Mechatronics 1
Weeks 5,6, & 7
Learning Outcomes
• By the end of week 5-7 session, students
will understand the dynamics of industrial
robots.
Course Outline
• General understanding of moment, force and
moment of inertia which occur in a robot.
• Interpretation of robot dynamic behaviour and
dynamic equation.
• Introduction to Newton-Euler dynamic equation.
• Newton Euler approach.
• Forward iteration & backward iteration
algorithms.
• Implementation of Newton Euler approach and
its physical interpretation.
Why Dynamics ?
• Instant response never happens in a real world.
• Various postures, tasks and payloads.
• Fundamental problem to solve, prior to
designing a control system  robot controller.
• Never expect to have a good control system if
we don’t know the dynamic behaviour of the
robot.
• There is no such task accomplishment if the
control system is poor.
Dynamics (1)
It concerns about forces/ torques excerted in a
robot movement.
Centre of mass of link i
O’
pi* + si = ri - pi-1
si
fi
(xi-1,yi-1,zi-1)
pi*
ni
o*
Link i-1
(xi,yi,zi)
ri
pi-1
pi
Z0
Y0
X0
fi+1
ni+1
Link i+1
Dynamics (2)
Dynamic components :
•Mass & Inertia
•Friction
•Inductance
Dynamics (3)
• Mathematical equations describes the
dynamic behavior of the manipulator
– For computer simulation
– Design of suitable controller
– Evaluation of robot structure
– Joint torques
Robot motion, i.e.
acceleration, velocity, position
Dynamics (4)





Forward Dynamics


Inverse Dynamics





Dynamics (5)
• Lagrange Euler
– Closed form.
– Tedious and heavy computing burden.
• Newton Euler
– Recursive Form.
– Efficient computing burden.
– Suitable for real time application.
Physics Quick Review
•
•
•
•
•
Mass
Moment of Inertia
Centre of Mass
Gravity
Physical interpretation in a robot case.
Physics Quick Review
•
•
•
•
•
Forces & Torques
Newton Law
Kinetic & Potential Energies
Physical interpretation in a robot case.
Posture of robot (link configuration)
Lagrange-Euler - Principle
• Lagrange-Euler Formulation
d L
L
( )
 i
dt qi
qi
– Lagrange function is defined
L K P
• K: Total kinetic energy of robot
• P: Total potential energy of robot
•  : Joint variable of i-th joint
• : first time derivative of
qi
•
: Generalized force (torque) at i-th joint
qi
i
q
i
• Physical interpretation
Newton - Euler
• Intuitively, more understandable.
• Composed of 2 steps
– Forward iteration to obtain information on
position, velocity and acceleration.
– Backward iteration to obtain information on
forces and torques.
• Frames attached to the corresponding
links are considered moving to each other
while the robot moves.
Basic Concept
• The movement of lower joints affect the
movement of upper (distal) joints. Hence, the
movement of upper joints are the functions of
lower joint movement.
• Forces and torques occur in the upper joints
affect the forces and torques occur in the lower
joint (lower joints suffer forces/ torques exerted
by the upper joints).
• All information are presented in vector
components.
Revealing Kinematic Information
zi-1
zi
Joint i
i
Link i
xi
zi-2
ai
O`
Link i -1
Pi*
yi
xi-1
Pi
Joint i + 1
i
z0
Joint i-1
ai-1
Pi-1
O
x0
O*
y0
yi-1
Forward Recursive w.r.t. Inertial Frame
qi   i , q i  i , qi  i
For i = 1 to n;
n = number of joints; Input :
q  d , q  d , q  d
i
i
i
Revolute Joint
ωi 1  z i 1qi
ωi  
ωi 1
Prismatic Joint

 i-1  z i 1qi  ωi 1  z i 1qi
Revolute Joint
ω

ωi  
 i 1
ω
Prismatic Joint

 i  pi * ωi  (ωi  pi *)  v i 1
 ω

v i  z i 1qi  ωi  pi * 2ωi  (z i 1qi )

 ωi  (ωi  pi *)  v i 1
ai 
Notes :
i
i
Revolute Joint
Prismatic Joint
 i si ωi  ωi si   v i
ω
 0  V0  0
ω0  ω
gx 
v 0  g   g y 
 g z 
g  9.8062 m/s 2
i
Physical meaning of forward
recursive.
Forces/ Torques Exerted in a
Robot
Centre of mass of link i
O’
pi* + si = ri - pi-1
si
fi
(xi-1,yi-1,zi-1)
pi*
ni
o*
Link i-1
(xi,yi,zi)
ri
pi-1
pi
Z0
Y0
X0
fi+1
ni+1
Link i+1
Backward Recursive w.r.t Inertial Frame
Centre of mass of link i
O’
pi* + si = ri - pi-1
For i = n to 1; n = number of joints.
Fi  mi ai
pi*
ni
 i  ωi Ii  ωi 
Ni  I i ω
o*
Link i-1
pi-1
pi
Z0
Y0
ni  ni 1  pi*  fi 1  pi*  si  Fi  Ni
X0
Revolute Joint
Prismatic Joint
fi+1
(xi,yi,zi)
ri
fi  Fi  fi 1
T

n i z i-1  bi qi
i   T

 fi z i-1  bi qi
si
fi
(xi-1,yi-1,zi-1)
ni+1
Link i+1
Physical meaning of backward
recursive.
Manipulator Dynamics
• Dynamics Model of an n-link Arm
  D(q)q  h(q, q )  C (q)
 D11  D1n 

D  

 The Acceleration-related Inertia
 Dn1  Dnn  matrix term, Symmetric
 h1 
h( q, q )     The Coriolis and Centrifugal terms (negligible
hn  for a slow movement)
 C1 
C (q)    
The Gravity terms
Cn 
 1 
     Driving torque
applied on each link
 n 
Physical meaning of a complete
dynamic model of an n link arm.
Forward Recursive w.r.t. Own
Frame
i
i
 i R i 1 (i 1 R 0ωi 1  z 0 qi )
R 0i   i

R i 1 (i 1R 0ωi 1 )
 i R i-1
i 
R 0ω



i-1
 i-1  z 0 qi 
R 0ω

i
R i 1

i 1
 
Revolute
Joint
Prismatic

i 1

R 0ωi 1  z 0 qi
 i 1
R 0ω

 
Joint

Revolute Joint
Prismatic Joint


 i 1 R 0ω
 i  i R 0p i  i R 0ω i  i R 0ω i  i R 0p i *

Revolute Joint
  i R i 1 i 1 R 0 v i 1
i
R 0 v i   i
i 1
i
i 1
Prismatic Joint



R
z
q

R
v

R
ω

R 0p i *
0 i 1
0 i
 i 1 0 i
  2 i R ω  i R z q  i R ω  i R ω  i R p *
0 i
i 1 0 i
0 i
0 i
0 i


i





  
   



 
 
 
 

 i  i R 0 si  i R 0ω i   ι R 0ω i  i R 0 si  i R 0 v i
R 0 ai  i R 0ω
Backward Recursive w.r.t. Own
Frame
i
i
R 0n i 
i
R i 1


R 0fi  i R i 1
i 1
R 0ni 1 
 i R 0I i 0 R i




i 1
i 1

R 0fi 1  mi i R 0 ai
 
R 0pi* 
i 1
 
 R ω   R ω   R I
i
i
0
i


i
0


 i R n T i R z  b q
i-1 0
i i
i   0 i T
 i R 0fi i R i-1z 0  bi qi
 
i
 
R 0fi 1  i R 0pi*  i R 0 si  i R 0Fi
0 i
0
Ri
 R ω 

i
0
i
Revolute Joint
Prismatic Joint

τ  D(θ)θ  H θ, θ  Gθ  B θ
τ  nx1
D  nxn  symmetric
H  nx1
G  nx1
B  nx1
matrix
Dynamic Simulation Process
θ0
θ 0
τ
Dynamic Equation
θ
Integration
(ODE)
θ θ