Estimation of the blood velocity for nanorobotics Matthieu Fruchard, Laurent Arcese, Estelle Courtial

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Matthieu Fruchard, Laurent Arcese, Estelle Courtial. Estimation of the blood veloc- ity for nanorobotics. IEEE Transactions on Robotics, IEEE, 2014, 30 (1), pp.93-102. ￿10.1109/TRO.2013.2288799￿. ￿hal-01071275￿

HAL Id: hal-01071275 https://hal.archives-ouvertes.fr/hal-01071275 Submitted on 3 Oct 2014

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Estimation of the blood velocity for nanorobotics

Matthieu Fruchard, Laurent Arcese, Estelle Courtial

Abstract—The paper aims at estimating the blood velocity to enhance the navigation of an aggregate in the human vascu- lature. The considered system is a polymer binded aggregate of ferromagnetic nanorobots immersed in a blood vessel. The drag force depends on the blood velocity and specially acts on the aggregate dynamics. In the design of advanced control laws, the blood velocity is usually assumed to be known or set to a constant mean value to achieve the control objectives. We provide theoretical tools to on-line estimate the blood velocity from the sole measurement of the aggregate position and combine the state estimator with a backstepping control law. Two state estimation approaches are addressed and compared through simulations: a high gain observer and a receding horizon estimator. Simulations illustrate the efficiency of the proposed approach combining on- line estimation and control for the navigation of an aggregate of nanorobots. Fig. 1. Aggregate of nanorobots in a blood vessel. (a) Aggregate binded by Index Terms—Blood velocity, high gain observer, receding ligand, including a payload; (b) aggregate binded by magnetic and interaction horizon estimator, magnetic nanorobots. forces.

I. INTRODUCTION Whatever the proposed design, these systems are subject There has been a growing interest in the development of to different forces whom modelling is necessary in order to therapeutic microrobots and nanorobots for some years [1]. estimate their dynamic behaviors in a fluidic environment Such systems have the potential to perform complex surgical [15], [16]. The blood velocity is usually assumed to be known procedures without heavy trauma for the patients [2], [3], or set to a constant mean value whilst it is a key nonlinear [4], [5], [6]. Among the numerous prototypes developed, parameter of the drag force which prevails at a small scale. those possessing a deported actuation are the most promising. The measurement of the blood velocity is a difficult task that Indeed these robots do not embed any energy source thus is often assigned to ultrasonic sensors, at least in vessels inducing smaller sizes. Untethered robots thus benefit from a close to the sensor. This solution calls for an end-effector possible better therapeutic targeting since they can access to servoing so as to track the aggregate position. Another hard-to-reach body’s area and also lessen medical side effects solution relies on a priori knowledge of the blood velocity. linked to surgical operations. Works related to the numerical resolution of the velocity profiles using Computational Fluid Dynamics software had The kind of deported actuators required to control such been reported in the literature [17], [18]. But these studies robots depends on the propulsion strategy. A variable magnetic can not be used for real-time purposes because they are field is necessary for robots with elastic flagellum [7], [8] based on the Navier Stokes equations whom resolution is or helical flagella [9]. The control of bead pulled robots or computational time consuming. In [15], the authors proposed swarm of robots [10], [11], [12] is provided by the magnetic analytical expressions of the blood velocity profiles. These field gradients of either Magnetic Resonance Imaging (MRI) expressions are valid in the neighborhood of a bifurcation devices or magnetic setups (e.g. from Aeon Scientific or but require an accurate knowledge of the vessel geometry. Stereotaxis). So as to benefit both from a large motive force The blood velocity could also be considered as a disturbance in the macrovasculature and from a possible break up in and be rejected in the control law, like in [19] for breathing the microvasculature (so as to avoid undesired thrombosis compensation. However the knowledge of the blood velocity and to improve the targeting), a promising approach is to is relevant for the navigation since the main force depends on consider aggregates. Such aggregates of magnetic nanorobots it. Finally the on-line estimation of the blood velocity seems are binded either by a biodegradable ligand [13] or by to be an interesting approach to avoid the drawbacks of the self-assembly properties [14] as shown on Figure 1. aforementioned methods.

The purpose of this paper is to propose an on-line state M. Fruchard and E. Courtial are with PRISME EA 4229, Univ. Orleans,´ 63 Av de Lattre de Tassigny, F18020, Bourges, estimation of the blood velocity. The latter is modeled by a France. [email protected], truncated Fourier series. The model of the nanorobot dynamics [email protected] taking into account the blood velocity behavior is proved to L. Arcese is with CReSTIC EA 3804, Univ. Reims Champagne- Ardenne, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France. be observable. Two different state observers are then designed [email protected] and combined with a backstepping law to achieve a reference TABLE I PHYSIOLOGICAL PARAMETERS[20], ACTUATION AND IMAGING. X STANDS FOR A COMPATIBLE DEVICE. Physiological Parameters (mean values) System Design Actuator (∇B in mT.m−1) Imager resolution in m Blood Clinical Pre-clinical Maxwell Cardiovascular Network Diameter Required µ-MRI MRI µ-MRI CT1 µ-CT2 OCT3,4 US5,4 velocity MRI MRI Coils −3 −4 −3 −5 −6 −3 (m) −1 gradient (≈ 450) < 10 < 10 < 10 < 510 < 10 < 10 (m.s ) (< 80) (≈ 100) Setup −2 −1 Arteries 10 2.510 60 X X X X X X X X X X

Small 10−3 210−2 90 - X XX X X X X X X Arteries Arterioles 10−4 10−3 250 - - XX - X - XX -

Capillaries 10−5 510−5 2800 - - -X - - - - X -

0.12 trajectory tracking. The paper consists in five sections. Section Blood velocity v0 = II details the need for a blood velocity estimation and proposes 0.10 v1 a generic blood velocity model. Then, in Section III, the v2 v ) 0.08 3 nonlinear model of an aggregate navigating in a blood vessel is 1 presented. Section IV is dedicated to the design of observers: − m.s a high gain observer and a receding horizon estimator are ( 0.06 addressed to on-line reconstruct the blood velocity. Finally, simulations illustrate the efficiency of the proposed approach 0.04 combining control and estimation in Section V. Conclusions 0.02 and discussions on open issues are summarized in Section VI. Blood velocity 0 II.WHY AND HOW TO ESTIMATE THE BLOOD VELOCITY? −0.02 0 0.20.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Magnetically actuated robots are classically described Time (s) [11] by the motive magnetic force induced by the magnetic field or gradient depending on the robot design [1], the Fig. 2. Blood velocity v (black solid line), mean value v0 (gray solid line) and nth order truncated Fourier series vn: v1 (blue dots), v2 (black dashes) apparent weight and the drag force induced by a non and v3 (green crosses). pulsatile blood flow. Hydrodynamic wall effects show that a partial vessel occlusion by the particles results in an optimal ratio between the microrobot and the vessel radii, the vasculature, the most disturbed by external time-varying denoted r and R respectively. As suggested in [14], a perturbations is the drag force because of the pulsatile first approximation for dimensioning the actuators relies blood flow. Several possibilities to deal with this periodic on the study of the optimal ratio of motive over external perturbation might be considered: direct measurement, forces. Table I summarizes these technical constraints and disturbance rejection or blood velocity estimation. the consequences on the choice of both actuators and imagers. Physiological periodic perturbations are usual issues in Optimality requires that the robot goes smaller as it medical robotics and are often addressed using periodic enters smaller vessels. Recently the idea of using microscale signal rejection: either using filtering of the output or using a carrier made up of a biodegradable aggregate of iron-cobalt precompensation in the control law, like in [19] for breathing nanoparticles and doxorubicin has been rendered possible rejection. For the present case, both solutions require the blood [13]. In large vessels, a millimeter-sized robot has an velocity measurement. However the pulsatile blood velocity important propelling force making possible to resist to the is a local information and is consequently more difficult to blood drag. In small vessels where the blood flow is lower, be measured. Studies have been carried out to measure the a micrometer-sized robot can ensure a sufficient thrust while blood velocity using the Doppler effect with ultrasound or avoiding any injury caused by embolism or thrombosis. MRI devices [21], [22]. The quality of the measurement Furthermore the motive force is volumetric whilst the drag depends on the ultrasound transducer orientation during the force is —at best— surfacic which implies an increasing acquisition and it would require to ensure the servoing of the prevalence of the drag force as the vessel radius decreases. devoted imager on the robot position as it navigates through Among the different forces affecting the robot navigation in the human body. When the robot penetrates deeper into the human body, the device resolution, as well as the perturbations 1 CT stands for Computed Tomography device induced by the different tissues separating the robot from the 2Cannot be used on living animals 3OCT stands for Optical Coherence Tomography device sensor, may impact the measurement reliability. Finally the 4Limited penetration depth poor spatial accuracy of such measurements is not compatible 5US stands for Ultrasound device with the precision required to discriminate the parabolic blood velocity profile; yet the robot does not face the same where A is the frontal area, the drag coefficient Cd is given drag depending on its position in the vessel. The proposed by [25], β is a dimensionless ratio related to the wall effect approach is different since it does not rely on the direct caused by the vessel occlusion by a spherical aggregate of measurement of this velocity but on the disturbance estimation. radius r [26], and v denotes the relative velocity between the aggregate and the fluid. η and ρf denote respectively the blood Since an imager is already necessary to localize the robot viscosity and density. In the case of a spherical aggregate, the and thus provides an output of the system, one can exploit drag force can be rewritten as: a model of the system to reconstruct unmeasured states. 2 2 v v Observers of the robot velocity have already been studied in Fd = a v + bv + c (4) [23]. We propose to extend this approach to estimate the blood − | | 1+ d v v | | || || velocity. It requires to have a dynamical model of the blood The parameters in (4) are inherited from (3): velocity behavior. Arterial pulsatile flow profiles are usually 6πηr 0.2ρ πr2 modeled using the Womersley model [24], which results in a a = , b = f truncated Fourier series as shown on Fig. 2. It is easy to show β β2  2 (5) 3ρf πr 2rρf that any velocity 1 expressed as a second-order truncated  c = , d = Fourier series is aℵ solution of the autonomous system:  β2 βη ˙ 2) Magnetic force: The propulsion of the magnetic robot is 1 = 2  ℵ˙ ℵ 2 provided by the magnetic force Fm induced by the magnetic ( ℵ1 ) 2 = ω ( 1 3) (1) S  ℵ˙ − ℵ − ℵ field gradients. This force can be written as:  3 = 0 ℵ where is the average value of the blood velocity and ω Fm = τmV (M. )B (6) ℵ3  ∇ is the pulsation of the blood flow assumed to be known. For where V is the aggregate volume, B the magnetic field, M a blood velocity 1 defined as a nth order truncated Fourier the magnetization of the nanorobots and τm = Vm/V the series, (1) can beℵ extended in the (2n + 1)-dimensional form: ferromagnetic ratio with Vm the ferromagnetic volume. ˙ 1 = 2 3) Apparent weight: The apparent weight of the aggregate ℵ˙ ℵ 2 results from the contribution of the weight and the buoyancy:  2 = ω ( 1 3) ℵ. − ℵ − ℵ  . W a = V (ρ ρf )g (7)  . −  ˙ ( ℵ)  2k−1 = 2k (2) with the aggregate density ρ = m/V where m is the aggregate S  ℵ˙ ℵ 2 2  2k = ω (k 2k−1 2k+1) mass. Since the aggregate is made of ferromagnetic nanorobots ℵ. − ℵ − ℵ . and a payload (ligand, drugs, micelle...), the aggregate density   ˙ can be decomposed as ρ = τmρm +(1 τm)ρp where ρm and  2n+1 = 0 −  ℵ ρp are respectively the density of the magnetic nanorobots and  and where the mean value is , up to a k 1,...,n 2n+1 the density of the payload. constant∀ ∈ { factor.} See the Appendix for technicalℵ details. B. State Space Representation The objectif is then to link the disturbance dynamic exten- sion (2) to the robot modeling. Let us consider the system derived from (4), (6) and (7):

p˙1 = p2 III. MODELING τmM ( 1) p˙2 = f(p2, 1)+ ρ u (8) Let us consider an aggregate navigating in the arterial net- S  ℵ  y = p1 work. Any system immersed in a moving fluid is subjected to where p and p denote respectively the aggregate position two forces: the drag force and the buoyancy force. This section 1 2  and velocity, u the control input and the blood velocity. is devoted to briefly introduce these forces and the magnetic 1 The output y is the position of the aggregateℵ measured by the force that controls the robot along a pre-planned reference imager. The expression of the function f(p , ) is given by: trajectory (see e.g. [15] for more details). The internal forces 2 ℵ1 affecting the aggregate cohesion are not taken into account. f(p , )= σ(p ) a p + b(p )2 2 ℵ1 − 2 − ℵ1 | 2 − ℵ1| 2 − ℵ1 2 (9) A. Forces (p2 1) 1 +c − ℵ + V (ρf ρ)g , 1) Hydrodynamic drag force: In a fluidic environment, any − m 1+ d p2 1 system is subjected to the drag force which opposes its motion: | − ℵ | where the sign functionσ is defined by: 1 v 2 v Fd = ρf ACd 1 if ξ < 0  −2 β v σ(ξ)= −0 if ξ = 0 (10) 24 6   C = + + 0.4 (3) 1 if ξ > 0.  d   Re 1+ √Re  2rρf v The pulsatile periodic blood velocity is modeled by a nth order Re =   βη truncated Fourier series given by the system (2). Note that  ℵ   T T T 2n+3 belongs to a compact set ℵ. Let x = [p , ] R the control law (12) needs to know the position and velocity denote an extended state vector.K From (2) andℵ (8), we∈ obtain of the robot and the velocity of the blood. The former, i.e. an extended system in the single output control-affine form the position, is measured by the imaging device whereas the x˙ = F (x)+ Gu, y = h(x), k 1,...,n : velocities are not and consequently require to be estimated. ∀ ∈ { } x˙ 1 = x2 τmM B. Observability of the Extended System x˙ 2 = f(x2,x3)+ ρ u  .  . Lemma 1 Let denote any compact subset of a neighbor-  . ˙K  hood of (Xr, Xr). x ℵ, the vector field F satisfies  x˙ 2k+1 = x2k+2 ∀ ∈K×K ( 2)  2 2 (11) the following properties: S  x˙ 2k+2 = ω (k x2k+1 x2k+3)  − − ∂Fi(x) . P1) i 1,..., 2n + 1 , j 2, = 0; . ∀ ∈ { } ∀ ≥ ∂xi+j   x˙ = 0 ∂Fi(x)  2n+3 P2) i 1,..., 2n + 2 , α ,β > 0: α β ;  y = h(x)= x . i i i i  1 ∀ ∈ { } ∃ ≤ ∂xi+1 ≤  In order to implement a feedback control law, the state  ∂Fi(x) vector of the system (11) is required to be known. Given P3) i 1,..., 2n + 3 , γi 0: γi, ∀ ∈ { } ∃ ≥ ∂x ≤ unmeasured variables, a state-observer has to be designed. j j i + 1. The next section is devoted to the synthesis of observers for ∀ ≤ estimating the blood velocity. Proof: From (11), it is straightforward that (P1) is satisfied. IV. MAIN RESULTS The controllability and the design of backstepping control The property (P2) is easily obtained for the linear part of the laws used to Lyapunov stabilize the system (8) have been system i.e. for i = 2 with addressed in [23]. We briefly recap these results to point α2k+1 = β2k+1 = 1, k 0,...n out the necessity of estimating both the unmeasured robot 2 ∀ ∈ { } (13) α k = β k = ω , k 1,...n . and blood velocities which are required for implementing the 2 +2 2 +2 ∀ ∈ { } control law. We then address the main results of the paper: α2 and β2 are detailed hereafter. We have the following even the observability of the extended system (11) and the design function by differentiating the expression in (11) and setting of two classes of observers using either a high gain observer v = x x : 2 − 3 or a receding horizon estimator. 3 ∂F (v) v cd v 2 1 2 = a + b v + 2c | | | | . A. Prerequisites ∂x3 | | 1+ d v − 2(1 + d v )2 m | | | | (14) ˙ ¨ ∂F v Proposition 1 Let Xref = Xr(t), Xr(t), Xr(t) denote any We consequently have 2( ) =a/m equivalent to ∂x3 continuous and bounded reference trajectory. The system (8) is locally controllable along the reference trajectory. cd v v 2b(1+ d v )2 + 2c(1+ d v ) | | = 0, (15) | | | | | | − 2 Proof: The controllability of system (8) is inherited from the controllability of its linearized time-variant system along whose solutions are the reference trajectory Xref (t) [27], [28]. See [23] for more v = 0 details. 3c 16b 3c (16)  2 1 9c 4b 2  v = − − − . Proposition 2 Let θ denote any linear uncertain parameter  | | 2bd in (9) such that f(p2, 1) = f0(p2, 1)+ ϕ(p2, 1)θ. Under 16b ℵ ℵ ℵ Since we have, from (5), c > 9 , it is not difficult to show assumptions of Proposition 1 and for 1 bounded, the adaptive ∂F2(v) ℵ that > a/m, v = 0. Hence we obtain α2 = a/m. backstepping control law ∂x3 ∀

ρ ∂F2(x) u= [ (k1 + k2)(p2 X˙ r) (1 + k1k2)(p1 Xr) The upper bound β = max is inherited from the τmM 2 ∂x3 − − − ˆ ¨ − x∈K f0(p2, 1) ϕ(p2, 1)θ + Xr] (12) ∂F2(v) R∗  − ℵ − ℵ continuous differentiability of ∂x on which implies its ˙ T 3 θˆ=Γϕ (p , )[(p X˙ r)+ k (p Xr)] local Lipschitz continuity. The property (P2) is consequently 2 ℵ1 2 − 1 1 − 0 satisfied i 1,..., 2n + 2 . stabilizes the system (8) along any reference trajectory for ∀ ∈ { } any initial estimate θˆ(0) with the controllerC gains k ,k > 0 1 2 The property (P3) is easily obtained for the linear part of the and a positive-definite matrix gain Γ. system: Proof: The proof, given in [23], relies on a two steps γ2k+1 = 0, k 0,...n constructive proof of the Lyapunov stabilizing controller. 2 2 ∀ ∈ { } (17) γ k = ω k , k 1,...n . 2 +2 ∀ ∈ { } If no uncertain parameter is considered, Proposition 2 stands Using the previous result and noticing that ∂F2(x) = ∂F2(x) , ˙ ∂x2 − ∂x3 with f0 = f and no updating that is θˆ = 0. In every instance, the property (P3) is satisfied with γ2 = β2. Remark 1 The property (P1) is related to the triangular ∆ǫ = diag(ǫ1,...,ǫ2n+3) respectively denote the gain and a structure of the system. The existence of constants βi and γi in normalization diagonal matrix. S is a tridiagonal symmetric (P2) and (P3) are linked to the Lipschitz property of functions definite positive matrix such that the following Lyapunov Fi, at least locally. In this case, one can use globally Lipschitz inequality is satisfied: 1 extensions of the functions Fi. C AT (t)S + SA(t) δ CT C I, t 0 (20) − 0 ≤ ∀ ≥ Proposition 3 The system (11) is uniformly observable. with δ > 0, (δ ) > 0, the parameter δ δ0 and 0 0 ≥ 2 Proof: The observability matrix is the matrix A(t) which nonzero entries are only ai,i (t) +1 ∈ i αiǫi+1 βiǫi+1 ∂LF h(x) [ , ], i 2n + 2. = , 0 i 2n + 2, 1 j 2n + 3 (18) ǫi ǫi ∀ ≤ O ∂xj ≤ ≤ ≤ ≤ Proof: We first perform a normalization step by defining where Li h(x) denotes the ith Lie derivative of the output h(x) F a new state vector z = ∆−1x where the choice of the nonzero of the system (11) along the vector field F . Lie differentiating ǫ parameters ǫ is related to both the conditioning matrix S the output of the system (11), we obtain, k 1,...,n : i ∀ ∈ { } in (20) and the resulting minimal gain L0. In the sequel we 0 consider the z-system z˙ = F (z,t)+ G u derived from (11): I ... O ... z z 2 2 0 k 1,...,n  . .  ∀ ∈ { } .. . ǫ2 . z˙1 = z2   ǫ1  2(k−1) ∂f(x) 1 0 0  τmM =  ... ω ...  z˙2 = fz(z2,z3,t)+ ρǫ u O  ∗ ∂x 1 0   2  3 ∗   .  .. .   .  .   ǫ2k+2 .  z˙ = z   ( 3)  2k+1 ǫ2k+1 2k+2  2n ∂f(x) S  2 2 2  ...... ω   ω ǫ2k+3 ω k ǫ2k+1    z˙2k+2 = k z2k+3 k z2k+1  ∗ ∗ ∂x3  ǫ2 +2 − ǫ2 +2   . where stands for bounded functions. It is straightforward that  . ∗ ∂f(x)  the observability matrix is full ranked as from the  z˙2n+3 = 0 = 0  ∂x3  (21) property (P2) of Lemma 1, hence the result.  1 with fz(z2,z3,t)= f(ǫ2z2,ǫ3z3, θˆ). ǫ2 Since it is possible to estimate the full state vector of the Let e =z ˆ z denote the observation error. Using (21) and extended system (11) using the sole output accessible to mea- − surement, observers for the system (11) can be designed. Two (19), we obtain: different kinds of observers are studied in the next sections: a −1 T e˙ = Fz(ˆz,t) Fz(z,t) δ∆S C Ce. (22) high gain observer and a receding horizon estimator. The first − − one is theoretically well founded and the second one is well The basic idea of the proof is to decompose the difference known for its easy implementation. Fz(ˆz,t) Fz(z,t) as follows: − ǫ2 ǫ2 e2 e C. High Gain Observer ǫ1 ǫ1 2 f (ˆz , zˆ ) f (z ,z ) Since the vector fields are locally Lipschitz continuous, z 2 3 z 2 3 fz(ˆz2, zˆ3) fz(ˆz2,z3)  −.   −.  high gain observers are a promising tool. Indeed, we have . .  ǫ2k+2   ǫ2k+2  already addressed high gain observers in [23] for system (8) e2k+2 e  ǫ2k+1   ǫ2k+1 2k+2   2    which can easily be written in the observability canonical form  ω ǫ2k+3  =   e2k+3 [29]. Yet writing system (11) in the observability canonical  ǫ2k+2     2 2   2   ω k ǫ2k+1   ω ǫ2k+3  form requires an exact linearization feedback which is very  k e2k+1   e2k+3   − ǫ2 +2   ǫ2k+2  sensitive to output noise. To avoid this drawback, we would  .   .   .   .  consequently rather develop a high gain observer that directly        0  exploits the triangular structure [30] –related to property(P1)–  0    of the system (11).    0  fz(ˆz ,z ) fz(z ,z )  2 3 − 2 3  Proposition 4 Let denote any compact subset of a neigh- . K . borhood of (Xr, X˙ r) and the compact set of admissible   U +  0  control inputs. Then x(0) ℵ, xˆ(0) ℵ,  2 2  ∀ ∈ K×K ∀ ∈ K×K  ω k ǫ2k+1 e  ˆ  ǫ2k+2 2k+1  u , θ(0), L0 > 1 such that L>L0,  −  ∀ ∈U ∀ ∃ ∀  .  ˙ ˆ −1 T −1  .  xˆ = F (y, x,ˆ θ)+ Gu δ∆ǫ∆S C (C∆ǫ xˆ y) (19)   − −  0    is a high gain observer for system (11) on the com- = A(t)e + B(e,t).  2n+3 pact set ℵ where ∆ = diag(L,...,L ) and (23) K×K TABLE II Using the mean value theorem, we get the matrix A defined PARAMETER VALUES in Proposition 4 with Blood viscosity η 16 × 10−3 [Pa.s] −3 Blood density ρf 1060 [kg.m ] −3 ǫi+1 ∂Fz,i Nanorobot density ρm 7500 [kg.m ] ai,i+1(t)= (ˆz1,..., zˆi,zi+1 + φ(t)ei+1,t) (24) −4 ǫi ∂zi+1 Aggregate radius r 2.5 10 [m] Vessel diameter D 3 10−3 [m] −3 for some (unknown) φ(t) [0, 1] and the vector B(e,t) whose Payload density ρp 1500 [kg.m ] nonzero components are even∈ ones: Ferromagnetic ratio τm 0.75 Magnetization M 1.23 × 106 [A.m−1] Controller gains (k1,k2) (7, 14) ∂Fz,2 High gain L 3 b2(t) = (ˆz1,z2 + φ(t)e2,z3,t) ǫ2∂z2  (25) 2 2  ω k ǫ2k+1 TABLE III  b2k+2 = , k 1,...,n . − ǫ2k+2 ∀ ∈ { } INITIALCONDITIONSFORTHESYSTEMANDTHEOBSERVER  (x1,x2,x3,x4,x5) (0 ; 0 ; 0.1075 ; 0 ; 0.05)  Bounds on ai,i+1(t) and on b2(t) γ2 are inherited from (xˆ1, xˆ2, xˆ3, xˆ4, xˆ5) (0.001 ; 0.01 ; 0 ; 0.001 ; 0.01) properties (P2) and (P3) of Lemma 1.≤

ei The usual change of coordinates ε = i leads to: i L D. Receding Horizon Estimation (RHE) − ε˙ = L(A(t) δS 1CT C)ε + D(t)ε (26) Among the different approaches of state estimation, RHE − presents an attractive alternative to the theoretically well- where the nonzero entries of the matrix D(t) are: founded observers by proposing a systematic way to design an optimization-based observer [31]. The estimation is formulated d , (t)= b (t), d k , k = b k /L k 1,...,n . 2 2 2 2 +2 2 +1 2 +2 ∀ ∈ { } as solving on-line a nonlinear optimization problem. The (27) principle of RHE is to minimize a cost function over a past The following upper bounds: finite time interval, usually named estimation horizon and denoted Ne. The difference between the measured output y d2,2(t) γ2 | | ≤ 2 2 and the estimated output yˆ over the estimation horizon defines ω k ǫ2k+1 γ2k+2ǫ2k+1 (28)  d2k+2,2k+1 = the cost function J. Due to the sampled measurements (with | | ≤ ǫ2k+2 ǫ2k+2  Te the sampling period), the state estimation problem is written are satisfied since L 1. in discrete-time. At the current time k, the function J is ≥ minimized with respect to the state x at time (k Ne). The − A candidate Lyapunov function can be given by: estimated output yˆ(k) is then computed thanks to the model ⋆ and the computed optimal state estimate xˆ (k Ne). At each − V (t)= εT Sε λ ε 2. (29) sampling time, the past estimation horizon moves one step ≤ forward and the whole procedure (model-based estimation and Differentiating (29) using (26) leads to: optimization) is repeated to guarantee the robustness of the approach in regard to disturbances and model mismatches. The V˙ (t)= LεT AT S + SA 2δCT C ε + 2εT SDε. (30) mathematical formulation of the RHE is given by: − We recap the Lemma of [30]: assuming that property (P2) is k min J(x)= [y(j) yˆ(j)]T Q [y(j) yˆ(j)] (34) satisfied, then δ0 > 0, > 0 and S a tridiagonal positive x(k−Ne) − − ∀ ∃ ∃ j=k−Ne definite matrix such that (20) holds. Using (20) in (30), we obtain: subject to: ˙ 2 2 T V (t) L ε (2δ δ0)L Cε + 2ε SDε. (31) xˆ(k +1) = F (ˆx(k)) + G u(k) ≤− || || − − || || k k (35) yˆ(k)= hk(ˆx(k)). Set 2δ δ leads to ≥ 0 Q is a symmetric definite positive matrix and the nonlinear V˙ (t) L ε 2 + 2εT SD(t)ε. (32) ≤− || || model (35) obtained by a second-order Runge-Kutta method describes the dynamics of the process (11) in discrete time. Set M(γ,S,ǫ)= SD(t) ∞ involves || || The main advantages of RHE are its capability to explicitly 2 V (t) V˙ (t) (L 2M) ε (L 2M) (33) take into account state constraints, its easy implementation ≤− − || || ≤− − λ requiring the setting of two parameters (Ne,Q) and its ro- 2M with λ the highest eigenvalue of S. Let L0 = max 1, , bustness in regard to disturbances because of the repeated it follows from (33) that L>L0, (19) is an exponential optimization procedure. On the other hand, it suffers from a observer for system (11). ∀ lack of convergence proof since the solution is numerical. 0.025 0.025 0.015 0.015

0.02 0.02 0.01 0.01 0.015 0.015 0.005 0.005 0.01 0.01

0.005 0.005 0 0

0 0 Position(m) Position(m) Position(m) −0.005 Position(m) −0.005 −0.005 −0.005 −0.01 −0.01 −0.01 Simulated −0.01 Simulated Simulated Simulated Reference Estimated Reference Estimated −0.015 −0.015 −0.015 −0.015 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

Time(s) −3 Time(s) −3 Time(s) −3 Time(s) x 10 x 10 x 10 0.015 1.5 1.5 1

0.01 1 1 0.5 0.005 0.5 0.5 Error (m) Error (m) Error (m) Error (m)

0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time(s) Time(s) Time(s) Time(s) (a) Aggregate position: reference and (b) Aggregate position: real and esti- (a) Aggregate position: reference and (b) Aggregate position: real and esti- real trajectories and tracking error. mated trajectories and estimation error. real trajectories and tracking error. mated trajectories and estimation error.

0.05 0.15 0.15 Real 0.1 Supposed 0.1 ) ) ) −1

−1 −1 0.1 0.08 0

) 0.05

0.06 −1 0.05 0 0.04 −0.05 −0.05 0.02 0 Blood velocity (m.s Blood velocity (m.s Robot’s velocity (m.s −0.1 0 Real Simulated −0.1 Estimated Estimated −0.02 −0.05 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

Time(s) Time(s) −3 Time(s) x 10 0.06 0.04 1 ) ) ) Magnetic Field Gradients (T.m −1 −1 −0.15 0.03 −1 0.04 0.02 0.5 0.02 u 0.01 Error (m.s Error (m.s Error (m.s 0 −0.2 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time(s) Time(s) Time(s) Time(s) (c) Blood velocity: real and estimated (d) Control input: ∇B. (c) Blood velocity: real and estimated (d) Aggregate velocity: real and esti- velocities and estimation error. velocities and estimation error. mated velocities and estimation error.

0.05 Fig. 3. Simulation without an estimation of the blood velocity.

0 ) −1

−0.05

V. SIMULATIONS −0.1

Magnetic Field Gradients (T.m −0.15

u −0.2 0 1 2 3 4 5 6 7 8 9 10 The first simulation illustrates that the tracking is highly Time(s) degraded when an incorrect a priori knowledge of the blood (e) Control input: ∇B. velocity is considered, and consequently emphasizes the need for an estimation of the blood velocity. In the two last simula- Fig. 4. Simulation with a high gain observer. tions the aggregate position is assumed to be measured within an accuracy of 100m consistent with the -MRI resolution. A Gaussian white noise with a standard deviation of 100m B. Simulation with a high gain observer is thus added to the measured output. The nominal values The estimation of the full state vector is performed by a of parameters are given in Table II and the initial conditions high gain observer with parameters δ = 2500 and S given by: of the system and the observer in Table III. Without loss of generality, the dynamic of the blood velocity is modeled by a 0.13 0.015 0 0 0 0.015− 0.01 0.022 0 0 second-order truncated Fourier series.   S = − 0 0.022− 0.1 1.5 0 (36) − −  0 0 1.5 100 50  − −   0 0 0 50 60   −    The aim of the observer is twofold. First, it reconstructs A. Simulation without an estimation of the blood velocity the aggregate velocity needed to compute the control inputs. The control input, based on Lyapunov functions, ensures In this simulation, we show that an error on the blood veloc- the stability of the system along the pre-planned reference ity estimation particularly affects the tracking. The simulation trajectory. Second, the observer on-line estimates the blood is performed assuming a 50% error on the blood velocity with velocity in order to compensate for modeling errors or respect to its nominal value (see Figure 3(c)). Figure 3(a) show incorrect a priori knowledge that affects the drag force. that the tracking is degraded because of a wrong estimation of the blood velocity. One can notice a tracking error greater than Figure 4(a) illustrates that the tracking of the reference 1cm and a maximum position estimation error about 1mm trajectory by the aggregate is efficient and the system is on the Figure 3(c). Although the control input ensures the stable. One can notice a transient phase between t = 0s and stability of the system (see Figure 3(d)), the position error is t = 1s due to the different initial conditions (see Table III). too large and compromises the access to a specific area of the Figures 4(b), 4(c) and 4(d) show that the convergence of the human body. The interest of estimating the blood velocity is observer is not affected by the Gaussian white noise added to demonstrated in the two next simulations. the measured output. After a brief transient phase, the blood 0.015 0.015 information is neither sufficient nor accurate enough for an 0.01 0.01 aggregate navigating in the vasculature. The drag force has a 0.005 0.005

0 0 major impact on the dynamic behavior of the system and this Position(m) −0.005 Position(m) −0.005 force strongly and nonlinearly depends on the relative velocity −0.01 Simulated −0.01 Simulated Reference Estimated between the aggregate and the time-varying blood flow. The −0.015 −0.015 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

−3 Time(s) −3 Time(s) x 10 x 10 1.5 6 blood velocity is pulsatile and periodic. Since any periodic 1 4

0.5 2 signal has a Fourier series decomposition, it is then possible Error(m) Error(m)

0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time(s) Time(s) to define a dynamic model of the blood velocity, yielding to (a) Aggregate position: reference and (b) Aggregate position: real and esti- an extending system. real trajectories and tracking error. mated trajectories and estimation error. The core idea of the paper was to estimate both the blood and the aggregate velocities using the sole measurement of 0.14 0.15

0.12 0.1 ) ) the aggregate position given by an imager. The extended 0.1 −1 −1 0.08 0.05 system (11) was proved to be observable and then two 0.06 0 0.04 different observers were proposed in order to estimate the 0.02 −0.05 Robot velocity(m.s

Blood velocity (m.s 0 Simulated −0.1 Simulated −0.02 full state vector required to implement the control law. Estimated Estimated −0.04 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time(s) Time(s) The state observer was combined with a control law based 0.2 0.1 ) ) −1 0.15 −1 on Lyapunov functions. The simulations demonstrated the 0.1 0.05

0.05 Error(m.s Error(m.s 0 0 benefits to on-line estimate the blood velocity in order to 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time(s) Time(s) ensure a precise stabilization of the aggregate of nanorobots (c) Blood velocity: real and estimated (d) Aggregate velocity: real and esti- along any reference trajectory. velocities and estimation error. mated velocities and estimation error.

0.05 The high gain observer whose convergence is theoretically well founded obtained satisfactory results. Yet the tuning of

0 ) −1 the design parameters was difficult because of their sensitivity

−0.05 in regard to noise and modeling errors. This phenomenon depends on the choice of the high gain L and the matrix S. A −0.1 good conditioning of the matrix S is not always easy to obtain Magnetic Field Gradients (T.m −0.15 and furthermore the matrix ∆ is formed in ascending powers u of L in (19). It is therefore necessary to make a compromise −0.2 0 1 2 3 4 5 6 7 8 9 10 Time(s) to ensure a fast convergence of the estimator with a high gain (e) Control input: ∇B. L not increasing too much. The receding horizon estimator, Fig. 5. Simulation with a Receding Horizon Estimation. based on a nonlinear minimization problem, was very easy to implement and provided an efficient on-line estimation but without proof of convergence. The tuning parameters were velocity is correctly reconstructed with an average error less the estimation horizon Ne and the symmetric matrix Q. The than 1mm.s−1. obtained results can largely be improved by using an efficient algorithm and also a time-varying matrix Q which can play C. Simulation with a Receding Horizon Estimation a role similar to a forgetting factor by weighting the recent measurements. In order to show that the efficiency of the RHE does not depend on the algorithm, we have deliberately chosen a This work can be very useful in therapeutic diagnosis where standard algorithm (fminsearch function in Matlab based on a the estimation of the blood velocity is a relevant information. simplex method). The past estimation horizon is chosen equal The use of aggregates can also be considered in the case of to 5 (with a discretization period of 2ms) and the matrix Q clogged arteries with two objectives: firstly, nanorobots will is the identity matrix. The estimated state vector converges remove these plaques and in a second time, the medical team to the simulated one although the algorithm is basic and the can on-line check that the plaques have been correctly removed measured output is noised (see Figures 5(b), 5(c) and 5(d)). by observing the evolution of the estimated blood velocity. The control law using the estimated state feedback is able to Some technical limitations of this approach have to be achieve the reference trajectory tracking (see Figure 5(a)). As noted. The access to hard-to-reach body’s area such as capillar- can be seen in the figures, the estimator needs to acquire Ne ies requires a substantial reduction of the aggregate diameter, measures before starting the optimization. This is illustrated yet the force of propulsion and the localization of the system by the variations at the beginning. Results could be largely directly depend on its size (see Table I). The usual imagers improved by using a more sophisticated algorithm (Levenberg- do not have sufficient resolution and the actuators encounter Marquardt) and consequently be more robust to noise and power limitations for this kind of application in small vessels. modeling errors. Spatial resolution of the imager can also be a limitation since the noise measurement affects the quality of the blood velocity VI. DISCUSSION AND CONCLUSION estimation. Yet preliminary results indicate that these observers Many studies on the measurement of the average blood are efficient even considering the modern clinical MRI or CT velocity are available in the literature. Unfortunately, this devices, i.e. for a 500m resolution. ACKNOWLEDGMENT thereby proving that the property (A.38) holds for k + 1. 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Matthieu Fruchard received the Dipl. Ing. and the Ph.D. degrees from Ecole Centrale of Lille, France, in 2001, and from the Ecole des Mines de Paris, Sophia-Antipolis, France, in 2005, respectively. He joined the laboratory PRISME (Laboratoire Pluridisciplinaire de Recherche en Ingenierie´ des Systemes,` Mecanique´ et Energetique),´ University of Orleans,´ Bourges, France, in 2007, where he is currently Associate Professor. His research interests include control and observer synthesis for classes of nonlinear systems using Lyapunov theory.

Laurent Arcese received the M.Sc. degree from the University of Lyon, France, in 2008, and the Ph.D. degree in Automatic Control from the University of Orleans,´ France, in 2011. He is currently an Associate Professor at the Centre de Recherche en Sciences et Technologies de l’Information et de la Communication (CReSTIC), University of Reims Champagne-Ardenne, Reims, France. His research interests include the synthesis of controllers from Lyapunov theory and observer design for classes of nonlinear systems.