METR 4202: Advanced Control & Robotics

Transcription

1 Position & Orientation & State t home with Homogenous Transformations METR 4202: dvanced Control & Robotics Drs Surya Singh, Paul Pounds, and Hanna Kurniawati Lecture # 2 July 30, 2012 metr4202@itee.uq.edu.au RoboticsCourseWare Contributor 2012 School of Information Technology and Electrical Engineering at the University of Queensland Continuing from last week Robotics Definition: robot is a reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks. (Robotics Institute of merica) It is a machine which has some ability to interact with physical objects and to be given electronic programming to do a specific task or to do a whole range of tasks or actions. (Wikipedia) Programmable electro-mechanical systems that adapt to identify and leverage a structural characteristic of the environment (Surya) METR 4202: Robotics 30 July

2 Types of Robotics Systems Manipulators Multiple Mobile Enabling Mathematics: - Computational Kinematics - Operational Space - ehaviour based Reflexive control rules -Probabilistic methods METR 4202: Robotics 30 July Schedule Week Date Lecture (M: 12-1:30, ) 1 23-Jul Introduction Representing Position & Orientation & State 2 30-Jul (Frames, Transformation Matrices & ffine Transformations) 3 6-ug Robot Kinematics and Dynamics 4 13-ug Robot Dynamics & Control 5 20-ug Obstacle voidance & Motion Planning 6 27-ug Sensors, Measurement and Perception 7 3-Sep Localization and Navigation 8 10-Sep State-space modelling & Controller Design 9 17-Sep Vision-based control 24-Sep Study break 10 1-Oct Uncertainty/POMDPs 11 8-Oct Robot Machine Learning Oct Guest Lecture Oct Wrap-up & Course Review METR 4202: Robotics 30 July

3 X Y Today s Lecture is about: Frames & Their Mathematics z Make one (online): SpnS Template roll!!! ± pitch y Peter Corke s template x yaw Z Roboti cs Toolbox for Matl ab http :/ / ax$s.pdf METR 4202: Robotics 30 July Don t Confuse a Frame with a Point Points Position Only Doesn t Encode Orientation Frame Encodes both position and orientation Has a handedness METR 4202: Robotics 30 July

4 Kinematics Definition Kinematics: The study of motion in space (without regard to the forces which cause it) ssume: Points with right-hand Frames Rigid-bodies in 3D-space (6-dof) 1-dof joints: Rotary (R) or Prismatic (P) (5 constraints) N links M joints DOF = 6N-5M If N=M, then DOF=N. The ground is also a link METR 4202: Robotics 30 July Kinematics Kinematic modelling is one of the most important analytical tools of robotics. Used for modelling mechanisms, actuators and sensors Used for on-line control and off-line programming and simulation In mobile robots kinematic models are used for: steering (control, simulation) perception (image formation) sensor head and communication antenna pointing world modelling (maps, object models) terrain following (control feedforward) gait control of legged vehicles METR 4202: Robotics 30 July

5 asic Terminology y Coordinate System Frame point axis x origin METR 4202: Robotics 30 July Coordinate System The position and orientation as specified only make sense with respect to some coordinate system k {} i j Z P Y X METR 4202: Robotics 30 July

6 Frames of Reference frame of reference defines a coordinate system relative to some point in space It can be specified by a position and orientation relative to other frames The inertial frame is taken to be a point that is assumed to be fixed in space Two types of motion: Translation Rotation METR 4202: Robotics 30 July Translation motion in which a straight line with in the body keeps the same direction during the Rectilinear Translation: long straight lines Curvilinear Translation: long curved lines 1 2 METR 4202: Robotics 30 July

7 Rotation The particles forming the rigid body move in parallel planes along circles centered around the same fixed axis (called the axis of rotation). Points on the axis of rotation have zero velocity and acceleration METR 4202: Robotics 30 July Rotation: Representations Orientation are not Cartesian Non-commutative Multiple representations Some representations: Rotation Matrices: Homegenous Coordinates Euler ngles: 3-sets of rotations in sequence Quaternions: a 4-paramameter representation that exploits ½ angle properties Screw-vectors (from Charles Theorem) : a canonical representation, its reciprocal is a wrench (forces) METR 4202: Robotics 30 July

8 Position and Orientation [1] position vectors specifies the location of a point in 3D (Cartesian) space O UT we also concerned with its orientation in 3D space. This is specified as a matrix based on each frame s unit vectors METR 4202: Robotics 30 July Position and Orientation [2] Orientation in 3D space: This is specified as a matrix based on each frame s unit vectors Describes {} relative to {} The orientation of frame {} relative to coordinate frame {} Written from {} to {} or given {} getting to {} O Columns are {} written in {} METR 4202: Robotics 30 July

9 Position and Orientation [3] The rotations can be analysed based on the unit components That is: the components of the orientation matrix are the unit vectors projected onto the unit directions of the reference frame METR 4202: Robotics 30 July Position and Orientation [4] Rotation is orthonormal The of a rotation matrix inverse = the transpose thus, the rows are {} written in {} METR 4202: Robotics 30 July

10 Position and Orientation [5]: note on orientations Orientations, as defined earlier, are represented by three orthonormal vectors Only three of these values are unique and we often wish to define a particular rotation using three values (it s easier than specifying 9 orthonormal values) There isn t a unique method of specifying the angles that define these transformations METR 4202: Robotics 30 July Position and Orientation [7] Shortcut Notation: METR 4202: Robotics 30 July

11 Position and Orientation [8] Rotation Formula about the 3 Principal xes by θ X: Y: Z: METR 4202: Robotics 30 July Euler ngles Minimal representation of orientation (α,β,γ) Represent a rotation about an axis of a moving coordinate frame : Moving frame w/r/t fixed The location of the axis of each successive rotation depends on the previous one! So, Order Matters (12 combinations, why?) Often Z-Y-X: α: rotation about the z axis β: rotation about the rotated y axis γ: rotation about the twice rotated x axis Has singularities! (e.g., β=±90 ) METR 4202: Robotics 30 July

12 Fixed ngles Represent a rotation about an axis of a fixed coordinate frame. gain 12 different orders Interestingly: 3 rotations about 3 axes of a fixed frame define the same orientation as the same 3 rotations taken in the opposite order of the moving frame For X-Y-Z: ψ: rotation about x (sometimes called yaw ) θ: rotation about y (sometimes called pitch ) φ: rotation about z (sometimes called roll ) METR 4202: Robotics 30 July Roll Pitch Yaw In many Kinematics References: In many Engineering pplications: z roll x pitch y roll pitch y x yaw yaw z e careful: This name is given to other conventions too! METR 4202: Robotics 30 July

13 Euler ngles [1]: X-Y-Z Fixed ngles (Roll-Pitch-Yaw) One method of describing the orientation of a Frame {} is: Start with the frame coincident with a known reference {}. Rotate {} first about X by an angle g, then about Y by an angle b and finally about Z by an angle a. METR 4202: Robotics 30 July Euler ngles [2]: Z-Y-X Euler ngles nother method of describing the orientation of {} is: Start with the frame coincident with a known reference {}. Rotate {} first about Z by an angle a, then about Y by an angle b and finally about X by an angle g. METR 4202: Robotics 30 July

14 Position and Orientation [6]: Proof of Principal Rotation Matrix Terms Geometric: y b θ a x c d θ y x METR 4202: Robotics 30 July Unit Quaternion (ϵ 0, ϵ 1, ϵ 2, ϵ 3 ) [1] Does not suffer from singularities Uses a 4-number to represent orientation Product: Conjugate: METR 4202: Robotics 30 July

15 Unit Quaternion [2]: Describing Orientation Set ϵ 0 = 0 Then p=(p x,p y,p z ) Then given ϵ the operation : rotates p about (ϵ 1, ϵ 2, ϵ 3 ) Unit Quaternion Rotation Matrix METR 4202: Robotics 30 July Direction Cosine Uses the Direction Cosines (read dot products) of the Coordinate xes of the moving frame with respect to the fixed frame It forms a rotation matrix! METR 4202: Robotics 30 July

16 Screw Displacements Comes from the notion that all motion can be viewed as a rotation (Rodrigues formula) Define a vector along the axis of motion (screw vector) Rotation (screw angle) Translation (pitch) Summations via the screw triangle! METR 4202: Robotics 30 July Coordinate Transformations [1] Translation gain: If {} is translated with respect to {} without rotation, then it is a vector sum {} Z {} Z P P P Y X Y X METR 4202: Robotics 30 July

17 Coordinate Transformations [2] Rotation gain: {} is rotated with respect to {} then use rotation matrix to determine new components NOTE: The Rotation matrix s subscript matches the position vector s superscript {} Z P Y X This gives Point Positions of {} ORIENTED in {} METR 4202: Robotics 30 July Coordinate Transformations [3] Composite transformation: {} is moved with respect to {}: {} Z P P P Y X METR 4202: Robotics 30 July

18 General Coordinate Transformations [1] compact representation of the translation and rotation is known as the Homogeneous Transformation This allows us to cast the rotation and translation of the general transform in a single matrix form METR 4202: Robotics 30 July General Coordinate Transformations [2] Similarly, fundamental orthonormal transformations can be represented in this form too: METR 4202: Robotics 30 July

19 General Coordinate Transformations [3] Multiple transformations compounded as a chain {} Z P C P P C P Y X METR 4202: Robotics 30 July Homogenous Coordinates ρ is a scaling value METR 4202: Robotics 30 July

20 Homogenous Transformation γ is a projective transformation METR 4202: Robotics 30 July METR 4202: Robotics Projective Transformations 30 July p.44, R. Hartley and. Zisserman. Multiple View Geometry in Computer Vision METR 4202: Robotics 30 July

21 METR 4202: Robotics Projective Transformations & Other Transformations of 3D Space 30 July p.78, R. Hartley and. Zisserman. Multiple View Geometry in Computer Vision METR 4202: Robotics 30 July Special Orthogonal & Special Euclidean SO(n): Rotations SE(n): Transformations of EUCLIDEN space ELEC 3004: Signals, Systems & Control 30 July

22 Forward Kinematics [1] Forward kinematics is the process of chaining homogeneous transforms together. For example to: Find the articulations of a mechanism, or the fixed transformation between two frames which is known in terms of linear and rotary parameters. Calculates the final position from the machine (joint variables) Unique for an open kinematic chain (serial arm) Complicated (multiple solutions, etc.) for a closed kinematic chain (parallel arm) METR 4202: Robotics 30 July Forward Kinematics [2] Can think of this as spaces : Operation space (x,y,z,α,β,γ): The robot s position & orientation Joint space (θ 1 θ n ): state-space vector of joint variables q f Forward Kinematics x f q i q i Joint Limits Inverse Kinematics x fi Workspace METR 4202: Robotics 30 July

23 Summary Many ways to view a rotation Rotation matrix Euler angles Quaternions Direction Cosines Screw Vectors Homogenous transformations ased on homogeneous coordinates METR 4202: Robotics 30 July Cool Robotics Share METR 4202: Robotics 30 July