DETC2000/MECH KINEMATIC SYNTHESIS OF BINARY ACTUATED MECHANISMS FOR RIGID BODY GUIDANCE

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1 Proceedings of DETC ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference Baltimore, Maryland, September -3, DETC/MECH-7 KINEMATIC SYNTHESIS OF BINARY ACTUATED MECHANISMS FOR RIGID BODY GUIDANCE Michael S. Hanchak Graduate Assistant Dept. of Mechanical and Aerospace Engineering University of Dayton Dayton, Ohio 9- Andrew P. Murray Assistant Professor Dept. of Mechanical and Aerospace Engineering University of Dayton Dayton, Ohio 9- ABSTRACT This paper presents a method for designing mechanisms composed of Revolute-Binary state prismatic-revolute (RBR) chains for rigid body guidance. Where a prismatic joint allows for any distance between two revolute joints, a binary state prismatic joint reaches two distances precisely. A single RBR chain can be designed to reach six positions. A parallel arrangement of three RBR chains can be assembled at the six positions but, in general, is not a viable kinematic solution. By requiring the arrangement of three RBR chains to share specific fixed and moving pivots, called an N-type arrangement, four positions are reachable. Further, the design space is quickly searchable for singularity-free solutions. Examples illustrate a solution to a four position synthesis problem and a ten position problem using a serial assembly of these mechanisms. INTRODUCTION A binary state prismatic joint (or B-joint), placed between a fixed and moving pivot, produces two precise lengths for the RBR chain. One goal of using binary actuation is to limit control requirements; the B-joint is simply expected to drive between the two lengths. As a result, a device driven by a set of binary actuators has a finite number of fixed states. Prior to the work of Chirikjian (99) for truss like manipulators and Waldron and Yang (998) for large parallel arrays, little can be found on the use of binary actuation (Anderson and Horn, 97; Pieper, 98; Roth et al. 973). A B-joint actuated mechanism, used for rigid body guidance, can move the body to a finite number of positions including orientations. In this paper, we present an algorithm for determining the revolute joint locations and two required lengths for the RBR chain. Additionally, we consider couplings of these chains to define a viable mechanism. Although we deal with low numbers of actuated joints, the kinematic synthesis with large numbers of actuators to achieve large sets of positions without regard to orientation is presented in Chirikjian (99). The design of mechanisms composed of RBR chains is similar to the design of fourbar mechanisms via classical Burmester Theory. Specifying up to five positions, Burmester Theory determines the fixed and moving pivot locations of a set of twodegree-of-freedom RR chains capable of guiding a body through the positions. Solving the simultaneous equations generated by writing the constant length condition (or crank constraint) for the unknown joint locations in each of the specified positions calculates these pivot locations. Any one-degree-of-freedom mechanism defined by coupling two of these chains can be assembled at the specified positions. The mechanism, however, cannot necessarily move between the positions by actuating a single joint requiring a separate discussion, typically called solution rectification, on the coupling of chains into mechanisms. Consult most machine theory texts for discussions of Burmester Theory and solution rectification (for example Erdman and Sandor, 997; Waldron and Kinzel, 999). The design of a mechanism defined by a grouping of RBR Copyright by ASME

2 chains proceeds analogously to design of a fourbar mechanism. Specifying up to six positions, we determine the pivot locations of a set of RBR chains by solving the simultaneous equations generated by writing the constant length condition for the unknown pivot locations in each of the specified positions. The constant length in this case can be either of two values due to the two possible states of the B-joint. Any mechanism defined by grouping three of these chains can be assembled at the specified positions. The mechanism, however, cannot necessarily move between the positions by actuating the three B-joints requiring a separate discussion on the grouping of chains into mechanisms. In addition to the design of an individual RBR chain to guide a body through six positions, three other design problems are considered. First, the grouping of three of these chains into a single mechanism capable of reaching the six positions is presented. Due to the difficulties in producing viable designs from the first technique, a second method details the design of platforms with an N-type (or truss-like) arrangement of the chains to achieve four positions. A common moving pivot location between the first and second chains and a common fixed pivot location between the second and third chains defines this N-type arrangement. The third design problem extends the second to synthesis for ten positions by serially connecting three of the N-type arrangements. All of the design routines mentioned are implemented and verified with Matlab software available upon request from the authors. THE BINARY CRANK CONSTRAINT Let A i and d i be the rotation matrix and displacement vector, respectively, of the moving frame M i relative to the fixed frame F. With G being the coordinates of a fixed pivot in F, and g i being the coordinates of the the pivot in M i, Figure. An RBR chain with fixed and moving pivots defined relative to the fixed frame, F, and the moving frame, M i. lengths, l or l. If we let positions through j be reachable at length l of the RBR chain, then by subtracting equation (3) evaluated at the first position from equation (3) evaluated at positions through j, Alternately, Z i Z G g i g z Z i Z i Z Z i j () g i g i g g i j () in moving frame coordinates. If we let positions j through n be reachable at length l of the RBR chain, then by subtracting equation (3) evaluated at position j from equation (3) evaluated at positions j through n, G A i g i d i () Z i Z j G Z i Z i Z j Z j!" # i " j $ #%%%# n% () See Figure. Likewise, the location of a moving pivot z of an RBR chain in M i is related to its location in F by Z i A i z d i () Alternately, g i g j! z & g i ' g i ( g j) ' g j) *,+ - i + j. -///- n (7) Given that the distance between G and Z i is l i, Z i G T Z i G l i (3) If we constrain all locations of M i to have equal values of l i, equation (3) is called the crank constraint. For an RBR chain, however, the distance between the two pivots can be one of two in moving frame coordinates. KINEMATIC SYNTHESIS OF AN RBR CHAIN We now present the method for designing an RBR chain to reach six positions. The design problem has six unknowns: G x, G y, z x, z y, l, and l. We write equation (3) at all six positions and recognize that the positions must be grouped into two sets. The first set is reached with the length l between the fixed and Copyright by ASME

3 E moving pivots, the second set with l between pivots. Grouping positions,, and 3 into the first set, we see from equation () that Z Z 3 G Z 3 Z 7 G ; < ; Z Z Z Z 798 and Z 3 Z 3 Z Z =9>? (8) Grouping positions,, and into the second set, we see from equation () that Z < Z =; G < Z B Z C A G B A B A F G F Z Z Z Z C9D and Z Z Z Z H9I J (9) Equations (8) and (9) are four bilinear equations in the four unknowns G and z. For the arbitrary case, the solution of four general bilinear equations in four unknowns is sixth order (Wampler, 99). Using a resultant, equations (8) and (9) reduce to a single fourth order equation in G x due to coefficients being equal. Hence, we determine four solutions to this synthesis problem. Because of the potential for pairs of roots to be imaginary, there are,, or real RBR chains that result from this procedure. Each grouping of the six positions produces a different set of up to four chains. For example, grouping positions, 3, and in the first set and positions,, and in the second set results in a different set of chains as compared to those determined above. Note that there are ten potential groupings of the six positions for this procedure. Figure. A 3-RBR mechanism that reaches the six positions shown. The B-joints lie between the darkened circles representing revolute joints. position 3 leg leg leg3 Table. Regarding the RBR chains as being one length at state and another at state, a potential design scheme for the chains of a 3-RBR mechanism PARALLEL CONNECTION OF THREE RBR CHAINS Three RBR chains designed via the previous procedure define a mechanism capable of being assembled at the six positions. Note that each chain must be designed from a different grouping of the six positions (see Table for example). Table also suggests the actuation scheme to move the mechanism through the six positions if the mechanism is a kinematically viable one. For example, to move from position to position, all three legs must change length. To move from position 3 to position, leg must be actuated. Figure illustrates one of these mechanisms designed via the grouping presented in Table. Testing of the method has revealed that designing a kinematically viable mechanism that reaches the six positions using the grouping and actuation scheme shown in Table is unlikely. Two issues need to be addressed. A mechanism, when located at each of the six positions, must assemble to the same side of a singularity. The joint velocities are related to the operational velocities by J x ẋ K J d ḋl () where ḋ, the vector of joint velocities, is the rates of change of the lengths of the RBR chains and ẋ is the vector of operational velocities. Each RBR chains varies between two lengths greater than. By virtue of this fact, the Jacobian J d can never be singular. We recognize singularities when the Jacobian J x loses rank or det M J xn K. Thus, we seek mechanisms in which either det M J xnpo at every position or det M J xnpq at every position. In our testing, only a small percentage of mechanisms pass this criteria. As we now argue, passing this criteria is necessary but not sufficient. Although the grouping of chains dictates a potential actuation scheme to move between two positions, this order is not unique. Further, the obvious order of actuation may not work 3 Copyright by ASME

4 V Z Z _ while a far less trivial ordering succeeds. As long as the chains that need to change length from one position to another are actuated an odd number of times and the chains that do not need to change length from one position to another are actuated an even number of times, many different actuation schemes are possible. For example, consider the two actuation schemes: () leg, leg3, leg, leg, leg 3 and () leg, leg, leg. Both of these result in a mechanism with the same leg lengths, however they may be in different configurations. This results from the ability of the mechanism to possess singularity-free paths from one configuration to another (Innocenti and Parenti-Castelli, 99). See Wenger and Chablat, 998 for a descriptive study of this notion. In Figure 3, we illustrate this fact by actuating two of the legs of the mechanism an even number of times. The mechanism completes the motion in a different configuration from its starting location (where the leg lengths are the same in both). Searching all possible actuation schemes for successful paths between positions is the challenge. Leg Leg Leg Leg Leg Leg Leg Leg DESIGN OF AN N-TYPE CHAIN In order to design singularity-free RBR platforms, we connect the chains in an N-type truss as seen in Chirikjian, 99. This constrains the first and second chains to share moving pivots and the second and third chains to share fixed pivots (see Figure ). With these simplifications, the N-type configuration is capable of reaching four desired positions. There are also two free parameters that can be specified in the design procedure, in our case the coordinates of the unshared fixed pivot, G. We choose a unique grouping of the positions for each of the three chains (see Table ). Leg Leg Table. The leg states at each position for the N-type 3-RBR mechanism. position 3 leg leg leg3 Since we are dealing with two distinct fixed pivots in this procedure, G and G, we use G j R A i g ji S d it i R T T 3T T j R T U () to solve for the fixed pivot location g ji in each desired position. Rewriting equation () and having selected the first fixed pivot Figure 3. A movement of the mechanism by actuating the chains in the following order: leg, leg 3, leg, and leg 3. Note that the length of leg is fixed throughout the motion. G, we solve two linear equations in two unknowns, g W g XY z W g \ g 3][ z \ for the first moving pivot z. We then use [ \ [ ` a ` g g g g ]9^ and g g g 3 g 3b9c d () Z ji c A i z j e d i d i c d d 3d d j c d f (3) to obtain the moving pivot s location in the fixed frame at each Copyright by ASME

5 g k p v v { k p position. Manipulating equation (), Z 3 h Z ij G h Z m Z nl G m l m l q r q Z 3 Z 3 Z Z n9o and Z Z Z Z s9t u () yields the second fixed pivot G. Finally, using equations () and () we find the second moving pivot z : g r g sq z r g 3 x g yw z x w x w } g g g g y9z and g 3 g 3 g g ~9 () for positions i 3. Equations () and (7) are evaluated at all four positions for each possible design. A viable mechanism will have either C i, i 3 or C i Œ, i 3, and either D i, i 3 or D i Œ, i 3. Note that the sign need not be the same between the two conditions C i and D i. All platforms that meet these criteria are singularity-free and we plot the associated fixed pivots as shown in Figure. Figure also shows a viable binary actuated mechanism that reaches the four positions found in Table 3 (θ in radians). Table lists the various leg lengths at the four positions. The fixed pivots are G and G Ž The moving pivots are z Ž 9 and z 9. Leg Leg Figure. An N-type 3-RBR mechanism is capable of reaching four prescribed positions. Figure. The shaded regions are acceptable locations of the fixed pivot G for a singularity-free N-type mechanism. EXAMPLE: A SINGULARITY-FREE N-TYPE MECHANISM To generate singularity-free designs, we grid the workspace for possible fixed pivot locations. A mechanism is designed at each fixed pivot location G. Each platform is tested at every position using two singularity conditions: Let G ƒ G x G y, G H x H y, Z i Z xi Z yi, and Z i W xi W yi, then and C i ˆ Z xi G xš ˆ H y G yš ˆ Z yi G yš ˆ H x G xš () position 3 x y θ Table 3. The four desired positions to be reached with the N-type 3- RBR mechanism. D i ˆ Z xi H xš ˆ W yi H yš ˆ Z yi H yš ˆ W xi H xš (7) Copyright by ASME

6 position 3 leg leg leg Table. Leg lengths at each position for the N-type 3-RBR mechanism example. EXAMPLE: POSITION SYNTHESIS We reach ten positions by serially connecting three N-type mechanisms. The first mechanism is designed to reach four of the positions. The second mechanism is designed to reach four positions, having one position in common with the first mechanism. The third is designed to reach four positions with one position in common with the second mechanism. Applying the same singularity check as previously required generates Figure. There are three separate regions dictated in Figure (dots, crosses, and asterisks). Each of these correspond to a single four position synthesis problem. A manipulator capable of reaching ten positions results from rigidly joining the fixed pivots of the first mechanism to the moving pivots of the second mechanism and likewise the fixed pivots of the second mechanism to the moving pivots of the third mechanism. Potentially, one could keep joining mechanisms in this fashion, each time adding three positions to the design task. Figure 7 shows the mechanism designed to reach the ten desired positions given in Table (θ in radians). Figure. Acceptable locations of the fixed pivots for each of the three mechanisms that reach four of the ten positions shown. position x y θ Figure 7. A singularity-free position mechanism Table. The ten desired positions to be reached with the three serially connected 3-RBR mechanisms. CONCLUSIONS This paper presented the kinematic synthesis of mechanisms driven by binary actuated prismatic joints for rigid body guid- Copyright by ASME

7 ance. Several synthesis problems were addressed. First, six positions, divided into two groups, were shown to define up to four RBR chains capable of guiding the body. Different groupings of the positions produced additional sets of four chains. Second, a collection of three of these chains define a mechanism that can be assembled at the six positions. Satisfactory mechanisms prove elusive, however, due to singularity conditions and non-trivial actuation schemes. Third, a simplification from the general to an N-type mechanism with shared fixed and moving pivots allowed for four position synthesis in a singularity-free fashion. Finally, serially connecting three of these N-type mechanisms solved a ten position synthesis problem using only binary actuated joints. All of the design routines were implemented in Matlab to verify the results and are available from the authors upon request. REFERENCES Anderson, V.C., and Horn, R.C., Tensor Arm Manipulator Design, ASME paper 7-DE-7, 97. Chirikjian, G. S., A Binary Paradigm for Robotic Manipulators, Proceedings of the 99 IEEE International Conference on Robotics and Automation, San Diego, CA, 99. Chirikjian, G., Kinematic Synthesis of Mechanisms and Robotic Manipulators with Binary Actuators, Journal of Mechanical Design, Vol. 77, pp.73-8, December 99. Erdman, A.G., and Sandor, G.N., Mechanism Design: Analysis and Synthesis, Vol., Prentice Hall, New Jersey, 997. Innocenti C. and Parenti-Castelli V., Singularity-Free Evolution From One Configuration to Another in Serial and Fully- Parallel Manipulators, Robotics, Spatial Mechanisms, and Mechanical Systems, ASME DE-Vol., pp.3-, 99. Pieper, D.L., The Kinematics of Manipulators under Computer Control, Ph.D. Dissertation, Stanford University, 98. Roth, B., Rastegar, J., and Scheinman V., On the Design of Computer Controlled Manipulators, First CISM-IFTMM Symposium on Theory and Practice of Robots and Manipulators, pp. 93-3, 973. Waldron, K.J., and Kinzel, G.L., Kinematics, Dynamics, and Design of Machinery, John Wiley & Sons, Inc., New York, 999. Waldron, K.J., and Yang, P-H., Parallel Arrays of Binary Actuators, Proceedings of the Conference on Advances in Robot Kinematics: Analysis and Control, Eds. Lenarcic, J. and Husty, M.L., Kluwer Academic Publishers, Boston, pp. 7-, 998. Wampler, C. W., Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics From a New Perspective, ASME Design Engineering Technical Conferences, 99. Wenger, Ph. and Chablat, D., Workspace and Assembly Modes in Fully-Parallel Manipulators: A Descriptive Study, Advances in Robot Kinematics: Analysis and Control, pp.7-, Copyright by ASME