Continuum robots, which are currently the subject of much research, have a continuous backbone structure. Unlike traditional rigid-link robots which are largely inspired by human (vertebrate) limbs, they lack rigid links and joints (like the elbow). An elephant trunk or a snake is more representative of the form of a continuum robot. These robots are usually inspired by nature and many robots which mimic continuum structures seen in nature have been built in recent years. Some of the work inspired by animals includes that by elephant trunks [1], [2], [3], octopus arms [4], [5], [6] snakes [7], and tails [8], [9]. Continuum robots have also been inspired by plants as seen in [10], [11].
Continuum and soft robotics have been the subject of rapid developments in the past few years, expanding their capabilities in medicine [12], [13], space [14], inspection [11], [15], [16] and other applications [17]. High precision and accuracy are expected from robots, especially in the medical field and commercial industry. To achieve this, accurate kinematic and dynamic models for continuum robots are essential. Various kinematics and dynamics modeling approaches have been applied to continuum robots. The constant curvature model [18] is by far the most widely used kinematic model. This model assumes that the continuum structure can be approximated as a serially connected set of constant curvature sections.
Chirikjian [19] introduced a general method for modeling continuum kinematics and dynamics. His approach uses modal functions to represent the curve shaping function.
Fig. 1: Euler (in red) vs constant curvature (in green) modeling on a section developed in [27] .
However, it has restrictions on the class of shapes it is able to approximate, due to the modal functions selected. Problems such as singularity in constant curvature models were addressed in [20] and a solution via an alternative modal model approach was presented. Another noteworthy model by Mochiyama [21] uses the Serret-Frenet formulae to develop the kinematics. These approaches support constant or variable curvature functions. A comparative study of five kinematic modeling methods -lumped system dynamic model [22], constant curvature [18], two-step modified constant curvature [23], variable curvature kinematics [24] with Cosserat rod and beam theory [25], and series solution identification -is detailed in [26]. The prevailing choice in continuum robot modeling remains constant curvature.
However, no continuum section has precisely constant curvature, and some exhibit highly variable curvature. The work in [28] proposes a model for a wire driven continuum manipulator with variable curvature. While this and other variable curvature modeling approaches, as in [24], [29], [30], have been shown to better approximate the shape of some continuum robot sections, they are generally computationally complex and non-intuitive, relative to constant curvature modeling. This paper introduces a new, computationally straightforward and intuitive approach to nonconstant curvature modeling of continuum robot sections.
In the following section, we review the constant curvature approach. Section III introduces an alternative approach to variable curvature modeling, based on Euler spirals. The approach is evaluated on the sections of numerous continuum sections, including two novel parallel systems. These systems are introduced in Section IV, with results from the variable curvature modeling following in Section V. Section VI presents discussion and conclusions. Continuum robots are not well modeled using the same techniques that are used for rigid-link robots which use discrete joint angles exploiting constant link lengths of the robot. Arc lengths and curvatures better describe the shape of a continuum section. Due to the lack of an accurate model for the theoretically infinite degrees of freedom of a continuum structure, the constant curvature approximation has been widely used.
Various approaches have been used to formulate the constant curvature approximation, as discussed in [31]. For example, continuum kinematics via virtual rigid-link kinematics extends the Denavit-Hartenberg (D-H) convention, the standard approach for rigid-link robots, to solve for the tip position by generating the transformation matrix [32], [33]. The result (consistent with alternative, more geometrically direct approaches to constant curvature modeling) is summarized below.
For a planar continuum section as shown in Fig. 2 with the orange curved shape originating from the frame origin such that the initial tangent is aligned with the X-axis, the tip position equations are given by ( 1) and ( 2), where, s ∈ R + is the arc length and k ∈ R + is the curvature (given by 1/R).
where (x ∈ R, y ∈ R) is the tip position for given values of s and k.
The Euler spiral, also termed the Clothoid spiral, is a curve whose curvature is linearly proportional to its arc length (shown in Fig. 3). They are widely applied in railroads as transition curves (to transition from straight segments to circular curves). A clothoid approximation technique and its application in path planning application is detailed in [34]. Hirose's serpenoid curves are inspired by Clothoids [7].
Notice the visual resemblance of the early (proximal) part of the Euler spiral in Fig. 3(a), notably its steadily increasing curvature, to the curve exhibited in the continuum section in Fig. 3(b). In Section V, we will show that the curvatures of this section, and others forming part of the parallel continuum robot systems introduced in the following section, are closely modeled by Euler spirals. We believe that this strong shape match (which extends to other continuum sections of variable curvature), together with the inherent simplicity of the underlying formulation, suggests that Euler spirals warrant further consideration in continuum robot curvature modeling. Note the simple extension of the constant curvature model in the previous section to Euler spirals in ( 3) -( 5) below allows them to inherit much of the convenience and low computational complexity of those models.
Recall that Euler spirals are defined to be those curves whose curvature increases linearly with arc length [35]. Given curvature k ∈ R + and arc length s ∈ R + this gives the equations below.
where r ∈ R is a constant. Substituting k by s/r in ( 1) and ( 2), Euler curve equations can be generated as follows:
The values for x and y are undefined at s = 0, so in all numerical examples we start with s = 0.01.
The analytical form of the equations for Euler curves presents advantages compared to the synthetic curves studied in [30]. The explicit inclusion of the curvature provides a more direct way to visualize the curves than using the control points in synthetic curve approaches.
We next introduce two continuum robot systems with strongly variable curvature sections. These sections will be used to implement, validate, and explore the applicability of Euler curves to variable curvature modeling. We introduce two continuum robot systems, named Separating Sections and Mawby. Each system features three degree of freedom sections operating in parallel.
Each section of the continuum robots discussed in this paper consists of three pneumatic McKibben actuators [36].
A key feature of both designs in the relatively long length of the sections, relative to their diameters. While the Separating Sections' sections have an average length of 45 cm and the muscles are connected by zip ties, Mawby's sections are about 56 cm each in length and connected with fishing line. To compare the size of these sections with those of an extensively studied serial continuum robot in our laboratory, the Octarm [37], the tip section of the Octarm has a lengthdiameter ratio of 6:1 whereas the ratio for the Mawby sections is close to 17:1 and that for Separating Sections is around 13.6:1. This higher length-diameter ratio for the Separating Sections and Mawby sections enables them to have a greater curl than the Octarm but this means they also demonstrate a considerably different curvature to the Octarm [37].
The underlying concept for the Separating Sections was to create a multi-purpose robot, which can operate as a single continuum robot with its three continuum "finger" sections aligned, but also have those fingers "open up" to form a parallel gripper. Motivation is provided by the way the squid deploys its arms and tentacles (see Fig. 4). The main design challenge was how to achieve the separation of the sections, which was done using fishing wire, as detailed below.
The design of Separating Sections is shown in Fig. 5(a). Zip ties were added to muscles 2,3,4,6,7 and 8 and fishing line was passed through these zip ties in a zig-zag fashion as shown in Fig. 5(b) to connect the three sections. The loose ends of the fishing line are terminated at the bottom and are fed back through the center of the 3 sections system and taken out from the top to be connected to servo motors. This was done to ensure the fishing line is loosened at the tip, where the sections branch out from, instead of the base.
When the fishing line is held taut, the sections work as a single unit. The system extends in length when all nine muscles are actuated. When a single section is actuated, the resulting motion mimics that of a three-muscle unit when a single muscle in that unit is actuated as shown in Fig. 5(c). When the fishing line was loosened, the three sections are not held together anymore and can branch out. Fig. 5(d) shows all three sections curving outwards. When the system is located directly above the object, the section can open up, hover around the object, and when in reach, the sections can close in and grasp objects.
Mawby consists of three continuum sections in parallel, with a single degree of freedom rotation about their common base. In order to be able to grasp a wide range of objects, the design of this robot was inspired by the Barrett Hand [38]. In this design there are three robotic fingers. One finger is fixed while the other two can move 180 • about the palm of the hand. The configurations to grasp objects can be made with a simple, single degree of freedom design. To construct a similar capability for Mawby, gears were printed using additive manufacturing. A servo motor was attached to the central gear and was able to move the two movable sections 180 • . Two different orientations of the sections are shown in Fig. 6.
The robot designs introduced in this section represent a novel approach to continuum robot design. The Separating Sections is the first continuum robot that can operate as both an independent serial section and a parallel continuum robot. Mawby is the first parallel continuum robot in which the section base locations can be varied. Both feature high length to diameter ratio sections.
Experiments with prototype hardware realizations of the designs indicate that they can be highly effective in a variety Fig. 6: Mawby in two different orientations. In the image on the left the two movable sections are 90 • apart from the fixed section. In the image on the right the movable sections are 180 • apart from the fixed section. of novel adaptive grasping operations. One feature common to both robots is the uneven, non-constant, curvature of their sections. This is a function of their long, thin design, which proves effective in grasping. However, it makes their modeling a challenge. This issue is addressed, using Euler spirals, in the following section of the paper.
In order to analyze and predict the behavior of the continuum robots introduced in Section IV and continuum robots in general, variable curvature models are necessary. In this section, we apply Euler spirals to modeling the curvature of both long and short continuum sections and support the analysis with experimental validation.
To analyze the ability of both constant and non-constant curvature models to fit the continuum robots considered in Section IV, numerous images were captured from a fixed camera position of a Mawby section, first under various pressure values in which only one muscle was pressurized. For the range of (0, s], using (1) and (2), constant curvature curves were subsequently matched to the shapes of the Mawby section shown in Fig. 7.
Due to the shape discrepancies clearly visible in the constant curvature images for Mawby sections there is a clear need for a better model. Using ( 4) and ( 5), Euler curves were generated to approximate the Mawby section curves. The input parameters for each curve in the example sets are seen in Table I and the corresponding results can be seen in Fig. 8. To calculate the error between the generated curve and Mawby section curve, 50 points were manually selected on the section image along the length of curve, starting from the base of the section and ending at the tip. The Euclidean distance between these points and the point closest to it in the generated Euler curve or constant curvature curve was calculated. The root-mean-square value of the error values for each point selected was calculated to achieve an error Fig. 7 6.
where e1 is the error from the constant curvature model and e2 is the error from the Euler model -both of which are shown in Table II.
Comparison of the Euler spiral model to the constant curvature model is seen in Table II
The effect of this model was further investigated on the shorter sections of the Separating Sections robot discussed in Section IV. The Separating Sections' section curves were generated here by actuating a single muscle with the other two muscles unpressurized. The input parameters are as shown in Table III and their corresponding comparisons can be seen in Fig. 9. Comparison of the Euler spiral model to the constant curvature model is seen in Table IV.
A new 'short' section was constructed with length 25.4 cm and a length to diameter ratio of 7.7:1. This section exhibits curvature visually close to constant. The effect of the Euler model was further investigated on this short section. The input parameters are as shown in Table V and their corresponding comparisons can be seen in Fig. 10. Comparison of the Euler spiral model to the constant curvature model is seen in Table VI. It can be seen that even for "almost" constant curvature sections, the Euler curve method generates significantly better shape approximations.
With the improvement of the Euler curves over the constant curvature model on the robots and sections designed in our laboratory, the effect of the Euler model was further investigated on curves generated by other soft/continuum robots designed outside our laboratory. Images were taken from references as noted or freely available on the internet. Fig. 11(a) shows the OctopusGripper designed by Festo. An 82% improvement in error was recorded by the Euler model compared to constant curvature model. Significant improvements were also recorded for sections developed in [27] (shown in 11(b)) and [39] (shown in 11(c)).
Numerous cases of the Mawby section and Separating Sections section curves were illustrated and analyzed in this paper. The effect of the Euler curve approach was investigated on 42 different curves generated by varying the pressures in the muscles of the robots discussed in Section IV [40]. It was found that the accuracy of the Euler curve approach is superior across a wide range of configurations.
This paper discussed the application of Euler spirals to the modeling of continuum robot sections exhibiting significant variable curvature. Euler spirals represent a straighforward extension from commonly used constant curvature models, with curvature increasing linearly as a function of length. Use of Euler spirals thus inherits many of the convenient properties of constant curvature approaches, and represents both an intuitive and a computationally simple approach to modeling variable curvature.
Euler spirals were used to model the curvature of two novel continuum manipulators introduced in this paper, Mawby and Separating sections. Mawby is a three section [27]. (c) Section developed in [39]. adaptive gripper, with its design inspired by the Barrett Hand, and is the first parallel continuum robot to feature adjustable relative section base positions. The three sections of Separating Sections can operate as a single serial section unit, but can also branch out into three separate and distinct units as the situation demands. A key enabling feature of both robots however, is the use of relatively long sections, which exhibit significant variability in their curvature.
Euler spirals were applied to 42 different curves generated by different combinations of pressure values in the three muscles of sections of Mawby and Separating Sections. When compared to the constant curvature model, the Euler spiral curves yielded a 93% improvement in predicting the shape in the sections of Mawby. The approach was applied to continuum sections of the Separating Sections robot and the results were also an improvement over the constant curvature model with an 86% decrease in error.
Euler spirals were also applied to a variety of other prominent continuum sections as well as one with close-toconstant curvature. The results again demonstrates the ability of Euler curves to significantly outperform constant curvature models.
For future work, we intend to work on the following: 1. Further develop the relationship between the input parameters and the shape parameters. 2. Comparison with other variable curvature techniques. 3. Extend analysis to multiple segments.
International Conference on Robotics and Automation (ICRA) Palais des congres de Montreal, Montreal, Canada, May 20-24, 2019 978-1-5386-6027-0/19/$31.00 ©2019 IEEE