CMPUT 412 Motion Control

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CMPUT 412Motion Control Wheeled robots

• Csaba Szepesvári
• University of Alberta

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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

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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 9 f ! X, x f(µ)
• Mobile robots
• (1 challenge!)
• Physics Wheel motion/constraints ? robot motion

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Kinematics Model

• Goal
• robot speed
• Wheel speeds
• Steering angles and speeds
• Geometric parameters
• forward kinematics
• Inverse kinematics
• Why not?

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Representing Robot Positions

• Initial frame
• Robot frame
• Robot position
• Mapping between the two frames
• Example Robot aligned with YI

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Example
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Differential Drive Robot Kinematics

• wheels spinning speed ? translation of P
• Spinning speed ? rotation around P

wheel 2
wheel 1
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Differential Drive Robot Kinematics II.
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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)

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Wheel Kinematic Constraints Fixed Standard Wheel
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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

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Wheel Kinematic Constraints Steered Standard
Wheel
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Wheel Kinematic Constraints Castor Wheel
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Wheel Kinematic Constraints Swedish Wheel
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Wheel Kinematic Constraints Spherical Wheel
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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 NNf Ns standard
  wheels
• We can develop the equations for the constraints
  in matrix forms
• Rolling
• Lateral movement

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Example Differential Drive Robot

• J1(s)J1fC1(s)C1f
• ? ?
• Right weel -¼/2, ¼
• Left wheel ¼/2, 0

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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

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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)

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Instantaneous Center of Rotation

• Ackermann Steering Bicycle

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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

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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, Ns2 ?
  common axle

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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

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Wheel Configurations M2

• Differential Drive Tricycle

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Basic Types of 3-Wheel Configs
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Synchro Drive
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Workspace Degrees of Freedom

• What is the degree of vehicles freedom in its
  environment?
• Car example
• Workspace
• how the vehicle is able to move between different
  configuration in its workspace?
• The robots independently achievable velocities
• differentiable degrees of freedom (DDOF)
• Bicycle
  DDOF1 DOF3
• Omni Drive DDOF3
  DOF3
• Maneuverability DOF!

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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

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Non-Holonomic Systems
s1s2 s1Rs2R s1Ls2L but x1 x2 y1 y2

• Non-holonomic systems
• traveled distance of 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

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Path / Trajectory Considerations Omnidirectional
Drive
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Path / Trajectory Considerations Two-Steer
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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

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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

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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

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Feedback Control

• Find a control matrix K, if exists
  with kijk(t,e)
• such that the control of v(t) and w(t)
• drives the error e to zero.

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Kinematic Position Control Differential Drive
Robot

• 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

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Coordinate Transformation

• Coordinate transformation into polar coordinates
  with origin at goal position
• System description, in the new polar coordinates

for
for
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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 t0. However
  this does not mean that a remains in I1 for all
  time t. a

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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.

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Kinematic Position Control Resulting Path
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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 -gt cosx 1, sinx xand
  the characteristic polynomial of the matrix A of
  all roots have negative real parts.

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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