Lecture Slides

Outline 

Velocity Kinematics 

EE 451 - Velocity Kinematics 

H.I. Bozma 

Electric Electronic Engineering 

Bogazici University 

November 5, 2012 

H.I. Bozma EE 451 - Velocity Kinematics

Outline 



Introduction 

Angular Velocity: Fixed Axis 

Skew-Symmetric Matrices 

Angular Velocity 

Linear Velocity 

Jacobian 

Tool Velocity 

Analytic Jacobian 

Singularities 

Force-Torque Relationships 

Inverse Velocity 



Outline 


Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Velocity in Configuration space C ⇔ Velocity in Workspace W 

◮ Representation of velocities 

◮ Revolute – angular 

◮ Prismatic – linear 

◮ Angular velocity about a fixed axis 

◮ Rotation around a moving axis 

◮ Instantaneous transformations btw n-vector joint velocities in 

C ⇔ 6-vector of angular and linear velocities in W → 

Jacobian (6-n matrix) 


Outline 


Pure Rotation About Fixed Axis 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Pure rotation → Every point moves in a circle. 

◮ Centers of circles – On the axis of rotation 

◮ Perpendicular to the axis – θ 

◮ Angular velocity ω = ˙θk where 

˙θ = dθ 

dt and 

k - unit vector in the axis of rotation 

◮ Linear velocity v = ω ×r 

r - Vector from origin (axis of rotation) to the point 


Goal 

Outline 


◮ Goal – The motion of a moving frame. 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ The motion of the origin of the frame 

◮ The rotational motion of the frame’s axes 


Outline 


Angular vs Linear Velocity 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Attach a frame rigidly to each object with an orientation 

◮ Each point on the object – Same angular velocity! 

◮ Angular velocity – Property of the frame attached to a body 

◮ Linear velocity – Property of the point, but rather the frame 


Outline 


Definition and Properties 

◮ Linearity 

◮ Relation to cross product 

◮ Similarity transformation 

◮ Quadratic form 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




Outline 


Derivative of a Rotation Matrix 

d 

dθ Rk,θ = S(k)Rk,θ 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




Outline 


Introduction 





Jacobian 

Tool Velocity 


Singularities 



Angular Velocity about a Moving Axis 

◮ Time varying rotation matrix R(t), R(t) ∈ SO(3) 


Outline 


Addition of Angular Velocities 

R 0 n = R 0 1 R1 2 ...Rn−1 

n 

˙R 0 n = S(ω 0 0,n )R0 n 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



ω 0 0,n = ω 0 0,1 +R 0 1ω 1 1,2 +R 0 2ω 2 2,3 +...+R 0 n−1ω n−1 

n−1,n 

= ω 0 0,1 +ω 0 1,2 +ω 0 2,3 +...+ω 0 n−1,n 


Outline 


Introduction 





Jacobian 

Tool Velocity 


Singularities 



Linear Velocity of a Point p Attached to a Frame 

Assume: p – Attached rigidly to o1x1y1z1 

◮ Case 1: o1x1y1z1 is rotating wrt o0x0y0z0 

◮ Case 2: Motion of o1x1y1z1 wrt o0x0y0z0 - Defined by 

H0 1 (t) = 

� R 0 1 (t) o 0 1 (t) 

0 1 

� 


Jacobian 

Outline 


◮ n link robotic system – q1,...,qn 

◮ T0 � 

R0 n = n(t) o0 � 

n(t) 

0 1 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ As robot moves around, qi, R 0 n and o 0 n – functions of time 

◮ Angular velocity of end effector ω 0 n(t) – Defined by 

S(ω 0 n(t)) = ˙R 0 n(t)(R 0 n(t)) T 

◮ Linear velocity of end effector v0 n = ˙o 0 n 

� 

v0 n 

ω0 � � 

Jv 

= J˙q where J = 

n Jω 

Goal: Find ξ = 

H.I. Bozma EE 451 - Velocity Kinematics 

� 

⇐ Jacobian

Outline 


Angular Velocity - Revolute Joint 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



If revolute joint, qi = θi with axis of rotation zi−1 

Let ω i−1 

i – Angular velocity of joint i wrt oi−1xi−1yi−1zi−1 Note 

that 

ω i−1 

i 

= ˙qiz i−1 

i−1 = ˙qik where k = 

⎡ 

⎣ 

0 

0 

1 

⎤ 

⎦ 


Outline 


Angular Velocity – Prismatic Joint 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



If prismatic joint, qi = di with axis of translation zi−1 

ω i−1 

i 

= 0 


Outline 


Angular Velocity - End effector 

Let ρi = 

� 1 if revolute 

0 otherwise 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



ω 0 n = ρ1˙q1k +ρ2˙q2R 0 1k +...+ρn˙qnR 0 n−1k 

= 

n� 

i=1 

ρi ˙qiz 0 i−1 


Outline 


Linear Velocity - End effector 

◮ Prismatic joint 

◮ Revolute joint 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




Outline 


Linear Velocity - Prismatic Joint 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




Outline 


Linear Velocity - Revolute Joint 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




Jacobian – Summary 

where 

Outline 


� 

Jv1 ... Jvn J = 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



Jω1 ... Jωn 

Jvi = 

� 

z0 i−1 ×(on −oi−1) if revolute 

z0 i−1 if prismatic 

Jωi = 

� 

z0 i−1 if revolute 

0 if prismatic 


�

Outline 


2 DOF RR Planar Robot 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




3 DOF RRR Robot 

Outline 


Introduction 





Jacobian 

Tool Velocity 


Singularities 




Tool Velocity 

Outline 


◮ TI transformation T6 tool = 

� 

R d 

0 1 

◮ ω tool = ω6 → ω tool 

tool = RT ω 6 6 

◮ v tool 

tool 

= vtool 6 +ωtool 6 ×rtool 

Introduction 





Jacobian 

Tool Velocity 


Singularities 




�

End Effector Frame 

◮ X = 

� d(q) 

α(q) 

Outline 


� 

∈ R3 ×SO(3) 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ ˙X = Ja(q)˙q ⇐ Analytic Jacobian 

◮ Assuming Euler angles R = Rz,φRy,θRz,ψ, 

� 

I 0 

Ja(q) = 

0 B−1 � � 

I 0 

J(q) = 

(α) 0 B−1 (α) 

⎡ 

where B(α) = ⎣ 

cosψsinθ −sinψ 0 

sinψsinθ cosψ 0 

cosθ 0 1 

⎤ 

⎦ 


�� ˙d 

ω 

�

Outline 


Singularities - Singular Configurations 

◮ ξ = J(q)˙q 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ J(q) = � J1(q) J2(q) ... Jn(q) � → ξ = � n 

i=1 Ji(q)˙qi 

◮ J(q) is a 6×n matrix → Rank(J(q)) ≤ min(6,n) 

◮ End effector velocity = � Arbitrary if Rank(J(q)) = 6 

◮ Rank(J(q)) is time-varying! 

◮ q ∈ C for which Rank(J(q)) < maxRank(J(q)) ← Singularity 

◮ For 6×6 matrix, det(J(q)) = 0 


Outline 


Problems of Singularities? 

◮ Certain velocities are not attainable 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Bounded end-effector velocities – Unbounded joint velocities 

◮ Often correspond to points on the boundary of workspace 

◮ Often correspond to points unreachable via small 

perturbations of link parameters 


Outline 


Decoupling of Singularities 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Arm singularities - Singularities resulting from the arm motion 

◮ Wrist singularities - Singularities resulting from the wrist 

motion 


RR Planar Robot 

Outline 


Introduction 





Jacobian 

Tool Velocity 


Singularities 




Outline 



Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Interaction with the environment → Forces and moments at 

the end effector F = [FxFyFznxnynz] T 

◮ [FxFyFznxnynz] → Joint torques τ where τ = J(q) T F 


Outline 


Inverse Velocity Problem 

Introduction 





Jacobian 

Tool Velocity 


Singularities 



◮ Problem Statement: Given ξ, find ˙q such that ξ = J(q)˙q 

◮ If J(q) is invertible (square and full rank), ˙q = J(q) −1 ξ 

◮ If n �= 6, J(q) is not invertible ! 

◮ If ξ ∈ Span(J(q)) ↔ Rank(J(q)) = Rank([J(q) |ξ]) 

(Gaussian Elimination) 

◮ If n > 6, Pseudoinverse J + = (J T J) −1 J → ˙q = J + (q) −1 ξ

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