Fractional-Order Differential Equations

PART I: Fractional Calculus: Stability and ControlFractional Chebyshev Collocation Method

Discretization Framework for Spectral Collocation MethodPART II: Stewart Platform For Lower Extremity Robotic Rehabilitation

PART I: Fractional-Order Differential Equations: Stability and Control

PART II: An Optimal Stewart Platform For Lower Extremity Robotic Rehabilitation

A. Dabiri, E. A. Butcher, M. Nazari, and M. Poursina,A. Dabiri, M. Poursina, S. Sabet, D. Armstrong, and P. Nikravesh

Aerospace and Mechanical EngineeringUniversity of Arizona

[email protected]

August 07, 2017

A. Dabiri, E. A. Butcher, M. Nazari, and M. Poursina, A. Dabiri, M. Poursina, S. Sabet, D. Armstrong, and P. Nikravesh University of Arizona 1 / 47



Overview

1 PART I: Fractional Calculus: Stability andControl

Historical: How and why was the fractionalcalculus introduced?Introduction: What are fractional operatordefinitions?Geometrical: How do fractional differentialequations look like?Motivation: Why should we be bothered byusing fractional operators?

2 Fractional Chebyshev Collocation MethodSpectral Methods vs Finite DifferenceMethods

3 Discretization Framework for SpectralCollocation Method

What is the idea?Fractional Differentiation MatrixNumerical StabilityConclusion

4 PART II: Stewart Platform For LowerExtremity Robotic Rehabilitation




Historical: How and why was the fractional calculus introduced?Introduction: What are fractional operator definitions?Geometrical: How do fractional differential equations look like?Motivation: Why should we be bothered by using fractional operators?

SectionPART I:

Spectral Collocation Methods For Fractional-Order Periodic Delay-Differential Equations: Stability andControl





Historical: How and why was the fractional calculus introduced?

The Evolution of Numbers

N ⇒ W ⇒ Z ⇒ Q ⇒ R ⇒ C

The Evolution of the factorial operator

n!, n ∈N ⇒ Γ(α), α ∈ R

The Evolution of the derivative operator

Dn ≡ dn

dxn , n ∈N ⇒ Dα, α ∈ R ⇒ Dα, α ∈ C ⇒ Dα(·), α ∈ Ω





Introduction: What are fractional operator definitions?

Integer Derivative ⇒ Fractional Derivative ⇒ Fractional Integral

i nth Order Backward Differences of f (x)

Dn f (x) = limh→0

1hn ∑

k=0(−1)k f (x− kh)

D Grunwald-Letnikov Fractional Derivative

GLDx f (x) ≡ limh→0

1hα ∑

k=0(−1)k f (x− kh)

a

aAnatoly A. Kilbas/H. M. Srivastava/Juan J. Trujillo: Theory and Applications of Fractional Differential Equations, vol. 204, New York, NY 2006.








Dn f (x) = limh→0

1hn

n

∑k=0

(nk

)(−1)k f (x− kh)



1hα ∑

k=0(−1)k f (x− kh)

a









Dn f (x) = limh→0

1hn

n∑k=0

(−1)k f (x− kh)



1hα ∑

k=0(−1)k f (x− kh)

a









Dn f (x) = limh→0

1hn

n∑k=0

(nk

)(−1)k f (x− kh)



1hα ∑

k=0(−1)k f (x− kh)

a









Dn f (x) = limh→0

1hn

n∑k=0

(nk

)(−1)k f (x− kh)


GLa Dα

x f (x) ≡ limh→0

1hα

k= x−ah

∑k=0

Γ (α + 1)k! Γ (α− k + 1)

(−1)k f (x− kh)

a









Dn f (x) = limh→0

1hn

n∑k=0

(nk

)(−1)k f (x− kh)


GLa Dx f (x) ≡ lim

h→0

1hα

k =x− a

h∑k=0

Γ (α + 1)k! Γ (α− k + 1)

(−1)k f (x− kh)

a









Dn f (x) = limh→0

1hn

n∑k=0

(nk

)(−1)k f (x− kh)


GLa D

αx f (x) ≡ lim

h→0

1hα

k =x− a

h∑k=0

Γ (α + 1)k! Γ (α− k + 1)

(−1)k f (x− kh)

a






Integer Integral ⇒ Fractional Integral ⇒ Fractional Derivative

i n-fold Cauchy’s integral

Let f (x) be a continuous function on the real line. The nth repeated integral of f (x) at a ∈ R is

Jn f (x) =∫ x

a

∫ σ1

a· · ·

∫ σn−1

af (σn)dσn · · · dσ2 dσ1 =

1

(n− 1)!

∫ x

a(x− ζ)n−1 f (ζ)dζ

D Riemann-Liouville Fractional Integral and Derivative

RLa J α

x f (x) ≡ 1

Γ(α)

∫ xa (x− ζ)α−1 f (ζ) dζ RL

a Dαx f (x) ≡ Ddαe a J

dαe−αx f (x) (1)

a


requires fractional order initial conditions at the terminal for fractional differential equations.gives a non zero fractional derivative of a constant, i.e. RL

a Dαxc 6= 0.





i RL Fractional Derivative

Ra Dα

x f (x) ≡ Ddαe aJ dαe−αx f (x)

D Caputo Fractional Derivative

CaDα

x f (x) ≡a J dαe−αx Ddαe f (x)

Initial conditions appear with integer order for fractional differential equations.

It also results in vanishing fractional derivative of a constant, i.e. CaDα

xc = 0.





Geometrical: How do fractional differential equations look like?





Figure: The solution of two FDEs by the use of the FCC Toolbox1

1A. Dabiri: Guide to FCC: Stability and solution of linear time variant fractional differential equations with spectral convergence using the FCC toolbox package in MATLAB,http://u.arizona.edu/~armandabiri/fcc.html, version 4.0.0, [Online; accessed 26-July-2017], 2017.


http://u.arizona.edu/~armandabiri/fcc.html




Motivation: Why should we be bothered by using fractional operators?

I A better mathematical tool for controlling

PIαDβ

PID ⇒ Three tuning parameters

PIαDβ ⇒ Five tuning parameters

Fractional Damped-Delayed Mathieu Equation

x (t) + (a + b cos (2πt)) x (t) =0.5x (t− 1) + u (t)

(x (t) , x (t)) = (1, 0) , −1 ≤ t < 0

Let the fractional delayed feedback control be chosen

u (t) = k11 x (t− 1) + k21C0D

βt x (t− 1) (2)





Motivation: Why should we be bothered by using fractional operators?

I A better mathematical tool for controlling

PIαDβ

PID ⇒ Three tuning parameters

PIαDβ ⇒ Five tuning parameters

Fractional Damped-Delayed Mathieu Equation

x (t) + (a + b cos (2πt)) x (t) =0.5x (t− 1) + u (t)

(x (t) , x (t)) = (1, 0) , −1 ≤ t < 0

Let the fractional delayed feedback control be chosen

u (t) = k11 x (t− 1) + k21C0D

βt x (t− 1) (2)

The stability chart for a fractional damped-delayed Mathieuequation by a feedback control with a fractional order β





The model of a second order mass-spring system with a damper is:

mx(t) + c(t)x(t) + kx(t) = u(t),

where m, k, and c(t) are the mass, spring stiffness, and damping function, respectively.

We consider the control variable u(t) with feedforward and feedback parts as

u(t) = xd(t) + xd(t) + xd(t)− kp e(t)− kiν(·)0+ I

β(t)t e(t)− kd

ν(·)0+ D

α(t)t e(t),

where α(t) = a + b exp(−ct), a > 1, b > 0, c > 0, so that a + b > 1, and β(t) = d + f exp(−gt), d, f , andg > 0.

J (θ) =∫ ∞

0τ eT(τ, θ)e(τ, θ) dτ,





II A better mathematical tool for modeling

0 0.2 0.4 0.6 0.8 1Time [hours]

-20

-10

0

10

20

30

40

50

60

ShearStrain[%

]

Reference dataNRMSEInt=25.78%NRMSEFra=93.80%

0 0.5 1Time [hours]

0

2

4Shear Stress [MPa]

(a)

(a) Viscoelastoplastic model of EC2216a ,

aA. Dabiri/M. Nazari/E. A. Butcher: The Spectral Parameter Estimation Method for Parameter Identification of Linear Fractional Order Systems, in: American Control Conference (ACC), Boston,MA 2016.






0 0.2 0.4 0.6 0.8 1Time [hours]

-20

-10

0

10

20

30

40

50

60

ShearStrain[%

]


0 0.5 1Time [hours]

0

2

4Shear Stress [MPa]

(b) (c)

(a) Viscoelastoplastic model of EC2216a, (b) Impact problems ,







0 0.2 0.4 0.6 0.8 1Time [hours]

-20

-10

0

10

20

30

40

50

60

ShearStrain[%

]


0 0.5 1Time [hours]

0

2

4Shear Stress [MPa]

(d) (e) (f)

(a) Viscoelastoplastic model of EC2216a, (b) Impact problems ,(c) the fractional Kelvin-Voiget modelb


bA. Dabiri/E. A Butcher/M. Nazari: Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation, in: Journal of Sound and Vibration 388 (2017),pp. 230–244.




Spectral Methods vs Finite Difference Methods

Section

Fractional Chebyshev Collocation Method





A system of multi-order FDDEs with multiple delays

C0D

(α)t [x(t)] = f(t, x(t), C

0D(β)t [x(t)], y(t− τ)) (3)

x(t) = [x1(t), · · · , xn(t)]T, xi(t) ∈ R, f(t, x(t), C0D

(β)t [x(t)], y(t− τ)) ∈ Rn

φ(t), −τs ≤ t ≤ 0, 0 < τ1 < τ2 < · · · < τs

y(t− τ) = col

C0D

(ν1)t [x(t− τ1)], · · · , C

0D(νs)t [x(t− τs)]

α = 0 < αi ≤ 1, i = 1, 2, · · · , n,β = 0 < βi ≤ 1, i = 1, 2, · · · , n,νi = 0 < νi,j ≤ 1, i = 1, 2, · · · , n, j = 1, 2, · · · , sC0D

(α)t [·] = diag

([C0D

α1t [·], C

0Dα2t [·], · · · , C

0Dαnt [·]

]), where the operator diag(·) cre-

ates a diagonal matrix.





Spectral Method

Finite difference methods:: Equispaced collocation points

xi−3 xi−2 xi−1 xi+1 xi+2 xi+3xi

P1(x)

Finite difference methods:: Nonequispaced collocation points

xi−3xi−2 xi−1 xi+1xi+2 xi+3xi

P1(x)





Spectral Method



P1(x)

Pn(x)



P1(x)





Spectral Method



P1(x)

Pn(x)



P1(x)

Pn(x)






Finite difference methods:

Using a local nth-degree polynomial on n local equispaced collocation points leadingto a maximum accuracy of O(hn).

Spectral methods:

Using a global Nth-degree polynomial on all N non-equispaced collocation points resultsin maximum accuracy.





Direct spectral collocation methods

A direct implementation of a well-known property of fractional differentiation of polynomial bases. This is

CaDα

t (t− a)β =Γ (β + 1)

Γ (β + 1− α)(t− a)β−α, β 6= 0. (4)

Spectral collocation methods (in fractional)

Direct Indirect

Power methodsa

Schur-Pade Schure-dec. Logarithmic Exponential

Recurrent methodb

aA. Dabiri/E. A Butcher: Efficient Modified Chebyshev Differentiation Matrices for Fractional Differential Equations, in: Communications in Nonlinear Science and Numerical Simulation 50.ISSN 1007-5704 (2017), pp. 584–310.

bA. Dabiri/E. A Butcher: Stable Fractional Chebyshev Differentiation Matrix for Numerical Solution of Fractional Differential Equations, in: Nonlinear Dynamics 2017, First Online: 19 July 2017, 17.





Section

Discretization Framework for SpectralCollocation Method





What is the idea?

Schematic representation of using the state transition operator to find the solution of a linear ordinaryperiodic time-delayed system in Banach space X.





Fractional Differentiation Matrix

Fractional Differentiation Matrix

The discretized matrix of Ct0Dα

t [·] at the collocation points t = [t0, t1, · · · , tN−1] is named

fractional differentiation matrix and denoted by Dαdt

. It is a linear map that maps the discretized

function xdt onto the discretized value of Ct0Dα

t [x(t)] at those points as

Dαdt

xdt =

[[Ct0Dα

t [x(t)]]

t=t0[Dα

t [x(t)]]t=t1· · ·

[Ct0Dα

t [x(t)]]

t=tN−2

[Ct0Dα

t [x(t)]]

t=tN−1

]T.





Theorem

The fractional finite difference differentiation matrix in the sense of Caputo at uniform collocation pointst = [t0 = a, t1, · · · , tN−1 = b] with the fixed step h = b−a

N−1 isa

Dαdt

:=1

hαΓ(2− α)

0 0 0 0 · · · 0 0

a1 1 0. . .

. . . 0 0

a2 b1 1 0. . .

. . . 0

a3 b2 b1 1 0. . . 0

.... . .

. . .. . . 1 0 0

aN−2 bN−3. . . b2 b1 1 0

aN−1 bN−2 bN−3 · · · b2 b1 1

(5)

where ak = (k− 1)1−α − k1−α and bk = (k− 1)1−α − 2k1−α + (k + 1)1−α.

aA. Dabiri/E. A Butcher: Efficient Modified Chebyshev Differentiation Matrices for Fractional Differential Equations, in: Communications in Nonlinear Science and Numerical Simulation 50.ISSN1007-5704 (2017), pp. 584–310.





D Chebyshev Polynomials

The Chebyshev polynomials of the first kind aregiven recursively as

TN+1(t) = 2tTN(t)− TN−1(t)

where the two first terms are T0(t) = 1 andT1(t) = t .





Theorem

The left-sided Caputo-fractional Chebyshev differentiation matrix at the N + 1 CGL points in[a, b] is given by the following stable recurrent relationsa

C+Dα

N+1 =β

Γ(1− α)(D +IN+1 (α))

T H, α ∈ (0, 1]. (6)

aA. Dabiri/E. A Butcher: Stable Fractional Chebyshev Differentiation Matrix for Numerical Solution of Fractional Differential Equations, in: Nonlinear Dynamics 2017, First Online:19 July 2017, 17.





Multi-Order Fractional Delay-Differential Equations with Multiple Delays

C0D

(α)t [x (t)] = A(t) x (t) + G(t) C

0D(ϑ)t [x (t)] +

s

∑i=1

Fi(t) C0D

(ζ)t [x (t− τi)] (7)

subject to the initial function φ(t) = [φ1(t), · · · , φn(t)]T, −τs ≤ t ≤ 0.

Proposition

The solution of FDDE (7) in [0, pτ1], p ∈N, is given by the following equation

xdti= 0Tdti

(s

∑k=1

Fk,dtiD

(ζ)dti

xdzi,k+

φ0

dt

), i = 1, 2, · · · , p,

where ti and zi,k include the N number of discretized points in [0, iτ1] and [−τk,−τk + iτ1], i = 1, 2, · · · , p,respectively, and

0Tdti=(

D(α)dti− Adti

− GdtiD

(ϑ)dti

+ In ⊗ J)−1

, det(

0T−1dti

)6= 0.





Definition

The numerical stability of a fractional spectral collocation method is tested by solving the following autonomouslinear time-invariant FDEa

C0Dα

t [x(t)] + λx(t) = 0, α ∈ (0, 1], (8)

with x(0) = x0 ∈ R and λ ∈ C. The exact solution of this equation is given by the Mittag-Leffler function asx(t) = x0Eα,α(−λtα) which is asymptotically stable if arg(λ) > α π

2b.

aA. Dabiri/E. A Butcher: Stable Fractional Chebyshev Differentiation Matrix for Numerical Solution of Fractional Differential Equations, in: Nonlinear Dynamics 2017, First Online: 19 July 2017, 17.

bDenis Matignon: Stability results for fractional differential equations with applications to control processing, in: Computational engineering in systems applications, vol. 2, IMACS, IEEE-SMC Lille, France, 1996,pp. 963–968.

State transition matrix Tdt

Tdt =(

Dαdt + λ IN

)−1J,

where IN is a N × N identity matrix with its first element sets to zero and J is a N × N zero matrix whose firstrow is modified as [0, 0, · · · , 0, 1]a.

aA. Dabiri et al.: Optimal Periodic-gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-delayed Systems, in: IEEE Transactions on Automatic Control 2017, Date of Publication:24 July 2017.





(g)

The stability region of the FCC method and the finite difference method for Eq. (8) for different valuesof N when α = 1 and τ = 1.





(h) (i)

The stability region of the FCC method and the finite difference method for Eq. (8) for different valuesof N when α = 1 and τ = 1.





The stability region of the spectral numerical method for different values of τ when α = 0.5 and N = 100.





Definition

The numerical stability of a fractional spectral collocation method for solving FDDEs is tested by solvingthe following linear time-invariant scalar FDDE with a single delaya

C0Dα

t [x(t)] = ax(t) + bx(t− τ), 0 ≤ α ≤ 1, (9)

subject to initial function φ(t), t ∈ [−τ, 0], where a and b ∈ R.The approximated monodromy matrix of FDDE (9) defined as

M =(

Dαdt − aIN

)−1(bIN + J) . (10)

aA. Dabiri/E. A Butcher: Numerical Solution of Multi-Order Fractional Differential Equations with Multiple Delays via Spectral Collocation Method, in: Applied Mathematical Modelling submitted.

Proposition

The FCC method is numerically stable for FDDE (9) if and only if the spectral radius of the approximatedmonodromy matrix locates in the unite circle for large enough values of N and τ, or α’s close to 1.a

aA. Dabiri/E. A Butcher: Numerical Solution of Multi-Order Fractional Differential Equations with Multiple Delays via Spectral Collocation Method, in: Applied Mathematical Modelling submitted.





(k) (l) (m) (n)

Figure: The stability region of the fractional Hayes equation by using 100 CGL collocation points fordifferent values of α and (a) τ = 0.2π, (b) τ = π, (c) τ = 2π, and (d) τ = 10π.





Example 1: Linear FDEs

x′′(t) + C0D0.5

t [x(t)] + x(t) = t2 + 4

√tπ

+ 2, 0 ≤ t ≤ 20,

with initial conditions x(0) = 0 and x′(0) = 0.It is easy to show that the exact solution of this problem is x(t) = t2.

FCC method P(EC)mE method

N εto CPU time (s) h εto CPU time (s)

10 2.84×10-13 0.000 10−1 1.17×10-02 0.091100 5.97×10-11 0.001 10−2 4.23×10-04 0.159500 4.11×10-10 0.006 10−3 1.40×10-05 1.219

1000 1.28×10-08 0.040 10−4 4.48×10-07 11.0122000 1.13×10-07 0.279 10−5 1.42×10-08 101.979





Example 2: Highly oscillatory problem

C0D2−ε

t [x(t)] + (100π)2x(t) = 0, 0 ≤ t ≤ 10,


N εto CPU time (s) h εto CPU time (s)

100 1.29×10-01 0.004 10−2 1.53×10+21 0.072200 2.22×10-03 0.002 10−3 1.69×10+67 0.163

1000 8.90×10-05 0.044 10−4 5.54×10+11 1.6461500 3.95×10-05 0.122 10−5 1.14×10-01 16.9662000 2.22×10-05 0.261 10−6 8.22×10-03 176.848





Example 2: The chaotic damped externally driven pendulum

Consider the chaotic damped externally driven pendulum with a dashpot

x + µ C0Dα

t x + ω2 sin x = F0 sin Ωt, (11)

where µ = 0.5 is the damping coefficient, and the natural and external frequencies of the systemare assumed to be ω = 1 rad/s and Ω = 2

3 rad/s, respectively.

The dynamics of the system have been studied for α = 0.8, and it has been shown that thesystem becomes chaotic after F0 > 1.052a.

aA. Dabiri/M. Nazari/E. A. Butcher: Chaos analysis and control in fractional-order systems using fractional Chebyshev collocation method, in: ASME 2016 International MechanicalEngineering Congress & Exposition (IMECE), Phoenix, AZ 2016.





Table: The convergence order of the solution of the chaotic damped externally driven pendulum by usingthe FCC and P(EC)mE method when F0 = 1.02 and α = 0.8.


N CO(I)i CO(I I)

i h CO(I)i CO(I I)

i

140 −9.16 −9.13 0.0071 1.10 1.04196 −9.89 −9.85 0.0051 1.13 1.08275 −18.67 −18.66 0.0036 1.15 1.11385 −20.78 −23.22 0.0026 1.16 1.14538 1.05 −3.97 0.0019 1.17 1.16753 −0.24 7.32 0.0013 1.18 1.17





Example 1: Beck’s column with a non-conservative periodic retarded follower force

(a)

Figure: (a) The Beck’s column with a (non-conservative) periodic retarded follower load (b) The discrete-mass model of the Beck’s column with the double inverted pendulum with two fractional dampers.





Example 1: Beck’s column with a non-conservative periodic retarded follower force

(a) (b)

Figure: (a) The Beck’s column with a (non-conservative) periodic retarded follower load (b) The discrete-mass model of the Beck’s column with the double inverted pendulum with two fractional dampers.





[3 11 1

][θ1(t)θ2(t)

]+ k[

2 −1−1 1

][θ1(t)θ2(t)

]+ c f

[2 −1−1 1

][C0Dα

t θ1(t)C0Dα

t θ2(t)

]= p (t, τ) (12)

and the follower force is in the form of

p (t, τ) = P[

1 00 1

][θ1(t)θ2(t)

]− P

[0 λ0 λ

][θ1(t−τ)θ2(t−τ)

]. (13)

where x (t) =[θ1(t), θ1(t), θ2(t), θ2(t)

]Tdenotes the state vector, k is the spring stiffness, c f

is the damping coefficient with fractional order α.





The numerical solution of the double inverted pendulum by using the FCC method for different valuesα. The dotted and dashed lines are the solution obtained by the dde23-solver in MATLAB when α = 0and α = 1, respectively.





Example 2

Consider the linear FDE

x(t) + x(t) + x(t) + ε C0Dα

t [x(t)] = 0, ε = 10−13, 0 ≤ t ≤ 100, (14)

with the initial conditions x(0) = 0 and x(0) = 100. Since ε is in the range of machine precision,its solution can be well approximated as x(t) ≈ 200√

3exp

(−t2)

sin( 3

2 t).





0 0.2 0.4 0.6 0.8 1

α

10-1

100

101

102

Com

putation

time(sec)

10-20

10-10

100

1010

||error||2

Numerical computation error (red lines) andcomputation time (blue lines) by using the FCC(dashed lines) and PECE (solid lines) method forExample 2.





Conclusion

P Pros of the proposed Fractional Chebyshev Collocation (FCC) Method

1 The FCC method has spectral convergence in solution of linear fractional order differential equations.2 The FCC method has high-order convergence in solution of linear fractional delayed differential equa-

tions.3 The FCC method can be used in non-canonical form.4 The FCC method can be used for commensurate and non-commensurate fractional order differential

equations.5 The FCC method is results in the minimum inner dimension state-space representation.

C Cons of the proposed Fractional Chebyshev Collocation (FCC) Method

1 Studying the numerical stability of the FCC method for nonlinear FDEs is complicated and dependent onthe nonlinear solver.

2 Studying the numerical stability of the FCC method for FDDEs with many delays is complicated anddependent on the nonlinear solver.




SectionPART II:

An Optimal Stewart Platform For Lower Extremity Robotic Rehabilitation




About one in 10 American adults has diabetes according to the centers for DiseaseControl and Prevention.

Based on the National Institute of Diabetes and Digestive and Kidney Diseases the totalcosts of diagnosed diabetes were 245 billion dollars in 2012.

Ulceration of the foot is one of the most common complications of diabetes mellitus anddiabetes-related cause of hospitalization and lower extremity amputations.




It has been shown among of all the proposed treatments, manually assisted physicaltherapy is effective in the treatment of diabetic foot wounds.

Manually assisted physical therapy is expensive, labor-intensive (the duration of the training isbased on the personal shortage and fatigue of the therapist), and limited in the repeatabilityand evaluating patients’ performance.




Five design parameters

The design variable column vector: d = [R, r, θb, θt, h]T where

1 The radius of the circumcircles of the bottom plate R2 The radius of the circumcircles of the top plate r3 The top offset angle θt4 The base offset angle θb5 The height of the platform h




(a) (b)

Figure: The maximum speed of the actuators in θt − θb −WU design variable space when R = 25 cm,r = 5 cm and (a) h = 0.2 cm (b) h = 0.5 cm.ab

aA. Dabiri et al.: An Optimal Stewart Platform for Lower Extremity Robotic Rehabilitation, in: American Control Conference (ACC), Seattle, WA 2017.

bS. Sabet et al.: Computed Torque Control of the Stewart Platform with Uncertainty for Lower Extremity Robotic Rehabilitation, in: American Control Conference (ACC), Seattle,WA 2017.




(a) (b)

Figure: The maximum force of the actuators in θt − θb −WU design variable space when R = 25 cm,r = 5 cm and (a) h = 0.2 cm (b) h = 0.5 cm.ab

aA. Dabiri et al.: An Optimal Stewart Platform for Lower Extremity Robotic Rehabilitation, in: American Control Conference (ACC), Seattle, WA 2017.

bS. Sabet et al.: Computed Torque Control of the Stewart Platform with Uncertainty for Lower Extremity Robotic Rehabilitation, in: American Control Conference (ACC), Seattle,WA 2017.




Section

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Fractional-Order Differential Equations

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