Navigation and control of endovascular helical swimming microrobot using dynamic programing and adaptive sliding mode strategy

CHAPTER SEVEN

Navigation and control of endovascular helical swimming microrobot using dynamic programing and adaptive sliding mode strategy Mohammad Javad Pourmanda and Mojtaba Sharifib a

School of Mechanical Engineering, Shiraz University, Shiraz, Iran Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

b

1 Introduction Microrobots and miniature robots have been recently designed, studied, and employed in different applications, including medical and biological operations. Several novel designs have been presented for microrobots; however, before introducing them it is needed to know about an untethered microagent. The untethered microrobotics is now the superior approach for delicate micromanipulations in inaccessible or hardly accessible environments with medical purposes. Exemplifications of these environment and areas are the circulatory system (Belharet et al., 2011), urinary system and prostate (Kristo et al., 2003), central nervous system (Duffner et al., 2003; Zaaroor et al., 2006), eye (Kummer et al., 2010; Yesin et al., 2006), ear (Nelson et al., 2010), and fetus (Flake, 2003). Microagents in these environments can be utilized for diversity of operations such as the biopsy (De Cristofaro et al., 2010), drug delivery (Khalil and Misra, 2017; Nelson et al., 2010), marking (Fluckiger and Nelson, 2007), occlusion ( Jeong et al., 2016), ablation (Kim et al., 2016), and so on (Nelson et al., 2010). Accordingly, the key point in designing microrobots is insightful knowledge of the operation environment as its target. Most of the proposed microrobots have been designed to operate in human body (generically mammals’ bodies); therefore, it is evident that these microrobots should swim inside body fluids in different parts. Relatively intricate and expensive local instruments (i.e., microsensors and microactuators) have made researchers prone to Control Systems Design of Bio-Robotics and Bio-mechatronics with advanced applications https://doi.org/10.1016/B978-0-12-817463-0.00007-1

© 2020 Elsevier Inc. All rights reserved.

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benefit from distant sensors, power supply, and actuators in miniature environments. The most promising distant navigation employed for microrobots is the magnetic navigation, used in the magnetically driven elastic flagellated propulsion (Evans and Lauga, 2010), magnetically driven rigid flagellated propulsion (Abbott et al., 2010), and OctoMag system (Kummer et al., 2010). The idea of controlling a microrobot within an intravascular environment can be involved with parametric uncertainties and unmodeled dynamics (Folio and Ferreira, 2017; Arcese et al., 2013; Hund et al., 2017; Pourmand et al., 2018), although in a highly viscous environment, using a bioinspired flagellated propulsion can accomplish an effective locomotion (Evans and Lauga, 2010) that relatively increases the structured and unstructured uncertainties (in comparison with other methods). Moreover, pulsatile flow causes remarkable disturbances in the system that requires an effective robust control strategy. Accordingly, several controllers have been proposed for such conditions. For instance, the model predictive (Belharet et al., 2011), adaptive (Arcese et al., 2013), and H∞ (Marino et al., 2014) control strategies are presented for microrobots, and they failed to perform an effective control in a realistic environment (Folio and Ferreira, 2017). Comparing magnetic resonance propulsion (MRP, particularly executed in the magnetic resonance imaging (MRI) device) and magnetically driven helical-flagellated propulsion (MHP) results in that MHP produces larger propulsive forces; however, it can elevate the uncertainty of the system. One of recently introduced methods to deal with the mentioned uncertainties is considering some of the unstructured uncertainties in the path planning of the robot inside endovascular system (Folio and Ferreira, 2017). The sliding mode control (SMC) as a robust strategy was employed in uncertain nonlinear systems (Slotine and Li, 1991; Sharifi and Moradi, 2017). The basic issues of this controller are two interconnected phenomenon: the chattering phenomenon and high control effort (Utkin and Poznyak, 2013; Aghajanzadeh et al., 2018; Sharifi et al., 2017). The magnitude of a discontinuous SMC input is proportional to the amplitude of chattering; therefore, the adaptive siding mode control was introduced to reduce the magnitude of SMC gain and the chattering issue (Utkin and Poznyak, 2013; Torabi et al., 2017). Moreover, the adaptive sliding mode control is one of the best robust strategies for the nonlinear systems that their dynamic models cannot be linearly parametrized or the parametrization is complicated.

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Microrobots have some other control issues due to the limitation of applicable equipment such as the precision of sensory data and the underactuation characteristic. Furthermore, microrobots in the fields of biorobotics (Nelson et al., 2010; Abbott et al., 2010) have currently encountered some considerations such as biocompatibility and a large number of uncertainties due to the interaction with biosystems. As a result, developed control models for varieties of these microrobots (Arcese et al., 2012) have been operated in the circulatory system involved with unmodeled dynamics (Belharet et al., 2011) and structured uncertainties (Arcese et al., 2012; Hund et al., 2017; Kenner, 1989). With respect to these issues, the sliding mode control (SMC) method with robustness property against structured and unstructured uncertainties seems to be an effective strategy for the control of endovascular microrobots. Accordingly, a nonlinear adaptive sliding mode controller is designed for the first time in this work for a magnetically driven monohelical flagellated microrobot. To resolve the chattering issue and autonomously overcome uncertainties, an adaptation law is presented for online adjustment of the controller gain. The global stability and tracking convergence of the proposed control method is proven via the Lyapunov theorem. In addition, a three-dimensional (3D) optimal path planning is employed based on the dynamic programming (DP) for generating the desired trajectory of the microrobot. Some simulations are executed to evaluate the objective achievement and the controller performance.

2 3D optimal path planning Path planning in 2D environment has been widely used and discussed for microrobots; however, a 3D path planning is necessary for the endovascular environment. Due to the lack of direct measurement of the microagent velocity using currently available imaging devices, an appropriate feedback controller has to be devised instead of previous approaches needed the velocity signal. Firstly, it is considered that we have a 3D image of the vascular system obtained from MRI, computed tomography (CT), or any other imaging devices. In the domain CD
ℝ3 that is in the permissible space of the microrobot operation, any path starts from p(0) 2 CD and ends at p(1) 2 CD can be expressed by p : l 2 ½0, 1 ! pðlÞ 2 CD

(1)

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Now, the control effort is defined as w ðvðpðlÞÞ, f m ðpðlÞÞÞ ¼ arg max ðf m ðpðlÞÞ  vðpðlÞÞÞ,

(2)

where fm and v are the control force and micororbot velocity, respectively, at each location p(l) of the path (Folio and Ferreira, 2017). Assumption. The induced magnetic force is controllable in any direction and the flow velocity is not directly measurable since conventional imaging devices cannot provide such data. Thus, the control effort metric is determined based on the velocity distribution obtained from the steady-state solution of Navier-Stokes equations for an uniform flow in the walled space (Munson et al., 2014). Fig. 1 shows an illustration of the scaled control effort metric in a 2D space (the result is comparable with the one in Folio and Ferreira, 2017). Then, a performance measure based on the control effort metric can be developed as Z 1 J ðpÞ ¼ w ðpðl ÞÞ  kp_ ðlÞkdl: (3) 0

Therefore, the optimum path with minimum effort can be found from p∗ ¼ arg min J ðpÞ:

(4)

Considering the image as a discrete domain, the first step for finding the solution of Eq. (4) by the dynamic programming (DP) approach (Kirk, 2012) is meshing this domain. After generating an adequately small mesh, each node has directly reachable neighbors. Thus, according to the

Fig. 1 An illustration for the magnitude of weighted objective function based on minimum effort, defined in Eq. (2), for a 2D image: The color bar demonstrates how this magnitude would be high or low. Note that the magnitude of this function is higher wherever the pressure is lower.

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optimality principle (Kirk, 2012), for a path that contains the nodes G, H, and I, there is a total optimum path as J ∗ GHI ¼ J ∗ GH + J ∗ HI . For this purpose: (1) either destination or start point is considered as the initial point of DP (as the solution is reversible), (2) the minimum cost of a path from each node to its neighborhood nodes is calculated, and (3) different paths between start and destination points in the domain are analyzed and the optimum total path with the minimum total cost is obtained. In the perspective of time complexity, it is noteworthy that gradient-based methods are superior to the proposed method if the search space of problem (4) is smooth (Pourmand et al., 2019).

3 Dynamic modeling The dynamic model of the monohelical swimming microrobot (shown in Fig. 2) is presented in this section. Study of the monohelical swimming microrobot is beneficial for generic acquaintance to microrobots dynamics, since large number of microrobots designed to have a bead (Belharet et al., 2011) and single (Dreyfus et al., 2005) or multiple flagella (Pourmand et al., 2018). The viscous force applied on each element of the helix (dsh) has normal and tangential components obtained as dF ? ¼ C? V? dsh dF k ¼ Ck Vk dsh ,

(5)

where from the geometry of the helix, normal and tangential velocities are formulated as _ cos βp V? ¼ Va sinβp + RΩ _ sinβp Vk ¼ Va cos βp + RΩ

(6)

Fig. 2 (A) Schematic picture of the proposed magnetically driven helical swimming microrobot: the spherical head can contain the operation essentials and the tail is just for exerting the propulsion, and (B) demonstration of the parameters and the tangential and normal forces on the helix.

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and the normal and tangential viscous drag coefficients are given in Abbott et al. (2010) as C? ¼

4πη 2πη > Ck ¼ 0:72πR 0:72πR ln ln + 0:5 d sinβp d sinβp

(7)

In Eqs. (6) and (7), Va is magnitude of the robot velocity along its approach _ is magnitude of the robot angular velocity around its approach vecvector, Ω tor, R is the helix radius, βp is the helix pitch angle. η and d are also the fluid viscosity and the microrobot filament diameter, respectively. Therefore, the propulsive force along the approach vector for each element along the helix is: dF ¼ dF k cosβp  dF ? sin βp Substituting Eq. (6) in Eq. (5) leads to h i _ cosβp dsh dF ? ¼ C? Va sin βp  C? RΩ h i _ sin βp dsh dF k ¼ Ck Va cos βp  Ck RΩ , and substituting these equations in Eq. (8) leads to nh i _ sin βp cos βp dF ¼ Ck Va cos βp  Ck RΩ h i _ cos βp  C? Va sinβp sinβp gdsh + C? R Ω

(8)

(9)

(10)

By integrating the resultant equation along the helix, we have:       _ cosβp  C? sinβp + Ck cos 2 βp cscβp Va , F ¼ 2πRN R C?  Ck Ω (11) where N denotes the number of helix turns. The hydrodynamic drag force of the microrobot head is obtained using the drag coefficient of spherical beads as 1 ! 2 ! Fd ¼ ρA v ∞ Cd v U 2 !

!

(12)

in which ρ, A, Cd, v ∞ , and v U denote the fluid density, the reference area, the superficial velocity in an infinitely uniform flow, and the unit vector of the superficial velocity. Since the proposed microrobot operates inside the circulatory system, it is essential to consider the wall effect (Kehlenbeck and Felice, 1999) of the vessels as

Navigation and control of endovascular helical swimming microrobot

! v  1  λp 2:2 !  ¼  2:2 ,  v ∞ λp 1+ λ0

207

(13)

where kk is Euclidean norm, and λ0 and λp are dimensionless parameters defined as λ0 ¼

0:27 + 0:06Re0:65 ; λp ¼ D=2r 1 + 0:05Re0:65

(14)

in which D, r, and Re are the robot spherical head diameter, the vessel radius, and the Reynolds number, respectively. One of the most accurate hydrodynamic drag coefficients of spherical beads in laminar flow was proposed recently in Barati et al. (2014) as       C2 C C4 C C6 C Cd ¼ C1 tanh + C3 tanh + C5 tanh kx_ k kx_ k kx_ k     (15) C8 C C10 C + C7 tanh + C9 tanh + C12 kx_ k kx_ k + C11 C with the following coefficient values: ½ C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12  ¼ ½ 5:4856 4:3774 0:0709 700:6574 0:3894 74:1539 0:1198 7429:0843 1:7174 9:9851 2:3384 0:4744

(16)

and C¼

η : Dρ

Using Eqs. (11)–(16), Eq. (12) can be rewritten as 0 0   ! 12:2 12 12 0 kx_ k 0:65 B D 1 + 0:05 C C CB B C C C π B D CB B C B B ! Fd ¼ ρB 1 + C C B  2:2  0:65 C C B A C B 8 @ D A @ x _ k k A @ 1 2r 0:27 + 0:06 2r C        C2 C C4 C C6 C C1 tanh + C3 tanh + C5 tanh kx_ k kx_ k kx_ k      C8 C C10 C + C7 tanh + C9 tanh + C12 kx_ kx_ kx_ k kx_ k + C11 C (17)

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where x is the robot position. A spherical magnetic bead attached to the helical tail produces a magnetic torque generating the corkscrew effect (Chwang !

!

!

and Wu, 1971) determined as Γ m ¼ Vm M  B , where Vm is the volume of !

!

magnetic material in the bead, and M and B are the bead magnetization vector and the oscillating magnetic field with the frequency of ω, respectively. Then, a feasible control input is defined as !

Γm  u ¼ ω !  : Γ  m

(18) !

!

Considering apparent weight as Wa ¼ Vb ðρb  ρÞ g with Vb, ρb, and g as the bead volume, the bead density, and gravitational acceleration, respectively, knowing F and Fd from Eq. (11) and Eq. (17), and using Eq. (18), the dynamics (equation of motion, i.e., mx€ ¼ F  Fd + Wa ) of the microrobot is formulated as   2πR2 N C?  Ck cos βp x€ ¼ u m 12 2 0 C 62πRN   πρ B D C 6 B C? sinβp + Ck cos 2 βp csc βp  B 6  2:2 C A 4 m 8m @ D 1 2r 1 0 0   ! 12:2 2 kx_ k 0:65 B B D 1 + 0:05 C C   B B C C C C2 C C B B C B1 + B   !C C C1 tanh kx_ k B @ kx_ k 0:65 A C A @ 2r 0:27 + 0:06 C 

     C4 C C6 C C8 C + C3 tanh + C5 tanh + C7 tanh kx_ k kx_ k kx_ k ! # !   C10 C + C9 tanh + C12 kx_ k x_ + Wa kx_ k + C11 C

(19)

Based on Eq. (19), it can be written that x€ ¼ βu + f ðx_ Þ,

(20)

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where β and f ðx_ Þ can be obtained by comparing Eqs. (19), (20). Other forces (e.g., van der Waals interaction) are not considered in this dynamic model (19) for the proposed microrobot because these forces are significantly smaller with lower order of magnitude compared to the hydrodynamic drag force.

4 Adaptive sliding mode control The presence of structured uncertainties (such as inaccuracy of viscosity and density parameters) is possible in the modeled dynamics. Moreover, unstructured uncertainties due to existence of the pulsatile flow and negligence of the low-order forces will affect the system performance. Therefore, the application of robust controllers seems to be inevitable for these microrobotic systems. Although the sliding mode control scheme has been recommended as an appropriate robust controller (Slotine and Li, 1991) for uncertain dynamical systems, to exert this control method it is necessary to cope with two interconnected issues: the chattering phenomenon and the high magnitude of control action (Utkin and Poznyak, 2013). Accordingly, an adaptive version of the sliding mode strategy is employed in this work to control the microrobot inside the circulatory system, handle the chattering properly, and reduce the high activity of control action. Based on boundedness of f and β in Eq. (20), one can write: ^ β f^ f < fB ; 1  < βB , (21) β where β^ and f^ are the estimated values of β and f in Eq.(20), respectively. The desired state (trajectory) of the microrobot is considered as

 xd x_ d … xd n1 , where n is the order of system dynamics, and the posi tion tracking error is defined as x ¼ x  xd . Since the microrobot has a second-order dynamics (19), i.e., n ¼ 2, the sliding surface is defined as  n1 d    +λ x ¼ x_ + λ x , where λ is a positive constant. Differentiating s¼ dt this surface and using the system dynamics (20), it can be expressed that  s_ ¼ βu + f ðx_ Þ  x€d + λx_

(22)

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To make s_ ¼ 0, the best approximation for the control law is   1 u^ ¼ β^ f^ðx_ Þ + x€d  λx_ . Now, by adding the robust term k^sgnðsÞ to the control input for converging to the sliding surface s ¼ 0 in the presence of modeling uncertainties, we have:  1  u ¼ f^ + x€d  λx_  k^sgnðsÞ , (23) β^ where k^ is a positive time-varying robust gain updating by the following adaptation law: ^k_ ¼ φjsj

(24)

in which φ is a positive constant. The closed-loop dynamics of the microrobot is obtained by substituting the control law (23) into Eq. (22) and doing some mathematical operations as follows:     β^ β^ β β _ s_ ¼  k sgnðsÞ + f  f +  1 x€d + 1  (25) λx β^ β^ β^ β^ To prove the stability and tracking convergence of the controlled microrobot, the following Lyapunov function is considered:   1 β  ^ 2 V ¼ s2 +  0, (26) kγ ^ 2 βφ where γ is a positive value that will be discussed later. The time derivative of the Lyapunov function is obtained:

V ¼ s_s +

β ^_  ^  k kγ : ^ βφ

(27)

Substituting the closed-loop dynamics (25) into Eq. (27), yields       β^ β β β _ β ^_  ^   1 x€d + 1  λx + k kγ V ¼ s  k sgnðsÞ + f  ^f + ^ β^ β^ β^ β^ βφ

(28) Using the adaptation law (24), the time derivative of the Lyapunov control function is obtained as        β^ β^ β β _ β   1 x€d + 1  V ¼ s  k sgnðsÞ + f  f + λx + jsj k^ γ β^ β^ β^ β^ β^ (29)

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After a few simplifications, Eq. (29) can be rewritten as          β β^ β^ β^  2 _ ^ ^ ^ sgnðsÞ f  f + 1  x€d + V ¼ jsj  1 λx + k  k ð sgnðsÞÞ  γ β β β β^

(30)

  The term k^ k^ ð sgnðsÞÞ2 is always zero unless s ¼ 0, in which case k^_ ¼ 0 based on the update rule (24), i.e., k^ remains finite. Therefore, if s ¼ 0 (the microrobot trajectory converged to the sliding surface), γ is considered such large in the Lyapunov function (26) that k^ γ

(31)



Thus, it is guaranteed that V 0. On the other hand, if s 6¼ 0, γ is considered such large that ! ! β^ ^ ^ β β  (32)  1 λx_ + jsj γ f  f^ + 1  x€d + β β β

Accordingly, V <0 in this case. Regarding the time derivatives of x that are assumed to be bounded, there exists a bounded (positive) value for γ, which is large enough to satisfy both inequalities (31) and (32). Therefore, by choosing a proper value for γ, the time derivative of the Lyapunov function is negative semidefinite. Consequently, Eqs. (30)–(32) result in β V wB ðtÞ ¼  s2 0 β^

(33)

  Since wB ðtÞ ¼ β=β^ s2 is a uniformly continuous function, integrating Eq. (33) from zero to t ! ∞ gives Z t wB ðτÞdτ V ð0Þ  V ð∞Þ (34) lim t!∞ 0



Based on the negative semidefiniteness of V 0, V(0)  V(∞) is positive and Rt finite in Eq. (34). Therefore, according to Eqs. (33), (34), lim t!∞ 0 wB ðτÞdτ exists and it is positive and finite. As a result, based on Barbalat’s lemma, it is obtained that β lim wB ðτÞ ¼ lim s2 ¼ 0: ^ t!∞ β

t!∞

(35)

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  Thus, as β=β^ is nonzero and positive, Eq. (35) implies that lim t!∞ s ¼ 0.   Regarding the definition of s ¼ x_ + λ x , the convergence to desired trajec

tory x ¼ ðx  xd Þ ! 0 is concluded. Due to the positive definiteness of the Lyapunov function and the negative semidefiniteness of its time derivative, the proposed adaptive sliding mode control method guarantees the global stability, robustness against the bounded modeling uncertainties, and the racking convergence of the microrobot to the desired trajectory xd.

5 Simulation results The proposed control strategy for the endovascular swimming microrobot is evaluated via some simulations in this section. The human body circulatory system in arterioles with three different disturbances is included in these simulations. Parameters of the helix tail and head of microrobot are presented in Table 1, which are compatible with the data presented in Nourmohammadi et al. (2016) and Folio and Ferreira (2017). One of the most challenging disturbances in arterioles is the pulsatile flow that is considered as (Munson et al., 2014; Belharet et al., 2011)  2 !   r 10π ! σ ðtÞ ¼ 0:001 1  1 + 1:15cos 2πt + (36) c , Rv 16 !

where c is the unit vector along the vessel centerline. To more evaluate the robustness of the proposed control method, sinusoidal disturbances are considered in the fluid density and viscosity and applied in the simulations as   2πt ηðt Þ ¼ 17:5 + 3 sin (37) 7   2πt ρðtÞ ¼ 1057 + 12 sin (38) 5

Table 1 Parameters of the helix tail and head of the microrobot. Parameters Dimensions and values

βp, helix angle N, helix revolutions d, filament diameter D, spherical head diameter R, helix amplitude

75 degrees 5 20 (μm) 700 (μm) 250 (μm)

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The constants of the blood viscosity and density magnitude are extracted from Nelson et al. (2010). After finding the optimal path, corresponding to the minimum effort, inside a 3D complex phantom of endovascular system (shown in Fig. 3), the trajectory tracking is achieved in the presence of artificial pulsatile flow, density, and viscosity variation as illustrated in Figs. 3–5. Fig. 4A demonstrates that the position error of the microrobot with respect to its desired trajectory that is controlled and converged to zero in the endovascular phantom with the mean diameter of 3 mm (Dong and Nelson, 2007; Nelson et al., 2010). In addition, Fig. 5A shows that after passing a bounded time, adaptive gains of the robust controller grow enough to overcome modeling uncertainties and disturbances and to control the robot position close to its desired trajectory (Fig. 3). Note that without any disturbance in the system, the robust gains are relatively small and autonomously reach small saturation levels as shown in Fig. 5B. The deviation of the microrobot velocity from the desired velocity in Fig. 4B remains small even in the frequency of artificial pulsatile flow and viscosity disturbance. The control inputs in 3 directions are also demonstrated in Fig. 4C. The magnitude of these control components is saturated to be <3 rad/s. For the sake of comparison and demonstration of the proposed adaptive sliding mode control method’s efficiency, the sliding mode gains (elements ^ fixated on the constant values of 1 and 3 without employing the update of k) rule (24). In these conditions, the control input, and trajectory tracking of the velocity and position in the presence of disturbances are illustrated in Fig. 6. Comparing Fig. 6A and 6D implies that the control effort is a little higher at intervals when the robust gains are 3 in comparison with the case

Fig. 3 Trajectory tracking of an optimal minimum-effort path in the presence of disturbances.

Fig. 4 (A) Convergence of the position error, (B) tracking of the desired velocity, and (C) control input of the swimming microrobot in the presence of artificial pulsatile flow, blood density, and viscosity disturbances.

Fig. 5 Adaptation of robust gains during the microrobot motion (A) in the presence of disturbances and (B) without any disturbance.

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Fig. 6 (A) The control input, (B) the velocity tracking, and (C) the position error for the steady sliding mode gains with the value of 1 and (D) the control input, (E) the velocity tracking, and (F) the position error for the steady sliding mode gains with the value of 3.

having the value of 1 for these gains. However, it could be mentioned that when the robust gains of the controller are larger, the position and velocity errors become smaller. The analogy of Figs. 4 and 6 clarifies how the online adjustment of the adaptive sliding mode gain k^ (Fig. 5) in the proposed control strategy is effective in decreasing the trajectory tracking errors by suggesting an efficient control effort. Assuming the initial value of the gains is equal to 1 in the adaptive control scheme, the obtained results of the closed-loop system are demonstrated in Fig. 7. Note that the initial value of the robust gains was zero in Figs. 4 and 5.

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Fig. 7 The simulation results when the initial values of the robust controller gains is considered 1: (A) the gains adaptation, (B) the position error, (C) the velocity tracking, and (D) the control inputs.

Based on Figs. 5A and 7A, it can be concluded that the adaptive performance of the controller in the presence of various uncertainties is achieved successfully that is in accordance with the control objectives and the Lyapunov stability analysis. Comparison of Fig. 4A with Fig. 7B and Fig. 4B with Fig. 7C intimates that the convergence of position and velocity trajectories is provided more appropriately when a larger initial value is assumed for the controller’s time-varying robust gains. Specifically, a faster convergence rate of the transient response is seen in the initial part of the microrobot’s motion. However, with reference to Figs. 4C and 7D, the control effort is lower in the case when the initial value of the robust gains is zero. From the control inputs of all four simulation studies presented in the presence of disturbances shown in Figs. 5–7, it is concluded that not only the control effort increases due to the rise of robust gains, but also the system chattering around the sliding surface occurs more often (the frequency and amplitude of this phenomenon are elevated). Therefore, it is recommended that the initial values of the controller’s robust gains are considered small when the appropriate steady-state levels of these gains are unknown.

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6 Conclusion The main goals of the present chapter were (1) introducing an accurate and well-clarified dynamic model of the microrobot, (2) deployment of an optimal 3D path planning, and (3) mathematical formulation and designing a proper robust controller with adaptive gains. In this work, an adaptive sliding mode control method was presented for a monohelical-flagellated endovascular microrobot, for the first time. Since this type of microrobots deals with more challenging uncertainties in comparison with the magnetic resonance propulsion (MRP) systems due to the disturbances and inaccuracy of dynamic parameters, a robust adaptive control strategy was designed and adopted in this study. An optimality principle was employed for the path planning of the microrobot with a minimum effort criterion using dynamic programming (DP) in three dimensions to obtain the desired robot trajectory for the controller. The dynamic model of the microrobot was developed and its physical interaction inside the circulatory vascular system as the most dominant force (i.e., the hydrodynamic drag) was thoroughly taken into account. It was proven using the Lyapunov theorem that the controller guarantees the stability and tracking convergence in the presence of disturbances. Robustness and performance of the proposed control method is evaluated by some simulations considering the most important uncertainties in vascular system (the pulsatile flow, blood density, and viscosity fluctuations and inaccuracies).

References Abbott, J.J., Peyer, K.E., Dong, L.X., Nelson, B.J., 2010. How should microrobots swim? In: Robotics Research. Springer. Aghajanzadeh, O., Sharifi, M., Tashakori, S., Zohoor, H., 2018. Robust adaptive Lyapunovbased control of hepatitis B infection. IET Syst. Biol. 12, 62–67. Arcese, L., Fruchard, M., Ferreira, A., 2012. Endovascular magnetically guided robots: navigation modeling and optimization. IEEE Trans. Biomed. Eng. 59, 977–987. Arcese, L., Fruchard, M., Ferreira, A., 2013. Adaptive controller and observer for a magnetic microrobot. IEEE Trans. Robot. 29, 1060–1067. Barati, R., Neyshabouri, S.A.A.S., Ahmadi, G., 2014. Development of empirical models with high accuracy for estimation of drag coefficient of flow around a smooth sphere: an evolutionary approach. Powder Technol. 257, 11–19. Belharet, K., Folio, D., Ferreira, A., 2011. Three-dimensional controlled motion of a microrobot using magnetic gradients. Adv. Robot. 25, 1069–1083. Chwang, A., Wu, T.Y., 1971. A note on the helical movement of micro-organisms. Proc. R. Soc. Lond. B: Biol. Sci. 178, 327–346.

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