The idea is something I can relate to largely. Robotics Education is only just taking off in most parts of the world. With this project, I designed something that students at my home university, Manipal Institute of Technology in India, would be able to build. For the design, we considered various constraints such as component availability, 3D printer access, and skill level.
The kit comprises a Tendon Driven Continuum Robot (TDCR), four actuators, two TI Launchpads, various complementary electronic components, and a custom casing to enclose the entire system. For the sake of brevity, I have omitted details thoroughly explained on our GitHub.
The manipulator of the TDCR consists of 10 Type-II spacer disks equally distributed along a Nitinol Backbone with a diameter of 1.4 mm. The robot is capable of 3 DOFs, namely bending across two planes and rotation about its axis. I used three independently actuated tendons made of Spectra Fiber and an actuator to rotate the backbone, enabling these degrees of freedom.
The body of the spacer disk is fixed to the backbone axially, while a bearing in the middle enables the rotational DOF. It is split into two parts to allow for the insertion of the bearing in the middle. I changed the regular ball bearing to a pulley to reduce the number of unique components. The profile of the two parts allows for easy assembly while ensuring that the actuation through the tendons is optimal. The grooves ensure the adhesive stays inside the segment and does not spill out. A small part, functioning as a spacer to make the pulley bore smaller, is designed with a unique profile that balances the required friction to grip the backbone and the need for a constant curvature.
Our actuation module was modified to suit this kit’s requirements. I replaced the multi-part casing with a single unit, such that the new case is easier to manufacture while maintaining adequate performance. The assembly process is simplified as the singular casing eliminates input and output shaft misalignment. The motor and the gears can easily fit into designated spaces using dedicated slots. Since the actuator will only actuate tendons, I introduced an extra bearing at the output shaft to handle the increased radial tension and eliminate shaft eccentricity.
I used the Launchpad TI LAUNCHXL-F28069M, a microcontroller development board, to control the motors with two motor Drivers, BOOSTXL-DRV8305EVM. Since four motors and each motor driver can control one motor, two Launchpads and four motor drivers are required to actuate the continuum robot.
All components are placed inside a hexagonal casing with windows made of transparent acrylic sheets to view the system’s working. The three tendon actuators are fixed horizontally to reduce tendon routing. Six pulleys are placed inside to help reduce the tendon offset and increase the efficiency of tendon control. One actuation module is placed at the bottom and controls backbone rotation. To secure the electric components in the case, I designed custom 3D-printed mounts. All components are visible from the windows to allow the user to see the robot’s functioning, allowing better understanding and enabling intuition development.
I used several techniques to adapt the design for manufacturing with an FDM 3D Printer:
Find a detailed assembly guide, printable STL Files and mechanical drawings on GitHub and watch the assembly video on YouTube: GitHub Video
I encourage you to try and fabricate this learning kit and contact us regarding any queries or further developments you make! We would love to hear about your experience and see your prototypes!
I took the design home and encouraged robotics enthusiasts at my home university to build and use OpenCR’s Learning Kit. The learning kit allows students to easily understand concepts, design, and implement various algorithms. I hope this kit or its successor will find its way into classrooms at different schools! Stay tuned for updates.
Reinhard M. Grassmann, Priyanka Rao, Quentin Peyron, Jessica Burgner-Kahrs: FAS — A Fully Actuated Segment for Tendon-Driven Continuum Robots. Front. Robot. AI 9:873446, 2022. doi: 10.3389/frobt.2022.873446 ↩
Reinhard M. Grassmann, Chengnan Shentu, Taqi Hamoda, Puspita Triana Dewi, Jessica Burgner-Kahrs. “Open Continuum Robotics – One Actuation Module to Create them All,” 2023, arXiv: 2304.11850 ↩
First, let’s discuss one of the primary differences between continuum robots and more “traditional” robots (for lack of a better term)—modeling.
As you’ve seen in prior (and future, I promise) blogs on this site, modeling the mechanics of continuum robots is fundamentally different from more traditional serial-link manipulator arms.
As a subset of mechanics, consider just the forward kinematics (FK) problem for continuum robots, i.e., mapping the robot’s joint state to the robot’s resultant shape in space. For concentric tube robots this may be mapping the robot’s tubes’ rotation/translation values to the robot’s shape and for tendon-driven robots this may be mapping the robot’s tendon tensions/displacements to the robot’s shape.
What’s actually happening in the physical world is complicated here. There are elastic elements interacting with (potentially) other elastic elements in complex ways. There is difficult-to-model friction between components of the robot, hysteresis, potentially difficult-to-predict material properties, etc.
The continuum robotics community has taken a variety of approaches to modeling, but a fundamental trend exists: a trade-off between accuracy and speed-of-computation. That is, models which may be fast to compute are typically fast because they neglect to account for some of the physical phenomenon present in the actual robot. On the flip side, models that account for those phenomenon are slow to compute. In reality, there’s a spectrum here.
Constant curvature models, for instance, may take a purely geometric approach to FK, which is fast to compute, but tend to be less accurate. On the other end of the spectrum, you could consider leveraging something like finite element analysis to simulate the physics of a continuum robot to compute its shape. This would improve accuracy but would be much, much slower to compute. Somewhere in the middle of these you may find things like the Cosserat rod/string models, which are a popular way to model continuum robots and will be discussed in other blog posts here (this is the approach we take in my lab, usually, when we aren’t using machine learning-based methods).
Let’s contrast this with serial-link manipulator arms. For these, FK typically require only the multiplication of a handful of matrices, which is comparatively VERY fast, and comparatively VERY accurate, as these robots are rigid.
Okay, so why is this a problem when performing motion planning for continuum robots?
Well, one way to think about motion planning is that we are reasoning in advance of actually moving about the best way to move the robot to achieve our goals. This means that we need our ability to reason about motion to be accurate, otherwise how we predict the robot will move doesn’t mean much. For this reason, model inaccuracy plays havoc with naïve motion planning methods (such as we’ve discussed in last week’s post), and continuum robots have inaccurate models.
Further, the ways in which we are reasoning in advance about moving require us to consider a whole bunch of potential motions for the robot. Every vertex in the grid/lattice-based and sampling-based methods represents an FK computation. Every edge between them requires multiple (often many) FK computations. In the optimization-based methods, the computation of objective and constraint gradients frequently require many FK computations. So the time it takes to compute FK has HUGE implications in motion planning.
This modeling tension, fast but inaccurate or (somewhat) accurate but slow, is one of the fundamental challenges of applying motion planning to continuum robots, and the one I’ll focus on for the rest of this post.
The continuum robotics community (myself included) has largely taken an extremely unsatisfying approach to overcoming modeling inaccuracy. This approach, also used by much of the non-continuum robotics planning world, boils down to a simple idea: don’t let the robot get close to things. That way, if the model is inaccurate, hopefully the robot is far enough away from its environment that it won’t collide with it in unexpected ways.
As an example, in one of my papers^{1}, we build into the objective function a notion of distance from obstacles. Specifically, We define the cost of a configuration $\boldsymbol{q}$ as:
\[\begin{split} &\texttt{cost}(\boldsymbol{q})=\left\{ \begin{array}{rcl} \cfrac{1}{\texttt{clear}(\boldsymbol{q})},&&\texttt{clear}(\boldsymbol{q}) > 0\\ \infty,&&\texttt{clear}(\boldsymbol{q}) \le 0, \end{array}\right. \end{split}\]where $\texttt{clear}$ is, informally defined here, the signed distance from the robot’s body at that configuration to the nearest obstacle.
The cost of a path $\boldsymbol{\xi}$ then becomes \(\texttt{Cost}(\boldsymbol{\xi}) = \int_0^1 \texttt{cost}(\boldsymbol{\xi}(s))ds .\)
where $s$ is a normalized path arclength parameter. Note here that I’ve used little ‘c’ cost for cost over configuration and big ‘C’ Cost for cost over a path. Creative, I know.
You should really read the paper if you want formality here, but this should give you the gist. Also note that in the paper the notation is slightly different, but I’ve changed it here to more-closely match what I’ve used in the last post.
In that method, we leverage both sampling-based motion planning and optimization-based motion planning in parallel (building on [13]), where the method attempts to minimize the $\texttt{cost}$ of a path. This results in paths that travel as far from obstacles as possible, while still being constrained at the start and end configurations.
The community has taken a variety of approaches to overcome the slow computation speed of continuum robot mechanical models. Here are a few, with some examples.
The first, and most obvious, is the use of a fast-but-inaccurate mechanical model: For instance, in Lyons et al. ^{2}, the authors simplify the mechanical model of concentric tube robots in two key ways. First, they assume that the tubes are torsionally rigid. Second, they assume that the stiffness of each tube dominates all of the tubes nested inside of it. This means that the shape of a tube is assumed to be independent of the motions of the tubes internal to it and that each tube deploys in a constant-curvature circular shape. The mechanical model that leverages these assumptions is sufficiently fast for planning, but again as discussed above ad nauseam, comes at the significant cost of accuracy.
Another approach is to lean on advances in software engineering and programming languages to speed up the computation: Leibrandt et. al.^{3} leverage template metaprogramming in C++, which (I’m oversimplifying this explanation) moves a lot of expensive run-time computation to compile-time computation for concentric tube robots. This enabled very fast kinematics computation. The method leverages this in a PRM-style motion planner. In our paper^{1} mentioned above, we use this template-based kinematic model, wrapping our parallel sampling and optimization method around it. Speaking of, our^{1} use of parallelism is another example of leveraging software engineering concepts to speed up computation in planning for continuum robots.
We can also consider leveraging precomputation: In the paper ^{4} (its extension ^{5}), the authors propose leveraging precomputation to build a roadmap for planning after the environment is known but in advance of when it is needed. This is a pretty specific (and restrictive) scenario, but makes sense in the application put forth by the papers, namely in surgery. These works present the case that pre-operative medical imaging provides the method with a 3D environment a day (or more) prior to when the surgery will happen. The method can then leverage that time to compute a dense, collision-free roadmap before it’s needed at surgery time. The proposed method utilizes the sampling-based method Rapidly-exploring Random Graph (RRG) (which is also presented in ^{6}, in addition to PRM* and RRT*). The method also utilizes one of, if not the most precise (at the time, at least) mechanical models available for concentric tube robots. This dense, collision-free roadmap is built over the course of many hours using this accurate model and the anatomical environment segmented from the pre-operative medical imaging. Then, at the time of the surgery, the method proposes an interactive-rate supervisory control-style scheme in which a haptic device is used by the surgeon to input a desired tip position for the robot in the patient. Leveraging a nearest-neighbor data structure, the vertex in the roadmap that corresponds to the configuration with the robot’s tip closest to the desired position is then identified. Graph search is performed to identify a path on the roadmap from the current configuration to that configuration, and then the method “steps off” of the roadmap and leveraging damped least-squares iterative inverse kinematics ^{7}^{8}, which is conceptually very similar to resolved-rates control, drives the tip of the robot as close to the desired tip position as it can. This then repeats in a loop.
In our paper^{9}, my group has built upon this concept in a few ways. Among other contributions, we adapt the concepts to work for tendon-driven continuum robots (^{4} and ^{5} are specific to concentric tube robots), introduce a modification to mechanical modeling and collision detection to improve speed, and remove the need to have the environment in advance of precomputing the dense roadmap. The method still heavily leverages precomputation, but only requires the environment immediately prior to when the supervisory control loop is started.
What are the open problems or areas to be explored in motion planning for continuum robots? Here are a few that I’m personally excited about.
There has recently been a large interest in machine learning-based mechanical models of continuum robots, both for their potential to learn unmodeled (or unknowable/unmodelable) effects and their potential for fast computation (e.g., frequently the shape computation of a learned model is faster than the physics-based equivalent). I’ll leave the details of these methods to another post, written by myself or someone else. However germane to THIS post, how to leverage these types of models for motion planning of continuum robots is as-yet underexplored.
Motion planning in a way that handles uncertainty in a principled way has been leveraged to great effect in other areas of robotics. Given the uncertainty associated with continuum robots, the application/exploration of using these types of methods/formulations in planning for continuum robots has great potential.
The fact that we have a variety of models for continuum robots that trade computation speed for accuracy implies exciting potential for the use of multi-fidelity planning. One can imagine, as maybe the most obvious example, using a fast but inaccurate model to prioritize edge evaluation in a lazy planning paradigm, look up Lazy PRM if that doesn’t make sense to you, where full edge evaluation is then done with a high-fidelity model.
Hopefully the above alongside last week’s Part 1 were a relatively gentle introduction to motion planning and how it relates conceptually to continuum robots. This wasn’t intended to be a comprehensive survey of planning for continuum robots. Rather, my intention was to give you the vocabulary and an understanding of the rough concepts sufficient for you to investigate the rest of the literature yourself, and to innovate on the state-of-the-art as you need for your specific problems.
A. Kuntz, M. Fu, R. Alterovitz: Planning High-Quality Motions for Concentric Tube Robots in Point Clouds via Parallel Sampling and optimization. IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 2205-2212, 2019. doi: 10.1109/IROS40897.2019.8968172 ↩ ↩^{2} ↩^{3}
L. A. Lyons, R. J. Webster, R. Alterovitz: Motion planning for active cannulas. IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 801-806, 2009. doi: 10.1109/IROS.2009.5354249 ↩
K. Leibrandt, C. Bergeles, G.-Z. Yang: Concentric Tube Robots: Rapid, Stable Path-Planning and Guidance for Surgical Use. IEEE Robotics & Automation Magazine, vol. 24, no. 2, pp. 42-53, 2017. doi: 10.1109/MRA.2017.2680546 ↩
L. G. Torres, C. Baykal, R. Alterovitz: Interactive-rate motion planning for concentric tube robots. IEEE International Conference on Robotics and Automation (ICRA), pp. 1915-1921, 2014. doi: 10.1109/ICRA.2014.6907112 ↩ ↩^{2}
L. G. Torres et al.: A motion planning approach to automatic obstacle avoidance during concentric tube robot teleoperation. IEEE International Conference on Robotics and Automation (ICRA), pp. 2361-2367, 2015. doi: 10.1109/ICRA.2015.7139513 ↩ ↩^{2}
S. Karaman and E. Frazzoli: Sampling-based algorithms for optimal motion planning. The International Journal of Robotics Research, 30(7):846-894, 2011. doi: 10.1177/0278364911406761 ↩
Y. Nakamura and H. Hanafusa: Inverse Kinematic Solutions With Singularity Robustness for Robot Manipulator Control. ASME. J. Dyn. Sys., Meas., Control., 108(3):163–171, 1986. doi: 10.1115/1.3143764 ↩
C. W. Wampler: Manipulator Inverse Kinematic Solutions Based on Vector Formulations and Damped Least-Squares Methods. IEEE Transactions on Systems, Man, and Cybernetics, 16(1):93-101, 1986. doi: 10.1109/TSMC.1986.289285 ↩
M. Bentley, C. Rucker, A. Kuntz: Interactive-Rate Supervisory Control for Arbitrarily-Routed Multitendon Robots via Motion Planning. IEEE Access, vol. 10, pp. 80999-81019, 2022. doi: 10.1109/ACCESS.2022.3194515 ↩
Motion planning stands as a central pillar in robotics. Planning enables robots and autonomous agents to move themselves and other parts of the world to a desired goal by choosing a sequence of appropriate actions. Like robotics in general, motion planning draws on a number of different fields in developing its tools. Approaches to motion planning have grown out of artificial intelligence, optimal control, and operations research.
That’s the first paragraph of my motion planning course syllabus. It’s a bit… grandiose, but I’ve found it to be effective in convincing the students that they are about to learn something interesting; hopefully it worked on you as well. It was also written by the professor who taught the course before I did, and whose syllabus I used as a starting point for my own (on the shoulders of giants etc etc etc). If that was sufficiently dramatic and you’re convinced to keep reading, let’s get to it.
More colloquially, motion planning can be broadly defined as the problem of computing motions, actions, or controls that move a robot through its environment in such a way that some task is performed while obeying some set of constraints.
In its most basic form, that task can be defined as moving from some start configuration to some goal configuration, with typical constraints being defined as obstacle avoidance over the motion, obeying joint limits, etc. Complexity in task definitions and constraint definitions builds from there.
I personally like to present motion planning as a generic optimization problem and then look specifically at how aspects of that problem are defined. Be forewarned, I’m going to grossly abuse notation/terminology and ignore nuance all over this post with the goal of imparting intuition rather than specifics.
Consider:
$\boldsymbol{\xi}^* = \underset{\boldsymbol{\xi}}{\texttt{argmin}}\ \texttt{cost}(\boldsymbol{\xi})$
$s.t. \ \ \ c(\boldsymbol{\xi}) \ge 0$ $\ \ \ \ \ \ \ \ d(\boldsymbol{\xi}) = 0$
where $\boldsymbol{\xi}$ represents some trajectory, path, or plan parameterization, $\texttt{cost}$ is some cost function defined over these, $\boldsymbol{\xi}^*$ is an optimal plan under the cost function, and $c$ and $d$ are generic, as-of-yet-unspecified inequality and equality constraints (or sets of such constraints).
This generic optimization formalization is sufficiently expressive to talk about most of the basics of motion planning, and by varying how aspects of this formalization are defined we can arrive at most of the ways people have historically thought about motion planning.
As an example, let’s consider the very simple (relative to other motion planning problems) task of moving a holonomic robot from some start state to some goal state while avoiding some set of obstacles in its environment.
If we conceptualize $\boldsymbol{\xi}$ as defining a space curve in the robot’s configuration space, we can parameterize $\boldsymbol{\xi}$ by the normalized path length such that $\boldsymbol{\xi}(0)$ is the robot’s start configuration and $\boldsymbol{\xi}(1)$ is the robot’s configuration after following the full plan.
Often people consider minimizing path length, so we can imagine defining $\texttt{cost}$ as the length of the path (under some appropriate definition of length).
We also want to constrain the trajectory such that the robot ends in a goal configuration (broadly as an element of some goal set), and such that the entire trajectory avoids obstacles.
We can then cast this simple problem into our optimization framework simply as:
$\boldsymbol{\xi}^* = \underset{\boldsymbol{\xi}}{\texttt{argmin}}\ \texttt{length}(\boldsymbol{\xi})$
$s.t. \ \ \ \boldsymbol{\xi}(0) = \textrm{start configuration}$
$\ \ \ \ \ \ \ \ \boldsymbol{\xi}(1) \in \textrm{goal set}$
$\ \ \ \ \ \ \ \ \boldsymbol{\xi}(s) \textrm{ is collision free } \forall s \in [0,1]$
As in most interesting things, key considerations lie in the details (which I’ve completely ignored above). How is your trajectory parameterized? How do you define your goal set? How do you represent your robot and environment geometry, enabling you to check for collision? Is length even the right objective to minimize? The answer to these questions depends on your specific problem/robot/etc. If I knew what this is for you, I’d just solve your problems myself and take all the credit.
Okay, so this is the first step—representing your problem. Next, let’s solve it.
In my head I broadly group modern motion planning methods into three groups, discrete grid/lattice-based search methods, sampling-based methods, and optimization-based methods. There’s an implied fourth group, “other” or maybe “miscellany”, but I’m going to focus on the first three in this post.
At their simplest, these methods consider a graph or tree (which is just a special kind of graph) embedded in the robot’s configuration space. If you are unsure what I mean by “graph or tree” stop now and read this and this before moving on.
Notably for these methods, the graph is usually implied by a discretization over the robot’s possible actions. Imagine vertices of the graph as representing configurations or states of the robot, and edges as actions between states, which when applied from the source state result in the robot transitioning to the destination state. This action discretization is frequently defined in advance (and you can imagine applying this concept to robot’s with continuous action spaces simply by discretizing those action spaces). Due to the constant discretization, you can conceptualize these as creating something like a grid or lattice as the graph.
With this conceptualization, motion planning then reduces to a search over that graph, seeking the “shortest” path from your start vertex to any vertex that represents a goal configuration or state.
How does this map back to our optimization formalization above? The cost function is minimized by applying a graph-search method that minimizes the accumulation of edge cost (so you need to define a cost over edges and recognize that without substantial modification you’re limited to considering objectives that can be modeled by costs that accumulate). The constraints are enforced simply by not considering edges that would result in a violation of the constraints during the search.
Cannonical graph search methods (not specific to motion plannning) include Uniform Cost Search (UCS) (closely related to Djikstra’s algorithm) and A* search.
The key to applying these methods in motion planning is frequently in coming up with an effective and admissible/consistent heuristic (see A* details for what that means). More advanced methods applied to motion planning specifically include D*^{1}, D*-Lite^{2}, and LPA*^{3}. It’s also very much worth noting that inadmissible heuristics can also be leveraged to potentially great effect (see Multi-Heuristic A*^{4}).
Perhaps the most popular class of motion planning methods, the sampling-based approaches are closely related to the graph-search methods. In fact, these methods frequently leverage graph search but on a graph constructed in a different manner.
Rather than a graph constructed via a discretization over the robot’s actions, the sampling-based methods leverage random sampling to construct the graph incrementally (this is why they are called sampling-based, as you may have guessed). The graph (or tree, which again, is just a special kind of graph) is embedded in the robot’s configuration space as above. It is constructed by iteratively randomly sampling states (or actions, sometimes) and attempting to connect the newly sampled state (or state resulting from the sampled action) to the graph/tree.
When adding states, (usually) the new state and the edge connecting it to the graph are checked for validity (e.g., joint-limit satisfaction and obstacle avoidance). If the state and/or edge are not valid, they aren’t added to the graph.
At some point, the hope is that the graph will contain a goal state, and then by running graph search (e.g., A*) on this randomly-generated graph the method will find a path from the start state to a goal. If the graph is a tree, this search is trivial as there’s only one path in the graph (ignoring some nuance, that’s what tree means).
The specifics of the graph, how the states/actions are sampled, how the connections are made, etc., are what distinguish the sampling-based methods from each other.
Okay, so let’s map these concepts back to our optimization formulation.
Cost: If there are multiple paths in the graph, then the graph search over the graph can provide the lowest cost path in the graph, however the lowest cost path possible may not be in the graph at any given iteration. Many of the earlier versions of these methods ignored cost and instead attempted to find any path that avoided obstacles, providing a property called probabilistic-completeness. Since then, other methods build upon this to provide a guarantee called asymptotic optimality, which intuitively means that their best path will approach an optimal path (in cost) as runtime progresses (only getting arbitrarily close to an optimal path in the limit though). This contrasts with the discrete grid/lattice-based search methods in that those methods typically only provide a property called resolution completeness/optimality which intuitively means the path is complete or optimal for the resolution of the discretization, but not necessarily in general.
Constraints: As with the discrete grid/lattice-based search methods, the sampling-based methods usually encode constraint satisfaction by only considering states/edges in the graph construction or search that satisfy the constraints, including obstacle avoidance.
Canonical probabilistically-complete methods include Probabilistic Roadmaps (PRM)^{5} and Rapidly-exploring Random Trees (RRT)^{6}. Popular asymptotically-optimal methods include PRM*/RRT*^{7}, Batch-Informed Trees (BIT)^{8}, and Asymptotically-Optimal-RRT (AO-RRT)^{9}.
Super-duper important point! Just because you’re using sampling-based methods does not mean you’re getting probabilistic completeness or asymptotic optimality. There are properties/assumptions you must satisfy in your specific application of these methods that are frequently nuanced, you’ll need to ensure that you satisfy these things. Read the papers detailing these methods for details!
Given the formulation we started with, i.e., casting the motion planning problem as constrained optimization, you may be asking yourself: “wait, why don’t we just use optimization methods from, you know, mathematics, to solve this problem?” Great question! Enter the optimization-based motion planning methods.
This class of methods leverages nonlinear, constrained optimization methods from the field of optimization itself explicitly to solve the motion planning problem using variations of our above formulation—rather than relying on the graph-search to minimize cost.
As of now, the majority of these methods (with an exception mentioned below) leverage gradients (and frequently higher-order derivatives) of the objective function and/or of the constraint functions in iterative optimization methods. Think penalty methods, augmented Lagrangian methods, interior point methods, etc. If these concepts are new to you, there is an amazing textbook on the subject by Wright and Nocedal ^{10} that is open almost perpetually in my office.
At a high level there isn’t much more to the story. These methods take exactly the optimization-based formulation above and apply optimization techniques that work directly on such formulations.
Is it just that easy? Unfortunately, no. The prior paragraph is true at a high level, but the low level is where the details lie. The specific ways in which these methods define cost and their constraint functions such that they work well with the optimization techniques—e.g., ensuring they have gradients that are well-behaved—can make these methods a bit tricky to use in practice. However in many cases they end up working very, very well, which can make it worth the complexity.
Canonical methods include CHOMP^{11}, which leverages a variation of gradient descent, and TrajOpt^{12} which leverages Sequential Quadratic Programming (SQP). A different method, that is not nearly as popular as those methods, combines sampling-based planning with interior point optimization^{13}. I include it in this list because it was part of yours truly’s PhD work, and so here is a shameless plug.
There is also a method called Cross-Entropy Motion Planning (CEMP)^{14} that leverages a gradient-free optimization method (the cross-entropy method, as you may have guessed from the name). This is notable because many difficulties one may encounter in leveraging optimization-based methods may come from the gradients, or lack-thereof.
This concludes Part 1 of my introduction to motion planning for continuum robots. In next week’s Part 2 we will look at the challenges of motion planning for continuum robots. Stay tuned!
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S. Karaman and E. Frazzoli: Sampling-based algorithms for optimal motion planning. The International Journal of Robotics Research, 30(7):846-894, 2011. doi: 10.1177/0278364911406761 ↩
J.D. Gammell, T.D. Barfoot, S.S. Srinivasa: Batch Informed Trees (BIT*): Informed asymptotically optimal anytime search. The International Journal of Robotics Research, 39(5):543-567, 2020. doi: 10.1177/0278364919890396 ↩
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My keynote presentation at this year’s ICRA is all about the Open Continuum Robotics Project.
Tuesday, 17:00 - 18:30 - Healthcare Robotics
Speaker: Jessica Burgner-Kahrs
Abstract: From robotically steered catheters and bronchoscopes to single port flexible instruments, continuum robotics has led to great advances in medical robotics over the past decade. Yet, research findings remain stubbornly difficult to reproduce because continuum robotics hardware and software are not readily available, and no common systems exist. Ultimately this slows the pace of innovation.
The Open Continuum Robotics Project combines hardware, software, and education with the goal of fostering more collaborative, cost-effective, and reproducible research. This enables rapid prototyping and iterative design, allowing researchers and clinicians to test and refine their ideas more quickly – including in resource-constrained settings. In this way, the Open Continuum Robotics Project accelerates the development of new technologies and treatments in medical robotics.
ICRA 2023 features an incredible rich set of workshops and tutorials this year, which makes it very hard to choose which one to attend (or when to rush from one room to the next #FOMO). The following ones feature aspects of continuum robotics and are worth a look!
The majority of oral presentations of ICRA 2023 are adressing continuum robot modeling, with a focus on kinetostatics and dynamics using Cosserat theory and optimal control theory. It is fantastic to see a trend towards open toolboxes and knowledge mobilization! There are also some exciting soft continuum robot designs!
09:50-10:00, Paper TuAT2.9
Authors: Ferrentino, Pasquale; Roels, Ellen; Brancart, Joost; Terryn, Seppe ; Van Assche, Guy; Vanderborght, Bram
Abstract: Soft robotics modeling is a research topic that is evolving fast. Many techniques are present in literature but most of them require analytical models with a lot of equations that are time-consuming, hard to resolve, and not so easy to handle. For this reason, the help of a soft mechanics simulator is essential in this field. In fact, this paper presents a tutorial on how to build a soft-robot model using an open-source Finite Element Analysis (FEA) simulator, called SOFA. This software is able to generate a simulation scene from a code written in Python or XML, so it can be used by people that with different fields of competence like mechanical knowledge, knowledge of material properties and programming skills. As a case study, a Python simulation of a cable-driven soft actuator that makes contact with a rigid object is considered. The basic working principles of SOFA required to make a scene are explained step by step. In particular, it shows how to simulate the mechanics and animate the bending behavior of the actuator. Furthermore, it will be shown also how to retrieve and save data from simulation, demonstrating that SOFA can easily adapt to a multi-disciplinary subject as the research in soft-robotics, but also be useful for teaching simulation and programming language principles to engineering students.
15:40-15:50, Paper TuBT2.5
Authors: Mathew, Anup Teejo; Ben Hmida, Ikhlas; Armanini, Costanza; Boyer, Frédéric; Renda, Federico
Abstract: Soft robotics has been a trending topic within the robotics community for almost two decades. However, available tools for the modeling and analysis of soft robots are still limited. This paper introduces a user-friendly MATLAB toolbox, SoRoSim, that integrates the Geometric Variable Strain model of Cosserat rods to facilitate the static and dynamic analysis of soft, rigid, or hybrid robotic systems. We present a brief overview of the design and structure of the toolbox and validate it by comparing its results with those published in the literature. To highlight the toolbox’s potential to efficiently model, simulate, optimize, and control various robotic systems, we demonstrate four sample applications. The demonstrated applications explore different actuator and external loading conditions of single-, branched-, open-, and closed-chain robotic systems. We think that the soft-robotics research community will significantly benefit from the SoRoSim toolbox for a wide variety of applications.
15:50-16:00, Paper TuBT2.6
Authors: Briot, Sébastien; Boyer, Frédéric
Abstract: There is a growing interest on the study of continuum parallel robots (CPRs) due to their higher stiffness and better dynamics capacities than serial continuum robots (SCRs). Several works have focused on the computation of their geometrico- and kinemato-static models, that can be sorted into two main categories: (i) models based on the continuous Cosserat equations: They are very accurate but assessing elastic stability with them is tricky; (ii) discretized models: They allow easily checking the elastic stability but they require a large number of elastic variables to be accurate. In this paper, we extend an approach based on assumed strain modes developed for the dynamics of SCRs to the statics of CPRs. This method is able to predict the robot configuration with an excellent accuracy with a very limited number of elastic variables, contrary to other discretization methods. The method is also more than 100 times faster than finite differences for a better prediction accuracy. Finally, it is possible to assess the robot elastic stability by only checking the Hessian of the potential energy as for any discretization method, thus making the analysis of this property simpler.
16:00-16:10, Paper TuBT2.7
Authors: Wu, Zicong; De Iturrate Reyzabal, Mikel; Sadati, Seyedmohammadhadi; Liu, Hongbin; Ourselin, Sebastien; Leff, Daniel Richard; Katzschmann, Robert Kevin; Rhode, Kawal; Bergeles, Christos
Abstract: Soft robots that grow through eversion/apical extension can effectively navigate fragile environments such as ducts and vessels inside the human body. This letter presents the physics based model of a miniature steerable eversion growing robot. We demonstrate the robot’s growing, steering, stiffening and interaction capabilities. The interaction between two robot-internal components is explored, i.e., a steerable catheter for robot tip orientation, and a growing sheath for robot elongation/retraction. The behavior of the growing robot under different inner pressures and external tip forces is investigated. Simulations are carried out within the SOFA framework. Extensive experimentation with a physical robot setup demonstrates agreement with the simulations. The comparison demonstrates a mean absolute error of 10–20% between simulation and experimental results for curvature values, including catheter-only experiments, sheath-only experiments and full system experiments. To our knowledge, this is the first work to explore physics-based modelling of a tendon-driven steerable eversion growing robot. While our work is motivated by early breast cancer detection through mammary duct inspection and uses our MAMMOBOT robot prototype, our approach is general and relevant to similar growing robots.
16:20-16:30, Paper TuBT2.9
Authors: Boyer, Frédéric; Lebastard, Vincent; Candelier, Fabien; Renda, Federico; Alamir, Mazen
Abstract: This paper explores the relationship between optimal control and Cosserat beam theory from the perspective of solving the forward and inverse dynamics (and statics as a subcase) of continuous manipulators and snake-like bio-inspired locomotors. By invoking the principle of minimum potential energy, and the Gauss principle of least constraint, it is shown that the quasi-static and dynamic evolution of these robots, are solutions of optimal control problems (OCPs) in the space variable, which can be solved at each step (of loading or time) of a simulation with the shooting method. In addition to offering an alternative viewpoint on several simulation approaches proposed in the recent past, the optimal control viewpoint allows us to improve some of them while providing a better understanding of their numerical properties. The approach and its properties are illustrated through a set of numerical examples validated against a reference simulator.
15:00-15:10, Paper WeBT2.1
Authors: Alatorre, David; Robles-Linares, Jose A.; Russo, Matteo; Elbanna, Mohamed A.; Wild, Samuel; Dong, Xin; Mohammad, Abdelkhalick; Kell, James; Norton, Andy; Axinte, Dragos
Abstract: The maintenance of critical industrial components is often hindered by limited access, tortuous passages, and complex geometries. In highly constrained environments, inspection tasks are currently performed with borescopes, but even skilled operators struggle with hard-to-reach targets and the limited mobility prevents in-situ repair when defects are identified. Thanks to an active shape control, snake-like and continuum robots can outperform borescopes for short range inspection as well as enable intervention. However, their actuation technology limits their scalability in length, as longer bodies pose control challenges due to their intrinsically low stiffness and space constraints. To overcome the limitations of both borescopes and continuum robots, we here propose a modular design at their intersection, with both active tendon-driven and passively flexible segments. The main elements of the novel design, including actuation and control interface, are described, and the system is demonstrated in scenarios for aerospace assets, nuclear installations, and robot-assisted surgery.
10:10-10:20, Paper ThAT2.8
Authors: Lin, Jui-Te;Girerd, Cedric; Yan, Jiayao; Hwang, John T.; Morimoto, Tania K.
Abstract: Concentric tube robots (CTRs) show particular promise for minimally invasive surgery due to their inherent compliance and ability to navigate in constrained environments. Due to variations in anatomy among patients and variations in task requirements among procedures, it is necessary to customize the design of these robots on a patient- or population-specific basis. However, the complex kinematics and large design space make the design problem challenging. Here we propose a computational framework that can efficiently optimize a robot design and a motion plan to enable safe navigation through the patient’s anatomy. The current framework is the first fully gradient-based method for CTR design optimization and motion planning, enabling an efficient and scalable solution for simultaneously optimizing continuous variables, even across multiple anatomies. The framework is demonstrated using two clinical examples, laryngoscopy and heart biopsy, where the optimization problems are solved for a single patient and across multiple patients, respectively.
10:20-10:30, Paper ThAT2.9
Authors: Lloyd, Peter Robert; Onaizah, Onaizah; Pittiglio, Giovanni; Chathuranga, Damith Suresh; Chandler, James Henry; Valdastri, Pietro
Abstract: Flexible catheters are used in a wide variety of surgical interventions including neurological, pancreatic and cardiovascular. In many cases a lack of dexterity and miniaturization along with excessive stiffness results in large regions of the anatomy being deemed inaccessible. Soft continuum robots have the potential to mitigate these issues. Due to its enormous potential for miniaturization, magnetic actuation is of particular interest in this field. Currently, flexible magnetic catheters often rely on forces of anatomical interaction to generate large deformations during navigation and for soft anatomical structures this could be considered potentially damaging. In this study we demonstrate the insertion of a high aspect ratio, 50 mm long by 2 mm diameter, soft magnetic catheter capable of navigating up to a 180 degree bend without the aid of interactive forces. This magnetic catheter is reinforced with a lengthwise braided structure and its magnetization allows it to shape form along tortuous paths. We demonstrate our innovation in a planar silicone pancreas phantom. We also compare our approach with a mechanically equivalent tip driven magnetic catheter and with an identically magnetized, unreinforced catheter.
15:00-15:10, Paper ThBT5.1
Authors: Lilge, Sven; Burgner-Kahrs, Jessica
Abstract: Tendon-driven parallel continuum robots consist of multiple individual continuous kinematic chains, that are actuated in bending utilizing tendons routed along their backbones. This work derives and proposes a Cosserat rod based kinetostatic modeling framework for such parallel structures that allows for efficiently solving the forward, inverse and velocity kinetostatic problems. Using this model, the kinematic properties such as reachable workspace, singularities, manipulability and compliance of tendon-driven parallel continuum robots are studied in detail. Experiments are conducted using a real robotic prototype to validate the derived modeling approach. Overall, a median pose accuracy of 4.9 mm, corresponding to 3.4% of the continuum robots’ lengths, and 6.2◦ is achieved. The median of the model’s computation time results in 0.51 s on standard computing hardware. Fast computations of below 100 ms can be achieved, if an appropriate initial guess for solving the kinetostatic model is available, making the model suitable for a range of different applications including optimization or control.
The poster presentations at ICRA 2023 cover various aspects of (soft) continuum robots, including design, control, and sensing.
08:30-10:10, Paper TuPO1S-01.2
Authors: Tao, Jian; Hu, Qiqiang; Luo, Tianzhi; Dong, Erbao
Abstract: Soft continuum robots have shown tremendous potential for medical and industrial applications owing to their flexibility and continuous deformability. However, their telescopic and bending capabilities and variable stiffness are still limited. This study proposes a novel origami-inspired soft continuum robot to possess large telescopic and bending capabilities while improving stiffness based on the principle of antagonistic actuation. The soft robot consists of dual origami structures. The inner forms an air chamber actuated by pneumatics, and the outer is controlled by nine tendon-driven actuators. The proposed design uses the advantages of a hybrid actuation to achieve motion and stiffness control. The performance of the soft robot is studied experimentally based on single and three robot modules. Results show that the robot has an excellent stretch ratio and a maximum bending angle of 180°. The robot can also increase stiffness to resist the bending deformation induced by self-weight and loads.
Tuesday 08:30-10:10, Paper TuPO1S-01.6, Room T8
Authors: Lu, Jingpei; Liu, Fei; Girerd, Cedric; Yip, Michael C.
Abstract: State estimation from measured data is crucial for robotic applications as autonomous systems rely on sensors to capture the motion and localize in the 3D world. Among sensors that are designed for measuring a robot’s pose, or for soft robots, their shape, vision sensors are favorable because they are information-rich, easy to set up, and cost-effective. With recent advancements in computer vision, deep learning-based methods no longer require markers for identifying feature points on the robot. However, learning-based methods are data-hungry and hence not suitable for soft and prototyping robots, as building such bench-marking datasets is usually infeasible. In this work, we achieve image-based robot pose estimation and shape reconstruction from camera images. Our method requires no precise robot meshes, but rather utilizes a differentiable renderer and primitive shapes. It hence can be applied to robots for which CAD models might not be available or are crude. Our parameter estimation pipeline is fully differentiable. The robot shape and pose are estimated iteratively by back-propagating the image loss to update the parameters. We demonstrate that our method of using geometrical shape primitives can achieve high accuracy in shape reconstruction for a soft continuum robot and pose estimation for a robot manipulator.
08:30-10:10, Paper TuPO1S-01.8
Authors: Franco, Enrico; Aktas, Ayhan; Treratanakulchai, Shen; Garriga-Casanovas, Arnau; Donder, Abdulhamit; Rodriguez y Baena, Ferdinando
Abstract: In this article we investigate the discrete-time model based control of a planar soft continuum manipulator with proprioceptive sensing provided by fiber Bragg gratings. A control algorithm is designed with a discrete-time energy shaping approach which is extended to account for control-related lag of digital nature. A discrete-time nonlinear observer is employed to estimate the uncertain bending stiffness of the manipulator and to compensate constant matched disturbances. Simulations and experiments demonstrate the effectiveness of the controller compared to a continuous time implementation.
08:30-10:10, Paper TuPO1S-02.1
Authors: McCandless, Max; Juliá Wise, Frank; Russo, Sheila
Abstract: Soft optical sensing strategies are rapidly developing for soft robotic systems as a means to increase the controllability of soft compliant robots. In this paper, we present a roughness tuning strategy for the fabrication of soft optical sensors to achieve the dual functionality of shape sensing combined with contact recognition within a single multi-modal sensor. The molds used to fabricate the soft sensors are roughened via laser micromachining to achieve asymmetrical sensor responses when bent in opposite directions. We demonstrate the integration of these sensors into a fully soft robotic platform consisting of a multi-directional bending module with integrated 3D shape sensing and a gripper with tip position monitoring along with contact force recognition. We show the accuracy of our sensing strategy in validation experiments and a pick-andplace task is performed to demonstrate the robot’s functionality.
08:30-10:10, Paper TuPO1S-03.7
Authors: Srivastava, Manu; Walker, Ian
Abstract: This paper discusses the effect of axial backbone compression on tendon-driven continuum robots. A new mechanics model for compensating for this effect that does not require tendon tension sensing or knowledge of manipulator material properties/stiffnesses is introduced and analyzed. In addition, we provide an analytical expression for the minimum preload on the tendons to achieve a given bend, a quantity determined empirically thus far. Our model is computationally efficient and achieves real time control on low cost hardware. The analysis is supported by experimental results demonstrating significant improvement over kinematics in open loop control of a tendon-driven continuum hose robot.
Tuesday 15:00-16:40, Paper TuPO2S-02.6, Room T8
Authors: Donat, Heiko; Mohammadi, Pouya; Steil, Jochen J.
Abstract: Concentric tube continuum robots (CTCRs) belong to the family of continuum robots with applications in minimally invasive surgeries. Because of this application domain, measuring the external forces along the body of the robot is paramount. CTCRs are made up of thin elastic rods and are intended to be applied inside the human body, where conventional sensor-based measurements are not feasible. Consequently, research is resorting to estimate the forces through geometric, numeric, or optimization methods. However, these methods often suffer from slow convergence. In this paper, we introduce a novel data-driven approach for estimating contact forces along the body of a CTCR that offers an estimation precision comparable to the current state-of-the-art optimization-based approaches, but exhibits nearly two orders of magnitude faster convergence. The proposed method is scalable and exhibits a significant performance in response to a wide range of external forces. The approach was evaluated in simulations and on a real 2-tube CTCR.
09:00-10:40, Paper WePO1S-01.6
Authors: Nayar, Namrata Unnikrishnan; Qi, Ronghuai; Desai, Jaydev P.
Abstract: Intracardiac transcatheter systems guided by advanced imaging modalities are gaining popularity in treating mitral regurgitation in non-surgical candidates. Robotically steerable transcatheter systems must use model-based control strategies to ensure safer and more effective transcatheter procedures with less trauma while using smaller control gains. In this paper, a 4-DoF robotically steerable tendon-driven robot was fabricated, and the relationship between the tendon displacement and the joint angle was derived. This relation was derived in two parts to make this approach applicable to any other catheter system. A model was derived to determine the tendon tensions needed to achieve desired joint angles. Then, the tendon characteristics were studied, and a tendon elongation (TE) model was derived as a function of tendon length. Executing the modeling process in two steps makes it easy to introduce additional parameters like length, friction, and pose, to characterize complex systems like catheters. The TE model was used to actuate the joints of the robot and RMSE was computed to characterize its performance. Also, PID control was used along with the TE model to improve the system’s performance, and the contribution of the model and the controller in the system was recorded.
09:00-10:40, Paper WePO1S-01.11
Authors: Sharma, Susheela; Park, Ji Hwan; Amadio, Jordan P.; Khadem, Mohsen; Alambeigi, Farshid
Abstract: In this paper, we present the design, fabrication, and evaluation of a novel flexible, yet structurally strong, Concentric Tube Steerable Drilling Robot (CT-SDR) to improve minimally invasive treatment of spinal tumors. Inspired by concentric tube robots, the proposed two degree-of-freedom (DoF) CT-SDR, for the first time, not only allows a surgeon to intuitively and quickly drill smooth planar and out-of-plane J- and U- shape curved trajectories, but it also, enables drilling cavities through a hard tissue in a minimally invasive fashion. We successfully evaluated the performance and efficacy of the proposed CT-SDR in drilling various planar and out-of-plane J-shape branch, U-shape, and cavity drilling scenarios on simulated bone materials.
09:00-10:40, Paper WePO1S-01.12
Authors: Pittiglio, Giovanni; Mencattelli, Margherita; Dupont, Pierre
Abstract: This paper introduces a novel class of hyperredundant robots comprised of chains of permanently magnetized spheres enclosed in a cylindrical polymer skin. With their shape controlled using an externally-applied magnetic field, the spherical joints of these robots enable them to bend to very small radii of curvature. These robots can be used as steerable tips for endoluminal instruments. A kinematic model is derived based on minimizing magnetic and elastic potential energy. Simulation is used to demonstrate the enhanced steerability of these robots in comparison to magnetic soft continuum robots designed using either distributed or lumped magnetic material. Experiments are included to validate the model and to demonstrate the steering capability of ball chain robots in bifurcating channels.
09:00-10:40, Paper WePO1S-03.2
Authors: Lawson, Jared; Chitale, Rohan; Simaan, Nabil
Abstract: Small catheters undergo significant torsional deflections during endovascular interventions. A key challenge in enabling robot control of these catheters is the estimation of their bending planes. This paper considers approaches for estimating these bending planes based on bi-plane image feedback. The proposed approaches attempt to minimize error between either the direct (position-based) or instantaneous (velocity-based) kinematics with the reconstructed kinematics from bi-plane image feedback. A comparison between these methods is carried out on a setup using two cameras in lieu of a bi-plane fluoroscopy setup. The results show that the position-based approach is less susceptible to segmentation noise and works best when the segment is in a non-straight configuration. These results suggest that estimation of the bending planes can be accompanied with errors under 30◦. Considering that the torsional buildup of these catheters can be more than 180◦, we believe that this method can be used for catheter control with improved safety due to the reduction of this uncertainty.
09:00-10:40, Paper WePO1S-03.4
Authors: Watson, Connor; Nguyen, Anna; Morimoto, Tania K.
Abstract: In this work, we address the problem of robust segmentation of a continuum robot from images without the need for training data or markers. We present a method that leverages information about the kinematics of these robots to produce an estimate of the robot shape, which is refined through optimization over global image statistics. Our approach can be straightforwardly applied to any continuum robot design and is able to handle partial occlusions of the robot body, as well as challenging background conditions. We validate our method experimentally for a concentric tube robot in a simulated surgical environment and show that our method significantly outperforms a naive projection of the robot shape and color thresholding, which is commonly used in current vision-based estimation algorithms for these robots. Overall, this work has the potential to improve the viability of vision-based state estimation for continuum robots in real-world settings.
09:00-10:40, Paper ThPO1S-12.4
Authors: Fan, Yeman; Liu, Dikai
Abstract: Obtaining the shape and size of a robot’s workspace is essential for both its design and control. However, determining the accurate workspace of a multi-segment continuum robot by graphic or analytical methods is a challenging task due to its inherent flexibility and complex structure. Existing numerical methods have limitations when applied to a continuum robot. This paper presents an Equivalent Two Section (ETS) method for calculating the workspace of multi-segment continuum robots. This method is based on the forward kinematics and a piecewise constant curvature (PCC) model to determine the boundaries of the workspace. In order to verify the proposed method, simulation experiments are conducted using six different maximum bending angles and seven different number of segments. Results of the ETS method are compared to the true workspaces of these configurations estimated by an exhaustive approach. The results show that the proposed ETS method is both efficient and accurate, and has small estimation errors. Discussions on the advantages and limitations of the proposed ETS method are also presented.
15:00-16:40, Paper ThPO2S-23.1
Authors: Luo, Xiaolei; Lin, Jui-Te; Morimoto, Tania K.
Abstract: Haptic feedback can provide operators of hand-held robots with active guidance during challenging tasks and with critical information on environment interactions. Yet for such haptic feedback to be effective, it must be lightweight, capable of integration into a hand-held form factor, and capable of displaying easily discernible cues. We present the design and evaluation of HaPPArray — a haptic pneumatic pouch array — where the pneumatic pouches can be actuated alone or in sequence to provide information to the user. A 3x3 array of pouches was integrated into a handle, representative of an interface for a hand-held robot. When actuated individually, users were able to correctly identify the pouch being actuated with 86% accuracy, and when actuated in sequence, users were able to correctly identify the associated direction cue with 89% accuracy. These results, along with a demonstration of how the direction cues can be used for haptic guidance of a medical robot, suggest that HaPPArray can be an effective approach for providing haptic feedback for hand-held robots.
]]>As you are reading this on our blog, you are already aware of our initiative called the OpenCR Project. With this week’s post, we are releasing our open actuation module ^{1} for the continuum robotics community to build torque-controlled continuum robots. The actuation module is capable of measuring the external torque as demonstrated with very simplified toy examples. Its versatile and modular characteristics are highlighted by building three different types of continuum robot prototypes. The presented prototypes are capable of torque control and fast trajectory tracking making them the first of a kind of the respective continuum robot type.
Stepping aside, what problem are we actually addressing? In short – the lack of torque-controlled continuum robot prototypes available to all researchers. In fact, this lack is grounded on two observations we made by observing our research community and relating to other research areas in robotics.
First, in our previous study ^{2}, we observed that precious time and focus are devoted to building custom prototypes. One can acknowledge that experiments on physical hardware in robotics are necessary and valuable for evaluations. However, over 60% of all prototypes are used for only a single publication. This demonstrates an inefficient habit of not reusing existing prototypes. Thus, a significant amount of time is devoted to creating proprietary hardware and software. This large and growing number of single-use prototypes imposes a barrier for the research community shifting away time and efforts from important research questions.
Second, on a more personal note, having the opportunity ^{3}, ^{4} to regularly work with torque-controlled manipulators such as the Franka Emika Panda^{5}, utilizing torque control for continuum robots seems to be a natural next stepping stone. Torque control has a rich history, especially for serial kinematic mechanisms ^{6} ^{7} ^{8} ^{5}. However, for some reason, the spread and success of torque-control used in pHRI (physical human-robot interaction) research has not reached our continuum robotics community. Perhaps, one reason is that not much time has been directed to build a continuum robot prototype capable of torque control.
To provide torque-controlled continuum robot prototypes, we propose a modular actuation module consisting of a high-torque brushless electric motor, a high-resolution optical encoder, and a low-gear-ratio transmission. This choice of hardware allows for torque control and fast trajectory tracking in real time. Furthermore, it enables proprioceptive torque sensing by measuring motor currents without relying on additional sensors, e.g., tension sensors or load cells. In addition, the actuation module and its coupling system can be used in different actuation modes for a wide range of continuum robots. For instance, tendon displacement/tension and backbone rotation/torque can be controlled and measured.
Recapping the technological trend towards torque-controlled robot systems in other research fields, we are forecasting a likely impact of the proposed actuation module and continuum robot prototypes built on it. The manuscript ^{1} takes initial steps towards advanced control methodologies. We are confident that the ability of continuum robots utilizing our actuation module (or one with similar capabilities) has a decisive advantage over previous existing prototypes. For example, a CTCR can detect external disturbance as shown here:
The actuation module is modular and as such allows you to build any (!?) continuum robot. In the nature of the OpenCR project, we build three different types of continuum robots – a parallel CR, a spatial TDCR, and a CTCR. The image below shows renderings and respective prototypes. By providing open-source hardware and software, we hope that this minimizes the repetitive tasks of developing prototypes, allowing researchers worldwide to focus on the main research challenges in continuum robotics. As a welcome side effect, robot-based evaluations can be performed quickly and become reproducible by other groups!
To improve our prototypes, we took great inspiration from the Open Dynamic Robot Initiative, where efforts are made to build cost-efficient actuators for torque-controlled legged robots ^{9} and manipulators ^{10}. As a matter of fact, open robot hardware projects proliferate progress in robotics ^{11}. Therefore, we encourage you – dear reader – to participate in the development of open-source software and hardware as well as to join the journey of leveraging torque-controlled prototypes. Let’s pave the way with new stepping stones toward untapped or under-investigated research directions.
We hope that the open source nature will bear fruit by opening untapped or under-investigated research directions related to dynamics and advanced control of continuum robots.
Reinhard M. Grassmann, Chengnan Shentu, Taqi Hamoda, Puspita Triana Dewi, Jessica Burgner-Kahrs. “Open Continuum Robotics – One Actuation Module to Create them All,” 2023, arXiv: 2304.11850 ↩ ↩^{2} ↩^{3}
R. M. Grassmann, S. Lilge, P. Le, and J. Burgner-Kahrs, “CTCR prototype development: An obstacle in the research community?” in Robotics Retrospectives Workshop at RSS, 2020. Online ↩
Sami Haddadin, Lars Johannsmeier, Johannes Schmid, Tobias Ende, Sven Parusel, Simon Haddadin, Moritz Schappler, Torsten Lilge, and Marvin Becker. “roboterfabrik: A pilot to link and unify german robotics education to match industrial and societal demands.” International Conference on Robotics and Education, 2018. DOI: 10.1007/978-3-319-97085-1_1 ↩
Kristy Strauss, “‘It’s a really cool place’: U of T Mississauga students get hands-on experience in new robotics teaching lab.” Accessed: April 25, 2023 Online ↩
S. Haddadin, S. Parusel, L. Johannsmeier, S. Golz, S. Gabl, F. Walch, M. Sabaghian, C. Jaehne, L. Hausperger, and S. Haddadin, “The Franka Emika Robot: A reference platform for robotics research and education,” IEEE Robotics & Automation Magazine, vol. 29, no. 2, pp. 46-64, 2022. DOI: 10.1109/MRA.2021.3138382 ↩ ↩^{2}
J. Luh, W. Fisher, and R. Paul, “Joint torque control by a direct feedback for industrial robots,” Transactions on Automatic Control, vol. 28, no. 2, pp. 153–161, 1983. DOI: 10.1109/TAC.1983.1103215 ↩
L. E. Pfeffer, O. Khatib, and J. Hake, “Joint torque sensory feedback in the control of a PUMA manipulator,” IEEE Transactions on Robotics and Automation, vol. 5, no. 4, pp. 418–425, 1989. DOI: 10.1109/70.88056 ↩
R. Bischoff, J. Kurth, G. Schreiber, R. Koeppe, A. Albu-Schäffer, A. Beyer, O. Eiberger, S. Haddadin, A. Stemmer, G. Grunwald, and G. Hirzinger, “The kuka-dlr lightweight robot arm-a new reference platform for robotics research and manufacturing,” in International Symposium on Robotics, 2010, pp. 1–8. online ↩
F. Grimminger, A. Meduri, M. Khadiv, J. Viereck, M. Wüthrich, M. Naveau, V. Berenz, S. Heim, F. Widmaier, T. Flayols, J. Fiene, A. Badri-Spröwitz, and L. Righetti, “An open torque-controlled modular robot architecture for legged locomotion research,” IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 3650–3657, 2020. DOI: 10.1109/LRA.2020.2976639 ↩
M. Wüthrich, F. Widmaier, F. Grimminger, J. Akpo, S. Joshi, V. Agrawal, B. Hammoud, M. Khadiv, M. Bogdanovic, V. Berenz, J. Viereck, M. Naveau, L. Righetti, B. Schölkopf, and S. Bauer, “Trifinger: An open-source robot for learning dexterity,” in Conference on Robot Learning, 2020. arXiv: 2008.03596 ↩
V. V. Patel, M. V. Liarokapis, and A. M. Dollar, “Open robot hardware: Progress, benefits, challenges, and best practices,” IEEE Robotics & Automation Magazine, pp. 2–9, 2022. DOI: 10.1109/MRA.2022.3225725 ↩
Looking at the history of continuum robotics, early research has focussed on larger scale continuum robots. Prototypes were built comparable in size to industrial serial arm robots. This probably resulted in expectations in terms of performance. If an industrial robot arm could achieve superior accuracy and repeatabilty with high payloads and at high speed, why would one need a continuum robot performing worse on this scale? Repeatability, accuracy, speed, and payload are clearly not unique selling points for continuum robots. Poised by the third industrial revolution in the late sixties and early seventies of the 20th century, roboticists naturally focussed on robots applicable to automation and its goals. Continuum robots were just not cutting it (yet). Consequently, continuum robotics remained a niche research area for a while.
Medical robotics was just on the onset in the mid eighties and was using repurposed industrial robots. The first medical robotics procedure used the PUMA 560 robot to perform a neurosurgical biopsy at the National Hospital for Neurology and Neurosurgery in London in 1985. The robot guided a needle into the brain to take a tissue sample for examination. Fun fact: The Puma 560 design is based on an earlier robot design by Victor Scheinman, the same Victor Scheinman who developed the first continuum robot.
The trend towards more minimally invasive surgey and ultimatley keyhole access to operate promises benefits for patient care, such as shorter hopsitalization and quicker recovery times. While industrial style robots certainly play an important role in medical robotics today, their large footprint and size naturally places them beside physicians and outside of the patient. But as soon as one wants to move away from large incisions and the need to open the body, thin, versatile, and dextrous robotics enters the scene. Small-scale continuum robots could sneak into a patient with more dextrous motion capabilties than conventional robots and the ability to reach surgical sites inaccessible by conventional robots or standard medical instruments! And that ability is more defining than accuracy and payload. In contrast to industrial automation, small-scale continuum robots promise significant benefit over serial robot arms!
With medical applications, continuum robots found a compelling use case. Motivated by unsought of minimally-invasie approaches and the ability to navigate within the human body through natural orifices or miniscule inscisions, research in continuum robotics excelled. Over the past decades we have witnessed a vast surge in medical continuum robots. In fact, the majority of continuum robot papers draw their motivation from medical applications. The prominent surveys in the field of continuum robotics are centered around medicine^{1} ^{2} ^{3} and I refer the interested reader to those surveys to learn about the breadth and depth of medical continuum robotics research.
One may argue that continuum robots found their sweet spot in the medical application domain. While this is certainly the case, the untapped potential of continuum robots in non-medical applications is enormous! Think about autonomous small-scale or medium-scale continuum robots in in-situ and non-destructive inspection, maintenance, repair, and operations.
Example: On-wing jet engine inspection provides an example of a keyhole procedure, where a technician inspects components such as turbine blades and the combustion chamber using a flexible camera, called a borescope, through tiny access points on the engine. This is a cumbersome procedure requiring an expert to fiddle with the borescope in hunched body postures, with limited time, as the plane stands ready for takeoff. If an issue is overlooked, e.g., a small crack in a blade, it may be disastrous and result in a worst-case scenario of the engine blowing off during flight. At the same time, full service of an engine means that it must be taken off-wing for maintenance – a costly endeavour. Only very few researcher worldwide look at this problem, with the University of Notthingham spearheading research with Rolls Royce on robotics for in-situ repair
This in-situ inspection is just one of many examples where we can build on our expertise in continuum robotics in the medical domain and innovate the next generation of autonomous continuum robot systems. In fact, non-medical applications for continuum robots are characterized by similar challenges:
At the same time, turning to new application domains opens new research questions challenging to solve:
I envision a new generation of continuum robots that are of small and medium scale, dextrous enough to enter an environment for in-situ inspection, maintenance, and repair through multiple keyholes, which act autonomously and are able to collaborate with one another to achieve the task.
I believe, that we as the continuum robotics community, should turn to non-medical use-cases to invent this next-generation of continuum robot systems with autonomous capabilities. By focussing on medical applications, we have naturally limited our design choices to certain continuum robot dimensions, materials, workspaces, control paradigms, etc. that are informed by anatomical constraints and clinical requirements. In fact, most medical continuum robots are rather short if they can are designed to work in free spaces. Longer medical continuum robots need environmental constraints for steering - such as vasculature for robotic catheters.
Drawing motivation from one application domain lead to narrow thinking and limited creativity. Continuum robotics researchers and innovators have become too focused on the specific needs and constraints of medicine. I argue that this prevented us from considering new and unconventional ideas!
Overall, while focusing on medicine as the dominant application domain was necessary to develop expertise in continuum robotics, it is important for the continuum robotics community to maintain a broad perspective and engage with other fields and industries to encourage innovation and avoid limiting the potential for new and unconventional ideas.
J. Burgner-Kahrs, D.C. Rucker, and H. Choset: Continuum Robots for Medical Applications: A Survey. IEEE Transactions on Robotics, 31(6), pp.1261–1280. doi: 10.1109/TRO.2015.2489500 ↩
T. da Veiga, J.H. Chandler, P. Lloyd, G. Pittiglio, N.J. Wilkinson, A.K. Hoshiar, R.A. Harris, and P. Valdastri: Challenges of continuum robots in clinical context: a review. Progress in Biomedical Engineering, 2(3): 032003, 2020. doi: 10.1088/2516-1091/ab9f41 ↩
P.E. Dupont, N. Simaan, H. Choset, and D.C. Rucker: Continuum robots for medical interventions. Proceedings of the IEEE, 110(7):847-870, 2022. 10.1109/JPROC.2022.3141338 ↩
Let’s recap: When we first looked at tendon-driven continuum robots, we looked at the most general design, i.e. with tendon routing disks at equidistant spacing afixed to a a flexible backbone. Tendons terminate at the end disk of each segment to actuate it. Most generally, these tendons are routed in parallel to the central backbone. For spatial bending either 3 separate tendons or 2 antagonistic tendon pairs are used. To achieve a tendon-driven continuum robot segment capable of all motion primitive, we will expand our understanding of tendon routing disks and its implications.
Let’s conceptually think about tendon routing disks. They provide channels to route the tendon actuating a continuum robot. We depict four conceptual tendon routing disk types below. The dotted arrows indicate the degree of freedom of the disks themselves and the solid arrows indicate the motion primitives achievable. We omit the tendon themselves in the drawings.
The most common type is afixed to a central backbone and routes the tendons with a fixed distance parallel to the backbone. We refer to this type of tendon routing disk as
Fixing the tendon routing disks to the central backbone is the most restrictive as they only allow for bending. Allowing the tendon routing disks to freely translate along the central backbone allows for extension.
Most realizations of this tendon routing disk type ensure equidistant spacing of the disks as the backbone extends and contracts. This is either achieved by compression springs between disks or magnets within the disks alternating polarity for repulsion.
Conceptually, tendon routing disks may also be freely rotating about the central backbone, but constrained in position along the backbone. Such a tendon routing would enable bending but also twist as the tendon route is not constrained to being parallel.
We fabricated a tendon-driven continuum robot segment using type-II disks. The end disk is rigidly attached to the backbone rod which can rotate about its axis. The base disk is fixed in orientation. The image series below shows different rotation angles $\alpha$.
Ultimately, the least restrictive tendon routing disk is free to rotate and translate along the central backbone. Consequently, the tendon routing can allow for parallel and non-parallel tendon paths for bending and twisting, while also allowing for extension.
Building on our extensible tendon-driven continuum robot design, we have fabricated a tendon-driven continuum robot segment with type-III disks. The image series below shows the prototype at various backbone rotation angles and applied tendon loads. The flag at the tip indicated the central backbone rotation angle. In addition to bending a twist can be observed.
The image series below shows the type-III tendon disk prototype contracting while also adjusting the rotation angle.
Further Reading: You want to read more on tendon routing disk types and the prototypes shown above? Check out our open access paper in Frontiers in Robotics & AI^{1}.
R.M. Grassmann, P. Rao, Q. Peyron, J. Burgner-Kahrs: FAS—A Fully Actuated Segment for Tendon-Driven Continuum Robots. Frontiers in Robotocs and AI, 9:873446, 2022. doi: 10.3389/frobt.2022.873446 ↩
robotindependentmapping(kappa, phi, ell, ptsperseg)
which generates a framed backbone curve using constant curvature arc parameters and the desired number of points (per segment) as input.
function [fig] = draw_ctcr(g,tube_end,r_tube,options)
% DRAW_CTCR Creates a figure of a concentric tube continuum robot (ctcr)
%
% Takes a matrix with nx16 entries, where n is the number
% of points on the backbone curve. For each point on the curve, the 4x4
% transformation matrix is stored columnwise (16 entries). The x- and
% y-axis span the material orientation and the z-axis is tangent to the
% curve.
%
% INPUT
% g(n,16): backbone curve with n 4x4 transformation matrices reshaped into 1x16 vector (columnwise)
% tube_end(1,m): indices of g where ctcr tubes terminate
% r_tube(1,m): radii of tubes
% options:
% tipframe (shows tip frame, default true/1)
% baseframe (shows robot base frame, default false/0)
% projections (shows projections of backbone curve onto
% coordinate axes, default false/0)
% baseplate (shows robot base plate, default false/0)
%
We provide an example file ctcr_draw_example.m
to create a figure of a CTCR with three tubes. The example also makes use of the new utility function robotindependentmapping(kappa, phi, ell, ptsperseg)
. This function takes constant curvature parameters and generates a backbone curve. This backbone curve $g$ is in the format required by draw_ctcr(g,tube_end,r_tube,options)
and draw_tdcr(g,seg_end,r_disk,r_height,options)
.
function g = robotindependentmapping(kappa, phi, ell, ptsperseg)
% ROBOTINDEPENDENTMAPPING creates a framed curve for given configuration parameters
%
% EXAMPLE
% g = robotindependentmapping([1/40e-3;1/10e-3],[0,pi],[25e-3,20e-3],10)
% creates a 2-segment curve with radius of curvatures 1/40 and 1/10
% and segment lengths 25 and 20, where the second segment is rotated by pi rad.
%
% INPUT: configuration parameters
% kappa (nx1): segment curvatures
% phi (nx1): segment bending plane angles
% l (nx1): segment lengths
% ptsperseg (nx1): number of points per segment
% if n=1 all segments with equal number of points
% OUTPUT: backbone curve
% g (n,16): backbone curve with n 4x4 transformation matrices reshaped into 1x16 vector (columnwise)
%
Find out more and test CRVisToolkit on GitHub
Happy continuum robot visualization!
The CRVisToolkit stems from the Continuum Robotics Laboratory MATLAB codebase that grew over the past decade of our research activities. Numerous students contributed in one way or the other to the CTCR visualization and are acknowledged here in no particular order: Josephine Granna, Sven Lilge, and Malte Bormann.
]]>Sears & Dupont refer to the idea as
a steerable needle technology using curved concentric tubes^{1}
and Webster, Okamura & Cowan refer to the idea as
continuously ﬂexible snake-like robots, called active cannulas, that consist of several telescoping pre-curved superelastic tubes^{2}
What is less known, is the fact that the concept of a robotic device using concentric precurved tubes was already introduced one year earlier, in 2005, by Furusho et al.^{3} at the IEEE International Conference on Mechatronics & Automation. They provide a conceptual drawing of what they call a
curved multi-tube catheter^{3}
consisting of an outer and inner guide, both precurved elastic tubes, and a third innermost tube/needle. Furusho et al. state the kinematics of this robotic device and present a motorized prototype that they teleoperated.
This is not to say that Furusho et al. was not recognized. Their work has been acknowledged, but just not as much. The number of citations (obtained from Google Scholar on Jan 27, 2023) show a quite drastic difference. It is hard to tell, and would be pure speculation, why the two papers from IROS 2006 are more often cited than Furusho et al. And yet, as we are all standing on the shoulders of giants in academia, we should not forget those who paved the way and published innovative ideas first.
In his PhD thesis, available open-access here, Hunter Gilbert provides a thorough comprehensive historical perspective on the origin of curved mutli-tube medical tools and how those relate to concentric tube continuum robots in Section 1.1. This section is also part of Gilbert, Rucker & Webster^{4} review paper on concentric tube continuum robots.
P. Sears & P. Dupont: A Steerable Needle Technology Using Curved Concentric Tubes. IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2850-2856, 2006. doi: 10.1109/IROS.2006.282072 ↩ ↩^{2}
R. J. Webster, A. M. Okamura, N. J. Cowan: Toward Active Cannulas: Miniature Snake-Like Surgical Robots. IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2857-2863, 2006. doi: 10.1109/IROS.2006.282073. ↩ ↩^{2}
J. Furusho, T. Ono, R. Murai, T. Fujimoto, Y. Chiba, H. Horio: Development of a curved multi-tube (CMT) catheter for percutaneous umbilical blood sampling and control methods of CMT catheters for solid organs. IEEE International Conference Mechatronics and Automation, pp. 410-415, 2005. doi: 10.1109/ICMA.2005.1626582 ↩ ↩^{2}
H.B. Gilbert, D.C. Rucker, R.J. Webster III: Concentric Tube Robots: The State of the Art and Future Directions. International Symposium on Robotics Research, pp. 253–269, 2016. https://doi.org/10.1007/978-3-319-28872-7_15. ↩
Under the assumption that the robot’s shape is composed of constant curvature segments, it can be expressed as a concatenation of multiple circular arcs. Each circular arc corresponds to one segment \(i\) and each segment is described by its length \(\ell_{i}\), curvature \(\kappa_{i}\), and bending plane angle \(\phi_{i}\). To determine the robot dependent mapping for a CTCR, we need to identify the number \(m\) of segments present. As a segment is characterized by constant curvature, we have to locate transitions points along the robot where the curvature changes.
We identify a tube within the CTCR with index \(j \in \lbrack 1,N_{t}\rbrack\) where \(N_{t}\) is the number of tubes, \(j = 1\) is the innermost tube, and \(j = N_{t}\) is the outermost tube. Each tube is characterized by it length \(L_{j} = L_{j}^{s} + L_{j}^{c}\) where \(L_{j}^{s}\) is the length of the straight section and \(L_{j}^{c}\) is the length of the curved section (see Figure 5 left). We assume that the curved section has constant curvature \(\kappa_{j}^{*}\). The precurvature of a tube is deﬁned in the xz-plane, where the z axis is tangential to the tube’s straight section pointing to the tip. Furthermore, each tube’s outer \(\text{OD}_{j}\) and inner \(\text{ID}_{j}\) diameter are specified. Lastly, the modulus of elasticity \(E_{j}\) (Young’s modulus) is determined by the tube’s material. Superelastic NiTi tubes typically come with elastic modulus of \(28 - 83\) GPa.
The CTCR joint space is given as \(\mathbf{q} = \lbrack\alpha_{1},\ldots,\alpha_{N_{t}},\beta_{1},\ldots,\beta_{N_{t}}\rbrack\) where \(\alpha_{j} \in \lbrack - \pi,\pi)\) is the rotation angle about the z-axis and \(\beta_{j} \in \lbrack - L_{j},0\rbrack\) is the translation along the z-axis of tube \(j\). If all \(\alpha_{j} = 0\), the centreline stays in the x-z-plane and curves toward the positive x axis. The translational joint variables are subject to the following constraints
\[\beta_{1} \leq \ldots \leq \beta_{N_{t}} \leq 0\]and
\[L_{N_{t}} + \beta_{N_{t}} \leq \ldots \leq L_{1} + \beta_{1}\]These arise from the telescoping nature of the tube assembly and the actuation principle, i.e., each tube can be retracted at maximum by its own length. Furthermore, we require the tubes to extend one another outside the actuation unit and to not be fully retracted within any more outer tube. In the robot’s zero position, all tubes are fully retracted and align with the front plate of the actuation unit. Below, we illustrate the joint space of a CTCR with \(N_{t} = 3\) and the base coordinate frame.
The robot dependent mapping is based on the principles of beam mechanics to solve the forward kinematic problem. We assume the tubes to deform only in bending and not in twisting, such that the resulting centerline curvature can be computed in closed-form based on a moment equilibrium by applying Bernoulli–Euler beam mechanics.
To describe the complete shape of a CTCR, we will ﬁrst develop a model for the shape of a single segment composed of several fully overlapping concentric curved tubes. Let’s first consider the example depicted here.
For a collection of two tubes with circular precurvatures \(\kappa_{1}^{*}\) and \(\kappa_{2}^{*}\), once inserted into each other, the resultant constant curvature \(\kappa\) is common to both tubes. Here, the tubes have not been axially rotated with respect to one another and their natural curvature planes are aligned. The bending moments \(M_{j}\) (in Nm) will be constant along the length of each tube and can be described by
\[M_{1} = (\kappa - \kappa_{1}^{*})EI_{1}\]and
\[M_{2} = (\kappa - \kappa_{2}^{*})EI_{2}\]where \(E\) is the Young’s modulus (in Pa), and \(I_{1}\) and \(I_{2}\) are the second area moment (in \(m^{4}\)) of the symmetric cross section of the respective tube. The second area moment \(I\) is a geometrical property of the tube’s cross section area which reflects how its points are distributed with regard to the central axis. As a tube’s cross section is an annulus, the second area moment can be determined from the inner and outer diameter as
\[I_{j} = \frac{\pi}{64}(\text{OD}_{j}^{4} - \text{ID}_{j}^{4})\]We can describe the resultant curvature of these two overlapping tubes by evaluating the moment equilibrium
\[M_{1} + M_{2} = 0\]and resolving for
\[\kappa = \frac{EI_{1}\kappa_{1}^{*} + EI_{2}\kappa_{2}^{*}}{EI_{1} + EI_{2}}\]This can be generalized to collections of \(N_{t}\) axially aligned tubes
\[\kappa = \frac{\sum_{j = 1}^{N_{t}}EI_{j}\kappa_{j}^{*}}{\sum_{j = 1}^{N_{t}}EI_{j}}\]Let’s now consider the example depicted here. When precurved tubes are rotated axially, their natural planes of curvature are no longer aligned and the direction of the bending moments they apply changes. The tubes exert a moment on one another about their respective y-axes, which is caused by their initial precurvature about this direction. The tubes are rotated by angles \(\alpha_{1}\) and \(\alpha_{2}\), and the equilibrium plane angle is \(\phi\). As before, each tube experiences a constant bending moment \(M_{1} = (\kappa - \kappa_{1}^{*})EI_{1}\) and \(M_{2} = (\kappa - \kappa_{2}^{*})EI_{2}\), but now these bending moments have two component projections on the base frame x and y axes. The equilibrium state \(M_{1} + M_{2} = 0\) yields
\(\begin{bmatrix} \kappa_{x} \\ \kappa_{y} \\ \end{bmatrix} = \frac{1}{EI_{1} + EI_{2}}\begin{pmatrix} EI_{1}\kappa_{1}^{*}\lbrack\begin{matrix} \cos\alpha_{1} \\ \sin\alpha_{1} \\ \end{matrix}\rbrack + EI_{2}\kappa_{2}^{*}\lbrack\begin{matrix} \cos\alpha_{2} \\ \sin\alpha_{2} \\ \end{matrix}\rbrack \\ \end{pmatrix}\).
For general number of concentric tubes, this can be generalized to
\[\begin{bmatrix} \kappa_{x} \\ \kappa_{y} \\ \end{bmatrix} = \frac{1}{\sum_{j = 1}^{N_{t}}EI_{j}}\overset{N_{t}}{\sum_{j = 1}}EI_{j}\kappa_{j}^{*}\begin{bmatrix} \cos\alpha_{j} \\ \sin\alpha_{j} \\ \end{bmatrix}(Eq. 1)\]Given the components of the resultant curvature, we can determine the bending plane angle \(\phi = \text{atan2}(\kappa_{y},\kappa_{x})\) and curvature \(\kappa = \sqrt{\kappa_{x}^{2} + \kappa_{y}^{2}}\). Note, that \(\phi\) is undefined for straight segments, i.e., for zero curvature as \(\text{atan}2(0,0) = \text{nan}\). In this special case, the rotation angle \(\alpha\) of the outermost present tube in the segment is used as \(\phi\).
The first step in determining the robot dependent mapping of a CTCR is determining the number of segments \(m\) and their lengths \(\ell_{i}\) for \(i \in \lbrack 1,m\rbrack\). These are deﬁned by transition point locations, where the component tubes either end or transition from their straight to curved section. The set \(T\) of transition points can be determined as algebraic functions of the tube geometry and the translational joint positions \(\beta_{j}\)
\[T = \text{sort}_{\text{asc}}\{ 0,\beta_{n} + L_{n}^{s},\beta_{n} + L_{n},\ldots\beta_{1} + L_{1}^{s},\beta_{1} + L_{1}\}\]From the resulting values of the set T, all duplicate elements are omitted, i.e., tube sections terminate at the same location, as well as all elements \(< 0\), i.e., the respective tube section is located within the actuation unit. While the maximum number of segments is \(2N_{t}\), the number of actual segments is dependent on the current joint values \(\mathbf{q}\) as some segments might be located within the actuation unit. Here, we show an example of a CTCR composed of three tubes and its transition points.
Using the transition points T, the length of the constant curvature segments of the CTCR can be determined as
\[\begin{matrix} \ell_{1} & = T_{1} - T_{0} \\ \ell_{2} & = T_{2} - T_{1} \\ \ldots & \\ \ell_{m} & = T_{m} - T_{m - 1} \\ \end{matrix}\]In a second step, we can determine \(\kappa_{x}\) and \(\kappa_{z}\) for each segment \(i \in \lbrack 1,m\rbrack\) enclosed by the transition points \(\lbrack T_{i - 1},T_{i}\rbrack\) using Equation (\(1\)), and then \(\kappa_{i}\) and \(\phi_{i}\). Note, that not all tubes are present in each segment, such that only the present ones with their respective parameters have to be considered (either zero curvature in straight tube sections or the respective precurvature in curved sections, alongside ID and OD, as well as \(E\)).
The constant curvature kinematics franmework assumes that \(\phi\) is expressed locally, i.e., w.r.t. the previous segment for the robot independent mapping. Consequently, after determining \(\phi_{i}\) expressed in the base frame for each segment using Equation (\(1\)), we will need to apply a final correction step
\(\phi_{i} = \phi_{i} - \phi_{i - 1}\).
The model introduced today makes multiple simplifying assumptions, i.e., pure bending, no torsional deformation, no friction, no effect of gravitation and external forces. The assumption of infinite torsional rigidity in the tubes is only reasonable as long as the relative rotation angles between the tubes are small and the straight sections of each tube are short. Considering torsional deformation is particularly important in the straight transmission sections of the tubes that lie between the actuators and the first curved link, since these transmissions are long relative to the curved sections. In fact, despite the simple actuation of component tubes, the resulting motion of CTCR is characterized by a highly non-linear behaviour due to the elastic interactions between the tubes.
An advanced modelling approach is to leverage the elasticity theory, in particular the Cosserat theory of elastic rods. This involves writing rod equation for each tube, and then enforce concentricity by requiring all tubes to conform to the same curvature as a function of arc-length, leaving them free to rotate axially with respect to each other. This results in a system of differential equations with mixed boundary conditions. The boundary conditions at the base of the robot are the axial angles of the tubes, and the boundary conditions at the tip are internal moments that vanish because there is no material beyond the tip to support them. Note that after this mechanics problem is solved, to determine the axial tube angles, one must still integrate along the robot to determine the space curve of the robot itself. Experimental testing of the model has shown that with calibration, mean error in the prediction of tip position can be as low as 1% to 3% of overall robot length. With typical CTCR length of 200-300mm, this means that we can estimate the pose of the tip with an error of about 2-9 mm. This is the most accurate model existing for CTCR to date. As this 101 category of the OpenContinuumRobotics Blog is an introduction to continuum robotics, we do not cover advanced modelling approaches for CTCR (yet!).
Webster, R.J. & Jones, B.A., 2010. Design and Kinematic Modeling of
Constant Curvature Continuum Robots: A Review. The International Journal
of Robotics Research, 29(13), pp. 1661–1683.
https://doi.org/10.1177/0278364910368147
Section 3.2.3
Mahoney, A.W., Gilbert, H.B., Webster, R.J., 2018. A Review of Concentric Tube Robots: Modeling, Control, Design, Planning, and Sensing. The Encyclopedia of Medical Robotics, Chapter 7, pp. 181-202. https://doi.org/10.1142/9789813232266_0007
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