types of robot modules, the fixed-dimension joint modules and the variable dimension link modules that ... planning, and application development of the parallel.
Autonomous Robots 10, 83–89, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. °
Kinematic Design of Modular Reconfigurable In-Parallel Robots GUILIN YANG Automation Technology Division, Gintic Institute of Manufacturing Technology, Singapore 638075 I-MING CHEN, WEE KIAT LIM AND SONG HUAT YEO School of Mechanical & Production Engineering, Nanyang Technological University, Singapore 639798
Abstract. This paper describes the kinematic design issues of a modular reconfigurable parallel robot. Two types of robot modules, the fixed-dimension joint modules and the variable dimension link modules that can be custom-designed rapidly, are used to facilitate the complex design effort. Module selection and robot configuration enumeration are discussed. The kinematic analysis of modular parallel robots is based on a local frame representation of the Product-Of-Exponentials (POE) formula. Forward displacement analysis algorithms and a workspace visualization scheme are presented for a class of three-legged modular parallel robots. Two three-legged reconfigurable parallel robot configurations are actually built according to the proposed design procedure. Keywords: 1.
kinematics, reconfigurable, parallel robot, modular design
Introduction
Parallel type robots possess high force loading capacity and fine motion characteristics because of the closed-loop mechanism. Recently, researchers try to utilize these advantages to develop novel multi-axis machining tools (Ryu, 1998) and precision assembly tools (Badano, 1994). However, the design, trajectory planning, and application development of the parallel robot are difficult and tedious because the closed-loop mechanism creates complex kinematics. To overcome this drawback, modular design concept is introduced in the development of parallel robots. A modular parallel robot system consists of a set of standardized modules, such as actuators, passive joints, rigid links (connectors), mobile platforms, and end-effectors, that can be rapidly assembled into a complete robot with various configurations. The concept of modularity has been used in the design of serial-type industrial robots for flexibility, ease of maintenance, and rapid deployment (Chen, 1994; Chen 1996; Cohen et al., 1992; Paredis, 1988). From our experience, a modularly designed reconfigurable parallel robot not only possesses the above advantages but also can shorten its development cycle time.
The focus of this paper is on the kinematic design issues of a class of three-legged modular inparallel robots, namely, configuration selection, kinematic analysis and workspace visualization. Because a modular robot may have unlimited assembly configurations, the selection of suitable module assembly configuration is important. In the forward kinematics aspect, no general closed-form solution algorithm has been developed for parallel robots except for certain specific robot configurations (Innocenti and ParentiCastelli, 1990; Lee and Roth, 1993; Lin et al., 1990). Most of the closed-form solution algorithms follow an algebraic analytical approach, and involve solving high order polynomial equations, which requires extensive computation effort. The sensor-based algorithm and the iterative numerical algorithm hence become feasible and practical. Previous works on such algorithms usually follow a pure geometrical approach (Notash and Podhorodeski, 1994; Cleary, 1993). Based on the local Product-Of-Exponential (POE) formula, a sensorbased solution algorithm and an iterative numerical algorithm are re-formulated here for modular parallel robots. It is shown that the local POE formula is a systematic and well-structured method for the kinematic analysis of parallel robots. Based on the POE formula,
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a general and simple visualization scheme is developed to generate the workspace of the mobile platform for design purpose. 2. 2.1.
Robot Modules and Configuration Design Robot Modules
The closed-loop structure of parallel robots imposes kinematic constraints on the dimensions of the robot sub-assembly, and makes the construction of a useful parallel robot configuration from entirely standard modules challenging. To overcome this, we propose to use two types of sub-assemblies to modularize a parallel robot, namely, standard fixed-dimension modules and variable-dimension modules that can be rapidly fabricated. • Fixed-Dimension Modules: The standard fixed-dimension modules include actuator modules, passivejoint modules and end-effector modules. For the sake of modularity, the actuator modules and endeffector modules must be compact self-contained intelligent mechatronic drive units. Each drive unit contains a built-in motor, a controller, an amplifier, and the communication interface. The inter-module communication and power transmission are through inter-connection cables and interfaces. Here, we utilize a series of intelligent mechatronic drives and end-effector units for rapid development. Both revolute and prismatic (Fig. 1) actuator modules are deployed. The connection between modules is through mechanical means. The inter-module communication is through CAN bus protocol and RS-485 interface. Three types of passive-joint modules (without actuators) are in-house designed and fabricated: the rotary joint, the pivot joint, and the spherical joint
(Fig. 1). The form factors of the passive joints conform with the actuator modules. Angular displacement sensors are built into the passive rotary and pivot joint modules for forward displacement sensing of the parallel robots. • Variable-Dimension Modules: Rigid links, module connectors, and the mobile platform are the modules that can be designed with customized dimensions. These modules usually have simple designs and can be rapidly fabricated based on functional requirements. Allowing for dimensional change in module design provides the end-users the ability to rapidly fine-tune the kinematic and dynamic performance of the completed robot, especially, the size and geometry of the workspace and the dexterity of the end-effector. A set of links with various geometrical shapes and dimensions and a circular mobile platform have been designed and fabricated. 2.2.
Robot Configuration Determination
In this project, we intend to develop a non-redundant 6-DOF in-parallel modular robot which can perform light machining tasks that require both rigidity and dexterity. An enumeration schemes for a class of parallel robots with 6-DOF, non-redundant topological structure is summarized in Table 1 (Podhorodeski, 1994). Ideally, one can use the module set to construct a modular parallel robot with any number of legs between two to six as shown in Table 1. However, we choose the Table 1. Enumeration of a 6-DOF non-redundant parallel robot (adapted from Podhorodeski et al., 1994). Number of legs in-parallel
Number of actuated single-degree of freedom joints in each leg
1
6
2
1, 5 2, 4 3, 3
3
3, 3, 0 1, 1, 4 1, 2, 3 2, 2, 2
4
1, 1, 1, 3 1, 1, 2, 2
Figure 1.
Actuator and passive joint modules.
5
1, 1, 1, 1, 2
6
1, 1, 1, 1, 1, 1
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three-legged robot geometry as the norm of modular parallel robot design because i. The number of legs in the parallel robot affects the occurrence of leg interference. With less number of legs in the robot, there is less chance of leg interference. ii. Leg symmetry is an advantage for parallel robots when uniform force distribution among the supporting legs and robot configuration control are considered. Considering weight reduction and minimization of the number of active and passive modules, the class of three-legged parallel robots will have two actuator joints (prismatic and/or rotary), one passive revolute joint and one passive spherical joint for each leg. 2.3.
Unique Parallel Robot Leg Structure
Based on the three-legged (2-2-2) robot geometry, we are able to enumerate all of its unique parallel robot configurations. When unnecessary offsets of the leg links are eliminated and the successive joint directions are parallel or at right angle, construction and kinematic modeling of the legs will be simplified. This is in accordance to our hardware design as the actuator modules are cubic in shape. In addition, to avoid joint redundancy and to maintain 6-DOF motion capability, successive joints cannot be parallel unless separated by a length. First, we focus on the structures where all the joints are revolute. Every leg structure will comprise of three revolute joints and a passive spherical joint at the leg end. In order to have 6-DOF leg end motion, the three revolute joint axes must not intersect at a common point. Avoiding unnecessary offsets, the three joints are dispersed amongst a fixed shoulder and an elbow (Fig. 2), i.e., there are either two joints at the shoulder
Figure 2.
Directions of leg joints.
Figure 3. joints.
Five unique leg structures—containing pure revolute
and one at the elbow or vice versa. To maintain 6-DOF motion, the joint axes should avoid certain positions. Considering the permutations of the possible joint distributions and directions yields a total of 36 potential leg layouts for the class described. After eliminating the equivalent structure, we are left with five unique leg structures (Fig. 3) (Podhorodeski, 1994). Theoretically, any revolute joint can be replaced with a prismatic joint. Due to the weight and actual construction, prismatic joint modules are mounted close to the base vertically and only one prismatic joint should be used in a leg. Based on this fact, we can generate the possible unique leg structures comprising of one prismatic joint and two revolute joints as shown in Fig. 4. In total, we will have 13 feasible unique leg structures for the (2-2-2) parallel robot with symmetric structure. Figure 5 shows two distinct 6-DOF threelegged (2-2-2) parallel robot configurations that have been constructed in our laboratory.
Figure 4. joint.
Eight unique leg structures—containing one prismatic
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Yang et al. • Local POE Formula for Open Chains: Based on the Dyad kinematics, Eq. (1) the forward kinematics for an open chain can be easily derived. Consider an open kinematic chain with n + 1 links, sequentially 0, 1, . . . , n (from the base 0 to the end link n). The forward kinematic transformation can be given by: T0,n (q1 , q2 , . . . , qn ) = T0,1 (q1 )T1,2 (q2 ) . . . T(n−1),n (qn ) = T0,1 (0) esˆ1 q1 T1,2 (0) esˆ2 q2 . . . T(n−1),n (0) esˆn qn (2) 4.
Figure 5. Two assembled three-legged modular reconfigurable parallel robots.
3.
Local POE Formula
Brockett (1983) showed that forward kinematics of a robot containing either revolute or prismatic joints can be uniformly expressed as a product of matrix exponentials. According to the coordinate frames used in expressing the joint axes, the POE formula can be written in different forms (Chen, 1996; Murray et al., 1994; Park, 1994). To serve our purpose here, only the local frame representation of the POE formula is introduced. • Dyad Kinematics: Let link i − 1 and link i be two adjacent links connected by joint i, as shown in Fig. 6. Link i and joint i are termed as link assembly i. Denote the body coordinate frame on link assembly i by frame i, then the relative pose (position and orientation) of frame i with respect to frame i − 1, under a joint displacement, qi , can be described by a 4 × 4 homogeneous matrix, an element of SE(3), such that Ti−1,i (qi ) = Ti−1,i (0) esˆi qi ,
Forward Displacement Analysis
The schematic diagram of the class of three-legged (22-2) parallel robot configurations is shown in Fig. 7. We assume that joint i j (ˆsi j ) are actuating joints (i = 1, 2, 3; j = 1, 2), and joint i3 (ˆsi3 ) are passive revolute joints (i = 1, 2, 3). Define frame A as the local frame attached to the mobile platform and frame B as the base frame. The coordinate of point Ai (i = 1, 2, 3), centers of the spherical joints, relative 0 0 0 T to frame A and frame Bi3 are pi0 = (xai , yai , z ai ) 00 00 00 00 T and pi = (xai , yai , z ai ) respectively. The forward displacement analysis is to determine the pose of frame A relative to the base frame B with given joint angles, qij (i = 1, 2, 3; j = 1, 2). 4.1.
Sensor-Based Solution Approach
The sensor-based forward kinematics utilizes displacements measured by the angular encoders in the passive joint modules to compute the forward displacement of the mobile platform. As shown in Fig. 7, the position
(1)
where sˆi ∈ se(3) is the twist of joint i expressed in frame i, and Ti−1,i (0) ∈ SE(3) is the initial pose of frame i relative to frame i − 1.
Figure 6.
Two consecutive links: a dyad.
Figure 7.
A three-legged (2-2-2) parallel robot.
Kinematic Design of Modular Reconfigurable In-Parallel Robots of point Ai (i = 1, 2, 3) can be directly determined as a function of the actuating joint displacements and the passive joint displacement in leg i. Treating each leg as an independent unit, this forward kinematics computation is identical to the forward kinematics of a serial robot. The positional vector, pi , of point Ai , can be given by " # pi = TB,Bi0 TBi0 ,Bi1 (0) esˆi1 qi1 TBi1 ,Bi2 (0) esˆi2 qi2 1 " # pi00 sˆi3 qi3 × TBi2 ,Bi3 (0) e , (3) 1 where TB,Bi0 is the fixed kinematic transformation from frame B (the base platform frame) to frame Bi0 . Let the forward kinematic transformation from the base frame B to the mobile platform frame A be TB,A . Then " # " # pi pi0 TB,A = (i = 1, 2, 3). (4) 1 1 We also have TB,A
"
0 0 × p23 p12 0
#
" =
# p12 × p23 , 0
p0 p0 p0 p0 × p0
Note that [ 11 12 13 12 0 23 ] is a constant matrix for a specific robot configuration, and is always invertible if points A1 , A2 , and A3 neither coincide with each other nor fall on the same line. 4.2.
Numerical Solution Approach
The sensor-based algorithm can only be implemented on the actual parallel robot. In situations where the passive joint displacements cannot be acquired, e.g., in off-line computations and simulations, the numerical method is useful. The passive joint angles will be determined through numerical iteration. For simplicity, Eq. (3) can also be written as Pi = Ti esˆi3 qi3 Pi00
(i = 1, 2, 3),
00
where Pi = [ p1i ], P00i = [ p1i ], and Ti = TB,Bi0 TBi0 ,Bi1 (0) esˆi1 qi1 TBi1 ,Bi2 (0) esˆi2 qi2 TBi2 ,Bi3 (0). Since −−→ k A1 A2 k2 = (P2 − P1 )T (P2 − P1 ),
(7)
(8)
The differential form of Eq. (8) can be written as (P2 − P1 )T −−→ dk A1 A2 k = −−→ (d P2 − d P1 ). k A1 A2 k
(9)
According to Eq. (7), we find d Pi = Ti esˆi3 qi3 sˆi3 Pi00 dqi3
(i = 1, 2, 3).
(10)
The actual distance from A1 to A2 is a constant, denoted by a12 . Then, (P2 − P1 )T −−→ a12 − k A1 A2 k = −−→ (d P2 − d P1 ) (11) k A1 A2 k We can derive similar relationships for the other two legs. Combining all three equations for the legs, we get
(5)
0 0 = p20 − p10 , p23 = p30 − p20 , p12 = p2 − p1 , where, p12 and p23 = p3 − p2 . Combining Eqs. (4) and (5), the pose of the mobile platform, TB,A , then can be found by # " p1 p2 p3 p12 × p23 TB,A = 1 1 1 0 #−1 " 0 0 × p23 p10 p20 p30 p12 . (6) × 1 1 1 0
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dq = J −1 da,
(12)
where dq = column[dq13 , dq23 , dq33 ], da = column −−→ −−→ −−→ [a12 − k A1 A2 k, a23 − k A2 A3 k, a31 − k A3 A1 k], J =
2 −P1 ) − (P− →
T
k A1 A2 k
P˙1
0 (P1 −P3 )T ˙ −→ P1 k A3 A1 k
(P2 −P1 )T ˙ −→ P2 k A1 A2 k T 3 −P2 ) ˙ − (P− → P2 k A2 A3 k
0
0
(P3 −P2 )T ˙ −→ P3 , k A2 A3 k T 1 −P3 ) P˙3 − (P− → k A3 A1 k
and P˙i = Ti esˆi3 qi3 sˆi3 Pi (i = 1, 2, 3). Equation (12) can be written in an iterative form, i.e., 00
q (k+1) = q (k) + (J −1 da)(k) ,
(13)
where k represents the number of iterations. Based on Eq. (13), Newton-Raphson method is employed to derive the numerical solution of q. The inverse of the Jacobian matrix J can be found in a symbolic form. Although the proposed algorithm is computationally equivalent to the method given by Cleary and Brooks (Clearly, 1993), it has a simple and explicit Jacobian matrix because of the use of local POE formula. After the joint displacement q is derived, the pose of the mobile platform can be easily determined by using the sensor-based algorithm.
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Workspace Visualization
The complete workspace of a 6-DOF parallel robot is a 6-D manifold which cannot be represented in a 3-D space. Hence, the workspace mentioned here is the reachable workspace, i.e., the region of the 3-D Cartesian space that can be reached by the mobile platform with a given orientation of the platform (Gosselin, 1990). Since the pose, TB,A , of the mobile platform p B,A frame A, has the form of [ R B,A ], Eq. (4) can be 0 1 written as: p B,A = pi − R B,A pi0
(i = 1, 2, 3).
generated through 3-D graphical functions provided by Maple. The orientation of the mobile platform is specified as a 3 × 3 identity matrix. The workspace of Leg 1 is shown in Fig. 8(a). The 3-D workspace (the intersection volume) of the platform generated by all three legs is shown in Fig. 8(b). Cross-sectional views of the workspaces along the x-y plane reveal the internal structure of the workspace. From the cross-sectional views we can identify the reachable regions in the composite workspace based on the plotting density. 6.
Summary
(14)
Equation (14) is an alternative expression of leg i’s inverse kinematic equation, where each leg is considered as an independent serial chain. For a given orientation, R B,A , the platform position, p B,A , can be derived by translating point Ai through a fixed vector (−R B,A pi0 ). In other words, the reachable workspace of the mobile platform determined by leg i is a fixed translation of point Ai ’s workspace. Since the motion of the three-legged parallel robot is the composite movement of all three legs, the position of the mobile platform must satisfy the three inverse kinematic equations, Eq. (14) simultaneously. The reachable workspace of the mobile platform is the intersection of the individual workspace of each leg. Substituting Eq. (3) into Eq. (14) for each leg, the individual workspace can be constructed by varying the three joint displacements within their joint limitations. Taking the intersection of the three individual workspaces, the actual reachable workspace of the mobile platform can be obtained. The workspace of the three-legged (2-2-2) robot in Fig. 5(b) is illustrated in Fig. 8. This workspace is
We have demonstrated the use of different types of modules, for the development of reconfigurable parallel robots. The clear separation of the topological design and the dimensional design can significantly shorten the development cycle of a reconfigurable parallel robot and hence reduce the complexity of the overall design to a manageable level. Based on the local POE formula, the sensor-based and the iterative numerical forward kinematics algorithms are developed for a class of three-legged modular in-parallel robots. With the POE formula, these two algorithms can uniformly deal with various three-legged robot configurations regardless of the joint types and the degrees of freedom. A simple and general workspace visualization scheme is also presented for the workspace analysis. Two three-legged (2-2-2) modular parallel robots are actually constructed. As indicated, the standard active and passive modules can be re-used in different robot configurations. Although some commercial type modular actuators are employed, the basic kinematic design methodology is generally applicable to all modularly constructed parallel robots. Acknowledgments This work is supported by Gintic Institute of Manufacturing Technology, Singapore, under Upstream project U-97-A006. References
Figure 8.
Workspace of the parallel robot platform.
Badano, F. 1994. Evaluation of exploration strategies in robotic assembly. In Proceedings of 4th IFAC Symposium on Robot Control, pp. 63–68. Brockett, R. 1983. Robotic manipulators and the product of exponential formula. In International Symposium in Math. Theory of Network and Systems, Beer Sheba, Israel, pp. 120–129. Chen, I.M. 1994. Theory and applications of modular reconfigurable robotic systems. Ph.D. thesis, California Institute of Technology, CA, USA.
Kinematic Design of Modular Reconfigurable In-Parallel Robots
Chen, I.M. and Yang, G. 1996. Configuration independent kinematics for modular robots. In Proceedings of IEEE Conference on Robotics and Automation, pp. 1440–1445. Cleary, K. and Brooks, T. 1993. Kinematic analysis of a novel 6-DOF parallel manipulator. In Proceedings of IEEE Conference on Robotics and Automation, pp. 708–713. Cohen, R., Lipton, M.G., Dai, M.Q., and Benhabib, B. 1992. Conceptual design of a modular robot. ASME Journal of Mechanical Design, 114:117–125. Gosselin, C.M. 1990. Determination of the workspace of 6-DOF parallel manipulators. ASME Journal of Mechanical Design, 112:331–336. Innocenti, C. and Parenti-Castelli, V. 1990. Direct position analysis of the Stewart platform mechanism. Journal of Mechanism and Machine Theory, 25(6):611–621. Lee, H.Y. and Roth, B. 1993. A closed-form solution of the forward displacement analysis of a class of in-parallel mechanisms. In Proceedings of IEEE Conference on Robotics and Automation, pp. 720–724. Lin, W., Duffy, J., and Griffis, M. 1990. Forward displacement analysis the 4-4 Stewart platforms. In Proceedings of ASME 21st Biennial Mechanism Conference, De-25, pp. 263–269. Merlet, J.P. 1990. Trajactory verification in the workspace for parallel manipulators. International Journal of Robotic Research, 13(4):326–333. Murray, R., Li, Z., and Sastry, S.S. 1994. A Mathematical Introduction to Robotic Manipulation, CRC Press: Boca Raton. Notash, L. and Podhorodeski, R.P. 1994. Complete forward displacement solutions for a class of three-branch parallel manipulators. Journal of Robotic System, 11(6):471–485. Paredis, C.J.J., Brown, H.B., and Khosla, P.K. 1997. A rapidly deployable manipulator system. Journal of Robotics and Autonomous Systems, 21:289–304. Park, F.C. 1994. Computational aspect of manipulators via product of exponential formula for robot kinematics. IEEE Transactions on Automatic Control, 39(9):643–647. Podhorodeski, R.P. and Pittens, K.H. 1994. A class of parallel manipulators based on kinematically simple branches. ASME Journal of Mechanical Design, 116:908–914. Ryu, S.J., Kim, J., Park, F.C., and Kim, J. 1998. Design and performance analysis of a parallel mechanism-based universal machining center. In Proceedings of 3rd International Conference on Advanced Mechatronics, Okayama, Japan, pp. 359–363.
Guilin Yang received the B.Eng. degree and M.Eng. degree from Jilin University of Technology, China, in 1985 and 1988 respectively, and Ph.D. degree from Nanyang Technological University in 1999. Currently he is with Gintic Institute of Manufacturing Technology, Singapore, as a Research Fellow. His research interests include mechanism and machine design, kinematics and dynamics of mechanical
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systems, application of modular reconfigurable robot system, and robot calibration. He is a member of ASME.
I-Ming Chen received the B.S. degree from National Taiwan University in 1986, and M.S. and Ph.D. degrees from California Institute of Technology, Pasadena, CA in 1989 and 1994 respectively. He has been with the School of Mechanical and Production Engineering of Nanyang Technological University in Singapore as Assistant Professor since 1995. He was JSPS Visiting Scholar in Kyoto University, Japan in 1999. His research interests are reconfigurable robotic manufacturing workcells, sensor-based mechanical system simulations, medical applications of robotic systems, the locomotion of biologically inspired robotics systems, and smart material based actuators. He is a member of IEEE and ASME.
Wee Kiat Lim received his Bachelor and Master of Engineering degrees in Mechanical Engineering from Nanyang Technological University in 1998 and 2000 respectively. He is now working in Defence Science Organization of Singapore as Research Engineer. His research interest is in robotics and automation.
Song Huat Yeo received both his B.Sc. and Ph.D. degrees from the University of Birmingham (UK) in 1983 and 1987 respectively, all in mechanical engineering. He continued to work in UK first as a Research Fellow & then as a Lecturer. Since 1992, he has been with the School of Mechanical and Production Engineering of Nanyang Technological University in Singapore. Currently he is an Associate Professor in the School. His current research interests include modular robotic systems, robot gripper design, smart material based actuators and mechanisms design.
