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Studying slippage on pushing applications with snake robots
 Fabian Reyes^{1}View ORCID ID profile and
 Shugen Ma^{1}Email author
 Received: 8 September 2017
 Accepted: 24 October 2017
 Published: 2 November 2017
Abstract
In this paper, a framework for analyzing the motion resulting from the interaction between a snake robot and an object is shown. Metrics are derived to study the motion of the object and robot, showing that the addition of passive wheels to the snake robot helps to minimize slippage. However, the passive wheels do not have a significant impact on the force exerted onto the object. This puts snake robots in a similar framework as robotic arms, while considering special properties exclusive to snake robots (e.g., lack of a fixedbase, interaction with the environment through friction). It is also shown that the configuration (shape) of the snake robot, parameterized with the polar coordinates of the robot’s COM, plays an important role in the interaction with the object. Two examples, a snake robot with two joints and another with three joints, are studied to show the applicability of the model.
Keywords
 Snake robot
 Contact
 Modeling
Background
Robots that are capable of locomotion in unstructured conditions are necessary for realistic applications. However, locomotion alone may not be sufficient when more dexterous interaction with the environment is needed. Therefore, robotic systems with capability to locomote and also interact dexterously with their surroundings are desirable, and indeed a natural extension of robotics research.
Snake robots have shown promise regarding locomotion [1]. Locomotion in planar environments has been probably the main topic of research for snake robots [2–4] and has been extended to motion in planar slopes [5, 6], motion in 3Dspace [7, 8], and more broad studies on locomotion [9]. An interesting idea that combines locomotion and interaction with the environment, called obstacleaided locomotion (OAL), has been proposed in [10] where obstacles in the environment are used as auxiliary sources for propulsion or to avoid jamming.
The lack of a fixedbase makes it difficult for a snake robot to manipulate an object as dexterously as a robotic arm. Another difference is that a snake robot has contact with the environment through friction at several points of its body. Additionally, mass becomes a very important parameter to study. Unlike research regarding robotic arms where it is assumed that the arm can lift the object and it is a matter of choosing an optimal input, snake robots may not be able to move the object due to its inertial properties.
Because the kinematic structure of a snake robot resembles a robotic arm, papers that deal with similar (but not exactly the same) situations can be found in existing literature. In [11] a hyperredundant serial robot was considered and both locomotion and manipulation of an object were considered. However, the analysis was purely kinematic while assuming a fixedbase robotic system. In other words, there was not force analysis showing the conditions for feasibility of the problem. In [12], the duality between locomotion and manipulation of a snake robot was considered under the assumption that the snake robot can be treated similarly to a robotic arm with a fixedbase when manipulating an object. This was achieved by making the first link of the snake robot behave similarly to fixedbase (due to its shape and mass), but the results cannot be extended to the case of a general snake robot. The problem of analyzing and controlling a snake robot under these conditions has been reported in [13], where it is shown that several assumptions made in previous published literature are not enough to guarantee accurate control of a planar snake robot with frictional contacts with the ground.
The main objective of this paper is to study the resulting interaction between a snake robot and an object, when the task is to push the object. We consider this to be a prelude to more interesting interactions like grasping or dexterous manipulation. However, it is important to understand the basics of the interaction first. This is an extension of previously published work [14, 15] where a more complete mathematical modeling of the problem has been presented. In [15], the optimal configurations of the snake robot to maximize the force exerted onto the object have been presented.
This paper focuses on the motion of the system, rather than forces. The main motivation for this study is that, as presented in [13], calculating an optimal input instantaneously (i.e., at one instant of time) is not enough to accurately control the system. We conjecture that understanding how the system will behave over time is also important. In other words, if the task is to push an object, then the motion of the object must be maximized, while the motion of the snake robot minimized.
 1.
All bodies in the system are rigid.
 2.
Contacts between the snake robot and objects or the environment (except the ground) are considered frictionless point contacts.
 3.
The snake robot has passive wheels or any other mechanical means to achieve anisotropic friction between the robot’s belly and the ground.
 4.
There is only one constraint per link of the snake robot.
 5.
The operational space is a plane (embedded in a full 3D space).
 6.
The snake robot has only one contact point with the object to be manipulated.
The paper is organized as follows. In “Mathematical background” section, the necessary mathematical background to understand this paper is presented along references necessary to develop the concepts further. “Motion of the system due to the interaction” section is the main body of the paper; the modeling of the system is presented, and metrics and quantities mentioned in this section are derived. In “Results” section, a specific example is studied to show the application of the proposed metrics. “Discussion and future applications” section includes several comments regarding the scope and limitations of the results presented in this paper. The paper concludes with some remarks in “Conclusion” section.
Mathematical background
In this section, we give a very brief introduction to the mathematical topics necessary to understand this paper. We recommend [20–22] for a more detailed treatment. In particular, the foundations of the model used in this paper have been presented in [15]; readers are encouraged to read this reference for a more detailed treatment of snake robots in the framework of articulatedbodies. As stressed in previous research, it is important to guarantee invariance of metrics in order for the results to be meaningful. Not only to avoid inconsistency of units, but also for the metric to be invariant to a change of coordinates. To derivate the analysis of the system to lead to meaningful metrics, we employ dual vectors [23] and basic differential geometry [21].
Differential geometry: twists, wrenches, and metrics
A twist (concatenation of linear and angular velocity) \({\vec { \varvec{\upsilon }} \in {\text{M}}^{n}}\) can be expressed w.r.t. a covariant basis \({\varvec{e} = [\vec {\varvec{e}}_{1}, \ldots , \vec {\varvec{e}}_{n}]^{\mathrm{T}}}\) as \({\vec { \varvec{\upsilon }} = \varvec{e}^{\mathrm{T}} \varvec{\upsilon }}\). The element \(\varvec{\upsilon }\in {\mathfrak{R}}^{n}\) can be interpreted as the (vector of) contravariant components of \(\vec {\varvec{\upsilon }}\). A wrench (concatenation of linear force and torque) \({\vec { \varvec{f}} \in \text{ F }^{n}}\) can be expressed w.r.t. a contravariant basis \({\varvec{e}^{*} = [\vec {\varvec{e}}_{1}^{*},\ldots ,\vec {\varvec{e}}^{*}_{n}]^{\mathrm{T}}}\) as \({\vec {\varvec{f}} = \varvec{e}^{*^{\mathrm{T}}} \varvec{f}^{*}}\) and \(\varvec{f}^{*} \in {\mathfrak{R}}^{n}\) can be interpreted as the covariant components of \(\vec {\varvec{f}}\).
Unconstrained model of the snake robot
The snake robot has a total of \({n_{s} \in \mathbb {N}}\) degrees of freedom (DOFs), and its generalized coordinates are encapsulated in the vector \(\varvec{q}_{s}(t) \in {\mathfrak{R}}^{n_{s}}\). The snake robot has \(n_{\ell }=n_{a}+1\) links each with mass \(m_{i}\).
Unconstrained model of an object
If all links of the snake robot have the same mass m, then the inertia matrix of the snake robot can be factored as \(\varvec{M}_{s} := m \bar{\varvec{M}}\), and the inertia matrix of an object as \(\varvec{I}_{\mathrm{obj}} = m_{\mathrm{obj}} \bar{\varvec{I}}_{\mathrm{obj}}\), where the new inertia matrices \(\bar{\varvec{M}}\) and \(\bar{\varvec{I}}_{\mathrm{obj}}\) correspond to inertia matrices with unitary mass.
Summary of constraints
Motion of the system due to the interaction
Slippage ratio

\(\hbox {sr} \rightarrow 1\) which implies that the acceleration of the snake robot is minimal (\({\vec {\ddot{\varvec{q}}}_{s}^{2} \ll \vec {\varvec{a}}_{\mathrm{obj}}^{2}}\) or \({\vec {\ddot{\varvec{q}}}_{s}^{2} \approx 0}\)).

\(\hbox {sr} \approx 0.5\) which implies a similar magnitude of acceleration for the two subsystems (\({\vec {\ddot{\varvec{q}}}_{s}^{2} \approx \vec {\varvec{a}}_{\mathrm{obj}}^{2}}\)).

\(\hbox {sr} \rightarrow 0\) which implies that the magnitude of the acceleration of the object is minimal (\({\vec {\ddot{\varvec{q}}}_{s}^{2} \gg \vec {\varvec{a}}_{\mathrm{obj}}^{2}}\) or \({\vec {\varvec{a}}_{\mathrm{obj}}^{2} \approx 0}\)).
Polar coordinates of the COM of the snake robot
Results

Scenario 1: The snake robot is in contact with an object but unconstrained in any other way. The friction between the snake robot and ground is negligible.

Scenario 2: The snake robot contacts one object and has passive wheels in all other links. The friction between the passive wheels and ground is bounded by its limit surface.

Scenario 3: The snake robot is contacting one object and has passive wheels in all other links. The passive wheels impose (unbounded and bilateral) nonholonomic constraints.
Case study 1: Snake robot with two joints
Parameters of the simulation for case study 1
Symbol  Value  Unit  Description 

n  5  Number of DOFs of the system  
\(n_{a}\)  2  Number of actuated joints  
\( {m}_{i}\)  1  (kg)  Mass of \(link_{i}\), \(i=1,\ldots ,n_{\ell }\) 
\(\ell _{i}\)  0.15  (m)  Length of \(link_{i}\), \(1=1,\ldots ,n_{\ell }\) 
\(I_{{\mathrm{com}},i}\)  0.002  (kg m^{2})  Rotational inertia for the ith link 
\(\varvec{\tau }_{\mathrm{act}} = [\tau _{a1}, \tau _{a2}]^{\mathrm{T}}\)  (N m)  Input joint torques  
\(\mu _{s}\)  0.1  Coefficient of (static) friction used for Scenario 2 
The slippage ratio (30) gives quantitative information about the movement of the system and can be studied in the same way as the previous norms. Figure 4c shows the value of the slippage ratio for all three scenarios with several values for \(\kappa \) for one configuration. It can be seen that \(\hbox {sr} \rightarrow 0\) in the region where there is no contact with the object (i.e., the snake robot can move freely and therefore \(\vec {\varvec{a}}_{\mathrm{obj}}^{2} \rightarrow 0\)). It is interesting to see that in all three scenarios it is always possible to make the object move. However, Scenario 2 (nonideal passive wheels) has a limited region where the slippage ratio is high, compared to Scenario 3, where a whole region seems to give high values of slippage ratio. These regions in the input space are highlighted in Fig. 4b, c. Regions where the slippage of the snake robot are minimized tend to have higher slippage ratio.
Case study 2: Snake robot with three joints
The proposed framework and metrics can be applied to a snake robot with any number of joints. In this section, a snake robot with three joints is studied. However, studying the threedimensional input space could be cumbersome. Instead, the norms \(\vec {\varvec{\lambda }}^{2}\), \(\vec {\varvec{a}}_{\mathrm{obj}}^{2}\) and slippage ratio (30) are studied as a function of the polar coordinates of the COM of the snake robot \((COM,\angle COM)\) w.r.t. the contact point, as discussed in previous sections.
Figure 5b reports the result for the norm of the object’s acceleration \(\vec {\varvec{a}}_{\mathrm{obj}}^{2}\). The polar plots report the results for scenarios 1, 2, and 3, respectively. It can be seen that the addition of passive wheels (even ideal ones) have little impact on the acceleration of the object. However, the configuration of the snake robot, parametrized with the polar coordinates of its COM, has a clear and meaningful impact on the acceleration of the object.
Although basic intuition would tell that the addition of constraints (i.e., passive wheels) should have an impact on the force applied to the object (through \(\vec {\varvec{\lambda }}\)), and consequently on its acceleration \(\vec {\varvec{a}}_{\mathrm{obj}}\), this study shows that is not the case (at least, not that simply).
An important addition of this paper w.r.t. [15, 24] is the study of the slippage ratio sr. By studying the relationship between motions of both systems (snake robot and object), we can understand how the additional constraints have an impact on the system. A snake robot without passive wheels will slip as it pushes the object. Therefore, minimizing this motion while keeping a steady force on the object (and therefore producing an acceleration) is desirable. Figure 5c shows the result of the slippage ratio (30). It can be seen that in Scenario 1 (without passive wheels) the same trend as with the object’s acceleration appears. However, passive wheels (even nonideal ones) have a big impact on the slippage ratio (take notice of the change of scale).
Results of the norms
Concept  Scenario 1  Scenario 2  Scenario 3 

Worst \(\vec {\varvec{a}}_{\mathrm{obj}}^{2} \)  0.000763215  0.00107454  0.00107454 
Best \(\vec {\varvec{a}}_{\mathrm{obj}}^{2} \)  0.00785302  0.00861168  0.00861168 
Worst sr  0.00138484  0.00185713  0.00185098 
Best sr  0.0267348  0.988638  0.988643 
Discussion and future applications
Several assumptions have been made in this line of research, especially the number of contacts considered between objects, and their rigidity. This is because we are interested in giving a solution that is mathematically rigorous while guaranteeing uniqueness of solution. To include more contact points means to loose this in favor of robustness. For example, a penalty method (aka. virtual springs) or barrier functions may be considered, which is common for wholearm body manipulation (WAM) tasks. Although our assumptions are restrictive, it allows us to give a solid foundation for the research. Other models or considerations can be used for more realistic scenarios, but the rigidbody assumption used here allows to have a clear basis for comparison. Considering the gaps in knowledge regarding snake robots (as highlighted in “Background” section), we consider the model and results presented to be useful for moving research forward.
The holonomic constraints (e.g., constraints due to joints of the snake robot) are already encoded in the kinematic model of the snake robot presented in “Mathematical background” section. These holonomic constraints are described by the use of the robot’s Geometric Jacobians. Further distinction between holonomic and nonholonomic constraints is not necessary, as both can be expressed in the same unified manner (13), as mentioned in [21, 33].
The metrics and general framework presented can be used to analyze more complex tasks involving snake robots (or similar robotic systems) interacting with an object or the environment. This gives an opportunity to study both snake robots and robotic arms in the same framework, since the analysis is similar to the oftenused force/manipulability ellipsoids used to study robotic arms or hands [34].
However, the metrics presented in previous research do not consider the motion of the robot itself, since a fixedbase was always assumed. The framework presented in this paper can be applied to other mobile systems in a more complete manner than reported in the literature [28, 29]. More specifically, the analysis presented extends the concept of force or manipulability ellipsoids [26, 34], from the case of a fixedbase robot with endeffector, to a mobile robot without an endeffector. For a given task and configuration, an analysis can be carried out to find the optimal input (vector of joint torques) to minimize or maximize slippage of the system.
A few conclusions can be drawn from analyzing norms (26) and (27) on the input space (c.f. Fig. 4). In all scenarios, \(\tau _{2}\) which is the joint furthest away from the object has almost no effect on the acceleration of the object. However, the addition of passive wheels helps to anchor the snake robot and couples the effect of \(\tau _{2}\) on the system.
Conclusion
In this paper, a modeling and analysis framework for snake robots in contact with an external body has been presented. Results show that the addition of passive wheels has little effect on the wrench applied to the object and therefore, little change in its acceleration. However, the passive wheels do have an effect on the motion of the robot itself. In other words, under certain conditions the slippage of the robot can be minimized while pushing the object. This could be beneficial for pushing or manipulation tasks. To the best of our knowledge, this problem has not been fully studied with snake robots.
Declarations
Authors' contributions
FR conducted mathematical analysis, programming to perform the simulations, and wrote the manuscript. SM supervised the research. Both authors read and approved the final manuscript.
Acknowledgements
This study was in part supported by “Strategic Research Foundation Grantaided Project for Private Universities (2013–2017) from Ministry of Education, Culture, Sports, Science and Technology, Japan, and RGIRO (Ritsumeikan Global Innovation Research Organization). We acknowledge and thank their support.
Competing interests
The authors declare that they have no competing interests.
Funding
This research did not receive funding from any particular organization.
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Authors’ Affiliations
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