Restriction of Transverse Feedback Linearization for Piecewise Linear Paths

This work presents a path-following controller for a unicycle robot. The main contribution of this paper is to demonstrate the restriction of transverse feedback linearization (TFL) to obtuse angles on piecewise linear paths. This restriction is experimentally demonstrated on a Kobuki mobile robot, where it is possible to observe, as a result of the limitation of the TFL, the convergence to another domain of attraction.


Introduction
Mobile robot applications have significantly increased in recent years. Following a general rule of technological evolution, mobile robotics has increasingly prioritized autonomy in relation to the environment, requiring greater efficiency of navigation and control systems. To ensure that a robot follows a reference path or to stabilize a robot at a desired point, a motion control system is needed. According to the type of reference to be followed by the mobile robot, the motion control systems are classified into three types: (1) Point stabilization (2) Trajectory tracking (3) Path following e objective of point stabilization is to stabilize the robot in a certain desired position. According to [1], this objective cannot be achieved with continuous or smooth state feedback control laws if the robot is subject to nonholonomic constraints. In this case, approaches such as control laws that vary slightly in time [2] and discontinuous and hybrid feedbacks [3] can be used. Another way to avoid the problem of nonholonomy is the use of Frenet-Serret and Triedro [4] to describe the state space.
In trajectory tracking, the control laws that act on the robot must follow a parameterized law of timing associated with the movement.
In path following, the robot converges and follows a predefined path.
is path does not have any temporal specification. e robot tracks a desired speed profile, and the controller acts on the orientation to drive the robot along the intended path [5].
Of the three categories described above, we chose the path-following technique for this work. In the path-following control, the forward speed of the robot tracks the desired speed profile, while the controller acts on the orientation of the robot in order to steer the vehicle along the path. Usually, smoother convergence to a path is achieved when path-following strategies are used compared with trajectory-tracking control laws [6].
Approach for path following is transverse feedback linearization (TFL). e main advantage of adopting TFL is that the controllers guarantee path invariance. Some pathfollowing control techniques, such as sliding control, do not have this property. Path invariance means that if the robot starts with tangential velocity to the path, the robot will remain on the path for all future time [11].
ere are many examples in the literature of the use of TFL to control robotics systems.
In Nielsen and Maggiore [12], the authors solve the TFL for a car-like robot with fixed translational velocity. Akhtar and Nielsen [13] propose a path-following controller for a kinematic model of car-like mobile robots using TFL and tangential dynamic extension. Akhtar et al. [14] propose a path-following controller for a quadrotor using TFL and tangential dynamic extension. In Hladio et al. [15], the authors propose a path-following controller for a five-bar robotic manipulator using TFL to put the system into a normal form for control design. Akhtar et al. [11] propose path-following controllers for the kinematic model of carlike mobile robots using TFL with dynamic extension. Dovgobrod [16] proposes an algorithm to provide a precision stabilization of a vehicle on a path. Finally, Nguyen et al. [17] design a controller combining model predictive control and TFL techniques.
is article analyzes the response of TFL in relation to the piecewise linear path and, as its main contribution, demonstrates analytically the restriction to an obtuse path. is limitation is experimentally verified on a Kobuki mobile robot.

Problem Formulation
In this section, we demonstrate path-following controllers for a kinematic model of a unicycle. e following properties of the path-following strategy are assumed: (1) e robot is able to plan a piecewise linear path from the starting position to the goal position (2) Each line segment of the path should be controlled invariant 2.1. Kinematic Model of Robot. Consider the kinematic model of a unicycle mobile robot shown in Figure 1: (1) Here, (x 1 , x 2 ) ∈ R 2 represents the robot's position in the plane, and x 3 ∈ S is the robot's heading. e control inputs are the translational speed v ∈ R and the angular velocity ω ∈ S. e output of the system is the robot's location in the plane, which is given by e configuration space of (1) is X � R 2 × S, and the output space of (2) is Y � R 2 . It is assumed that only the position coordinates of a robot are measured and continuous updates are provided in its state vector x. e heading vector is defined as τ(x 3 ) � (sin(x 3 ), cos(x 3 )); its Euclidean norm equals 1 and it points in the direction the robot is facing Figure 1.

Workspace.
e unicycle robot model (1) operates in a workspace W ⊂ R 2 . e set W is assumed to be compact and path-connected with a nonempty interior. e output space has a finite number of obstacles O i ⊂ R 2 , i ∈ 1, . . . , n { }. Each obstacle O i is assumed to be a closed, convex set. e free workspace is defined as e differential drive robot moves in the environment under the assumption that models (1) and (2) have a map of W free stored in its memory, which contains two distinguished points p start , p goal ∈ W free .

Configuration Map.
In order to model the fact that the robot is not simply a point with no area in its workspace, this section introduces the notion of a map.

Definition 1.
ere exists a set-valued function B: ⇉ called the configuration map that maps each configuration x ∈ X of robot (1) to the set of all points B(x) ⊂ Y belonging to the robot.
As depicted in Figure 1, it is assumed that the differential drive robot has a polygonal shape. erefore, B(x) is a polygonal region in the plane, for B(x(t)) ⊂ W free is necessary to ensure that there are no collisions, during the period of time in which the robot is moving from the start to the goal position.

Path-Following Outputs for Path Planners.
e pathfollowing output associated to a smooth parameterized curve σ: R ⟶ R p was defined in [18]. It consists of two complementary objects: a transversal output and a tangential output. e transversal output is a smooth function μ: U⊆R p ⟶ R p− 1 with the property that zero is a regular value of the function and e tangential output is a projection ϖ: U⊆R p ⟶ R which, given an output location y ∈ U, returns the value of λ ∈ R with the property that σ(λ) is the closest point to y on the given curve.
e resulting function h PF : U⊆R p ⟶ R p , and y↦(ϖ(y), μ(y)) is the path-following output. is virtual output has a clear physical meaning for path-following control design and can be viewed as a local diffeomorphism on the output space of the robot. In this section, keeping in mind that the paths will be generated as a sequence of waypoints, a procedure is presented to generate a set of linear path-following outputs from a sequence of waypoints.
Suppose that a sample-based path planner has returned a path from p start to p goal as a sequence of points in W free which is a subset of a p-dimensional real vector space Y (because the results of this section apply equally well in higher dimensions, we do not assume that the workspace of our robot is a subset of R 2 as in the case of the differential drive robot). at is, given a set p 1 the insertion map. e insertion map is represented by any matrix whose column vectors form a basis for V i relative to the given basis for Y. Finally, denote its associated dual map by A zero-level set representation of the affine subspace containing ℓ i is obtained using where and so the value of the transversal part of the path-following output associated to the line segment ℓ i has been calculated. Repeating this for each line segment, a set of pairs S � is generated from the set of waypoints.
To compute the tangential part of the path-following output associated to ℓ i , a constrained optimization problem is solved. To begin, a parametrization of ℓ i is generated. Next, l i and m i are defined: Given a point y ∈ Y, consider the constrained optimization problem.

Problem 1
minimize : Using Lagrange multipliers, the minimizing argument is given by e value of λ * ∈ [0, l i ] for which σ i (λ * ) � y * can now be computed yielding the second part of the path-following output for line segment ℓ i : In summary, equations (5)-(8) define the path-following output h i PF (y) � (μ i (y), ϖ i (y)) associated to each line segment ℓ i � p i p i+1 . is output happens to be a linear affine function of y: Using the fact that m ⊤ i V i � 0 to simplify notation, let can be written: Mathematical Problems in Engineering 3 e set is further defined as follows: and then, each line segment ℓ i is associated with the pairs

Transverse Feedback Linearization
Consider the differential drive robot configurations (1) and (2). Assume that the robot's translational speed v is a fixed positive constant. In addition, as in Section 2, a samplebased path planner has returned a path from p start to p goal as a sequence of points p 1 , Following the construction of Section 2.4, the sets S and T are known. en, for each i ∈ N N− 1 , the robot's path-following output is given by e path-following manifold of ℓ i is the largest controlled invariant set containing the lift of the path It is verified that y belongs to the line that contains ℓ i if and only if x ∈ Γ i . Of course, Γ i is not controlled invariant: if the robot is initialized on ℓ i but with τ(x 3 (0)) parallel Γ i to V i , then the robot must leave the set before possibly returning to the set. From a planning point of view, such behaviors are obviously undesirable because they could lead to large deviations from the planned path and therefore collisions. e direction in which the robot travels along the line depends on its initial conditions, which determine which component Γ i * (Γ i ( * ,+) or Γ i ( * ,− ) ) the robot approaches. In the path-planning framework, the robot must move in a prescribed direction along each segment ℓ i . is ensures that the robot always approaches Γ i ( * ,+) so that it moves in the correct direction. e path-following manifold of the line σ i (λ) � n i λ + p i , λ ∈ R, with respect to the differential drive robot (1) and (2) is given by a union of disjoint closed sets: where and Sets (16) and (17) are disjoint because x 3 ∈ S, and set (15) is closed because its complement is open. e complement is given by Γ c � R p \Γ � y ∈ R p : V ⊤ i Cx + d i ≠ 0 and the empty set is open by definition. Γ * is the controlled invariant set. Physically, it consists of all motions of the unicycle robot (1) for which the output signal (2) can be made to remain on the curve σ i .
To prove, it is necessary to show that V ⊤ i Cx + d i has a well-defined relative degree.

Controller (20) is well defined on the open set
3 ) ) � (π/2) , the feedback controller will result in a closed-loop system whose trajectories approach Γ i * , and therefore, the output y will approach the line σ(λ).

Supervisory Control of TFL Path-Following Controllers
As shown in Section 3, on each line segment ℓ i , controller (20) drives the robot to its path-following manifold. is section describes how the robot is switched from one line segment to the next and also how the convergence to Γ i is tangential state describes the robot's position on ℓ i . When η i � 0, the closest point to the robot's output on the path defined by p 1 , . . . , p N− 1 is p i . When η i 1 � l i , the closest point is p i+1 . With this in mind, the algorithmic state machine shown in Figure 2 is used to supervise our path-following controllers. 4 Mathematical Problems in Engineering In state i, the robot's control signal is given (10) by e supervisory control monitors how far the robot is from the next waypoint. When that distance is sufficiently small, controller (20) changes its parameters to (V ⊤ i+1 , d i+1 ) ∈ S, (m ⊤ i+1 , n i ) ∈ T. In principle, controller (20) can use a different set of gains k i ∈ R 2 ≥ 0 on each line segment. However, as the next proposition shows, a single set of gain values is chosen to ensure that the robot transitions from Γ i ( * ,+) to Γ i+1 ( * ,+) and thus moves in the correct direction along the paths.

Proposition 1. With respect to system (18), a set U + i is defined by
Proof. e Laplace transform is used to prove the existence of a domain of attraction for the system under analysis. ose domains of attraction, shown in Figure 3, are defined for Γ * as (23) It can be stated that path following is an output zeroing problem. erefore, it is necessary to show that ξ 2 (t) ≤ ξ 2 (0), ∀t ≥ 0. ξ 2 can be described as follows: If ‖V ⊤ i ‖ is normalized so that it is equal to 1, ‖τ(x)‖ � 1; therefore, it can be deduced that ξ 2 � 0 if and only if ϕ � ± (π/2). If ϕ � kπ for k ∈ N.
For this case, it is sufficient to prove that ξ 2 (t) ≤ ξ 2 (0), see Figure 4 _ From expression (25), Developing the above differential equation using the Laplace transform yields the following: ; if ε > 1, there exist two real poles: , Recall k � (1/k 1 ) � (1/ab): Because the transversal proportional controller is given by v ⋔ (0), ξ � 0 is an equilibrium point of the closed-loop transversal system. As a consequence, path invariance is achieved, meaning that if the robot is initialized in the path, it will remain in the path at all future points.
□ us, any initial condition in U + i results in trajectories that approach Γ i ( * ,+) , assuming the gains in (20) are chosen properly.    As a result when the supervisory control switches at time, from state i to state i + 1, the controller will cause the robot to traverse the path in the desired direction so long as Because controller (20) exponentially stabilizes Γ i * , it is reasonable to assume that x(T) ∈ Γ i ( * ,+) or is very close to it. As such, the next result says that if the robot is traversing segment ℓ i in the correct direction, it will continue to traverse ℓ i+1 in the desired direction if Γ i ( * ,+) ∩ U + i+1 ≠ ∅.

Results and Conclusion
In this section, simulation results dealing with the TFL and experiments on Kobuki mobile robot are presented. e limitation of the TFL is illustrated by showing the robot's inability to follow paths with obtuse angles.
In order to demonstrate the robot's inability to correctly follow a path with obtuse angles, a control was implemented to follow the points [0 1; 25 30; 40 10] whose distance between them is Euclidean. e angle between the yellow and red straight segments is 102.37 ∘ , and the robot's initial state was x 0 � (0, 1, (π/4)). As a result, a convergence to another attraction domain was observed; consequently, the robot followed the opposite trajectory, as shown in Figure 5. In this figure, the red straight segment indicates the desired path and the purple straight segment indicates the path followed by the robot.
Circles were drawn at each point. e radius of these circles, ε � 0.5, defines the path switching area; if the robot gets close enough as to the target point, the controller can then guide the robot through the next path. Such a transition can be observed in Figure 6.
Tests were performed on a robotic platform called Kobuki, made by Yujin Robot. It is controlled by an embedded Raspberry Pi 3 microcontroller running ROS (Robotic Operational System). Matlab software on a second computer was used to run the TFL controller, which communicated with the robot through an IEEE 802.11 wireless network. e Kobuki mobile robot was asked to follow a set of points whose distance is Euclidean between them. To demonstrate the robot's inability to follow an obtuse path, points [(0, 0), (0.5, 0), (0.45, 1)] were selected as the robot's path, with the initial position given by x 0 � (0, 0, 0) and the gains given by k 1 � 2 and k 2 � 2.84. As expected, the robot followed another attraction domain, as shown in Figure 7.
On the contrary, when the trajectory has only acute angles, the robot remains on the path. Figure 8(a) shows that the robot remains on the desired path Figure 8(b) and follows it. e initial position is given by x 0 � (0, 0, 0); the points followed by the robot were (0, 0), (1, 0), (2, 1), and (3, 1.25); the gains were given by k 1 � 2 and k 2 � 2.84.
In this paper, a path-following controller has been designed for a Kobuki mobile robot, using transverse feedback linearization for piecewise linear paths. e restriction of TFL was demonstrated using a kinematic model of a unicycle mobile robot. It has been shown that when the path has obtuse angles, the robot follows another domain of attraction, making the TFL unfeasible in these cases.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 7