Mobile Robotics. Marcello Restelli. Dipartimento di Elettronica e Informazione Politecnico di Milano tel:

Transcription

1 Marcello Restelli Dipartimento di Elettronica e Informazione Politecnico di Milano restelli@elet.polimi.it tel: Mobile Robotics Robotica for Computer Engineering students A.A. 2006/2007

2 From Industrial Robotics to Mobile Robotics Industrial robotics complex robots simple world manipulation in known environment Mobile robotics simple robots complex world navigation in dynamic environments 2

3 Robot Dynamics vs World Dynamics In industrial robotics we need to model the robot functioning this may be hard according to the simplifications adopted In mobile robotics we need to model the environment this may be impossible and depends from possible events external sensors 3

4 Development of Mobile Robotics The first autonomous robot was Shakey (1969) 1970s: JPL Lunar rover: planetary exploration late 1970s: CART followed a line on the road 1980s: Automated Guided Vehicles (AGV) used in factories, based on magnetic and optic guides 1994: Dante II. a six-legged robot explored a volcano 1997: the Sojourner rover explored Mars 1997: Honda presented the wonderful humanoid P3 1997: the RoboCup competitions began 2001: Sony presented the humanoid Q-rio 4

5 Autonomous Land Vehicle (ALV) We focus our attention on ALV vehicle that autonomously move on a plane surface may be equipped with robotic arms we will focus on the navigation problem Three basic questions Where am I? Where am I going? How do I get there? To answer these questions the robot has to have a model of the environment perceive and analyze the environment find its position within the environment plan and execute the movement 5

6 Industrial Robotics vs Mobile Robotics There are several differences between industrial robotics and mobile robotics A manipulator, typically, is an open kinematic chain, while a wheeled robot is a closed multiple chain, since it has at least two wheels on the ground. Also legged robots are closed multiple chains, but it opens when legs are risen Manipulator joints have only 1 DOF, while wheels may have 2 (or even 3) DOFs In manipulators all the joints are actuated with the aim of moving the end-effector, while mobile robots may have passive wheels To control the trajectory of the end-effector we need to control position, speed, and acceleration of each joint, while for mobile robots we need to control each DOF of each wheel 6

7 Applications of Mobile Robotics Indoor Structured worlds transportation: industry and services customer support: museums and shops research, entertainment, toys cleaning of large buildings surveillance buildings Outdoor Environments Unstructured worlds space forest demining fire fighting agriculture construction mining sewage tubes underwater military air 7

8 DOF for Mobile Robots If a robot has an actuator for each DOF, then all DOF are controllable Mobile robots often cannot control all the DOF related to their position Let us consider the example of a car A car has 3 DOF Only two of them can be controlled Some motions cannot be done The two available DOF can get to any position and orientation in 2D 8

9 Holonomicity A robot is holonomic if the number of controllable DOF is equal to the number of DOF of the robot A robot is non-holonomic if the number of the controllable DOF is smaller than the number of DOF of the robot A robot is redundant if the number of the controllable DOF is larger than the number of DOF of the robot Examples a car is non-holonomic a human arm (7 DOF) is a holonomic, redundant system 9

10 Effectors in Mobile Robotics In mobile robotics effectors are mainly used for locomotion legs walking, crawling, climbing, jumping, hopping wheels rolling arms swinging, crawling, climbing flippers swimming and many others taken from biological examples 10

11 Locomotion Concepts found in nature difficult to imitate technically Biped walking mechanism not far from real rolling rolling of a polygon small steps -> circle Fully rotating joints are not developed in nature 11

12 Characterization of Locomotion Concept Locomotion physical interaction between the vehicle and its environment Locomotion is concerned with interaction forces and the mechanisms and actuators to generate them The most important issues in locomotion are stability characteristics of contact type of environment 12

13 Mobile Legged Robots The fewer the legs the more complicated becomes the locomotion 3 legs are required for static stability During walking some legs are lifted the stability may be loosed For static walking at least 6 legs are required 13

14 Number of Joints of Each Leg A minimum of two DOF are required to move a leg forward a lift and a swing motion sliding free motion in more than only one direction is not possible Three DOF for each leg in most cases Fourth DOF for the ankle joint might improve walking additional DOF increase the complexity of the design and especially of the locomotion control Each leg may be considered as a manipulator 14

15 Gaits The gait is characterized as the sequence of lift and release events of the individual legs it depends on the number of legs the number of possible events N for a walking machine with k legs is N=(2k-1)! For a biped (k=2) the number of possible events is 6 For a robot with 6 legs N=39,916,800 15

16 Most Obvious Gaits 16

17 Examples of Walking Robots 17

18 Walking or Rolling? Number of actuators Structural complexity Control expense Energy efficient terrain (flat ground, soft ground, climbing...) Movement of the involved masses walking/running includes up and down movements of COG some extra losses 18

19 Wheeled Mobile Robots (WMR) A WMR is a robot capable of locomotion solely through the actuation of wheel assemblies mounted on the robot and in contact with the surface. A wheel assembly is a device which provides or allows motion between its mount and surface on which it is intended to a single point of rolling contact Most robots use wheels for locomotion simplicity of control stability If so, why don't animals have wheels? some do!!! some bacteria have wheel-like structure however, legs are more prevalent in nature Most robots have three or four wheels (recently robots with two wheels have been produced) three wheels are sufficient to guarantee stability with more than 3 wheels a flexible suspension is needed selection of wheels depends on the application 19

20 Wheels Types Standard wheel: two DOF rotation around the motorized wheel axle rotation around the contact point Castor wheel: three DOF rotation around the wheels axle rotation around the contact point rotation around the castor axle Swedish wheel: three DOF rotation around the motorized wheel axle rotation around the contact point rotation around the rollers Ball or spherical wheel: three DOF suspension technically not solved 20

21 Stability Robots need to be stable to get their job done Stability can be Static: the robot can stand still without falling over is achieved thorugh the mechanical design of the robot Dynamic: the body must actively balance or move to remain stable is achieved through control For stability the Center Of Gravity (COG) of the body needs to be above the polygon support 21

22 Different Arrangement of Wheels Two wheels Three wheels Four Wheels Six Wheels 22

23 Characteristics of Wheeled Robots and Vehicles Stability of the vehicle can be guaranteed with 3 wheels center of gravity must fall inside the triangle formed by the ground contact points of the wheels Stability is improved by 4 and more wheels these arrangements are hyper-static and require a flexible suspension system Bigger wheels allow to overcome higher obstacles but they require higher torque or reductions in the gear box Most arrangements are non-holonomic require high control effort Combining actuation and steering on one wheel makes the design complex and adds additional errors for odometry 23

24 Examples of Wheeled Robots 24

25 Motion Control To control the movement of a wheeled robot we need kinematic/dynamic model of the robot model of the interaction between the wheel and the ground definition of the required motion (speed and position control) control law that satisfies the requirements 25

26 Mobile Robot Kinematics Aim description of mechanical behavior of the robot similar to manipulator kinematics mobile robots can move unbound w.r.t. the environment we cannot measure the robot's position position must be integrated over time lead to inaccuracy in the position estimation To understand mobile robot motion we need to understand wheel constraints 26

27 Kinematics Model Goal given the geometric parameters of the robot, wheel speeds, steering angles, and steering speeds we want to establish the robot speed Forward kinematics [] x = y = f 1,, n, 1,, m, 1,, m Inverse kinematics [ 1 n 1 m 1 m ] = f 1 x, y, T why not using the following formulation? [] x = y = f 1,, n, 1,, m 27

28 Representing Robot Position Fix an initial frame initial frame: {XI,YI} robot frame: {XR,YR} robot position: [] x = y mapping between the two frames [] x R =R I =R y [ cos sin 0 R = sin cos ] 28

29 Wheel Kinematic Constraints: Assumptions Movement on a horizontal plane Point contact of the wheels Wheels not deformable Pure rolling (v=0 at contact point) No slipping, skidding or sliding No friction for rotation around contact point All the steer axes are orthogonal to the ground Wheels connected by rigid frame (chassis) 29

30 Kinematic Constraints: Fixed Standard Wheel Pure rolling constraint [sin cos lcos ] R I r =0 Lateral movement constraint [cos sin l sin ] R I =0 30

31 Kinematic Constraints: Steered Standard Wheel Pure rolling constraint [sin cos lcos ] R I r =0 Lateral movement constraint [cos sin l sin ] R I =0 31

32 Kinematic Constraints: Castor Wheel Pure rolling constraint [sin cos lcos ] R I r =0 Lateral movement constraint [cos sin d l sin ] R I d =0 32

33 Kinematic Constraints: Swedish Wheel Pure rolling constraint [sin cos lcos ] R I r cos =0 Lateral movement constraint [cos sin l sin ] R I r sin r sw sw =0 33

34 Kinematic Constraints: Spherical Wheel Pure rolling constraint [sin cos lcos ] R I r =0 Lateral movement constraint [cos sin sin ] R I =0 34

35 Robot Kinematic Constraints Given a robot with M wheels each wheel imposes zero or more constraints only fixed and steerable and standard wheels impose constraints What is the maneuverability of a robot considering a combination of different wheels? Suppose we have a total of N = Nf + Ns standard wheels f t J 1f t = Rolling J 1 s = J 1s s s t J 1 s R I J 2 =0 [ ] N f N s 1 [ ] N f N s 3 J 2=diag r 1 r N Lateral movement C 1 s R I =0 [ ] C 1f C 1 s = C 1s s N f N s 3 35

36 Mobile Robot Maneuverability The maneuverability of a mobile robot is the combination of the mobility available based on the sliding constraints plus additional freedom contributed by the steering Three wheels are sufficient for static stability additional wheels need to be synchronized this is also the case for some arrangements with three wheels It can be derived using the equation seen before Degree of mobility δm Degree of steerability δs Robots maneuverability δm = δm + δs Two robots with same δm are not necessary equal 36

37 Instantaneous Center of Rotation Imposing the absence of lateral movement is the same as requiring the existence of the Instantaneous Center of Rotation (ICR) The ICR is the point of intersection between the wheel axles For any robot with δm = 2 the ICR is always constrained to lie on a line For any robot with δm = 3 the ICR is not constrained an can be set to any point on the plane 37

38 Three-Wheel Configurations 38

39 Mobile Robot Workspace Workspace how the vehicle is able to move between different configurations in its workspace? DOF: Degrees of freedom robots ability to achieve various poses DDOF: Differentiable degrees of freedom robots ability to achieve various path DDOF δm DOF Holonomic robots a holonomic kinematic constraint can be expressed as an explicit function of position variables only a non-holonomic constraint requires a different relationship, such as the derivative of a position variable Fixed and steered wheels impose non-holonomic constr. 39

40 Differential Drive Robots It is the easiest configuration 2 actuated wheels on same axle 1 passive wheel Input desired velocities Output wheel velocities v r v L v r v L L R =v R = = 2 L L L v r v L L v R v L R =v L R= v= 2 2 v R v L 2 when vr = vl, R goes to infinity -> straight movement when vr = -vl, R is zero, the robot rotates on itself 40

41 Kinematics of Differential Drive Robots Forward kinematics given the wheel speeds find the robot position 1 v L t v R t cos t d t 2 1 y t = v t sin t dt = v L t v R t sin t d t 2 1 t = t dt = v R t v L t dt L x t = v t cos t dt = Inverse kinematics find the speeds to reach a given destination the problem has infinite solutions the equations for the constraints in the velocities cannot be integrated in a constraint for the position particular cases when the wheel velocities are the same the robot moves straight 41

42 Synchronous Drive Robots It is a robot with simple kinematics three actuated steering wheels all the wheels receive the same actuation two motors one to make the wheels run one to make the wheels steer the wheels have always the same direction rotations are around the robot center it is possible to directly control θ 42

43 Kinematics of Synchronous Drive Robots Forward kinematics rotation around the center is equal to the angular speed ω the movement speed is equal to the linear velocity v CIR is always at infinity when the wheels steer CIR direction changes Inverse kinematics we use the special case when v = 0 and the angular speed is ω during the interval δt, the robot turns of an angle equal to ω*δt when ω = 0 and the linear velocity is v then the robot moves forward for v*δt 43

44 Kinematic Control The objective of a kinematic controller is to follow a trajectory described by its position and/or velocity profiles as function of time Motion control is not straightforward since mobile robots may be non-holonomic systems some solutions have been proposed open loop control feedback control kinematic position control Most controllers do not consider the dynamics of the system 44

45 Open Loop Control The trajectory is divided in motion parts of clearly defined shape straight lines arcs of a circle Control problem pre-compute a smooth trajectory Disadvantages not easy to pre-compute a feasible trajectory limitations to robot velocities and accelerations does not adapt to dynamical changes trajectory are not smooth 45

46 Feedback Control Find a control matrix K (if it exists), with kij = k(t,e) [ K= k 11 k 12 k 13 k 21 k 22 k 23 ] such that the control of v(t) e ω(t) [ ] [] x v t =K e= K y t drives the error e to 0 lim e t =0 t 46

47 Kinematic Position Control The kinematic of a differential drive mobile robot described in the initial frame {xi, yi, θ} is given by [ ] [ ][ ] cos 0 x v y = sin where the first two components are the linear velocity in the direction of xi and yi let α be the angle between the xr axis and the vector connecting the center of the axle of the wheels with the final position 47

48 Kinematic Position Control Transform the coordinates into polar coordinates with its origin at goal position 2 2 = x y = arctan x y = System in the new coordinates [][ [][ ] ][ ] cos 0 sin v = for cos 0 sin v = for, [], 2 2 ] ] ],

49 Kinematic Position Control It can be shown that with v=k =k k the feedback controlled system [][ k cos = k alpa k k sin k k ] will drive the robot to the final position The control signal v has always constant sign the direction of the movement is kept positive or negative during movement parking maneuver is performed always in the most natural way and without ever inverting its motion 49

50 Non-Holonomic Systems Differential equations are not integrable to the final position The measure of the traveled distance of each wheel is not sufficient to compute the final position of the robot We need to know also how this movement was executed as a function of time s x y = cos sin t t t d s=d x cos d y sin s s s d s= d x d y d x y s s s =cos ; =sin ; =0 x y v t = non-holonomic 2 s 2 s = sin 0= x x 50