Arboreal concertina locomotion of snake robots on cylinders

Abstract

This article proposes a novel gait on cylinders for a snake robot as well as arboreal concertina locomotion gait, including the generation method. The gait on the cylinder of a snake robot is a kind of spatial motion planning that is difficult in manual design. In this article, a planar gait of a snake robot is first constructed, followed by cylindrical gait via a transformation mechanism. In addition, a gait generating scheme of arboreal concertina locomotion gait of snake robots on cylinders is constructed and verified via a simulation platform. Such gait makes advantages on cylindrical motion capability of snake robots and the proposed generation approach for gait construction can be promoted to other cylindrical gaits, reducing design difficulty.

Introduction

After millions of years of revolution, animals achieved amazing adaptability which inspired researchers to create new inventions. Snakes are one of the most versatile animals in nature; it can move in diverse environments, such as grasslands, deserts, rivers, and trees. As being simple though, the elongated bodies of snakes enable free movement within various environments by utilizing different gaits.

The artificial structure composed of rods or pipes can be found everywhere in our daily life, and related discovery still poses a challenge in the design and application of robots. Because of its special body shape, snake robot has great advantages in exploring rod or pipe structure. Hatton and Choset proposed the helix rolling gait of pole climbing based on the method of annealed chain fitting and Keyframe wave extraction (ACFKWE). ¹ Kamegawa et al. realized the climbing motion of snake robots with passive wheels on cylinders by analyzing the continuum model of snake robots. ² Melo et al. studied the relationship between climbing speed and geometry of the robot body and cylinders when snake robots climb on a horizontal cylinder. ³ Enner et al. proposed an estimation method for position and posture of snake robots based on Extended Kalman Filter. ⁴ Based on this estimation method, Rollinson and Choset realized the compliant control method, which provides snake robots the abilities of self-adaption of cylinder diameter and blocked stopping. ⁵

Above climbing gaits of snake robots on cylinders were proposed from engineers’ perspectives. The snake robot surrounds the cylinder, which restricts the robot, allowing it to only move on a pole rather than a more complicated structure, such as a truss or a tree with many branches. There are two main gaits of snake moving in trees: arboreal lateral undulation locomotion (ALUL) and arboreal concertina locomotion (ACL). ⁶ ALUL gait has a higher speed of movement but is limited to a driving force add-in due to branches. ACL gait has a better adaptability to environment, including a simple rod and a complex truss structure.

The difficulty of designing gaits on cylinders is the contact surface of snake robots from the horizontal plane onto a curved surface, making the entire snake robot space configuration unimaginable. Currently, gaits generation method can be divided into four categories, sine-based method, model-based method, central pattern generation-based (CPG-based) method, and curve-based method.

Sine-based method uses sine-based function as joint reference angle for snake robots. Based on the observation of motion of snakes on grounds, Hirose accomplished lateral undulation gait with sine functions on the snake robot assembling passive wheels. ⁷ By setting different sine functions as reference angle for pitch and yaw joints, and adjusting the amplitude, frequency, and phase difference of those sine functions, different gaits of snake robots can be obtained, including side-winding gait, ^8,9 rectilinear gait, ^10,11 and helix rolling gait. ¹ The advantages of sine-based method are easy to achieve and there is a strong correlation between parameters of sine function and the motion performance.

Model-based method can improve the motion performance with kinematics and dynamics analysis of robots. Hicks and Ito proposed an optimal control strategy for snake robots based on a series of reduced dynamics equations of robot system. ¹² To realize the obstacle-aided motion without wheels, Transeth et al. developed a hybrid dynamic system for snake robots to simultaneously deal with the contact force from obstacles and friction force using set-value force laws. ¹³ The limitations of model-based method are the dependency on accurate model and the computation complexity increasing with the number of joints.

CPG-based method uses a special neural network or coupled oscillators as gaits generators for snake robots. One interesting feature of this method is the ease of incorporating sensory feedback. Using sensory information, Crespi and Ijspeert realized the amphibious motion of snake robots in water and land and adjusted swing amplitude to obtain the optimal moving speed by Powell’s method. ¹⁴ Tang et al. constructed multi-phase CPG to realize a variety of gaits and designed the optimal gaits selection strategy based on the speed sensory signal. ¹⁵ CPG-based method has great potential in improving the environmental adaptability, but its model design and parameter adjustment still remain a challenge.

Curve-based method is different from abovementioned gaits generation methods. No matter what method used, the sine-based, model-based, or CPG-based method, the control objects are the joint angle or joint torque of snake robots. Curve-based method designs the body curve of snake robots and the mechanism for converting the body curve to the joint angle. Hatton and Choset proposed the approach of ACFKWE. ¹ In this approach, the motion is decomposed into a series of curves, and then the control function of the joint angle is extracted. Liljebäck et al. proposed a control framework to solve the problem of shape control in obstacle-aided motion. ¹⁶ This control framework sets the contact points between the snake robot and obstacle as shape control points (SCP) and then connects these points with Bézier curves. The control function joint angle is obtained by progressing a virtual snake along the body curve. This control framework can also generate lateral undulation gait and concertina gait on ground.

For generation of ACL gait on cylinders, the sine-based, model-based, and CPG-based methods cannot guarantee the shape of snake robots to meet the gait requirements. Liljebäck’s control framework is not suitable for generating ACL gait, because the Bézier curve is a plane curve. The approach of ACFKWE is a possible solution but is suffered from complexity. Therefore, in this article, a novel gait design method for ACL is proposed. The basic idea is to design concertina locomotion (CL) gaits on horizontal planes first and then to translate gaits on planes into ACL gaits on cylinders by transformation mechanism. The process of design CL gaits is similar to the method of ACFKWE. It is easy to build CL gaits since they occur on plane. By coordinate transformation, the mechanism lies in transformation of control functions of joint angles of CL gaits on planes into the control functions of joint angles of ACL gaits on cylinders. The process of coordinate transformation mainly refers to the kinematics of manipulators. ¹⁷ Then, in order to guarantee the feasibility of the proposed ACL gait, the kinematics and dynamics of the gait are analyzed to avoid the geometric interference and slipping from cylinders. Most of the researches about the gaits of snake robots involve kinematics and dynamics analysis, partly for the motion of the snake robots on the plane ^18,19 and partly for the motion on the cylinder. ^4,5 In this article, the force analysis of snake robots winding the cylinder is analyzed to obtain the optimization design parameters of ACL gait.

The article is organized as follows. “Design of CL gait on a plane” section analyzes the CL gait of snake robots on horizontal planes. “Planar–cylindrical gaits transformation mechanism” section presents the transformation mechanism of CL gait on a plane to ACL gait on a cylinder. “Gaits analysis and parameter determination” section introduces the parameters analysis of ACL gait based on kinematic and dynamic models of snake robots. “The design scheme of ACL gait of snake robots” section provides the complete design scheme of ACL gait and its verification with simulation platform. Finally, the last section contains the conclusion and future work.

Design of CL gait on a plane

It is much simpler to plan CL gaits on the plane than to directly plan ACL gaits of snake robots on cylinders. Based on two different winding types, this section designs two different CL gaits. The process of building CL gaits is similar to the method of ACFKWE.

There are two winding types for snake robots on cylinder: H-type winding and S-type winding. In the H-type winding, snake robots wrap around the entire circumference of the cylindrical surface, while the S-type winding does not. The CL gaits of H-type winding and S-type winding are planned first on the plane, and then converted to gaits on the cylinder using a transforming mechanism introduced in “Planar–cylindrical gaits transformation mechanism” section. The sketches of H-type and S-type winding of snake robots are shown in Figure 1.

Figure 1.

Sketches of H-type and S-type winding of snake robots: (a) H-type winding and (b) S-type winding.

The Planning of CL gait of H-type winding on a plane

The body structure of the snake robot was divided into three sections: head section, middle section, and tail section. Head section and tail section were used to affix on the body of snake robots and the length of the middle section is related to the moving distance of each movement period. The length of each link making up the snake robot is equal, so the length of each section is determined by the number of links within each section: the number within the head section n_H, the number within the middle section n_M, and the number within the tail section n_T. The larger the value of n_H and n_T, the safer the snake robot is during climbing. The larger the value of n_M, the longer the moving distance within each period.

There are two types of links; they are segment links and transient links. The orientation of segment links in each segment remained consistent. The transient links connected the adjacent segment links and the orientation of segment links was different from the segment links at the start of every motion period. As shown in Figure 2, solid bar represents segment links and blank bar illustrates transient links.

Figure 2.

H-type winding gait planning on plane.

The period of movement of snake robots is divided into four phases: head opening phase (HO), head folding phase (HF), tail opening phase (TO), and tail folding phase (TF).

As illustrated in Figures 2, steps (b) and (c) are head opening phases. The number of steps in the head opening phase is equal to n_M. In this phase, the snake robot raises parts of its body to move forward. Steps (d) to (f) are head folding phases. The number of steps in the head folding phase is equal to n_H. In this phase, the snake robot affixes itself with tail section and the head section moves forward. Steps (g) to (j) are tail opening phases. The number of steps of the tail opening phase is equal to n_T. In this phase, the snake robot affixes itself with head section, the middle and tail sections move forward. Steps (k) and (l) are part of the tail folding phase. The number of steps in the tail folding phase is equal to n_M. In this phase, the snake robot folds up the links within its tail section and finishes one period movement. If the number of links making up the snake robot is n_B = n_H + n_M + n_T, the number of steps in each movement period is n_n = n_H + 2n_M + n_T.

There are three types of joints: the normal joint (NJ), which is illustrated with blank circles in Figure 2; the transition-start joint (TSJ), which is presented with green circles in Figure 2; and transition-end joint (TEJ), which is shown with red circles in Figure 2.

The angle curve of ith NJ is presented in Figure 3(a). The joint moves the links forward at the peak of the curve and returns to the original direction at the trough of the curve. Figure 3(b) and (c) shows the angle curve of ith TSJ and TEJ. The behaviors of TSJ and TEJ are similar to NJ. However, the initial angle of TSJ and TEJ are not zero, because of this, the peak and trough of the curve of TSJ and TEJ are not symmetric.

Figure 3.

The ith joint angle curve of H-type winding gait in one period.

The anticlockwise direction is defined as positive. The right end of the snake robot is defined as the head, so the angle is positive when the ith joint moves in the first half period. If the left end of the snake robot is its head, the angle is negative.

The planning of CL gait of S-type winding on a plane

In the process of S-type winding gait, the snake robot curves its body and grasps the cylinder like a hand. The links making up the snake robots are divided into two categories: segment links and transition links. Segment links are used to secure the robot and transition links are added to reduce the angle of adjacent links. Each segment has n_s links. In Figure 4, segment links are illustrated as solid bars and transition links are illustrated as blank bars.

Figure 4.

S-type winding gait planning on plane.

The body of the snake robot body is also divided into three sections: head section, middle section, and tail section. The function of each section is similar to that of the H-type winding gait. The head section has n_Hs segments, the middle section has n_Ms segments, and the tail section has n_Ts segments. There is one transition link between two adjacent segments; therefore, there are n_tl(n_tl = n_Hs + n_Ms + n_Ts − 1) transition links.

The period of S-type winding gait also has four phases: head opening phase (HO), head folding phase (HF), tail opening phase (TO), and tail folding phase (TF). The function of each phase is comparable to the H-type winding gait. The difference lies in the fact that S-type type winding gait has two head folding directions corresponding to the position of the head section at the last step of the head opening phase. If n_ms is an odd number, the head section folds toward the opposite direction compared to the initial pose. If n_ms is an even number, the head section folds into the same direction compared to the initial pose.

There are three types of joints, normal joints (NJ), left transition joints (LTJ), and right transition joints (RTJ). As shown in Figure 4(a), NJs are presented with blank circles, LTJs are expressed as green circles, and RTJs are shown as red circles.

In S-type winding gait, the snake robot folds its body to make an S shape. The head of the snake robot alternately points toward the left or right or the same direction, which is determined whether the $n_M(n_M=n_{Ms}-n_s)$ is an odd or even number. The S-type winding has two different periods, which is determined by the uniformity of n_M. The period is (n_B + n_M), if n_M is an even number, and the period is 2(n_B + n_M), if n_M is an odd number.

Figure 5(a) shows the angle curve of ith NJ of S-type winding gait, which is similar to H-type winding gait. But there are two different curves controlled by the parity of n_M. When n_M is even, the joint curve of NJ is red; it has one peak and one trough in one period. When n_M is odd, the joint curve of NJ is green. It has two peaks and two troughs in one period. The Figure 5(b) and (c) illustrates the joint curve of ith LTJ and RTJ. They are not similar to the curve of NJ. They have a nonzero initial joint angle and do not have an independent peak or trough within one period.

Figure 5.

The ith joint angle curve of S-type winding gait in one period.

The total number of all segment links is $n_{Bs}=(n_{Hs}+n_{Ms}+n_{Ts})\cdot n_s$ ; the total number of links of snake robot is $n_B=n_{Bs}+n_{tl}$ ; and the number of steps in each motion period is $n_n=n_{Bs}+n_{Ms}\cdot n_s$ .

It should be clearly noted that the change of joint angle is illustrated as linear in Figures 3 and 5 but the change process of the angle is arbitrary even though the start and stop angles are determined.

Planar–cylindrical gaits transformation mechanism

The section “Design of CL gait on a plane” illustrates the gait planning of snake robots on a plane. This section presents how to transform the reference joint angles of the snake robot from a plane to a cylinder. This includes three parts: first part illustrates the attitude of every link of snake robot on cylinder; second part gives the revolute angle of each joint on the snake robot; and third part presents the constraints of angle velocity of each joint for avoiding the slipping of the snake robots along the direction of a cylinder axis.

The process of establishing the transformation mechanism mainly uses coordinate transformation, which is similar to the kinematics analysis of manipulator. In section 2, joint angles of snake robot are classified into different joints, such as NJ, TSJ, and TEJ, but they are not distinguished when performing planar to cylindrical gait transformation.

In this section, all calculations are based on the assumption that every joint of the snake robot have the capability of moving in pitch and yaw directions.

Notation defining

p_i means the ith vector in frame $o_0{x}_0{y}_0{z}_0$

$p_i^j$ represents the ith vector in frame $o_j{x}_j{y}_j{z}_j$ .

${\mathbf{R}}_i^j$ means a rotation matrix from frame $o_i{x}_i{y}_i{z}_i$ to frame $o_j{x}_j{y}_j{z}_j$ .

$\mathbf{R}(x^i,{\mathit{\theta }}_i)$ represents a rotation matrix which makes a vector rotate θ_i around the x axis of frame $o_i{x}_i{y}_i{z}_i$ .

The position of the snake robot on a cylinder

The position of links in each moment of the snake robot has been illustrated on a plane as shown in Figure 6(a). The snake robot is imagined as a curve on a paper, and then this paper is rolled as clarified in Figure 6(b). A reference $o_0{x}_0{y}_0{z}_0$ is defined, which obeys the right-hand rule. The axis of the cylinder is the z₀ axis of reference $o_0{x}_0{y}_0{z}_0$ . The direction of x₀ is normal to the surface of the cylinder. The origin of $o_0{x}_0{y}_0{z}_0$ locates is at the starting point of link 1.

Figure 6.

The orientation of links of the snake robot: (a) links on plane and (b) links on cylinder.

Two hypotheses are premised. One is that the length of every link of the snake robot projecting on plane $o^{\prime }{x}^{\prime }{y}^{\prime }$ is equal. The plane $o^{\prime }{x}^{\prime }{y}^{\prime }$ is vertical to the cylinder climbed by the snake robot; the other is that each link makes contact with the cylinder at its center point.

Two rotation angles are defined. The first rotation angle is θ_hi, which represents the angle between adjacent links of the snake robot moving on a plane. The second rotation angle is θ_vi, which symbolizes the projection angle on the plane of $o^{\prime }{x}^{\prime }{y}^{\prime }$ between adjacent links of snake robots moving on a cylinder.

The p₀ is defined along the direction of the y-axis of frame $o_0{x}_0{y}_0{z}_0$ . The p_i is used to represent the position of the ith link in frame $o_0{x}_0{y}_0{z}_0$

$\begin{array}{c}{p}_i=\mathbf{R}(x^0,{\alpha }_i)\mathbf{R}(z^0,{\beta }_i)p_0\\ {\alpha }_i={\displaystyle {\sum }_{j=1}^i{\mathit{\theta }}_{hi}}\\ {\beta }_i={\displaystyle {\sum }_{j=1}^i{\mathit{\theta }}_{vi}}\end{array}$

1

Revolute angle of every joint of snake robots

Each joint of the snake robot has pitch and yaw degrees of freedom. The axis of pitch and yaw joints has a point of intersection o_i. The point o_i is also the origin of the ith local frame $o_i{x}_i{y}_i{z}_i$ . The axis of pitch and yaw joints are the x-axis and z-axis of ith local frame $o_i{x}_i{y}_i{z}_i$ .

The position of every link of the snake robots p_i is calculated in first part; the relationship between the ith pitch joint revolute angle θ_pi, the ith yaw joint revolute angle θ_yi, and the placement of ith link p_i is presented in equation (2).

$\begin{array}{c}{p}_i^0={\mathbf{R}}_{i-1}^0\mathbf{R}(x_i,{\mathit{\theta }}_{pi})\mathbf{R}(z_i,{\mathit{\theta }}_{yi})p_i^i\\ {\mathbf{R}}_i^0={\mathbf{R}}_{i-1}^0\mathbf{R}(x_i,{\mathit{\theta }}_{pi})\mathbf{R}(z_i,{\mathit{\theta }}_{yi})\end{array}$

2

Equation (2) is a recurrence formula, so ${\mathbf{R}}_1^0$ is still unknown. The orientation of the first link p₁ is defined in first part, but there is still uncertainty for no driver between the first link and the ground. The orientation of the first link p₁ can be measured and by adjusting the revolute angles θ_p1 and θ_y1, the position of the second link p₂ will be ensured. Assuming that p₁ can be measured and be noted as ${\left[\begin{array}{ccc}{k}_x& k_y& k_z\end{array}\right]}^T$ and the 2-norm of p₁ equals one. ${\mathit{R}}_1^0$ is calculated by equation (3).

$\begin{array}{l}{\mathit{R}}_1^0=\left[\begin{array}{ccc}{C}_{\alpha }{C}_{\beta }& -S_{\alpha }& C_{\alpha }{S}_{\beta }\\ S_{\alpha }{C}_{\beta }& C_{\beta }& S_{\alpha }{S}_{\beta }\\ -S_{\beta }& 0& C_{\beta }\end{array}\right]\\ \{\begin{array}{l}{S}_{\alpha }=\frac{k_y}{\sqrt{k_x^2+k_y^2}}\hfill \\ C_{\alpha }=\frac{k_x}{\sqrt{k_x^2+k_y^2}}\hfill \\ S_{\beta }=\sqrt{k_x^2+k_y^2}\hfill \\ C_{\beta }=k_z\hfill \end{array}\end{array}$

3

$S_{\alpha }$ means sin (α) and $C_{\alpha }$ means cos (α). $S_{\beta }$ and $C_{\beta }$ have the same definition.

Now, ${\mathit{R}}_i^0$ is known, and the revolute angle of each pitch and yaw joint ${\mathit{\theta }}_{pi}$ and ${\mathit{\theta }}_{yi}$ can be calculated by equation (4).

Defining the components of ${\mathit{R}}_i^0$ and ${\mathit{p}}_i$ at first.

$\begin{array}{c}{\mathit{R}}_i^0=\left[\begin{array}{ccc}{\mathit{\gamma }}_1& {\mathit{\gamma }}_2& {\mathit{\gamma }}_3\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\\ {\mathit{p}}_i=\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]\end{array}$

The domain of ${\mathit{\theta }}_{\mathit{p}i}$ and ${\mathit{\theta }}_{yi}$ is $\left[\frac{-\mathit{\pi }}{2},\frac{\mathit{\pi }}{2}\right]$ .

$\{\begin{array}{c}{\mathit{\theta }}_{\mathit{p}i}={\text{tan}}^{-1}\left(\frac{-{\mathit{\gamma }}_3^T{\mathit{p}}_i}{{\mathit{\gamma }}_2^T{\mathit{p}}_i}\right)\\ {\mathit{\theta }}_{yi}={\text{sin}}^{-1}\left({\mathit{\gamma }}_1^T{\mathit{p}}_i\right)\end{array}$

4

Constrains of revolute velocity of pitch and yaw joints

In the second part, the position of the snake robot has been determined, but the body of snake robot will have a sliding movement along the axis of the cylinder. This sliding motion makes the snake robot unable to hold the required position. Because of this, the components of velocity within each link of the snake robot along the z-axis of frame $o_0{x}_0{y}_0{z}_0$ must equal zero.

The angular velocities of the ith pitch and yaw joints are ${\dot{\mathit{\theta }}}_{pi}$ and ${\dot{\mathit{\theta }}}_{yi}$ which are the derivatives of ${\mathit{\theta }}_{pi}$ and ${\mathit{\theta }}_{yi}$ . The linear velocity of the ith link in frame $o_j{x}_j{y}_j{z}_j$ is noted as $v_i^j=\left[\begin{array}{ccc}{v}_{ix}^j& v_{iy}^j& v_{iz}^j\end{array}\right]$ , and the superscript is omitted if the linear velocity is presented in frame $o_0{x}_0{y}_0{z}_0$ . The linear velocity of the ith link v_i is the composite result of the ith pitch and yaw joints. The l is the length of link of the snake robot.

$v_i=\left({\dot{\mathit{\theta }}}_{pi}{z}_i+{\dot{\mathit{\theta }}}_{yi}{x}_i\right)l$

5

The component of velocity of each link of the snake robot along the z-axis of frame $o_0{x}_0{y}_0{z}_0$ equals zero, so the constraint of the revolute velocity of pitch and yaw joints is given below:

$a_{31}{\dot{\mathit{\theta }}}_{yi}+a_{33}{\dot{\mathit{\theta }}}_{pi}=0$

6

equation (6) means the angular velocity of the ith pitch and yaw joints must be proportional. Constraints of the pitch and yaw joints’ angular velocity are practical, angular velocity being the main control parameter of most motors.

Gaits analysis and parameter determination

After the gait planning of snake robots on a cylinder, gait parameters are studied considering the geometry of cylinder, the geometry of snake robots, and the drive force of motors. Gait parameter analysis contains two aspects: kinematic analyses and dynamic analyses. Kinematic analysis checks collision between the links of the snake robot. The analysis includes the limitations of the cylinder radius, the length and radius of links of snake robots on the joint angles, and the number of links of the snake robot. Dynamic analysis assures the gaits work. On one hand, the influence of winding angle and number of winding links on fixing force is obtained with Newton–Euler equations and convex optimization. On the other hand, it analyzes the joint torques required to ensure that a snake robot can perform the ACL gait.

Kinematic analyses

First, the relationship between ${\mathit{\theta }}_{vi}$ and the number of links in a single loop of the snake robot winding on cylinder n_s is discussed.

Assuming the snake robot has a consistent ${\varphi }_i({\varphi }_i=\varphi )$ , which is the angle between links of the snake robot and the plane normal to the cylinder. The length of snake robots links is l, the angle of snake robots body winding on the cylinder is θ_c, and the angle between adjacent links is θ_s. The parameters θ_c and θ_s are shown in Figure 7. If θ_c and θ_s are known, the needed number of links n_s is:

$\{\begin{array}{l}{l}^{\prime }=l\text{cos}\varphi \hfill \\ {\mathit{\theta }}_s=2{\text{tan}}^{-1}\left(\frac{l^{\prime }}{2R}\right)\hfill \\ n_s=\lfloor \frac{{\mathit{\theta }}_c}{{\mathit{\theta }}_s}\rfloor +1\end{array}$

7

Figure 7.

Snake robots winding on cylinder with one loop.

The angle between adjacent links project to plane of $o_0{x}_0{y}_0$ is ${\mathit{\theta }}_{vi}$ , the angle is equal to θ_s. The radius of cylinder R, the length of links l, and the angle ϕ determines it. If both the snake robot and cylinder are fixed, the judgment conditions of whether θ_c satisfies the requirements of no less than π can be given by equation below:

$\{\begin{array}{l}{l}^{\prime }=l\text{cos}\varphi \hfill \\ {\mathit{\theta }}_{vi}=2{\text{tan}}^{-1}\left(\frac{l^{\prime }}{2R}\right)\hfill \\ {\mathit{\theta }}_c=n_s{\mathit{\theta }}_{vi}\hfill \end{array}$

8

Second, the relationship between angle θ_r and the geometry of the snake robot is explored. The maximum rotation angles of joints of snake robots θ_r will influence the flexibility of snake robots’ motion. It is determined by the maximum rotation angle of the motor and the body structure of snake robots. The main parameters of geometry of snake robot bodies relating to ${\mathit{\theta }}_{r}r$ are the radius of links r and the width of the gap between adjacent joints l_w. The parameters θ_r, r, and l_w are shown in Figure 8, and their relationship is listed below:

Figure 8.

The constrain of geometry of links of the snake robot to the revolution angle.

${\mathit{\theta }}_r=2{\text{tan}}^{-1}\left(\frac{l_w}{2R}\right)$

9

Third, the relationship between the value domain of angle ϕ and the parameters of snake robots is studied. In H-type winding gait, the improper choice of angle will cause interference in the snake robots body. The height occupied by the snake robot body on a cylinder is h_s. It should be larger than the rising height of snake robots around the cylindrical circle h. The parameters ϕ, h_s, and h are shown in Figure 9. The body of the snake robot is illustrated with a yellow band.

Figure 9.

The sketch of geometry interference of snake robots on a cylinder.

The constraints for the angle ϕ is listed below:

$\{\begin{array}{l}h=2\mathit{\pi }R\text{tan}\varphi \hfill \\ h_s=\frac{2r}{\text{cos}\varphi }\hfill \\ h\ge h_s\hfill \end{array}\Rightarrow \varphi \ge {\text{sin}}^{-1}\left(\frac{r}{\mathit{\pi }R}\right)$

10

Dynamic analyses

The dynamic analyses of ACL gaits focus on the condition where the snake robot winds along a cylinder with a single loop. Considering the condition of a multi-loop enhances the difficulty of dynamic analyses, losing efficient analysis results. The purpose of dynamic analyses in this article is providing a foundation for choosing a motor by providing the maximum torque demanded. Motors on most of these snake robots are designed for position control or velocity control. One just needs to estimate the torque range of motors used on snake robots. Figure 10 shows the force diagram of the snake robot winding on a cylinder with a single loop. N_i is the normal force exerted by a cylinder to the ith link of the snake robot. τ_i is the torque produced by the ith joint of the snake robot. If the length of links of the snake robot is equal, the attitude of the snake robot is symmetric. So, Figure 10 only illustrates the right half of the whole snake robot.

Figure 10.

The force diagram of the snake robot winding on cylinder.

By momentum equivalence, the relationship between normal force N_i and the motor torque τ_i is established.

$\{\begin{array}{l}{\mathit{\tau }}_n=N_n\frac{l^{\prime }}{2}\hfill \\ {\mathit{\tau }}_{n-1}=N_n\text{cos}({\mathit{\theta }}_{vi})l^{\prime }+N_{n-1}\frac{l^{\prime }}{2}+{\mathit{\tau }}_n\hfill \\ {\mathit{\tau }}_i=\left({\displaystyle {\sum }_{j=i+1}^n\text{cos}\left[(n-j+1){\mathit{\theta }}_{vi}\right]}\right)+N_i\frac{l^{\prime }}{2}+{\mathit{\tau }}_{i+1}\hfill \end{array}$

11

By simplifying:

${\mathit{\tau }}_i=\left({\displaystyle \sum _{j=i}^{i-1}\frac{\text{sin}\left[(i-j+05){\mathit{\theta }}_{vi}\right]N_j}{2\text{sin}\left(\frac{{\mathit{\theta }}_{vi}}{2}\right)}}\right)l^{\prime }+\frac{N_i{l}^{\prime }}{2}$

12

T is a vector of τ_i, N is a vector of N_i, and M is a coefficient matrix. To rewrite equation (11):

$T=MN$

13

M is a positive definite matrix, so an inverse matrix must exist. The expression of N is:

$N=M^{-\text{1}}T=KT$

14

There are infinite combinations of τ_i for the output of motors being within a domain from zero to maximum τ_max. There must be a combination satisfying these two requirements: the first is resultant force of all N_i being zero, ensuring the safety of snake robots; the second is the sum of all the absolute values of N_i being maximum. Linear programming can solve this. The cost function is:

$J={\displaystyle \sum |}{N}_i|={\displaystyle \sum (c_i{\mathit{\tau }}_i)}$

15

The constraints should be classified into two situations: the number of single loop links of snake robot n_s is either even or odd. If n_s is even, the sum of projection of all normal pressure force $\{N_{\text{1}},\cdots ,N_n\}$ on the axis of z₀ is zero. All other constraints are, N_i is not less than zero and all motor torques τ_i are not less than zero and not greater than τ_max. τ_max is defined as one in this paper. The expression is:

$\begin{array}{c}\text{max\hspace{0.17em}\hspace{0.17em}}J={\displaystyle {\sum }_{i=1}^n\left|N_i\right|={\displaystyle {\sum }_{i=1}^n(c_i{\mathit{\tau }}_i)}}\\ st\{\begin{array}{l}{N}_z=0\hfill \\ \forall N_i\ge 0\hfill \\ 0\le {\mathit{\tau }}_i\le {\mathit{\tau }}_{\text{max}}\hfill \end{array}\\ N_z={\displaystyle \sum _{j=1}^n\left(\text{sin}\left[(n-i+1){\mathit{\theta }}_{vi}-\frac{{\mathit{\theta }}_{vi}}{2}-\frac{\mathit{\pi }}{2}\right]N_i\right)}\end{array}$

16

If n_s is odd, there is an additional force N₀, its quantity is not less than zero. Compared to the even situation, constraints reduce an equation. The expression is:

$\begin{array}{c}\text{max\hspace{0.17em}\hspace{0.17em}}J={\displaystyle {\sum }_{i=0}^n\left|N_i\right|={\displaystyle {\sum }_{i=1}^n(d_i{\mathit{\tau }}_i)}}\\ s.t.\{\begin{array}{l}\forall N_i\ge 0\hfill \\ 0\le {\mathit{\tau }}_i\le {\mathit{\tau }}_{\text{max}}\hfill \end{array}\\ N_0={\displaystyle {\sum }_{i=1}^n(\text{sin}\left[(n-i+1){\mathit{\theta }}_{vi}\right]N_i)}\end{array}$

17

The force of fixing the snake robots body to avoid falling from the cylinder is the force of friction between the robots body and the cylinder. The quantity of friction force is determined by the sum of all normal force N_i, which is represented with J. The maximum of fixing force of snake robots F_max with different winding range and winding links can be obtained by calculating J_max and multiplying friction coefficient μ. The minimum of winding range of the snake robot is no less than π and the number of winding links is no less than three for force balance. Figure 11 shows the relationship between the maximum of the fixing force F_max, winding angle θ_c, and the number of winding links n_s. The winding angle is from π to $\text{11}\mathit{\pi }/\text{6}$ , and the number of winding links is from 3 to 48.

Figure 11.

The relationship between fixing force, winding angle, and number of winding links.

As shown in Figure 11(a) and (b), the fixing force F_max increases with winding angle θ_c when n_s is greater than five. When n_s is not greater than five, the relationship between F_max and θ_c should be calculated, respectively. As presented in Figure 11(a) and (c), the fixing force F_max increases with the number of winding links n_s when θ_c is greater than n. The parity of n_s influences the increasing trend of F_max. When n_s is greater than ten, the F_max tends to be a constant value. This means that the greater number of n_s is not helpful to the fixing force F_max when n_s is greater than ten.

The maximum torque of motors in the process of lifting the snake robot body is another factor to be considered. Based on static analysis, two cases of the motor torque of the highest requirement are studied. As shown in Figure 12, one case is where the cylinder is laid vertically and the snake robot winds around the cylinder with the head section; however, the winding angle is less than π, so the head section cannot supply the fixing force. The motor raising the body of the snake robot bares the maximum torque. This motor is named Mo and shown in Figure 12 with a red circle.

Figure 12.

The case of where the motor bears the maximum torque when the cylinder is vertical.

The moment applied to the motor Mo caused by the gravity of the snake robot can be simplified into three parts: one is the moment cause by the gravity of the link near the motor Mo; another is the moment caused by the gravity of middle section of the snake robot which marked with n_M in Figure 12; the last one is the moment caused by the gravity of head section of the snake robots. The rated torque of motor should not be less the sum of the three parts mentioned above. The links of the three parts are shown with red bars in Figure 12. The parameter l_cm means the distance between the center of gravity of the head section of the snake robot and the axis of the cylinder. The parameter $n_{\theta }$ is the number of links winding around the cylinder in the range of ${\mathit{\theta }}_c$ .

$\mathit{\tau }\ge mg·\frac{l}{2}+n_{M}mg·l+n_{\theta }mg·(l+\frac{R{\text{sin}}^2({\mathit{\theta }}_c/2)}{{\mathit{\theta }}_c/2})$

18

Another case is that the cylinder is laid horizontally and the snake robot acts similar to the last case. In this case, the main factor influencing the quantity of τ consists of two parts, the moment caused by the gravity of the middle section and the head section of the snake robot. The rated torque of motor should satisfy the expression below:

$\mathit{\tau }\ge n_{M}mg·n_M\frac{l}{2}+n_{\theta }mg·n_{M}l$

19

The design scheme of ACL gait of snake robots

After choosing the geometry parameters of the cylinder and the friction coefficient between the cylinder and the snake robot body, the design scheme can be made based on the analyses in the fourth section. The content of design scheme includes the geometry parameters of the snake robot, the rated torque of motors, and control parameters of ACL gait.

Analyses in the fourth section give some suggestions toward the design scheme. The radius of the snake robot body should be as small as possible, which is determined by mechanical design. The parameter n_H should be equal to n_T. The setting of parameters of n_H and n_T considers if the friction force produced by the head section or the tail section of the snake robot is solely greater than the gravity of the whole snake robot. The number of links of the middle section n_M determines the forward distance in each period, so it should be as large as possible. But the rated torque of motors restricts the setting of n_M.

Supposing the relationship between the mass and the length of links of the snake robot is linear, that is:

$m=\alpha l+\beta$

20

The number of payload links of a single loop of the snake robot $n_x(l,n_s)$ is the payload of additional links of the snake robot with the links winding cylinder; this is explored based on the analyses in “The design scheme of ACL gait of snake robots” section. The expression ${\displaystyle \sum N_i}\mu$ is fixing force of the single loop of the snake robot. The parameter m is the mass of a single link of the snake robot. The parameter n_s is the number of links within a single loop of the snake robot. The single loop of links in the head and tail sections may act separately. Fixing force generated by the single loop of links should bear two times the weight compared to the single loop. Figure 13 shows the relationship between the number of payload links n_x and the number of links of single loop n_s, which produced the n_x optimal procedure based on the dynamic analysis in “Dynamic analyses” section. Different lengths of links of the snake robot are shown with different types of line in Figure 13.

Figure 13.

The relationship between the number of payload links and the number of links of single loop.

$n_x(l,n_s)=\frac{{\displaystyle \sum N_i}\mu -2n_{s}mg}{mg}$

21

As shown in Figure 13, n_x reaches its maximum value when l goes to $l_{\text{opt}}(l_{\text{opt}}=4R/5)$ and n_s goes to $n_{s\_\text{opt}}$ (8). The parameters μ, α, and β are set as 0.2, 0.2, and 0.1, respectively. The above procedure gives the maximum value of n_x, which is noted as $n_{x\_\text{max}}$ . The number of links of all segments of S-type winding gait n_s can be set as $n_{x\_\text{max}}$ . The winding angle involved by $n_{x\_\text{max}}$ links may be a complete or incomplete circle, making the optimal value of the number of links winding around the cylinder, as a whole circle $n_{c\_\text{max}}$ needs an additional calculation. The number $n_{c\_\text{max}}$ can be calculated based on the procedure computing $n_{x\_\text{max}}$ by fixing winding angle to 2n. In this case, the number of links winding around the cylinder as a whole circle is noted as $n_{c\_\text{opt}}$ . In H-type winding gait, the relationship between the number of head or tail section n_H or $n_T(n_H=n_T)$ and the number of links of middle section n_M is listed below:

$n_M=\lfloor n_x+\frac{(n_H-n_{s\text{–opt}})n_{c\text{–opt}}}{n_{c\text{–opt}}}$

22

In S-type winding gait, the relationship between the number of segments of head or tail section $n_{Hs}$ or $n_{Ts}(n_{Hs}=n_{Ts})$ and the number of links of middle section n_M is listed below:

$n_M=\lfloor n_{Hs}{n}_x\rfloor$

23

The larger the number of n_M, the greater number of links within the snake robot that move forward during each period, and the more efficient the ACL gait. But the parameter n_M is constrained by the rated torque of motors τ_max, which derived from equations (18) and (19). Constraints are listed below:

$\{\begin{array}{l}{n}_M\le \frac{{\mathit{\tau }}_{\text{max}}}{mgl}-0.5-n_{\theta }(1+\frac{2R{\text{sin}}^2({\mathit{\theta }}_c/2)}{{\mathit{\theta }}_{c}l})\hfill \\ n_M\le \sqrt{{N_{\theta }}^2+\frac{2{\mathit{\tau }}_{\text{max}}}{mgl}}-N_{\theta }\hfill \end{array}$

24

From the above calculation, the structural parameters and control parameters of ACL gait are obtained. Two examples are given to verify the design scheme of ACL gait of snake robots. Proposing the radius of the cylinder R is 50, the rated torque of motors τ_max is 20, the friction coefficient μ is 0.5, the parameters α and β in equation (17) are 0.05 and 0.1, respectively, and V-rep is applied as simulation platform.

First, the ACL gait of H-type winding is simulated. The parameters n_H, n_M, and n_T are all set as 6, which are obtained through the n_x optimal procedure. The number of single loop n_s is also equal to 6 which satisfies equation (24). Figure 14 shows the climbing process of the ACL gait of H-type winding. The four phases of the gait: head opening, head folding, tail opening, and tail folding are illustrated in Figure 14(a) to (f). The whole process of ACL gait can be compared with Figure 2. Figure 14(a) shows the initial state of the snake robot winding on a cylinder, which corresponds to Figure 2(a). In the head opening phase, the snake robot extends its head section upward along the cylindrical axis, which is illustrated in Figure 14(b) and Figure 2(b) and (c), respectively. The snake robot winding cylinder with its head section in head folding phase as shown in Figure 14(c) and Figure 2(d) to (f). As presented in Figure 14(d) and Figure 2(g) to (j), the snake robot simultaneously opens the tail section and folds the other parts of body in tail opening phase. In tail folding phase, the snake robot folds the tail section and restore the body posture to its initial state, which is shown in Figure 14(e) and (f) and Figure 2(k) and (l).

Figure 14.

The climbing process of H-type winding gait.

Second, the S-type winding gait is simulated. The parameter n_s, n_Hs, n_Ms, and n_Ts are set as 4, 1–5, 1, and 1–5, which are also obtained through the n_x optimal procedure and equation (24). Figure 15 shows the climbing process of the S-type winding gait. The four phases of the gait: head opening, head folding, tail opening, and tail folding are illustrated in Figure 15(a) to (f). The whole process of ACL gait of S-type winding can be compared with Figure 3. Figure 15(a) shows the initial state of the snake robot winding on a cylinder, which corresponds to Figure 3(a). In the head opening phase, the snake robot extends its head section upward along the cylindrical axis, which is illustrated in Figure 15(b) and Figure 3(b) to (d), respectively. The snake robot winding cylinder with its head section in head folding phase as shown in Figure 15(c) and Figure 3(e) to (g). As presented in Figure 15(d) and Figure 3(h) to (j), the snake robot simultaneously opens the tail section and folds the other parts of body in tail opening phase. In tail folding phase, the snake robot folds the tail section and restore the body posture to its initial state, which is shown in Figure 15(e) and (f) and Figure 3(k) to (m).

Figure 15.

The climbing process of S-type winding gait.

Conclusion and future work

This article proposed ACL gait of snake robots on a cylinder. Based on parameter analyses, the gait design scheme is presented and verified by a simulation platform. The problem of a hard-to-design gait on a cylinder is solved by a method where the gait is designed on a plane and is then transformed to ACL gait on a cylinder. To prove the safety of the snake robot in motion, the kinematic and dynamic analyses of ACL gait are executed. To assure the applicability of the gait, the design scheme of ACL gait is proposed and verified by two examples of H-type and S-type winding gaits. The proposed ACL gaits only achieve the motion of a snake robot on a single pole. In the next step, the turning and avoidance motion of ACL gait will be designed to enhance the motion capability of snake robots on a complex truss structure.