two search
graph formulations and a path planning approach. Simulation
experiments, based on structures and robots created for the
Innovate UK Connect-R project, examine scalability to large
grid spaces while maintaining performances close to optimal.
Weight efficiency, high stiffness, and strength of lattice-
like cellular structures have made them the focus of an
increasing body of research in the fields of self-assembly
and autonomous construction [1]–[4]. As illustrated in Fig. 1,
often, in these applications, the system design includes a
robot that needs to traverse a lattice to undertake assembly
or inspection tasks. To do that, the robot must find a feasible
path from one part of the structure to another, giving rise to a
unique path planning problem. While the modular nature of
the structure lends itself to discrete search, motion planning
is carried out in a continuous joint space. In this paper, we
propose a novel methodology that brings these two aspects
together, satisfying the main assumptions of discrete search
and leveraging the kinematic capabilities of modern robotics.
A significant work that explores robotic assembly of lattice
structures is the project TERMES [5]. Inspired by termites, it
features a team of small robots building a rectilinear structure
from modular magnetic blocks. The robots can pick up the
blocks, climb an existing structure and attach new blocks
according to a building plan. The research explores assembly
sequencing while maintaining structure traversability at any
point during construction. A similar project by Jenett et al.
[6], [7] propose a Material-Robot system that leverages a
complex multi Degree of Freedom (DoF) robot assembling
cub-octahedral and octahedron lattice elements. Here, multi-
ple bipedal robots traverse and construct the structure simul-
taneously. However, both of these works adopt a simplified
primitives-based approach for robot motion. In the case of
Both authors contributed equally to this work.
1{Santiago.FrancoAixela},{Sara.Bernardini}, Royal Hol-
loway University of London., Ross Robotics Ltd
Fig. 1: Lattice Structure Traversal Problem: Lattice components are
shown in grey, obstacles in white and connectors (sockets) in green.
Robot starting connector marked in yellow, end connector in red.
TERMES, the robot relies on mechanical solutions to assure
motion success and can only move orthogonally, climbing
up or down. The Material-Robot system uses only a handful
of human prescribed and pre-computed primitive motions.
While use of primitives simplifies the path finding problem
and allows the use of standard discrete search algorithms
such as A, it fails to leverage the capabilities of multi-DoF
robot kinematics and requires strong assumptions such as
an obstacle free environment and a completely rigid lattice
structure. These assumptions compromise the applicability
of self-assembly structures to real-world applications.
This paper is inspired by recent work on self-assembling
structures for nuclear decommissioning [8]. In this setting,
information about the environment and structure is available,
but obstacles might be present. Furthermore, as exposure to
radiation is undesirable, minimizing the time the robot takes
to traverse the structure becomes crucial. The lattices in this
paper are based on the structural robot prototype discussed
in [8]. Therefore, we propose a methodology on how joint
space robot motion planning and discrete lattice search can
be coupled, allowing complex collision-aware motions. The
contributions of this paper are as follows: 1) we propose
two graph formulations for path finding in lattice structures,
requiring different computational efforts; 2) we propose a
discrete planning approach for these graphs; and 3) we carry
out an extensive evaluation of our approach in simulation.
The structure of this paper is as follows. Section II reviews
literature on self assembly systems, climbing robots and
discrete search. In Sec. III, we describe how the lattice, the
robot and the planning problem is modelled. We then propose
graph formation in Section IV and search approaches in
Section V. Section VI offers an evaluation of our techniques
in simulation and discusses their qualitative and quantitative
properties. Conclusions are presented in Section VII.
1) Climbing Robots and Lattice Traversal: Together with
self-assembly and construction, systems similar to ours have
been used for inspection [9]–[11] and maintenance [12]. A
common type of robot considered for these problems is the
Inchworm archetype [13]. These robots consist of a single
kinematic chain that is symmetrical about a revolute joint.
The two ends of the kinematic chain are used to attach the
robot to the environment creating a biped-like motion.
Considering structure traversal, leveraging structure dis-
creteness could help find paths over it using established
search methodology. The Material-Robot system [6], for
example, after defining motion primitives, uses Awith the
Manhattan heuristic. However, further work by Jenett et al.
[14] examines compliance and deformation within the lattice
structures and thus highlights the necessity to plan online.
When motion primitives are not assumed, finding optimal
intermediate states or gait sequences for the inchworm robot
becomes a challenge. A work by Yang et al. [15] describes
how a graph can be constructed by considering the end-
effector landing locations as the base for future steps. To find
a feasible path, they use Breadth First Search and validate
each edge via an online call to motion planning. Determining
these gaits or attachment locations is also explored by Zhu
et al. [16], who focus on biped robots climbing over truss-
like structures. The robot’s kinematic capabilities determine
its ability to perform a collision free transition from one
truss to another, in turn informing global path planning. The
search space is small as only a few trusses are considered,
however determining the grip locations as part of the problem
increases its complexity. The related problem of climbing
tree branches is studied by Lam et al. [17], [18]. The robot
used in these works has a flexible continuum body, but
locomotion-wise is similar to the inchworm archetype. These
works shows how tree branches can be described as a graph
and locations along the branches are discretized to allow
for dynamic programming search. However, the problem
is simplified by preventing the robot switching between
branches, which reduces the graph’s size.
Pagano et al. [19] proposes a more complex scenario
where incomplete information about the world is assumed.
In this work, an inchworm robot uses depth sensors to allow
an iterative Line of Sight Tree approach to find intermediate
robot states. Meanwhile, a potential field motion planning is
used to find the joint space motion that the robot needs to
follow. While this approach runs online and is highly flexible,
it is far from optimal as it requires the robot to backtrack
frequently to gather additional information.
2) Discrete Search: The robotics works discussed above
tend to split the discrete and continuous parts of the problem
and apply different methods to each part. Divide and conquer
approaches are also seen in discrete search literature, e.g. the
TIM planner [20], where a generic planner uses a specialized
planner to solve a subset of the problem. Semantic attach-
ments follow a similar strategy [21], [22].
In this paper, we run A* multiple times as we learn the real
cost of actions. LPA* [23] is an incremental type of A* that
repeatedly calculates an plan from a fixed initial state to the
goal state, when some of the action costs change. The first
search is identical to A*, following searches place nodes in a
priority queue for reevaluation if a node becomes inconsistent
due to changes to the predecessor edge’s costs. Expansions
are ordered by f-value and tie-breaked by min(g,rhs), where
rhs is a look-ahead value based on the minimum gvalues
of the node’s predecessors. LPA* is guaranteed to expand
every state at most twice per search episode. LPA* is not
guaranteed to be correct (always find optimal plan) when
using admissible but inconsistent heuristics, unlike A*. In
our application, the A* search effort is insignificant for most
problems, however for the largest discrete search spaces
LPA* should be considered.
D* [24] can be used to generate optimal paths in a partially
known environment. In our case, the discrete environment is
fully known, but edges need to be validated because obsta-
cles or deformations require ad hoc solutions. We ignore
obstacles in the discrete search space because validating all
expanded edge costs would be too expensive.
In this section, we outline how the lattice structure and the
robot are modeled and state the path planning problem we
tackle. We also describe the software framework we used.
A. The Lattice Structure
To capture the construction and self-assembly use cases
presented in the literature [6], [8], we generalize the lattice
structure as a 3-dimensional grid of homogeneous cubes of
side d(d=0.35m in our experiments). Fig. 1 shows how
cubes form an overall lattice with obstacle cubes in white.
Each exposed face of each lattice cube has a connection
socket that a robot can attach itself onto. Both lattice and
obstacle cubes have an associated collision box, but there is
a gap (1cm) between the socket frame and the collision box
to allow easy motion planning when attaching to the socket.
A socket coordinate can be represented as a vector of
integers s= (x,y,z,f), where x,y,zZ,f {0,1,2,3,4,5}.
The x,y,z components translate into multiples of cube widths.
The face value fdenotes the index of cube faces expressed
as the rotations {I,Ry(π
where Iis the identity matrix and Rx,Ryare rotation matrices
about the x and y axes. Fig. 2 illustrates the socket naming
and frame orientation conventions.
Fig. 2: Socket frame x,y,z axes shown in red, green and blue.
The conventions above allow a compact description of
socket coordinate in grid-world. The homogeneous transfor-
mation matrix denoting the socket frame can be found as the
following multiplication of translations and rotations:
Ts=Trans(d·(x,y,z))Rot(Rf)T rans(d
2·(0,0,1)) (1)
B. The Multi-Task Bot
Following literature, we adopt the inchworm robot
archetype. For our experiments we simulated the inchworm
type robot created for the Connect-R project [25]. We refer to
the robot as Multi-Task Bot (MTB). We model the MTB as a
symmetrically arranged 5-DOF arm. The robot is illustrated
in Fig. 3. In our experiments, we use a robot with link lengths
of d1=11.015 cm, d2=5.065 cm, d3=33.6 cm. Robot
joints 2-4 rotate about their y axes, while joints 1 and 5
rotate about z axis. Rotation limits for joints 1-5 are ±180,
±95,±121,±95,±180, respectively. We also assume all
joints have the same velocity limit, set to ±0.5 rads1and
gear reduction ratio of 90.
Fig. 3: Connector frames and joints of the MTB robot. Joints 1 and
5 rotate about the z-axis, while joints 2-4 rotate about the y-axis.
The robot is designed to be symmetrical about joint 3,
making the ends of the kinematic chain interchangeable.
This also means that we only require one kinematic model
to describe the robot both ways along the chain. For the
robot to fix itself to the structure, a connector frame must
fully overlap a lattice socket frame, orientation included. As
a convention, we say root connector to mean the one that
remains fixed during robot motion. Given a homogeneous
transform to a root connector TRand a joint state j=
(j1,j2,j3,j4,j5), the leaf connector frame can be found via
Forward Kinematics TL=FK (TR,j). To model the reverse,
we define a robot joint inverse function as follows:
inverse(j) = (j5,j4,j3,j2,j1),then
if TL=FK(TR,j),it follows:
C. Software Framework
The lattice and robot described in this section are modeled
using tools from the Robot Operating System (ROS) ecosys-
tem. All transformation calculations are carried out using
the ROS tf and PyKDL frameworks. Robot motion planning
uses the MoveIt! [26] software package, where the lattice
structure populates the planning scene of the robot. The robot
inverse kinematics are computed via the Trac-ik solver [27].
We use two joint-space planners, RRTConnect [28] and
RRTStar [29], as they are implemented by the Open Motion
Planning Library (OMPL) [30]. Both planners are sampling-
based and adopt the Rapidly exploring Random Tree (RRT)
methodology. The RRTStar planner is asymptotically optimal
and we set its optimization objective as to minimize the
maximum element of joint distance traveled.
D. Path Planning Problem Statement
Consider the problem in Fig. 1. A multi-task robot is
attached to the lattice structure at a starting socket s0and
its joint configuration is known. The position and size of the
lattice cubes, the obstacles and the set Sof all free sockets
on the lattice are also assumed to be known. The robot is
tasked with connecting to any free socket on a goal cube
g= (gx,gy,gz)located on the lattice, i.e. the set of goal
sockets is Sg={(gx,gy,gz,0...5)} S. The planner must
find a sequence of sockets siand corresponding joint space
trajectories Ji,i+1taking the robot from s0to one of the goal
state sgSgoal , such that each Ji,i+1is feasible and collision
In this section we tackle the lattice structure path planning
problem. We first examine the problem statement presented
in the previous section and propose two alternatives of
how lattice and robot descriptions can lead to a discrete
search graph construction. The vertices of these graphs will
correspond to robot state, while edges - to robot motion.
Then, in Section V, we show that calling continuous joint
space planners to retrieve graph edge costs during discrete
search leads to poor performance and scalability. To address
this, we describe how computing an offline cache can provide
asymptotically-minimal estimated edge costs and in doing so
allow an iterative search-validation approach.
A. Single Socket ab-Graph
A naive way to form a search graph is by choosing on
the neutral joint configuration that the MTB takes when
attached to a single socket. We use the all-zero configuration
j=0, corresponding to the MTB fully stretching out (see
Figure 4a). In this way, tuple v= (s,0)corresponds to
placing the MTB root connector at Tsand setting all joints
to zero. This tuple describes the MTB state completely and,
in turn, defines a graph vertex v. An edge of such a graph
corresponds to the MTB hopping from one socket to another,
starting and ending with j=0, see Fig. 4. Only vertices with
mutually reachable sockets share an edge, this is discussed in
Sec. IV-C. Computing the joint space trajectory for an edge
between sockets aand bis thus composed of four steps:
1) Find the inverse kinematics solution j=IK(Ta,Tb).
2) Compute the collision aware trajectory J1taking the MTB
from state (a,0)to (a,j).
3) Compute the collision aware trajectory J2taking the MTB
from state (b,inverse(j)) to (b,0).
4) Concatenate the trajectories: Ja,b= [J1,J2].
We call this graph formulation the Single Socket ab-graph,
signifying that each edge corresponds to two sockets, while
vertices to one. Note that the joint inverse function allows us
to use a single robot kinematics model during computation.
For Ja,bto be performed on real hardware, the J2component
needs to be inverted back.
The ab-graph’s main advantage is that each vertex in the
graph can be described by using a single socket. However,
two expensive joint-space motion plans must be formulated
to compute the edge costs. Also, as the neutral state has
the same configuration for all sockets no matter of their
surroundings, there might be cases in which the MTB
ends up colliding with nearby obstacles, invalidating the
corresponding vertices.
B. Double Socket abc-Graph
A graph can also be built by using two sockets for a
vertex and three for an edge (see Figure 4b). We call this
graph Double Socket abc-graph, signifying three sockets per
edge where socket bis common to both the initial and
final vertex. Here, a vertex is a tuple v= (a,b,ja,b), s.t.
ja,b=IK(Ta,Tb). The pair of sockets a,bmust be mutually
reachable. Assuming that, for each socket sS, we have
access to the set of reachable sockets CsS, the pairs then
are (a,b),aS,bCa. We explain in the next section
how the set Csis obtained. Note that there can be multiple or
even infinite (in case of high-DoF robots) inverse kinematics
solutions between two sockets. For this reason, when we
create the graph data structure, we find a single ja,bof each
vertex. A path planning algorithm operating on this graph
will not consider alternative ja,band as such, solutions found
or optimality properties are restricted to the specific set of
ja,bsassociated with the vertices. Furthermore, the vertices
(a,b,ja,b)and (b,a,inverse(ja,b)) correspond to the exact
same robot state. To avoid unnecessary vertices, we store
unique vertices only by using alphanumerical sorting.
The abc-graph dominates the ab-graph in terms of com-
pleteness. The ab-graph requires to fully extend the MTB
after connecting to a new socket while the abc-graph only
requires that a physical connection is possible between two
sockets. Hence, there are multiple cases where a socket
is reachable in the abc-graph while unreachable in the
ab-graph. Any solution in the ab-graph has at least one
equivalent solution in the abc-graph while the opposite is not
necessarily true. This is critical when dealing with obstacles.
Any particular set of reachable sockets Cscan be obtained
by appropriately translating and rotating sockets inside C0.
In our implementation, we compute the homogeneous trans-
formations for all sockets in the CM, perform the frame
transformation Tsand cast the transforms back into socket
tuples. The resulting set is then intersected with the set of
free sockets Son a lattice, obtaining CsS.
Finally, an edge of the abc-graph corresponds to a single
hopping motion between any two reachable vertices that
share a socket, with the robot start and end joint configura-
tions being the ones associated with the vertices, see Fig. 4.
The joint space trajectory for the motion can be computed
as follows:
1) Find the matching socket bbetween (a,b)and (b,c)vertices.
2) Appropriately invert the joint configurations associated to
(a,b)and (b,c)vertices to obtain jb,aand jb,c.
3) Compute the collision aware trajectory J(a,b),(b,c)taking the
robot from state (b,jb,a)to (b,jb,c)
(a) ab vertex and edge (b) abc vertex and edge
Fig. 4: Robot motions for ab and abc graphs. Final vertex positions
are shown in full opacity, intermediate configurations are transpar-
ent, and the edge motions are show by the arrows.
C. Connectivity Map
The above descriptions on how the ab and abc graphs
can be constructed rely on knowing a set of reachable
sockets CsSto which the robot can connect while still
being attached to socket sS. To find these sockets, we
compute the robot’s Connectivity Map (CM) C0empirically.
We create a fully populated grid world of size ncentered
at world origin. Grid size is chosen chose so it encapsulates
robot reach volume. In our case, n=7, i.e. all cubes between
(3,3,3)to (3,3,3)exist. All six faces of all cubes
in the grid have sockets. Collision detection for this setup
is disabled. We attach the robot to socket (0,0,0,0)and
compute the inverse kinematics IK(T0,TsSgrid )to all sockets
in this grid. The sockets for which an IK solution is found
form the robot connectivity map C0. In our case, the CM is
comprised of 53 sockets.
Any particular set of reachable sockets Cscan be obtained
by appropriately translating and rotating sockets inside C0.
In our implementation, we compute the homogeneous trans-
formations for all sockets in the CM, perform the frame
transformation Tsand cast the transforms back into socket
tuples. The resulting set is then intersected with the set of
free sockets Son a lattice, obtaining CsS.
D. Graph Validation
For both types of graphs, a collision check is performed so
only valid vertices are considered. The Connectivity Map is
used to determine the existence of the graph edges. For single
socket ab-graphs, the edges exist for all vertices v,w= (s,0)
such that wsCvs. For double socket, abc-graphs, edges
exist for all vertices v,w= (a,b,ja,b)such that waor wb
is in CvaCvb. Note that this check also guarantees that v
and wshare a socket. Finally, we define the edge cost as
the time taken by the collision free edge motion, which is
dependent on the structure and obstacles. Thus, at the graph
construction stage, edge costs are left unset.
In Section IV, we describe how the ab and abc graphs can
be constructed to capture the lattice structure and a robot’s
ability to traverse it. We can now define a global search
problem that captures the path planning problem stated in
Sec. III-D. Given the set of free sockets S, the goal cube
gthat defines the goal sockets Sgand a starting socket
s0, the global search problem can be defined as the tuple
Π=hV,v0,Vg,Ji, where Vis the set of the graph vertices
constructed from S,v0is the initial vertex corresponding to
s0,VgVis a set of goal vertices such that the robot is
attached to any of the goal sockets in Sgand Jrepresents all
the available actions, i.e. the joint trajectories that a robot can
perform. A plan π=J0,J1,..., Jnis made up of consecutive
actions that take the robot from v0to a state vgVg. A
plan is optimal if there is no alternative plan reaching any
vgVgwith a smaller cost. The function c(J)represents the
cost of each action, that is the time of the joint trajectory
J. The search problem seeks to minimize c(π) = Jπc(J).
To distinguish between the two types of graphs, we use
v= (s,0)Vab and v= (a,b,ja,b)Vabc.
A. Awith Online Motion Planning
To grasp an intuition on the search problem, we first
examine how a naive Aapproach performs on both graphs.
The graph connectivity is computed offline, but every time
Aattempts to obtain the edge cost of a new edge, the
RRTConnect joint space planner is called to find a joint
trajectory Jtaking the robot from one vertex state to the
other as described in Sections IV-A and IV-B. The RRT-
Connect planner is fast but not optimal and is used only
for illustration. Using an optimal planner for this test would
lead to very poor performance (we obtained no meaningful
data in our experiments). Also, efficient Aplanning requires
an informed admissible heuristic. Here, we use a Weighted
Euclidean heuristic. We describe and prove the admissibility
of this heuristic in an upcoming section. The results of testing
Awith online motion planning are shown in Fig. 5.
Fig. 5: Number of Solved and OutOfTime (2 h) instances over 100
trials with online validation. Lattice sizes range from 6 to 21.
Figure 5 shows that online edge cost validation scales up
poorly, even without obstacles, as the lattice size increases.
Section VI shows how a cache-based approach scales better.
Note that using greedy search algorithms would generally
reduce the number of node expansions and hence speed
up finding solutions. However, we next present an iterative
method which preserves optimality while greatly reducing
average node expansion costs. If optimality is not required,
the proposed iterative approach could easily be adapted to
greedy search methods as well.
B. Iterative AWith Cached Costs
Figure 5 shows that calculating a joint space solution for
each node expansion is impractical even in the best case
scenario (without any obstacles). Instead, we propose an
iterative Awith cached costs approach. Firstly, we compute
an offline cache of asymptotically optimal costs associated
with the ab and abc motions. This cache can then be used
to obtain admissible and consistent cost estimates during
Avertex expansion. We also use these costs to derive the
weights for the Weighted Euclidean heuristic. To ensure the
paths are feasible, Awill be called iteratively, performing
path validation between iterations.
1) Costs Cache Computation: We compute the cache by
performing multiple joint space plans for ab and abc motions
using the RRTStar planner. We use the same grid setup as
in Section IV-C, disable collisions with obstacles, but keep
MTB self collisions. Vertex pairs considered for ab cache
are shown in Eq. 3, for abc in Eq. 4:
((0,0,0,f),b)f {0..5},bC0(3)
((0,0,0,f),a),((0,0,0,f),b)f {0..5},a,bC0(4)
We do not assume a single IK solution for ((0,0,0,f),a)
and ((0,0,0,f),b). IK is called every time, this way many
IK solutions are tried and the approach is applicable to
higher DoF robots. Furthermore, as described in Sec. III,
cube sockets have specific orientations. Therefore, the whole
cache data structure cannot be computed only for f=0. For
example, I·Ry(π
2). This does not affect
the CM computation, but when planning a motion, joints
could travel a different distance and so all sockets on cube
(0,0,0)are tested. Structure-dependent symmetries could be
exploited to speed up cache computations. A total of 100
trials are computed for each edge, in order to capture all
possible IK solutions, and the fastest is taken. The resulting
cache thus stores the minimum costs found for any edge
on v,v0on the (0,0,0)cube K0(v,v0) = min(c(v,v0)).K0
is computed once for ab and abc motions and is specific
to robot root connector similarly to CM. The cache can
be applied to any edge on the lattice by expressing the
x,y,zcomponents of the vertex sockets relatively to the root
connector socket. However, unlike CM, computing the K0is
expensive. The ab cache involves 318 vertex combinations
and takes about 25h using a consumer grade laptop. The
abc cache involves 8268 vertex combinations and takes about
650h. To perform these computations we used the computing
cluster described in Section VI. The cache is asymptotically-
minimal because we are using the RRTStar planner, which
is asymptotically optimal. Cache optimality is subject to
number of trials and the RRTStar time out threshold (we
use 5 s). Fig. 6 shows how during both types of cache
computation the minimum costs over increasing number of
trials converges to the min over all trials.
2) Heuristics: In addition to edge costs, the cache allows
us to derive an informative Weighted Euclidean heuristic
hεto be used with Asearch. A heuristic h(vV)is an
estimation of the minimum cost of a feasible plan π, taking
(a) ab-cache (b) abc-cache
Fig. 6: % of edges whose mincosts over num trials are within an
error margin of 1% and 0.1% of min over all trials.
Algorithm 1: Iterative A*
Data: Π=hV,v0,Vg,Ki,RRTC
Result: π
1ValActions ;c(A)EstimatedCosts();Sol utionFound ;
2while ¬SolutionFound do
3π,SolutionFound Astar(Π);
4if SolutionFound then
5validated =>;
6for aπdo
7if aValActions then
9f easible,cost RRTC onnect(hv,v0i a);
10 if f easible then
11 c(a)cost;
12 ValActions a;
13 else
14 AA\a;validated ;
15 break;
16 if validated then
17 return π
18 return // No Feasible Solution found
vto Vg. A heuristic is admissible if h(v)h(v)where h(v)
is the costs of a true optimal plan. A heuristic is consistent if
both h(v)0|∀vVgand h(v)c(v,v0) +h(v0)|∀vV\Vg
where v0and vshare an edge. Consistency implies admissibil-
ity. Both properties are desirable as admissibility guarantees
optimality and consistency leads to fewer node expansions
and shorter search times. First we define a distance function
between two sockets s,s0. We then extend this definition to
define hεfor ab and abc graphs:
dist(s,s0) = ||(sx,sy,sz)(s0
hε,ab(vVab ,g) = ω·dist(vs,g)(6)
hε,abc(vVabc ,g) = ω·min(dist(va,g),dist(vb,g)) (7)
The weight ωin Eqs. (6, 7) is determined using the cache K0.
For both graphs, it is the minimum (over all edges ([v,v0]
K0) ratio between edge costs K0(v,v0)and the maximum
distance between sockets involved with the edge. That is,
s,s0 {vs,v0
s}for ab cache and s,s0 {va,vb,v0
b}for abc.
In other words, Eq 8 describes capturing the minimal costs
to travel the longest Euclidean distance.
Due to the triangle inequality, an unweighted Euclidian
distance is a consistent heuristic if costs(v,v0)d(v,v0).
Therefore, assuming the costs c(v,v0) = K0(v,v0)are mini-
mal, we only need to ensure that |hε(v0)hε(v)| c(v,v0)
to keep consistency. The trivial solution would be to make
ω0, but that would not be informative. Eq. (8) ensures that
(a) S6O5 lattice (b) S11O5 lattice
(c) S16O5 lattice (d) S21O5 lattice
Fig. 7: Cuboid lattice structure used for testing. For example, S11O5
corresponds to lattice of side length 11 and 5% of bounding 17 ×
17 ×17 cube volume being obstructed by obstacles. A video of
experiments is available here
there is no action for which the change in heuristic values
from vto v0is bigger than the cost of moving from vto v0.
The same logic applies to both graphs because we defined the
distance function for abc vertices as the pair-wise minimum
distance between any sockets ((0,0,0,f),va,vb) {v,v0}.
The admissibility and consistency of the heuristic using these
weights is only guaranteed up to the correctness of the
asymptotically-optimal cost estimates.
3) Iterative A:To reduce the number of calls to the joint
space planner, we perform iterative search and validation
using the following steps: 1) The graph is populated with
vertices and edges. Edge costs are estimated using K. 2) An
Asearch is performed on the graph. 3) We then validate
all edges in the path found using the RRTConnect planner.
During this step, the collision-free joint-trajectory Jis stored
with the edge and costs are updated with the duration of J.
This also includes removing edges, when joint state planner
finds no solution. 4) We repeat steps 2,3 using a persistent
graph data structure. That is, the edges are validated or
removed in every iteration. We are finished once we have
found a feasible path such that all edges in the path are
validated. Please see Algorithm 1 for details.
4) Optional Replanning Window: The first time Asearch
is performed, the path is found using the estimated edge
costs, let the costs of this path be c(π). The algorithm 1
then goes on to validate the edges and call Aagain until the
whole path is feasible, resulting in a valid plan π. However, π
is not guaranteed to be the cheapest plan, because there might
be alternative cheaper paths using yet to be validated edges.
On the other hand, searching and validating edges takes time.
Hence, optimally, one could keep calling Iterative Afor the
time duration c(π)c(π). This replanning window allows
to only search for more optimal paths if a more optimal plan
is possible when taking into account the replanning costs. In
Section VI, we show that in the majority of experiments the
first feasible plan returned by iterative A* is also the best
one once the replanning time effort is taken into account.
1) Experimental Setup: To evaluate our approach, we use
four lattice-structures as presented in Fig. 7. The cuboid
lattices range in size form 6 to 21 and are made up of
repeating sub-lattices. In order to obstruct inside and outside
the structure, we first define a bounding cube. We do this
by adding 3 cube units along each direction, e.g. a bounding
cube for lattice of size 6 would be size 12. Then, for each
free cube-point we place an obstacle with a probability 0% to
50% in 5% increments, creating different obstacle densities.
Obstacle sizes are also chosen randomly using uniform
distribution between 0.035 to 0.28 m. Two random seeds
are used to generate worlds with >0% obstacle density. The
heuristic weights (eq. 8) were ωab =7.7 and ωabc =2.43
All experiments have a randomized initial connector s0=
(sx,sy,sz,sf)and goal cube gwhich defines the goal con-
nectors Sg. We make sure s06∈ Sg. We run suites of 100
experiments per world size, obstacle density and obstacle
seed. This means data points with 0% obstacles have 100
trials, while others have 200. To run the experiments we
used a cluster with Intel Xeon E5-2640 processors running
at 2.60GHz. The memory limit by process was always set to
4 GBs and the time limit was 7,200 seconds.
2) Comparing ab and abc Graphs: Figure 8 shows that
the double socket abc hop model solves significantly more
problems than the ab approach. This is due to two factors:
Firstly, the start and end joint-configurations of the robot
when using abc are more compact and are less likely to
be in collision with an obstacle hovering above the socket.
Secondly, not having to fully extend abc motions offer higher
maneuverability when avoiding obstacles, hence enabling
more feasible paths than ab which always needs to extend
fully between hops.
Fig. 8: Average Percentage of Solved, Unsolvable and Out of time
outcomes out of trials tested.
Despite abc-graph solving more problems, both methods
are worth considering as the double socket abc-graph relies
on a much more expensive cache computation. Also, the two
sockets abc-graphs are much bigger, see table I.
Lattice ab vertices ab edges abc vertices abc edges
S6O0 216 1056 503 3824
S6O25 82 170 211 940
S6O50 27 10 55 136
S11O0 918 4824 2219 17816
S11O25 386 892 963 4844
S11O50 99 40 222 434
S16O0 2400 13056 5879 48080
S16O25 1023 2532 2649 13948
S16O50 293 234 894 2730
S21O0 4950 27480 12203 100664
S21O25 2058 5096 5389 28312
S21O50 602 444 1570 4316
TABLE I: Vertex and Edge counts of graphs used in testing. Data is
for one random seed. The graph sizes shrink as obstacles obstruct
sockets or force collision with the robot.
3) Search Effort And Path Costs: Path execution costs
over a 100 trials are compared to the search effort spent
performing Aand joint space validation in Fig. 9.
Fig. 9: Stacked average plan execution and search costs (split into
Aand validation costs). The black line shows the average plan
costs of abc-graph only for problems solved by ab-graph as well.
As we add obstacles, and hence the graph size shrinks,
the robot must perform more complex motions, leading to
quickly increasing validation costs. Asearch time increases
for larger obstructed lattices as well. In general, to reduce
the computational effort, better validation strategies are re-
quired. The Aeffort could be reduced by using re-planning
algorithms like LPA, but that would not affect validation.
The double socket abc-graph not only solves all problems
that are solved via the single socket ab graph, but also finds
shorter paths. This is shown via the superimposed black lines
denoting abc performance on problems solved by ab. This
is expected as the ab motion involves the robot returning
to an all-zero joint state. Thus moving longer distances in
joint-space and encountering more obstacles. Furthermore,
for both graphs, the plan costs increase with obstacle density,
but start decreasing as it approaches 50%. This is consistent
with the graph size results in Table I as only problems with
start and end sockets that are close together are solvable.
4) Time Saved By Replaning: In Fig. 10 we show how
re-planning performs, see Sec. V-B.4. In the figure, savings
mean the difference between the first feasible plan and the
best ever found feasible plan. All results are normalized
against the first feasible plan cost. For most problems the
first found feasible solution is also the best one when taking
into account the re-planning search effort. However, actual
results are specific to the robot capabilites.
Fig. 10: Average time savings and re-planning effort, in terms of
the ratio against the first feasible plan, over all trials.
In this paper, we have presented two ways to formulate
a graph to perform hybrid discrete-continuous path planning
for an inchworm robot on obstructed lattice structures. We
showed that for both graphs combining iterative A* search
and a pre-computed cache structure scales significantly better
than purely online validation strategies.
The double socket abc graph formulation clearly outper-
forms the single socket ab, both in terms of problems solved
and path costs. Although, the abc-graph requires a more
expensive offline cache computation, this is reasonable con-
sidering target applications are heavy duty tasks like nuclear
decommissioning. Furthermore, the offline cache is subject
to exploitation of structure-specific symmetries and can be
further optimized if limited inverse kinematics solutions are
considered combinatorially instead of stochastically.
For both graphs, the search effort is mostly dominated by
continuous motion planning costs. Future work could explore
optimization-based planners like CHOMP [4] to address this.
Additionally, algorithms like LPAshould be used when the
search effort is significant compared to the validation effort.
However, the most interesting research direction is to com-
bine robot path planning with structure assembly planning.
The ability to accurately estimate traverse costs midway
through assembly could prove invaluable for solving the
assembly sequencing problem.
This work is partially supported by InnovateUK Connect-
R (TS/S017623/1).
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