Robotic systems are expected to engage in various types of tasks, such as housework, nursing and welfare work, and industrial work done by skilled workers. Although fully automated robots are desirable, it appears difficult to produce such robots from the viewpoints of cost efficiency and the technologies available currently. Human-operated robotic systems are a good compromise, and hence are widely studied. Objectives of these robots include extending human mechanical power (Kazerooni & Steger, 2006), providing precise and smooth operation for human workers in difficult tasks (Bettini et al., 2001) (Peshkin et al., 2001), and executing a task in remote or hazardous environment (Anderson & Spong, 1989) (Lawrence, 1993).
In human-operated robotics systems, controllers are required to incorporate the human operator's command and compensate for the operator's mistakes without reducing the ease of operation. For this purpose we propose a model reference control approach, in which the reference model generates a desired trajectory according to the operator's input and constraints such as collision avoidance. This approach is applied to a two wheeled mobile robot that transports an object. This type of robot has various applications in many areas. Because transporting objects is a fundamental task of robotic systems, we realize a function to prevent slip and tumble of the transported object even when the human operator makes mistakes during operation. Fixing the transported object to the robotic system to prevent the object from tumbling requires extra time to transport the object and reduces the operational ease. This is because fixing is a time-consuming and inconvenient task. In particular, supposing that the robot is operated by elderly or disabled people, this function will be necessary for providing easy and safe operations. In addition, a collision function is implemented by the proposed model reference approach.
Many studies have been conducted into the obstacle avoidance of mobile robots (Bonnafous & Lefebvre, 2004) (Fox et al., 1997) (Khatib, 1986 gren, P. & Leonard, 2005). Most of the existing approaches provide sophisticated algorithms that minimize some objective functions, such as required time to reach the goal and moving distances. However, these methods assume the fully automatic motion of robotic systems, and hence, the human operator's commands cannot generally be incorporated in real-time. In addition, tumble avoidance of the transported object is not considered in current methods. In the case of human-operated robot, a simple algorithm for real-time calculation, rather than optimization, is required because the time-consuming processing required may reduce the robot's operativity. The effectiveness of the proposed approach is demonstrated by experimental results, where ten unskilled operators operate the robot with/without the proposed method.
2. Human-Operated Mobile Robot
In this chapter, we consider a control problem of a general type two-wheeled mobile robot that transports an object as shown in Fig. 1. Human operators are enabled to handle the robot using control sticks. They can give command signals for driving forces of each wheel and by inclining left and right sticks, respectively. The magnitudes of driving forces are proportional to the inclined angles of the sticks. The robot dynamics is given as follows:
where and are the inertia and the mass of the robot, respectively. The symbol is the half distance between the two wheels. The symbols and are the translational speed and rotation angle of the robot, respectively. The slip of wheels is not considered in this study. The shape of the robot is assumed as a circle for simplicity. Distance sensors to detect obstacles are located symmetrically with respect to the centreline parallel to the translational direction of the robot as shown in Fig. 1. The distance from the centre of the robot to each sensor is denoted by . The located direction of each sensor from a line that links wheel centres is denoted by . Note that has a positive value.
3. Controller Design
3.1. Model reference control for obstacle avoidance
To consider the nonholonomicity of the robot and incorporate the operator's command, we propose an obstacle avoidance algorithm based on the model reference approach as shown in Fig. 2, where the reference model generates the desired angles of each wheel, and , according to the operator's command input and distance sensor information. The reference model, which has a similar dynamics with the mobile robot except for an obstacle avoidance function, is given as follows:
where and are the virtual viscous friction coefficients. The viscous friction terms generally exist due to the actuator viscous friction and increase the system stability. We use these terms to increase the control system stability as shown in the analysis in Section 3.2. The symbols and are the distances between sensors at angle and the obstacle, where the subscripts and mean that the sensor is located at the left and right wheel side, respectively. Only sensors that are located in the same half side of the robot body with moving direction are active. The symbol denotes the half number of the active sensors. The last two terms on the right-hand side of Eq. (3) give an effect of steering. The magnitude of the steering depends on the distances to the obstacle and . The last two terms on the right-hand side of Eq. (4) play a role of brake. The magnitude of braking force also depends on the distances to the obstacle. The symbols and are constant parameters for changing the effects of these steering- and brake-like functions. The th roots of the distances are employed in these terms for varying the response to the obstacle, and their effects are shown in Fig. 3. Decreasing the value of increases the effects of steering- and brake-like functions. The reference motion of the robot is obtained by numerically integrating Eqs. (3) and (4). The values of and in Eqs. (3) and (4) are converted into wheel reference signals as follows:
where is the radius of wheel.
3.2 Stability analysis based on linear model
This section presents a stability analysis based on a linear model of the proposed reference model in Eqs. (3) and (4). In this analysis, we consider the case in Fig. 4, where the two parallel walls are obstacles. It is assumed that the mobile robot moves almost along the centerline between the two walls with a velocity , where is a desired constant and is a small-sized variable. Because the mobile robot is in an almost straight line motion with a constant velocity, it is reasonable to assume that the input from the operator satisfies the relation and . We also assume that both the shift from the centerline and the inclination from the lateral line in Fig. 4 are small-sized variables.
The distance between each sensor and walls are given by
where is the half distance between the walls. Because and are small-sized variables, the following linear approximation is reasonable:
Note that and are positive constants.
The following linear approximation is also reasonable because and have small magnitudes:
where and are positive constants.
The dynamics on is given as:
Equation (4) is linearized as:
Because Eq. (11) has no coupling term on and , we consider Eqs. (9) and (10) in the stability analysis. It should be noted that Eq. (11) is stable because and is positive and the right-hand side is bounded.
Defining a vector , we have the following linear dynamics from Eqs. (9) and (10):
The characteristic polynomial of the system Eq. (12) is
Because all the coefficients of the right-hand side of Eq. (13) are positive, the stability condition is given by:
From Eqs. (7) and (14), the following sufficient condition for the stability is derived:
By assigning the coefficient such that Eq. (15) is satisfied, the stability of the linearized dynamics in Eq. (12) is guaranteed.
3.3 Object transportation control
Because transporting objects is a fundamental task of robotic systems, we include a function to prevent slip and tumble of the transported object in the reference model block in Fig. 2 even when the human operator makes mistakes during operation. Fixing the transported object to the robotic system to prevent the object from slip and tumble requires extra time to transport the object and reduces the operational ease.
Because the value of in Eq. (4) does not necessarily have an exact value of the mass of robot, we change this value in real time to adjust the reference acceleration to prevent the object from slip and tumble. Increasing this value reduces the magnitude of the reference acceleration.
In this study, we assume that the slip and tumble of the transported object is caused mainly by the translational acceleration, although the acceleration normally includes the centrifugal and the Coriolis terms. The slip of the object is prevented if the inertial force is smaller than the static friction force as follows:
where is the mass of the transported object, is the static friction coefficient of the contacting surface between the object and the robot, and is the gravitational acceleration. Figure 5 is the schematic of this relation. Hence, we have the allowable acceleration to avoid the slip as follows:
Next, we consider the allowable acceleration to avoid the tumble. We assume that the object starts to rotate at the end point of the contacting surface with the robot as shown in Fig. 5. Considering the equation around the centre of rotation, we obtain the following condition for preventing the object from starting to tumble.
where is the angle from the contacting surface line to the centre of gravity of the transported object, and is the distance between centres of rotation and gravity, as shown in Fig. 5. Hence, we obtain the acceleration limit for avoiding the tumble as follows:
To avoid the tumble, we propose to adjust the mass coefficient as follows from Eq. (4) :
where is the initial value of the mass coefficient
The effectiveness of the proposed controller is experimentally verified in a corridor-like space shown in Fig. 6. Parameter values for the experiment are given in Table 1. Parameters for obstacle avoidance and are determined in a trial and error manner. DC servo motors (20 [W]) are employed for each wheel motion. Rotary encoders (500 [PPR]) attached to the motors are used for measuring the position and orientation of the robot. Infrared distance sensors, whose measurable ranges are 4 - 30 [cm], are employed to measure the distance to the obstacle.
To verify the effect for the operational easiness, ten unskilled persons (students) are employed to operate the robot with the transported object in Fig. 1 under the following conditions:
|0.056 [kgm 2]||3.0 [Nms/rad]||4 [Nm 5/4]|
|5.4 [kg]||10.0 [Ns/m]||10.0[Nm 1/4]|
|0.11 [m]||0.029 [m]||4|
(a1) Manual control
(a2) Control with the obstacle avoidance function presented in section 3.2.
(a3) Control with the obstacle avoidance and the tumble avoidance functions in section 3.3.
In (a3), only the tumble is considered because in this experiment.
Figures 7 - 9 show the obtained robot trajectories by one operator under conditions (a1) – (a3), respectively. In case (a1), as negative values of and are shown in Fig. 7 (a), backward motions were required to pass through the course. The backward motion is confirmed in Fig. 7 (d). In addition, both collision and tumble occurred in this case. The latter is caused by the large magnitude of acceleration as shown in Fig. 7 (b).
In case (a2), although and were almost constant during operation as shown in Fig. 8 (a), the robot changed its orientation in Fig. 8 (c) by the obstacle avoidance function. In addition, no backward motion was required as shown in Fig. 8 (d). However, a large magnitude of the acceleration in Fig. 8 (c) caused the tumble of the transported object.
In case (a3), the robot was enabled to smoothly pass through the course by an almost constant inputs in Fig. 9 (a) without requiring a large magnitude of acceleration as shown in Fig. 9 (b).
Table 2 summarizes experimental results by ten unskilled operators (students), where no collision occurs in cases (a2) and (a3), and no tumble occurs in case (a3), for all operators. Figure 10 summarizes the control time required to pass through the course. The control time is largely reduced for almost all operators by the model reference control approach, because they do not have to consider the obstacle or tumble avoidance during operation. The control time in case (a3) increases little compared to case (a2), although the acceleration magnitude is reduced to avoid the tumble.
: Not occur, ×: Occur
|Operator×fs No.||(a1) Manual control||(a2) Obstacle avoidance||(a3) Obstacle and tumble avoidance|
This chapter presents a model reference control approach for a human-operated mobile robot that transports an object. This type of robot has wide applications in industrial and household tasks. The operational easiness of the robot is verified by experiments where operators are required to operate the robot with a transported object to pass through a corridor-like space. Because even young students failed to operate the robot, a function to support the operation is obviously required. The operational easiness is improved by the proposed approach, with which all operators succeeded in transporting the object without collision nor tumble of the object.