International Journal of Advanced Robotic Systems
ARTICLE
Motion Planning of Kinematically Redundant 12-tetrahedral Rolling Robot Regular Paper
Xingbo Wang1*, Xiaotao Wang2, Zhongpeng Zhang2 and Ying Zhao3 1 Nanjing University of Posts and Telecommunications, Nanjing, China 2 Nanjing University of Aeronautics and Astronautics, Nanjing, China 3 Aerospace System Engineering Shanghai, Shanghai, China *Corresponding author(s) E-mail:
[email protected] Received 21 July 2015; Accepted 19 December 2015 DOI: 10.5772/62178 © 2016 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 The 12-tetrahedral robot is an addressable reconfigurable technology (ART)-based variable geometry truss mecha‐ nism with 26 extensible struts and nine nodes arranged in a tetrahedral mesh. The robot has the capability of config‐ uring its shape to adapt to environmental requirements, which makes it suitable for space exploration. This paper considers the motion planning problem for the robot in terms of gait planning and trajectory planning. First, a gait planning method is developed that limits the forward falling angles to only 25 degrees. Then, according to the given gait, the jerk-bounded method and inverse kinemat‐ ics are utilized to calculate the trajectories of the nodes and the struts, respectively. A robot system model was built in ADAMS and simulations were conducted to demonstrate the feasibility of the motion planning method. Keywords 12-tetrahedral Robot, Kinematics, Gait Plan‐ ning, Trajectory Planning
1. Introduction In space exploration, some terrains can be difficult or dangerous for humans to access, yet can provide valuable data for scientific research, e.g, valleys or caves on the
surface of Mars. Developing autonomous systems for such complicated terrestrial environments has received consid‐ erable attention and various types of mobile robots have been designed. Wheeled robots and legged robots are two types of the most common autonomous mechanisms used for terrestrial exploration. Generally, wheeled robots can adapt to relatively smooth terrains, but is inefficient in passing over rough terrains [1]. A legged robot can walk on various types of terrain in an adaptive manner, but can easily lose stability of its gravity of centre when moving fast [2]. Crawler robots can be applied to irregular environ‐ ments but are relatively cumbersome and have poor flexibility [3]. The newly proposed spherical robot has a fully symmetrical structure with the capability of omnidir‐ ectional mobility. This robot can adapt well to rough terrains, but lacks stability when passing over slops and other road conditions [4]. In addition to the complexity of terrains, space exploration generally includes diverse detection tasks; therefore, the above- noted mechanisms often cannot satisfy the require‐ ments of exploration. In order to increase the capability of passing over complex terrains, researchers propose combining legged, wheeled, crawler and other robots into a novel mobile robot, e.g., a wheeled-crawler robot. Another approach is to design a mechanism that can reconfigure itself to adapt to different terrains such as the reconfigurable robot Int J Adv Robot Syst, 2016, 13:23 | doi: 10.5772/62178
1
and metamorphic robot. A reconfigurable robot takes modular structures that can be reorganized to adapt to different task requirements and environments. Although this type of robot can adapt to some extent to a variety of tasks and environments, its structure is constrained by the size of its basic building blocks and a lack of transforma‐ tion patterns. A metamorphic robot is designed based on metamorphic theory and can change it’s configuration and degrees of freedom (DOFs) automatically to adapt to functional requirements [5]. The above mobile robots have the ability to pass through general obstacles but have difficulty passing through steep slops, large gullies and narrow caves. The tetrahedral rolling robot proposed by NASA is an addressable, reconfigurable technology (ART)-based variable geome‐ try truss mechanism that consists of nodes and extensi‐ ble struts arranged in a tetrahedral mesh. This type of robot can reconfigure its shape to adapt to different types of terrain environments, which makes it very suitable for irregular and complex environments encountered in space exploration. Three generations of tetrahedral rolling robot prototypes have been proposed and designed by NASA since 2004 [6]. The first generation was a 1-tetrahedral robot consisting of four nodes and six struts, with an extension ratio of about 2:1. The robot can move its centre of gravity from its base and achieve a tumble by changing the strut lengths. The second generation was a 12-tetrahedral robot with 26 struts featuring nested square tubes, inside of which were steel cable and pulleys driven by a power electric motor with a planetary gearhead and worm gear. The third generation was also a 12-tetrahedral robot, the struts of which were made up of new material and adopted nested screws with an exoskeleton, which greatly reduced the weight of the robot. The struts can achieve a 5:1 extension ratio with the expectation of climbing a 40 degree slope. Since the 12-tetrahedral robot is a complex system with multi-struts, multi-nodes and highly dynamic, the motion planning problem is a challenging one that has received considerable attention. In [7, 8, 9], the authors designed and analysed gaits for a 4-, 6-, and 8-tetrahedral robot based on a simulating walker model. Specifically, in [10], the authors developed a gait planning method for a 12-tetrahedral robot and proposed three types of gait planning proce‐ dures. Although the proposed method provided important references to the gait planning problem, its simplified geometrical computation can only be used by the reduced model. In [11, 12], the authors designed a prototype for the single tetrahedral robot, then proposed a motion planning strategy and presented the toppling condition for the robot based on its kinematics. In [13], the authors analysed the kinematics of the different motion phases of a 1-tetrahedral robot and presented the rolling critical condition. In this paper, we present a detailed motion planning method for the 12-tetrahedral robot. First, a gait is planned 2
Int J Adv Robot Syst, 2016, 13:23 | doi: 10.5772/62178
for the robot, the forward falling angles of which are constrained to be only 25°, which greatly reduces the impact force of the ground on the robot during its falling. Then, the jerk-bounded method [14] is adopted to plan the trajectories of the robot nodes and the inverse kinematics is solved to obtain the trajectories of the corresponding driving struts. A model of the 12-tetrahedral robot was established in ADAMS and simulations were conducted to verify the feasibility of the proposed motion planning method.
The remainder of the paper is organized as follows. In section 2, we present the model and the kinematic model for the 12-tetrahedral robot. In this section, we also present the rolling condition and the external contact model. In Since the 12-tetrahedral robot is a complex system with multi-struts, multi-nodes and section 3 we propose a motion planning approach, includ‐ planning problem is a challenging one that has received considerable attention. In [7, ing gait planning and trajectory planning. Simulations are analysed gaits for a 4-, 6-, and 8-tetrahedral robot based on a simulating walker mode conducted in asection 4. Finally, conclusion are presented developed gait planning method for a 12-tetrahedral robot andin proposed three type Section 5. the proposed method provided important references to the gait planning p Although
computation can only be used by the reduced model. In [11, 12], the authors designed tetrahedral robot, then proposed a motion planning strategy and presented the toppl
2. Model and kinematics analysis of the 12-tetrahedral on its kinematics. In [13], the authors analysed the kinematics of the different motion robot and presented the rolling critical condition.
In this of paper, we present a detailed 2.1 Model the 12-tetrahedral robotmotion planning method for the 12-tetrahedral ro
robot, the forward falling angles of which are constrained to be only 25°, which great ground on the robot during its falling. Then, the jerk-bounded method [14] is adopte The 12-tetrahedral robot is an over-constrained, close-chain robot nodes and the inverse kinematics is solved to obtain the trajectories of the corre mechanism consisting of 26 struts, eight external nodes and of the 12-tetrahedral robot was established in ADAMS and simulations were conduct one proposed central motion node, planning as shown in Figure 1. The robot is method.
initialized as a cube, with one strut along each edge and of theone paper is organized as follows. section 2, we present the mode onlyThe oneremainder strut along diagonal of each facet.InThe central In this eight section, we also present nodethe is 12-tetrahedral connected torobot. the other external nodesthe viarolling eightcondition and the 3 we propose a motion planning approach, including gait planning and trajectory pla struts. The central node provides space for payloads such in section 4. Finally, conclusion are presented in Section 5. as telemetry and communication modules, and the external nodes can provide accommodation The robot 2. Model and kinematics analysis offor thesensors. 12-tetrahedral external nodes are highly symmetrical and can be used as 2.1. Model thestrut 12-tetrahedral robot landing points. of Each is driven by two DC motors via planetary gearheads and worm gears. The struts extend or The 12-tetrahedral robot is an over-constrained, close-chain mechanism consisting of and one node, as shown in Figure 1. Theto robot initialized contract to central collaboratively drive the robot fallisover and as a cube, with on one strutThis alongtype one diagonal eachan facet. The centralof node is connected to the other roll about. of robotofhas abundance geomet‐ The central node provides space for payloads such as telemetry and communi ricalstruts. shapes and can achieve a variety of gaits, e.g., walking, nodes can provide accommodation for sensors. The external nodes are highly symme rolling over and climbing, which improves its capacity to points. Each strut is driven by two DC motors via planetary gearheads and worm gea traverse across extreme terrain topographies such as This steep to collaboratively drive the robot to fall over and roll about. type of robot has an and and deep canyons, cancraters, achieve agullies, variety of gaits, e.g.,etc. walking, rolling over and climbing, which imp across extreme terrain topographies such as steep and deep craters, gullies, canyons,
Figure 1. The 12-tetrahedral rolling robot
2.2 Kinematics of the 12-tetrahedral robot
Let pij = xij yij zij
The first step in the design of the robot’s motion is to specify its kinematics. Thus, a systematically analytical method is needed for acquiring the node positions from the strut lengths, and vice verse. In the following, we present the direct and inverse kinematic models for the robot.
T
be the position of the hinge fixed at N i and
connected to N j expressed in frame M i , and Pij = X ij Y ij Z ij
T
be its position expressed in frame B . Then, pij is a constant
vector, since the position of each hinge remains unchanged relative to the fixed node. Given positions Pi of N i and Pj
of N j , the strut from N i to N j can be represented with a vector Lij :
Figure 1. The 12-tetrahedral rolling robot
2.2.1 Inverse kinematics
2.2. Kinematics of the 12-tetrahedral robot
B Lij = Pji - P = Pj +method Rp jiis- Pi - MB Rpij InverseThe kinematics isdesign usedofto theis lengths ofkinematics. the ij M first step in the thecalculate robot’s motion to specify its Thus, a systematically analytical i i the node positions from strut lengths, and vice verse. In the following, we present the direct struts, needed given for theacquiring positions of the nodes. Intheorder to analyse and inverseakinematics models for theisrobot. the kinematics, simplified model given(see Figure 2) for with length the 12-tetrahedral robot, where N 0 is the central node and 2.2.1. Inverse kinematics N 1, … ,N 8 denote the external nodes. A set of frames is Inverse kinematics is used to calculate the lengths of the struts, given the positions of the nodes. In order to analyse the attached to the model. Frame B , positioned on node N 1, 2 2 kinematics, a simplified model is given(see Figure 2) for the 12-tetrahedral robot, where lij = N0Xisji the - Xcentral +node Yji and - Yij + Z ji - Zij ij denote Nthe reference frame and frames M (i = 0,2,...,8) , fixed i , , N 8 denote the external nodes. A set of frames is attached to the model. Frame B , positioned on node N1 , 1 on the denote centretheofreference the mass of N , denote local frames. frame and iframes M (i = 1,2,...,9) , fixed on the centre of the mass of N , denote local frames.
(
i
) (
)
2
(4)
i
where i and j(i < j) are the subscript indices of the nodes connected with one strut. Therefore, 26 explicit functions can be achieved to formulate the relations between the strut lengths and the node positions.
N7
N8
) (
(3)
N6
N5
2.2.2 Direct kinematics Direct kinematics is used to find the poses of the nodes, given the lengths of the struts. The basic unit of the 12tetrahedral robot is shown in Figure 3 and consists of four nodes connected together with six struts, where ΔABC denotes the base plane and D is the vertex. Frame A - xyz is fixed at the tetrahedral, where A is the origin, the x -axis is along AB , the z -axis is perpendicular to ΔABC and points towards D , and the y -axis is given by y = z × x .
N0
Z
N3 x
N4 Y
y M 2
N1 X
B
z
N2
Figure 2. A simplified model of the 12-tetrahedral robot and definition of Figure 2. A simplified model of the 12-tetrahedral robot and definition of the coordinate frames the coordinate frames
'
The position and orientation of N i in frame B can be denoted by the position vector Pi and the rotation matrix
B The position and orientation of N i in frame B can be R, M
i
denoted by the position vector Pi and the rotation matrix B M i R,
B M i
R RZ i RY i RX i
RX ( i )
RY ( i )
( ) ( ) ( )
B Dummy Text R =where R g
lAD
z
(1)
RZ ( i )
are the elementary rotations of frame about the X -, Y - and ai RY , b i RX and (1) M Z i i ( i , i , i ) Z -axes of frame B through angles i , i and i , respectively. The X - Y - Z fixed angle denotes the $ Mi i orientation of frame relative to frame B , where is the yaw rotation about the X -axis of frame B , i is the
where RX (αi ), RY (βi ) and RZ (γi ) are the elementary rotations i
is the roll rotation about the Z -axis. Hence, the rotation matrix pitch rotation about the Y -axis and M i about the X -, Y - and Z -axes of frame B of frame
x
abbreviations s := sin and c := cos is given as follows:
through angles αi , βi and γi , respectively. The X - Y - Z fixed
B M
R , with
i
of frame B , βi is the pitch rotation about the Y -axis and γi
is the roll rotation about the Z -axis. Hence, the rotation B R , with abbreviations sθ : = sinθ and cθ : = cosθ is matrix M i
given as follows: écb i cg i ê B R = ê cb i sg i M i ê - sb i ë
sa i sb i cg i - ca i sg i sa i sb i sg i + ca i cg i cb i sa i
ca i sb icg i + sa i sg i ù ú ca i sb i sg i - sa icg i ú ú ca i cb i û
lAC
& lBC
lAB
angle (αi ,βi ,γi ) denotes the orientation of frame M i relative to frame B , where αi is the yaw rotation about the X -axis
lCD
lBD
y
Mi
% Figure 3. The coordinate frame fixed at a tetrahedron
Let pa, pb, pc and pd be the coordinates of A, B , C and D
expressed in frame A - xyz , respectively. Given the strut lengths, l AB , l AC , l AD , l BC , l BD and lCD , pa, pb and pc can be easy obtained from the geometrical relationships between A, B
(2)
and
C
as
pa = 0 0 0 T ,
pb = l AB 0 0
T
and
T
pc = l AC cosθ l AC sinθ 0 , while pd is given by the geomet‐
rical relationships between A, B , C and D , as follows,
Xingbo Wang, Xiaotao Wang, Zhongpeng Zhang and Ying Zhao: Motion Planning of Kinematically Redundant 12-tetrahedral Rolling Robot
3
é l2 + l2 - l2 pd = ê AB AD BD 2lAB ëê
where cosθ =
2 2 - lCD - 2 xc xd xc2 + yc2 + lAD 2 yc
ù 2 - xd2 - yd2 ú lAD ûú
T
2 2 2 l AB + l AC − l BC ,(0 < θ < π ). 2l AB l AC
corresponding nodes connected by the strut, respectively. Then, the mass of N 0 will change to be m0 = mc + 4ms and the 7
mass of N 1, N 3, N 6 and N 8 will be m7n = mn + 2 ms , respec‐ tively, while the mass of N 2, N 4, N 5 and N 7 will be m4n = mn + 2ms . Let G be the coordinate of the centre of gravity
Given the coordinates A, B and C in the reference frame, Pa,
of the robot, given as follows:
Pb and Pc , the transformation matrix from frame A - xyz to
the reference frame, can be represented as follows: éX Y Z P ù T=ê ú ë 0 0 0 1û
G=
(5)
where P = Pa denotes the translation matrix, R = X Y Z denotes
the
rotation
matrix,
X=
Pb − Pa
| Pb − Pa | ,
(Pb − Pa) × (Pc − Pa) | | Z= | (Pb − Pa) × (Pc − Pa) | and Y = Z × X , with ⋅ , denotes
the norm of a vector. The coordinate D in the reference frame can be obtained using the transformation matrix T , Pd = T × pd
In this case, the direct kinematics of the entire robot can be achieved by sequentially choosing another adjacent tetrahedral cell and computing the unknown node posi‐ tions of the latter tetrahedral using the above method. 2.3 Rolling analysis of the 12-tetrahedral robot The 12-tetrahedral robot can achieve different motion patterns such as rolling and walking in the manner of a four-legged robot [15] by using a classic criterion of static balance (the centre of mass should project inside the convex hull of contact points) or a dynamic balance method [16, 17]. The dynamic balance method can achieve relatively high robot speed, but is sensitive to environmental varia‐ bility; alternatively, the static balance method is required to keep the centre of gravity of the robot in a stable region, which renders it extremely suitable for walking at lowspeeds and obstacle avoidance. More than three contact nodes are needed to keep the robot in equilibrium and the centre of gravity of the robot must be kept stable, which is achieved via the contact nodes on the ground. When stretching and/or contracting the struts to deviate its centre of gravity from the stable region, the robot will roll about, thus achieving the rolling motion of the robot. Assume that each strut has equal mass and a multi-segment design, and that the centre of gravity of the strut is coinci‐ dent with its geometrical centre. Let ms be the mass of the
strut, mn the mass of the external node and mc the mass of
the central node. For the sake of simplicity, the mass of each strut is split into two halves, which are added to two
4
Int J Adv Robot Syst, 2016, 13:23 | doi: 10.5772/62178
m0 P0 + m7 n ( P1 + P3 + P6 + P8 ) + m4n ( P2 + P4 + P5 + P7 ) m0 + 4 m7 n + 4 m4 n
(6)
When the projection of G onto the ground deviates from the stable region formed by the contact nodes with the ground, the robot will fall over and begin rolling about. 2.4 The external contact model of the robot When the robot falls over, certain nodes will collide with the ground and their velocities will change greatly in a short period of time, since their movements are constrained by the ground. Hence, their accelerations will be large, as will be the contact forces [18, 19]. It is assumed that the positions of the contact nodes will remain unchanged from the beginning to the end of the collision, that is, their displacements during collision can be ignored. According to the above analysis, the impact function is adopted to model the contact between the nodes with the ground. Therefore, the contact process can be considered as a linear spring-damper system. The contact model [20, 21] of N i (i = 1,2,3, ⋯ 8) with the ground can be represented as: ì F = 0, ï ci í ïî Fci = Kd i + Cd&i ,
(d (d
z i z i
) > 0)
£0
(7)
where δi = δix δiy δiz is the depth of N i penetrating into
the ground; the contact stiffness K(N/mm) and the damping coefficient C(N⋅sec/mm) are given, respectively, as follows: ék x ê K=ê0 ê ëê 0
0 k
y
0
éc x 0ù ú ê 0 ú ,C = ê 0 ú ê k z ûú ëê 0
0 c
y
0
0ù ú 0ú ú c z ûú
3. Motion and gait planning of the 12-tetrahedral robot As a kind of autonomous mobile system, the 12-tetrahedral robot requires substantial intelligence and/or autonomy. In this paper, the 12-tetrahedral robot is considered an endeffector with underlying reflective intelligence that re‐ ceives commands from the global task planning layer and moves according to the optimal gait chosen. The control system considered in this paper is given in Figure 4 and includes gait query and selection and a sensor-actuator loop. When the robot controller receives control instruc‐
tions from the above layer (or a human), it first queries the gait library and selects the current gait, then transmits the instruction to the strut controllers. Finally, the strut controllers regulate the corresponding struts to extend or contract in the form of planned trajectories.
Control commands Executing
Gait Library
Gait Generation
Current Gait
Planning Player
Path Generation
drons increases, the method becomes too complicated to compute; (ii) plan the positions of each node and then calculate elongation for each of the links. Clearly, the second option is more feasible than the first, since the number of nodes is significantly less than that of the links. In this paper, we only considered the case of the robot rolling about and restoring the status quo ante on a flat surface. The gait planning method for the 12-tetrahedral robot is presented in Figure 5, where the dark circle ’●’ denotes the nodes that contact the ground, the white circle ’○’ denotes the nodes that are not in contact with the ground, the thick lines denote the struts, the lengths of which change during robot motion and the thin lines denote the strut lengths that remain unchanged. The motion of the robot can be divided into the following steps: Step 1: The robot is at the initial state, with nodes N 1, N 2,
Link Controller
Execution Layer
Robot Figure 4. The control system of the 12-tetrahedral robot
Motion planning is an important aspect of robot control and addresses the fundamental robotics task of planning collision-free motion from a start to a goal position among a collection of obstacles. Meanwhile, the 12-tetrahedral robot is a complex system with multi-struts, multi-nodes and highly-dynamic, presenting challenges in terms of the robot’s motion planning. For the sake of simplicity, in this paper, we only considered the case of selecting gait from the predetermined gait library when the controller received commands from the above layer. In this way, the motion planning problem was divided into two parts: (1) the gaits of the robot, i.e., the way in which the robot opts to move, such as tumbling or walking; (2) the motion of the struts, i.e., how to drive the struts to accomplish the chosen gait and achieve stable motion for the robot. In [10], the authors developed a gait planning method for a 12-tetrahedral robot and proposed three types of gait planning procedures. Although the proposed method provided important references for the gait planning problem, its simplified geometrical computation can only be used in a reduced model. In this paper, we focus on a realistic model, where the obtained strut displacements can be directly input into the control system. In the following, we first present a gait planning method and then provide trajectory planning for the struts and nodes. 3.1 Gait planning for the 12-tetrahedral rolling robot According to the properties of the 12-tetrahedral robot, there are two ways in which to plan the robot’s motion: (i) directly determine the elongations of each link, similar to a 1-tetrahedral robot. However, as the number of tetrahe‐
N 3 and N 4 in contact with the ground, as shown in Figure
5(a) and the robot will move towards the right side.
Step 2: Struts l06, l07, l16, l56, l68 and l78 contract or expand to
drive quadrilateral N 2 N 3 N 7 N 6 to rotate to the right around the axis N 2 N 3 through an angle θ61 = 25 ° , as shown in Figure
5(b). Nodes N 6 and N 7 move synchronously and the other
nodes remain stationary.
Step 3: Struts l01, l04, l05, l08, l15, l16, l18, l38 and l48 drive nodes N 5, N 8, N 6, N 7 and N 0 to rotate to the right side around
N 2 N 3, as shown in Figure 5(c), where N 5 and N 8 rotate
synchronously through θ5 = 60 ° , N 0 rotates through θ0 = 40 °
and N 6 and N 7 follow their motions and rotate through
θ62 = 51 ° .
Step 4: Following the above motions, the centre of gravity of the robot will deviate from the stability regions accord‐ ing to equation (6), then the robot realizes tumbling and nodes N 6 and N 7 fall to the ground, as shown in Figure 5(d). Step 5: The robot contracts its struts as shown in Figure 5(e) and finally restores the status quo ante, as shown in Figure 5(f). Following the above steps, the robot will realize rolling about. According to the above gait planning method, the forward falling angle of the robot is limited to be only 25 ° , which greatly reduces the impact force on the robot during falling. 3.2 Trajectory planning for the 12-tetrahedral robot Given a gait for the robot, the trajectory planning problem focuses on generating trajectories for the nodes and struts. In this paper, the jerk-bounded method proposed in [14] is adopted to calculate the trajectories of the robot nodes. Since the 12-tetrahedral robot is a special parallel robot and proposes a unique solution to the inverse kinematics, the inverse kinematics can be solved to obtain the motion trajectories for corresponding driving struts. Xingbo Wang, Xiaotao Wang, Zhongpeng Zhang and Ying Zhao: Motion Planning of Kinematically Redundant 12-tetrahedral Rolling Robot
5
acements In the hod and d nodes.
800
N8
Node6&7 Node0 Node5&8 Node1&4 Node2&3
N8
N7
N5
700
N5
N6
N7
N0
N0
N4
N6
N1
N3 N1
Ɵ
500
Ɵ61
N2
(a)
z/mm
N2
al robot, robot’s of each r, as the becomes s of each he links. the first, n that of ase of the ante on a
600
N4
N3
(b)
400
300 N8
N8 N4 Ɵ5
N0
N0
N4
N7
Ɵ0
N1
N6
Ɵ62
0
N2
N2
N6
(c)
(e)
N6
800
900
1000
strut16 strut36 strut68 strut07 & strut06 strut56 & strut78 strut04 & strut01 strut05 &strut08 strut18 strut15 & strut48
150
(f)
3.2. Trajectory planning for the 12-tetrahedral robot In the jerk-bounded method, a concatenation of fifth-order polynomials is presented to provide a smooth Given a gait for the robot, the trajectory planningtrajectory problem betweenontwo waypoints and a for sinethewave is focuses generating trajectories nodestemplate and struts. utilized to calculate the end conditions for proposed ramps from In this paper, the jerk-bounded method inzero [14] acceleration nonzerotheacceleration. According this is adopted toto calculate trajectories of the robotto nodes. Since thethe 12-tetrahedral special parallel robot method, position Pi ofrobot node isN iacan be formulated as a and proposes a unique solution to the inverse kinematics, function of time, the inverse kinematics can be solved to obtain the motion trajectories for P6 (corresponding t ) = R(0,q 6 (t ),0) driving pd6 + P26 struts. (8) In the jerk-bounded method, a concatenation of fifth-order polynomials Pis(tpresented ) = R(0,q 5 (tto ),0)provide pd + P25 a smooth trajectory (9) between two 5 waypoints and a5 sine wave template is utilized to calculate the end conditions for ramps from P0 (t ) =toR(0, q 0 (t ),0) pd + P20 zero acceleration nonzero acceleration. According to (10) 0 this method, the position Pi of node Ni can be formulated as a function of time, where R(α,β,γ) = RZ (γ)RY (β)RX (α) is the rotation matrix 2.2, distance from the centre defined in subsection P6 (t) = R (0,pd θ6i(is t)the , 0) pd (8) 6 + P26 of node N i to the virtual axis defined by nodes N 2 and N 3, P5 (t) = R(0, θ5 (t), 0) pd5 + P25 (9) and P2i is the corresponding supporting point of N i on the P0 (t) = R(0, θ0 (t), 0) pd0 + P20 (10) virtual axis; θi (t) is the rotation angle of N i , defined as a where (α,time, β, γ)which = RisZobtained (γ) RY ( β)using R X (α)theisjerk-bounded the rotation functionRof matrix defined in subsection 2.2, pd is the distance from i trajectories planning method. the centre of node Ni to the virtual axis defined by nodes Given in subsection 3.1, the x- and zN N3 ,gait and provided P2i is the corresponding supporting point 2 andthe coordinates each node the reference of Ni on theof virtual axis; expressed θi (t) is theinrotation angle frame of Ni , defined as in a function time,we which is obtained usingrolls the are shown Figure 6, of where assume that the robot jerk-bounded trajectories planning along the y - axis, while the x and method. z coordinates of nodes N 1 andthe N 4 are by (0,0)in . Furthermore, the Given gaitgiven provided subsection according 3.1, the x-to and node trajectories, the trajectories of theincorresponding z-coordinates of each node expressed the reference drivingare struts are obtained frame shown in Figureby6,solving where the we inverse assumekinemat‐ that the robot rolls along theshown y− axis, while the ics equation (4), as in Figure 7. x and z coordinates of nodes N1 and N4 are given by (0, 0). Furthermore, 6
700
200
Figure 5.5. The rolling gaitsgaits of theof12-tetrahedral robot robot. Figure The rolling the 12-tetrahedral
e rolling hod, the only 25◦ , ot during
600
N7
N2
Length/mm
N7 N6
hown in ante, as
500 x/mm
250
N3
N3
f gravity regions realizes ound, as
400
N0
N2
l48 drive ight side e N5 and 0 rotates motions
300
N5
N0
r expand the right = 25◦ , N7 move tionary.
200
Strut Drive Plan
N8 N1
N1
odes N1 , shown in the right
100
N4
N5
robot is notes the denotes the thick e during t lengths ot can be
0
Figure 6. The trajectories of the nine nodes in the Cartesian frame
(d) N8
N4
100
N7 N3
N3 N1
200
N5
N5
50
0
−50
0
5
10
15
20
25 time/s
30
35
40
45
50
Figure 7. The trajectories of the moving struts
Although the Jacobian matrix of the robot is used to obtain the strut velocities and the Jacobian matrix and its deriva‐ tive are used to achieve their accelerations, in real-world applications, the velocities v(t) and accelerations a(t) are generally obtained using numerical differential equations from their position trajectories, d (t), as follows:
v(t ) =
d(t ) - d(t - d t ) dt
(11)
a(t ) =
v(t ) - v(t - d t ) dt
(12)
where δt is the computing interval; l15 and l48 are the struts
with the longest displacement, as shown in Figure 7. Their velocities and accelerations are obtained using equations (11-12) and are illustrated in Figure 8. Figure 8 shows that the change in the lengths of struts l15
and l48 exhibits symmetry, which reflects stretching of the
struts first, followed by shrinking of the struts. These symmetries are also exhibited in the velocities and acceler‐ ations of the struts.
Int J Adv Robot Syst, 2016, 13:23 | doi: 10.5772/62178
: Motion planning of kinematically redundant 12-tetrahedral rolling robot
100
5
Prior to conducting simulations, we needed to set param‐ eters such as units, gravity, etc., and add constraints as follows,
Lengthn(mm)
300 200 100 0
0
5
10
15
20
25
30
35
40
45
50
Velocity(mm/s)
50
0
Acceleration(mm/s2)
−50 0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25 time(s)
30
35
40
45
50
20
0
−20 0
Figure 8. The trajectories of the displacement, velocity and acceleration of struts l15 and l48
The above method can be generalized to other trajectory planning approaches of the 12-tetrahedral robot. When a new gait is given, only the trajectories of the moving nodes need to be re-calculated and their trajectories can be utilized to achieve the trajectories for corresponding struts, based on the inverse kinematics. 4. Motion simulations of the 12-tetrahedral robot 4.1 ADAMS model setup of the robot To illustrate the results of the above presented motion planning method for the 12-tetrahedral robot, a 3D model was built in Pro/E software and then imported into ADAMS software. In order to reduce the model’s complex‐ ity and computation times yet maintain the performance of the model, certain elements were omitted that had little effect on the robot’s performance, such as motor, screws, etc. Thus, the reduced model consisted primarily of nodes, struts and universal joints fixed on nodes. The physical properties and other specifications of the model are summarized in Table 1. Parameter
Value
Mass of N 0
0.90(kg)
Rotational inertia of N 0
(703.81, 705.08, 5515.16)
(Ixx , Iyy , Izz )
(kg⋅mm2)
Mass of N1 N8
1.41(kg)
Rotational inertia of N1 N8
(6316.94, 5916.40, 5515.16)
(Ixx , Iyy , Izz )
(kg⋅mm2)
Mass of central links
0.84(kg)
length of central links
294(mm)
Mass of border struts
0.90(kg)
Length of border struts
330(mm)
Mass of diagonal struts
1.33(kg)
Length of diagonal struts
568(mm)
Table 1. Primary dimensions of the 12-tetrahedral robot
• A rectangular box was added below the robot to indicate that the ground was fixed to the earth through a fixed joint. • Two orthogonal revolute joints were added to each junction where the node was connected with the corre‐ sponding strut; a revolute joint was added to each junction at which the node is connected with the diago‐ nal strut and a revolute and fixed joint were added to each junction at which the node was connected with the central strut. • A translational joint was added to each strut and translational motion, with expression 0.0 * time , was added to the joint. • Contacts were added between the eight external nodes with the ground, where the impact function method was used to compute contact force; a Coulomb model was used to calculate the friction. 4.2 Simulation results and analysis The trajectory data for each strut from the above motion planned method in section 3 were imported into the model. The simulation time was set to be 49.9s. The rolling process of the robot is illustrated in Figure 9. It is shown that the robot followed the planned gait and completed actions from stretching to rolling to tumbling, and finally restored the status quo ante, illustrating the feasibility of the above motion planning method. It is also shown in the figures (Figure 9a- 9i) that two parts of a strut stretch or shrink synchronously, which keeps the centre of mass of the strut consistent with its geometrical centre. Figure 9a–Figure 9c show the procedure during which the struts stretch, while the centre of gravity of the robot remains in its stable region. Figure 9d–Figure 9f illustrates the process from the centre of gravity deviating from its stable region, to the robot tumbling and then to nodes N 6 and N 7 landing. The process of the robot restoring to its initial state is illustrated in Figure 9g–Figure 9i.
The velocities of nodes N 6 and N 7 are illustrated in Figure
10, and the touching forces of the two nodes with the ground are illustrated in Figure 11. It is shown that the touching forces are equal to zero before the two nodes achieve contact with the ground. When the nodes collide with the ground (about t = 22s ) their velocities change greatly, so that the corresponding forces are also very large. Following contact, their velocities and the corresponding touch forces are not zero, since the two nodes bounce on the ground until their potential energy is depleted. It is also shown that a higher speed will result in a larger touching force, validating the impact force model presented in section 2.4. In order to reduce the force, the external nodes Xingbo Wang, Xiaotao Wang, Zhongpeng Zhang and Ying Zhao: Motion Planning of Kinematically Redundant 12-tetrahedral Rolling Robot
7
Aerospace System Engineering Shanghai, ShanghaiChina 2016132107201519122015© 2016 Author(s). Licensee InTech.2016This 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.The 12-tetrahedral robot is an addressable reconfigurable technology (ART)-based variable geometry truss mechanism with 26 extensible struts and nine nodes arranged in a tetrahedral mesh. The robot has the capability of configuring its shape to adapt to environmental requirements, which makes it suitable for space exploration. This paper considers the motion planning problem for the robot in terms of gait planning and trajectory planning. First, a gait planning method is developed that limits the forward falling angles to only 25 degrees. Then, according to the given gait, the jerk-bounded method and inverse kinematics are utilized to calculate the trajectories of the nodes and the struts, respectively. A robot system model was built in ADAMS and simulations were conducted to demonstrate the feasibility of the motion planning method.12-tetrahedral RobotKinematicsGait PlanningTrajectory Planning
the robot to earth through
dded to each ted with the was added to nnected with d fixed joint he node was (a)
ch strut and 0 ∗ time, was
(b) 1. Introduction Figure 11. The
5. Conclusion
xternal nodes ction method ulomb model
above motion rted into the 9s. The rolling
(c)
nned gait and to tumbling, lustrating the method. It is (h)) that two nously, which nsistent with 9(c) show the h, while the stable region. ess from the region, to the landing. The e is illustrated
illustrated in wo nodes with It is shown efore the two hen the nodes heir velocities orces are also cities and the since the two tential energy er speed will g the impact der to reduce d with elastic he nodes and t from sliding tempt to find robot with a
In this paper, we presented motion planning for the 12tetrahedral robot, including gait planning and trajectory planning. First, we proposed and analysed a type of rolling gait based on the actual model of the robot. We then utilized an acceleration-bounded trajectory planning method to compute the expected position, velocity and acceleration for each strut. The obtained data could be directly input into the corresponding strut controller. The simulations were conducted using ADAMS to verify the feasibility of the proposed gait.
(d)
6. Acknowledgements
(e)
This work is supported by the National High Technology Research and Development Program ("863" Program) of China (2015AA7046411), the National Natural Science Foundation (NNSF) of China under grant no. 61403162, the NUPTSF (grant no. NY212027) and the Shanghai Academy of Space Technology (SAST201318).
(f)
7. References 10.5772/1International Journal of Advanced Robotic SystemsInt J Adv Robot Syst1729-8814InTech10.5772/621780Motion Planning Of Kinematically Redundant 12-tetrahedral Rolling Robot
[email protected] (g) (h) WangXiaotao ZhangZhongpeng ZhaoYing Figure 9. The rolling process of theof 12-tetrahedral robot robot. Nanjing University of Posts Telecommunications, NanjingChina Figure 9. Theand rolling process the 12-tetrahedral Nanjing University of Aeronautics and Astronautics, NanjingChina Aerospace System Engineering Shanghai, ShanghaiChina 2016132107201519122015© 2016 Author(s). Licensee InTech.2016This 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.The 12-tetrahedral robot is an addressable reconfigurable technology (ART)-based variable geometry truss mechanism with 26 extensible struts and nine nodes arranged in a tetrahedral mesh. The robot has the capability of configuring its shape to adapt to environmental requirements, which makes it suitable for space exploration. This paper considers the motion planning problem for the robot in terms of gait planning and trajectory planning. First, a gait planning method is developed that limits the forward falling angles to only 25 degrees. Then, according to the given gait, the jerk-bounded method and inverse kinematics are utilized to calculate the trajectories of the nodes and the struts, respectively. A robot system model was built in ADAMS and simulations were conducted to demonstrate the feasibility of the motion planning method.12-tetrahedral RobotKinematicsGait PlanningTrajectory Planning
can be covered with elastic material to increase the friction between the nodes and the ground, and to further prevent the robot from sliding on the ground. Furthermore, we can also attempt to find better gait planning methods to endow the robot with a soft landing capability.
Figure 10. The actual velocities of nodes N6 and N7 .
: Motion planning of kinematically redundant 12-tetrahedral rolling robot 1.Figure Introduction 10. The
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contact forces of nodes N6 and N7 with the ground
In space exploration, some terrains can be difficult or dangerous for humans to access, yet can provide valuable data for scientific research, e.g, valleys or caves on the surface of Mars. Developing autonomous systems for such complicated terrestrial environments has received considerable attention and various types of mobile robots have been designed. Wheeled robots and legged robots are two types of the most common autonomous mechanisms used for terrestrial exploration. Generally, wheeled robots can adapt to relatively smooth terrains, but is inefficient in passing over rough terrains [1]. A legged robot can walk on various types of terrain in an adaptive manner, but can easily lose stability of its gravity of centre when moving fast [2]. Crawler robots can be applied to irregular environments but are relatively cumbersome and have poor flexibility [3]. The newly proposed spherical robot has a fully symmetrical
7
actual velocities of nodes N6 and N7
In space exploration, some terrains can be difficult or dangerous for humans to access, yet can provide valuable data for scientific research, e.g, valleys or caves on the surface of Mars. Developing autonomous systems for such complicated terrestrial environments has received considerable attention and various types of mobile robots have been designed. Wheeled robots and legged robots are two types of the most common autonomous mechanisms used for terrestrial exploration. Generally, wheeled robots can adapt to relatively smooth terrains, but is inefficient in passing over rough terrains [1]. A legged robot can walk on various types of terrain in an adaptive manner, but can easily lose stability of its gravity of centre when moving fast [2]. Crawler robots can be applied to irregular environments but are relatively cumbersome and have poor flexibility [3]. The newly proposed spherical robot has a fully symmetrical
Int J Adv Robot Syst, 2016, 13:23 | doi: 10.5772/62178
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University of Aeronautics and Astronautics. 2011; 37(4):415-420. DOI: 10.13700/j.bh.1001-5965.2011. 04.010. [14] Macfarlane S, Croft E A. Jerk-bounded manipulator trajectory planning: design for real-time applica‐ tions. IEEE Transactions on Robotics and Automa‐ tion. 2003; 19(1): 42-52. DOI: 10.1109/TRA.2002. 807548. [15] Marsh R, Ogaard K. 12-TET Walker Using a Quadrupedal Walking Algorithm. In: Proceedings of the 2008 International Conference on Artificial Intelligence (ICAI 2008); 14-17 July 2008; Las Vegas, Nevada, USA. Georgia: CSREA Press; 2008. p. 680-685. [16] Huang B, Zhao J, Sun L. Straight Walking and Stair Climbing Gait of Quadruped Robot Based on Static Balance. Robot. 2010;32(2):226-232. DOI: 10.13973/ j.cnki.robot.2010.02.008. [17] Siciliano B, Khatib O, editors. Springer Handbook of Robotics. Berlin Heidelberg: Springer-Verlag; 2008. 1611 p. DOI: 10.1007/978-3-540-30301-5. [18] Goyal S, Pinson E N, Sinden F W. Simulation of dynamics of interacting rigid bodies including friction I: General problem and contact model. Engineering with computers. 1994;10(3):162-174. DOI: 10.1007/BF01198742. [19] Goyal S, Pinson E N, Sinden F W. Simulation of dynamics of interacting rigid bodies including friction II: Software system design and implemen‐ tation. Engineering with computers. 1994;10(3): 175-195. DOI: 10.1007/BF01198743. [20] Lee W H, Sanderson A. Dynamic rolling locomotion and control of modular robots. IEEE Transactions on Robotics and Automation. 2002;18(1):32-41. DOI: 10.1109/70.988972. [21] Lee W H, Sanderson A C. Dynamic rolling of modular robots. In: Proceedings of IEEE Interna‐ tional Conference on Robotics and Automation, Robotics and Automation (ICRA’00); 24-28 April 2000; San Francisco, CA. New York: IEEE; 2000. p. 2840-2846.
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