0International Journal of Advanced Robotic Systems
OPEN V!/ ACCESS ARTICLE
Hierarchical Kinematic Modelling and Optimal Design of a Novel Hexapod Robot with Integrated Limb Mechanism
Regular Paper
Guiyang Xinu, Hua Deng1-2*, Guoliang Zhongu and Hengsheng Wang1-2
1 School of Mechanical and Electrical Engineering, Central South University, Changsha, China
2 The State Key Laboratory of High-performance and Complex Manufacturing, Central South University, Changsha, China Corresponding author(s) E-mail: hdeng@mail.csu.edu.cn
Received 03 August 2014; Accepted 23 November 2014
DOI: 10.5772/59989
© 2015 The Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper presents a novel hexapod robot, hereafter named PH-Robot, with three degrees of freedom (3-DOF) parallel leg mechanisms based on the concept of an integrated limb mechanism (ILM) for the integration of legged locomotion and arm manipulation. The kinematic model plays an important role in the parametric optimal design and motion planning of robots. However, models of parallel mechanisms are often difficult to obtain because of the implicit relationship between the motions of actuated joints and the motion of a moving platform. In order to derive the kinematic equations of the proposed hexapod robot, an extended hierarchical kinematic modelling method is proposed. According to the kinematic model, the geometrical parameters of the leg are optimized utilizing a comprehensive objective function that considers both dexterity and payload. PH-Robot has distinct advantages in accuracy and load ability over a robot with serial leg mechanisms through the former's comparison of performance indices. The reachable workspace of the leg verifies its ability to walk and manipulate. The results of the trajectory tracking experiment demonstrate the correctness of the kinematic model of the hexapod robot.
Keywords Integrated limb mechanism, parallel mechanism, kinematic modelling, hierarchical modelling, optimal design
1. Introduction
Mobile robots possess tremendous potential for displacing workers and extending the available environment of humans. Their enhanced mobility render mobile robots appealing for grabbing, transporting, welding, drilling and operating functions. In terms of mobility, hexapod walking robots are superior and more practical for use in uneven terrain or irregular environments with obstacles and gaps [1, 2].
Most conventional robots are designed to have both handling mechanisms and mobile mechanisms. Inspired by some insects that use legs as arms to manipulate their food, hexapod robots can offer superior flexibility using integrated leg and arm mechanisms [3-9]. In [3, 4], the concept of ILM (integrated limb mechanism) was presented. A six-bar linkage mechanism with 4-DOF used for manipulation and locomotion was verified [3, 4].
Later, two prototypes, "MELMANTIS-1" and "MEL-MANTIS-2" were developed [5, 6]. In reference to [7, 8], a new walking robot named "ASTERISK" based on the concept of ILM was developed. In [9], a modular and reconfigurable robot "MiniQuad" was developed, which has the ability to pack objects. All of the previously mentioned robots have one aspect in common: their leg mechanisms are serial mechanisms.
It is well-accepted that parallel manipulators yield better performance than serial manipulators in terms of accuracy, rigidity and load ability [10, 11]. Due to these advantages, many researchers have applied parallel mechanisms to walking robots. Takanishi Laboratory of Waseda University of Japan developed the world's first biped robot with parallel leg mechanism WL-15 and is currently developing the latest generation of WL-16RIV [12]. The leg of WL-16RIV has 6-DOF. Qi et al. introduced a reconfigurable quadruped/biped walking robot in [13].
Inspired by the studies noted above, we attempted to implement ILM hexapod robots with leg- and arm-utilizing parallel mechanisms. However, parallel manipulators, especially 6-DOF parallel manipulators, always have smaller workspaces compared to serial manipulators. In reference to [14], a novel 3-DOF parallel mechanism known as a 1-UP&2-UPS manipulator (UPS = Universal, Prismatic and Spherical, respectively), was presented, which has wide workspace and high motion performance. Consequently, in this paper, a new hexapod robot named PH-Robot (Parallel Hexapod Robot) with a parallel leg mechanism, i.e., 1-UP&2-UPS is proposed.
The legs of the PH-Robot can be used for both locomotion and manipulation, rendering the proposed robot compact and light, as two functions are integrated into one linkage, while the robot retains the advantages of parallel mechanisms. In addition, the PH-Robot is easily protected against water and other risks of damage, thanks to the base-mounted actuation arrangement, which is very important for field robots. The performance capabilities of the PH-Robot will be verified by kinematic analysis.
The kinematic model plays an important role in evaluating and optimizing the parameters of mechanisms [15-19]. The calculations of Jacobian and Hessian matrices of parallel mechanisms are extremely complex as per the directly differentiating method caused by the multi-closed loop topology. In reference to [20], the hierarchical approach was utilized to derive the Jacobian matrices of multi-body systems. In [21], the modular modelling approach was presented to derive dynamic models of multi-body systems. We extended the hierarchical or modular modelling approach to obtain the kinematics of the provided parallel robot PH-Robot, which included not only the velocity analysis, but also the displacement and acceleration analyses. Thus, the proposed hierarchical modelling approach can simplify the process of kinematic modelling.
After completing kinematic modelling, the dimensional synthesis will be implemented according to the kinematic model. It is well-accepted that the condition number of the velocity of the Jacobian matrix is one of the most suitable local performance measures for evaluating the dexterity, accuracy and velocity isotropy characteristics between the actuated joint variables and the reference point of the moving platform [22-25]. In addition to the condition number of the velocity of the Jacobian matrix, payload index according to the force of the Jacobian matrix was proposed in [22, 26]. We present a comprehensive optimal function for optimizing the geometric parameters of the parallel leg mechanism by considering both kinematic performances and payload capability, because the target of the design is to implement the concept of ILM.
The remainder of the paper is organized as follows. In section 2, we present the topology structure of the robot. In section 3, we use the hierarchical modelling approach to derive the kinematic model. Section 4 is based on the Jacobian matrix and provides optimized geometrical parameters according to a comprehensive cost function. In section 5, we analyse the reachable workspace of the integrated limb mechanism of the leg and arm; additionally, the trajectory of the foot and body is calculated.
2. Topology structure of the PH-Robot
Figure 1 shows a virtual PH-Robot prototype. The schematic diagrams for the PH-Robot are shown in Figures 2 and 3. Different to most insects that present directivity when walking, this robot's legs are arranged on the vertices of a regular hexagonal body. Thus, the robot has omnidirectional mobility and manipulability. Each leg is composed of three chains: 1 UP and 2 UPS. Here, P pair denotes the actuated joint driven by an actuator. The DOF of one leg can be calculated using the Grubler-Kutzbach criterion:
M=6(n - g -1) + £ f = 6 x (7 - 8 -1) +15 = 3 (1)
where M denotes the number of DOF, n the number of links, g the number of joints and fi the DOF corresponding to the ith joint. Here, the links of one leg consist of seven parts, including the static platform (the body) and the movable platform (the part of the foot) relative to the body. Three-DOF parallel mechanisms are more suitable for use in the integrated leg and arm mechanism, because they usually have wider motion spaces than 6-DOF parallel mechanisms. The DOF of the overall robot also can be calculated by the Grubler-Kutzbach criterion:
M = 6((1 +1 + 61) - 91-1) f. ° 6 (2)
where A is the number of legs that touch the ground.
Figure 1. Geometry model and photograph of PH-Robot; V. body, 2: universal joint, 3: linear actuator, 4: spherical joint
the swinging legs and the body is considered as the same part. As for the full robot, the ground is the relative static platform and the body is the movable platform relative to the ground.
3. Kinematic modelling
3.1 Hierarchical modelling approach
The kinematic model concerns the motion relationship between the end-effector and the actuated joints. The displacement relationship can be represented as follows:
Figure 2. Schematic diagram of PH-Robot
p = / (q)
Thus it can be seen that, no matter how many legs support the body, the DOF of the entire robot is 6. It is worth noting that the spherical foot is considered to be an S pair, while
where, p is the vector of the end-effector's position in a Cartesian coordinate system and q is the vector of the actuated joint's variables. For the parallel mechanism, obtaining the inverse kinematic model is always easier than obtaining the forward kinematic model, which is generally difficulttoderive Eq. (3)directly. Ifthereexistsa group of vectors for the joint's variables qx , q2 ,..., qn and if the followingequationscanbefound,
p = /1 (qi X qi =fi (q 2 q„ = /n(q) (4)
we can derive the relational expressions between p and q indirectly. In other words, the procedure for the calculation of the forward kinematic and inverse kinematic is a step-by-step procedure, rather than derived from one equation. Additionally, the following equation need not be composited,
p = /i (/2 (... /n(q))) (5)
because the Jacobian matrix and Hessian matrix are easier to derive according to Eq. (4) than according to Eq. (5). The velocity and acceleration relationships have the following equations,
p = Jiq 1/ qi = Lq2> .. , q„ = J„q
p = Lqr+q iT Hiq ^
qi = Lq2 + q2H2«i2, .. , q„ = J„q+<iTH„q
O4 - x4 y4z4 was completely described by 01 , 02 and li ; thus, we defined the transition hierarchy 0i 02 liY . Firstly, we attempted to obtain the relationship between 0po and qi . Secondly, we obtained the relationship
between qi and q . Then, the relationship between 0p and
qi could be derived. This denoted a hierarchical modelling approach. According to the definition for a transformation matrix between coordinate frames, we obtained
where Jt(i = i, 2,..., n) denotes the Jacobian matrix of the subsystem and H^i = i, 2,..., n) denotes the Hessian matrix of the subsystem [27]. According to Eq. (6), the following equation can be obtained recursively
0 ry i ry 2 ry 3 r"|
si "CiS2 CiC2
" 0 R 0 pO4 -ci "SiS2 SiC2
0 i 0 -C2 _S2
p=JiJ2 ■■■ J„q
so that the Jacobian matrix can be expressed as
" 0 xO4 " (li + m)CiC2
J = JJ2 .. Jn (9) 0 p04 = 0 y0i = (li + m)siC2
0 ZO4 -(li + m)s2 _
where si , ci , s2 and c2 refer to sin0i , cos0i , sin02 and cos02 , respectively, and where m denotes the fixed range between O3 and O4 . The following equation thus follows:
Similarly, according to Eq. (7), we can obtain the acceleration relationship recursively. It should be noted that the vector of generalized coordinates, i.e., q,- (i = i, 2,..., n) e Rn , may include some actuated joint variables. The Hessian matrix, i.e., H,(i = i, 2,..., n) e RN xN xN is a three-dimensional tensor with N layers, in which each layer is a N x N matrix, where R denotes the set of real numbers.
3.2 Kinematic modelling of one leg
Figure 4 shows the coordinate frames of the PH-Robot, where B - XYZ is an inertial base frame (global frame) fixed on the ground and C - xyz is a local frame fixed on the body. We separated U i into two revolute pairs, but the distance between the two revolute pairs was 0, i.e., d =0 . The rotational angles of U i were defined as 0i and 02 ; U i - x0 y0z0 was fixed on the body while the coordinate origin coincided with the geometrical centre of U i . The other coordinate systems were similar to what is usually defined in serial mechanisms. O4 was the end point of the leg and was considered as the foot. It should be noted that the plane constructed by the three points O3 , S2 and S3 was perpendicular to the axis of Pi . U iO3=li , U 2S2=l2 and U3S3=l3 were defined in the coordinate frame U i - x0 y0z0 .
0p =[°xo °y0 °zo]T was the position vector of O4 in
O4 444
U i- x0 y0z0 .
Directly deriving the equation of 0p and q = [li l2 T was
problematic. It is noted that the position and pose of
Figure 4. Coordinate frames of the leg mechanism
The inverse kinematic problem between 0p and qi could
then be solved through the following constraint equation:
d2 = - arcsin(
qi = arcsin(
li + m 0 yo.
(li + m)c
The following equation was derived from the geometrical relationship:
Similarly, differentiating between Eq. (17) and q J2q , the acceleration relationship can be derived.
p2 " |_ 0 = 0 T = 3T 3 PS2 1 - : 0 PU2 " 1
-13 ■ _ 0 = 0 T = 3T : 3 PS3 1 1 - : 0 PU3 " 1
where 3p =[0 Oy , 3p =[0 „ -^f ,
lU2 U3 J
3p =[ - K - by 0 T , 3p = [" hx by 0T . We were then able to obtain the following equations:
0 v =q iTHi«i i+Jiq i =(J2q)T h,( Lq)+Ji&i 1 (20)
&i 1=J 2q+Lq=(Lq )T H2q+Lq (2i)
where, H1 and H2 are the Hessian matrices and H1, H2 e R3x3x3 . Substituting Eq. (21) in Eq. (20) yielded
0 a04=qT J2T * HiLq+qT JiJ2T * H2q+JiLq (22)
-h s, + h c,s, + i,c,c,
x 1 y 12 112
h c. + h s.s2 + 1.s.c2 - a
x 1 y 12 112 y
h c, - ls2 - a
y 2 12 z
where * denotes a generalized scalar product [10]. The generalized scalar product of two matrices X e Rmxn and Y e Rnxnxn is defined as follows:
-h s. - h c.s2 + 1,c.c2
x 1 y 12 112
h c, - h ss + Is.c. - a
x 1 y 12 112 y
-h c2 - Ls2 + a
y 2 12 z
[X * YL=2XHY,:e Rn*n k = 1, 2, ...,
where X * Y e Rmx" ; Xk :l denotes the element on the kth row and l th column of matrix X .
a2 + a2 + h2 + h2 +l2 - 2a h c1 - 2a h s1s2
1 = y z x y 1 y x 1 y y 12
2 \ -2a l.s,c. - 2a h c. + 2a l,s.
V y 112 z y 2 z 12
a2 + a2 + h2 + h2 +l2 - 2a h c, + 2a h ss
l = y z x y 1 y x 1 y y 12
3 y-2al,s,c, -2ah c2 -2a ls
¥ y 112 z y 2 z 12
Thus, we can see that the inverse kinematic process is as follows: when 0p is known, according to Eq. (12), we can
derive q1 , then derive q according to Eq. (16). Similarly, the process of forward kinematics is: when q is known, by solving Eq. (16) we can derive q1 , then obtain 0p easily by
referring to Eq. (11). Next, instantaneous kinematic relationships can be derived by differentiating between Eq. (11) and (16); thus, the following equations are solved,
0 V04 = Jiq i
q=J-1q i
The complete expressions for Eq. (17) and (18) are given in Appendix A. According to Eq. (17) and (18), we can obtain the following equation,
0 vo4 = M = JiJ2q
3.3 Kinematic modelling of the rohot hody
In order to implement the robot body's motion, we needed to derive the inverse kinematic model, which can in turn obtain the motion for the supporting legs' actuated joints when Bp , CR and Bp have been planned. The following
equation can be derived according to Figure 5,
B P04 = B Pc + C R C P04 (24)
According to Eq. (24), we obtain
C P04 = C RT (B P04 - B Pc ) (25)
In addition, the following equation exists,
C P04 = C Pu, + C Pu,04 = C Pu,+ C0R 0 P04
Integrating Eq. (25) and (26) yields
0 P04 = C RT (C RT (B P04 - B Pc ) - C Pu,) (27)
As noted previously, q can be derived from Eq. (12) and (16). Similarly, the other supporting legs can repeat this process to obtain their respective joint variables.
XaT \z4
Figure5. Vectordiagramforsupportinglegandbody
Next, the velocity relationship between body motion and supportinglegsmotionwas derived. We set the velocity ve ctor of the body as
t vc=[vc be v
Diifrsentiatmg Eq. (e4) and (26)yielded
0 =B vc + Bw x C RC po4 + C RC p 04
c p o4 =C J i
According to Eq. (29), Eq. (30), we obtained q.o-O2 XS^ + g^x SSRC°0 4)) Substituting ei= J2q in Eq. (3i) yielded
q=-og9e1oPOB vc+e«^xm rcPo4S)
Defining D = [Bi Bj Bk Bi x A Bj x A Bkx A
mgRc p / Ecg. (31)capbe representddas
q = -J2J B0 RTDB Vc
Thus, we obtained
(vc=Ldy q = (-L a RdD a V iq
It is necessary to point out that
C R = Bw x C R = S(W) C R (35)
where S(w) is the angular velocity operator, i.e.,
0 -Bw c zC Bw c yc
S(w) = Bw c C 0 - Bw c xC. (36)
-Bm c yc Bm c J.C 0
After substituting S(w) for Bwc , differentiating Eq. (3i) yielded,
B RJiq i + B RJiq i + B RJi&ii = -B v c - S(«C RC p
C ^vw/C" fo4
-s» B RC po4 - S») BR C p 04
Substituting (j i= J2q and Eq. (2i) to Eq. (37), we finally obtained
Vc = -D 1 (S (•) C RX + 2S(W)0 RJiJ2q
+ qT B0 RJ2t * HiJ2q + B RJi(qT J2T * H2q + J2cJ))
4. Optimization based on global performance indices
Since the design is aimed at presenting a robot with integrated legs and arms, kinematic performance is very important to this robot. In this section, the optimum design of the PH-Robot will be carried out with an emphasis being placed on the determination of the geometric parameters that enable a global kinematic performance to be achieved.
4.1 Dimensional variables
Dimensional synthesis primarily refers to the geometric parameters of one leg of this six-legged robot. Figure 6 shows the side elevation of one leg; h i , ax , ri , r2 and r3 are provided according to the requirements of different tasks assigned to the robot, which are the objective variables of its design. Additionally, ai and a2 influenced the magnitude of the force generated by linear actuators. The component of ground reaction force on the leg in the direction of li increased alongside an increase of a2 , while ai in turn has been defined. However, the value of a2 cannot be too small, because of the premature interference between the lifted leg and the robot body. We set a2=60 by taking into account both the force and interference factor. The length ranges of li , l2 and l3 were derived roughly according to h i , ai , ri and r2 . Meanwhile, let the geometry of the base, i.e., AU iU 2U 3 , be an equilateral triangle, where ay is the altitude of the triangle and let the geometry of the movable platform, i.e., AO3S2S3 , be an equilateral triangle, where bx is the altitude of the triangle. Consequently,
a$=%H?ai
Next the variables cy , az , bx and by will be optimized by performance indices.
4:2Performanceindicesandoptimization
4.2.1 Condition number of the Jacobian matrix velocity
According to Eq. (19), the condition number for the velocity of the Jacobian matrix can be defined as follows :
¡t = ||j, IIIT = _Jmax
teg teg _
where omax and o • are the maximum singular value and
max ^Qin cj
the minimum singular value of the Jacobian Matrix and || • || denotes the two-norm of the matrix. As we know, the sin%lar valuerepresentsthe kinematictransmission ability in the corresponding direction. Consequently, the ratio k reflects the velocity isotropy. Similarly, k represents the transmission accuracy and error sensitivity in the mapping between the joint space and the task space. The reciprocal of the condition number, i.e., i/ k , is always defined as the local conditioning index (LCI), as follows:
L - L J k
JfeJ Jle
where 0< kj < i and the bigger the kj is, the better the dexterity. However, k and kj are simply local indices that are solely dependent on the configuration of the system; based on this, a global dexterity index (GDI) is proposed by [28] as follows:
_ f kJdV
b _ JvJ_
V3 \ r \ tt \ h W bx
Figure 6. Schematic diagram of design variables
4.2.2 Payload index
In order to realize locomotion and manipulation, the leg should have high performance in force transmission. As is iwell-known, the mapping relation expression between forces applied at the movable platform under static conditions and forces of actuators can be derived from Eq. (i9) according to the principle of the virtual work, as follows,
f = Jn t
where f=[fx fy fz]T is the external force at the toe and t = [ Ti T2 T3]T is the vector of actuator-generalized forces, and J N =(JiegT )-i . Using the Lagrange equation with the multiplier AF [22, 26], we can derive
l =tt Jnt jnt-1 (tt t -1)
When the vector t is unity, the extreme of f can be derived, that is,
= ^max(|1 )
where V is the total workspace volume. As there exist no closed-form solutions for Eq. (43), the integral of the dexterity must be calculated numerically, which can be approximated by a discrete sum:
|=V min(l 1 )
where AFi(i = i, 2, 3) are the eigenvalues of the matrix
J NT J N . 11 fmaxll and 11 fminll can be used to evaluate the payload capability of the mechanism and the larger the forces are, the better the payload capability is. The local payload index (LPI) is defined as follows,
where N denotes the points distributed over the total workspace.
and hF
Similarly, a global payload index (GPI) that can estimate the payload capability in the total workspace can be defined as follows:
^ - hF max ^ - hF n
hF = —-and hF . = —-
IF max n Fmln N
where N is the same variable as in Eq. (44).
4.2.3 Comprehensive optimum ohjective function
In order to consider both dexterity and payload, a global and comprehensive objective function f (ay, hx) is employed for maximization:
f(a ,h ) = u,k, + Uhr + Mhr ® max
J \ y' x' '1 J > 2 IF max '3 IF min
where fit(i = 1, 2, 3) is the weight coefficient. Eq. (51) is a class of optimization problem for constrained nonlinear multivariable functions.
4.3 Case study
Utilizing the design methodology proposed in the present study, an example for dimensional synthesis was carried out. Preparatory dimensional ranges were determined according to the requirements for expected work capabilities according to section 4.1, as follows:
480mm < I, < 730mm, 540mm < l2 = l3 < 790mm (52)
Except for the length range constraints, universal joints and spherical joints were limited to allowable angle ranges. That is,
According to Eq. (53) and the definition of 01 and 02 in subsection 3.2, yields,
-52° <G1 < 52°, - 52° <02 < 52'
Then, the discourse domain of two independent variables were set,
100mm < a < 250mm, 10mm < h < 80mm
Now, when ay and hx are given, the workspace can be derived according to Eqs. (52) and (53), which means that the points distributed over the total workspace and condition number corresponding to every point, i.e., N and kj , can be solved. Firstly, the effects of ay and hx on kj , r\F max and n F min ,respectively, were observed. Figure 7 (a), (b) and
(c) show the variations of kj , r\F max and r\F min versus cy and hx . It can be seen that kj and r\F min increased alongside an increase of ay . Contrarily, with the increase of hx , kj and nF min decreased slightly, meaning ay had a larger effect on motion performance than hx . However, r\F max changed with ax and hy , which is contrary to the change regulation of kj and i]F min . Therefore, there exist optimal values of ax and hy for gaining the maximum of Eq. (51), i.e., f max .
0.1 250
Figure7. Variationsof kj , TJf max and TJf min versus a^ ,
-52° <ju < 52°, - 22° <j < 22
According to Figure 7, kj , 77F max and 77F min are not in the same magnitude. In order to let the three indices have the same weight, the weight coefficients should be set as follows :
^=10, m2=1 and m = 5
In order to compare the performance indices for the robot with serial and parallel leg mechanisms, we calculated kj , ñ F max and i] F min of a general 3-DOF serial leg mechanism, as shown in Figure 8. To keep the results comparable between the two cases, the dimension of the serial leg mechanism was set according to our prototype, as follows,
Then, the constrained nonlinear multivariable function yields,
L1 = 100mm, L2 = 320mm, L3 = 380mm
max f(a ,b ) = 10k +h + 5h
J V y' x' J 'Fmax 'J
s.t. 100 < a < 250
10 < b < 80
Equation (57) can be solved by the function "fmincon" in Matlab. The result derived by Matlab is:
fmax = f (250,38.1) = 5.092
It should be noted that hx=38.1mm cannot be achieved in practice. Table 1 shows the geometrical parameters of our prototype following the above optimization and by considering the physical interference of three linear actuators.
Parameter Value Parameter Value
ay 200mm b x 65mm
az 100mm b y 37.5mm
l1min 485mm l1max 735mm
l2min l3min 542mm l2max l3max 792mm
a2 60° r3 321mm
m 136mm
Table 1. Geometrical parameters of prototype
The result was
k Serial
0.0353
hFr* = 0.068
• F max
ñsrerid = 0.0013
•F min
Figure 8. Diagram of a 3-DOF serial leg mechanism
Referring to Figure 7, the performance indices of the serial leg mechanism were smaller than the minimum values of the three indices of our proposed robot, respectively, which means that the parallel leg mechanism exhibited distinct advantages in terms of accuracy and load ability.
5. Workspace analysis and trajectory planning
5.1 Workspace of the integrated leg and arm
After the geometrical parameters are provided, the workspace of the leg can be determined via several algorithms [28-30]. In this paper, due to computing the configuration corresponding to every point over the entire reachable workspace during the process of the dimensional optimization, a discrete method via solving forward kinematics was employed to determine the reachable workspace in three-dimensional Cartesian space. The algorithm consisted of three nested loops that traversed the strokes of three limbs at the same step length, respectively, while the forward kinematic Eq. (16) was solved using the numerical method. If Eq. (53) was satisfied, the point evaluated by Eq. (11) was recorded as one point of the set of the reachable workspace. The proposed algorithm was implemented in Matlab using the date given in Table 1.
Figure 9 (a) shows the workspace of leg 1 as represented by a scatter diagram, in three-dimensional Cartesian space of the inertial base frame and referring to the location of the robot shown in Figure 12. Here, the constraint of the ground was not taken into account. The diagram of workspace described by scattered points indicates that the condition number, i.e., k , was bigger on the edge of workspace, because the points turn sparse when near the edge, which means the kinematic performance declines near the edge. Figure 9 (b) shows a vertical view of the workspace. According to Figure 9 (b), we can obtain the shared workspace of two neighbouring legs. The virtual prototype and physical prototype both demonstrated that two neighbouring legs can pick up an object through coordina-
(a) Reachable workspace of the first leg (b) Vertical view of the first leg's reachable workspace
Figure 9. Workspace of the integrated leg and arm
tion,asshownin F igure i0. Therefore, the wide workspace allows the leg to have both the abilities of manipulation and locomotion.
5.2 Trajectory planning
With respect to the movement of a walking robot, there remains important work to complete using inverse kinematics, which is, according to the trajectory planning of the foot, derived from the displacements, velocities and accelerations of actuators. Sending the results of inverse kinematics to servo motors, the trajectory tracking control is implemented without regard for frictions, clearance and sliding.
The cycloid curve was chosen as the foot trajectory due to its smooth motion virtue. Let us consider the tripod gait motion for example, as shown in Figure ii. Set the trajectory function of the foot and body, respectively, as follows,
x(t) = 0, t e [0, T /2]
y(t) = 2St / T - (S/2p)sin(4P / T), t e [0, T/2]
z(t) = 4Ht / T - (H / 2p) sin(4p / T), t e [0, T / 4]
z(t) = 2H - 4Ht / T + (H/2p)sin(8pt / T), t e[T/4, T/2]
x(t) = 0, te [0, T/2]
i/(t) = 2St/T-(S/2a)sin(4;rt/T), te[0,T/2] (62)
z(t) = 0, 6e[0,T/2]
where [x(t) y(t) z(t^T is defined as the coordinate of reference point on the foot and body in an inertial base frame B - XYZ , H and S denotes the step height and step side, respectively and T denotes the gait cycle. Here, we set
S = 150mm, H = 80mm and T = 4s.
Figure 11. Schematic diagram of the tripod gait
(a) Foot trajectory (b) Motion of the body
Figure 12. Motion profile of the foot and body
Parumeter Value Parameter Value
!iiyl 578.5mm 642.5mm
tU csd5mm sp I rC 1 t=0 -f, t- 5t5.7sg
Table 2. Initialparameters oftripodgait
Figure 1°. Shows the motion paofile, i.e., ehe functional image op E^t..(poei^n.ai6ao).]n aact,iwingingthelegsand bodly to move at the sime time cause s the foot trajectory to not 'ook the eerne as m Figure 12 (a). Prior to computing the acheator motioho, sume necessarn iiiOia) parame)ert lied So
be provided, as shown in Table 2. Other geometrical parameters are shown in Table 1.
Bo substituting Eqs. (61) and (62) and their first-order dorioa11 vhs aed seconF-arder derivativos mto tho kinematic mo del derivedin section 3, we ob° oined thn disp lacement, veiocitd and aoceSeration -or every leg't linecr actuator.Figurelti f^V^r-wslt-e trsultr for knlofanenSare gait eyclu. I^e^; 1 fiastiy guncgnns as a swanninvteg, while in she foUowing oalffCyde tccms into a supporting leg. Besidec ithe upplicationof ^giia resudtwitycn Oopjectory inFigptrnCnare alsouoeful for choosing motop'g rotg^^y^ncliip^r^il. nnp ppwpy.
The offline gait motion planning is able to verify the correotneso of the kinematic modei if the robot The control computer of the PH-Robot is an embedded CX2030 Beckhoff controller that can read the offline motion planning lists computed by Matiab. Here, only the lists of displocements and velocities are required by the controller. Sin ce °he servo eye lei s 1m s, the 0 iscrete period of inver se kinematiccalcslationisequ al to the servscgcle. Accgf ding to the offline gait planning, the walking experiment was implemented sstceespalty. Feeucs Oil tgposr fhe trajectory of leg 1 in the swing phase. The result is very similar to the profile sMowhin hlguao go (a). Theoafore, the experiment illustrates the reliability of the proposed method and the correctness of the kinematic model of the PH-Robot.
(c) Actuator fistlcce-fonti cl log f (is) Aytaetos vclaatil'ti of log f (c) Ac.uclos eddclcsclions ol teg f
Figuree3. rhe neeti tn slactustors of leglin sgattcyde
(a) Record of swing-by time-lspse sVsstidg
tU- Record of swing Uq West ysntinusus sVssting
Figure 14. Photograph of trajectory tracking experiment
6. Conclusion
A novel, six-legged robot with parallel leg mechanisms is proposed in this paper. The kinematic analysis verified that the legs of the robot have ILM characteristics that hold advantages for both the robot's legs and arms. A detailed and complete kinematic model of the robot was provided. An extended approach based on a hierarchical modelling approach was presented for solving complex kinematic problems, utilizing some passive joints variables that led to an explicit expression for the mapping matrices of velocities and accelerations. Based on the Jacobian matrix, two global performance indices, i.e., GDI and GPI were proposed for optimizing geometrical parameters. The computed results for the performance indices indicated that the PH-Robot with parallel leg mechanism had better performance than a robot with a serial leg mechanism in terms of motion accuracy and payload capacity. The workspace analysis verified the transformability from leg posture into arm posture of the parallel mechanism. The shared workspace of two neighbouring legs was analysed for dual arm coordination work.
The developed kinematic model has been implemented and demonstrated by walking experiments. In the future, the dynamics of the robot and control system should be studied systematically.
7. Acknowledgements
This work was supported by the National Basic Research Program of China (Grant No. 2013CB035504). and the National Natural Science Foundation of China (Grant No. 51405515).
8. References
[1] Agheli M, Qu L, Nestinger S (2014) SHeRo: scalable hexapod robot for maintenance, repair, and operations. Robotics and Computer-Integrated Manufacturing 30: 478-488.
[2] Asif U (2012) Improving the navigability of a hexapod robot using a fault-tolerant adaptive gait. International Journal of Advanced Robotic Systems 9(33): 1-12.
[3] Koyachi N, Arai T, Adachi H (1995) Hexapod with integrated limb mechanism of leg and arm. Proceedings of 1995 IEEE International Conference on Robotics and Automation. 1995 May 21-27; Nagoya, Japan. IEEE. 2: 1952-1957.
[4] Arai T, Koyachi N, Adachi H, Homma K (1995) Integrated arm and leg mechanism and its kinematic analysis. Proceedings of 1995 IEEE International Conference on Robotics and Automation. 1995 May 21-27; Nagoya, Japan. IEEE. 1: 994-999.
[5] Koyachi N (1997) Mechanical design of hexapods with integrated limb mechanism: MELMANTIS-1
and MELMANTIS-2. Proceedings of 1997 8th International Conference on Advanced Robotics (ICAR'97). 1997 Jul 07-09. Monterey, CA. IEEE. pp. 273-278.
[6] Koyachi N (2002) Control of walk and manipulation by a hexapod with integrated limb mechanism: MELMANTIS-1. Proceedings of 2002 IEEE International Conference on Robotics and Automation (ICRA'02). pp. 3553-3558.
[7] Takahashi Y, Arai T, Mae Y, Inoue K, Koyachi N (2000) Development of multi-limb robot with omnidirectional manipulability and mobility. Proceedings of 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2000). 2000 Oct 31-Nov 05. Takamatsu, Japan. 3: 2012-2017.
[8] Theeravithayangkura C, Takubo T, Ohara K, Mae Y, Arai T (2011) Dynamic rolling-walk motion by the limb mechanism robot ASTERISK. Advanced Robotics 25: 75-91.
[9] Chen X, Pu H, Wang X, Sun Yi, Jia W (2007) Control System of a Modular and Reconfigurable Multileg-ged Robot. Proceedings of 2007 International Conference on Mechatronics and Automation (ICMA 2007). 2007 Aug 5-8. Harbin, China. IEEE. pp. 1926-1931.
[10] Ding W, Deng H, Li Q, Xia Y (2014) Control-orientated dynamic modeling of forging manipulators with multi-closed kinematic chains. Robotics and Computer-Integrated Manufacturing 30: 421-431.
[11] Li Y, Xu Q (2007) Kinematic analysis of a 3-PRS parallel manipulator. Robotics and Computer-Integrated Manufacturing 23: 395-408.
[12] Ugahara Y, Carbone G, Hashimoto K, et al. (2007) Experimental stiffness measurement of WL-16RII biped walking vehicle during walking operation. Journal of Robotics and Mechatronics 19: 272-280.
[13] Qi Z, Wang H, Huang Z, Zhang L (2009) Kinematics of a quadruped/biped reconfigurable walking robot with parallel leg mechanisms. ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots (ReMAR 2009). London, England. IEEE. pp. 558-564.
[14] Huang T, Li M, Zhao X, Mei J, Chetwynd D, Hu S (2005) Conceptual design and dimensional synthesis for a 3-DOF module of TriVariant-a novel 5-DOF reconfigurable hybrid robot. IEEE Transactions on Robotics 21: 449-456.
[15] Li Y, Xu Q (2008) Stiffness analysis for a 3-PUU parallel kinematic machine. Mechanism and Machine Theory 43: 186-200.
[16] Asif U (2012) Design of a parallel robot with a large workspace for the functional evaluation of aircraft dynamics beyond the nominal flight envelope. International Journal of Advanced Robotic Systems 9(51): 1-13.
[17] Liu X, Wang J, Zheng H (2006) Optimum design of the 5R symmetrical parallel manipulator with a surrounded and good-condition workspace. Robotics and Autonomous Systems 54(3): 221-233.
[18] Chablat D, Wenger P (2003) Architecture optimization of a 3-DOF translational parallel mechanism for machining applications, the orthoglide. IEEE Transactions on Robotics and Automation 19(3): 403-410.
[19] Ruggiu M, Kong X (2012) Mobility and kinematic analysis of a parallel mechanism with both PPR and planar operation modes. Mechanism and Machine Theory 55: 77-90.
[20] Eberhard P, Schiehlen W (1998) Hierarchical modeling in multibody dynamics. Archive of Applied Mechanics 68: 237-246.
[21] Kubler R, Schiehlen W (2000) Modular simulation in multibody system dynamics. Multibody System Dynamics 4: 107-127.
[22] Liu X, Wang J, Pritschow G (2006) On the optimal kinematic design of the PRRRP 2-DOF parallel mechanism. Mechanism and Machine Theory 41: 1111-1130.
[23] Li Y, Xu Q (2006) Kinematic analysis and design of a new 3-DOF translational parallel manipulator. ASME Journal of Mechanical Design 128: 729-737.
[24] Zhang P, Yao Z, Du Z (2014) Global performance index system for kinematic optimization of robotic mechanism. Journal of Mechanical Design 136(3): 031001.
[25] Saglia J, Dai J, Caldwell D (2008) Geometry and kinematic analysis of a redundantly actuated parallel mechanism that eliminates singularities and improves dexterity. ASME Journal of Mechanical Design 130 (12): 124501.
[26] Liu X (1999) The relationships between the performance criteria and link lengths of the parallel manipulators and their design theory. [PhD Thesis]. Qinhuangdao, China: Yanshan University (In Chinese).
[27] Gosselin C, Angeles J (1991) A global performance index for the kinematic optimization of robotic manipulators. ASME Journal of Mechanical Design 113 (3): 220-226.
[28] Bonev I, Gosselin C (2006) Analytical determination of the workspace of symmetrical spherical parallel mechanisms. IEEE Transactions on Robotics 22 (5): 1011-1017.
[29] Bonev I, Ryu J (2001) A new approach to orientation workspace analysis of 6-DOF parallel manipulators. Mechanism and Machine Theory 36: 15-28.
[30] Kim D, Chung W, Youm Y (1997) Geometrical approach for the workspace of 6-DOF parallel manipulators. Proceedings of 1997 IEEE International Conference on Robotics and Automation. 1997 Apr 20-25. Albuquerque, NM. pp. 2986-2991.
Appendix A
Detailed expressions for some equations:
-(l1 + m)sic2 -(li + m)cis2 cic2 (l1 + m)cic2 -(li + m)sis2 sic2 0 -(l1 + m)c2 -s2
jii j12 j13 j21 j22 j23 j31 j32 j33
j12 = 0 j13 = 1
ah s. - ah c.s2 - a l,c,c,
y x 1 y y 12 y 112
-a h sx2 + a l.ss + a h s2 + a lx2
yy 12 y 112 z y 2 z 12
l1 - aysic2 + azs2
ah s.+ah c.s2 - a l,c,c,
y x 1 yy12 y112
ayhys1c2 + ayl1s1s2 + azhys2 - azl1c2 yy 12 y 112 z y 2 z 12
l1 - ays1c2 - azs2
1 y 12 z 2
jii = 0
