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Mobile Robot Navigation – some issues in controller design and implementation L. Huang School of Engineering and Advanced Technology Massey University Seminar at Monash University, Sunway Campus, 14 Dec 2009 Outlines 1. Introduction 2. Target tracking control schemes based on Lyapunov method Potential field method 3. Speed control 4. Conclusion Seminar at Monash University, Sunway Campus, 14 Dec 2009 Introduction A wheeled mobile robot (WMR) can be driven by wheels in various formations: Differential Omni Directional Seminar at Monash University, Sunway Campus, 14 Dec 2009 Steering Differential Wheel Robot Omni Wheel Robot Seminar at Monash University, Sunway Campus, 14 Dec 2009 Two basic issues: 1. How to move a robot from posture A to posture B stand alone ? B A 2. How to determine postures A and B for a robot when a group of robots performing a task (such as soccer playing) ? Seminar at Monash University, Sunway Campus, 14 Dec 2009 Strategy Planning A, B Point Path Generation xi , y i Line Trajectory Generation Motion Control Speed Control x(t ), y(t ) vd (t), ωd (t) v(t), ω(t) Line+Time Desired Velocity Actual Velocity Seminar at Monash University, Sunway Campus, 14 Dec 2009 Differential wheel driven robot (no-holonomic): Robot’s posture (Cartesian coordinates) cannot be stablized by time-invariant feedback control or smooth state feedback control (Brockett R. W. etc.). Stabilization problem was solved by discontinuous or time varying control in Cartesian space (Campion G. B., Samson C. etc.) Asymptotic stabilization through smooth state feedback was achieved by Lyapunov design in Polar coordinates – the system is singular in origin, thus avoids the Brockett’s condition (Aicardi M. etc ). Trajectory tracking control is easier to achieve and is more significant in practice (desired velocity nonzero) (Caudaus De Wit, De Luca A etc.). Seminar at Monash University, Sunway Campus, 14 Dec 2009 Omni-wheel driven robot It is fully linearisable for the controller design (D’AndreaNovel etc.) Dynamic optimal control was implemented ( KalmarNagy etc.) Robot modeled as a point-mass Potential field method was used for robot path planning (Y.Koren and J. Borenstein) Issues to be addressed Application of Lyapunov-based and potential field based methods in the development of target tracking control scheme Seminar at Monash University, Sunway Campus, 14 Dec 2009 General control approaches • Kinematic Model x cos 0 v y sin 0 0 1 • Nonholonomic Constraint (rolling contact without slipping) Differential Wheel Robot x sin y cos 0 Nonhonolonic (No-integrable) and under actuated (2-input~3-output) cannot be stabilized by time-invariant or smooth feedback control Seminar at Monash University, Sunway Campus, 14 Dec 2009 Trajectory tracking (Cartesian coordinates based) Given find to make xd , yd , xd and y d v and x xd , y y d Seminar at Monash University, Sunway Campus, 14 Dec 2009 It can be proved (due to Lyapunov and Barbalat), the following control can meet the objective : v vd cos( d ) k1[cos ( xd x) sin ( yd y )] d k2 sgn(vd )[sin ( xd x) cos ( yd y)] k3 ( d ) d yd xd xd y d xd2 y d2 d ATAN 2( y d , x d ) k Desired linear velocity (along the trajectory) Desired angular velocity Desired direction k1 k3 2 d2 bvd2 , k2 b vd Note: The trajectory needed to be specified in prior; the controller fails when vd 0 Seminar at Monash University, Sunway Campus, 14 Dec 2009 From the planned trajectory vd xd2 y d2 with nonlinear modifications to adjust angular motion: v vd cos( d ) k1[cos ( xd x) sin ( yd y)] sin( d ) d k 2 vd [sin ( xd x) cos ( yd y )] k3 (vd , d )( d ) d , where k2 b Seminar at Monash University, Sunway Campus, 14 Dec 2009 Goal / target tracking (Polar coordinates based) Control task: move the robot from its original posture: ( x p , y p , θ p ) to the target posture ( xg , y g , θ g ) ( g 0 : parallel p arking). Seminar at Monash University, Sunway Campus, 14 Dec 2009 The system model described in polar coordinates: v cos , v sin sin , v ( x g x) 2 ( y g y ) 2 tan ( 1 yg y xg x ) The model is singular at Seminar at Monash University, Sunway Campus, 14 Dec 2009 0 Let v k1 cos ω = k2 a2 γ + k1 sin 2γ (a2 γ + a3δ ) 2a2 γ It can be proved that ( due to Lyapunov and Barbalat) 0, 0, 0 with the Lyapunov function candidate V 1 1 1 a1 2 a 2 2 a 3 2 , a1 , a 2 , a 3 > 0 2 2 2 2 V a1k1 cos 2 2 k 2 a2 2 0 • large control effort or fluctuation when the angle tracking error is near zero or the linear tracking error is big • the target is assumed to be stationary Seminar at Monash University, Sunway Campus, 14 Dec 2009 Potential field approach (point mass model) Attractive and repulsive fields: 1 U att 1 prtT prt 2 1 2 ( 1 01 ) 2 , if 0 U rep 2 0 else Robot move along the negative gradient of the combined field: v pU att ( p) pU rep ( p) 1 prt 2 ( 1 01 ) 2 p , if 0 1 prt , else •The law only specifies the direction of the robot velocity • target is assumed to be stationary • local minima Seminar at Monash University, Sunway Campus, 14 Dec 2009 Lyapunov based target tracking controller with limited control efforts System model (extended from the conventional one by including the velocity of the target): v cos v cos t sin sin vt v sin sin v v t , 0 t , t Seminar at Monash University, Sunway Campus, 14 Dec 2009 Controller 1: Extension of the general control approach v (vt cos v ) cos sin 2 sin sin 2 ( cos )vt t v ( ) 2 2 It can be proved with Lyapunov method, that under the controller, 0, 0 and 0 1 2 (Lyapunov function candidate: V ( 2 2 2 ) ) Note: • target motions directly affects the control efforts • sinusoidal functions of the systems states attenuate the magnitude of control • tracking errors appear as the denominators in the terms of the controller • linear tracking and angular tracking errors are treated equally – too demanding ? Seminar at Monash University, Sunway Campus, 14 Dec 2009 Controller 2: Improvement from Controller 1 Prioritise and change the control objectives: 0, 0 (or bounded), 0 (or bounded) and reflect them in the definition of the Lyapunov function: 1 1 1 V 2 2 ( ) 2 2 2 2 New controller: v (vt cos v ) cos v sin 2 sin 2 ( t ( cos sin ) v ) t 2 2 2 2 which can also achieve the convergence of the tracking errors, but with less control efforts Seminar at Monash University, Sunway Campus, 14 Dec 2009 Comparison: control efforts of Controllers 1 and Controller 2 1 2 or sin 2 sin sin 2 ( cos )vt t v ( ) 2 2 v sin 2 sin 2 ( t ( cos sin ) v ) t 2 2 2 2 1 1 2 2 vt sin 2 sin 2 ( cos sin ) v 2 2 1 k t , 1 k , 2 2k k Seminar at Monash University, Sunway Campus, 14 Dec 2009 • By observation, the magnitude of controller 2 is less than that controller 1 • Analysing the factors ( ) affecting the controller magnitude, it is obvious that, except for the region near k 2 that affecting Controller 1 is larger in magnitude than that affecting Controller 2. 1 1 ____ k Seminar at Monash University, Sunway Campus, 14 Dec 2009 Simulation Results (tracking a target Moving along a circle) xt 3 15 cos(0.08t ), yt 47 15 sin( 0.08t ), vt 1.2 v 0.075, 0.15 Linear tracking Seminar at Monash University, Sunway Campus, 14 Dec 2009 Angular tracking Linear velocity Seminar at Monash University, Sunway Campus, 14 Dec 2009 Angular velocity Experiments Robot trajectory under Controller 1 Robot trajectory under Controller 2 Seminar at Monash University, Sunway Campus, 14 Dec 2009 Under Controller 1: Tracking errors Velocities Under Controller 2: Tracking errors Seminar at Monash University, Sunway Campus, 14 Dec 2009 Velocities Demonstrations Controller 1 Seminar at Monash University, Sunway Campus, 14 Dec 2009 Controller 2 Conclusions: It is feasible to reduce the control efforts through prioritization of control objectives defining of Lyapunov function to reflect that priority attenuation of controller outputs with some special functions of the system states (like sinusoidal functions etc.) while achieving the same or better control results in comparison with the conventional controllers The performance of the controller is affected by the noises of the sensors for state feedback (esp. velocity). Seminar at Monash University, Sunway Campus, 14 Dec 2009 Potential field based control approach for robot’s target tracking System model: prt [ xrt yrt ]T x rt vtar cos tar v cos y tar vtar sin tar v sin Potential fields: U U att U rep 1 U att 1 prtT prt 2 1 2 ( 1 01 ) 2 , if 0 U rep 2 0 else Seminar at Monash University, Sunway Campus, 14 Dec 2009 Case 1: Moving target free of obstacles Minimization of the angle between the gradient of the field and the direction of robot motion relative to the target. Seminar at Monash University, Sunway Campus, 14 Dec 2009 • Direction Minimisati on of U att p rt 3 ( U att p U att rt p rt p rt U att U att x y y rt x rt )( rt y rt rt x rt ) 0 xrt y rt arcsin( vtar sin( tar ) ), v 2 Robot direction is adjusted around the directional line pointing to the target Seminar at Monash University, Sunway Campus, 14 Dec 2009 • Speed Intuitively v vtar sin( tar ) It is chosen to decrease U att , or U att 1 prtT p rt 1 prt (vtar cos( tar ) v cos ) 1 2 1 prt (vtar cos( tar ) (v v sin ( tar )) ) 0 2 2 tar 2 One of the choices is: v (v 21vtar prt cos( tar ) prt ) 2 tar It leads to: 2 1 2 1 2 U att U att (0)e2 t 0 1 prt 0 The speed determined by the relative linear distance, the target velocity and there directional relationship. Seminar at Monash University, Sunway Campus, 14 Dec 2009 Comparison of the robot and target speeds: 1 prt v 2 2 2 (1 21 cos( tar ) 1 ) , vtar vtar 1 0.8 1 0.4 1 0.1 ( tar ) The robot does not need to be always faster than the target (e.g. . when (tar ) ) 2 Seminar at Monash University, Sunway Campus, 14 Dec 2009 Case 2: Moving target with moving obstacles The approach can be extended to solve the path/speed planning of the robot surrounded by multiple obstacles. vtar sin( tar ) arcsin v n 2 v v tar cos( tar ) i vobsi cos( obsi roi ) 1 prt ) 2 vtar sin 2 ( tar ) i 1 n arctan sin i sin roi i 1 n cos i cos roi i 1 i i proi , i 2 i 2 proi 1 prt 1 ( i1 01 ) Seminar at Monash University, Sunway Campus, 14 Dec 2009 Simulation Results: xt 3.0 sin t , yt 2.0 cos t tar t , vtar 1.0, Trajectories 1 1 Relative Distance Solid line: target Dashed line : robot under the proposed controller Dotted line :robot under the conventional potential field controller Seminar at Monash University, Sunway Campus, 14 Dec 2009 Speed Angle Solid line: target Dashed line : robot under the proposed controller Dotted line :robot under the conventional potential field controller Seminar at Monash University, Sunway Campus, 14 Dec 2009 Performance of the conventional field method with a high gain Trajectories Seminar at Monash University, Sunway Campus, 14 Dec 2009 Speed Conclusion: • the speed as well as the direction of the robot motion are determined with potential field method • the velocity of the moving target is taken into consideration • the proposed approach maintains or improves tracking accuracy and reduce control efforts, in comparison to the traditional approaches • further study on the determination of the optimum speed of the robot can be done by specifying additional performance requirements. Seminar at Monash University, Sunway Campus, 14 Dec 2009 Speed control considering dynamic coupling between the actuators • Synchronisation of the wheels’ motion affects the robot’s trajectory • Coupling between the actuators needs to be considered Seminar at Monash University, Sunway Campus, 14 Dec 2009 Model based adaptive control M C ( ) Dynamic model: [r l ]T r2 2 2 (mb I ) I w M 4b 2 r (mb 2 I ) 4b 2 0 C ( ) l r r2 2 (mb I ) I w 2 4b r2 2 ( mb I) 4b 2 r l 0 mc dr 3 , m mc 2mw 2 4b I mc d 2 2m w b 2 I c 2 I m mc , mw , I m , I c are the inertia parameters of the robot and the wheels b, d , r are the geometric parameters Seminar at Monash University, Sunway Campus, 14 Dec 2009 Introducing new variables then T , T , T 1 2 T , 1 2 T 1 r l , 2 r l 1 r l , 2 r l 1 1 1 2 1 1 Dynamic model is transformed to a more compact form : M 2C 0 0 1 M T 1 MT 1 , C 1 0 0 2 mr 2 Ir 2 1 Iw, 2 2 Iw 2 2b Seminar at Monash University, Sunway Campus, 14 Dec 2009 Based on the transformed dynamic model, the adaptive speed controllers are derived: k1 k 2 k k2 1 ( rd r ) 1 ( ld l ) (ˆ 1 ld ˆ 2 2 d ) ˆ 2 ld 2 2 2 k k k k 1 l 1 2 ( ld l ) 1 2 ( rd r ) (ˆ 1 ld ˆ 2 2 d ) ˆ 2 rd 2 2 2 ˆ1 1d e1 , ˆ 2 2 d e2 , ˆ 2 ( 2 1d 1 2 d ) r e1 1d 1 , e2 2 d 2 Modified to reduce the amplitudes of the control outputs: k1 k 2 k k 1 (rd r ) 1 2 (ld l ) (ˆ1 ld ˆ 2 2 d ) kˆl 2 2 2 k k k k 1 l 1 2 (ld l ) 1 2 (rd r ) (ˆ1 ld ˆ 2 2 d ) kˆr 2 2 2 ˆ k ( e e ) r 2 1 1 2 k 2 k (2 e1 1e2 ) Seminar at Monash University, Sunway Campus, 14 Dec 2009 Simulation results r l l rad/s Nm t (sec) Seminar at Monash University, Sunway Campus, 14 Dec 2009 t (sec) Model free PID control A loop for the coupling of the wheels’ speeds is added. l ld - + ul + K l (s) + + + rd + + l + Gl (s) K a (s ) + K r (s) - u r r Gr (s) - r Seminar at Monash University, Sunway Campus, 14 Dec 2009 When Gl (s) Gr (s) G(s), K l ( s) K r (s) K (s) Transfer functions : Gind ( S ) G ( s ) K ( s )( K ( s ) 2 K s ( s )) , G ( s ) K ( s )( K ( s ) 2 K a ( s )) K ( s ) K a ( s ) Gind ( S ) K s ( s) , G ( s ) K ( s )( K ( s ) 2 K a ( s )) K ( s ) K a ( s ) l ( s) Gind ( s)ld ( s) Gsyn ( s)r ( s) r ( s) Gind ( s)rd ( s) Gsyn ( s)l ( s) G ( s) km , m JRa / K t2 1 ms • First order motor model is adopted: • PID controller is used for the speed control • Implemented with one PIC18F252 microcontroller Seminar at Monash University, Sunway Campus, 14 Dec 2009 Speed Control of an Omni-wheel robots Modeling (Kinematics) 1 v1 v r Yr Y 2 1 O Or Xr v3 b 3 X 2 v2 Omni Wheel Robot Seminar at Monash University, Sunway Campus, 14 Dec 2009 3 Inverse kinematic model: ri b vr vi (i 1, 2, 3) v1 [1 0]T v2 [cos v3 [cos sin 3 sin 3 3 3 ]T ]T 1 r 1 (b vrx ) 2 r 1 (b vrx cos 3 r 1 (b vrx cos 3 3 vry sin vry sin vr [vrx 3 3 ) ) vry ]T vrx v x cos v y sin vry v x sin v y cos Seminar at Monash University, Sunway Campus, 14 Dec 2009 • Chooped fed motors with drivers to drive the wheels • PID controller implemented with one one 80296 microcontrollers (three PWM outputs) • Encoder resolution 512 ppr • Sampling time 1 ms • Control loop completed within 0.5ms This is achieved through: • codes written in an assembly language without using floating point libraries (too slow) • fixed point notation and a look up table of whole numbers to represent a floating point number with reasonable accuracy • only the simple operations like addition, substration, multiplication and bits-shifting are used. Seminar at Monash University, Sunway Campus, 14 Dec 2009 Implementation Seminar at Monash University, Sunway Campus, 14 Dec 2009 Demonstrations Seminar at Monash University, Sunway Campus, 14 Dec 2009 Conclusion Lyapunov and potential field based target tracking controllers, and speed controller for dynamically coupled wheels for mobile robots were presented Both position and velocity of the target were considered in the target tracking controller design Functions of the system states, especially those of the target, are are designed to moderate the magnitude or fluctuation of the control effort The states of the system were assumed to be available; sensor noises affect the performance of the controller. To get a good system states estimation and prediction from the sensor data is another big issue to be addressed together with the controller design (Kalman filtering, Bayesian method etc.) Further study can be undertaken on integrating open-loop optimal control, closed-loop control and system states estimation and prediction Seminar at Monash University, Sunway Campus, 14 Dec 2009