Analysis. Control of a dierential-wheeled robot. Part I. 1 Dierential Wheeled Robots. Ond ej Stan k

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1 Control of a dierential-wheeled robot Ond ej Stan k SRH Hochchule Heidelberg, Mater IT, Advanced Control Engineering project Abtract Thi project for the Advanced Control Engineering cla invetigate a movement control ytem of a dierential-wheeled robot. A model of the robot drive i deigned in Matlab Simulink, and a control ytem i analyzed, implemented a tuned. The work i baed on author' previou experience in robot control. The control ytem preented in thi paper wa already teted on two real robotic platform. Thi time, the author invetigate the theoretical background behind hi particular control ytem, that wa originally developed without any prior kill in Control Engineering theory. Part I Analyi 1 Dierential Wheeled Robot Dierential drive i very popular in mobile robotic. Dierential wheeled robot uually feature two individually propelled wheel 1 [2]. By adjuting the power applied to motor, the robot can be operated to go forward, rotate in place or perform movement on any arbitrary curve in plane. Such wheelframe i very exible, yet till eay to control, o it i often prioritized over other teering ytem, uch a Ackerman drive. Thank to high movement exibility, the planning of robot trajectory i much eaier compared to car-like teering ytem. If the robot i intended for operation on at urface (indoor condition), the dierential drive i denitely good way to go. 1 and one or more omni-directional upportive wheel that only enure tability of the wheelframe 1

2 2 Goal and Objective of the Control Sytem The control ytem enure accurate movement of the robot. It ue feedback from wheel encoder that meaure wheel rotation. Information from encoder i ued to control peed and poition of the robot. The control ytem calculate power applied to motor terminal to meet the peed and poition requirement. The control ytem hould compenate for any outer diturbance, caued by urface and tire friction, external force to the robot and colliion with maller object that might temporary block robot' wheel. Part II Deign 3 Robot Model A model of dierential drive i deigned in Matlab Simulink. The model take into account: outer diturbance caued by external force (f.e. maller object, blocking of a wheel..) wind, colliion with tire lipping 2 (driving on wet urface/now/ice/and) lightly dierent tranfer function for left and right motor+gearbox+wheel ubytem i upported propagation delay of wheel encoder 4 Movement Controller The original controller wa deigned to upport variou type of movement (traight movement, rotation in place, circular movement..). However, in thi paper, the function of the controller wa implied a to only upport traight movement and peed control. To keep thing imple, the poition control i not comprehended in the imulation. The poition control would introduce another PID regulator that govern the whole peed/balance control ytem preented in thi paper. That mean the complete control ytem can be build in a modular manner. But we will only imulate the peed and wheel balance control for clarity. 2 the controller cannot compenate for thi becaue the only feedback i the information about rotation of the wheel, not the actual poition of the robot in it environment 2

3 Part III Implementation 5 Speed and Balance Controller The movement control ytem i deigned a two chained PID controller. There i a Speed PID Controller, which calculate the PWM 3 duty for motor according to the deired and actual peed of the robot. The calculated PWM duty i then proportionally ditributed to left and right motor, which i done by the Wheel Balance PID Controller. Thi controller adjut the ratio of left and right motor power, o that a contant trajectory ratio between left and right wheel i maintained. Obviouly, for a traight movement, the deired trajectory ratio i 1:1 (trajectory of left wheel : trajectory of right wheel). For a rotation movement, the ratio would be -1:1. And nally, any arbitrary ratio of wheel trajectorie yield circular movement of the robot. 6 Robot Speed Feedback The robot peed i calculated a an average of the peed of it two wheel. The peed of wheel i the derivative of the trajectory meaured by the wheel encoder. It i not advied to ue the derivative block in Simulink model a it may introduce extreme numerical noie, o intead of the derivative block we ue the tranfer function G d = Tranfer Function of the Robot' Propeller Sytem The behavior of the motor, gearbox and wheel i modeled by a two-pole tranfer function: 1 G m () = ( p 1 )( p 2 ) The pole of the left motor are choen to be p 1 = 2, p 2 = 3 and for the right motor we aume a light change in the tranfer function due to inaccurate fabrication proce, o that the econd pole i lightly hifted to p 2 = 3.1. Thi imulate the variance in motor/gearbox characteritic. However, for the calculation of the PID control parameter we aume that the pole are at their expected value p 1 = 2, p 2 = 3. 3 pule width modulation 3

4 Part IV PID Contrller Tuning The PID regulator i commonly ued in indutry becaue it can be eaily implemented and work reaonably well for many procee. However, chooing the proper value of the P, I and D parameter i often a challenge. There are everal method how to do thi manually, uing trial-and-error approach. The tuning proce can be upported with ome rule of the thumb method 4. Plant / Proce Figure 1: Block diagram of a PID regulator 8 Calculation of the PID Parameter for Speed Control For our model, however, we aume that we already know the tranfer function G m () of the robot propeller ytem. In reality, the tranfer function ha to be obtained from the real robot by performing a et of tet and meaurement. I did not do thi, I jut choe appropriate 5 1 tranfer function G m = (+2)(+3). The PID control block can be expreed by it tranfer function [1]: 1 G c () = K p + K i + K d = K p + K i + K d 2 = K ( + 1)( + 2 ) where C and K p, K i, K d R are the real parameter of the PID controller. Thee equation hold for the PID parameter: (1) K p = K( ) (2) K i = K 1 2 (3) 4 ZieglerNichol method i the popular one 5 thi two-pole tranfer function i recognized a a motor tranfer function 4

5 K d = K (4) Thi can be eaily veried ubtituting (2), (3), (4) to the equation (1). For the cloed-loop ytem we receive the tranfer function 6 : Y () = G cg m G d 1 + G c G m G d R(S) (5) Where G d hould be the derivative of the wheel trajectory, however, we choe G d = +1 to avoid numerical noie during imulation. 1 Once we have the motor tranfer function G m = ( p 1)( p 2), we chooe the zero of the PID tranfer function to compenate for the pole of the motor tranfer function G m, o that 1 = p 1 = 2 and 2 = p 2 = 3: G c G m = K ( + 1)( + 2 ) 1 ( p 1 )( p 2 ) = K ( p 1)( p 2 ) ( p 1 )( p 2 ) = K 1 Then we ubtitute G c G m = K 1 to the tranfer function of the ytem (5): Y () = G cg m G d R(S) = K 1 G d 1 + G c G m G d 1 + K 1 G R() = KG d R() d + KG d From thi equation we can ee that K i a free parameter that can be adjuted to move the pole of the complete feedback ytem on the real axi. By increaing parameter K, we move the pole away from the zero origin, making the ytem repone fater. Finally, we calculate the PID parameter given by equation (2), (3), (4): K p = K( ) = K( p 1 + ( p 2 )) = K(2 + 3) = 5K K i = K 1 2 = K( p 1 )( p 2 ) = K 2 3 = 6K K d = K 9 Calculation of the PID Parameter for Balance Controller The peed and balance PID controller are dependent on each other and therefore it i not eay to calculate the PID parameter. However, we will make imilar implication a in previou ection; for the balance regulator we aume that the power ignal (output of the peed PID regulator) i contant. Then, the 6 for implication we aume that the peed PID controller control only one wheel, and thu the balance PID regulator doen't have to be conidered 5

6 Balance PID regulator can be een a a controller for the wheel path 7 and the tranfer function of the ytem i: Y () = G cg m R(S) = K G c G m 1 + K 1 R() = K + K R() Thi i almot the ame a the tranfer function for peed regulation loop (5), except there the G d derivation term i miing. We can chooe the Balance PID parameter exactly the ame a for the peed PID regulator. The gain coecient K ha to be tuned eparately for the balance regulator, of coure. Part V Simulation 10 Simulink Model The imulation i done in Matlab Simulink. Block diagram of the imulation etting i hown in Figure 2 on the following page. The proportional power divider i a cutom made block that balance the power calculated by Robot peed PID Controller proportionaly to left and right motor according to the output of the Wheel balance PID Controller. When the balance adjutment ignal i zero, 50% of the power ignal i ent to left motor and 50% to the right motor. The manual wheel blocking block imulate a condition when the left wheel i blocked (due to external force that eectively block the left wheel from rotation) for a few econd. The eect of thi heavy intervention to the ytem i then evaluated in the following ection. The noie level of diturbance and wheel lipping block wa et to DiturbanceNoie = 0.5; WheelSlippingNoie = 0.1; in the imulation. 11 Speed Control Simulation The prole of the deired peed wa choen to imulate mooth take-o and mooth breaking of the robot. Thi i a good practice in robot control that prevent tire lipping and propeller ytem degradation. The peed etpoint i plotted in yellow and the actual robot peed i the pink line. There wa noie added to the meaurement to imulate real operation condition. In Figure 3 on page 8 the gain of the PID controller wa low (K = 100). The performance of the controller i not bad, but we can go even better if we increae the gain up to K = 500. Figure 4 how very accurate peed control. 7 again, I made another big implication here 6

7 7 Figure 2: Simulink model of the complete control ytem

8 Figure 3: K_peed = 100 (uboptimal peed control) Figure 4: K_peed = 500 (accurate peed control) 8

9 12 Balance Control Simulation The Balance Controller wa evaluated on plot that how the trajectory traveled by the left and right wheel with repect to time. To recall, the purpoe of the Balance Controller i to maintain the trajectorie the ame if poible, even if there are external force that may block the wheel. In imulation, a reitance to the left wheel wa applied for a period of time. If there wa no Balance Regulator, the left wheel would top for a while (yellow line in the plot), which would ultimately change the robot orientation, ince the right wheel till rotate (pink line). After the intervention, the robot doe not move in a line anymore. Thi cenario wa plotted in Figure 5, the Balance PID regulator wa eectively diabled by etting the gain K = 0. Figure 5: K_balance = 0 (without balance control) Once the Balance Controller i enabled (K = 0.001), it compenate for the external force and the robot till move traight forward, with jut a light error in it trajectory. Thi i hown in Figure 6. Finally, we can increae the gain up to K = 0.01 to get perfect balance control, a hown in Figure 7. 9

10 Figure 6: K_balance = (moderate balance control) Figure 7: K_balance = 0.01 (accurate balance control) 10

11 Concluion A model of the movement control ytem for a dierential driven robot wa deigned and veried in Matlab Simulink. When we are given the tranfer function of the robot' motor, we can calculate the PID parameter for the Speed and Balance regulator. The imulation how very good reult, even if there are extreme external intervention to the ytem, the controller till perform well. Future work The model can be further extended to upport accurate poitioning (ditance control) and another type of movement, for example rotation in place, movement on a circular trajectory or a pline curve. Reference [1] Jan Chritoph Schlake: Advanced Control Engineering, Block 5 PID Control and Control Structure [lecture lide] [2] 11