1. Introduction to Robotics notes 2.pdf - Google Drive

Chapter 4 Robot Dynamics and Control

Summer School-Math. Methods in [email protected] 13-31 July 2009

1 Lecture Notes for A Mathematical Introduction to Robotic Manipulation

Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator

By Z.X. Li∗ and Y.Q. Wu#

Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

∗

Dept. of ECE, Hong Kong University of Science & Technology # School of ME, Shanghai Jiaotong University

24 July 2009


Summer School-Math. Methods in [email protected] 13-31 July 2009

2

Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

Chapter 4 Robot Dynamics and Control 1 Lagrangian Equations 2 Inertial Properties of Rigid Body 3 Dynamics of an Open-chain Manipulator 4 Newton-Euler Equations 5 Coordinate-invariant algorithms for robot dynamics 6 Lagrange’s Equations with Constraints


4.1 Lagrangian Equations ◻ A Simple Example: Chapter 4 Robot Dynamics and Control Lagrangian Equations

m

Newton’s Equation: m¨x = Fx

Dynamics of an Open-chain Manipulator

Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

m¨y = Fy − mg Momentum: Px = m˙x d dt Px

=

Py = m˙y d Fx , dt Py = Fy − mg

Fy

F Fx

mg

♢ Review:

Inertial Properties of Rigid Body

Newton-Euler Equations

3

y

⇔

Lagrangian Equation: d ∂L ∂L − = Fx dt ∂˙x ∂x d ∂L ∂L − = Fy dt ∂˙y ∂y Lagrangian function: L = T − V, Px = ∂L ∂˙x , Py = Kinetic energy: T = 21 m(˙x2 + y˙2 )2 Potential energy: V = mgy

x

∂L ∂˙y


4.1 Lagrangian Equations 4 ◻ Generalization to multibody systems: Chapter 4 Robot Dynamics and Control

y m3

Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

q3

m2 q2 m1

qi , i = 1, . . . , n: generalized coordinates Kinetic energy: T = T(q, q˙ ) Potential energy: V = V(q) Lagrangian: L(q, q˙ ) = T(q, q˙ ) − V(q) τi , i = 1, . . . , n: external force on qi Lagrangian Equation:

q1 x

d ∂L ∂L − = τi , i = 1, . . . , n dt ∂˙qi ∂qi


4.1 Lagrangian Equations 5 ◇ Example: Pendulum equation Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

y Generalized coordinate: 1 θ∈S Kinematics: x x = l sin θ, y = −l cos θ ˙ y˙ = l sin θ ⋅ θ˙ ⇒ x˙ = l cos θ ⋅ θ, Kinetic energy: θ 1 1 2 ˙2 2 2 ˙ ˙ T(θ, θ) = m(˙x + y ) = ml θ 2 2 Potential energy: mg V = mgl(1 − cos θ) Lagrangian function: 1 ∂L ˙ ∂L = −mgl sin θ L = T − V = ml2 θ˙ − mgl(1 − cos θ), ⇒ = ml2 θ, 2 ∂θ ∂ θ˙ Equation of motion: d ∂L ∂L − = τ ⇒ ml2 θ¨ + mgl sin θ = τ dt ˙ ∂θ


4.1 Lagrangian Equations 6 ◇ Example: Dynamics of a Spherical Pendulum Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

⎡ l sin θ cos ϕ ⎤ ⎢ ⎥ r(θ, ϕ) = ⎢ l sin θ sin ϕ ⎥ ⎢ −l cos θ ⎥ ⎣ ⎦ 1 1 T = m∥˙r∥2 = ml2 (θ˙2 + (1 − cos2 θ)ϕ˙2 ) 2 2 V = −mgl cos θ 1 L(q, q˙ ) = ml2 (θ˙ + (1 − cosθ )ϕ˙2 ) + mgl cos θ 2

θ

ϕ

mg


4.1 Lagrangian Equations 7


⎧ d ∂L d ⎪ ˙ = ml2 θ, ¨ ∂L = ml2 sin θ cos θ ϕ˙2 − mgl sin θ ⎪ = (ml2 θ) ⎪ ⎪ ⎪ ˙ ∂θ ⎪ dt ∂ θ dt ⎨ d d ∂L ⎪ ⎪ ˙ = ml2 sin2 θ ϕ¨ + 2ml2 sin θ cos θ θ˙ϕ, ˙ ∂L = 0 ⎪ = (ml2 sin2 θ ϕ) ⎪ ⎪ ˙ ⎪ ∂ϕ ⎩ dt ∂ ϕ dt [

ml2 0 ][ 0 ml2 s2θ

θ¨ ] + [ −ml2 sθ cθ ϕ˙2 ] + [ mglsθ ] = [ 0 ] 0 0 ϕ¨ 2ml2 sθ cθ θ˙ ϕ˙


4.2 Intertial Properties of Rigid Body 8 ◻ Kinetic energy of a rigid body: Chapter 4 Robot Dynamics and Control

r

Lagrangian Equations Inertial Properties of Rigid Body

B

ra A

gab

Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

Volume occupied by the body: Mass density: Mass: Mass center: Relative to frame at the mass center:

V ρ(r) m= ρ(r)dV V 1 r≜ ρ(r)rdV m V r=0

∫

∫


4.2 Intertial Properties of Rigid Body 9 In A-frame Chapter 4 Robot Dynamics and Control

1 ˙ 2 dV = 1 ˙ + ∥Rr∥ ˙ 2 )dV ρ(r)∥p˙ + Rr∥ ρ(r)(∥p˙ ∥2 + 2p˙ T Rr 2 V 2 V 1 1 ˙ ˙ 2 dV = m∥p˙ ∥2 + p˙ T R ρ(r)rdV + ρ(r)∥Rr∥ 2 2 V ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

Lagrangian Equations


∫

∫

Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator

∫

T=

∫

=0

1 2

∫

V

1 = 2 1 = 2

˙ 2 dV ρ(r)∥Rr∥ 1

1

˙ dV = ˆ dV = ∫ ρ(r)∥ˆr ω∥ dV ρ(r)∥ωr∥ ∫ ρ(r)∥R Rr∥ 2 ∫ 2 1 1 ∫ ρ(r)(−ω ˆr ω)dV = 2 ω ( ∫ (−ρ(r)ˆr) dV) ω ≜ 2 ω I ω T

2

T 2

2

T

2

2

T


4.2 Intertial Properties of Rigid Body 10 where Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

I =− with Ixx =

∫

∫ ρ(r)(y

2

⎡ Ixx Ixy Ixz ⎢ ρ(r)ˆr2 dV ≜ ⎢ ⎢ Ixy Iyy Iyz ⎢ Ixz Iyz Izz ⎣ + z2 )dxdydz, Ixy = −

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

∫ ρ(r)xydxdydz

1 1 1 1 T = m∥p˙ ∥2 + (ωb )T I b ωb = m∥RT p˙ ∥2 + (ωb )T I b ωb 2 2 2 2 1 b T mI 0 = (V ) [ ] Vb 0 Ib 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ Mb

˙ and vb = RT p˙ , M b is the For Vˆ b = g −1 ⋅ g˙ , with ωˆ b = RT ⋅ R Generalized inertia matrix in B frame.


4.2 Intertial Properties of Rigid Body 11 ◇ Example: M b for a rectangular object Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

z

m lωh Ixx = ρ(y2 + z2 )dxdydz

ρ=

∫

y

V

=ρ

h 2

ω 2

l 2

− h2

− ω2

− 2l

∫ ∫ ∫

= ρ(

h

1 m (lω3 h + lωh3 )) = (ω2 + h2 ) 12 12

Ixy = − = −ρ

(y2 + z2 )dxdydz

∫

V

ρxydv = −ρ

h 2

ω 2

− h2

− ω2

∫ ∫

l

h 2

ω 2

l 2

− h2

− ω2

− 2l

∫ ∫ ∫

y 2 2 x ∣ dydz = 0 2 −l 2

xydxdydz

x l w


4.2 Intertial Properties of Rigid Body


⎡ ⎢ ⎢ I =⎢ ⎢ ⎢ ⎣

m 2 12 (w

+h ) 0 0 2

Vˆ 1 = g1−1 ⋅ g˙1 , M1 1 T = V1T M1 V1 2 V1 = Adg0 V2

m 2 12 (l

0 + h2 ) 0

⎤ ⎥ ⎥ ⎥ , M = [ mI03×3 I0 ] m 2 2 ⎥ ⎥ 12 (w + l ) ⎦ 0 0

g0 A

B

g1 (t)

g2 (t)

g0

1 1 1 T = (Adg0 V2 )T M1 (Adg0 V2 ) = V2T AdTg0 M1 Adg0 V2 ≜ V2T M2 V2 2 2 2

◻ M under change of frames: M2 = AdTg0 M1 Adg0

12


4.2 Intertial Properties of Rigid Body 13 ◇ Example: Dynamics of a 2-dof planar robot Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

⎡ Ixxi ⎢ Ii = ⎢ 0 ⎢ 0 ⎣

0 Iyyi 0

0 0 Izzi

⎤ ⎥ ⎥ , i = 1, 2 ⎥ ⎦

˙ = 1 m1 ∥v1 ∥2 + 1 ωT I1 ω1 T(θ, θ) 2 2 1 1 1 + m2 ∥v2 ∥2 + ωT2 I2 ω2 2 2

l2 r 2 θ2

l1

y r1

x

0 ω1 = [ 0 ] θ˙1

ω2 = [

0 0 ] θ˙1 + θ˙2

θ1


4.2 Intertial Properties of Rigid Body 14 Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

xi Pi = [ yi ]: Mass center γi : Distance from joint i to mass center 0 x˙ 1 = −r1 s1 θ˙1 x1 = r1 c1 ⇒ y1 = r1 s1 y˙1 = r1 c1 θ˙1 x˙ 2 = −(l1 s1 + r2 s12 )θ˙1 − r2 s12 θ˙2 x2 = l1 c1 + r2 c12 ⇒ y2 = l1 s1 + r2 s12 y˙1 = (l1 c1 + r2 c12 )θ˙1 + r2 c12 θ˙2 ˙ = 1 m1 (x˙ 2 + y˙2 ) + 1 Iz θ˙2 + 1 m2 (x˙ 2 + y˙2 ) + 1 Iz (θ˙2 + θ˙2 ) T(θ, θ) 1 2 2 1 2 2 2 1 1 2 2 2 1 ˙ 1 α + 2βc δ + βc2 = [θ˙1 θ˙2 ] [ δ + βc 2 ] [ θ˙ 1 ] δ 2 θ2 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ α=

Iz1 + Iz2 + m1 r12

+ m2 (l12

¨ ⇒ M(θ) [ θ¨ 1 θ2

M(θ) + r22 ), β

= m2 l1 r2 , δ = Iz2 + m2 r22 , L = T ˙ −βs2 θ˙2 −βx ]+[ ] [ θ˙ 1 ] = [ ττ21 ] ˙ βs2 θ 1 0 θ2


4.3 Dynamics of Open-chain Manipulator 15 ◻ Dynamics of open-chain manipulator: Chapter 4 Robot Dynamics and Control

Definition: ˆ ˆ Li : frame at mass center of link i, gsli (θ) = expξ1 θ 1 ⋯ expξi θ i gsli (o) θ1 θ2

Lagrangian Equations

l2 L3

L2

Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator

θ3 l1

r1 l0

r2

L1 r0 S


⎡ ⎢ ⎢ ⎢ ⎢ b b † † † Vsli = Jsli (θ)θ˙ = [ξ1 ξ2 ⋯ ξi 0 ⋯ 0] ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

θ˙1 ⋮ θ˙i θ˙i+1 ⋮ θ˙n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ = Ji (θ)θ˙ ⎥ ⎥ ⎥ ⎥ ⎦


4.3 Dynamics of Open-chain Manipulator 16 ξj† = Ad−1 (eξj+1 θ j+1 ⋯eξi θ i gsli (0))ξj , j ≤ i ˆ


ˆ

˙ = 1 (V b )T M b V b = 1 θ˙T J T (θ)M b Ji (θ)θ˙ Ti (θ, θ) i sli i i 2 sli 2 n ˙ = 1 θ˙T M(θ)θ, ˙ T(θ) = ∑ Ti (θ, θ) 2 i=1

1 n ∑ Mij (θ)θ˙i θ˙j 2 i,j=1 Vi (θ) = mi ghi (θ), V(θ) = ∑ mi ghi (θ)

M(θ) = ∑ JiT (θ)Mib Ji (θ) = i

hi (θ): Height of Li ,

Lagrange’s Equation: d ∂L ∂L − = Γi , i = 1, . . . , n, dt ∂ θ˙i ∂θ i

i=1

⎞ n d ∂L d ⎛n ˙ ij θ˙j = Mij θ˙j = ∑ Mij θ¨j + M ∑ dt ∂ θ˙i dt ⎝ j=1 ⎠ j=1


4.3 Dynamics of Open-chain Manipulator 17 Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

∂L 1 n ∂Mkj ˙ ˙ ∂V = ∑ θkθj − ∂θ i 2 j,k=1 ∂θ i ∂θ i n

˙ ij = ∑ M k

∂Mij ˙ θk ∂θ k

n

∂Mij ˙ ˙ 1 ∂Mkj ˙ ˙ ∂V ⇒ ∑ Mij θ¨j + ∑ ( θj θk − θk θj) + = Γi 2 ∂θ i ∂θ i j=1 j,k=1 ∂θ k n n ∂V ⇒ ∑ Mij θ¨j + ∑ Γijk θ˙k θ˙j + = Γi ∂θ i j=1 j,k=1

Γijk =

1 ∂Mij ∂Mik ∂Mkj ( + − ) 2 ∂θ k ∂θ j ∂θ i

θ˙i ⋅ θ˙j , i ≠ j : Coriolis force

θ˙i2 : Centrifugal force n n ˙ = ∑ Γk θ˙k = 1 ∑ ( ∂Mij + ∂Mik − ∂Mkj )θ˙k Define: cij (θ, θ) ij 2 k=1 ∂θ k ∂θ j ∂θ i k=1 ˙ θ˙ + N(θ) = τ ⇒ M(θ)θ¨ + C(θ, θ)


4.3 Dynamics of Open-chain Manipulator 18 Property 1: Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

1 2

M(θ) = M T (θ), θ˙T M(θ)θ˙ ≥ 0, θ˙T M(θ)θ˙ = 0 ⇔ θ = 0 ˙ − 2C ∈ Rn×n is skew symmetric M

Proof : ˙ − 2C)ij = M ˙ ij − 2cij (θ) (M n ∂M ∂Mkj ˙ ∂Mij ˙ ∂Mik ˙ ij ˙ =∑ θk − θk − θk + θk ∂θ k ∂θ j ∂θ i k=1 ∂θ k n

=∑ k=1

∂Mkj ˙ ∂Mik ˙ θk − θk ∂θ i ∂θ j

˙ − 2C)T = −(M ˙ − 2C) Switching i and j shows (M


4.3 Dynamics of Open-chain Manipulator 19 ◇ Example: Planar 2-DoF Robot (continued)


m11 (θ) = α + 2β cos θ 2 , m22 = δ

m12 (θ) = m21 (θ) = δ + β cos θ 2 ˙ = −β sin θ 2 ⋅ θ˙2 , c12 (θ, θ) ˙ = −β sin θ 2 (θ˙1 + θ˙2 ) c11 (θ, θ)

˙ = β sin θ 2 ⋅ θ˙1 , c22 (θ, θ) ˙ =0 c21 (θ, θ)

⎧ ⎪ ⎪ Γ111 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Γ2 = ⎪ ⎪ ⎪ 11 ⎨ ⎪ ⎪ 1 ⎪ Γ12 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Γ12 = ⎪ ⎪ ⎩

1 ∂M11 ∂M11 ∂M11 1 ∂M11 ( + − )= =0 2 ∂θ 1 ∂θ 1 ∂θ 1 2 ∂θ 1 1 ∂M11 ∂M12 ∂M21 1 ∂M11 ( + − )= = −β sin θ 2 2 ∂θ 2 ∂θ 1 ∂θ 1 2 ∂θ 2 1 ∂M11 1 ∂M12 ∂M11 ∂M12 ( + − )= = −β sin θ 2 2 ∂θ 1 ∂θ 2 ∂θ 1 2 ∂θ 2 1 ∂M12 ∂M12 ∂M22 ∂M12 1 ∂M22 ( + − )= − = −β sin θ 2 2 ∂θ 2 ∂θ 2 ∂θ 1 ∂θ 2 2 ∂θ 1



⎧ 1 ⎪ ⎪ Γ21 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Γ ⎪ ⎪ ⎪ 21 ⎨ ⎪ ⎪ 1 ⎪ Γ22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Γ22 ⎪ ⎪ ⎩

1 ∂M21 ∂M21 ∂M11 ∂M21 1 ∂M11 + − )= − = β sin θ 2 = ( 2 ∂θ 1 ∂θ 1 ∂θ 2 ∂θ 1 2 ∂θ 2 1 ∂M21 ∂M22 ∂M21 1 ∂M22 = ( + − )= =0 2 ∂θ 2 ∂θ 1 ∂θ 2 2 ∂θ 1 1 ∂M22 ∂M21 ∂M12 1 ∂M22 = ( + − )= =0 2 ∂θ 1 ∂θ 2 ∂θ 2 2 ∂θ 1 1 ∂M22 ∂M22 ∂M22 1 ∂M22 = ( + − )= =0 2 ∂θ 2 ∂θ 2 ∂θ 2 2 ∂θ 2 ˙ ˙ ˙ − 2C = [ −2β sin θ 2 ⋅ θ 2 −β sin θ 2 ⋅ θ 2 ] M −β sin θ 2 ⋅ θ˙2 0 −2β sin θ 2 ⋅ θ˙2 −2β sin θ 2 (θ˙1 + θ˙2 ) −[ ] 2β sin θ 2 ⋅ θ˙ 1 0 0 β sin θ 2 (2θ˙1 + θ˙2 ) =[ ] ⇐ skew-symmetric ˙ ˙ −β sin θ 2 (2θ 1 + θ 2 ) 0


4.3 Dynamics of Open-chain Manipulator 21 ◇ Example: Dynamics of a 3-dof robot Chapter 4 Robot Dynamics and Control

θ1

⎡ ⎢ ⎢ Lagrangian ⎢ Equations ξ 1 = ⎢ ⎢ ⎢ ⎢ Inertial ⎣ Properties of Rigid Body

Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

0 0 0 0 0 1

⎤ ⎡ 0 ⎥ ⎢ −l0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ , ξ 2 = ⎢ −1 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 1 ⎦ ⎣

⎤ ⎡ 0 ⎥ ⎢ −l0 ⎥ ⎢ ⎥ ⎢ l1 ⎥ , ξ 3 = ⎢ −1 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎦ ⎣ 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

θ2

θ3 l1

l2 L3

L2 r1 l0

r2

L1 r0 S

⎡ ⎢ I gsl1 (0) = ⎢ ⎢ ⎢ 0 ⎣

⎤ ⎡ 0 ⎢ ( 0 ) ⎥ ⎥, gsl (0) = ⎢ I ⎥ 2 ⎢ r0 ⎥ ⎢ 0 1 ⎦ ⎣

⎤ ⎡ 0 ⎢ ( r1 ) ⎥ ⎥, gsl (0) = ⎢ I ⎥ 3 ⎢ l0 ⎥ ⎢ 0 1 ⎦ ⎣

0 ⎤ ( l 1 + r2 ) ⎥ ⎥ ⎥ l0 ⎥ 1 ⎦



⎡ mi 0 0 ⎤ ⎢ 0 mi 0 ⎥ 0 ⎢ ⎥ ⎢ 0 0 mi ⎥ Mi = ⎢ Ixi 0 0 ⎥ ⎢ ⎥ ⎢ 0 Iy i 0 ⎥ 0 ⎢ ⎥ ⎢ 0 0 Iz i ⎥ ⎣ ⎦ mi : The mass of the object Ixi : The moment of inertia about the x axis Γ112 = (Iy2 − Iz2 − m2 r12 )c2 s2 + (Iy2 − Iz3 )c23 s23 − m3 (l1 c2 + r2 c23 )(l1 s2 + r2 s23 ) Γ113 = (Iy3 − Iz3 )c23 s23 − m3 r2 s23 (l1 c2 + r2 c23 ) 1 Γ12 = (Iy2 − Iz2 − m2 r12 )c2 s2 + (Iy3 − Iz3 )c23 s23 − m3 (l1 c2 + r2 c23 )(l1 s2 + r2 s23 ) 1 Γ13 = (Iy3 − Iz3 )c23 s23 − m3 r2 s23 (l1 c2 + r2 c23 ) 1 Γ21 = (Iz2 − Iy2 + m2 r12 )c2 s2 + (Iz3 − Iy3 )c23 s23 + m3 (l1 c2 + r2 c23 )(l1 s2 + r2 s23 ) 3 2 3 Γ22 = −l1 m3 r2 s3 , Γ23 = −l1 m3 r2 s3 , Γ23 = −l1 m3 r2 s3 2 Γ311 = (Iz3 − Iy3 )c23 s23 + m3 r2 s23 (l1 c2 + r2 c23 ) , Γ32 = l 1 m3 r2 s 3


4.3 Dynamics of Open-chain Manipulator 23 ˙ = ∂V , V(θ) = m1 gh1 (θ) + m2 gh2 (θ) + m2 gh3 (θ) N(θ, θ) ∂θ Chapter 4 Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints

gsli (θ) = eξ1 θ 1 ⋯eξi θ i gsli (0) ⇒ ˆ

ˆ

h1 (θ) = r0 , h2 (θ) = l0 − r1 sin θ, h3 (θ)

= l0 − l1 sin θ 2 − r2 sin(θ 2 + θ 3 )gsli (θ)

= eξ1 θ 1 ⋯eξi θ i gsli (0) ⇒ ˆ

ˆ

h1 (θ) = r0 , h2 (θ) = l0 − r1 sin θ, h3 (θ)

= l0 − l1 sin θ 2 − r2 sin(θ 2 + θ 3 )



⎡ ⎢ ⎢ ⎢ b J1 = Jsl1 (θ) = ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ J2 = Jslb2 (θ) = ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ b J3 = Jsl3 (θ) = ⎢ ⎢ ⎢ ⎢ ⎣

0 0 0 ⎤ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ 1 0 0 ⎥ ⎦ −r1 c2 0 0 ⎤ 0 0 0 ⎥ ⎥ 0 −r1 0 ⎥ 0 −1 0 ⎥ ⎥ −s2 0 0 ⎥ c2 0 0 ⎥ ⎦ −l2 c2 − r2 c23 0 0 ⎤ 0 l1 s3 0 ⎥ ⎥ 0 −r2 − l1 c3 −r2 ⎥ 0 −1 −1 ⎥ ⎥ −s23 0 0 ⎥ ⎥ c23 0 0 ⎦



M11 M12 M13 M(θ) = [ M21 M22 M23 ] = J1T M1 J1 + J2T M2 J2 + J3T M3 J3 M31 M32 M33 2 M11 = Iy2 s2 + Iy3 s223 + Iz1 + Iz2 c22 + Iz3 c223 + m2 r12 c22 + m3 (l1 c2 + r2 c23 )2

M12 = M13 = M21 = M31 = 0

M22 = Ix2 + Ix3 + m2 l12 + M2 r12 + m3 r22 + 2m3 l1 r2 c3 M23 = Ix3 + m3 r22 + m3 l1 r2 c3 M32 = Ix3 + m3 r22 + m3 l1 r2 c3 M33 = Ix3 + m3 r22

n n ˙ = ∑ Γk θ˙k = 1 ∑ ( ∂Mij + ∂Mik − ∂Mkj ) θ˙k Cij (θ, θ) ij 2 k=1 ∂θ k ∂θ j ∂θ i k=1



◻ Additional Properties of the dynamics in terms of POE: Define:

⎧ ⎪ Ad−1ξˆj+1 θ j+1 ξˆ θ i > j ⎪ ⎪ ⋯e i i ⎪ ⎪ e Aij = ⎨I i=j ⎪ ⎪ ⎪ ⎪ ⎪ i

1. Introduction to Robotics notes 2.pdf - Google Drive

1. Introduction to Robotics notes 2.pdf - Google Drive

Suggest Documents

Introduction To Computer Networks-1 Notes 1.pdf - Google Drive

Introduction to Sericulture Notes 1.pdf - Google Drive

Introduction to Levelling Notes 1.pdf - Google Drive

Introduction to Levelling notes 1.pdf - Google Drive

Short Introduction to Robotics

Introduction to Robotics - Mechatronics

Introduction to Robotics

Introduction to Robotics

1 Introduction - CMU Robotics Institute

Introduction to AI Robotics - Murphy R.R.pdf - Google Drive

[PDF] Introduction to Robotics - Google Sites

PDF Download Introduction to Robotics - Google Sites

[PDF] Download Introduction to Robotics - Google Sites

[PDF] Download Introduction to Robotics - Google Sites

Online PDF Introduction to Robotics - Google Sites

Read PDF Introduction to Robotics - Google Sites

[PDF] Introduction to Robotics - Google Sites

PDF Introduction to Robotics - Google Sites

DownloadPDF Introduction to Robotics: Analysis ... - Google Sites

PDF Online Introduction to Robotics - Google Sites

Plant introduction and acclimatization notes 1.pdf - Google Drive

Welcome Introduction to Mobile Robotics

An Introduction to Swarm Robotics

Introduction to AI Robotics