gait analysis of a radial symmetrical hexapod robot based

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 27,aNo. 5,a2014

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DOI: 10.3901/CJME.2014.0619.115, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Gait Analysis of a Radial Symmetrical Hexapod Robot Based on Parallel Mechanisms

XU Kun* and DING Xilun

Robotics Institute, Beihang University, Beijing 10083, China

Received February 18, 2014; revised June 3, 2014; accepted June 19, 2014

Abstract: Most gait studies of multi-legged robots in past neglected the dexterity of robot body and the relationship between stride

length and body height. This paper investigates the performance of a radial symmetrical hexapod robot based on the dexterity of parallel

mechanism. Assuming the constraints between the supporting feet and the ground with hinges, the supporting legs and the hexapod body

are taken as a parallel mechanism, and each swing leg is regarded as a serial manipulator. The hexapod robot can be considered as a

series of hybrid serial-parallel mechanisms while walking on the ground. Locomotion performance can be got by analyzing these

equivalent mechanisms. The kinematics of the whole robotic system is established, and the influence of foothold position on the

workspace of robot body is analyzed. A new method to calculate the stride length of multi-legged robots is proposed by analyzing the

relationship between the workspaces of two adjacent equivalent parallel mechanisms in one gait cycle. Referring to service region and

service sphere, weight service sphere and weight service region are put forward to evaluate the dexterity of robot body. The dexterity of

single point in workspace and the dexterity distribution in vertical and horizontal projection plane are demonstrated. Simulation shows

when the foothold offset goes up to 174 mm, the dexterity of robot body achieves its maximum value 0.1644 in mixed gait. The

proposed methods based on parallel mechanisms can be used to calculate the stride length and the dexterity of multi-legged robot, and

provide new approach to determine the stride length, body height, footholds in gait planning of multi-legged robot.

Keywords: hexapod robot, parallel mechanism, kinematics, stride length, dexterity, weight service sphere

1 Introduction

Multi-legged robots possess much better adaptability to rough terrain than wheeled and tracked robots because they do not need continuous support on the ground[1]. This type of robots includes biped robots, quadruped robots, hexapod robots and more legged ones. In nature most arthropods have six legs so they can easily maintain static stability while they are moving. It has been proved that there is no speed advantage in having more than 6 legs[2], and hexapod robots also show good robustness in case of leg faults[3–7]. For these reasons, hexapod robot has become an intensive research topic in the field of mobile robot in recent years. The key problem of multi-legged robots mostly focuses on kinematics and dynamics analyses and gait planning.

Gait planning, which is the methodology followed in order to determine the sequence for lifting off and placing the feet, is an important topic for multi-legged robots locomotion. Because hexapod robot has six legs statically stable gaits are more suitable for it. Typical hexapod robots can be divided into two types: rectangular hexapods and hexagonal hexapods. Rectangular hexapod has a rectangular

* Corresponding author. E-mail: [email protected] Supported by National Science Foundation for Distinguished Young

Scholar, China(Grant No. 51125020), National Natural Science Foundation of China(Grant No. 51305009), and CAST Foundation

© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2014

body with 6 legs distributed symmetrically on the two sides[8–9]. Hexagonal hexapod has a circular or hexagonal body with six legs distributed radial symmetrically around[3]. 3+3 tripod gait[4,10], 4+2 quadruped gait[5–6], 5+1 one by one gait[7] are common gaits of hexapods. In a 3+3 tripod gait cycle, there are always 3 legs to support the robot body. The 4+2 quadruped gait has been demonstrated its fault-tolerant ability[5–6]. For the fault-tolerance of hexapods, YANG, et al[5–6, 11–13], analyzed the fault-tolerant gaits for hexapod. Free gait which was aperiodic to adapt different terrain was proposed in Refs. [14–17]. Free gait is suit for complex uneven terrain. Comparison between the rectangular model and the hexagonal model on mobility, fault-tolerance, and stability was given in Ref. [18], and result showed that the hexagonal model has better turning ability, a higher margin of stability during the fault-tolerant gait, and greater stride length in certain conditions. According to the special structure of hexagonal robots, WANG, et al[19–20], proposed insect-wave tripod gait, mammal kick-off tripod gait and mixed tripod gait, they also studied its fault-tolerance when one or two legs were damaged.

Performance of multi-legged robots was usually investigated as serial mechanisms in Refs. [21–23]. The robot body was taken as a basis, each leg was regarded as a serial manipulator to study stride length and dexterity. In fact when the multi-legged robots are moving on the

Y XU Kun, et al: Gait Analysis of a Radial Symmetrical Hexapod Robot Based on Parallel Mechanisms

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ground, the system composed of the ground, supporting legs and the robot body is more similar to parallel mechanism than the serial one. In recent years there are some works to study this problem from the view of parallel mechanism. DING and XU[24] designed a novel metamorphic wheel-legged robot and analyzed its DOF by taking the ground, supporting legs and robot body as parallel mechanism. WANG, et al[19–20] investigated the stride length and the stability margin of hexapod robot using parallel mechanism method. But most of them neglected the influence of different foothold points on walking performance of the robot. Meanwhile, in different periods while hexapod robot moves on the ground, it may have different equivalent mechanisms. Hence it is not suitable to take the system as a single specific mechanism. The robot and the ground can be considered as a series of hybrid serial-parallel mechanisms while it walks on the ground.

Based on current research, a new method is proposed to analyze the walking performance of multi-legged robots in this paper. In this method, one gait cycle is divided into several stages. In a specific stage, the constrains between the footholds and the ground are considered as hinges, so the ground, supporting legs and the robot body form a parallel mechanism, each swinging(transferring) leg is taken as a series manipulator, so the whole system is a hybrid serial-parallel mechanism when the robot moves on the ground. In different periods the equivalent mechanisms of the system are not same. By taking NOROS robot, as shown in Fig. 1, built in space robot lab of BUAA as an example, a new method is proposed to analyze the maximum stride length and the dexterity of its body while it walks on the ground in 3+3 gaits.

Fig. 1. Small prototype of NOROS robot

This paper is organized as follows. First the structure and the main parameters of NOROS robot are briefly introduced and the typical 3+3 tripod gaits are described in section 2. Secondly the kinematics of swing leg, supporting leg and the whole system are established in section 3. Thirdly the workspace of hexapod body and the stride length are deduced in section 4. Then in section 5 the orientation dexterity of hexapod body is estimated and simulated. Finally section 6 makes the conclusion.

2 NOROS Robot Structure and Its Typical

3+3 Tripod gaits NOROS robot was designed for exploring in

unstructured environments. It has six legs, each with a wheel attached. It combines both advantages of multi- legged and wheeled robots. NOROS robot can use its legs to walk out of excessively soft, obstacle, steep or other extreme terrain, meanwhile it can use wheels for efficient driving over stable, flat terrain. The wheel system of NOROS is not discussed in this paper.

The body of NOROS robot is composed of a hemispherical shell and a double-deck chassis. Each leg has five elements: foot, wheel, calf, thigh and hip, as shown in Fig. 1. Calf, thigh and hip are linked together by two parallel rotary joints, knee and hip joint. The leg is connected to the chassis by waist joint whose rotating axis is perpendicular to the chassis. Main length parameters of small prototype of NOROS robot, which will be used for subsequent analysis and simulation, are as follows.

Body radius R is 150 mm; hip length l1 is 50 mm; thigh length l2 is 120 mm; calf length l3 is 130 mm.

Typical 3+3 tripod gaits mainly includes insect-wave tripod gait, mammal kick-off tripod gait and insect-mammal mixed tripod gait[21]. Motion sequences of these three 3+3 tripod gaits mentioned in Ref. [21] are shown in Fig. 2, Fig. 3 and Fig. 4. “●” means supporting foot, “○” means transporting foot, and “→” denotes the walking direction of robot.

Fig. 2. Insect-wave tripod gait I

Fig. 3. Mammal kick-off tripod gait

Fig. 4. Mixed tripod gait

CHINESE JOURNAL OF MECHANICAL ENGINEERING

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However in nature most insect gaits are not like insect-wave gait I shown in Fig. 2 while they are walking. Fig. 5 shows a typical six-legged beetle in nature. From the figure we can see that the stance legs of insect are not parallel but distributed divergently, it is different from the stance posture shown in Fig. 2. Therefore, insect-wave tripod gait II shown in Fig. 6 is more accord with bionic principle.

Fig. 5. Six-legged beetle in nature

Fig. 6. Insect-wave tripod gait II

Fig. 2–Fig. 4 and Fig. 6 show initial stance posture and

the sequence for lifting off and placing the feet in one gait cycle. Insect-wave tripod gait I and mammal kick-off tripod gait have the same initial stance posture when the robot begins to walk, but their walking directions are mutually perpendicular. Insect-wave tripod gait II and mixed tripod gait have the same initial stance posture too, and their directions form a 30°angle. Our study here focuses on mixed tripod gait and insect-wave gait II.

Ground supports feet of stance legs while multi-legged robot moves. Because of the particular contact between ground and feet, there must be some hypothesizes to restrict the relationship between ground and stance foot. Similar to other studies, several assumptions are given for analysis carried out in the following sections:

I: point-contact between ground and stance foot; II: no slipping between ground and stance foot; III: the chassis of robot body is always parallel to the

supporting plane formed by stance feet; IV: The triangle composed of three footholds in mixed

tripod gait and insect-wave gait II is an equilateral triangle. The triangle composed of three footholds in insect-wave tripod gait I and mammal kick-off tripod gait is an isosceles triangle.

According to above assumptions, the contact between ground and stance foot can be considered as a virtual spherical joint. So the system composed of ground, stance legs and body is a parallel mechanism. Each swinging leg is considered as a serial mechanism with 3 DOFs. Therefore, while hexapod moves on the ground it is

considered as a constantly changing serial-parallel mechanism.

Because of the same initial stance posture of mixed tripod gait and insect-wave tripod gait, their footholds have the same position under the initial stance posture. Although the footholds are certain while the robot walks according assumption II, the robot can choose different footholds to get different stride length and static stability margin. Footholds can be close to the body and can be away from the body, so in analysis the contact between ground and each stance foot can be considered as a virtual spherical joint and a virtual sliding joint, as shown in Fig. 7, all six legs are stance legs. The mechanism of hexapod with 3 swinging legs and 3 supporting legs is shown in Fig. 8. The parameter lw shown in Fig. 8 means the offset between the stance foot point lA and the hip point lB in supporting plane. According to the assumption IV, three sliding joints are not independent. If one of them is given the other two are certain. If the parameter w is given the workspace of the hexapod body is certain.

Fig. 7. Mechanism of hexapod with six legs supporting in mixed tripod gait and insect-wave tripod gait II

Fig. 8. Mechanism of hexapod with three legs supporting in mixed tripod gait and insect-wave tripod gait II

Similarly, the mechanism of initial stance posture in insect-wave tripod gait I or mammal kick-off tripod gait is shown in Fig. 9. The mechanism of hexapod with three

Fig. 9. Mechanism of hexapod with six legs supporting in mammal kick-off tripod gait and insect-wave tripod gait I


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stance legs in insect-wave tripod gait I or mammal kick-off tripod gait is shown in Fig. 10.

Fig. 10. Mechanism of hexapod with three legs supporting in mammal kick-of tripod gait and insect-wave tripod gait I

In particular moment the equivalent mechanism of

hexapod robot is certain. But at different moments its equivalent mechanism may be different. The locomotion of the robot can be seen as the motion of this series of equivalent mechanisms.

3 Kinematics of the Robot

This section deals with the kinematics of hexapod robot which include swinging leg, supporting leg and the hybrid serial-parallel mechanism. The forward kinematics equations are established through product of exponentials formula, and the inverse kinematics are derived by geometric method.

3.1 Forward kinematics of swinging leg

As shown in Fig. 11 and Fig. 12, four frames are established for kinematics analysis. the global reference frame {O} and body reference frame {C} which is fixed on the geometrical center of the robot body and moves with the body, the single leg reference frame {Bi} which is established at the waist joint and fixed to the chassis of the hexapod, and foot reference frame {Ai} which is located at the end of the swinging leg i (foot of swinging leg i). The configuration when all legs fully extended is chosen as the home configuration. In this position, all the joint variables are taken as zero. All parameters concerned with swinging leg i are shown in Fig. 11.

Fig. 11. Mechanism of swinging leg

Fig. 12. Projection diagram of the robot in chassis plane

Because frame {C} and frame {Bi} are all fixed on the

robot body, the transformation matrix from frame {C} to frame {Bi} is constant and can be given as follows:

πˆexp ( 1)

,311

i i ii

c B B BCB

ig

æ öæ ö ÷ç ÷ æ öç ÷- - ÷ç ÷ç ç÷÷ç ç ÷ç÷è ø= = ÷ç ÷ ç ÷ç ÷ ç ÷çç è ø÷÷çè ø

ω p R p

00

(1)

where

iBR is a rotation 3´3 matrix,

cos(( 1)π 3) sin(( 1)π 3) 0

sin(( 1)π 3) cos(( 1)π 3) 0 ,

0 0 1iB

i i

i i

æ ö- / - / ÷ç ÷ç ÷ç ÷= - - / - /ç ÷ç ÷ç ÷÷çè ø

R

and

iBp is a 3´1 position vector,

sin(( 1)π 3)

cos(( 1)π 3) .

0iB

R i

R i

æ ö- / ÷ç ÷ç ÷ç ÷= - /ç ÷ç ÷ç ÷÷çè ø

p

The transformation matrix from frame {Bi} to frame {Ai}

at home configuration can be given as follows:

3 3 1 2 3

0

( ) .0

1

i iB A

l l lg ´

æ ö÷ç ÷ç ÷ç ÷+ +ç ÷ç ÷= ç ÷÷ç ÷ç ÷ç ÷ç ÷çè ø

I0

0

(2)

Twist ij for each joint is introduced to describe the

screw motion of the jth joint of the ith leg. The twist for a rotation joint can be represented by

,ij

ijij ij

æ ö÷ç ÷= ç ÷ç ÷´çè ø

ωξ

r ω (3)

where T( , , )ij ij ij ij =ω is the direction of screw axis, T( , , )ij ij ij ijr r r=r is the vector of any point on the axis. According to the parameters shown in Fig. 11, we can get the twist of each revolution joint.

Then, the forward kinematics of swing leg i is given by the product of exponentials:

c c c

c

c

c

c

x y z

x y z


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1 2 3ˆ ˆ ˆ

( ) e e e ( ),i i i i i ii i i iOA OC CB B Ag g g g =θ 0 (4)

where i is the waist joint variable, i is the hip joint variable, i and is the knee joint variable， and OCg is the transformation matrix from frame {O} to frame {C},

1C C

OCgæ ö÷ç ÷= ç ÷ç ÷÷çè ø

R p

0, with CR a 3´3 rotation matrix,

which points out the orientation of the hexapod body

relative frame {O}, and Cp is a 3´1 position vector T( , , )C c c cx y z=p , which denotes the position of point

C relative to frame {O}.

3.2 Forward kinematics of supporting leg Similar to section 3.1, this section deals with the forward

kinematics of supporting leg. Three mutually perpendicular rotation joints take the place of the virtual spherical joint. Four frames are established, as shown in Fig. 13. Frame {O} is built as the global reference frame. The body reference frame {C} is fixed on the robot body and moves with the body. The single leg reference frame {Bi} which is established at the waist joint and fixed to the chassis of hexapod, and foot reference frame {Ai} which is located at the foot of supporting leg i. The configuration when the leg is fully extended is taken as the home configuration. In this position, all the joint variables are set to zero.

Fig. 13. Mechanism of supporting leg Because l4, l5 and l6 are parameters of the virtual sphere

joint and sliding joint, they are equal to zero. The transformation matrix from frame {Ai} to frame {C} at the home configuration is represented by

3 3 1 2 3

0

( )( )

0

1

iA C

R l l lg ´

æ ö÷ç ÷ç ÷ç ÷- + + +ç ÷ç ÷=ç ÷÷ç ÷ç ÷ç ÷ç ÷çè ø

I0

0

.

According to Eq. (2), all joint twists can be obtained and

the forward kinematics of supporting leg i is given in the following:

7 1 2 3 4 5 6

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) e e e e e e e ( ),i i i i i i i i i i i i i i

ii

wOC A COBg g g

¢=θ 0 (5)

where i is the waist joint variable, i is the hip joint variable, i is the knee joint variable. i , i and i are the variables of the virtual sphere joint. iw is the variable of virtual sliding joint. In fact the position of foothold is known, not the position of point iB¢ . So the forward kinematics of supporting leg i can be rewrote as

1 2 3 4 5 6

ˆ ˆ ˆ ˆ ˆ ˆ( ) e e e e e e ( )i i i i i i i i i i i i

i iOC OA A Cg g g =θ 0 , (6)

where

iOAg is the transformation matrix from frame {O} to frame {Ai},

1i i

i

A AOAg

æ ö÷ç ÷ç= ÷ç ÷ç ÷çè ø

R p

0,

with iAR a 3´3 rotation matrix, which points out the orientation of the frame {Ai} relative to frame {O}, and

iAp is a 3´1 position vector T( , , )i i i iA A A Ax y z=p ,

which denotes the position of point Ai relative to frame {O}.

3.3 Forward kinematics of hybrid

serial-parallel mechanism For statically stable locomotion of hexapod, there must

be 3 or more legs standing on the ground to support the hexapod body at any time. If hexapod is in static stability and all the joint variables have certain value, the position and orientation of the body are determined. If there are more than 3 supporting legs, the joint variables of the redundant legs are determined by the other 3 supporting legs.

As shown in Fig. 8, there are 3 legs (j, l and n) supporting the body, and three swinging legs (i, k and m). Using Eq. (6), a set of forward kinematics equations can be obtained:

( ) e e e e e e e ( ),

( ) e e e e e e e ( ),

( ) e e e e e e e ( )

j s j j j j jj j j j j j j j

l s l l l l l ll l l l l l l

n s n n n n n nn n n n n n n

wj jOC j OC j

wl lOC l OC l

wn nOC n OC n

g g

g g

g g

=

=

=

θ 0

θ 0

θ 0 ,

ìïïïïïïïïíïïïïïïïïî(7)

where jw is the offset of jth supporting leg.

( )jOC jg θ , ( )l

OC lg θ and ( )nOC ng θ describe the body

position and orientation, hence we can get another set of equations:

( ) ( ),

( ) ( ).

j lOC j OC l

l nOC l OC n

g g

g g

θ θ

θ θ (8)

In Eq. (8) there are 21 variables. When the hexapod

o oo

o

o oo o

o o

o

o oo oo


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moves on the ground, contact points of footholds are certain. Therefore, jw , lw , nw are determined by the position of the foothold points of supporting feet. The other 18 variables are not all independent. In fact only 6 of them are independent. Given the knee joint, hip joint variables of each supporting leg, all the other variables can be calculated by solving Eq. (8). Further more, we can get the body position and orientation by Eq. (7). So we can get

( ) ( ) ( ) ( )j l nOC OC j OC l OC ng g g g θ θ θ θ . (9)

3.4 Inverse kinematics

Once the body position and orientation are known, the position and orientation of swinging legs can be easily solved by Eq. (4).

Here we used the geometric approach to find the inverse kinematics solution.

In a geometric approach, the spatial geometry of the robot leg is decomposed into several plane-geometry problems. Because of the simple structure of the robot leg this can be done quite easily. Variables of each leg joints (joint angles) can be solved by using the tools of plane geometry. For leg i with three degrees of freedom shown in Fig. 14, the vectors iBp and iAp which are the position of points Bi and Ai relative to coordinate system {O} are known, and the orientation of frame {Bi} relative to the frame {O}, R is given. We can get the position of point Ai relative to {O}:

i i i iA B B A= +p p p , (10)

where i iB Ap is the description of vector i iB A relative to {O}. By using the rotation transformation, Eq. (10) can be written as follows:

i i iA B A= +p p R p , (11)

where iAp is the position of point Ai relative to the frame { Bi }. So we can get iAp as follows,

1( ) ( )

i i i i iA A B A B-= - = -p R p p R p p . (12)

Fig. 14. Mechanism sketch of leg i in space

Because the axis of knee joint is parallel to the axis of vertical swing hip joint and is perpendicular to the axis of yawing hip joint, so the leg is planar and the projection in the body chassis plane of the leg is a straight line. According to the geometrical relationship shown in Fig. 14 we can get

1 2 3

2 3

cos cos( ),

sin sin( ),

i i i i

i i i i

L l l l

H l l

ì = + + +ïïíï = + +ïî (13)

where Li is the projection length of leg i in the xy plane of frame {O} and Hi is the height from foot Ai to the plane of frame {Bi}.

Fig. 15 shows the projection of leg i onto xy plane of {Bi}. Li is a positive value when it locates in I quadrant and II quadrant, and it is a negative value when locating in III quadrant and IV quadrant. So we can get the following equations:

2 2 ,

,

tan(π 2 ) ,

i i

i

i i

i A A

i A

i A A

L x y

H z

y x

ìï = +ïïïïï =-íïïï / + = /ïïïî

( 0)iAy ≥ (14)

2 2 ,

,

tan( π) ,

i i

i

i i

i A A

i A

i A A

L x y

H z

y x

ìï =- +ïïïïï =-íïïï - = /ïïïî

( 0)iAy < (15)

where ( )

i i iA A Ax y z is given to specify the position of point Ai relative to frame {Bi}.

Fig. 15. Leg i projection onto xy plane of frame {Bi}

If the position of point Ai relative to the frame {O} and

the position and the orientation of robot body relative to the frame {O} are known, the variables of leg i, ,i i and

i can be solved by Eqs. (12)–(15).

4 Workspace and Stride Length of the Robot

In the following, the mixed tripod gait and insect-wave gait II are taken as examples to investigate the performance of the hexapod robot with the same initial posture.

o o

o

i

oB

o

o o

o o bi

oB

i

oB

b

b

b o o o oiBo

b b b

b b

b

b b

b

b b

b

b b

b


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The mechanism composed of ground and the hexapod robot is varying during locomotion of the hexapod robot. Fig. 16(a) shows the transformation process of the hexapod robot in one mixed gait cycle. Where “●” indicates supporting foot and “○” indicates transporting foot. One gait cycle is divided into four periods: initial transition status, early half-period, intermediate transition status and later half-period. Fig. 16(b) presents the simplified diagram of the system in initial transition status, Fig. 16(c) shows the simplified diagram of the system in early half-period, Fig. 16(d) presents the intermediate transition status when transporting feet just contact the ground and supporting feet do not lift up yet. Fig. 16(e) shows the simplified diagram of the system in later half-period. In all the different 3+3 static stability gaits, there must be four periods mentioned above.

Fig.16 Diagram of one mixed gait cycle

In Fig. 16, S denotes the stride length of robot and stroke

S1 means the moving distance of the parallel mechanism composed of ground and supporting legs(j, l, n) and the body in early half-period, and stroke S2 means the moving distance of the parallel mechanism composed of ground and supporting legs(i, k, m) in later half-period. Stride length S is consisted of two moving distance stroke S1 and stroke S2. Thus if we want to get maximum stride length we must know the moving distance of each parallel mechanism. Maximum stride length can be calculated by

1 2max max maxS S S= + . (16)

Similarly the stride length of hexapod robot, while it

moves on the ground using 4+2 gaits, can be got:

1 2 3S S S S= + + , (17)

where S1, S2, S3 mean the moving distance of three parallel mechanisms because there are three moving stages in 4+2

gaits. In all the different 4+2 static stability gaits, there must be 6 periods: 3 transition status and 3 moving stages.

Workspace is an important index for parallel mechanism. Geometrical methods and numerical methods are main two types to analyze the workspace of parallel mechanism[25–29]. The geometrical methods take each leg of the parallel mechanism as a simple serial mechanism, and deduce its workspace Wi under its constraint, the intersection of all the Wi is the workspace of the parallel mechanism. Numerical methods construct a set of equations that the boundary points must satisfy. A set of points on the boundary can be found by solving the equations numerically. Here, we used both methods to determine the workspace of parallel mechanism composed of ground, supporting legs and the body.

In order to walk, the supporting legs must stand on the ground. So the angle between the calf of supporting leg and the ground must greater than a certain angle, min i . Because of the structure limit, there are certain ranges of knee joint, hip joint and waist joint. The range of knee joint is from –140° to 40°. The range of hip joint is from –90° to 100°. The range of waist joint is from –60° to 65°. By using the forward kinematics established in previous section the workspace of the supporting leg which has a certain foothold can be obtained. Fig. 17 shows the range of activity of calf i, where the point Ai is the contact point between supporting leg i and the ground.

Fig. 17. Range of calf

Using the geometrical methods the workspace of parallel

mechanism can be determined. The intersection of each supporting leg’s constant orientation workspace is the constant orientation workspace of the parallel mechanism. Meanwhile because the hexapod is in a static stability, the parts of workspace whose vertical projection onto ground plane exceeds the triangle composed by three footholds, must be removed. The left parts are the workspace which satisfies the static stability. Fig. 18 shows the constant orientation workspace of the hexapod body when min 0 = and w equals 0 mm, 50 mm, 100 mm, 150 mm, 200 mm, 250 mm, respectively. Fig. 19 shows the constant orientation workspace of the hexapod body when min 30 = and w equals 0 mm, 50 mm, 100 mm, 150 mm, 200 mm, 250 mm, respectively. We can see from the figures, the workspace varies by the change of foothold offset. The workspace under constraint min 0 = is larger than that under constraint min 30 = while their foothold offsets are equal.


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Fig. 18. Workspace of the body while min =0°

Fig. 19. Workspace of the body while min =30°

When the triangle composed of foothold points is equilateral, the parallel mechanism consisting of the ground, supporting legs and hexapod body is radial symmetrical because each leg has the same structure. Thus the projection of the parallel mechanism onto the ground is radial symmetrical and has three axes of symmetry AiAl, AjAm, and AkAn, as shown in Fig. 20.

Fig.20. Projection diagram of the mechanism onto the ground

Therefore the workspace of parallel mechanism has three

planes of symmetry. When we use the numerical methods to determine the workspace we can just seek out 1/6 of the boundary of the workspace and deduce the other 5/6 by the symmetric relation. Hence the computing time is saved. For radial symmetrical mechanism there are n branches, there must be n planes of symmetry, so we just determined 1/2n of the whole workspace and the others can be obtained by symmetric relation.

Fig. 21 shows the constant orientation workspace of the robot body by using the numerical method. As can be seen from the figures, the workspace under constraint min 0 = is larger than that under constraint min 30 = while their foothold offsets are equal.

Fig. 21. Workspace of the body while The workspace of hexapod body varies against the

foothold offset w. The curves shown in Fig. 22 indicate the relationship between the constant orientation workspace and the foothold offset. The solid line represents the relationship when min 0 = and the dash line gives the relationship when min 30 = . As it can be seen from Fig. 22, in initial stage the constant orientation workspace becomes larger with the increase of foothold offset w. It attains the largest value 6 139 513 mm3 when w equals 144 mm under the constraint min 0 = , and it attains the largest value 5 541 039 mm3 when w equals 142 mm under the constraint min 30 = . After the constant orientation workspace reaches the largest value it becomes smaller with the increase of foothold offset w.

Fig. 22. Workspace of robot body with different foothold offset Fig. 23 shows the relationship between the workspaces

of two parallel mechanisms in two adjacent moving periods of mixed gait. The triangles in figures are formed by foothold points. In Fig. 23(a) and Fig. 23(b) the two workspaces are disjoint so the hexapod can not walk in a static stability gait. In this case the hexapod must be airborne to realize moving. The only difference between two figures is that the legs may conflict in Fig. 23(a), so in any periods there are 3 supporting legs at most. There is


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intersection in Fig. 23(c), so the hexapod can walk in the static stability gait. If the hexapod robot wants to move in 3+3 static stability gaits, the workspaces of the two parallel mechanisms in two moving periods must joint together, as shown in Figs. 24 and 25.

Fig. 23. Relationship between the workspaces of two parallel mechanisms of mixed tripod gait

Figs. 24 and 25 represent the vertical sections of

workspaces. Using geometric method to analyze the vertical section we can find the regions where the hexapod can move on the ground by static stability gaits. If we want to calculate the maximum stride length we must determine the height of hexapod body and the boundary of two workspaces are exactly connect to each other. Using this method the maximum stride length can be deduced if the foothold offset and the height of robot body are certain.

Fig. 24. Workspaces of two parallel mechanisms in two adjacent moving periods of static stability mixed tripod gait

Fig. 25. Workspaces of two parallel mechanisms in two adjacent moving periods of static stability insect-wave tripod gait II

Figs. 26 and 27 show the maximum stride length of the

hexapod robot. As seen in the figures the trends of the maximum stride size by using two different gaits are similar. When the foothold offset w becomes small, the stride size can reach an appropriate value in higher height of robot body. When the foothold offset w becomes large the stride size can reach an appropriate value in lower height of robot body. The hexapod can get suitable stride size in a wider range while the foothold offset w equals to 150mm. This agrees with Fig. 22 because the workspace of equivalent parallel mechanism gets a larger value when the foothold offset w equals 150 mm.

Fig. 26. Maximum stride size of mixed tripod gait

Fig. 27. Maximum stride size of insect-wave tripod gait II

5 Orientation Dexterity of Robot Body

Orientation dexterity is a critical performance index of robot. It indicates the capability of robot to achieve different orientation at a certain point in workspace. At a certain point in workspace, the more the orientation solutions are, the better the orientation dexterity is. In order to study the dexterity of robot, several interesting quantitative measures of robot dexterity have been proposed. Refs. [30–33] proposed the concepts of service point, service angle and service sphere, and used the service factor and dexterity index to estimate the dexterity of the robot.

For the robot whose end effector has only one rotary degree of freedom, when its end effector locates on a certain point q, its end effector can be compared to a bar linked to the point by one virtual rotational joint, as shown in Fig. 28(a). The trajectory of tip point p can be a circle if the virtual rotational joint can freely rotate. Thus the orientation dexterity of robot at point q can be defined by the ratio of the arc length and the whole circle perimeter:

1 2 ,D = / (18)


·876·

where is the service angle. For the robot whose end effector has two rotary degrees

of freedom, when its end effector locates on certain point q, its end effector can be compared to a bar linked to the point by two virtual orthogonal rotational joint, as shown in Fig. 28(b). The trajectory of end joint point p can be a spherical surface if the two virtual orthogonal rotational joint can freely rotate. Thus the orientation dexterity of robot at point q can be defined by the ratio of the reachable service region and the whole service sphere:

2D = ,ps s/ (19)

where ps is the area of reachable service region and s is the whole area of service sphere.

For the robot whose end effector has three rotational degrees of freedom, when its end effector locates on a certain point q, its end effector can be compared to a bar linked to the point by three virtual mutually orthogonal rotational joint, as shown in Fig. 28(c). The trajectory of tip point p can be a spherical surface if the three virtual mutually orthogonal rotational joint can freely rotate. It is different from the spherical surface which has two virtual rotational joints. At any point in the sphere surface vector qp can rotate about itself. Each point in the sphere surface has a weight coefficient 2 = / , where is the possible range of the rotation angle. So the sphere surface is also to be weighted. Thus the orientation dexterity of robot at point q can be defined by the ratio of the reachable weight service region and the whole weight service sphere,

3D = ò (20)

where ps is the area of reachable weight service region

and s is the whole area of weight service sphere.

Fig. 28. Virtual equivalent mechanism of robot end effector

From Eqs. (18), (19) and (20) we can get that Eq. (20) contains the case of Eqs. (18) and (19), and Eq. (19) includes the case of Eq. (18). For Eq. (20), if the rotary degrees of freedom reduce to 2, the weight coefficient equals 1. So Eq. (20) can be written as Eq. (19). For Eq. (20), if the rotary degrees of freedom reduce to 1, the reachable service region sp drops down to a reachable arc and the service sphere s comes down to the whole cycle perimeter. So Eq. (19) can be written as Eq. (18).

We used z-y-x Euler angles ( ) to describe the

orientation of the robot body. means the yaw angle of the robot body, means the roll angle of the robot body, and means the pitch angle of the robot body. The angles , and can vary within the range [0 2]. According to Eq. (18), we introduce Yaw dexterity, Roll dexterity and Pitch dexterity, which can be written as follows:

Yaw ,2

D

D= Roll ,

2D

D= Pitch ,

2D

D=

where , and are the possible range of variation of yaw, roll and pitch angles for any point in workspace.

Fig. 29 shows the 3 dimension model of hexapod NOROS robot. We attach a vector p to the z axis of body frame {C} whose origin locates on the center of the robot chassis. When the point C of hexapod body locates at one certain point of the workspace the trajectory of vector p can form a weight service sphere if the hexapod body can freely rotate. According to Eq. (20), we can get the dexterity of hexapod robot:

.

D

s=ò

(21)

Fig. 29. 3D model of hexapod NOROS

But to multi-legged robots, when they move on the

ground, their body must be hold up by supporting legs. Thus the vector p impossibly has the downward component. Therefore the dexterity of hexapod body can be modified as

.

2

D

s=

/ò

(22)

By using discretization methods Eq. (23) can be written

as

yaw1 1

1 1

( , , , , )

=

1

a b

i j

a b

i j

f x y z D

D

= =

= =

¢åå

åå， (23)

where 0 unreachable

( , , , , ) ,1 reachable

f x y z ìïï=íïïî

yawD¢

¢ = , x,

d ps

,pds s /

d ps


·877·

y, z constrain the body position of the hexapod, the angles and determine the orientation of the vector p and is the possible rotation range of vector p under

orientation and . Parameter a means the dot number of angle , which is chosen by fixed step in range ( 2 2) / / , and parameter b means the dot number of angle , which is chosen by fixed step in range ( 2 2) / / .

This algorithm can be applied for each point of the workspace. Using method mentioned above we simulate the dexterity of single point and specific plane in workspace. Fig. 30 shows the dexterity of single point while w equals 145 mm, and Fig. 31 shows the dexterity of single point while w equals 100 mm. In the graph “•” denotes unreachable points and “*” denotes reachable points. The color of the point implies the weight of the point.

Fig. 30. Dexterity index simulation of single point

while w=145 mm

Fig. 31. Dexterity index simulation of single point

while w=100 mm

Fig. 32 shows the dexterity of horizontal plane of workspace while foothold offset w equals 145 mm and height z equals 100 mm. Fig. 33 shows the dexterity of horizontal plane of workspace while foothold offset w equals 100 mm and height z equals 150 mm. As it can be seen from the figures, the dexterity of the parallel mechanism presents a radial symmetrical distribution because of the radial symmetrical structure. And the point which is near to the center of the workspace has better dexterity.

From the simulation results, we can find that if the mechanism has radial symmetrical structure, its dexterity has radial symmetrical distribution. And the closer to the

center of the workspace, the more dexterous the hexapod robot body becomes. That is to say, the hexapod robot body can get more postures when it locates near to the center of the workspace, and it can easily yaw, pitch and roll.

Fig. 32. Dexterity distribution in horizontal plane while w=145mm, h=100mm

Fig. 33. Dexterity distribution in horizontal plane while w=100 mm, h=150 mm

Fig. 34 shows the dexterity of vertical plane of workspace while x equals 0 mm.

Fig. 34. Dexterity distribution in horizontal plane while x=0 mm

If the foothold offset w is determined, the hexapod robot body will have a certain workspace, and the maximum dexterity can be determined. Fig. 35 represents the relationship between the maximum dexterity D and foothold offset w. In the initial part of the curve, the maximum dexterity increases by the increasing of the offset w. When the offset w goes up to 174 mm, the dexterity D achieves its maximum value 0.164 4. After that the maximum dexterity decreases by the increasing of the offset w.


·878·

Fig. 35. Relationship between the maximum dexterity and the foothold offset

6 Conclusions

(1) A new method is proposed to determine the maximum stride length of multi-legged robot, considering the influence of body height and foothold offset. In this method one gait cycle is divided into several different periods. In one particular period the system is taken as specific mechanism. The maximum stride size can be obtained by analyzing the workspaces of adjacent equivalent parallel mechanisms.

(2) Considering the change of body height and foothold offset, the maximum stride length is obtained by analyzing the workspaces of two adjacent equivalent parallel mechanisms in one gait cycle. Simulation results show that body height and foothold offset have considerable influence on stride length.

(3) Weight service sphere and weight service region are defined to estimate the body dexterity of the hexapod, according to service angle, service region and service sphere. This method contains the methods mentioned in previous studies, it is easy to evaluate the dexterity of robot, no mater how many rotation degrees of freedom the robot end effector has. Some simulations have been done to verify our methods proposed and all of them agree with each other perfectly.

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Biographical notes XU Kun, born in 1981, is currently a postdoctoral at School of Mechanical Engineering and Automation, Beihang University, China. He received his master degree and PhD degree also from Beihang University, China, in 2008 and 2012 respectively. His research interests include mechanical design and kinematics. Tel: +86-10-82339055; E-mail: [email protected]

DING Xilun, born in 1967, is currently a professor and a PhD candidate supervisor at Robotics Institute, Beihang University, China. He received his PhD degree from Harbin Institute of Technology, China, in 1997. His main research interests include mechanism and robotics. Tel: +86-10-82338005; E-mail: [email protected]

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