Cartesian Coordinates

Transcription

1 Cartesian Coordinates Cartesian coordinates are rectilinear two-dimensional or three-dimensional coordinates. The three axes of threedimensional Cartesian coordinates, conventionally denoted the x-, y-, and z-axes. n three dimensions, x, y, and z may lie anywhere in the interval (, ). Wolfram and Wiipedia Polar Coordinates Each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point is the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth. The polar coordinates r (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by x = r cos y = r Where r is the radial distance from the origin, and is the counterclocwise angle from the x-axis. n terms of x and y, r x y Wolfram and Wiipedia y tan x

2 Mobile Robot Kinematics A description of mechanical behavior of the robot for design and control is required. Remember that mobile robots can move unbounded with respect to the environment. A ey point: there is no direct way to measure the robot s position Position must be integrated over time, which can lead to inaccuracies of the position (motion) estimate. n order to understand mobile robot motion, we must understand the constraints wheels place on the robot s mobility. Kinematics Kinematics is the study of the geometrical motion of a mechanical system. Two types Forward Kinematics nverse Kinematics

3 Forward Kinematics Forward Kinematics (angles to position) Robot Manipulators: Start with the length of each lin and the angle of each joint and determine the position of any point. Mobile Robots Start with the robots geometry and the speeds of the wheels and determine how the robot moves. Predicts the robot s overall speed. nverse Kinematics nverse Kinematics (position to angles) Robot Manipulator: Start with the length of each lin and the position of some point on the robot and determine the angles of each joint needed to obtain the position. 3

4 Mobile Robot Kinematics Differential Drive, Goal: Determine the robot s speed x, y as a function of the wheel speeds i and the geometric parameters of the robot (configuration coordinates). forward inematics x y f l, r,,, ) ( Mobile Robot Position Global reference frame: {X, Y } Local reference frame: {X R, Y R } Robot position: x, y, T Must map between the two frames Global reference frame and the robot R R( ) R( ) x, y, T where local reference frame cos R( ) cos Robot aligned to the global reference frame 4

5 Wheel Kinematics: Assumptions Assumptions for inematics of individual wheels. Movement on a horizontal plane A gle point contact with the ground Wheels not deformable Pure rolling v c = at the contact point No slipping, sidding or sliding No friction for rotation around contact point Steering axes orthogonal to the surface Wheels connected by rigid frame (chassis) Fixed Standard Wheel 5

6 Fixed Standard Wheel Rolling Constraint: cos ( l)cos R( ) r Sliding Constraint: cos l R( ). y (. x. cos( ẏ. -l). -l)cos(. ẋ y -cos(. x (. l ( v =r. Steered Standard Wheel Rolling Constraint: cos ( l)cos R( ) r Sliding Constraint: cos l R( ) (t) 6

7 Castor Wheel Rolling Constraint: cos ( l)cos R( ) r Sliding Constraint: cos d l R( ) d Swedish Wheels Rolling Constraint: cos ( l)cos( ) R( ) r cos Sliding Constraint: cos l ( ) R( ) r r sw sw 7

8 Spherical Wheel Roll Rate: cos ( l)cos R( ) r Wheel Rotation Orthogonal to direction of motion: cos l R( ) Kinematic Constraints A robot has M wheels connected to the chassis. Each wheel imposes zero or more constraints on the robot motion. Only standard wheels that are fixed or steerable impose constraints on the motion. f a robot has different types of wheels what is the robots maneuverability? Assume N = N f + N s standard wheels Rolling J( s ) R( ) J ( t) f ( t) ( t) s ( N f N ) Lateral Movement C( s ) R( ) s J f J( s ) J ( ) s s ( N f N ) 3 C f C( s ) C ( ) s s ( N f N ) 3 s s J = diag(r r N ) 8

9 Mobile Robot Maneuverability Robot maneuverability is a combination of The mobility based on the sliding constraints. The freedom created by steering. Robot maneuverability ( M ) = Degree of mobility ( m ) + Degree of Steerability ( s ) Degree of Mobility n order to avoid lateral slip, the motion vector R( ) must satisfy the following constraints: C ( ) f R Cs ( s ) R( ) C( s ) C Cs f ( s ) Mathematically: R ) must belong to the null space of the projection matrix Null space of C ( ) is the space N st for any vector n N ( C ( ) s C( s ) n Geometrically this can be shown by the nstantaneous Center of Rotation (CR) s 9

10 nstantaneous Center of Rotation Acermann Steering Bicycle Degree of Mobility Robot chassis inematics is a function of the independent constraints of all the standard wheels. ranc C f R( ) ( s ) f C( ) C s C ( ) s s Cs ( s ) R( ) ndependence is related to the ran of the matrix, which is the smallest number of independent rows and columns. Mathematical definition of Degree of Mobility m dim N C ( s ) 3 ran C( s ) ran C( s ) No standard wheels ran C ( ) 3 s All directions constrained ranc ( s ) 3 Examples Unicycle: One gle fixed standard wheel Differential drive: two fixed standard wheels Wheels on same axle Wheels on different axle

11 Degree of Steerability The Degree of Steerability ( s ) focuses on the number of independently controllable steering parameters. ( ) s ran C s s s The particular orientation at any moment in time imposes a inematic constraint. The ability to change the orientation can lead to additional maneuverability. Degree of Maneuverability The degree of maneuverability ( M = m + s ) represents the overall degrees of freedom that the robot can manipulate. Multiple robots can have the same M, but they are not necessarily the same. Differential drive (a) and tricycle (c)

12 Degree of Maneuverability The CR of any robot with M = is always constrained to lie on a line. The CR of any robot with M = 3 is not constrained and can be set to any point on the plane. Mobile Robot Worspace Need to determine how the robot can position itself in and move about the environment. Maneuverability is equivalent to the vehicle s degree of freedom (DOF) Ability to achieve different poses The robot s independently achievable velocities is the differentiable degrees of freedom (DDOF) = m Ability to achieve various paths DDOF m DOF Bicycle: M = m + s = + = DDOF = ; DOF=3 Omni Drive: M = m + s = 3 + = 3 DDOF=3; DOF=3

13 Holonomic Robots A holonomic inematic constraint can be expressed as an explicit function of position variables only. A robot is holonomic iff DDOF = DOF. A non-holonomic constraint requires a different relationship, such as the derivative of a position variable Robots with fixed and steered standard wheels have non-holonomic constraints. Motion Control The objective of a inematic controller is to follow a trajectory described by a position and/or velocity as a function of time. 3

14 Open Loop Motion Control The path (trajectory) is decomposed into motion segments straight lines and segments of a circle. The control problem requires pre-computing a smooth overall trajectory. Disadvantages: t can be difficult to pre-compute a feasible trajectory. Must consider the robot s limitations and constraints on velocity and acceleration. f dynamic environmental changes occur, the trajectory is not adapted or corrected. The resulting trajectories are usually not smooth. Open-Loop Motion Control nput: Desired Velocity Controller Control: u(t) Vehicle Motor Output: y(t) Beey, Autonomous Robots 4

15 Closed-loop Motion Control Use real-state feedbac in the controller nput: Desired Velocity Error: e(t) Controller Control: u(t) Vehicle Motor Output: y(t) Sensor Beey, Autonomous Robots Closed-Loop Motion Control Actual pose error in robot reference frame {X R, Y R, } is: e R x, y, The controller has to find a control matrix K 3 K with ij ( t, e) 3 Such that the control of v(t) and R (t) is x v( t) e K y ( t) Drives the error e towards zero. lim t e( t) T 5

16 6 Kinematic Position Control: Differential Drive The inematics can be described in the initial frame {X, Y, } by: where and are the linear velocities in the direction of the X and Y of the initial frame. Let denote the angle between the x R axis of the robot s reference frame and the vector connecting the center of the axle of the wheels with the final position. v y x cos x y Kinematic Position Control: Differential Drive Transformation of the Coordinates Transform to polar coordinates with the origin at the goal position. The system description in polar coordinates. ), tan ( x y a y x, for cos v,, for cos v

17 7 Kinematic Position Control: Differential Drive The coordinate transformation is not defined at x = y = ; with such a point, the determinant of the Jacobian matrix of the transformation is not defined, i.e., it is unbounded. For, the forward direction of the robot points toward the goal, for, it is the bacward direction. By properly defining the forward direction of the robot at its initial configuration, it is always possible to have at t =. However, this does not mean that remains in for all time t., Kinematic Position Control: The Control Law A control law is required to drive the robot from its starting position to the goal position. The linear control law This closed-loop system is locally exponentially stable if, for cos v v Ug cos We get