J Control Theory Appl 2013 11 (2) 207–214 DOI 10.1007/s11768-013-1037-y

PD-based trajectory tracking control in automatic cell injection system Wenkang XU, Chenxiao CAI† , Yun ZOU School of Automation, Nanjing University of Science and Technology, Nanjing Jiangsu 210094, China

Abstract: Trajectory tracking technology has been the focus of industrial manipulatory applications for many years, and its research has been found in micromanipulation in bioengineering recently. In this paper, a hybrid vision and force control method is applied to the automatic cell injection. The three-dimensional cell injection process involves the trajectory tracking in free space and the force control in contact space. A PD plus feedforward compensation control method is applied to the trajectory tracking in 3D space. Further, a PD-based robust controller is introduced into trajectory tracking while the systemic uncertainty of the cell injection is additionally considered. Both of the two control methods are theoretically proved to be exponential convergent. Finally, the effectiveness of the proposed method is verified as compared with other control methods by its application to trajectory tracking problem. Keywords: Cell injection; Trajectory tracking; PD-based robust control

1 Introduction As an important embodiment of biomanipulation, injection of foreign materials into individual cells has found significant applications in areas of gene injection [1], vitro fertilization [2], drug development, intracytoplasmic sperm injection [3] and so on. The existing equipment for manipulation is still operated manually, and cannot meet the commercial demands. Studies have declared that even skilled operators can only achieve a very low success rate, about 15% success ratio during the whole transgenic operation [4]. Besides, the low ratio of precision has also been a difficulty in cell injection tasks. Hence, there exists an urgent demand that accurate and reliable cell injection system and methodology to guarantee large batch biomanipulation production should be realized automatically and successfully. Generally, the automatic suspended cell injection process can be divided into two independent steps: the trajectory tracking before the manipulator interacts with the cell, and the injection-force regulation (whcih is actually a position controller) after the manipulator-cell contact is made. Methods for trajectory tracking control of parallel robots can be loosely categorized into two groups: model based and model free [5]. The commonly used PD and PID are widely employed control methods in many industrial applications [6–8], fuzzy control [9], and neural nerwork control [10]. All these algorithms belong to the model free control methods, which do not rely on the system model but rather the empirical rules or neural network training. These controllers also can exhibit robustness to large load disturbance and provide satisfactory dynamic performance but are time-consumed. There are still many a model-based control method utilized in manipulator trajectory tracking, such as computed torque control [11–12], sliding mode con-

trol [13], and adaptive control [14–15]. An accurate systemic model helps the design of the controller and undoubtedly contributes to the satisfactory dynamic performance. In practice, however, we cannot get a clear view of the system model, and this explains why the system uncertainties should be considered in the development of control schemes. A microrobotic system for fully automated zebrafish embryo injection is proposed by Sun et al. [4–17], which overcomes the problems inherent in manual operation, such as human fatigue and large variations in success rates due to poor reproducibility. In papers presented by Sun et al., a hybrid vision and force control method was proposed, which divided the out-of-plane cell injection task into two relatively independent processes: the visual (position) control in X-Y plane and the force control in Z-axis direction, and correspondingly, two control algorithms, a position trajectory tracking controller and a visual-based impedance force controller, are designed as to X-Y plane and Z-axis direction, respectively. When discussing the two relatively independent manipulation processes, and correspondingly, the two different controllers, we can make a division and consider separately, and this plan has been proved to be feasible when the two parts are considered separately or combined together [18]. Moreover, the precision and performance of impedance force controller in robotic manipulator largely depends on the accuracy of the position trajectory controller, and thus, there is a necessity to investigate the robotic trajectory tracking issue as an independent branch. In this paper, we make a detailed analysis on the trajectory tracking control in X-Y plane, which is the first step in the whole cell injection process, and also the preliminary step in impedance force control, especially when a position-

Received 23 February 2011; revised 17 December 2011. † Corresponding author. E-mail: [email protected]. Tel.: +86-13813987611; fax: +86-25-84313809. This work was supported by the National Natural Science Foundation of China (No. 60874007), the Research Foundation for the Doctoral Program of Higher Education under Grant (No. 20070288055), the China Postdoctoral Science Foundation (No. 20090440084), and the Nanjing University of Science and Technology Special Research Program (No. 2010GJPY017). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013 

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based impedance control [19–20] method is applied. As to the position control methods in [4,16], three traditional control algorithms, a PID controller, a computed torque controller and a visual-based controller, are discussed. However, the former could only yield poor performance, especially when the system uncertainties are considered, and the latter method not only requires a precise knowledge of the system model, which is often difficult in practice, but also necessitates large on-line computation, which means the real-time response cannot be guaranteed. Compared with the work of H. Huang and Sun et al. [18, 21], two simple and effective control methods are proposed during the X-Y plane trajectory tracking in free space. Both of the two control methods not only have a simpler structure, but with less on-line calculation. Given the relatively independent relationship between the X-Y plane and the Z-axis, either of the two control methods draw from this paper can be readily introduced into the hybrid vision and force control method [18] as an alternative or good choice for replacement. Furthermore, trajectory tracking with system uncertainties is further discussed, and correspondingly, a robust controller is designed to maintain the desirable performance. Both of the two trajectory controllers are simulated numerically and it is shown that each scheme is very general and computationally efficient, even in the presence of the bounded disturbances. Hence, the object of this note is twofold. we first develop a PD-based position control scheme for the uncertainties-free cell injection system. Then, a robust trajectory tracking controller is presented under the system uncertainties, and both of the two schemes ensure that the output tracking error asymptotically converges to zero as time tends to infinity.

2 System modeling and motion design 2.1 Motion design A trajectory motion plan should be first designed for the cell injection tasks [1, 21]. Define qa as the position of the tip to the injector pipette at the very beginning of cell injection. The pipette makes a transportation from qa to qb , a place which is within the focal range of the microscope. The point qs , qc and qd are specified as the starting point, contacting point and the destination point, respectively. Three steps are required to implement a single cell injection process. First, the prepiercing step: the injector pipette is accelerated from the starting injection place qs to the contacting place qc , where the pipette exactly contacts with the cell membrane. In order to pierce the cell with the minimal harm, the velocity of the pipette is designed to be its maximum value when the pipette reaches qc . Second, the piercing period: the pipette pierces the cell and decelerates to the desired place qd . Then, several seconds are needed to inject the genetic materials into the cell at qd . Third, withdrawing period: the pipette is accelerated in the opposite direction to move out of the cell and finally returns to the starting place qs . The above process can be illustrated in Fig. 1 and the detailed information can be found in [21]. Obviously in Fig. 1, the process from point qs to qc , which denotes the general trajectory tracking problem, motivates the study in this article.

Fig. 1 Motion plan in X-Y plane.

2.2 Modeling of cell injection system Consider the n-joint manipulator model taking the form of ˙ q˙ + g(q) = τ , M (q)¨ q + N (q, q)

(1)

where q(t) is the n × 1 vector of joint angular positions, M (q) is the n × n symmetric positive definite inertia ma˙ denotes the n × n matrix containing coriolis trix, N (q, q) and centrifugal forces, g(q) is the n × 1 vector of gravity, and τ is the n × 1 vector of applied joint torques. Therefore, the dynamic equation of four DOF motion stage belonging to the automatic cell injection system in Cartesian coordinate often has the following form [4–5,21]: ⎤⎡ ⎤ ⎡ ¨ X 0 0 0 mx + my + mp ⎥ ⎢ ⎢ 0 my + mp 0 0 ⎥ ⎢ Y¨ ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎣ 0 0 mz 0 ⎦ ⎣ Z¨ ⎦ 0 0 0 I θ¨ ⎡ ⎤ ⎡ ⎤ X˙ 0  ⎢ Y˙ ⎥ ⎢ 0 ⎥ fe ⎢ ⎥ ⎢ ⎥ , (2) +N  ⎢ ˙ ⎥ + ⎢ ⎥=τ− ⎣ Z ⎦ ⎣ −mz g ⎦ 0 θ˙ 0 where mx , my , mz are masses of the positioning table in X, Y and Z direction, respectively, mp is the mass of the working plate, Ip is the inertia matrix of the rotational axis and the working plate and θ is the rotational angle of the pisitioning table. N  denotes the diagonal matrix of the positioning table that reflects damping and viscous friction effects, τ = [τx τy τz τr ]T is the 4×1 vector of actuator joint torques, and f e = [fex fey fez ]T is the interaction force between the actuator and the environment. Note that f e = 0 in free space where the injector does not make contact with the cell [21]. For the sake of brevity, (2) can be rewritten in X, Y , and Z axes as ⎡ ⎤ ⎡ ⎤ ¨ X X˙ ⎢¨⎥ ⎢ ˙ ⎥ M xyz ⎣ Y ⎦ + N xyz ⎣ Y ⎦ + Gxyz = τ − f e . (3) Z¨ Z˙ According to the entire cell injection manipulation, system (3) can be divided into two relatively independent parts: the position serve control in free space and the force control in Z-axis, with subsystems respectively defined as   ¨ X X˙ τ xy = M xy ¨ + N xy ˙ + f exy , (4) Y Y

W. Xu et al. / J Control Theory Appl 2013 11 (2) 207–214

τz = Mz Z¨ + Nz Z˙ + fez .

(5)

System (1) contains some important structural properties which can be exploited to good advantage for developing control algorithms [7, 22]. Some of these properties will be discussed here. Property 1 (The positive definite property) The inertial matrix M (q) is positive definite for any q. Property 2 (The boundedness property) The system ˙ are uniformly bounded for all matrices M (q) and N (q, q) ˙ namely, there exist positive constants λm , λM and posq, q, ˙ for any q, q˙ such that itive definite function η(q), λm I  M (q)  λM I, ˙ T N (q, q) ˙  η(q)I, ˙ ˙ N (q, q) ∀q, q. (6) Property 3 (The skew symmetry property) The matrix ˙ (q) − 2N (q, q) ˙ is skew-symmetric for any q, q˙ namely, M there exists vector ξ such that ˙ (q) − 2N (q, q)}ξ ˙ = 0. (7) ξ T {M Property 4 (The exchangeable property) As to the cen˙ there exist any trifugal and coriolis force matrix N (q, q), vector ϑ1 , ϑ2 ∈ Rn , such that (8) N (q, ϑ1 )ϑ2 = N (q, ϑ2 )ϑ1 . Property 5

There exists positive k, k  max |Nijk (q)|, i,j,k

such that the following holds: N (x, y)z  kyz, x, y, z ∈ Rn .

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3.1 PD control with feedforward compensation In the control algorithms to robots, PD or PID is the simplest but very effective controlling modes, which are still widely used in industrial robots currently. Experience has proved that the traditional PD or PID controlling methods can also be very effective when applied to the strong coupled and nonlinear robotic systems. Takegaki et al. [24] was the first to propose PD control with gravity compensation utilized in robotic control. Based on the PD control with gravity compensation, PD control with feedforward compensation was presented by Koditschek for the first time [25]. Thus, it then follows our work about the trajectory tracking of the automatic cell injection under PD plus feedforward compensation control scheme. Fig. 2 illustrates the framework of the PD control with feedforward compensation. The output of the PD controller and the feedforward part constitute the input of the automatic cell injection system. The proposed controller is formally similar to the computed torque control (CTC) which also belongs to the robotic controlling methods. The computed torque control was used by Sun et al. in trajectory tracking and achieved good performance [4, 16–17]. In this paper, the PD control with feedforward compensation is analyzed and applied to the trajectory tracking in the cell injection process. The simulation results have shown that compared with CTC or common PID control modes, PD control with feedforward compensation has more accurate and effective capability in this note.

(9)

3 Controller design and stability analysis Two different control methods are designed to verify the effectiveness in position serve control. First of all, the PD control with feedforward compensation is proposed towards the uncertainties-free automatic cell injection model, and then the PD-based robust controller is designed when the structured uncertainties are involved. Finally, both of the two control methods will be proved to be global exponential stable. As mentioned before, the external force fe = 0 in free space, and the dynamic equation (4) takes the form ˙ q˙ = τ xy , q + N xy (q, q) (10) M xy (q)¨ T ˙ T ¨ ˙ ˙ ¨ where the vectors q = [X Y ] , q = [X Y ] , q = [X Y¨ ]T are the joint angular position, the joint angular velocity, and the joint angular acceleration in X-axis, Y -axis, respectively. An uniformly final-value-bounded lemma should be introduced firstly before the stability analysis [23]. Lemma 1 Considering the following nonautonomous system: x˙ = f (x(t), t). (11) If there exists a positive definite function V (x, t) satisfying the following two requirements: λ1 x2  V (x, t)  λ2 x2 , ∀x, ∀t  0, (12) (13) V˙ (x, t)  −λ3 x2 + ε, ∀x, ∀t  0, where λ1 > 0, λ2 > 0, λ3 > 0 are fixed positive values, as to any initial state x(0), the following inequality holds: 1 ε λ2 (1 − e−λt )} 2 , (14) x(t)  { x(0)2 e−λt + λ1 λ1 λ λ3 > 0. where λ = λ2

Fig. 2 PD control with feedforward compensation.

Theorem 1 The following control law is proposed to control system (10): ˙ q˙ d , (15) τ = −K p e − K v e˙ + M xy (q)¨ q d + N xy (q, q) where e(t) = q(t)  − q d (t) denotes the trajectory trackXd is the desired trajectory, K p , K v ing error, q d = Yd are suitably chosen gains of the PD controller, which are utilized to perform perfect decoupling effect. The last two ˙ q˙ d are q d and N xy (q, q) terms on right-hand side M xy (q)¨ feedforward compensation parts which are utilized to compensate for the nonlinearity contained in (10). Then, the tracking error e and e˙ converge to zero exponentially when ¨ d are bounded. the desired velocity q˙ d and acceleration q Proof Substituting (15) into the systemic equation (10) derives the closed-loop error equation ˙ e˙ + K P e + K D e˙ = 0. M (q)¨ e + N (q, q) Define the Lyapunov function as 1 1 ˙ = e˙ T M (q)e˙ + eT K P e V (e, e) 2 2 (16) +αe˙ T M (q)f (e), where f (e) = βe, 0 < α < 1, 0 < β < 1. ˙  0 can be The proof of the Lyapunov function V (e, e) found in [23], and we only give the details of the remaining

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proof. The Lyapunov derivative along the solution to the closedloop system is given as 1 ˙ (q)e˙ + e˙ T K P e ˙ = e˙ T M (q)¨ e + e˙ T M V˙ (e, e) 2 ˙ (q)f (e) +α¨ eT M (q)f (e) + αe˙ T M +αe˙ T M (q)f˙ (e) ˙ e˙ − K P e − K D e) ˙ = e˙ T (−N (q, q) 1 T ˙ eT M (q)f (e) + e˙ M (q)e˙ + e˙ T K P e + α¨ 2 ˙ (q)f (e) + αe˙ T M (q)f˙ (e) +αe˙ T M 1 ˙ (q) − 2N (q, q)} ˙ e˙ − e˙ T K D e˙ = e˙ T {M 2 ˙ (q)f (e) +α¨ eT M (q)f (e) + αe˙ T M (17) +αe˙ T M (q)f˙ (e). As xT y = y T x, ∀ x, y ∈ Rn , and based on Property 3, we have ˙ e˙ ˙ = −e˙ T K D e˙ + αf (e)T (−N (q, q) V˙ (e, e) T ˙ ˙ + αe˙ M (q)f (e) −K P e − K D e) T ˙ (18) +αe˙ M (q)f (e). T ˙ Based on Property 3, we have M − N = N , and we can rewrite (18) ˙ (e) − αf (e)T K P e ˙ = −e˙ T K D e˙ + αe˙ T N (q, q)f V˙ (e, e) (19) −αf (e)T K D e˙ + αe˙ T M (q)f˙ (e). According to Properties 1–2, the following inequalities are true: ⎧ T ˙ 2, −e˙ K D e˙  −λmin (K D )e ⎪ ⎪ ⎪ ⎨ −αf (e)T K P e  −αβλmin (K P )e2 , (20) ⎪ ˙ 2, αe˙ T M (q)f˙ (e)  αβλmax (M (q))e ⎪ ⎪ ⎩ ˙ −αf (e)T K D e˙  −αβλmin (K D )ee. Moreover, consider Properties 4–5 and Cauchy inequality |xT y|  xy, we have ˙ (e) = αβ e˙ T N (q, q˙ d + e)e ˙ αe˙ T N (q, q)f ˙ ˙  αβeN (q, q˙ d + e)e ˙ ˙  αβeN (q, e)(q˙ d + e) ˙ ˙  αβkee(sup( q˙ d ) + e) 2 ˙ + αβγ2 e ˙ , (21)  αβγ1 ee where γ1 = k · sup(q˙ d ) and γ2 = k · sup(e). Now, using the results from (20)–(21), we can rewrite (19) as ˙ 2 − αβλmin (K P )e2 ˙  −λmin (K D )e V˙ (e, e) ˙ 2 − αβλmin (K D )ee ˙ +αβλmax (M (q))e ˙ + αβγ2 e ˙ 2 +αβγ1 ee ˙ 2  −(λmin (K D ) − αβλmax (M (q)) − αβγ2 )e ˙ −(αβλmin (K D ) − αβγ1 )ee (22) −αβλmin (K P )e2 . 1 ˙  (e2 + e ˙ 2 ), (22) now would be As ee 2 ˙ 2 ˙  −(λmin (K D ) − αβλmax (M (q)) − αβγ2 )e V˙ (e, e) 1 ˙ 2) + (αβγ1 − αβλmin (K D ))(e2 + e 2 −αβλmin (K P )e2  −{λmin (K D ) + αβλmin (K D )

1 ˙ 2 −αβλmax (M (q)) − αβγ2 − αβγ1 }e 2 1 −αβ{λmin (K P ) + λmin (K D ) − γ1 }e2 2 ˙ 2  −{(1 + αβ)λmin K D − γ3 }e −αβ(λmin K P − γ4 )e2 ˙ 2 − 2 e2 , = − 1 e (23) where 1 γ3 = αβλmax (M (q)) + αβγ2 + αβγ1 , 2 1 (24) γ4 = αβγ1 − αβλmin (K D ). 2 Therefore, by suitably choosing the gains K P and K D , we can find 1 > 0, 2 > 0. Consequently, the tracking error e and e˙ converge to zero exponentially, namely, ˙ = 0. lim |e(t)| = 0, lim |e(t)| t→∞

t→∞

Now, the global exponential stability is achieved. Ideally, we ignore the uncertainties of the system, and the PD control with feedforward compensation can often exert excellent performance. Actually, the robotic system is always a complex nonlinear coupled system which includes various uncertainties, such as measuring error, friction, random disturbance and so on. All the neglected but does exist uncertainties not only baffle us to obtain an accurate mathematical model, but probably become the origins of the systemic instability, and further lead to difficulty in trajectory tracking. Robust control is a modern control theory which is proposed against the systemic uncertainties. Robust control can guaranty the systemic stability and certain performance index while the uncertainties belong to a variation range [8], [23–26]. Therefore, the development of high quality robust tracking control methods which can guarantee strong robustness and desirable system performance become an important topic in the robotic control field. And we emphasize that no information about the realization of the uncertainty is ever assumed and used, only its possible bound is needed for the control design. In this paper, a robust PD control is proposed when the modeling error of the cell injection system is considered. 3.2 PD-based robust control Once the system uncertainty is involved, we transform the dynamical equation (10) into the form: ˙ q˙ + f exy + Δ(q, q) ˙ = τ xy , (25) q + N xy (q, q) M xy (q)¨ ˙  ρ(e, e), ˙ Δ(q, q) (26) where f exy = 0 in free space and we assume that the uncer˙ is bounded by a known scalar function ρ( · ), tainty Δ(q, q) which belongs to a specified set [23]. Theorem 2 Based on the controller (15), the robust PD controller can be written in the following form: ˙ q˙ d − [αM xy (q) + K D ]e˙ q d + N xy (q, q) τ = M xy (q)¨ ˙ + K P ]e + u, (27) −[αN xy (q, q) where ˙ − ν, u = αN (q, q)e ˙ ˙ 2 (e, e) eρ , (28) ν= ˙ ˙ + ε − βe2 eρ(e, e) and α, ε, β are fixed positive values, K P and K D are the diagonal and positive definite matrices which are the same as the PD control plus control plus feedforward compensation.

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As to any initial error e(0), the the final value of trajectory tracking error e(t) is uniformly bounded. Namely, we can find positive constants k1 , k2 , and k3 , such that

From the conclusion (38), we can find that     e  ε      e˙  λ1 λ

1

x(t)  {k1 x(0)2 e−λt + k2 e−λt + k3 } 2 , (29) ˙ T. where x(t) = [e e] Proof As to the system in (25)–(26), substituting the auxiliary control variable u and ν in (28) into (25)–(26) derives the dynamical equation of error: ˙ e˙ + αe) M (q)(¨ e + αe) + N (q, q)( +K D e˙ + K P + Δ − u = 0. (30) Define the Lyapunov function as 1 1 ˙ = e˙ T M (q)e˙ + eT K P e. V (e, e) (31) 2 2 According to the positive definite boundness of M (q) and K P , there exists positive constants λ1 and λ2 , and  2  2  e   e      ˙  λ2  λ1    V (e, e)  , ∀e, e˙ (32)  e˙   e˙ 

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as t → ∞. Thus, the proposed robust PD controller ensures that the position error is uniformly final value bounded as to the bounded modeling error. Besides, in order to obtain a better control effect, the parameter ε should be as small as possible when the robust control algorithm (28) is designed [23].

4

Simulation

In this section, the simulation results obtained from the application of the proposed controllers for tracking the desired trajectories in cell injections are given. In order to approximate the real system to the maximum, the modeling parameters in simulation are obtained from Dong Sun et al. [17]. Through a series of identification experiments, the computed estimates of M xy and N  xy can be obtained. The system inertia matrix M xy = holds. 0.022180 0 kg·m2 , and the system damping and Differentiating V w.r.t. time t, we have 0 0.011386 1 ˙ 1.146 ˙ ˙ = eM ˙ (q)¨ ˙ Pe V˙ (e, e) e + e˙ M (q)e˙ + eK +0.00265X− viscous frictional matrix N xy = diag{ 2 X˙ 0.172 ˙ e˙ − αN (q, q)e ˙ = e˙ T (−αM (q)e˙ − N (q, q) − 0.001473Y˙ + 0.04023} kg·m2 ·s−1 . De0.04749, Y˙ −K D e˙ − K P e − Δ + u) fine thedesired injection trajectory in X- and Y -axis as 1 ˙ (q)e˙ + e˙ T K P e. (33) + e˙ T M cos(πt) 2 ˙ qd = m, and the initial value of q(0), q(0): sin(πt) ˙ (q) − 2N (q, q)} ˙ e˙ = 0, (33) can be According to e˙ T {M   shown 0 0 ˙ q(0) = m, q(0) = m·s−1 . T T T ˙ 0 0 ˙ − e˙ K D e˙ ˙ = −αe˙ M (q)e˙ − αe˙ N (q, q)e V (e, e) Simulations of the injection trajectory tracking control −e˙ T Δ + e˙ T u T T T with and without the systematic uncertainties are carried ˙ ˙ ˙ ˙ ˙ ˙ = −αe M (q)e − αe N (q, q)e − e K D e out, respectively. Comparisons of the accuracy in trajectory T T ˙ − ν). (34) tracking have been made between the outcomes in H. Huang −e˙ Δ + e˙ (αN (q, q)e T As Cauchy inequality |x y|  x y, we rewrite (34) et al. [18, 21] and the simulation conclusions in this paper. as 4.1 PD with feedforward control ˙ · Δ − e˙ T ν ˙  −αe˙ T M (q)e˙ − e˙ T K D e˙ + e V˙ (e, e) The PD control gains are chosen as   ˙ ˙ − e˙ T ν. e)  −αe˙ T M (q)e˙ − e˙ T K D e˙ + eρ(e, 3000 0 20 0 , KD = . KP = (35) 0 3500 0 25 Consider (28), we have Figs. 3 and 4 illustrate the trajectory tracking in X- and Y ˙  −αe˙ T M (q)e˙ − e˙ T K D e˙ V˙ (e, e) axis, respectively. ˙ ρe 2 From Figs. 3 and 4, we can see that the actual trajectory + (ε − βe ) ˙ + ε − βe2 ρe can track the desired trajectory well under the PD feedfor −αe˙ T M (q)e˙ − e˙ T K D e˙ − βe2 + ε. (36) ward control, and the PD control with feedforward compensation method executes good performance in trajectory According to the positive definiteness of M (q), K D and tracking. assigned ε, β, there exists λ3 > 0, Figs. 5 and 6 illustrate the position errors in X, Y -axis  2  e  with PD feedforward control. As seen from the results,   ˙  λ3  (37) the largest trajectory error in X-axis direction is about V˙ (e, e)  +ε  e˙  ±0.8 μm, and in Y -axis direction ±1.8 μm, either of which is much smaller than those in [17], where the largest error holds. is about ±4 μm under the computed torque control or the Thus, according to equations (32)–(37) and Lemma 1, visual based impedance force control. As a result, the PD 1 λ2 ε control with feedforward compensation implements a better x(t)  { x(0)2 e−λt + (1 − e−λt )} 2 λ1 λ1 λ performance than the CTC and the visual based impedance holds. force control in trajectory tracking in this note.

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4.2

Robust PD control

The control gains of K P and K D in the robust PD control are the same as the gains in PD feedforward compensation control. The modeling error Δ = ˙ = 0.3, α = 60, ε = [0.3sgnX˙ 0.3sgnY˙ ]T , and ρ(e, e) 0.001, β = 100. The trajectory tracking in X-Y plane under the proposed PD-based Robust control is illustrated in Figs. 7 and 8.

Fig. 3 X-axis PD trajectory tracking.

Fig. 7 X-axis robust trajectory tracking.

Fig. 4 Y -axis PD trajectory tracking.

Fig. 8 Y -axis robust trajectory tracking.

Fig. 5 X-axis PD trajectory error.

Fig. 6 Y -axis PD trajectory error.

As depicted in Figs. 7 and 8, the robust PD control method can restrain the systemic uncertainties and show strong capability in trajectory tracking. For comparison purpose, the previous PD control with feedforward compensation is also used to control (25) and (26) into which the modeling error is added. The position errors of trajectory tracking in X-Y plane under the common PD feedforward compensation control are shown in Figs. 9 and 10. As shown in Figs. 9 and 10, the largest position error under the common PD control with feedforward compensation is about | ± 100| μm, which illustrates that the PD control with feedforward compensation mode cannot closely track the desired reference signals when the modeling error is added. Figs. 11 and 12 show the position errors in X- and Y -axis under the proposed robust PD control.

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Fig. 9 X-axis PD control trajectory error.

It can be seen from Figs. 11 and 12 that the largest tracking errors in both axes are around ±1.5 μm, which are much smaller than those of Figs. 9 and 10. Compared with the former PD control with feedforward compensation, the proposed robust PD control scheme guarantees good performance in trajectory tracking and has a strong robustness property with respect to system uncertainties. Therefore, we can see that under the proposed robust control, required convergence is achieved even in the presence of system uncertainties, and good tracking performance is obtained. As the preparatory work of the cell injection impedance force control, the proposed methods exhibit good performance in trajectory tracking, and this is good news for an impedance force controller as the accuracy of the force controller largely depends on the precision of the position controller, especially when a further study in force control is implemented within the framework of a position-based impedance control, the proposed methods in this paper can be readily introduced into the inner-loop for trajectory control. Consequently, the position serve control in this note is just an initial part within the entire cell injection process and further work will be on the interaction in contact space, where the desired injection force should be maintained while following the desired trajectories [27].

5

Fig. 10 Y -axis PD control trajectory error.

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Conclusions

In this paper, two simple and effective methods are discussed on trajectory tracking during the automatic cell injection for batch suspended cells. The whole cell injection process can be divided into two relatively independent steps: the trajectory tracking in X-Y plane and the force control in Z-axis. The PD control with feedforawrd compensation method is applied to the trajectory tracking in XY plane. Further, the systemic uncertainty is considered in trajectory tracking, and correspondingly, a PD-based robust control is proposed for the modified system. Both methods are proved to guarantee stability and exponential convergence. Finally, the effectiveness of the proposed approaches are demonstrated by simulation. References [1] J. Kuncova, P. Kallio. Challenges in Capillary pressure microinjection. IEEE Conference on Engineering in Medicine and Biology Society (EMBS). San Francisco: IEEE, 2004: 4998 – 5001.

Fig. 11 X-axis robust trajectory error.

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Fig. 12 Y -axis robust trajectory error.

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Chenxiao CAI was born in Shandong, China, in 1975. She received her Ph.D. degree in Control Theory and Control Engineering from Automatic School, Nanjing University of Science and Technology in 2004. Currently, she is an associate professor at Department of Automation, Nanjing University of Science and Technology. Her research interests cover analysis and synthesis about singularly perturbed system. E-mail: [email protected]. Yun ZOU was born in Jiangsu, China, in 1962. He received his B.S. degree from Northwestern University, China, in 1983, majored in Numerical Mathematics, and his Ph.D. degree from Nanjing University of Science and Technology, China, in 1990, majored in Automatic Control. He is currently a professor of the Department of Automation, Nanjing University of Science and Technology, and a mathematical reviewer of Mathematical Reviews. His research interests include singular systems, 2-D systems, nonlinear systems and power systems. E-mail: [email protected].