# Control Based on Linear Algebra for Trajectory Tracking and Positioning of Second-Order Chained Form System

Authors: Leandro Rodriguez; Emanuel Serrano; Mabel Cristina Sánchez; Gustavo Scaglia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The development of controllers for underactuated systems with nonholonomic constraints has been a topic of significant interest for many researchers in recent years. These systems are hard to control because their linearization transform them into uncontrollable systems. The proposed approaches involve the use of a permanent excitation in the reference trajectory; coordinate transformation; discontinuities; or complex calculations. This paper proposes the design of the controller of the second-order chained form system for trajectory tracking by using a simpler approach based on linear algebra. Up to the present time, no controllers based on this approach have been designed for that system. The control problem is solved by setting two of the three systems variables as a reference, while the remaining variable is calculated imposing the condition that the equations system has an exact solution to ensure that tracking errors go to zero. The stability of the proposed controller is theoretically demonstrated, and simulations results show a suitable control system performance. Also, no coordinate transformation is necessary.


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## Body

## 1. Introduction

In recent years, the interest in the control of underactuated systems with nonholonomic constraints has increased [1]. Underactuated systems are mechanical systems with fewer control inputs than degrees of freedom and nonintegrable acceleration constraints [2]. This paper considers the control for trajectory tracking and positioning of second-order chained form systems. These are hard to control because their linearization transform them into uncontrollable systems. The well-known paper of Brockett highlights the difficulty of solving these problems [3].Several articles addressed the control of underactuated systems with nonholonomic constraints. Aneke et al. [4, 5] resolved the problem of trajectory tracking by using a three-stage method. In the first step, they transformed the original system into an extended chained form system with a cascade arrangement. They applied a linear feedback controller to the first subsystem. After that, the second subsystem is exponentially stabilized by employing a backstepping procedure. In the third step, they showed the closed-loop tracking dynamics stability of the system. More recently, other authors used the backstepping procedure to develop control methodologies for trajectory tracking [6, 7]. A permanent excitation in the reference trajectory is also necessary for system stability. This implies that the signals must not converge to zero, limiting their applications.Also, in [8], smooth time-varying controllers are proposed to address the exponential tracking of second-order nonholonomic systems. These techniques use coordinate transformation based on time-varying dilation with a cascade-design strategy. In contrast to the previous approach, it does not require the excitation condition in the reference trajectory. This kind of control laws frequently involves complex calculations; their design is not intuitive and only applies for special nonholonomic systems. Another method used a discontinuous coordinate transformation to map the initial system into the space of discontinuous nonholonomic systems. In this new space, the control problem can be solved [9, 10].An adaptive dynamic sliding mode controller with integrator was also found in the literature to control nonholonomic systems [11]. This scheme uses a modified backstepping kinematic controller for generating the kinematic trajectories and the velocity errors to drive the sliding surface. The normalized least-square mean was used to control the switching of a sliding mode controller in the presence of uncertainties and parameter variations. Also, other authors developed sliding mode control methodologies for trajectory tracking and positioning of nonholonomic second-order chained form systems [12–14]. Sarfraz and ur Rehman [15] addressed the positioning problem of a particular nonholonomic planar underactuated mechanical system using adaptive sliding mode control. The control law assures the derivative of a suitable Lyapunov function becomes strictly negative.The aforementioned controllers use a permanent excitation in the reference trajectory [4, 5]; coordinate transformation [8]; discontinuities [9, 10]; and high computational load calculations [11–15] to solve the trajectory tracking problem. To overcome these drawbacks, this paper aims to solve the problem of trajectory tracking of a second-order chained form system with nonholonomic constraints by using a novel control law inspired by the design method Linear Algebra-Based Controller (LABC) [16]. This technique computes the control actions by solving a system of linear equations and has been applied to different nonlinear systems [17–23].Until now, LABC has not been applied to second-order chained form systems with nonholonomic constraints in the space of the original variables. For this reason, the major contribution of this work is to extend the LABC technique to solve the tracking and positioning problem for these systems. The system should follow the reference coordinates of two of three variables while the other one is unknown. The key point of this strategy is to find the conditions for which the system of linear equations has an exact solution, ensuring the calculus of the remaining variable, named sacrificed variable. In this way, the reference trajectories of the selected variables and the sacrificed one form an admissible trajectory of the system and the tracking error goes to zero [16].This paper is organized as follows. The Materials and Methods section describes the second-order chained form system with nonholonomic constraints and solves the problem of trajectory tracking and positioning using a controller based on linear algebra. The stability of the closed-loop system is demonstrated by applying the Khalil lemma. In the Results and Discussion section, simulation results show the effectiveness of the proposed controllers. Conclusions are given at the end of this manuscript.


## 2. Materials and Methods

### 2.1. Problem Formulation

This section presents the special canonical form of an underactuated system with nonholonomic constraints. The control of this system remains a problem under study [1]. One of the most important issues is the transformation to a chained form.Consider the second-order chained form system,Sξ, given by following set of equations:(1)ξ¨1=u1,(2)ξ¨2=u2,(3)ξ¨3=ξ2u1,where ξ=ξ1;ξ2;ξ3T∈R3 are the generalized coordinates of the second-order chained form system and u=u1u2T represents the inputs such that u1∈R and u2∈R. Also, the reference trajectory is Φξ=ξ1,ref;ξ2,ref;ξ3,ref.The tracking errorse=e1;e2;e3T∈R3 are the differences between the reference paths and the variables:(4)e1=ξ1,ref−ξ1,e2=ξ2,ref−ξ2,e3=ξ3,ref−ξ3.This work proposes the design of a controller for trajectory tracking and positioning of a second-order nonholonomic system represented by (1)–(3).


### 2.2. Problem 1: Trajectory Tracking

Given the systemSξ, the trajectory tracking control problem is solvable if it is possible to design an appropriate controller of the form(5)u1=u1t,ξ,Φξ,(6)u2=u2t,ξ,Φξ,such that the closed-loop system constituted by (1)–(3) and the control actions (5) and (6) is globally uniformly asymptotically stable.


### 2.3. Problem 2: Positioning

Given the systemSξ, the positioning control problem is solvable if it is possible to design an appropriate controller of the form(7)u1=u1t,ξ,Φξ,(8)u2=u2t,ξ,Φξ,such that the closed-loop system constituted by (1)–(3) and the control actions (7) and (8) is globally uniformly and asymptotically stable. Here, the reference trajectory Φξ is zero.To the best of our knowledge, LABC technique has not been applied directly to systems of second-order equations without considering a model in state variables; therefore, we extend its use to these systems. The control method is based on determining the conditions for which the system has an exact solution. Knowing the system model, it is only necessary to fix the desired trajectory ofξ1ξ1,ref and ξ3ξ3,ref to find the solution [16].


### 2.4. Controller Design for Trajectory Tracking

Given an initial stateξ0, the system represented by (1)–(3), Sξ, should follow the desired trajectory Φξ with a low tracking error. Therefore, it is necessary to find a control law capable of generating a signal u=u1u2T such that it satisfies this objective.The systemSξ is expressed in matrix form as follows:(9)1001ξ20u1u2=ξ¨1ξ¨2ξ¨3.To calculate the control actionsu1 and u2, (9) should have an exact solution, that is,(10)u2=ξ¨2,(11)1ξ2u1=ξ¨1ξ¨3.If a proportional control approach is assumed, the tracking errors are the differences between the reference and real trajectories. Thus, the second-order derivatives of the variablesξ1 and ξ3, i.e., ξ¨1 and ξ¨3, are ξ¨1=ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1 and ξ¨3=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3, respectively [16].If the internal variableξ2,ez is set to the value of ξ2 for which the above system has an exact solution, then (11) is rewritten as(12)1ξ2,ezu1=ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3,and the tracking errors in variables 1 and 3 tend to zero.Variableξ2,ez is computed according to(13)ξ2,ez=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1.Remark 1.
Variableξ2 follows ξ2,ez and this one tracks ξ2,ref.
The first-order derivatives of the tracking errors aree˙1=ξ˙1,ref−ξ˙1 and e˙3=ξ˙3,ref−ξ˙3, and the control action u1 is obtained using the linear least-squares approximation method to (12) is(14)u1=ξ¨1,ref+k11e˙1+k10e1+ξ2,ezξ¨3,ref+k31e˙3+k30e31+ξ2,ez2.
We can write the second-order derivative of the variableξ2ξ¨2 as a function of the sacrificed variable ξ2,ez:(15)ξ¨2=ξ¨2,ez+k21ξ˙2,ez−ξ˙2+k20ξ2,ez−ξ2.
Furthermore, the first-order derivative of the tracking error of variable 2 ise˙2=ξ˙2,ez−ξ˙2, and the control action u2 is(16)u2=ξ¨2,ez+k21e˙2+k20e2.Remark 2.
If the numerator and denominator of (13) tend to zero simultaneously, the division is undetermined and ξ2,ez cannot be calculated. To avoid this problem, it is necessary to choose adequately the coefficients k11, k10, k31, and k30 forcing the errors e3 and e˙3 to go to zero more quickly than e1 and e˙1. This avoids that ξ2,ez tends to infinity following trajectories whose second derivatives are zero (linear and constant reference trajectories).


### 2.5. Stability of the Tracking Error Dynamics: Trajectory Tracking Case

In this section, the stability of the closed-loop system [(1)–(3), (14)–(16)] is demonstrated.Theorem 1.
Given the second-order chained form systemSξ, expressed in (1)–(3), and the control actions u1 and u2 formulated by (14)–(16), e12+e22+e32⟶0 when t⟶∞.Proof.
The following steps are proposed to prove Theorem1.Step 1Variableξ2,ez is substituted in (14), and after performing algebraic manipulations, u1 is obtained:(17)u1=ξ¨1,ref+k11e˙1+k10e12+ξ¨3,ref+k31e˙3+k30e32/ξ¨1,ref+k11e˙1+k10e1ξ¨1,ref+k11e˙1+k10e12+ξ¨3,ref+k31e˙3+k30e32/ξ¨1,ref+k11e˙1+k10e12,(18)u1=ξ¨1,ref+k11e˙1+k10e1.From (1), u1=ξ¨1, also we define the second-order derivative of the tracking error of variable 1 as e¨1=ξ¨1,ref−ξ¨1; in consequence, (18) is rewritten as follows:(19)0=e¨1+k11e˙1+k10e1.Step 2Replacing (16) with (10),(20)0=ξ¨2,ez−ξ¨2+k21e˙2+k20e2,and defining the second-order derivative of the tracking error of variable 2 ase¨2=ξ¨2,ez−ξ¨2, the following equation arises:(21)0=e¨2+k21e˙2+k20e2.Step 3Given thatξ2=ξ2,ez−e2, the algebraic manipulation of (3) gives(22)ξ¨3=ξ2,ez−e2u1.
Furthermore, from (13) and (14), the following expression is obtained: ξ2,ezu1=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3. We define the second-order derivative of the tracking error of variable 3 as e¨3=ξ¨3−ξ¨3,ref and rewrite (22) as(23)0=e¨3+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3−e2u1.
By replacing the control actionu1=ξ¨1,ref+k11e˙1+k10e1 in the above equation, the following second-order equation arises:(24)0=e¨3+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3−e2ξ¨1,ref+k11e˙1+k10e1.
Then, (19), (21), and (24) are rewritten according to (5.32) and (5.33) on page 221 of [24], where x˙1=f1t,x1,x2 and x˙2=f1t,x2, being x1=ξ˙1;ξ1;ξ˙3;ξ3 and x2=ξ˙2;ξ2. By defining x11=e1; x12=e˙1; x21=e2; x22=e˙2; x13=e3; x14=e˙3, the following system of equations is obtained:(25)x˙11x˙12x˙13x˙14=0−k10001−k1100000−k30001−k31x11x12x13x14−x21000ξ¨1,ref+k11e˙1+k10e1,x˙21x˙22=01−k20−k21x21x22.
Finally, by applying Lemma 5.6 of Khalil, the system is globally uniformly asymptotically stable and satisfies the conditione12+e22+e32⟶0 when ⟶∞.Remark 3.
If the controller parameters are adjusted such that the roots of the polynomial (19), e¨1+k11e˙1+k10e1=0, have a negative real part closer to the imaginary axis than the roots of the polynomial (24), e¨3+k31e˙3+k30e3=0, then e3 will go to zero faster than e1 and ξ2,ez does not go to infinity.


### 2.6. Controller Design for Positioning

The positioning problem for the system,Sξ, consists in computing the control inputs, u1=u1t,ξ,Φξ and u2=u2t,ξ,Φξ, such that the origin of the system is stable for any initial state ξ0. Positioning is a particular case of trajectory tracking, where ξ˙1,ref, ξ˙3,ref, ξ¨1,ref, and ξ¨3,ref are zero, and ξ2,ez is the sacrificed variable.The systemSξ is expressed in matrix form as follows:(26)1001ξ20u1u2=ξ¨1ξ¨2ξ¨3.To calculate the control actionsu1 and u2, (26) should have an exact solution; that is,(27)u2=ξ¨2,1ξ2u1=ξ¨1ξ¨3.If a proportional control approach is assumed, the tracking errors are the differences between the reference (zero) and real trajectories. Thus, the variablesξ¨1 and ξ¨3 are ξ¨1=−k11ξ˙1+k10e1 and ξ¨3=−k31ξ˙3+k30e3.Ifξ2,ez is the value of ξ2 for which the above system has an exact solution, then(28)1ξ2,ezu1=−k11ξ˙1+k10e1−k31ξ˙3+k30e3,and the tracking errors in variables 1 and 3 tend to zero.Variableξ2,ez is computed according to(29)ξ2,ez=−k31ξ˙3+k30e3−k11ξ˙1+k10e1.Remark 4.
Variableξ2 follows ξ2,ez and this one tracks ξ2,ref.
The control actionu1 obtained using linear least-squares approximation method to (28) is(30)u1=−k11ξ˙1+k10e1+ξ2,ez−k31ξ˙3+k30e31+ξ2,ez2.
Furthermore, the control actionu2 is a function of variables ξ¨2,ez, e˙2, and e2:(31)u2=ξ¨2,ez+k21e˙2+k20e2.Remark 5.
Equations (30) and (31) for positioning are a particular case of (13) and (14) for trajectory tracking of the system, Sξ.


### 2.7. Stability of the Tracking Error Dynamics: Positioning Case

In this section, the stability of the closed-loop system [(1)–(3) and (29)–(31)] is demonstrated.Theorem 2.
Given the second-order chained form system (1)–(3), and the control actions u1 and u2 formulated in (29)–(31), the variables ξ1, ξ2, ξ3⟶0 and ξ2,ez, ξ˙2,ez, ξ¨2,ez⟶0 when t⟶∞.Proof.
The following steps are proposed to prove Theorem2:Step 1Variableξ2,ez is substituted in (30), and after doing algebraic manipulations, u1 is obtained:(32)u1=−k11ξ˙1+k10e1.From (1), u1=ξ¨1. Then, (32) is rewritten as follows:(33)0=−ξ¨1−k11ξ˙1+k10e1.Step 2Substituting (31) with (10), and given that e¨2=ξ¨2,ez−ξ¨2, the following equation arises:(34)0=e¨2+k21e˙2+k20e2.Equation (34) for positioning is coincident with (21) for trajectory tracking.Step 3Given thatξ2=ξ2,ez−e2, the algebraic manipulation of (3) gives(35)ξ¨3=ξ2,ez−e2u1.
Furthermore, from (29) and (30), the following expression is obtained: ξ2,ezu1=−k31ξ˙3−k30ξ3. We rewrite (35) as(36)0=ξ¨3−k31ξ˙3−k30ξ3−e2−k11ξ˙1+k10e1.
Then, (33), (34), and (36) for positioning are rewritten according to (5.32) and (5.33) of [24] where ξ˙1,ref=0; ξ˙3,ref=0; ξ¨1,ref=0; and ξ¨3,ref=0; e˙1=−ξ˙1; e˙3=−ξ˙3; e¨1=−ξ¨1 and e¨3=−ξ¨3, being x1=ξ˙1;ξ1;ξ˙3;ξ3 and x2=ξ˙2;ξ2. By defining x11=e1; x12=e˙1; x21=e2; x22=e˙2; x13=e3; and x14=e˙3, the following equations arise:(37)x˙11x˙12x˙13x˙14=0100−k10−k1100000100−k30−k31x11x12x13x14−x21000k11e˙1+k10e1,x˙21x˙22=01−k20−k21x21x22.
Finally, according to the stability lemma of Khalil [24], the system is globally uniformly asymptotically stable and verifies the conditions (ξ1, ξ2, ξ3⟶0) and (ξ2,ez, ξ˙2,ez, ξ¨2,ez⟶0) when t⟶∞.Remark 6.
By adjusting the controller parameters in such a way the roots of the polynomial (33) have a negative real part closer to the imaginary axis than the roots of the polynomial (36), the tracking error e3 will go to zero faster than e1 and ξ2,ez will not tend to infinity.


## 2.1. Problem Formulation

This section presents the special canonical form of an underactuated system with nonholonomic constraints. The control of this system remains a problem under study [1]. One of the most important issues is the transformation to a chained form.Consider the second-order chained form system,Sξ, given by following set of equations:(1)ξ¨1=u1,(2)ξ¨2=u2,(3)ξ¨3=ξ2u1,where ξ=ξ1;ξ2;ξ3T∈R3 are the generalized coordinates of the second-order chained form system and u=u1u2T represents the inputs such that u1∈R and u2∈R. Also, the reference trajectory is Φξ=ξ1,ref;ξ2,ref;ξ3,ref.The tracking errorse=e1;e2;e3T∈R3 are the differences between the reference paths and the variables:(4)e1=ξ1,ref−ξ1,e2=ξ2,ref−ξ2,e3=ξ3,ref−ξ3.This work proposes the design of a controller for trajectory tracking and positioning of a second-order nonholonomic system represented by (1)–(3).


## 2.2. Problem 1: Trajectory Tracking

Given the systemSξ, the trajectory tracking control problem is solvable if it is possible to design an appropriate controller of the form(5)u1=u1t,ξ,Φξ,(6)u2=u2t,ξ,Φξ,such that the closed-loop system constituted by (1)–(3) and the control actions (5) and (6) is globally uniformly asymptotically stable.


## 2.3. Problem 2: Positioning

Given the systemSξ, the positioning control problem is solvable if it is possible to design an appropriate controller of the form(7)u1=u1t,ξ,Φξ,(8)u2=u2t,ξ,Φξ,such that the closed-loop system constituted by (1)–(3) and the control actions (7) and (8) is globally uniformly and asymptotically stable. Here, the reference trajectory Φξ is zero.To the best of our knowledge, LABC technique has not been applied directly to systems of second-order equations without considering a model in state variables; therefore, we extend its use to these systems. The control method is based on determining the conditions for which the system has an exact solution. Knowing the system model, it is only necessary to fix the desired trajectory ofξ1ξ1,ref and ξ3ξ3,ref to find the solution [16].


## 2.4. Controller Design for Trajectory Tracking

Given an initial stateξ0, the system represented by (1)–(3), Sξ, should follow the desired trajectory Φξ with a low tracking error. Therefore, it is necessary to find a control law capable of generating a signal u=u1u2T such that it satisfies this objective.The systemSξ is expressed in matrix form as follows:(9)1001ξ20u1u2=ξ¨1ξ¨2ξ¨3.To calculate the control actionsu1 and u2, (9) should have an exact solution, that is,(10)u2=ξ¨2,(11)1ξ2u1=ξ¨1ξ¨3.If a proportional control approach is assumed, the tracking errors are the differences between the reference and real trajectories. Thus, the second-order derivatives of the variablesξ1 and ξ3, i.e., ξ¨1 and ξ¨3, are ξ¨1=ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1 and ξ¨3=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3, respectively [16].If the internal variableξ2,ez is set to the value of ξ2 for which the above system has an exact solution, then (11) is rewritten as(12)1ξ2,ezu1=ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3,and the tracking errors in variables 1 and 3 tend to zero.Variableξ2,ez is computed according to(13)ξ2,ez=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3ξ¨1,ref+k11ξ˙1,ref−ξ˙1+k10ξ1,ref−ξ1.Remark 1.
Variableξ2 follows ξ2,ez and this one tracks ξ2,ref.
The first-order derivatives of the tracking errors aree˙1=ξ˙1,ref−ξ˙1 and e˙3=ξ˙3,ref−ξ˙3, and the control action u1 is obtained using the linear least-squares approximation method to (12) is(14)u1=ξ¨1,ref+k11e˙1+k10e1+ξ2,ezξ¨3,ref+k31e˙3+k30e31+ξ2,ez2.
We can write the second-order derivative of the variableξ2ξ¨2 as a function of the sacrificed variable ξ2,ez:(15)ξ¨2=ξ¨2,ez+k21ξ˙2,ez−ξ˙2+k20ξ2,ez−ξ2.
Furthermore, the first-order derivative of the tracking error of variable 2 ise˙2=ξ˙2,ez−ξ˙2, and the control action u2 is(16)u2=ξ¨2,ez+k21e˙2+k20e2.Remark 2.
If the numerator and denominator of (13) tend to zero simultaneously, the division is undetermined and ξ2,ez cannot be calculated. To avoid this problem, it is necessary to choose adequately the coefficients k11, k10, k31, and k30 forcing the errors e3 and e˙3 to go to zero more quickly than e1 and e˙1. This avoids that ξ2,ez tends to infinity following trajectories whose second derivatives are zero (linear and constant reference trajectories).


## 2.5. Stability of the Tracking Error Dynamics: Trajectory Tracking Case

In this section, the stability of the closed-loop system [(1)–(3), (14)–(16)] is demonstrated.Theorem 1.
Given the second-order chained form systemSξ, expressed in (1)–(3), and the control actions u1 and u2 formulated by (14)–(16), e12+e22+e32⟶0 when t⟶∞.Proof.
The following steps are proposed to prove Theorem1.Step 1Variableξ2,ez is substituted in (14), and after performing algebraic manipulations, u1 is obtained:(17)u1=ξ¨1,ref+k11e˙1+k10e12+ξ¨3,ref+k31e˙3+k30e32/ξ¨1,ref+k11e˙1+k10e1ξ¨1,ref+k11e˙1+k10e12+ξ¨3,ref+k31e˙3+k30e32/ξ¨1,ref+k11e˙1+k10e12,(18)u1=ξ¨1,ref+k11e˙1+k10e1.From (1), u1=ξ¨1, also we define the second-order derivative of the tracking error of variable 1 as e¨1=ξ¨1,ref−ξ¨1; in consequence, (18) is rewritten as follows:(19)0=e¨1+k11e˙1+k10e1.Step 2Replacing (16) with (10),(20)0=ξ¨2,ez−ξ¨2+k21e˙2+k20e2,and defining the second-order derivative of the tracking error of variable 2 ase¨2=ξ¨2,ez−ξ¨2, the following equation arises:(21)0=e¨2+k21e˙2+k20e2.Step 3Given thatξ2=ξ2,ez−e2, the algebraic manipulation of (3) gives(22)ξ¨3=ξ2,ez−e2u1.
Furthermore, from (13) and (14), the following expression is obtained: ξ2,ezu1=ξ¨3,ref+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3. We define the second-order derivative of the tracking error of variable 3 as e¨3=ξ¨3−ξ¨3,ref and rewrite (22) as(23)0=e¨3+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3−e2u1.
By replacing the control actionu1=ξ¨1,ref+k11e˙1+k10e1 in the above equation, the following second-order equation arises:(24)0=e¨3+k31ξ˙3,ref−ξ˙3+k30ξ3,ref−ξ3−e2ξ¨1,ref+k11e˙1+k10e1.
Then, (19), (21), and (24) are rewritten according to (5.32) and (5.33) on page 221 of [24], where x˙1=f1t,x1,x2 and x˙2=f1t,x2, being x1=ξ˙1;ξ1;ξ˙3;ξ3 and x2=ξ˙2;ξ2. By defining x11=e1; x12=e˙1; x21=e2; x22=e˙2; x13=e3; x14=e˙3, the following system of equations is obtained:(25)x˙11x˙12x˙13x˙14=0−k10001−k1100000−k30001−k31x11x12x13x14−x21000ξ¨1,ref+k11e˙1+k10e1,x˙21x˙22=01−k20−k21x21x22.
Finally, by applying Lemma 5.6 of Khalil, the system is globally uniformly asymptotically stable and satisfies the conditione12+e22+e32⟶0 when ⟶∞.Remark 3.
If the controller parameters are adjusted such that the roots of the polynomial (19), e¨1+k11e˙1+k10e1=0, have a negative real part closer to the imaginary axis than the roots of the polynomial (24), e¨3+k31e˙3+k30e3=0, then e3 will go to zero faster than e1 and ξ2,ez does not go to infinity.


## 2.6. Controller Design for Positioning

The positioning problem for the system,Sξ, consists in computing the control inputs, u1=u1t,ξ,Φξ and u2=u2t,ξ,Φξ, such that the origin of the system is stable for any initial state ξ0. Positioning is a particular case of trajectory tracking, where ξ˙1,ref, ξ˙3,ref, ξ¨1,ref, and ξ¨3,ref are zero, and ξ2,ez is the sacrificed variable.The systemSξ is expressed in matrix form as follows:(26)1001ξ20u1u2=ξ¨1ξ¨2ξ¨3.To calculate the control actionsu1 and u2, (26) should have an exact solution; that is,(27)u2=ξ¨2,1ξ2u1=ξ¨1ξ¨3.If a proportional control approach is assumed, the tracking errors are the differences between the reference (zero) and real trajectories. Thus, the variablesξ¨1 and ξ¨3 are ξ¨1=−k11ξ˙1+k10e1 and ξ¨3=−k31ξ˙3+k30e3.Ifξ2,ez is the value of ξ2 for which the above system has an exact solution, then(28)1ξ2,ezu1=−k11ξ˙1+k10e1−k31ξ˙3+k30e3,and the tracking errors in variables 1 and 3 tend to zero.Variableξ2,ez is computed according to(29)ξ2,ez=−k31ξ˙3+k30e3−k11ξ˙1+k10e1.Remark 4.
Variableξ2 follows ξ2,ez and this one tracks ξ2,ref.
The control actionu1 obtained using linear least-squares approximation method to (28) is(30)u1=−k11ξ˙1+k10e1+ξ2,ez−k31ξ˙3+k30e31+ξ2,ez2.
Furthermore, the control actionu2 is a function of variables ξ¨2,ez, e˙2, and e2:(31)u2=ξ¨2,ez+k21e˙2+k20e2.Remark 5.
Equations (30) and (31) for positioning are a particular case of (13) and (14) for trajectory tracking of the system, Sξ.


## 2.7. Stability of the Tracking Error Dynamics: Positioning Case

In this section, the stability of the closed-loop system [(1)–(3) and (29)–(31)] is demonstrated.Theorem 2.
Given the second-order chained form system (1)–(3), and the control actions u1 and u2 formulated in (29)–(31), the variables ξ1, ξ2, ξ3⟶0 and ξ2,ez, ξ˙2,ez, ξ¨2,ez⟶0 when t⟶∞.Proof.
The following steps are proposed to prove Theorem2:Step 1Variableξ2,ez is substituted in (30), and after doing algebraic manipulations, u1 is obtained:(32)u1=−k11ξ˙1+k10e1.From (1), u1=ξ¨1. Then, (32) is rewritten as follows:(33)0=−ξ¨1−k11ξ˙1+k10e1.Step 2Substituting (31) with (10), and given that e¨2=ξ¨2,ez−ξ¨2, the following equation arises:(34)0=e¨2+k21e˙2+k20e2.Equation (34) for positioning is coincident with (21) for trajectory tracking.Step 3Given thatξ2=ξ2,ez−e2, the algebraic manipulation of (3) gives(35)ξ¨3=ξ2,ez−e2u1.
Furthermore, from (29) and (30), the following expression is obtained: ξ2,ezu1=−k31ξ˙3−k30ξ3. We rewrite (35) as(36)0=ξ¨3−k31ξ˙3−k30ξ3−e2−k11ξ˙1+k10e1.
Then, (33), (34), and (36) for positioning are rewritten according to (5.32) and (5.33) of [24] where ξ˙1,ref=0; ξ˙3,ref=0; ξ¨1,ref=0; and ξ¨3,ref=0; e˙1=−ξ˙1; e˙3=−ξ˙3; e¨1=−ξ¨1 and e¨3=−ξ¨3, being x1=ξ˙1;ξ1;ξ˙3;ξ3 and x2=ξ˙2;ξ2. By defining x11=e1; x12=e˙1; x21=e2; x22=e˙2; x13=e3; and x14=e˙3, the following equations arise:(37)x˙11x˙12x˙13x˙14=0100−k10−k1100000100−k30−k31x11x12x13x14−x21000k11e˙1+k10e1,x˙21x˙22=01−k20−k21x21x22.
Finally, according to the stability lemma of Khalil [24], the system is globally uniformly asymptotically stable and verifies the conditions (ξ1, ξ2, ξ3⟶0) and (ξ2,ez, ξ˙2,ez, ξ¨2,ez⟶0) when t⟶∞.Remark 6.
By adjusting the controller parameters in such a way the roots of the polynomial (33) have a negative real part closer to the imaginary axis than the roots of the polynomial (36), the tracking error e3 will go to zero faster than e1 and ξ2,ez will not tend to infinity.


## 3. Results and Discussion

The closed-loop system (1)–(3) and (13), (14), (16) for trajectory tracking is simulated and analyzed in the first test. In the same way, the closed-loop system (1)–(3) and (29)–(31) for positioning is simulated, and the results are discussed in the second test.


### 3.1. Trajectory Tracking Test

The aim of this section is to show that given a reference trajectory, the proposed controller can calculate a control law that allows following effectively. The reference trajectories ofξ1; ξ2; and ξ3 are simulated considering ξ10=40 and ξ20=ξ30=0 as the initial state of the system, and ξ1,ref and ξ3,ref follow a parabolic path, while ξ2,ref follows an ascending sine wave. The controller parameters, set by trial and error, are k10=0.72; k11=0.72; k20=0.55; k21=0.55; k30=0.90; and k31=0.90. With these values, ξ1; ξ2; and ξ3 converge satisfactorily to the reference trajectory. Figure 1 shows the trajectory of variable 1. Variable ξ1 (blue line) initially fluctuates around its reference ξ1,ref (red-dashed line) reaching it after 40 seconds.Figure 1
Trajectory tracking ofξ1.The trajectories ofξ2(blue solid line); ξ2,ez (green solid line); and ξ2,ref (red dashed line) are shown in Figure 2. Variable ξ2 follows the sacrificed variable ξ2,ez, and after 60 seconds, both variables follow the reference ξ2,ref. In this way, the tracking errors e1 and e3 tend to zero. This remarks the performance of the proposed controller.Figure 2
Trajectory tracking ofξ2.Figure3 shows the trajectory of variable 3. Variable ξ3 (blue line) perfectly follows ξ3,ref (red-dashed line) without undesirable oscillations.Figure 3
Trajectory tracking ofξ3.Figures4 and 5 present the control actions u1 and u2, respectively. Its initial values are u10=u20=0, respectively. The first one, u1, evolves quickly from 0 to 10−4 remaining constant. The second one, u2, follows a sine wave path with mean 0, amplitude 0.8, and frequency 1/8π.Figure 4
Control actionu1.Figure 5
Control actionu2.


### 3.2. Positioning Test

The closed-loop system (1)–(3) and (29)–(31) is simulated considering ξ10=1; ξ20=0; and ξ30=10−3 as the initial state. The controller parameters are k10=0.15; k11=0.15; k20=0.45; k21=0.45; k30=0.95; and k31=0.95. Figures 6 and 7 show the trajectories of system variables ξ1 and ξ3 (blue solid lines). The red and magenta dashed lines indicate the zero reference. Figure 8 presents the control action u1, calculated by (30). In Figure 9, variables ξ2 and ξ2,ez are represented by a blue and orange solid lines, respectively. Finally, Figure 10 displays the trajectory of the control action u2 calculated by (31).Figure 6
Trajectory ofξ1.Figure 7
Trajectory ofξ3.Figure 8
Control actionu1.Figure 9
Trajectory ofξ2.Figure 10
Control actionu2.


## 3.1. Trajectory Tracking Test

The aim of this section is to show that given a reference trajectory, the proposed controller can calculate a control law that allows following effectively. The reference trajectories ofξ1; ξ2; and ξ3 are simulated considering ξ10=40 and ξ20=ξ30=0 as the initial state of the system, and ξ1,ref and ξ3,ref follow a parabolic path, while ξ2,ref follows an ascending sine wave. The controller parameters, set by trial and error, are k10=0.72; k11=0.72; k20=0.55; k21=0.55; k30=0.90; and k31=0.90. With these values, ξ1; ξ2; and ξ3 converge satisfactorily to the reference trajectory. Figure 1 shows the trajectory of variable 1. Variable ξ1 (blue line) initially fluctuates around its reference ξ1,ref (red-dashed line) reaching it after 40 seconds.Figure 1
Trajectory tracking ofξ1.The trajectories ofξ2(blue solid line); ξ2,ez (green solid line); and ξ2,ref (red dashed line) are shown in Figure 2. Variable ξ2 follows the sacrificed variable ξ2,ez, and after 60 seconds, both variables follow the reference ξ2,ref. In this way, the tracking errors e1 and e3 tend to zero. This remarks the performance of the proposed controller.Figure 2
Trajectory tracking ofξ2.Figure3 shows the trajectory of variable 3. Variable ξ3 (blue line) perfectly follows ξ3,ref (red-dashed line) without undesirable oscillations.Figure 3
Trajectory tracking ofξ3.Figures4 and 5 present the control actions u1 and u2, respectively. Its initial values are u10=u20=0, respectively. The first one, u1, evolves quickly from 0 to 10−4 remaining constant. The second one, u2, follows a sine wave path with mean 0, amplitude 0.8, and frequency 1/8π.Figure 4
Control actionu1.Figure 5
Control actionu2.


## 3.2. Positioning Test

The closed-loop system (1)–(3) and (29)–(31) is simulated considering ξ10=1; ξ20=0; and ξ30=10−3 as the initial state. The controller parameters are k10=0.15; k11=0.15; k20=0.45; k21=0.45; k30=0.95; and k31=0.95. Figures 6 and 7 show the trajectories of system variables ξ1 and ξ3 (blue solid lines). The red and magenta dashed lines indicate the zero reference. Figure 8 presents the control action u1, calculated by (30). In Figure 9, variables ξ2 and ξ2,ez are represented by a blue and orange solid lines, respectively. Finally, Figure 10 displays the trajectory of the control action u2 calculated by (31).Figure 6
Trajectory ofξ1.Figure 7
Trajectory ofξ3.Figure 8
Control actionu1.Figure 9
Trajectory ofξ2.Figure 10
Control actionu2.


## 4. Conclusion

The major contribution of this paper is to extend the LABC technique for trajectory tracking and positioning of a second-order nonholonomic system. This controller is based on determining the conditions for which the system model has an exact solution. Knowing the model equations, it is necessary only to fix the desired trajectory to find the control action. Variables 1 and 3 were set as references, while variable 2 is the sacrificed one to ensure that the tracking errors go to zero. An important advantage of the proposed controller is the low computational load of the required mathematical operations. The stability of the system is demonstrated by applying the Khalil lemma [24].Simulation results show the performance of the proposed control algorithm. The controller parameters were empirically tuned. For trajectory tracking, these parameters ensure that the system variables quickly reach the reference and follow it without undesirable oscillations. In the case of the positioning problem, these parameters were tuned in such a way variableξ3 goes to zero more quickly than ξ1.



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*Source: 6082586-2020-08-01.xml*