A comprehensive study on the locomotion

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A comprehensive study on the locomotion characteristics of a metameric earthwormlike robot Hongbin Fang, Chenghao Wang, Suyi Li, K. W. Wang & Jian Xu

Multibody System Dynamics ISSN 1384-5640 Volume 35 Number 2 Multibody Syst Dyn (2015) 35:153-177 DOI 10.1007/s11044-014-9428-5

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Author's personal copy Multibody Syst Dyn (2015) 35:153–177 DOI 10.1007/s11044-014-9428-5

A comprehensive study on the locomotion characteristics of a metameric earthworm-like robot Part B: Gait analysis and experiments Hongbin Fang · Chenghao Wang · Suyi Li · K.W. Wang · Jian Xu

Received: 4 March 2014 / Accepted: 15 July 2014 / Published online: 23 September 2014 © Springer Science+Business Media Dordrecht 2014

Abstract This is the second part of a comprehensive study on the locomotion characteristics of a metameric earthworm-like robot. Three major contributions from the first part paper are used as the foundations of this investigation: (a) the novel analytical models describing the kinematic and dynamic characteristics of the robot locomotion, (b) a gait generation algorithm based on the mechanism of retrograde peristalsis wave, and (c) the discovery of critical conditions for actuator-overload and anchor-slippage. This paper focuses on the issues of gait analysis and its experimental verification. Analysis on the kinematic model attains the optimal gait that corresponds to the maximum ideal average speed. However in practical applications, such ideal speed might not be achievable due to actuator overload and anchor slippage. Therefore, the dynamic model is exercised to survey the relationship between gait designs and the occurrence of the overload and slippage, and this survey identifies five types of locomotion with distinct dynamic characteristics. Then, this research branches out to investigate two important topics. The first topic is to understand how changes in the gait and physical parameters, such as the number of robot segments and the friction coefficients, affect the robot’s locomotion behavior. The second topic is to optimize the gait designs in consideration of the effects from actuator overload and anchor slippage, such that the maximum achievable average speed can be obtained for

B

H. Fang · J. Xu ( ) School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China e-mail: [email protected] H. Fang e-mail: [email protected] H. Fang · C. Wang · S. Li · K.W. Wang Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA C. Wang e-mail: [email protected] S. Li e-mail: [email protected] K.W. Wang e-mail: [email protected]

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different types of locomotion. The observations from these two topics of investigation reveal many insights regarding the physics of the metameric earthworm-like robot locomotion and provide comprehensive guidelines for the robot design and control. Finally, an eight-segment earthworm-like robot prototype is fabricated to experimentally verify the gait analysis, especially the relationship among gait design, average speed, and the occurrence of anchor slippage. Keywords Earthworm-like robot · Metastructure · Locomotion · Gait analysis · Gait optimization · Average speed · Dynamics 1 Introduction The kinematic and dynamic models of the metameric earthworm-like robot presented in Part A (modeling and gait generation) are the foundations of study discussed in this part of the paper. A schematic of the model is shown in Fig. 1 (both the kinematic and dynamic models share the same configuration of segments, N is the total number of segments, k is the number of driving modules, nA and nR are the numbers of anchoring and relaxing/contracting segments in each driving module, respectively, a global numbering system from #1 to #N is adopted). The motivation of this research (Part B) comes from two aspects. On one hand, the average speed is one of the most important indexes to evaluate the robot’s locomotion performance. However, in previous studies on earthworm-like robots, the effects of physical parameters and gait parameters on the average speed have not been systematically investigated. On the other hand, as is shown in Part A, not only the average speed but also the critical conditions of actuator overload and anchor slippage are related to the physical parameters and gait parameters. In addition, some previous experiments [1–4] have pointed out that the backward slippage of anchoring segments is a major reason for the speed loss. However, these two dynamic behaviors of the earthworm-like robot, that is, actuator overload and anchor slippage, have not been rigorously analyzed. Therefore, taking into account of these two aspects, the interrelationship among the physical parameters, gait designs, and dynamic behavior of the robot are investigated and reported in this paper (Part B). Aiming at maximizing the average speed of the robot, optimization problems are also discussed. Experimental verification is another important task of this research. In previous work, a number of prototypes of earthworm-like robots have been developed [1–8], where the major concern is the effectiveness of the actuation mechanism. Some general problems like how the gait influences the robot’s locomotion speed and dynamic behavior have not been experimentally studied. In this research, a metameric earthworm-like robot prototype is developed to verify the effectiveness of the generated gaits and the results on gait optimization and gait analysis. The Part B paper includes five sections. Section 2 discusses the gait optimization problem of the N -segment kinematic model, with maximum ideal average speed as the optimization objective. Section 3 discusses gait analysis on the N -segment dynamic model from two aspects. First, motion of the robot is categorized based on whether anchor overload or anchor slippage occurs; then the interrelationships among physical parameters, gait designs, and dynamic behavior of the robot are explored. Second, gait optimization is studied to obtain the maximum achievable average speed. The optimization results are presented and thoroughly interpreted. Section 4 briefly explains how the prototype of the metameric earthworm-like robot is built and shows the validation of the theoretical analysis on gaits. Section 5 summarizes the contributions of the investigations presented in both Part A and Part B of this two-part paper.

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Fig. 1 The general N -segment model with k driving modules (each driving module contains nA anchoring segments, nR relaxing and nR contracting segments) and N − k(nA + 2nR ) unactuated segments

2 Gait optimization of the kinematic model In Part A, it is shown that changes in gaits will directly alter the global displacement of the robot in a specified time period and, hence, alter the average speed (see Fig. 8 in Part A). In this section, gait optimization based on the N -segment kinematic model of the earthwormlike robot is discussed. 2.1 Optimization objective and constraints In robotics, the average speed is an important index to evaluate the locomotion ability of the robot, and the normalized average speed in terms of body length (with unit “body length per second”) is always employed as an index to evaluate the efficiency of the robot’s motion. As a result, these two indexes are selected as the object functions of gait optimization in this study. In Part A, expression for the average speed of the general N -segment kinematic model of the robot has been derived (see Eq. (9) in part A): N − k(nA + nR ) l , V¯ = N/nR t

(1)

where l is the stroke of a robot segment, and t is the time for one transition. Dividing the average speed by the body length of the robot yields the normalized average speed V¯˜ =

1 N − k(nA + nR ) l V¯ = , N lmax N lmax (N/nR ) t

(2)

where lmax is the longest length of a segment (lmax = lmin + l), and therefore, N lmax is the body length of the robot. It should be noticed that with the assumption of ideal actuator and ideal anchor, the average speed (1) and the normalized average speed (2) are in fact the ideal speeds of the robot. Following the kinematics law, the four gait parameters (N, k, nA , and nR , which are all positive integers) have to satisfy the following kinematic constraints (obtained in Sect. 4.2 of Part A): k ≥ 1,

nA ≥ 1,

nR ≥ 1,

N ≥ k(nA + 2nR ).

(3)

Hence, in this optimization, the goal is to maximizing the objective functions (1) and (2) under constraints (3).

Author's personal copy 156 Table 1 Optimal gaits and the corresponding maximum ideal average speed V¯K (in l/t ) and maximum ideal normalized average speed V¯˜K (in l/(lmax t)) for N = 3, . . . , 10

H. Fang et al. N

k

nA

nR

l ) V¯K ( t

V¯˜K ( l lt ) max

3

1

1

1

0.333

0.111

4

1

1

1

0.500

0.125

5

1

1

2

0.800

0.160

6

1

1

2

1.000

0.167

7

1

1

3

1.286

0.184

8

1

1

3

1.500

0.188

9

1

1

4

1.778

0.198

10

1

1

4

2.000

0.200

Fig. 2 The optimal gaits (solid dots) for N = 3, . . . , 10 in the nA − nR plane, where the right triangle regions are determined by constraints (3)

2.2 Optimization results The Karush–Kuhn–Tucker (KKT) conditions are effective methods to solve the optimization problem with inequality constraints. In this research, KKT conditions are adopted to perform the optimization. Details about the KKT conditions are omitted here and can be referred at [9]. Assuming constant physical parameters (l, t , and lmax ), the KKT condition gives the maximum ideal average speed V¯K and the maximum ideal normalized average speed V¯˜K corresponding to different values of N (Table 1) Analytically, the optimal gait corresponding to both V¯K and V¯˜K are the same:   N OPTK : k = 1, nA = 1, nR = − 1, (4) 2 where · denotes the ceiling functiony = min{n ∈ Z|n ≥ y}, and the subscript K stands for the kinematic model. In fact, in the nA − nR plane, the optimal gait OPTK always locates at the upper vertex of the right triangle determined by constraints (3), see Fig. 2. Equation (4), i.e., OPTK , reveals that the maximum values of both V¯K and V¯˜K are realized when there is only one driving module in the robot. The only driving module is made up of one anchoring segment and the largest admissible number of relaxing and contracting segments. Such optimal gait for maximum ideal speeds is intuitive because it corresponds to the largest actuating forces (both pushing and pulling). Substituting the optimal gaits OPTK into the object functions (1) and (2) yields the analytical expression of V¯K and V¯˜K . Considering the case that the value of N is fairly large, one

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Fig. 3 The optimization results of the kinematic model (N ranges from 3 to 100): (a) the maximum ideal average speed V¯K (m/s) and (b) the maximum ideal normalized average speed V¯˜K (body length/s). V¯K increase linearly with respect to N , with an almost constant slope 0.00311, coinciding with the analytical result ∂ V¯K /∂N = l/(4t) = 0.00312; V¯˜K = 0.049 when N = 100, agreeing with the limiting value l/(4lmax t) = 0.05

gets N − (1 + N/2) l N→∞ N l V¯K = , −→ N/N/2 t 4 t V¯˜K =

1 N − (1 + N/2) l N→∞ 1 l . −→ N lmax N/N/2 t 4 lmax t

(5)

Besides, the derivatives of V¯K and V¯˜K with respect to N are calculated when N is fairly large, which gives   ∂ V¯˜K  1 l ∂ V¯K  , = = 0. (6) ∂N N→∞ 4 t ∂N N→∞ From Eqs. (5) and (6) we know that V¯K will increase monotonously with respect to the total number of segments N , without an upper limit. However, although V¯˜K will increase with respect to N as well, it has an upper limit. To verify the above analysis, as an example of optimization, the following parameters of the kinematic model are used in simulation: lmax = 0.0625m,

l = 0.0125m,

t = 1 s.

(7)

The optimization results corresponding to V¯K and V¯˜K are shown in Figs. 3(a) and (b), respectively, with N ranging from 3 to 100. Physically, it can be seen from the above analysis that with the increase of the total number of segments, the maximum ideal average speed V¯K follows a linear growth. However, such rise in V¯K is accompanied by the elongation of the robot’s body length. As a result, in terms of body length, the increase of the maximum ideal normalized average speed V¯˜K flattens when the total number of segment increases, and finally V¯˜K converges to a constant. In other words, the maximum ideal average speed of the robot can be advanced by increasing the total number of segments, but the robot’s motion efficiency cannot be infinitely improved by this way.

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3 Gait analysis on dynamic model In this section, the N -segment dynamic model of the earthworm-like robot will be considered. For convenience, we list all the physical parameters characterizing the dynamic model, which have been used and explained in Part A: lmax and lmin are the longest and shortest lengths of a segment, respectively; l is the stroke of a segment (l = lmax − lmin ); t is the time for one transition; m is the mass of each segment; g is the gravitational acceleration; f+ and f− are the friction coefficients for forward and backward motions, respectively; η is a positive ratio characterizing the additional pressure induced by the radial expansion of a segment; α and β characterize the accelerations of strain change, with a discontinuity point ξ ∈ (0, t); ρ is the maximum velocity of strain change of the relaxing segments; and Qmax is the maximum actuating force of a segment. In what follows, the earthworm-like robot’s motion will be categorized based on whether actuator overload and backward/forward anchor slippage occurs. Then relationship among physical parameters, gait designs, and dynamic behavior of the robot will be studied. After that, with the same purpose as the kinematic model, that is, maximizing the average speed of the robot, gait optimization will be discussed. 3.1 Categorization of the robot’s motion In the dynamic model, the kinematic constraints (3) still have to be satisfied. Besides, in Part A, critical conditions for actuator overload and anchor slippage have been derived (see Eqs. (35), (37), and (38) of Part A), which depend on both the physical parameters (including the actuator parameters and the environment parameters) and gait parameters. For simplicity, let   mαlmin  + (nR − 1) + N − k(nA + 2nR ) mαlmin 2 N − k(nA + 2nR ) + f+ mg + f+ mg − Qmax , nR   c.c.2 = N − k(nA + nR ) nR αlmin + (N − knA )f+ g − knA (1 + η)f− g,   c.c.3 = − N − k(nA + nR ) nR βlmin + (N − knA )f+ g + knA (1 + η)f+ g. c.c.1 =

(8) (9) (10)

Then the critical condition for actuator overload is c.c.1 > 0. Providing that all actuators work within the output range, that is, c.c.1 ≤ 0, the critical conditions for backward and forward slippage of anchoring segments are c.c.2 > 0 and c.c.3 < 0, respectively. Hence, based on whether these critical conditions are satisfied, the robot’s motion can be categorized into five types (from type I to type V), with their conditions and distinct characteristics listed in Table 2. The situation that actuator overload occurs, that is, motion of type V, will not be studied in detail. 3.2 Effects of the total number of segments N Fixing all the physical parameters (Table 3), the effects of the total number of segments N on the critical conditions and the dynamics of the robot are studied. When N is relatively small, anchor overload and slippage will not happen due to small actuation and anchoring force requirements. Increasing the total number of segments causes higher demands on the actuating force and the anchoring force, and actuator overload and

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Table 2 Conditions and characteristics of the five types of motion Type

Kinematic constraints

Actuator-overload condition

Anchor-slippage conditions

Characteristics

I

(3)

c.c.1 ≤ 0

c.c.2 ≤ 0, c.c.3 ≥ 0

No anchor-slippage

II

(3)

c.c.1 ≤ 0

c.c.2 > 0, c.c.3 ≥ 0

Backward, but no forward anchor slippage

III

(3)

c.c.1 ≤ 0

c.c.2 > 0, c.c.3 < 0

Both forward and backward anchor slippage

IV

(3)

c.c.1 ≤ 0

c.c.2 ≤ 0, c.c.3 < 0

Forward, but no backward anchor slippage

V

(3)

c.c.1 > 0

/

Actuator overload

Table 3 Physical parameters of the dynamic model

Parameters Values

Parameters Values

Parameters Values

m

0.1 kg

η

0.5

α

0.67 m/s2

t

1.0 s

f+

0.1

β

2.0 m/s2

g

9.8 m/s2 f−

0.5

ρ

lmin

0.05 m

0.0125 m Qmax

l

0.5 m/s 1.5 N

anchor slippage may occur. To illustrate the changing process with respect to N , with parameters shown in Table 3, the kinematic constraints (3) and the critical conditions c.c.1 = 0, c.c.2 = 0, and c.c.3 = 0 on the nA − nR planes are plotted in Figs. 4(a) to (f) for N = 4, 5, 9, 43, 62, and 72, respectively. The number of driving modules is assumed to be one (k = 1). It reads from Fig. 4 that with the increasing of the total number of segments, multiple types of motions are possible with different gait designs. When N = 3 or 4, only type I motion is possible (see Fig. 4(a)), and both anchors and actuators work normally. When N = 5, actuator overload (type V motion) first appears (see Fig. 4(b)), which means that for certain gaits, some actuators may fail. Failure of actuators is always possible for N ≥ 5. When the value of N is increased to 9, type II motion, that is, backward slippage of anchoring segments, first turns up and coexists with type I (see Fig. 4(c)). Such scenario will continue until N is increased to 43, at which type III motion appears (see Fig. 4(d)). In type III, anchoring segments can slip both forward and backward during a transition. Keeping increasing the value of N , the region of type III will continue to expand. When N = 62, the region of type III first outstrips the region of type II, inducing a new type of motion, type IV, that only forward slippage occurs on anchoring segments (see Fig. 4(e)). All the five types of motion are possible when N = 62. Such situation will continue until N increases to 72. During the increase of N from 62 to 72, regions of type III and type IV keep enlarging. At N = 72, the region of type II is completely covered by the region of type III, which means that type II motion cannot be achieved when N = 72 and k = 1. Similar analysis can be carried out, and similar figures can be drawn as well when there are multiple driving modules (k > 1) in the robot. In sum, the above analysis manifests that the total number of segments N has significant effects on the robot’s motion. With the increase of N , actuators may overload, and anchoring segments may slip backward or (and) forward for some gaits. For relatively large values of N , multiple types of motion are possible to occur. In many applications, actuator overload and anchor slippage are undesired; for this purpose, region plots like Fig. 4 provide very insightful guideline for gait design.

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Fig. 4 With parameters shown in Table 3, regions of different types of motion on the nA − nR plane for k = 1: (a) N = 4; (b) N = 5; (c) N = 9; (d) N = 43; (e) N = 62; (f) N = 72. The regions are divided by curves of critical conditions, denoted by c.c.1 , c.c.2 , and c.c.3

3.3 Effects of the physical parameters Notice that the critical condition also depends on the physical parameters. In this subsection, effects of the environmental parameters (friction coefficients f+ and f− ) and the actuator parameters (actuator accelerations α and β) on the critical conditions and dynamic behaviors of the robot are studied. The other actuator parameters are assumed to be constant because they are either not relevant to the critical conditions or uncontrollable for a fixed type of actuator. The signs of the derivatives of c.c.1 , c.c.2 , and c.c.3 with respect to f+ , f− , α, and β are calculated, respectively: (c.c.1 ) f+ > 0,

(c.c.1 ) f− = 0,

(c.c.1 ) α > 0,

(c.c.1 ) β = 0.

(11)

(c.c.2 ) f+ > 0,

(c.c.2 ) f− < 0,

(c.c.2 ) α > 0,

(c.c.2 ) β = 0.

(12)

(c.c.3 ) f+ > 0,

(c.c.3 ) f− = 0,

(c.c.3 ) α = 0,

(c.c.3 ) β < 0.

(13)

First, we consider the effects of friction coefficients. According to Eqs. (11)–(13), with the increase of the friction coefficient for forward motion f+ , the regions of actuator over-

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Fig. 5 The effects of friction coefficient f+ on the regions of different types of motion, with N = 50 and k = 1 as an example: (a) f+ = 0.1; (b) f+ = 0.3. The other parameters are the same with those in Table 3. The upward, downward, and horizontal arrows indicate that the region is increased, decreased, and remains the same during the change of parameters, respectively

Fig. 6 The effects of friction coefficient f− on the regions of different types of motion, with N = 50 and k = 1 as an example: (a) f− = 0.5; (b) f− = 0.7. The other parameters are the same with those in Table 3

load (type V) and backward anchor slippage (type II) expand, simultaneously diminishing the region of motion without anchor slippage (type I), while the region of both forward and backward anchor slippage (type III) shrinks. Figure 5 clearly shows the changing trend of regions when f+ is increased from 0.1 (Fig. 5(a)) to 0.3 (Fig. 5(b)). Similar analysis can be carried out for the friction coefficient f− . Figure 6 illustrates the change of the critical conditions and the regions of different types of motion when f− is increased from 0.5 to 0.7. The region of actuator overload (type V) and region of both forward and backward anchor slippage (type III) remain the same. However, the region of backward anchor slippage (type II) shrinks, leading to the increase of the region of motion without anchor slippage (type I). Then we study the actuator accelerations α and β. Notice that the values of α and β ρα are constrained by β = αt−ρ . Hence, a larger value of α will necessarily lead to a smaller value of β. The derivatives in Eqs. (11)–(13) and the changing trend of regions with respect to α and β shown in Fig. 7 reveal that a larger value of α (smaller value of β) corresponds to larger regions of actuator overload (type V) and backward anchor slippage (type II), but smaller regions of motion without anchor slippage (type I) and motion with both forward and backward anchor slippage (type III). The above parametric study provides us another way to avoid undesired phenomena (actuator overload or anchor slippage). Adjusting the physical parameters (including the friction coefficients and actuator acceleration) is an effective and global way to tailor the regions of different types of motion on the gait-parameter plane. For example, in order to reduce the region of actuator overload, smaller values of f+ and α, but larger value of f− should be taken.

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Fig. 7 The effects of actuator accelerations α and β on the regions of different types of motion, with N = 50 and k = 1 as an example: (a) α = 0.667, β = 2; (b) α = 2, β = 0.667. The other parameters are the same with those in Table 3

3.4 Optimization objectives and constraints In Sect. 2, corresponding to the maximum ideal average speed, the optimal gait OPTK is always obtained at the upper vertex of the right triangle regions for different N (see Fig. 2). However, in the dynamic model with gait OPTK , the temporary anchor may not be strong enough and as a result may slip backward or (and) forward, which is always regarded as a negative phenomenon in application. However, the average speed of motion with anchor slippage may not be inherently lower than that of motion without anchor slippage. Therefore, it is interesting and necessary to study the optimal gaits of each type of motion. During the interpretation of optimization results, both the maximum average speed and the dynamic behavior of the robot with optimal gaits are focused. The average speed and the normalized average speed in terms of body length are selected as the object functions of the dynamic model, and to achieve their maximum values is the optimization objective. However, instead of explicit expressions as Eqs. (1) and (2), the average speed of the dynamic model cannot be analytically obtained because the transition instants between fixing anchors and slipping anchors cannot be determined. Performing numerical simulation on the dynamic model, the average speed of the robot can be expressed by Xc (t2 ) − Xc (t1 ) , V¯ = V¯c ≈ t2 − t1

(14)

and the normalized average speed can be expressed by V¯˜ =

V¯ 1 Xc (t2 ) − Xc (t1 ) = . N lmax N lmax t2 − t1

(15)

The time period (t2 − t1 ) in Eqs. (14) and (15) should be long enough to cover multiple transitions. The constraints for the four types of motion (I, II, III, and IV) are listed in Table 2, including the kinematic constraints and the critical conditions for actuator overload and anchor slippage. 3.5 Optimization results Based on KKT conditions and numerical simulation, the optimization procedure for the dynamic model is given in Appendix for interested readers. Following the optimization procedure and using the parameters in Table 3 again, the maximum achievable average speed and the maximum achievable normalized average speed (which corresponds to the same optimal gait) for each type of motion are obtained and are shown in Figs. 8 and 9, respectively.

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Fig. 8 The maximum average speed of each type of motion. V¯K stands for the maximum ideal average speed of the kinematic model, V¯D−K , V¯D−B , V¯D−BF and V¯D−F stand for the maximum achievable average speeds of the motions of type I, II, III and IV, respectively

Fig. 9 The maximum normalized average speed of each type of motion in terms of body length. V¯˜K stands for the maximum ideal normalized average speed of the kinematic model, V¯˜D−K , V¯˜D−B , V¯˜D−BF , and V¯˜D−F stand for the maximum achievable normalized average speeds of the motions of type I, II, III, and IV, respectively

The optimization results of the kinematic model (V¯K and V¯˜K ) are also plotted in Figs. 8 and 9 as reference. V¯D−K (V¯˜D−K ), V¯D−B (V¯˜D−B ), V¯D−BF (V¯˜D−BF ), and V¯D−F (V¯˜D−F ) are the maximum achievable (normalized) average speeds of motions of type I, II, III, and IV, respectively. Here, the subscript D means dynamic model, K, B, BF, and F stand for no anchor slippage, backward anchor slippage, both backward and forward anchorslippage, and forward anchor slippage, respectively. Comparing Figs. 8, 9 with Fig. 4, the critical values of N are consistent in revealing the appearance and disappearance of certain type of motion. Besides, Figs. 8 and 9 provide us with a wealth of information: (1) From Fig. 8, one reads that before N = 9, there is no optimal gait for the motions with anchor slippage, that is, V¯D−B , V¯D−BF , and V¯D−F do not exist. The maximum achievable average speed V¯D−K of the motion without anchor slippage is exactly the same as the maximum ideal average speed of the kinematic model V¯K when N < 9. In fact, as is mentioned, when there is no anchor slippage, the dynamic model degenerates into the kinematic model, and the same optimization method, KKT condition, is used, which necessarily leads to the same optimal gait OPTK and the same maximum average speed. (2) Figure 8 reveals that for N ∈ [9, 42], V¯D−BF and V¯D−F do not exist, whereas V¯D−K and V¯D−B exist and are obviously lower than V¯K . The relatively low speed of V¯D−K and V¯D−B can be explained from two aspects. On one hand, due to the occurrence of motion with backward anchor slippage (type II) when N ≥ 9 (see Fig. 4(c)), the optimal gait for motion without

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Fig. 10 For N = 11, the time histories of the velocity (m/s) of the mass center of the dynamic model X˙ Dc (solid), the velocity of the mass center of the kinematic model X˙ Kc (dashed), and the velocity of the anchoring segments x˙A (dot-dashed) with gaits (a): k = 1, nA = 2, nR = 4 and (b): k = 1, nA = 1, nR = 5

anchor slippage (type I) cannot be obtained at the upper vertex of the right triangle anymore; instead, it has to locate at some point with larger value of nA and (or) smaller value of nR . As a result, V¯D−K is necessarily lower than V¯K . On the other hand, although the upper vertex of the right triangle can be taken as the optimal gait for type II motion, V¯D−B is still lower than V¯K because backward slippage of the anchoring segments happens, which reduces the average speed of the robot as a whole. To illustrate the above two aspects, two gaits are simulated when N = 11 : k = 1, nA = 2, nR = 4 and k = 1, nA = 1, nR = 5; see Fig. 10. The two gaits correspond to the maximum achievable average speeds of the type I motion V¯D−K and type II motion V¯D−B , respectively. Particularly, the second gait is also the optimal gait OPTK of the kinematic model. It reads from Fig. 10(a) that for the first gait, the velocity of the mass center of the kinematic model X˙ Kc and the velocity of the mass center of the dynamic model X˙ Dc coincide with each other, and the velocity of the anchoring segments x˙A is zero, which mean that no anchor slippage takes place, and the dynamic model degenerates into the kinematic model. However, comparing Figs. 10(a) and (b), X˙ Kc of the first gait (nA = 2, nR = 4) is lower than that of the second gait (nA = 1, nR = 5), which explains why V¯D−K (0.0227 m/s) is lower than V¯K (0.0284 m/s). For the second gait (Fig. 10(b)), owing to the negative velocity of the anchoring segments x˙A , X˙ Dc is lower than X˙ Kc , which explains why V¯D−B (0.0224 m/s) is lower than V¯K (0.0284 m/s). (3) It can be read from Fig. 8 that for N ∈ [43, 61], V¯D−K , V¯D−B , and V¯D−BF exist and are lower than V¯K , whereas V¯D−F is still absent. The reasons for the relatively low magnitudes of V¯D−K , V¯D−B , and V¯D−BF are similar to that stated in (2), that is, fewer relaxing/contracting segments but more anchoring segments, and backward slippage of the anchoring segments. To validate the above analysis, three gaits are simulated when N = 45 : k = 1, nA = 7, nR = 19; k = 1, nA = 4, nR = 20; and k = 1, nA = 1, nR = 22, corresponding to the maximum achievable average speeds of the motions of type I (V¯D−K ), II (V¯D−B ), and III (V¯D−BF ), respectively. In Fig. 11(a), the anchoring segments keep still; in (b), backward velocity of anchoring segments can be found; in (c), large backward velocity and some forward velocity of the anchoring segments are observed. The maximum achievable average speeds V¯D−K , V¯D−B , and V¯D−BF are 0.100 m/s, 0.102 m/s, and 0.097 m/s, respectively, which are all lower than the maximum ideal average speed of the kinematic model V¯K = 0.134 m/s. (4) It reads from Fig. 8 that the maximum achievable average speeds for the four types of motions coexist when N ≥ 62 but are all lower than the maximum ideal average speed of the kinematic model V¯K . Especially, at N = 70, a sudden drop of V¯D−B is observed. This is because at N = 70, motion with backward anchor slippage (type II) cannot be realized if there is only one driving module (k = 1). To achieve motion of type II, two driving modules (k = 2) are required, inducing a limitation on the forward displacement and, hence, an abrupt decline of the average speed. Notice that the break point N = 70 is a little different with that

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Fig. 11 For N = 45, the time histories of the velocity (m/s) of the mass center of the dynamic model X˙ Dc (solid), the velocity of the mass center of the kinematic model X˙ Kc (dashed), and the velocity of the anchoring segments x˙A (dot-dashed) with gaits (a): k = 1, nA = 7, nR = 19, (b): k = 1, nA = 4, nR = 20, and (c): k = 1, nA = 1, nR = 22

Fig. 12 For N = 75, the time histories of the velocity (m/s) of the mass center of the dynamic model X˙ Dc (solid), the velocity of the mass center of the kinematic model X˙ Kc (dashed), and the velocity of the anchoring segments x˙A (dot-dashed) with gaits (a): k = 1, nA = 17, nR = 26, (b): k = 2, nA = 4, nR = 16, (c): k = 1, nA = 13, nR = 31, and (d): k = 1, nA = 10, nR = 32

shown in Fig. 4(f) (N = 72) because at N = 70 or 71, although the region of type II has not been completely covered by the region of type III, no integer values of (nA , nR ) can be taken at a very tiny region of type II. It can be predicted that such kind of drop will continue to occur, for example, at some large value of N , type II motion cannot be realized at k = 2 anymore, and the feasible gait can only be sought for k ≥ 3. Similarly, for N = 75, four gaits corresponding to the maximum achievable average speeds of the motions of type I (V¯D−K ), II (V¯D−B ), III (V¯D−BF ), and IV (V¯D−F ) are simulated, respectively. These four gaits are: k = 1, nA = 17, nR = 26; k = 2, nA = 4, nR = 16; k = 1, nA = 13, nR = 31; and k = 1, nA = 10, nR = 32. As predicted, zero velocity, backward velocity, both backward and forward velocity, and forward velocity of the anchoring segments are observed for the four gaits, respectively (Fig. 12). The maximum achievable average speeds are calculated: V¯D−K = 0.139 m/s, V¯D−B = 0.086 m/s, V¯D−BF = 0.167 m/s, and V¯D−F = 0.172 m/s, which are all lower than the maximum ideal average speed of the kinematic model V¯K = 0.228 m/s. Fewer relaxing/contracting segments and backward slippage of anchoring segments are still the two main reasons for the low average speed. (5) It is noticed that for N = 75, V¯D−BF > V¯D−K and V¯D−F > V¯D−K . A similar phenomenon is also observed in the simulation for N = 45, where V¯D−B > V¯D−K . Hence, it is worth pointing out that although forward and/or backward anchor slippages exist in motions of type II, III, and IV, due to the smaller number of anchoring segments but larger number of relaxing/contracting segments, the maximum achievable average speed of the motions with

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anchor slippage (i.e., V¯D−B , V¯D−F , V¯D−BF ) can be higher than that of the motion without anchor slippage (i.e., V¯D−K ). Generally, when N is very large, we have V¯D−BF > V¯D−F > V¯D−K .

(16)

Here, due to the jump of V¯D−B with respect to different numbers of driving modules, ¯ VD−B is not put into comparison. As a result, in terms of maximum achievable average speed only, motions with anchor slippage (i.e., types II, III, and IV) are not necessarily worse than motion without anchor slippage (type I). (6) As stated in the kinematic model, the normalized average speed in terms of body length is a key index to evaluate the locomotion efficiency of the robot. In this dynamic analysis, it reads from Fig. 9 that the maximum achievable normalized average speed of each type of motion (V¯˜D−K , V¯˜D−B , V¯˜D−BF , and V¯˜D−F ) is no higher than the maximum ideal normalized average speed of the kinematic model (V¯˜K ). On the other hand, compared to Fig. 8, we see a very different scenario on the limiting situation of the maximum achievable normalized average speed when N is very large. Figure 8 reads that the increases of V¯D−BF and V¯D−F do not have an upper limit, but the increase of V¯D−K levels off when N ≥ 62. However, in Fig. 9, V¯˜D−BF and V¯˜D−F also increase with respect to N , but with upper limits; V¯˜D−K increases to a constant and then begins to decline at N = 62, indicating a loss of motion efficiency. Predictably, when N keeps increasing, V¯˜D−K will keep decreasing and will be much lower than the almost constant values of V¯˜D−BF and V¯˜D−F ; V¯˜D−B will tend to several different constants, corresponding to different number of driving modules. Generally, the same relationship of the maximum achievable normalized average speed when N is very large can be obtained (V¯˜D−B is not put into comparison): V¯˜D−BF > V¯˜D−F > V¯˜D−K .

(17)

Based on the above observations of Figs. 8 and 9, it is clear that the change of gait parameters nA and nR significantly affects the maximum achievable average speed of the robot. Notice that the values of nA and 2nR represent the number of anchoring segments and the number of actuating (relaxing and contracting) segments in a robot, respectively. Hence, it is meaningful to study the relationship between the ratio 2nR /nA and the maximum achievable average speed of each motion type. Especially, when N is large enough, the N segment robot can be approximately considered as a continuum model, and the ratio 2nR /nA represents the length ratio of actuating parts and anchoring parts of an artificial earthworm. Figure 13 illustrates the correlations between N and the ratio 2nR /nA corresponding to the optimal gait of each type of motion. When a specific type of motion first appears, its correspondence 2nR /nA ratio scatters by a large range due to the discrete nature of the segmented robot. However, as N increases to large enough, the robot can be treated as a continuous structure, and the 2nR /nA ratio gradually shows regularity. Different types of motion exhibit similar trend, but they converge to slightly different values when N is large enough:



2nR 2nR 2nR > > . (18) nA D−BF nA D−F nA D−K The ratio (2nR /nA )D−B is not put into comparison because it jumps with respect to different number of driving modules (k ≥ 2).

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Fig. 13 For different values of N , ratio 2nR /nA corresponding to the maximum achievable average speed of each type of motion

Comparing Eqs. (16), (17), and (18), one concludes an intrinsic relation that a higher 2nR /nA limit ratio leads to a higher average speed of the robot. Such observation offers valuable insight behind the physics of the earthworm-like robot’s locomotion and provides guidelines for gait design improvement of both segmented models and continuum models. Consequently, the following results of gait optimization can be concluded, based on which, gait design and locomotion prediction of the earthworm-like robot can be achieved. (1) Based on the regional division shown in Fig. 4, the maximum achievable (normalized) average speed for each type of motion and the corresponding optimal gaits are obtained. It is shown that the maximum achievable (normalized) average speed of the dynamic model is no higher than the maximum ideal (normalized) average speed of the kinematic model for two reasons: first, there are fewer actuating segments but more anchoring segments in the obtained optimal gait; second, anchoring segments have some backward slippage. (2) The maximum achievable average speeds V¯D−K , V¯D−BF , and V¯D−F (corresponding to motions of type I, III, and IV, respectively) increase with respect to the total number of segments N , and especially, V¯D−BF and V¯D−F increase without upper limits. However, as to the maximum achievable normalized average speed in terms of body length, V¯˜D−BF and V¯˜D−F increase with upper limits, but V¯˜D−K declines when N is large. Sudden drops of V¯D−B and V¯˜D−B (corresponding to type II motion) appear because of the switch of the number of driving modules. It manifests that increasing the total number of segments of a robot cannot improve the robot’s locomotion ability unboundedly. (3) If aiming at the average speed only, motion without anchor slippage is not necessarily the optimal one. Notice that strong anchors can be maintained at the expense of increasing anchoring segments and (or) reducing actuating segments, which as a result reduces the average speed and brings down the motion efficiency. Instead, allowing the existence of some anchor slippage, the loss of average speed can be reduced. (4) The ratio 2nR /nA represents the length ratio of the actuating parts to the anchoring parts of a segmented or continuous earthworm-like robot. It is found that a larger 2nR /nA limiting ratio can result in a higher average speed of the robot, regardless of whether anchor slippage occurs.

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H. Fang et al. Table 4 Admissible gaits for the eight-segment earthworm-like robot

4 Experimental investigations In this section, the gait generation algorithm and the results on theoretical gait analysis and gait optimization are experimentally validated based on an eight-segment earthwormlike robot prototype. The earthworm-inspired robot is designed and fabricated based on the modeling ideas presented in Part A, which is briefly introduced in Appendix. On the LabView platform, a gait generator is developed to automatically generate the locomotion gaits and to control the robot segments. More detailed discussions of the experimental setup and the program interface can be found in [10]. 4.1 Gait generation and locomotion To investigate and validate the effectiveness of the generated gait, a locomotion test is carried out by operating the eight-segment earthworm-like robot in a horizontal pipe (see Fig. 14), detailed setting of the test can be found in [10]. A gait N = 8, k = 1, nA = 3, nR = 2 is used in the test. Figure 14 displays the video frames snapped at each instant when a transition is over. The robot’s displacement in two periods is marked. It is shown that the eight-segment earthworm-like robot can effectively move in the horizontal pipe. Between each two frames, three segments are anchored with the pipe. In front of the anchoring segments, two segments relax, pushing the forward segments forward; behind the anchoring segments, two segments contract, pulling the backward segments forward. The anchoring segments propagate backward, forming a retrograde peristalsis wave. The test results illustrate that the design of the robot is reasonable, and the general gait generation algorithm as well as the gait generator are effective to construct gaits following the retrograde peristalsis waves. 4.2 Verification of gait analysis Notice that for the eight-segment earthworm-like robots, there are 14 admissible gaits that satisfy the kinematic constraints (3) (listed in Table 4). In this section, the average speed corresponding to each gait will be experimentally measured and compared to the theoretical predictions, from which the optimal gait will be validated. After that, the anchor-slippage phenomena are examined to verify the results on gait analysis. 4.2.1 Average speed Average speed is a key index to evaluate the locomotion ability of a robot. In previous discussions, it has been shown that by optimizing the gait parameters, the average speed of the

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Fig. 14 Test of the eight-segment earthworm-like robot moving in a horizontal pipe with gait N = 8, k = 1, nA = 3, nR = 2. The test runs two periods (2 × 4 transitions). The net displacement (toward right) is marked by x. At each frame, the fully contracted segments are indicated by vertical lines

robot can be maximized. By testing a single robot segment (see details in [10]), the stroke of a robot segment l and the time for one transition t are obtained (rotation angle of the servomotor is set as 130◦ ), t = 1.5 s, l = 1.67 cm. Substituting l and t into Eq. (1) yields the theoretical average speed of the kinematic model. Notice that the obtained theoretical value is the ideal prediction without considering the possible anchor slippage. The experimental values of the average speed are obtained by dividing the displacement by the period (8 × 1.5 s). To validate the optimization results, both the theoretical and experimental values of the average speed corresponding to each gait are plotted in Fig. 15(a). To clearly observe the trend and make comparison, for each value of k (k = 1, 2) and nR (nR = 1, 2, 3), the corresponding curves are displayed in Figs. 15(b)–(e). Figure 15 reveals that the experimental values and trends of the average speed qualitatively follow the theoretical predictions. Furthermore, for most gaits, they also match well quantitatively. The results not only demonstrate that the major characteristics of the kinematic model are well reflected in the robot prototype, but also verify that the kinematics analysis on the locomotion speed is correct. Notice that for some gaits, the differences be-

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Fig. 15 Experimental and theoretical values of the average speed (cm/s) corresponding to each gait of the eight-segment earthworm-like robot. (a) All the gaits, (b) k = 1, nR = 1, (c) k = 1, nR = 2, (d) k = 1, nR = 3, (e) k = 2, nR = 1. The maximum values of the experimental and theoretical average speed are denoted by Vmax _Expre and Vmax _Theore , respectively

tween the theoretical and experimental values are large, which will be analyzed in the next subsection. According to the discussion in Sect. 2.2, OPTK (Eq. (4)) is the optimal gait of the N segment kinematic model, corresponding to the maximum ideal average speed. Hence, we know that the maximum average speed of the eight-segment earthworm-like robot should be achieved with gait No. 11, k = 1, nA = 1, nR = 3. Reviewing Fig. 15(a), one notices that the experimental value of the average speed do reach the peak with gait k = 1, nA = 1, nR = 3, illustrating that the results of gait optimization is correct. 4.2.2 Anchor slippage Notice from Fig. 15 that for the gaits with small number of anchoring segments nA but large number of relaxing/contracting segments nR , relatively large differences between the theoretical and experimental values exist. Specifically, for gaits No. 1 (k = 1, nA = 1, nR = 1), No. 7 (k = 1, nA = 1, nR = 2), No. 11 (k = 1, nA = 1, nR = 3), and No. 12

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Fig. 16 Time histories of the position of each segment of the robot corresponding to (a) gait No. 1, (b) gait No. 7, (c) gait No. 11, (d) gait No. 2, (e) gait No. 8, and (f) gait No. 12, respectively. Transitions are denoted by dashed vertical lines, and anchoring segments are denoted by shaded areas. Obvious backward slippages are indicated by arrows

(k = 1, nA = 2, nR = 3), the experimental values of the average speed are obviously lower than the theoretical values, denoted in Figs. 15(b), (c), and (d) with shaded rectangle. To explain the reason for these differences, for gaits No. 1, 2, 7, 8, 11, and 12 in Table 4, time histories of the position of each segment are plotted in Figs. 16(a)–(f), respectively. The six gaits can be divided into two groups: Gaits No. 1, 7, and 11 possess only one anchoring segment, but have 1, 2, and 3 relaxing/contracting segments, respectively; gaits No. 2, 8, and 12 have two anchoring segments but have 1, 2, and 3 relaxing/contracting segments, respectively. Shaded rectangles are placed on the anchoring segments during each transition for easily identification. It shows in Fig. 16 that for gaits No. 1, 7, 8, and 12, backward slippages occur on some anchoring segments. Recall that in Fig. 15 it is also with these gaits, the experimental average speeds are obviously lower than the theoretical average speeds. In addition, experiments on the other gaits (k = 1, nA = 3, 4, 5, 6 and k = 2, nA = 1, 2) show that backward anchor slippage does not happen anymore, in keeping with the fact that there is little difference

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Fig. 17 For the eight-segment earthworm-like robot, regions of the two types of motion in the nA − nR plane with k = 1. Motion with gaits in region I (dots) does not have backward anchor slippage; motion with gaits in region II (squares) has backward anchor slippage

between the experimental and theoretical average speeds. As a result, we have reason to believe that the lower experimental average speeds are caused by the backward slippage of anchoring segments. Comparing Figs. 16(a), (b), and (c), with the increasing of the number of relaxing/contracting segments nR from 1 to 3, backward anchor slippages become more significant. For gait No. 1, the backward slippages of anchoring segments do exist but are very insignificant; for gait No. 7, the backward slippages are more obvious; for gait No. 11, anchor slippages occur on almost all segments, and especially, significantly large backward slippages are observed on segments #4, #5, and #6. Similarly, trend is also observed when comparing Figs. 16(d), (e), and (f). For gaits No. 2 and No. 8, anchor slippage does not appear at all, whereas for gait No. 12, obvious backward slippages occur on some anchoring segments. Therefore, we know that with the same number of anchoring segments, backward anchor slippage is more likely to happen when there are more relaxing/contracting segments in the robot. Comparing Fig. 16(a) with (d), Fig. 16(b) with (e), and Fig. 16(c) with (f), one notices that with the same number of relaxing/contracting segments, backward anchor slippage is more likely to happen when there is only one anchoring segment in the robot. It is worth pointing out that the performances of the anchoring segments may be different due to the differences among segments. For example, in Fig. 16(c), that is, with gait No. 11, segments #4, #5, and #6 have significant backward-slippages when serving as anchoring segments; however, backward slippage on other segments is relatively small. Figure 17 shows the admissible gaits for the eight-segment earthworm-like robot on the nA − nR plane for k = 1. The gaits corresponding to motions with and without backward anchor slippage are denoted by squares and dots, respectively. It can be estimated that there must exist a boundary that divide the right-triangle region into two parts, similarly as in Fig. 4. In summary, by testing all the admissible gaits of the eight-segment earthworm-like robot, the following results can be concluded. (1) Both qualitatively and quantitatively, the average speeds measured experimentally match well with the theoretical predictions, showing that the robot prototype can be well described by the kinematic model, the kinematic analysis approach is valid, and the theoretical results are correct. (2) The kinematic-model-based optimal gait OPTK is experimentally verified to correspond to the maximum average speed. (3) The differences between the theoretical and experimental average speeds are induced by the backward slippage of the anchoring segments.

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(4) The relationship between the backward anchor slippage and gait design is experimentally verified. Providing there is only one driving module in the robot, backward anchor slippages are more likely to happen when the robot contains less anchoring segments and/or more actuating segments. (5) A region plot that divides the motions of the eight-segment earthworm-like robot is experimentally derived.

5 Conclusion This two-part paper, for the first time, presents a comprehensive investigation on the locomotion characteristics of a metameric earthworm-like robot, which belongs to the general class of metastructure. It quantitatively addresses many critical issues related to the robot locomotion, including the multibody kinematic/dynamic models and behavior, generalized gait design, effects of actuator overload and anchorslippage, average speed, and gait optimization. In Part A, the discussion mainly focuses on the derivation of kinematic and dynamic models and the general gait generation algorithm; Part B is devoted to the interrelationship among physical parameters, gaits and locomotion performance of the robot, as well as experimental verification. The overall lessons learned and the major contributions of this research are summarized as follows: (1) General and rigorous kinematic and dynamic models are developed to describe the locomotion characteristics of an earthworm-like robot of arbitrary number of segments. The dynamic model incorporates the actuator overload and anchor slippage so that their effects on the locomotion performance can be studied in great details (Part A). (2) A gait generation algorithm is synthesized based on the earthworm’s retrograde peristalsis wave mechanism. This algorithm is capable of generating all kinematically admissible gaits for an N -segment robot (Part A). A gait generation program is developed, through which gait can be constructed and used to control the multisegment earthwormlike robot (Part B). (3) With the help of KKT optimization on the kinematic model, the maximum ideal average speed and the corresponding optimal gait for the N -segment robot are obtained. The optimal gait consists of only one anchoring segment but largest admissible number of relaxing/contracting segments. The maximum ideal normalized average speed, in terms of body length per unit time, increases as the number of segments increases, but eventually converges to a constant value that is related to physical parameters (Part B). (4) In practical applications, the maximum ideal average speed is sometimes unachievable due to the actuator overload and anchor slippage; therefore, critical conditions are derived for the occurrence of these two negative effects based on the dynamic model (Part A). Based on these conditions, the dynamic locomotion of the N -segment robot can be categorized into five types: (I) motion without anchor slippage, (II) motion with backward anchor slippage, (III) motion with both forward and backward slippage, (IV) motion with forward anchor slippage, and (V) actuator overload. Generally speaking, when N is sufficiently large, the segmented robot can exhibit different types of motion by switching to different gaits (Part B). (5) The critical conditions are correlated to the physical parameters, such as the number of segments, the anisotropic contact friction coefficient, and the actuator accelerations. Parametric analysis is conducted to investigate these correlations. It is shown that adjusting physical parameters provides a global way to tailor the dynamic behavior of the robot (Part B).

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(6) With the help of numerical simulations on the dynamic model, the maximum achievable average speeds are found for different types of motion. The achievable speeds of the dynamic model are no higher than the ideal speed of the kinematic model. Interestingly, the maximum achievable average speed for motion without anchor slippage (type I) is sometimes lower than that for motions with anchor slippage (types II, III, and IV) (Part B). (7) Investigation of the optimal gaits corresponding to the aforementioned maximum achievable average speeds reveals that the ratio of actuating (relaxing and contracting) segment number over anchoring segment number is closely related to the maximum average speed. A large length ratio of the actuating parts to the anchoring parts can lead to higher average speed of the robot. (8) Finally, experimental investigations on an eight-segment earthworm-like robot validate the kinematic model and the gait generation algorithm. The optimal gait corresponding to the maximum average speed is quantitatively verified. The interrelationship among the locomotion gaits, the anchor-slippage phenomenon, and the average speed are qualitatively confirmed. This comprehensive study provides a wealth of knowledge for the development of metameric earthworm-like robots regarding modeling, gait generation, gait analysis, gait optimization, and locomotion performance. The outcome of this investigation could significantly advance the state of the art of the related academic research and engineering development. Acknowledgements This research was partially supported by the National Natural Science Foundation of China under grant No. 11272236, the US Air Force Office of Scientific Research under grant number FA955013-1-0122, the US National Science Foundation Emerging Frontier and Research Innovation (EFRI-BSBA) program under award number 0937323, and the scholarship from China Scholarship Council (CSC).

Appendix A.1 Optimization procedure for the dynamic model For given physical parameters and a fixed number of segments N , we first investigate whether actuator overload will happen for all possible gaits. Referring to Table 2 of Part A, one notices that the limiting case corresponding to the minimum requirement on segment #(nR + nA + 1, k) is that the driving module only contains one contracting and one relaxing segment (nR = 1) and at the same time there is no unactuated segment in the model. In such a situation, segment #(nR + nA + 1, k) demands the lowest actuating force to provide itself with acceleration and to overcome the friction force, that is, Q#(n

R +nA +1,k)

=

mαlmin + f+ mg. 2

(A.1)

If Q#(n +n +1,k) is still larger than the maximal output Qmax , actuator overload will necesR A sarily take place for all possible gaits, which yields the first limiting case Limiting case (1): If it is satisfied, no feasible gait exists.

mαlmin + f+ mg > Qmax . 2

(A.2)

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Providing that the limiting case (1) is not satisfied, we consider a situation that for any possible gait, anchor slippage does not happen. Notice that the optimal gait of the kinematic model OPTK corresponds to the situation that the only anchoring segment has to assume the largest reaction force. If for gait OPTK , the anchoring segment of the dynamic model does not slip backward/forward, anchor slippage will not happen for any gait. Substituting OPTK into c.c.2 and c.c.3 and letting (c.c.2 )OPTK ≤ 0, (c.c.3 )OPTK ≥ 0 yields the second limiting case:    

N N Limiting case (2): N− − 1 αlmin + (N − 1)f+ g − (1 + η)f− g ≤ 0 2 2    

N N & − N− − 1 βlmin + (N − 1)f+ g + (1 + η)f+ g ≥ 0. 2 2 (A.3) If it is satisfied, the dynamic optimization problem degenerates into the kinematical one, and the optimal gait OPTK is also the optimal gait of the dynamic model, corresponding to the maximum ideal average speed V¯K and maximum ideal normalized average speed V¯˜K . Providing that neither the limiting cases (1) nor (2) are satisfied, optimization has to be carried out on the four types of motion case by case (motion of type V is not studied here). For motion of type I, since no actuator overload and anchor slippage happens, it degenerates into the kinematic model again, and optimization can be performed through KKT conditions, as was used in the kinematic model. For motions of type II, III, and IV, numerical simulations on all legal gaits that satisfy corresponding constraints are performed to find out the optimal gaits that correspond to the maximum achievable average speed and the maximum achievable normalized average speed. Figure 18 shows the optimization procedure of the dynamic model. The optimization can lead to six results. If limiting case (1) is satisfied, actuator overload will always happen, and no feasible gait exists; if limiting case (1) is not satisfied, but limiting case (2) is satisfied, OPTK in Eqs. (4) is the optimal gait; if both limiting cases (1) and (2) are not satisfied, OPTD−K , OPTD−B , OPTD−BF , and OPTD−F are the optimal gaits corresponding to the maximum achievable (normalized) average speed of motions of type I, II, III, and IV, respectively. A.2 Design and prototype of an eight-segment earthworm-like robot In paper Part A, a modeling idea that integrates actuator and anchor into an individual segment is proposed. In the kinematic model, when a segment is in its fully relaxed state, it has the longest length but the smallest diameter, similar as the earthworm’s body segment when the circular muscles are contracted; when a segment is in its fully contracted state, it has the shortest length but the largest diameter, similar as the earthworm’s body segment when the longitudinal muscles are contracted. The fully contracted segment can have extra contact with the working environment, forming an anchorage. Such modeling idea accords well with the earthworm’s morphology structure and as a result makes it possible to adopt earthworm’s locomotion mechanism (i.e., retrograde peristalsis waves) in the robot. Figure 19 shows the earthworm’s body segment, the SolidWorks design, and the prototype of a robot segment. For specifications of the segment, one can refer to [10]. The key components in a segment are the active servomotor, the servomotor-driven cords, and the passive spring-steel belts. When the servomotor is actuated, the horn rotates by a prescribed angle (less than 180 degree) and tensions the cords, thus reducing the distance between the two plates and bending the spring-steel belts. The segment is as a result contracted in the

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Fig. 18 Optimization procedure for the dynamic model

Fig. 19 (a) Cross section of an earthworm’s body segment, (b) SolidWorks design, and (c) prototype of an earthworm-like robot’s segment. The servomotor-driven cord and the spring-steel belts play a similar role to the earthworm’s longitudinal muscles and circular muscles

axial direction and expanded in radial direction. When the servo motor is reversely actuated, the horn rotates back to its original position and loosens the cords, leading to the recovery of the space between the two plates and the shapes of the spring-steel belts. The segment is then extended in the axial direction and narrowed in the radial direction. Comparing the earthworm’s body segment with the robot’s segment, the longitudinal muscles are mimicked by the servomotor-driven cords, and the circular muscles are mimicked by the spring-steel belts. Thus, by alternating the forward and reverse actuation of the servomotor, the segment can repeat the axial contraction (radial expansion) and axial extension (radial contraction) in a similar way as an earthworm’s body segment.

Author's personal copy A comprehensive study on the earthworm-like locomotion

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Fig. 20 (a) Longitudinal section of an earthworm, (b) SolidWorks design, and (c) prototype of the eight-segment earthworm-like robot. The earthworm’s body segment is functionally mimicked by robot’s segment

Assembling eight segments together though screw rods, we obtain an eight-segment earthworm-like robot shown in Fig. 20. Each segment of the robot can be actuated independently, similar to the independent motion of earthworm’s body segment.

References 1. Kim, B., Lee, M.G., Lee, Y.P., Kim, Y., Lee, G.: An earthworm-like micro robot using shape memory alloy actuator. Sens. Actuators A, Phys. 125, 429–437 (2006) 2. Seok, S., Onal, C.D., Wood, R., Rus, D., Kim, S.: Peristaltic locomotion with antagonistic actuators in soft robotics. In: Proc. 2010 IEEE Int. Conf. Robotics and Automation (ICRA), Anchorage, Alaska, USA, pp. 1228–1233. IEEE, New York (2010) 3. Mangan, E.V., Kingsley, D.a., Quinn, R.D., Chiel, H.J.: Development of a peristaltic endoscope. In: Proc. 2002 IEEE Int. Conf. Robotics and Automation (ICRA), Washington, DC, USA, pp. 347–352. IEEE, New York (2002) 4. Boxerbaum, a.S., Shaw, K.M., Chiel, H.J., Quinn, R.D.: Continuous wave peristaltic motion in a robot. Int. J. Robot. Res. 31, 302–318 (2012) 5. Jung, K., Koo, J.C., Nam, J., Lee, Y.K., Choi, H.R.: Artificial annelid robot driven by soft actuators. Bioinspir. & Biomim. 2, S42–S49 (2007) 6. Omori, H., Nakamura, T., Yada, T.: An underground explorer robot based on peristaltic crawling of earthworms. Ind. Robot 36, 358–364 (2009) 7. Kishi, T., Ikeuchi, M., Nakamura, T.: Development of a peristaltic crawling inspection robot for 1-inch gas pipes with continuous elbows. In: Proc. 2013 IEEE/RSJ Int. Conf. Intell. Robot. Syst. (IROS), Tokya, Japan, pp. 3297–3302. IEEE, New York (2013) 8. Saga, N., Nakamura, T.: Development of a peristaltic crawling robot using magnetic fluid on the basis of the locomotion mechanism of the earthworm. Smart Mater. Struct. 13, 566–569 (2004) 9. Bhatti, M.A.: Practical Optimization Methods: With Mathematical Application. Springer, Berlin (2000) 10. Fang, H., Wang, C., Li, S., Wang, K.W., Xu, J.: Design and Experimental Gait Analysis of a MultiSegment In-Pipe Robot Inspired by Earthworm’s Peristaltic Locomotion. In: Proc. SPIE 9055, Bioinspiration, Biomimetics, and Bioreplication. 90550H, San Diego, CA, USA. SPIE, Bellingham (2014)