Dynamics of Haptic and Teleoperation
Systems
Jee-Hwan Ryu
School of Mechanical Engineering
Korea University of Technology and Education
Teleoperation System Overview
xm
fm
fs
xs
Dynamic Model of Teleoperation Systems
1. Mechanical Model
2. Mechanical and Electrical Analogy
3. Electrical Model
4. Understanding of the behavior of
Teleoperation Systems
One-DOF Schematic Diagram
Operator
f op
bm
fm
k op
bop
Slave
Master
bs
fs
ke
ms
mm
m op
Environment
τm
τs
xm
me
be
xs
The dynamics of the master arm and slave arm is given by the
following equations
Master : mm &x&m + bm x&m = τ m + f m
Slave : ms &x&s + bs x& s = τ s − f s
Definition of Parameters
Master : mm &x&m + bm x&m = τ m + f m
Slave : ms &x&s + bs x& s = τ s − f s
xs : displacement of master arm
xs : displacement of slave arm
mm : mass coefficient of the master arm
bm : viscous coefficient of the master arm
ms : mass coefficient of the slave arm
bs : viscous coefficient of the slave arm
f m : force that the operator applies to the master arm
f s : force that the slave arm applies to the environment
τ m : actuator driving forces of master
τ s : actuator driving forces of slave
Dynamics of the Environment
The dynamics of the environment interacting with the
slave arm is modeled by the following linear system:
f s = me &x&s + be x& s + k e x s
where
me : mass coefficient of the environment
be : damping coefficient of the environment
ke : stiffness coefficient of the environment
The displacement of the object is represented by x s because the
slave arm is assumed to be rigidly attached with the environments
or slave arm firmly grasping the environments, in such a way that
it may not depart from the object, once the slave arm contact the
environments.
Dynamics of the Operator
It is also assumed that the dynamics of the operator can be
approximately represented as a simple spring-damper-mass system
f op − f m = m op &x&m + bop x& m + k op x m
where
mop : mass coefficient of the operator
bop : damping coefficient of the operator
kop : stiffness coefficient of the operator
f op : force generated by the operator’s muscles
The displacement of the operator is represented by x m because it
is assumed that the operator is firmly grasping the master arm
and operator never releases the master arm during the operation.
Master/Operator Cooperative System
f op
mop s + bop s + kop
2
xm
−
fm
1
mm s 2 + bm s
+
+
τm
+
Slave/Environment Cooperative System
me s 2 + be s + ke
xs
fs
−
1
ms s 2 + bs s
+
Total System
f op
mop s 2 + bop s + kop
xm
−
fm
1
mm s 2 + bm s
+
+
τm
+
me s 2 + be s + ke
xs
fs
1
2
ms s + bs s
τs
−
τs
+
Let’s Do This
•
•
Simulate dynamics behavior
How to make stable simulation ?
“Y. Yokokohji and T. Yoshikawa, “Bilateral Control of
Master-slave Manipulators for Ideal Kinesthetic CouplingFormulation and Experiment,” IEEE Trans. Robotics and
Automation, Vol. 10, No. 5, pp. 605-620, 1994.”
Constitutive Relation
The environment defines a “constitutive relation,” a relation
between force and position or one of its derivatives.
Examples:
Spring
F
Damper F
K
Inertia
B
x
F
M
x&
&x&
Electrical Analogy
These relations for mechanical systems are directly analogous
to similar relations for electrical systems
Capacitor V
Resistor V
Inductor V
1/C
R
∫ idt
L
i
Force
Velocity
Voltage (effort)
Current (flow)
Example 1
Convert an environment defined by the mechanical system
F = m&x& + bx& + kx
to the equivalent electrical circuit.
We can make the substitutions
V ↔ F , i ↔ x&
giving
V =m
t
di
+ bi + k ∫ idt
0
dt
The parameters m, b, k correspond to the electrical parameters
m ↔ L, b ↔ R , k ↔
1
C
di
dt
Example 1
k
F
F = m&x& + bx& + kx
m
b
L
R
V
V =m
t
di
+ bi + k ∫ idt
0
dt
C
Mechanical to Electrical and vice versa
Consider two equations which correspond physical laws:
∑ F = 0, Point mass
∑ x& = 0, mechanical loop
The analogous electrical laws are
∑V = 0, electrical loop
∑ i = 0, circuit node
We MUST equate a point mass with an electrical loop and
circuit node with a mechanical loop. In other words, we must
map series mechanical connections to parallel electrical ones
and vice versa.
Example 2
Convert the following mechanical system to an equivalent electrical
network:
x1
x2
k
F
M1
M2
b
1) point mass = electrical loop
Example 2
2) A force generator is connected to the first mass. Thus we insert
a voltage source in the first loop:
Vf
3) M1, M2 correspond to inductors. Each inductor should have a
current which corresponds to the correct velocity therefore:
L1
Vf
L2
Example 2
4) b,k, are connected to both masses. The velocity/position which
determines their forces is the difference between the two masses’
velocities. They thus correspond to resistance and capacitance
connected into the common branch of the two loops since
x&2 − x&1 → i2 − i1
L2
L1
Vf
R
C
where R=b, and C=1/K
Example 3: Contact
• Discontinuous contact is harder to model, but more
important since contact always begins with and impact
between the robot and environment. Consider a robot which
is predominantly an inertia. Contact with a rigid wall could
be modeled by a switch:
Robot
i1
open : Contact → i1 = 0
closed : free motion → i1 ≠ 0
Example 3: Contact
or, for a non-rigid environment:
LE
Robot
i1
RE
Environment
CE
The switch can be controlled by the position:
∫ i (t )dt
t
0
1
Electrical Conversion
Z op
Contact point
=
Circuit port
Zm
Two-port Network
Operator
f op
k op
fs
ke
ms
mm
m op
Environment
bs
bm
fm
bop
Slave
Master
me
τs
τm
xm
be
xs
Teleoperation system have two-contact point
Thus, two-circuit port
Operator
Im
Z op
Vop
2-port Network
+
Vm
Is
Environment
+
Zm
Zs
-
Vs
Ze
-
Correspondence btw. Mech. Elec.
velocity of the master arm x& m
↔
current
Im
velocity of the slave arm − x& s
↔
↔
current
Is
↔
↔
voltage Vm
operator’s force
f op
force at the master side
fm
force at the slave side
fs
voltage Vop
voltage Vs
Two-port Mapping
The relationship between efforts and flows is commonly described
in terms of an immittance matrix (P).
Immittance mapping : y = Pu
Admittance matrix
Impedance matrix
⎡ f m ⎤ ⎡ z11
⎢ f ⎥ = ⎢z
⎣ s ⎦ ⎣ 21
z12 ⎤ ⎡ v m ⎤
z 22 ⎥⎦ ⎢⎣− v s ⎥⎦
y12 ⎤ ⎡ f m ⎤
y 22 ⎥⎦ ⎢⎣ f s ⎥⎦
Alternate hybrid matrix
Hybrid matrix
⎡ f m ⎤ ⎡ h11
⎢− v ⎥ = ⎢h
⎣ s ⎦ ⎣ 21
⎡ v m ⎤ ⎡ y11
⎢− v ⎥ = ⎢ y
⎣ s ⎦ ⎣ 21
h12 ⎤ ⎡v m ⎤
h22 ⎥⎦ ⎢⎣ f s ⎥⎦
⎡v m ⎤ ⎡ g11
⎢ f ⎥ = ⎢g
⎣ s ⎦ ⎣ 21
g12 ⎤ ⎡ f m ⎤
g 22 ⎥⎦ ⎢⎣− v s ⎥⎦
All of the immittane mapping satisfy the following condition
y T u = f m vm − f s vs
Hybrid Parameter Interpretation
h11 =
h12 =
h21 =
h22 =
Vm
Im
Vm
Vs
Is
Im
Is
Vs
⇔
Vs = 0
⇔
I s =0
⇔
Vs = 0
⇔
I m =0
fm
vm
⇔
Free motion input impedance
⇔
Force feedback gain λ f
⇔
Forward velocity gain − λ p
⇔
Output admittance w/ clamped input
f s =0
fm
fs
vs
vm
vs
fs
vs = 0
f s =0
vm = 0
λf ⎤
⎡ Z In
H =⎢
⎥
−
λ
Z
1
/
Out ⎦
⎣ p
Dynamic Model of Haptic Interfaces
1. Mechanical Model
2. Electrical Model
3. Discrete Model with ZOH
Haptic Interaction System Overview
3D Graphics Processor
60Hz
Virtual Environment
or Model
Mechanism
Haptics Processor
1000 Hz
Power Converter
Mechanical and Electrical Model
Operator
f op
Master
bm
fm
k op
mm
m op
bop
Virtual
Environment
τm
xm
− vs
vm
Human
Operator
+
fm
−
Haptic
Interface
Mechanical Model
mv&a + bva = f h − f a , va = vh
⎡ f h ⎤ ⎡ms + b 1⎤ ⎡ vh ⎤
⎢− v ⎥ = ⎢ − 1
0⎥⎦ ⎢⎣ f a ⎥⎦
⎣ a⎦ ⎣
+
fs
−
Virtual
Environment
Discrete Model with ZOH
Z d (z ) = (ms + b ) s → 2 ⎛⎜ z −1 ⎞⎟
T ⎝ z +1 ⎠
ZOH ( z ) =
1 ( z + 1)
2 z
⎡ f h ⎤ ⎡ Z d ( z ) ZOH ( z )⎤ ⎡ vh ⎤
⎢− v ∗ ⎥ = ⎢ − 1
⎥⎢ f ∗ ⎥
0
⎦⎣ c ⎦
⎣ c⎦ ⎣

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