Stability of Robotic Path Tracking 

Part I. One-Dimensional Scalar Models 

by Michael Grinfeld and Scott Schoenfeld 

ARL-TR-3646 September 2005 

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ARL-TR-3646 September 2005 

Stability of Robotic Path Tracking 

Part I. One-Dimensional Scalar Models 

Michael Grinfeld and Scott Schoenfeld 

Weapons and Materials Research Directorate, ARL 

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Stability of Robotic Path Tracking Part I. One-Dimensional Scalar Models 

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14. ABSTRACT 

Stability of robotic path tracking can be violated because of different reasons. One of those is the time lag in the control 

system. Like in many other engineering problems, it is desirable to use the regimes of exploitation of robotic systems with 

near-critical values of control parameters. When crossing boundaries of stability domains (in the space of control parameters), 

the system can deviate from the desired regime either slightly or strongly. In the former case, the boundary of the stability 

domain is called safe, in the latter—unsafe. When dealing with robotic systems working in the near-critical regimes, it is 

important to design relevant algorithms and control systems having safe boundaries only. In this report, the concept of safe 

boundaries of the stability domain is discussed regarding the time-lag driven destabilization of control systems. The first part 

of this report deals with the simplest models described by a single, first-order Ordinary Differential Equation with a retarded

argument. More complex and sophisticated models will be analyzed in the forthcoming parts of the report. We demonstrate 

that when using currently popular Pure Pursuit algorithm of robot path tracking, the stability domain has unsafe boundary. 

The problem of the boundary safety can be handled by switching to another algorithm of path tracking, which we coined as a 

“Hit-the-Road” algorithm. 

15. SUBJECT TERMS

unmanned vehicles, path planning, time lag 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. The Simplest Models of Path Tracking 2

2.1 The “Pure Pursuit” for Boating (PPB-Algorithms).........................................................3 

2.2 The “Hit-the-Road” (HR) Algorithms.............................................................................5 

3. A Delay-Free Control 5

4. Delay-Triggered Instability of Deterministic Control 8

4.1 Linear Control Without Instantaneous Controllers .........................................................8 

4.2 Numerical Analysis .......................................................................................................10

4.3 Stabilization of Path Tracking by Adding an Instantaneous Controller........................11 

5. Nonlinear Effects in Delay-Triggered Unstable Control 12

5.1 The “Pure Pursuit for Boating” Controls With Delay...................................................12 

5.2 The PPB-1 Deterministic Control With Delay..............................................................13 

5.3 The PPB-2 Deterministic Control With Delay..............................................................13 

5.4 The PPB-3 Deterministic Control With Delay..............................................................14 

5.5 How to Choose an Appropriate Look-Ahead Distance.................................................15 

6. The “Hit the Road” Deterministic Control With Delay 17

7. Conclusion 19

8. References 21

Appendix. Stability Analysis of Control Based on the Lyapunov Direct Method 23

Distribution List 25 
iv

List of Figures 

Figure 1. The geometry of robotic vehicle path tracking................................................................2 

Figure 2. Three algorithms of a “pure pursuit for boating.” ...........................................................4 

Figure 3. Trajectories for three exponential deterministic delay-free controls...............................7 

Figure 4. An UGV’s trajectory for a linear stochastic delay-free control. .....................................8 

Figure 5. A linear determistic control with delay for three rates of approaching.........................11 

Figure 6. The PPB-2 deterministic control with delay for three rates of approaching. ................14 

Figure 7. The PPB-3 deterministic control with delay. ................................................................15 

Figure 8. Towards choosing an appropriate look-ahead distance in the PPB-3 algorithm...........16 

Figure 9. Maximum total velocity as function of maximum allowable deviation in the PPB-3 

algorithm..................................................................................................................................17 

Figure 10. Three graphs of F(z) = H arctan(Γz) .........................................................................18 

Figure 11. HR deterministic control with delay for three rates of approaching. ..........................19 

Figure 12. Maximum deviation from targeted path in the Hit the Road control. .........................20 
1

1. Introduction 

Stable path tracking is one of the central problems in robotics. The topic of robotic path tracking 

includes many branches with hundreds of academic and engineering publications. Suggested 

numerical algorithms of path tracking have already become so complex and sophisticated that 

rigorous mathematical analysis of the underlying engineering ideas is hardly possible. There is, 

however, a necessity to have several mathematically manageable models which can be used as 

benchmarks for validating numerous suggested algorithms. 

Like in many other engineering problems, it is desirable to use the regimes of exploitation of 

robotic systems with near-critical values of control parameters. When crossing boundaries of 

stability domains (in the space of control parameters), the system can deviate from the desired 

regime either slightly or strongly. In the former case, the boundary of the stability domain is 

called safe, in the latter it is called unsafe. When dealing with robotic systems working in the 

near-critical regimes, it is important to design relevant algorithms and control systems having 

safe boundaries only. For the simplest models of mechanical systems, the concept of the safety 

of stability boundaries was developed in the 1940s by Russian mathematician and engineer N. N. 

Bautin. (See Andronov et al. (1) and references therein.) 

One of the sources of complexity of suggested models stems from the fact that the robotic 

vehicles are modeled as solid bodies of finite size (bicycles or cars) rather than material points. 

Obviously, the models of finite size are closer to real vehicles than their point-like counterparts. 

In many respects, point-like models cannot compete with the finite-size models. On the other 

hand, some important experimental and theoretical phenomena have minor relations to the finitesize features of the model and they show up in the point-like models as well. In these cases, the 

point-like models have a serious advantage by providing the possibility of deep theoretical 

analysis. In the first part of the report, we review certain problems of this sort. Clear 

understanding of the problems and possibilities for the point-like models can help us to shape a 

realistic picture for the potential achievements concerning the finite size models of robots and of 

their path tracking abilities. 

The simplest models of systems with delay are based on Ordinary Differential Equations 

(ODEs). These models have been intensively studied during several decades. Relating 

mathematical methods and results are summarized in (2–11). Some of these methods and 

models have been used for developing algorithms of path tracking (12–18). The simplest and 

most popular of these algorithms, called Pure Pursuit (PP), has been intensively studied in paper 

(12) and in many publications triggered by it. More recently, the PP-based algorithms faced 

serious obstacles triggered by the time-lag in the control. We discuss this instability and suggest 

another algorithm – called “Hit-the-Road” (HR) – which will hopefully be helpful to overcome 

the instability.
2

2. The Simplest Models of Path Tracking 

In what follows, Unmanned Ground Vehicles (UGVs) and their paths are modeled as material 

points and mathematical curves, respectively, in a plane. In the (, ) x z plane, let us consider a 

point P that has an assignment of moving along the x-axis with the constant speed V (see 

figure 1). We assume that the coordinates are dimensionless and are measured in appropriate 

units. For instance, it is natural to measure z in terms of a typical width of the road. 

Figure 1. The geometry of robotic vehicle path tracking. 

In fact, because of various uncontrolled disturbances, its path experiences some perturbations 

which should be kept as small as possible. Different algorithms of control have been suggested 

to keep the perturbations small enough. Let us consider the simplest control algorithm, which 

we will refer to as “normal return.” According to this algorithm, the vehicles maintain a constant 

horizontal velocity, V dx dt = / , along the x-axis, whereas its vertical velocity dz/ dt is always 

directed towards the z-axis and depends on the distance z from this axis. 

( ( )) ( ) a z t

dt

dz t = . (1) 

The function a(z) can be either linear (linear control) or nonlinear (nonlinear control). In fact, 

there is a natural time-delay τ in any control system. Therefore, instead of the ODE equation 1 

we have to deal with the differential-delay equation (DDE) 

( ( )) ( ) = a z t −τ

dt

dz t . (2) 

There is always a stochastic component in the vehicle motion that can be either big or small. 

This component can be modeled by adding a stochastic perturbation ) π (t to the RHS of 

equation 2.
3

( ( )) ( ) ( ) a z t t

dt

dz t = −τ + π . (3) 

Alternatively, certain stochastic behavior can be triggered by indeterminacy in the available 

information of the terrain or in the time-lag characteristics τ . 

The simplest deterministic control functions of interest are the two-parameter power functions 

defined as 

z sign( )z

T

a z

1 m

( ) = − , (4) 

with a positive constant T and non-negative m . 

Because of the time-lag τ , the equations similar to equation 2 demand the initial data specified 

on the whole interval (−τ ≤ t ≤ 0) rather than at a single point: 

 0 ( ) ( ) z t = Z0 t at −τ ≤ t ≤ , (5) 

where ) ( 0 z t is a given function. At zero time-lag τ = 0 the initial condition equation 5 reduces 

to the standard one: 0 z(0) = Z . 

Another nonlinear model of interest is called ATAN-model. We define it by the equation 

 ) a(z) = −H arctan(Γz , (6) 

where H and Γ are certain positive constants. 

2.1 The “Pure Pursuit” for Boating (PPB-Algorithms) 

The pure-pursuit algorithms for UGVs have been intensively studied in many publications 

(12–19). They rely on the so-called “bicycle” model of UGV. The algorithm is formulated in 

terms of an appropriate choice of the radius of curvature of the trajectory to be chosen. The key 

element of any pure-pursuit algorithm is the notion of the look-ahead distance L . In our onedimensional point-like model we dwell on several analogies of the pure-pursuit algorithms. We 

call them “Pure Pursuit for Boating” (PPB). In order to justify this name, one has to imagine a 

power boat with a motor in the rear end and the front end raised high above the sea level. In this 

situation, the keel plays secondary role. The direction and speed of propulsion of the boat is 

determined by the power of the motor and by direction of its thrust. We consider three schemes 

of PPB; they are justified in figure 2. 

The first PPB algorithm is shown schematically in the first column of figure 2. The motor 

enforces the constant total speed of V and the constant look-ahead distance L between the current 

point and the “rabbit” point on the targeted path: z = 0 . Then, simple trigonometry leads to the 

following formula of the vertical velocityVz :
4

V Vd = V

z 

Vx 

L L 

L 

V Vd z 

Vx = V

V Vd = V 

z 

Vx 

1) 2) 3) 

Figure 2. Three algorithms of a “pure pursuit for boating.” 

z

L

V V a z z = ( . (7) ) = −

The second PPB algorithm is shown schematically in the second column of figure 2. There is 

only one difference of PPB-2 from PPB-1: in PPB-2 the motor enforces the constant propulsion 

speed . V V x = Then, simple trigonometry leads to the following formula of the vertical velocity 

Vz : 

2

2

1

( )

L

z

z

L

V V a z z

−

= = − . (8) 

The third PPB algorithm is shown schematically in the third column of figure 2. In PPB-3, the 

look-ahead L is the distance between the “rabbit” and the projection of the current position 

(rather than the position itself) on the targeted x-axis. Then, simple trigonometry leads to the 

following formula of the vertical velocity Vz : 

2

2

1

( )

L

z

z

L

V V a z z

+

= = − . (9)
5

The same trigonometry leads to the following formula of the propulsion velocity Vx : 

2

2

1

L

z

V

z

L Vx Vz

+

= = . (10) 

The PPB-2 and PPB-3 algorithms appear to be nonlinear in z. For sufficiently small values of 

z, they both reduce to the linear PPB-1 algorithm. For sufficiently large ,z however, all three 

algorithms describe qualitatively different behaviors. 

2.2 The “Hit-the-Road” (HR) Algorithms 

The following suggested HR algorithms differ qualitatively from the PPB algorithms. It does not 

include anything like the look-ahead distance parameter. The main idea of HR algorithms is 

extremely simple: the propulsion velocity Vx has no relation whatsoever to the delay-driven 

destabilization of the path-tracking control. On the contrary, the transverse component of the 

velocity Vz is of key importance in what concerns the delay-driven destabilization. Thus, an 

appropriate algorithm of it should be chosen very carefully. In what follows, for this component 

we will choose one of the two-parameter sets of the ATAN functions. In other words, this sort of 

control is described by the ODE 

arctan( ) ( ) ( ) H z t

dt

dz t = − Γ (11) 

or by its delay-sensitive version 

arctan( ) ( ) ( ) = −H Γz t −τ

dt

dz t . (12) 

There are two main sources of noise in such a system. One, ( ), z τ t is in the indeterminacy of the 

z coordinate; the other, ( ), τ t ∆ is in the indeterminacy of the delay τ . The corresponding ODE 

reads 

arctan( ) ( ( )) ( ) ( ) H z t t t

dt

dz t

π π z = − Γ −τ − ∆ + . (13) 

3. A Delay-Free Control 

When dealing with the delay-free control, we arrive at the equation 

( ( )) ( ) ( ) a z t t

dt

dz t = +π . (14) 
6

Example 1: Nonstochastic control 

In this case, we arrive at the initial-value problem 

0 ( ( )), (0) ( ) a z t z Z

dt

dz t = = . (15) 

For the power-function models (equation 4), the last equation reads 

( ) () 1 , dz t m

z sign z dt T = −

or else 

() 1 at 0

() 1 ( ) at 0

m

m

dz t

z z

dt T

dz t

z z

dt T

⎧ =− > ⎪⎪

⎨

⎪ = −− < ⎪⎩

 . (16) 

At m = 1, the Cauchy problem for equation 16 has the following solution: 

T

t

z t Z e

−

= 0 ( . (17) )

The point returns back to its path (i.e., to the point z = 0 ) at z → ∞ . 

At 

2

1

m = , we get for 0 Z0 >

2

1

0 ( ) ⎟

⎠

⎞ ⎜

⎝

⎛ = −

T

t

z t Z . (18) 

The point returns back to its path within finite time-interval at . T Z0 t →

At m = 2, we get 

T

t Z

z

+

= −1

0

1 . (19) 

The point returns back to its path at t → ∞, but it does this slower than at . m = 1.

More generally, at , m ≠ 0 the solution reads 

( ) 1

1

1

0 0 1 1 − + −

⎥

⎦

⎤ ⎢

⎣

⎡ = + − m m

T

t

z Z m Z . (20)
7

At 0 1, < < m the point returns back at the moment 

m

Z

t T

m

return − =

−

1

(1 )

0 . (21) 

The graph in figure 3 shows numerical solutions of equation 16 for , T = 1, Z0 = 3 and 

m = 1/ 2 (series 1), m = 1 (series 2), and m = 2 (series 3). 

NO DELAY DETERMINISTIC SOLUTIONS

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15

t

z

Series1

Series2

Series3

Figure 3. Trajectories for three exponential deterministic delay-free 

controls. 

Example 2: Stochastic control at m = 1 . 

Stochastic control at m = 1 is described by the following system: 

0 ( ) ( ) , (0) ( ) 1

z t t z Z

dt T

dz t = − + π = , (22) 

that can be integrated explicitly to read 

( ) ( ) 0 ξ π ξ

ξ

∫

− −

= + T

x

T

t

z t Z e d e . (23) 

The graph in figure 4 shows the solution of equation 22 for the case , T = 1, Z0 = 3 and the 

stochastic perturbation π (t) with the amplitude 0.1. 
8

STOCHASTIC CONTROL WITHOUT DELAY

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12

Figure 4. An UGV’s trajectory for a linear stochastic delay-free control. 

4. Delay-Triggered Instability of Deterministic Control 

4.1 Linear Control Without Instantaneous Controllers 

A linear stochastic-free control is described by the system 

() 1 ( ) at 0 dz t

zt t

dt T = −− ≥ τ

0 zt Z t ( ) at 0 = −<< τ . (24) 

At 0 Z0 = , the system (equation 24) has the solution z = 0 , usually called trivial solution. The 

stability concerns the behavior of the solutions at 0 Z0 ≠ . If, for sufficiently small initial values 

of 0 | | Z0 ≠ , the corresponding solutions ) z(t are close to the trivial solution on the whole semiinfinite interval −τ < t < ∞ , then, the trivial solution is called stable. Otherwise, the trivial 

solution is called unstable. 

In the absence of delay, the linear system ) ( )/ ( 1 dz t dt T z t − = − has the general solution 

t T

z Ae− / = with the stable trivial solution . z = 0 The presence of delay changes the situation 

dramatically. In systems with sufficiently large delay the trivial solution 0 z = appears to be 

unstable. Let us remind the classical facts of the theory of linear systems of ODE with delay. 

They can be found in any of the monographs (2–11). 

Let us consider solutions of equation 24 of the form 

pz z( , (25) t) = Ae
9

where A and p are certain constants. Combining equations 24 and 25, we arrive at the following 

transcendental equation for the exponent : p

 0 + = − p Tp e τ . (26) 

Generally speaking, the roots p of the characteristic equation 26 are complex numbers 

depending on the parameters T and τ : p = p(T,τ ). The function p = p(T,τ ), generally 

speaking, has infinitely many branches. If one those roots has a positive real part, the solution 

equation 25 grows exponentially and the trivial solution z = 0 appears to be unstable. 

Appearance of the roots with destabilizing real parts depends of the parameters T and τ . It is 

important to find in the parameter space ) (T,τ the domains corresponding to the unstable 

behavior of the trivial solution, as well as boundaries of these domains. The boundary of the 

stability domain is defined by the equation 

Re p(T,τ ) = 0 . (27) 

Equation 26 has infinitely many roots which cannot be calculated explicitly. In almost all 

practically sound engineering problems, the transcendental characteristic equations are even 

harder to solve. There is no chance whatsoever to solve them explicitly. Fortunately, it is much 

easier to make conclusions about the appearance of the unstable roots of the characteristic 

equations than to solve those equations explicitly. Some rather general mathematical results and 

engineering procedures in this direction have been established by Nyquist, Neiman, and 

Chebotarev (3, 5–7). For our transcendental equation 26, the boundary of the stability domain 

can be found by rather elementary calculations without any complex algebraic analysis. Using 

notation 

τp = u, , (28) τ /T = γ

we can rewrite the original equation 26 as 

 0 + = −u u γe . (29) 

The classical equation 29 has been analyzed in hundreds of papers and several monographs 

including those of Pinney (2), Bellman and Cooke (3), Kolmanovskii and Myshkis (7), and 

others. 

Looking for complex solutions of equation 29 in the form , u x iy = + we reduce this complex 

equation to the following pair of the real ones: 

 0 + cos = 0 , − sin = − − x e y y e y x x γ γ . (30) 

Let us find the boundary of the stability domain of the equation 29. According to the equation 

27, the boundary can be determined from the condition 
10

x = 0 , (31) 

allowing to rewrite the system equation 30 as follows: 

 0 γ cos y = 0 , y − γ sin y = . (32) 

The system equation 32 allows one to find infinitely many (countable) number of critical values 

n γ of the single parameterγ : 

n πn

π

γ 2

2 = + , (33) 

where n is an arbitrary (positive or negative) digit. Corresponding critical imaginary parts are 

equal to 

yn πn

π 2

2 = ± + . (34) 

Coming back to the original variables ( , ), T τ we arrive at the following conditions of neutral 

stability: 

n

T n

π

τ π 2

2 ⎟ = + ⎠

⎞ ⎜

⎝

⎛ . (35) 

By its physical meaning, only positive values of τ /T are of interest. The smallest of it is the 

following: 

2

τ π ⎟ = ⎠

⎞ ⎜

⎝

⎛

T crit

. (36) 

The criterion equation 36 implies the following strategy of stabilization of control: we have 

either to decrease the time-lag τ or, alternatively, to increase the constant T . Both approaches 

are in full agreement with engineering solutions in various applications, including robotic path 

tracking (12–17). 

4.2 Numerical Analysis 

The three graphs of the figure 5 show the trajectories of the controlling system obtained 

numerically for three different rates of approaching the desired position at z = 0 . In principle, 

the results visualized in the figure could be obtained analytically for this simplest case. The 

possibility of the full theoretical analysis is rather an exception than the rule, especially when 

dealing with nonlinear control. Theoretical consideration, similar to that previously given, 

almost always permits analysis of the bifurcation and of the initial (linear) fraction of the 

unstable trajectory only. Further analysis of the unstable trajectory seems impossible without 

numerical methods.
11

LINEAR DETERMINISTIC CONTROL WITH DELAY

-20

-10

0

10

20

0 5 10 15

Series1

Series2

Series3

Figure 5. A linear determistic control with delay for three rates of approaching. 

The rate of approach for all three shown graphs is the same: 0.2 1 = − T . Also, the initial 

deviation for all three graphs is the same: 3 z0 = . The only difference is in the choice of the 

delayτ . For the three series shown, it was chosen as 0.2 τ 1 = , 0.3 τ 2 = , and 5 0. τ 3 = , 

respectively. Per the stability criterion equation 36, for the first two regimes the solution z = 0

is stable, and for the last one it is unstable. The graphs clearly demonstrate these theoretical 

facts. 

4.3 Stabilization of Path Tracking by Adding an Instantaneous Controller 

Consider the following generalization of the ODE control described by the system equation 24: 

( ) ( ) ( ) at 0 dz t Az t Bz t t

dt = +− ≥ τ

0 zt Z t ( ) at 0 = −<< τ . (37) 

Constant A defines the instantaneous component of control, whereas B defines the delay 

component of control. At A = 0 , we arrive at the situation analyzed earlier. 

When looking for exponential solutions of the equation 37 instead of equation 24, then, instead 

of equation 25, we get 

 0 − − = − p Be A τ p . (38) 

Following the procedure of the previous section, instead of equation 30 we arrive at the 

following system of algebraic equations: 

 0 − cos − = 0 , − sin = − − x B e y A y Be y τx τx γ . (39) 
12

Let us find those An and Bn for which roots pn appear to be purely imaginary. In this case, we 

get the following equations defining the boundary of neutral stability: 

cos = − , − sin = 0 n n n

n

n

n y B y B

A

y , (40) 

implying 

n

B

A

yn arccos ⎟ + 2π ⎠

⎞ ⎜

⎝

⎛ ± = ± − . (41) 

The system equation 40 has no real solutions provided A/ B > 1, and it does have real solutions 

otherwise. In other words, the delay-driven instability can be suppressed by adding to the 

control a sufficiently strong instantaneous component. More than that, given the amplitude B of 

the “delayed” ingredient of the controller, we can choose the appropriate amplitude A of the 

“instantaneous” component of controller which is able to suppress the instability with respect to 

any time delay. 

5. Nonlinear Effects in Delay-Triggered Unstable Control 

5.1 The “Pure Pursuit for Boating” Controls With Delay 

Below we describe three different algorithms of pure pursuit which we call “Pure Pursuit for 

Boating” or PPB algorithms. From one hand, these algorithms resemble algorithms implemented 

earlier for UGV in several publications (12–16). On the other hand, they resemble control of a 

power boat when nose of the boat raised highly out of water and the motor dictates both: the 

direction of propulsion and the speed. Another analogy of the vehicle would be a circus 

“monocycle” with one wheel (as opposed to a standard bicycle with two wheels). The PPB-1 

algorithm is linear, the PPB-2 and PPB-3 algorithms are nonlinear. In the linear approximation, 

however, all three of them are described by the same linear equation 

( ) ( ) = − z t −τ

L

V

dt

dz t , (42) 

where V is the vehicle speed and L is the so-called look-ahead distance. The stability criterion 

equation 36, then, dictates the constraint 

2

τ π ≤

L

V . (43) 

or 
13

τ

π

2 max

L V = . (44) 

Thus, the only ways to increase the maximum speed of stable propulsion are to decrease the 

time-lag τ or increase the look-ahead distance L . Both ways have their natural limitations. The 

time-lag is dictated by various engineering constraints. Choice of appropriate look-ahead 

distance is limited by the spatial scale of curvature of the preplanned path. These are conceptual 

limitations of various versions of the pure pursuit algorithms. To avoid these limitations, we 

suggest a conceptually different algorithm called “Hit the Road” or HR algorithm. This 

algorithm does not use the concept of lookahead distance, and, in principle, it permits a stable 

path tracking with arbitrary propulsion speed V.

5.2 The PPB-1 Deterministic Control With Delay 

The PPB-1 control is described by the linear function equation 7: ( ) z

V V az z

L = =− . Thus, the 

full theoretical analysis of the PPB-1 control with delay reduces to the analysis of the linear 

equation 24. Given the delay, this control appears to be stable for sufficiently large look-ahead 

distance L or for sufficiently small velocity V. Otherwise, the targeted solution 0 z(t) =

appears to be unstable. In this case, the actual trajectory resembles the Series-3 of figure 5. 

However, not the entire curve-3 describes the unstable PPB-1 solution but only its initial 

fraction. This fraction covers the interval * 0 , < t t < where *

t is the moment of time at which the 

Series-3 crosses one of the horizontal lines z L = ± . Obviously, at * t > t the PPT-1 algorithm 

fails. 

5.3 The PPB-2 Deterministic Control With Delay 

The PPB-2 control with delay is described by the function equation 8: 

L

V

L

z V a z z z

1/ 2

2

2

( ) 1

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = = − − . For small deviation from the targeted solution z = 0 , the 

nonlinear PPB-2 control reduces to the linear control PPB-1. Hence, the stability criteria for the 

two controls are the same. Contrary to the PPB-1 control, the analysis of unstable regimes for 

the PPB-2 requires numerical methods. Figure 6 shows three different trajectories for the PPB-2. 

All these trajectories start at the same initial position 01 0. z = z0 = , they have the same delay 

factor τ = 0.32 and all of them have the same value of the parameter A = V / L = 5 . For these 

parameters, the solution of the linear approximation is unstable. 

The difference in the three graphs is implied by the choice of the parameter 2 h ≡ 1/ L . For the 

red Series-1, the value of h is equal to 1. This solution disappears at t ≈ 1.22 (see the red 

vertical line). For the Series-2, the value of h is equal 0.1. The solution disappears at a later 

moment of time (this moment is greater than 15, and it is not visible in the graph). At last, 
14

A "PURE PURSUIT"-2 CONTROL WITH DELAY

-4

-2

0

2

4

0 5 10 15

t

z

Series1

Series2

Series3

Figure 6. The PPB-2 deterministic control with delay for three rates of approaching. 

for the Series-3, the value of h is equal to 0.01. Surprisingly enough, this solution seems to be 

stable. We do not have any justification for this nonlinearity-driven stabilization of the linearly 

unstable solution. 

5.4 The PPB-3 Deterministic Control With Delay 

The PPB-3 control with delay is described by the function equation 9: 

L

V

L

z V a z z z

1/ 2

2

2

( ) 1

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = = − + . Again, all these trajectories start at the same initial position 

z = z0 = 0.01, they have the same delay factor τ = 0.4 , and all of them have the same value of 

the parameter A = V / L = 5 . The three series shown correspond the values h = 1, 0.1, 0.01. All 

these solutions remain finite. The ratio V V z L z h ρ ≡ z / x = / = diminishes when h decreases. 

Still, it can be of a considerable magnitude. This means that a considerable loss of time and 

power can be implied by the PPB-3 control. 

Let us consider regimes in which the UGV has the speed 

τ

π ε

2

( )L V + = slightly exceeding the 

maximum stable speed 

τ

π

2 max

L V = ( 0 < ε << 1). In this case, we get 

1/ 2

2

2

1

2 ( )

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

+ = = − L

z V a z z z τ

π ε . In this situation, the delay-triggered instability shows up for 

any look-ahead distance L . Figure 7 demonstrates that the PPB-3 control implies appearance of 

oscillations of finite amplitude.
15

A "PURE PURSUIT"-3 CONTROL WITH DELAY

-10

-5

0

5

10

0 10 20 30 40

t

z

Series1

Series2

Series3

Figure 7. The PPB-3 deterministic control with delay. 

5.5 How to Choose an Appropriate Look-Ahead Distance 

The PPB-3 control with delay is closest of all three PPB to the traditional Pure Pursuit control for 

UGV. We can see that the PPB-3-control keeps the vehicle on a finite distance from the desired 

path after small accidental excess of the critical speed. We remind that destabilization of PPB-1 

control leads to an unlimited growth of the deviation at t → ∞ . PPB-2control leads to an 

unlimited growth at t approaching certain (finite) moment of time. In fact, if the boundary of 

the stability region was crossed, both PPB-1 and PPB-2 controls fail after finite time interval: 

this happens because deviation from the pre-assigned path exceeds eventually the look-ahead 

distance. Thus, the stability boundary for the PPB-3 control is obviously safer than for PPB-1 

and PPB-2 controls. On the other hand, the stability boundary for the PPB-3 control is still 

unsafe. For safety, small spontaneous deviation should imply arbitrarily small, but not finite, 

deviations of the robot in the interval t ≤ ∞ . This demand is certainly violated for the PPB-3 

control. 

To characterize this situation quantitatively, let us explore the maximal deviation of the UGV 

with PPB-3 control more carefully. The main question should be formulated as follows: given 

the look-ahead distance L, what is the maximum deviation of the UGV from the pre-assigned 

path when the velocity slightly exceeds its critical value Vmax ? The goal of this section is to 

explore the dependence of the amplitude upon the look-ahead distance. The maximum deviation 

of the vehicle from the desired path cannot be made arbitrarily small no matter how small the 

excess of the critical speed. 
16

This problem can be addressed by means of direct numerical experiments based on numerical 

integration of the nonlinear equation 

1/ 2

2

2

( ) ( ) 1

2

( ) −

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − − +

+ = − L

z t

z t

dt

dz t τ

τ

τ

π ε . (45) 

How do we choose an appropriate or optimal look-ahead distance? First, we have to ensure that 

the UGV should be able to move as fast as possible. But there is the time-lag stability limitation. 

According to the formula (5b) – π / 2τ Vmax = L , the bigger the desirable propulsion speed the 

bigger should be the look-ahead distance L . However, when the UGV incidentally (or 

deliberately) exceeds Vmax the delay-induced instability implied considerable deviation from the 

desired path. When destabilization occurs that the vehicle moves along an oscillatory path with 

growing amplitude and a specific frequency. We have a conjecture that the value of the 

frequency can be used as indicator of occurrence of the delay-triggered instability. This 

indicator can be used for subsequent adjustment of path-tracking parameters. It seems crucially 

important though to ensure that, until the adjustment takes place, the vehicle will not deviate too 

far from the pre-assigned path. The graphs on figure 3 suggest that the maximum instabilityinduced deviations grow with growth of the look-ahead distance L (or equivalently of Vmax ). In 

other words, the bigger the look-ahead distance the more unsafe is the boundary of the stability 

domain in the parameter space. The implication of this discussion is very simple. The maximum 

look-ahead distance L should be chosen based on the maximum allowable deviation from the 

path. We determined the dependence of maximum deviation vs. look-ahead distance by making 

a series of numerical simulations. The results are shown in the figure 8. 

MAXIMUM DEVIATION VERSUS LOOKAHEAD 

DISTANCE

0

10

20

30

0 20 40 60 80 100

Look-ahead distance

Maximum deviation

Figure 8. Towards choosing an appropriate look-ahead 

distance in the PPB-3 algorithm. 
17

Figure 8 shows that for the PP-3 algorithm, the maximum deviation is approximately one third of 

the look-ahead distance. 

Figure 9 gives the maximum velocity as function of maximum allowable deviation. 

Maximum Velocity as a Function of 

Maximum Allowable Deviation

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

Maximum allowable deviation

Maximum velocity

Figure 9. Maximum total velocity as function of maximum 

allowable deviation in the PPB-3 algorithm. 

6. The “Hit the Road” Deterministic Control With Delay 

As shown in the previous section, the demand of stability of the pure-pursuit algorithms imposes 

undesirable constraints on the admissible speed of stable propulsion of an UGV. In order to 

avoid those constraints, we suggest using conceptually different algorithms. The suggested HR 

algorithm does not use the notion of look-ahead distance at all. Also, this algorithm does not 

deal with the propulsion (tangential) component of the velocity—it can be arbitrarily large. 

Instead, it deals only with the normal component of the vehicle’s velocity, and the normal 

deviation z(t) of the vehicle from the pre-assigned trajectory is the only quantity that matters. 

The HR deterministic control is a nonlinear ODE-based algorithm of control with delay. We 

suggest choosing a two-parameter function— ) H arctan(Γz —for the vehicle’s normal velocity. 

In other words, the suggested HR algorithm is described by the following first order ODE with 

delay: 
18

arctan( ( )) ( ) = − Γ −τ

∂

∂ H z t

t

z t , (46) 

where H and Γ are certain positive constants. 

Figure 10 shows the function ) H arctan(Γz for 1 H = and three different values of Γ : Γ = 0.3

(Series 1), 1 Γ = (Series 2), and Γ = 10 (Series 3). 

Function F = H arctan(Γz)

0

0.5

1

1.5

2

0 10 20 30 40 50

Series1

Series2

Series3

Figure 10. Three graphs of F(z) = H arctan(Γz) .

In linear approximation, equation 47 reads 

( ) ( ) = − Γ −τ

∂

∂ H z t

t

z t . (47) 

The stability condition for equation 47 can be obtained from equation 36. It reads 

2 ( ) π

τHΓ crit ≤ . (48) 

Figure 11 shows possible trajectories for the HR algorithm. All graphs are calculated for the 

same delay timeτ = 0.36 , the same value of the parameter 4 H = , and the same initial position 

z = z0 = 0.01. The three series shown correspond to their different values of Γ : Γ = 0.3 

(Series 1), 1 Γ = (Series 2), and Γ = 10 (Series 3). 

The first regime is obviously stable, while the other two are unstable and lead to finite amplitude 

oscillations about the pre-planned path. 
19

AN "ATAN" CONTROL WITH DELAY

-3

0

3

0 5 10 15

t

z

Series1

Series2

Series3

Figure 11. HR deterministic control with delay for three rates of approaching. 

Similar to the PPB-3 algorithm, for planning an optimal choice of the parameters Γ and H we 

have to investigate maximum deviations of the unstable paths when intersecting the stable 

domain in the space of parameters (τ ,Γ,Η) . Regimes, close to the critical one, are described by 

the following equations: 

arctan( ( )) 2

( ) τ

τ

π ε Γ − Γ+ = − ∂

∂

z t

t

z t , (49) 

or equivalently 

⎟

⎠

⎞ ⎜

⎝

⎛ − + = − ∂

∂ ( ) 2

arctan ( ) τ

τ

π ε

z t

H

H

t

z t , (50) 

where ε is a small parameter. 

The graph in figure 12 shows the dependence of maximum normal deviation from the targeted 

path as function of the parameter H . The graph was built by means of numerical solving of 

equation 50 for 3.3

2

5, = + Γ = ≡ τ

π ε A . 

7. Conclusion 

Time lag in control leads to destabilization of path-tracking algorithms when parameters of the 

algorithms leave stability domains. Boundaries of stability domains in the parameter space for 

different algorithms can be either safe or unsafe. When crossing safe boundaries, the disturbed 
20

MAXIMUM DEVIATION FOR THE "HIT THE 

ROAD" CONTROL

0

4

8

12

0 20 40 60 80 100 120

Parameter H

Maximum deviation

Figure 12. Maximum deviation from targeted path in the Hit the 

Road control. 

path appears to be close to the pre-assigned one if the parameters stay close to the boundaries. 

When crossing unsafe boundaries, the path abruptly and strongly deviates far from the preassigned path. It is highly desirable to use algorithms of path tracking with the safe boundaries 

only. The previously mentioned analysis has demonstrated that traditional Pure-Pursuit 

algorithms, based on the concept of look-ahead distance, lead to the stability domains with 

unsafe boundaries. This fact imposes sensitive constraints on the maximum allowable 

propulsion velocity of UGV. To cope with this difficulty, we suggest using a novel HR 

algorithm. This algorithm does not use the concept of look-ahead distance and does not apply 

any stability constraints on the maximum propulsion velocity. This avenue of thought requires 

further studies and various efforts—mathematical, computational, and engineering.
21

8. References 

1. Andronov, A. A.; Vitt, A. A.; Khaikin, S. E. Theory of Oscillators. Pergamon Press: 

Oxford, NY, 1966. 

2. Pinney, E. Ordinary Difference-Differential Equations. University of California Press: 

1958. 

3. Bellman, R. E.; Cooke, K. L. Differential-Difference Equations. Academic Press: New 

York, NY, 1963. 

4. Diekmann, O.; van Gils, S. A.; Verduyn-Lunel, S. M.; Walther, H. O. Delay Equations, 

Functional, Complex and Nonlinear Analysis, Applied Math, Sciences Series, 110; SpringerVerlag: New York, NY, 1995. 

5. Gu, K.; Kharitonov, V. L.; Chen, J. Stability and Robust Stability of Time-Delay Systems. 

Birkhauser: Boston, MA, 2003. 

6. Hale, J. K.; Verduyn Lunel, S. M. Introduction to Functional Differential Equations, 

Applied Math, Sciences, 99, Springer-Verlag: New York, NY, 1993. 

7. Kolmanovskii, V. B.; Myshkis, A. D. Applied Theory of Functional Differential Equations. 

Kluwer, Dordrecht: The Netherlands, 1992. 

8. Kuang, Y. Delay Differential Equations With Applications in Population Dynamics. 

Academic Press: Boston, MA, 1993. 

9. Krasovskii, N. N. Stability of Motion. Stanford University Press: 1963. 

10. MacDonald, N. Biological Delay Systems: Linear Stability Theory; Cambridge University 

Press: Cambridge, 1989. 

11. Niculescu, S. I. Delay Effects on Stability: A Robust Control Approach, LNCIS; 269; 

Springer-Verlag: Heidelberg, 2001. 

12. Amidi, O. Integrated Mobile Robot Control, M.S. Thesis, CMU, 1990. 

13. Ollero, A.; Amidi, O. Predictive Path Tracking of Mobile Robots. Applications to the CMU 

Navlab. Proceedings of the IEEE 5th International Conference on Advanced Robotics, Pisa, 

1991, pp 1081–1086. 

14. Murphy, K. N. Analysis of Robotic Vehicle Steering and Controller Delay. Proceedings of 

the 5th International Symposium on Robotics and Manufacturing, Wailea, Maui, HI, 1994.
22

15. Murphy, K.; Legowik, S. GPS Aided Retro-Traverse for Unmanned Ground Vehicles. 

SPIE 10th Annual AeroSense Symposium, Orlando, FL, 1996. 

16. Ollero, A.; Heredia, G. Stability Analysis of Mobile Robot Path Tracking. In Proceedings 

of the 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems, Part 3 of 

3. A Study of Polynomial Curvature Clothoid Paths for Motion Planning for Car-like 

Robots, Pittsburgh, PA, 1995, pp 461–466. 

17. Pivtoraiko, M. A Study of Polynomial Curvature Clothoid Paths for Motion Planning for 

Car-Like Robots; CMU-RI-TR-04-68; Robotics Institute: Carnegie Mellon University, 

December 2004. 

18. Yang, H.; Yang, S.; Mittal, G. Tracking Control of a Nonholonomic Mobile Robot by 

Integrating Feedback and Neural Dynamics Techniques. In Proceedings of the 2003 IEEE 

International Conference on Intelligent Robots and Systems, Las Vegas, NV, October 2003. 

19. Wit, J.; Crane, C. D. Autonomous Ground Vehicle Path Tracking. Journal of Robotic 

Systems 2004, 21 (8), 439–449.
23

Appendix. Stability Analysis of Control Based on the Lyapunov Direct 

Method 

The analysis of stability of control systems with delay presented in this report and based on 

consideration of a characteristic function is applicable for linear control system only. There is 

another, alternative method of stability analysis called the Second, or Direct, method of 

Lyapunov. This method is a rather heuristic one and it does not have any systematic 

methodology. On the other hand, its applicability to various systems is not as sensitive to 

linearity as the system under study as the First method based on analysis of the characteristic 

equation. However, we demonstrate this method using the linear system and following the 

monograph of Krasovskii.1

Let us consider the following time-dependent functional L{z(t)} of the trajectory ) z(t : 

{ } ( ) ( ) ( ) 2 0 2 ξ τ τ = + + ∫− L z t Iz t D d z t , (A-1) 

where I > 0 and D > 0 are certain positive constants. Treated as a function of t, the function 

L{ } z(t) is obviously positive. 

L{z(t)} > 0 if z(t) ≠ 0 . (A-2) 

This is the first key relation required for validity of the Lyapunov direct method. The second one 

is the relationship 

L{ } () z(t) < 0 if z t ≠ 0

dt

d . (A-3) 

With some additional (secondary) mathematical demands about L{z(t)}, the inequalities of 

equation A-2 and equation A-3 guarantee stability of the solution z(t) = 0 . 

Let us find the derivative { }

dt

dL z(t) along the trajectory of the system equation 37. Using the 

equation A-1, we get the following sequence of the relations: 

 1

Krasovskii, N. N. Stability of Motion. Stanford University Press: 1963. 
24

{ ( )}

( )

( )

( )

( )

0

0

2 0

0 2

2 2

1 () ( ) () ( ) 2

( ) () () ( )

( ) () () 2

() () ( ) 2

() () () ( ) 2

dL z t dz t dz t Iz t D d z t

dt dt dt

dz t Iz t Az t Bz t D d z t

dt

D dz t Iz t Az t Bz t d

dt

D Iz t Az t Bz t z t

D Iz t Az t Bz t z t z t

τ

τ

τ

τ

ξ ξ ξ

ξ τ ξξ

ξ τ ξ

τ ξ

τ τ

−

−

−

−

+

=+ +

+

= + −+ + ⎡ ⎤ ⎣ ⎦

+

= + −+ ⎡ ⎤ ⎣ ⎦

= + −+ + ⎡ ⎤ ⎣ ⎦

= + −+ − − ⎡ ⎤ ⎡ ⎣ ⎦

∫

∫

∫

⎤ ⎣ ⎦

2 2 () () ( ) ( ) 2 2

D D AI z t BIz t z t z t τ τ

⎛ ⎞ = + + −− − ⎜ ⎟ ⎝ ⎠ . (A-4) 

The second stability inequality equation A-3 will be satisfied if one can find such positive 

constants as I and D such that 

B I ≤ −(2AD + D)D 2 2 . (A-5) 

Such positive constants as I and D can be found if the “instantaneous control coefficient” A is 

negative and sufficiently large in the absolute value and the “lag control coefficient” B is 

sufficiently small in absolute value. 

The direct method of Lyapunov delivers some sufficient conditions of stability which might be 

(and quite often are) very far from the necessary conditions of stability. The success depends 

quite often on luck in finding an appropriate Lyapunov function. From that standpoint, when 

dealing with linear systems, the analysis based on the study of the corresponding characteristic 

equation is much deeper. Unfortunately, the concept of characteristic equation fails when 

dealing with nonlinear systems, whereas the direct method of Lyapunov permits direct 

generalization for nonlinear cases.
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