Type Synthesis of Walking Robot Legs

Abstract

Walking robots use leg structures to overcome obstacles or move on complicated terrains. Most robots of current researches are equipped with legs of simple structure. The specific design method of walking robot legs is seldom studied. Based on the generalized-function (GF) set theory, a systematic type synthesis process of designing robot legs is introduced. The specific mobility of robot legs is analyzed to obtain two main leg types as the goal of design. Number synthesis problem is decomposed into two stages, actuation and constraint synthesis by name, corresponding to the combinatorics results of linear Diophantine equations. Additional restrictions are discussed to narrow the search range to propose practical limb expressions and kinematic-pair designs. Finally, all the fifty-one leg structures of four subtypes are carried out, some of which are chosen to make up robot prototypes, demonstrating the validity of the method. This paper proposed a novel type synthesis methodology, which could be used to systematically design various practical robot legs and the derived robots.

1 Introduction

Walking robots with multiple legs have been wildly applied to tough missions associated with uneven terrains. It is the leg structure that separates the body locomotion from the feet motion while overcoming obstacles. As a result, the body could maintain balance, leading to a good adaption of most common terrains. Walking robots become the hot spots recently at the leading edge of robot research. Various robot prototypes were tested with specific tasks, including BigDog [1, 2], FROG-I [3, 4], B-elepht [5, 6], Hector [7], Crabster [8,9,10,11], PPHex [12,13,14] and many other biomechanics [15, 16]. Legs are the key parts of the walking robot structure. On the one hand, exquisitely-designed legs make it possible to match the corresponding control algorithm to achieve high operating efficiency [17]. On the other hand, most of the leg designs in up-to-date researches are simply of serial structure or of pantograph mechanism [18]. The oversimplified type of legs, to a certain extent, increases difficulty in gait controlling or mechanism protecting. A specific type synthesis method of designing walking robot legs is a powerful tool for improving the operating performance.

Type synthesis is fundamental to mechanism design. The types of the mechanisms primarily determine several basic characters, such as the degree of freedom (DoF), coupling relation between input and output, singularity, workspace. A large number of researchers have done tremendous work on type synthesis, especially the relative difficult topics on parallel mechanisms. Many theoretical synthesis methods have been proposed and applied on numerous mechanisms basic on various mathematical concepts, such as differential manifolds [19], translational topology [20], screw theory [21], set theory [22]. But in previous literatures, few of them has been involved in systematic design of walking robot legs.

The theory of generalized-function sets (GF set theory) was proposed by Gao et al. [23], to mathematically express the topological-performance property of the end-effectors of robotic mechanisms. The sequence and interaction effects between rotational and translational DoF of the end-effector were analyzed to describe the exact kinematic mobility. GF set theory was applied in type synthesis by Yang et al. [24], and Meng et al. [25, 26], where number synthesis formula and rules of operation were integrated. The systematic number synthesis solution, which led to the equation of structure parameters and end-effector characteristics, was creatively proposed in GF set theory to open up new possibilities for mechanism design.

Most of the robot legs in recent researches have no less than three DoF. Theoretically, the leg’s end-effector (i.e., the foot) is capable of moving arbitrarily in 3-D environment only if it has, at least, three active DoF to locate itself. While other DoF (active or passive) could be used to adapt to changing terrains or handle jobs like a manipulator. For this reason, this paper uses a systematic type synthesis method for 3-DoF walking robot legs based on GF set theory. The typical results of the design process are classified into two main types and four subtypes by motion characteristics and connection types. The listed results in this paper could be further studied to be combined into whole walking robots. With matched control algorithm, these robots could achieve high operating performance. The main contributions of the paper are as follows.

• (1) Two main types of robot legs are analyzed and proposed.

• (2) The equivalent number-synthesis equations of actuations and constraints are analyzed and solved by corresponding solutions in combinatorial mathematics.

• (3) Various restrictions are set in every stage of design process to realize a systematic type-synthesis method based on GF set theory, which has never been referred to in previous literatures.

• (4) Deign results are listed by four subtypes, several examples of which are compared with available prototypes to demonstrate the validity of the method.

This paper is organized as follows. Section 2 sets the goals of synthesis, based on the analyses of the two main robot leg types. Section 3 provides the solutions of the number-synthesis equations of actuations and constraints, which determine the GF expressions of limbs in Section 4. Section 5 raises some recommended limb designs of kinematic pairs. Typical design results of different limbs are illustrated in Section 6. Section 7 gives the conclusions.

2 Analyses of the Leg Mobility

Walking robots realize space locomotion via discontinuous contact with ground. This specific motion requires at least three DoF which could be either translational or rotational despite the conditions as follows.

Translational condition: At least one translational DoF is required to adapt to the terrains (i.e., RRR excluded).

Spatial condition: 3-D mobility of the end-effector is required (i.e., planar 3-DoF mechanism excluded).

Based on above application requests as well as the relevant studies around the world, the DoF of legs could be divided into two main types.

Type R: The end-effector has RRT DoF, as shown in Figure 1(a). Two rotational DoF about the hip and one translational DoF along the leg appear to be low level of anisotropy in horizontal directions. This type of leg fits omnidirectional experimental platforms or highly stable manipulator robots, e.g., RAIBERT Hopper [17, 27] and PPHex [14].

Figure 1

Moving ability of the two main types of legs

Type T: The end-effector has RTT DoF, as shown in Figure 1(b). Foot moves in the sagittal plane which is located by an R DoF about the hip. The relatively high performance in the sagittal plane makes it a splendid biomimetic mechanism, e.g., BigDog [2] and HyQ [28].

Table 1 gives the 6-D GF expressions of the two main types. GF expression, such as $G_{\text{F14}}^{\text{II}} \left( {{\text{R}}_{\upalpha} , {\text{R}}_{\upbeta} , {\text{T}}_{\text{a}} , 0 , 0 , 0} \right)$, means it is the fourteenth GF set of class II, with two rotational and one translational DoF. All the GF sets used in this paper are listed in Ref. [25].

Table 1 GF expressions of type R & type T

3 Number Synthesis

In GF set theory, mechanism parameters should satisfy the following integrated number-synthesis equation [29], see Eq. (1):

$$\left\{ {\begin{array}{*{20}l} { 2F_{\text{D}} + Q_{\text{r}} - C_{\text{o}} - \sum\limits_{{i{ = 1}}}^{N} {\left( {q_{i} - 1} \right) + \sum\limits_{{i{ = 1}}}^{N} {\left( {c_{i} - 1} \right){ = 6,}} } } \hfill \\ {c_{i} \le 6- F_{\text{D}} , \quad q_{i} \le F_{\text{D}} { + }Q_{\text{r}} ,} \hfill \\ \end{array} } \right.$$

(1)

where F_D is the dimension of the end-effector’s characteristics, i.e., the quantity of nonzero elements in the GF set expression (F_D and DoF are equal in this paper); Q_r is the number of redundant actuations; C_o is the number of over-constraints; N is the total number of limbs; c _i is the number of constraints within the ith limb; q _i is the number of actuations within the ith limb.

Here we decompose Eq. (1) into two sub-equations for practical convenience. The end-effector’s DoF, numbers of limbs and actuations should obey the actuation equation:

$$\left\{ {\begin{array}{*{20}l} {N = F_{\text{D}} { + }Q_{\text{r}} - \sum\limits_{{i{ = 1}}}^{N} {\left( {q_{i} - 1} \right) ,} } \hfill \\ {q_{i} \le F_{\text{D}} { + }Q_{\text{r}} .} \hfill \\ \end{array} } \right.$$

(2)

The end-effector’s DoF, numbers of limbs and constraints should obey the constrain equation:

$$\left\{ {\begin{array}{*{20}l} {N = C_{\text{D}} { + }C_{\text{o}} - \sum\limits_{{i{ = 1}}}^{N} {\left( {c_{i} - 1} \right) ,} } \hfill \\ {c_{i} \le C_{\text{D}} = {6} - F_{\text{D}} .} \hfill \\ \end{array} } \right.$$

(3)

Limbs with total number N connect with each other in series, parallel or hybrid connection. It is worth mentioning that limbs connect in series or in parallel (actually series is a special case of parallel when N = 1) should obey Eqs. (2), (3) as a whole. While limbs in hybrid connection should obey the equations as local serial part or parallel part.

3.1 Number Synthesis of Actuations

Eq. (2) is the number-synthesis equation of actuations, where $q_{i} = 0$ indicates the limb is passive (i.e., without actuators). Passive limbs have two main specific impacts.

Constraining: Passive limbs provide constraints and rigidity to the end-effector if $c_{i} > 0$. The number of passive limb constraints is maximized in practice:

$$c_{i} = {6} - F_{\text{D}} , \, i \in \left\{ {i|q_{i} = {0}} \right\} .$$

(4)

Output: Passive limbs need not make room for actuator installation. Moreover, it has the same mobility as the end-effector if Eq. (4) is obeyed. As a result, it could be further designed to operate as the output of the leg (see more in Section 6).

In normal conditions, there is no more than one passive limb within the leg structure. Without loss of generality, set zero redundant actuation and Eq. (2) has the following equivalent form:

$$\left\{ {\begin{array}{*{20}c} {\sum\limits_{j = 1}^{6} {j \cdot n_{qj} = n_{q1} + 2n_{q2} + \ldots + 6n_{q6} = F_{\text{D}} ,} } \\ {\sum\limits_{j = 0}^{6} {n_{qj} = n_{q0} + n_{q1} + n_{q2} + \ldots + n_{q6} \,=\, N,} } \\ \end{array} } \right.$$

(5)

where $n_{qj}$ is the number of limbs with j actuations, $n_{q0} = 0,1$.

Then we have:

$$N \le n_{q 0} + F_{\text{D}} .$$

(6)

F_D is known as the goal of design. So far the solution of Eq. (5) could be expressed as an eight-dimension vector:

$$\left( {N,n_{q0} ,n_{q1} ,n_{q2} ,n_{q3} ,n_{q4} ,n_{q5} ,n_{q6} } \right) .$$

(7)

Eq. (5) is called an constrained linear Diophantine Equation in combinatorics, solvability of which is included in Hilbert’s tenth problem [30]. There have been numerous studies on this topic [31, 32]. In this paper, the solution Eq. (7) could be listed in sequence because of the specific parameters in the equation, and the analytical solving process is omitted.

Solutions of most common cases have been given in several articles of type synthesis [29, 33]. For parallel/serial legs, $\sum {q_{i} } = F_{\text{D}} = 3$, solutions are listed in Table 2, Q1–Q6. Similarly, for multi-DoF part in hybrid legs, the local parameters satisfy $\sum {q_{i} } = F_{\text{D}} = 2$. The local solutions are listed in Table 2, Q7–Q10.

Table 2 Solutions of actuation number synthesis for parallel/serial legs

3.2 Number Synthesis of Constraints

Similar to number synthesis of actuations, constraint Eq. (3) has the following equivalent form:

$$\left\{ {\begin{array}{*{20}l} {\sum\limits_{j = 1}^{6} {j \cdot n_{cj} = n_{c1} + 2n_{c2} + \ldots + 6n_{c6} = { 6} - F_{\text{D}} + C_{\text{o}} ,} } \hfill \\ {\sum\limits_{j = 0}^{6} {n_{cj} = n_{c0} + n_{c1} + n_{c2} + \ldots + n_{c6} { = }N,} } \hfill \\ \end{array} } \right.$$

(8)

where $n_{cj}$ is the number of limbs with j constraints.

F_D and N are given as the design goal (Table 1) and the actuation number synthesis (Table 2), respectively. The solution of Eq. (8) could be expressed as an eight dimension vector:

$$\left( {C_{\text{o}} ,n_{c0} ,n_{c1} ,n_{c2} ,n_{c3} ,n_{c4} ,n_{c5} ,n_{c6} } \right) .$$

(9)

The equations are analyzed for different types (see Table 1) and connections (parallel/serial, hybrid) in the rest of the section.

3.2.1 Type R in Parallel/Series (Type R-P)

According to the analysis at the beginning of Section 3, in the process of synthesis, connection in series is a special case of that in parallel when N = 1. So here type R-P represents legs of type R in both parallel and serial connections.

The end-effector’s DoF is expressed in GF sets as ${\text{G}}_{\text{F14}}^{\text{II}} \left( {{\text{R}}_{\upalpha } , {\text{R}}_{\upbeta } , {\text{T}}_{\text{a}} , 0 , 0 , 0} \right)$ in Table 1. The corresponding constraint condition is as follows.

Condition R0: The total constraints—two translational and one rotational constraints—are given in GF expression as $\overline{{{\text{G}}_{\text{F}} }} \left( {\overline{{{\text{T}}_{\text{b}} }} ,\overline{{{\text{T}}_{\text{c}} }} ,\overline{{{\text{R}}_{\upgamma } }} , 0 , 0 , 0} \right)$.

The expression ${\text{G}}_{\text{F14}}^{\text{II}} \left( {{\text{R}}_{\upalpha } , {\text{R}}_{\upbeta } , {\text{T}}_{\text{a}} , 0 , 0 , 0} \right)$, denotes GF14, belongs to the second class of GF sets [34]. Since GF14 has two incomplete rotations, leading to the analyses below.

Analysis 1: According to intersection rules, all the intersecting limbs should contain the end-effector’s DoF. Table 3 gives the possible limb expressions.

Table 3 Intersection limb expressions of GF14

Analysis 2: All the incomplete rotations should be co-axis if the limbs intersect. But this could easily raise interference problem during assembly as well as decrease the stiffness of the whole structure, which is to be avoided.

The above analyses give the following conditions.

Condition R1: GF expression of the end-effector is the intersection set of all the limbs’ expressions.

Condition R2: Only one limb has two incomplete rotations as GF14.

Condition R3: Despite those referred to in Condition R2, rotations of limbs (if exist) must be complete.

All limb expressions meet Condition R3 have three translations, i.e., no translational constraints exist (see first two cases in Table 3). Therefore the only solution is that the two translational constraints in Condition R0 exist in the limbs of Condition R2, which comes to the condition below:

Condition R4: Only one translation is permitted in the limbs of Condition R2 (GF10, GF23 and GF 25 are all excluded).

All the candidate limbs are marked in Table 3.

Candidates in Table 3 give the restrictions of Eq. (8).

Firstly, either GF8 or GF14 must be chosen to be the most constrained limb, i.e., $n_{c2} + n_{c3} = 1.$ In addition, passive limb (if exists) is the most constrained as Eq. (4) indicates, i.e.,

$$c_{i} = \hbox{max} \left\{ c \right\} , { }i \in \left\{ {i|q_{i} = 0} \right\}.$$

(10)

Secondly, the only possible over-constraint is $\overline{{{\text{R}}_{{{\upgamma }}} }} ,$ which comes to $C_{o} \le N - 1$.

The solutions are listed in Table 4.

Table 4 Solutions of constrain number synthesis for type R-P

Every group of limbs in Table 4 matches a proper actuation solution in Table 2.

3.2.2 Type R in Hybrid Connection (Type R-H)

For every set of solution in Table 4, if one limb of which is decomposed into serial part and parallel part, a hybrid leg is synthesized, as shown in Figure 2. In particular, if one limb is multi-actuated, decomposition to hybrid connection would rearrange its actuators. The location of the actuators is optimized to get a lower inertia and higher stiffness. For this reason, transformation into hybrid form changes the actuations and limb structures, which takes advantage of both series and parallel connections.

Figure 2

Decomposition of a leg into hybrid connection

For type R-P, the total number of actuations is $\sum {q_{i} } = F_{\text{D}} = {3,}$ which indicates the existence of multi-actuation limb in Table 4 if N < 3. Here this multi-actuation limb is chosen to take a DoF rearrangement to transform into hybrid form.

If N = 1, it is exactly serially connected. The high isotropy in horizontal directions requires the equally good performance of the two rotational pairs. Hence the rotations are transformed to parallel here:

$$\begin{aligned} {\text{ G}}_{\text{F14}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{T}}_{\text{a}} , 0 , 0 , 0} \right) = \hfill \\ {\text{ parallel G}}_{\text{F18}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0 , 0 , 0 , 0} \right) \cup {\text{series G}}_{\text{F7}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , 0 , 0 , 0 , 0 , 0} \right) .\hfill \\ \end{aligned}$$

(11)

Similar to type R-P, the parallel part of hybrid limb, GF18, has two incomplete rotational DoF. Considering analyses R1–R4 for type R-P, candidate limbs are derived from a similar process, see Table 5.

Table 5 Candidate limbs of GF18

Here the local DoF F_D = 2. In addition, the spatial rotational ability of GF18 requires 3-D stability, suggesting at least 3 limbs. Then local passive limb is included that $n_{q0} = 1$. Take the equal sign in Eq. (6) so that N = 3.

The solutions are listed in Table 6.

Table 6 Solutions of constrain number synthesis for GF18

Every group of limbs in Table 6 matches a two-actuation solution, Q10 in Table 2. Another actuation is left for the serial part GF7 in Eq. (11).

If N = 2, the more constrained limb restricts the end-effector’s mobility. Consequently, it sustains larger constraining force. Here this limb is chosen to be the passive one according to Eq. (10). Thus the other actuating limb with less constraints is to be decomposed into hybrid form.

Referring to Table 3, the actuating limb for N = 2 in Table 4 is expressed as GF1 (6-DoF) or GF2 (5-DoF), indicates that the total DoF of the single limb (local F_D) is more than the number of actuations. The difference of local F_D and actuation quantity is called passive DoF, which must be constrained by the passive limb. Generally, the passive DoF locates at the end of the limb. The passive DoF could be easily integrated into one kinematic pair, as the serial part of the hybrid limb. In the case here, rotations of GF1 or GF2 (see Table 3) are selected to be the passive DoF, operating as a universal joint or a spherical joint (see Section 5).

If the passive 2-rotational DoF (GF18 as a universal joint, see C-RP-3/4 in Table 4) is implemented, a 3-translational DoF (GF4) is required to get the specific characteristic of the end-effector:

$$\begin{aligned} {\text{ G}}_{\text{F2}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , {\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0} \right) = \hfill \\ {\text{parallel G}}_{\text{F4}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , 0 , 0 , 0} \right) \cup {\text{series G}}_{\text{F18}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0 , 0 , 0 , 0} \right) .\hfill \\ \end{aligned}$$

(12)

Pure multi-translational limb as GF4 is relatively complex in structure, and easily to import assembly error or interference problem. The design process here applies only one pure 3-translational limb. All the candidate limbs are listed in Table 7, using similar GF-set-theory analyses in Section 3.2.1.

Table 7 Candidate limbs of GF4

Here the local DoF F_D=3 and the local passive limb is not included that $n_{q0} = 0$. Take the equal sign in Eq. (6) so that N=3. The solutions are listed in Table 8. Every group of limbs in Table 8 matches a three-actuation solution, Q3 in Table 2.

Table 8 Solutions of constrain number synthesis for GF4

If the passive 3-rotational DoF (GF12 as a spherical joint, see GF1 in Table 3 and C-RP-2 in Table 4) is implemented, all GF sets with spatial mobility are equivalent to the 3-translational limb GF4, see the decomposition expressions below:

$${\text{G}}_{\text{F1}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , {\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\upgamma }}} } \right) =$$

$${\text{G}}_{\text{F4}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F12}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\uplambda }}} , 0 , 0 , 0} \right) =$$

(13)

$${\text{G}}_{\text{F14}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{T}}_{\text{a}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F12}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\uplambda }}} , 0 , 0 , 0} \right) =$$

(14)

$${\text{G}}_{\text{F16}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F12}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\uplambda }}} , 0 , 0 , 0} \right) .$$

(15)

Take the isotropy of type R into consideration, Eq. (14) is implemented as the same form as the passive limb (GF14 of case C-RP-2 in Table 4). The results have been listed already, see N = 3 cases in Table 4. Every group of which matches a three-actuation solution, Q3 in Table 2.

Above all analyses in this subsection, it is easily concluded that synthesis of hybrid structure could be transformed to those of parallel/series connections.

3.2.3 Type T in Parallel/Series Connections (Type T-P)

The end-effector’s DoF is expressed in GF sets as GF16 (Table 1). The corresponding constraint condition is as follows.

Condition T0: The total constraints of the end-effector is given in GF expression as $\overline{{{\text{G}}_{\text{F}} }} \left( {\overline{{{\text{T}}_{\text{c}} }} ,\overline{{{\text{R}}_{{{\upbeta }}} }} ,\overline{{{\text{R}}_{{{\upgamma }}} }} , 0 , 0 , 0} \right)$, which has one translational and two rotational constraints.

The expression GF16 belongs to the second class of GF sets like GF14 (Section 3.2.1). The incomplete rotation in GF16 imply the similar analyses of condition R1–R4 in Section 3.2.1. The counterpart candidate limbs are listed in Table 9.

Table 9 Candidate limbs of type T-P

Table 9 imposes restrictions on Eq. (8). Firstly, if GF16 is excluded in a synthesis, then both GF10 and GF3 should be implemented together to constrain $\overline{{{\text{T}}_{\text{c}} }}$ and $\overline{{{\text{R}}_{{{\upbeta }}} }} ,$ i.e., $n_{c2} \ge 2, \, if \, n_{c3} = 0$. Secondly, either $\overline{{{\text{R}}_{{{\upbeta }}} }}$or $\overline{{{\text{R}}_{{{\upgamma }}} }}$ could be over-constrained that $C_{\text{o}} \le 2\left( {N - 1} \right)$.

The solutions are listed in Table 10, each group of which matches a proper actuation solution in Table 2.

Table 10 Solutions of constrain number synthesis for type T-P

3.2.4 Type T in Hybrid Connection (Type T-H)

Similar to type R-H in Section 3.2.2, solutions of N < 3 in Table 10 show the feasibility of transformation into hybrid limbs. Choose the most-actuated limb to be decomposed into serial and parallel parts.

If N=1, it is exactly serially connected. Take into consideration the high performance in sagittal plane of type-T leg, the two translational DoF in the plane are transformed to be the parallel part:

$$\begin{aligned} {\text{ G}}_{\text{F16}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , 0 , 0 , 0} \right) = \hfill \\ {\text{series G}}_{\text{F21}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , 0 , 0 , 0 , 0 , 0} \right) \cup {\text{parallel G}}_{\text{F6}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , 0 , 0 , 0 , 0} \right) .\hfill \\ \end{aligned}$$

(16)

For planer mechanism, to increase stiffness and to reduce the unnecessary kinematic pairs, only planer limbs are considered here without local passive ones [35, 36]. Here are two candidates listed in Table 11. The solutions are listed in Table 12. Every group of limbs matches a two-actuation solution, Q8 in Table 2. Another actuation is left for the series part GF21 in Eq. (16).

Table 11 Candidate limbs of GF6

Table 12 Solutions of constrain number synthesis for GF6

If N = 2, referring to the analyses of type R-H in Section 3.2.2, the more constrained limb is chosen to be passive. The other actuating limb is to be decomposed into hybrid form.

Referring to Table 9, the actuating limb for N=2 in Table 10 is expressed as GF1 (6-DoF), GF2 (5-DoF), or GF3 (4-DoF), indicates that the total DoF of the single limb (local F_D) is more than the number of actuations. Here we again combine the rotations of the limbs into passive joints. The problem is transformed to 3-DoF mechanism synthesis. The union formula goes as below:

$$\begin{aligned} {\text{ G}}_{\text{F1}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , {\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\upgamma }}} } \right) = \hfill \\ {\text{G}}_{\text{F16}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F12}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , {\text{R}}_{{{\uplambda }}} , 0 , 0 , 0} \right) ,\hfill \\ \end{aligned}$$

(17)

$$\begin{aligned} {\text{ G}}_{\text{F2}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , {\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0} \right) = \hfill \\ {\text{G}}_{\text{F4}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F18}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , {\text{R}}_{{{\upbeta }}} , 0 , 0 , 0 , 0} \right), \hfill \\ \end{aligned}$$

(18)

$$\begin{aligned} {\text{ G}}_{\text{F3}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , {\text{R}}_{{{\upalpha }}} , 0 , 0} \right) = \hfill \\ {\text{G}}_{\text{F4}}^{\text{I}} \left( {{\text{T}}_{\text{a}} , {\text{T}}_{\text{b}} , {\text{T}}_{\text{c}} , 0 , 0 , 0} \right) \cup {\text{G}}_{\text{F21}}^{\text{II}} \left( {{\text{R}}_{{{\upalpha }}} , 0 , 0 , 0 , 0 , 0} \right). \hfill \\ \end{aligned}$$

(19)

Decomposition Eq. (17) requires to synthesize GF16, which is already done in Table 10 (N = 3 cases). Decomposition Eqs. (18), (19) require to synthesize GF4, which is listed in Table 8.

4 Limb Decomposition and Expression

Results of number synthesis are listed in Tables 4, 6, 8, 10, 12, each of which corresponds to one or more GF expressions in Tables 3, 5, 7, 9 and 11. If a mapping like this satisfies the restricted conditions below, it would represent a valid limb decomposition.

Condition 4.1: Intersection rules of GF sets (as shown in the “Condition” column of Tables 13, 14, 15, 16) [33]. The rules are about axes’ locations of associated DoF between limbs, which were ever discussed in conditions R1–R4 in Section 3.2.

Table 13 Limb expression for type R-P (referring Table 4)

Table 14 Limb expression for type R-H (referring Tables 4, 6 and 8)

Table 15 Limb expression for type T-P (referring Table 10)

Table 16 Limb expression for type T-H (referring Tables 8, 10 and 12)

Condition 4.2: Other considerations based on practical use, including assembly interference (condition R2 and R3 in Section 3.2), structure stability (discussions in Sections 3.2.2 and 3.2.4), spatial symmetry, etc.

It is worth mentioning that, theoretically, the first condition above is necessary while the second is not. The unnecessary condition should be practically analyzed to find the required synthesis solutions in a relatively small search range.

Generally, legged robots have symmetric mobility. Here we take the symmetric limb expressions as examples, and set N ≥ 3 for spatial parallel mechanisms (or parallel part of hybrid legs). For the rest results, the expressing process is similar and omitted here.

For different leg mobility characteristics (Table 1), the qualified limb expressions are listed in Tables 13, 14, 15, 16. $\parallel {\text{R}}_{{{\upalpha }}}^{j} {\text{ or }}\parallel {\text{T}}_{\text{a}}^{j}$ means all the specified rotational or translational DoF axes in limbs j are parallel to each other. The indexes in the last column indicate the number synthesis in referred tables (Section 3).

5 Limb Design

A specific kinematic limb is designed by deciding the type, quantity and consequence of various kinematic pairs, which is based on the relation between the end-effector’s mobility and the kinematic-pair axes.

Kinematic pairs include simple pairs and composite ones [33], see Table 17. Composite pairs assemble linkages with simple pairs to get compact multi-DoF units.

Table 17 GF expressions of kinematic pairs used in this paper

Using composite pairs in Table 17 and intersection rules of GF sets (e.g., Eqs. (11)–(19)), it is convenient to simplify the limb design, as well as to improve the structure stiffness, stability, etc. Table 18 lists the GF expressions employed in this paper, and the suggested compact kinematic limb designs as well. The subscript “//” or “//” in Table 18 means the rotation axes are parallel or unparalleled.

Table 18 Limb design of GF expressions

Actuation matching is another topic in robot design. In type synthesis stage, we focus on the total numbers and assembly locations of the actuators in a limb. The actuation numbers are matched during the discussion about Tables 4, 6, 8, 10 and 12 in Section 3. The actuation locations are underlined in the last column of Table 18. Different actuation systems could convert to each other for practical use, e.g., translational actuators could output rotations via linkage-slider mechanisms, while rotors could output translations by means of screws, see prototypes in the next section.

6 Examples of Synthesis Results

For each limb expression in Section 4, take the corresponding kinematic form in Section 5 to get a final type of walking robot leg. Here several typical examples of the four subtypes (i.e., type R-P, R-H, T-P and T-H) are illustrated and compared with some real robots to show the validity of the synthesis methodology. Actuators, joints, robot frames and fixed adjacent linkages are distinguished by colors.

6.1 Examples of Type R-P

Two examples of legs with RRT DoF in parallel/series are listed in Table 19.

Table 19 Example legs of type R-P

Figure 3(a) illustrates the leg model in series, which is a simple and common case of type R-P. All kinematic pairs install actuators (present as brown discs or golden cylinders in Figure 3) to perform as a lightweight experimental platform. Passive DoF could be implemented between the foot and the ground to adapt to terrains. A famous typical robot using this kind of leg is the one-legged hopper by Raibert et al. [17].

Figure 3

Leg model of type R-P

Figure 3(b) illustrates the second set in Table 19. Three actuators distributed on each parallel limb perform as muscles. Actuations are easily matched to get high load capacity. A six-legged robot prototype with isotropy in horizontal plane [13, 37] is shown in Figure 4, which was designed by the authors’ research team.

Figure 4

Six-legged robot of R-P legs

6.2 Examples of Type R-H

Hybrid legs of type R-H take advantage of those in series and in parallel, which should be specifically designed in practice. Some examples are shown in Table 20.

Table 20 Example legs for type R-H

Figure 5(a) illustrates the first example in Table 20, which is of 3-DoF hybrid limb. It is the basic hybrid form.

Figure 5

Leg model of type R-H

Figure 5(b) illustrates the second set of leg in Table 20. It has a 6-DoF hybrid limb with a 3-DoF-active part and 3-DoF passive joints. The actuation part is isolated from the passive leg linkages to get better protection or insulation. In Figure 5(b), a revolute pair of the hybrid limb is integrated into the prismatic pair of the passive limb to combine a cylinder pair. A possible robot using this kind of leg is shown in Figure 6.

Figure 6

Robot model of R-H legs

6.3 Examples of Type T-P

Examples of legs with RTT-DoF in parallel/series are listed in Table 21.

Table 21 Example legs for type T-P

Figure 7(a) illustrates a simple serial model. The parallel revolute pairs perpendicular to the sagittal plane are used to substitute for the translational DoF.

Figure 7

Leg model of type T-P

Figure 7(b) introduces a type of one passive and three active limbs. Three active actuating “muscles” are centro-symmetric to improve the lateral performance. The actuators are installed separately from leg linkages to make it possible to work in extreme conditions (earthquake, fire, etc.). Passive prismatic pairs in the sagittal plane are replaced with a ^U pair. Corresponding walking robot model with this type of legs is shown in Figure 8.

Figure 8

Robot model of T-P legs

6.4 Examples of Type T-H

Hybrid legs of type T derive from serial/parallel ones, see Table 22 for some examples.

Table 22 Example legs for type T-H

Figure 9(a) shows the first model of 3-DoF hybrid limb in Table 22. The parallel revolute pairs are used to create the derivative translational characteristics in the sagittal plane. Figure 10 is a quadruped prototype using this kind of leg [6]. It was designed and tested by the authors’ research group, leading by Prof. GAO. Additional parallel links are used to increase the planer rigidity.

Figure 9

Leg model of type T-H

Figure 10

Quadruped BabyElephant of T-H legs

Figure 9(b) shows the second model in Table 22 with a passive leg and a 6-DoF actuating hybrid limb. Actuations are completely isolated and installed on the robot frame, to reduce the moment of inertia. Legs of this type could also be realized using a 5- or 4-DoF hybrid limb (last two rows in Table 22). Figure 11 shows a hexapod prototype of the authors’ research team, using T-H legs. The rotational input of this kind of leg is generated by screws and linkages, protected in the bodyshell.

Figure 11

Hexapod robot of T-H legs

7 Conclusions

• (1) Two main types of walking robot legs are proposed. Type R fits omnidirectional moving platforms while type T plays a role of biomimetic mechanism.

• (2) A decoupled solving method of number synthesis equation in GF set theory is proposed, which uses combinatorics to get the solution for both actuation part and constraint part.

• (3) Reasonable application of GF set rules is presented to narrow the search range and successfully derives all the fifty-one kinds of walking robot legs, which could be combined to form various robots of practical value.

• (4) The proposed design results are illustrated with 3D models and corresponding real prototypes, which show the practical validity of the design method.