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An adaptive dynamic controller for autonomous mobile robot rajectory tracking
abstract
This paper proposes an adaptive controller to guide an unicycle-like mobile robot during trajectory tracking. Initially, the desired values of the linear and angular velocities are generated, considering only the kinematic model of the robot. Next, such values are processed to compensate for the robot dynamics, thus generating the commands of linear and angular velocities delivered to the robot actuators. The parameters characterizing the robot dynamics are updated on-line, thus providing smaller errors and better performance in applications in which these parameters can vary, such as load transportation. The stability of the whole system is analyzed using Lyapunov theory, and the control errors are proved to be ultimately bounded.
Simulation and experimental results are also presented, which demonstrate the good performance of the proposed controller for trajectory tracking under different load conditions.
1. Introduction
Among different mobile robot structures, unicycle-like platforms are frequently adopted to accomplish different tasks, due to their good mobility and simple configuration. Nonlinear control for this type of robot has been studied for several years and such robot structure has been used in various applications,such as surveillance and floor cleaning. Other applications, like industrial load transportation using automated guided vehicles (AGVs) automatic highway maintenance and construction, and autonomous wheelchairs, also make use of the unicycle-like structure. Some authors have addressed the problem of trajectory tracking, a quite important functionality that allows a mobile robot to describe a desired trajectory when accomplishing a task.
An important issue in the nonlinear control of AGVs is that most controllers designed so far are based only on the kinematics of the mobile robot.
However, when high-speed movements and/or heavy load transportation are required, it becomes essential to consider the robot dynamics, in addition to its kinematics. Thus, some controllers that compensate for the robot dynamics have been proposed.
As an example, Fierro and Lewis (1995) proposed a combined kinematic/torque control law for nonholonomic mobile robots taking into account the modeled vehicle dynamics. The control commands they used were torques, which are hard to deal
with when regarding most commercial robots. Moreover, only simulation results were reported. Fierro and Lewis (1997) also proposed a robust-adaptive controller based on neural networks to deal with disturbances and non-modeled dynamics, although
not reporting experimental results. Das and Kar (2006) showed an adaptive fuzzy logic-based controller in which the uncertainty is estimated by a fuzzy logic system and its parameters were tuned on-line. The dynamic model included the actuator dynamics, and the commands generated by the controller were voltages for the robot motors.
The Neural Networks were used for identification and control, and the control signals were linear and angular velocities, but the realtime implementation of their solution required a high-performance computer architecture based on a multiprocessor system.
On the other hand, De La Cruz and Carelli (2006) proposed a dynamic model using linear and angular velocities as inputs, and showed the design of a trajectory tracking controller based on their model. One advantage of their controller is that its
parameters are directly related to the robot parameters.
However, if the parameters are not correctly identified or if they change with time, for example, due to load variation, the performance of their controller will be severely affected.
To reduce performance degradation, on-line parameter adaptation becomes quite important in applications in which the robot dynamic parameters may vary, such as load transportation.
It is also useful when the knowledge of the dynamic parameters is limited or does not exist at all.In this paper, an adaptive trajectory-tracking controller based on the robot dynamics is proposed, and its stability property is proved using the Lyapunov theory.
The design of the controller was divided in two parts, each part being a controller itself. The first one is a kinematic controller, which is based on the robot kinematics, and the second one is a dynamic controller, which is based on the robot dynamics. The dynamic controller is capable of updating the estimated parameters, which are directly related to physical parameters of the robot. Both controllers working together form a complete trajectory-tracking controller for the mobile robot. The controllers have been designed based on the model of a unicycle-like mobile robot proposed by De La Cruz and Carelli A s-modification term is applied to the parameter-updating law to prevent possible parameter drift.
The asymptotic stability of both the kinematic and the dynamic controllers is proven. Simulation results show that parameter drift does not arise even when the system works for a long period of time. Experimental results regarding such a controller are also presented and show that the proposed controller is capable of updating its parameters in order to reduce the tracking error. An experiment dealing with the case of load transportation is also presented, and the results show that the proposed controller is capable of guiding the robot to follow a desired trajectory with a quite small error even when its dynamic parameters change.
The main contributions of the paper are: (1) the use of a dynamic model whose input commands are velocities, which is usual in commercial mobile obots, while most of the works in the literature deals with torque commands; (2) the design of an adaptive controller with a s-modification term, which makes it robust, with the corresponding stability study for the whole adaptive control system; and (3) the presentation of experimental results showing the good performance of the controller in a typical industrial application, namely load transportation.
2. Dynamic model
In this section, the dynamic model of the unicycle-like mobile robot proposed by De La Cruz and Carelli (2006) is reviewed. Fig. 1depicts the mobile robot, its parameters and variables of interest. u and o are the linear and angular velocities developed by the robot, respectively, G is the center of mass of the robot, C is the position of the castor wheel, E is the location of a tool onboard the robot, h is the point of interest with coordinates x and y in the XY plane, c is the robot orientation, and a is the distance between the point of interest and the central point of the virtual axis linking the traction wheels (point B). The complete mathematical model
is written as
where and are the desired values of the linear and angular velocities, respectively, and represent the input signals of the system.
A vector of identified parameters and a vector of parametric uncertainties are associated with the above model of the mobile robot, which are, respectively,
where dx and dy are functions of the slip velocities and the robot orientation, du and do are functions of physical parameters as mass, inertia, wheel and tire diameters, parameters of the motors and its servos, forces on the wheels, etc., and are considered as disturbances.
The equations describing the parameters h were firstly presented in, and are reproduced here for convenience. They are
It should be stressed that i=1,2,4,6,Parameters y3 and y5 will be null if, and only if, the center of mass G is exactly in the central point of the virtual axis linking the traction wheels。 In this paper it is assumed that b6=0.
The robot’s model presented in (1) is partitioned into inematic art and a dynamic part, as shown in Fig. 2. Therefore, two controllers are implemented, based on feedback linearization, or both the kinematic and dynamic models of the robot.
3. The kinematic controller
3.1. Design
The design of the kinematic controller is based on the
kinematic model of the robot, assuming that the disturbance
term in (1) is a zero vector. From (1), the robot’s kinematic model s given by
whose output are the coordinates of the point of interest, thus
meaning . Hence
Note 2: The stability of the whole system will be revisited in the next section, in which an adaptive dynamic controller is added to the kinematic controller in order to implement the whole control scheme of Fig. 2.
4. The adaptive dynamic controller
4.1. Design
The dynamic controller receives from the kinematic controller the references for linear and angular velocities, and generates another pair of linear and angular velocities to be delivered to the robot servos, as shown in Fig. 2.
The design of the adaptive dynamic controller is based on the parameterized dynamic model of the robot. After neglecting the disturbance terms du and do the dynamic part of Eq. (1) is
By rearranging the terms, the linear parameterization of the dynamic equation can be expressed as
which can also be rewritten as
Note 4: It is important to point out that a nonholonomic mobile robot must be oriented according to the tangent of the trajectory path to track a trajectory with small error. Otherwise, the control errors would increase. This is true because the nonholonomic platform restricts the direction of the linear velocity developed by the robot. So, if the robot orientation is not tangent to the trajectory, the distance to the desired position at each instant will increase. The fact that the control errors converge to a bounded value shows that robot orientation does not need to be explicitly controlled, and will be tangent to the trajectory path while the control errors remain small.
5. Experimental results
To show the performance of the proposed controller several experiments and simulations were executed. Some of the results are presented in this section. The proposed controller was implemented on a Pioneer 3-DX mobile robot, which admits
linear and angular velocities as input reference signals, and for which the distance b in Fig. 2 is nonzero.
In the first experiment, the controller was initialized with the dynamic parameters of a Pioneer 2-DX mobile robot, weighing
about 10 kg (which were obtained via identification). Both robots are shown in Fig. 3, where the Pioneer 3-DX has a laser sensor weighing about 6 kg mounted on its platform, which makes its dynamics significantly different from that of the Pioneer 2-DX.
In the experiment, the robot starts at x=0.2m and y=0.0 m, and should follow a circular trajectory of reference. The center of the reference circle is at x =0.0m and y= 0.8 m. The reference trajectory starts at x=0.8m and y=0.8m and follows a circle having a radius of 0.8 m. After 50 s, the reference trajectory suddenly changes to a circle of radius 0.7 m. After that, the radius of the reference trajectory alternates between 0.7 and 0.8m each 60 s.
Fig. 4 presents the reference and the actual robot trajectories for a part of the experiment that includes a change in the trajectory radius. In this case, the parameter updating was active.
Fig. 5 shows the distance errors for experiments using the proposed controller, with and without parameter updating, to follow the described reference trajectory. The distance error is defined as the instantaneous distance between the reference and the robot position. Notice the high initial error, which is due to the fact that the reference trajectory starts at a point that is far from the initial robot position. First, the proposed controller was tested with no parameter updating. It can be seen in Fig. 5 that, in this case, the trajectory tracking error exhibits a steady-state value of about 0.17 m, which does not vary even after the change in the radius of the reference trajectory. This figure also presents the distance error for the case in which the dynamic parameters are updated. By activating the parameter-updating, and repeating the same experiment, the trajectory tracking error achieves a much smaller value, in comparison with the case in which there is no ig. 3. The robots used in the experiments.
Fig. 4. Part of the reference and real circular trajectories.
Fig. 5. Distance errors for experiments with and without parameter updating.
6. Conclusion
An adaptive trajectory-tracking controller for a unicycle-like mobile robot was designed and fully tested in this work. Such a controller is divided in two parts, which are based on the kinematic and dynamic models of the robot. The model on sidered takes the linear and angular velocities as input reference signals, which is usual when regarding commercial obile robots. It was considered a parameter-updating law for the dynamic part of the controller, improving the system performance.
A s-modification term was included in the parameter up dating law to prevent possible parameter drift. Stability analysis based on Lyapunov theory was performed for both kinematic and dynamic controllers. For the last one, stability was proved considering a parameter-updating law with and without the s-modification term. Experimental results were presented, and showed the good performance of the proposed controller for trajectory tracking when applied to an experimental mobile robot.
A long-term simulation result was also presented to demonstrate that the updated parameters converge even if the system works for a long period of time. The results proved that the proposed controller is capable of tracking a desired trajectory with a small distance error when the dynamic parameters are adapted. The importance of on-line parameter updating was illustrated for the cases where the robot parameters are not exactly known or might change from task to task. A possible application for the proposed controller is to industrial AGVs used for load transportation, because on-line parameter adaptation would maintain small tracking error even in the case of important changes in the robot load.
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