The lack of structural support inherent in tendril robot designs tends to generate sagging and buckling when the robot has to support external loading with its structure alone. Thin plants such as vines encounter the same difficulties. However, they have evolved to solve these problems by various means, in particular by actively seeking contact with their environment, bracing against or firmly attaching to it to increase stability. Notice that this is counter to the usual mode of robot navigation, where collisions with the environment are avoided as much as possible.
2.1.1. Plant-Inspired Environmental Contact Hardware
From a physical (hardware) point of view, the way vines exploit their environment, supporting their flexible structures via fixed environmental support, is by the use of specialized structures along their backbones. Different plants have evolved a variety of attachment types, including prickles, thorns, roots or pad structures [
21]. Correspondingly, in this paper, we augment the tendril robot of
Section 2 with artificial prickles (
Figure 3). We reported on initial results with tendril robots using simple artificial prickles made from hooks previously [
13]. In that work, we showed that attaching to the environment with these prickles improved stability by reducing coupling proximal to the point of contact [
13].
The simple prickle design above was effective once attachment was made [
13], but sometimes, attachment is hard to achieve in a complex environments. Precise orientation was required to consistently engage the environment with these prickle hooks, located 180
apart from each other. The prickles could only attach well when the environment offered a sufficiently small diameter, and at the proper orientation, allowing a good fit for the prickle. Furthermore, the prickles became stuck in the environment at times; good for plants, but for robots, we seek the ability to detach, as well as attach. Hence, two improved designs for attachment hardware are introduced in this paper.
Specialized hardware for adaptive attachment of tendril robots for environmental support increases mechanical complexity and diminishes its ability to maneuver in congested environments. Hence, we designed attachment hardware with the same general scale of the hook prickles, but with better performance. The overall design is still sized to fit around a spacer and has several variations depending on which section’s spacer is is meant to be placed. The hooks are small diameter metal nails bent into the appropriate shape, which is typically between 90
and 120
. The size of the hooks is empirically determined to be wide enough to grasp meaningful environmental features, while small enough to allow the robot to enter the environments, particularly the Space Station equipment racks, which were our application focus for this work. One new design, shown in
Figure 3a, features four hooks, rather than two. These hooks are separated by approximately ninety degrees, in order to minimize the necessity of precisely orienting the robot in order to engage the environment; thereby facilitating a higher probability of making contact with un-sensed environmental features, which we found to be the case during experiments. The second design, shown in
Figure 3b, was a different approach that was meant to improve the contact quality. In either design, attachment and detachment are achieved by manipulating the length of the section that has the set of prickles. For the second design, where a set of prickles is on the first spacer of the tip section, and the last spacer of the middle section, the middle section is extended, while the tip section compresses. Both designs were manufactured in our laboratory by 3D-printing a structure to fit around a spacer with holes through which metal nails can be placed for the prickles.
Bracing against the environment to gain stability has been used as a strategy for conventional rigid link robots [
34,
35,
36]. The notion is quite new for continuum robots, however, and our
Section 3 will show how the use of the new prickles introduced in this paper simplifies environmental contact with tendril robots. However, as with plants, an efficient way to explore the environment to seek such contact needs to be found. This issue is addressed in the following subsection.
2.1.2. Plant-Inspired Motion Planning: Circumnutation
As noted earlier, circumnutation is an elliptical pattern of growth displayed by the tips of many plants [
24] and some roots [
18]. Circumnutation movements have been studied in some detail in the botanical literature, both qualitatively [
37] and more quantitatively [
38]. This “existence proof” for how to move thin, compliant structures has been shown to be an optimal, energy-efficient strategy in the plant world. In the following, we take advantage of the understanding developed and generate a new kinematic model of circumnutation suitable for implementation with tendril-like robots to explore.
Here, we present a new kinematic formulation for extensible continuum robots. The key innovation in the model is that it is explicitly driven by the extension (“growth”) of the outer layer (outer surface of the plant, the layer containing the tendons for continuum robots), which is motivated by similar analyses for plants. This makes it particularly suitable for analyzing plant-inspired movements inherently involving growth such as circumnutation. Note that the approach is different from the kinematics underlying the follow-the-leader movements in [
26], in which a Cosserat rod model was used. The kinematics in this paper lacks the inclusion of materials’ properties, but is simpler and, we believe, somewhat more intuitive. The analysis herein extends that in [
27,
28] in providing a more complete derivation, further insight into the constraints imposed by tendon-actuated continuum robots and a new Jacobian formulation taking into account those constraints. The resulting model is then used, later in this section, as a basis for the synthesis of circumnutation motions in continuum robots.
In order to arrive at a kinematic model of circumnutation suitable for robot simulation and motion generation, the underlying motion needs to be expressed in terms of variables directly related to robot coordinates. To achieve this, in the following, we utilize a kinematic model developed in the botanical literature to model plant circumnutation [
25]. That model results in differential kinematics relating changes in local backbone (plant stem) shape (curvature
and orientation of curvature
) to changes in backbone length. In the case of plants, this change in length is achieved by growth, which can occur at all points around the circumference of the stem.
In the case of extensible continuum robots, however, “growth” corresponds to increases in actuator lengths and can only occur at finite (with a number equal to the number of independently actuated tendons) and fixed locations around the backbone circumference; the radial angles at which the tendons are routed down the backbone. Therefore, to be applicable to continuum robotics, the plant growth kinematics needs to be restricted to these finite, fixed, locations of “growth” (notice that robots, unlike plants, can “ungrow”, with negative backbone length changes when the tendons decrease, and in this sense, the work in this paper both generalizes and provides an alternative perspective to the plant kinematics in [
25]).
Figure 4 gives a brief summary of the notation used in this section.
Assumptions made in formulating the model:
the radius of the section is R, a constant value
the backbone section considered has constant curvature with respect to s at any given time t
the initial curvature of the section is
the initial length along center line of the section is
the initial orientation of maximum curvature is , with respect to a fixed coordinate frame fixed at the section base with the z-axis aligned with the section tangent at that point
Note that, at orientation
, the constant curvature assumption implies that the section centerline is a circular arc with radius
, and defining
, the angle subtended by the arc, via:
we can find the length of all lines lying along the section exterior, parallel to the center line, as follows. The length
of any line along the exterior of the section, parallel to the center line and at a constant separation
R from it, and at a constant angle
with respect to
as measured in the plane orthogonal to the section tangent, is:
If the section is now deformed, producing a new center line of length
, new curvature
and new orientation of maximum curvature
, at orientation
about the exterior, the exterior arc length is, correspondingly,
We next define the growth rate (elongation strain)
at a given orientation
around the section exterior as the ratio between the elongation length
and the original length
:
Subsequently, an average growth rate can be defined as:
Then, the deformed center line
, and Equation (
3) becomes:
Combining (3), (4) and (6):
We expand the above formula, introducing an infinitesimal time step and deformations
,
,
and
, neglecting second order infinitesimal terms, to obtain a differential equation for the kinematics:
Equation (
8) relates the change of exterior length, at orientation
radially about the section, to the rate of change of configuration
and average elongation rate
.
The above approach was introduced in [
25], to analyze the observed growth and circumnutation behavior in plants. A key observation that we exploit here is as follows: defining differential growth
as the difference between elongation strain rates on opposite sides of the structure, divided by the average strain rate,
when expanding the expressions for
and
, the model in (7) simplifies to:
(in which the small quadratic terms are neglected), and:
Equations (10) and (11) give insight into how section configuration shape rates are related to differential growth () and average elongation rate . Note that is the difference in growth rates on opposite sides of the section in the plane of maximum curvature, producing in-plane changes in curvature, and is the equivalent growth rate difference in the plane orthogonal to this, producing out-of-plane changes in section orientation. The configuration rate changes are in each case also linearly related to the average elongation strain rate .
The core modeling approach above was introduced in [
25] to model plant growth, specifically circumnutation. However, in this form, it is not suited to adaptation to robotics. For the case of modeling tendon-driven continuum robots, the assumption that the structure can grow at arbitrary locations of
about the structure is invalid. This is because the prototype described in the beginning of
Section 2 can only independently change its length at a finite number of angles, namely where the tendons are located about its central axis. For the tendril robot in this paper and most other tendon-driven continuum robots, there are three tendon actuators per section, spaced at
apart in
-space (some tendon-driven continuum robots feature four tendons spaced at
apart).
For a continuum section actuated by three tendons arranged at
intervals, as for the tendril in this paper, the average elongation strain rate
is given by:
where
is the strain at the orientation of the
i-th tendon. Methods of analyzing and generating motions, including circumnutations, in such continuum robots need to take this into account.
To achieve this, we note that in [
25], the differential growth rates can be found as projections of a (as yet unknown) direction
of maximal principal growth
:
If the direction of maximal principal growth is aligned with one of the three actuators in a continuum robot and the direction is fixed at this angle, then it can be found, using (9)–(11), that the orientation of the plane of maximum curvature is driven to align with the direction of maximum growth, and (using (11)) will remain there. In other words, driving the section with a dominant, single actuator ( will thus become the radial angle corresponding to its tension) will rotate it to align the direction of maximum curvature with that actuator, a result that makes intuitive sense. This observation suggests a new approach to the modeling and generation of continuum robot circumnutations: the elongation of tendons in sequence (note that the tendril is spring loaded, so tendon tension can decrease), successively cycling from tendons i–, and switching when the plane of curvature matches the tendon location. This insight is used in the next section to directly generate both the rotation and extension motions underlying circumnutation in a tendril robot.
The above kinematic model also yields new insight into the underlying capabilities of continuum robots, when the constraints of finite tendons are taken into account. Consider the most common situation in current hardware, where each section of a continuum robot is driven by three tendons equally radially spaced about the backbone at 120° apart. If we refer to a fixed coordinate axis at the base of a section, with the
z-axis aligned with the backbone tangent (positive direction distal) and the (
x,
y) axes (in the plane of a slice through the backbone) fixed such that the
x-direction is aligned with Tendon 1 (with corresponding elongation
), we can exploit the kinematic model and underlying geometry to derive the following:
These expressions, combined with the derivative of (12):
define the Jacobian relating task space shape change (extension and bending in two dimensions) to tendon length changes:
This result provides interesting insight. From the perspective of an orientation of one of the tendons (Tendon 1 in this case), bending in the (x) plane is seen to be purely a function of movement of that tendon (note that, unlike the plant growth situation, for this orientation, no “growth” is possible at the orientation opposite Tendon 1, as there is no tendon there).
Furthermore, from this perspective, we see that bending in the perpendicular (y) direction is achieved (even though there are no tendons physically present at the backbone at this orientation, ())= by differential motion between the other two tendons, with the effect attenuated (via the multiplier) by their relative orientational displacement from . Similar relationships can be easily derived for any finite number of tendons and any arbitrary, but known orientations about the backbone. Note that the Jacobian is nonsingular at all times; hence the three-tendon design is inherently stable. This new insight allows for constraints to be placed in the model, based on the locations about the cross-section of tendon-driven continuum robots, where length can be independently changed.
In the next section, we demonstrate the utility of the above new approach to continuum robot kinematics discussed in this subsection, via simulations and experiments using the tendril robot hardware of
Section 2.1.1 and exploiting the novel environmental attachment hardware introduced in the previous subsection.