Design and optimization of a lightweight and compact waist mechanism for a robotic rat

Mechanism and Machine Theory 146 (2020) 103723

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Research paper

Design and optimization of a lightweight and compact waist mechanism for a robotic rat Chang Li a,b, Qing Shi a,b,∗, Zihang Gao a,b, Mengchao Ma a,b, Hiroyuki Ishii c, Atsuo Takanishi c, Qiang Huang b,d, Toshio Fukuda b,d a

School of Mechatronical Engineering, Beijing Institute of Technology, Beijing, 100081, China Beijing Advanced Innovation Center for Intelligent Robots and Systems, Beijing Institute of Technology, Beijing, 100081, China c Faculty of Science and Engineering, Waseda University, Tokyo, 162–8480, Japan d Key Laboratory of Biomimetic Robots and Systems, Beijing Institute of Technology, Ministry of Education, Beijing 100081, China b

a r t i c l e

i n f o

Article history: Received 10 August 2019 Revised 28 October 2019 Accepted 23 November 2019 Available online xxx Keywords: SIMO Mechanism Mechanism optimization Robotic rat Biomimetics

a b s t r a c t A multi-degree-of-freedom (multi-DOF) waist mechanism is required for robotic rats to perform species-typical behaviors, but the existing waist mechanisms are heavy and cumbersome. To solve this problem, we propose a linkage-based slider-coupled symmetric swing (S3 ) mechanism, which features a single-input multiple-output structure, allowing it to couple multiple DOFs. The linkage mechanism has a variety of forms and linkage curves, making the S3 mechanism suitable for multiple-constraint optimization. Based on kinematic and dynamic analyses, we constrain the S3 mechanism in terms of bending angle, transmission angle, symmetry, and its existence, and then we optimize its dimensions using an interior-point method to make it compact. Compared with an existing waist mechanism, the proposed waist mechanism has only 47.8% of the weight and smaller dimensions, making it more lightweight and compact. Experiments on a robotic rat show that the proposed waist mechanism enables a robotic rat to perform rat-like upright rearing in 0.5s and tail grooming behavior in 0.4s, indicating its good biomimetic and dynamic performances. Comparisons between two generation robotic rats also reveal that the robot with new waist mechanism has similar (sometimes superior) pitching and bending abilities with the former one. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Various types of biomimetic robots have been used to study the corresponding animals [1]. For example, Wang et al.[2] used a robot to study the extraordinary adsorption capabilities of the family Echeneidae. Similarly, a robotic bat with silicone-membrane-covered flexible wings was used to study bat flight [3]. Nyakatura et al.[4], with the help of an OroBOT robot, quantitatively examined plausible gaits of Orobates pabsti, an extinct animal. Rat-inspired robots have drawn increasing attention in the field of biomimetic robotics because of the rat’s rapid mobility and extremely agile movements in complex and dynamic environments [5–10]. For example, the PiRat, which is the same size as a rat and has artificial whiskers, has been used for rat social behaviors studies [5]. Even mutual rescue behaviors have been realized between a rat and a robot [6]. Lucas et al. designed a small quadruped rodent robot, whose weight is similar to that of an actual rat, that has ∗

Corresponding author at: Beijing Institute of Technology, School of Mechatronical Engineering, Beijing 100081, China. E-mail address: [email protected] (Q. Shi).

https://doi.org/10.1016/j.mechmachtheory.2019.103723 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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C. Li, Q. Shi and Z. Gao et al. / Mechanism and Machine Theory 146 (2020) 103723

Nomenclature AA BL dymin DOF FI Fi FN FE H hmax p L6 lci lei , Le lmi lpi , Lp lsi , Ls MA Mi MGi mi N Q rymin S3 Sphi SIMO T1 x

αi α max p γi  λ μ ωi ψ ψˆ  ϕi φi φi σ

Abduction-adduction Body length Minimum yaw distance Degree of freedom Input force on the slider Drive force of the ith coupler Supporting force on the slider Flexion-extension Normalized half stroke of the slider Maximum pitch height Last lumbar vertebrae Length of the ith crank Offset distance of the ith slider and its normalized value Distance between the center of mass of the load and the pivot of the ith output rod Length of the ith coupler and its normalized value Initial distance between ith slider and corresponding pivot and its normalized value Additional torque on the slider Drive torque of the ith crank Additional torque on the ith crank caused by gravity Load mass on the ith output rod Number of equidistant points Optimization variables vector Minimum yaw radius Slider-coupled symmetric swing ith Spherical joint of the waist mechanism Single-input multiple-output First thoracic vertebrae Displacement of the slider or its normalized value Angular acceleration of the ith crank Maximum pitch angle Transmission angle of the ith slider-crank mechanism Threshold value for γ i Weight factor for optimization objective function Friction coefficient Angular velocity of the ith crank Bending angle of the rat Bending angle of the S3 mechanism Threshold value for ψˆ ith output angle of the S3 mechanism, i = 1, 2 Angle of the ith crank Angle of the ith coupler Threshold value for allowable difference between ϕ 1 and ϕ 2

various walking patterns [7]. However, most robotic rats have a rigid body and thus can have only a few primitive interactions with rats [10]. To provide better insight into the rat, a robotic rat should have a multi-degree-of-freedom (multi-DOF) body to let it perform more species-typical behaviors with rats [11]. To this end, we previously developed a robotic rat with a multi-DOF body called WR-5M. Because of its agile multi-DOF waist mechanism, WR-5M can perform rat-like pitching and bending movements [12], and thus outperforms the majority of other robotic rats. However, the waist mechanism limits its use. Because a differential gear structure is used [13], the whole waist part is, to some degree, too bulky for the robotic rat. The weight of the waist part is 45% of the whole robot (270 g/600 g) and its height is 80% of the total height (62 mm/80 mm). Because of this cumbersome waist mechanism, the center of gravity of WR-5M must be very carefully designed while the final result shows the robot is still prone to overturn. A new multi-DOF but lightweight and compact waist mechanism is thus required. To achieve this purpose, the DC motors, which account for most of the mass in the waist mechanism, can be replaced with lighter versions or their number can be reduced while maintaining the number of DOFs. Recently, biomimetic robots that use piezo-motors, which are light and can be customized, have been developed. For example, an insect-sized microrobot that can perform untethered flight using piezo-actuated wings and solar energy has been reported [14]. A quadrupedal

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microrobot [15] that combines a piezo-motor and the electroadhension force can invert itself and climb vertical surfaces. However, these alternative motors are unsuitable for our waist mechanism because of their high drive voltage. Other types of motor, such as servo and stepper motors, are also unsuitable because of their high weight or low rated power. To reduce the number of motors while maintaining the number of DOFs, a single-input multiple-output (SIMO) mechanism can be applied to our waist mechanism. Because the bending of the rat’s waist is C-shaped in most cases, this approach does not influence biomimetic performance. A typical SIMO mechanism is a planetary gear train, which allows stable transmission and for which power flow can be specified [16]. However, planetary gear trains have a very complicated structure and they are not suitable for space transmission but for planar transmission. Using a tendon-driven SIMO mechanism, Xu et al. implemented a single-motor-actuated prosthetic hand with 11 DOFs [17] that can realize various daily life grasping patterns. However, a tendon-driven mechanism has several drawbacks, such as low transmission efficiency, poor load capacity, and variable pre-tension in the cable, which limit its use. Among SIMO mechanisms, the multi-linkage-based mechanism has great potential. Van der Wijk et al. presented a SIMO mechanism in [18] that drives two sliders with a single crank. Two couplers move symmetrically if a certain condition is satisfied. A dual-rod slider rocker mechanism has been applied to realize tristate rigid active docking in a compact mobile robot [19]. Because the linkage-based mechanism has a variety of forms and linkage curves, it can be easily optimized for various requirements, such as having the desired end-effector path and smallest drive torque [20], maximum output torque [21], or smallest possible size [22]. This allows the multi-DOF waist mechanism to be optimally designed for lightweight and compactness. In this paper, we propose a multi-DOF biomimetic waist mechanism for use in a robotic rat. The main contributions of this paper are: •

•

•

This waist mechanism is based on the analysis of a rat’s waist and consists of two slider-coupled symmetric swing (S3 ) mechanisms so that it can bend in flexion-extension and abduction-adduction directions. The S3 mechanism has a singleinput double-output structure able to symmetrically swing which allows it to couple two DOFs of the waist mechanism. Particularly, we optimized the size of the waist mechanism under multiple constrains by the interior-point method [23]. Thereby, the optimized waist mechanism is compact, and capable of bending to a designated angle and maintaining a sufficiently large transmission angle during bending; its two output angles are almost symmetrical. Compared with the existing waist mechanism, the proposed waist mechanism has less than half (47.8%) the weight and smaller dimensions of the former. According to the results of experiments and comparisons, the newly designed waist mechanism enables the robotic rat to achieve excellent biomimetic and dynamic performance. The robot with a new waist is able to finish rat-like upright rearing in 0.5s and tail grooming in 0.4s; it also shows similar (sometimes superior) pitching and bending abilities compared with the former robot.

The rest of the paper is organized as follows. An analysis of the rat’s waist, a description of the proposed SIMO mechanism, including its kinematics and dynamics, and the conceptual design of the proposed biomimetic waist mechanism are described in Section 2. In Section 3, the optimization of the proposed mechanism is presented. In Section 4, the design of the biomimetic waist mechanism and some comparison results are given. Conclusions are summarized in Section 5. 2. Design In this section, we first analyze the waist of rats with the help of an X-ray machine and summarize some requirements for a waist mechanism. We then describe the proposed SIMO mechanism and analyze its kinematics and dynamics. Finally, the conceptual design of the waist mechanism is shown. 2.1. Bending analysis of rat’s waist Although we previously analyzed rat motion and configured DOFs for a robotic rat [12], we did not focus on the bending angle of the rat’s waist, which is important for the design of a waist mechanism. We thus first analyzed X-ray images of rat movement, as shown in Fig. 1. We define the bending angle ψ as the angle between the axis of the first thoracic vertebrae T1 and the axis of the last lumbar vertebrae L6 , as indicated in Fig. 1(a). We marked the bending angles during the rat’s upright rearing and tail grooming movements. These two movements respectively represent the rat’s maximum bending capability in two orthogonal directions, namely flexion-extension (FE) and abduction-adduction (AA). They can also be seen as primitive motions because they often appear in turning, exploring, and play fighting. Fig. 1(b) and (c) show the results of the rat’s bending angle. The bending angle for upright rearing ψ P changes from 0◦ to 110◦ and that for tail grooming ψ G changes from 0◦ to 140◦ . Note that because a rat can groom both sides, the grooming bending angle is twice the value shown above. According to our analysis results, the following requirements were derived for a waist mechanism: 1. 2. 3. 4.

It should be able to bend in two orthogonal directions (FE and AA) For FE bending: the range of motion should be not less than 110◦ For AA bending: the range of motion should be not less than 140◦ The waist mechanism should have multiple DOFs but be lightweight and compact

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Fig. 1. (a) Rat skeleton (from Mieke Roth) and its waist bending angle ψ , which is defined as the angle between the axis of the first thoracic vertebrae T1 and the axis of the last lumbar vertebrae L6 . The highlighted part includes all the thoracic and lumbar vertebrae. (b) X-ray images of rat pitching movement and corresponding bending angles ψ P . (c) X-ray images of rat grooming movement and corresponding bending angles ψ G .

2.2. S3 Mechanism As mentioned in Section 1, a linkage structure may be suitable for a lightweight and compact waist mechanism. To minimize weight while maintaining the number of DOFs, we propose an S3 mechanism. A kinematic diagram of the S3 mechanism is shown in Fig 2. The mechanism consists of two slider-crank mechanisms, A1 B1 C1 D1 and A2 B2 C2 D2 , whose sliders are coupled (D1(2) ). A1 O1 and A2 O2 are the respective output rods of A1 B1 and A2 B2 . As slider D1(2) moves a distance x to the left from its initial position (indicated by the gray line), A1 O1 counterclockwise swings through angle ϕ 1 and A2 O2 clockwise swings through angle ϕ 2 . With A1 O1 and A2 O2 taken as representations of the axes of T1 and L6 , this S3 mechanism is a good approximation to the rat’s waist, as shown in Fig. 2(b). The equivalent bending angle ψˆ of the mechanism can be calculated using Eq. (1). This equation makes it convenient to compare the bending similarities between the rat spine and the S3 mechanism. The advantage of the S3 mechanism derives from its structure. The S3 mechanism has two output rods but is driven by only one slider. This SIMO structure allows the use of fewer motors, the main contributors to waist mass, to realize biomimetic and controllable bending. Although the S3 mechanism can couple two non-parallel output shafts, we focus on the parallel case because the rat’s waist movements are symmetrical and we can easily transform the non-parallel case into the parallel case.

ψˆ = ϕ1 + ϕ2

(1)

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Fig. 2. (a) Kinematic diagram of the S3 mechanism. Gray dashed lines represent the initial posture. A1 B1 C1 D1 and A2 B2 C2 D2 are two slider-crank mechanisms whose sliders are coupled. A1 O1 and A2 O2 are the output rods of links A1 B1 and A2 B2 , respectively. (b) Displacements and equivalent bending angle. When D1(2) slides a distance x to left from its initial position, A1 O1 counterclockwise swings through angle ϕ 1 and A2 O2 clockwise swings through angle ϕ 2 . The equivalent bending angle ψˆ is defined as the angle between A1 O1 and A2 O2 . The shape behind the diagram represents an approximation of the rat spine.

Fig. 3. (a) Angle definitions for the S3 mechanism. φ i is for the crank Ai Bi and φi is for the coupler Bi Ci . x represents the displacement of the slider. The positive direction of each variable is shown in this figure. (b) Singularities of the S3 mechanism. When this mechanism is in its singularity postures, i equals zero.

2.2.1. Kinematics Because the S3 mechanism consists of two slider-crank mechanisms, there exist two closed vector loops. These vector loops can be represented by Eq. (2) after defining a coordinate system XY on the initial position of the slider. In this equation, lci , lpi , andlei respectively represent the lengths of the crank Ai Bi , coupler Bi Ci , and offset Ci Di , (i = 1, 2 ). The initial distance between the slider and the pivot is denoted by lsi . The length of Ai Di is thus lsi − (−1 )i x. The angles of the crank and the coupler are denoted by φ i and φ i , respectively, as shown in Fig. 3(a).





lci eφi j + l pi eφ i j = lsi − (−1 )i x e(i−1)π j + lei e

(2i−1 )π 2

j

, (i = 1, 2 )

(2)

By decomposing Eq. (2) into X and Y directions, we get the coordinate representation of the vector loop, as shown in Eq. (3). By taking the first- and second-order derivatives of Eq. (3), we get the angular velocity and acceleration of the crank, as shown in Eqs. (4) and (5), respectively. The velocity and acceleration can be solved only when i does not equal zero, which means that the S3 mechanism should not be at its singularity postures, as shown in Fig. 3(b).



lci cos φi + l pi cos φ i = (−1 )i+1 lsi + x lci sin φi + l pi sin φ i = (−1 )i+1 lei

    Ai v ωi = ωi i 0    2   lci A2i Ai a αi ω i = + αi i −ωi2 i 0 where:

  i = (−1 )i+1 l ei cos φi − (−1 )i+1 lsi + x sin φ i

(3)

(4)

(5)

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Fig. 4. (a) Dynamics model of the S3 mechanism. (b) Forces and torque acting on the slider. Additional torque MA is caused by F1 and F2 . (c) Contact model of the slider in the slot. d represents the distance between the components of the supporting force couple FN . μ is the friction coefficient.

(−1)i+1 l

si +x

lci

Ai =

(−1 )i+1 lei lci

− cos φi

− sin φi

− cos φi



− sin φi

2.2.2. Dynamics The loads on the S3 mechanism are shown in Fig. 4(a). Two blocks, whose masses are m1 and m2 , respectively, are respectively connected to two output rods. The distances between the centers of gravity of the mass and the pivot are lm1 and lm2 , respectively. The torques on the two output rods, M1 and M2 , are generated by forces F1 and F2 , respectively. The dynamics of the S3 mechanism can be formulated as Eq. (6), where MGi , the torque caused by gravity, is a conditional term that only exists in FE bending. 2 Mi − mi lmi αi (+MGi ) = 0

(6)

where

Mi = Fi lci sin γi

⎡ ⎢ γi = cos−1 ⎣



lci2 + l 2pi − lei2 − lsi − (−1 ) x i

2lci l pi

2 ⎤ ⎥ ⎦

(7)

Because of F1 and F2 , an additional torque MA will arise on the slider (shown in Fig. 4(b)). This torque slightly tilts the slider, as shown in Fig. 4(c). It is then balanced by the supporting force couple FN . Given the friction coefficient μ between the slider and the slot, the input force FI on the slider can be written as



FI =



2μle1 − 1 F1 cos φ¯ 1 − d





2μle2 + 1 F2 cos φ¯ 2 d

(8)

2.3. Waist mechanism A robotic rat’s waist should be able to bending in two orthogonal directions, as mentioned in Section 2.1. However, the S3 mechanism can only realize bending in one direction. We propose a waist mechanism that uses two S3 mechanisms placed in two orthogonal planes to solve this problem. The conceptual diagram is shown in Fig. 5(a). The S3 mechanism in the sagittal plane is responsible for FE bending and that in the frontal plane is responsible for AA bending. Their output rods are jointly connected to spherical joints Sph1 and Sph2 , allowing the proposed waist mechanism to perform complicated spatial movements. Compared with our original waist mechanism, this waist uses half the number of motors while maintaining the number of DOFs. Fig. 5 (b) shows an oriented topological graph, drawn according to [24], to show the arrangement of links and joints. The top and bottom halves of the graph show the topology of the S3 mechanism in the sagittal and frontal planes, respectively. In both halves, sliders split the movement into two parts and transmit them to Sph1 and Sph2. Then, these two spherical joints merge the split movements and output spatial rotational movements. 3. Optimization of S3 mechanism As mentioned in Section 2.1, the proposed waist mechanism should be able to bend over certain angles in two directions. To this end, the parameters of the S3 mechanism need to be carefully designed. This slider-driven mechanism must avoid its singularity postures. Moreover, the mechanism should ensure a sufficiently large transmission angle as well as a compact size within its working range. We optimized the mechanism based on these considerations.

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Fig. 5. (a) Conceptual diagram of the waist mechanism. Two S3 mechanisms are placed in the sagittal and frontal planes, respectively. Their output rods are jointly connected to spherical joints Sph1 and Sph2 . (b) Oriented topological graph of the waist mechanism. The top and bottom halves show the topology of the S3 mechanism in the sagittal and frontal planes, respectively.

3.1. Preconditions There are two preconditions for optimizing the S3 mechanism. First, the linkage lengths of the S3 mechanism must satisfy Eq. (9). Second, the linkages must be normalized by the crank’s length lci , as shown in Eq. (10).

l p1 ls1 le1 lc1 = = = =1 lc2 l p2 ls2 le2 Lp =

l pi , lci

Ls =

lsi , lci

Le =

(9)

lei , lci

x=

x lci

(10)

The first precondition states that the two slider-crank mechanisms in the S3 mechanism must be identical. This makes the S3 mechanism totally symmetrical, which reduces the additional torque MA , resulting in smaller friction on the slider. This precondition also makes the positive and negative travels of the slider symmetrical. With the half stroke of the slider defined as H, x should be in [−H, H]. The number optimization variables are reduced under this precondition, making it easier to converge to a reasonable optimal solution. The second precondition makes the linkage lengths nondimensional. Because scaling the S3 mechanism does not influence the performance of the mechanism, this precondition allows the number of optimization variables to be further reduced. So, the following variables are optimized:

Q = [L p , Ls , Le , H] To make the slider displacement x better correspond to the output angles, we redefine the crank angle as ϕ i (x) (shown in Fig. 3(a)), which can be calculated using Eq. (11). In this equation, φ i (0) is a constant that is defined by the first three elements of Q. With this definition, ϕ i (x) is positive when x is positive, and vice versa. Based on the preconditions, ϕ 2 (x) always equals −ϕ1 (−x ).

ϕi (x ) = (−1 )i [φi (x ) − φi (0 )] ⎛ ⎜ ⎝

= (−1 )i ⎜tan−1



Le Ls − (−1 ) x i

− tan−1

Le + cos−1 Ls

Ls − (−1 ) x i



2

2

⎞ + L2e + 1 − L2p

Ls − (−1 ) x i

2

+

L2e

− cos−1

L2s + L2e + 1 − L2p ⎟



2 L2s + L2e

⎟ ⎠

(11)

3.2. Constraints and the objective of optimization Before the S3 mechanism is optimized, the following constraints should be taken into consideration.  Constraint on bending angle The waist mechanism should be able to bend to a certain angle. To this end, the constraint on the bending angle is considered. The S3 mechanism reaches its maximum bending angle when its slider moves to H or −H. Therefore, according to Eq. (1) and our preconditions, this constraint can be written as

ϕ1 (H ) − ϕ1 (−H ) ≥ Ψ where Ψ is a positive threshold value. It is 140◦ for AA bending and 110◦ for FE bending.

(12)

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C. Li, Q. Shi and Z. Gao et al. / Mechanism and Machine Theory 146 (2020) 103723 Table 1 Optimization parameters and variables for two orthogonal bending directions . Optimization parameters

FE S3 mechanism AA S3 mechanism

Optimized variables Q

Ψ

σ

N

Γ

λ

Lp

Ls

Le

H

lci

120◦ 150◦

1.8◦ 1.8◦

5 5

40◦ 20◦

0.8 0.2

1.976 3.350

1.471 2.117

1.370 2.385

0.990 1.345

5 mm 5 mm

 Constraint on symmetry The two slider-crank mechanisms in the S3 mechanism are identical (first precondition). However, their output angles, namely ϕ 1 and ϕ 2 , are not necessarily equal to each other. This non-equality may cause unequal force conditions between the two output rods, increasing friction on the slider. Therefore, we set the following symmetry constraint

ϕ1 (x ) − ϕ2 (x ) = ϕ1 (x ) + ϕ1 (−x ) = 0, ∀x ∈ [−H, H]

(13)

However, this constraint is too strict (i.e., there are no feasible solutions under this constraint). Therefore, we appropriately ease this constraint. As shown in Eq. (14), we constrain N equidistant points of the left side of Eq. (13) so that its value is under σ , a sufficiently small positive number. Because the left side of this formula is an even function, we only need to consider the case for x > 0.



ϕ1

k H N





+ ϕ1

k − H N



≤ σ,

∀k ∈ [1, 2, · · · , N](σ > 0 )

(14)

 Constraint on transmission angle The transmission angle is defined as the smaller angle between the crank Ai Bi and the coupler Bi Ci [25]. It reflects the quality of force or torque transmission of a mechanism [26] and a transmission angle close to 90◦ will have better force transmission performance. It is thus necessary to constrain the transmission angle of the S3 mechanism to a proper value. With Γ defined as the lower limit of the transmission angle, this constraint can be written as



H ≤ min

1+

L2p

−

L2e

+ 2L p cos  − Ls ,





Ls −

1+

L2p

−

L2e

− 2L p cos  ,

Γ ∈ (0◦ , 90◦ )

(15)

 Constraint on existence of S3 mechanism This constraint guarantees that the S3 mechanism can be physically configured with the given variables in Q. In such a scenario, every element of Q should be a positive or non-negative real number, expressed as

L p > 0,

Ls > 0,

Le ≥ 0,

H>0

(16) S3

Here, only Le is allowed to equal zero; in such a scenario, the mechanism is allowed to consist of two in-line slider-crank mechanisms. Furthermore, we should also constrain the root components of Eq. (15) so that H is not a complex number. This constraint can be written as





1 + L2p − L2e − 2L p cos  ∈ 0, L2s





1 + L2p − L2e + 2L p cos  ∈ L2s , +∞

(17)

So far, the first three requirements in Section 2.1 can be implemented based on the properties of the mechanism and the four constraints stated above. However, the last requirement is still not met. To make the waist part sufficiently compact, the span and width of the S3 mechanism should be minimized. To this end, we set the objective function as Eq. (18). The first and second terms in the equation reflect the length and width of the S3 mechanism, respectively. We use the factor λ to weight these two terms to indicate their importance.

f = λ ( L s + H ) + ( 1 − λ ) Le

(18)

3.3. Optimization results In this section, we conduct optimizations considering the various requirements for FE and AA bending. The respective optimization parameters for these two types of bending are shown in Table 1. Here, we slightly extend the bending angle (10◦ more than that of the rat) to allow our mechanism to tolerate errors caused by the system or the assembly process. The difference σ between the two output angles of the S3 mechanism is controlled to be less than a hundredth of π , and N is set to 5. The two output rods can thus almost symmetrically swing. Furthermore, transmission angle threshold  is set to 40◦ and 20◦ for FE and AA bending, respectively. FE bending has a larger transmission angle than that of AA bending because the former bending must overcome the force of gravity caused by its loads. Finally, factors λ used in the optimization objective function are selected according to the design of the waist mechanism given in Section 4. For example, λ is set to 0.8 for the FE S3 mechanism because the height of the waist mechanism is more limited than its length.

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Fig. 6. Variation of angular displacements ϕ (top row), velocities ω (middle row), and accelerations α (bottom row) with respect to slider displacement for the optimized FE S3 mechanism (left column), the optimized AA S3 mechanism (middle column), and a manually designed S3 mechanism (right column).

To find the optimal solution for both mechanisms, an interior-point optimization method [23] was used in Matlab (R2018b). The final optimization results are shown in Table 1. We used an interior-point method because its initial variables and the variable generated during iterations are within the feasible region, avoiding complex numbers during the intermediate calculation process. Other optimization methods, such as exterior-point methods, genetic algorithms, and particle swarm optimization, generate complex numbers during their iteration process. We compared the differences between the two output rods in terms of angular displacement, velocity, and acceleration. In addition to the optimized FE and AA S3 mechanisms, a manually designed S3 mechanism was compared to show the difference between optimized and unoptimized mechanisms. The following variables were used for the manually designed mechanism:

Qmanual = [1.5, 1.48, 0.75, 0.8] Because it was difficult to manually design a mechanism that satisfies all four constraints, we attempted to make it perform similarly to the optimized FE S3 mechanism. The comparison results are shown in Fig. 6. In this figure, the top, middle, and bottom rows of charts respectively show the variation of angular displacements ϕ , velocities ω, and accelerations α with respect to slider displacement. The left, middle, and right columns are for the optimized FE S3 mechanism, optimized AA S3 mechanism, and manually designed mechanism, respectively. According to the figure, the angular displacements of the optimized S3 mechanisms almost linearly change with respect to slider displacement. This is also reflected in the fact that their displacement error between the two output rods is almost zero (within ± σ ). However, the case for angular velocity and acceleration is different. As the slider gradually deviates

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Fig. 7. Variation of transmission angles (γ ) with respect to slider displacement for (a) the optimized FE S3 mechanism, (b) the optimized AA S3 mechanism, and (c) a manually designed S3 mechanism.

from the initial position, the velocity and acceleration differences of the two output rods become larger. These differences are caused by incomplete equality of angular displacement. Despite this, the velocity and especially acceleration errors are at considerably lower levels across most of the slider’s stroke ([−H, H]) compared with those for the manually designed S3 mechanism. Although the manually designed mechanism is more compact, its displacement and acceleration errors are almost twice those of the optimized mechanisms, which means that it has comparatively poor dynamic performance. The two optimized S3 mechanisms could bend to their designated angle, whereas the manually designed one could bend to only 120◦ . A comparison of the transmission angle was also conducted. This angle was calculated using Eq. (7). The results are shown in Fig. 7. Because of the symmetry of the mechanism parameters, the two transmission angles about the respective output rods show symmetrical variations. For the optimized mechanism, the transmission angles are above their prescribed threshold value, indicating the effectiveness of our optimization. However, the minimum transmission angle for the manually designed mechanism is less than 40◦ , which does not satisfy the prescribed threshold value for the FE mechanism. In other words, it is difficult to control the transmission angle when manually designing an S3 mechanism. In summary, the optimized mechanisms both met their design requirements, and outperformed the manually designed mechanism. Of note, the variable x in Figs. 6 and 7 is normalized by the crank length lci , as shown in Eq. (10). Therefore, the result may be a little different when lci does not equal 1 mm. For Fig. 6, the curves will, for example, grow along the vertical axis if lci is greater than 1 mm. For Fig. 7, if lci is greater than 1 mm, the ranges of transmission angles will remain unchanged except for a decrease in their slopes. 4. Evaluations and comparisons To verify the performance of the optimized S3 mechanism and the proposed waist mechanism, we modeled this waist mechanism in SolidWorks (2016SP3). Because the optimized variables Q are normalized by lci , we first scaled these variables by factor lci when designing the waist mechanism. The factor values for the two S3 mechanisms used in the waist mechanism are shown in the last column of Table 1. To drive the slider by a DC motor, we connected the slider and the motor by two linkages to form another crank-driven slider-crank mechanism. In such a scenario, the power flows from the DC motor to the slider, is then split into two parts by the S3 mechanism, and flows into the corresponding output rods. The motor used in this waist mechanism is a Faulhaber 1516 series DC motor with a reduction gearbox and an encoder. We verified that the motor is strong enough to drive the waist and its loads. The final model of the waist mechanism is shown in Fig. 8(a). Furthermore, we made a robotic rat with the newly designed waist mechanism (as the model shown in Fig. 9(a)) to test the waist’s performance. This robotic rat consists of 4 parts: the head, the forelimb, the waist, and the hip, and it can communicate with a computer through the WIFI module such that the robot can be wirelessly controlled. 4.1. Performance evaluations We evaluated the biomimetic and dynamic performance of the robotic rat to show the effectiveness of the new waist ), maximum pitch height (hmax mechanism. For biomimetic performance, we used the maximum pitch angle (α max p p ), minimin min mum yaw radius (ry ), and minimum yaw distance (dy ), the same definition as used in [12], as the parameters to characterize the upright rearing and tail grooming between the robotic rat and the rat. As shown in Fig. 9(b), the maximum pitch angle of the robot is 100◦ , bigger than that of the rat; the maximum pitch height is 1.2BL, close to that of the rat (1.5BL), where BL is the abbreviation of body length. Both parameters indicate that the robot has a similar (sometimes superior) pitching capability of the rats. In terms of grooming, the robot’s minimum yaw radius reaches 0.22BL, close to the value of rats (0.2BL). The minimum yaw distance of the robot (0.35BL), however, cannot reach a similar level of rats because rats

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Fig. 8. Designs of (a) proposed lightweight and compact waist and (b) WR-5M waist.

min ) and maximum pitch height (hmax Fig. 9. (a) Robotic rat with new waist mechanism; (b) maximum pitch angle (α max p p ); (c) minimum yaw radius (ry ) and minimum yaw distance (dymin ).

can touch their hip when grooming (dymin =0BL). The cause for this difference lies in the rat’s soft body which enables better extension ability than the robotic rat. For dynamic performance, we tested the minimum time when the robotic rat stably pitched or bended to its limit position. As shown in Fig. 10, the robotic rat with a new waist mechanism is able to rear up to the limit position within 15 frames (around 0.5s) and bend to the minimum bendable position within 12 frames (around 0.4s). According to our observations, rats can finish upright rearing in 0.7s or bend its body to groom tail in 0.5s, both of which are longer than those of robotic rat. Therefore, the robotic rat has good dynamic performance with our new wait mechanism.

4.2. Comparisons between the new and the former waist As a comparison, the waist mechanism used in WR-5M is shown in Fig. 8(b). The height and length of the new waist mechanism are 46.6mm and 49mm, respectively, much lower than those of WR-5M (62mm and 68.2mm, respectively). Besides, the new waist mechanism weighs only 129g, less than half the weight of WR-5M (270g). These results show that the proposed waist mechanism meets the requirement for compactness and lightweight. Moreover, the robot with a former waist is not as good in the dynamic performance as the one with a new waist. Under the stable condition, the robot with the former waist requires 1.2s to rear up to its limit pitching positions, or 0.5s to bend its body to the minimum radius. We also compared the biomimetic performance between the two robotic rats. For WR-5M robot, the four extreme pamin min = 65◦ , hmax rameters are measured as α max p p =1.2BL, ry =0.2BL, and dy =0.22BL. Clearly, the new robotic rat outperforms the WR-5M in the pitching capability since it has a larger extreme pitch angle, while its bending capability is slightly decreased compared with WR-5M. However, the decreasing of bending capability is not caused by our waist mechanism but by the interference between different parts of the robot. According to Fig. 9(c), the bending angle is only 110◦ in the extreme bendable position, much less than the designed value. This is because the forelimb and the hip parts collided with the waist

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Fig. 10. Snapshots of (a) robot upright rearing, and (b) robot tail grooming.

part and, as a result, blocked the bending. However, the decrease of the bending capability is so considerably small that we can say that those two robots have similar bending abilities. 5. Conclusions In this paper, we presented an optimal design of a multi-DOF, lightweight, and compact waist mechanism for a robotic rat. Based on the rat’s waist analysis, we proposed an S3 mechanism and investigated its kinematics as well as dynamics. Then, by defining the optimization function and constraints, we optimized both the FE and AA S3 mechanisms by using an interior-point method to make them compact without influencing their functionality. Comparisons show that the optimized mechanisms outperform a manually designed S3 mechanism. Besides, the new waist mechanism has only 47.8% the weight and smaller size of the former, showing its lightweight and compactness. To verify the effectiveness of the designed waist mechanism, we made a robotic rat and measured its four extreme parameters: maximum pitch height, maximum pitch angle, minimum yaw radius, and minimum yaw distance to quantify the robot’s pitching and bending abilities. Experimental results proved that our new waist mechanism endows the robot with similar pitching and bending abilities of rats. Besides, the new robotic rat shows excellent dynamic performance because it can stably pitch or bend to its extreme position in 0.5s, a relatively short time compared with rats and WR-5M. Although some comparisons show that the new robot’s bending capability is slightly decreased compared to that of the WR-5M, we proved that the decrease is not caused by the waist mechanism but by the interference between different parts of the robot, and the decrease is considerably small. In general, our newly designed waist mechanism outperforms the former one and enables the robotic rat to achieve biomimetic movements with high dynamic performance. In the future, we will use the robotic rat with the proposed waist mechanism to interact with real rats, and further optimize the waist or other parts of the robot according to the interaction feedback. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported in part by the National Key R&D Program of China under the grant No. 2017YFE0117000, the National Nature Science Foundation of China (NSFC) under grant No. 61773058 and 61627808. We thank Adam Przywecki, B.Eng, from Liwen Bianji, Edanz Editing China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript.

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