Dimensional Synthesis of Bennett Linkages

Dimensional Synthesis of Bennett Linkages DETC 00 000 ASME Design Engineering Technical Conferences 10-13 September 000 Baltimore, Maryland, USA Alba Perez (maperez@uci.edu), J. M. McCarthy (jmmccart@uci.edu) Robotics and Automation Laboratory Department of Mechanical and Aerospace Engineering University of California Irvine, California 9697 Introduction The Bennett linkage The workspace of the RR chain The cylindroid The design equations Bennett linkage coordinates Solving the design equations Examples Conclusions 1

Literature review Synthesis Bennett, G.T., (1903) A New Mechanism. Engineering, vol. 76, pp. 777-778. Veldkamp, G.R., (1967) Canonical Systems and Instantaneous Invariants in Spatial Kinematics. Journal of Mechanisms, vol. 3, pp. 39-388. Suh, C.H., (1969) On the Duality in the Existence of R-R Links for Three Positions. J. Eng. Ind. Trans. ASME, vol. 91B, pp. 19-134. Tsai, L.W., and Roth, B.,(1973) A Note on the Design of Revolute-Revolute Cranks. Mechanisms and Machine Theory, vol. 8, pp. 3-31. Perez, A., McCarthy, J.M., (000), Dimensional Synthesis of Spatial RR Robots. 7th. International Symposium on Advances in Robot Kinematics, Piran-Portoroz, Slovenia. Analysis Suh, C.H., and Radcliffe, C.W., (1978), Kinematics and Mechanisms Design. John Wiley. Yu, H.C., (1981) The Bennett Linkage, its Associated Tetrahedron and the Hyperboloid of its Axes. Mechanism and Machine Theory, vol. 16, pp. 105-114. Huang, C., (1996), The Cylindroid Associated with Finite Motions of the Bennett Mechanism. Proc. of the ASME Design Engineering Technical Conferences, Irvine, CA. Baker, J.E., (1998), On the Motion Geometry of the Bennett Linkage. Proc. 8th Internat. Conf. on Engineering Computer Graphics and Descriptive Geometry, Austin, Texas, USA, vol., pp. 433-437. Linear Systems of Screws Hunt, K.H.,(1978), Kinematic Geometry of Mechanisms. Clarendon Press. Parkin, I.A., (1997), Finding the Principal Axes of Screw Systems. Proc. of the ASME Design Engineering Technical Conferences, Sacramento, CA.

The Bennett Linkage - Spatial closed 4R chain W α H a γ g G U Ground link Formed by connecting the end links of two spatial RR chains {G, W} and {H, U} to form a coupler link. Mobility: For a general 4R closed spatial chain: M =6(n 1) m p k c k =6.3 4.5= k=1 However, the Bennett linkage can move with one degree of freedom. Special geometry: Link lenght and twist angle (a, α) and (g, γ) must be the same for opposite sides. Ratio of the sine of the twist angle over the link length: sin α a = sin γ g The joints of the Bennett linkage form the vertices of a tetrahedron 3

The Design Theory - RR Chain The Workspace of the Synthesis Theory- Find the kinematic chain that reaches exactly a number of specified positions {M 3 } {M } Moving Axis {M 1 } {F} Fixed Axis End Effector The specified positions must lie on the workspace of the chain. The kinematics equation for the RR chain defines its workspace 4

The Kinematics Equation for the RR Chain {M 3 } {M } {F} [G] a θ α φ {M 1 } [H] The kinematics equation in matrix form:the set of displacements [D(θ, φ)] of the RR chain. [D] =[G][Z(θ, 0)][X(α, a)][z(φ, 0)][H] If we choose a reference configuration [D 1 ], we can write the workspace of the relative displacements [D 1i ]=[D i ][D 1 ] 1. [D 1i ]=[T(θ i,g)][t (φ i, W)] where [T ( θ, G)]=[G][Z(θ, 0)][Z(θ 0, 0)] 1 [G] 1, [T ( φ, W)]=([G][[Z(θ 0, 0)][X(ρ, r)])[z(φ, 0)][Z(φ 0, 0)] 1 ([G][[Z(θ 0, 0)][X(ρ, r)]) 1 5

The Kinematics Equation for the RR Chain Dual quaternion formulation We can also formulate the workspace using dual quaternions to express the relative displacements. The dual quaternion form of the workspace is given by: ˆD 1i = Ĝ( θ)ŵ ( φ) cos( ˆψ 1i θ φ θ φ ) = cos cos sin sin G W, sin( ˆψ 1i )S 1i = sin θ φ φ cos G + sin θ θ φ cos W + sin sin G W where S 1i is the screw axis of the relative displacement and ˆψ 1i =(φ 1i,d 1i ) is the associated rotation about and slide along this axis for each displacement in the workspace. Every pair of values θ and φ defines a screw axis S 1i that represents a relative displacement from position 1 to position i. 6

The Workspace of the Bennett Linkage α W G γ θ H φ γ φ α U Restriction to a Bennett linkage: The angles θ and φ are not independent. There exist the input/coupler angular relation: tan φ γ+α = sin sin γ α tan θ = K tan θ The workspace of the Bennett linkage: The set of screw axes obtained applying the input/coupler relation to the workspace of the RR chain, generates a cylindroid tan( ˆψ 1i )S 1i = G + K W 1 + K tan θ G W1 cot θ K tan θ G. W1 7

The cylindroid Simply-Ruled surface that has a nodal line cutting all generators at right angles. z(x + y )+(P X P Y )xy =0 It appears as generated by the real linear combination of two screws. The cylindroid has a set of principal axes located in the midpoint of the nodal line. K Y S 13 z 0 σ S 1 X The principal axes can be located from any pair of generators. tan σ = (P b P a ) cot δ + d (P b P a )+dcot δ z 0 = 1 (d (P b P a ) cos δ sin δ ) 8

Bennett linkage coordinates P 1 b W 1 K Q X c G Y c κ C 1 B a Yu, 1981: The Bennett linkage can be determined using a tetrahedron defined by four parameters (a, b, c, κ). The principal axes are located in the middle of the tetrahedron. The joint axes G and W 1 are given by the cross product of the edges. This ensures that the chosen points B, P 1 are on the common normal. ( ) G = K g (Q B) (P 1 B)+ɛK g B (Q B) (P 1 B) W 1 = K w (B P 1 ) (C 1 P 1 )+ɛk w P 1 ( (B P 1 ) (C 1 P 1 ) ) We obtain: and G = K g { bc sin κ bc cos κ 4ab cos κ sin κ W 1 = K w { ac sin κ ac cos κ 4ab cos κ sin κ } { } b cos κ (4a sin κ + c ) + ɛkg b sin κ (4a cos κ + c ) abc(cos κ sin κ ) } + ɛkw { a cos κ (4b sin κ + c ) a sin κ (4b cos κ + c ) abc(cos κ sin κ ) } Using the principal axes and the tetrahedron formulation, we can write the coordinates of the joint axes of the Bennett linkage with only four parameters. 9

The design equations for an RR dyad The constant dual angle constraint: ˆα =(α, a),the angle and distance between the fixed and moving axes,must remain constant during the movement. G [ ˆT 1i I]W 1 =0,i=,3, Usign the equivalent screw triangle formulation and separating real and dual part, 1. The direction equations. The distance equations tan ψ 1i = G (S 1i W 1 ) (S 1i G) (S 1i W 1 ), i =,3. (B P 1 ) S 1i t 1i =0, i =,3. The normal constraints: The normal line to G andw, P i B,remains the same. G ([T 1i ]P 1 B) =0, W 1 (P 1 [T 1i ] 1 B)=0,i=1,,3. Total equations: (n 1)+n Total parameters: 10 Number of positions needed for a finite number of solutions: n =3. The standard algebraic formulation of the synthesis problem consists on solving ten equations in ten parameters. 10

Solving the design equations in the principal axes frame The six common normal constraints are automatically satisfied. We solve system of four equations in four parameters a, b, c, κ. tan ψ 1 = G (S 1 W 1 ) (S 1 G) (S 1 W 1 ) tan ψ 13 = G (S 13 W 1 ) (S 13 G) (S 13 W 1 ) (1) () (B P 1 ) S 1 t 1 (B P 1 ) S 13 t 13 =0 (3) =0 (4) Solution for a and b :the distance equations (3) and (4) are linear in a, b. t 1 +(a b) cos δ 1 cos κ +(a+b) sin δ 1 sin κ =0 t 13 +(a b) cos δ cos κ +(a+b) sin δ sin κ =0 Defining the constraints: K s = t 1 cos δ t 13 cos δ 1 sin(δ 1 δ ) We obtain: K d = t 13 sin δ 1 t 1 sin δ sin(δ 1 δ ) a = K s sin κ + K d cos κ b = K s sin κ K d cos κ. 11

Solution for c : Substitute the values of a and b in the direction equations (1) and () and make the algebraic substitution y = tan κ. tan ψ 1 ( K s K d ψ y )+c tan 1 K d ( K s K d y )+ (y (cos δ 1 1) + cos δ 1 +1) cy K d (cos δ 1 + Ks sin δ 1 K d ) c K d (y (cos δ 1 1) + cos δ 1 +1) =0, tan ψ 13 ( K s K d ψ y )+c tan 13 K d ( K s K d y )+ (y (cos δ 1) + cos δ +1) cy K d (cos δ + Ks sin δ K d ) c K d (y (cos δ 1) + cos δ +1) =0, The numerator and denominator share the roots associated with c =0, which are not a solution of the spatial problem. Eliminate them from the numerator forcing the linear system to have more solutions than the trivial. [ tan ψ 1 tan ψ 13 tan ψ 1 K d tan ψ 13 K d (y c(cos δ 1 1) + cos δ 1 +1) y K d (cos δ 1 + Ks sin δ 1 K d ) (y c(cos δ 1) + cos δ +1) y K d (cos δ + Ks sin δ K d ) ] { (K Making the determinant of the matrix equal to zero we obtain a linear equation in c. Define the constants ( ) K 1 = t 1/ 1 tan ψ 1 sin δ 1 sin δ We obtain the expression for c: K 13 = t 13/ tan ψ 13 ( ) 1 sin δ 1 sin δ s /K d y ) c } =0. c =(K 13 K 1 ) sin κ 1

Solution for κ : Substitute the expressions for a, b, c in one of the direction equations, (1) or (). We obtain a cubic polynomial in y. The coefficients are: P : C 3 y 6 + C y 4 + C 1 y + C 0 =0 C 3 = K d, C = K s K d +4(K 1 K 13 )(K 13 sin δ 1 K 1 sin δ ), C 1 =K s K d 4(K 1 K 13 )(K 13 cos δ 1 K 1 cos δ ), C 0 = K s. Solve the cubic polynomial for z = y. This polynomial has one and only one real positive root z 0 : P (0) = K s P ( ) = K d P( 1) = 4(K 1 K 13 ) The square root of the positive root gives the two solutions for κ. tan κ = ± z 0 13

The Solutions The two sets of solutions (a, b, c, +κ) and ( b, a, c, κ) correspond to both dyads of the Bennett mechanism: Solution 1 Solution G( a, b, c, κ) = H( b, a, c, κ) W( a, b, c, κ) = U( b, a, c, κ) H( a, b, c, κ) = G( b, a, c, κ) U( a, b, c, κ) = W( b, a, c, κ) The synthesis procedure yields the two RR dyads that form a Bennett linkage. 14

Example Tsai and Roth positions (Tsai and Roth, 1973) The specified positions: x y z θ φ ψ M 1 0.0 0.0 0.0 0 0 0 M 0.0 0.0 0.8 0 0 40 M 3 1.11 0.66 0.05 18.8-8.0 67. The joint axes in the initial frame: Axis G W 1 H U 1 Line coordinates (0.36, 0.45, 0.81), (0.6, 1.05, 0.70) (0.60, 0.36, 0.7), (0.87, 0.83, 1.14) (0.60, 0.36, 0.7), (0.87, 0.83, 1.14) (0.36, 0.45, 0.81), (0.6, 1.05, 0.70) 15

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Conclusions Using the geometry of the RR chain to formulate the problem leads to a simple convenient set of equations. The design procedure for three positions for an RR chain yields a Bennett linkage. A Mathematica notebook with the complete synthesis procedure can be downloaded from: http://www.eng.uci.edu/ mccarthy/pages/resprojects.html The synthesis routine is to be used in a robot design environment for continuous tasks. 0