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Medical Robotics
Islam S. M. Khalil, Omar Shehata, and Nour Ali
German University in Cairo
March 28, 2018
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
Any dynamical system consists of energy storage elements andenergy dissipation elements. Energy flows along the powerconserving interconnections of a system.
Figure: Network representation of a dynamical system. Σi represents theith dynamical subsystem. ui and yi are the ith input and output of thesystem, respectively.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
Energy storage elements consists of:
Input (e.g., force and velocity)
State variable (e.g., momentum and position)
Scalar energy function (e.g., kinetic and potential)
Output (e.g., force and velocity)
Figure: Energy storage element consists of an input (u(t)), state variable(q(t)), scalar energy function (E (q)), and an output (y(t)).
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
Mass (Σm) is a kinetic energy storage element, whereas spring(Σs) is a potential energy storage element.
Figure: Energy storage element consists of an input (u(t)), state variable(q(t)), scalar energy function (E (q)), and an output (y(t)).
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
Energy flows through the power conserving interconnection of themass and spring energy storage elements. The effort (es) and flow(f s) variables are the force and velocity, respectively.
Figure: Energy storage element consists of an input (u(t)), state variable(q(t)), scalar energy function (E (q)), and an output (y(t)).
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
Energy flows through the power conserving interconnections of thelumped mass spring system.
Figure: Energy-based representation of a lumped mass spring system withthree degrees of freedom. Σm and Σs represent the kinetic and potentialenergy storage elements, respectively. e and f represent the effort andflow variables of the power conserving interconnections.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
lumped parameter elements: physical entities whos energy(storage elements) or power (dissipative elements) is definedby a scalar (e.g., mass, resistor)
network: a system described by lumped parameter elementsconnected in series and parallel (e.g., RLC circuit)
port: a location where energy can move into or out of anetwork (e.g., contact point between human and hapticinterface)
analysis of energy flow in a network provides key insights
the derivative of energy is power and can be expressed as theproduct of two variables: effort ξ and flow Feffort and flow are linked together by the behavior of systemsat their ports through
impedance Z = F/ξ admittance A = ξ/F = 1/Z
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Analysis of energy flow
LTI continuous systems can be described by the relationshipsbetween effort and flow variables
effort variable flow variablemechanical applied force/torque linear/angular velocity
electrical voltage current
electricalvoltage v(t)current i(t)resistance Rinductance Lcapacitance 1/Cone-portimpedance Z
mechanicalforce f (t)velocity V (t)viscous friction binertia Mstiffness Kseries/parallel ofprevious elements f (s) = Z (s)V (s)
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
One-port network
effort and flow define positive or negative power going intothe network at a single location where energy exchange takesplace
signs of effort and flow are usually defined so that power ispositive flowing into the port (effort of interest: on the port)
examples: mechanical (MSD system), electrical (RCL)
Figure: one-port network.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Two-port network
two-ports have separate effort and flow variables, defined foreach port, and a separate coordinate system (sign convention)for each port
example: electric network
Figure: two-port network.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Two-port model of teleoperator
bilateral teleoperation systems can be viewed as a cascadeinterconnection of two-port (master, communication channeland slave) and one-port (operator and environment) blocksusing mechanical/electrical analogy and network theory, theteleoperation system is described as interconnection of oneand two-port electrical elements
Figure: F ∗h and F ∗
e are the exogenous force inputs by the operator andthe environment. Zt is the transmitted impedance (seen by the human),we assume passive environment F ∗
e = 0.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Two-port model of teleoperator
in the two-port model the behavior of the system iscompletely characterized by measurements of the forces andvelocities at the two ports
Fe = ZeVe Fh = ZtVh
of these four involved variables, two may be chosen asindependent and the remaining two dependent
dependent variables are related to independent ones throughthe
impedance (Fh,Fe , t) = Z(Vh,Ve , t)
admittance (Vh,Ve , t) = Y(Fh,Fe , t)
hybrid (Fh,Ve , t) = H(Vh,Fe , t)
scattering (F − bV , t) = S(F + bV , t)
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
transparency
For the same forces Fe = Fh we want the same motionVe = Vh
Transparency condition
Zt = Ze
for the analysis consider the linearized behavior around contactoperating point using the hybrid matrix formulation[
Fh(s)Vh(s)
]=
[H11(s) H12(s)H21(s) H22(s)
] [Ve
−Fe
]Fh = (H11 − H12Ze) (H21 − H22Ze)−1 Vh
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Lawrence’s general 4-channel architecture
Figure: .Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Particular control schemes
Block position-position position-forcemaster impedance Zm Mms Mms
master controller Cm Bm + Kms/s Bm
slave impendence Zs Mss Mss
slave controller Cs Bs + Ks/s Bs + Ks/s
velocity channel C1 Bs + Ks/s Bs + Ks/s
force channel C2 not used Kf
force channel C3 not used not used
Velocity channel C4 −(Bm + Km/s) not used
operator impedance Zh not a function of control architecture
task impedance Ze not a function of control architecture
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Example: position-position vs. the 4-channel architecture
Master
impedance Zm = Mms
controller Cm = Bm + Km/s
Communication channels
C1 = Bs + Ks/s
C4 = −(Bm + Km/s)
C2 = not used
C3 = not used
Slave
impedance Zs = Mss
controller Cs = Bs + Ks/s
Master and slave
Fmc = −CmVh − C4Ve
= (Bm + Km/s)(Ve − Vh)
Fsc = −CsVe − C1Vh
= (Bs + Ks/s)(Vh − Ve)
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Example: position-position vs. the 4-channel architecture
solving for the transfer functions between master and slave forcesand velocities
H11 = (Zm + Cm)D(Zs + Cs − C3C4) + C4
H12 = −(Zm + Cm)D(I − C3C4) + C2 − C2
H21 = D(Zs + Cs − C3C4)− C2
H22 = −D(I − C3C2)
D = (C1 + C3Zm + C3Cm)−1
these expressions can be used to
design suitable control laws Ci (i = 1, . . . , 4,m, s)
design suitable master and slave (Zm,Zs)
compare transparency performance of different teleoperationarchitectures
improve or optimize transparency
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Optimizing for transparency
Since Zt = (H11 − H12Ze) (H21 − H22Ze)−1
Perfect transparency (Zt = Ze) can be obtained by choosingCi such that H22 = 0, H11 = 0, and H12 = I
C3C2 = I , C4 = −(Zm + Cm), C1 = (Zs + Cs), C2 = I
but C4 = −(Zm + Cm) and C1 = −(Zs + Cs) requireacceleration measurements
at low frequencies good transparency can be achieved withposition and velocity measurements
in any case, stability is an issue
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
passivity: a useful tool for stability analysis
necessary and sufficient conditions for stability are difficult toobtain with signal-based approach
an energy-based approach is more suited
passivity provides sufficient conditions for stability
transmission delays destroy passivity
suitable encoding of the transmitted information preservespassivity
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
passivity: a useful tool for stability analysis
Consider the dynamics
x = f (x) + g(x)u
y = h(x)
Assume f (0) = 0 and h(0) = 0, the system is dissipative if thereexist:
a continuous lower bounded function of the state (storagefunction)
V (x) ∈ C : Rn → R+
a function of the input/output pair (supply rate)
w(u, y) : Rm ×Rm → R
V (x(t))− V (x(t0)) ≤∫ t
t0
w(u(s), y(s))ds
V (x(t)) ≤ w(u(s), y(s))
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
passivity: a useful tool for stability analysis
V (x(t))− V (x(t0)) ≤∫ t
t0
w(u(s), y(s))ds
V (x(t)) ≤ w(u(s), y(s))
Stored energy: V (x(t))− V (x(t0)). Energy supplied to thesystem:
∫ tt0w(u(s), y(s))ds
when the supply rate is the scalar product between the in/out pair
w(u, y) = yTu
dissipativity is named passivity and the system is said passive.V (x(t)) can be interpreted as the energy of the system: physically,a passive system cannot produce energy
V (x(t)) ≤ V (x(t0)) +
∫ t
t0
y(s)Tu(s)ds
current energy V (x(t)) is at most equal to the initial energyV (x(t0)) + supplied energy from outside
∫ tt0y(s)Tu(s)ds
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
passivity and Lyapunov stability
assume V (0) = 0, i.e., 0 is a (local) minimum of the storagefunction
then V (x) is a Lyapunov candidate around 0 and
1 if u ≡ 0 then V ≤ 0, i.e., 0 is a (Lyapunov) stableequilibrium (passivity) stability of the unforcedequilibrium)
2 if y ≡ 0 then V ≤ 0, i.e., the zero dynamics of thesystem is (Lyapunov) stable
3 the system can be stabilized by a static outputfeedback
u = −φ(y), yTφ(y) > 0 ∀ y 6= 0
u = −ky , k > 0
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Passivity in feedback interconnection
very useful properties to analyze stability of teleoperationsystems
given two passive systems with appropriate in/out dimensionsand storage functions V1(x1) and V2(x2)
x1 = f1(x1) + g1(x1)u1
y1 = h1(x1)
x2 = f2(x2) + g2(x2)u2
y2 = h2(x2)
their feedback interconnection is passive with respect toV1(x1) + V2(x2) with (new) input v = (v1, v2) and outputy = (y1, y2) pairs
u1 = ±y2 + v1
u2 = ∓y1 + v2
ThereforeV = V1 + V2 ≤ yT1 v1 + yT2 v2
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
scattering representation: wave variables
consider the total power flow P in a two-port element as composedof two terms: the input power Pin and output power Pout ; denote
the effort F =[Fh Fe
]Tand the flow V =
[Vh −Ve
]T
P = Pin − Pout = FTV
= FhVh − FeVe
= 1/2(uTu − vTv)
u =[uTh uTe
]uh =
1√2b
(Fh + bVh)
vh =1√2b
(Fh − bVh)
v =[vTh vTe
]ue =
1√2b
(Fe − bVe)
ve =1√2b
(Fe − bVe)
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
scattering representation: wave variables
u =[uTh uTe
]uh =
1√2b
(Fh + bVh)
vh =1√2b
(Fh − bVh)
v =[vTh vTe
]ue =
1√2b
(Fe − bVe)
ve =1√2b
(Fe − bVe)
u is the input wave and v is the output wave
from physical intuition in stable systems the amplitude of v isless than the amplitude of u:
1
2
∫vTvdt ≤ 1
2
∫uTudt
i.e., the gain of the system is smaller than one
There is no requirements on phase. Delay does not destroystability.
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
scattering representation: wave variables
the scattering operator (or matrix) relates the input/outputwave variables, u and v , at each port of the teleoperatorinstead of the power variables (velocities and forces)
given an n-port system, the scattering matrix (or scatteringoperator) is defined as the operator which relates input andoutput wave variables as:
v(t) = S(t)u(t)⇒ F (t)− bV (t) = S(t) [F (t) + bV (t)]
For LTI systems
v(s) = S(s)u(s)⇒ F (s)− bV (s) = S(s) [F (s) + bV (s)]
if the power variables are related by the hybrid matrix H(s):[Fh(s)−Ve(s)
]= H(s)
[Vh(s)Fe(s)
]Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
scattering representation: wave variables
scattering and hybrid representation are related by
S(s) =
[I 00 −I
][H(s)− I][H(s) + I]−1
theorem
an LTI n-port element with scattering matrix S(s) is passive if andonly if
‖S(s)‖ ≤ 1
an LTI n-port element with scattering matrix S(s) is passive if andonly if
‖S(s)‖ = supωλ1/2max (S∗(jω)S(jω)) ≤ 1
where λmax(S(jω)) is the maximum eigenvalue and S∗(jω) thetranspose conjugate of S(jω)
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Example: force reflection
master and slave dynamics (including local controller)
Mhxh(t) = −Fe(t − T ) + Bhxh(t)− Khxh(t)
Me xe(t) = Kp (xh(t − T )− xe(t))− Be xe(t)
where Kh is the human operator stiffness (model). Ve = xe andVh = xh
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation
Thanks
Questions please
Islam S. M. Khalil, Omar Shehata, and Nour Ali Bilateral Teleoperation