Chapter 6. Screw theory for instantaneous kinematics. 6.1 Introduction. 6.2 Exponential coordinates for rotation

Screw theory for instantaneous kinematics 6.1 Introduction Chapter 6 Screw theory was developed by Sir Robert Stawell Ball [111] in 1876, for application in kinematics and statics of mechanisms (rigid body mechanics). It is a way to express displacements, velocities, forces and torques in three dimensional space, combining both rotational and translational parts. Rotational motion of a rigid body is represented by an element of the special orthogonal group. SO 3 = R R 3 3 R T R = I, det R = 1 (6.1) Other parameterizations of SO(3) include fixed and Euler angle sets and unit quaternions. The change in position as well as orientation of a rigid body is equivalent to a screw motion, a simultaneous translation along and rotation about the some axis in space. A pure screw is simply a geometric concept which describes a helix. A screw with zero pitch looks like a circle. A screw with infinite pitch looks like a straight line. A zero pitch screw is a pure rotation (as a screw) or a pure force (as a wrench). Conversely, an infinite pitch screw is a pure translation (as a screw) or a pure torque (as a wrench) for statics [112,113]. 6.2 Exponential coordinates for rotation Rotation (R) is function of unit direction vector of rotation (ω) and angular rotation θ in radians. Rotation of a body with constant unit velocity about the axis ω, the velocity of point can be expressed as, Integrating time invariant linear differential equation, q t = ω q t = ωq t (6.2) q t = e ωt q 0 (6.3) Where, q 0 is initial position of point and e ωt is matrix exponential e ωt = I + ωt + ωt 2 2! + ωt 3 3! The net rotation about axis ω at unit velocity for θ unit times, R ω, θ = e ωθ = I + ωθ + ωθ 2 2! + (6.4a) + ωθ 3 3! + 6.4b 158

Infinite series is not useful for computation purpose and its closed form formulation is referred as Rodrigues formula [114], R ω, θ = e ωθ = I + ωsinθ + ω 2 1 cosθ, ω = 1 (6.5) The exponential coordinates are called the canonical coordinates of the rotation group and ω is skew symmetric matrix. The angular velocity vector of a rigid body can be expressed with respect to a certain basis as, ω = ω 1 ω 2 ω 3 ω = 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0 (6.6) Let us consider a rotation about x-axis, The x-axis is 1, 0, 0, which is in a skew symmetric matrix form as, ω = e ωθ = I + ωsinθ + ω 2 1 cosθ 0 0 0 0 0 1 0 1 0 (6.6a) = 1 v θ ω 2 2 2 + ω 3 ω 1 ω 2 v θ ω 3 s θ ω 1 ω 3 v θ + ω 2 s θ ω 1 ω 2 v θ + ω 3 s θ 1 v θ ω 2 2 1 + ω 3 ω 2 ω 3 v θ ω 1 s θ ω 1 ω 3 v θ ω 2 s θ ω 2 ω 3 v θ + ω 1 s θ 1 v θ ω 2 2 1 + ω 2 = ω 2 1 v θ + c θ ω 1 ω 2 v θ ω 3 s θ ω 1 ω 3 v θ + ω 2 s θ ω 1 ω 2 v θ + ω 3 s θ ω 2 2 v θ + c θ ω 2 ω 3 v θ ω 1 s θ (6.7) ω 1 ω 3 v θ ω 2 s θ ω 2 ω 3 v θ + ω 1 s θ ω 2 3 v θ + c θ Where, v θ = 1 cosθ, s θ = sinθ and c θ = cosθ The general rotation matrix, R = r 11 r 12 r 13 r 21 r 22 r 23 (6.8) r 31 r 32 r 33 R x = e xθ = Similarly, other representations are, R y = e yθ = R z = e xθ = 1 0 0 0 cθ sθ 0 sθ cθ cθ 0 sθ 0 1 0 sθ 0 cθ cθ sθ 0 sθ cθ 0 0 0 1 (6.8a) (6.8b) (6.8c) 159

6.3 Finite rotations The purpose of this chapter to make aware the possible ways to tackle kinematics and dynamics with large rotations by rigid and flexible bodies and finally computing angular velocities and accelerations associated them. Ways of finite rotations Geometrical Approach - Euler Angles - Bryan Angles - Euler-Chasles representation - Euler and Rodrigues parameters Matrix Approach - Based on orthonormal property of rotation operator - Rotation operator is expressed by either Euler or Rodrigues parameters Algebraic Approach - Quaternion algebra - Matrix algebra (Based on differential geometry) Figure 6.1 Various approaches for finite rotations 6.3.1 Body fixed to reference frame transformation Let A(x 1, y 1, z 1 ) is any point on the rigid body B (in-deformable body hypothesis). The coordinates (x 1, y 1, z 1 ) are denoted with an absolute reference frame O; L, M, N and (X 1, Y 1, Z 1 ) are denoted with respect to relative reference frame O ; P, Q, R rigidly attached to body B as shown in figure 6.2. z A z a a 0 R P A O Q B y N O L M x y x Figure 6.2 Body fixed to reference frame transformation 160

Decomposition of position vector A should be in form of, Where, In matrix form, a = a 0 + A (6.9) a = OA, a 0 = OO and A = O A a T = x 1 y 1 z 1 a T 0 = x 01 y 01 z 01 [A] T = X 1 Y 1 Z 1 The vectors a, a 0 and A represent unicolumn matrices as above. Here, components of vector A are expressed in form of body fixed reference frame O ; P, Q, R. The corresponding components P in the inertial frame can be obtained by, A = R A (6.10) Where R represents transformations from relative reference frame to fixed reference frame (i.e. frame O ; P, Q, R O; L, M, N ) The stated transformation preserves the length of vector A A T A = A T R T R A = A T A (6.11) The R T R = 1 is an algebraic condition of this transformation matrix. These are known as unitary or orthogonal matrices with R = 1. If R = +1 then called proper orthogonal matrices used for rigid body rotations but R = 1 are improper orthogonal matrices employed for reflections. Thus, P = R P with R 1 = R T represents the body reference frame coincides with fixed reference frame. So, a = a 0 + RA (6.12) The above expression depicts the position and orientation transformation resulting from a rigid body rotation and translation. The orthonormality property of R can be expressed in terms of 3 independent parameters like normal, orient and approach. Actually, R is 3 3 matrix with 9 components as expressed in equation (6.8). The orthonormality property of the vectors r j imposed six constraints to each other. r T ir j = δ ij, (i = 1,2.. j, j = 1,2,3 and δ ij = 0 or 1) (6.13) 161

The vectors are linked together by six constraints are, n x 2 + n y 2 + n z 2 = 1 o x 2 + o y 2 + o z 2 = 1 a x 2 + a y 2 + a z 2 = 1 n x o x + n y o y + n z o z = 0 o x a x + o y a y + o z a z = 0 6.3.2 Translational and rotational velocities a x n x + a y n y + a z n z = 0 (6.14) The matrix form of the any position vector A on the rigid body B, a = a 0 + R A (6.15) The time differentiation of the above expression gives the velocity vector, Where, a = a 0 + R A + R A (6.16) a 0 = Velocity vector at reference point O. A = 0 (As rigid body) Hence, velocity of any point on the rigid body, Using the equation (6.15), The velocity expression becomes, a = a 0 + R A (6.17) A = R T a a 0 (6.18) a = a 0 + R R T a a 0 (6.19) The absolute velocity of point P can be expressed in matrix form using extrinsic expression of vector in Cartesian coordinates, da dt = da 0 dt + ω a a 0 (6.20) Where, ω = Angular velocity vector from frame O ; P, Q, R to frame O; L, M, N Owing to orthonormal property of R, R R T = 1 R R T + R R T = 0 (6.21) R R T = R R T, which is a skew symmetric The matrix of the angular velocities can be expressed using equation 6.6 as, ω = R R T = 0 ω 3 ω 2 ω 3 0 ω 1 ω 2 ω 1 0 162

Where, ω 1, ω 2 & ω 3 are the components of the vector ω with respect to a frame O; L, M, N. The matrix analog of the above equation is, 6.3.3 Translation and rotational accelerations in the form of da dt = da 0 dt + ω a a 0 (6.22) The second time derivatives of a general spatial motion equation is expressed a = a 0 + R A a = a 0 + R R T a a 0 (6.23) The matrix R R T can be obtained using time differentiation of the angular velocity matrix ω. Rigid body differential motion of equation 6.22 may also be interpreted as screw motion. The screw motion can be described as combination of translation and rotation motion about the screw axis s in a space. Hence, the concept of screw, screw axis, pitch of screw, twist and wrench of screw are explained in next section. 6.4 The concept of screw, screw axis and pitch of screw 6.4.1 The screw The screw S is nothing but a line l together with a scalar pitch h. A screw is a six dimensional vector composed by a primal part s and dual part s O and can be represented in form of $ = s s O (6.24) Where, s is a unit vector along the screw axis, where as s O is the moment produced by s about a point O fixed to the reference frame, which is calculated according to the pitch of the screw and vector r O, denoting the position of the point O with respect to any point on the screw axis. s O = r O s + s (6.24a) A screw is a line in space with an associated pitch, which is a ratio of linear to angular quantities. Zero pitch is a rotation about the line, then the screw represents a revolute joint and using Plu cker coordinates, $ = s r O s (6.24b) 163

Infinite pitch is a translation, then the screw pair reduces to prismatic joint and represented by $ = 0 s (6.24c) Where, 0 = 0 0 0 T is a three dimensional null vector. If a screw passes through the origin of the coordinate system, the screw coordinates can be denoted as: 6.4.2 Screw axis $ = s s (6.24d) The line l is called the (screw) axis of the motion. For a pure translation, this axis is at infinity. Screw is a line (q, q 0 ) + a pitch. Define the screw coordinates to be (s, s 0 ), where s = q s O = q 0 + q (6.25) The pitch p can be determined back using q q O = 0, s s O = q q O + q. q (6.26) 6.4.3 Pitch of Screw The ratio of the translation parallel to the axis (d) to a rotation about the axis (θ) is known as a pitch of the screw () with a finite motion. It is a coordinate independent property of motion. = d θ (6.27) Where, Using equation (6.26), d = translational displacement θ = rotation around the axis = s s O s s 164 (6.28) [A norm is invariant to the choice of a coordinate system although it can depend on the choice of length scale. A norm is similar to the concept of a distance and distance does not depend on coordinate system used. The distance measured can be in m, cm, mm etc. and hence the number signifying distance will be different. Hence the pitch is invariant to the choice of the coordinate system] The pitch measures linear advancement along the axis per unit rotation about the axis. The magnitude of pitch is zero, the point on nut move in circular path that

provides pure rotation similar to hinge or revolute joint motion. Pitch magnitude is very large, the point on nut move in straight line path that provides pure translation similar to sliding or prismatic joint. For finite displacements the pitch of the twist is not unique since the rotation angle may differ by multiples of 2π. 6.5 Twist and wrench of screw It is well known that screw theory underlies the foundation of both instantaneous kinematics and statics. The concept of twist and wrench are also explained by Sir Robert Stawell Ball [111] for kinematic and statics of mechanism in 1876. 6.5.1 Twist The infinitesimal version of a screw motion is called a twist and it provides a description of the instantaneous velocity of a rigid body in terms of its linear and angular components. ω v A(t) A A(t) A B (a) Revolute joint (b) Prismatic joint Figure 6.3 Revolute joint and prismatic joint Twists represent velocity of a body. For example, climbing up a spiral staircase at a constant speed can also be represented in form of twist. In general, the velocities of each particle within a rigid body define a helical field called the velocity twist. A twist contains 6 quantities (three linear and three angular). Another way of decomposing a twist is by 4 line coordinates (see Plücker coordinates), 1 scalar pitch value and 1 twist magnitude. The velocity of a point A(t) is as shown in figure 6.3(a), A t = ω A t B = ω A t ω B = ω A ω B (6.29a) 165

Using homogeneous coordinates, A 0 = ω ω B 0 0 A 1 = ξ A 1 Rigid body transformations can be represented as the exponentials of twists: Where, ξ = ω v 0 0, ω SO 3, v R3, θ R, ξ R 4 4 The twist coordinates are ξ = (v, ω) R 6. (6.29b) Conversely, given a screw one can write the associated twist. Two special cases are pure rotation about an axis l = b + ω by an amount θ and pure translation along an axis= 0 + v. ξ = ξ = v 0 ω q ω θ pure rotation (6.29c) θ pure translation (6.29d) The velocity of a point attached to a prismatic joint moving with unit velocity as shown in figure 6.3(b), A t = v and ξ = 0 v 0 0 (6.30) a ω a b b θ d b + e ωθ (a b) b + e ωθ a b + θω Figure 6.4 Generalized screw motions [114] A twist ξ = (v, ω) is associated with screw motion having attributes, Pitch: Axis: Magnitude: = ωt v ω 2 (6.31) l = M = ω v + ω R, if ω 0 ω 2 0 + v R, if ω = 0 ω, if ω 0 v, if ω = 0 166 (6.32) (6.33)

6.5.2 Wrench Wrenches represent forces and torques. One way to conceptualize this is to consider someone who is fastening two wooden boards together with a metal screw. The person turns the screw (applies a torque), which then experiences a net force along its axis of rotation. In screw notation force wrenches transform with a 6 6 transformation matrix. A finite or infinite motion (displacement) of a rigid body in 3D space is defined by using the concept of twist. It is possible to reduce a system of forces and couples to a resultant force and couple about any point of interest. In general to resultant force and couple are not collinear, however, one can show that it is possible to find a unique axis of which both the resulting force f and the resulting couple c could be acted along. This force-couple combination is called a wrench. A generalized force acting on a rigid body consists of a linear component (pure force) and an angular component (pure moment) acting at a point. Wrenches are represented as a force, moment pair as a vector in R 6. $ w = f, M R 6 (6.34) The unique axis is called the wrench axis or screw axis of the system of forces and couples. Unit wrench is defined as, $ w = s r r o s r + P r s r (6.35a) Where, s r is a unit vector in direction of the screw axis and r o is a position vector of any point on the axis. The vector r o s r defines moment of screw axis about the origin. For a pure Force, the P r = 0 For a pure couple, the P r = $ w = s r r o s r (6.35b) $ w = 0 (6.35c) s r A wrench of intensity ρ can be expressed as, $ w = ρ$ w (6.36) The pitch of the wrench is defined as the ratio of the couple to the force. P r = c f (6.37) 167

First three components of the wrench represent the force f, and the last three represent the resulting moment due to the combined effect of the force f and the couple c about the origin of the reference frame. Wrenches are dual to twists, so that many of the theorems which apply to twists can be extended to wrenches. A wrench F = f, τ is associated with a screw having attributes, Pitch: Axis: Magnitude: = ft v f 2 (6.38) l = M = f τ + f R, if f 0 f 2 0 + τ R, if f = 0 f, if f 0 τ, if f = 0 Table 6.1 Instantaneous kinematics and statics description Instantaneous Kinematics Pitch Statics Twist ω, r ω + ω Wrench f, r f + f Screw S, r S + S Screw S, r S + S Rotation ω, r ω = 0 Force f, r f Translation 0, v = Couple 0, c (6.39) (6.40) Screw theory is suitable for kinetostatic analysis of rigid body. It is based on two well known fundamental theorems: (a) Chasles theorem and (b) Poinshot s theorem. 6.5.3 Chasles theorem Rigid body motion is equivalent to twist on a screw, rotation about a unique axis and translation parallel to this axis. (Lipkin and Duffy, [115]) Any motion of a rigid body can be reproduced as a rotation of the body about a unique line in space and a translation along that same line. 168

ω ω A B A r rω B A ω v a ω B v a v b ω AB Figure 6.5 Representation of motion at different points of body ω is an angular velocity about an axis passing through point A and translational velocity due to rotation for point B is ω AB. $ t = ω x ω y ω z v x v y v z (6.41) 6.5.4 Poinshot Theorem Rigid body action is equivalent to a wrench on a screw, force along a unique line and a couple parallel to the line. (Lipkin and Duffy, [115]) Any set of forces and couples applied to a body can be reduced to a single force acting along a specific line in space, and a pure couple acting in a plane perpendicular to that line. A f A B B r f A f PQ m A B m A f m B f AB Figure 6.6 Poinshot theorem A wrench is a screw describing the resultant force and moment of a force system acting on a rigid body. $ w = f x f y f z M x M y M z (6.42) Constraints analysis: Each twist has a reciprocal called a wrench expressed in form of screw coordinates as [f x f y f z M x M y M z ]. It represents all the forces and torques that the feature can transmit to a mating part. Where motion is allowed, no force or torque can be transmitted, and vice versa. The intersection of all wrenches acting on a part shows the amount of constraint on the part provided by those features. If a part is constrained 169

in some direction, then every feature can provide that constraint. Constraint analysis is depicted for revolute and prismatic joints in next section of reciprocal screws. Point lying on the screw axis: Assumption: Point lying on the axis of the screw and nut is fixed. Using Chasle s theorem, Let ω be the angular velocity of the screw, any point A on the axis of the screw at any instance moves with a linear velocity v a = ω, where h is the pitch of the screw. If any point A is known, then the vector pair ω, v a is used to specify the twist. Using poinsot s theorem, Let f be the force on the body along the screw axis of wrench, any point on the axis have net parallel moment M a = f. If any point A is known, then the vector pair f, M a is used to specify the wrench. Point lying on the body: Let ω be the angular velocity and f be the force on the body, then for any point on the body the vector pairs specify the twist and wrench in that case are: ω, v b = ω, r b ω + ω (6.43a) f, M b = f, r b f + f (6.43b) Where, r b is a vector with magnitude of perpendicular distance between point on body and an axis. 6.6 Representation of screws in different forms Nowadays, Screws can be described in several different forms: Vector Representations Dual Numbers Plu cker Coordinates 6.6.1 Vector Representation A screw is presented as two classical vectors: a) angular quantity vector b) total effect vector (total velocity + total force) as expressed in equation 6.43, ω, v p = ω, r p ω + ω f, M p = f, r p f + f Where, r = The vector directed from point P to the screw f and M = The resultant and moment vector ω and v = The resultant angular and linear velocity vector = Pitch 170

6.6.2 Dual Number The dual number algebra was originally developed by W K Clifford [116]. A screw is represented by an operator called as ε (Clifford s operator). It is analogous to imaginary operator i, where i 2 = 1. Existence of the complex numbers is to cater the need of roots of polynomial equations. Imaginary unit i (i.e. i 2 = 1) was invented and the complex field C can be defined as: C = a + ib a, b R, i 2 = 1 (6.44) Dual numbers are other types of complex numbers of the form: x + εy. ε is Clifford operator (ε 2 = 0) (i.e. whose square is zero and yet is not itself zero, nor is it small ). First term is scalar and second term is accompanied by ε, is called dual number. θ d Figure 6.7 Dual angle between two screw axes is θ + εd, where d is the directed distance along common normal and θ is the angle measured along common normal [115] The function of dual number has an extraordinarily simple Taylor s series expansion about x. f x + εy = f x + εy df(x) dx Since, higher order terms vanish as ε 2 = ε 3 = ε 4 = = 0. For example, (6.45) e x+εy = e x 1 + εy (6.46) Table 6.2 Comparison between complex number and dual number Cartesian Form Polar Form Exponential Form Complex Number a + ib r(cosθ + isinθ) re iθ Dual Number x + εy R(1 + εt) Re εt θ = Angle between x-axis and the radial line from origin to point a + ib. T = Distance along the line x=1 from axis to the radial line from the origin to point x + εy. 171

Modulus: R = x Argument: T = y x for x 0 Figure 6.8 Interpretation of complex numbers as points in a plane: (a) ordinary complex number, and (b) dual number [117] 6.6.2.1 Dual number algebra Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. The algebraic interpretation of dual numbers can be same as two dimensional geometric senses of planar operators. Addition: x + εy + x + εy = x + x + ε y + y (6.47a) Subtraction: x + εy x + εy = x x + ε y y (6.47b) Multiplication: x + εy x + εy = xx + ε yx + xy (6.47c) 0ε = ε0 = 0 and yε = εy If x 1 + εy 1 = x 2 + εy 2 ten x 1 = x 2, y 1 = y 2 Thus addition, subtraction and multiplication exist for any pair and they are commutative, associative and distributive. Multiplication of complex number re iθ with a unit complex number e iα rotates the point a + ib by an angle α in counter clockwise direction about an origin. e iα re iθ = re i θ+α (6.48) Hence, the complex number e iα is planar rotational operator (Rotating the points in the plane without affecting their distance from the origin. Multiplication of dual number Re εt with a unit dual number e ετ translates the point (x, y) along the y axis through distance Rτ The net effect of multiplication operator is to shear the entire plane parallel to y axis through the shear angle of arctan τ. Multiplications of dual numbers correspond to shear mapping. 172

Figure 6.9 Plane transformations under multiplications by (a) an ordinary complex number e iα (b) dual number e ετ [117] If the dual numbers are considered as functional arguments, using Taylors series expansions for the dual argument x + εy Therefore, f x + εy = f x + εyf x (6.49) sin x + εy = sinx + εycosx cos x + εy = cosx εysinx cot x + εy = cotx εy sin 2 x, sinx 0 x + εy 1 2 = x 1 2 + εy 2x 1 2, a > 0 (6.50) If p = p x, p y, p z and q = q x, q y, q z are two vectors in Euclidean space then three components of the dual vector p + εq are also dual numbers. p + εq = p x + εq x, p y + εq y, p z + εq z (6.51) A dual vector multiplication with a dual scalar a + εb p + εq = ap + ε aq + pb Scalar product: p + εq r + εs = p r + ε p s + r q Cross product: p + εq r + εs = p r + ε p s + q r 6.6.2.2 Dual Unit Quaternion (6.52a) (6.52b) (6.52c) A unit quaternion of the form q = cos θ 2 + nsin θ 2, when applied to an arbitrary vector r in the form of transformation, effects a fixed point rotation through an angle of θ about the spatial axis n. 173

Dual quaternions are defined as 4-tuples of dual numbers, q = q 1 + εq 01 + i q 2 + εq 02 + j q 3 + εq 03 + k q 4 + εq 04 = q 1 + εq 01, q 2 + εq 02, q 3 + εq 03, q 4 + εq 04 = q 1 + q 2 + q 3 + q 4 + ε q 01 + q 02 + q 03 + q 04 = q + εq 0 (6.53) Where, q and q 0 are real quaternions. Applying the principle of transference, the following expression for a general screw displacement of a dual vector l through the dual angle θ = θ + εd along the dual spatial axis n : l = cos θ 2 + nsin θ 2 l cos θ 2 nsin θ 2 (6.54) Moreover, n in the above equation must be a unit line vector to ensure that the screw operator Q is of unit magnitude, i.e. Q = q 1 + εq 2 01 + q 2 + εq 2 02 + q 3 + εq 2 03 + q 4 + εq 2 04 = 1 (6.55) This requirement translates into two distinct conditions q 2 1 + q 2 2 + q 2 3 + q 2 4 = 1 (6.56a) q 1 q 01 + q 2 q 02 + q 3 q 03 + q 4 q 04 = 0 (6.56b) Which, imposed on the eight parameters of a general dual quaternion, effectively reduce the number of degrees of freedom of its parameter space to six. Hamilton [118] has introduces the algebra of quaternion. Dual Angles and screws: A screw is described as dual vector as shown in figure 6.9. The relationship of two directed skew straight lines a and b in space can be expressed uniquely as: Figure 6.10 Parameters describing relative location of two skew lines in a space [117] 174

the common normal vector n = a b between the two lines the distance d between the lines along the common normal, and the twist angle α, measured positive from a to b Above figure shows parameters describing relative location of two skew lines in space. The set of parameters defined above now allows us to specify the position and orientation of an arbitrary line l 2 in space with respect to a given line l 1 by a single rotation (through the twist angle α ) about a unique spatial axis (the common normal n), combined with a unique translation (through the distance d ) along that same axis. In summary, a spatial screw displacement can be thought of as a dual angular displacement α = α + εd about a spatial axis. The six plu cker coordinates of the screw axis along with the rotation angle α and the translation parameter d completely describe a general rigid body displacement. 6.6.3 Plu cker coordinates Plu cker Coordinates are very concise and efficient for numerous chores in field of robot mechanics. They are also directly describing lines at infinity, which describe pure translations. Point: A point in a space has three degrees of freedom and consequently three dimensional space is commonly treated as an aggregate of 3 points. Line: At least 4 parameters are required to completely specify a line in space. Slope and intercept of the projection of the line onto two orthogonal planes. It is considered that the space should be composed of an aggregate of 4 lines. Line has one more degree of freedom than a point, but more convenient to use for spatial mechanisms as rotation axis, link orientation axis etc. Only 4 parameters are needed to completely and uniquely define a line in a space. A line is represented by a 6 parameters as a reason of mathematical convenience called Plu cker Coordinates. The relation between Plu cker Coordinates with Cartesian space will be described using dual vector in space as shown in figure 6.11, First term is scalar and second term is accompanied by ε, is called dual number. E = e + εe (6.57) = e + ε(ρ e) Above equation defines a unit screw in space. Screw algebra is vector algebra of this dual vector. 175

Figure 6.11 A unit screw with a radius vector a space A screw can be defined using three dual coordinates in the space as Where, E = e = Unit line vector Unit vector of screw axis. (Resultant Vector) ε = Dual number with properties ε 2 = 0, ε 0 e = E = L, M, N (6.58) Moment of e with respect to the origin of fixed coordinate system (moment vector). It is vector product of the radius vector and unit vector as: e = ρ e (6.59) ρ = ρ(x, y, z) is the radius vector to an arbitrary point on the screw axis E. Using equation (6.88), ρ e = det 176 i j k x y z L M N Pi + Qj + Rk = Ny Mz i + Lz Nx j + Mx Ly k or P = Ny Mz Q = Lz Nx R = Mx Ly (6.60) Equations (6.57) are called as the equations of the screw axis, which are the equations of line in space. The equations of the axis of the screw in Plu cker coordinates are homogeneous with respect to their coordinates. One can write the equations in form of: (λn)y (λm)z (λp) = 0 (λl)z (λn)x (λq) = 0 (λm)x (λl)y (λr) = 0 If (L, M, N, P, Q, R) satisfies the line equations then λl, λm, λn, λp, λq, λr also satisfies the line equations. Since screw E is unit, E 2 = 1 + ε 0

Using equation (6.57) and definition (6.58), each dual coordinate of the screw can be divided in two parts, L = L + εp M = M + εq N = N + εr (6.61) The six real coordinates of screw of equation (6.61) are called as Plu cker Coordinates of a unit screw E (L, M, N, P, Q, R). L, M, N = Components of the unit vector e P, Q, R = Components of the moment vector e Velocity and acceleration of a unit screw in the space can be obtained from the first and the second differentials of (6.61) with respect to time as: L = L + εp M = M + εq N = N + εr (6.62) L = L + εp M = M + εq N = N + εr (6.63) Hence, the time derivatives of Plu cker coordinates of a unit screw as: E = (L, M, N, P, Q, R ) E = (L, M, N, P, Q, R ) In coordinate form, L 2 + M 2 + N 2 = 1 + ε 0 (6.64) L 2 = L 2 + 2εPL M 2 = M 2 + 2εQM N 2 = N 2 + 2εRN (6.65) Substituting (6.65) into (6.64), a resulting equation is in form of, L 2 + M 2 + N 2 + 2ε PL + QM + RN = 1 + ε 0 (6.66) Using the components of e 0 and e as: e. e 0 = 0 LP + MQ + NR = 0 (6.67a) It means orientation vector e and the moment vector e 0 are mutually perpendicular. Also, since the vectors e and ke (for k 0) define the same spatial orientation, one can render the representation of a line unique by restricting the orientation vector to be of unit magnitude, i.e., e. e = 1 L 2 + M 2 + N 2 = 1 (6.67b) The above two equations provide necessary two constraints regarding Plu cker coordinates. So, out of six Plu cker coordinates only four are independent. The remaining two can be found out using equations (6.65). (e, e 0 ): e 0, e 0 0: general line e, e 0 : e 0, e 0 = 0: line through origin e, e 0 : e = 0, e 0 0: [not allowed] 177

Note: Plu cker Coordinates are dependent. The axis of screw is defined synonymously with e (L, M, N) and e (P, Q, R). Time derivatives of e as: - P = N y + Ny M z Mz - Q = L z + Lz N x Nx - R = M x + Mx L y Ly - P = N y + Ny M z Mz + 2(N y M z ) - Q = L z + Lz N x Nx + 2(L z N x ) - R = M x + Mx L y Ly + 2(M x L y ) (6.68) Using (6.61) and (6.68), one can determine the magnitude of components of the moment vector e (P, Q, R) and its time derivatives from known vector ρ(x, y, z) and e L, M, N. 6.6.3.1 Plu cker Coordinates of a line, twist and wrench: The line is defined in a plane with q 1 and q 2 with direction ratios L and M as shown in figure 6.12, A line vector F 12 from P 1 to P 2 whose direction ratios are L and M, L = x 2 x 1 = P 12 cos θ 6.69 M = y 2 y 1 = P 12 sin θ The moment of F 12 about z-axis as R and using right hand rule, Using equation 6.69 and 6.70, R = F 12 OA = x 1 M y 1 L = x 1 y 1 L M (6.70) R = x 1 y 1 x 2 x 1 y 2 y 1 = Three determinants of single matrix can be defined as, The gradient of line is given by, x 1 y 1 x 2 y 2 (6.71) R L M = 1 x 1 y 1 1 x 2 y 2 (6.72) θ = tan 1 M L (6.73) The shortest distance from line to origin is given by, R OA = (6.74) L 2 + M 2 178

Y F 12 M O A θ L X Figure 6.12 A line in a plane with q 1 and q 2 with direction ratios L and M In homogeneous form, equation (6.71) is represented as, Hence, R L M = w 1 x 1 y 1 w 2 x 2 y 2 (6.75) L = w 1 x 1 w 2 x 2, M = w 1 y 1 w 2 y 2, R = x 1 y 1 x 2 y 2 (6.76) Similarly, Plu cker coordinates representations of line in R 3. The directed line containing q 2 and q 1 in affine 3-Dimensional space has a third direction ratio N. For a orthogonal XYZ system, N = w 1 z 1 w 2 z 2 (6.77) L = x 2 x 1 = s 1 M = y 2 y 1 = s 2 N = z 2 z 1 = s 3 (6.78) The moment about X, Y and Z axis are P, Q and R respectively, The Plu cker coordinates of 3D line will be, P = y 1 z 2 y 2 z 1 = y 1 N z 1 M = y 2 N z 2 M Q = z 1 x 2 z 2 x 1 = z 1 L x 1 N = z 2 L x 2 N R = x 1 y 2 x 2 y 1 = x 1 M y 1 L = x 2 M y 2 L (6.79) s 1 = w 1 x 2 w 2 x 1 s 2 = w 1 y 2 w 2 y 1 s 3 = w 1 z 2 w 2 z 1 s 4 = y 1 z 2 y 2 z 1 s 5 = x 2 z 1 x 1 z 2 s 6 = x 1 y 2 x 2 y 1 = ω = q 2 q 1 = Direction of line = v = q 2 q 1 = Moment of line 179

The six coordinates of line can be written as L M N P Q R or s 1 s 2 s 3 s 4 s 5 s 6. The first three elements of such a sextuple represent a direction, and the last three a moment. Thus, the Plu cker line coordinates are given by L i, where L i = s i, i = 1,2,3 L i = s i s i 3, i = 4,5,6 The notations can be reduced further by combing each term from two sets of numbers s 1 s 2 s 3 and s 4 s 5 s 6 into three pairs. s = s 1, s 4, s 2, s 5, s 3, s 6 In short, six-dimensional real vector which contains the Plu cker coordinates for the line joining q 1 and q 2 with respect to frame i: l i = x 1 x 2 w 1 w 2, y 1 y 2 w 1 w 2, z 1 z 2 w 1 w 2, y 1 y 2 z 1 z 2, z 1 z 2 x 1 x 2, x 1 x 2 y 1 y 2 can be defined using 2-by-2 determinants of the coordinates of the points. It is easily possible to see that above equation can be written as: l i = w 2q 1 w 1 q 2 q 1 q 2 = w 2q 1 w 1 q 2 q 1 q 2 q 1 T (6.80) (6.81) From this last expression, it is possible to consider two cases: the first one when the defining points are finite (w1, w2 0) and the second when the points are at infinity (w1, w2 = 0) and therefore describe a line at infinity. 6.6.3.2 Plu cker screw coordinates A screw $ is defined by straight line with an associated pitch h and is conveniently defined by six homogeneous six Plu cker coordinates, $ = s s o + s = L M N s 1 s 2 s 3 (6.82) Where s denotes direction ratios pointing along screw axis, s o = r s defines the moment of screw axis about origin of the coordinate system, r is position vector of any point on screw axis with respect to coordinate system. Screw axis is denoted by Plu cker homogeneous coordinates, s $ axis = s o 180

Assume Figure 6.13 The screw s = L M N T s o + s = P Q R T (6.83) Considering s. s o + s = s. s o + s 2 = s 2 assuming s 0, Pitch of screw: = s. s o + s s 2 = The axis of screw can be denoted as LP + MQ + NR L 2 + M 2 + N 2 (6.84) $ axis = L M N P L Q M R N (6.85) Assume that vector of projective point of the origin on the screw axis is represented by r Op. The screw axis and vector r Op are perpendicular to each other. Using concept of vector triple product, s r Op s = s s r Op + s. r Op s = s 2 r Op (6.86) Using equation (6.82) and (6.83), s = L M N T r Op s = P L Q M R N (6.87) Using above equations, r Op = s r O p s s 2 = 1 M R N N Q M L 2 + M 2 + N 2 N P L L R N L Q M M P L If the Plu cker coordinates of screw are given, then 1) Unit direction vector (s) 2) The pitch (h) 3) The screw axis ($ axis ) 6.88 4) The vector of the projective point of origin on the axis (r Op ) are easily obtained. 181

If the pitch of screw equals to zero, then $ = s s o is just Plu cker homogeneous coordinates of the screw axis. Assume that point O P is the projective point of the origin on a line l and point A is any other point on the line. Then, r A = r OP + r OP A = r O P + as s (6.89) Where s is direction vector of line l and a is length of line segment O P A. The moment of line l about the origin at point A will be, s 0 = r A s = r OP + as s s (6.90a) One obtains that the moment of a line about the origin is irrelevant to the point s selection on the line. A screw associated with a revolute pair is a twist of zero pitch ( = 0) pointing along the pair axis, s $ = r s Hence, the revolute joint is represented as a line in plucker coordinates. (6.90b) A screw associated with a prismatic pair is a twist of infinite pitch ( = ) pointing in the direction of the translational guide line of the pair. $ = 0 s (6.90c) Hence, a prismatic Joint (p = ) is represented as a line at infinity contained in a plane perpendicular to the line represented at (6.88). Of course the magnitude of the vector along such a line in the plane at infinity must be taken to be zero, r at the best infinitesimal By virtue of the first three of its line coordinates being zero but its moment about the origin can be regarded as meaningful and finite [Hunt, 1978]. The kinematic screw is often denoted in the form of Plücker homogeneous coordinates: $ = L M N P Q R T (6.91) Where, the first three components denote the angular velocity, the last three components denote the linear velocity of a point in the rigid body that is instantaneously coincident with the origin of the coordinate system. The displacement of a rigid body cannot be completely determined without the specification of the amplitude or intensity of the screw axis is specified. Let q be the intensity of a twist, then the twist can be expressed as: $ = q $ (6.92) 182

Where, q = θ for revolute joint (angular velocity) and q = d for prismatic joint (linear velocity) 6.6.4 Kinematics of two unit screw in a space A rigid body in space can be described by two unit vectors E 1 and E 2. Figure 6.14 shows two unit screws placed arbitrarily in space. The angle and the distance between these two screws is defined with the dual angle, A = α + ε. a Where, α = Twist angle a = Shortest distance between two axes E 1 L 1, M 1, N 1 α a E 2 L 2, M 2, N 2 Figure 6.14 Two arbitrary unit screws in space The following equations describe the physical situation of the above two unit screws, From screw algebra, E 1 2 = 1 + ε. 0 E 2 2 = 1 + ε. 0 E 1 E 2 = cosa L 1 2 + M 1 2 + N 1 2 = 1 + ε 0 L 2 2 + M 2 2 + N 2 2 = 1 + ε 0 L 1 L 2 + M 1 M 2 + N 1 N 2 = cosa (6.93) cos A = cosα ε asinα (6.94) From six dual coordinates L 1, M 1, N 1, L 2, M 2, N 2 that describe the position of rigid body in space, just three of them are independent out of six constraints. From equations 6.93 and 6.94, using the real Pluckers coordinates of the screws E 1 and E 2, L 2 1 + M 2 1 + N 2 1 + 2ε L 1 P 1 + M 1 Q 1 + N 1 R 1 = 1 + ε 0 L 2 2 + M 2 2 + N 2 2 + 2ε L 2 P 2 + M 2 Q 2 + N 2 R 2 = 1 + ε 0 L 1 L 2 + M 1 M 2 + N 1 N 2 + ω P 1 L 2 + L 1 P 2 + Q 1 M 2 + Q 2 M 1 + R 1 N 2 + R 2 N 1 = cosα 12 ε a 12 sinα 12 It is clear that unit vectors e 1 and e 2, moment vectors e 1 and e 2 can be written using their components as, 183

e 1 e 1 = 0 L 1 P 1 + M 1 Q 1 + N 1 R 1 = 0 e 1 e 1 = 1 L 2 1 + M 2 1 + N 2 1 = 1 e 2 e 2 = 0 L 2 P 2 + M 2 Q 2 + N 2 R 2 = 0 e 2 e 2 = 1 L 2 2 + M 2 2 + N 2 2 = 1 e 1 e 2 = cosα 12 L 1 L 2 + M 1 M 2 + N 1 N 2 = cosα 12 (6.95) e 0 1 e 2 + e 1 e 0 2 = a 12 sinα 12 P 1 L 2 + L 1 P 2 + Q 1 M 2 + M 1 Q 2 + R 1 N 2 + N 1 R 2 a 12 sinα 12 The equation 6.95 describes the relative moment of two screws E 1 and E 2 in screw theory. 6.7 Reciprocal screws The concept of reciprocal screw was first introduced by Ball [111]. Mohamed and Duffy [119] applied the theory of reciprocal screws for the Jacobian analysis of parallel manipulators. The main idea of this concept is that if a wrench acts on a rigid body in such a way that that it produced no work while the body undergoes an infinitesimal twist, then both screws representing the twist and the wrench are said to be reciprocal to each other as shown in figure 6.15. Body B α $ r Wrench: $ w a z s ro x O s O y $ Twist: $ t Figure 6.15 Twist and wrench in 3D-Space [120] One of them represents the motion of the rigid body, the other is constraint force exerted on the body. The reciprocal product of screws can be defined as the instantaneous virtual power developed by a force screw along the kinematic screw. Therefore, a reciprocal screw $ r, denotes a wrench which produces no power for a kinematic screw $ (or a twist). Lipking and Duffy [115] have mentioned that reciprocity is a geometric quantity relating two screws, $ 1 = L 1 M 1 N 1 P 1 L 2 M 2 N 2 P 2 Q 2 R 2 Q 1 R 1 and $ 2 = and can be identified using following condition, 184

Where, $ 1 $ 2 = $ 1 T $ 2 = 0 (6.96) = 0 3 3 I 3 3 I 3 3 0 3 3 In other form, two screws, $ 1 and $ 2 are called to be reciprocal if they satisfy the equation, L 1 P 2 + M 1 Q 2 + N 1 R 2 + L 2 P 1 + M 2 Q 1 + N 2 R 1 = 0 (6.97) If the screws $ 1 and $ 2 represent revolute joints, then these screws are reciprocal when their primal parts intersect or are parallel. Reciprocal screws emerges if the primal part of $ 1 representing a revolute joint is perpendicular to the dual part of the screw $ 2, representing a prismatic joint as shown in figure 6.17. For a screw system, twist system represents mobility and wrench system represents the constraints. The screw is a twist if it represents an instantaneous motion of a rigid body, and a wrench if it represents a system of forces and couples acting on a rigid body. First three components of a wrench represent the resultant force and the last three components represent the resultant moment about the origin of the reference frame as discussed earlier. Twist matrix and its corresponding wrench matrix are reciprocal to each other. Each kinematic chain consists of (m) twist screws. Normally, each screw forms a row vector of the screw matrix. The rank of this screw matrix is known as dimension of screw space. In short, dimension of screw matrix and dimension of reciprocal screw matrices are always six. Moreover, two screw systems $ 1 and $ 2 are reciprocal if all the screws in $ 1 are reciprocal to all the screws in $ 2. Particular Instances [111]: Parallel or intersecting screws are reciprocal when the sum of their pitches is zero. Screws at right angles are reciprocal either when they intersect or when one of the pitches is infinite. Two screws of infinite pitch ($ and $ ) are reciprocal, because a couple could not move a body which was only susceptible of translation. A screw whose pitch is zero or infinite is reciprocal to itself. A zero pitch screw $ 0 is reciprocal to an infinite pitch screw $ if only if their directions are orthogonal to each other. 185

Geometrical method to determine the reciprocal screws was explained by Jianguo Zhao, Bing Li, Xiaojun Yang and Hongjian Yu [121]. The approach is free of algebraic manipulation and completely based on three observations for determination of reciprocal screw system for any kinematic pairs and chain. 6.8 Calculating twist and wrench for common kinematic pairs: Revolute pair: Associated screw of revolute pair is line vector. It provides a rotary movement between two bodies about an axis in a space. Suppose, vector r = x r, y r, z r represents the point of rotational axis and unit vector along the direction of rotation is represented by ω = a b c. Hence, $ t = ω v, where, v = r ω = d e f. The twist for the joint can be calculated with the following formula: The corresponding wrench matrix will be, $ t = d e f a b c $ w = 1 0 0 0 0 d 0 1 0 0 0 e 0 0 1 0 0 f 0 0 0 g 0 0 0 0 0 i f Where, vectors g 0, 0 i f and a b c are mutually perpendicular. (6.98) The plucker coordinates for the joint for a coordinate frame as shown in figure 6.16 is represented as, $ t = 0 0 1 0 0 0 Using reciprocal conditions stated in equation (6.97), following are the different conditions for reciprocal screw system for a revolute joint axis in direction of coordinate frame z-axis direction as shown in figure 6.16, Screws at right angles are reciprocal either when they intersect, $ 1 r = 1 0 0 0 0 0 $ 2 r = 0 1 0 0 0 0 A screw whose pitch is zero or infinite is reciprocal to itself. $ 3 r = 0 0 1 0 0 0 A zero pitch screw $ 0 is reciprocal to an infinite pitch screw $ if only if their directions are orthogonal to each other. $ 4 r = 0 0 0 1 0 0 $ 5 r = 0 0 0 0 1 0 186

z z $ $ y y x x (a) Revolute pair (b) Prismatic pair Figure 6.16 Revolute and prismatic pair axis aligned with z-axis of coordinate frame Prismatic pair: The associated screw of a prismatic pair is a couple vector. The first triplet is always null as no angular velocity with this joint. The second triplet describes vector direction ratios of linear movement of joint in a space. Hence, $ t = 0 u. $ t = 0 0 0 a b c The wrench matrix can be derived using the reciprocity property and it consists of five rows. Each row of the wrench matrix represents vector of the base of the reciprocal space of the twist. It can be represented in a simplest form as, $ w = d e f 0 0 0 g i 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 Where, vectors a b c, d e f and g i are mutually perpendicular. (6.99) One can choose any five linearly independent ones to form the reciprocal 5 system in same way as explained for revolute joint pair. Twist screw can be determined for prismatic pair as shown in figure 6.16 as, Using reciprocal condition, $ t = 0 0 0 0 0 1 $ r 1 = 0 0 0 1 0 0 $ r 2 = 0 0 0 0 1 0 $ r 3 = 0 0 0 0 0 1 $ r 4 = 1 0 0 0 0 0 $ r 5 = 0 1 0 0 0 0 (6.100) 187

The zero-pitch reciprocal screws lie on all planes perpendicular to the axis of prismatic joint. Figure 6.17 Reciprocal screws Cylindrical Pair: Kinematically, any cylindrical joint can be replaced by a revolute joint plus a coaxial prismatic joint as shown in figure 6.18. y θ z x Figure 6.18 Cylindrical pair Two screws are associated with cylindrical pair, $ t = 0 0 1 0 0 0 $ t = 0 0 0 0 0 1 The constraint in form of wrench form four screw system, $ r 1 = 1 0 0 0 0 0 $ r 2 = 0 1 0 0 0 0 $ r 3 = 0 0 0 1 0 0 $ r 4 = 1 0 0 0 1 0 (6.101) Spherical Pair: The associated unit screws to spherical joint form a three system of zero pitch screws (three intersecting line vectors) passing through the center of the joint. $ t = 0 0 1 0 0 0 $ t = 0 1 0 0 0 0 $ t = 1 0 0 0 0 0 (6.102) 188

By observation, there are no reciprocal screws of couple vectors. Hence, constraints in form of wrench form three screws system of line vectors. In short, reciprocal screws form a three-system of zero pitch passing through the center of the sphere. In short, number of twist screws represents degrees of freedom of kinematic pair and number of reciprocal screws of wrenches indicates constraint imposed on it. Application of reciprocal screw system: To carry out mobility analysis of serial or parallel manipulators It is used to identify number of degrees of freedom and associated types of motion. Mobility analysis of 3-PRS configuration is presented in next section. To investigate constraint and motions in assembly A constraint condition of parts into assembly is possible to distinguish by forming twist and wrench matrix for an exact kinematic state of the assembly. To analyze Jacobian of serial or parallel manipulator [122] Generalized Jacobian of parallel manipulators with n limbs can be formulated using reciprocal screw concept. Instantaneous twist of moving platform can be expressed as linear combination of m instantaneous twists of i t limb. m $ P = q j,i$ j,i, i 1,2, n (6.103) j =1 Where, q j,i and $ j,i indicates intensity and unit screw associated with j t joint of a i t limb. Equation (6.103) contains many un-actuated joint rates, which should be eliminated by applying the theory of reciprocal screws. Actuated joints are identified as g screws, $ r,j,i for j = 1, 2, g of each limb. Each of them is reciprocal to the screw system of all un-actuated joints of i t limb. Then, the orthogonal product of both sides of equation 6.103 with each reciprocal screw results in g equations, which can be written in a matrix form as follows: J x,i $ P = J q,i q i (6.104) Where, q i = q 1,i q 2,i q g,i $ T r,1,i $ T r,1,i $ 1,i $ T r,1,i $ 2,i $ T r,1,i $ g,i J x,i = $ T r,2,i $ T r,g,i and J q,i = $ T r,2,i $ 1,i $ T r,2,i $ 2,i $ T r,2,i $ g,i $ T r,g,i $ 1,i $ T r,g,i $ 2,i $ T r,g,i $ g,i Equation (6.104) is written for all n limbs of the manipulator. 189

To understand singularities of parallel manipulator Singularity analysis of a parallel manipulator is often very complicated. Screw theory helps to analyze the relationship between different criterions of singular configurations. Singularity analysis of 3-PRS is also carried out in next section. In this way, screw and reciprocal screw system provides a new approach in analysis and synthesis of mechanisms. Recently, many researchers have used this tool for various applications as stated above. 6.9 Mobility analysis of 3-PRS using screw theory A spatial parallel manipulator configuration consists of three limbs. Each limb as shown in figure 6.19 has prismatic, revolute and spherical joint. Thus, each kinematic chain has two single DOF pairs and one three DOF pair. Screw axis method is quite different and each joint axis is identified as screw. This section will investigate the mobility of a PRS kinematic chain using screw theory. So, the kinematic screws for each kinematic chain can be expressed as: $ j i or $ j,i is used to represent the unit twist associated with the j t kinematic pair in the i t limb. $ r,3 S 3 S31 S23 S51 S 2 S 1 S41 $ r,1 S S22 $ r,2 U R S21 S13 z P A3 y A1 r ri S11 x S12 A2 represented by Figure 6.19 Screw coordinate system for 3-PRS configuration Infinite pitch is a translation, then the screw pair reduces to prismatic joint and $ t 1,i = 0 s 1,i (6.105) 190

Zero pitch is a rotation about the line, then the screw represents a revolute joint and twist of revolute joint is represented using Plu cker coordinates as, $ t 2,i = s 2,i r ri s 2,i (6.106) Cartesian coordinates r ri = x r1 y r1 z T r1 are denoted for i t limb plane revolute joint as shown in figure 6.19. For the spherical kinematic pair S 1, its centre point cartesian coordinates are denoted by r si = x s1 y s1 z T s1, spherical joint with three DOF is equivalent to three single-dof revolute pairs and the twists for the three orthogonal axes can be expressed as: $ t j,i = s j,i r si s j,i j = 3, 4, 5 (6.107) Where, s j,i, j = 3,4,5 indicates three orthogonal directions of spherical joints S i and i indicates the i th limb under consideration. As far as 3-PRS configuration as shown in figure 6.19 has r ri = R ix R iy R iz and r si = S ix S iy S iz. In fact, the Cartesian coordinates of spherical joints can also be represented in form of, r si = r ri + Us 5,i (6.108) Direction of s 5,i is in same direction of axis of connecting link in limb plane -1. Thus, the kinematic screw system of the first limb consists of k=5 system. The reciprocal screw system for i th limb motion screw system is a 6 k system in which any one screw is reciprocal to all the screws in the given system. It is found from the literature that a reciprocal screw for a kinematic chain is reciprocal to the screws associated with all pairs in the chain. Hence, there is only one reciprocal screw in existence for i t limb of 3-PRS configuration. Passive revolute joint axes are reciprocal to the zero pitch wrench (pure force) applied via the actuated prismatic joint. Hence, all the passive joint axes are reciprocal to the zero pitch wrench (pure force) applied via the actuated prismatic joint. Using reciprocal screw theory, one can obtain the limb constraint system of limb i as, $ rw c,i = s 2,i r si s 2,i (6.109) Zero pitch screw is passing through the centre of spherical joint and its direction is parallel to revolute joint axis of i t limb. It is orthogonal to all unit twist screws in i t limb. In fact, the intersection of moving platform plane and another 191

plane passing from centre of spherical joint and revolute joint axis define the reciprocal screw axis at any instance. Case Study: Assumed structural parameters of 3-PRS configuration are: - Initially position of prismatic joint P i from fixed base =160 mm - Connecting link length U =482 mm - The prismatic joint axis and revolute joint axis offset (b) = 41 mm - Distance between two prismatic joint axis on fixed base (p) = 750 mm - Distance between two spherical joint centre points q = 300mm - Fixed Base Coordinates: B 1x B 1y B 1z B 2x B 2y B 2z B 3x B 3y B 3z = - Prismatic Joint Coordinates: P 2 P 2 0 P 2 3 0 P 2 3 0 P 3 0 = 375 216.506 0 375 216.506 0 0 433.013 0 P 1x P 1y P 1z P 2x P 2y P 2z = P 3x P 3y P 3z - Revolute Joint Coordinates: B 1x B 1y B 1z + T 1 B 2x B 2y B 2z + T 2 = B 3x B 3y B 3z + T 3 375 216.506 160 375 216.506 160 0 433.013 160 R 1x R 1y R 1z R 2x R 2y R 2z = R 3x R 3y R 3z - Spherical joint coordinates: P 1x + 3 2 b P 1y + b 2 P 1z P 2x 3 2 b P 2y + b 2 P 2z P 3x P 3y b P 3z = 339.4930 196.0064 160 339.4930 196.0064 160 0 392.0127 160 S 1x S 1y S 1z S 2x S 2y S 2z = S 3x S 3y S 3z 150 86.6025 589.473 150 86.6025 589.473 0 173.2051 589.473 The five twist screws form a limb-1 motion screw system can be represented using Plucker s coordinates, $ t 1,1 = 0 0 0 0 0 1 $ t 2,1 = 0.866 0.5 0 80 138.56 339.4825 $ t 3,1 = 0.866 0.5 0 294.7365 510.4836 149.9978 $ t 4,1 = 0.4455 0.7716 0.3931 420.7939 203.6452 77.1586 192

$ t 5,1 = 0.3931 0.227 0.891 210.9732 365.3718 0.0066 The one wrench screw as a reciprocal screw of a limb-1 as constraint screw of moving platform $ rw 1,1 = 0.866 0.5 0 294.7365 510.4836 149.9978 5-twist screws system of limb -2 motion screw system using Plucker s coordinates is, $ t 1,2 = 0 0 0 0 0 1 $ t 2,2 = 0.866 0.5 0 80 138.56 339.4825 $ t 3,2 = 0.866 0.5 0 294.7365 510.4836 150 $ t 4,2 = 0.4455 0.7716 0.3931 420.7939 203. 6452 77.1586 $ t 5,2 = 0.3931 0.227 0.891 210.9732 365.3718 0.0066 The one wrench screw as a reciprocal screw of a limb-1 as constraint screw of moving platform $ rw 1,2 = 0.866 0.5 0 294.7365 510.4836 150 The five twist screws form a limb-3 motion screw system can be represented using Plu cker s coordinates, $ t 1,3 = 0 0 0 0 0 1 $ t 2,3 = 1 0 0 0 160 392.0127 $ t 3,3 = 1 0 0 0 589.4730 173.2051 $ t 4,3 = 0 0.6504 0.5591 480.2322 0 0 $ t 5,3 = 0.5141 0.5591 0.6504 216.9218 303.0481 89.0447 The one wrench screw as a reciprocal screw of a limb-3 as constraint screw of moving platform, $ rw 1,3 = 1 0 0 0 589.473 173.2051 Reciprocity condition of two screw systems using equation (6.101) is also verified for all limbs i 1,2,3 as, $ t j,i $ rw 1,i = 0, j = 1,2 5 (6.110) Each reciprocal screw of each limb imposes constraint force (zero pitch screw) on the moving platform. The wrench screw system as a platform constraint system can be represented as, $ w 1 = 0.866 0.5 0 0 0 0.866 $ w 2 = 0.866 0.5 0 0 0 0.866 $ w 3 = 1 0 0 0 0 1 193

y S 3 v 3 $ rw 1,3 x $ rw 1,1 30 S 2 v 1 v 2 S 1 $ rw 1,2 Figure 6.20 Moving platform constraint screw system The corresponding platform motion screw system which is reciprocal to platform constraint system is obtained using reciprocity condition as expressed in equation (6.101), $ rt 1 = 0 0 0 0 0 1 $ rt 2 = 0.866 0.5 0 0 0 0.866 $ rt 3 = 0.866 0.5 0 0 0 0.866 Translation along x, translation along y and rotation along z are three constraints imposed by three reciprocal screws of platform motion screw system as observed from platform motion screw system. Hence, 3-PRS manipulator of proposed configuration allows three degrees of freedom. It verifies the same results as mentioned in mobility analysis of chapter 2. The 3-PRS manipulator has two rotary degrees of freedom along x and y - directions and one translational DOF along z - direction. 6.10 Singularity analysis of a manipulator using screw theory The following steps to be followed for a singularity analysis of 3-DOF parallel manipulator using screw theory. Step1: Each PRS kinematic chain contains 3 kinematic pairs of prismatic, revolute and spherical pair and each pair has κ i DOF. The twist screw system of i t chain is represented as, T i = $ t 1,i $ t 2,i $ t 3,i $ t 4,i $ t T 5,i (111) Here, twist screw system for i t chain has 5 6 matrix dimension. The number of reciprocal screws in i t chain has 6 n linearly independent screws. It is observed that there is only one constraint wrench screw for an individual limb. Moreover, they are satisfying the condition of reciprocity as per equation (6.107), $ t j,i $ rw 1,i = 0, j = 1,2 5 and i 1,2,3 (112) 194