Type synthesis of a class of novel 3-DOF single-loop parallel leg mechanisms for walking robots

Mechanism and Machine Theory 145 (2020) 103695

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Research paper

Type synthesis of a class of novel 3-DOF single-loop parallel leg mechanisms for walking robots Luquan Li, Yuefa Fang∗, Sheng Guo, Haibo Qu, Lin Wang School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, PR China

a r t i c l e

i n f o

Article history: Received 21 October 2019 Accepted 5 November 2019

Keywords: Quadruped robot Type synthesis Single-loop leg mechanism Screw theory Equivalent drive mechanism

a b s t r a c t This paper presents a class of novel 3-degree-of-freedom (DOF) single-loop parallel leg mechanisms for walking robots. An approach for constructing the 3-DOF single-loop parallel leg mechanism is proposed. The single-loop parallel leg mechanism is composed of a serial limb with two actuated joints and a serial limb with one actuated joint, where two serial actuated joints of a limb are replaced by an equivalent drive mechanism. The novel legs can generate three translation (3T), two rotation and one translation (2R1T), and one rotation and two translation (1R2T) motions, which make the workspace of the foot-endpoint three-dimensional in volume, and the robots can walk flexibly and omnidirectionally. Compared with existing leg mechanisms, the novel leg mechanisms have the potential advantages of larger step length and step height. This will enable the walking robot to move faster and step over larger obstacles. A quadruped robot is built with novel leg mechanisms, and its performances are analyzed and omnidirectional walk is demonstrated by simulation. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Legged robots have been a focus in the field of mobile robotics in recent decades. Researchers have presented many different legged robot structures [1]. These leg mechanisms can be classified into five types according to their structures. The first type of leg mechanism is 1-DOF linkages. For instance, Plecnik and McCarthy [2] presented a design procedure for Stephenson I-III six-bar linkages and used six-bar linkage leg mechanisms to construct a six-leg walker. Zhang et al. [3] designed a locust-inspired robot with a 1-DOF jumping leg based on the jumping stability mechanism of locusts. Wu and Yao [4] proposed a novel six-bar closed-chain leg mechanism with variable topological structures that acts as a single leg to generate different gaits. Jansen’s leg mechanism [5] is a 1-DOF planar 8-link leg mechanism that can be used in mobile robots. Chen et al. [6] designed a six-bar linkage leg mechanism that is derived from a four-bar linkage. All of the abovementioned leg mechanisms reduce the number of actuators, but the robot’s flexibility is decreased because the trajectory of the foot-endpoint is fixed. The second type of leg mechanism is 2-DOF planar linkages, which allow step length and height to be varied. For instance, Guo et al. [7] presented a parallel actuated pantograph leg with an approximately decoupled configuration. Li and Ceccarelli [8] designed a 2-DOF modified planar LARM leg that can overcome obstacles. Kenneally et al. [9] used a symmetric planar five-bar linkage with a coaxial mechanism as the leg mechanism to develop a quadruped robot. Planar 2-DOF leg mechanisms have enhanced ability to overcome obstacles, but the trajectory of the ∗

Corresponding author. E-mail address: [email protected] (Y. Fang).

https://doi.org/10.1016/j.mechmachtheory.2019.103695 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

foot-endpoint is a two-dimensional surface, which makes it difficult to change the direction of motion. The DOFs of the first and second types of leg mechanisms are less than three, allowing the robots to have few motors and a simple control system. Although the abovementioned robots can also turn by differential steering, they cannot walk omnidirectionally. In certain narrow and uneven environments, these robots must choose from a limited set of motions when maintaining posture and stability. The third type of leg mechanism is multi-DOF serial open chain leg mechanisms. For example, MIT designed a Cheetah robot [10]. Boston Dynamics presented a quadruped robot named SpotMini [11]. Because of the open chain structure, the carrying capability of the robot is low. Additionally, some motors mount on distal joints that create large swing inertias. To improve the carrying capability of walking robots, a fourth type of leg mechanism was introduced based on parallel topologies. Giewont and Sahin [12] designed an omnidirectional quadruped robot that used a 3-RSR parallel mechanism as the leg mechanism. Gao and Pan [13,14] presented a hexapod robot named ‘Octopus’ whose legs are 1UP+2UPS parallel mechanisms. Russo et al. [15,16] developed a tripod parallel leg mechanism based on the 3-UPR parallel mechanism. Wang et al. [17] investigated a biped locomotor with two 3-UPU parallel leg mechanisms. Parallel leg mechanisms with at least 3-DOF allow a large selection of gaits and increase load capacity. Due to the geometric restrictions and interferences between limbs and the moving platform, parallel leg mechanisms generally have small workspaces, which limit step length and height. Combining the advantages of serial and parallel mechanisms, a fifth type of leg mechanism is based on hybrid topologies. Zhang et al. [1] synthesized a series of 3-mixed-DOF protectable leg mechanism based on the hybrid mechanism, where the parallel mechanisms are used as the driving mechanism, and the serial mechanisms are used as the walking mechanism. Other serial-parallel hybrid leg mechanisms are proposed in [18,19]. Although hybrid leg mechanisms have many merits, such as high flexibility and the capacity to carry heavy loads, most of them are complex because they consist of both serial and parallel topologies. Therefore, it is necessary to design a class of novel leg mechanisms to overcome the abovementioned drawbacks. To avoid the drawbacks of complex structures and the small workspace of parallel mechanisms, a class of fewer-limb multi-DOF parallel mechanisms were proposed [20–29] in which the number of limbs is less than the number of DOFs. These mechanisms reduce the number of limbs connecting the moving platform to the fixed base to decrease the chances of link interferences with each other and the geometric restrictions on the moving platform. In fewer-limb multi-DOF parallel mechanisms, each limb may contain more than one actuated joint and all motors are mounted on the fixed base [20,21,23–26]. This leads to a simple structure and a large workspace when compared with a general parallel mechanism (the limb number is not less than the DOF number) [23,24,26]. For instance, Lee and Lee [22] proposed three novel twolimbed parallel Schönflies-motion generators. The Monash Epicyclic Parallel Manipulator (MEPaM) is a three-legged 6-DOF parallel mechanism [23–25]. Other three-limbed 6-DOF parallel mechanisms are introduced in [26–29]. To avoid moving actuators, an equivalent drive mechanism is proposed to replace two serial actuated joints on a limb. Abeywardena and Yoon et al. [24,29]applied cable-driven and linkage-driven equivalent drive mechanisms to replace two serial actuated joints with parallel axes. Harada et al. [21] designed a type of C-drive (cylindrical drive). Eskandary and Angeles [20] proposed a series of cable-driven C-drives with an arbitrarily large pitch. Considering the existing drawbacks of the abovementioned leg mechanisms and the advantages of the fewer-limb multiDOF parallel mechanisms, in this work, a class of novel single-loop parallel leg mechanisms with two limbs and 3-DOFs are presented. With only two limbs, the structure of the novel leg mechanisms is simpler. Limb interference can be avoided effectively, and the workspace of the foot-endpoint of the mechanisms will be increased. As a result, the maximum step length and maximum step height of the walking leg mechanism will be increase. The novel leg mechanism reduces the number of limbs and moving parts; furthermore, all motors are mounted on the robot’s body. In addition, novel leg mechanisms enable the quadruped robot to walk more flexibly and omnidirectionally. This paper is organized as follows: Section 2 details the approach for constructing novel single-loop leg mechanisms. Section 3 determines the motion types of novel leg mechanisms. Section 4 synthesizes single-loop parallel leg mechanisms based on the constraint synthesis method and the principle of equivalent substitution. Section 5 analyses the performances of a novel leg mechanism and compares it with other three-limb parallel mechanism. Finally, the virtual prototype of a representative quadruped robot is built, and the omnidirectional walk performance of the quadruped robot is verified by simulation. 2. Approach to constructing novel 3-DOF single-loop parallel leg mechanisms The goal of this paper is to construct a class of novel leg mechanisms for walking robots with simple structures and flexible and omnidirectional walks. The 3-DOF leg mechanism can meet these requirements. Generally, 3-DOF parallel mechanisms have three limbs. The fewer-limb 3-DOF parallel mechanisms have only two limbs, that is, fewer-limb 3-DOF parallel mechanisms are 3-DOF single-loop parallel mechanisms. Therefore, the aim of this work is to construct a class of novel 3-DOF single-loop parallel leg mechanisms. The steps for constructing a novel single-loop parallel leg mechanism are as follows: Step 1. Determine the motion types of the leg mechanism’s output link. The motion of the output link must guarantee that the workspace of the foot-endpoint is a three-dimensional volume. Step 2. Synthesize various serial limbs that can be used to construct single-loop mechanisms and select the feasible serial limbs that are suitable for constructing the single-loop parallel leg mechanism. Step 3. Construct the 3-DOF single-loop parallel mechanisms with feasible serial limbs.

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Step 3

3

Step 2 Serial kinematic limbs Joint Single-loop closed chain

Link

Limb-2

Limb-1 Step 4

Output link

Step 5

Equivalent Two serial actuated joints drive mechanism Two serial actuated joints

Actuated joint Fixed base Fig. 1. Construction process of a novel single-loop parallel leg mechanism.

(a)

z

o

z

(b)

y

o

y

x

y

o

Rigid body

Rigid body

x

z

(c)

x

Rigid body

Fig. 2. 3D spatial motion types of a rigid body: (a) 3T motion (b) 2R1T motion (c) 1R2T motion.

Step 4. Design the equivalent drive mechanisms that can equally substitute for two serial actuated joints. Step 5. Construct the single-loop leg mechanisms by using equivalent drive mechanisms to replace the two serial actuated joints of one limb. According to the above five steps, the entire process for constructing a novel 3-DOF single-loop parallel leg mechanism is shown in Fig. 1. 3. Motion types of leg mechanism For a quadruped robot to walk flexibly and omnidirectionally, the workspace of the foot-endpoint of the robot must be a three-dimensional volume. There are three motion types that satisfy the above requirement, namely, 3T(Tx Ty Tz ), 2R1T(Rx Ry Tz ) and 1R2T(Rx Tx Tz ), as shown in Fig. 2. In general, a rigid body, such as the output link of a leg mechanism, is considered to be formed by a number of closely linked points [1]. When a rigid body undergoes these motion types, the workspace of a point on the rigid body will be a three-dimensional volume. Therefore, for the output link that generates these 3D motions in a parallel leg mechanism, the workspace of the foot-endpoint is also a three-dimensional volume. Three typical serial limbs that can generate these three motion types are considered to show the workspace of the endpoints and to prove that the proposed three motion types meet the requirements, as illustrated in Fig. 3. Hence, in this work, we construct the novel leg mechanism with 3T, 2R1T and 1R2T motions. 4. Type synthesis of novel 3-DOF single-loop parallel leg mechanism In this section, according to the construction process shown in Fig. 1, feasible serial limbs are synthesized, and then equivalent drive mechanisms are designed. Finally, a class of novel single-loop leg mechanisms are synthesized.

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L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

(a)

(b) P

(c)

R

P

P

R

P

R

R

Endpoint

Endpoint

R

Endpoint

Fig. 3. Three typical serial limbs: (a) PPP limb (b) RRR limb (c) PRR limb.

$dr srd

αd

z

ad rrd rri

sri

αi

ai

a1 s

α1

r rr1 o

$ir

$ sr1

$1r y

Rigid body

x Fig. 4. A d-dimensional wrench system on a rigid body.

4.1. Constraint synthesis of feasible serial limbs When synthesizing single-loop parallel leg mechanisms, feasible serial limbs need to be constructed first. Both displacement group theory and screw theory can be used to construct the serial limbs [30–35]. Compared with the displacement group theory, the screw theory is more complex, but the relationships among joint axes, the constraints of limbs and the output link are presented in detail. As a consequence, the serial limbs are constructed by screw theory and the virtual work principle based on the constraint synthesis method in the following section. Then, the feasible limbs are selected according to the corresponding requirements. 4.1.1. Constraint synthesis of serial limbs A d-dimensional wrench system $r exerts on a rigid body with the infinitesimal twist $, as shown in Fig. 4, where d stands for the dimensions of the wrench system, namely, the number of independent constraints in the wrench system. In Fig. 4, s and r are the corresponding direction and location vectors in the twist $, respectively. rri and sri represent the corresponding direction and location vectors in the ith constraint screw $ri , respectively. α i and ai stand for the corresponding twist angles and common normal vectors between s and sri , respectively. Note that the constraints include the constraint force (F) and constraint couple (C). We use an additional subscript F or C to represent the corresponding vectors and parameters of a constraint force or a constraint couple. For example, rrFi , srFi , aFi and α Fi are vectors and parameters of a constraint force, and srCj , aCj and α Cj are vectors and parameters of a constraint couple. The virtual work [36–38] produced by the wrench system can be written as follows



δW = τ

k  i=1

[(h+hrF i )cosαF i − aF i sinαF i ]+

m  

(h+hrC j )cosαC j − aC j sinαC j





, (k + m = d )

(1)

j=1

where δ W is the virtual work of the rigid body; τ is the product of wrench intensity and twist rates; k and m are the number of constraint forces and constraint couples, respectively; hrFi , hrCj and h represent the pitch of the ith constraint force, the pitch of the jth constraint couple and the pitch of the twist, respectively; and aFi and aCj are modules of common normal vectors aFi and aCj , respectively. Additionally, for a constraint couple with an infinite pitch, hrCj =∞, aCj =0. The pitch of the constraint force is equal to zero, i.e., hrFi =0. In general, τ is not equal to zero. Hence, the reciprocal condition can be satisfied by adjusting α Fi , α Cj and aFi . The output link of a mechanism is restricted by serial limbs and generates corresponding motions. According to the constraint characteristics of the output link, there may exist a LkFmC -limb, denoting a serial limb, that exerts k(k = 0,1,2,3)

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

5

L0F1C

L0F2C L3F2C

Fig. 5. Construction process of L0F1C -limb.

constraint forces and m(m = 0,1,2,3) constraint couples on the output link. The right subscript ‘kFmC’ represents the number of constraint forces and the constraint couples exerted on the output link by the serial limb. The constraint force is denoted as F, and the constraint couple is denoted as C. The right subscript ‘k’ represents the number of constraint forces, and the right subscript ‘m’ represents the number of constraint couples. As shown in Fig. 2(a), three linear independent constraint couples can be applied to a rigid body, as a result, the rigid body can generate the Tx Ty Tz -motion type. These constraint couples can be supplied by combinations of two limbs among the L0 F 1 C , L0 F 2 C and L0 F 3 C -limbs. In Fig. 2(b), two linear independent constraint forces and one constraint couple can be applied to the rigid body such that the rigid body can generate the Rx Ry Tz -motion type. These constraints can be supplied by combinations of two limbs among the L1 F 0 C , L0 F 1 C , L2 F 0 C , L1 F 1 C and L2 F 1 C -limbs. In Fig. 2(c), two linear independent constraint couples and one constraint force are exerted on the rigid body, causing the rigid body to exhibit the Rx Tx Tz motion type. These constraints can be supplied by combinations of two limbs among the L1 F 0 C , L0 F 1 C , L0 F 2 C , L1 F 1 C and L1 F 2 C -limbs. Hence, to construct single-loop parallel mechanisms, the abovementioned limbs are constructed first. Note that the unconstrained LD -limbs can also be used to construct single-loop parallel mechanisms. When d = 5, five constraints are exerted on the output link, and the twist of the serial limb can be formed by a single revolute (R) joint or a single prismatic (P) joint. The single R joint and the single P joint correspond to the L3 F 2 C -limb and the L2 F 3 C -limb, respectively, and are the most basic limbs. All limbs can be composed of L3 F 2 C -limbs and L2 F 3 C -limbs according to the geometric relationships between the joints and the constraints associated with limbs. When the dimension of the wrench system is four, i.e., d = 4, the joint screw of the limb is reciprocal to three constraint forces and one constraint couple, two constraint forces and two constraint couples, or one constraint force and three constraint couples. Hence, the limbs can be L3 F 1 C , L2 F 2 C and L1 F 3 C -limbs. L3 F 1 C -limbs can consist of two linearly independent R joint without P joint. There is no R joint in the L1 F 3 C -limbs, but two linearly independent P joints can form the L1 F 3 C -limbs [37]. For L2 F 2 C -limbs, based on Eq. (1), the virtual work is produced by two constraint forces and two constraint couples as follows

δW = τ · [hcosαF 1 − aF 1 sinαF 1 + hcosαF 2 − aF 2 sinαF 2 + (h + hrC1 )cosαC1 + (h + hrC2 )cosαC2]

(2)

For an R joint in the limbs, h = 0. To satisfy the reciprocal condition, the parameters in Eq. (2) need to satisfy α C 1 =α C 2 = 90° and aF 1 =aF 2 =0, or α C 1 =α C 2 =90°, aF 1 (aF 2 ) =0 and α F 2 (α F 1 ) =0. According to the implications of α Cj , α Fi and aFi , the geometric relations between the R joint axes and constraints associated with an L2 F 2 C -limb can be deduced: all R joint axes must be parallel to the common normal of two constraint couples and must intersect simultaneously with two constraint forces, or all R joint axes must be parallel to the common normal of two constraint couples and R joint axes intersect one constraint force while parallel to another constraint force. For the P joint, α F 1 = α F 2 =90° must be satisfied, that is, all the P joint axes are parallel to the common normal of two constraint forces. According to the above analysis, the L2 F 2 C -limbs with RP and RR architectures can be obtained by combining L3 F 2 C -limbs and L2 F 3 C -limbs. According to the above analysis of the geometric relations between joint axes and constraints associated with L2 F 2 C limbs, when d = 3, 2 and 1, respectively, the geometric relations between joint axes and constraints associated with L3 F 0 C , L2 F 1 C , L1 F 2 C , L0 F 3 C , L2 F 0 C , L1 F 1 C , L0 F 2 C , L1 F 0 C and L0 F 1 C -limbs can be derived in the same way. Table 1 shows the parameter conditions that Eq. (1) needs to satisfy, and the geometric relations between the R (P) joint axes and the constraints associated with the LkFmC -limb. In Table 1, ‘—’ indicates that there are no corresponding joints, and ‘ࢬ’, ‘’ and ‘⊥’represent intersecting, parallel and perpendicular, respectively. The LkFmC -limbs can be obtained by the manner in which the L( k + + 1) FmC -limbs and the LkF ( m + + 1) C -limbs combine with an L3 F 2 C -limb or an L2 F 3 C -limb. For example, the L0 F 1 C -limb can be constructed by connecting an L3 F 2 C -limb at the end of the L0 F 2 C -limb, as shown in Fig. 5. The positions of the P joint in the limb are changed, but the geometric relations between the axes of the R joints are unchanged. The limb will have the same constraint characteristics as the original limb. We call these limbs isomorphic limbs. For example, in Fig. 6, two limbs are isomorphic limbs in terms of the constraint characteristics. Hence, without loss generality, only one of the isomorphic limbs will be considered in the following synthesis process. The procedure of constructing the LkFmC -limb is shown in Fig. 7. First, construct the LkFmC -limb with the L( k + + 1) FmC limb and L2 F 3 C -limb, and then determine whether the limb can generate the specified constraints. If so, the limb is an LkFmC limb. Second, construct the LkFmC -limb with the L( k + + 1) FmC -limb and L3 F 2 C -limb, and then determine whether the limb can

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L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Table 1 The geometric relations between joint axes and constraints associated with LkFmC -limbs. Limb type

Parameters for R joints

R joint axes

Parameters for P joints

P joint axes

L3 F 2 C

—

—

—

L2 F 3 C

—

Rthe common normal of constraint couples —

—

L1 F 3 C

—

—

α F 1 =90°

L3 F 1 C L2 F 2 C

α C 1 =90° α C 1 =α C 2 =90° and aF 1 = aF 2 =0, or α C 1 =α C 2 =90°, aF 1 (aF 2 )=0 and α F 2 (α F 1 ) =0

Linear independence Rthe common normal of constraint couples, and Rࢬorconstraint forces Linear independence — R⊥constraint couple, and Rࢬorconstraint forces Rthe common normal of constraint couples, and Rࢬorconstraint force Rࢬorconstraint forces

Pthe common normal of constraint forces P⊥constraint force, Linear independence — Pthe common normal of constraint forces

L3 F 0 C L0 F 3 C L2 F 1 C L1 F 2 C

L2 F 0 C L0 F 2 C L1 F 1 C L1 F 0 C L0 F 1 C LD

a F 1 = a F 2 = a F 3 =0 — α C 1 =90° and aF 1 = aF 2 =0, or α C 1 =90° and α F 1 = α F 2 =0 α C 1 =α C 2 =90° and aF 1 = 0, or α C 1 =α C 2 =90° and α F 1 =0 aF 1 = aF 2 =0, or α F 1 = α F 2 =0 α C 1 =α C 2 =90°

α C 1 = 90° and aF 1 = 0, or α C 1 = 90° and α F 1 =0 aF 1 = 0 or α F 1 =0 α C 1 = 90° —

—

α F 1 = α F 2 =90° — —

α F 1 = α F 2 =90° α F 1 =90° α F 1 = α F 2 =90°

Rthe common normal of constraint couples R⊥constraint couple, and Rࢬorconstraint force Rࢬorconstraint force

—

α F 1 =90° α F 1 =90°

R⊥constraint couple At least three R joints are linear independence

LD

— —

— Linear independence Pthe common normal of constraint forces P⊥constraint force, and linear independence Pthe common normal of constraint forces Linear independence P⊥constraint force, and linear independence P⊥constraint force, and linear independence Linear independence Linear independence

LD

Fig. 6. Two isomorphic limbs.

generate specified constraints. If so, judge whether it is isomorphic to the limbs obtained in the first step. If not, the limb is a new LkFmC -limb. Third, construct the LkFmC -limb with the LkF ( m + + 1) C -limb and L3 F 2 C -limb (L2 F 3 C -limb), then determine whether the limb can generate the specified constraints. If so, judge whether it is isomorphic to the limbs obtained in the first and second steps. If not, the limb is a new LkFmC -limb. where ‘+’ represents two limbs connected in series. The cylinders with two circles and the cuboids represent the R joints and the P joints, respectively. The solid and dashed lines represent the links and axes of R joints, respectively. Based on the above procedure, various limbs associated with d-dimensional constraints are constructed, as shown in Table 2. For structural simplicity, fewer-link limbs (two/three-link limbs) are preferred in the leg mechanisms. In Table 2, only the L3 F 1 C (1) -limb and L3 F 0 C (1) -limb are used to construct other limbs because the single-DOF joints of these limbs can be replaced by universal (U) joint and spherical (S) joint. We use the additional subscripts ‘1, 2, 3…’ to represent limbs with different structures. 4.1.2. Selection of feasible limbs for leg mechanisms Since not all limbs can be used to construct single-loop parallel leg mechanisms, the feasible limbs are selected with the following requirements. First, the limb should have a relatively small number of links. Two-link limbs are preferred for structural simplicity. As shown in Fig. 3(b), the serial limb has only two links; however, it enables the foot-endpoint to generate an arbitrary threedimensional trajectory. When the limbs meet the constraint requirements, the smaller the number of links is, the fewer moving parts are required for the legs, and the lower is the inertia of the legs. Second, to avoid the substantial friction possibly introduced by prismatic joints, we limit the number of prismatic joints to no more than two.

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Table 2 Serial kinematic limbs.

7

8

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Construct a LkFmC -limb

L(k+1)FmC +L3F2C Does the limb generate the specified constraints? Yes

L(k+1)FmC +L2F3C

LkF(m+1)C +L3F2C (and L2F3C)

Does the limb generate the specified constraints?

Does the limb generate the specified constraints?

Yes

Isomorphism ?

Yes

No Isomorphism ? No A new LkFmC -limb Fig. 7. Construction of the LkFmC -limb.

Third, the workspace of the endpoint of a limb must be a three-dimensional volume because the workspace of the parallel mechanism can be regarded as the intersection of several workspaces of the limb’s endpoint. If the workspaces of the limb’s endpoint are not three-dimensional volumes, the workspace of the parallel mechanism cannot be a threedimensional volume. Note that a U joint is equivalent to two intersecting R joints. A cylindrical (C) joint is equivalent to one P joint and one concentric R joint. An S joint is equivalent to three R joints intersecting at a common point. Therefore, the fewer-link limbs can be constructed by equivalent substitution. Thus, the feasible limbs are selected, as shown in Table 3. Using the C joint, U joint and S joint as substitutes for the corresponding joints, fewer-link limbs are obtained, as shown in Table 4. In Table 4, the black cylinder, cross-shaped cylinder and circular represent the C joint, U joint and S joint, respectively. 4.2. Design of equivalent drive mechanisms Some limbs of the fewer-limb multi-DOF parallel mechanisms include at least two actuated joints. To mount all motors on the fixed base, the researchers adopted an equivalent drive mechanism to replace two serial actuated joints of a limb [20,21,23–26]. When constructing equivalent drive mechanisms, the kinematic characteristics of serial actuated joints must be analyzed first. Then, the corresponding equivalent drive mechanisms are designed based on the kinematic characteristics. Two types of equivalent drive mechanisms are designed in the following section. 4.2.1. The RR-type equivalent drive mechanisms When two R actuated joints are connected serially, there are two common types of arrangements. One is that two joint axes intersect, as shown in Fig. 8. This arrangement can be replaced by a U joint. The other is that two axes are parallel. When fixing one axis (along the X-axis) of a U joint, the other axis of the U joint will locate in the plane perpendicular to the fixed axes (namely, the YOZ-plane). The kinematic representation of a 2-DOF spatial mechanism is shown in Fig. 9 [39]. The axes of two revolute joints of each link are orthogonal, and all axes of R joints concentrate at the O-point. The mechanism can generate independent two-dimensional rotations along two orthogonal axes. The base frame {G}: O-XYZ and the reference frame {P}: o-uvw are built, as shown in Fig. 9. u, v and w are the unit vectors along the axes of the frame {P}. The kinematics are derived based on screw theory. The twists of the mechanism are given by

$11 = [s11 ; 0] = [0, 1, 0; 0], $12 = [s12 ; 0] = [cα1 , 0, −sα1 ; 0] $21 = [s21 ; 0] = [1, 0, 0; 0], $22 = [s22 ; 0] = [0, cα2 , cα2 ; 0], $23 = [s23 ; 0] where the symbols c and s represent the cosine and sine functions, respectively.

(3)

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

9

Table 3 Feasible serial limbs.

Fixed axis Actived axis Z

θ1

θ2 O Y

X

Fig. 8. A schematic diagram of the U joint.

According to the geometric relationships among links, the equations are derived as

s23 = (s12 × s22 )/l

(4)

v = (w × u ) = (s23 × s12 )/m

(5)

where l = s12 × s22 , m = s23 × s12 . Therefore, the orientation matrix of the frame {P} is obtained as



T =

cα1 cα1 sα1 sα2 /m cα2 sα1 /l

0 cα2 /m −cα1 sα2 /l

−sα1 c α1 sα2 /m cα1 cα2 /l 2



(6)

According to Eq. (6), the u-axis is consistently located in the XOZ-plane; only when α 1 (or α 2 ) =0, is the v-axis located in the YOZ-plane. As a consequence, when using the equivalent drive mechanism to replace the U joint, the direction of the

10

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695 Table 4 Feasible fewer-link limbs.

w

Z

$21

α2

$23

$11

$22 α1 θ1

u

O

θ2

Output link

$12

v Y

X Fig. 9. A schematic diagram of the 2-DOF spatial mechanism.

fixed axis is the Y-axis direction. One type of equivalent U-drive mechanism is designed, as shown in Fig. 10. The inputs of the equivalent U-drive mechanism are α 1 and α 2 , and the outputs are θ 1 and θ 2 , where θ 1 and θ 2 represent the output angles of the fixed axis and active axis, respectively. According to the vector operations, the equation is derived as

cθ2 = s11 · v = cα2 /m

(7)

Therefore, the relationships among the inputs and outputs of the equivalent U-drive mechanism are obtained as

θ1 = α1 ,θ2 = acos(cα2 /m )

(8)

For a serial chain with two R actuated joints whose axes are parallel, its equivalent drive mechanisms (i.e., the timing belt-, cable-and linkage-drives) are presented in [23,24,29]. One type of equivalent R// R-drive mechanism is shown in Fig. 11. The relationships among the inputs and outputs of the equivalent R// R -drive mechanism are obtained as

θ1 = α1 ,θ2 = α2

(9)

4.2.2. The RP-type equivalent driven mechanisms When an R actuated joint and a P actuated joint are connected serially, there are two common types of arrangements. One is the concentric of joint axis, which can be replaced by a C joint. The other is the perpendicular joint axes. Referencing

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

11

Actived axis

θ2

α1 α 2

θ1 Output link

Fixed axis

Fig. 10. The U-drive mechanism.

θ2

Actived axis

Output link α2

α1 θ1

Fixed axis Fig. 11. The R// R-drive mechanism.

[20], one type of C-drive mechanism is designed, as shown in Fig. 12. The mechanism consists of two parallel right-hand and left-hand screws and gear sets. The C-drive mechanism can generate independent rotation and translation along the concentric axis. The displacement relations of the C-drive mechanism are derived as

q=

p (ψL − ψR ),θ = (ψL + ψR )/2 4π

(10)

where q and θ are the translation and rotation variables of the C-drive mechanism, ψ L and ψ R are the rotation of the motor connected to the right-hand and left-hand screws, and p is the pitch of the screws. To generate the identical motion that is produced by the serial chain with a R actuated joint and a P actuated joint, where the axes of the R joint and the P joint are perpendicular, the R⊥ P-drive mechanism is designed, as shown in Fig. 13. The mechanism consists of two parallel right-hand and left-hand screws and a bevel gear set. The displacement relations of the R⊥ P-drive and the C-drive mechanism are the same. 4.3. Construction of single-loop parallel leg mechanisms The single-loop parallel mechanism is composed of two serial limbs. From Tables 3 and 4, we can select a set of feasible serial limbs to assemble into a single-loop parallel mechanism with the specified motion. Note that in assembling the two limbs, the following condition must be met: the result of the linear combination of the constraints of the two limbs is that the output link is constrained. The output link must be able to generate the specified motion. In this work, we construct a class of single-loop parallel mechanism with 3T, 2R1T and 1R2T motion types. For the single-loop parallel mechanism in which the output link generates the 3T motion, the constraints exerted on the output link are three linear independent

12

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Output link

q

θ

Left-hand nut Left-hand screw ψL

Right-hand screw Right-hand nut

ψR

Fig. 12. The C-drive mechanism.

Output link

Left-hand screw

Left-hand nut

θ

q ψL

Right-hand screw Right-hand nut

ψR

Fig. 13. The R⊥ P-drive mechanism.

constraint couples. Therefore, the result of the linear combination of the constraints of the two limbs must be three linear independent constraint couples. For the single-loop parallel mechanism in which the output link generates the 2R1T motion, the constraints exerted on the output link are two constraint forces and one constraint couple. Therefore, the result of the linear combination of the constraints of the two limbs must be two constraint forces and one constraint couple. For the single-loop parallel mechanism in which the output link generates the 1R2T motion, the constraints exerted on the output link are one constraint force and two constraint couples. Therefore, the result of the linear combination of the constraints of the two limbs must be one constraint force and two constraint couples. Thus, the detailed arrangement conditions of three types of single-loop parallel mechanisms are determined as follows: For the 3T single-loop parallel mechanism, the limbs must be arranged such that the following conditions are satisfied: C1. The constraint couples of the limbs are not parallel and in different planes. For the 2R1T single-loop parallel mechanism, the limbs must be arranged such that the following conditions are satisfied: C2. The constraint forces of the limbs are coplanar but not parallel and collinear. C3. The constraint couples are parallel to each other and to the common normal of the constraint forces. For the 1R2T single-loop parallel mechanism, the limbs must be arranged such that the following conditions are satisfied: C4. The constraint couples are coplanar but not parallel. C5. The constraint forces are concentric and perpendicular to the common normal of the constraint couples. C6. The constraint forces are parallel, and the common normal of the constraint forces is parallel to the common normal of the constraint couples. Note that the conditions C5 and C6 are different cases of the constraint forces exerted on different positions of the output link. Thus, only one of the conditions C5 and C6 needs to be met. Following the above arrangement conditions and the concept of joint substitution, three types of single-loop parallel mechanisms are constructed by using the feasible limbs in Tables 3 and 4, as shown in Tables 5–7. A coaxial P joint and R joint are replaced by a C joint, for example ‘y Py R’ is replaced by ‘y C’. Two orthogonal R joints are replaced by a U joint, for example ‘y Rxoz R’ is replaced by ‘xoz U’. In Table 5, the symbol ‘+’ represents that the single-loop parallel mechanism is composed of two limbs; ‘_’ represents the actuated joints; the left superscript ‘x’ and ‘y’ represent the initial mounted directions of the R and P joint axes; ‘xoz P’ and ‘xoz R’ represent the axes parallel to the xoz-plane; and ‘xoz U’ represents the common normal of the U joint parallel to the xoz-plane but is not parallel to the z-axis.

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13

Table 5 Single-loop parallel mechanisms with 3T motion. Combination of limbs L0F2C + L0F2C

Single-loop parallel mechanism y y

xoz y

x x

yoz x

P R P R- P R P R P Ry Ry R-x Px Rx Rx R y y y y P R R R-x Px Ryoz Px R y y xoz xoz y R R R R R-x Rx Ryoz Px P y y xoz xoz y R R R R R-x Rx Rx Rx P y y

L0F1C + L0F2C

y

Cxoz Py R-x Ryoz Px Rx P Cy Ry R-x Rx Rx Rx P y y y C R R-x Ryoz Px Rx P y xoz xoz R U U-x Ryoz Px C y xoz xoz R U U-x Rx Rx C y

Table 6 Single-loop parallel mechanisms with 2R1T motion. Combination of limbs

Single-loop parallel mechanism

L1F1C + L2F1C

x x

P Ry Rxoz P-xoz Py Rx R x x y y P R R R-xoz Py Rx R x y y y R R R R-xoz Py Rx R x y xoz x R R P P-y Ry Rx R x y y x R R R P-y Ry Rx R x y y y R R R R-y Ry Rx R

x z

P Uxoz P-xoz Pz U P Uy R-xoz Pz U z y y U R R-xoz Py Rx R(z U) x z xoz P U P-y Rz U x z y P U R-y Rz U z y y U R R-y Ry Rx R(z U) x z

Table 7 Single-loop parallel mechanisms with 1R2T motion. Combination of limbs L0F2C + L1F2C

L0F1C + L1F2C L1F1C + L0F2C

Single-loop parallel mechanisms x

Rx Pyoz Px R-yoz Px Px R x x x x P R R R-yoz Px Px R x x yoz x R P P R-x Rx Px R x x x x P R R R-x Rx Px R x x yoz yoz y R R R R R-yoz Px Rx P x x yoz x yoz R R R R R-x Rx Rx P x x yoz yoz R R R R-x Ryoz Px Rx P x x yoz yoz R R R R-x Rx Rx Rx P

x

Cyoz Px R-yoz Px Px R Cx Rx R-yoz Px Px R x yoz x C P R-x Rx Px R x x x C R R-x Rx Px R x yoz yoz R U U-yoz Px C x yoz yoz R U U-x Rx C x yoz yoz R U R-x Ryoz Px C x yoz yoz R U R-x Rx Rx C x

When constructing the single-loop parallel mechanisms with 3T-motion, to simplify the mechanism, we do not consider the single-loop parallel mechanism containing the fully constrained joint. According to Section 4.1, the L0F1C (5) -limb can be obtained by connecting an R joint at the end of the L0F2C (2) -limb, as shown in Table 3. Hence, the L0F1C (5) -limb has one more R joint than the L0F2C (2) -limb. Combining the L0F2C (2) -limb with the L0F2C (2) -limb, makes it possible to obtain a single-loop parallel mechanism. We call it M1. In addition, a single-loop parallel mechanism can also be obtained by combining the L0F1C (5) -limb with the L0F2C (2) -limb. We call it M2. Compared with M1 and M2, the output motions of the output link are identical, however, in M2, the redundant joint of M2 compared with M1 is fully constrained and cannot generate motion. Hence, such a single-loop parallel mechanism will not be listed in Tables 5–7. In Table 6, ‘z U’ represents the initial direction of the common normal of the U joint along the z-axis. In Tables 5–7, when the joints of the limb are replaced by the C joint and U joint, two serial actuated joints will become a 2-DOF actuated joints, namely, the C joint and U joint. The single-loop parallel leg mechanisms are constructed by using the equivalent C-drive mechanism and the equivalent U-drive mechanism to replace the C joint and the U joint of the singleloop parallel mechanism in Tables 5–7. To show the substitution process, three examples are given, as shown in Fig. 14. In Fig. 14(a), the leg mechanism is composed of a CRR limb and an RUU limb, forming the CRRUUR single-loop parallel leg mechanism with 3T motion output. The axes of the C joint and R joint are orthogonal. When the C joint is replaced by the C-drive mechanism, the output axis of the C-drive mechanism should be coaxial with the axis of the C joint. In Fig. 14(b), the leg mechanism consists of a URR limb and a UR limbs, forming the URRRU single-loop parallel leg mechanism with 2R1T motion output. The fixed axes of two U joints are coaxial. Two revolute joints in one U joint need to be driven, and only one revolute joint in the other U joint needs to be driven. In Fig. 14(c), the leg mechanism includes a CR limb and an RUU limb, forming the CRUUR single-loop parallel leg mechanism with 1R2T motion output. The axes of the C joint and R joint are parallel.

5. Performance analysis of a novel single-loop leg mechanism In this paper, we used the above synthesized single-loop parallel leg mechanisms to design a novel quadruped robot. Each leg mechanism is a CRRUUR single-loop parallel leg mechanism with 3T motion. It is composed of a CRR limb and an RUU limb, the 3D model is shown in Fig. 15. To verify that the single-loop leg mechanism has simpler kinematics and larger workspace than other three-limb parallel mechanism, a three-limb mechanism with 3T motion output is constructed by three RUU limbs, as shown in Fig. 16.

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C-joint U-joint

C-drive foot-endpoint

U-drive

(a)

foot-endpoint

(b) C-joint

C-drive

foot-endpoint

(c) Fig. 14. Novel leg mechanisms: (a) CRRUUR mechanism (b) URRRU mechanism (c) CRUUR mechanism.

Fig. 15. CRRUUR single-loop parallel mechanism.

5.1. Kinematics analysis The kinematics are the most basic parts for the mechanisms. In the following section, the kinematics of two kinds of mechanisms are derived. The computational cost is calculated to compare the conciseness of the kinematics. 5.1.1. CRRUUR single-loop parallel leg mechanism The geometrical model of the CRRUUR single-loop parallel leg mechanism is depicted in Fig. 17. The leg reference frame {G}: o-xyz is attached at the intersection of the axis of the C joint and the R joint. The C joint and the R joint are mounted on the robot’s body, and their axes are orthogonal. The x-axis is along the sliding direction of the C joint, and y-axis is along the axis of the R joint. The output link frame {P}: oP -xP yP zP is attached to the midpoint P of the output link MD, with the xP -axis along line MD and the zP -axis vertical upward. According to Section 4.3, using CRR and RUU limbs, three pure constraint couples are exerted on the output link, one along the y-axis, one along the z-axis, and another along the common normal of the U joint. The output link MD possesses

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

15

Fig. 16. 3-RUU parallel mechanism.

z Limb-2 a

q

Limb-1 C θ

x

e F

A

R

U

t

b

α

y E

o

n

f

B R

zP

c

U M

P R xP

D Fp

h

yP

Fig. 17. The geometrical model of the CRRUUR single-loop parallel leg mechanism.

3T motion. The number of common constraints is three, the number of overconstraints is zero. The resulting leg mechanism is sufficiently constrained and thus fulfills the modified Grübler–Kutzbach [40] mobility criterion

M = 6 (n − g − 1 ) +

g 

f i + μ = 6 (6 − 6 − 1 ) + 9 + 0 = 3

(11)

i=1

where M is the number of DOF of the mechanism, n is the number of links, g is the number of kinematic pairs, fi is the number of DOF of the ith joint, and μ is the number of overconstraints. The vector loop equation of the mechanism can be derived with link vectors

q n + l 1 b + l 2 c = t + l 3 n/ 2

(12)

da + l4 e + l5 f = t − l3 n/2

(13)

where q and θ are the translation and rotation variables of the C actuated joint and α is the input angle of the R actuated joint; n=[1, 0, 0]T and a=[0, 1, 0]T are the unit vectors of the x and y-axes, respectively; b=[0, -sθ , cθ ]T , c,e=[sα , 0, cα ]T and f are the unit vectors of links AB, BD, EF and FM, respectively; l1 , l2 , l4 and l5 denote the lengths of links AB, BD, EF and FM, respectively; l3 is the distance between two joints of the output link MD; and u is the distance between the R actuated joint and origin o of the frame{G}. The distance between the foot-point Fp and point P is h. The vector t=[x, y, z]T represents the absolute position of the midpoint P of the output link MD in reference frame o-xyz. The absolute position of the foot-point

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L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

Fp is (xFp , yFp , zFp ). Note that the position relations between the midpoint P of the output link MD and the foot-endpoint Fp can be expressed as

x = xFP ,y = yFP ,z = zFP +h

(14)

The translation of the C actuated joint can be obtained directly as

q = x + l3 /2

(15)

Squaring both sides of Eqs. (12) and (13) gives

(t − xn )2 − 2l1 (t − xn )b + l1 2 = l2 2

(16)

(t − l3 n/2 − ua )2 − 2l4 (t − l3 n/2 − ua )e + l4 2 = l5 2

(17)

Substituting b=[0, -sθ , cθ ]T , e=[sα , 0, cα ]T , and Eq. (15) into Eqs. (16) and (17), respectively, results in the following equations

(A3 − A2 )k2 + 2A1 k + (A2 + A3 ) = 0

(18)

(B3 − B2 )g2 + 2B1 g + (B2 + B3 ) = 0

(19)

where k/=/tan(θ /2), g = tan(α /2), p1 = t − xn, A1 = 2l1 p1 ·η2 , A2 = −2l1 p1 ·η3 , A3 = p1 + l1 − l2 p2 = t − l3 n/2 − ua, B1 = −2l4 p2 ·η1 , B2 = −2l4 p2 ·η3 , B3 = p2 2 + l4 2 − l5 2 . Moreover η1 = [1, 0, 0]T , η2 = [0, 1, 0]T and η3 = [0, 0, 1]T are unit coordinate vectors. Then the inverse kinematics relations of the CRRUUR single-loop parallel leg mechanism are derived as 2

2

2.

q = x + l3 /2

θ = 2tan

−1

α = 2tan

−1

(15) A1 ±

B1 ±



A1 2 + A2 2 − A3 2 A2 − A3



B1 2 + B2 2 − B3 2 B2 − B3

 (20)

 (21)

Substituting Eqs. (15) and (20) into Eq. (10), the input rotation angles of the C-drive mechanism can be derived as

ψL = 2π q/p + θ ,ψR = −2π q/p + θ

(22)

The directional vectors of link BD and link FM can be formed as

c = ( p1 − l1 b )/l2

(23)

f = ( p2 − l4 e )/l5

(24)

The passive joint angle, namely, the angle between link AB and link BD can be derived as

β = arccos (b · c )

(25)

Two U joint angles are derived as

ϕU1 = arccos(e · f )

(26)

ϕU2 = arccos (n · f )

(27)

5.1.2. 3-RUU parallel leg mechanism The 3-RUU parallel leg mechanism consists of three identical RUU limbs arranged in cyclic symmetry. The 3-RUU parallel mechanism with 3T motion is one type of Delta robot, and its mobility is demonstrated in [41]. The kinematic diagram of the 3-RUU parallel mechanism is illustrated in Fig. 18. The fixed frame {O}: o-xyz is attached at the center of the fixed platform. The x-axis crosses the point A1 , and the z-axis is perpendicular to the fixed platform. The moving frame {P}: oxP yP zP is established at the center of the moving platform, and the frame {P} is parallel to the fixed frame {O}. The link frame {Ai }: o-xi yi zi is attached at the ith R actuated joint, the yi -axis along the axis of the R actuated joint, and the zi -axis along the link Ai Bi . The vector loop equation about point Ci can be derived with link vectors as

t + r bi = Rbi + la ei + lb f i

(28)

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

z

A2

z

y

y

αi+1

o o

R

θi

Ai

A1 B2

yi

x

bi

x

A3

17

ei zi

t

la

B1 B3

zp

C2

zp

yp

r P C1

C3

P

xp

fi

yp bi

lb

Ci xp

Fig. 18. Kinematic diagram of the 3-RUU parallel mechanism.

Where α i = 2π (i−1)/3, i = 1, 2, 3, ei and fi are the unit direction vectors of the actuated link Ai Bi and the passive link Bi Ci , respectively. The terms la and lb denote the lengths of link Ai Bi and link Bi Ci , respectively. The vector bi =[cα i , sα i , 0]T represents the position vector of joint Ai in the fixed frame {O}. R and r are the parameters of the fixed and moving platforms. Squaring both sides of the Eq. (28) leads to

(t − (R − r )bi )2 − 2la (t − (R − r )bi )ei + la 2 = lb 2 By substituting bi =[cα i , sα i , the equation can be changed to

0]T

(29)

and ei =[cα i sθ i , sα i sθ i , cθ i

]T

into Eq. (29) and using a triangular function transformation,

(Ni − Qi )ki 2 + 2Mi ki + (Ni + Qi ) = 0

(30)

where ki = tan(θ i /2), ࢞R = R−r, wi = t−࢞Rbi ,Ni = wi + la − lb Mi = −2la wi ·(κ1 cα i + κ2 sα i ), Qi = −2lb wi ·κ3 . Moreover κ1 = [1, 0, 0]T , κ2 = [0, 1, 0]T and κ3 = [0, 0, 1]T are unit vectors along the axes of the fixed frame {O}. Then, the inverse kinematics relations of the 3-RUU parallel mechanism are derived as 2



θi = 2tan−1

−Mi ±



Mi 2 − Ni 2 + Qi 2 Ni − Qi

2

2,



(31)

The link vector fi can be written as

fi = (wi − la ei )/lb

(32)

The joint angle of two U joints of ith limbs can be derived as

γiU1 = arccos (ei · fi )

(33)

γiU2 = arccos (bi · fi )

(34)

When calculating the workspace, the joint angles will be the constraint conditions. 5.1.3. Computational cost The above position analysis of inverse kinematics involves basic calculating operations, including addition (Add), multiplication (Mul), square roots (Sqrt), trigonometric functions (Tri) and inverse-trigonometric functions (Inv-Tri) operations. According to the abovementioned kinematics, the number of various operations on related terms involved in the inverse kinematic positions analysis of the two mechanisms listed in Table 8. Table 8 shows that the number of operations of the CRRUUR mechanism is less than that of the 3-RUU mechanism. Hence, the computational cost for the CRRUUR mechanism is lower than that of the 3-RUU mechanism. 5.2. Workspace analysis Workspace is an extremely important factor to reflect the walking capacity of a mechanism. For leg mechanisms, footendpoint trajectory planning depends on the shape of the workspace. The size of the workspace of a mechanism depends

18

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695 Table 8 Number of various operations for the CRRUUR and 3-RUU mechanism. CRRUUR mechanism Terms

q p1 A1 A2 A3 p2 B1 B2 B3

θ α ψL ψR Total

3-RUU mechanism

Number of operations

Terms

Add

Mul

Sqrt

Tri

Inv-Tri

1 3 2 2 2 6 2 2 2 4 4 1 1 32

1 3 5 5 5 7 5 5 5 4 4 3 3 55

0 0 0 0 0 0 0 0 0 1 1 0 0 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 2

Number of operations Add

Mul

Sqrt

Tri

Inv-Tri

θi

1 0 1 3 2 1 2 4

4 0 0 3 5 7 5 4

0 0 0 0 0 0 0 1

0 2 0 0 0 2 0 0

0 0 0 0 0 0 0 1

Total

42

84

3

12

3

αi bi

R wi Ni Mi Qi

Note that for the 3-RUU mechanism, i = 1, 2, 3. Hence, when calculating the total amount of calculation, each term needs to be calculated three times.

Fig. 19. Workspaces of two kinds of mechanisms: (a) CRRUUR mechanism and (b) 3-RUU mechanism.

on the dimensions of the links that construct the mechanism. Hence, under the conditions of identical link lengths and joint angles, the workspaces of the 3-RUU parallel mechanism and the CRRUUR single-loop parallel leg mechanism are comparable. The architecture parameters of the CRRUUR leg mechanism are l1 =190 mm, l2 =330 mm, l3 =84 mm, l4 =190 mm, l5 =330 mm and u = 330.28 mm. The relevant architecture parameters of the 3-RUU mechanism are la =190 mm, lb =330 mm, r = 42 mm and R = 100 mm. The workspaces of the two kinds of mechanisms are obtained based on the abovementioned inverse kinematics analysis, as shown in Fig. 19. The workspace of the CRRUUR single-loop parallel leg mechanism can be regarded as the intersection of the workspaces of two limb endpoint. The workspace of the CRR limb can be seen as the reachable area of the RRR limb endpoint moving along the R joint axes to form a three-dimensional volume. Hence, the workspace of the CRR limb is part of a cylinder. In the leg mechanism, the RUU limb further restricts the workspace of the CRR limb endpoint. Therefore, the workspace of the CRRUUR leg mechanism is a part of the workspace of the CRR limb endpoint. For leg mechanisms, the maximum step length and step height directly reflect the size of the workspace. Hence, we use the maximum step height that the leg mechanism can reach under the maximum step length to reveal the size of the workspaces. For example, three foot-endpoint trajectories are planned on the vertical plane (x = 360 mm) as shown in Fig. 20. These trajectories show that the foot can reach the maximum step heights (Hmax ) at different maximum step lengths (Lmax ). For the CRRUUR single-loop leg mechanism and the 3-RUU parallel leg mechanism, on different vertical planes that

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19

0 -50 -100

-150 Hmax

z/mm

-200 -250 -300 -350 Lmax

-400 -450

0

100

200

300 y/mm

400

500

600

Fig. 20. Three different trajectories on the x = 360 mm plane.

350

400

300

300

250

Hmax/mm

Hmax/mm

500

200 100

0 -100 0

200 150 100

100

200

100

300

Lmax/mm

400

500 400

300

200 x/mm

(a)

0

50 0

100

200 Lmax/mm

300

400 500

300 400 y/mm

200

100

(b)

Fig. 21. Relative representations of Hmax and Lmax on different planes of the CRRUUR mechanism: (a) the planes are parallel to the yoz-plane (b) the planes are parallel to the xoz-plane.

are parallel to the yoz-plane (xoz-plane) within the whole workspace, the maximum step heights that the foot can reach at different maximum step lengths are shown in Fig. 21 and Fig. 22, respectively. According to Fig. 22, in the fixed frame {O} of the 3-RUU mechanism, the maximum values of Hmax are obtained on the x = 0 plane (y = 0 plane), and then Hmax decreases along the negative x-axis (y-axis) direction and the positive x-axis (y-axis) direction. To obtain more straightforward and convincing comparison results, the workspace of the 3-RUU parallel mechanism is divided into x ≤ 0 (y ≤ 0) and x>0 (y>0), and then compared with the workspace of the CRRUUR mechanism. According to the principle of comparing large values with large values and small values with small values, the comparison results are obtained by corresponding coordinate transformation, as illustrated in Fig. 23. As shown in Fig. 23, under identical values of Lmax , the CRRUUR leg mechanism can generally reach a higher Hmax than the 3-RUU mechanism in the whole workspace. Hence, for the leg mechanism, the workspace of the CRRUUR single-loop parallel leg mechanism is larger than that of the 3-RUU parallel mechanism.

6. Walking simulation of a quadruped robot A quadruped robot can be constructed by using CRRUUR single-loop parallel leg mechanisms, and a 3D model of a quadruped robot is illustrated in Fig. 24. In the following section, the trajectories are planned and walking simulations are performed.

20

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

400

400

300

300 Hmax/mm

Hmax/mm

200 100 0

200

100 0

-100

-100

-200 0 200

-200

400 Lmax/mm 600

200

100

-100

0 x/mm

0 200

-200

400 600

Lmax/mm

(a)

200

100

-200

-100

0 y/mm

(b)

500

500

400

400

300

Hmax/mm

Hmax/mm

Fig. 22. Relative representations of Hmax and Lmax on different planes of the 3-RUU mechanism: (a) the planes are parallel to the yoz-plane (b) the planes are parallel to the xoz-plane.

200 100 0

300

200 100 0

-100 0

100 200 300 400 500 400 Lmax/mm (a)

300

200

100

0

0

400 Lmax/mm

x/mm

600 400

400

300

300

100

200

300

0

x/mm

(b)

400

200

200

100

100

0

0

-100

-100 -200 0 100

200

Hmax/mm

Hmax/mm

-200

-100

200

300 400 500 500 Lmax/mm (c)

200

400

300 y/mm

100

-200 0

200

200 400 Lmax/mm

600 500

400

100

300 y/mm

(d)

Fig. 23. Comparison of Hmax of the CRRUUR and the 3-RUU mechanisms: (a) x ≤ 0, (b) x>0, (c) y ≤ 0, and (d) y>0 part.

6.1. Trajectory planning of the leg mechanism The trajectory of the foot-endpoint of the robot can be an arbitrary spatial curve allowing the quadruped robot to walk flexibly and omnidirectionally. An example trajectory of the foot-endpoint is designed, as shown in Fig. 25(a). This trajectory consists of a curve and a level line and is located in the vertical plane(y = 310 mm), such that the quadruped robot walks along the x-direction. The rectangle represents the obstacle. The legs have two actions: lifting and supporting to ensure that the robot can walk. When a leg is lifted, the foot-endpoint moves along the curve section of the trajectory. The span of the curve represents the step length, and the height of the curve represents the step height. The robot can walk at different

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21

Fig. 24. A 3D model of the quadruped robot.

0 -50

0

-100

T3

-100

-150 z/mm

H

-200

-200

-300

H

z/mm

-250 -300

-400

-350 -400 -450

L

-500 0 T1

T2

L

0

100

200

300

x/mm (a)

400

ζ

600

200 x/mm 400

400 600

0

200 y/mm

(b)

Fig. 25. The foot-endpoint trajectories in different vertical planes: (a) y = 310, ζ =0° (b) ζ =45°

speeds and cross obstacles with different sizes by adjusting the step length and step height. When the leg supports the body of the robot, the foot-endpoint moves along the level line. In reality, the foot-endpoint is stationary relative to the ground instead of moving, and the robot body moves forward or backward. Four legs coordinate the above two actions and achieve flexible and stable walking by the quadruped robot. In this work, the parabolic function and line function are used to plan the foot trajectory. The points T1 , T2 and T3 are characteristic points on the foot trajectory, the line T1 T2 is a level line, and the curve T1 T2 T3 is a para-curve. The positions of points T1 , T2 in frame {G} are respectively T1 (xT1 ,yT1 ,zT1 ) and T2 (xT2 ,yT2 ,zT2 ). The height of para-curve T1 T2 T3 , i.e., the step height is H. The length of the level line T1 T2 , i.e., the step length is L = xT2 − xT1 . The position of point T3 is derived as T3 (xT1 +L/2,yT3 ,zT1 +H ). The trajectory is located in the vertical plane (y = 310 mm). In the base frame {G}, the planning functions of the foot trajectory are derived as follows In the lifting phase:



x(t ) ∈ (xT1 , xT2 ) TL (t ) : y(t ) = 310 2 z (t ) = k1 x(t ) + k2 x(t ) + k3

(35)

In the supporting phase:



x(t ) ∈ (xT1 , xT2 ) TS (t ) : y(t ) = 310 z (t ) = zT1

(36)

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L2 lifts and moves L

L3

L2 L1 lifts and moves 2L/3

L3 lifts and moves L/3

L4

L1

Body moves L/3

State:1

State: Initial L1 lifts and moves L Body moves L/3

L4 lifts and moves L

Body moves L/3

Body moves L/3

State:2

State:3

L3 lifts and moves L

L2 lifts and moves L/3

Body moves L/3

Body moves L/3

State:5

State:4 L4 lifts and moves L/3

State:6

State: End

State:7

Fig. 26. Tripod gait planning along a certain direction.

Y

C

X

D

C

A L3

D

L2

A y

B

L4 (a)

x

(b) Y

L1

Y

C

C

X

X L2

L3 L2

L3

y

L1

L4 y

D

D A

A

L4 (c)

B

B L1

x B

x (d)

Fig. 27. Omnidirectional walk of quadruped robot: (a) Trace of L4 (b) t = 1.5 s (c) t = 8.5 s (d) t = 17.5 s.

where k1 = −4xT1 H /L2 , k2 = 4H (xT1 + xT2 )/L2 , k3 = (zT1 L2 − 4H xT1 xT2 )/L2 . The foot trajectory can be adjusted by changing the positions of points T1 and T2 , and step height H. The foot-endpoint trajectory in other vertical plane is planned similarly, as shown in Fig. 25(b). In Fig. 25(b), the footendpoint trajectories are located in a vertical plane ζ =45°, namely, the plane and the yoz-plane are at an angle of 45°. Hence, the quadruped robot walks along a 45° diagonal direction. In the process of planning the trajectory, the trajectories can be curves of other shapes, and they can be planned according to different task requirements. Note that when the leg supports the robot’s body, the body can move forward stably if the trajectory is a level line.

L. Li, Y. Fang and S. Guo et al. / Mechanism and Machine Theory 145 (2020) 103695

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6.2. Simulation of the quadruped robot To verify the omnidirectional walk, a tripod gait is planned. Fig. 26 illustrates the gait planning for the quadruped robot to walk along a certain direction. The gray regions represent the support polygon. The square symbol represents the robot body, the filled circles indicate that the foot contacts the ground, and the empty circle represents the foot leaving the ground. L1, L2, L3 and L4 are legs 1, 2, 3 and 4, respectively. From the initial state to the end state, the entire process is described by nine states, where states 2–5 represent four states of a complete gait cycle. When a robot moves from one state to another, it is realized by switching the motions of four legs and moving the robot body. For example, from state 2 to 3, L1, L3 and L4 support robot body, and L2 lifts and moves L/3 relative to the robot body. At the same time, the body moves L/3 relative to the ground. When the robot walks in the tripod gait, the robot always has three foots in contact with the ground, and the projection of the gravity center is always in the support polygon. As a result, the robot can maintain stability when walking. Similarly, other directional gaits can be planned. Combining the inverse kinematics, trajectory planning and tripod gait planning, the omnidirectional walk is realized by simulation. The walking process is shown in Fig. 27, and the foot-endpoint of leg-4 is traced as shown in Fig. 27(a). The robot walks along the three directions and forms a broken-line ABCD path composed by a line AB along the Y-direction, a line BC along the X-direction and an oblique line CD forms an angle with the Y-direction, as illustrated in Fig. 27(b)-(d). The robot can realize an omnidirectional walk and maintain the orientation of its body when walking in different directions. In addition, the quadruped robot can also turn flexibly and smoothly. 7. Conclusion This paper presents a class of novel 3-DOF single-loop parallel leg mechanisms for walking robots. A synthesis approach for single-loop parallel leg mechanisms has been proposed. Based on screw theory and the virtual work principle, feasible limbs are synthesized. Two types of equivalent drive mechanisms are designed. Moreover, novel single-loop parallel leg mechanisms are constructed based on the constraint synthesis method and the principle of equivalent substitution. The performances of the novel leg mechanism are analyzed and compared with that of other three-limb parallel mechanism. Finally, a quadruped robot virtual prototype is built. An omnidirectional walk showing the performance of the robot has been simulated. The novel leg mechanisms can be used to construct various types of multi-legged robots. The approach proposed in this paper can also be used to synthesize other fewer-limb multi-DOF parallel mechanisms. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 51675037 and 51875033). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103695. References [1] J.Z. Zhang, Z.L. Jinn, H.B. Feng, Type synthesis of a 3-mixed-DOF protectable leg mechanism of a firefighting multi-legged robot based on GF set theory, Mech. Mach. Theory 130 (2018) 567–584. [2] M.M. Plecnik, J.M. McCarthy, Design of stephenson linkages that guide a point along a specified trajectory, Mech. Mach. Theory 96 (2016) 38–51. [3] Z.Q. Zhang, Q. Yang, S. Gui, et al., Mechanism design for locust-inspired robot with one-DOF leg based on jumping stability, Mech. Mach. Theory 133 (2019) 584–605. [4] J.X. Wu, Y.A. Yao, Design and analysis of a novel walking vehicle based on leg mechanism with variable topologies, Mech. Mach. Theory 128 (2018) 663–681. [5] L. Patnaik, L. 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