ANALYTICAL AND EXPERIMENTAL PREDICTOR-BASED TIME...

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Proceedings of the ASME 2018 Dynamic Systems and Control Conference DSCC 2018 September 30-October 3, 2018, Atlanta, Georgia, USA DSCC2018-9101 ANALYTICAL AND EXPERIMENTAL PREDICTOR-BASED TIME DELAY CONTROL OF BAXTER ROBOT Mostafa Bagheri Research Assistant Dynamic Systems and Control Lab. Dept. of Mechanical and Aero. Eng. UC San Diego & San Diego State La Jolla, California 92093 Email: [email protected] [email protected] Miroslav Krsti´ c Daniel L. Alspach Endowed Chair in Dynamic Systems and Control Dept. of Mechanical and Aero. Eng. University of California, San Diego La Jolla, California 92093 Email: [email protected] Peiman Naseradinmousavi Assistant Professor Dynamic Systems and Control Lab. Dept. of Mechanical Eng. San Diego State University San Diego, California 92115 Email: [email protected] [email protected] ABSTRACT In this paper, a predictor-based controller for a 7-DOF Bax- ter manipulator is formulated to compensate a time-invariant in- put delay during a pick-and-place task. Robot manipulators are extensively employed because of their reliable, fast, and precise motions although they are subject to large time delays like many engineering systems. The time delay may lead to the lack of high precision required and even catastrophic instability. Using com- mon control approaches on such delay systems can cause poor control performance, and uncompensated input delays can pro- duce hazards when used in engineering applications. Therefore, destabilizing time delays need to be regarded in designing con- trol law. First, delay-free dynamic equations are derived using the Lagrangian method. Then, we formulate a predictor-based controller for a 7-DOF Baxter manipulator, in the presence of input delay, in order to track desirable trajectories. Finally, the results are experimentally evaluated. 1 Introduction Robot manipulators are extensively employed because of their reliable, fast, and precise motions [1] although they are sub- ject to large time delays like many engineering systems. Moti- vation of studying delay, as a common dynamic phenomenon, is induced by applications in control of traffic systems [2], teleop- erators [3], and robot manipulators [4,5], to name only a few. Time delay may lead to the lack of high precision required and even catastrophic instability. The large input delays emanate from communication delays in sensor-actuator networks, or from the time-consuming computational burden of multi-agent net- works. Note that the robustness margin to small input delays can be optimized by applying finite-dimensional feedback laws when the input delay is short relative to the plant’s time scales. However, a large input delay needs to be compensated through controller design [6]. Tackling the challenge of control problems with input de- lays has always been significantly important. In 1959, Smith pre- sented the compensator known as the Smith predictor [7]. How- ever, the Smith predictor may fail to achieve closed-loop stabil- ity when the plant is unstable, even though a nominal controller was designed to stabilize the delay-free system [6]. Many efforts were carried out for linear systems subject to the input delays [8–16]. Tsubakino et al. [17] proposed a predictor-based state feedback controller for multi-input linear time-invariant (LTI) systems with different time delays in each input channel using the modified backstepping transformation due to the differences among delays. In addition to the recent developments of predictor-based control laws for nonlinear systems with input delays [18–28], Bekiaris-Liberis and Krsti´ c [29] addressed the problem of sta- bilization of multi-input nonlinear systems with distinct arbi- trary large input delays, and developed a nonlinear version of 1 Copyright c 2018 by ASME

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Proceedings of the ASME 2018 Dynamic Systems and Control ConferenceDSCC 2018

September 30-October 3, 2018, Atlanta, Georgia, USA

DSCC2018-9101

ANALYTICAL AND EXPERIMENTAL PREDICTOR-BASED TIME DELAY CONTROLOF BAXTER ROBOT

Mostafa BagheriResearch Assistant

Dynamic Systems and Control Lab.Dept. of Mechanical and Aero. Eng.UC San Diego & San Diego State

La Jolla, California 92093Email: [email protected]

[email protected]

Miroslav KrsticDaniel L. Alspach Endowed Chair in

Dynamic Systems and ControlDept. of Mechanical and Aero. Eng.University of California, San Diego

La Jolla, California 92093Email: [email protected]

Peiman NaseradinmousaviAssistant Professor

Dynamic Systems and Control Lab.Dept. of Mechanical Eng.

San Diego State UniversitySan Diego, California 92115

Email: [email protected]@gmail.com

ABSTRACTIn this paper, a predictor-based controller for a 7-DOF Bax-

ter manipulator is formulated to compensate a time-invariant in-put delay during a pick-and-place task. Robot manipulators areextensively employed because of their reliable, fast, and precisemotions although they are subject to large time delays like manyengineering systems. The time delay may lead to the lack of highprecision required and even catastrophic instability. Using com-mon control approaches on such delay systems can cause poorcontrol performance, and uncompensated input delays can pro-duce hazards when used in engineering applications. Therefore,destabilizing time delays need to be regarded in designing con-trol law. First, delay-free dynamic equations are derived usingthe Lagrangian method. Then, we formulate a predictor-basedcontroller for a 7-DOF Baxter manipulator, in the presence ofinput delay, in order to track desirable trajectories. Finally, theresults are experimentally evaluated.

1 IntroductionRobot manipulators are extensively employed because of

their reliable, fast, and precise motions [1] although they are sub-ject to large time delays like many engineering systems. Moti-vation of studying delay, as a common dynamic phenomenon, isinduced by applications in control of traffic systems [2], teleop-erators [3], and robot manipulators [4, 5], to name only a few.

Time delay may lead to the lack of high precision requiredand even catastrophic instability. The large input delays emanatefrom communication delays in sensor-actuator networks, or fromthe time-consuming computational burden of multi-agent net-works. Note that the robustness margin to small input delayscan be optimized by applying finite-dimensional feedback lawswhen the input delay is short relative to the plant’s time scales.However, a large input delay needs to be compensated throughcontroller design [6].

Tackling the challenge of control problems with input de-lays has always been significantly important. In 1959, Smith pre-sented the compensator known as the Smith predictor [7]. How-ever, the Smith predictor may fail to achieve closed-loop stabil-ity when the plant is unstable, even though a nominal controllerwas designed to stabilize the delay-free system [6]. Many effortswere carried out for linear systems subject to the input delays[8–16]. Tsubakino et al. [17] proposed a predictor-based statefeedback controller for multi-input linear time-invariant (LTI)systems with different time delays in each input channel usingthe modified backstepping transformation due to the differencesamong delays.

In addition to the recent developments of predictor-basedcontrol laws for nonlinear systems with input delays [18–28],Bekiaris-Liberis and Krstic [29] addressed the problem of sta-bilization of multi-input nonlinear systems with distinct arbi-trary large input delays, and developed a nonlinear version of

1 Copyright c© 2018 by ASME

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(a)

q4

q6

q2

(b)

q1

q5

q3

q7

(c)

FIGURE 1. (a) The 7-DOF Baxter manipulator; The joints’ configu-ration: (b) sagittal view; (c) top view

the predictor-based control law.Motivated by the harmful consequences of input delays on

the stability and performance of such control systems, we formu-late and implement a predictor-feedback controller for the com-pensation of large input delays of the 7-DOF Baxter manipulator(a multi-input highly nonlinear system). We assume that all inputchannels induce the same delay, since all commands are sent tothe joints simultaneously.

The paper is organized as follows. We begin with the math-ematical modeling of the system in Section II, along with deriv-ing equations in order to formulate the predictor-feedback con-trol law in the presence of time delay. In Section III, we presentthe global asymptotic stability of the closed-loop system usingpredictor-feedback control law and necessary assumptions. Fi-nally, Section IV is devoted to the results of experiments (pick-and-place task) in order to reveal the significance of the predictor.

2 Mathematical ModelingThe redundant manipulator, which is being studied here,

has 7-DOF as shown in Fig. 1. The manipulator’s Denavit-Hartenberg parameters are shown in Table 1 provided by themanufacturer.

The mass, Coriolis, and gravitational (stiffness) matrices aresymbolically derived using the Euler-Lagrange equation. We

TABLE 1. Baxter’s Denavit-Hartenberg Parameters

Link ai di αi θi

1 0.069 0.27035 −π/2 θ1

2 0 0 π/2 θ2 +π/2

3 0.069 0.36435 −π/2 θ3

4 0 0 π/2 θ4

5 0.010 0.37429 −π/2 θ5

6 0 0 π/2 θ6

7 0 0.3945 0 θ7

employed the symbolic toolbox in MATLAB to obtain the cou-pled nonlinear dynamic equations as follows [30–32],

M(q)q+C(q, q)q+φ(q) = τ (1)

M(q)q+C(q, q)q =U, U = τ−φ(q) (2)

where, q, q, q ∈ R7 are angles, angular velocities and accelera-tions of joints, respectively, and τ ∈ R7 indicates the vector ofjoints’ driving torques. Also, M(q) ∈ R7×7 and C(q, q) ∈ R7×7,and φ(q) ∈R7 are the mass, Coriolis, and gravitational matrices,respectively; which are symbolically derived using the Euler-Lagrange equation.

The multi-input nonlinear system (2) can be rewritten as14th-order Ordinary Differential Equations (ODE) with the fol-lowing general state-space form,

X = f (X ,U) (3)

where X =[q1, · · · ,q7, q1, · · · , q7

]T∈ R14.

3 Designing the Predictor-Based ControllerDealing with unmolded dynamics including joints’ friction

and backlash, along with strong dynamic interconnections wouldcause a complicated problem of designing robust and computa-tionally efficient control schemes to avoid the large destabilizingtime delays. Therefore, in this section, we formulate a predictor-based controller for a multi-input nonlinear system in the pres-ence of input delay.

In order to demonstrate the generality of our approach, con-sider the following general multi-input nonlinear system withconstant input delay,

X(t) = f(X(t),U1(t−D), · · · ,Um(t−D)

)(4)

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where, X ∈ Rn is the vector of states, U1, · · · ,Um ∈ R are thecontrol inputs, D is input delay, and f : Rn ×Rm → Rn is alocally Lipschitz vector field. We assume that a feedback lawUi(t) = κi(X) is known such that the functions κi : Rn→R glob-ally asymptotically stabilize the delay-free system. On the otherhand, X(t) = f (X(t),κ(X(t))) is globally asymptotically stablein the absence of delay. Therefore, in delay system the controllaw needs to be as follows,

Ui(t−D) = κi(X(t)) (5)

which can alternatively be written as,

Ui(t) = κi(X(t +D)) = κi(P(t)) (6)

where P(t) is the D-time units ahead predictor of X(t). The pre-dictor feedback law for the system (4) is given by,

P(t) = X(t)+∫ t

t−Df (P(θ),U1(θ),U2(θ), · · · ,Um(θ))dθ (7)

with initial conditions for the integral (7),

P(θ) = X(0)+∫

θ

−Df (P(s), ,U1(s),U2(s), · · · ,Um(s))ds (8)

where θ ∈ [−D,0].Note that P(t) is calculated based on its past values, however

a solution P(t) to (7) does not always exist since the control ap-plied after t = 0 has no effect on the plant over the time interval[0,D]; system (4) can consequently exhibit finite escape over thatinterval. Therefore, in order to ensure the global existence of thepredictor state, we assume that, for all initial conditions and alllocally bounded input signals, the system’s solutions exist for alltime. This property is called forward completeness.

Definition 1. A system is forward-complete if it has bounded so-lutions (and a suitable continuous gain function) for any boundedinput.

Forward-complete systems include all linear systems bothstable and unstable, as well as various nonlinear systems thathave linearly bounded nonlinearities, such as systems contain-ing trigonometric nonlinearities, as a result of rotational motions(e.g. robotic manipulators). Therefore, the following assumptionfor our system is made,

Assumption 1. The system X = f (X ,U1, · · · ,Um) is forwardcomplete with respect to U = [U1, · · · ,Um]

T .

Assumption 1 guarantees that the system does not escape infinite time and, in particular, before the input reaches the systemat t = D. It ensures that for every initial condition and every lo-cally bounded input signal the corresponding solution is definedfor all t ≥ 0.

Forward complete systems yield global stability whenpredictor feedback is applied to them. In order to havethe global asymptotic stability of closed-loop system X(t) =f (X(t),κ(X(t))), at the expense of not having a Lyapunov func-tional available, it can be shown that the closed-loop systemshould be Input-to-State Stable (ISS). Note that we use the com-mon definition of class K , K∞ and K L functions from [33].

Definition 2. A system is ISS if there exist γ ∈K ,β ∈K Lsuch that for all initial conditions x(t0), u, and t ≥ 0

|x(t)| ≤ β (|x0|, t− t0)+ γ(‖u‖∞) (9)

where ‖u‖∞ is the L∞ norm of u. An equivalent condition forISS is the existence of a smooth (continuously differentiable) V :Rn → R (ISS-Lyapunov function) such that α1(|x|) ≤ V (x) ≤α2(|x|) for some α1,α2 ∈K∞ and satisfying for all x, u:

∂V∂x

f (x,u)≤−α(|x|)+σ(|u|) (10)

where σ ∈K and α ∈K∞

The Input-to-State Stability of our system can be proved us-ing the following theorem [34].

Theorem 1. A system is ISS provided that:

1. it admits a quasi ISS Lyapunov function:

V1 ≤−γ(|y|)+σ1(|u|) (11)

2. it admits an Input-Output to State Stable (IOSS) Lyapunovfunction:

V2 ≤−α(|x|)+σ2(|u|)+λ (|y|) (12)

3. lim supr→∞ λ (r)/γ(r)<+∞

Note that in (2), M(q) is a periodic function. In particular,M(q) is a uniformly bounded matrix and ∃α1,α2 such that α1I ≤H(q)≤ α2I. Therefore, we have the following inequalities,

|H(q)| ≤ c1 (13)

|H| ≤ c2|q| (14)

|C(q, q)| ≤ c3|q| (15)

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In order to control the system (2), U is chosen as

U =−KP(q−qd)−KDq (16)

where, KP and KD are symmetric positive definite matrices, qd isthe vector of desired trajectories, and qd indicates a generic func-tion of time to be regarded as input of the closed-loop system.Considering the following Lyapunov function, we can establishthat our system holds the first assumption of Theorem 1,

V1(q, q) =12

qT H(q)q+12

qT KPq (17)

Note that V1 is not a proper ISS-Lyapunov function for the systemsince V1 is only negative semi-definite [34].

V1 =−qT KDq+ qT KPqd ≤−α1|q|2 +α2|qd |2 (18)

where α1 and α2 are sufficiently small and large constants, re-spectively. Defining the system output y as q, V1 becomes aquasi ISS-Lyapunov function. We hence define the followingLyapunov function,

V2(q, q) =12

qT H(q)q+12

qT KPq+ εqT H(q)q√

1+qT q(19)

which is a positive definite function for a sufficiently small valueof ε . The derivative of V2 with respect to time is [34],

V2(q, q) = − qT

(KD− ε

H√1+qT q

)q− ε

qT KPq√1+qT q

+ qT KPqd + εqT KPqd√

1+qT q+ ε

qT(H−C−KD

)q√

1+qT q

− ε

(qT q)(

qT Hq)

(1+qT q)3/2

≤ 12

qT KDq− 12

εqT KPq√1+qT q

+M1

(|qd |+ |qd |2

)+M2|q|2 (20)

where ε > 0 is sufficiently small and M1,M2 > 0 are sufficientlylarge. The last inequality follows due to bounded H, H, and C asin (13) - (15), and exploiting the fact that ∀v,w∈R −v2+vw≤−v2/2+2w2. Consequently, the following is held:

12

qT KDq− 12

εqT KPq√1+qT q

≤−ε1|q|2 + |q|2√

1+ |q|2 + |q|2(21)

Defining α(r) = r2/√

1+ r2 and x = [qT qT ]T , we observethat V2 is an IOSS Lyapunov function [34],

V2 ≤−α(|x|)+M1|qd |2 +M2|y|2 (22)

Finally, according to (18) and (22), λ (r)/γ(r) is a constant,and the assumption 3 of Theorem 1 is satisfied,

lim supr→∞

M2r2

α2r2 < ∞ (23)

Therefore, using Theorem 1, the system (2) is ISS indicatingthat the following assumption for our system is held,

Assumption 2. The system X = f (X ,ω1 + κ1(X), · · · ,ωm +κm(X)) is Input-to-State Stable (ISS) with respect to ω =[ω1, · · · ,ωm]

T .

The stability proof of the closed-loop system with input de-lay is based on an equivalent representation of plant (4), usingtransport PDEs (first-order hyperbolic PDEs) for the actuatorstates, as well as an equivalent PDE representation of the pre-dictor states (7). Therefore, we present the equivalent represen-tations for the plant and the predictor states using transport PDE.

3.1 Equivalent Representation of the Plant UsingTransport PDEs for the Actuator States

Delay, as a common dynamic phenomenon, can be repre-sented by a partial differential equation (PDE) of transport type,which evolves in one spatial dimension, with one derivative inspace and one derivative in time.

ODEs with delays are interconnected systems of ODEs andtransport PDEs. A control system with ODE plant in the pres-ence of input delay, has a cascade PDE-ODE structure, wherethe control signal enters through a boundary condition of thePDE [6]. System (4) can be written equivalently as the followingPDE system,

X(t) = f (X(t),u1(0, t), · · · ,um(0, t)) (24)∂tui(x, t) = ∂xui(x, t), x ∈ (0,D), i = 1, · · · ,m (25)ui(D, t) =Ui(t) i = 1, · · · ,m (26)

Note that the solutions to (25) and (26) are given by

ui(x, t) =Ui(t + x−D), x ∈ [0,D] i = 1, · · · ,m (27)

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3.2 Transport PDE Representation of the PredictorStates

The predictor states P(θ) (8) can be written as

p(x, t) = X(t)+∫ x

0f(

p(y, t),u1(y, t), · · · ,um(y, t))

dy (28)

where x ∈ [0,D]. Based on (28), the functions p(x, t) satisfyingthe following ODE in x:

∂x p(x, t) = f(

p(x, t),u1(x, t), · · · ,um(x, t))

x ∈ [0,D] (29)

with initial solution,

p(0, t) = X(t) (30)

We intend to reveal that the solution to (29) and (30) is

p(x, t) = X(t + x), x ∈ [0,D] (31)

Note that the function X(t + x) satisfies the ODE in x (29)due to the (4):

X ′(t + x) = f(X(t + x),U1(t + x−D), · · · ,Um(t + x−D)

),

(32)for all t ≥ 0 and 0≤ x≤ D. It is resulted from the uniqueness ofsolutions to the ODE (4). Therefore, by defining

p(D, t) = P(t) (33)

and using the fact that p is a function of one variable, namelyx+ t, one can conclude that

P(t + x−D) = p(x, t), x ∈ [0,D] (34)

Doing the change of variables x = θ +D− t in (28), and utilizing(27), (30), and (34) results in

P(θ) = X(t)+∫

θ

t−Df(P(s),U1(s), · · · ,Um(s)

)ds (35)

where t −D ≤ θ ≤ t. Finally, with this representation, the fol-lowing holds

Ui(t) = ui(D, t) = κi(p(D, t)), i = 1, · · · ,m (36)

We define the backstepping transformation of ui and its in-verse backstepping based on the following lemmas:

Lemma 1. The direct backstepping transformations of ui, i =1, · · · ,m, are defined by

wi(x, t) = ui(x, t)−κi(p(x, t)) (37)

with x ∈ [0,D], p(x, t) is defined in (28), and transform system(24)-(26) to the following “target system”:

X(t) = f(

X(t),w1(0, t)+κ1(X(t)),wm(0, t)+κm(X(t)))

(38)

∂twi(x, t) = ∂xwi(x, t), x ∈ (0,D) (39)wi(D, t) = 0 (40)

Lemma 2. The inverse backstepping transformations of (37) aredefined by

ui(x, t) = wi(x, t)+κi(π(x, t)), i = 1, · · · ,m (41)

with x ∈ [0,D] and π(x, t) is defined as follows

π(x, t) = X(t)+∫ x

0f(

π(y, t) , w1 +κ1(π(y, t)), · · · ,

wm +κm(π(y, t)))

dy (42)

Therefore,

∂xπ(x, t) = f(π(x, t),κ1(x, t)+w1(x, t), · · · ,κm(x, t)+wm(x, t)

)(43)

where x ∈ [0,D], with the following initial solution,

π(0, t) = X(t) (44)

Also, based on assumption 2, we can mention the followinglemma,

Lemma 3. There exist a class K L function β1 such that thefollowing holds

Ξ(t)≤ β1(Ξ(0), t

), for all t ≥ 0 (45)

where

Ξ(t) = |X(t)|+m

∑i=1‖wi(., t)‖∞ (46)

The following lemmas help the proof of closed-loop systemstability [29],

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Lemma 4. There exist class K∞ functions ρ such that

‖p(., t)‖∞ ≤ ρ(Ξ(t)) (47)

where

Ξ(t) = |X(t)|+m

∑i=1‖ui(., t)‖∞ (48)

Lemma 5. There exist class K∞ functions ρ such that

‖π(., t)‖∞ ≤ ρ(Ξ(t)) (49)

where Ξ is defined in (46).

Lemma 6. There exist class K∞ functions ρ and ρ such that

Ξ(t)≤ ρ(Ξ(t)) (50)Ξ(t)≤ ρ(Ξ(t)) (51)

Finally, based on the mentioned technical lemmas 1-6, thefollowing theory can be proved [29].

Theorem 2. Consider the closed-loop system consisting of theplant (24) - (26) and the control laws (36). Under Assumptions1 and 2 there exists a β function belongs to class K L suchthat for all initial conditions X0 ∈ Rn and ui0 ∈ C[0,D], i =1, · · · ,m, which are compatible with the feedback laws, thatis, they satisfy ui0(D) = κi(p(D,0)), i = 1, · · · ,m, the closed-loop system has a unique solution X(t) ∈C1[0,∞) and ui(x, t) ∈C([0,D]× [0,∞)

), i = 1, · · · ,m, and the following holds for all

t ≥ 0,

Ξ(t)≤ β(Ξ(0), t

), for all t ≥ 0 (52)

where Ξ(t) defined in (48).

Proof of Theorem 2: By combining (51) with (45), weget that Ξ(t) ≤ ρ

(β1(Ξ(0), t)

), for all t ≥ 0, and hence, with

(51) we arrive at (52) with β (s, t) = ρ(β1(ρ(s), t)

). The proof

of existence and uniqueness of a solution X(t) ∈ C1[0,∞) andui(x, t) ∈ C([0,D]× [0,∞)), i = 1, · · · ,m, is shown as follows.Using relation (29) for t = 0, the compatibility of the initial con-ditions ui0 , i = 1, · · · ,m, with the feedback laws (36) guaranteethat p(x,0) ∈C1[0,D]. Hence, using relations (28), (30), and thefact that ui0 ∈C[0,D], i = 1, · · · ,m, it also follows from (37) thatwi0 ∈ C[0,D], i = 1, · · · ,m. The solutions to (39) and (40) are

given for all i = 1, · · · ,m by

wi(x, t) =

wi0(t + x), 0≤ x+ t ≤ D0, x+ t ≥ D

(53)

There is a unique solution for (53) due to the unique-ness of the solution to (39) and (40). Hence, the compati-bility of the initial conditions ui0 , i = 1, ...,m, with the feed-back laws (36) guarantee the existence of a unique solutionwi(x, t) ∈ C

([0,D]× [0,∞)

), i = 1, · · · ,m. Also, it follows that

X(t) ∈ C1[0,∞) from the target system (38). The fact thatπ(x, t) = X(t + x), for all t ≥ 0 and x ∈ [0,D], along with the in-verse backstepping transformations (41) guarantee that ui(x, t) ∈C([0,D]× [0,∞)

), i = 1, · · · ,m. The proof is completed [29].

By using (27), Theorem 2 can be written in the form of stan-dard delay notation as follows [29],

Corollary 1 (Theorem 2 in Standard Delay Notation). Considerthe closed-loop system consisting of the plant (4) and the controllaws (6)-(7). Under Assumptions 1 and 2, the following holds forall t ≥ 0,

Ω(t)≤ β(Ω(0), t

)(54)

where

Ω(t) = |X(t)|+n

∑i=1

supt−D≤θ≤t

|Ui(θ)| (55)

As Assumptions 1 and 2 are held for our system, we canemploy Theorem 2 and Corollary 1, and formulate a predictor-based controller for our system to compensate any large inputdelays.

Experimental ResultsWe implement the predictor-based controller for Baxter

robot through a pick-and-place task, whereas there is a similar in-put delay in all input channels. We formulate the control schemefor the highly interconnected nonlinear system and then experi-mentally validate.

First, in order to reveal the significant destabilizing effectof delay for the robot (Fig. 2), we intentionally apply 0.05s in-put delay and then control the manipulator without any predictor.The join-space trajectories are demonstrated in Fig. 4.

Shown in Fig. 4 reveals that, for a small input delay, theclosed-loop system becomes unstable. It is worth mentioningthat we operate the robot using joint torque control mode, as anadvanced control scheme. By operating the robot in the torquecontrol mode, access is granted to the lowest control levels. On

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FIGURE 2. The failing robot without the predictor

FIGURE 3. A stable obstacle-avoidance pick-and-place task using thepredictor-based controller

the other hand, we cannot capture more and accurate data, sinceBaxter is moving stochastically leading to the catastrophic mal-function. The AVI files of the experiments are accessible throughour Dynamic Systems and Control Laboratory (DSCL) website.

Then, we intentionally apply 0.5s input delay and controlthe system to compensate the destabilizing delay, as shown inFig. 3. As all the assumptions are held for the system, we takethe advantage of the predictor-based controller using Theorem 2and Corollary 1, in order to globally asymptotically stabilize therobot. The results are shown in Fig. 5.

As can be seen, the robot does not have any motion at thefirst 0.5s, since there is no control input to the system. Notethat the gravity compensation torques need to be applied on theBaxter manipulator in order to oppose the effect of gravitationalforce; note that this is a basic mode which is, by default, activefor the onset of the robot operation. Then, the robot begins track-ing the desired join-space trajectories just after 0.5s. As can beobserved, the joints are tracking, almost perfectly, the desirabletrajectories. As shown in Fig. 5, the negligible error mainly rootson the inaccuracy of sensors and actuators.

0 0.5 1 1.5 2 2.5 3

Time (sec)

0

10

20

30

40

50

60

70

θ1(deg.)

Real TrajectoryDesired Trajectory

(a)

0 0.5 1 1.5 2 2.5 3

Time (sec)

-50

-40

-30

-20

-10

0

10

20

θ2(deg.)

Real TrajectoryDesired Trajectory

(b)

0 0.5 1 1.5 2 2.5 3

Time (sec)

0

5

10

15

20

25

30

θ3(deg.)

Real TrajectoryDesired Trajectory

(c)

0 0.5 1 1.5 2 2.5 3

Time (sec)

30

40

50

60

70

80

90

100

110

θ4(deg.)

Real TrajectoryDesired Trajectory

(d)

0 0.5 1 1.5 2 2.5 3

Time (sec)

-50

-40

-30

-20

-10

0

10

20

30

θ5(deg.)

Real TrajectoryDesired Trajectory

(e)

0 0.5 1 1.5 2 2.5 3

Time (sec)

-70

-60

-50

-40

-30

θ6(deg.)

Real TrajectoryDesired Trajectory

(f)

0 0.5 1 1.5 2 2.5 3

Time (sec)

-40

-30

-20

-10

0

10

20

30

θ7(deg.)

Real TrajectoryDesired Trajectory

(g)

FIGURE 4. Inability to track the desired trajectories without any pre-dictor in the presence of 0.05s input delay

ConclusionThrough this paper, we presented the formulation and im-

plementation of the predictor-based controller for a highly inter-connected nonlinear system, Baxter manipulator as a case study,with the time-invariant input delay. The results reveal the sig-nificance of predictor used in the feedback control law subjectto the destabilizing time delay. We established that our systemis forward complete and Input-to-State Stable, and then formu-lated the predictor-based controller utilizing Theorem 2. The re-sults demonstrate that the manipulator becomes stable and, asexpected, tracks the desired joint-space trajectories despite the

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0 1 2 3 4 5 6 7

Time (sec)

0

10

20

30

40

50

60

70

θ1(deg.)

Real TrajectoryDesired Trajectory

(a)

0 1 2 3 4 5 6 7

Time (sec)

-50

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0

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θ2(deg.)

Real TrajectoryDesired Trajectory

(b)

0 1 2 3 4 5 6 7

Time (sec)

0

5

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30

θ3(deg.)

Real TrajectoryDesired Trajectory

(c)

0 1 2 3 4 5 6 7

Time (sec)

30

40

50

60

70

80

90

100

110

θ4(deg.)

Real TrajectoryDesired Trajectory

(d)

0 1 2 3 4 5 6 7

Time (sec)

-50

-40

-30

-20

-10

0

10

20

30

θ5(deg.)

Real TrajectoryDesired Trajectory

(e)

0 1 2 3 4 5 6 7

Time (sec)

-70

-65

-60

-55

-50

-45

-40

θ6(deg.)

Real TrajectoryDesired Trajectory

(f)

0 1 2 3 4 5 6 7

Time (sec)

-40

-30

-20

-10

0

10

20

30

θ7(deg.)

Real TrajectoryDesired Trajectory

(g)

FIGURE 5. Tracking the desired trajectories using predictor-basedcontroller in the presence of 0.5s input delay

destabilizing input delay. The negligible tracking error mainlyroots on sensors and actuators inaccuracy.

We are currently focusing our efforts on the creation ofa nonlinear adaptive time-delay controller for highly inter-connected multi-agent systems; with application to high-DOFrobots.

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