Motion planning and specifically robotic navigation in obstacle-cluttered environments is a fundamental problem in the field of robotics [lumelsky2005sensing]. Several techniques have been developed in the related literature, such as discretization of the continuous space and employment of discrete algorithms (e.g., Dijkstra, ), probabilistic roadmaps, sampling-based motion planning, and feedback-based motion planning [lavalle2006planning]. The latter, which is the focus of the current paper, offers closed-form analytic solutions by usually evaluating appropriately designed artificial potential fields, avoiding thus the potential complexity of workspace discretization and the respective algorithms. At the same time, feedback-based methods provide a solution to the control aspect of the motion planning problem, i.e., the correctness based on the solution of the closed-loop differential equation that describes the robot model.
Early works on feedback-based motion planning established the Koditschek-Rimon navigation function (KRNF) [koditschek1990robot, rimon1992exact], where, through gain tuning, the robot converges safely to its goal from almost all initial conditions (in the sense of a measure-zero set). KRNFs were extended to more general workspaces and adaptive gain controllers [filippidis2011adjustable], to multi-robot systems [dimarogonas2006feedback, verginis2017decentralized, verginis2017robust, roussos2013decentralized], and more recently, to convex potential and obstacles [paternain2017navigation]. The idea of gain tuning has been also employed to an alternative KRNF in [tanner2005towards]. Tuning-free constructions of artificial potential fields have also been developed in the related literature; [panagou2017distributed] tackles nonholonomic multi-robot systems, and in [loizou2017navigation, vlantis2018robot] harmonic functions, also used in [szulczynski2011real], are combined with adaptive controllers to achieve almost global safe navigation. A transformation of arbitrarily shaped worlds to points worlds, which facilitates the motion planning problem, is also considered in [loizou2017navigation, vlantis2018robot] and in [loizou2014multi] for multi-robot systems. The recent works [loizou2017navigation], [vrohidis2018prescribed] guarantee also safe navigation in predefined time.
Barrier functions for multi-robot collision avoidance are employed in [wang2017safety] and optimization-based techniques via model predictive control (MPC) can be found in [filotheou2018, mendes2017real, verginis2018communication, morgan2016swarm]; [van2011reciprocal] and [alonso2012image] propose reciprocal collision obstacle by local decision making for the desired velocity of the robot(s). Ellipsoidal obstacles are tackled in [stavridis2017dynamical] and [Grush18obstacle] extends a given potential field to nd-order systems. A similar idea is used in [montenbruck2015navigation], where the effects of an unknown drift term in the dynamics are examined. Workspace decomposition methodologies with hybrid controllers are employed in [arslan2016exact], [arslan2016coordinated], and [berkane2019hybrid], and [slotine19avoidance] employs a contraction-based methodology that can also tackle the case of moving obstacles.
A common assumption that most of the aforementioned works consider is the simplified robot dynamics, i.e., single integrators/unicycle kinematics, without taking into account any robot dynamic parameters. Hence, indirectly, the schemes depend on an embedded internal system that converts the desired velocity signal to the actual robot actuation command. The above imply that the actual robot trajectory might deviate from the desired one, jeopardizing its safety and possibly resulting in collisions. Second-order realistic robot models are considered in MPC-schemes, like [mendes2017real, filotheou2018, verginis2018communication], which might, however, result in computationally expensive solutions. Moreover, regarding model uncertainties, a global upper bound is required, which is used to enlarge the obstacle boundaries and might yield infeasible solutions. A nd-order model is considered in [stavridis2017dynamical], [Grush18obstacle], without, however, considering any unknown dynamic terms. The works [koditschek1991control, dimarogonas2006feedback, loizou2011closed, arslan2017smooth] consider simplified nd-order systems with known dynamic terms (and in particular, inertia and gravitational terms that are assumed to be successfully compensated); [montenbruck2015navigation] guarantees the asymptotic stability of nd-order systems with a class of unknown drift terms to the critical points of a given potential function. However, there is no characterization of the region of attraction of the goal. Adaptive control for constant unknown parameters is employed in [cheah2009region], where a swarm of robots is controlled to move inside a desired region.
In this paper, we consider the robot navigation in an obstacle-cluttered environment under nd-order uncertain robot dynamics, in terms of unknown mass and friction/drag terms. Our main contribution lies in the design of a novel nd-order smooth navigation function as well as an adaptive control law that guarantees the safe navigation of the robot from almost all initial conditions. We also show how the proposed scheme can be applied to star-worlds, i.e., workspaces with star-shaped obstacles [rimon1992exact], as well as to decentralized multi-robot navigation. Adaptive control for multi-robot coordination was also employed in our previous works [verginisLCSS, verginis2017robust]. The results in [verginis2017robust], however, are only existential, since we do not provide an explicit potential function that satisfies the desired properties, while [verginisLCSS] focuses on the multi-agent ellipsoidal collision avoidance, without guaranteeing achievement of the primary task.
The rest of the paper is organized as follows. Section 2 provides the notation used throughout the paper. Section 3 describes the tackled problem and Section 4 provides the main results. Sections 5 and 6 extend the proposed scheme to star worlds and multi-agent frameworks, respectively. Finally, simulation studies are given in Section 7 and Section 8 concludes the paper.
The set of natural and real numbers is denoted by , and , respectively, and , , , are the -dimensional sets of nonnegative and positive real numbers, respectively. The notation implies the Euclidean norm of a vector . The identity matrix is denoted by
implies the Euclidean norm of a vector
. The identity matrix is denoted by, the matrix of zeros by and the -dimensional zero vector by . The gradient and Hessian of a function are denoted by and , respectively.
3 Problem Statement
Consider a spherical robot operating in a bounded workspace , characterized by its position vector , and radius , and subject to the dynamics:
where is the unknown mass, is the constant gravity vector, is the input vector, and is an unknown friction-like function, satisfying the following assumption:
The function is analytic and satisfies
, where is an unknown constant.
The aforementioned assumption is inspired by standard friction-like terms, which can be approximated by continuously differentiable velocity functions [makkar2005new]. Constant unknown friction terms could be also included in the dynamics (e.g., incorporated in the constant gravity vector). Note also that implies , and . The workspace is assumed to be an open ball centered at the origin
where is the workspace radius. The workspace contains closed sets , , corresponding to obstacles. Each obstacle is a closed ball centered at , with radius , i.e., . The analysis that follows will be based on the transformed workspace:
and set of obstacles where the robot is reduced to the point . The free space is defined as
also known as a sphere world [koditschek1990robot]. We consider the following common feasibility assumption [koditschek1990robot, vrohidis2018prescribed] for :
The workspace and the obstacles satisfy and , .
Assumption 3 implies that we can find some such that
This paper treats the problem of navigating the robot to a destination while avoiding the obstacles and the workspace boundary, formally stated as follows:
Consider a robot subject to the uncertain dynamics (1), operating in the aforementioned sphere world, with . Given a destination , design a control protocol such that
4 Main Results
We provide in this section our methodology for solving Problem 3. Define first the set as well as the distances , , with , , and . Note that, by keeping , , we guarantee that 111A safety margin can also be included, which needs, however, to be incorporated in the constant of (6).. We also define the constant
as the minimum distance of the goal to the obstacles/workspace boundary. We introduce next the notion of the nd-order navigation function:
A nd-order navigation function is a function of the form
where is a (at least) twice contin. differentiable function and are positive constants, with the followings properties:
is strictly decreasing, , and , , , for some ,
has a global minimum at where ,
if and for some , then , for all .
The function , with is strictly decreasing.
By using the first property we will guarantee that, by keeping bounded, there are no collisions with the obstacles or the free space boundary. Property will be used for the asymptotic stability of the desired point . Property places the rest of the critical points of (which are proven to be saddle points) close to the obstacles, and the last property is used to guarantee that these are non-degenerate. An example for that satisfies properties 1) and 4), is
Note that is essentially a reciprocal barrier function [wang2017safety]. We prove next that, by appropriately choosing , only one , affects the robotic agent for each , and furthermore that . Hence, properties 2) and 3) of Def. 4 are satisfied.
Moreover, it holds for the desired equilibrium that
and hence , .
Intuitively, the obstacles and the workspace boundary have a local region of influence defined by the constant , which will play a significant role in determining the stability of the overall scheme later. This robot interaction with only one obstacle at a time has also been demonstrated in the feedback control-based related literature, e.g., [arslan2016exact, vrohidis2018prescribed, filippidis2011adjustable, lionis2007locally, paternain2017navigation], which deals with simplified single-integrator models, as well as in the more discrete decision making bug algorithms [lumelsky2005sensing], which involve circumnavigation of obstacles and can handle in general complex unknown environments.
The expressions for the gradient and the Hessian of , which will be needed later, are the following:
Given the aforementioned definitions, we design a reference signal for the robot velocity as
Next, we design the control input to guarantee tracking of the aforementioned reference velocity as well as compensation of the unknown terms and . More specifically, we define the signals and
as the estimation terms ofand (see Assumption 3), respectively, and the respective errors , . We design now the control law as , with
where , and , are positive gain constants. Moreover, we design the adaptation laws for the estimation signals as
with , positive gain constants, , and arbitrary finite initial condition . As will be verified by the proof of Theorem 4, the choices for the control and adaptation laws are based on Lyapunov techniques, and follow standard adaptive control methodologies (see, e.g., [lavretsky2013robust]).
Consider the Lyapunov candidate function
Since , there exists a constant such that , , and hence there exists a finite positive constant such that . By considering the time derivative of and using and Assumption 3, we obtain after substituting (12):
which, by substituting (11) and using , becomes
Hence, we conclude that is non-increasing, and hence , , which implies that collisions with the obstacles and the workspace boundary are avoided, i.e.,
. Moreover, (9) implies also the boundedness of , . In addition, the boundedness of implies also the boundedness of , , , , and hence of , , , . More specifically, by letting , , we conclude that , , with
Therefore, by invoking LaSalle’s invariance principle, we conclude that the solution will converge to the largest invariant set in , which, in view of (10), becomes . Consider now the closed-loop dynamics for :
Note that, in view of the aforementioned discussion and the continuous differentiability of , the right-hand side of (14b) is bounded in . Note also that (9) implies the boundedness of in . Moreover, by differentiating , using the closed loop dynamics (14) and (9), we conclude the boundedness of and the uniform continuity of in . Hence, since , we invoke Barbalat’s Lemma to conclude .
Therefore, the set consists of the points where , , and by also using the property we obtain and . Note also that is a monotonically increasing function and it converges thus to some constant positive value , since and . Therefore, we conclude that the system will converge to an equilibrium satisfying . Since , the system converges to the critical points of , i.e., we obtain from (9) that at steady state:
where , . According to the choice of in Prop. 4, implies that , , and hence the desired equilibrium satisfies (15). Other undesired critical points of consist of cases where the two sides of (15) cancel each other out. However, as already proved, only one can be nonzero for each . Hence, the undesired critical points satisfy one of the following expressions:
and hence and have the same direction. Therefore, since , for , , (16b) is not feasible.
We proceed now by showing that the critical points satisfying (16a) are saddle points, which have a lower dimension stable manifold. Consider, therefore, the error , where represents the potential undesired equilibrium point that satisfies (16a). Let also , where , whose linearization around zero yields, after using (14) and ,
We aim to prove that the equilibrium
has at least one positive eigenvalue. To this end, consider a vector, where is a positive constant, and is an orthogonal vector to , i.e. . Then the respective quadratic form yields
which, after employing (9) with , and , becomes
From (16a), by recalling that , we obtain that
Therefore by defining , we obtain
which is rendered positive by choosing a sufficiently large . Hence, has at least one positive eigenvalue. Next, we prove that has no zero eigenvalues by proving that its determinant is nonzero. For the determinant of , in view of (9) that
By using the property
, for any invertible matrixand vectors , we obtain
Note that, since and decreases to , , the derivatives satisfy