# Synthesis of Constrained nr Planar Robots to Reach Five Task Positions

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3 nr τ τ τ τ τ τ Base Pulleys Planetary Gear Train θ i, i =,,..., n- τ, τ Fig. 3. Schematic for a constrain nr Planar Robot driven by cables, mechanically constraint by planetary gear train and RR cranks. to reduce the tendon driven robot to a three degree-of-freedom system. Two RR constraints were then added to obtain a one degree-of-freedom system. The first RR chain serves to couple the planetary gear train inputs τ and τ together, and the purpose of the second RR chain is to couple the nth joint, θ n, to τ and hence τ. As shown in Figure 3, if we denote τ as the input that drive the first joint angle, we can see that τ is related to the input τ through the four bar linkage formed from the attachment of an RR constraint to the second link. The planetary gear train was designed in such a way that it impose symmetry condition on the joint angles so that θ = θ 3 = = θ n = τ. Hence τ get reflected through the tendons to serve as the input to the floating four bar linkage from the RR attachment between the end-effector and the (n )th link. The output from this floating four-bar linkage is the joint angle θ n. In other words, if we drive the system with input τ, τ and θ n will be driven by the input τ through the mechanical coupling of the planetary gear train, and the two RR constraints. VI. THE COUPLING BETWEEN JOINT ANGLES AND INPUTS The desired coupling between joint angles, Θ, and the inputs, Γ, of a nr serial chain can be described by matrix [A] as follows, Θ = [A]Γ. (6) Our goal here is to constrain the nr serial chain to a three degree-of-freedom system first using tendons and a planetary gear train before we add RR chainss to constrain it to one degree-of-freedom. Hence Γ = (τ, τ ) T in particular if we ignore θ n for now. The desired coupling between the inputs and the joint angles for a nr serial planar robot can be obtained by specifying the tendon routing, the size of the base pulleys, and the transmission coupling the input to the base pulleys. This is equivalent to selecting the matrices [B], [S], and [C] τ θ n so we obtain the desired coupling matrix [A] in the relation [A] = [B] [S][C]. (7) [B] is a matrix that describes the relationship between the relative joint angles and the rotation of the base pulleys, see (3). [S] is a diagonal matrix that scales the sum of the row elements of matrix [C] to unity. This matrix was included to account for the effects of changing the size of the base pulleys and is the ratio of the i th base pulley to the i th joint pulley. [C] is a matrix that describes the relationship between the rotation of the base pulleys to the inputs of the transmission system. VII. THE TRANSMISSION SYSTEM The transmission that couples the nr serial robot is composed of two stages. The bottom stage connects the controlled inputs to the base pulleys using simple and planetary gear trains and the top stage connects the base pulleys to the joint pulleys using tendons. The relationship between the inputs, Γ and the base pulley rotations Θ, which defines the bottom stage of the transmission is given by: Θ = [C]Γ. (8) The matrix [C] is called the mechanical coupling matrix. Its elements determine the gear train design which couples the inputs to the tendon drive pulleys. In this paper, our goal is to first design a transmission system that couples the nr serial chain to three degree-of-freedom, and then design two RR cranks to constraint the system to one degree-of-freedom. The equation relating the base pulley rotation, θi to the inputs at the transmission τ and τ is: θ i = c i τ + d i τ. (9) Three cases exist for (9). They correspond to a simple gear train if either c i or d i = 0, a planetary gear train if both c i and d i are non-zero, and un-driveable if both c i and d i are zero. An example of a planetary gear train with sum of c i = and d i = 3 is as shown in Figure 4. Fig. 4. Carrier Sun Gear ( UNIT DIA.) Annular Gear τ τ (3 UNIT DIA.) Planetary Gear Train VIII. NR CHAIN DESIGN METHODOLOGY Our design of constraining nr robots using tendon, planetary gear train and RR chainss to reach five specified task positions proceed in three steps. The following describes the design process.

4 A. Step - Solving the Inverse Kinematics of the nr Serial Chain Given five task positions [T j ], j =,..., 5 of the endeffector of the nr chain, we seek to solve the equations [D] = [T j ], j =,..., 5, (0) to determine the joint parameter vectors Θ j = (θ,j, θ,j,..., θ n,j ) T, where θ i,j represents the i th joint angle at the j th position. Because there are three independent constraints in this equation, we have free parameters when n > 3. In what follows, we show how to use these free parameters to facilitate the design of mechanical constraints on the joint movement so the end-effector moves smoothly through the five task positions. In the design of serial chain robots, it is convenient to have near equal length links to reduce the size of workspace holes. For this reason, we assume that the link dimensions of our nr planar robot satisfy the relationship a = a 3 = = a n,n = l. () This reduces the specification of the nr robot to the location of the base joint C in G, and the end-effector joint C n relative to the task frame H. In this paper, we take the task frame H to be the identity matrix to simplify the problem even more. The designer is free to chose the location of the base joint C and the link dimensions l. We define the input crank joint angle to be, θ,j = σ j, j =,..., 5, () and if the planar nr robot has n > 3 joints, then we impose a symmetry condition θ,j = θ 3,j = = θ n,j = λ j, j =,..., 5, (3) in order to obtain a unique solution to the inverse kinematics equations. B. Step - Designing of a planetary gear train The second step of our design methodology consists of designing a two-degree-of-freedom planetary gear train to drive the base pulleys to achieve the symmetry conditions listed in (3). Note that we ignore the degree of freedom of the n th joint at this stage. The gear train controls the shape of the chain through the two inputs. Proper selection of the routing of the tendons from their joints to their base pulley can simplify gear train design. We now generalize the routing of the n tendon such that we only require a planetary gear train with inputs τ and τ to drive the first n joint angles. Consider the following (n ) desired coupling matrix 0 0 [A] =... (4) 0 If we chose the (n ) (n ) structure matrix [B] with elements such that starting from the second column, we have + on the even columns and - on the odd columns to get the following lower triangular matrix, [B] = (5) ± Refer to Figure 7 and 8 for a graph and schematic representation of such a routing system. Using (7), we get 0 0 [B][A] = [S][C] =. (6) 0.. c n, c n, If we introduce a (n ) (n ) diagonal scaling matrix, [S] = , (7) s n with element s n = if n is even, and s n = if n is odd. It would result in a mechanical coupling matrix [C], with row elements adding to unity, of the form [C] = c n,. c n,. (8) Note that the rows of matrix [C] consists of elements (, 0) at odd rows and (, ) at even rows. The values at the last row would depend on if n- is odd or even. Because of this special alternating structure, we only require a two degree-of-freedom planetary gear train to drive the n- tendon base pulleys. The mechanical coupling matrix [C] indicates that the planetary gear train used should have the property such that τ drives odd base joints, and τ and τ drives even base joints. A planetary gear train that fits this characteristic would be a bevel gear differential since an ordinary planetary gear train would result in a locked state. Also, the scaling matrix indicates that the ratio of the even k th base pulley to the even k th joint pulley is to. See Figure 5 and Figure 6 for an example of such a bevel gear differential and tendon routing on a 6R planar robot.

5 θ θ 3 θ 5 θ θ 4 3R 3 UNIT DIA. UNIT DIA. τ τ 4R 3 4 Tendon : Fig. 5. θ Planetary Gear Train for a 6R Planar serial Robot 5R Tendon : θ Tendon 3: Tendon 4: θ 3 θ R Tendon 5: θ C C C 3 C 4 C 5 Fig. 6. Tendon Routing for a 6R Planar Robot nr n- n- n C. Step 3 - Synthesis of an RR Drive Crank The last step of our design methodology consists of sizing two RR chainss that constrains the nr robot to one degree-offreedom. Figure 7 and 8 shows how we can systematically attach two RR chainss to constraint the three degree-offreedom nr cable driven robot to one degree-of-freedom. In this section, we expand the RR synthesis equations, Sandor and Erdman (984) [0] and McCarthy (000) [], to apply to this situation. Let [B k,j ] be five position of the (k )th moving link, and [B k,j ] be the five positions of the kth moving link measured in a world frame F, j =,..., 5. Let g be the coordinates of the R-joint attached to the (k )th link measured in the link frame B k, see Figure 9. Similarly, let w be the coordinates of the other R-joint measured in the link frame B k. The five positions of these points as the two moving bodies move between the task configurations are given by G j = [B k,j ]g and W j = [B k,j ]w (9) Now, introduce the relative displacements [R j ] = [B k,j ][B k, ] and [S j ] = [B k,j ][B k, ], so these equations become G j = [R j ]G and W j = [S j ]W (0) where [R ] = [S ] = [I] are the identity transformations. The point G j and W j define the ends of a rigid link of length R, therefore we have the constraint equations ([S j ]W [R j ]G ) ([S j ]W [R j ]G ) = R () Fig. 7. This shows the graph representation of mechanically constrained serial chain. Each link is represented as a node and each R joint as an edge. Parallel type routing is denoted by a double-line edge and cross type routing is denoted by a heavy edge. These five equations can be solved to determine the five design parameters of the RR constraint, G = (u, v, ), W = (x, y, ) and R. We will refer to these equations as the synthesis equations for the RR link. To solve the synthesis equations, it is convenient to introduce the displacements [D j ]= [R j ] [S j ]=[B k, ][B k,j ] [B k,j ][B k, ], so these equations become ([D j ]W G ) ([D j ]W G ) = R () which is the usual form of the synthesis equations for the RR crank in a planar four-bar linkage, see McCarthy (000)[]. Subtract the first of these equations from the remaining to cancel R and the square terms in the variables u, v and x, y. The resulting four bilinear equations can be solved algebraically, or numerically using something equivalent to Mathematica s Nsolve function to obtain the desired pivots. The RR chain imposes a constraint on the value of τ as a function of τ given by b sin ψ a sin τ τ = arctan( ) + β, (3) g + b cos ψ a cos τ where β is the joint angle between pivots C W and C C 3,

8 TABLE VI SOLUTION OF G W WITH A HEXAGON INITIAL CONFIGURATION Pivots G W (0, 0) (0.5, 0.866) ( 0.09, 0.030) (3.966, 4.384) 3 (0.05, 0.006) (0.396, 0.44) 4 (0.79, 0.070) ( 0.70, 0.4) TABLE VII SOLUTION OF G W WITH A HEXAGON INITIAL CONFIGURATION Pivots G W (.5, 0.866) (, 0) (.379, 0.50) (.8, 0.3) 3 (. 0.0i, i) ( i, i) 4 ( i, i) ( i, i) state to its extended state. We solve the inverse kinematics of the 6R chain at each of the task positions by equating the forward kinematics equation (5). Once we solve the inverse kinematics, the positions of its links was used to design RR chain G W, and G W. We obtained four real solution for G W, and two real solutions for G W. Table VI, and VII show the various possible pivots. Again, we use the same transmission design and tendon routing as shown in Figure 5 and 6 respectively. We follow the analysis procedure described in IX to animate the movement of the constrained 6R robot. See Figure XI. CONCLUSIONS In this paper, we show that the design of a cable drive train integrated with four-bar linkage synthesis yields new opportunities for the constrained movement of planar serial chains. In particular, we obtain a 6R planar chain through five arbitrary task positions with one degree-of-freedom. The same theory applies to nr chains with more degrees of freedom. Two examples show that the end-effector moves through the specified positions, while the system deploys from square and hexagonal initial configurations. REFERENCES [] J.J. Lee, Tendon-Driven Manipulators: Analysis, Synthesis, and Control, PhD. dissertation, Department of Mechanical Engineering, University of Maryland, College Park. MD; 99. [] M. Arsenault and C. M. Gosselin, Kinematic and Static Analysis of a Planar Modular -DoF Tensegrity Mechanism, Proceedings of the Robotics and Automation, Orlando, Florida, 006, pp [3] M. J. Lelieveld and T. Maeno, Design and Development of a 4 DOF Portable Haptic Interface with Multi-Point Passive Force Feedback for the Index Finger, Proceedings of the Robotics and Automation, Orlando, Florida, 006, pp [4] G. Palli and C. Melchiorri, Model and Control of Tendon-Sheath Transmission Systems, Proceedings of the Robotics and Automation, Orlando, Florida, 006, pp [5] Krovi, V., Ananthasuresh, G. K., and Kumar, V., Kinematic and Kinetostatic Synthesis of Planar Coupled Serial Chain Mechanisms, ASME Journal of Mechanical Design, 4():30-3, 00. [6] A. S. K. Kwan and S. Pellegrino, Matrix Formulation of Macro- Elements for Deployable Structures, Computers and Structures, 5():37-5, 994. Fig.. This mechanically constrained 6R chain guides the end effector through five specified task positions with a hexagon initial configuration [7] S. O. Leaver and J. M. McCarthy, The Design of Three Jointed Two-Degree-of-Freedom Robot Fingers, Proceedings of the Design Automation Conference, vol. 0-, pp. 7-34, 987. [8] S. O. Leaver, J. M. McCarthy and J. E. Bobrow, The Design and Control of a Robot Finger for Tactile Sensing, Journal of Robotic Systems, 5(6):567-58, 988. [9] L.W. Tsai, Robot Analysis: The Mechanics of Serial and Parallel Manipulators, John Wiley & Sons Inc, NY; 999. [0] G.N. Sandor and A.G. Erdman, A., Advanced Mechanism Design: Analysis and Synthesis, Vol.. Prentice-Hall, Englewood Cliffs, NJ; 984. [] J. M. McCarthy, Geometric Design of Linkages, Springer-Verlag, New York; 000.

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