In the dynamic model of the P U M A 560 robot (1.25), D(i, j) = D(j, i)
5 IMPLEMENTATION OF EVENT-BASED PLANNING AND CONTROL 49
a n d
C(i, j, k) = C(i, k,j) C(i, k, k) = - C(k, j, i),
C(i,j,j) = 0 , i >~j
i , k > ~ j
After n e g l e c t i n g the less s i g n i f i c a n t t e r m s [67], t h e n o n z e r o t e r m s are g i v e n as follows.
I n e r t i a l t e r m s (K9 - m 2 ) :
D(1, 1) = 2.57 + 1 . 3 8 C 2 C 2 n t- 0 . 3 S 2 3 S 2 3 n t- 0 . 7 4 C 2 S 2 3
D(1, 2) = S 2 - - 0 . 0 0 5 7 S 2 3 - 0.1367C23 D(1, 3) = - 0 . 0 0 5 7 S 2 3 - 0.1367C23 D(2, 2) = 6.79 + 0.74S 3
D(2, 3) = 0.3679 + 0.3922S 3 - 0 . 0 1 3 4 C 3 D(3, 3) = 1.16
D(4, 4) = 0.2
D(5, 5) =0.18
D(6, 6) = 0.19 C o r i o l i s t e r m s ( K g - m 2 ) :
C(1, 1, 2) = 0.0174 - 1.362C2S 2 -+- 0 . 3 5 6 2 C 3 S 3 - 0.7124S2S3S23 -+- 0.0268C2S23 n t- 0.3922C2C23 - 0.3922S2S23 - 0.046C22 - 0.0347C23C23 - 0.0112S 3 C(1, 1, 3) = 0.0174 + 0 . 3 5 6 2 C 2 S e + 0 . 3 5 6 2 C 3 S 3 - 0.7124SES3S23
+0.0134C2S23 + 0.3922C2C23 - 0.0347C23C23 C(1, 2,2) = 1.8181C 2 + 0.1367S23 - - 0 . 0 0 5 7 C 2 3
C(1, 2, 3) = 0.1367S23 - 0.0057C23 C(2, 2, 3) = 0 . 3 9 2 2 C 3 + 0.0134S 3 G r a v i t y t e r m s ( N - m):
G(1) - - 0
G(2) = - 9 9 . 8 C 2 - 3 . 7 3 S 2 + 1 . 0 8 C 2 3 - 26.64S23 G(3) = 1.08C23 - 26.64S23
G(4) = 0.085S23S4S 5
G(5) = - 0 . 0 8 5 ( C 2 3 8 5 -Jr- 8 2 3 C 4 C 5 )
G(6) = 0
M o s t of t h e e x p e r i m e n t s w e r e c a r r i e d o u t b y u s i n g t h e p r e c e d i n g d y n a m i c m o d e l . O u r
experimental results, however, suggest that when the velocity of the robot is confined within the limit set by the manufacturer, neglecting the Coriolis and centripetal forces does not cause performance to deteriorate [35].
Sticktion and Friction Compensation
The joint sticktion and friction are not incorporated in the robot dynamic model. They are compensated by augmenting the voltage commands to the controllers.
P U M A 560 is driven by brush-type joint motors. They typically have a breakaway friction that is about 4 to 6% of the full rated torque of the motor that sets a lower bound on the friction values. The sticktion and friction parameter values were experimentally determined [68].
The compensation of sticktion plays an especially important role in event-based planning and control. It can be seen in Figure 1.3 that the initial feedforward and feedback are zero, since the desired acceleration is zero with no position and velocity errors according to the event-based plan (Figure 1.3). Therefore, the sticktion must be well compensated, and some initial error has to be given in order to start a motion. The modified motion plan creates a velocity error about 10% of the maximum planned velocity at the beginning of the motion.
Experimental results show that this scheme can start the motion smoothly and also can overcome the incomplete compensation of the sticktion.
Joint Velocity Measurement Filtering
The joint velocity measurements are derived from the joint angle measurements by differen- tiation. As a result, the joint velocity measurements are very noisy and cannot be directly used for feedback control.
A multistep velocity filter is designed to estimate the joint velocity measurement. Each step calculates the estimate of joint velocity as
i = k - n + l
b(i)+v(k)- max V k - min Vk)/(n -- 2)
where
y~ = { v ( k ) , ~(k - 1) . . . . ~(k - n + 1)}
The estimate of the velocity is a running average of prior estimates and new measurements after removing the maximum and minimum values. The window of filtering is n.
Each filter can run independently without synchronizing with the others. It is convenient to implement filters in the distributed computing architecture as shown in Figure 1.25.
In the experiments presented in Sections 3 and 4, three filter units were used for each joint velocity. One unit was implemented in the U M C user processor. The other two units were implemented in SGI.
The main focus of this section has been the implementation of an event-based planning and control scheme in a distributed computing architecture. The system that is described in this section has proved to be an extremely successful research tool for dual-arm coordinated control.
Apart from being versatile, robust, and open ended in its architecture, it can provide sensory information at various levels, according to the requirements of the control strategy.
6 CONCLUSIONS 51
In particular, the experiments reported in previous sections conclusively demonstrate the effectiveness of the system. Another important feature of the system is that it can also accommodate a variety of control strategies besides the event-based scheme. All control strategies may be implemented easily by coding in high-level languages such as the C programming language.
The computing and control architecture demonstrated here can clearly pave the way for a new generation of commercial robot controllers that are more responsive, more flexible, more efficient, and more robust.
6 CONCLUSIONS
In this research an integrated event-based planning and control method has been developed using a motion reference variable other than time. It has been successfully applied to single-robot arm motion planning and control, as well as multirobot coordination planning and control. The important contributions of this research are as follows:
1. A new planning and control scheme--event-based planning and c o n t r o l - - h a s been developed. Instead of time, the events of a system are used as a motion reference to describe the motion plan and to drive the system. The time is implicitly included inside the motion plan and control process. Therefore, the planning along with the feedback control becomes a real-time dynamic process. The planner becomes an investigation- decision component of the system. As a result, the system has the ability to deal with unexpected and uncertain events. The event-based planning and control method can be an important step toward the development of intelligent planning and control theory.
2. The event-based planning and control theory has been applied to single-robot motion planning and control. An event-based representation of robot arm motion in the task space is proposed. The time and energy optimization techniques are used to determine event-based trajectories. A new event-based error definition and computation scheme has been introduced and combined with a nonlinear feedback control law, which linearizes and decouples the control in the task space.
The significance of the event-based motion planning and control scheme is its compatibility with sensor-based planning and control, because sensed events in robotic manipulation rarely, if ever, occur on a precise time scale. Obstacle avoidance is an important example. Therefore, it can significantly improve the safety and reliability of robotic systems.
The event-based planning and control schemes were experimentally implemented and tested on the 6-DOF position and orientation control of a P U M A 560 robot arm with very good results.
3. Based on the event-based planning and control theory, a new coordination scheme for a multirobot system has been developed. The event-based motion reference for the planning and control of coordinated robots has been introduced. It drives the system to achieve optimal coordination. After introducing the general task space, the task- independent controllers for each robot have been designed. The significance of the event-based coordination scheme is that the controls of robots are carried out on the task level and the structure of the control system is task independent. This makes it possible for the multirobot system to work on complex tasks. In addition, the scheme
can be implemented in a distributed computing architecture, which increases the efficiency and flexibility of multirobot system operation.
The event-based coordination scheme has been experimentally implemented and tested for the coordinated control of two 6-DOF P U M A 560 robots with very good results.
4. A new hybrid position-force controller has been developed. It incorporates the robot dynamics as well as the dynamics of robot joint motors. A new stability conclusion has been obtained. It requires the consideration of joint motor dynamics for the design of a stable and improved hybrid position-force controller. The inclusion of the joint motor dynamics improves the force control performance, in particular in the case of tracking the desired variable force.
5. The event-based multirobot coordination scheme is developed with consideration of the implementation. It naturally lends itself to a distributed computing architecture. Based on the existing computing facility at the Center for Robotics and Automation in Washington University, a distributed computing and control architecture has been developed to implement the event-based coordination scheme. It combines the tightly and loosely coupled schemes. High-speed information exchange (1000 Hz) has been achieved. In addition, all processors can be programmed separately. This provides a convenient user interfacing method.
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