ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 5 VELOCITY AND STATIC ANALYSIS OF MANIPULATORS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian Institute of Science Bangalore 56 12, India Email: asitava@mechengiiscernetin NPTEL, 21 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 1 / 13

1 CONTENTS 2 LECTURE 1 Introduction Linear and Angular Velocity of Links 3 LECTURE 2 Serial Manipulator Jacobian Matrix 4 LECTURE 3 Parallel Manipulator Jacobian Matrix 5 LECTURE 4 Singularities in Serial and Parallel Manipulators 6 LECTURE 5 Statics of Serial and Parallel Manipulators 7 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 2 / 13

OUTLINE 1 CONTENTS 2 LECTURE 1 Introduction Linear and Angular Velocity of Links 3 LECTURE 2 Serial Manipulator Jacobian Matrix 4 LECTURE 3 Parallel Manipulator Jacobian Matrix 5 LECTURE 4 Singularities in Serial and Parallel Manipulators 6 LECTURE 5 Statics of Serial and Parallel Manipulators 7 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 3 / 13

INTRODUCTION Position kinematics Position & orientation of links, workspace, mobility etc Change of position and orientation with respect to time Velocity kinematics Linear velocity as derivative of position vector Angular velocity in terms of derivative of rotation matrix Topics in velocity kinematics include Linear and angular velocities of links Manipulator Jacobian(s) Singularities in velocity domain Static equilibrium Relation between external forces & moments and joint torques & forces Singularities in force domain ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 4 / 13

LINEAR VELOCITY OF RIGID BODY The linear velocity of O i with respect to {} is defined as V Oi = d dt O i (t + t) O i (t) O i (t) = lim t t (1) {} Ẑ Rigid body at t + t denote the coordinate system {} where the limit is taken O i (t) {i}(t) {i}(t + t) O i The linear velocity vector can be described in {j} as ˆX Figure 1: O i (t + t) Ŷ O i Rigid body at t Linear velocity of a rigid body j ( V Oi ) = j [R] V Oi (2) Two different coordinate system involved: where differentiation done, and where described! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 6 / 13

ANGULAR VELOCITY OF RIGID BODY Angular velocity cannot be obtained as a time derivative of 3 quantities representing orientation Angular velocity from time derivative of rotation matrix Recall i [R] i [R] T = [U], Differentiate with respect to time t [U] is a 3 3 identity matrix i [R] i [R] T + i [R] i [R]T = [] Note: derivative of a matrix implies derivative of all components of the matrix Above equation can be written as i [R] i [R] T + ( i [R] i [R] T ) T = [] Define a 3 3 skew symmetric matrix i [Ω] R = i [R] i [R] T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 7 / 13

ANGULAR VELOCITY OF RIGID BODY SKEW SYMMETRIC MATRIX Skew-symmetric matrix in detail ωz s ω s y i [Ω] R = ωz s ωx s (3) ωy s ωx s Product of O i [Ω] R and (p x,p y,p z ) T R 3 is a cross-product ω i [Ω] R (p x,p y,p z ) T y s p z ω s z p y = ωz s p x ωx s p z = ω s ωx s p y ωy s i p p x (4) i [Ω] R called angular velocity matrix ω s i : angular velocity vector of {i} with respect to {} In contrast to linear velocity, angular velocity vector is not a straightforward differentiation of orientation variables! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 8 / 13

ANGULAR VELOCITY IN TERMS OF EULER ANGLES Angular velocity in terms of Z-Y-Z Euler angles Recall for α, β and γ as the Z-Y-Z Euler angles c α c β c γ s α s γ c α c β s γ s α c γ c α s β A B [R] = s α c β c γ + c α s γ s α c β s γ + c α c γ s α s β (5) s β c γ s β s γ c β Obtain A B [R] A B [R]T The X, Y and Z components of the angular velocity vector ω s x = γ cosα sinβ β sinα ω s y = γ sinα sinβ + β cosα (6) ω s z = γ cosβ + α ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 9 / 13

ANGULAR VELOCITY LEFT AND RIGHT INVARIANT i [Ω] R called right-invariant Derived from right multiplication i [R] i [R]T = [U] ω s i called the space-fixed angular velocity Superscript s i [R]T i [R] = [U] Another skew-symmetric matrix ω i [Ω] L = i [R] T z b ω b y i [R] = ωz b ωx b (7) ωy b ωx b Define an angular velocity vector ω b i components (ωx b,ωy b,ωz b ) from the three ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 1 / 13

ANGULAR VELOCITY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω b x = α cosγ sinβ + β sinγ ω b y = α sinβ sinγ + β cosγ (8) ω b z = α cosβ + γ i [Ω] L called left-invariant angular velocity matrix ω b i called body-fixed angular velocity vector of {i} with respect to {} Superscript b The two skew-symmetric matrices are related like two tensors i [Ω] R = i [R] i [Ω] L i [R] T (9) The two angular velocities are related as ω s i = i [R] ω b i (1) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 11 / 13

ANGULAR VELOCITY OF RIGID BODY (CONTD) {i} t ˆX {} Ẑ O i i p Rigid Body at t Rigid Body at t + t Ŷ Consider rigid body undergoing pure rotation about a fixed point Points O i (t) and O i (t + t) are coincident and only the elements of the rotation matrix i [R] change with time Point P located by i p, and fixed in {i} {i} t+ t Figure 2: Angular velocity of a rigid body ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 12 / 13

ANGULAR VELOCITY OF RIGID BODY (CONTD) Location of P in {} p = i [R]i p Since P is fixed in {i} ṗ = Vp = i [R] i p and since i [R] 1 = i [R]T, V p = i [R] i [R] T p = i [Ω] R p = ω s i p (11) The coordinate system {i} does not appear except in denoting that rigid body {i} is being considered Space-fixed angular velocity vector is said to be independent of the choice of the body coordinate system ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 13 / 13

ANGULAR VELOCITY OF RIGID BODY (CONTD) Using relation between i [Ω] R and i [Ω] L V p = i [R] i [Ω] L i [R] T p = i [R] i [Ω] L i p to get i [R] 1 V p = i [Ω] L i p Yielding i V p = i [Ω] L i p = ω b i i p (12) Again except for denoting the reference (fixed) coordinate system, the coordinate system {} does not appear! Body-fixed angular velocity vector is said to be independent of the choice of the fixed coordinate system Unless explicitly stated, space-fixed angular velocity vector derived from i [R] i [R]T is normally used in kinematic analysis ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 14 / 13

ANGULAR VELOCITY IN SERIAL MANIPULATOR ROTARY (R) JOINT For two links connected by a rotary (R) joint (see Module 2, Lecture 2, Slide # 29) The time derivative operation i [R] = i 1[R] i 1 i [R(ˆk,θ i )] i [R] i [R] T = d dt ( i 1[R] i 1 i [R(ˆk,θ i )]) ( i 1 i [R(ˆk,θ i )] T i 1[R] T ) Rewrite above equation as i [Ω] R = i 1[Ω] R + i 1[R] ( i 1 [Ṙ(ˆk,θ i )] i 1 i [R(ˆk,θ i )] T ) i 1[R] T i To simplify, use the result i 1 i [R(ˆk,θ i )] = e (i 1 i [K ]θ i ) where i 1 i [K ] is the skew-symmetric form of the rotation axis vector ˆk and θ i is the rotation at the R joint (see also Module 2, Lecture 1, Slide # 7) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 15 / 13

ANGULAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT ˆk is fixed in {i 1} and {i} d dt e(i 1 i [K ]θ i ) = i 1 i [K ] θ i e (i 1 From above and properties of a rotation matrix, i [K ]θ i ) i [Ω] R = i 1[Ω] R + i 1[R] i 1 i [K ] i 1[R] T θ i = i 1[Ω] R + i [K ] θ i and in terms of the space-fixed angular velocity ω ( ) ω i = ω i 1 + ˆki θ i Serial manipulators R joint axis is along Z axis Pre-multiply both sides by i [R] and simplify to get i ω i = i i 1[R] i 1 ω i 1 + θ i ( 1) T (13) i ω i denotes i [R] ω i i ω i not necessarily Equation (13) gives the angular velocity propagation in links of a serial manipulators connected by R joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 16 / 13

LINEAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT For two consecutive links in a serial manipulator (see Module 2, Lecture 2, Slide # 29) O i = O i 1 + i 1[R] i 1 O i Taking derivatives on both sides V Oi = V Oi 1 + ω i 1 i 1[R] i 1 O i Simplify and rewrite above as i V i = i i 1[R]( i 1 V i 1 + i 1 ω i 1 i 1 O i ) (14) Note: i V i and i 1 V i 1 denote i [R] V i and i 1 [R] V i 1, respectively They are not necessarily! Equation (14) gives the linear velocity vector propagation in links of a serial manipulator connected by rotary joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 17 / 13

VELOCITY PROPAGATION PRISMATIC (P) JOINTS Two links connected by a prismatic (P) joint (see Module 2, Lecture 2, Slide # 31) Prismatic joint allows relative translation between {1 i} and {i} Angular velocity is same Relative translation is along Z axis ḋi( 1) T Velocity propagation for P joint Angular velocity i ω i = i i 1[R] i 1 ω i 1 (15) Linear velocity i V i = i i 1[R]( i 1 V i 1 + i 1 ω i 1 i 1 O i ) + ḋi( 1) T (16) where i i 1 [R]i 1 ω i = i ω i and i i 1 [R]i 1 V i = i V i ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 18 / 13

VELOCITY PROPAGATION PLANAR 3R MANIPULATOR ˆX 3, ˆX Tool {Tool} Ŷ Tool Link 3 l 3 θ 3 ˆX 2 O 3 All joint axis are parallel and coming out of page {} is fixed Ŷ 1 Ŷ Ŷ 2 {2} Ŷ 3 {3} l 2 Link 2 θ 2 ˆX1 ω = V = {1} {} l 1 θ 1 Link 1 O 2 ˆX Links connected by rotary (R) joint Equations (13) and (14) give velocities of all links O 1 Figure 3: The planar 3R manipulator revisited ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 19 / 13

VELOCITY PROPAGATION PLANAR 3R MANIPULATOR For i=1 For i=2 1 ω 1 = ( θ 1 ) T 1 V 1 = 2 ω 2 = ( θ 1 + θ 2 ) T c 2 s 2 2 V 2 = s 2 c 2 1 For i=3 l 1 θ 1 = 3 ω 3 = ( θ 1 + θ 2 + θ 3 ) T (l 1 s 23 + l 2 s 3 ) θ 1 + l 2 s 3 θ 2 3 V 3 = (l 1 c 23 + l 2 c 3 ) θ 1 + l 2 c 3 θ 2 l 1 s 2 θ 1 l 1 c 2 θ 1 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 2 / 13

VELOCITY PROPAGATION PLANAR 3R MANIPULATOR For i = Tool Tool ω Tool = ( θ 1 + θ 2 + θ 3 ) T (l 1 s 23 + l 2 s 3 ) θ 1 + l 2 s 3 θ 2 Tool V Tool = (l 1 c 23 + l 2 c 3 + l 3 ) θ 1 + (l 2 c 3 + l 3 ) θ 2 + l 3 θ 3 Linear and angular velocity in {} ω Tool = ( θ 1 + θ 2 + θ 3 ) T (17) and V Tool = l 1 s 1 θ 1 l 2 s 12 ( θ 1 + θ 2 ) l 3 s 123 ( θ 1 + θ 2 + θ 3 ) l 1 c 1 θ 1 + l 2 c 12 ( θ 1 + θ 2 ) + l 3 c 123 ( θ 1 + θ 2 + θ 3 ) (18) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 21 / 13

SUMMARY Linear velocity of a point on rigid body Time derivative of position vector Angular velocity of a rigid body in terms of derivative of rotation matrix Two kinds Right invariant angular velocity (skew-symmetric) matrix: i [Ω] R = i [R] i [R]T Obtained from time derivative of i [R] i [R]T = [U] i [Ω] R gives rise space fixed angular velocity vector ω s i Left invariant angular velocity (skew-symmetric) matrix: i [Ω] L = i [R]T i [R] Obtained from time derivative of i [R]T i [R] = [U] i [Ω] L gives rise body fixed angular velocity vector ω b i Space fixed angular velocity vector used typically Propagation of linear and angular velocities between links Can easily obtain linear and angular velocity of any serial manipulator connected with rotary (R) and prismatic (P) joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 22 / 13

SERIAL MANIPULATOR JACOBIAN MATRIX Linear and angular velocity of {Tool} (equations (17) and (18)) can be written in a compact form as l 1 s 1 l 2 s 12 l 3 s 123 l 2 s 12 l 3 s 123 l 3 s 123 l 1 c 1 + l 2 c 12 + l 3 c 123 l 2 c 12 + l 3 c 123 l 3 c 123 θ 1 V Tool = θ 2 θ 3 1 1 1 (19) V Tool V Tool is a 6 1 entity V Tool = ω Tool ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 24 / 13

SERIAL MANIPULATOR JACOBIAN MATRIX V Tool is not a 6 1 vector 1 Contains linear velocity and the angular velocity which have different units! Use or ; to separate the linear and angular velocities & to remind that V Tool or ( V Tool ; ω Tool ) T is not a vector Matrix in square brackets, Tool [J(Θ)], is called the Jacobian matrix for the planar 3R manipulator Tool [J(Θ)] relate the linear and angular velocities of the Tool with the joint velocities Jacobian matrix is for the end-effector or the {Tool} See subscript Tool Linear and angular velocities are in {} See leading superscript 1 In theoretical kinematics, ( ω Tool ; V Tool ) is called twist (see Additional Material in Module 2) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 25 / 13

PROPERTIES OF MANIPULATOR JACOBIAN MATRIX Tool [J(Θ)] is not a true Jacobian matrix Tool [J(Θ)] is not obtained by direct differentiation of a vector-valued function, The first and the last three rows represent linear and angular velocity, Elements of the first three rows have units of length, elements of last three rows have no units Similar to V Tool, top and bottom halves of Tool [J(Θ)] matrix are separated by Many matrix operations on Tool [J(Θ)] makes no sense Finding the condition number 2 of this matrix is meaningless since it changes with the choice of length units 2 The condition number of a matrix is the ratio of the absolute value of the largest to the smallest eigenvalues ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 26 / 13

PROPERTIES OF MANIPULATOR JACOBIAN MATRIX (CONTD) Tool [J(Θ)] is best thought of as a map Tool [J(Θ)] : Θ V Tool The manipulator Jacobian matrix can be derived 3 for any serial manipulator with rotary and prismatic joints Compute the linear and angular velocities using propagation equations Rearrange in a matrix equation as done for the planar 3R manipulator Tool [J(Θ)] is very important in velocity kinematics of serial manipulators 3 The Jacobian matrix is defined for any differentiable vector-valued function X = Ψ(θ 1,,θ n ) The Jacobian matrix, [J(Θ)], is the matrix of first partial derivatives of Ψ with respect to θ i The i th column of [J(Θ)] is Ψ θ i ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 27 / 13

PROPERTIES OF MANIPULATOR JACOBIAN MATRIX (CONTD) The elements of the Jacobian matrix are non-linear functions of the joint variables Θ Manipulator in motion Tool [J(Θ)] is time varying At instant with Θ known, Tool [J(Θ)] relates linear and angular velocities to joint rates The relationship between joint rates and linear/angular velocities is linear! The Jacobian matrix can be obtained for any link Most often obtained for end-effector The Jacobian matrix is always with respect to a coordinate system Where the linear and angular velocities are obtained Most often Jacobian matrix is with respect to fixed {} Jacobian matrix can be written in any coordinate system using rotation matrices ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 28 / 13

PROPERTIES OF MANIPULATOR JACOBIAN MATRIX (CONTD) The Jacobian matrix is m n, where m is dimension of the motion space 4 and n is the number of actuated joints If Tool det( Tool [J(Θ)] is square, ie, m = n, and if the determinant [J(Θ)]), then Θ = Tool [J(Θ)] 1 V Tool (2) Above relationship gives joint velocities required for a desired linear and angular velocities of {Tool} Direct velocity kinematics V Tool = Tool [J(Θ)] Θ Inverse velocity kinematics Θ = Tool [J(Θ)] 1 V Tool 4 Same as λ in the definition of DOF in Module 3, Lecture 1, Slide # 6 m = 6 for R 3 and m = 3 for plane ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 29 / 13

GEOMETRIC INTERPRETATION OF MANIPULATOR JACOBIAN MATRIX ˆX 2 Ŷ {} O 1 l 1 (x, y) θ 1 Link 1 l 2 θ 2 O 2 ˆX Link 2 Consider a planar 2R manipulator shown in in Figure 4 The linear velocity V of the end-effector (point (x,y)) is ( ) V = ẋ ẏ ˆX 1 = [ l1 s 1 l 2 s 12 l 2 s 12 l 1 c 1 + l 2 c 12 l 2 c 12 where θ 1, θ 2 are joint rates ]( The matrix in square brackets is the Jacobian matrix in {} θ 1 θ 2 ) Figure 4: A planar 2R manipulator ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 3 / 13

GEOMETRIC INTERPRETATION OF MANIPULATOR JACOBIAN MATRIX Magnitude of linear velocity vector V 2 = V V = g11 θ 2 1 + 2g 12 θ 1 θ 2 + g 22 θ 2 2 (21) g ij, i,j = 1,2, are the elements of a matrix [ g ] = [J(Θ)] T [J(Θ)] For the planar 2R manipulator the g ij s are g 11 = l 2 1 + l 2 2 + 2l 1 l 2 c 2 g 12 = g 21 = l2 2 + l 1 l 2 c 2 g 22 = l2 2 (22) The elements g ij s are functions of θ 2 alone and g 22 is a constant g ij s could in general be function of all joint variables ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 31 / 13

GEOMETRIC (CONTD) INTERPRETATION Maximum and minimum V 2 subject to constraint θ 2 1 + θ 2 2 = 15 Solve V 2 / θ i =, i = 1,2, where V 2 = g 11 θ 2 1 + 2g 12 θ 1 θ 2 + g 22 θ 2 2 λ( θ 1 1 + θ 2 2 1) Partial differentiation reduces to an eigenvalue problem The eigenvalues are [ g ] Θ λ Θ = (23) λ 1,2 = (1/2){(g 11 +g 22 )±[(g 11 +g 22 ) 2 4(g 11 g 22 g 2 12)] 1/2 } 5 Without any constraint V R 2 and fills up R 2 The constraint θ 1 2 + θ 2 2 = 1 is similar to the unit speed constraint in differential geometry of space curves ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 32 / 13

GEOMETRIC (CONTD) INTERPRETATION [ g ] real, symmetric and positive definite Eigenvalues are always real and positive For λ 1 > λ 2, V max = λ 1, V min = λ 2 For square Jacobian matrix, eigenvalues of [J(Θ)] are λ 1 and λ 2 (see Strang 1976) Maximum and minimum V for 2R manipulator are λ 1 and λ 2 If θ 2 1 + θ 2 2 = k2 is used Maximum and minimum V are scaled by k ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 33 / 13

GEOMETRIC (CONTD) INTERPRETATION From V = [J(Θ)] Θ, [J] T V = [ g ] Θ For non-singular [ g ], V T ([J][ g ] 1 )([J][ g ] 1 ) T V = Θ T Θ For a planar 2R manipulator, ([J][ g ] 1 )([J][ g ] 1 ) T is symmetric and of rank 2 Hence for Θ T Θ = 1, (ẋ,ẏ) T ([J][ g ] 1 )([J][ g ] 1 ) T (ẋ,ẏ) = 1 x T [A]x = 1, with [A] symmetric and non-singular, describes an ellipse The tip of the linear velocity vector traces an ellipse and the semi-major and semi-minor axes of the ellipse are λ 1 and λ 2, respectively For Θ T Θ = k 2, size of ellipse is scaled by k, but shape of ellipse does not change with k ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 34 / 13

GEOMETRIC INTERPRETATION (CONTD) Eigenvalues of [ g ] are only functions of θ 2 Shape and size of ellipse will change with θ 2 Can plot ellipses at all points in the workspace Recall: workspace of a planar 2R is the area between two circles of radii l 1 + l 2 and l 1 l 2 Ellipse independent of θ 1 All ellipses at a chosen radius (in the annular region) are same! Ŷ {} O 1 l 1 (x, y) θ 1 Link 1 Link 2 l 2 O 2 Figure 5: Velocity ellipse for a planar 2R manipulator ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 35 / 13 ˆX θ 2

GEOMETRIC (CONTD) INTERPRETATION The shape of the velocity ellipse indicates which directions are easier to move for given joint rates V is larger along major axis Easier to move along major axis Less easier to move along the minor axis Ellipse reduces to a circle Equally easy to move in all directions All points in the workspace, where the ellipse is a circle, are called isotropic (see Salisbury, 1982) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 36 / 13

GEOMETRIC (CONTD) INTERPRETATION Isotropic configuration Eigenvalues of [J(Θ)] (or [ g ]) are equal For planar 2R, eigenvalues equal only if g 11 = g 22 and g 12 = From the expressions of g ij s above conditions imply that l 2 1 + 2l 1 l 2 c 2 = and l 2 2 + l 1 l 2 c 2 = and this is only possible if l 1 = 2l 2 and c 2 = 1 2 A planar 2R manipulator can posses isotropic configurations only if the link lengths have a ratio of 2, and θ 2 = 135 Since θ 1 [,2π], all the isotropic configurations lie on a circle Degenerate form of velocity ellipse Singular configuration (see Lecture 4) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 37 / 13

GEOMETRIC (CONTD) INTERPRETATION Spatial motion & 2 degree-of-freedom Velocity vector on tangent plane to a surface Velocity ellipse Spatial motion & 3 degree-of-freedom Velocity vector lies in R 3 Velocity ellipsoid Same ideas can be extended to angular velocity vector Extension to 6 6 manipulator Jacobian matrix More complicated since not a dimensionally homogeneous matrix (see Ghosal and Ravani (1998), Bandyopadhyay and Ghosal (24b) and references in them) Need to use notions of screws and twists (see Hunt, 1976) Velocity ellipse Cylindroid & Two screw system Velocity ellipsoid Hyperboloid & Three screw system Extension to parallel manipulators using parallel manipulator Jacobian ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 38 / 13

RESOLUTION OF REDUNDANCY AT VELOCITY LEVEL For square Jacobian Matrix can be inverted to obtain joint rates Redundant systems (see Module 3, Lecture 3, Slide # 48) Jacobian matrix is not square Number of joint variables more than 6 (for R 3 ) or more than 3 (for R 2 ) Jacobian matrix cannot be inverted to obtain joint rates given linear and angular velocity of end-effector Use of pseudo-inverse (Strang, 1976) to resolve redundancy Pseudo-inverse of m n (n > m) matrix [J(Θ)] [J(Θ)] # = [J(Θ)] T ([J(Θ)][J(Θ)] T ) 1 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 39 / 13

RESOLUTION OF REDUNDANCY AT VELOCITY LEVEL Some properties of pseudo-inverse Dimension of [J(Θ)] # is n m Not square! Left inverse [J(Θ)][J(Θ)] # = [U] Identity matrix Not a right inverse [J(Θ)] # [J(Θ)] [U] General solution to V Tool = Tool [J(Θ)] Θ is Θ = [J(Θ)] # V Tool + ([U] [J(Θ)] # [J(Θ)]) W ([U] [J(Θ)] # [J(Θ)]) W lies in the null-space of [J(Θ)] Pseudo-inverse, without null-space, minimises Θ T Θ The null-space term used to avoid obstacles, joint limits and to maximise a manipulability index det([[j(θ)][j(θ)] T ] 1/2 )(see Nakamura, 1991) Disadvantages Local numerical scheme No global or analytical results Velocity level and not a position and orientation level scheme for resolution of redundancy ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 4 / 13

SUMMARY Propagation of velocities can be used to obtain linear and angular velocity of end-effector in terms of joint rates Manipulator Jacobian matrix relate end-effector linear and angular velocity to joint rates Manipulator Jacobian matrix is not dimensionally homogeneous! Geometric interpretation of manipulator Jacobian for linear and angular velocity can be done separately More complicated description in terms of screw cylindroid etc when rigid body motion is considered! Velocity ellipse and ellipsoid Ease of motion direction Resolution of redundancy at velocity level uses pseudo-inverse of manipulator Jacobian ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 41 / 13

PARALLEL MANIPULATOR JACOBIAN MATRIX Parallel manipulators has both actuated and passive joints q = (θ,ϕ) T Loop-closure equations do not contain all joint variables No natural choice of end-effector {Tool} No velocity propagation Platform type parallel manipulator Position of centroid & orientation of platform {Tool} is of interest Linear and angular velocity of centroid and {Tool} ω Tool = d dt ( Tool [R]) Tool [R]T = Tool [J ω (q)] q V Tool = 1 3 ( p 1 + p 2 + p 3 ) = Tool [J V(q)] q (24) Tool [J ω (q)], Tool [J V(q)] Angular and linear velocity Jacobian q Time derivatives of configuration variables q ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 43 / 13

ELIMINATION OF PASSIVE JOINT RATES Linear and angular velocity function of all q and q Only the actuated joints θ i, i = 1,2,,n are specified The m passive ϕ i s can be obtained from direct kinematics Need expression for ϕ i and obtain linear and angular velocities in terms of only θ i s Derived from the m loop-closure or constraint equations ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 44 / 13

ELIMINATION OF PASSIVE JOINT RATES For m passive variables, m constraint equations η i (q 1,,q n+m ) =,i = 1,,m or in a vector form η(q) = η(θ,ϕ) = (25) Differentiate equation (25) with respect to t, and rearrange [K(q)] θ + [K (q)] ϕ = (26) Columns of the m n matrix [K(q)] are the partial derivatives of η(q) with respect to the actuated variables θ i, i = 1,,n, Columns of m m matrix [K (q)] are the partial derivatives of η(q) with respect to the passive variables ϕ i, i = 1,,m [K (q)] is always an m m square matrix [K(q)] and [K (q)] are functions q = (θ,ϕ) R n+m ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 45 / 13

ELIMINATION OF PASSIVE JOINT RATES If det([k ]), ϕ = [K ] 1 [K] θ (27) The angular and linear velocity can partitioned as ω Tool = [Jω ] θ + [J ω ] ϕ, V Tool = [J V ] θ + [J V ] ϕ Substitute ϕ to get ω Tool = ([Jω ] [J ω ][K ] 1 [K]) θ V Tool = ([J V ] [J V ][K ] 1 [K]) θ Define equivalent [Jω ] eq and [J V ] eq [J V ] eq = [JV ] [J V ][K ] 1 [K] (28) [Jω ] eq = [J ω ] [J ω ][K ] 1 [K] (29) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 46 / 13

EQUIVALENT JACOBIAN MATRIX IN PARALLEL MANIPULATORS Using [J V ] eq and [Jω ] eq V Tool V Tool = ω Tool = Tool [J eq] θ (3) The 6 n matrix, Tool [J eq], consists of 3 n rows from [J V ] eq and 3 n rows from [Jω ] eq The matrix Tool [J eq] is the Jacobian matrix 6 for parallel manipulators At a known q, equation (3) relate actuated joint rates θ to the linear and angular velocity of chosen end-effector {Tool} 6 As in serial manipulators, not a true Jacobian as it is not obtained from the derivative of a vector valued function ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 47 / 13

EQUIVALENT JACOBIAN MATRIX IN PARALLEL MANIPULATORS The matrix Tool [J V] eq can be used define a [ g V ] eq for parallel manipulators [ g V ] eq = ([J V ] [J V ][K ] 1 [K]) T ([J V ] [J V ][K ] 1 [K]) (31) [ g V ] eq is symmetric and positive definite Similar to a serial manipulator, the tip of the linear velocity vector lies on an ellipse or an ellipsoid Much more complicated than in serial manipulators! [ gω ] eq defined using [Jω ] eq Angular velocity ellipse or ellipsoid The above geometrical description is valid if det[k ] ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 48 / 13

EXAMPLE PLANAR 4-BAR MECHANISM Constraint equation of a four-bar (see Module 4, Lecture 1, Slide #16 ) η 1 (q) η 2 (q) = l 1 cosθ 1 + l 2 cos(θ 1 + ϕ 2 ) l l 3 cosϕ 1 = = l 1 sinθ 1 + l 2 sin(θ 1 + ϕ 2 ) l 3 sinϕ 1 = θ 1 is the actuated joint variable and (ϕ 1,ϕ 2 ) are the passive joint variables Derivative of constraint equations with respect to time t gives ( ) l1 sinθ 1 l 2 sin(θ 1 + ϕ 2 ) l 1 cosθ 1 + l 2 cos(θ 1 + ϕ 2 ) ( l3 sinϕ 1 l 2 sin(θ 1 + ϕ 2 ) l 3 cosϕ 1 l 2 cos(θ 1 + ϕ 2 ) θ 1 + )( ϕ 1 ϕ 2 ) = ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 49 / 13

EXAMPLE PLANAR 4-BAR MECHANISM [K] and [K ] matrices for the planar 4-bar are ( ) l1 sinθ [K] = 1 l 2 sin(θ 1 + ϕ 2 ) l 1 cosθ 1 + l 2 cos(θ 1 + ϕ 2 ) [ [K l3 sinϕ ] = 1 l 2 sin(θ 1 + ϕ 2 ) l 3 cosϕ 1 l 2 cos(θ 1 + ϕ 2 ) The matrix [K ] is a square 2 2 matrix [K] and [K ] matrices are functions of the actuated and passive variables Fairly simple to calculate for planar 4-bar Multi-degree-of-freedom spatial mechanisms Use symbolic algebra software such as MAPLE R ] ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 5 / 13

EXAMPLE 3-RPS PARALLEL MANIPULATOR For the 3-RPS manipulator, loop-closure equations are η 1 (q) = 3 3a 2 + l 2 1 + l 2 2 + l 1l 2 c 1 c 2 2l 1 l 2 s 1 s 2 3l 1 c 1 3l 2 c 2 = η 2 (q) = 3 3a 2 + l 2 2 + l 2 3 + l 2l 3 c 2 c 3 2l 2 l 3 s 2 s 3 3l 2 c 2 3l 3 c 3 = η 3 (q) = 3 3a 2 + l 2 3 + l 2 1 + l 3l 1 c 3 c 1 2l 3 l 1 s 3 s 1 3l 3 c 3 3l 1 c 1 = Perform the derivative of η i (q), i = 1,2,3, with respect to time and rearrange to obtain [K] and [K ] [K] involves derivative with respect to the actuated variables l 1, l 2 and l 3 2l 1 3c 1 + l 2 c 1 c 2 2l 2 s 1 s 2 2l 2 3c 2 + l 1 c 1 c 2 2l 1 s 1 s 2 2l 2 3c 2 + l 3 c 2 c 3 2l 3 s 2 s 3 2l 3 3c 3 + l 2 c 2 c 3 2l 2 s 2 s 3 2l 1 3c 1 + l 3 c 1 c 3 2l 3 s 1 s 3 2l 3 3c 3 + l 1 c 1 c 3 2l 1 s 1 s 3 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 51 / 13

EXAMPLE 3-RPS PARALLEL MANIPULATOR [K ] involves derivative with respect to passive joint variables, θ 3l 1s 1 l 1 l 2 s 1 c 2 2l 1 l 2 c 1 s 2 3l 2 s 2 l 1 l 2 c 1 s 2 2l 1 l 2 s 1 c 2 3l 2 s 2 l 2 l 3 s 2 c 3 2l 2 l 3 c 2 s 3 3l 3 s 3 l 2 l 3 c 2 s 3 2l 2 l 3 s 2 c 3 3l 1 s 1 l 1 l 3 s 1 c 3 2l 1 l 3 c 1 s 3 3l 3 s 3 l 1 l 3 c 1 s 3 2l 1 l 3 s 1 c 3 For the centroid, [J V ] and [J V ], are c 1 (1/2)c 2 (1/2)c 3 [J V ] = (1/3) ( 3/2)c 2 ( 3/2)c 3 s 1 s 2 s 3 and [J V ] = (1/3) l 1 s 1 (1/2)l 2 s 2 ( 3/2)l 2 s 2 ( 1/2)l 3 s 3 ( 3/2)l 3 s 3 l 1 c 1 l 2 c 2 l 3 c 3 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 52 / 13

[Jω ] = EXAMPLE 3-RPS PARALLEL MANIPULATOR To obtain [Jω ] and [Jω ], compute d dt (Base Top [R]) Base Top [R]T and then rearrange Expressions are too large For l 1 = 2/3, l 2 = 3/5, l 3 = 3/4 and corresponding θ 1 = 7593, θ 2 = 2851, θ 3 = 828 radians, 644 281 9548 11519 1361 4815 1953 5339 383, [Jω ] = 37 6686 3398 3961 4256 1713 369 38 1353 Expressions for [J V ] eq and [Jω ] eq are more harder to obtain as [K ] 1 is needed For above numerical values [J V ] eq = 2313 5372 114 722 6758 1951 11765 1683 9223, [Jω ] eq = 2149 64331 4665 72 41216 1648 1565 457 3285 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 53 / 13

EXAMPLE 3-RPS PARALLEL MANIPULATOR For a = 1/2, and (l 1,l 2,l 3 ) = (5,1,2) meters (θ 1,θ 2,θ 3 ) = (4,7535,242) radians by direct kinematics (see Module 4, Lecture 2, Slide # 34) Tip of linear velocity vector of centroid lies on an ellipsoid Shown in Figure 6 as three sectional views and a 3D plot Maximum, intermediate, and minimum velocities along the principal axes of the ellipsoid are 3724, 3162, 231 m/sec, respectively The directions of principal axes are (9921, 394,1187) T, (1166,6338, 7646) T and ( 452,7724,6335) T, respectively ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 54 / 13

EXAMPLE 3-RPS PARALLEL MANIPULATOR 4 4 2 2 Vy Vz 2 2 4 4 2 2 4 Vx 4 4 2 2 4 Vx 4 2 5 Vz Vz 2 4 4 2 2 4 Vy 5 5 Vy 5 5 Vx 5 Figure 6: Velocity ellipsoid at a non-singular point ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 55 / 13

SUMMARY Parallel manipulator Jacobian in terms of active and passive joint variables Two more matrices, [K] and [K ], arise from derivative of constraint equations Can solve for passive joint rates ϕ i and obtain equivalent Jacobian matrix Can obtain equivalent Jacobian only if det[k ] Can obtain geometric interpretation as in serial manipulators Ellipse and ellipsoids More difficult to obtain numerical results Elimination of passive variables! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 56 / 13

SERIAL MANIPULATORS REVIEW Direct velocity kinematics V Tool = Tool [J(Θ)] Θ For known Θ and Θ, linear and angular velocity of end-effector obtained from above equation V Tool always exists Inverse velocity kinematics Θ = Tool [J(Θ)] 1 V Tool Joint rates can be obtained when Jacobian matrix is square, and det( Tool [J(Θ)]) det( Tool [J(Θ)]) = Loss of rank of Tool [J(Θ)] Singular configuration At singular configuration, Θ cannot be obtained for given linear and angular velocity ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 58 / 13

SINGULARITY IN PLANAR 2R MANIPULATOR For a planar 2R manipulator ( ẋ ẏ The Jacobian matrix is ) [ l1 s = 1 l 2 s 12 l 2 s 12 l 1 c 1 + l 2 c 12 l 2 c 12 ]( ) θ 1 θ 2 [ Tool [J(Θ)] = l1 s 1 l 2 s 12 l 2 s 12 l 1 c 1 + l 2 c 12 l 2 c 12 det( Tool [J(Θ)]) = sinθ 2 = This implies θ 2 =,π Second link is stretched completely or folded on top of first link ] ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 59 / 13

SINGULARITY IN PLANAR 2R MANIPULATOR Possible directions of velocity vector Ŷ {} O1 radius = l1 + l2 θ1 radius = l1 l2 l1 Link 1 O2 ˆX l2 Link 2 Figure 7: Singular configurations for a planar 2R manipulator (l 1 > l 2 ) Planar 2R manipulator for θ 2 =,π End-effector can only move perpendicular to the line O 1 O 2 connecting the two rotary joints The end-effector cannot have a velocity component along the second link Instantaneous loss (only at this configuration) of one degree of freedom ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 6 / 13

SINGULARITY IN PLANAR 2R MANIPULATOR At θ 2 =,π, the velocity ellipse degenerates to a line Along the possible direction of motion, see Figure 7 For the 2R planar example, the Jacobian matrix can be inverted easily, and ( θ 1 θ 2 ) = 1 l 1 l 2 s 2 ( l 2 c 12 l 2 s 12 l 1 c 1 l 2 c 12 l 1 s 1 l 2 s 12 )( ẋ ẏ As θ 2 or π, s 2, and ( θ 1, θ 2 ) T Knowledge of singularity is important When det( Tool [J(Θ)]) is close to zero, joint velocities tend to become large and cause problems for controller of robot Singularities occurs in all serial manipulator and not only in planar 2R For planar 2R, singularity only at workspace boundaries In other manipulators can happen elsewhere also! ) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 61 / 13

PARALLEL MANIPULATORS RE- VIEW In parallel manipulators, equivalent Jacobian need to be used For parallel manipulators, the linear and angular velocity Jacobians are Equivalent Jacobian [J V ] eq = [JV ] [J V ][K ] 1 [K] [Jω ] eq = [J ω ] [J ω ][K ] 1 [K] Tool [J] eq = [J V ] eq [Jω ] eq det( Tool [J eq]) = End-effector loses one or more degrees of freedom Actuated joint rates Θ Similar to serial manipulators At singularity velocity ellipse or ellipsoid degenerates ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 62 / 13

PARALLEL MANIPULATORS GAIN SINGULARITY In addition to loss singularity, there exists a second kind of singularity in parallel manipulators θ = Actuated joints locked Mechanism becomes a structure Equation [K(q)] θ + [K (q)] ϕ = becomes [K (q)] ϕ = From linear algebra, non-zero solution ϕ exists when det([k ]) = ϕ is the eigenvector corresponding to the zero eigenvalue of [K ] ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 63 / 13

PARALLEL MANIPULATORS GAIN SINGULARITY For a non-zero ϕ, and θ =, ω Tool = [Jω ] ϕ V Tool = [J V ] ϕ Even with actuators locked the linear and angular velocity are non-zero The end-effector of the parallel manipulator can instantaneously gain one or more degrees of freedom Termed as gain singularity ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 64 / 13

PARALLEL MANIPULATORS GAIN SINGULARITY Geometric picture: Non-singular configuration At non-singular configurations, θ = ϕ = V Tool = At a non-singular position velocity ellipsoid is of zero size Geometric picture: Gain singularity configuration Loss of rank of [K ] If rank is (m 1) There exists non zero eigenvector ϕ 1 for the zero eigenvalue of [K ] C 1 ϕ also an eigenvector with C 1 a scaling constant For θ = V Tool = C 1 [J V ] ϕ 1 There can be motion along [J V ] ϕ 1! The zero velocity ellipsoid grows into a line ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 65 / 13

PARALLEL MANIPULATORS GAIN SINGULARITY If rank of matrix [K ] is (m 2), then V Tool = C 1 [J V ] ϕ 1 + C 2 [J V ] ϕ 2 ϕ 1, ϕ 2 are eigenvectors from the two zero eigenvalues of [K ] C 1, C 2 are the two scaling constants For C 2 1 + C 2 2 = 1, tip of velocity vector traces an ellipse7 If rank of [K ] is (m 3), then tip of velocity vector will lie on an ellipsoid If rank is less than (m 3) and only V Tool is of interest Similar to a redundant serial manipulator 7 C 1 and C 2 are similar to θ 1 and θ 2 and C1 2 + C 2 2 = 1 is similar to the constraint θ 1 2 + θ 2 2 = 1 used in the planar 2R example Using same reasoning as in 2R case, the tip of V Tool for a parallel manipulator lies on an ellipse ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 66 / 13

GAIN SINGULARITY Gain singularity occurs in parallel/hybrid manipulators In fully-parallel six DOF manipulators (end-effector directly connected to base by one actuated joint Stewart-Gough platform) only gain singularity possible (Hunt, 1991) In six DOF hybrid/parallel manipulator (example three-fingered hand, Module 2, Lecture 3, Slide # 66 and Module 4, Lecture 2, Slide # 38) both loss and gain singularity possible Gain singularity is related to capability of resisting external force or moments (see Lecture 5) Large amount of literature on singularity analysis of parallel manipulators (for example, Hunt (1986), Litvin et al (199), Merlet (1991), Gosselin and Angeles (199), Zlatanov (1995), Park and Kim (1999)) Singularity analysis & uses of singularity is an active topic of research (see Bandyopadhyay and Ghosal (29), Ranganath et al (24)) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 67 / 13

GAIN SINGULARITY (CONTD) Special link lengths and geometry Gain over finite range of motion Passive link can show instantaneous and finite dwell O3 l2 ŶL {L} Link 2 O2 l1 Link 1 θ1 ˆXL l3 Link 3 Link 4 l4 ŶR {R} θ2 Link 2 and link 3 can rotate from to 2π with θ 1 and θ 2 locked (see Bandyopadhyay and Ghosal (24a) for details) l OR ˆXR OL Figure 8: Finite motion at gain singularity ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 68 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES O2 φ2 Link 2 ŶL {L} l1 Link 1 l2 O3 ŶR {R} l3 Link 3 φ1 θ1 ˆXL l ˆXR OR OL, O1 Figure 9: Singular configuration for a planar four-bar mechanism det([k ]) = gives l 2 l 3 sin(θ 1 + ϕ 2 ϕ 1 ) = θ 1 + ϕ 2 ϕ 1 = nπ ϕ 3 = 2π, Link 2 and 3 are parallel Instantaneous gain: θ 1 locked, point O 2 is fixed Link 2 and link 3 along a straight line O 3 can have instantaneous velocity along the common tangent ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 69 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example The 3-RPS parallel manipulator (see Basu and Ghosal, 1997) For the 3-RPS parallel manipulator, det([j V ] eq ) = Linear velocity ellipsoid described by the centroid of the top platform degenerates to an ellipse 8 For (l 1,l 2,l 3 ) = (5,1,1971) meters and (θ 1,θ 2,θ 3 ) = (11691,4781,2355) radians det([j V ] eq ) = The linear velocity ellipse at this configuration is shown in sectional and a 3D view in Figure 1 Not a contradiction to result by Hunt (1991) The 3-RPS parallel manipulator is not a six DOF manipulator, and Only the linear velocity vector of the centroid is considered 8 See Ghosal and Ravani (21) for more details ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 7 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example The 3-RPS parallel manipulator 2 2 1 1 Vy Vz 1 1 2 2 1 1 2 Vx 2 2 1 1 2 Vx 2 1 2 Vz Vz 1 2 2 1 1 2 Vy 2 2 Vy 2 2 Vx 2 Figure 1: Linear velocity ellipse at a loss singular point ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 71 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example 3-RPS parallel manipulator Gain one or more degrees-of-freedom when det([k ]) = ie, det([k ]) = (3l 1 s 1 l 1 l 2 s 1 c 2 2l 1 l 2 c 1 s 2 ) (3l 2 s 2 l 2 l 3 s 2 c 3 2l 2 l 3 c 2 s 3 ) (3l 3 s 3 l 1 l 3 c 1 s 3 2l 1 l 3 s 1 c 3 ) +(3l 1 s 1 l 1 l 3 s 1 c 3 2l 1 l 3 c 1 s 3 ) (3l 2 s 2 l 1 l 2 c 1 s 2 2l 1 l 2 s 1 c 2 ) (3l 3 s 3 l 2 l 3 c 2 s 3 2l 2 l 3 s 2 c 3 ) = det([k ]) is a function of all (θ,ϕ) det([k ]) = and three loop-closure equations Four equations in six variables A 2D surface Difficult to eliminate (see Module 3, Lecture 4) and get analytical expression ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 72 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example 3-RPS parallel manipulator For (l 1,l 2,l 3 ) = (575,483,544), and (θ 1,θ 2,θ 3 ) = ( 3441, 138,232) radians, det[k ] The eigenvalues of [K ] are approximately 5565, and 459 The three corresponding eigenvectors are ( 898,3571, 4656) T, ( 319, 8743, 3727) T and ( 877, 4781, 8739) T Gained velocity of centroid is V Tool = 647 184 θ 1 + 11 19 161 θ 2 + 28 361 1763 ( θ 1, θ 2, θ 3 ) T = α ( 319, 8743, 3727) T with α arbitrary θ 3 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 73 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example 3-RPS parallel manipulator 5 1 5 Vy Vz 5 5 1 5 5 1 Vx 1 1 5 5 1 Vx 1 Vz 5 Vz 1 5 1 5 5 Vy 1 5 Vy 5 1 Vx 1 Figure 11: Velocity at a gain singular point ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 74 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example 3-RPS parallel manipulator At (l 1,l 2,l 3 ) = (19363,29998,19363) meters Corresponding (θ 1,θ 2,θ 3 ) = (1396,9817,1396) radians, det[k ] Eigenvalues are approximately,, 3968 At this configuration, gain of two degrees of freedom The singularities corresponding to gain of two degrees of freedom lie on a curve in R 3 Difficult to get analytical expression ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 75 / 13

SINGULARITIES IN PARALLEL MANIPULATORS EXAMPLES Example 3-RPS parallel manipulator Vy 15 1 5 5 1 15 4 2 2 4 Vx Vz 1 5 5 1 4 2 2 4 Vx 1 Vz 5 Vz 1 5 1 2 1 1 2 Vy 1 2 Vy 2 5 Vx 5 Figure 12: Velocity ellipse at a gain singular point ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 76 / 13

SUMMARY Singularity in serial manipulators Loss of rank of manipulator Jacobian Loss of one or more DOF Velocity ellipsoid/ellipse degenerates Not possible to move along singular direction(s) Singularity in parallel manipulators Loss singularity Loss of DOF as in serial manipulators Can also gain one or more DOF with actuators locked Gain of DOF due to loss of rank of [K ] and det[k ] = Examples of a planar serial 2R and a 3-RPS parallel manipulator ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 77 / 13

STATICS OF SERIAL AND PARALLEL MANIPULATORS Joints of a serial manipulator are locked Manipulator becomes a structure Forces and moments acting at joints when manipulator structure is subjected to external forces and moments External forces and moments on end-effector if pushing some object or carrying a payload Useful to know joint forces or torques which can maintain the static equilibrium Use free-body diagram ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 79 / 13

SERIAL MANIPULATORS STATICS n i+1 {i + 1} Ẑi+1 f i+1 Ẑ i Ŷ i+1 {i} i O i+1 n i f i Figure 13: Ŷ i O i Link i ˆX i Free-body diagram of a link O i+1 ˆX i+1 Two intermediate rotary (R) joints and a link of a manipulator f i and n i denote the forces and moments exerted on link {i} by link {i 1} ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 8 / 13

SERIAL MANIPULATORS STATICS For static equilibrium of {i}, ΣF = i f i i f i+1 = f i+1 is the force on link {i + 1} exerted by link {i} Force on link {i} exerted by link {i + 1} will be equal and of opposite sign The leading superscript i signifies that the vectors are described in {i} For static equilibrium of {i}, ΣM = i n i i n i+1 i O i+1 i f i+1 = i O i+1 is the vector from O i to O i+1 Negative sign due to same reason as for forces ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 81 / 13

SERIAL MANIPULATORS STATICS Equilibrium equations can be written as i f i = i i+1[r] i+1 f i+1 i n i = i i+1[r] i+1 n i+1 + i O i+1 i f i (32) Inward recursion for forces and moments on each link Forces and moments at the end-effector: n+1 f n+1 = n+1 n n+1 = if not in contact with environment n+1 f n+1, n+1 n n+1 known otherwise Recursively compute i f i, i n i for i : n to 1 using equation (32) Joint can only apply force or moment along Ẑ axis; all other components resisted by structure/bearings Torque required at joint i to maintain equilibrium τ i = i n i i Ẑ i (joint i is rotary (R)) τ i = i f i i Ẑ i (joint i is prismatic (P)) (33) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 82 / 13

STATICS EXAMPLE 3R PLANAR MANIPULATOR 3R planar manipulator applying force f Tool = (f x, f y, ) T n Tool = (,, n z ) T f ˆX3, ˆXTool {Tool} ŶTool n Link 3 lf3 θ3 τ3 Link 2 l2 ˆX2 O3 In {Tool} coordinate system f x c 123 s 123 f x f y = s 123 c 123 f y 1 Ŷ {} τ1 θ1 l1 Link 1 τ2 θ2 O2 ˆX1 and (,, n z) T = (,, n z ) T O1 ˆX Figure 14: A 3R manipulator applying force and moment ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 83 / 13

STATICS EXAMPLE 3R PLANAR MANIPULATOR i=3 i=2 i=1 3 f 3 = (f x, f y, ) T 3 n 3 = (,, n z + l 3 f y ) T 2 f 2 = (c 3 f x s 3 f y, s 3 f x + c 3 f y, ) T 2 n 2 = (,, n z + l 2 (s 3 f x + c 3 f y ) + l 3 f y ) T 1 f 1 = (c 23 f x s 23 f y, s 23 f x + c 23 f y, ) T 1 n 1 = (,, n z + l 1 (s 23 f x + c 23 f y ) + l 2 (s 3 f x + c 3 f y ) + l 3 f y ) T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 84 / 13

STATICS EXAMPLE 3R PLANAR MANIPULATOR Finally, the joint torques required to maintain equilibrium τ 1 = 1 n 1 1 Ẑ 1 = n z + f x(l 1 s 23 + l 2 s 3 ) + f y (l 1 c 23 + l 2 c 3 + l 3 ) τ 2 = 2 n 2 2 Ẑ 2 = n z + f xl 2 s 3 + f y (l 2 c 3 + l 3 ) τ 3 = 3 n 3 3 Ẑ 3 = n z + f y l 3 Above equations can be re-arranged as τ = f x l 1 s 1 l 2 s 12 l 3 s 123 l 1 c 1 + l 2 c 12 + l 3 c 123 1 f y l 2 s 12 l 3 s 123 l 2 c 12 + l 3 c 123 1 l 2 s 123 l 3 c 123 1 n z (34) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 85 / 13

FORCE TRANSFORMATION MATRIX Term in the square bracket is the transpose of the Jacobian matrix (see equation (19)) As in velocities, denote forces and moments acting on the end-effector by f Tool F Tool = = (f x f y f z ; n x n y n z ) T (35) n Tool Note: F Tool is not a 6 1 vector since forces and moments have different units F Tool is called a wrench in theoretical kinematics, and a wrench can be thought of as screw with a magnitude which has units of force ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 86 / 13

FORCE TRANSFORMATION MATRIX Consider an infinitesimal Cartesian displacement of the end effector δx Tool 9 and the virtual work done by F Tool Equating virtual work done by external force/moment and at joints F Tool δx Tool = ftool δx + n Tool δθ = τ δθ From definition of Jacobian, δx Tool = Tool [J(Θ)]δΘ, F Tool Tool [J(Θ)]δΘ = τ δθ Above equations hold true for all δθ, hence τ = Tool [J(Θ)]T F Tool (36) Not surprising transpose of Jacobian appears in statics! 9 The entity δx Tool is 6 1 and it is like a twist The infinitesimal change in position and orientation could be (δx;δθ) T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 87 / 13

PARALLEL MANIPULATORS STATICS For serial manipulators τ = Tool [J(Θ)]T F Tool Principle of virtual work equally applicable for parallel manipulator τ = Tool [J eq] T F Tool {Tool} is a chosen end-effector Tool [J eq] is the equivalent Jacobian Function of q, and τ is the vector of forces or torques applied at the actuated joints only Difficult to compute Tool [J eq] due to [K ] 1 Inverse problem: Obtaining forces/moments applied by {Tool} F Tool = Tool [J(q) eq ] T τ Inverse of Jacobian even more difficult! Simpler approach for Gough-Stewart platform ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 88 / 13

PARALLEL MANIPULATORS STATICS {P } P p i P i S Joint P Joint The vector along the leg, B S i, B t B b i l i Ẑ Ŷ B i ˆX B S i = B P [R] P p i + B t B b i Unit vector along leg B s i = B S i l i {B } U Joint Figure 15: A leg of a Stewart-Gough platform revisited ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 89 / 13