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13th World Congress in Mechanismand Machine Science, Guanajuato, México, 19-25 J une, 2011 IMD-123 1 Workspace of a 3-RRPS Parallel Robot Leg with a Constant Orientation C. K. Qi * X. C. Zhao † Z. L. J in ‡ Shanghai J iao Tong University Shanghai J iao Tong University Yanshan University Shanghai, China Shanghai, China Qinhuangdao, Chi
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  13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 1 Workspace of a 3-RRPS Parallel Robot Leg with a Constant Orientation C. K. Qi *  X. C. Zhao †  Z. L. Jin ‡  Shanghai Jiao Tong University Shanghai Jiao Tong University Yanshan University Shanghai, China Shanghai, China Qinhuangdao, China X. X. Zhang § C. X. Ma **  F. Gao ††  Shanghai University Shanghai Jiao Tong University Shanghai Jiao Tong University Shanghai, China Shanghai, China Shanghai, China Abstract  —  The leg design is very important for a multi-legged rescue robot which can carry heavy payload and walk in complex unstructured terrain. For parallel mechanism, the allowed payload is usually larger than that of serial mechanism. However, the workspace and dexterity of parallel mechanism (e.g., 6-limb SPS Stewart mechanism) may be reduced due to the limb interference.  In this study, to balance the payload and the workspace, the less-limb parallel mechanism 3-R RPS will be used for the leg design of a six-legged rescue robot. After the kinematic model is established, the workspace of the 3- R RPS parallel leg will be analyzed with a constant orientation. The actuated and passive joint constraints and the limb interference are considered in the analysis. This leg workspace analysis will be useful for gait  planning and motion control of the rescue robot. Keywords: 3-RRPS parallel mechanism, limb interference, multi-legged robot, rescue robot, workspace analysis I Introduction Rescue robots are fast developing fields in the robotics  because of the rescue demanding for the disasters in the world. However, rescue robots are still not mature to handle all tasks. Many theoretical and technical aspects need to solve or improve, e.g., better mechanical design,  better sensing system, better modeling, control and  planning, better system integration, more robustness and intelligence. The mechanical design is a first step for a rescue robot development. Considering various rescue requirements, we will focus on the rescue robot that can carry heavy payload and perform the rescue tasks in complex terrain   . Compared with the wheeled and tracked robots, the legged robots are more suitable for complex terrain though it could be more difficult to design and control. The allowed payload of legged robots is usually smaller than that of wheeled and tracked robots. To improve the  payload capability, more powerful actuators are often *   chenkqi@sjtu.edu.cn †   xczhao@sjtu.edu.cn ‡   zljin@ysu.edu.cn §  xianxia_zh@shu.edu.cn **  cxma@sjtu.edu.cn ††  fengg@sjtu.edu.cn required if the leg is designed with the serial mechanism. However, it is not always possible due to the limited  power of the actuator. Therefore, with the reasonable  power of the actuator, we will use the parallel mechanism for the leg design in order to carry the heavy payload. To achieve the flexible adaption to complex terrain, the leg is expected to have 6 degree-of-freedom (6-DOF). However, if the leg is designed with the parallel mechanism, its workspace may become small due to the mechanical constraints and the limb interference. For example, though a 6-SPS Stewart parallel mechanism has 6 DOF, it may result in a reduced workspace due to the 6-limb interference. There are some studies on the dexterity improvement using better mechanical design [1]. In this study, to balance the payload and workspace, we will use a less-limb parallel 6-DOF mechanism for the leg design. Various 6-DOF parallel mechanisms with 3 limbs have  been studied. For example, 3-PPSP [2][3], 3-PRPS [4-6], 3-U r  RS [7][8], 3-RRRS [9], 3-PPSR [6][10-12], 3-RRR [13] and 3-limb mechanism with five-bar linkages [14], where the number denotes the number of kinematic chains linking the moving platform to the base, and the set of letters defines the sequence of joints used in each kinematic chain, the letters P, R and S denote the  prismatic, revolute and spherical joints. Specially, there are a few studies on the 3-UPS mechanism. The direct kinematics of 3-UPS is solved in [15], and the singularity analysis is performed in [16]. The 3-UPS parallel machine tool is established and analysized in [17][18]. In [19-21], the U pair is a spatial five-bar 2-DOF mechanism, which is chosen as the actuation input. However, in [22][23], the U pair is decomposed into one active R joint and one passive R joint, the P pair is active and the S joint is passive. The inverse kinematics, operational capability and workspace analysis of this type of 3-UPS are also studied. Until now, there are few studies reported on the applications of parallel mechanism to the leg design of the multi-legged robot. Though some studies on the kinematics and workspace of the 3-UPS mechanism have  been conducted, there are still some problems due to the complexity of the 3-UPS leg for a practical multi-legged robot. Specially, the workspace analysis should consider the various constraints. To make it more clear, here the 3-  13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 2 UPS will be named as the 3-RRPS, where the actuated  joints are indicated by underlining. The workspace of various parallel robots has been studied [24]. Several kinds of workspaces can be defined, e.g., orientation workspace and positional workspace. With different orientation requirements, the positional workspace can be further classified into: constant orientation workspace, reachable workspace, inclusive workspace, total orientation workspace, and dextrous workspace [25]. Positional workspace with a constant orientation has been widely studied. For example, a geometrical method is proposed to determine the  positional workspace of 6-SPS and 6-RTS mechanisms [26]. However, the limb interference is not considered. There are also some studies on the orientation workspace [27-30] and the 6-dimensional positional/orientation workspace [31]. For simplicity, we only study the workspace for a constant orientation, but the similar  procedure can also be applicable for the case of varying orientation. In the workspace analysis, there are mainly three types of kinematic constraints, i.e., the actuator constraint, the  passive joint limitations, and the link interference conditions [27][32][33]. The actuator constraints and  passive joint limitations can be easily handled while the link interference constraints are complicated. The interference of two limbs can be identified by comparing the nearest distance of two limbs and the diameter of the limbs [33]. For example, based on this idea, a geometrical method is proposed to identify the limbs interference of 6-SPS Stewart platform [34]. The numerical method is used to investigate the workspace of 6-DOF 3-PPSR parallel manipulator [12] by considering the limbs interference, revolute and spherical joint limitations. The leg length limits, angle limits, leg interference are considered in [35]. Guo, Shan, Chen & Chen [36] studied the workspace of 3/6-SPS Stewart  platform. The interference checking over a given workspace or trajectory is studied in [37]. In this study, the constant orientation workspace will be obtained by considering all these three constraints. The workspace can be determined by at least four methods: geometrical method, continuation method, discretization method and optimization method [38]. In this study, we will follow the discretization method  because it is an easy and stable method, though the computational expense may increase with the higher accuracy required. Using a numerical search algorithm, we can obtain the reachable workspace of the 3-R RPS  parallel mechanism for a given orientation, from which the movement capability of foots of the rescue robot can  be obtained. The results of this paper will lay the foundation for the successful applications of the rescue robot. Especially it will play an important role for the gait  planning and control of the rescue robot. The rest of the paper is organized as follows. The 3-RRPS parallel leg is introduced in Section 2. In Section 3, the kinematics and workspace analysis are presented. Section 4 contains numerical simulations. Finally, a few conclusions are presented in Section 5. II. The 3-RRPS parallel leg The parallel mechanism is structurally more rigid than the serial mechanism, so it is expected to take more  payload. Currently, we are developing a six-legged rescue robot whose leg consists of a 6-DOF 3-R RPS parallel mechanism as shown in Fig. 1. The underlined “R” and “P” joints are actuated joints and others are passive joints. With the limited power of actuators, this rescue robot can still carry heavy payload due to this kind of leg design. Foot Passive R joint  Actuated R joint Passive S joint  Actuated P joint BaseLimb 1 Limb 2 Limb 3 Fig. 1. Prototype of the 3-RRPS parallel leg III. Workspace analysis of the 3-RRPS parallel leg  A. Coordinate system P    L 1    L 2    L 3  A 1    A 2    A 3  B 1    B 2  X Y     Z     X’   Y’    Z’   OF    G 3  B 3 G 1   G 2   P S R R Fig. 2. Coordinate system of the 3-RRPS parallel leg As shown in Fig. 2, the base and moving platform of the 3-RRPS parallel mechanism are connected by three RRPS chains. Each chain is connected to the base  13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 3 through a U pair. This U pair is further decomposed into an actuated R joint and a passive R joint. The motion centers 1  B , 2  B  and 3  B  of three U pairs form an equilateral triangle 123  BBB   ( 122331  BBBBBB   ). On the other hand, each chain is also connected to the moving platform through an S joint. However, the motion centers 1  A , 2  A  and 3  A  of three S joints is designed to form an isosceles triangle 123  AAA   ( 121323  AAAAAA   ). Obviously, these two triangles are not similar. The universe coordinate system OX   is defined on the base, where the srcin O  is located at the center of the equilateral triangle YZ   123  BBB  . The axis OY    is along with the direction . The axis is perpendicular to the base plane and the upward direction is positive. The axis OX    is then determined by the right-hand screw rule. The coordinate system of the moving platform is defined as  1 OB  OZ   ''' PXY    Z  , however due to a special design the srcin P  is not located at the center of the isosceles triangle 123  AAA ''  . Similarly, the axis PY    is along with the direction . The axis is perpendicular to the plane of the moving platform and the upward direction is  positive. The axis is then determined by the right-hand screw rule. When the leg is in the initial states, the axis directions of the coordinate systems OX   and ' '  1 PA  '  PZ   ' PX   YZ   PXYZ    are same (see Fig. 2).  B. Inverse kinematics The distance between the S joint and the center of U  pair is also the length of the P pair, which is denoted as 1  L , 2  L  and 3  L  respectively. The coordinates of the vertices 1  A , 2  A  and 3  A  of the triangle 123  AAA   with respect to the moving coordinate system are denoted as 1 P  A , 2 P  A  and 3 P  A . Similarly, the coordinates of the vertices 1  B , 2  B  and 3  B  of the triangle 123  BBB   with respect to the base coordinate system are denoted as 1 O  B , 2 O  B  and 3 O  B . The coordinate of the srcin of the moving  platform with respect to the base coordinate system is expressed as . O P Given the position O F   and the orientation   ,     nd a     -Y-X Euler angles) of the foot (Z F   with respect to the coordinate system, the problem of inverse kinematics is to obtain all actuated inputs: the length of three P OXYZ    pairs 1  L , 2  L , 3  L  and the angle of three active R  joints 1   , 2   , 3   . First, using the Z-Y-X Euler angles the rotation matrix of the moving frame relative with respect to the base frame can be expressed by ccccc O p ssscscssTssssccssccsscscc                                        Then, the coordinates of P  with respect to the OXYZ    can be obtained as OOOP p PFTF     , (2) and the coordinates of the vertices 1  A , 2  A , 3  A  with respect to the can be calculated by OXYZ     1 OOOP p 1  APTA    , (3) 2 OOOP p 2  APTA    , (4) 3 OOOP p 3  APTA    . (5) For each chain, the distance between the S joint and the U pair can give the length of the P pair, which can be determined by 11111 |||||||| OO  LBAAB     , (6) 22222 |||||||| OO  LBAAB     , (7) 33333 |||||||| OO  LBAAB     . (8) If the position and the orientation of the moving frame are given, the length of the P pair can be calculated by (6)-(8). On the other hand, we can calculate the direction vector ii  BA   of the P pair with respect to the OXYZ    coordinate system OOiiii  BAAB    . (9) Considering the relationship between the P pair and the U    pair, the angle between direction vectors ii  BA   and i  BO   will give the passive angle i    of the R joint in the U pair. This angle can be easily calculated using the triangle functions which are ingored here. To calculate the angle i    of the actuated R joint in the U    pair, the normal direction vector ii  BG   of the plane will be used ii OBA   iiiii  BGOBBA      , (10) where the “  ” is the cross product. Suppose (,,) iiiii  BGiljmkn     , then the angle i    between ii  BG   and the base plane 123  BBB  can be obtained by 222 arcsin iiiii nlmn      . (11) The equations (6)-(8) and (11) are the solutions of inverse kinematics of the 3-RRPS parallel leg. C. Kinematic constraints C.1 Actuated R joint and P pair constraints  For each leg, the length of three actuated P pairs should satisfy   minmax iii  LLL   , , 1,2,3 i   . (1) where min i  L  and max i  L  are the minimal and maximal length of the i th -link respectively. The angle of actuated R joints in three U    pairs (see Fig. 3) should satisfy minmax iii       , , 1,2,3 i    13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 4 where and  max  are the minimal and maximal angles of the actuated R  min i     i    joints respectively. C.2 Passive R joint constraints  As shown in Fig. 3, the passive R joint angle in the U    pair is the angle i    between ii  BA   and i  BO  . The constraints of three passive R joints can be expressed by minmax iii       , , 1,2,3 i   where min i    and are the minimal and maximal values respectively. max i    L 1    L 2    L 3    A 1    A 2    A 3    B 1    B 2   O   G 3    B 3   G 1 G 2   1     2   3     3   2     1     Fig. 3. Actuated R joint and passive R joint constraints C.3 Passive S joint constraints  For easy implementation, the passive S joint is designed as a serial RRR mechanism (see Fig. 4a). The middle R joint angle i   , i.e., the angle between lines ii  AB   and (see Fig. 4b), will have the following constraints: 1 PA  minmax iii       , , 1,2,3 i   where and are the minimal and maximal values respectively. min i   max i     R R R (a) P    L 1    L 2    L 3  A 1    A 2    A 3  B 1    B 3   2   1     3        B 2   (b) Fig. 4. Passive S joint constraints   C.4 Link interference constraints . As shown in Fig. 5, each link consists of two cylinders and ii  EA ii  BE  , and their diameters are denoted by and i d  i  D  respectively ( i dD i  ). The shortest distance between the links i  L  and  j  L  over two sublinks and ii  EA  jj  EA  are . Similarly, the shortest distance over another two sublinks ij dis ii  BE   and  jj  BE   are ij  DIS  . Then the conditions to avoid the link interference can be expressed by ()/2 iji dddis    j j , ()/2 iji  DDDI    S  . The shortest distance between the center-lines of two sublinks and ij dis ii  EA  jj  EA  can be calculated as below    the length of the common perpendicular ij n  , if their common perpendicular has intersection points and i C   j C   within two links ii  EA  and  jj  EA  respectively (see Fig. 5 for an example).    the shortest distance between the endpoint i  A  of the link ii  EA  and the link  jj  EA , if only the intersection  point i C   of the link ii  EA  and the common  perpendicular of the two links is beyond the link ii  EA .    the distance between the two endpoints i  A  and  j  A , if two intersection points i C   and  j C   are both  beyond the links ii  EA  and  jj  EA . The shortest distance ij  DIS   between the center-lines of two sublinks ii  BE   and  jj  BE   can also be obtained in a same way.
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