1

Safety-Control of Mobile Robots Under Time-DelayUsing Barrier Certificates and a Two-Layer

PredictorAzad Ghaffari and Manavendra Desai

Abstract—Performing swift and agile maneuvers is essential forthe safe operation of autonomous mobile robots. Moreover, thepresence of time-delay restricts the response time of the systemand hinders the safety performance. Thus, this paper proposesa modular and scalable safety-control design that utilizes theSmith predictor and barrier certificates to safely and consistentlyavoid obstacles with different footprints. The proposed solutionincludes a two-layer predictor to compensate for the time-delay inthe servo-system and angle control loops. The proposed predictorconfiguration dramatically improves the transient performanceand reduces response time. Barrier certificates are used todetermine the safe range of the robot’s heading angle to avoid col-lisions. The proposed obstacle avoidance technique convenientlyintegrates with various trajectory tracking algorithms, whichenhances design flexibility. The angle condition is adaptivelycalculated and corrects the robot’s heading angle and angularvelocity. Also, the proposed method accommodates multipleobstacles and decouples the control structure from the obstacles’shape, count, and distribution. The control structure has onlyeight tunable parameters facilitating control calibration andtuning in large systems of mobile robots. Extensive experimentalresults verify the effectiveness of the proposed safety-control.

I. INTRODUCTION

Obstacle avoidance has become an integral part of non-holonomic mobile robots, self-driving cars, unmanned aerialvehicles, and surface vehicles [1]–[5]. Prominent methodsinclude potential field [6], [7], collision cone [8], [9], pathplanning [10], [11], receding horizon control [12]–[14], andfuzzy neural networks [15]. Recently, barrier certificates andcontrol barrier functions have gained attention to design safety-controllers [16]–[19].

However, analytical and numerical complexity restricts thepracticality and scalability of existing safety methods for large-scale applications. For example, methods based on potentialfields and barrier functions tie the safety-control law to theshape of the functions that model the obstacles. On the otherhand, methods based on receding horizon optimization arenumerically demanding. Thus, when the number of obstaclesincreases, one faces the cumbersome task of obstacle modelingand control implementation, which rapidly depletes processingresources. Also, simultaneous proof of stability and safetybecomes an involved task.

Furthermore, time-delay in mobile robots causes long tran-sients in trajectory tracking and delayed avoidance maneuvers

The authors are with the Department of Mechanical Engineering,Wayne State University, MI 48202, USA, aghaffari@wayne.edu,manavendradesai@wayne.edu.

which may cause collisions. In the presence of time-delay,any attempt to achieve fast transient by implementing high-gain controls often leads to oscillatory responses and insta-bility. Hence, advanced techniques such as receding horizonoptimal control [14], nonlinear predictor-based control [20],adaptive sliding mode control [21], and nonlinear trackingalgorithm [22] have been proposed to compensate time-delayfor nonholonomic mobile robots.

Moreover, the Smith predictor is effective for time-delaycompensation in linear time-invariant systems [23]–[25]. Thenonholonomic mobile robot investigated in this work com-prises DC motors and linear gear-boxes. Also, the wheel slipis negligible. Thus, one can accurately model the servo-systemand heading angle using linear differential equations. Hence,a two-layer predictor, comprising three independent Smithpredictors, is proposed to compensate for time-delay. First, thetime-delay is compensated in the servo-system loops, whichtransfers the time-delay into the heading angle control. Second,another Smith predictor is embedded in the heading anglecontrol to compensate for the transferred time-delay in steeringthe mobile robot.

Modular safety-control using barrier certificates has beenproven effective to enforce safety nets around unmanned aerialvehicles [26], [27]. Here, collision avoidance for mobile robotswith time-delay is addressed such that control scalability isalso achieved. Thus, the implementation and calibration effortremains minimal regardless of the number of robots andobstacles. Design objectives include high precision trajectorytracking, obstacle avoidance, and control scalability. Thus, amodular control structure is proposed. First, the two-layerpredictor compensates time-delay and guarantees a desirabletransient performance. The proposed predictor comprises aninner loop for the servo-system and an outer loop for angletracking. Trajectory tracking is based on the vector-field-orientation (VFO) control, which guarantees accurate trackingby modifying the linear speed and heading angle [28].

The safety-control is achieved by calculating the safe rangeof the heading angle using an exponential barrier certificate.Multiple obstacles with different shapes and arrangementscan be modeled as a barrier certificate. The number anddistribution of the obstacles only affect the barrier certificate’sequation, and the rest of the algorithm remains unchanged.Each obstacle has an avoidance region with a tunable radius.The safety-control pushes the robot away from the avoidanceregion as soon as the actual heading angle projects a collisionwith the obstacle. The safety condition provides an inequality

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that ties the position, safe heading angle, and translationalvelocity of the robot to the obstacle’s location and avoidanceradius. The proposed safety algorithm provides a dynamicestimate of the safe heading angle. Moreover, when the robot isfar enough from the obstacle, the safety algorithm is inactive.

The proposed modular safety-control dramatically improvescontrol scalability by utilizing linear control components,Smith predictor, and safe heading angle estimate. Regardlessof the number, shape, and distribution of the obstacles orthe reference trajectory, the proposed algorithm maintains safetrajectory tracking for the nonholonomic mobile robot. Also,the algorithm’s analytical complexity and needed processingpower are independent of the number of obstacles. Theproposed design is flexible and easy to tune and calibrate.Therefore, improved scalability is a byproduct of the proposedalgorithm.

This paper is built upon the work by Ghaffari [29]. Thiswork presents three main contributions: 1) the two-layer pre-dictor to compensate time-delay, 2) embedded modular safety-control structure, and 3) extensive experimental results toverify the effectiveness of the proposed algorithm when facingobstacles of different size and shape. Tracking accuracy isachieved as high as 98%, and the response time is improved bya factor of four. Hence, unlike the preliminary work [29], thispaper does not utilize the contour error estimate and feedbackmodification to improve trajectory tracking precision. The two-layer predictor and the experimental results are reported herefor the first time.

The rest of this paper is presented in the following order.Dynamic model and trajectory tracking using the VFO al-gorithm is explained in Section II. The two-layer predictoris described in Section III. The safety-control and allowableheading angle are presented in Section IV. Section V presentsexperimental results to verify and validate the effectiveness ofthe proposed method. Section VI concludes the paper.

II. DYNAMIC MODEL AND TRAJECTORY TRACKING

The nonholonomic mobile robot is driven by a differential-drive system comprised of two identical electric wheels. Fig. 1shows the schematic of the robot and inertial and body-fixedreference frames. The body-fixed reference frame is attachedto the robot at the center of mass. The heading angle is θ,and is measured with respect to the x-axis. The kinematicequations of the robot are given as

x= v cos θ (1)y = v sin θ (2)θ= ω, (3)

where [x y]T is the robot’s position in the inertial referenceframe, θ is the heading angle, v is linear velocity, and ω isangular velocity.

The force and torque generated by the electric wheels con-trol the linear and angular velocity of the robot, respectively.Assume that the effect of friction and wheel slip on thedynamic model is negligible, and the left and right wheel’smovements do not affect each other. Thus, one can decouplethe dynamic equation of the left and right wheels. Since the

Inertial frame x

y xbyb

θ

d

Body frame

Fig. 1. A schematic of a nonholonomic mobile robot in an inertial referenceframe x-y. The body-fixed reference frame is xb-yb and θ is the headingangle.

mobile robot is symmetric with identical left and right wheelsand motors, it is reasonable to assume that both wheels havethe same dynamic equation given as

Vi(s)

Ui(s)=G(s)e−τs (4)

where G(s) is the a transfer function, τ is the constant time-delay, Vi(s) = L(vi), and Ui(s) = L(ui), where L denotesthe Laplace transform. The wheel linear velocity is vi andthe wheel voltage is ui, where i = R,L for the right andleft wheel, respectively. The voltage of the electric wheels arebounded as |ui| ≤ umax for i = R,L, where umax is a positiveconstant. The robot’s linear and angular velocity are related tothe wheels’ velocity through the following equations

v =(vR + vL

)/2 (5)

ω =(vR − vL

)/d, (6)

where d is the distance between the center of the wheels.Therefore, the equations of linear and angular velocity can bewritten as the following

V (s)

Wi(s)=G(s)e−τs, (7)

where V (s) = L(v), Ω(s) = L(ω), Wi(s) = L(wi) for i =v, ω, where

wv =(uR + uL

)/2 (8)

wω =(uR − uL

)/d. (9)

The vector-field-orientation (VFO) method proposed byMichalek and Kozlowski [28] is utilized to determine adjustedlinear and angular velocities, va and ωa, for the servo-system(inner loop) of the mobile robot. Given the instantaneousposition [x y]T and heading angle θ of the mobile robot, theessence of the VFO lies in orienting the mobile robot along anadjusted direction θa, and pushing it along this direction withadjusted velocity va, to eventually reach the reference position[xr yr]

T with a reference orientation θr. Thus, the differential-drive nature of the mobile robot model is effectively reducedto that of a unicycle.

The reference trajectory is constructed such that the non-holonomic condition is satisfied

xr = vr cos θr (10)yr = vr sin θr (11)θr = ωr. (12)

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The proper selection of vr and ωr creates a variety ofreference trajectories. Also, this paper only considers forwardmovement. Thus, vr is a positive real value. However, ωr cantake a positive or negative real value. The VFO control isused for trajectory tracking control. The algorithm calculatesthe adjusted linear velocity, va, and heading angle, θa, fromthe position error.

Denote the position error variables as ex(t) = xr(t)− x(t)and ey(t) = yr(t) − y(t). The adjusted linear velocity andheading angle are obtained as the following

va(t) = hx(t) cos θ + hy(t) sin θ (13)θa(t) = atan2c

(hy(t), hx(t)

), (14)

where

hx(t) = kex(t) + xr(t) (15)hy(t) = key(t) + yr(t), (16)

where k > 0, and h(t) = [hx(t) hy(t)]T is the convergencevector field, which can be interpreted as the desired linearvelocity of the mobile robot. Note that q(θ) = [cos θ sin θ]T

interprets the instantaneous heading of the mobile robot. Thus,the adjusted linear velocity va(t) is defined as the projectionof h(t) on q(θ). This ensures that the mobile robot is pushedonly in proportion to the extent of collinearity between θand θa(t). Moreover, the function atan2c (a, b) is the fourquadrant arctangent of a and b, which is implemented suchthat θa(t) provides a continuous and differentiable curve andis not wrapped between [−π, π]. For more information pleasesee the work of Michalek and Kozlowski [28].

For situations where the time-delay is negligible, one canobtain the adjusted angular velocity as the following

ωa =KP,θeθ(t) +KI,θ

∫ t

0

eθ(η)dη + θa(t), (17)

where KP,θ > 0 and kI,θ > 0 are control gains, eθ(t) =θa(t)− θ(t), and

θa(t) =hy(t)hx(t)− hy(t)hx(t)

h2x(t) + h2y(t), (18)

where

hx(t) = k(xr(t)− va cos θ

)+ xr(t) (19)

hy(t) = k(yr(t)− va sin θ

)+ yr(t). (20)

The integral action is added to (17) to further reduce thetracking error in steady-state. Using (5) and (6), the adjustedwheel velocity values for the servo-system control loop arefound using the following relationships

vR,a = va +d

2ωa (21)

vL,a = va −d

2ωa. (22)

Since the wheel voltage is limited, the obtained values of vR,aand vL,a are scaled to avoid control saturation

vi,sc =

vi,a/µ if µ > 1vi,a if otherwise , i = R,L, (23)

where µ = max (|uR|, |uL|) /umax, and vi,sc is the scaledwheel voltage, for i = R,L.

The success of the VFO heavily depends on the performanceof the angle tracking and servo-system. If the servo-systemand angle tracking are not enough fast and accurate, theVFO performance deteriorates, and the trajectory tracking mayfail. Since time-delay limits the system’s response time, oneneeds to compensate for the effect of time-delay to achievea desirable trajectory tracking performance. Thus, the nextsection presents a two-layer predictor that guarantees enoughfast transient responses.

III. TWO-LAYER PREDICTOR

The robot is driven using DC motors and linear gear-boxes,and the wheel slip is negligible. Moreover, reasons such ascommunication lags and actuator or sensor properties maycause time-delay in the servo-system. Here, a constant time-delay is considered in the input of the servo-system. Thus,each wheel is modeled as (4), which is a combination of atransfer function and constant time-delay. If the time-delayis comparable to the response time of the servo-system, oneneeds to compensate for the effect of time-delay to improvethe transient performance of the system.

As shown in Fig. 2, the Smith predictor (SP) compensatestime-delay by acting on a nominal model of the system toprovide a controlled response that is unaffected by time-delay. Furthermore, the Smith predictor compares the actualsystem output to the nominal delayed-output to eliminatedrifts and external disturbances in the system response. Sinceheading angle determines the robot’s translational motion, alayer of predictor only in the servo-system won’t achievethe best tracking precision. Thus, another layer of predictoris considered for the heading angle. The two layers shownin Fig. 2 work in tandem to achieve a desirable transientperformance for trajectory tracking.

Since the servo-system of the wheels are assumed identical,the velocity control for the two wheels are also identical.Consider the nominal model of each wheel is given as

Vi(s)

Ui(s)= G(s)e−τs, i = R,L, (24)

where Vi is the estimate of wheel velocity, and G(s) and τ arefound using mathematical modeling or system identification,which might be different from the actual values G(s) and τ .One can use (24) to obtain an estimate of the wheel velocity,Vi(s) = G(s)e−τsUi(s), where Ui(s) is the input voltage ofthe wheel, where i = R,L. Moreover, one can predict thefuture output as Vi(s)eτs. Thus, the feedback signal can becorrected as the following

Ui(s) = C(s)(Vi,sc(s)−

(Vi(s)− Vi(s) + Vi(s)e

τs))

,

(25)where C(s) is the control and Vi,sc is the scaled adjustedvelocity obtained from the VFO, where i = R,L. Using (24),an implementable realization of the (25) is obtained as

Ui(s) = C(s)

(Vi,sc(s)−

(Vi(s) + Z(s)Ui(s)

)), i = R,L,

(26)

4

va

θa

θa + Cθ(s) +

Scaling

+ C(s)

Robot

+ C(s)

G(s) e−τs +

+

G(s) e−τs +

+

Gθ(s)

e−τs

++

ωa

VR,sc uR

VL,sc uL

−

−

−

−

vL

vR

−

−

θ

SP forangle loop

SP forleft wheel

SP forright wheel

Servo-system

Fig. 2. Block diagram of the proposed two-layer predictor. The innermost loops (highlighted in red and yellow) compensate for the time-delay in theservo-system, and the outer-loop (highlighted in green) compensates the transferred time-delay to the heading angle control from the servo-system.

whereZ(s) = G(s)− G(s)e−τs. (27)

The controller with the Smith predictor can be described as

Csp(s) =C(s)

1 + C(s)Z(s). (28)

Hence, the closed-loop transfer function of the servo-systembecomes

Gcl =Csp(s)G(s)e−τs

1 + Csp(s)G(s)e−τs

=C(s)G(s)e−τs

1+C(s)G(s)−C(s)G(s)e−τs + C(s)G(s)e−τs. (29)

For the ideal case where the nominal model and delay areperfectly known, i.e., G(s) = G(s) and τ = τ , the closed-looptransfer function of the servo-system simplifies to

Vi(s)

Vi,sc= Gv,cl(s)e

−τs, (30)

where

Gv,cl(s) =C(s)G(s)

1 + C(s)G(s)(31)

Therefore, the Smith predictor removes the effect of time-delayon the closed-loop poles of the servo-system and moves thetime-delay to outside of the servo-system loop. For a detailedlook inside the Smith predictor, please see [23], [24].

The first layer of the Smith predictor transfers time-delay toangle and position loops. Achieving proper orientation is keyfor the success of the VFO. Thus, the effect of transferredtime-delay on the angle control needs to be compensated.Assume that the wheel voltages are not saturated, vi,sc = vi,afor i = R,L. Using (6) and (30), one obtains

Ω(s)

Ωa(s)= Gv,cle

−τs, (32)

where Ωa(s) = L(ωa), where (21) and (22) give

ωa =vR,a − vL,a

d. (33)

As mentioned earlier, the trajectory tracking performancedepends on the servo-system and the angle control to a moreconsiderable extent. Furthermore, the VFO convergence rateis limited by the servo-system and angle control convergencerate. In other words, the inner-loop control must achieve anadequately fast transient response in comparison with theVFO. Hence, another Smith predictor is designed for the anglecontrol loop.

Using equation (3) and (32), one obtains

Θ(s)

Ωa(s)= Gθe

−τs, Gθ = Gv,cl/s, (34)

where Θ(s) = L(θ(t)). Thus, the angle control with the Smithpredictor correction is implemented as the following

Ωa(s) = Cθ(s)(

Θa(s)−(Θ(s) + Zθ(s)Ωa(s)

)), (35)

whereZθ(s) = Gθ(s)− Gθ(s)e−τs, (36)

where Gθ(s) = Gv,cl/s, where

Gv,cl(s) =C(s)G(s)

1 + C(s)G(s). (37)

The block diagram of the two-layer predictor is shown inFig. 2. The adjusted values of, va, θa, and θa, are produced bythe VFO control. The angle control and servo-system controlis shown as Cθ(s) and C(s), respectively.

Remark 1: The time-delay determines the response time ofthe servo-system and the angle control. Moreover, the VFOis required to act slower than the inner-loops, including theangle control and the servo-system. In other words, the inner-loops need to settle down long before the VFO settles down.Thus, the response time of the VFO will be enough larger

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than the sample-time. Therefore, an additional predictor forthe VFO may neither be an appropriate design nor can lead toa noticeable improvement in trajectory tracking performance.

IV. SAFE HEADING ANGLE AND COLLISION AVOIDANCE

Popular obstacle avoidance control techniques such as arti-ficial potential fields, collision cones, and receding horizonoptimization tie the control law to the obstacle properties.Hence, any changes in the number, shape, and distribution ofthe obstacles lead to control recalculation, which complicatescontrol design and prolongs the control calibration. Thus, inthis paper, exponential barrier certificates are used to isolatethe safety-control from obstacle properties. The proposedalgorithm does not interfere with the trajectory tracking con-trol, simplifies the collision avoidance control design, andminimizes the processing power required to implement thesafety-control.

Consider the following system

X = F (X), X ∈ X ⊆ Rn, (38)

where F (X) is smooth enough. A set of initial conditionsX0 ∈ X and a set of unsafe states Xu ⊂ X are given. Thesafety is achieved if all the state trajectories initiated insideX0 avoid the unsafe set for all t > 0. The following lemma isintroduced to use barrier certificates to obtain the safe limitsof the robot’s heading angle.

Lemma 1 (Exponential Safety Condition [16]): Considerthe system (38) and the corresponding sets X ,X0, and Xu.For any given α ∈ R, if there exists a barrier certificate,i.e., a continuously differentiable function B(X) : X → Rsatisfying the following conditions:

∀X ∈ X0 : B(X) ≤ 0 (39)∀X ∈ Xu : B(X) > 0 (40)∀X ∈ X : (∂B(X)/∂X) f(X) ≤ −αB(X) (41)

then the safety property is satisfied by the system (38), i.e.,B(X(t)) ≤ 0 for all t > 0.

Note that B(X) = 0 shows the avoidance boundary. Also,for α > 0, the system can be steered very close to theavoidance boundary without violating the safety condition.A negative or zero value of α creates a repelling avoidanceboundary leading to a conservative control design, which is notsuitable to augment with the trajectory tracking control. Thus,throughout this paper, positive values of α are considered.

Consider an avoidance boundary modeled as a barrier cer-tificate function B(x, y) satisfying conditions (39)–(41). Notethat the barrier certificate’s choice is optional, and multipleobstacles with different shapes could be modeled using asingle barrier certificate. Denote the gradient and Hessian ofthe barrier certificate as

g =

[∂B(x, y)

∂x

∂B(x, y)

∂y

](42)

H =

[∂2B(x,y)∂x2

∂2B(x,y)∂x∂y

∂2B(x,y)∂x∂y

∂2B(x,y)∂y2

]. (43)

Using the kinematic equations (1) and (2), the following isobtained using (41)

g1x+ g2y ≤ −αB(x, y) (44)g1v cos θ + g2v sin θ ≤ −αB(x, y). (45)

Since v > 0, inequality (45) can be written as

g1 cos θ + g2 sin θ≤−αB(x, y)

v(46)

g1‖g‖

cos θ +g2‖g‖

sin θ < c, (47)

where c = −αB(x, y)/(v‖g‖), and ‖·‖ denotes the Euclideannorm.

Denote

cosβ =g1‖g‖

(48)

sinβ =g2‖g‖

. (49)

Thus, inequality (47) gives the angle condition

cos(θ − β) ≤ c. (50)

Note that c > 0 outside the avoidance region because v > 0and B(x, y) < 0. Moreover, if c > 1, the angle condition (50)gives

cos (θ − β) ≤ 1, (51)

which is true for any value of θ. Thus, the robot heading angleis not restricted, and the robot can safely track the referencetrajectory. On the other hand, if c < 1, there is a range ofsafe heading angles prohibiting the robot from entering theavoidance region.

To calculate the safe limits of the heading angle, first, angleδ is introduced such that

cos δ = c. (52)

Recall that c > 0 outside the avoidance region. Thus, it isobtained that 0 ≤ δ ≤ π/2. Hence, one can calculate δ asshown in the following

δ = arccos (c) . (53)

Therefore, the angel condition (50) gives

cos (θ − β) ≤ cos δ (54)

The safety result about the heading angle is summarized inthe following proposition.

Proposition 1: Consider the kinematic equation of the mo-bile robot is given as (1)–(3), where the linear velocity ispositive. Assume obstacles are described as a barrier certificateB(x, y), where B(x, y) > 0 inside the avoidance zone ofobstacles, and B(x0, y0) ≤ 0, where [x0 y0]T is the initialposition of the mobile robot. There exist a positive α and aset of angular velocities such that the heading angle satisfies(54) for all t > 0. Then, the mobile robot does not enter theavoidance zone of any obstacle.

Condition (54) gives the unsafe range of heading angle as2k′π−δ ≤ θ−β ≤ 2k′π+δ for k′ = 0,±1,±2, · · · . However,the heading angle cannot change abruptly. Therefore, one can

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neglect non-zero values of k′, and express the unsafe range ofthe heading angle as the following set

Θu = θ ∈ R : −δ ≤ θ − β ≤ δ= θ ∈ R : β − δ ≤ θ ≤ β + δ (55)

Since β can take any real values, applying the inversetrigonometric functions to (48) and (49) is not a viable methodto calculate β in real-world applications. Here, a dynamicupdate law is provided to ensure robust calculation. Takingthe derivative of both sides of tanβ = g2/g1 gives

β(1 + tan2 β

)=g2g1 − g1g2

g21(56)

β

(1 +

g22g21

)=g2g1 − g1g2

g21(57)

β =g2g1 − g1g2‖g‖2

. (58)

The barrier certificate function only depends on the robot’sposition. Thus, g1 and g2 depend on x and y. Also, recall thatx and y are expressed as (1) and (2), respectively. Hence, g1and g2 depend on x, y, v, and θ. Therefore, it is obtainedthat g1 = φ1(x, y, v, θ) and g2 = φ2(x, y, v, θ), where

φ1(x, y, v, θ) =H11v cos θ +H12v sin θ (59)φ2(x, y, v, θ) =H21v cos θ +H22v sin θ (60)

Thus, (58) can be rewritten as

β =vφ(x, y, v, θ)

‖g‖2, (61)

where

φ(x, y, v, θ) = g1(H21 cos θ +H22 sin θ)−− g2(H11 cos θ +H12 sin θ). (62)

Recall that β is calculated for c < 1. Thus, the initial valueof β is reset to the value of the heading angle at the momentwhere c < 1 for the first time.

Since the heading angle must avoid the set Θu, the adjustedangle, θa, obtained from the VFO is further modified to avoidthe unsafe set. The robot can either turn left or right toavoid the obstacles. Here, left turn avoidance maneuver isconsidered. Thus, the θa is modified as the following

θs =

θa if θa /∈ Θu

β + δ if θa ∈ Θu, (63)

where θs is the safe heading angle. Right turn avoidancemaneuver is obtained using the following logic

θs =

θa if θa /∈ Θu

β − δ if θa ∈ Θu. (64)

The low and high limits of the unsafe heading angle changedepending on the robot’s location and translational velocity.Thus, the values of β and δ are calculated in real-time.Moreover, the angle control requires the derivative of the safeangle. A high-pass filter is utilized to create an estimate of θs

z =s

Ts+ 1θs, (65)

(a)

Groundcontrolstation

Wirelessrouter

Motioncapturesystem

USB connection

Ethernetconnection

Mobile robot

(b)

Fig. 3. (a) Components of the experimental testbed. (b) Quanser’s QBot 2eused in the experiments [30]. The white spheres are passive markers.

TABLE ITECHNICAL PARAMETERS OF THE MOBILE ROBOT [30]

Parameter Value

Diameter 35 cmTotal mass 3.82 kgMaximum translational velocity 70 cm/sMaximum rotational velocity 180 /sDistance between right and left wheels 23.50 cmEncoder 11.73 ticks/mm

where T is small enough in comparison to the angle controlresponse time.

If the robot’s heading angle falls inside the unsafe anglerange, the safety control is activated. Then, the safe angle,θs, and its estimated time derivative, z, replace the adjustedheading angle, θa, and its time derivative, θa. Therefore,the adjusted rotational speed, ωa, is accordingly modified.However, the adjusted linear velocity, va, is calculated usingthe position error. The produced values by the VFO for va arenot reliable during avoidance maneuver. Thus, the adjustedlinear velocity is replaced by the reference linear velocity, vr,during the avoidance maneuver, i.e.,

vs =

va if θa /∈ Θu

vr if θa ∈ Θu, (66)

The details of the conducted experiments are explained inthe next section. It is shown that the combination of the VFO,two-layer predictor, and safety-control can achieve adequatelyprecise trajectory tracking performance with guaranteed safetybehavior.

V. EXPERIMENTAL RESULTS

Extensive experiments are carried out to verify the effective-ness of the proposed safety-control. As shown in Fig. 3, theexperimental testbed includes a ground control station, whichis used to code, compile, and download the executable files tothe mobile robot. The ground control station also acts as a dataacquisition system. The position and orientation informationis acquired using a motion capture system, which comprises

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eight Flex 13 infrared cameras. The cameras are connected tothe ground control station through two USB hubs. A wirelessrouter is used to communicate with the mobile robot. Themobile robot is equipped with a processing board that allowsrunning the control loops locally. The servo-system of eachwheel is comprised of a DC motor and a gearbox, whichamplifies the produced torque of the DC motor. The velocity ofeach wheel is numerically derived from the angular positionsof the respective axle, that are measured using encoders.Technical information of the mobile robot is given in Table I.

As shown in Fig. 3(b), the robot has six passive markerswhich allow the motion capture system to measure the robot’sposition and orientation in the operating environment. Thelinear and rotational velocities are produced using the wheelvelocities and from (5) and (6). The control sample rate is setto 1 ms throughout this paper.

System identification is carried out to obtain the dynamicequation of the vehicle. The results show that the wheelsbehave independently and a linear model with time-delay canmodel each wheel. A quantitative system identification showsthat a second-order transfer function with input delay fits theestimation data with accuracy above 85%. The wheel modelfrom the DC motor voltage to the wheel linear velocity isobtained as

Vi(s)

Ui(s)=

5.94s+ 1.45

s2 + 7.40s+ 1.42e−0.50s, i = R,L, (67)

where the input and output units are V and m/s, respectively.The wheel dynamic has a pole at s = −0.20 and a zero

at s = −0.24, which causes a lengthy transient. Moreover,time-delay restricts a controller’s ability to reduce the responsetime of the system to a desirable level. Numerical simulationswere conducted a priori to design the control and initializethe control parameters, evaluate the proposed safety-controlcapabilities, and troubleshoot the implementation issues. Ingeneral, the conducted numerical simulations align with theexperiments. However, modeling error and system uncertaintycause a noticeable deviation between numerical simulationsand experimental results. Hence, the experimental setup is usedto calibrate the control parameters. The subsequent analysisonly reports the experimental results.

The experiments are presented as follows: 1) the effect ofthe two-layer predictor on the trajectory tracking is inves-tigated, 2) the safety algorithm is tested with two circularobstacles in the operating environment, and 3) the obstacleavoidance maneuver is tested for a large obstacle with non-circular avoidance zone in the operating environment.

A. First Experiment—Effect of Two-Layer Predictor

Numerous experiments are carried out to evaluate the ef-fectiveness of the two-layer predictor to improve trajectorytracking. Control calibration is done empirically. Each wheel’sservo-system is controlled using a PI control, i.e., C(s) =2 + 1/s. Moreover, the angle control is also designed as aPI, i.e., Cθ(s) = 0.6 + 0.1/s. The integral action noticeablyreduces the steady-state tracking error.

First, a circular reference trajectory is generated as xr =R sin(ωrt), yr = −R cos(ωrt), where R = 1 m, and ωr =

-150 -100 -50 0 50 100 150

-200

-150

-100

-50

0

50

100

Fig. 4. Trajectory-tracking of a circular reference path. Experimentalperformance with (in solid red) and without (in dash-dotted blue) a two-layerSmith predictor.

0 10 20 30 40

0

20

40

60(a)

w/o two-layer SP

w/ two-layer SP

0 10 20 30 40

Time (s)

-80

-60

-40

-20

0

20

40(b)

w/o two-layer SP

w/ two-layer SP

Fig. 5. Trajectory-tracking of a circular reference path. Variation of (a)position error and (b) angle error versus time, with (in solid red) andwithout (in dash-dotted blue) the two-layer Smith predictor. With the two-layer predictor, the transient is passed in less than 7 s.

2π/20 rad/s. The VFO without the two-layer predictor isunstable with C(s) = 2 + 1/s. Thus, the servo-system controlis modified as C(s) = 0.5+0.1/s when the two-layer predictoris not present. The initial condition is set to x(0) = 0.05 m,y(0) = −1.50 m, and θ(0) = −3. The effect of the two-layer Smith predictor (SP) is shown in Fig. 4 and 5. Despitethe large initial error, the proposed algorithm brings the robotto steady-sate in less than 7 s, which improves the convergencetime of the VFO by a factor of four. Contour error is definedas the closest distance from the actual position to the referencecurve, directly measuring the tracking precision. The RMS andaverage value of the steady-state contour error for the proposedalgorithm are 1.69 cm and 1.57 cm, respectively.

Moreover, the proposed method is tested with a figure-8reference trajectory as given by xr = ax sin(2ωrt) and yr =−ay cos(ωrt), where ax = 0.5 m and ay = 1.5 m, and ωr =2π/30 rad/s. The initial condition is set to x(0) = 0.08 m,y(0) = −1.52 m, and θ(0) = 14 . The experimental result

8

-100 0 100-200

-100

0

100

200

(a)

0 20 40 600

2

4

6

8

10

(b)

0 20 40 60

Time (s)

-50

-40

-30

-20

-10

0

(c)

Fig. 6. Performance of the proposed method for a figure-8 reference path.(a) Robot accurately tracks the reference trajectory. (b) Variation of positionerror versus time. (c) Variation of angle error versus time. The transient ispassed in less than 5 s.

-150 -100 -50 0 50 100

-150

-100

-50

0

50

100

Fig. 7. Experimental validation with two circular obstacles (in solid blue) ona reference circular path (in dashed-green). The QBot 2e consistently avoidsthe obstacles over multiple revolutions.

is shown in Fig. 6. The algorithm passes the transient in lessthan 5 s. The RMS and mean contour error are 1.28 cm and1.16 cm.

Remark 2: The PI controller has limited ability to maintainsteady-state angle error below a desirable range. If the ref-erence angle exhibits complicated behavior, the steady-stateerror may increase. For example, in the circular reference,the angle increases with a fixed ramp. Thus, as shown inFig. 5(b), the PI controller keeps the RMS value of the steady-state angle error about 4. However, the reference angle ofthe figure-8 oscillates between ±2.13 rad. Thus, as shown inFig. 6(c), the RMS value of the steady-state angle error isabout 10. Advanced angle control techniques require specialinvestigation, which is not in the scope of this work.

B. Second Experiment—Avoidance Maneuver with MultipleCircular Obstacles

The proposed algorithm can handle single or multipleobstacles in the operating environment. The control structure,including the VFO, the two-layer predictor, and the safe angleestimate, remains the same. The barrier certificate can alsobe formed using a general methodology, where the number,

0 10 20 30 40-100

0

100

200

300

400 (a)

0 10 20 30 40

Time (s)

-0.6

-0.4

-0.2

0(b)

Fig. 8. Experimental validation of safety-control of the QBot 2e with twocircular obstacles. (a) The unsafe angle range is bounded between θL andθH . The algorithm keeps the heading angle outside the unsafe angle range.(b) Negative value of B(x, y) shows that the robot stays away from theobstacles.

0 5 10 15 20 25 30 35 40

Time (s)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

BC

F

Fig. 9. Effect of parameter α on the avoidance maneuver. As α increases,the robot moves closer to the avoidance radius. Thus, B(x, y) can experiencevalues close to zero. The safety-control may fail if α is increased beyond acertain level.

position, and dimension of the obstacles can be modified arbi-trarily. For example, consider the following barrier certificatefunction (BCF)

B(x, y) = −B0 +

m∑j=1

exp(−d2j/σj

), (68)

where B0 and σj are positive constants, m is the number ofobstacles, and dj is the distance of the robot from the obstaclej calculated as

dj =

√(x− xoj)2 + (y − yoj)2, (69)

where [xoj yoj ]T is the position of obstacle j and [x y]T is

the position of the robot.An appropriate selection of B0 and σj can model arbitrary

avoidance radii for all the obstacles. For example, two ob-stacles with different avoidance radii are considered for thisexperiment. The obstacles are located at [0.85 0.85]T m and[−1.25 0]T m, where σ1 = 0.4, σ2 = 0.3, and B0 = 0.6. Thedesigner is free to choose any barrier certificate function forthe obstacle avoidance as long as the conditions (39)–(41) aresatisfied.

9

The experimental result is shown in Fig. 7 and 8. Thereference trajectory is generated as xr = R sin(ωrt), yr =−R cos(ωrt), where R = 1 m, and ωr = 2π/40 rad/s.The initial position and heading angle are x(0) = 0.07 m,y(0) = −1.48 m, and θ(0) = 3. The algorithm providesconsistent avoidance performance. The robot stays away fromthe obstacles shown as blue circles. Additional obstaclescan be included by properly modifying the barrier certificatefunction (68). The calculated range of the unsafe heading angleis shown in Fig. 8(a), where the adjusted heading angle, θa,is not allowed to take any value between θL = β − δ andθH = β + δ. In other words, the safe angle is kept outsidethe unsafe range during the experiment. As Fig. 8(b) shows,the barrier certificate function stays below zero, which provesthat safety is achieved.

The selection of parameter α in (41) affects the obstacleavoidance maneuver. Small values of α cause a conservativeobstacle avoidance, which creates a long detour around theobstacle. On the other hand, a large value of α causes thecorresponding value of δ to remain near zero, which means thesafe angle estimate is not accurate. Thus, large values of α maycause the robot to collide with the obstacle. The experimentalresult shown in Fig. 9 verifies the effect of α on the variation ofthe barrier certificate function. If one increases the value of α,the robot may narrowly evade the obstacle. Hence, the barriercertificate experiences values near zero, as shown in Fig. 9.Thus, the upper limit of α needs to be selected carefully.

C. Third Experiment—Non-Circular ObstaclesThe proposed safety algorithm can also handle non-circular

obstacles. The components of the algorithm remain the same.A modified barrier certificate is needed to model the obstacleproperly. For example, a square obstacle can be expressed as

B(x, y) = −B0 + exp

(−(x− xoσx

)2n

−(y − yoσy

)2n),

(70)where n is positive integer larger than one, B0 is a positive realnumber, σx and σy are positive real numbers which specify thedimensions of the obstacle along x- and y-axis, respectively.Note that n = 1 models an ellipse. Additional circular orsquare obstacles can be modeled by adding similar exponentialterms with appropriate values for σx, σy , and n for eachobstacle. Here, a square obstacle is considered at [0 1.2]T mwith σx = σy = 1 and α = 1. A circular reference trajectoryis generated as xr = R sin(ωrt), yr = −R cos(ωrt), whereR = 0.75 m, and ωr = 2π/40 rad/s. The initial condition isset to x(0) = −0.1 m, y(0) = −0.87 m, and θ(0) = 2. Theexperimental results are shown in Fig. 10 and 11. The headingangle is kept outside the unsafe angle range. Thus, the robotsuccessfully avoids the obstacle. When the robot is enoughfar from the obstacle, the trajectory tracking is accurate. Also,the barrier certificate function is kept below zero during theexperiment.

VI. CONCLUSIONS

A scalable and modular trajectory tracking with a two-layer predictor and integrated barrier certificates for obstacle

-100 -50 0 50

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0

20

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60

80

Fig. 10. Experimental validation with a square obstacle (in solid blue) ona reference circular path (in dashed green). The QBot 2e consistently avoidsthe obstacles over multiple revolutions.

0 10 20 30 40

0

100

200

300

(a)

0 10 20 30 40

Time (s)

-1

-0.5

0(b)

Fig. 11. Experimental validation of safety-control of the QBot 2e witha square obstacle. (a) The heading angle is kept outside the unsafe rangespecified by θL and θH . (b) The barrier certificate stays negative. Hence, therobot does not collide with the obstacle.

avoidance is proposed for the safe and agile operation ofnonholonomic mobile robots with time-delay. The two-layerpredictor dramatically improves the transient performance ofthe heading angle control and servo-system loops. Barriercertificate functions are used to maintain a safe heading angleand thus avoid obstacles. The barrier certificate can modelan arbitrary number of obstacles with different footprints. Thestructure of the proposed control is fully modular and indepen-dent of the obstacle properties. Thus, the proposed algorithmguarantees precision tracking and fast, successful avoidancemaneuvers. Hence, the control design is dramatically simpli-fied. The safety-control has been implemented in the formof an independent adaptive saturation block determining therobot’s allowable heading angle, which steers the mobile robotclear of in-path obstacles. This block does not interfere withother components of the control system and hence makes thedesign modular and compatible with a wide class of controlarchitectures. The conducted experiments verify that desirableclosed-loop performance is achieved using linear control com-

10

ponents. Future work addresses the extension of the proposedscalable and modular control architecture to provide safety-control of mobile robots in uncontrolled environments in thepresence of moving obstacles.

REFERENCES

[1] T. Peng, L. Su, R. Zhang, Z. Guan, H. Zhao, Z. Qiu, C. Zong, andH. Xu, “A new safe lane-change trajectory model and collision avoidancecontrol method for automatic driving vehicles,” Expert Systems withApplications, vol. 141, p. 112953, 2020.

[2] X. Liang, X. Qu, N. Wang, Y. Li, and R. Zhang, “Swarm control withcollision avoidance for multiple underactuated surface vehicles,” OceanEngineering, vol. 191, p. 106516, 2019.

[3] A. Behjat, S. Paul, and S. Chowdhury, “Learning reciprocal actions forcooperative collision avoidance in quadrotor unmanned aerial vehicles,”Robotics and Autonomous Systems, vol. 121, p. 103270, 2019.

[4] H. Dong, C.-Y. Weng, C. Guo, H. Yu, and I.-M. Chen, “Real-timeavoidance strategy of dynamic obstacles via half model-free detectionand tracking with 2d lidar for mobile robots,” IEEE/ASME Transactionson Mechatronics, 2020.

[5] J. Lim, S. Pyo, N. Kim, J. Lee, and J. Lee, “Obstacle magnification for2-D collision and occlusion avoidance of autonomous multirotor aerialvehicles,” IEEE/ASME Transactions on Mechatronics, vol. 25, no. 5, pp.2428–2436, 2020.

[6] Z. Pan, D. Li, K. Yang, and H. Deng, “Multi-robot obstacle avoidancebased on the improved artificial potential field and PID adaptive trackingcontrol algorithm,” Robotica, vol. 37, no. 11, pp. 1883–1903, 2019.

[7] M. Karkoub, G. Atınc, D. Stipanovic, P. Voulgaris, and A. Hwang,“Trajectory tracking control of unicycle robots with collision avoidanceand connectivity maintenance,” Journal of Intelligent & Robotic Systems,vol. 96, no. 3-4, pp. 331–343, 2019.

[8] Z. Qu, J. Wang, and C. E. Plaisted, “A new analytical solution to mobilerobot trajectory generation in the presence of moving obstacles,” IEEETransactions on Robotics, vol. 20, no. 6, pp. 978–993, 2004.

[9] J. Alonso-Mora, P. Beardsley, and R. Siegwart, “Cooperative collisionavoidance for nonholonomic robots,” IEEE Transactions on Robotics,vol. 34, no. 2, pp. 404–420, 2018.

[10] R. Fareh, M. Baziyad, M. H. Rahman, T. Rabie, and M. Bettayeb,“Investigating reduced path planning strategy for differential wheeledmobile robot,” Robotica, vol. 38, no. 2, pp. 235–255, 2020.

[11] K. Chu, M. Lee, and M. Sunwoo, “Local path planning for off-road autonomous driving with avoidance of static obstacles,” IEEETransactions on Intelligent Transportation Systems, vol. 13, no. 4, pp.1599–1616, 2012.

[12] B. Pinkovich, E. Rivlin, and H. Rotstein, “Predictive driving in anunstructured scenario using the bundle adjustment algorithm,” IEEETransactions on Control Systems Technology, 2020.

[13] X. Zhang, A. Liniger, and F. Borrelli, “Optimization-based collisionavoidance,” IEEE Transactions on Control Systems Technology, 2020.

[14] Y. Gao, C. G. Lee, and K. T. Chong, “Receding horizon tracking controlfor wheeled mobile robots with time-delay,” Journal of mechanicalscience and technology, vol. 22, no. 12, p. 2403, 2008.

[15] C.-J. Kim and D. Chwa, “Obstacle avoidance method for wheeled mobilerobots using interval type-2 fuzzy neural network,” IEEE Transactionson Fuzzy Systems, vol. 23, no. 3, pp. 677–687, 2014.

[16] H. Kong, F. He, X. Song, W. N. N. Hung, and M. Gu, “Exponential-condition-based barrier certificate generation for safety verification ofhybrid systems,” in International Conference on Computer Aided Veri-fication, 2013.

[17] M. Z. Romdlony and B. Jayawardhana, “Stabilization with guaranteedsafety using Control Lyapunov-Barrier Function,” Automatica, vol. 66,pp. 39–47, 2016.

[18] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrierfunction based quadratic programs for safety critical systems,” IEEETransactions on Automatic Control, vol. 62, pp. 3861–3876, 2017.

[19] P. Glotfelter, I. Buckley, and M. Egerstedt, “Hybrid nonsmooth barrierfunctions with applications to provably safe and composable collisionavoidance for robotic systems,” IEEE Robotics and Automation Letters,vol. 4, no. 2, pp. 1303–1310, 2019.

[20] K. Kojima, T. Oguchi, A. Alvarez-Aguirre, and H. Nijmeijer, “Predictor-based tracking control of a mobile robot with time-delays,” IFACProceedings Volumes, vol. 43, no. 14, pp. 167–172, 2010.

[21] S. Roy and I. N. Kar, “Adaptive sliding mode control of a class ofnonlinear systems with artificial delay,” Journal of the franklin Institute,vol. 354, no. 18, pp. 8156–8179, 2017.

[22] B. S. Park and S. J. Yoo, “A low-complexity tracker design for uncertainnonholonomic wheeled mobile robots with time-varying input delay atnonlinear dynamic level,” Nonlinear Dynamics, vol. 89, no. 3, pp. 1705–1717, 2017.

[23] T. G. Molnar, D. Hajdu, and T. Insperger, “The smith predictor,the modified smith predictor, and the finite spectrum assignment: Acomparative study,” in Stability, control and application of time-delaysystems. Elsevier, 2019, pp. 209–226.

[24] A. Ingimundarson and T. Hagglund, “Robust tuning procedures of dead-time compensating controllers,” Control Engineering Practice, vol. 9,no. 11, pp. 1195–1208, 2001.

[25] H. Xing, J. Ploeg, and H. Nijmeijer, “Smith predictor compensating forvehicle actuator delays in cooperative acc systems,” IEEE Transactionson Vehicular Technology, vol. 68, no. 2, pp. 1106–1115, 2018.

[26] A. Ghaffari, I. Abel, D. Ricketts, S. Lerner, and M. Krstic, “Safety verifi-cation using barrier certificates with application to double integrator withinput saturation and zero-order hold,” in American Control Conference,2018.

[27] A. Ghaffari, “Analytical design and experimental verification of ge-ofencing control for aerial applications,” IEEE/ASME Transactions onMechatronics, 2020.

[28] M. Michalek and K. Kozlowski, “Vector-field-orientation feedback con-trol method for a differentially driven vehicle,” IEEE Transactions onControl Systems Technology, vol. 18, no. 1, pp. 45–65, 2009.

[29] A. Ghaffari, “Modular safety control for mobile robots using barriercertificates and modified feedback,” in American Control Conference,2021.

[30] Quanser, “QBot 2e user manual,” Quanser, Tech. Rep., 2019.