Criteria of type complexity for legged robots

Mechanism and Machine Theory 144 (2020) 103661

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Research paper

Criteria of type complexity for legged robots Da Xi, Feng Gao∗ State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e

i n f o

Article history: Received 21 July 2019 Revised 30 September 2019 Accepted 6 October 2019

Keywords: Legged robot Type complexity Type criteria General-Function set Incidence relation Drive distribution

a b s t r a c t Type characteristics limit the performance of legged robots. In this paper, the typecomplexity criteria for legged robots are proposed to evaluate the structure-related performance at the conceptual-design stage. Firstly, the incidence relation (IR) is used to represent the structural features of legged robots, where the General-Function (GF) set is used to characterize the end-effector of the robot and the incidence-relation matrix is used to characterize the input-output transformation. Secondly, the theoretical basis of type complexity is proposed and demonstrated. On this basis, two types of complexity criteria are defined according to the incidence relation of legged robots. The output complexity reflects the magnitude of the robot drive load, while the input complexity reflects the uniformity of the drive load. These two kinds of complexity point out the distribution of the robot actuation under the specific operation requirements, which is determined by the robot topology, reflecting the advantages and disadvantages of the robot structure. Finally, the theoretical calculations, model simulations, and prototype experiments are carried out for three typical legged-robot types. The correctness and usefulness of the type-complexity criteria are validated and demonstrated. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Legged robots can flexibly adapt to various irregular terrains and complete specific tasks in a non-structural environment, which have become hotspots of the robot research. A number of multi-legged robot prototypes of different types have been developed, including Atlas [1], Cheetah [2], HyQ [3], Hector [4], PPHex [5] and many other bionic-robots [6,7]. These legged robots have type differences, mainly in the layout, number, and topological structure of the mechanical legs: The legs are usually arranged in a certain central symmetry [8] or plane symmetry [9] on the body frame. In terms of leg quantity, it can be divided into biped [10], quadruped [7], hexapod [11] or eight-legged [12] robots. The topological structure of robot legs usually includes the bionic serial-connection [13], pantograph mechanism [14], hybrid limb [15], or special-shaped structure [16,17]. The type-synthesis is a preliminary design stage of low design cost for legged robots. However, the direction of this predesign stage directly affects the subsequent design route [18]. If the selection is slightly inadvertent, design defects that do not meet the requirements of the robot’s tasks or the robot-environment interaction are introduced [19]. In the later design phase, the iterative design is often carried out at a great cost, with various efforts to make up for the defects [20]. The upto-date research seems to prefer similar legged-robot structures: Detailed parameters are re-designed to complete different tasks (even the scope of these tasks is widely divergent), or the robot is combined with other existing mechanisms to create ∗

Corresponding author. E-mail address: [email protected] (F. Gao).

https://doi.org/10.1016/j.mechmachtheory.2019.103661 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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D. Xi and F. Gao / Mechanism and Machine Theory 144 (2020) 103661

a novel type. This also indicates the trade-off difficulties of the legged-robot type synthesis: On the one hand, each type candidate of the pre-design stage will experience a costly post-iterative design (even trial and error) to form an applicable parametric mechanism; On the other hand, modifying the detailed parameters of an existing robot according to new tasks often saves a lot of design cost compared to redesigning a different robot type, even if the original one is not optimal [21]. Therefore, legged-robot designers starve for a type evaluation method to obtain suitable task-specific robot types through simple and rapid calculation. The resultant types should have been screened or properly optimized to avoid the inconvenience of “birth defects” at the later design stage. In current research, commonly used robot indices include workspace geometry, dexterity, the kinetostatic, the dynamic, the elastostatic, the elastodynamic performance, etc. Specific to the legged robot, the criteria include stability margin, duty factor, Froude number, and specific resistance, etc. Jorge Angeles [22] and Shuuji Kajita [23] have respectively introduced and interpreted these criteria. However, most of these indicators or criteria are proposed for robots of determined parameters. There are few criteria for robots of a certain structure type but of undetermined parameters. Complexity is a commonly-used indicator that shows the input and output characteristics of a mechanism. Hervé [24] introduced the concept of complexity in the study of the kinematic bond. Khan et al. [19] embodied the concept and applied it to the robotic field. At the conceptual-design stage, the researchers combined the kinematic parameters into one indicator named complexity. These parameters include the number of joints, the number of loops, the types of kinematic pairs, the diversity in geometric constraints, the type of actuators, the diversity in actuators, etc. Caro et al. [25] used the complexity definition to establish a concept-evaluation framework for different Schönflies topologies. The framework filtered-out less promising Schönflies architectures in the conceptual-design phase. In reference [26], design was described as a mapping between the field of design parameters and the field of functional requirements, and complexity was defined as the difficulty of establishing such a mapping. However, there is little research on the specific complexity pertaining to the requirements and characteristics of legged robotic mechanisms. For legged robots, the lower the complexity of the mechanism, the easier it is to present the kinematic advantage of the structure. As a result, it is easier to dig out the kinematic "potential" of a specific robot type, indicating the direction for the later design process. In order to apply the complexity criteria to evaluate the legged-robot types, a suitable expression is first needed to describe the structural features of the robots. The general-function (GF) set theory is one of the ideal theoretical tools to synthesize and analyze the mechanism based on the end-effector’s motion. This theory was originally proposed by Gao et al. [27] to represent a special collection of end-effectors’ qualitative motion. The researchers considered the interaction and sequence of translational and rotational motions of the mechanism, accurately describing the movement ability of the endeffector. Yang [28] and Meng et al. [29] further studied number synthesis formulae and algorithms to develop a systematic mechanism analysis methodology based on the GF set theory. The concept of incidence relation (IR) [30] is proposed in GF set theory, which provides a clear direction for the analysis of input-output transformations. In this paper, the GF set expression is used to characterize the end-effectors’ motion of the legged robots, and the IR matrix is used to represent the input-output relationship of the mechanism. Based on these expressions, two complexity criteria are proposed to complete the type evaluation process of the legged robots. The study comprises the following steps: (1) The representations of the task requirements and input-output characteristics are proposed for the legged robots. (2) The theoretical basis for the evaluation of the legged-robot types with complexity criteria is proposed. The complexity is used to characterize the actuators’ operating state, which is related to the number of drives and the duration of the operation. The conclusion is demonstrated by the relevant theorem in the field of probability theory. (3) Based on the incidence relation of the legged-robot motion and the corresponding three complexity parameters, two kinds of complexity criteria are derived. The meaning of the criteria is indicated from the perspective of input-output characteristics. (4) As an application, three typical legged-robot types are evaluated using two types of complexity criteria. The assessment results are compared and analyzed. (5) Model simulations and prototype experiments are carried out on the evaluated legged-robot types. The validity and correctness of the complexity criteria are demonstrated by the simulation results and experimental data analysis. The rest sections of this paper are organized as follows. In Section 2, the IR expression of the legged-robot types and the corresponding complexity parameters are proposed. Section 3 defines two types of complexity criteria and indicates their practical implications. In Section 4, three specific legged-robot types are evaluated as an implementation of the complexity criteria. Section 5 separately analyzes the simulation results and the experimental data, and compares them with the complexity evaluation results to demonstrate the feasibility and correctness of the criteria. Section 6 concludes the paper and gives the future research directions. 2. Incidence relation of legged robots 2.1. Operating mode of legged robots Design was described as a mapping between the field of design parameters and the field of functional requirements, and complexity was defined as the difficulty of establishing such a mapping. In our application, design parameters refer to the inputs and types of the robot. The functional requirements refer to the outputs and the purpose of the robot. Thus, the complexity evaluation method in this paper is closely related to the robots’ operating tasks. For a unique walking robot, if the purpose of the robots’ operation changes, the complexity evaluation result will make a big difference. Without loss of generality, this section is concerned with the rigid body motion of the robot frame. Therefore, in this paper, the operating

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Table 1 Operating modes of common legged robots. No.

GF sets (end-effector)

1 2 3 4 5 6 7 8 9 10

GIF7 Ta , 0, 0, 0, 0, 0 GIF6 Ta ,Tc , 0, 0, 0, 0 GIF5 Ta ,Tb ,Rγ , 0, 0, 0 GIF5 Ta ,Tc ,Rβ , 0, 0, 0 GIF4 Ta ,Tb ,Tc , 0, 0, 0 GIIF21 Rα , 0, 0, 0, 0, 0 GIIF18 Rα ,Rβ , 0, 0, 0, 0 GIIF12 Rα ,Rβ ,Rλ , 0, 0, 0 GIF11 Ta ,Tb ,Rα ,Rβ , 0, 0 GIIF22 Ta ,Tb ,Rα ,Rβ ,Rγ , 0

( ( ( ( ( ( ( ( ( (

Operating mode

) )

) ) ) ) )

) )

)

Reciprocating patrol Repeated load Area patrol Continuous climbing Mobile load Field operation Field operation Field operation Sequential operation Sequential operation

mode refers to the degree-of-freedom (DoF) requirements of the legged robot’s body frame during operation. If researchers are concerned with the motion of a different end-effector, such as the end of a manipulator fixed on the robot’s body frame, it can be studied using a similar method in this paper. Usually, the body frame of the legged robot is supported by mechanical legs on various complex terrains, with 6-DoF motion ability. The velocity vector of the coordinate system fixed to the body frame is mathematically represented as a 6-D screw. However, type evaluation only cares about the qualitative motion of the legged robot and does not care about the specific values. For this purpose, the GF set expression is represented as a set of 6-D elements that clearly reflect the qualitative motion capabilities of the end-effector (such as the number of DoF, the direction of motion, the sequence of kinematic pairs, etc.). This kind of motion ability is determined by the robot’s type and does not involve specific structural parameters (such as link length, actuation power, control signal bandwidth, etc.). Here we use the GF set as the description of the end-effector of the walking robot, which is considered based on the convenience of the evaluation method in this paper. Researchers could also use other theoretical expressions (such as the instantaneous screw representation [31], finite screw expression [32,33], displacement submanifold [34], etc.) to get a similar evaluation process as this paper introduces. There are totally twenty-five kinds of GF sets. This paper pertains to the general DoF expression during the operation of legged robots. Here we ignore the order of motion, but focus on the category of specific moving directions. In Table 1, we hereby pick out the following ten expressions from the whole twenty-five GF sets to express the operating modes of common legged robots. Ten operating modes fall into seven categories, namely reciprocating patrol [35], area patrol [36], repeated load, continuous climbing, mobile load [37], field operation [38], and sequential operation [39].

2.2. Representation of the operating state The parameters of each legged-robot actuator approximately satisfy the independent and identical distribution (iid), because the operating environment, specific model, design parameters and assembly requirements are similar to one another. In addition, the number of legged-robot actuators is large enough to provide input redundancy. These two points of view show both the similarity and the complexity of legged robotic system. According to this description, this paper proposes two lemmas about the complexity and the legged-robot operating state: A. The more actuators that operate together, the more complex the robotic system and the worse the operating state. B. The longer a specific actuator operates, the more complex the robotic system and the worse the operating state. These two lemmas are generally obvious when we use complexity to represent the operating state of a legged robotic system. Here we could also facilitate this understanding from the perspective of probability theory. The similarity and the redundancy of legged-robot actuators imply that the actuator parameters satisfy Lindbergh’s condition. Under this premise, it can be considered that the probability of the legged robot staying within the reasonable operating state during a test period could be derived from the Lindeberg-Feller central limit theorem (CLT). Let the probability be as follows:

FN ( x ) = P

 N

Xi − Nμ ≤x √ Nσ

i=1

 (1)

where N is the number of working actuators. X is a variable related to the sum of the number of actuators. It is collectively referred to herein as the integrated operating state of the robotic system, including but not limited to: average load, reliability, mechanical life, mean motor power, and battery consumption of the drive system. The magnitude of X indicates the complexity of the robotic system [19,26]. x is the corresponding threshold of X. As N increases, both the expectation and

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Fig. 1. An overview of the input-output expressions of a legged robot: (a) the mechanism schematic shows the relationship between the end-effector, actuators, and ground; (b) the arrow diagram of the input-output incidence relation; (c) the sparse matrix indicating the input-output interaction.

the standard deviation (Nμ,

lim FN (x ) = lim P

N→∞

N→∞

√ Nσ ) increase, we have:

 N



Xi − Nμ ≤x √ Nσ

i=1

1 = √ 2π



x

−∞

t2

e− 2 dt = (x ).

(2)

Most samples in nature require a minimum sample size. We choose N no less than 32 in the universal case, and 20 or less in most fields of mechanism [40]. In the range of legged robots discussed in this paper, each mechanical leg has multiple drives, showing the actuation redundancy, not to mention the drives of manipulator and other mechanical parts (waist, head, etc.). The requirement of N could always be met. That is, variable X as a whole satisfies the normal distribution √ with expectation and standard deviation (Nμ, Nσ ). Here N could be interpreted as the number of effective drives or the time duration of the operation, which corresponds with the lemma A and B respectively. When we start pondering the question about the meaning of N, note that we only consider whether the relevant actuator works, which can be determined by the tasks and the mechanical characteristics after a given robot type, almost independent of the detailed parameters. We also did not consider other constraints, such as power limit, load, etc. Because these parameters can be optimized or compensated based on the classical mathematical programming methods at the parameter design stage [19]. In summary, the operating state of the legged robotic system is inversely related to the number of drives and the duration of the operation, as lemmas A and B mentioned above. It should be pointed out that because of the specificity of the tasks and the operating environment, the definition of the effective actuators of legged robots may be different from that of other robots (such as fixed manipulators in factories). Legged robots usually have a quasi-periodic operating state when walking: The gait phase of the mechanical legs is repeated, but the specific parameters may be adjusted with the terrain. And the robot will be ready to respond to unexpected situations and change the gait. So in fact, even if an actuator does not participate in the current job, it will be in standby at any time. It still participates in each control loop, but is temporarily assigned a constant or initial value, kept in a low-load operation. However, the actuating operation referred in lemmas A and B does not include this generalized standby working state, but is limited to the cyclic dynamic high-load working state when the driver participates in the task. We will elaborate on this limitation in Section 4.1 later. In fact, according to the experimental data of the related research, actuators in high-load working state are the most prone to operational errors [41]. The corresponding operating state essentially needs special study compared to the normal standby state. 2.3. Incidence relation As stated above, we could obtain the probability distribution of the operating state variable X of a given robot type, while specific parameters keep unknown. By comparing the probability distributions for different robot types, the potential performance of these types could be indirectly evaluated. Thus, less promising types could be filtered-out at the conceptualdesign stage. Therefore, how to construct the indicators that determine the distribution of the robot type becomes the key to the evaluation process. As discussed in Section 2.2, the robot operating state is related to the number of drives and the duration of the operation. This section introduces the incidence-relation (IR) matrix of GF set theory, and thus establishes the input-output relationship for the robot type. Furthermore, we can construct the complexity criteria using the specific summation operations of the IR matrix, which satisfies the requirements of type evaluation. The incidence-relation (IR) matrix represents the correspondence between the actuating joint space and the task space of the mechanism. Fig. 1 depicts the natural expressible logic of this correspondence. We can use the mechanism schematic Fig. 1(a) to indicate the correspondence among the end-effector, actuators, and ground. However, for complex robot mechanisms, it is difficult to perceive the specific interactions. In Fig. 1(b), the actuators are considered as an input, and the end-effector’s motion as an output. The dashed arrow lines indicate the mutual incidence relation, which is a specific qualitative expression. Fig. 1(c) digitizes the arrow lines into a tabular form, with 0 elements indicating no interaction between the corresponding input and output components. This tabular form could be transformed into a mathematical expression of

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a sparse matrix, which can be regarded as a bivariate distribution of the operating state from the drive space to the task space. Specifically, it could be expressed as:

Im×n vm×1 ⇐⇒sn×1

(3)

where v represents the m-dimensional output characteristic of the legged robot, which can be defined by the GF set expression in Table 1. s represents the n-dimensional input of generalized drive. I denotes the incidence-relation matrix derived in Fig. 1(c). If the corresponding input and output components do not interact with each other, the value of element in I should be 0, otherwise the value is 1. In order to facilitate writing and expressing, it could be written as follows:

vm×1 ⇐ Im×n ⇒ sn×1 .

(4)

In the IR matrix, the 0 element shows an intrinsic property of the mechanism, which represents an aspect that cannot be optimized. When a parametric design is available in later design phase, the non-zero element could be optimized, but the zero element does not change. From the perspective of mechanism, the sparsity of the IR matrix reflects the specific correspondence between the input and output, which indicates the specialty of the mechanism typology. As long as the specific mechanism type is determined, the IR matrix could be derived to perform the complexity analysis proposed in this paper. And no detailed parameters are needed. The above statement is also consistent with our common sense. For example, we can think that a basketball player may be more powerful and flexible than a young child. But the basketballer’s body structure is neither better-optimized nor more complicated than that of a little child. A child has great potential to be as sturdy as a basketballer. In this case, the same structure of a creature determines the complexity of the morphology between individuals, even if the detailed structural parameters are quite different. Back to the issue of the mechanism, the IR matrix could be used to establish the complexity criteria. The elements in the matrix are only relevant to the mechanism topology and not affected by specific dimensional parameters. This is a good fit for our intuitive requirements for type criteria. In this paper, Eq. (4) is defined as the incidence relation of legged robots. The input vector, output vector and incidence-relation matrix constitute the three elements of the IR. Thus these three elements are called the complexity parameters. Based on the relation Eq. (4), this paper establishes criteria of type complexity for legged robots from both output and input perspective.

3. Definition of complexity criteria The following two subsections analyze Eq. (4) of the legged robotic mechanism from two aspects. We hereby focus on the establishment and the corresponding mathematical understanding of the criteria.

3.1. Output complexity If we examine Eq. (4), we look from left to right and make the following understanding: The motion in the direction of each DoF at the end-effector (each component of the output vector v) is controlled by one or several drive DoF (the components of the input vector s). In the workspace for walking robots, usually there is an intuitive explanation for each DoF at the end-effector. It represents a certain direction of movement (such as the translation or rotation of the body frame in Cartesian coordinates), or a specific task (such as the wrist rotation or hand clamping of a simple robotic manipulator carried on the body frame). In other words, the output vector can directly reflect the purpose of the legged robot’s movement. Therefore, given a robotic mechanism, we can use Eq. (4) to establish mathematical expressions with practical meanings of complexity from the purpose of robot tasks. Investigate a component of the output vector in Eq. (4). Without loss of generality, it is set to the i th component, which is affected by the i th row and the input vector of the IR matrix:

vi ⇐ Ir=i ⇒ S.

(5)

Expand the above formula:

⎡ ⎤ 

vi ⇐ Ii,1

Ii,2

···

Ii,n



s1

⎢s 2 ⎥ ⇒ ⎢ . ⎥, 1 ≤ i ≤ m. ⎣ .. ⎦ sn

(6)

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Further, if the output vector of interest is a k-dimensional in a certain operating mode P (see Table 1) of the walking robot, the incidence relation can be expressed as:

⎡ ⎤ ⎡ vi1 Ii1,1 ⎢vi2 ⎥ ⎢Ii2,1 ⎢ . ⎥⇐⎢ . ⎣ .. ⎦ ⎣ .. vik Iik,1

Ii1,2 Ii2,2 .. . Iik,2

··· ··· .. . ···

⎤

⎡ ⎤

Ii1,n s1 Ii2,n ⎥ ⎢s2 ⎥ ⎢ ⎥ .. ⎥ ⎦ ⇒ ⎣ ... ⎦, 1 ≤ i1 < i2 < · · · < ik ≤ m. . Iik,n sn

(7)

Here, the sum of the IR matrix elements corresponding to all k output components of P is:

NVP =

ik n  

Ii, j .

(8)

i=i1 j=1

Obviously, the above result will not be larger than the total number of elements of the IR matrix. Thus we normalize the result and define the output complexity:

NP

 VP

P KOC =

=

max NV |∀P

ik n   1  1  I P  . Ii, j = 0 mn mn

(9)

i=i1 j=1

The output complexity (OC) indicates the driving distribution density on the output components for mode P (corresponding to this, there is also an important indicator — the driving distribution dispersion, which will be detailed in the next section). The larger the value of output complexity, the more effective actuators are required to complete the corresponding motion to fulfill the task assigned by P. According to lemma A in Section 2.2, this means a worse operating state. From the perspective of matrix theory, the sparsity of the row vector of the IR matrix in Eq. (4) reflects the average load state of the drive. For practical purposes, each output DoF may be assigned different weights for different importance. For example, for a walking robot, when performing area patrol tasks (see Table 1), the moving ability in the horizontal plane is far more important than the ability to move up and down. Because the moving ability in the horizontal plane (two translational directions and one rotational direction, see No.3 in Table 1) determines whether the robot can complete the task, while the ability to move in other directions is only related to the flexibility and robustness. Thus, we can define the overall output complexity of the legged robot. Assuming that there are totally q operating modes for the robot output vector, the average output complexity could be expressed as:

KOC =

q 

Pk ωk KOC =

k=1

q q   1   ωk IPk 0 , ωk = 1 mn k=1

(10)

k=1

where ωk is the weight of the k-th operating mode. The weights are determined based on the purpose of the robot’s work (see Table 1). When the robot’s input and output are strongly coupled, the above expression can take the maximum value of 1, and each output DoF is affected by all the actuators. It is worth noting that the criterion mentioned in this section, Eq. (10), is based on the analysis of the output v of the incidence relation model (Eq. (4)). For this reason, this criterion is collectively referred to as output complexity.

3.2. Input complexity Section 3.1 examines Eq. (4) from the left-to-right direction, and proposes the concept of output complexity to analyze the actuating intensity of the mechanism. In this section, we revisit Eq. (4) from right to left: Each drive of the mechanism (each component of s) contributes to one or several DoF at the end-effector (the components of v). For walking robots, the contribution of each drive to the output of the mechanism is different. Examine the j-th component of the input vector in Eq. (4), which affects the output vector via the j-th column of the IR matrix. Decompose Eq. (4) by column:

V ⇐ Ic=j ⇒ Sj .

(11)

Expand this relation:

⎡ ⎤ v1

⎡

⎤

I1, j ⎢ v2 ⎥ ⎢ I2, j ⎥ ⎢ . ⎥ ⇐ ⎢ . ⎥ ⇒ sj , 1 ≤ j ≤ n. ⎣ .. ⎦ ⎣ .. ⎦ vm Im, j

(12)

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If the output vector of interest is a k-dimensional in a certain operating mode P of the walking robot, the incidence relation can be expressed as:

⎡ ⎤ ⎡ ⎤ vi1 Ii1, j ⎢vi2 ⎥ ⎢Ii2, j ⎥ ⎢ . ⎥ ⇐ ⎢ . ⎥ ⇒ sj , 1 ≤ i1 < i2 < · · · < ik ≤ m. ⎣ .. ⎦ ⎣ .. ⎦ vik Iik, j

(13)

Similar to the definition of Eq. (8) above, the driving intensity of the j-th input component of P could be defined as follows:

NsPj =

ik 

Ii, j .

(14)

i=i1

Eq. (14) could represent the magnitude of the contribution of a single drive to the robot’s output. Combined with Eq. (9), it is easy to know the relationship between the input driving intensity and the output complexity: n 

P NsPj = mnKOC .

(15)

j=1

Assuming that there are totally q operating modes for the robot output vector, the overall input complexity of the robot is defined as:

n n

KIC =

 

j=1 Nsi −  2n nj=1 Nsj

i=1



Nsj 

(16)

where:

Nsj =

q  k=1

ωk NsPjk =

q  k=1

q P   k  ωk Ic=j , ωk = 1. 0

(17)

k=1

The weights are determined based on the purpose of the robot’s work (see Table 1). Eq. (16) can be referred to as the input complexity (IC) of legged robots. It indicates the uniformity of the incidence relation, reflecting the relative dispersion of driving distribution (corresponding to the driving intensity in Section 3.1). Hirschman [42] first proposed a continuous form of this representation and applied it in interdisciplinary fields such as economic dynamics. Gini introduced this coefficient to indicate the degree of income inequality [43]. Since the incidence relation matrix Eq. (4) is discrete, the corresponding input complexity Eq. (16) should be a discrete expression form of the Hirschman coefficient. The upper limit of Eq. (16) is 1, and the lower limit is 0. The larger the value, the more uneven the driving distribution. According to lemma B in Section 2.2, this means a possibility of a worse operating state. From the perspective of matrix theory, the 0-norm difference of the columns of IR matrix reflects the uniformity of the driving load. Here is an intuitive explanation for the input complexity Eq. (16). In order to improve the operating state of the mechanism and reduce the maintenance cost, we tend to equalize the working intensity of each driver, i.e.

Nsi = Nsj , ∀i ∈ [1, n], ∀ j ∈ [1, n].

(18)

This ideal operating state avoids some drivers that keep working at all, while others run occasionally. Still we give a biological example: Muscular well-balanced animals usually have strong sustained moving ability, while muscle groups with less exercise are prone to lose function or even atrophy in the end [44]. The laws of nature seem to imply how we can filter-out some of the less promising “types”. In summary, the uniformity of the incidence exerted by each input components exactly reflects the complexity of the driving load. That’s why we introduce the input complexity Eq. (16) as the second type of complexity criteria. For practical purposes, since the input complexity indicates the uniformity of the driving distribution, it is applicable to robots working under extreme conditions. When designing such mechanisms, the input complexity criterion should be considered with high priority. 4. Examples of implementation Since the two types of complexity criteria based on the IR matrix are proposed in Section 3, the mathematical expressions Eqs. (10) and (16) are universal for multi-rigid bodies. However, these indicators were originally proposed for legged robots. In other words, the analysis and calculation of these two types of indicators could reflect the characteristics of the complexity of legged robots. On the other hand, some features of the walking robots help to simplify the implementation of the above indicators. This realizes the fast evaluation of the complexity criteria in the absence of specific dimensional parameters at the beginning of the design, which exploits the potential of robot types and reduces the cost of the postiteration design. This section will give specific examples of this point of view.

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D. Xi and F. Gao / Mechanism and Machine Theory 144 (2020) 103661

Fig. 2. Leg type L1: (a) the schematic diagram of the leg with the structural parameters including the leg coordinate system, the body frame position, the limb topology, the input s and the output v; (b) the rendering model of the mechanical leg.

4.1. Evaluation objects This section first describes the difficulties of the legged-robot mechanisms in kinematic analysis, and then proposes a reasonable simplified analysis method. Through this method, the incidence relation Eq. (4) corresponding to the mechanism could be easily obtained, thereby calculating the complexity criteria. A legged robot is usually intricately constructed, with a number of drives which is much larger than the end-effector’s DoF. Such a robot could be considered as a hyper-redundant mechanism [45]. One of the characteristics of the hyperredundancy is that when a part is selected as an end-effector, there is a multi-solution or an infinite solution in driving space (input vector) to a task space point (output vector). This case demonstrates the difficulty for mathematical calculation as: (Q1) The incidence relation matrix is hard to calculate, or there are no obvious zero elements. (Q2) The incidence relation is uncertain. The above two points will lead to the extreme complexity of the IR matrix form. Therefore, in order to improve the practical operability of the complexity criteria, it is necessary to impose some restrictions on the mechanism and its operating conditions. These restrictions include the type and quantity of the mechanical legs, the layout of the legs on the body frame, the leg configuration, the reference gait, etc. This series of restrictions constitutes the objects of evaluation in this paper. A brief description will be given below. 4.1.1. Leg type The type of the mechanical legs is the premise for determining the IR matrix. Therefore, it’s the basis for solving the two problems Q1 and Q2 in Section 4.1. At the early design stage, according to the operating tasks and DoF requirements of the legged robot, several leg type candidates have been identified. For example, in the authors’ laboratory, the leg types could be usually obtained based on synthesis methods of GF set theory [15]. Two examples are directly proposed here as the evaluation objects in this paper, and the relevant synthesis steps are omitted. The detailed synthesis methods for these leg types are derived in the reference [15]. Leg type L1: As shown in Fig. 2, it’s a three-chain parallel mechanism with UP/2UPS limbs. It could be classified as an R-type mechanical leg, which is suitable for omnidirectional mobile platforms [5,8]. Leg type L2: As shown in Fig. 3, it’s a multi-linkage hybrid structure, with R-2RR limbs. It could be classified as a T-type mechanical leg, which is suitable for bionic robots [3,13]. 4.1.2. Body frame layout If a specific leg type is used, we could also develop various robot types by adjusting the position or the number of mechanical legs on the body frame. Therefore, determining the body frame layout is also one of the prerequisites for solving the two problems Q1 and Q2 in Section 4.1. In order to facilitate the subsequent simulation or experimental verification, this paper takes the existing prototypes of the authors’ laboratory as evaluation examples. Prototype I: As shown in Fig. 4, it’s a centrally symmetrical hexapod robot. The leg structure adopts the leg type L1. Prototype II: As shown in Fig. 5, it’s a hexapod with plane-symmetry. The leg structure adopts the leg type L1 as Prototype I. Prototype III: As shown in Fig. 6, it’s a hexapod with plane-symmetry. But the leg structure adopts the leg type L2. 4.1.3. Reference configuration The IR matrix reveals the characteristics of robot type. The matrix values are determined by the reference coordinate system of the robot, while the robot coordinate is usually established based on the configuration of the mechanical legs (see

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Fig. 3. Leg type L2: (a) the schematic diagram of the leg with the structural parameters including the leg coordinate system, the limb topology, the input s and the output v; (b) the rendering model of the mechanical leg.

Fig. 4. Prototype I: (a) the rendering model of the hexapod robot, with the DoF of the body frame labeled; (b) top view of the prototype I, with the coordinate system positions of six mechanical legs and the body frame. Legs with the same background color indicates the same set of tripod gait.

Fig. 5. Prototype II: (a) the rendering model of the hexapod robot, with the DoF of the body frame labeled; (b) top view of the prototype II, with the coordinate system positions of six mechanical legs and the body frame. Legs with the same background color indicates the same set of tripod gait.

Figs. 2 and 3). Therefore, the selection of the reference configuration should help to simplify the calculation and analysis of the coordinate values. In general, the reference configuration coincides with the central or extreme poses of the body frame or the mechanical leg for the following reasons: (1) The central pose corresponds to the core region of the workspace. Legs move in the vicinity of this pose for most cases. It means more cooperative actuators and longer operating duration. Therefore, the central pose is the key configuration for common tasks. (2) The central pose corresponds to the high kinematic or dynamic performance, with stable control and good mechanical properties. But it’s usually accompanied by a large driving load during operation.

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D. Xi and F. Gao / Mechanism and Machine Theory 144 (2020) 103661

Fig. 6. Prototype III: (a) the rendering model of the hexapod robot, with the DoF of the body frame labeled; (b) top view of the prototype III, with the coordinate system positions of six mechanical legs and the body frame. Legs with the same background color indicates the same set of tripod gait.

(3) The extreme pose often means poor structural stability. It’s close to the singular position, the mechanical boundary, or the control boundary, where the probability of failure during operation is large. Mechanical parts are susceptible to damage in this configuration and might require special maintenance. (4) The extreme pose limits the motion of the robot. It determines the robot’s mission capability, and it’s also crucial to the mathematical model calibration. To ensure that the robot can reach the upper limit of the expected ability, it is usually necessary to carry out trial and error in extreme poses before putting into application. Thus, a good design for this pose guarantees the ability to complete the tasks, which is crucial at both the design stage and the experiment stage. (5) There are certain specialties (symmetry or extremity) of these two poses. On the incidence relation matrix, the symmetry, antisymmetry, or sparsity of the matrix is presented, which simplifies the analysis and calculation of the complexity criteria and thereby reducing the overall evaluation cost. As previously analyzed in Section 2.2, the operating state X in Eq. (1) is the goal of type evaluation. The reasons of (1)-(4) above indicate that there are a number of factors affecting the operating state near the central or the extreme pose. Furthermore, the determination of the reference configuration also helped solve the difficulty Q2 in Section 4.1. The statement (5) uses the characteristics of these poses to solve the difficulty Q1 of the kinematic analysis of multi-rigid body mechanism, which greatly improves the feasibility of the type evaluation method. Therefore, the calculations below are based on the reference configuration near these two poses. 4.1.4. Reference gait This paper deals with complexity-based performance estimates. One of the most commonly used patrol gaits can be considered as a reference gait, such as the trot gait of a quadruped or the tripod gait of a hexapod. There are two reasons for this: (1) This is the most commonly used gait when the robot is walking, and it also means the longest working duration of the actuators. The complexity criteria obtained for this gait can be used as a benchmark for design. (2) This is the basic gait used by various robots with different types and tasks. The analysis results obtained can be used as a general indicator to compare the performance of different robots. More specifically, the purpose of complexity evaluation is not to redesign a new robot type, but to compare various existing types to pick out the optimal structures for subsequent development. A universal gait that is used by various robots for most tasks is a good choice. By determining a certain reference gait, it is possible to work out the incidence relation Eq. (4), thus solved the problem Q2 in Section 4.1. 4.2. Implementation procedure 4.2.1. Operating pattern According to the discussion in Section 2.1, the general twenty-five GF sets could be reduced to 10 commonly used operating modes for legged robots. In this section, two sets of operating patterns are determined for the type evaluation of legged-robot implementations. The weight coefficient combinations of the two patterns are shown in Table 2. The two sets of weights will be included in the theoretical complexity evaluation, as well as the analysis of simulation results or experimental data, to demonstrate the correctness of the criteria. The first set, pattern A, focuses on the omnidirectional mobility in the horizontal plane. It’s suitable for general-purpose area-patrolling robots or mobile transport platforms [5,8]. The second set, pattern B, focuses on the sagittal mobility of the robot, and is suitable for biomechanics or routine patrolling security robots [3,9,13].

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Table 2 Operating patterns for implementation examples. GF sets (end-effector)

( ( ( ( ( ( ( ( ( (

Operating mode

) )

GIF7 Ta , 0, 0, 0, 0, 0 GIF6 Ta ,Tc , 0, 0, 0, 0 GIF5 Ta ,Tb ,Rγ , 0, 0, 0 GIF5 Ta ,Tc ,Rβ , 0, 0, 0 GIF4 Ta ,Tb ,Tc , 0, 0, 0 GIIF21 Rα , 0, 0, 0, 0, 0 GIIF18 Rα ,Rβ , 0, 0, 0, 0 GIIF12 Rα ,Rβ ,Rλ , 0, 0, 0 GIF11 Ta ,Tb ,Rα ,Rβ , 0, 0 GIIF22 Ta ,Tb ,Rα ,Rβ ,Rγ , 0

) ) ) ) )

) )

)

Reciprocating patrol Repeated load Area patrol Continuous climbing Mobile load Field operation Field operation Field operation Sequential operation Sequential operation

Weight (pattern A)

Weight (pattern B) 0.6

0.6

0.1 0.1

0.1 0.2 0.1

0.2

Fig. 7. The step block diagram of a top-down design methodology based on GF set theory.

4.2.2. Implementation flow chart According to the evaluation objects given in Section 4.1, the output vector, input vector and IR matrix of each prototype for a specific task could be worked out. From this we have obtained the required data for Eq. (4). Fig. 7 shows a general design process of legged robots within the framework of GF set theory. The arrow in the figure indicates the decision direction: the design object at the end of the arrow is mainly affected by the object at the beginning of the arrow. All the evaluation objects and evaluation conditions described in Section 4.1 are depicted in this block diagram. The evaluation methodology described in this paper is mainly used in the type evaluation process in the figure. For the specific evaluation based on complexity criteria, a logical calculation routine should be considered, and a relatively linear flow chart is needed to ensure the operability of the evaluation methodology. In order to clearly demonstrate how to calculate the criteria described in this paper, a flow chart Fig. 8 is given substituting for the block diagram Fig. 7, which reflects the specific calculation steps of the entire evaluation process.

4.3. Application and results If we examine the mathematical basis of the evaluation method above, we can find that our method utilizes simple and effective mathematical tools, including the intuitive representation of the GF set, the relevant basis of probability theory, and the use of the incidence-relation matrix. These theoretical tools have almost no limitations on the structure and specific

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D. Xi and F. Gao / Mechanism and Machine Theory 144 (2020) 103661

Fig. 8. The flow chart of the type evaluation for legged robots (Different background colors correspond to different design stages in the step block diagram, see Fig. 7).

parameters of the mechanism itself, thus showing a strong application universality. In this section we give the demonstration results of various aspects of the legged robots with different types in our lab.

4.3.1. Criteria calculation This section gives specific calculation results for the complexity criteria. Note that although the final results are specific values, we need to understand that this is only a pre-design estimate. In other words, the numerical results in this paper reflect the performance potential, rather than the ability of the final real robots. As described in Section 1, the specific performance is further affected by the post-iterative design. Consider the two mechanical leg types mentioned in Section 4.1.1, with the relevant markings in Figs. 2 and 3. The input vector of leg type L1 corresponds to three linear actuators:

 s

L1

=



sL11 sL21 . sL31

(19)

The output vector is the three DoF of the main limb (UP), defined as:



vL11 L1 v = vL21 vL31

 (20)

where vL11 is the translational DoF along the direction of the main limb actuator (the prismatic pair), and obviously, vL11 = sL11 . vL21 and vL31 correspond to the rotational DoF of the near frame and the far frame of the universal pair, respectively. Taking the central pose as the reference configuration, the coordinate system is defined in Fig. 2(a). The origin is located in the center of the U-pair. The z-axis coincides the main limb UP. The negative-z direction is pointing to the toe. The XZ plane is the symmetry plane of the mechanical leg, and the shafts of the U-pairs are placed along the x-axis and the y-axis respectively. The incidence relation matrix could be obtained by means of parameter substitution method or mobility deduction [30], and the calculation process is briefly given here.

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Let sL11 = 0, sL21 = sL31 = 0, then the toe moves in the XZ plane, which can be derived as:

 L1    v1 1 vL21 ⇐ 0 ⇒ sL11 . 1 vL31

(21)

Let sL21 = 0, sL11 = sL31 = 0, then the toe moves parallel to the XY plane near the central pose, which can be derived as:

 L1    v1 0 vL21 ⇐ 1 ⇒ sL21 . 1 vL31

(22)

Similarly, sL31 = 0, sL11 = sL21 = 0, by symmetry, we have:

 L1    v1 0 vL21 ⇐ 1 ⇒ sL31 . 1 vL31

(23)

At this point, the incidence relation expression of leg type L1 is:

 L1   v1 1 vL21 ⇐ 0 1 vL31



0 1 1

0 1 1



⇒



sL11 sL21 . sL31

(24)

The incidence relation expression of leg type L2 can be obtained in a similar manner:

 L2   v1 0 vL22 ⇐ 0 1 vL32



1 1 0

1 1 0



⇒

sL12 sL22 sL32



(25)

see Fig. 3(a) for specific markings. The input vector of L2 is three rotational drives, and the reference coordinate system is attached to the drive intersection. sL12 drives the side-swing motion. sL22 and sL32 drive in the sagittal plane. And the z-axis is directed to the toe. The y-axis is determined according to the right-hand rule. The three output vectors are respectively along the two directions in the sagittal plane and the lateral orthogonal direction thereto. Next, we consider the three kinds of layouts mentioned in Section 4.1.2. Note that the body frame of the prototype could be understood as the moving platform of a parallel mechanism. Correspondingly, each supporting leg could be treated as a mechanical limb, and the ground is the base of the parallel mechanism (the fixed platform, see Fig. 1). For the prototype I, the tripod gait is introduced as the reference gait (see Section 4.1.4). Fig. 4(a) defines the body frame coordinate system. We obtain the incidence relation from 18 leg actuators to the body frame:

vI ⇐ II ⇒ sI

(26)

where the output vector:



vI = x

y

α

z

β

γ

T

(27)

and the input vector:

sI =



sLLF1

T



1 sLRM

T



sLL1H

T



sLR1F

T



1 sLLM

T



1 sLRH

T T

(28)

The subscript represents the leg location on the body frame, i.e., L, R, F, M, and H represent the left, right, front, middle, and hind legs, respectively, as shown in Fig. 4(b). Note that the mechanical legs with the same background color in Fig. 4(b) (LF, RM, LH or RF, LM, RH) are the same set of legs in a tripod gait. And the overall incidence relation matrix is obtained by expanding Eq. (26) according to Eq. (24):

⎡

1 ⎢1 ⎢1 II = ⎢ ⎢1 ⎣ 1 0

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 0 0

1 1 0 0 0 1

1 1 0 0 0 1

1 1 1 1 1 0

1 1 0 0 0 1

1 1 0 0 0 1

1 1 1 1 1 0

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 0 0

1 1 0 0 0 1

1 1 0 0 0 1

1 1 1 1 1 0

1 1 0 0 0 1

⎤

1 1⎥ 0⎥ ⎥. 0⎥ ⎦ 0 1

(29)

So far, we have established the overall IR expression of the prototype I. Through similar steps, the IR expressions of the other prototypes could be obtained. For prototype II, Fig. 5 shows the definition of the coordinates. We obtain the incidence relation from the actuators to the body frame:

vII ⇐ III ⇒ sII

(30)

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D. Xi and F. Gao / Mechanism and Machine Theory 144 (2020) 103661 Table 3 Results of the type evaluation. Operating pattern

Prototype I OC IC

Prototype II OC IC

Prototype III OC IC

Pattern A Pattern B

0.41 0.17

0.40 0.14

0.32 0.13

0.06 0.11

0.06 0.13

0.12 0.29

where the output is the same as prototype I:



vII = x

y

α

z

β

γ

T

(31)

the input vector:



sII =

sLLF1

T



1 sLRM

T



sLL1H

T



sLR1F

T



1 sLLM

T



1 sLRH

T T

(32)

and the overall incidence relation matrix is obtained by expanding Eq. (30) according to Eq. (25):

⎡

0 ⎢1 ⎢1 III = ⎢ ⎢1 ⎣ 1 1

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 0 0

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 1 1

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 1 1

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 0 0

1 1 0 0 0 1

1 1 0 0 0 1

0 1 1 1 1 1

1 1 0 0 0 1

⎤

1 1⎥ 0⎥ ⎥. 0⎥ ⎦ 0 1

(33)

For prototype III, Fig. 6 shows the definition of the coordinates. We obtain the incidence relation from the actuators to the body frame:

vI I I ⇐ II I I ⇒ sI I I

(34)

where the output is the same as other prototypes:



vI I I = x

y

α

z

β

γ

T

(35)

the input vector:

sI I I =



2 sLHL

T



2 sLFM

T



2 sLHR

T



sLF2L

T



2 sLHM

T



sLF2R

T T

(36)

and the overall incidence relation matrix is obtained by expanding Eq. (34) according to Eq. (25):

⎡

II I I

0 ⎢1 ⎢0 =⎢ ⎢0 ⎣ 0 1

1 0 1 1 1 1

1 0 1 1 1 1

0 1 0 0 0 1

1 0 1 0 1 0

1 0 1 0 1 0

0 1 0 0 0 1

1 0 1 1 1 1

1 0 1 1 1 1

0 1 0 0 0 1

1 0 1 1 1 1

1 0 1 1 1 1

0 1 0 0 0 1

1 0 1 0 1 0

1 0 1 0 1 0

0 1 0 0 0 1

1 0 1 1 1 1

⎤

1 0⎥ 1⎥ ⎥. 1⎥ ⎦ 1 1

(37)

So far, all the incidence-relation expressions of the three prototypes have been obtained, as Eqs. (26), (30) and (34). This allows calculation of the corresponding output complexity and input complexity. According to Eqs. (10), (16) and the weight sets in Table 2, all six complexity criteria could be calculated. Here we directly list the results as follows: 4.3.2. Analysis and assessment In order to visualize the complexity analysis, the results in Table 3 are scaled with respect to prototype I, and the normalized dot chart is shown in Fig. 9: According to Fig. 9, we make the following discussion: (1) Consider the output complexity (OC, the red lines in Fig. 9(a) and (b)). For the operating pattern A of planar omnidirectional mobility, the OC of the prototype I and the prototype II are almost equivalent (3% difference), and the OC of the prototype III is much smaller (19% difference). It indicates that the overall actuating intensity of the prototypes I and II is equivalent, and the driving load of the prototype III is lower. For the operating pattern B of linear working mobility, the OC of the prototype I is larger than the other two (17% difference), indicating that the driving load of the prototype I is higher. (2) Consider the input complexity (IC, the blue lines in Fig. 9(a) and (b)). For the operating pattern A, the IC of the prototype III is significantly larger than other prototypes (100% larger). Similarly, for the operating pattern B, the IC of the prototype III is also the largest.

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Fig. 9. The theoretical values and simulation results of the three legged robots (scaled with prototype I): (a) results of operating pattern A (b) results of operating pattern B.

(3) For the prototype I, the IC is the smallest in both operating modes, indicating that the driving load of the prototype I is uniform. While the OC of the prototype I is relatively large, indicating that the average driving load is large. These two points imply that the prototype I exhibits an isotropic nature and is suitable as an omnidirectional moving platform. So far, lots of research [5,8] has proved the advantages of such structures in omnidirectional tasks. (4) For the prototype III, the IC is the largest in both modes, indicating the uneven distribution of operating state. Meanwhile, the OC of the prototype III is relatively small, indicating that the average driving load is small. These two points imply that the prototype III has the characteristics of anisotropy and is competent for tasks with clear directionality. This is also evidenced by the fact [3,13] that most bionic robots use this structure to perform sagittal tasks, such as rushing forward, reciprocating patrolling, etc. (5) The task adaptability of Prototype II is between Prototype I and Prototype III. It could be utilized as an omnidirectional platform combined with field/directional tasks. This type of legged robot has been studied to complete tasks such as opening a door or turning a valve for disaster relief [46]. According to the above analysis, for different job requirements, we should consider specific robot types for the subsequent design process. As a result, these three robot types are the original prototypes that have been designed and manufactured in the authors’ laboratory. Each prototype has its own unique positioning, which is proposed as a differentiated design methodology. It’s like the “specialization” of human society. Robots with similar legged-shape need to consider different work targets and operating environment to achieve the best performance. 5. Simulation and experiment The essential problem discussed in this paper is how to calculate the complexity criteria of the legged robots in the absence of specific parameters at the early stage of design, and to use this as a reference to screen the robot types. See the flow chart Fig. 8 above for the details. As verification of these criteria of type complexity, theoretically it should start with a series of prototype design: Complexity calculation is performed for each original type, and then each type is put into the next design stage until the final prototype is produced for testing. Then the experimental data are compared with the previous complexity criteria to verify the effectiveness of this evaluation method. However, in doing so, some robots with poor performance are also designed and manufactured, just to verify a theoretical model, resulting in a waste of resources. So in this section, we take another low-cost approach. We have selected some of the existing prototypes in the authors’ lab (see Section 4.1.2 above), with different leg types and body layouts depending on the specific operating tasks. The complexity calculation of these selected robot types is carried out. The comparison between the theoretical results and the data of simulations/experiments can endorse the validity of the complexity criteria. Thus it proves to be practical guidance that is clear and efficient for the preliminary conceptual-design stage. 5.1. Simulation and results To further evaluate the efficacy of the proposed method of complexity evaluation with the characteristic of “definite type and few parameters”, all the prototypes were modeled and tested in the multibody dynamics simulation software. Prototype simulation is based on virtual modeling techniques with idealized 3-dimensional models. The digital models were simulated both kinematically and dynamically to realize the 6-DoF mobility of the body frame. The input vectors and output vectors satisfying the requirements of Section 4.1 were sampled. And the transformation matrix from the input to the output vectors was fitted using the least squares method [47]. After normalization, the incidence relation matrix via the fitting procedure

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Fig. 10. Snapshots of the simulation of Prototype I. The hexapod robot walked with tripod gait with gait period of 1.5s: (a) x-direction moving forward; (b) z-direction turning; (c) y-direction side shifting; (d) z-direction lifting; (e) x- and y-directions tilting.

Fig. 11. Scatter plot of input-output incidence relation matrix of prototype I. Color-depth dots were fitted with simulation data. Each row corresponds to a specific DoF of the robot end-effector, and each column corresponds to a specific actuation of the mechanical legs.

was obtained. As an example, Fig. 10 shows the motion of the various DoF of the end-effector during the simulation of the prototype I. The other two prototype simulation processes are similar and omitted here. Figs. 11, 12 and 13 show the scatter plots of the fitted IR matrix. Each block in the scatter plot represents the corresponding position (row and column) in the IR matrix. Each scatter dot in the block represents the data point fitting the specific input and output that meet the requirements of Sections 4.1.3 and 4.1.4 for a period of gait. Normalize the data point to get a value in the range 0 to 1, expressed in the corresponding color depth. The dots in each block are dispersed for visual effect. In order to compare the results, the corresponding theoretical values of the IR matrix (Eqs. (29), (33) and (37)) are labeled in the blocks. The scatter plots (Figs. 11–13) indicate that the input-output relation obtained by the simulation is basically consistent with the theoretical results (Eqs. (29), (33) and (37)). The IR matrix is a high-level overview of the input-output transformation for legged robots. To demonstrate the connection between the complexity criteria and the robot’s operating state (as discussed in Section 2.2), we start with a simple case. As mentioned in Section 2.3, the complexity discussed in this paper refers to

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Fig. 12. Scatter plot of input-output incidence relation matrix of prototype II. Color-depth dots were fitted with simulation data. Each row corresponds to a specific DoF of the robot end-effector, and each column corresponds to a specific actuation of the mechanical legs.

Fig. 13. Scatter plot of input-output incidence relation matrix of prototype III. Color-depth dots were fitted with simulation data. Each row corresponds to a specific DoF of the robot end-effector, and each column corresponds to a specific actuation of the mechanical legs.

the correspondence between the input and output of the mechanism. If we keep the output vector unchanged (the robot tasks fixed), the complexity of the mechanism could be reflected by analyzing the input-related data, such as average motor power, battery consumption, reliability, etc. In the simulation environment, it is difficult to simulate the failure rate and reliability of the actuators. Therefore, the average value of the motor output power and the battery consumption (input power) were selected as the reference data, which is shown in Fig. 9. The motor power was extracted near the reference configuration (see Section 4.1.3), and the energy consumption was obtained according to the built-in default motor model of the simulation software. What calls for special attention is that the complexity indicators defined above (Eqs. (10) and (16)) are all absolute values between 0 and 1. The lower limit indicates the worst operating state and the upper limit indicates the best, which defines the ideal baseline for comparison, while the simulation results do not have such a theoretical comparison base. Without loss of generality, the prototype I is used as a reference prototype in Fig. 9 and the relevant data is used as a baseline, i.e., we acquiesce in the fait accompli that data I is consistent with the theoretical calculation result. Accordingly, the data of the other prototypes is scaled with the base to obtain a relative value:

X=

X0

vmg

Xreg =

X XtypeI

(38)

(39)

where X0 is the original simulation data (motor power or battery consumption), v and m are the speed and the whole machine mass respectively, X is the data normalized with the speed and mass, XtypeI is the normalized data of the prototype I, Xreg is the final data scaled by XtypeI . For prototype I, the scaled data is naturally consistent with the theoretical calculation results in Fig. 9, while we could see the implication of the complexity criteria in action as the scaled data goes in a specific shape for other prototypes. As can be seen from Fig. 9, the curve shape of simulated motor power or energy consumption is basically consistent with the calculated output complexity. This is in line with the practical meaning of OC, as discussed about Eq. (9) in Section 3.1. The output complexity indicates the driving distribution density on the output components for the specific task. The larger the value of output complexity, the more effective actuators are required to complete the corresponding motion. And more drives mean higher power and more energy. 5.2. Experiment and results Corresponding to the simulation models, three physical prototypes were tested in the authors’ laboratory. The test was divided into two parts. The first part was to verify the rationality of each prototype’s complexity parameter definition. In this part, we mainly demonstrated the correctness of the input-output incidence relation matrix. In the second part, the test

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Fig. 14. Snapshots of the experiment of Prototype I. The hexapod robot walked with tripod gait with gait period of 1.5s: (a) x direction moving forward; (b) z direction turning.

Fig. 15. Scatter plot of input-output incidence relation matrix of prototype I. Color-depth dots were fitted with experimental data. The two rows correspond to x-direction moving and z-direction turning of the robot end-effector, respectively. Each column corresponds to a specific actuation of the mechanical legs.

Fig. 16. Scatter plot of input-output incidence relation matrix of prototype II. Color-depth dots were fitted with experimental data. The two rows correspond to x-direction moving and z-direction turning of the robot end-effector, respectively. Each column corresponds to a specific actuation of the mechanical legs.

Fig. 17. Scatter plot of input-output incidence relation matrix of prototype III. Color-depth dots were fitted with experimental data. The two rows correspond to x-direction moving and z-direction turning of the robot end-effector, respectively. Each column corresponds to a specific actuation of the mechanical legs.

is repeated for each prototype to measure its power, energy consumption and reliability. This part is aimed at verifying the theoretical calculation results of the complexity index. In order to more intuitively reflect the significance of the complexity evaluation indicators described in this paper, the experimental settings were simplified. During the experiment, the end-effector motion of the prototype was simplified to the two most common modes: straight walking and in-situ turning, namely mode 1 and mode 6 in Table 1. These two modes are in line with the most basic functions of the walking robot. In the simplified modes, it is also convenient to run the robot for a long time in a limited test site to measure the experimental data. The experimental data was sampled and analyzed using a method similar to that in Section 5.1. When the robot performed two kinds of operating modes: straight walking and in-situ turning, the input vectors and output vectors satisfying the requirements of Section 4.1 were sampled, and the transformation relationship of the input-output was fitted by the least squares method. The incidence relation matrix was then obtained via the fitting and normalization procedure. As an example, Fig. 14 shows the motions of the two operating modes of the end-effector during the test of the prototype I. The other two prototype experiment processes are similar and omitted here. Figs. 15, 16 and 17 show the scatter plots of the fitted IR matrix for x-direction patrolling and z-direction turning. Each block in the scatter plot represents the corresponding position (row and column) in the IR matrix. Each scatter dot in the block represents the data point fitting the specific input and output that meet the requirements of Sections 4.1.3 and 4.1.4 for a period of gait. Normalize the data point to get a value in the range 0 to 1, expressed in the corresponding color depth. The dots in each block are dispersed for visual effect.

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Fig. 18. The theoretical values and experiment results of the three legged robots (scaled with prototype I): (a) results of x-direction straight walking; (b) results of z-direction in-situ tuning.

In order to compare the results, the corresponding theoretical values of the IR matrix (Eqs. (38), (45) and (52)) are labeled in the blocks. Within the scope of the experimental settings, the IR matrix qualitatively decodes the input-output transformation for the legged robots in the sense that the fitted data points were highly consistent with the theoretical values. In the experiment, the mean motor power, battery consumption and failure rate of the actuators were selected to characterize the operating state of the robot (see Section 2.2). And all the data was extracted in comparison with the theoretical complexity criteria. The results were normalized and scaled with prototype I based on the method introduced in Section 5.1, see Fig. 18. The experiment was carried out in a series of time duration, and the total length of the robot test was 10 h. The mean output power of the motors was measured by the speed and torque feedback. The energy consumption was converted by the number of times the rechargeable battery was replaced during the total operating time. The failure rate reflects the reliability of the robot, and the number of failures occurring in the total operation time was used as the statistical result. From Fig. 18 we make the following discussion: (1) The experiment motor power and energy consumption are basically consistent with the calculated output complexity, which is in line with the practical meaning of OC (see Section 3.1). The larger the value of output complexity, the more effective actuators are required to complete the corresponding motion or fulfill the task, which is with close ties to higher power and severer energy depletion. (2) It could be further stated that the absolute difference in energy consumption (16% and 80% for moving forward and in-situ tuning, respectively) is less than that of motor power (25% and 120% for moving forward and in-situ tuning, respectively), in the sense that the curve shape is flatter. This case has by far broadly attracted our attention in the simulation as well but with less statistical significance, see Fig. 9. One of the main reasons is that the motor power was taken with respect to the reference pose (see Section 4.1.3), reflecting the input-output incidence of highload operating state, which has been already discussed in the last paragraph in Section 2.2. However, the energy consumption was measured throughout the entire period of the robot’s motion, including other leg configurations (leg phase) such as land-off, swing, touch-down, etc. The difference in robot type mainly affects the operating state near the reference pose (the state of high driving intensity), while it exerts little influence on other poses. Therefore, if the energy consumption is measured for the entire motion cycle, the difference between the experimental values of the reference pose will be “diluted” by the data of other poses. As a result, the battery consumption curve tends to be gentle. However, it can still be seen from Fig. 9 that the data of the reference pose is dominant, which proves that this pose as an evaluation condition is reasonable and correct. (3) Within the whole test, the measured failure rate was consistent with the theoretical result of input complexity. The robot failure in the experiment includes mechanical assembly failure (connector failure, wear, etc.), electrical failure (driver burnout, controller crash, etc.), and program failure (program bug, memory leak, etc.), and mainly focused on the first two types. This could be interpreted as follows: If the average driving intensity meets the design requirements, the reliability of the robot is positively correlated with the uniformity of driving load. To be more specific, uneven load or severe load changes can cause an increase in the failure rate. This explanation was discussed in detail about Eq. (18) in Section 3.2 above, and it has also been confirmed by many studies [48]. This comes as a result that at the evaluation stage of type synthesis (see Fig. 7), the consideration of input complexity is also essential, most notably to the robot fault detection and control. (4) Simplified test settings were used in the experiment to achieve long-term operation of the robot in a restricted lab area. Fig. 18(a) shows the extracted data of x-direction moving, which corresponds to the state of Fig. 14(a), and is also

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consistent with the data of Fig. 9(b). This indicates that the performance of the forward-backward traveling can reflect the motion characteristics of the robot’s sagittal plane (pattern B in Table 2). Fig. 18(b) shows the extracted data of z-direction turning, which corresponds to the state of Fig. 14(b), but it is in a diverse way to the data in Fig. 9(a). This indicates that the characteristics of the transverse-plane motion (pattern A in Table 2) should not only be judged by the unique performance of the in-situ turn, but also consider the x- and y-direction moving performance. This detailed opinion we get from the data is also in line with the intuition. After all, the most important feature of the legged robot is “walking”. This further demonstrates the correctness and intuitiveness of the type evaluation method in this paper. (5) Based on the comprehensive test, simulation, and theoretical data, the relative difference in output complexity is inversely related to the input complexity for all the three prototypes. That is, a low OC usually means a high IC. In practical terms, it can be interpreted as: a low driving intensity means an uneven driving state of all the actuators for a legged robot. This kind of dual dilemma in both mathematical and physical fields implies a trade-off for robot design. And a more in-depth study of this trade-off will further optimize the design method of legged robots in future research. 6. Conclusion The main objective of the presented research is to develop a general and practical evaluation method on the structurerelated performance for legged robots at the conceptual-design stage. This study proposes the concept of criteria of type complexity for legged robots, which describes the density and dispersion of the robotic drive distribution. Specifically, the general-function (GF) set expression is used to represent the output motion of the walking robot, and the incidence relation (IR) matrix is used to represent the input-output transformation. The definition of the complexity is proposed intuitively and logically, and the calculation method independent of the specific structural parameters is given. These definitions and calculations take advantage of the characteristics of the walking robots, helping the designer to assess the potential capabilities of the robot at the conceptual-design stage and to reduce the risk and cost of later iterative design. The criteria of type complexity for legged robots include output complexity (OC) and input complexity (IC). The average output complexity signifies the mean driving load of the robot for the task DoF. It is especially applicable to robots with output equalization, short-time operation, low load, or economical reliability. The input complexity shows clearly the uniformity of the driving load. From the perspective of the actuating-joint space, the dispersion of driving load is pointed out, which is essential to the occasions with high maintenance cost. Prototypes developed in the authors’ laboratory are taken as examples of the type-complexity evaluation. The implementation procedure of the complexity criteria is established, and the calculation data is analyzed. Using the analysis results, the applicable scene of the prototypes can be clarified, or the existing robots can be reversely engineered and modified. The simulated and experimental data of the prototypes demonstrate the correctness, intuitiveness, and rationality of the type-complexity criteria. Output complexity is related to the average load, power, and energy consumption of the actuators. Input complexity is related to drive damage, reliability, and maintenance cost. The complexity criteria proposed in this paper are based on the operating state of the walking robot and only relate to the correlation between the input and output. The presented work does not address the specific performance of the robot, such as decoupling, stiffness, speed characteristics, and load bearing. While the topological structure of the legged robot determines the input-output correlation, it also defines the specific relating forms, i.e., parallel/serial/hybrid connection, number of branches, constraint allocation, etc. Considering these specific forms, the complexity indicator can be further expanded, and the estimation/evaluation of the overall performance of the walking robot can be improved in detail. This will be discussed as a more in-depth topic in future related research. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in the manuscript entitled Criteria of Type Complexity for Legged Robots. Acknowledgements This work was funded by the National Key Research and Development Plan of China (No.2017YFE0112200), the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734575, the National Natural Science Foundation of China (No. U1613208) and the Natural Science Foundation of Jiangsu (No. BK20171250). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103661.

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