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IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 4, AUGUST 2014
Design of Efficient Propulsion for Nanorobots Xinghua Jia, Xiaobo Li, Scott C. Lenaghan, and Mingjun Zhang, Senior Member, IEEE
Abstract—Due to the constraints imposed at low Reynolds number, the design of efficient propulsive systems for nanorobots has proven challenging. In this paper, an approach for the design of an efficient nanorobotic propulsive system was proposed. First, resistive force theory was used to develop a dynamic model for the propulsion of nanorobots, accounting for the fluid dynamics generated by the propeller (flagellum). Second, an optimal control problem was formulated and solved to balance the tradeoff between energy utilization and tracking efficiency. Finally, simulations were conducted to analyze the effect of different body to flagellum ratios (BFR) on propulsive efficiency. It was found that the optimal flexural rigidity of the nanorobot propeller was 5.8 × 10−1 9 N·m2 , within the range of sperm flagellum, 0.7 × 10−1 9 −74.0 × 10−1 9 N·m2 . Simulations of multiple BFRs demonstrated that multipoint actuation of the nanopropeller was more efficient at BFRs of less than 1.0, while single actuation was only effective for nanorobots with a BFR >0.2. The results from this study could provide useful insights for the design of efficient nanorobotic propulsive systems, in terms of energy efficiency and trajectory tracking accuracy. Index Terms—Nanorobots, optimal control, propulsion.
I. INTRODUCTION INCE the inception of nanorobots in 1981 [1], it has been a dream to use them for medical applications. Nanorobots could potentially carry medical devices that could perform operations, inspections, and treat diseases inside the body and achieve ultrahigh accuracy in drug delivery [2]. However, there are great challenges in designing a highly controllable and efficiently propelled nanorobot. One major challenge is that most medical diagnosis and treatments (such as neoplasms, epatitis, and diabetes) involve conducting curative and reconstructive
S
Manuscript received July 18, 2013; revised December 26, 2013; accepted January 25, 2014. Date of publication February 21, 2014; date of current version August 4, 2014. The paper was recommended for publication by Associate Editor S. Martel and Editor B. J. Nelson upon evaluation of the reviewer’s comments. This work was supported by the Office of Naval Research Young Investigator Program award (ONR-N00014-11-1-0622) under the supervision of Dr. T. McKenna. X. Jia is with the Department of Mechanical, Aerospace and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996 USA (e-mail:
[email protected]). X. Li was with the Department of Mechanical, Aerospace, and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996 USA. He is now with Nurotron Biotechnology Inc., Irvine, CA 92618 USA (e-mail:
[email protected]). S. C. Lenaghan was with the Department of Mechanical, Aerospace, and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996 USA. He is now with the Center for Renewable Carbon, The University of Tennessee, Knoxville, TN 37996 USA (e-mail:
[email protected]). M. Zhang was with the Department of Mechanical, Aerospace and Biomedical Engineering, The University of Tennessee, Knoxville, TN 37996 USA. He is now with the Department of Biomedical Engineering and the Dorothy M. Davis Heart and Lung Research Institute, Ohio State University, Columbus, OH 43210 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TRO.2014.2303834
treatment at the cellular and subcellular levels. At these levels, highly targeted treatment is critical for the patient’s survival [2]. Another challenge is that due to the small size of nanorobots, the space available for onboard energy storage is very limited. This makes energy-efficient propulsion an essential goal for propulsion system design in nanorobots [3], [4]. Additionally, although the robotic design theory has been well developed in the macroscale, in low Reynolds number microscale regime, robotic design theory remains an open field [5]. Design theory for nanorobots is highly sought after. Over the past several decades, many research efforts in nanorobotics have been conducted. Due to the limitations of nanotechnology, many early studies focused on the fundamental understanding of behavior in the nanoworld [6]. Using the atomic force microscopy (AFM) manipulation approach, dynamics motion of a nanoparticle moved by the AFM cantilever was theoretically analyzed [7]. Based on the evolutionary approach and artificial neural networks, the fixed point delivery theory was developed to transport biomolecular pieces for nanorobot self-assembly through the circulatory system [8]. In early 2000, control methods allowing nanoparticles to start from an initial point to a desired location (without reference tracking) were developed, and realized by using magnetic field-guided nanoparticles [2]. In recent years, more advanced methods became available for nanorobot design and control. An artificial bacterial flagellum fabricated from the magnetic stacked thin films has been demonstrated to provide more options for nanorobot propeller design with comparable geometries and dimensions to their organic counterparts [9]. The nanorobot, consisting of a red blood cell body and a DNA-bound magnetic beads propeller, was capable of following simple reference trajectories under a short path controller [10], [11]. However, the relationship of design parameters, energy utilization, and controllability of the nanorobot remains unclear. To address the aforementioned challenges, a biologically engineered nanorobot was proposed using a bioengineering approach [4]. Included in the proposed design was a unique propulsive system inspired from the flagellum of swimming microorganisms, which have an unparalleled ability to maneuver at low Reynolds numbers. The first step of the approach is to choose proper parameters for the nanorobot design. Previous work has been proposed for determining the optimal flexural rigidity of a rotary bacterial flagellum with respect to maximization of the mean forward speed of the structure using regularized stokeslets method [12]. However, it remains an open question on how to choose a proper optimization approach to design a propeller that is capable of high energy utilization and good controllability, especially for propellers with undulating motion. The second question is how to design a controller to realize accurate tracking and energy efficient propulsion. Some
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JIA et al.: DESIGN OF EFFICIENT PROPULSION FOR NANOROBOTS
Fig. 1. Schematic of the proposed nanorobot. The inertial frame of reference, o − xy, is illustrated by the solid lines, while the body frame, o − tn, is illustrated by the dashed line.
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Fig. 2. Chain link model for flagellum. The inertial frame of reference, o − xy, is illustrated by the solid lines, while the flagellar link frame, o − ti n i , is illustrated by the dashed line. TABLE I DEFINITION OF PARAMETERS FOR NANOROBOT MODEL
optimal controllers have been proposed for reference tracking through external magnetic field regulation [10]. It may be interesting to see how the energy utilization is being optimized for the approach. Unfortunately, there is currently no approach for optimizing both for nanorobot design. In this paper, to balance the tradeoffs between energy utilization and trajectory tracking errors for a nanorobot, we extend the analysis of bio-inspired nanorobot propulsion by developing a two-step optimal control strategy. First, a mathematical model for describing the fluid dynamics of nanorobots was derived based on inspiration from swimming microorganisms. Next, a two-step optimal design approach was proposed to account for the tradeoff between tracking efficiency and energy consumption. In the first step of the optimization, a numerical method was derived to determine the optimal parameters for a given initial torque input. In the second step, an analytical solution was used to determine the time-dependent torque necessary to efficiently track a given trajectory. The approach was further validated using experimental data obtained from human sperm. Finally, simulations were conducted to analyze the effectiveness of the proposed nanorobot design approach. This paper is organized as follows. The dynamics model for the nanorobot is proposed in Section II. In Section III, the optimal control is formulated and solved. The results and discussion are illustrated in Sections IV and V. II. DYNAMICS MODEL FORMULATION Considering the severe penalties of poor designs through natural selection, microorganisms have evolved a variety of effective propulsive structures to maneuver at low Reynolds number. Of these, cilia and flagella represent the robust propellers from which human-made nanorobotic propulsive systems may be inspired. In this paper, we first analyzed the propulsion of a sperm-like nanorobot with an ellipsoidal body and polar flagellum (see Fig. 1). With the design, the viscous forces on the body can be approximated as an ellipsoid in Stokes flow (see Section II-A) [13], while the flagellum can be modeled as a chain of rigid links (see Fig. 2) [14], [15], and the viscous forces on the flagellum can be determined using resistive force theory (see Section II-B) [16]. The forces from the body and flagellum were combined to form the dynamics model for nanorobot locomotion described in Section II-C. Parameters used in the nanorobot model are shown in Table I. Based on resistive force theory, fluid forces can be described in the body and flagellar link frames. However, the dynamic
equations are usually formulated in the inertial frame. In order to conduct transformations between different frames, we define the following transformation matrices: cos θb − sin θb s := sin θb cos θb transformation from the body to inertial frame, cos θi − sin θi sθ i := sin θi cos θi transformation from the ith flagellar link frame to the inertial frame, and Cθ −Sθ Ωθ := Sθ Cθ transformation from all flagellar links to the inertial frame, where Sθ := diag(sin θ1 , . . . , sin θn ), Cθ := diag(cos θ1 , . . . , cos θn ). To apply “addition” and “subtraction” operations on neighboring flagellar links, we define the corresponding operators as ⎡ ⎤ 1 ⎢1 1 ⎥ A := ⎣ ⎦ ··· ··· 1 1 n ×n
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and
⎡ ⎢ B := ⎣
−1
1 −1 · · · ···
from the fluid and thrust/moment from the flagellum. Thus, the body model in the inertial frame can be described by
⎤ 1 −1
⎥ ⎦
sct sT [ z˙x
.
−
n ×n
In order to distribute the effects from the body to each flagellar link, we define the distribution matrix as e 0 E := 0 e where e := [ 1 · · · 1 ]Tn . In addition, an augment transformation matrix was derived to transform velocities between the inertial frame and all flagellar link frames. The velocity transformation matrix is Nθ := [ F S θ −F C θ ]T , where
−1 ALf , and the flagellar length matrix is Lf := F := B T diag(Δl, . . . , Δl). Finally, to calculate the fluid forces for all links along the flagellum, we define the fluid coefficient matrix for each flagellum Ct 0 Γ := 0 Cn where Ct := diag(ct , . . . , ct ) and Cn := diag(cn , . . . , cn ). A. Body Model 1) Body Fluid Force: Standard equations for the motion of an ellipsoid in Stokes flow were used to establish the model of the body motion [13]. The fluid forces (Fx , Fy ) and moment (MG ) on the body (see Fig. 1) in the inertial frame are described as [ Fx
Fy ] = scbt sT [ z˙x MG = −cbo θ˙b
z˙y ]T
(1) (2)
T
where [ z˙x z˙y ] is the body velocity in
the inertial frame. The fluid coefficient matrix, cbt = −diag cbl , cbn , is applied to calculate the fluid force of the body, which is transformed to the inertial frame by transformation matrix s. The angular velocity of the body in the inertial frame, θ˙b , and the rotational drag coefficient of the body, cbo , are used to calculate the moment of the body MG . The drag coefficients of the body are as follows [13]: 1 + eb −1
8 3 b 2 cl = 6πμa × eb −2eb + 1 + eb ln 3 1 − eb
2 1 + eb −1 16 3 b cn = 6πμa × eb 2eb + 3eb − 1 ln 3 1 − eb and
1 + eb −1
2 −2eb + 1+eb ln 1 − eb where μ is the fluid viscosity, and eb = 1 − d2 /a2 is the eccentricity of the body. 2) Body Model Formulation: Since the inertial effects are negligible at low Reynolds number (i.e., 10−5 –10−4 ), the equation for the body motion is a balance of viscous forces/moment cbo =
4 8πμad × e3b 3 2
2 − e2b 1 − e2b
cbo θ˙b
z˙y ]T + [ hx1
+ [ −ry
T
hy 1 ]T = 0, and
rx ] s [ hx1
(3)
T
hy 1 ] + b1 + u1 = 0. (4)
B. Flagellum Model The dynamic model of the flagellum can be formulated by balancing the force and moment applied to each flagellar link. A similar modeling approach has been used to describe the flagellar motion of Giardia lamblia [17]. 1) Flagellum Fluid Force: The fluid forces acting on the flagellum can be calculated using resistive force theory. For each link, the fluid forces can be calculated as [ fx i
fy i ]T = −sθ i diag (ct , cn ) ΔlsTθi [ x˙ i
y˙ i ]T
(5)
where [ x˙ i y˙ i ]T is the velocity vector in the inertial frame for the geometrical center of the ith flagellar link, sTθi transforms the velocity to the flagellar link frame, and sθ i transforms the fluid forces to the inertial frame. The tangential and normal drag coefficients of the flagellum (ct , cn ) can be calculated from the flagellar radius (bf ), wavelength (λf ), and fluid viscosity (μ) [18] ct =
2πμ 4πμ , and cn = . (6) ln (2λf /bf ) − 1/2 ln (2λ/bf ) + 1/2
Finally, to combine the fluid forces from each flagellar link (see 5) over the whole length of the flagellum, the following equation was used: [ fx i
fy i ]T = −Ωθ ΓΩTθ (E[ z˙x + E θ˙b s[ −ry
z˙y ]T
˙ rx ]T + Nθ θ).
(7)
2) Flagellum Model Formulation: For each flagellar link, the balanced equations for the force and moment in the inertial frame are as follows: hx i + 1 hx i fx i + − = 0, and (8) fy i hy i + 1 hy i
bi+1 − bi + ui+1 − ui − hx i + hx i + 1 Δl sin θi
+ hy i + hy i + 1 Δl cos θi = 0. (9) Assembling the forces (8) and moment (9) for each flagellar link over the entire length of the flagellum, leads to the following dynamics model: fx B hx + =0 (10) B fy hy Bb + Bu + [ −Lf AT Sθ
Lf AT Cθ ] [ hx
hy ]T = 0. (11)
3) Internal Force and Thrust Generated by Flagellum: The internal force on the flagellum can be derived by substituting flagellar fluid force (7) into (10) −1 hx B =− Ωθ ΓΩTθ hy B −1
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795
z˙x −ry b ˙ ˙ × E + Eθ s + Nθ θ . z˙y rx
(12)
The total thrust generated by the flagellum can be obtained by summing the fluid force along the entire flagellum (7)
−ry z˙ hx 1 = −E T Ωθ ΓΩTθ E x + E θ˙b s + Nθ θ˙ . hy 1 z˙y rx (13) C. Integrated Model for Swimming Nanorobot By substituting the internal forces (12) into the flagellar moment (11), the complete dynamics model of the flagellum can be obtained as A11 θ˙ + A12 θ˙b + A13 z˙ = But .
(14)
Furthermore, by substituting the thrust generated by the flagellum (13) into body model in (3) and (4), dynamics equations can be derived to account for the force (15) and moment (16) on the body, as shown below AT12 θ˙ + A22 θ˙b + A23 z˙ = ut1
(15)
AT13 θ˙ + A23 θ˙b + A33 z˙ = 0
(16)
where K = diag ( k1 k2 , . . . , kn ) , ut = u − [KB θ + Ke1 θb ], e1 = [ 1 0 , . . . , 0 ]T , and ut1 is the first element of ut . The coefficient matrices for the flagellum and body integrated model are derived as follows: T
A11 = NθT Ωθ ΓΩTθ Nθ , A12 = NθT Ωθ ΓΩTθ Es [ −ry A13 = NθT Ωθ ΓΩTθ E, A22 = [ −ry × [ −ry A23 = [ −ry
rx ]T
rx ] sT E T Ωθ ΓΩTθ Es
rx ]T + cbo rx ] sT E T Ωθ ΓΩTθ E, A33
= E T Ωθ ΓΩTθ E − scbt sT . The swimming nanorobot model in (14), (15), and (16) can be further grouped into the augmented matrix form A¯X˙ = [ But where
ut1
⎡
A11 A12 A¯ := ⎣ AT12 A22 AT13 AT23 ⎡ ⎤ θ˙ ˙ ˙ ⎣ X := θb ⎦ z˙
0 ]T
(17)
⎤ A13 A23 ⎦ A33
energy to the proposed nanorobot, and the small amount of onboard storage, it is critical that energy usage be maximized. Furthermore, the ability to closely follow a given trajectory is crucial for targeted delivery. We chose to optimize the tradeoff between energy consumption and tracking efficiency. A. Optimal Design Formulation To formulate the above as an optimization problem, the swimming nanorobot model in (17) can be rewritten as X˙ = g (X) + h (X) u
where variable definitions are in Appendix. The cost function established in this paper evaluates the performance of the nanorobot by comparing energy consumption with the ability to accurately follow a given trajectory. Since energy consumption is closely related to the actuation torque, T we use En := 0 [u (t)]T [u (t)] dt to represent the energy from time 0 to T . The tracking accuracy can be controlled by maintaining a minimum tracking error e (t) := X (t) − Xr (t), where Xr := [ θr θrb zr ]T is the reference trajectory. Using the two evaluators, the cost function can be formulated by choosing a tradeoff between these two objectives T (Xr − X)T Q (Xr − X) + uT Ru dt J (X, p, u) = 0
(19) where Q and R are two semidefinite matrices defining a tradeoff between tracking efficiency and energy consumption, and p represents the geometric or material properties of the flagellum and body. Encompassed in p are geometric parameters that affect the locomotion performance of the nanorobot, such as the length of the major (a) and minor (d) axis of the body, the flagellum length (l) and radius (bf ), and the number of flagellar links (n). Encompassed by p can also be the material properties of the flagellum, such as flexural rigidity (K). Considering the importance of the flexural rigidity of the flagellum on propulsion [12], [19], [20], we chose to illustrate the proposed approach by allowing p to represent the flexural rigidity of the flagellum. In addition to the close relationship of the actuation torque to energy consumption, tracking of the reference trajectory can also be achieved through regulation of the torque (u(t)) on the propeller (i.e., flagellum). We chose torque as the other parameter for optimization. Using these two terms (flexural rigidity and actuation torque) as the parameters for optimization, the following optimization problem was derived: min J (X, p, u) s.t. (18).
u (t),p
and
z˙x z˙ := . z˙y III. PROPULSION OPTIMIZATION
The movement of nanorobots can be evaluated by several metrics, such as energy consumption, translational and turning speeds, ability to track a reference trajectory, and thrust generation. However, considering the inability to provide external
(18)
(20)
B. Computational Approach The aforementioned problem (20) is related to the optimization of the time-independent flexural rigidity and the timedependent actuation torque. Unfortunately, it is difficult to find an analytical solution for the two parameters simultaneously. We separate the optimization problem into a two-step optimization. 1) Step One (Use a Jacobi and Gradient Method to Find the Optimal Flagellar Flexural Rigidity): The optimization for
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Fig. 4. Fig. 3.
Calculation flow chart for the second step optimization.
Calculation flow chart for the first step optimization.
the optimization min J (X, p∗ , u) s.t.(18).
Step One can be formulated as
u (t)
min J (X, p, u) s.t. (18) p
(21)
where u must be assigned with a specific function so that (18) becomes an autonomous system. Here, we chose a typical sinusoidal actuation torque [17] u = α sin (ωt + β) + γ.
H (X,λ) := E + λT (g + hu) = (Xr − X)T Q (Xr − X)
The parameters α, β, ω, and γ can be chosen based on the specific situation. In most cases, such as forward locomotion, we let γ = 0. Hence by substituting the assigned u, the optimization problem in (21) can be simplified to min J (X, p) s.t. (18) . p
In order to solve the aforementioned differential equation constrained problem, we propose a new optimization algorithm. Consider the following optimization problem: T L (X, p)dt (22) min J (X, p) := min p
p
0
s.t. X˙ = f (X, p)
(23)
where the constraint (23) leads to t X (p,t) = X0 + f (X, p)dt.
+ uT Ru + λT (g + hu) where λ is the Lagrange multiplier. According to Pontryagin’s maximum principle, we have the following two-point-boundaryvalue conditions: ∂H X˙ ∗ = = g ∗ + h∗ u, X ∗ (0) = X0 (25) ∂λ ∂H = 2Ru∗ + λ∗T h∗ (X, p) = 0, and (26) ∂u
T ∂g λ∗ λ˙ ∗ = −2Q (Xr − X) − ∂X
T ∂h − uT λ∗ , λ∗ (T ) = 0. (27) ∂X Since R in (26) is a scalar, the optimal actuation torque is
(24)
0
By substituting X (p, t) (24) into (22), the optimization problem in (22) and (23) can be converted into a nonconstraint optimization problem as min J (X (p, t) , p) = min J (p) . p
Since the actuation torque is time-dependent, an optimal actuation torque can be obtained by minimizing the cost function under the differential equation (18). This problem can be solved using Pontryagin’s maximum principle. Initially, we define a Hamiltonian function
p
The optimization can then be solved by the following gradient optimization algorithm. The description of the algorithm used for Step One (see Fig. 3) is as follows: a) Choose initial condition: X0 , and the initial parameter p0 . b) Compute differential (23) to obtain X (p, t), given X0 and pk . c) Compute ∇p k J (pk ) (see Appendix). If |∇p k J (pk )| < ε (ε is a small positive constant), output pk as optimal parameter, and then stop; Otherwise, go to the next step. d) Choose step length ls >0. Compute the next parameter pk +1 = pk − ls ∇p k J (pk ). e) Go to Step b, and repeat. 2) Step Two (Use Pontryagin’s Maximum Principle to Calculate the Time-Dependent Torque): If we substitute the optimal flexural rigidity p∗ obtained from Step One into the objective function (19) and constraint (18), we obtain the second step for
u∗ = −0.5R−1 λ∗T h∗ (X, p) . Substituting (28) into (25), we have X˙ ∗ = g ∗ (X, p) − h∗ (X, p) 0.5R−1 λ∗T h∗ (X, p) .
(28)
(29)
To obtain the optimal actuation torque numerically, we need to solve (27) and (29), and then apply the forward and backward sweeping algorithm [21]. The description of algorithm used for Step Two (see Fig. 4) is as follows: a) Make an initial guess u (0) for an optimal input. b) Solve forward by (29) to obtain X ∗ based on the initial condition X ∗ (0) = 0. c) Solve backward using (27) to obtain λ∗ , based on the final condition λ∗ (T ) = 0. d) Update control law (28) by entering X ∗ and λ∗ . e) Check convergence by comparing the obtained torque with the previous value. If yes, then we take u∗ as the optimal control. If not, then go to Step b), and repeat the process. IV. VALIDATION FOR THE PROPOSED APPROACH In order to validate the proposed optimization method, we have tested the dynamics model and optimal control by using a nanorobot with the same geometric parameters as human sperm (see Table II) [22], [23]. The actuation torque was applied along
JIA et al.: DESIGN OF EFFICIENT PROPULSION FOR NANOROBOTS
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TABLE II PARAMETERS USED IN NANOROBOT SIMULATIONS
Fig. 6. 5, 5).
Fig. 5. Nanorobot straight line tracking, where R = 1 and Q = diag(0, 0, 0, 0, 0, 0, 2.5, 2.5).
the length of the nanorobot flagellum, similar to sperm, and the initial value for the torque was set as 44000 pN·nm, which was within the torque range specified for sperm (103 –104 pN·nm) [24]. Additionally, to mimic the design of human sperm, the first 4 μm of the nanorobot propeller was considered ten times more rigid than the other links. Based on these initial conditions, simulation was conducted for the sperm-based nanorobot, and the results were then compared with experimental data collected for human sperm. A. Analysis of Tracking Errors Based on this optimization, the flexural rigidity of the flagella was determined to be 5.8 ×10−19 N·m2 . In previous studies, the flexural rigidity of sperm flagella was 0.07 × 10−20 −7.4 × 10−20 N·m2 [25], [26], with bacterial flagella being much stiffer, 10−16 N·m2 . Therefore, the optimized value is within a reasonable range for flagella in general. For the second step, two cases were considered. The first case examined the ability of the nanorobot to track a linear trajectory. We assumed that, in the inertial frame, the trajectory of the mass center of the nanorobot was a straight line along the x-axis with a starting position of 0 that moved in the negative x-direction, with a reference speed of −74 μm/s. In this case, the weighting parameters were R = 1 and Q = diag(0, 0, 0, 0, 0, 0, 2.5, 2.5). The simulation was conducted for 2 s. Fig. 5 shows the simulated trajectory of the nanorobot along the linear reference trajectory. Due to the structure constraint of the nanorobot, it cannot move directly along the reference trajectory but instead oscillates around the trajectory, similar to sperm [23]. While this
Nanorobot curve tracking where R = 1 and Q = diag(0, 0, 0, 0, 0, 0,
may inherently appear to be an energy inefficient strategy, the design limitations at the micro and nanoscale dictate this form of motion. Simulations were further conducted with a curved trajectory. A curve, zxr =−0.95t0.5 and zy r =−0.55 t, was chosen. The larger curved reference trajectory requires more energy to achieve the same tracking accuracy, and larger |Q| for more sensitive output (zxr , zy r ) regulation. Since this trajectory was more complex than the original linear trajectory, the weighting parameter for tracking, Q, was increased to Q = diag(0, 0, 0, 0, 0, 0, 5,5). Fig. 6 shows the trajectory of the nanorobot compared with the curved reference trajectory. When following the curved trajectory, the nanorobot achieved a speed of 72.3 μm/s, compared with the reference speed, 74 μm/s. The slower speed can be explained as the result of the tradeoff between tracking accuracy and energy consumption. When comparing the results of the simulation with experimental data obtained from human sperm, the nanorobot performed with similar efficiency. Human sperm, with a flagellum length of 40 –90 μm and beat frequency of 6.6–10.4 Hz, achieved speeds of 55.7 ±24.9 μm/s for linear trajectories and speeds of 88 ±28.7 μm/s when swimming in curved trajectories [23]. It is not possible to experimentally measure the energy expenditure of sperm while moving in different trajectories for comparison. However, as described previously, the ability to follow similar trajectories over the same range of speeds demonstrates the effectiveness of the model and the proposed optimization method. B. Analysis of Energy Expenditure Relative to Tracking What can also be observed from the oscillating nature of the trajectory is the need to balance the energy input versus the ability to track the trajectory. Clearly the nanorobot, in a nonoptimized system, could reach the same endpoint through numerous paths; however, it would require a much larger energy expenditure. To inspect the tracking efficiency and energy efficiency after optimization in terms of uT u, an increased weight was given to the energy efficiency by increasing R to 5. As shown in Fig. 7, with more weight placed on energy efficiency, the nanorobot can not maintain the accurate tracking as shown
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Fig. 7. Results from increasing the weight of energy efficiency. (a) Actuation torque at each link of the flagellum. (b) Trajectory tracking with an increased weight given to energy efficiency, R = 5 and Q = diag(0, 0, 0, 0, 0, 0, 1, 1).
Fig. 8.
Small head size nanorobot actuated by single point torque.
in Fig. 6. However, there is a reduction in the torque over the first three links of the flagella. In essence, by weighting the energy efficiency in this way, the overall energy input is reduced by 16.5%. This data demonstrates how the optimal controller can effectively balance the tradeoff between energy consumption and reference tracking. Through modification of the weighting parameters Q and R, multiple simulations can be conducted to determine the best balance for a specific application, thus showing the strength and tunability of this approach. V. NANOROBOT DESIGN ANALYSIS After validating the controller, we applied the approach to analyze varying nanorobot designs. For the purposes of this study, we analyzed the effect of body to flagellum ratios (BFRs) (BFR = 2 a/l, where a is the length of the major axis of the body, and l is the length of the flagellum) on propulsion. We compared three categories of nanorobots: 1) BFR 1, inspired from sperm; 2) BFR ∈ [ 0.2 1.0 ], inspired from Giardia lamblia and Tritrichomonas foetus [27], and 3) BFR 1, inspired from ciliates. A. Small Body to Flagellum Ratio (BFR 1) As we have demonstrated, actuation at multiple points along the flagellum can effectively propel nanorobots. At present, however, this form of actuation has proven difficult for implementation, and requires the development of an electrically active flexible filament. Considering this constraint, we analyzed the effect of a single actuation at the proximal segment of the flagella, and the translation of this energy through a passive filament. From a design perspective, this represents a simplification of the biological system and increases the possibility of implementation. However, when a torque of 44 000 pN·nm was applied at the first segment only, the nanorobot was not able to achieve forward propulsion, and instead oscillated around a stable position [see Fig. 8(a)]. As shown in Fig. 8(b), with a small BFR of 0.05, the nanorobot body can oscillate within the range of [−15◦ , 15◦ ]; however the flagellum oscillation is almost zero. The lack of effective flagellum oscillation means that the energy cannot be transmitted through the flagellum, at a small BFR. Based on the data obtained from the simulation, if a nanorobot was designed with this small BFR, then it would be necessary
Fig. 9. Curve tracking actuated by (a) single torque and (b) multiple torque. All conditions were the same as the sperm simulation except the body size (BFR = 0.7). R = 1, Q = diag(0,0,0,0,0,0,5,5), and reference trajectory zx r =−0.88t0 . 6 , zy r = −0.27 t.
to actuate the flagellum at multiple points. While practically this is challenging, much research is focused on the design and fabrication of novel electroactive materials that may one day fulfill this need [28]. B. Medium Body to Flagellum Ratio (BFR ∈ [0.2 1.0]) Unlike the small BFR, when the ratio increases to 0.2–1.0, both of the actuation strategies (multipoint actuation and single actuation) are capable of propelling the nanorobot. Fig. 9 shows the ability of a nanorobot with BFR = 0.7 to be propelled by both types of actuation to follow a given curved reference trajectory. All other parameters for the nanorobot are the same as Fig. 6. As shown in Fig. 9(a), the nanorobot driven with single actuation at the proximal link has a more complex swimming motion, but is still able to track the reference trajectory. The complicated trajectory is caused by oscillation of the body induced by the application of the torque at this proximal link. In Fig. 9(b), it can be seen that the multipoint actuation is capable of tracking the reference trajectory at this BFR, but with less deviation from the reference trajectory. While both the actuation strategies are capable of following the reference trajectory, there are obvious differences between the torque applied in these cases. In order to further identify the
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Despite the aforementioned advantages for the multipoint actuation, feasibility of the single actuation design, and its ability to follow the reference trajectory validates this method as a potential strategy for nanorobot design at BFR ∈ [ 0.2 1.0 ]. C. Large Body to Flagellum Ratio (BFR 1)
Fig. 10. Single and multipoint actuation compared in terms of power dissipation and internal force generated along the flagellum (a) average power and (b) average internal force distribution along the flagellum.
optimal actuation approach for the nanorobot design, the power consumption and internal force generated along the flagellum in each case were inspected. The power consumed at each flagellar link can be obtained by integrating the product of the force and velocity in the x- and y-direction along the flagellum for each link, and can be expressed as Pf i = [ x˙ i
y˙ i ] sθ i cLf sTθi [ x˙ i
y˙ i ]T
where c = diag (ct , cn ) . The average power consumed by each link during a given T time period can be computed as, 0 Pf i dt T . Furthermore, the internal force is fj = fx2j + fy2j , where fx j and fy j are the internal forces generated for the jth link, which can be expressed as fx j =
n
{−ct [cos θi x˙ i + sin θi y˙ i ] cos θi
i=j +1
− cn [sin θi x˙ i − cos θi y˙ i ] sin θi } and fy j =
n
Simulations were conducted with BFR = 10 to determine the effect of relatively short flagella on a large body. Based on the simulations, we determined that the speed of the body was 0.6% body length/s for both actuation methods. While at first the slow speed seems to indicate that the large BFR nanorobot design is an insignificant strategy for propulsion, the propulsion performance in fact is often improved by incorporating more flagella in nature. Some microorganisms with this BFR often have thousands of short flagella termed cilia that covered the outer surface of the cell to generate significant propulsion. While this may be an effective strategy for microorganisms, the fabrication and control over thousands of propulsive structures at the microscale represents a significant challenge for nanorobot design. VI. CONCLUSION In this paper, we have developed and validated a design approach for energy efficient propulsion in nanorobots. The approach has been used to determine the optimal flexural rigidity of a flagellum-like propeller for a given torque, and was found to be in agreement with the known range of sperm flagella. While the optimal flexural rigidity was used as an example for this research, the approach could also be used to optimize the flagellum radius and length, and the body shape and length. It is possible to design more energy efficient propulsive systems for nanorobots. The proposed model and optimal control were used to analyze the effect of single and multipoint actuations along the length of flagella. It was determined that for the simplest single actuation method, it was necessary for the nanorobot to have a BFR > 0.2. At a BFR below this limit, single actuation was not able to propel the nanorobot. Additionally, multipoint actuations were more effective at propelling a nanorobot of similar geometry and providing greater maneuverability.
{−ct [cos θi x˙ i + sin θi y˙ i ] sin θi
APPENDIX A BENDING MOMENT DERIVATION
i=j +1
+ cn [sin θi x˙ i − cos θi y˙ i ] cos θi }. For each link, the average force generated in each cycle is T computed as 0 fj dt T . From Fig. 10(a), we note that in both actuation strategies, the maximum power dissipation to the fluid, i.e., thrust, occurred at the most distal link. However, in the multipoint actuation strategy, the thrust generated at each link is greater than the single actuation, with the exception of the first link. In the single actuation strategy, the higher energy concentration at the proximal link leads to the larger oscillation of the nanorobot, which reduces the maneuverability. Additionally, the larger average internal force as shown in Fig. 10(b) increases the difficulty for the propeller design.
Relationship between the passive bending moment and the deviation angle of the flagellum can be expressed as bi = −EIi Ci , where Ci is the local curvature of the flagellum [29]. Furthermore, the curvature can be computed as 2 Ci (s) = (1 − cos (θi − θi−1 )) Δl2 =
n (θi − θi−1 ) 2 sin ((θi − θi−1 ) /2) n ≈ . l l
Therefore, the bending moment can be computed as bi = −nEIi /l (θi − θi−1 ) = −ki (θi − θi−1 ) where the flexural rigidity can be redefined as ki = nEIi /l.
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APPENDIX B TRANSFORMATION RELATIONSHIP DERIVATION A. Derivation of the Fluid Forces Along the Flagellum The fluid forces along the whole flagellum can be derived by assembling the flagellar fluid force (5): [ fx
fy ]T = −Ωθ ΓΩTθ [ x˙ y˙ ]T
(30)
where Ωθ , − Γ, ΩTθ , [ x˙ y˙ ]T are the assembled expressions of sθ i , − diag (ct Δl, cn Δl) , sTθi , [ x˙ i y˙ i ]T along the flagellum. Substituting the flagellar link velocity [ x˙ y˙ ]T = E [ z˙x
z˙y ]T + E θ˙b s [ −ry
rx ]T + Nθ θ˙
to (30), we can obtain the assembled fluid forces along the whole flagellum. B. Optimization Model Derivation For the right hand side of (17), we can rewrite each term in the column vector as But = Bu − BKB T θ + BKe1 θb , ut1 = eT1 u − eT1 KB T θ + k1 θb . Hence, the RHS of (17) can be further rewritten as ⎡ ⎤ ⎡ ⎤ B BKB T + BKe1 RHS = ⎣ eT1 ⎦ Gu − ⎣ eT1 KB T + k1 ⎦ X 0 0 by introducing the actuation torque regulation matrix G (e.g., G = diag( 1 . . . 1 ), when all links have actuation torque.) Inversing the coefficient matrix A¯ for the previous equation, and leaving X˙ on the left hand side, we obtain the following model for optimization: X˙ = g (X) + h (X) u where
¯ −1 D ¯ g(X) = A 0
¯ 0 B −1 B ¯ X, h(X) = A G, B = T 0 0 e1
and
¯ =− D
BKB T eT1 KB T
−BKe1 . k1
APPENDIX C ∇p J (p) CALCULATION Since X is a function of p in (22), the gradient can be derived as
∇p J (p) = 0
T
∂L (X (p) , p) ∂X ∂L (X (p) , p) + ∂x ∂p ∂p
dt.
Using (24), we generate the following relationship to calculate in the previous gradient equation: t ∂f (X, p) ∂X f (X, p) ∂X (t) = + dt S (t) := ∂p ∂X ∂p ∂p 0
∂X ∂p
which is equivalent to the differential equation [30] ∂f (X, p) ∂f (X, p) S˙ = S+ . ∂X ∂p
Therefore, we can obtain S by solving the previous differential equation. If we recall the cost function (19), we can calculate the gradient ∇p J (p) as follows: T ∂X dt. (31) ∇p J (p) = 2 (Xr − X) Q ∂p 0 REFERENCES [1] K. E. Drexler, “Molecular engineering: An approach to the development of general capabilities for molecular manipulation,” in Proc. Nat. Acad. Sci., 1981, vol. 78, pp. 5275–5278. [2] P. Vartholomeos, M. Fruchard, A. Ferreira, and C. Mavroidis, “MRIguided nanorobotic systems for therapeutic and diagnostic applications,” Annu. Rev. Biomed. Eng., vol. 13, pp. 157–184, 2011. [3] M. Sitti, “Microscale and nanoscale robotics systems [grand challenges of robotics],” IEEE Robot. Autom. Mag., vol. 14, no. 1, pp. 53–60, Mar. 2007. [4] S. Lenaghan, Y. Wang, N. Xi, F. Toshio, T. Tarn, W. Hamel, and M. Zhang, “Grand challenges in engineering life sciences and medicine: Bio-engineered nanorobots for cancer therapy,” IEEE Trans. Biomed. Eng., vol. 60, no. 3, pp. 667–673, Mar. 2013. [5] E. M. Purcell, “Life at low Reynolds number,” Amer. J. Phys., vol. 45, pp. 3–11, 1977. [6] A. Cavalcanti, L. Rosen, L. C. Kretly, M. Rosenfeld, and S. Einav, “Nanorobotic challenges in biomedical applications, design, and control,” in Proc. IEEE Int. Conf. Electron. Commun. Syst., Tel-Aviv, Israel, Dec. 2004, pp. 447–450. [7] M. Sitti and H. Hashimoto, “Tele-nanorobotics using atomic force microscope,” in Proc. IEEE Int. Conf. Intell. Robots Syst., Victoria, BC, Canada, Oct. 1998, pp. 1739–1746. [8] A. Cavalcanti and R. A. Freitas Jr, “Nanorobotics control design: A collective behavior approach for medicine,” IEEE Trans. Nanobiosci., vol. 4, no. 2, pp. 133–140, Jun. 2005. [9] L. Zhang, J. J. Abbott, L. Dong, B. E. Kratochvil, D. Bell, and B. J. Nelson, “Artificial bacterial flagella: Fabrication and magnetic control,” Appl. Phys. Lett., vol. 94, pp. 064107-1–064107-3, 2009. [10] C. Mavroidis and A. Ferreira, Nanorobotics: Current Approaches and Techniques. New York, NY, USA: Springer, 2013. [11] R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, “Microscopic artificial swimmers,” Nature, vol. 437, pp. 862– 865, 2005. [12] E. J. Lobaton and A. M. Bayen, “Modeling and optimization analysis of a single-flagellum micro-structure through the method of regularized stokeslets,” IEEE Trans. Control Syst. Technol., vol. 17, no. 4, pp. 907– 916, Jul. 2009. [13] A. T. Chwang, T. Y. Wu, and H. Winet, “Locomotion of Spirilla,” Biophys. J., vol. 12, pp. 1549–1561, 1972. [14] I. R. Gibbons, “Cilia and flagella of eukaryotes,” J. Cell Biol., vol. 91, pp. 107–124, 1981. [15] H. Lodish, A. Berk, C. A. Kaiser, and M. Krieger, Molecular Cell Biology. New York, NY, USA: W. H. Freeman, 2000. [16] J. Gray and G. Hancock, “The propulsion of sea-urchin spermatozoa,” J. Exp. Biol., vol. 32, pp. 802–814, 1955. [17] C. Jun, S. C. Lenaghan, and Z. Mingjun, “Analysis of dynamics and planar motion strategies of a swimming microorganism-Giardia lamblia,” in Proc. IEEE Int. Conf. Rob. Autom., 2012, pp. 4204–4209. [18] C. J. Brokaw, “Spermatozoan motility: A biophysical survey,” Biol. J. Linn. Soc., vol. 7, pp. 423–439, 1975. [19] H. Flores, E. Lobaton, S. M´endez-Diez, S. Tlupova, and R. Cortez, “A study of bacterial flagellar bundling,” Bull. Math. Biol., vol. 67, pp. 137– 168, 2005. [20] C. H. Wiggins, D. Riveline, A. Ott, and R. E. Goldstein, “Trapping and wiggling: Elastohydrodynamics of driven microfilaments,” Biophys. J., vol. 74, pp. 1043–1060, 1998. [21] X. Li, S. C. Lenaghan, and M. Zhang, “Evolutionary game based control for biological systems with applications in drug delivery,” J. Theor. Biol., vol. 326, pp. 58–69, 2013. [22] D. Smith, E. Gaffney, H. Gadˆelha, N. Kapur, and J. Kirkman Brown, “Bend propagation in the flagella of migrating human sperm, and its modulation by viscosity,” Cell Motil. Cytoskel., vol. 66, pp. 220–236, 2009. [23] T.-W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3-D tracking of human sperms reveals rare statistics of helical trajectories,” in Proc. Nat. Acad. Sci., 2012, pp. 16018–16022.
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[24] M. Gudipati, J. D’Souza, J. Dharmadhikari, A. Dharmadhikari, B. Rao, and D. Mathur, “Optically-controllable, micron-sized motor based on live cells,” Opt. Exp., vol. 13, pp. 1555–1560, 2005. [25] S. Ishijima and Y. Hiramoto, “Flexural rigidity of echinoderm sperm flagella,” Cell Struct. Funct., vol. 19, pp. 349–362, 1994. [26] J. E. Schoutens, “Prediction of elastic properties of sperm flagella,” J. Theor. Biol., vol. 171, pp. 163–177, 1994. [27] S. C. Lenaghan, C. A. Davis, W. R. Henson, Z. L. Zhang, and M. J. Zhang, “High-speed microscopic imaging of flagella motility and swimming in Giardia lamblia trophozoites,” in Proc. Nat. Acad. Sci., Aug. 2011, vol. 108, pp. E550–E558. [28] W. Liu, X. Jia, F. Wang, and Z. Jia, “An in-pipe wireless swimming microrobot driven by giant magnetostrictive thin film,” Sens. Actuators A Phys., vol. 160, pp. 101–108, 2010. [29] M. C. Lagomarsino, F. Capuani, and C. Lowe, “A simulation study of the dynamics of a driven filament in an Aristotelian fluid,” J. Theor. Biol., vol. 224, pp. 215–224, 2003. [30] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ, USA: Prentice Hall, 1996.
Scott C. Lenaghan received the B.S. degrees in marine science and biology from the University of Miami, Miami, FL, USA, in 2000, followed by the Ph.D. degree in biological sciences from Auburn University, Auburn, AL, USA, in 2008. In 2009, he joined the Department of Mechanical, Aerospace, and Biomedical Engineering, University of Tennessee, Knoxville, TN, USA, as a Postdoctoral Research Associate in the Nano Bio-Systems and Bio-Mimetics Lab. In 2011, he joined the faculty of the University of Tennessee as a Research Assistant Professor, within the same lab. Currently, he is a Research Assistant Professor with the Center for Renewable Carbon, University of Tennessee, where he is developing materials for biomedical applications from cellulosic waste. His research interests include nanorobotics, bio-inspired material development, advanced diagnostics, cellular biology, and biomedical imaging. He has authored over 30 journal publications. Dr. Lenaghan received distinguished NSF STEM Fellowship in 1998 for excellence in teaching.
Xinghua Jia received the B.S. degree in mechanical engineering from North University of China, Shaanxi, China, in 2008, and the M.S. degree in mechanical engineering from Dalian University of Technology, Dalian, China, in 2010. He is currently working toward the Ph.D. degree in mechanical engineering with the University of Tennessee, Knoxville, TN, USA. His research interests include modelling and experimental study of efficient propulsion and the application in bioinspired robot design for biomedical
Mingjun Zhang (S’98–M’01–SM’06) received the Doctor of Science degree from Washington University in St. Louis, St. Louis, MO, USA, in 2000, the M.S. degree in bioengineering and electrical engineering from Stanford University, Stanford, CA, USA, in 2007, and the Ph.D. degree from Zhejiang University, Hangzhou, China, in 1996. He is currently a Professor and an Investigator with the Department of Biomedical Engineering and the Dorothy M. Davis Heart and Lung Research Institute, Ohio State University, Columbus, OH, USA. After working seven years in industry, he started his academic career first as an Associate Professor with University of Tennessee, Knoxville, TN, USA, in 2008 and then as a Professor with Ohio State University in 2014. He and his colleagues made several original contributions to naturally occurring and bio-inspired nanoparticles, including ivy nanoparticles, fungus nanoparticles, tea nanoparticles, and sundew nanoscaffolds for applications in drug delivery, tissue engineering, and cancer therapy. They also discovered the unique energy-efficient propulsion mechanisms for microorganisms, including giardia, T. foetus, and whirligig beetles for applications in bio-inspired robotics. His research interest includes biologically engineered nanoparticle-based robots with flagella-enabled propulsion for applications in medicine. His research is funded by ARO, ONR, NSF, ORNL, National Academies’ Keck Future Initiative, and industries. Research results from his work have been published in PNAS, Nano Letters, Advanced Functional Materials, and PLoS Computational Biology and have drawn international media attention from sources such as Science, Nature, AAAS Science Update, BBC News, Science Daily, Science News, and the National Science Foundation of the USA. Dr. Zhang received the Young Investigator Program Award from the Office of Naval Research and an Early Career Award from the IEEE Robotics and Automation Society.
use.
Xiaobo Li was born in Changde, China. He received the B.S. degree in electrical engineering from the Central South University, Changsha, China, in 2003 and the Ph.D. degree in electrical engineering from the Louisiana State University, Baton Rouge, LA, USA, in 2009. He held Postdoctoral positions at the University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada, and at the University of Tennessee, Knoxville, TN, USA. He is currently an embedded System Engineer with Nurotron Biotechnology Inc., Irvine, CA, USA. His research interests include robust control, fault diagnosis, and fault tolerant control.
