# ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

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1 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 , India NPTEL, 2010 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

2 ...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 MODULE 5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

3 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 MODULE 5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

4 INTRODUCTION REVIEW 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, / 98

5 INTRODUCTION REVIEW 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, / 98

6 INTRODUCTION REVIEW 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, / 98

7 INTRODUCTION REVIEW 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, / 98

8 INTRODUCTION REVIEW 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, / 98

9 INTRODUCTION REVIEW 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, / 98

10 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 MODULE 5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

11 LINEAR AND ANGULAR VELOCITY OF RIGID BODY LINEAR VELOCITY OF RIGID BODY The linear velocity of O i with respect to {0} is defined as 0 V Oi = d dt 0 0 O i (t + t) 0 O i (t) O i (t) = lim t 0 t (1) {0} 0 O i (t) Ẑ {i}(t) {i}(t + t) O i Rigid body at t + t 0 denote the coordinate system {0} where the limit is taken. The linear velocity vector can be described in {j} as ˆX Figure 1: 0 O i (t + t) Ŷ O i Rigid body at t Linear velocity of a rigid body j (0 V Oi ) = j 0 [R]0 V Oi (2) Two different coordinate system involved: where differentiation done, and where described! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

12 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF A RIGID BODY Angular velocity cannot be obtained as a time derivative of 3 quantities representing orientation. Angular velocity can be derived from time derivative of rotation matrix. Recall 0 i [R] 0 i [R] T = [U], Differentiate with respect to time t [U] is a 3 3 identity matrix 0 i [R] 0 i [R] T + 0 i [R] 0 i [R]T = [0] where derivative of a matrix implies derivative of all components of the matrix. Above equation can be written as 0 i [R] 0 i [R] T + ( 0 i [R] 0 i [R] T ) T = [0] Define a 3 3 skew symmetric matrix 0 i [Ω] R = 0 i [R] 0 i [R] T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

13 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF A RIGID BODY Angular velocity cannot be obtained as a time derivative of 3 quantities representing orientation. Angular velocity can be derived from time derivative of rotation matrix. Recall 0 i [R] 0 i [R] T = [U], Differentiate with respect to time t [U] is a 3 3 identity matrix 0 i [R] 0 i [R] T + 0 i [R] 0 i [R]T = [0] where derivative of a matrix implies derivative of all components of the matrix. Above equation can be written as 0 i [R] 0 i [R] T + ( 0 i [R] 0 i [R] T ) T = [0] Define a 3 3 skew symmetric matrix 0 i [Ω] R = 0 i [R] 0 i [R] T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

14 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF A RIGID BODY Angular velocity cannot be obtained as a time derivative of 3 quantities representing orientation. Angular velocity can be derived from time derivative of rotation matrix. Recall 0 i [R] 0 i [R] T = [U], Differentiate with respect to time t [U] is a 3 3 identity matrix 0 i [R] 0 i [R] T + 0 i [R] 0 i [R]T = [0] where derivative of a matrix implies derivative of all components of the matrix. Above equation can be written as 0 i [R] 0 i [R] T + ( 0 i [R] 0 i [R] T ) T = [0] Define a 3 3 skew symmetric matrix 0 i [Ω] R = 0 i [R] 0 i [R] T ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

15 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY SKEW SYMMETRIC MATRIX Skew-symmetric matrix in detail 0 ωz s ω s y 0 i [Ω] R = ωz s 0 ωx s (3) ωy s ωx s 0 The product of O i [Ω] R and a vector (p x,p y,p z ) T R 3 is a cross-product ω 0 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 = 0 s ω i 0 p (4) ωx s p y ωy s p x 0 i [Ω] R called angular velocity matrix 0 s ω i called angular velocity vector of {i} with respect to {0}. In contrast to linear velocity, angular velocity vector is not a straightforward differentiation! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

16 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY SKEW SYMMETRIC MATRIX Skew-symmetric matrix in detail 0 ωz s ω s y 0 i [Ω] R = ωz s 0 ωx s (3) ωy s ωx s 0 The product of O i [Ω] R and a vector (p x,p y,p z ) T R 3 is a cross-product ω 0 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 = 0 s ω i 0 p (4) ωx s p y ωy s p x 0 i [Ω] R called angular velocity matrix 0 s ω i called angular velocity vector of {i} with respect to {0}. In contrast to linear velocity, angular velocity vector is not a straightforward differentiation! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

17 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY SKEW SYMMETRIC MATRIX Skew-symmetric matrix in detail 0 ωz s ω s y 0 i [Ω] R = ωz s 0 ωx s (3) ωy s ωx s 0 The product of O i [Ω] R and a vector (p x,p y,p z ) T R 3 is a cross-product ω 0 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 = 0 s ω i 0 p (4) ωx s p y ωy s p x 0 i [Ω] R called angular velocity matrix 0 s ω i called angular velocity vector of {i} with respect to {0}. In contrast to linear velocity, angular velocity vector is not a straightforward differentiation! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

18 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY SKEW SYMMETRIC MATRIX Skew-symmetric matrix in detail 0 ωz s ω s y 0 i [Ω] R = ωz s 0 ωx s (3) ωy s ωx s 0 The product of O i [Ω] R and a vector (p x,p y,p z ) T R 3 is a cross-product ω 0 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 = 0 s ω i 0 p (4) ωx s p y ωy s p x 0 i [Ω] R called angular velocity matrix 0 s ω i called angular velocity vector of {i} with respect to {0}. In contrast to linear velocity, angular velocity vector is not a straightforward differentiation! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

19 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY 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 x s ω y s ω z s = γ cosα sinβ β sinα = γ sinα sinβ + β cosα (6) = γ cosβ + α ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

20 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY 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 x s ω y s ω z s = γ cosα sinβ β sinα = γ sinα sinβ + β cosα (6) = γ cosβ + α ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

21 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY 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 x s ω y s ω z s = γ cosα sinβ β sinα = γ sinα sinβ + β cosα (6) = γ cosβ + α ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

22 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY 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 x s ω y s ω z s = γ cosα sinβ β sinα = γ sinα sinβ + β cosα (6) = γ cosβ + α ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

23 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY - LEFT AND RIGHT INVARIANT 0 i [Ω] R called right-invariant derived from right multiplication 0 i [R] 0 i [R]T = [U]. 0 ω i s called the space-fixed angular velocity superscript s. 0 i [R]T 0 i [R] = [U] another skew-symmetric matrix 0 ω 0 i [Ω] L = 0 i [R] T z b ωy b 0 i [R] = ωz b 0 ωx b ωy b ωx b 0 (7) Define an angular velocity vector 0 ω i b from the three components (ω b x,ω b y,ω b z ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

24 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY - LEFT AND RIGHT INVARIANT 0 i [Ω] R called right-invariant derived from right multiplication 0 i [R] 0 i [R]T = [U]. 0 ω i s called the space-fixed angular velocity superscript s. 0 i [R]T 0 i [R] = [U] another skew-symmetric matrix 0 ω 0 i [Ω] L = 0 i [R] T z b ωy b 0 i [R] = ωz b 0 ωx b ωy b ωx b 0 (7) Define an angular velocity vector 0 ω i b from the three components (ω b x,ω b y,ω b z ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

25 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY - LEFT AND RIGHT INVARIANT 0 i [Ω] R called right-invariant derived from right multiplication 0 i [R] 0 i [R]T = [U]. 0 ω i s called the space-fixed angular velocity superscript s. 0 i [R]T 0 i [R] = [U] another skew-symmetric matrix 0 ω 0 i [Ω] L = 0 i [R] T z b ωy b 0 i [R] = ωz b 0 ωx b ωy b ωx b 0 (7) Define an angular velocity vector 0 ω i b from the three components (ω b x,ω b y,ω b z ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

26 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY - LEFT AND RIGHT INVARIANT 0 i [Ω] R called right-invariant derived from right multiplication 0 i [R] 0 i [R]T = [U]. 0 ω i s called the space-fixed angular velocity superscript s. 0 i [R]T 0 i [R] = [U] another skew-symmetric matrix 0 ω 0 i [Ω] L = 0 i [R] T z b ωy b 0 i [R] = ωz b 0 ωx b ωy b ωx b 0 (7) Define an angular velocity vector 0 ω i b from the three components (ω b x,ω b y,ω b z ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

27 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω x b ω y b ω z b = α cosγ sinβ + β sinγ = α sinβ sinγ + β cosγ (8) = α cosβ + γ 0 i [Ω] L called left-invariant angular velocity matrix. 0 b ω i called body-fixed angular velocity vector of {i} with respect to {0} superscript b. The two skew-symmetric matrices are related like two tensors The two angular velocities are related as 0 i [Ω] R = 0 i [R] 0 i [Ω] L 0 i [R] T (9) 0 ω i s = 0 i [R] 0 ω i b (10) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

28 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω x b ω y b ω z b = α cosγ sinβ + β sinγ = α sinβ sinγ + β cosγ (8) = α cosβ + γ 0 i [Ω] L called left-invariant angular velocity matrix. 0 b ω i called body-fixed angular velocity vector of {i} with respect to {0} superscript b. The two skew-symmetric matrices are related like two tensors The two angular velocities are related as 0 i [Ω] R = 0 i [R] 0 i [Ω] L 0 i [R] T (9) 0 ω i s = 0 i [R] 0 ω i b (10) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

29 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω x b ω y b ω z b = α cosγ sinβ + β sinγ = α sinβ sinγ + β cosγ (8) = α cosβ + γ 0 i [Ω] L called left-invariant angular velocity matrix. 0 b ω i called body-fixed angular velocity vector of {i} with respect to {0} superscript b. The two skew-symmetric matrices are related like two tensors The two angular velocities are related as 0 i [Ω] R = 0 i [R] 0 i [Ω] L 0 i [R] T (9) 0 ω i s = 0 i [R] 0 ω i b (10) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

30 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω x b ω y b ω z b = α cosγ sinβ + β sinγ = α sinβ sinγ + β cosγ (8) = α cosβ + γ 0 i [Ω] L called left-invariant angular velocity matrix. 0 b ω i called body-fixed angular velocity vector of {i} with respect to {0} superscript b. The two skew-symmetric matrices are related like two tensors The two angular velocities are related as 0 i [Ω] R = 0 i [R] 0 i [Ω] L 0 i [R] T (9) 0 ω i s = 0 i [R] 0 ω i b (10) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

31 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY LEFT INVARIANT For the Z-Y-Z rotation the three components are ω x b ω y b ω z b = α cosγ sinβ + β sinγ = α sinβ sinγ + β cosγ (8) = α cosβ + γ 0 i [Ω] L called left-invariant angular velocity matrix. 0 b ω i called body-fixed angular velocity vector of {i} with respect to {0} superscript b. The two skew-symmetric matrices are related like two tensors The two angular velocities are related as 0 i [Ω] R = 0 i [R] 0 i [Ω] L 0 i [R] T (9) 0 ω i s = 0 i [R] 0 ω i b (10) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

32 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) {0} Ẑ Rigid Body at t i p More on two forms of angular velocity matrix and vectors. Pure rotation 0 O i (t) and 0 O i (t + t) are coincident and {i} t ˆX O i Rigid Body at t + t Ŷ only the elements of the rotation matrix i 0 [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, / 98

33 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Location of P in {0} and since P is fixed in {i} and since 0 i [R] 1 = 0 i [R]T, 0 p = 0 i [R] i p 0 p = 0 V p = 0 i [R] i p 0 V p = 0 i [R] 0 i [R] T 0 p = 0 i [Ω] 0 R p = 0 ω i s 0 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, / 98

34 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Location of P in {0} and since P is fixed in {i} and since 0 i [R] 1 = 0 i [R]T, 0 p = 0 i [R] i p 0 p = 0 V p = 0 i [R] i p 0 V p = 0 i [R] 0 i [R] T 0 p = 0 i [Ω] 0 R p = 0 ω i s 0 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, / 98

35 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Location of P in {0} and since P is fixed in {i} and since 0 i [R] 1 = 0 i [R]T, 0 p = 0 i [R] i p 0 p = 0 V p = 0 i [R] i p 0 V p = 0 i [R] 0 i [R] T 0 p = 0 i [Ω] 0 R p = 0 ω i s 0 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, / 98

36 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Using relation between 0 i [Ω] R and 0 i [Ω] L 0 V p = 0 i [R] 0 0 i [Ω] L i [R] T 0 p = 0 i [R] 0 i [Ω] i L p and get which yields 0 i [R] 1 0 V p = 0 i [Ω] L i p i V p = 0 i [Ω] L i p = 0 ω i b i p (12) Again except for denoting the reference coordinate system, the coordinate system {0} 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 0 i [R] 0 i [R]T is normally used. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

37 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Using relation between 0 i [Ω] R and 0 i [Ω] L 0 V p = 0 i [R] 0 0 i [Ω] L i [R] T 0 p = 0 i [R] 0 i [Ω] i L p and get which yields 0 i [R] 1 0 V p = 0 i [Ω] L i p i V p = 0 i [Ω] L i p = 0 ω i b i p (12) Again except for denoting the reference coordinate system, the coordinate system {0} 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 0 i [R] 0 i [R]T is normally used. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

38 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Using relation between 0 i [Ω] R and 0 i [Ω] L 0 V p = 0 i [R] 0 0 i [Ω] L i [R] T 0 p = 0 i [R] 0 i [Ω] i L p and get which yields 0 i [R] 1 0 V p = 0 i [Ω] L i p i V p = 0 i [Ω] L i p = 0 ω i b i p (12) Again except for denoting the reference coordinate system, the coordinate system {0} 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 0 i [R] 0 i [R]T is normally used. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

39 LINEAR AND ANGULAR VELOCITY OF RIGID BODY ANGULAR VELOCITY OF RIGID BODY (CONTD.) Using relation between 0 i [Ω] R and 0 i [Ω] L 0 V p = 0 i [R] 0 0 i [Ω] L i [R] T 0 p = 0 i [R] 0 i [Ω] i L p and get which yields 0 i [R] 1 0 V p = 0 i [Ω] L i p i V p = 0 i [Ω] L i p = 0 ω i b i p (12) Again except for denoting the reference coordinate system, the coordinate system {0} 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 0 i [R] 0 i [R]T is normally used. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

40 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY IN SERIAL MANIPULATOR ROTARY (R) JOINT For two links connected by a rotary (R) joint (see Module 2, Lecture 2) The time derivative operation 0 i [R] = 0 i 1[R] i 1 i [R(ˆk,θ i )] 0 i [R] 0 i [R] T = d dt (0 i 1[R] i 1 i [R(ˆk,θ i )]) ( i 1 i [R(ˆk,θ i )] T 0 i 1[R] T ) Rewrite above equation as 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] ( i 1 [Ṙ(ˆk,θ i )] i 1 i [R(ˆk,θ i )] T ) 0 i 1[R] T To simplify, use the result i 1 i i i 1 i [R(ˆk,θ i )] = e (i 1 i [K ]θ i ) [K ] is the skew-symmetric form of the rotation axis vector ˆk and θ i is the rotation at the rotary joint (see Module 2, Lecture 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

41 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY IN SERIAL MANIPULATOR ROTARY (R) JOINT For two links connected by a rotary (R) joint (see Module 2, Lecture 2) The time derivative operation 0 i [R] = 0 i 1[R] i 1 i [R(ˆk,θ i )] 0 i [R] 0 i [R] T = d dt (0 i 1[R] i 1 i [R(ˆk,θ i )]) ( i 1 i [R(ˆk,θ i )] T 0 i 1[R] T ) Rewrite above equation as 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] ( i 1 i [Ṙ(ˆk,θ i )] i 1 i [R(ˆk,θ i )] T ) 0 i 1[R] T To simplify, use the result i 1 i i 1 i [R(ˆk,θ i )] = e (i 1 i [K ]θ i ) [K ] is the skew-symmetric form of the rotation axis vector ˆk and θ i is the rotation at the rotary joint (see Module 2, Lecture 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

42 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY IN SERIAL MANIPULATOR ROTARY (R) JOINT For two links connected by a rotary (R) joint (see Module 2, Lecture 2) The time derivative operation 0 i [R] = 0 i 1[R] i 1 i [R(ˆk,θ i )] 0 i [R] 0 i [R] T = d dt (0 i 1[R] i 1 i [R(ˆk,θ i )]) ( i 1 i [R(ˆk,θ i )] T 0 i 1[R] T ) Rewrite above equation as 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] ( i 1 i [Ṙ(ˆk,θ i )] i 1 i [R(ˆk,θ i )] T ) 0 i 1[R] T To simplify, use the result i 1 i i 1 i [R(ˆk,θ i )] = e (i 1 i [K ]θ i ) [K ] is the skew-symmetric form of the rotation axis vector ˆk and θ i is the rotation at the rotary joint (see Module 2, Lecture 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

43 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY IN SERIAL MANIPULATOR ROTARY (R) JOINT For two links connected by a rotary (R) joint (see Module 2, Lecture 2) The time derivative operation 0 i [R] = 0 i 1[R] i 1 i [R(ˆk,θ i )] 0 i [R] 0 i [R] T = d dt (0 i 1[R] i 1 i [R(ˆk,θ i )]) ( i 1 i [R(ˆk,θ i )] T 0 i 1[R] T ) Rewrite above equation as 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] ( i 1 i [Ṙ(ˆk,θ i )] i 1 i [R(ˆk,θ i )] T ) 0 i 1[R] T To simplify, use the result i 1 i i 1 i [R(ˆk,θ i )] = e (i 1 i [K ]θ i ) [K ] is the skew-symmetric form of the rotation axis vector ˆk and θ i is the rotation at the rotary joint (see Module 2, Lecture 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

44 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY PROPAGATION IN SERIAL MANIPULATORS R JOINT ˆk is fixed in {i 1} and {i} d dt e(i 1 i [K ]θ i ) = i 1 i From above and properties of a rotation matrix, [K ] θ i e (i 1 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] i 1 i [K ] 0 i 1[R] T θ i = 0 i 1[Ω] R + 0 i [K ] θ i and in terms of the space-fixed angular velocity 0 ω ( ) 0 ω i = 0 ω i ˆki θ i i [K ]θ i ) Serial manipulators R joint axis is chosen along the Z axis. Pre-multiply both sides by i 0 [R] and simplify to get i ω i = i i 1[R] i 1 ω i 1 + θ i (0 0 1) T (13) i ω i denotes i 0 [R]0 ω i i ω i not necessarily 0. Angular velocity propagation in serial manipulators links connected by R joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

45 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY PROPAGATION IN SERIAL MANIPULATORS R JOINT ˆk is fixed in {i 1} and {i} d dt e(i 1 i [K ]θ i ) = i 1 i From above and properties of a rotation matrix, [K ] θ i e (i 1 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] i 1 i [K ] 0 i 1[R] T θ i = 0 i 1[Ω] R + 0 i [K ] θ i and in terms of the space-fixed angular velocity 0 ω ( ) 0 ω i = 0 ω i ˆki θ i i [K ]θ i ) Serial manipulators R joint axis is chosen along the Z axis. Pre-multiply both sides by i 0 [R] and simplify to get i ω i = i i 1[R] i 1 ω i 1 + θ i (0 0 1) T (13) i ω i denotes i 0 [R]0 ω i i ω i not necessarily 0. Angular velocity propagation in serial manipulators links connected by R joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

46 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY PROPAGATION IN SERIAL MANIPULATORS R JOINT ˆk is fixed in {i 1} and {i} d dt e(i 1 i [K ]θ i ) = i 1 i From above and properties of a rotation matrix, [K ] θ i e (i 1 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] i 1 i [K ] 0 i 1[R] T θ i = 0 i 1[Ω] R + 0 i [K ] θ i and in terms of the space-fixed angular velocity 0 ω ( ) 0 ω i = 0 ω i ˆki θ i i [K ]θ i ) Serial manipulators R joint axis is chosen along the Z axis. Pre-multiply both sides by i 0 [R] and simplify to get i ω i = i i 1[R] i 1 ω i 1 + θ i (0 0 1) T (13) i ω i denotes i 0 [R]0 ω i i ω i not necessarily 0. Angular velocity propagation in serial manipulators links connected by R joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

47 LINEAR AND ANGULAR VELOCITY OF LINKS ANGULAR VELOCITY PROPAGATION IN SERIAL MANIPULATORS R JOINT ˆk is fixed in {i 1} and {i} d dt e(i 1 i [K ]θ i ) = i 1 i From above and properties of a rotation matrix, [K ] θ i e (i 1 0 i [Ω] R = 0 i 1[Ω] R + 0 i 1[R] i 1 i [K ] 0 i 1[R] T θ i = 0 i 1[Ω] R + 0 i [K ] θ i and in terms of the space-fixed angular velocity 0 ω ( ) 0 ω i = 0 ω i ˆki θ i i [K ]θ i ) Serial manipulators R joint axis is chosen along the Z axis. Pre-multiply both sides by i 0 [R] and simplify to get i ω i = i i 1[R] i 1 ω i 1 + θ i (0 0 1) T (13) i ω i denotes i 0 [R]0 ω i i ω i not necessarily 0. Angular velocity propagation in serial manipulators links connected by R joints ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

48 LINEAR AND ANGULAR VELOCITY OF LINKS LINEAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT For two consecutive links in a serial manipulator (see Module 2, Lecture 2) 0 O i = 0 O i i 1[R] i 1 O i Taking derivatives on both sides 0 V Oi = 0 V Oi ω i 1 0 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 0 [R]0 V i and i 1 0 [R] 0 V i 1, respectively. They are not necessarily 0! Linear velocity vector propagation in links of a serial manipulator Rotary joint. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

49 LINEAR AND ANGULAR VELOCITY OF LINKS LINEAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT For two consecutive links in a serial manipulator (see Module 2, Lecture 2) 0 O i = 0 O i i 1[R] i 1 O i Taking derivatives on both sides 0 V Oi = 0 V Oi ω i 1 0 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 0 [R]0 V i and i 1 0 [R] 0 V i 1, respectively. They are not necessarily 0! Linear velocity vector propagation in links of a serial manipulator Rotary joint. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

50 LINEAR AND ANGULAR VELOCITY OF LINKS LINEAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT For two consecutive links in a serial manipulator (see Module 2, Lecture 2) 0 O i = 0 O i i 1[R] i 1 O i Taking derivatives on both sides 0 V Oi = 0 V Oi ω i 1 0 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 0 [R]0 V i and i 1 0 [R] 0 V i 1, respectively. They are not necessarily 0! Linear velocity vector propagation in links of a serial manipulator Rotary joint. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

51 LINEAR AND ANGULAR VELOCITY OF LINKS LINEAR VELOCITY PROPAGATION IN SERIAL MANIPULATOR R JOINT For two consecutive links in a serial manipulator (see Module 2, Lecture 2) 0 O i = 0 O i i 1[R] i 1 O i Taking derivatives on both sides 0 V Oi = 0 V Oi ω i 1 0 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 0 [R]0 V i and i 1 0 [R] 0 V i 1, respectively. They are not necessarily 0! Linear velocity vector propagation in links of a serial manipulator Rotary joint. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

52 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PRISMATIC JOINTS Two links connected by a prismatic (P) joint (see Module 2, Lecture 2) Prismatic joint allows relative translation between {1 i} and {i} angular velocity is same Relative translation is along Z axis ḋi(0 0 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(0 0 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, / 98

53 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PRISMATIC JOINTS Two links connected by a prismatic (P) joint (see Module 2, Lecture 2) Prismatic joint allows relative translation between {1 i} and {i} angular velocity is same Relative translation is along Z axis ḋi(0 0 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(0 0 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, / 98

54 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PRISMATIC JOINTS Two links connected by a prismatic (P) joint (see Module 2, Lecture 2) Prismatic joint allows relative translation between {1 i} and {i} angular velocity is same Relative translation is along Z axis ḋi(0 0 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(0 0 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, / 98

55 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PRISMATIC JOINTS Two links connected by a prismatic (P) joint (see Module 2, Lecture 2) Prismatic joint allows relative translation between {1 i} and {i} angular velocity is same Relative translation is along Z axis ḋi(0 0 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(0 0 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, / 98

56 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR ˆX 3, ˆX Tool Ŷ 0 {Tool} Ŷ Tool Link 3 l 3 Ŷ 3 {3} θ 3 ˆX 2 O 3 Link 2 All joint axis are parallel and coming out of page. {0} is fixed 0 ω 0 = 0 Ŷ 1 Ŷ 2 {2} l 2 θ 2 ˆX1 0 V 0 = 0 {1} {0} l 1 θ 1 Link 1 O 2 Links connected by rotary (R) joint Equations (13) and (14) give velocities of all links. ˆX 0 O 1 Figure 3: The planar 3R manipulator revisited ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

57 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR (CONTD.) For i=1 For i=2 For i=3 1 ω 1 = (0 0 θ 1 ) T 1 V 1 = 0 2 ω 2 = (0 0 θ 1 + θ 2 ) T c 2 s V 2 = s 2 c l 1 θ 1 0 = 3 ω 3 = (0 0 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 0 l 1 s 2 θ 1 l 1 c 2 θ 1 0 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

58 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR (CONTD.) For i=1 For i=2 For i=3 1 ω 1 = (0 0 θ 1 ) T 1 V 1 = 0 2 ω 2 = (0 0 θ 1 + θ 2 ) T c 2 s V 2 = s 2 c l 1 θ 1 0 = 3 ω 3 = (0 0 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 0 l 1 s 2 θ 1 l 1 c 2 θ 1 0 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

59 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR (CONTD.) For i=1 For i=2 For i=3 1 ω 1 = (0 0 θ 1 ) T 1 V 1 = 0 2 ω 2 = (0 0 θ 1 + θ 2 ) T c 2 s V 2 = s 2 c l 1 θ 1 0 = 3 ω 3 = (0 0 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 0 l 1 s 2 θ 1 l 1 c 2 θ 1 0 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

60 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR (CONTD.) For i = Tool Tool ω Tool = (0 0 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 0 Linear and angular velocity in {0} 0 ω Tool = (0 0 θ 1 + θ 2 + θ 3 ) T (17) and 0 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 ) 0 (18) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

61 LINEAR AND ANGULAR VELOCITY OF LINKS VELOCITY PROPAGATION PLANAR 3R MANIPULATOR (CONTD.) For i = Tool Tool ω Tool = (0 0 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 0 Linear and angular velocity in {0} 0 ω Tool = (0 0 θ 1 + θ 2 + θ 3 ) T (17) and 0 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 ) 0 (18) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

62 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 MODULE 5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

63 SERIAL MANIPULATOR JACOBIAN MATRIX INTRODUCTION Linear and angular velocity of {Tool} (Equations (17) and (18)) can be written in a compact form as 0 V Tool = 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 V Tool is a 6 1 entity 0 V Tool = 0 V Tool 0 ω Tool θ 1 θ 2 θ 3 (19) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

64 SERIAL MANIPULATOR JACOBIAN MATRIX INTRODUCTION Linear and angular velocity of {Tool} (Equations (17) and (18)) can be written in a compact form as 0 V Tool = 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 V Tool is a 6 1 entity 0 V Tool = 0 V Tool 0 ω Tool θ 1 θ 2 θ 3 (19) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

65 SERIAL MANIPULATOR JACOBIAN MATRIX INTRODUCTION 0 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 0 V Tool or ( 0 V Tool ; 0 ω Tool ) T is not a vector. Matrix in square brackets, 0 Tool [J(Θ)], is called the Jacobian matrix for the planar 3R manipulator. 0 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 {0}, leading superscript 0. 1 In theoretical kinematics, ( 0 ω Tool ; 0 V Tool ) is called twist (see Additional Material in Module 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

66 SERIAL MANIPULATOR JACOBIAN MATRIX INTRODUCTION 0 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 0 V Tool or ( 0 V Tool ; 0 ω Tool ) T is not a vector. Matrix in square brackets, 0 Tool [J(Θ)], is called the Jacobian matrix for the planar 3R manipulator. 0 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 {0}, leading superscript 0. 1 In theoretical kinematics, ( 0 ω Tool ; 0 V Tool ) is called twist (see Additional Material in Module 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

67 SERIAL MANIPULATOR JACOBIAN MATRIX INTRODUCTION 0 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 0 V Tool or ( 0 V Tool ; 0 ω Tool ) T is not a vector. Matrix in square brackets, 0 Tool [J(Θ)], is called the Jacobian matrix for the planar 3R manipulator. 0 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 {0}, leading superscript 0. 1 In theoretical kinematics, ( 0 ω Tool ; 0 V Tool ) is called twist (see Additional Material in Module 2). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

68 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX 0 Tool [J(Θ)] is not a proper matrix. 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 0 V Tool, top and bottom halves of a Jacobian matrix are separated by. Many matrix operations makes no sense the condition number 2 of this matrix changes with the choice of length units. 0 Tool [J(Θ)] is best thought of as a map 0 Tool [J(Θ)] : Θ 0 V Tool 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, / 98

69 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX 0 Tool [J(Θ)] is not a proper matrix. 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 0 V Tool, top and bottom halves of a Jacobian matrix are separated by. Many matrix operations makes no sense the condition number 2 of this matrix changes with the choice of length units. 0 Tool [J(Θ)] is best thought of as a map 0 Tool [J(Θ)] : Θ 0 V Tool 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, / 98

70 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX 0 Tool [J(Θ)] is not a proper matrix. 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 0 V Tool, top and bottom halves of a Jacobian matrix are separated by. Many matrix operations makes no sense the condition number 2 of this matrix changes with the choice of length units. 0 Tool [J(Θ)] is best thought of as a map 0 Tool [J(Θ)] : Θ 0 V Tool 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, / 98

71 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX 0 Tool [J(Θ)] is not a proper matrix. 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 0 V Tool, top and bottom halves of a Jacobian matrix are separated by. Many matrix operations makes no sense the condition number 2 of this matrix changes with the choice of length units. 0 Tool [J(Θ)] is best thought of as a map 0 Tool [J(Θ)] : Θ 0 V Tool 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, / 98

72 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) The Jacobian matrix can be derived 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. Jacobian can be defined for any differentiable vector function. Direct kinematics equations differentiable vector function X = Ψ(Θ) Θ = (θ 1,θ 2,...,θ n ) denotes the n joint variables Position and orientation of end-effector are denoted by X 3. [J(Θ)] is the matrix of first partial derivatives of Ψ with respect to θ i i th column of [J(Θ)] is the partial derivatives of Ψ with respect to θ i. [ ] Ψ Ψ Ψ [J(Θ)] =... θ 1 θ 2 θ n 3 For example, X denotes the three Cartesian position variables (x, y, z) and the three Euler angles (α, β,γ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

73 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) The Jacobian matrix can be derived 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. Jacobian can be defined for any differentiable vector function. Direct kinematics equations differentiable vector function X = Ψ(Θ) Θ = (θ 1,θ 2,...,θ n ) denotes the n joint variables Position and orientation of end-effector are denoted by X 3. [J(Θ)] is the matrix of first partial derivatives of Ψ with respect to θ i i th column of [J(Θ)] is the partial derivatives of Ψ with respect to θ i. [ ] Ψ Ψ Ψ [J(Θ)] =... θ 1 θ 2 θ n 3 For example, X denotes the three Cartesian position variables (x, y, z) and the three Euler angles (α, β,γ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

74 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) The Jacobian matrix can be derived 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. Jacobian can be defined for any differentiable vector function. Direct kinematics equations differentiable vector function X = Ψ(Θ) Θ = (θ 1,θ 2,...,θ n ) denotes the n joint variables Position and orientation of end-effector are denoted by X 3. [J(Θ)] is the matrix of first partial derivatives of Ψ with respect to θ i i th column of [J(Θ)] is the partial derivatives of Ψ with respect to θ i. [ ] Ψ Ψ Ψ [J(Θ)] =... θ 1 θ 2 θ n 3 For example, X denotes the three Cartesian position variables (x, y, z) and the three Euler angles (α, β,γ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

75 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) The Jacobian matrix can be derived 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. Jacobian can be defined for any differentiable vector function. Direct kinematics equations differentiable vector function X = Ψ(Θ) Θ = (θ 1,θ 2,...,θ n ) denotes the n joint variables Position and orientation of end-effector are denoted by X 3. [J(Θ)] is the matrix of first partial derivatives of Ψ with respect to θ i i th column of [J(Θ)] is the partial derivatives of Ψ with respect to θ i. [ ] Ψ Ψ Ψ [J(Θ)] =... θ 1 θ 2 θ n 3 For example, X denotes the three Cartesian position variables (x, y, z) and the three Euler angles (α, β,γ). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

76 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) 0 Tool [J(Θ)] very important in velocity kinematics of serial manipulators. The elements of the Jacobian matrix are non-linear functions of the joint variables Θ. Manipulator in motion 0 Tool [J(Θ)] is time varying. At instant with Θ known, 0 Tool [J(Θ)] relates linear and angular velocities to joint rates. The relationship is linear! The Jacobian matrix can be obtained for any link most often 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 {0}. Jacobian matrix can be written in any coordinate system using rotation matrices. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

77 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) 0 Tool [J(Θ)] very important in velocity kinematics of serial manipulators. The elements of the Jacobian matrix are non-linear functions of the joint variables Θ. Manipulator in motion 0 Tool [J(Θ)] is time varying. At instant with Θ known, 0 Tool [J(Θ)] relates linear and angular velocities to joint rates. The relationship is linear! The Jacobian matrix can be obtained for any link most often 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 {0}. Jacobian matrix can be written in any coordinate system using rotation matrices. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

78 SERIAL MANIPULATOR JACOBIAN MATRIX PROPERTIES OF JACOBIAN MATRIX (CONTD.) 0 Tool [J(Θ)] very important in velocity kinematics of serial manipulators. The elements of the Jacobian matrix are non-linear functions of the joint variables Θ. Manipulator in motion 0 Tool [J(Θ)] is time varying. At instant with Θ known, 0 Tool [J(Θ)] relates linear and angular velocities to joint rates. The relationship is linear! The Jacobian matrix can be obtained for any link most often 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 {0}. Jacobian matrix can be written in any coordinate system using rotation matrices. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 98

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