Emese Szádeczky-Kardoss

Tracking control of the orbitally flat kinematic car with a new time-scaling input

The paper defines a time-scaling scheme such that the scaling function depends on the state variables of the system and on an external variable referred to as the time-scaling input. The system evolving according to the scaled time has thus one more input variable compared to the original system. This time- scaling scheme is applied to the kinematic car with one input, namely the steering angle. The longitudinal velocity of the car is a measurable external signal and cannot be influenced by the controller. Using an on-line time-scaling, which is driven by the longitudinal velocity of the car, by a time-scaling output of the tracking controller, and by their time derivatives up to the second order, one can achieve exponential tracking of any sufficiently smooth reference trajectory with non-vanishing velocity, similar to the two-input differentially flat model of the kinematic car. The time derivatives of the longitudinal velocity and the time-scaling output are no longer necessary if one designs the tracking controller for the linearized error dynamics around the reference path.

Path Planning and Tracking Control for an Automatic Parking Assist System

This paper presents two major components of an automatic parking assist system (APAS). The APAS maps the environment of the vehicle and detects the existence of accessible parking place where the vehicle can park into. The two most important tasks the APAS must then realize are the design of a feasible path geometry and the tracking of this reference in closed loop such that the longitudinal velocity of the vehicle is generated by the driver and the controller determines the steering wheel angle which is realized by an Electronic Power Assist Steering (EPAS).

Tracking Error Based On-line Trajectory Time Scaling

This paper presents an online time scaling concept for robotic manipulators. The task is to ensure the fastest possible motion along a trajectory with a given geometry in the robot workspace. The other requirement is tracking accuracy. Robots should move not only with high speed but with high precision as well. The speed of the motion cannot be arbitrary fast. It is delimited by the saturation of the actuators and by some other (e.g. security) constraints. Offline time scaling solutions change the speed of the motion without changing its geometry, thus these constraints are not violated. If the exact torque bounds are not known in advance, another method should be used. This work gives an online time scaling algorithm which is based on the tracking error. Simulations demonstrate the usefulness of the algorithm proposed

Extension of the Rapidly Exploring Random Tree Algorithm with Key Configurations for Nonholonomic Motion Planning

The rapidly exploring random tree (RRT) algorithm is a randomized path planning method specifically designed for robots with nonholonomic constraints. This method builds a tree during the calculations and the path is searched in this tree. To solve special, more involved problems (e.g. moving through a narrow passage) the usage of a key configuration is suggested. A key configuration helps to find the path from the start to the goal position. This paper gives a method to use key configurations in the RRT method

On-Line Trajectory Time-Scaling to Reduce Tracking Error

Summary. The paper describes an on-line trajectory time-scaling control algorithm for wheeled mobile robots. To reduce tracking errors the controller modifies the velocity profile of the reference trajectory according to the closed loop behavior of the robot. The geometry of the reference trajectory is unchanged, only the time distribution varies during the motion. We give a control algorithm which uses time-scaled reference and a feedback calculated from the linearized error dynamics. The closed loop behavior is also studied together with the controllability of the linearized error dynamics.

On-line time-scaling control of a kinematic car with one input

This paper reports a time-scaling scheme to realize a tracking controller for the non-differentially flat model of the kinematic car with one input which is the steering angle or the angular velocity of the steering angle. The longitudinal velocity of the car is a measurable external signal and cannot be influenced by the controller. Using an on-line time-scaling, driven by the longitudinal velocity of the car, by a scaling output of the tracking controller, and by their time derivatives up to the second order, one can achieve exponential tracking of any sufficiently smooth reference trajectory, similar to the differentially flat case with two control inputs. The price to pay is the modification of the finite traveling time of the reference trajectory according to the time-scaling.

Time-scaling in the Control of Mechatronic Systems

Time-scaling is not a new concept in the theory of dynamical systems. It has been used to modify the time distribution along the reference paths and also to transform a system by changing the clock with which it evolves. The motivation to introduce time-scaling is often to gain useful properties for the system which evolves according to the modified time. It is shown in (Sampei & Furuta, 1986) that such a property to gain with time-scaling may be feedback linearizability. The notion of orbital flatness introduced by Fliess et al. (Fliess et al., 1995, 1999) involves also time-scaling to define an equivalence between a class of nonlinear systems and finite chains of integrators. The problem to check the orbital flatness of single input systems is addressed in (Respondek, 1998) and (Guay, 1999) together with the well known example of the kinematic car with constant longitudinal velocity which is shown to be orbitally flat. The time-scaling introduced by these concepts involves the state variables to express the relation between the different time scales, hence the time-scaling does not involve any new input or variable external to the system. Another concept for time-scaling is to use the tracking error in closed loop to modify the time-scaling of the reference path (Levine, 2004). Such methods change the traveling time of the reference path according to the actual tracking error by decelerating if the motion is not accurate enough and by accelerating if the errors are small or vanish. This chapter presents a new time-scaling scheme which is not driven only by the state variables of the system but also by a new input, referred to as the time-scaling input. In the setup suggested in this chapter, the new input variable, which is not an input of the original physical system, is also used to drive the time-scaling of the reference in closed loop. The usefulness of our approach is demonstrated for the nonholonomic model of the kinematic car with one input. Notice that solutions to the motion planning and tracking algorithms are reported to the kinematic car with two inputs (Cuesta and Ollero, 2005) exploiting its differentially flatness property or using other methods (Dixon et al., 2001). We show that the kinematic car with one input, such that the longitudinal velocity does not vanish, can track any smooth trajectory with non-vanishing longitudinal velocity such that the tracking error is reduced exponentially along the path. This is achieved using timescaling and a dynamical feedback similar to the differentially flat case.

On Drift Neutralization of Stratified Systems

Several mechanical systems possess stratified configuration space [1], which means that each contact combination implies a distinct nonholonomic constraint dividing the configuration space into intersecting submanifolds called strata. The equations of motion may be different on each submanifold and change discontinuously when the system steps across to another submanifold. The submanifold with the lowest dimension (highest number of constraints) is called the bottom stratum and it plays a distinguished role in the MPP, because the subsystem defined on it is not controllable (in the sense that the Lie Algebra Rank Condition is not satisfied). Therefore, one has to switch to other strata to find feasible path between any two points of the bottom stratum. To achieve this goal, successive or cyclic switching between strata is required. Such a strategy supports legged robots in the gaiting [2].

Velocity obstacles for Dubins-like mobile robots

The paper deals with the motion planning problem for Dubins-like mobile robots in dynamic environment. Velocity obstacles (VO) method and its non-linear version (NLVO) can be applied for planning a collision-free trajectory for a robot moving among static and dynamic obstacles, whose positions and velocity vectors are supposed to be known. VO and NLVO algorithms determine a velocity vector for the robot which corresponds not necessarily to the orientation of the robot, hence a Dubins-like mobile robot cannot apply it exactly. In this paper we present a method similar to NLVO, but it results motions feasible for Dubins-like robots. Longitudinal velocities and radii of turning circles are calculated, which ensure collision-free motion if the movement of the obstacles are known for some time-horizon.