A testbed is developed for testing Multi-AGent Intelligent Coordination and Control (MAGICC). The robots act as agents which make decisions and act upon them via classical control theory. The testbed consists of sensors( camera and encoders), embedded controllers, modems and a computer network which are integrated into a hierarchical control which has decision makers, controllers, estimators and predictors. The testbed has successfully been applied to initializing the robots into geometric formation. Simulation and Hardware results are presented

Beard, Randal W.

Young, Brett J.

English

DigitalCommons@USU

I I I I i I I I I I i I I I I I I I I I I I I I I I I A Testbed Architecture for MAGICC Applications Brett J. Young Randal W. Beard Electrical and Computer Engineering Department Brigham Young University Abstract A testbed is developed for testing Multi-AGent Intelligent Coordination and Control (MAGICC). The robots act as agents which make decisions and act upon them via classical control theory. The testbed consists of sensors( camera and encoders), embedded controllers, modems and a computer network which are integrated into a hierm;chical control which has decision makers, controllers, es-timators and predictors. The testbed has success-fully been applied to initializing the robots into geometric formation. Simulation and Hardware results are presented. 1 Introduction A fascinating problem emerging in engineering is that of having multiple agents working together as a group to achieve some goal or mission. Exam--ples of multiple agents include satellites flying in formation as well as robots working cooperatively to achieve some planetary exploration. The math-ematics of having satellites move in formation or robots cooperate traditionally falls under game theory. The problem becomes complex with more than two players and the payoff matrices can liter-ally take hours to calculate [Randal Beard, 1998] Also, conditional constraints may be imposed on the problem such as limited sensor range and com-munication. Having global coordination in such situations may be difficult. However, humans work in such environments everyday. The follow-ing list specifies some of the expectations for mul-tiple coordinating agents: Independence- Each agent should be able to as-sess the situation and act upon what data it has available, especially when communication is lim-ited. 1 Robustness- As given in [Brooks, 1986), the agents should act rational despite altered con-straints. Stability- Where applicable control theory should be applied. For example, if a robot de-cides to go to point A from point B, it should be able to guarantee convergence to its decision. Achievability- The agents should be able to guarantee some measure of success with respect to their group goals. For example, formation ma-neuvers should have guaranteed success. The multi-agent problem has started to re-ceive some attention in robotics [Brooks, 1986]. purposes a layered architecture which has low level controls as well a heuristic decision maker. Another approach used specifically for for-mation control relies completely on heuristics [Balch and Arkin, 1998], where the robots at-tempt to imitate flocking birds. Such ad hoc ap-proached work reasonably well for specific appli-cations. Yet, they often fail in general and new heuristics must be developed. Also the mathe-matical analysis of heuristics do not develop issues such as stability and convergence. Recently, a new tool for controlling mul-tiple agent has been developed at BYU [Stirling and Goodrich, 1999]. The theory uses a mathematical tool for decision making based on Isaac Levi's ideas on rational decision making. The advantage is that each agent can make coordinating decisions independently. This is different from other schemes since each agent infers what the actions of his neighboring agents. They relate to each other through an interdependence function. This approach does not try to find the "best solution" or minimize a pay-off matrix. Instead it tries to find good solutions with respect to some criteria (e.g. having robots all go to different formation targets in a reasonable amount of time). The remainder of this paper outlines a hardware architecture which has been constructed to implement and test Figure 1: The robots built for the MAGICC LAB multiple agent coordinated control via multiple wheeled robots. The overall structure developed for the testbed will be presented. To demonstrate the functionality of the testbed, simulation and experimental results will illustrate our solution to the problem of regulating a robot to a specified location 2 Basic Hardware First the robots had to be designed and built physically. The design agreed upon was a two wheeled mobile robots which had two casters: one in the front and one in the back. The two main wheels are driven by Pitman motors. The Pitman motors also have encoders mounted to them. We built our own decoder board to read the decoder chips. The Handy Board developed at M.l. T. was used as an embedded controller on the robot [Martin, 1998]. Encoder counts were sent to a computer via a radio modem, at an update rate of 0.2 samples/second. A camera was mounted above the testbed to provide global sensing capabilities. Colored dots were placed on top of the robots so that the camera could spot the robots via an edge detection algorithm. Also an algorithm was de-veloped and used to undo the distortion via edge detection. A picture of one robots can be seen in Figure 1. The picture of the testbed can be seen 2 Figure 2: The robots and their playing field in Figure 2. 3 Robot Model The governing equations of motion for the robot are important. A two wheeled mobile robot is a non-holonomic system which means that the ve-locity of the robot is constrained to be in its direc-tion of motion. Let x,y denote the Cartesian coor-dinates of the robot, 8 represents the orientation of the robot, v the linear velocity in the direction of motion and w the angular velocity of the robot. Thus the position vector of the robots is given by Xpos = (x,y,8) This can be seen in Figure 3. The full state vector is given by x = (x, y,8, v,w) After applying some basic physics to the prob-lem the equations are of the form: Kinematic equations: ± = vcos(8), (1) iJ = vsin(8) (2) il=w (3) Dynamic equations: mil= F (4) Jw=r, (5) where m is the mass, J the inertia, F is force and r the torque. Thus the equations are sep-arable into linear and rotational motion, a fact I I I I I I I I I I I II I I I I I I II L_------------------------------------------------------------------------------------------------------~ ' I i -I i ' I I I I -! I I ~ ! ! Robot Coordinate System Figure 3: The generalized position coordinate which shall be exploited in the control. The kinematic equations are the non-holonomic equa-tions for multiple robots and can be found in [Kolmanovsky and McClamroch, 1995]. The lin-ear and rotational forces must be translated into a force on the wheels. The equation which decom-poses these forces into wheel forces is given by: F T !wheel= -2 - ( )2 Trobot (6) The wheel forces are then further converted into Volts for motor voltage, given by Vin = kmp1(fwheel) + kmp2(vwheel) (7) where kmpl,kmp2 are motor constants, and Vwheel is the velocity of the wheel. However, the robots were controlled by an embedded controller which uses pulse width modulation and accepts discrete values between -100 to 100, where 100 corresponds to 12V. A look-up table was used to convert from Volts to motor command values. This table is non-linear and was developed by empirical testing. 4 Control Architecture All of the components in the hardware combine to make a hierarchical structure for the robots. Figure 4 presents this overall structure. The agent or rational decision maker decides where to go. The formation control generates the desired tra-jectories. Note that for regulation these are just set points and orientation. The desired values are 3 Figure 4: The hierarchy structure fed into the low-level control which produces mo-tor commands to the robots. 5 Estimator /Predictor The overall structure of the sensors is similar to that of missiles which use GPS(slow) and ac-celerometers(fast) to determine position and ac-celeration. In our case, the camera acts like a GPS(system) getting updates roughly every 2 sec and the encoders measure velocity and are analo-gous to accelerometers with updates every 0.2 sec. One difference is that the encoders only measure counts of wheel rotations. This is equivalent to measuring changes in x,y, and 8. So, the observer is not just a simple integrator. The velocity and Figure 5: Block Digram of Predictor /Estimator. angular velocity were estimated from propagating 'Ir .. I I I 0 100 3» :m 400 500 iO 700 Tme 11L .. 0 100 3» :m 400 500 700 Tme i'l-1 --1 -~--~-===~ 0 ~ 3» ~ 400 500 iO line Figure 6: Block Digram of Predictor/Estimator. a model of the robot using a Runge-Kutta tech-nique and updates to position and angle were a weighted sum of the encoder data and the ob-server model. The gains used to weight encoder and model data are equivalent to Kalman Gains. \Vhen camera updates are received they are 2 sec old, so the old encoder data is weighted with the camera data and propagated thorough a Kalman predictor to update the estimate. The estimator then feeds into a predictor to predict the current state x needed in the control. The reason a predic-tor is necessary is because the control is delayed 0.2 sec by the radio modems. Figure 5 shows a block diagram of the Estimator/ Predictor con-trol. Figure 6 shows how the predictor matched up to robot states in simulating robot regulation. 6 The control The non-holonomic nature of the robot makes the control difficult. Simple PD control will not work due to Brockett's Theorem which states that it is impossible to regulate a non-holonomic sys-tem with a Continuous Time Invariant Control [Brockett, 1983]. H simple PD control is applied the robots gets stuck in a limit cycle about the desired position and just spins about in a circle around the desired position. Finally a discontin-uous control was implemented which put the dis-continuity in the angle. The control is similar to that of [Astolfi, 1995]. The basic control Law used was: 4 Linear Control: miJ = -k1(x - Xd) - k2v (8) Rotational Control: Jw = -k3(8- 8d)- k4w,8d = tan-1(Yd) (9) Xd This control is given without proof. However, sim-ulation and hardware results are present below in Figure 7. The x,y position converge rapidly. How-ever there is a difference between desired steady angle. One reason is that the robot has static fric-tion which makes it hard to rotate at low veloci-ties. The forces to the motors become low, making it hard for the robot to reach its desired orienta-tion. This is because the gains on the linear and rotational control cannot be simultaneously high or the asymptotic convergence is long and often takes bizarre paths. So when the robot regulates to the right position the rotational control is low and the robot has difficulty overcoming frictional effects. Thinking about what the control is do-ing, it can be thought of as trying to park your car while driving to the destination. The robot is trying to simultaneously drive the x,y and 8 to some desired state. An alternative approach is to have the robot orient itself in the direction it wants to go, move there and re-orient. This has the added advantage that the rotational and lin-ear gains can separately be made large-helping to eliminate problems with friction. This is the basic idea behind the next control presented. 7 A Hybrid Control The hybrid control transitions between different control laws via a state machine which reaches a final state[Branicky, 1998]. In this case the final state is II X II::; fl, II y u::; f2, II B II::; fJ. Simulation results are presented in Figure 8. With a good control and estimator in place the testbed can be used for multi-agent coordination problems. The next step in application will be in having the robots do simple formation maneu-vers such as expansions and contractions. The low level part of the testbed is in place. Now the deci-sion making and supervisor control levels will be implemented. 8 Conclusion In this paper a testbed for studying Multi-Agent problems was presented. The testbed was built mainly out of off the shelf components and for a modest price. With the generalized framework in Ill I I I I il I I il I I I I I I I I I I I I I I I, I I I -! I I I I i I I ' I I I I I I I ' * ' t:-1 I ~ I "3-2 I Q, 1000 Sdid Lila- Experimen1al Results Dashed Lila- Sim~atioo Results 1500 T1me 2000 2500 >-a ---------------------------500 1000 1500 2000 2500 :JlOO lime 10 t: 0 ~ 5 Q, • '61 c o( 0 ; '------too 200 300 400 500 600 lime Figure 7: Simulation vs Hardware results for reg-ulation place, the MAG ICC lab is ready to test algorithms for the multi-agent problem. The framework is sig-nificant in that it will allow mathematical analysis of stability and achievability while being robust and flexible. The MAGICC lab hopes to create general tools for multi-agent problems and use the testbed to demonstrate some specific applications; namely, formation control and material handling. References [Astolfi, 1995] Astolfi, A. (1995). Exponential sta-bilization of a wheeled mobile robot via disconti-nous control. Nonlinear Control System Design Symposium, pages 7 41-7 46. [Balch and Arkin, 1998] Balch, T. and Arkin, R. C. (1998). Behavior-based formation con-trol for multirobot teams. IEEE Transactions on Robotices and Automation, 14 No 6:926-939. [Branicky, 1998] Branicky, M.S. (1998). Multiple lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transac-tions on Automatic Control, 43 No4:475-482. 5 -0.5 -1 -t.SL------.1.----'---.....1.---..l....-----'-----l 0 100 300 Tme 400 500 600 ~-1 'jj 0 Q, >-2 -3 0 100 200 300 400 500 600 rme 1.5 c ~ 8 I Q, tOO 200 300 400 500 600 Tme Figure 8: Simulation of a Hybrid Control for reg-ulation. [Brockett, 1983] Brockett, R. (1983). Asymptotic stability and feedback stabiliaztion. Differential Geometric Control Theory, pages 181-191. [Brooks, 1986] Brooks, R. (1986). A robust lay-ered control system. IEEE Journal of Robotics and Automation, RA-2 No 1:14-23. [Kolmanovsky and McClamroch, 1995] Kolmanovsky, I. and McClamroch, N. H. (1995). Developments in nonhlonomic control problems. IEEE Control Systems, pages 20-36. [Martin, 1998] Martin, F. G. (1998). The handy board techinial referecne. http:// el. www. media. mit. edujprojectsjhandy-board. [Randal Beard, 1998] Randal Beard, Wynn Stir-ling, R. F. (1998). A hierarchical coordina-tion scheme for satellite formation initialization. AIAA Guidance, Navigation and Control Con-ference. [Stirling and Goodrich, 1999] Stirling, W. C. and Goodrich, M. (March 1999). Satisfycing games. Infonnation Sciences, 144:255-280. *~--------~--------

A Testbed Architecture for MAGICC Applications

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A Testbed Architecture for MAGICC Applications

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