Parameters for various snake robot behaviors.
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Article Type: Review Paper
Date of acceptance: January 2024
Date of publication: February 2024
DoI: 10.5772/acrt.32
copyright: ©2024 The Author(s), Licensee IntechOpen, License: CC BY 4.0
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This paper presents a modern mathematical method to analyze snake robot dynamics. The method is rooted in three facets of contemporary mathematics: Cartan’s concept of endowing all moving bodies with their own reference frames, Lie group theory with its associated algebra, and a compact notation. Building upon previous work with cranes, this paper presents a new kinematic variable for determining the equations of motion for any number of rigid bodies linked in a tree structure with revolute joints. The core equations simplify the analysis and introduce a notation for the coordinate transformation matrix that directly ports to coding. The resulting equations can be readily applied using symbolic math packages and direct numerical solvers. In addition to its primary role as a research document, this paper also serves as an expository educational resource, presenting the methodology in a semi-tutorial format. The culmination of this work yields a comprehensive 3D forward-kinematics analytical model for analyzing the multi-body dynamics of a snake robotic system.
Engineering computing
snake robots
search and rescue
Lie algebra and group theory
robotics
multi-body dynamics
Hamilton’s principle
principle of virtual work
Author information
Snake robots have a wide range of potential applications due to their unique and flexible design. Such robots can move in ways that traditional wheeled or legged robots cannot, making them suitable for various tasks in different fields.
Early developments in snake robotics can be traced back to the work of Hirose [1], who introduced the concept of biologically inspired snake-like robots. Ali Asadian [2] provided insights into the design and control of snake robots and offered a comprehensive review of state-of-the-art techniques and methodologies. Continuing the applications, Rieber
This paper supplements these achievements by applying a new mathematical method to analyze snake robots. In this new method, both 3D and 2D analyses and single and multi-bodies analyses share the same notation. A precedent examination was conducted by Rykkje
Élie Cartan (1869–1951) [7] introduced the concept of assigning a reference frame to each object, be it a curve, a surface, or even Euclidean space itself. He then employed an orthonormal expansion to express the rate of frame change in terms of the frame; this expansion motivated this work. We leverage this by placing a reference frame on each moving link in a snake robot. However, this necessitates a means to relate these moving frames to each other; and for this, we turn to Lie group theory and associate algebra.
Marius Sophus Lie (1842–1899) developed the theory of continuous groups and their associated algebras, which we use to relate the moving frames. This work leverages the mathematics of rotation groups and their algebras but distills them to straightforward matrix multiplications. Yet, to maintain simplicity, a new notation is also required.
Ted Frankel [8] (1929–2017) devised a compact notation in his work on geometrical physics, which we adopt to ensure a uniform notation for both 2D and 3D analyses. This notation remains consistent whether dealing with single bodies or multi-body linked systems, making it accessible to undergraduate students embarking on advanced research endeavors.
The pedagogical effectiveness of this method in the undergraduate curriculum has been assessed by Impelluso [9]. The authors of this paper have previously applied the method to model various systems (not listed for vainglorious citations but to underscore the versatility of the method); these include studies on ROV motion [10], friction [11], gyroscopic wave energy converters [12], a knuckle boom crane [13], ship stability [14], and car dynamics [15].
Figure 2 presents two bodies not yet aligned in a snake-like linked tree (for the sake of clarifying the frames). At the center of mass of each moving body (iconified by a solid circle), a reference body frame is placed.
The parenthetical superscript on the frame references the body. The subscript references the frame axes’ coordinate directions. The third axis (not shown) conforms to the right-hand rule (but only for figure clarification; the method does not preclude 3D motion). The frame is time-dependent, translating and rotating:
Express a
Add the absolute position vector of the 𝛼-frame
Define the skew-symmetric angular velocity matrix as follows, noting that this element is a member of the associated Lie algebra,
The equations below formally define the properties of the Lie Group used, and its associated algebra:
Group Closure properties hold for sequential rotations (e.g.: the product of two rotation matrices is also a rotation matrix):
500 voiced, animated instructional PowerPoint slides where modified equations slide in to replace the previous ones;
100 problems (2D, 3D, single and multi-bodies) solved in a consistent manner to elicit patterns as a way to bridge conceptual understanding and unfortunate memorization; several problems solved incorrectly
50 3D web-based interactive WebGL animations (to assist with visualization);
5000 pages of pedagogical content written in a style called Swipe Learning: single concepts on each page, equations colored to show changing terms, figure repeated with additional information, all content repeated as necessary;
roadmaps for implementation in the curriculum from the undergraduate to the graduate.
Denavit and Hartenberg [16] presented homogenous transformation matrices and are widely used in computer science and robotics. However, this paper exploits their associated algebra. While even this continuation may also be familiar, we request the reader to be patient for a bit longer.
Define a
The body-1 frame connection relates to the inertial frame connection though a frame connection matrix.
Translate to Joint 1 from the inertial frame (bold vector in the figure), using the inertial frame:
All three transformations, in the form of three sequential members of SE(3) (where dimensionality of the block form will be shown only in the first term of this first equation) present as:
Define the frame connection rate (the product of the inverse rate of the member of SE(3)):
First, extract the column of components of the angular velocity matrix in the same frame:
State the body 2 and body 1 frame connections and the relative frame connection matrix:
Translate to Joint 2 from body 1:
Rotate body 1 into body 2:
Translate to body 2 from Joint 2:
Take the rate and inverse:
Consulting Figure 4, we can now extend the chain to three bodies, working out the mathematics or simply observing the pattern.
Generalizing to
Finally, we reassert that patterns appear throughout the MFM. The revelation of patterns reduces the pathological temptations to memorize equations. This, in turn, reduces student attrition while clarifying the discipline.
The Cartesian velocities and angular velocities for 𝛼-bodies are grouped in a 6
In accordance with Hamilton’s Principle, the trajectory of a system is defined by the finding the variation of the action (the integral of the mechanical Lagrangian: the difference between the kinetic, K, and potential, U, energy) and setting it to zero:
After performing the machinery of the Calculus of Variations, we obtain a second order coupled differential equation:
The power of the MFM lies in its simplicity; avoidance of the cross product and the use of matrix multiplications. This enables rapid analysis of a multitude of real world 3D machines and mechanism. This is now taught in an undergraduate class. This paper, however, presented further patterns that the method reveals, this time, with a focus on snake robots. In this way, the method opens vistas for new research and development.
Before finalizing, we note that in addition to constructing the B matrix for this analysis, it is incumbent to also build the B-dot matrix.
To construct the
We time differentiate the
Three-dimensional interactive physics-based simulations in engineering can sometimes provide a heightened level of verification compared to traditional 2D plots; and while this paper provides both, we strongly espouse the 3D animations (soon, PDF will incorporate 3D). These simulations enable engineers to replicate real-world scenarios with greater fidelity, considering multiple dimensions and complex interactions among physical forces. Unlike static 2D plots, 3D simulations allow engineers to observe dynamic behaviors, such as material stress, fluid flow, or structural deformations, in real time. This dynamic insight not only enhances the understanding of system behavior but also facilitates the identification of potential issues and design flaws that may be overlooked in 2D plots.
This is a first pass at the analysis for this new research group. The next steps are to present more proper validation of the method. To this end, we solicit guidance and collaboration from all interested readers. Until that time, we feel that 3D simulations pose an attractive way to quasi-validate the work, and we discuss that now.
WebGL (Web Graphics Library) is a JavaScript interface for rendering interactive 2D and 3D computer graphics. https://home.hvl.no/prosjekter/rykkje/Projects/robot/
To validate and test the MFM, we used the resulting Equations (89) through (92) to develop the system of equations that governs the motion of the system. We then provided second order time-dependent polynomial functions, as presented in Equation (93) and Table 1 to the
| |
Body number | Motion function factors |
3, 5, 7 | |
13, 15, 17 | |
0, 1, 2, 4, 5, 8, 9, 10, 11, 12, 14, 16, 18, 19 | |
| |
Bodies | Motion function factors |
2, 6, 14, 18 | |
4, 8, 12, 16 | |
0, 1, 3, 5, 7, 9, 10, 11, 13, 15, 17, 19 | |
| |
Bodies | Motion function factors |
2, 3, 4, 5, 6, 7, 9, 11, 13, 18 | |
12, 14, 16 | |
0, 1, 8, 10, 15, 17, 19 |
The accordion and cobra movement (ref.: Website) were given polynomial functions to simulate and validate 2D motion, as in both these cases, the bodies of the robot moves strictly in the same plane.
To simulate 3D movement, a spiral function was given as an input on the website. Figure 5 presents a snapshot of the animation with circular helix motion.
The resulting angles for each body were exported and analyzed using MATLAB.
The reader is encouraged to visit the website which presents the motion in full 3D. In its absence, static 2D snapshots Figures 6 and 7 show the motion of body 20 i.e. the head of the snake. Figure 6 shows the simulated relative rotation with respect to body 19, while Figure 7 shows the absolute translation relative to the inertial frame situated at body 1.
The difference in desired and replicated angles were calculated and summed up to
The motion desired is replicated by the solver, showing that the algorithm works as intended and simulates the motion of a 20 linked system of rigid bodies given the external forces.
This paper has presented an extensible and efficient way of developing the equations of motion for a robotic snake structure for an arbitrary number of links. The resulting kinematic transfer variable deployed in Equations (89) to (92) lays the groundwork for further software development and parallelization of the systems of equations generated by the method.
The equation of motion was validated using polynomial input to calculate the needed applied moments. Using these moments, the system replicates the motion desired with a total error sum reduced to numerical inaccuracies in the integration, not calculation errors. Thus, the system presented works for an arbitrary number of linked rigid bodies with known rotational axis.
By applying the kinetic transfer variable, the equation of motion for a system of
The equations of motion (85), and Equations (89) through (92) pose a significant efficiency increase, in the sense, that they can be calculated in parallel. The columns require the information of the input above in the
The systems of equations could also work as a digital twin. With feedback from a robot, it could simulate orientation and position of the structure for an operator. By visualizing the state of the robot, it would enhance the operators control and make the passage through narrow and complicated areas easier. However, this would require a contact algorithm, and perhaps an extension of the method into inverse dynamics. The authors are happy to share guidance so others can take on this challenge.
Finally, the authors reiterate their invitation to contact them for access to the learning materials that present the method in way accessible to students and faculty. Readers may contact the lead author for a theoretical discussion of the paper, and the second author for access to the pedagogical content.
The authors declare no conflict of interest.
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Article Type: Review Paper
Date of acceptance: January 2024
Date of publication: February 2024
DOI: 10.5772/acrt.32
Copyright: The Author(s), Licensee IntechOpen, License: CC BY 4.0
© The Author(s) 2024. Licensee IntechOpen. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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