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Autonomous Mobile 3D Printing of Large-Scale Trajectories
Julius Sustarevas1, Dimitrios Kanoulas1, Simon Julier1
Abstract— Mobile 3D Printing (M3DP), using printing-in-
motion, is a powerful paradigm for automated construction.
A mobile robot, equipped with its own power, materials and an
arm-mounted extruder, simultaneously navigates and creates its
environment. Such systems can be highly scalable, parallelizable
and flexible. However, planning and controlling the motion
of the arm and base at the same time is challenging and
most deployments either avoid robot-base motion entirely or
use human prescribed robot-base paths. In a previous paper,
we developed a high-level planning algorithm to automate
M3DP given a print task. The generated robot-base paths avoid
collisions and maintain task reachability. In this paper, we
extend this work to robot control. We develop and compare
three different ways to integrate the long-duration planned path
with a short horizon Model Predictive Controller. Experiments
are carried out via a new M3DP system — Armstone. We
evaluate and demonstrate our algorithm in a 250 m long multi-
layer print which is about 5times longer than any previous
physical printing-in-motion system.
I. INTRODUCTION
The sustainability goals of the 21st century are driving
the need to automate the building and construction indus-
try [1], [2]. Whilst a variety of digital robotic manufacturing
solutions have already been widely adopted for off-site
component prefabrication [3], [4], in-situ automation remains
mostly limited to site inspection and monitoring [5], [6].
However, a major exception is the use of large-scale Additive
Manufacturing (AM) – also known as 3D printing – systems.
AM methods have demonstrated increased efficiency and
waste reduction [7]–[9], however sustainable deployment and
material use challenges are still open [10]. Despite this,
such systems are already being commercialised [11]–[13]
and deployed to tackle sustainability and housing crises [14].
Most existing AM construction solutions use static
gantries or arms. They are highly restrictive and costly to
deploy [8]. An alternative is a Mobile 3D Printing (M3DP)
system, shown in Fig. 1, which uses smaller mobile manipu-
lator robots. Such systems have unlimited planar range, high
agility to track more complex print trajectories [15] as well
as potential to parallelise the printing task [16].
Despite their advantages, there are very few works that ex-
plore M3DP applications [16]–[22]. Autonomously tracking
print paths, which are much larger than the robot, requires
a robust motion and path planning approach. This remains
challenging and, although these works demonstrate the fea-
sibility of M3DP, they often make restricting assumptions
1Department of Computer Science, University College London, Gower
Street, WC1E 6BT, UK. julius.sustarevas@cs.ucl.ac.uk
This work was partially supported by the UKRI Future Leaders Fel-
lowship [MR/V025333/1] (RoboHike). Optitrack tracking system has been
provided by UKRI AABM Research Grant [EP/N018494/1].
Fig. 1: The Armstone autonomous mobile 3D printing system.
such as human-prescribed robot-base paths or decoupled
robot-base and arm control. In turn, this reduces system
autonomy and agility. Consequently, to date there has been
no demonstration of an M3DP system that can autonomously
plan and 3D print a structure given just a print task.
In this paper, we extend our previous work on Task
Consistent Path Planning (TCPP) to hardware by using
TCPP-derived robot-base paths to complement a whole-
body Model Predictive Controller (MPC). Our approach is,
thus, globally informed and benefits from locally optimal
combined base-arm motion planning. We compare three
ways in which the robot-base references can be added to the
MPC formulation. Finally, we demonstrate printing of long
(250m+) trajectories, over multiple layers, via a purpose-
built mobile manipulator, named Armstone.
The remainder of this work is organised as follows. The
next section overviews related work in mobile 3D printing
and MPC. Sec. III briefly covers the Armstone robot hard-
ware design and the MPC setup used in this work. Sec. IV
presents how we integrated robot-base reference trajectories
into the MPC formulation. Next, Sec. V and VI show large-
scale printing experiments. Lastly, a discussion of results and
future work can be found in Sec. VII.
II. REL ATED WORK
A. The Challenges of Mobile 3D Printing
Conceptually, the feasibility of Mobile 3D Printing has
been demonstrated. In our earlier work [17], we showed that
tracing print trajectories, from a moving mobile manipulator,
results in a marginal end-effector error increase, compared to
a stationary approach. Another work, by Tiryaki et al. [19],
realised the first concrete printing mobile manipulator sys-
tem. However, both of these works used a manually-specified
robot-base path to keep the printing task within the reach of
the robot.
This robot-base path finding problem was tackled in our
work on Task-Consistent Path planning (TCPP) [18]. We
proposed a methodology to compute the robot-base path
from the desired print trajectory and the knowledge of robot
kinematics. The TCPP algorithm took into account M3DP-
specific challenges, such as evolving obstacles created by the
printing process and manipulator reachability preservation.
However, these existing efforts decouple the robot-base
and end-effector control. The local manipulator trajectory is
obtained by subtracting the print path from the print task.
In that way, the error that was generated by following the
prescribed robot-base trajectory was directly translated to the
end-effector without any mitigation by the manipulator.
B. Path Planning for M3DP
Early research exploring the path planning problem for
mobile manipulators with given end-effector paths, tended
to treat the robot systems as redundant manipulators and
used sampling-based planners, such as PRM or RRT [23]–
[27]. In those planners, the task-constraint was inverted and
a path was formed from randomly sampled admissible robot
configurations [28], [29]. To help reduce the dimensionality
of the problem, Nagatani et al. [30] proposed planning
the robot-base path separately, while assuring that the task
remains in a manipulable region in the local robot arm frame.
We used Nagatani et al.’s approach to develop our TCPP
algorithm specifically for mobile 3D printing [18]. We used
the Inverse Reachability Map (IRM) [31] to help sample and
validate robot-base poses along a print path during the RRT ∗
search. Given a robot-base pose and an associated point on
the print path, the IRM provides a Reachability Index (RI)
for the print point. RI is a metric [32], [33] associated with
a voxel in a manipulator’s local workspace. For many SE(3)
poses inside a voxel, it is the fraction of poses with existing
Inverse Kinematics (IK) solutions. The IK computation takes
into account both the robot self-collisions and its joint limits.
Intuitively, RI relates to robots ability to position its end-
effector at a given point. Thus, by imposing a lower bound
on RI when validating robot-base poses, we were able to find
robot-base paths for long print trajectories, while assuring the
print task remains well reachable by the arm.
C. Control for M3DP
Given the robot-base and end-effector paths, an arm joint
trajectory can be computed. However, following both robot-
base and arm trajectories still requires coordination and ro-
bustness. Tiryaki et al. [19] used independent robot-base and
arm controllers. The base used a Model Predictive Controller
following a prescribed path with respect to an on-board
localisation system. The arm used a low-level joint trajectory
controller. This meant that the end-effector error was entirely
reliant on the robot-base trajectory tracking error. However,
as 3D printing is a relatively slow process, M3DP requires
slow and steady robot-base velocities. This is often difficult
to achieve due to breakaway friction, torque requirements,
or uneven terrain. A similar problem was also identified
in [22]. Using a separate robot-base and arm controllers led
to increases in end-effector error, especially at the start of
the print trajectories as the robot-base struggled to render
accurate motion from full stop.
Recently, there has been a great deal of research interest
in whole-body motion control for various quadruped, mo-
bile manipulator, and humanoid robots [34]–[37]. Specif-
ically, Pankert et al. [38] proposed a control strategy for
continuous process tasks. They used a Sequential Linear
Quadratic Model Predictive Control (SLQ-MPC) formulation
to demonstrate end-effector trajectory planning, admittance
control, and collision avoidance. The SLQ-MPC controller
rolls-out the current control policy over a prediction horizon
T, linearizes the system about the resulting states, and
minimises a cost function Lintegral over T:
J(x,ˆ
x,u) = Zt0+T
t0
L(x(τ),ˆ
x(τ),u(τ))dτ,
L(x,ˆ
x,u) = ∑Ci+∑Bi+u⊺Ru
(1)
Here, xis the system state, ˆ
xis the reference trajectory,
and uare the inputs. Lconsists of cost terms Ci, control
barrier terms Bi, and the input penalty term u⊺Ru. Pankert et
al. [38] used a quadratic end-effector tracking cost term and
imposed soft inequality constraints, using Bi, for manipulator
joint position and velocity limits, as well as the collision
avoidance and admittance control features. The computed
policies were evaluated using the state estimates while the
inputs were passed to the motor controllers. For further
details, e.g. derivative approximation, see [38].
In Pankert et al. [38], a failure case occurred when the
collision avoidance and end-effector tracking objectives were
competing. Such cases might occur often in 3D printing
applications. Print paths are vastly longer than the predictive
horizon and, thus, obstacles being created by printing can
prevent the robot from completing the task. To illustrate
this, consider the scenario in Fig. 2. The robot prints, left
to right, a trajectory that carries on beyond immediate reach.
The predictive horizon fails to see the need for the robot-
base to move rightwards before such motion would cause a
collision or compromise end-effector tracing accuracy.
Fig. 2: Trajectories substantially longer than predictive horizon may
lead to infeasible states.
Therefore, in this work, we supplement the SLQ-MPC
controller with a TCPP-derived robot-base reference tra-
jectory to provide foresight and allow tracking very long
trajectories. We use an open-source implementation for SLQ-
MPC [38]. This is based on the OCS2 [36], [39] library for
optimisation, RobCoGen [40] for kinematics and CppAD for
auto differentiation. The formulation of costs and constraints
used is presented in Sec. III-B and IV. We next describe
the Armstone robot we developed to carry out our M3DP
experiments.
III. ARMSTONE M3DP SYSTEM
In this section, we present the development of a new
robot, named Armstone, which is shown in Fig. 3. Armstone
is a low-cost mobile manipulator designed for M3DP. It
is a research platform and is not intended for realistic
deployment. It is material and power independent, allowing
it to leverage the advantages of mobile manipulators for
AM applications. Here, we cover hardware and software
components as well as present how the robot is modelled
by the SLQ-MPC controller.
A. Robot Hardware
The robot-base platform has a 0.8m ×0.62m ×0.3 m
footprint, uses Mecanum wheels for holonomic motion and
weighs approximately 20kg. The robot-base is powered by
a 48V,20A h battery and uses four Maxon EC45 70 W
motors with 91:1 reduction gearbox. The motors are run
using VESC motor controllers. Armstone features a low-cost,
6DoF, 5 kg payload, 0.7 m reach uFactory xArm manipulator.
The manipulator is mounted at the bottom of the base to
maximise reach at ground level.
Fig. 3: Armstone robot’s major components.
The robot carries two Intel Nuc (2-cores and 4-cores
@4GHz) computers as well as the xArm DC control box.
Internal communications pass through a 100Mbps network
switch and the robot hosts a wireless network through an
on-board router.
B. SLQ-MPC Setup
The system model closely follows that of Pankert et
al. [38]. The robot state xconsists of the manipulator joint
values xa= (φ1, .., φ6)and the robot-base pose, xb∈SE(3),
expressed as a position and a quaternion in the world
frame. The robot input uis composed of the robot-base
translational velocities vx,vy(in the robot-base frame), robot-
base rotational velocity ωb, and manipulator joint velocities.
x= [xb,xa]⊺
u= [vx,vy,ωb,˙
φ1, .., ˙
φ6]⊺(2)
For a given task trajectory ˆ
p, the end-effector tracking cost
function Cee is defined as follows:
Cee(t,x,ˆ
x) = [exyz,er py]⊺Qee[exyz ,erpy ]
exyz(x,ˆ
pxyz) = F K(x)xyz −ˆ
p
erpy (x,ˆ
pq) = FK(x)q⊖ˆ
pq
(3)
where, ˆ
pis a part of reference trajectory ˆ
xand is composed
of positions and quaternion orientations prescribing the print
path. The Cee is computed as the weighted sum of squared
translational and rotational end-effector pose (as provided
by forward kinematics) errors exyz and erpy . The quaternion
distance error is computed as in [41]. Qee is a diagonal matrix
of weights.
State position and velocity limits are implemented as soft
constraints by penalising constraint penetration zi≥0 via a
relaxed barrier function [42]:
Bi(zi) = (−µln(zi),zi>δ
µ β (zi,δ),zi≤δ(4)
where, βis a quadratic function such that Bis twice
differentially continuous, and µand δare tuning parameters.
In this work, µ=5·10−3and δ=10−4. We do not use any
admittance control or collision avoidance constraints present
in Pankert et al. [38].
In this work, the prediction horizon T, see Eq. 1, is
always 4 s. The MPC runs at 30 Hz and resulting policies
are evaluated at 100 Hz.
C. Localisation and Control
Localisation is achieved either using an external OptiTrack
system or two Hokuyo 04lx LiDAR sensors mounted on the
sides of the Armstone. Scans of these sensors are merged
using the ira-laser-tools package [43] and the output
used by the slam-toolbox localisation package [44].
The Armstone base runs a holonomic controller at 50 Hz,
translating robot-base velocities vx,vy,ωbto wheel angular
velocities. The xArm manipulator accepts desired joint ve-
locity inputs at 100Hz. Robot-base and arm controllers are
combined using ROS Control [45] hardware interface API.
D. Extrusion System
The Armstone’s on-board printing system extends our
previous work [22]. It consists of an on-board compressor
pressuring a 3L material cylinder. The pressure drives mate-
rial from the cylinder to the extruder via a PTFE tube which
has been heat treated to aid bending along with manipulator
motion. A Stoneflower3D 6mm nozzle extruder is mounted
to the xArm flange and is controlled by one of the on-board
computers. The material we use is porcelain with paper fibres
as we found it most reliably travels through the feeding tube.
IV. BASE-PATH I NFO RME D SLQ-MPC
Mobile 3D Printing requires the robot to accurately follow
a long end-effector trajectory. However, M3DP does not dic-
tate a precise robot-base pose throughout printing. As long as
the robot-base navigation supports the dynamic nature of the
evolving environment and maintains print path reachability,
the print task can be achieved successfully. Therefore, in this
section we describe how a SLQ-MPC motion controller can
be globally guided by a robot-base path reference that assures
feasibility to complete a long print task.
A. Task-Consistent Path Planning
In our previous work, we describe how the robot-base
path, can be obtained from the RRT ∗-based Task-Consistent
Path Planner (TCPP). The result is an ordered list of tuples
bi= (xb,i,si), where xb,i= (xi,yi,θi)∈SE(2)and siis a
process variable corresponding to the length along the print
path line integral. The validation routine, isValid(bi), uses an
Inverse Reachability Map to assure that the print path at siis
reachable from base pose xb,i, while accounting for obstacles
created by printing up to si. For details, please see [18].
Our original implementation did not model dynamic con-
straints and, as a result, the computed base paths were often
jagged. However, smoother paths are preferred. In this work,
we introduce the following post processing steps. Firstly,
after the TCPP algorithm reaches the goal and produces a
robot-base path, we continue running the RRT ∗loop until the
size of the underlying graph has doubled. This increases the
number of available samples and provides the opportunity
for the RRT ∗rewire function to shorten the path. Next,
we iterate through biin a random order and attempt to
bypass unnecessary robot-base poses [46]. Namely, biis
removed and bi−1is connected directly to bi+1, if densely
sampled nodes along this connection all satisfy the validation
function. Finally, we relax the robot-base poses as follows.
We iterate through bi, except for the first and last nodes, in
a random order and sample mrandom tuples b∗
i= (x∗
b,i,s∗
i)
in a small radius about bi. During sampling we assure that
si−1<s∗
i<si+1. In this work, considering physical robot and
task dimensions, we use radius 0.05 and m=15. Then, we
replace biwith bi,new by minimising a cost l(bi):
bi,new =argmin
b∗
i∪bi
l(bi),where
l(bi) = δxb,i+1+δxb,i
si+1−si−1
+
δxb,i+1
δsi+1
−δxb,i
δsi
(5)
where δxb,i=dist(xb,i,xb,i−1)is computed using RRT ∗’s
distance function. The minimised function l(bi)is similar
to how dynamic constraints are imposed in [47]. As print
speed is constant, the first term of lis related to the robot-
base velocity over the whole segment si−1to si+1. Similarly,
the second term of lis related to acceleration. The new bi,new
leads to lower and more constant velocity throughout the
segment. This relaxation step is carried out 10 times.
B. SLQ-MPC Base Trajectory Reference
We use the following procedure to add the robot-base path
bito the SLQ-MPC reference trajectory ˆ
xFirst, xb,i∈bi
elements are transformed to SE(3) and stored as positions
and quaternions assuming that the robot base motion is
constrained to the xy plane. Using the process variable si,xb,i
is interpolated and time-stamped to match the print trajectory
ˆ
p. Finally, ˆ
xis extended to include both the print task and
the robot-base reference ˆ
x(t) = (ˆ
p(t),ˆ
xb(t)).
We explored three ways to include ˆ
xb(t)in the MPC
controller cost function: quadratic tracking functions, time-
varying soft constraints, and a mixture of the two.
a) Quadratic tracking functions: The base tracking cost
Cbcan be computed the same way as the end-effector
tracking costs Cee (Eq. 3), but only penalising the x,yand
yaw elements of the robot-base state error:
Cb(t,x,ˆ
x) = [exy,eyaw ]⊺Qb[exy,eyaw](6)
The shortcoming of this approach is that Cbdirectly com-
petes with Cee. If the weights Qbare of comparable size
to the Cee weights Qee, the SLQ-MPC will effectively only
solve the inverse kinematics problem between the print and
robot-base trajectories. Also, this could mean compromising
the end-effector error over robot-base path error. On the other
hand, if Qbis small, the controller could leverage robot-base
for increased agility, but it could lead to a steady state error,
while following the robot-base trajectory. As in this work,
collision with the print task is taken into account only by the
TCPP-derived robot-base path, this could mean driving over
the printed structure.
b) Time-varying soft constraints: The base reference
trajectory can also be modelled as a soft constraint. To
do this, we first derive feasible regions around each biby
further leveraging the functionality of TCPP. For each bi,
we incrementally perturb the base pose xb,i= (xi,yi,θi)along
positive and negative SE(2) axes. We store the largest total
increment along each axis for which the perturbed xb,ican
still be validated using the TCPP isValid routine. That is, if
robot-base is not in collision and print path at siis reachable.
For example:
rx,i=max
N(N)·ε,such that,
isValid((x±nε,y,θ,si))∀n≤N
(7)
where the increments ε=0.03 are determined experimen-
tally. These maximum deviations rxyθ,i= (rx,i,ry,i,rθ,i)ef-
fectively define an ellipsoid around biin SE(2) of feasible
robot poses. rxyθis interpolated and time-stamped along with
biand added to the reference trajectory ˆ
x= (ˆ
p,ˆ
xb,rxyθ). In
turn, the violation of leaving this region zis then:
z=σ(rxyθ(t)− |xb(t)−ˆ
xb(t)|).(8)
Leaving this region is penalised using a relaxed barrier
function alongside other constraints as in Eq. 4. In this way,
a robot-base reference pose is provided as a moving region,
which still carries assurances from TCPP, but does not force
the controller to compromise between competing tracking
costs. σis a tuning parameter. We use σ=25 (determined
experimentally).
c) Hybrid constraint: Quadratic cost draws the robot-
base to nominal pose, whereas soft constraints allow un-
restricted movement within a feasible region. These are
complementary behaviours. Therefore, we introduce the hy-
brid method which sets the base tracking cost to be the
sum of the quadratic and soft constraint terms. This creates
bounded flexibility by only gently drawing the robot towards
the TCPP-derived path, but penalising leaving the feasible
region.
Fig. 4: Armstone robot printing the first 3 layers of the bowtie task. Note, the point of view of these images is facing along the negative
y-axis when comparing to Fig. 6. Full video of experiments is available at youtu.be/uBzdOuvIp9c
V. TUNING AND HARDWARE ERROR SOURCES
The SLQ-MPC end-effector tracking and system input
weights are set as follows. The translational elements of Qee
(see Eq. 3) are set to 250, while roll and pitch elements are
set to 100. As the end-effector orientation about the extruder
nozzle does not matter, the yaw weight is 0. Furthermore, the
input costs Rare set to 0.1 for all arm joint velocities, while
base input weights are 10. The task-orientation ˆ
pqremains
constant pointing the end-effector along negative Zaxis.
In tuning the system, we found two hardware-related
sources of end-effector error. As the Armstone manipulator
stretches away from the robot-base, mechanical components,
e.g., aluminium extrusion frame or front wheel axles, expe-
rience increased load. In turn, this causes a slight drop in
end-effector position. Using the OptiTrack we estimated the
end-effector is about 4mm lower at full arm stretch than
when close to robot-base. Moreover, at the time of writing,
the xArm manipulator firmware struggles to render smooth
motion at the low joint velocities we use to perform print-
ing. This effect is most pronounced and visible as vertical
oscillations. To estimate the magnitude these oscillations, we
performed straight line motions along x,yand zaxes in the
end-effector frame while the base is stationary. From the
tracking data we found the mean and maximum L2norms of
errors are 0.6mm and 1.9 mm. We do not account for these
errors and they are present in the results.
VI. EVAL UATI ON
To evaluate the effectiveness of our approach we carried
out two sets of large-scale printing experiments. In the first
set, we empirically evaluate the three different ways that the
robot-base trajectory can be integrated into SLQ-MPC. The
second set of experiments explored the effects of using on-
board sensors (odometry and SLAM) to localise the platform.
A. Long-Term Printing Experiments
In Sec. IV we introduced three ways to integrate the robot-
base trajectory into SLQ-MPC: the quadratic cost Cb, the
time-varying constraint Bb, and a hybrid approach which
summed both together. In this experiment, we compared
these different approaches for the large-scale print task
shown in Fig. 4. The task is a bowtie-shaped task trajectory. It
is formed of 5 layers of 6cm wide cross-hatching fill patterns
following a 2m ×1 m bowtie shape. The shape was chosen
as it has inside and outside corners as well as straight seg-
ments. Each layer is 51.93m long and is sampled at 2.5 mm
intervals along the path integral. Therefore, the total path is
approximately 260 m. The printing speed is 3 cm s−1. Each
layer takes 28 min to print. To isolate controller behaviour
from localisation errors an OptiTrack system was used to
provide the base pose estimate.
When only Cbis used, the weights Qbwere set to 20 in
order to assure close following of the base-path reference.
When Cbis used together with Bb, values of Qbare reduced
to 2. Tuning of Bbwas kept constant as in Sec. IV. The
end-effector tracking costs Cee are active at all times and are
weighted as in Sec. III-B. Physical printing was only carried
out for the hybrid costs formulation and is seen in Fig. 4. The
reference trajectory ˆ
xand is sent to the SLQ-MPC controller
one layer at a time.
1) Base Reference Tracking: Fig. 6 shows the first three
layers of the task solved by the TCPP algorithm. When
Layer 1 is printed, the space gradually becomes obstructed
by printing. When printing Layers 2 and 3, the print pattern is
effectively a static obstacle. The task differs slightly between
even and odd layers as the cross-hatching print pattern tilts in
alternating directions to aid area coverage. This is commonly
used technique in additive manufacturing. Fig. 5 shows a
close up segment of layer 3 and illustrates the behaviour of
the different cost function formulations. In Fig. 5a the robot-
(a) Robot-base position (b) Robot-base orientation
Fig. 5: A segment of robot-base paths for layer 3 of the bowtie
task.
base pose falls behind the reference trajectory when only Bb
is used. The small constraint violation seen in Fig. 5b is
pushing the robot-base as the boundary moves. The nature
of the task requires robot to move around it. This results in
an often small orientation constraint violations. The addition
of Cbleads to the base path following the reference trajectory
closer, but still allows deviations within the moving rxyθ
region.
(a) Layer 1
(b) Layer 2
(c) Layer 3. Highlighted region is the source of Fig. 5
Fig. 6: Armstone-base motion when printing the bowtie task. Base
path reference ˆ
xband base paths as followed by the controller. Light
green region illustrates the barrier function tolerance region rxyθ
Furthermore, Fig. 6 shows the rxyθregions shown in light
green. The regions are more generous during the first layer as
the robot prints behind itself and has more obstacle free space
to move. The Bb-only exploits this by deviating further from
reference trajectory. However, as the reference trajectory
makes a turn and rxyθfollows, Bb-only paths make a sharp
adjustment. This is seen in upper left region in Fig. 6a.
The combined Bband Cbpaths deviate further from the
reference trajectory than a Cb-only, but rarely come close
to Bbconstraint boundary. This indicates that Qbcould be
reduced further. However, as presented next, we did not find
it to be of large significance to the end-effector error.
2) End-Effector Trajectory Tracking: The end-effector
error was computed using the OptiTrack ground truth data.
Fig. 7 shows a close up of the traced print path when both
cost terms are active. The distance between step function-like
infill lines seen in Fig. 7 is 1cm. The end-effector is seen
occasionally missing the line and overlapping a previous one.
Fig. 7: Close up of end-effector path when Cband Bbare active
Table I presents layer-by-layer average translational end-
effector error (
ˆ
pxyz −˜
pxyz
2) for all experiments where ˜
p
is the independently measured end-effector ground truth. It
also shows an Iterative Closest Point (ICP) [48] derived
error. This is also computed as an average L2error, but after
firstly point matching and aligning ˆ
pxyz to ˜
pxyz. The data sug-
gests the hybrid cost-constraint approach has slightly lower
end-effector average error than quadratic costs only. When
printing with Bbconstraint only, the average and maximum
error varies between layers likely due to interactions with Bb
boundary. When printing layer 4 using combined costs, the
end-effector made a small aggressive motion penetrating the
print. This is seen in the highlighted 36 mm maximum error.
As this was a single occurrence and we were able to continue
printing. The ICP error shows less significant impact between
the robot-base path tracking costs and the resulting end-
effector path tracking error. The discrepancy between the two
error calculations presented indicates either a presence of a
steady-state trajectory tracking error or experimental setup
limitations arising from poor measurement synchronisation.
The average roll and pitch error throughout all experiments
was about 0.03rad.
ICP (mm) mean (std) (mm) max (mm)
L1, Cb2.37 8.57 (1.96) 16.57
L2, Cb2.73 8.40 (1.76) 13.88
L3, Cb2.68 8.35 (2.07) 16.34
L1, Bb2.39 6.66 (1.85) 16.39
L2, Bb2.47 8.63 (1.83) 20.23
L3, Bb2.61 2.78 (1.20) 10.57
L1, Cb,Bb2.09 6.75 (1.58) 19.04
L2, Cb,Bb2.6 6.87 (1.64) 22.37
L3, Cb,Bb2.46 2.99 (1.32) 12.97
L4, Cb,Bb2.47 6.85 (2.72) 36.69
L5, Cb,Bb2.5 6.87 (2.65) 20.12
TABLE I: End-effector translational error
The final result of physically printing of the bowtie task
is presented in Fig. 8. In total about 10 L of clay was used
and the Armstone material cylinder was refilled after every
(a) Top view
(b) Close up side view
Fig. 8: Final result of printing the long bowtie print task. The task
consists of five 51.93m long layers spanning 2 m ×1 m in scale.
layer. As fine tuning material viscosity, extrusion rate and
layer height is out of scope for this work, these parameters
were adjusted experimentally. The layer height was set to
6mm and print path was elevated by 1cm above the layer to
allow for inaccuracies in nozzle motion.
B. Printing with On-Board Localisation
High accuracy nozzle localisation is crucial to 3D printing,
but highly challenging to achieve with a mobile manipulator
system. To examine how printing fidelity is affected when
using on-board SLAM or wheel odometry we carried out a
second set of experiments. For this we used a 36.8 m long,
2.5 m ×0.3m scale, two layer sine wave task composed of
the same infill pattern as the bowtie task. Printing was carried
out using the hybrid cost formulation.
The SLAM implementation from Sec. III-A was used to
provide wheel odometry correction in the form of map to
odometry frame offsets. This correction could vary several
centimetres from update to update and could cause discon-
tinuities in SLQ-MPC state feedback. Therefore we used a
5 s moving average filter to smooth it.
Fig. 9: End-effector errors for different localisation methods.
Fig. 9 shows that the end-effector error remains bounded
within 0.1m when printing using SLAM. This is expected
(a) Localisation using external tracking
(b) Localisation using wheel odometry
(c) Localisation using SLaM
Fig. 10: Comparison of printing performance using different local-
isation methods. Printed task is 2.5 m ×0.3m scale. Task starts on
the left, proceeds right and comes back.
given the map resolution and Armstone’s LiDAR error of
3cm. The print in Fig. 10 confirms this. However, the
print also shows that using SLAM while printing can cause
problems. As the map is updated with more observations, the
localisation errors change. As a result, the different layers do
not precisely overlap one another and the infill pattern shows
higher deformation.
VII. CONCLUSIONS AND FUTURE WORK
In this paper we have demonstrated Autonomous Mobile
3D Printing of large-scale trajectories. Given only the print
task and robot kinematic model, the TCPP algorithm was
used to derive a feasible robot base path and in turn inform
a SLQ-MPC motion controller. To the best of authors knowl-
edge, the over 250 m long bowtie print experiment carried
out in Sec. VI is the largest and highest autonomy print done
with a moving mobile manipulator to date.
Notably, the nominal end-effector accuracy of 6 mm -
8 mm was achieved via external tracking. Reliance on exter-
nal tracking can be avoided by adopting fiducial marker [19]
or end-effector mounted laser range finding [49].
In future work, we will also explore simultaneous printing
and mapping to localise the end-effector with respect to the
print task [50]. Furthermore, granted end-effector localisa-
tion, the MPC motion controller and the TCPP path planner
can both already support printing of more complex, non-
planar structures. This, together with scaling up the Armstone
robot for deployment outside lab conditions has the potential
to drastically increase system applicability.
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