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CMPUT 412 Motion Control – Wheeled robots

CMPUT 412 Motion Control – Wheeled robots. Csaba Szepesv ári University of Alberta. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A. Motion Control (wheeled robots). Requirements Kinematic/dynamic model of the robot

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CMPUT 412 Motion Control – Wheeled robots

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  1. CMPUT 412Motion Control – Wheeled robots Csaba Szepesvári University of Alberta TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

  2. Motion Control (wheeled robots) • Requirements • Kinematic/dynamic model of the robot • Model of the interaction between the wheel and the ground • Definition of required motion  • speed control, • position control • Control law that satisfies the requirements

  3. Mobile Robot Kinematics • Kinematics • Actuator motion  effector motion • What happens to the pose when I change the velocity of wheels/joints? • Neglects mass, does not consider forces (dynamics) • Aim • Description of mechanical behavior for design and control • Mobile robots vs. manipulators • Manipulators: 9f: £!X, x = f(µ) • Mobile robots: (#1 challenge!) • Physics: Wheel motion/constraints  robot motion

  4. Kinematics Model • Goal: • robot speed • Wheel speeds • Steering angles and speeds • Geometric parameters • forward kinematics • Inverse kinematics • Why not?

  5. Representing Robot Positions • Initial frame: • Robot frame: • Robot position: • Mapping between the two frames • Example: Robot aligned with YI

  6. Example

  7. Differential Drive Robot: Kinematics • wheel’s spinning speed  translation of P • Spinning speed  rotation around P wheel 2 wheel 1

  8. Differential Drive Robot: Kinematics II.

  9. 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 • Steering axes orthogonal to the surface • Wheels connected by rigid frame (chassis)

  10. Wheel Kinematic Constraints:Fixed Standard Wheel

  11. Example • Suppose that the wheel A is in position such that • a = 0 and b = 0 • This would place the contact point of the wheel on XI with the plane of the wheel oriented parallel to YI. If q = 0, then this sliding constraint reduces to:

  12. Wheel Kinematic Constraints:Steered Standard Wheel

  13. Wheel Kinematic Constraints:Castor Wheel

  14. Wheel Kinematic Constraints:Swedish Wheel

  15. Wheel Kinematic Constraints:Spherical Wheel

  16. Robot Kinematic Constraints • Given a robot with M wheels • each wheel imposes zero or more constraints on the robot motion • only fixed and steerable 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 • We can develop the equations for the constraints in matrix forms: • Rolling • Lateral movement

  17. Example: Differential Drive Robot • J1(¯s)=J1fC1(¯s)=C1f • ® = ? ¯ = ? • Right weel:® = -¼/2,¯ = ¼ • Left wheel:® = ¼/2, ¯ = 0

  18. Mobile Robot Maneuverability • Maneuverability = • mobility available based on the sliding constraints+ • additional freedom contributed by the steering • How many wheels? • 3 wheels  static stability • 3+ wheels  synchronization • Degree of maneuverability • Degree of mobility • Degree of steerability • Robots maneuverability

  19. Degree of Mobility • To avoid any lateral slip the motion vector has to satisfy the following constraints: • Mathematically: • must belong to the null space of the matrix • Geometrically this can be shown by the Instantaneous Center of Rotation (ICR)

  20. Instantaneous Center of Rotation • Ackermann Steering Bicycle

  21. Degree of Mobility++ • Robot chassis kinematics is a function of the set of independent constraints • the greater the rank of , the more constrained is the mobility • Mathematically • no standard wheels • all direction constrained • Examples: • Unicycle: One single fixed standard wheel • Differential drive: Two fixed standard wheels • wheels on same axle • wheels on different axle

  22. Degree of Steerability • Indirect degree of motion • The particular orientation at any instant imposes a kinematic constraint • However, the ability to change that orientation can lead additional degree of maneuverability • Range of : • Examples: • one steered wheel: Tricycle • two steered wheels: No fixed standard wheel • car (Ackermann steering): Nf = 2, Ns=2  common axle

  23. Robot Maneuverability • Degree of Maneuverability • Two robots with same are not necessary equal • Example: Differential drive and Tricycle (next slide) • For any robot with the ICR is always constrained to lie on a line • For any robot with the ICR is not constrained an can be set to any point on the plane • The Synchro Drive example:

  24. Wheel Configurations: ±M=2 • Differential Drive Tricycle

  25. Basic Types of 3-Wheel Configs

  26. Synchro Drive

  27. Workspace: Degrees of Freedom • What is the degree of vehicle’s freedom in its environment? • Car example • Workspace • how the vehicle is able to move between different configuration in its workspace? • The robot’s independently achievable velocities • = differentiable degrees of freedom (DDOF) = • Bicycle: DDOF=1; DOF=3 • Omni Drive: DDOF=3; DOF=3 • Maneuverability · DOF!

  28. Mobile Robot Workspace: Degrees of Freedom, Holonomy • DOF degrees of freedom: • Robots ability to achieve various poses • DDOF differentiable degrees of freedom: • Robots ability to achieve various path • Holonomic Robots • A holonomic kinematic constraint can be expressed a 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 standard wheels impose non-holonomic constraints

  29. Non-Holonomic Systems • Non-holonomic systems • traveled distanceof each wheel is not sufficient to calculate the final position of the robot • one has also to know how this movement was executed as a function of time! • differential equations are not integrable to the final position s1=s2 ; s1R=s2R ; s1L=s2L but: x1 = x2 ; y1 = y2

  30. Path / Trajectory Considerations: Omnidirectional Drive

  31. Path / Trajectory Considerations: Two-Steer

  32. Beyond Basic Kinematics • Speed, force & mass  dynamics • Important when: • High speed • High/varying mass • Limited torque • Friction, slip, skid, .. • How to actuate: “motorization” • How to get to some goal pose: “controllability”

  33. Motion Control (kinematic control) • Objective: • follow a trajectory • position and/or velocity profiles as function of time. • Motion control is not straightforward: • mobile robots are non-holonomic systems! • Most controllers are not considering the dynamics of the system

  34. Motion Control: Open Loop Control • Trajectory (path) divided in motion segments of clearly defined shape: • straight lines and segments of a circle. • Control problem: • pre-compute a smooth trajectory based on line and circle segments • Disadvantages: • It is not at all an easy task to pre-compute a feasible trajectory • limitations and constraints of the robots velocities and accelerations • does not adapt or correct the trajectory if dynamical changes of the environment occur • The resulting trajectories are usually not smooth

  35. Feedback Control • Find a control matrix K, if exists with kij=k(t,e) • such that the control of v(t) and w(t) • drives the error e to zero.

  36. Kinematics: a: angle between the xR axis of the robots reference frame and the vector connecting the center of the axle of the wheels with the final position Dy Kinematic Position Control: Differential Drive Robot

  37. Dy Coordinate Transformation Coordinate transformation into polar coordinates with origin at goal position: System description, in the new polar coordinates for for

  38. Remarks • The coordinates transformation is not defined at x = y = 0; as in 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 backward direction. • By properly defining the forward direction of the robot at its initial configuration, it is always possible to have at t=0. However this does not mean that a remains in I1 for all time t. a

  39. The Control Law • It can be shown, that withthe feedback controlled system • will drive the robot to • The control signal v has always constant sign, • the direction of movement is kept positive or negative during movement • parking maneuver is performed always in the most natural way and without ever inverting its motion.

  40. Kinematic Position Control: Resulting Path

  41. Kinematic Position Control: Stability Issue • It can further be shown, that the closed loop control system is locally exponentially stable if • Proof: for small x -> cosx = 1, sinx = xand the characteristic polynomial of the matrix A of all roots have negative real parts.

  42. Summary • Kinematics: actuator motion  effector motion • Mobil robots: Relating speeds • Manipulators: Relating poses • Wheel constraints • Rolling constraints • Sliding constraints • Maneuverability (DOF) = mobility (=DDOF) + steerability • Zero motion line, instantaneous center of rotation (ICR) • Holonomic vs. non-holonomic • Path, trajectory • Feedforward vs. feedback control • Kinematic control

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