- Email: [email protected]
Classification of Sitting States for the Humanoid Robot SJTU-HR1
Journal of Bionic Engineering 8 (2011) 49–55
Classification of Sitting States for the Humanoid Robot SJTU-HR1 Jialun Yang, Feng Gao State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
Abstract The classification of sitting issues is investigated since detailed state classification for humanoid robots plays a key role in the practical application of humanoid robots, particularly for the humanoid robots doing complicated tasks. This paper presents the concept, the characteristics tree, and the prototype of the humanoid robot SJTU-HR1. The basic states for humanoid robots are proposed, including lying, sitting, standing, and handstanding. Moreover, the sitting states are classified into several states from the viewpoint of topology. The GF (generalized function) set theory is applied to achieve the kinematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1. Finally, the results indicate that a large number of the sitting states can be represented by the meaningful notations systematically. Furthermore, the one-to-one correspondence between the state and kinematic characteristics of the interested end-effectors of the SJTU-HR1 leads to deeper insight into the capabilities of the humanoid robot SJTU-HR1. Keywords: humanoid robots, generalized function sets, characteristics tree, sitting states, classification Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60005-X
1 Introduction The capabilities of humanoid robots have been enhanced dramatically thanks to the advancement of technologies and theories, including the issues of design, control, modeling, and manufacturing for humanoid robots[1–4]. It is reasonable to expect that the era human beings and humanoid robots can share the same environment simultaneously will come in the future. It is not difficult to see that the motion planning issue is essential for the successful implementation of humanoid robots. So far a lot of methodologies have been proposed and some of them have already been used in humanoid robots platforms. Nakaoka et al.[5] made use of a motion capturing system to acquire human dance motions. In the work of Lim et al.[6], the movement primitives, such as reaching, swinging, lifting light objects and lifting heavy objects for the forearm, were represented and stored as a set of joint trajectory basic functions which were extracted via a principal component analysis of human motion capture data. Through doing linear combination of those basis functions and optimization according to some criteria, the reasonable input data for humanoid robots were achieved. Kuffner Corresponding author: Feng Gao E-mail: [email protected]
et al.[7], employed a randomized search strategy based on rapidly-exploring random trees for the planning algorithm in order to generate dynamically-stable collision-free trajectories for humanoid robots. Kim et al.[8] focused on the motion planning issue when the humanoid robot walk on uneven and inclined floor. Potkonjak and Vukobratovic[9] proposed a general framework for the dynamic modeling issue of human and humanoid motion. Yamamoto and Kuniyoshi[10] proposed an approach for the rising motion of humanoid robots. Several states were introduced in their work using figures. Hirukawa et al.[11] made use of several figures to represent the states of humanoid robot HRP-2P for the process of lying down and getting up. Only typical states were shown and other intermediate states were omitted to make the description more concise. In the work of Stilman et al.[12], the humanoid robot HRP-2 interacted with the obstacles actively to generate a possible path for the desired task. In the work of Yoshida et al.[13], the resolved momentum control was used to finish the motion planning issue for the tasks that require whole body motion of humanoid robot. Ohashi et al.[14] described a collision avoidance method for a biped robot with an upper body.
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Journal of Bionic Engineering (2011) Vol.8 No.1
We can see that the process of solving this issue contains two main steps, namely, topological state planning (see Refs. [9–14]) and detailed motion planning (see Refs. [5–8]). In fact, the tasks introduced in Refs. [5–8] are in such an easy manner that the topological state planning is negligible. In contrast, the tasks mentioned in Refs. [9–14] are so complicated that the topological state planning is mandatory. Along with the further development of humanoid robots, it is straightforward to see that the topological state planning will become more and more important for the application of humanoid robots. The reason is that in some cases, especially when the tasks for humanoid robots are complicated, the humanoid robots should make use of both the environment and the body of humanoid robots themselves. However, the problem of topological state planning has been less studied. In this paper, we focus on the topological state planning for the humanoid robot SJTU-HR1, particularly the classification of the basic state of sitting for the SJTU-HR1. For the systematic description of the large amounts of the sitting states, the theory of GF set is employed to achieve the kinematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1.
2 GF set theory The GF set theory has been developed for solving the issue of type synthesis of parallel mechanisms[15,16] and classification of states for humanoid robots[17–19] and human fingers[20]. It is worth noting that detailed classification of sitting states taking the self-connection into account has not been studied, which will be covered in this paper. In this section, we will recall the GF set theory introduced in Refs. [17–19]. Definition: The GF set is such a set contains one and only one element which represents the generalized function of the end-effectors of mechanisms. Note that the generalized function is the mapping between the specific structure of mechanisms and the kinematic characteristics of the end-effectors of mechanisms. The operation rules of GF sets are summarized as follows: • Union operation of GF sets: Let GF1, GF2... GFN represent N GF sets. The union of those GF sets can be expressed as:
GF = GF 1 ∪ GF 2
∪ GFN .
(1)
• Intersection operation of GF sets Let GF1, GF2 ... GFN represent N GF sets. The intersection of those GF sets is
GF = GF 1 ∩ GF 2
∩ GFN .
(2)
• Reversion operation of GF sets The concept of the reversion operation of GF sets is
( GF )
R
= GFR .
(3)
• Reversion operation of the union of GF sets The reversion operation of the union of GF sets can be computed as: (GF 1 ∪ GF 2 ∪ R = GFN ∪ GFR( N −1) ∪
∪ GFN ) R ∪ GFR2 ∪ GFR1 .
(4)
The theorems regarding the application of the GF set theory are as follows. • Theorem for serial mechanisms For a serial mechanism, the kinematic characteristics of the end-effectors of the mechanism are the union of all of the GF sets with such a sequence that from the start of the mechanism to the end. • Theorem for parallel mechanisms The kinematic characteristics of the moving platform of a parallel mechanism are the intersection operation of the GF sets of each kinematic chain of the parallel mechanism.
3 Humanoid robot platforms SJTU-HR1 3.1 Characteristics tree of the SJTU-HR1 The human skeleton contains 206 bones and has more than 300 Degrees Of Freedom (DOFs) driven by more than 600 muscles. Generally, humanoid robots do not need to feature as many DOFs as the human skeleton. From the biological point of view, we have been developing a humanoid robot platform SJTU-HR1 whose CAD model is shown in Fig. 1a and the prototype is shown in Fig. 1c. There exist three types of joints for the SJTU-HR1, according to the number of DOFs the joints have. For instance, the shoulder, waist and hip joints feature 3 DOFs, with a structure of 3_RRR spherical parallel mechanism which is the same as the mechanism of agile eye[21]. The neck, elbow, wrist, and ankle joints have 2 DOFs, with a structure of 2_RR spherical
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
mechanism which is the same as the spherical 5R mechanism[22]; The knee joints feature 1 DOF, with a structure of revolute joints. Most of the motors can be mounted on or near the base, thanks to the introduction of parallel mechanisms. Moreover, multiple joints have the same structure could lead to modularity design for the SJTU-HR1. As a result, the SJTU-HR1 features the advantages of both parallel mechanisms and serial mechanisms.
Fig. 1 Sketch of the SJTU-HR1 (a), its characteristics tree (b), and the prototype (c).
Fig. 1b is the characteristics tree of the SJTU-HR1, which is the sketch of the SJTU-HR1 represented by GF sets. Particularly, in Fig. 1b, the empty small circle represents the joints featuring three DOFs, the small circle with cross indicates the joints with two DOFs and the small circle with bar depicts the joints with one DOF. Moreover, the letters A, B, C, D, E, F, G in Fig. 1b represent pelvis, thorax, right hand, left hand, right foot, left foot, and head, respectively. As stated before, it is very important to distinguish the sequence of the joints when describing the specific GF set for end-effectors of mechanisms. Consequently, we let the GF sets from A to E, F, B and the GF sets from B to C, D, G be GF1, GF2, GF5, GF3, GF4 and GF6, respectively. Making use of the characteristics tree of the SJTU-HR1, we can achieve the kinematic characteristics of the interested end-effectors of the SJTU-HR1 via the GF set theory. 3.2 Basic states of the SJTU-HR1 Basically, several types of contact exist between humanoid robots and the environment, including point contact, line contact, free contact, and fixed contact[17]. We only take the fixed contact into account, meaning that once the link contacts the environment, no relative motions exist between the link and the environment. From Fig. 1b, we can see that the SJTU-HR1 con-
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sists of 15 links connected by 14 joints. In fact, each link can contact the environment or other links of the SJTU-HR1. Consequently, the total number of the contact states of the humanoid robot would be quite huge from the topological point of view. Moreover, for practical application of the humanoid robot, it is impossible and unnecessary to function all of the states. As a result, we abstract several basic states which are commonly used during daily exercises, such as lying, sitting, standing, and handstanding. The typical states are shown in Fig. 2. Obviously, the main supports of each state in Fig. 2 from (a) to (h) are thorax and pelvis, pelvis, left foot, right foot, both feet, left hand, right hand, and both hands, respectively. We define the main support as primary support for humanoid robots and other supports which are responsible for balance as secondary support. On the one hand, if we consider the state only from the topological point of view, then the states of Fig. 2d and Fig. 2g are the same, which shows the effectiveness of the employment of the concept of primary support. On the other hand, the secondary support could be used for distinguishing the different topological structures within the same basic state.
Fig. 2 Basic states of the humanoid robot SJTU-HR1.
Theoretically, each link can contact the environment or other links of the humanoid robots. However, from the viewpoint of practical application of humanoid robots, only the head, thorax, pelvis, and four limbs are considered as the interested end-effectors of humanoid
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robots. Moreover, the possible secondary support includes four limbs and thorax. And the secondary support may vary a little bit depending on the specific basic state. In the following sections, the sitting states of the SJTU-HR1, including those having self-connection loops, will be addressed in detail. Particularly, the kinematic characteristics of the interested end-effectors of the SJTU-HR1 for each state will be achieved via the GF set theory, which is able to lead to deeper insight into the capabilities of the humanoid robot SJTU-HR1.
4
Sitting states for the humanoid robot SJTU-HR1
4.1 Sitting states without self-connection loops for the SJTU-HR1 For the sitting states without self-connection loops, the four limbs and thorax can function as the secondary support. As a result, we have six typical categories of states for this case from the topological point of view, which are classified according to the number of secondary support ranges from zero to five. In Fig. 3, the sketch for each typical category is illustrated and the notation for each state is shown below the specific state. Furthermore, the terms in the notations have their meanings. For instance, the term S indicates the sitting states; the terms fl, fr, hl, hr denote left foot, right foot, left hand, right hand, respectively; the same for their uppercase counterparts; the terms N and A represent that none and all of the possible secondary supports function as the secondary support. Consequently, the notation Shl
indicates that the left hand is the secondary support for the sitting state; the term Sflhr means that both the left foot and right hand function as the secondary support for the sitting state; the term SHRHL denotes that all of the possible secondary supports except for right hand and left hand function as the secondary support; SHR means all of the possible secondary supports except for right hand function as the secondary support. Note that other sitting states exist and are not shown in Fig. 3 for the sake of conciseness. However, the meaningful notations can be assigned for other sitting states following the same rules stated above. In this case, the interested end-effectors include four limbs and the head, thorax, and pelvis of the SJTU-HR1. Making use of the characteristics tree of the SJTU-HR1, all of the kinematic characteristics of the interested end-effectors of the SJTU-HR1 can be achieved and shown in Table 1. It is important to note that the kinematic characteristics of such end-effectors that contact the environment are Null GF set and not listed in Table 1. Table 1 Kinematic characteristics of the interested end-effectors of the SJTU-HR1 for the sitting states without self-connection loops Notation SN
Shl
Sflhr SN
Shl
Secondary support
Left hand
SHR
Kinematic characteristics
Right foot
GF1
Left foot Right hand
GF2 GF 5 ∪ GF 3
Left hand
GF 5 ∪ GF 4
Head
GF 5 ∪ GF 6
Thorax
GF5
Right foot
GF1
Left foot
GF2
Right hand
(GF 5 ∩ GFR4 ) ∪ GF 3
Head
(GF 5 ∩ GFR4 ) ∪ GF 6
Thorax
GF 5 ∩ GFR4
Left foot
Right foot
GF2
Right hand
Left hand
(GF 5 ∩ GFR3 ) ∪ GF 4
Head
(GF 5 ∩ GFR3 ) ∪ GF 6
thorax
GF 5 ∩ GFR3
Right foot
Right hand
GF3
Left foot
Left hand
GF4
Thorax
Head
GF6
Both feet
Right hand
GF3
Left hand
Head
GF6
Head
GF6
Sflhr SHRHL
Interested end-effectors
Thorax SA SHRHL
SHR
SA
Fig. 3 Sketches of the sitting states without self-connection loops.
Both feet Both hands Thorax
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
4.2 Sitting states with one self-connection loop for the SJTU-HR1
For the sitting states with one self-connection loop, the four limbs and thorax function as the secondary support. Moreover, only the four limbs are able to contact the body of the SJTU-HR1. Note that only the four limbs, head, and thorax which have no connection with the environment can be contacted by the secondary support. As a result, four categories exist for such kind of sitting states, including the right hand contacts the body, the left hand contacts the body, the right foot has connection with the body and the left foot has connection with the body, respectively. One state for each category is illustrated in Fig. 4. The terms in the notations have the same meanings except for those in the superscript. Actually, the superscript includes two parts: the first part indicates which link functions as the secondary support and the second part indicates the connected link by the secondary support. As a result, the hrhe notation Sfrfl represents such a state that both feet contact the environment and the right hand has connection with the head of the SJTU-HR1. Similarly, the kinematic characteristics of the interested end-effectors for the sitting states with one self-connection loop shown in Fig. 4 are listed in Table 2, excluding those with a Null GF set. Note that the sitting states with more than one self-connection loops can also be analyzed using the same procedure.
nematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1. Consequently, we can get deeper insight into the capabilities of the humanoid robot SJTU-HR1. Furthermore, the meaningful notations can represent the states for the SJTU-HR1 in a systematic, concise, and informative manner. Table 2 Kinematic characteristics of the interested end-effectors of the SJTU-HR1 for the sitting states with one self-connection loop Notation
Interested end-effectors
Kinematic characteristics
hrhe Sfrfl
Right hand
GF 5 ∪ (GF 3 ∩ GF 6 )
hlhe SHRHL
frth Sflhr
S
flth hr
Shlhrhe
S Nhrfr
hrhe Sfrfl
hlhe SHRHL
53
Left hand
GF 5 ∪ GF 4
Head
GF 5 ∪ (GF 3 ∩ GF 6 )
Thorax
GF5
Right hand
GF3
Left hand
GF 4 ∩ G F 6
Head
GF 4 ∩ G F 6
Right foot
GFR1 ∩ GFR3 ∩ GF 5
Left hand
(GFR1 ∩ GFR3 ∩ GF 5 ) ∪ GF 4
Head
(GFR1 ∩ GFR3 ∩ GF 5 ) ∪ GF 6
Thorax
GFR1 ∩ GFR3 ∩ GF 5
Right foot
GF 1
Left foot
GF 2 ∩ GFR3 ∩ GF 5
Left hand
(GF 2 ∩ GFR3 ∩ GF 5 ) ∪ GF 4
Head
(GF 2 ∩ GFR3 ∩ GF 5 ) ∪ GF 6
Thorax
GF 2 ∩ GFR3 ∩ GF 5
Right foot
GF 1
Left foot
GF 2
Right hand
(GF 5 ∩ GFR4 ) ∪ (GF 3 ∩ GF 6 )
Head
(GF 5 ∩ GFR4 ) ∪ (GF 3 ∩ GF 6 )
Thorax
GF 5 ∩ GFR4
Right foot
G1 ∩ (G5 ∪ G3 )
Left foot
GF 2
Right hand
G1 ∩ (G5 ∪ G3 )
Left hand
(GF 5 ∩ (GF 1 ∪ GFR3 )) ∪ GF 4
Head
(GF 5 ∩ (GF 1 ∪ GFR3 )) ∪ GF 6
Thorax
GF 5 ∩ (GF 1 ∪ GFR3 )
frth Sflhr
5 An application example
Shrflth
Shlhrhe
SNhrfr
Fig. 4 Sketches of the sitting states with one self-connection loop.
From the above introduction, we can see that the GF set theory can be utilized for achieving the one-to-one correspondence between the specific state and the ki-
For humanoid robots functioning in the human-centered environment, it is desirable that the humanoid robots are able to negotiate with the environment and robots themselves. As mentioned before, in some cases, it is necessary to make use of the environment or the robot body to finish the required tasks. Consequently, we can image that one potential task for the SJTU-HR1 is illustrated in Fig. 5 consisting of 6 steps. It is not difficult to see that steps 1–3 do not have self-connection loops and steps 4–6 feature one
Journal of Bionic Engineering (2011) Vol.8 No.1
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self-connection loop. Obviously, it is better to show this process using figures rather than language description. However, there are some problems of using figures to show the state changing process of the SJTU-HR1. Firstly, it is possible that the style of the Fig. 5 is inconsistent if another person uses figures to illustrate the same process. The fact is that apart from the stick-figures employed in this paper, computer generated simulation figures can also be used[11]. Moreover, even one person can make different figures for the same task. Secondly, the figures may be interpreted differently by different people. For instance, without the filled circles which are responsible for indicating the contact between the SJTU-HR1 and the environment, we can’t distinguish if the feet contact the environment. So some misunderstandings may arise due to poor quality drawings. Moreover, the task of making figures is cumbersome, especially when the number of the contact points is bigger.
SN Step 1
S hl Step 2
SNhrfr
S hlhrhe
Step 6
Step 5
SHRHL Step 3
hrhe S frfl
Step 4
Fig. 6 Moving process of the SJTU-HR1 illustrated using notations.
It is worth noting that apart from the basic states, there also exist other states which have not been covered in this paper. In case we need to classify such kind of states for the application of humanoid robots, it may be possible to introduce number and variables which can be added in the subscript or superscript of the corresponding GF sets to identify the classified and unclassified states, which will be discussed in detail in our future work.
6 Conclusions
Step 1
Step 6
Step 2
Step 5
Step 3
Step 4
Fig. 5 Moving process of the SJTU-HR1 illustrated using figures.
To eliminate the potential misunderstandings and decrease the complexity of the presentation of the moving process of the SJTU-HR1, we use the notations associated with the specific states to describe the state changing of the SJTU-HR1. Then the same process described in Fig. 5 can be illustrated in Fig. 6 from which we can see that the style of the Fig. 6 can be consistent following the same rule of assigning notations for the sitting states of the SJTU-HR1. Furthermore, from the notations in Fig. 6 and the corresponding characteristics of the interested end-effectors listed in Table 1 and 2, we can get deeper insight of the capabilities of the SJTU-HR1.
In this paper, we have addressed the classification issue for the humanoid robot SJTU-HR1, particularly the sitting states. The main contributions of our work are as follows: • The characteristics tree of the humanoid robot SJTU-HR1 is proposed. • The basic states of humanoid robots are proposed according to the primary support, including lying, sitting, standing, and handstanding. • The sitting states are classified further into those without self-connection loops and those with one self-connection loop from the view point of topology. • The meaningful notation for each sitting state is given for the systematic description. • The kinematic characteristics of the interested end-effectors for each sitting state are achieved via the GF set theory.
• One application example is given to show the effectiveness of the proposed methodology.
Acknowledgement This work was supported by the National Basic Research Program of China (2006CB705402), the National Natural Science Foundation of China (30770538, 50821003), and the Joint Research Fund for Overseas Natural Science of China.
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
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Classification of Sitting States for the Humanoid Robot SJTU-HR1 Jialun Yang, Feng Gao State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
Abstract The classification of sitting issues is investigated since detailed state classification for humanoid robots plays a key role in the practical application of humanoid robots, particularly for the humanoid robots doing complicated tasks. This paper presents the concept, the characteristics tree, and the prototype of the humanoid robot SJTU-HR1. The basic states for humanoid robots are proposed, including lying, sitting, standing, and handstanding. Moreover, the sitting states are classified into several states from the viewpoint of topology. The GF (generalized function) set theory is applied to achieve the kinematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1. Finally, the results indicate that a large number of the sitting states can be represented by the meaningful notations systematically. Furthermore, the one-to-one correspondence between the state and kinematic characteristics of the interested end-effectors of the SJTU-HR1 leads to deeper insight into the capabilities of the humanoid robot SJTU-HR1. Keywords: humanoid robots, generalized function sets, characteristics tree, sitting states, classification Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60005-X
1 Introduction The capabilities of humanoid robots have been enhanced dramatically thanks to the advancement of technologies and theories, including the issues of design, control, modeling, and manufacturing for humanoid robots[1–4]. It is reasonable to expect that the era human beings and humanoid robots can share the same environment simultaneously will come in the future. It is not difficult to see that the motion planning issue is essential for the successful implementation of humanoid robots. So far a lot of methodologies have been proposed and some of them have already been used in humanoid robots platforms. Nakaoka et al.[5] made use of a motion capturing system to acquire human dance motions. In the work of Lim et al.[6], the movement primitives, such as reaching, swinging, lifting light objects and lifting heavy objects for the forearm, were represented and stored as a set of joint trajectory basic functions which were extracted via a principal component analysis of human motion capture data. Through doing linear combination of those basis functions and optimization according to some criteria, the reasonable input data for humanoid robots were achieved. Kuffner Corresponding author: Feng Gao E-mail: [email protected]
et al.[7], employed a randomized search strategy based on rapidly-exploring random trees for the planning algorithm in order to generate dynamically-stable collision-free trajectories for humanoid robots. Kim et al.[8] focused on the motion planning issue when the humanoid robot walk on uneven and inclined floor. Potkonjak and Vukobratovic[9] proposed a general framework for the dynamic modeling issue of human and humanoid motion. Yamamoto and Kuniyoshi[10] proposed an approach for the rising motion of humanoid robots. Several states were introduced in their work using figures. Hirukawa et al.[11] made use of several figures to represent the states of humanoid robot HRP-2P for the process of lying down and getting up. Only typical states were shown and other intermediate states were omitted to make the description more concise. In the work of Stilman et al.[12], the humanoid robot HRP-2 interacted with the obstacles actively to generate a possible path for the desired task. In the work of Yoshida et al.[13], the resolved momentum control was used to finish the motion planning issue for the tasks that require whole body motion of humanoid robot. Ohashi et al.[14] described a collision avoidance method for a biped robot with an upper body.
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Journal of Bionic Engineering (2011) Vol.8 No.1
We can see that the process of solving this issue contains two main steps, namely, topological state planning (see Refs. [9–14]) and detailed motion planning (see Refs. [5–8]). In fact, the tasks introduced in Refs. [5–8] are in such an easy manner that the topological state planning is negligible. In contrast, the tasks mentioned in Refs. [9–14] are so complicated that the topological state planning is mandatory. Along with the further development of humanoid robots, it is straightforward to see that the topological state planning will become more and more important for the application of humanoid robots. The reason is that in some cases, especially when the tasks for humanoid robots are complicated, the humanoid robots should make use of both the environment and the body of humanoid robots themselves. However, the problem of topological state planning has been less studied. In this paper, we focus on the topological state planning for the humanoid robot SJTU-HR1, particularly the classification of the basic state of sitting for the SJTU-HR1. For the systematic description of the large amounts of the sitting states, the theory of GF set is employed to achieve the kinematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1.
2 GF set theory The GF set theory has been developed for solving the issue of type synthesis of parallel mechanisms[15,16] and classification of states for humanoid robots[17–19] and human fingers[20]. It is worth noting that detailed classification of sitting states taking the self-connection into account has not been studied, which will be covered in this paper. In this section, we will recall the GF set theory introduced in Refs. [17–19]. Definition: The GF set is such a set contains one and only one element which represents the generalized function of the end-effectors of mechanisms. Note that the generalized function is the mapping between the specific structure of mechanisms and the kinematic characteristics of the end-effectors of mechanisms. The operation rules of GF sets are summarized as follows: • Union operation of GF sets: Let GF1, GF2... GFN represent N GF sets. The union of those GF sets can be expressed as:
GF = GF 1 ∪ GF 2
∪ GFN .
(1)
• Intersection operation of GF sets Let GF1, GF2 ... GFN represent N GF sets. The intersection of those GF sets is
GF = GF 1 ∩ GF 2
∩ GFN .
(2)
• Reversion operation of GF sets The concept of the reversion operation of GF sets is
( GF )
R
= GFR .
(3)
• Reversion operation of the union of GF sets The reversion operation of the union of GF sets can be computed as: (GF 1 ∪ GF 2 ∪ R = GFN ∪ GFR( N −1) ∪
∪ GFN ) R ∪ GFR2 ∪ GFR1 .
(4)
The theorems regarding the application of the GF set theory are as follows. • Theorem for serial mechanisms For a serial mechanism, the kinematic characteristics of the end-effectors of the mechanism are the union of all of the GF sets with such a sequence that from the start of the mechanism to the end. • Theorem for parallel mechanisms The kinematic characteristics of the moving platform of a parallel mechanism are the intersection operation of the GF sets of each kinematic chain of the parallel mechanism.
3 Humanoid robot platforms SJTU-HR1 3.1 Characteristics tree of the SJTU-HR1 The human skeleton contains 206 bones and has more than 300 Degrees Of Freedom (DOFs) driven by more than 600 muscles. Generally, humanoid robots do not need to feature as many DOFs as the human skeleton. From the biological point of view, we have been developing a humanoid robot platform SJTU-HR1 whose CAD model is shown in Fig. 1a and the prototype is shown in Fig. 1c. There exist three types of joints for the SJTU-HR1, according to the number of DOFs the joints have. For instance, the shoulder, waist and hip joints feature 3 DOFs, with a structure of 3_RRR spherical parallel mechanism which is the same as the mechanism of agile eye[21]. The neck, elbow, wrist, and ankle joints have 2 DOFs, with a structure of 2_RR spherical
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
mechanism which is the same as the spherical 5R mechanism[22]; The knee joints feature 1 DOF, with a structure of revolute joints. Most of the motors can be mounted on or near the base, thanks to the introduction of parallel mechanisms. Moreover, multiple joints have the same structure could lead to modularity design for the SJTU-HR1. As a result, the SJTU-HR1 features the advantages of both parallel mechanisms and serial mechanisms.
Fig. 1 Sketch of the SJTU-HR1 (a), its characteristics tree (b), and the prototype (c).
Fig. 1b is the characteristics tree of the SJTU-HR1, which is the sketch of the SJTU-HR1 represented by GF sets. Particularly, in Fig. 1b, the empty small circle represents the joints featuring three DOFs, the small circle with cross indicates the joints with two DOFs and the small circle with bar depicts the joints with one DOF. Moreover, the letters A, B, C, D, E, F, G in Fig. 1b represent pelvis, thorax, right hand, left hand, right foot, left foot, and head, respectively. As stated before, it is very important to distinguish the sequence of the joints when describing the specific GF set for end-effectors of mechanisms. Consequently, we let the GF sets from A to E, F, B and the GF sets from B to C, D, G be GF1, GF2, GF5, GF3, GF4 and GF6, respectively. Making use of the characteristics tree of the SJTU-HR1, we can achieve the kinematic characteristics of the interested end-effectors of the SJTU-HR1 via the GF set theory. 3.2 Basic states of the SJTU-HR1 Basically, several types of contact exist between humanoid robots and the environment, including point contact, line contact, free contact, and fixed contact[17]. We only take the fixed contact into account, meaning that once the link contacts the environment, no relative motions exist between the link and the environment. From Fig. 1b, we can see that the SJTU-HR1 con-
51
sists of 15 links connected by 14 joints. In fact, each link can contact the environment or other links of the SJTU-HR1. Consequently, the total number of the contact states of the humanoid robot would be quite huge from the topological point of view. Moreover, for practical application of the humanoid robot, it is impossible and unnecessary to function all of the states. As a result, we abstract several basic states which are commonly used during daily exercises, such as lying, sitting, standing, and handstanding. The typical states are shown in Fig. 2. Obviously, the main supports of each state in Fig. 2 from (a) to (h) are thorax and pelvis, pelvis, left foot, right foot, both feet, left hand, right hand, and both hands, respectively. We define the main support as primary support for humanoid robots and other supports which are responsible for balance as secondary support. On the one hand, if we consider the state only from the topological point of view, then the states of Fig. 2d and Fig. 2g are the same, which shows the effectiveness of the employment of the concept of primary support. On the other hand, the secondary support could be used for distinguishing the different topological structures within the same basic state.
Fig. 2 Basic states of the humanoid robot SJTU-HR1.
Theoretically, each link can contact the environment or other links of the humanoid robots. However, from the viewpoint of practical application of humanoid robots, only the head, thorax, pelvis, and four limbs are considered as the interested end-effectors of humanoid
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robots. Moreover, the possible secondary support includes four limbs and thorax. And the secondary support may vary a little bit depending on the specific basic state. In the following sections, the sitting states of the SJTU-HR1, including those having self-connection loops, will be addressed in detail. Particularly, the kinematic characteristics of the interested end-effectors of the SJTU-HR1 for each state will be achieved via the GF set theory, which is able to lead to deeper insight into the capabilities of the humanoid robot SJTU-HR1.
4
Sitting states for the humanoid robot SJTU-HR1
4.1 Sitting states without self-connection loops for the SJTU-HR1 For the sitting states without self-connection loops, the four limbs and thorax can function as the secondary support. As a result, we have six typical categories of states for this case from the topological point of view, which are classified according to the number of secondary support ranges from zero to five. In Fig. 3, the sketch for each typical category is illustrated and the notation for each state is shown below the specific state. Furthermore, the terms in the notations have their meanings. For instance, the term S indicates the sitting states; the terms fl, fr, hl, hr denote left foot, right foot, left hand, right hand, respectively; the same for their uppercase counterparts; the terms N and A represent that none and all of the possible secondary supports function as the secondary support. Consequently, the notation Shl
indicates that the left hand is the secondary support for the sitting state; the term Sflhr means that both the left foot and right hand function as the secondary support for the sitting state; the term SHRHL denotes that all of the possible secondary supports except for right hand and left hand function as the secondary support; SHR means all of the possible secondary supports except for right hand function as the secondary support. Note that other sitting states exist and are not shown in Fig. 3 for the sake of conciseness. However, the meaningful notations can be assigned for other sitting states following the same rules stated above. In this case, the interested end-effectors include four limbs and the head, thorax, and pelvis of the SJTU-HR1. Making use of the characteristics tree of the SJTU-HR1, all of the kinematic characteristics of the interested end-effectors of the SJTU-HR1 can be achieved and shown in Table 1. It is important to note that the kinematic characteristics of such end-effectors that contact the environment are Null GF set and not listed in Table 1. Table 1 Kinematic characteristics of the interested end-effectors of the SJTU-HR1 for the sitting states without self-connection loops Notation SN
Shl
Sflhr SN
Shl
Secondary support
Left hand
SHR
Kinematic characteristics
Right foot
GF1
Left foot Right hand
GF2 GF 5 ∪ GF 3
Left hand
GF 5 ∪ GF 4
Head
GF 5 ∪ GF 6
Thorax
GF5
Right foot
GF1
Left foot
GF2
Right hand
(GF 5 ∩ GFR4 ) ∪ GF 3
Head
(GF 5 ∩ GFR4 ) ∪ GF 6
Thorax
GF 5 ∩ GFR4
Left foot
Right foot
GF2
Right hand
Left hand
(GF 5 ∩ GFR3 ) ∪ GF 4
Head
(GF 5 ∩ GFR3 ) ∪ GF 6
thorax
GF 5 ∩ GFR3
Right foot
Right hand
GF3
Left foot
Left hand
GF4
Thorax
Head
GF6
Both feet
Right hand
GF3
Left hand
Head
GF6
Head
GF6
Sflhr SHRHL
Interested end-effectors
Thorax SA SHRHL
SHR
SA
Fig. 3 Sketches of the sitting states without self-connection loops.
Both feet Both hands Thorax
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
4.2 Sitting states with one self-connection loop for the SJTU-HR1
For the sitting states with one self-connection loop, the four limbs and thorax function as the secondary support. Moreover, only the four limbs are able to contact the body of the SJTU-HR1. Note that only the four limbs, head, and thorax which have no connection with the environment can be contacted by the secondary support. As a result, four categories exist for such kind of sitting states, including the right hand contacts the body, the left hand contacts the body, the right foot has connection with the body and the left foot has connection with the body, respectively. One state for each category is illustrated in Fig. 4. The terms in the notations have the same meanings except for those in the superscript. Actually, the superscript includes two parts: the first part indicates which link functions as the secondary support and the second part indicates the connected link by the secondary support. As a result, the hrhe notation Sfrfl represents such a state that both feet contact the environment and the right hand has connection with the head of the SJTU-HR1. Similarly, the kinematic characteristics of the interested end-effectors for the sitting states with one self-connection loop shown in Fig. 4 are listed in Table 2, excluding those with a Null GF set. Note that the sitting states with more than one self-connection loops can also be analyzed using the same procedure.
nematic characteristics of the interested end-effectors of the humanoid robot SJTU-HR1. Consequently, we can get deeper insight into the capabilities of the humanoid robot SJTU-HR1. Furthermore, the meaningful notations can represent the states for the SJTU-HR1 in a systematic, concise, and informative manner. Table 2 Kinematic characteristics of the interested end-effectors of the SJTU-HR1 for the sitting states with one self-connection loop Notation
Interested end-effectors
Kinematic characteristics
hrhe Sfrfl
Right hand
GF 5 ∪ (GF 3 ∩ GF 6 )
hlhe SHRHL
frth Sflhr
S
flth hr
Shlhrhe
S Nhrfr
hrhe Sfrfl
hlhe SHRHL
53
Left hand
GF 5 ∪ GF 4
Head
GF 5 ∪ (GF 3 ∩ GF 6 )
Thorax
GF5
Right hand
GF3
Left hand
GF 4 ∩ G F 6
Head
GF 4 ∩ G F 6
Right foot
GFR1 ∩ GFR3 ∩ GF 5
Left hand
(GFR1 ∩ GFR3 ∩ GF 5 ) ∪ GF 4
Head
(GFR1 ∩ GFR3 ∩ GF 5 ) ∪ GF 6
Thorax
GFR1 ∩ GFR3 ∩ GF 5
Right foot
GF 1
Left foot
GF 2 ∩ GFR3 ∩ GF 5
Left hand
(GF 2 ∩ GFR3 ∩ GF 5 ) ∪ GF 4
Head
(GF 2 ∩ GFR3 ∩ GF 5 ) ∪ GF 6
Thorax
GF 2 ∩ GFR3 ∩ GF 5
Right foot
GF 1
Left foot
GF 2
Right hand
(GF 5 ∩ GFR4 ) ∪ (GF 3 ∩ GF 6 )
Head
(GF 5 ∩ GFR4 ) ∪ (GF 3 ∩ GF 6 )
Thorax
GF 5 ∩ GFR4
Right foot
G1 ∩ (G5 ∪ G3 )
Left foot
GF 2
Right hand
G1 ∩ (G5 ∪ G3 )
Left hand
(GF 5 ∩ (GF 1 ∪ GFR3 )) ∪ GF 4
Head
(GF 5 ∩ (GF 1 ∪ GFR3 )) ∪ GF 6
Thorax
GF 5 ∩ (GF 1 ∪ GFR3 )
frth Sflhr
5 An application example
Shrflth
Shlhrhe
SNhrfr
Fig. 4 Sketches of the sitting states with one self-connection loop.
From the above introduction, we can see that the GF set theory can be utilized for achieving the one-to-one correspondence between the specific state and the ki-
For humanoid robots functioning in the human-centered environment, it is desirable that the humanoid robots are able to negotiate with the environment and robots themselves. As mentioned before, in some cases, it is necessary to make use of the environment or the robot body to finish the required tasks. Consequently, we can image that one potential task for the SJTU-HR1 is illustrated in Fig. 5 consisting of 6 steps. It is not difficult to see that steps 1–3 do not have self-connection loops and steps 4–6 feature one
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self-connection loop. Obviously, it is better to show this process using figures rather than language description. However, there are some problems of using figures to show the state changing process of the SJTU-HR1. Firstly, it is possible that the style of the Fig. 5 is inconsistent if another person uses figures to illustrate the same process. The fact is that apart from the stick-figures employed in this paper, computer generated simulation figures can also be used[11]. Moreover, even one person can make different figures for the same task. Secondly, the figures may be interpreted differently by different people. For instance, without the filled circles which are responsible for indicating the contact between the SJTU-HR1 and the environment, we can’t distinguish if the feet contact the environment. So some misunderstandings may arise due to poor quality drawings. Moreover, the task of making figures is cumbersome, especially when the number of the contact points is bigger.
SN Step 1
S hl Step 2
SNhrfr
S hlhrhe
Step 6
Step 5
SHRHL Step 3
hrhe S frfl
Step 4
Fig. 6 Moving process of the SJTU-HR1 illustrated using notations.
It is worth noting that apart from the basic states, there also exist other states which have not been covered in this paper. In case we need to classify such kind of states for the application of humanoid robots, it may be possible to introduce number and variables which can be added in the subscript or superscript of the corresponding GF sets to identify the classified and unclassified states, which will be discussed in detail in our future work.
6 Conclusions
Step 1
Step 6
Step 2
Step 5
Step 3
Step 4
Fig. 5 Moving process of the SJTU-HR1 illustrated using figures.
To eliminate the potential misunderstandings and decrease the complexity of the presentation of the moving process of the SJTU-HR1, we use the notations associated with the specific states to describe the state changing of the SJTU-HR1. Then the same process described in Fig. 5 can be illustrated in Fig. 6 from which we can see that the style of the Fig. 6 can be consistent following the same rule of assigning notations for the sitting states of the SJTU-HR1. Furthermore, from the notations in Fig. 6 and the corresponding characteristics of the interested end-effectors listed in Table 1 and 2, we can get deeper insight of the capabilities of the SJTU-HR1.
In this paper, we have addressed the classification issue for the humanoid robot SJTU-HR1, particularly the sitting states. The main contributions of our work are as follows: • The characteristics tree of the humanoid robot SJTU-HR1 is proposed. • The basic states of humanoid robots are proposed according to the primary support, including lying, sitting, standing, and handstanding. • The sitting states are classified further into those without self-connection loops and those with one self-connection loop from the view point of topology. • The meaningful notation for each sitting state is given for the systematic description. • The kinematic characteristics of the interested end-effectors for each sitting state are achieved via the GF set theory.
• One application example is given to show the effectiveness of the proposed methodology.
Acknowledgement This work was supported by the National Basic Research Program of China (2006CB705402), the National Natural Science Foundation of China (30770538, 50821003), and the Joint Research Fund for Overseas Natural Science of China.
Yang and Gao: Classification of Sitting States for the Humanoid Robot SJTU-HR1
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