Lecture 1 Wheeled Mobile Robots (WMRs)

Transcription

1 Lecture 1 Wheeled Mobile Robots (WMRs) Course Chair: Prof. M. De Cecco Teaching: A. Cesarini Mechatronics Department, University of Trento andrea.cesarini@unitn.it

2 A- Introduction to Wheeled Mobile Robots (WMRs) B- WMRs - WMR design - Guidelines for WMR robotic system design - Environments C- Kinematic systems - Wheels and mechanical configuration - Configuration space and workspace - Holonomic and nonholonomic - Degrees of freedom - Mobility, Steerability and Manouvering - Assumptions - Unicycle - Car-like - Unicycle + N trays - 3-Omni WMR Contents

3 Curiosity Sojourner Examples of autonomous robotic vehicles

4 Flexy AGILE Examples of autonomous robotic vehicles (2)

5 Information via sensor fusion : odometry, remote localization and environmental ancillary datasets Path/trajectory and (hybrid) planning/control (offline - global strategy) Execution with robustness to perturbations and flexibility (online - local strategy) WMR design requirements

6 Mobile robotics provides methods allowing robotic systems to carry out predetermined tasks To design a robotic system is a refining process. The ultimate design of a robotic system does not exist by itself. The optimal design changes optimizing the design constraint. In general, providing a task, the optimal design is achieved maximizing for the Pareto-optimality within mechanical reliability, costs and simplicity 1 The task The overall strategy is selected via refining process 2 The environment (objects in the work space) 3 The dynamical characteristics of the robots Wheeled Mobile Robotics

7 Structured Environments Complete knowledge of the environment allowing an offline global planning of the motion. Semistructured environments Environmental uncomplete knowledge allowing motion global planning with local flexibility in presence of dynamical obstacles Unstructured enviroments Very limited environmental knowledge and/or wide variety of circumstances and conditions. o Environments Spatial categorizations of 2D/3D environment characteristics o o Indoor (2D) Outdoor (3D) o o o Structured environment (Known) o a priori map Semi-structured environment o Static and Dynamic Unstructured environment (Unknown)

8 Legs Wheels Directional Omnidirectional Trolley/Tracks Hybrid Robot mechanical configuration

9 Directional fix and stearable wheels: a) Fixed wheel (fixed axis and support) One fixed axis and fixed support. Stearable centered. b) Steerable off-centered (Caster/castor) wheel: Actuated or passive Image from [Siegwart 2011] Wheel mechanics

10 Omnidirectional wheels: c) Swedish wheel or Mecanum Rollers on the wheel circle, mounted with an angle (90,45 ) Holonomic Difficult construction Limited robustness Limited speed d) Sperical wheel Holonomic Difficult construction Limited robustness Image from [Siegwart 2011] Limited speed Wheel mechanics (2)

11 Examples of omnidirectional wheels: Swedish 90 (Omnidirectional): Honda Wheels (Ux3) Swedish 45 (Mechanum): Airtrax Mechanum principle Wheel mechanics (3)

12 Image from [Siegwart 2011] WMR configurations

13 Image from [Siegwart 2011] WMR configurations (2)

14 Image from [Siegwart 2011] WMR configurations (3)

15 Kinematic assumptions: Each wheel does not experience slipping (longitudinal direction) Each wheel can t skid (transversal direction) Single point of contact (no deformable) Vertical plane for the wheel v t = rf r v n = 0 Image from Siciliano et al., 2009 Wheel motion mechanics

16 Thus, it is always possible to individuate an instantaneous centre of rotation (ICR) It is determined by the intersection of all the wheel rotation axes (at a given time). ICR ICRs Inf ICR not defined ICR not defined Curvilinear motion Any motion Prevented motion Straight motion ICRs might be defined or not. Moreover it might be one or a set of infinite ICRs Instantaneous centre of rotation (ICR)

17 The parameters defining the configuration of a system are called generalized coordinates and the vector space defined by those coordinates is called configuration space Let be a system with generalized configuration coordinates q C, we assume that the n configuration space C (at least locally) coincides with R, since dim c n. x x q y, dim C 3 y q, dim C 4 Wheeled mobile robots: unicycle and car-like

18 The intersections of the wheel rotation axes provide the ICRs Unicycle Car-like ICR ICR In the case of the unicycle and in the case of the car-like, the ICR are defined and unique. The caster a passively re-orientering wheel and does not participate and does not steer the WMR motion. However, it can be used as an observer since it has memory of the system states. Wheeled mobile robots: unicycle and car-like (2)

19 N C y x q N 4 dim, Tricycle (equivalent to car-like) Trolleys Wheeled mobile robots: N-trailer system

20 A few considerations about the configuration space C and the workspace W in which any movement of the WMR (and eventually its end-effector) is described: The configuration space C provides the set of all possible configurations If a robot is modeled as a 0-dimensional material point in a 2D W-space, its configuration space C can still be a 2D space and its related configuration can be described with 2 parameters (or generalized coordinates, or DOFs). E.g. (x,y) in Cartesian coordinates If a robot is modeled as a 2D-object in a 2D workspace in W, its end-effector is univocally identified with position + orientation (obtained by translation + rotation), e.g. (x,y, θ). x q y, dim C 3 x y q, dim C 4 Unicycle and car-like cases Take a closer look: how many parameters are required for univocally identify a configuration of the C-space??? q i Note: The parameters describing the system configuration are called generalized coordinates and the vector space defined by those coordinates is called configuration space C Special Euclidean Groups SE(2) and SE(3)

21 If a robot is modeled as a 2D-object in a 2D workspace in W, any configuration of the robot can be described by the Special Euclidean group : SE 2 = R where SO(2) is the Special Orthogonal group of rotations in a 2D space and any single configuration in the W-space can be expressed by 3 parameters (x,y,θ) 2 SO 2 Similarly, any robot modeled as a 3D-object (volumetric) can be described with its translations and rotations in a special 3D C-space as SE 3 = R 3 SO 3 and in which any single configuration requires 6 parameters (x,y,z) for describing translations and 3 Euler angles (α,β,γ) for describing rotations. SE 3 = R 3 SO 3 World (3D) q i Note: The parameters describing the system configuration are called generalized coordinates and the vector space defined by those coordinates is called configuration space C Special Euclidean Groups SE(2) and SE(3) (2)

22 Any state in the C-space is directly related to a position and orientation of the robot (with its end-effector) in the workspace W Obstacles in the workspace W can be modeled as unreachable regions (or robot configurations) of the C-space Robot W Repr. DOFs Control DOFs C Circular 2D 2 2 R2 Unicycle 2D SE(2) = R2 x SO(2) Car-like 2D 2+1 (+1) 3 (+1) SE(2) = R2 x SO(2) Car-like (3D) 3D SE(3) = R3 x SO(3) Submarine (3D) 3D SE(3) = R3 x SO(3) N act. rot. joints 3D N+N N SE(N) = RN x SO(N) If the robot is able to translate and rotate, it has 3 DOFs (x,y,θ); its C-space is 3D (2R + 1 toroidal (wrapping on itself, 0-2PI). A planar 2R arm robot fixed at the base has 2 DOFs. Its C-space will be a 2D toroidal space. If the robot is a fixed-base manipulator with N actuated/active rotational joints (no closedloops), C is N-dimensional. Special Euclidean Groups SE(2) and SE(3) (3) 1, 2

23 The Degree of Mobility (DOMo) is the capability of a WMR to reorient itself (INSTANTANEOUSLY) OPERATIVELY: The mobility degree is related to the number of ICR associated with the WMR mechanical configuration ICR not defined m 0 (3) Constant arc for any fix steering m 1 (4) This configuration provides a variable arc of motion (Inf ICRs in line) m 2 (2) Note: See notation 0 (0) ; The first value is the mobility degree value and the second is the number of wheels. The value of the mobility degree can t be larger than the total number of wheels (which is the value between round brackets). Degree of Mobility (DOMo)

24 The Degree of Steerability (DOS) relies on the WMR indirect degree of motion which are provided by reconfiguring (only centered) wheels (NOT INSTANTANEOUSLY) The presence of any wheel imposes the presence of a kinematic specific constraint and the relocation/reorientation of any wheel changes the associated constraint itself OPERATIVELY: The steering degree is the number of steering centered wheels which can be independently reoriented s 0 (3) No steering degrees s 0 (2) Two centered dependent steering wheels s 1 (4) Two centered independent steering wheels s 2 (4) Special note: The eventual presence of passive steering wheels (e.g., castors or casters) does not provide any further steering degree and kinematic constraint!! Degree of Steerability (DOS)

25 A Three caster wheels m 3, 0 s B One caster + 2 stearable independent wheels m 1, s 2 C Two centered independent steering wheels m 1, s 2 D One caster and 2 fixed wheels m 2, 0 s E 3 omnidirectional wheels m 3, 0 Two centered dependent steering wheels m 1, 1 The Degree of Mobility (DOM) is the capability of a WMR to reorient itself (INSTANTANEOUSLY) provided by ICRs. The Degree of Steerability (DOS) relies on the WMR indirect degree of motion which are provided by reconfiguring (centered) wheels (NOT INSTANTANEOUSLY). Degree of Maneouverability (DOMa) =DOMo + DOS s F s

26 The Maneuverability degree is the number of DOFs provided by the steering and mobility: M m s Robot Mechanical Conf. A B C D E F Degree of Mobility Degree of Steerability Degree of Manouverability The Degree of Mobility (DOM) is the capability of a WMR to reorient itself (INSTANTANEOUSLY) provided by ICRs. The Degree of Steerability (DOS) relies on the WMR indirect degree of motion which are provided by reconfiguring (centered) wheels (NOT INSTANTANEOUSLY). Degree of Maneouverability (DOMa) =DOMo + DOS (2)

27 To be continued

28 Bibliography 1. Siegwart R., Nourbakhsh I, Scaramuzza D., Introduction to Autonomous Mobile Robots 2. A. De Luca, Dispense del corso di Robotica I/II, Universita La Sapienza, Roma Siciliano and Sciavicco, Robotica Industriale, Siciliano, Sciavicco, Villani and Oriolo, Robotics: modeling, planning and control, Barraquand and Latombe, Controllability of Mobile Robots with Kinematic Constraints, CIFE Center for Integrated Facility Engineering, G. Oriolo, Control of Nonholonomic Systems, Dispense del Corso di Dottorato di Ricerca in Ingegneria dei Sistemi. 7. A. De Luca e G. Oriolo, Introduction to the chained form, Chapter A. De Luca, Dispense del Corso di Controlli Automatici II, Roma Advanced Navigation, Edufill Lecture Slides 10.Sohani, Katsilieris and Bayraktar, Mobile Robot Kinematics, KTH. 11.AAVV, Notes freely available on internet.