Medical Robotics - MNRLab...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

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.



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)).



Mass (Σm) is a kinetic energy storage element, whereas spring(Σs) is a potential energy storage element.




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.




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.



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



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)


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.


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.


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.


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)


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


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


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)


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


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


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



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))



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


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


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


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)



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.



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 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)


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


Thanks

Questions please


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Medical Robotics - MNRLab...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