Edited by: S. M. Hadi Sadati, University of Bristol, United Kingdom
Reviewed by: Surya Girinatha Nurzaman, Monash University Malaysia, Malaysia; Chongjing Cao, University of Bristol, United Kingdom
This article was submitted to Soft Robotics, a section of the journal Frontiers in Robotics and AI
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Growing robots are a new class of robots able to move in the environment exploiting a growing from the tip process (movement by growing). Thanks to this property, these robots are able to navigate 3D environments while negotiating confined spaces and large voids by adapting their body. During the exploration of the environment, the tip of the robot is able to move in any direction and can be kinematically considered as a non-holonomic mobile system. In this paper, we show the kinematics of robot growing at its tip level. We also present the affordable workspace analyzed by an evaluation of feasible trajectories toward target poses. The geometrical key parameters imposing constraints on growing robots' workspace are discussed, in view of facing different possible application scenarios. The proposed kinematics was applied to a plant-inspired growing robot moving in a 3D environment in simulation, obtaining ~2 cm error after 1 m of displacement. With appropriate parametrization, the proposed kinematic model is able to describe the motion from the tip in robots able to grow.
The ability of robots to move and interact with the environment is of fundamental importance for the accomplishment of demanded tasks in out-of-factory scenarios. Several kind of locomotion have been studied and adopted for different applications: in-pipe inspection (Mirats Tur and Garthwaite,
Being growth a new topic in robotics, the kinematics of such kind of movement is still poorly described in literature. Yet, to a certain extent, particularly from a kinematics point of view, a growing robot shows some similarities with systems implementing a follow-the-leader strategy, similarly to serpentine or hyper-redundant manipulators (Choset and Henning,
In the scenario of a robot growing at the tip and moving in space, the main question to be addressed is related to the path that the robot can take toward the target point, rather than the trajectory that the end effector makes to reach a specific point. In this view, we can compare the motion of this growing robot to that of a mobile robot able to navigate in a three dimensional space. To this end, it is important to describe the geometric configurations, or poses, of a growing robot and the potential environments that it can be able to navigate. Another important consideration is that a growing robot at the tip is a non-holonomic system, having a total of five DoF in configuration space (tip position in 3D space, and heading, and pitch angles) but only three controllable DoF at joint space, which are: two degrees of steerability (for tip orientation), plus one degree of mobility (the system velocity - in this case called growth velocity). These three degrees of maneuverability define together the space of possible configurations of a growing robot in 3D. For mobile robots, analyzing the workspace includes definition of how the robot moves between different poses, as well as of possible trajectories that the robot takes to reach a desired position with a specific orientation. The kinematic control of a system moving from a pose to another along a desired trajectory is often done by dividing the path in motion segments composed by straight lines and segments of a circle (Siegwart et al.,
In Del Dottore et al. (
In the following, we first describe the kinematics and the key design parameters affecting the behavior of growing robots (section Methods); then, we present the strategy proposed for defining 3D trajectories (section Results); and, finally, we discuss results of the simulations (section Discussions), followed by conclusive remarks (section Conclusions).
The characterization of the motion of a robot requires the definition of its kinematics and strategy to move from a point A to a point B in its configuration space. Based on the Chasles' theorem, any robot starting from (point) A, with a certain orientation, can reach (point) B, with another orientation, by means of a translation followed by a rotation of the body about its initial position (Siciliano and Khatib,
In Del Dottore et al. (
A generic point ^{i}
which can also be written as:
where the first factor of the right hand is the homogenous transformation matrix ^{j}
where the first subscript of
As in mobile robotics, we can identify a path from
To approach this problem, we describe the kinematics in configuration space for the tip of a growing robot with:
where
From a geometric point of view, the
where
It should be noted that the parameters
The workspace of a growing robot can now be described by evolving Equation (6) and imposing the constraint (8). Formally, there is always a path from any two points in 3D space (free of obstacles) that the robot can perform, with a desired destination and orientation; the only limiting factor on the workspace, when only geometric parameters are considered, is basically imposed by the material available to grow.
Let's define an initial state
An example of Dubin's path in 2D, from a starting configuration in position
Similarly, we addressed the problem of finding a suboptimal solution in 3D by dividing the problem into two optimal problems with curvature constraints: find the optimal path in 2D over two selected planes,
Schematic representation for Dubin's path approach used for 3D resolution.
We adopted a similar approach to Babaei and Mortazavi (
To select
and analogously,
where
This way,
In (13), the ϵ is a small quantity (which can ϵ → 0) introduced just to overcome possible numerical approximation errors.
From now, the problem is divided in two 2D problems. We take the projection of
Vector
Analogously, to obtain the 2D coordinates of
By definition Equations (17) and (21), we have the heading angles of starting poses equal to 0, while we can obtain the heading angles for the target poses as:
Therefore, the parameters of the minimum path problem on
Once we get the sequence of path segments, we can identify for each segment the action represented by the triple 〈α, β,
then defining the two possible deposition angles, which should lie on the perpendicular line respect to
The triple 〈α, β,
To evaluate the proposed kinematics we simulated the growth of a robot, implementing the equations of section Methods in MATLAB. We parametrized the simulations to fit the physical parameters of the growing robot implemented and deeply presented in Sadeghi et al. (
Four different groups of simulations were performed, with 50 repetitions each. The groups were composed setting the Euclidean distance between starting and target position (||_{Ps − Pe}||_{2}) of, respectively 4, 8, 16, and 32 times
and the errors in the heading (ε_{θ}) and pitch angles (ε_{γ}) as the distance between the target and achieved angles:
where θ_{t} and γ_{t} are, respectively the heading and pitch angle achieved by the simulated robotic tip. The normalized error in position seems to slightly grow with the distance (
Performance achieved with the plant-inspired growing robot (Sadeghi et al.,
To verify the accuracy of the model, we introduced a random noise component to perturb the system. From each of the previously obtained group of simulations (4
Representative paths extracted from each of the four groups (paths having the positional error close to the median error value). First row shows the Dubins' path obtained by the proposed path planner, while the second row shows the final configuration reached by the simulated robot with the corresponding time required to grow.
Positional error obtained with four different level of noise by the plant-inspired growing robot (Sadeghi et al.,
Additionally, we varied robot parameters to verify how robot dimensions and speed could affect model accuracy. Robot dimensions come into play in the kinematic model in the form of curvature radius (
Different parameterization of robot speed, curvature radius, and maximal intensity of bending (
Robot A | 18 | 0.0043 | 10 | 0.0004 | 0.0077 |
Robot A1 | 1.8 | 0.0430 | 10 | 0.0043 | 0.0077 |
Robot A2 | 0.18 | 0.4300 | 10 | 0.0430 | 0.0077 |
Robot A3 | 18 | 0.0043 | 30 | 0.0001 | 0.0026 |
Robot A4 | 1.8 | 0.0430 | 30 | 0.0014 | 0.0026 |
Robot A5 | 0.18 | 0.4300 | 30 | 0.0143 | 0.0026 |
Robot A6 | 18 | 0.0043 | 3.33 | 0.0013 | 0.0232 |
Robot A7 | 1.8 | 0.0430 | 3.33 | 0.0129 | 0.0232 |
Robot A8 | 0.18 | 0.4300 | 3.33 | 0.1290 | 0.0232 |
Mean positional error (±SD) achieved by each robot parameterization, over the four groups of path, with 50 repetition each.
Robot A | 0.0084 ± 0.0069 | 0.0119 ± 0.0052 | 0.0160 ± 0.0079 | 0.0198 ± 0.0105 |
Robot A1 | 0.0086 ± 0.0045 | 0.0135 ± 0.0065 | 0.0150 ± 0.0087 | 0.0179 ± 0.0106 |
Robot A2 | 0.0089 ± 0.0063 | 0.0126 ± 0.0064 | 0.0129 ± 0.0058 | 0.0164 ± 0.0110 |
Robot A3 | 0.0031 ± 0.0019 | 0.0025 ± 0.0010 | 0.0038 ± 0.0022 | 0.0059 ± 0.0028 |
Robot A4 | 0.0026 ± 0.0013 | 0.0031 ± 0.0028 | 0.0042 ± 0.0026 | 0.0059 ± 0.0039 |
Robot A5 | 0.0031 ± 0.0016 | 0.0031 ± 0.0021 | 0.0045 ± 0.0026 | 0.0059 ± 0.0040 |
Robot A6 | 0.0223 ± 0.0152 | 0.0411 ± 0.0181 | 0.0459 ± 0.0243 | 0.0445 ± 0.0223 |
Robot A7 | 0.0250 ± 0.0156 | 0.0557 ± 0.0427 | 0.0472 ± 0.0397 | 0.0440 ± 0.0332 |
Robot A8 | 0.0291 ± 0.0224 | 0.0370 ± 0.0183 | 0.0447 ± 0.0290 | 0.0399 ± 0.0277 |
Parameters adopted for simulating the growth of robots having different size and growth velocity.
Δ |
18 s | 522 s | 0.0020 s |
0.0043 cm/s | 0.012 cm/s | 1000 cm/s | |
2.2 cm | 5.75 cm | 1.9 cm | |
10 cm | 68 cm | 3.8 cm | |
0.0004 rad/s | 0.01 rad/s | 4.47 rad/s | |
0.0077 | 0.0921 | 0.5263 |
Paths having the positional error closer to the median value for the group of simulations having
Results demonstrate that heading and pitch errors are unaffected by variation of parameters, showing a not significantly different behavior among all the simulations obtained with parameters as in
Comparison of mean heading
The motion obtained by growing from the tip is becoming an attractive ability in robotics since it can enable robots to navigate their environments by adapting their bodies and morphologies to the constraints of the surrounding. The body is built in real-time by the robot, according to environmental and task demand, through the addition of new material at the tip, driving in this way the tip navigation. This means that the robot's path is not predictable
Navigation of unstructured environments cannot rely on classic map-based path planning strategies; the robot in those cases should move with a higher level behavior control, i.e., a stimuli-oriented control (Sadeghi et al.,
The key parameter defining the path and the ability of a growing robot to adapt through different unknown patterns is the minimum curvature radius. This parameter is affected, and consequently, the space of maneuverability may be limited, by the design of the robot and particularly by the size of mechanical components (e.g. motors, other actuators, and components). Here, a parameterization of the growing system mechanical design is presented and formulation of the curvature radius in terms of that parameters proposed, giving a good agreement with experimental results, i.e., we geometrically evaluated the minimum curvature radius of our growing robot as
Yet, our analysis has been limited to a geometric evaluation aimed at characterizing the motion of growing from the tip robots. By looking at the kinematics, we evaluated the theoretical workspace of growing robots, however, when deepening in the analysis of the motion, dynamics of each specific system should be also considered. For instance, when a growing robot moves in the air, the weight of the tip and the suspended part of the built body should be carefully taken into account in the control dynamics, in order to prevent the structural collapse. In fact, speed and forces acting on a robotic system play a relevant role which could address the features of the robot from one application to another. Also, when designing the robot, the selection of the growth mechanism is particularly important when talking about applications. For instance, for biomedical applications, in the design of a growing robot, the reversibility of the system and the biocompatibility of the growth strategy and building material are fundamental, whereas, in a rescue scenario, the speed and robustness become much more relevant.
This paper formalizes the kinematics model for growing robots, setting the analogy with mobile non-holonomic systems, and shows the ability of the model to describe the motion of a plant-inspired growing robot. Given a starting and a final pose in the 3D space, here we defined a kinematic control to connect them. We propose to split the global movement into two optimal planar paths based on Dubin's solution and we formalize our approach finding the two planes and the trajectories above them. We verified our strategy with different poses in simulation demonstrating the ability of a plant-inspired growing robot to reach the expected final position with the desired orientation (maximal positional error of ~6 cm in 320 cm of path length and ~1.8° in orientation errors). We also evaluated the effects of different level of noise, and the effects of different model parametrization. We noted that not only the curvature radius but also the specific discretization of the robot affect its ability in reaching, with high or low accuracy, the desired point and thus must be taken into account when defining a feasible path. However, our analysis generally shows the accuracy of the proposed strategy, when considering an almost continuous growth of the robot, the efficacy of the model and its applicability over different sizes, curvature radius, and growth speeds.
However, when moving from simulation to physical implementation, the kinematic analysis is not enough to correctly analyze robot motion. Future steps will focus on formalizing the optimal path considering specific characteristics of the robot into the model, particularly, evaluating how the dynamics (considering self-weight and other forces exercised in interaction with the environment during growth) would affect the path, and implementing the strategy on the robo-physical model (Sadeghi et al.,
ED conceived and formalized the model. ED and AM discussed model and experiments. ED, AM, AS, and BM discussed results and wrote the paper.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest
The Supplementary Material for this article can be found online at: