Mobile Robotics: 11. Kinematics 2 Dr. Brian Mac Namee (www.comp.dit.ie/bmacnamee) Mobile Robotics: 11. Kinematics 2

The book can be bought at: The MIT Press and Amazon.com Acknowledgments These notes are based (heavily) on those provided by the authors to accompany “Introduction to Autonomous Mobile Robots” by Roland Siegwart and Illah R. Nourbakhsh More information about the book is available at: http://autonomousmobilerobots.epfl.ch/ The book can be bought at: The MIT Press and Amazon.com

More Kinematics Today we will continue our discussion of kinematics and movement of robots through a workspace

Wheel Kinematic Constraints: Assumptions We will make the following assumptions about wheels: Movement on a horizontal plane Point contact of the wheels Wheels are 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)

Wheel Kinematic Constraints: Fixed Standard Wheel The fixed standard wheel has a fixed angle to the robot chassis Motion is limited to: Back and forth along the wheel plane Rotation around the contact point with the ground plane Robot Chassis XR YR P l A β v α Example: For example, suppose that wheel A is in a position such that a=0, 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 sliding constraint (3.6) reduces to:

Wheel Kinematic Constraints: Fixed Standard Wheel (cont…) The first constraint states that all motion along the wheel plane is accompanied by the appropriate amount of wheel spin Which, through some maths jiggery-pokery we can write as: movement along wheel plane movement due to wheel spin

Wheel Kinematic Constraints: Fixed Standard Wheel (cont…) The second constraint is that motion at right angles to the wheel plane must be zero Which, through some maths jiggery-pokery we can write as:

Wheel Kinematic Constraints Similar equations can be determined for steerable standard wheels, but we won’t worry about those There are no constraints for Swedish wheels, castor wheels or spherical wheels - why?

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?

Instantaneous Center of Rotation Each wheel has a zero motion line through its horizontal axis perpendicular to the wheel plane At any moment wheel motion through this line must be zero So the wheel must be moving along some circle of radius R such that the centre of this circle is on the zero motion line The centre point is called the instantaneous centre of rotation (ICR) When R is at infinity the wheel moves in a straight line

Instantaneous Center of Rotation (cont…) Zero motion lines

Instantaneous Center of Rotation (cont…) What about these configurations? Differential Drive Tricycle

Mobile Robot Maneuverability Maneuverability can be considered a combination of: The mobility available based on the sliding constraints The additional freedom contributed by the steering (steerability) Equations based on the constraints we spoke about earlier can be derived to calculate mobility and steerability Maneuverability is simply the sum of mobility and steerability

Maneuverability Of Three-Wheel Configurations Where δM is manoeuvrability, δm is mobility and δs is steerability

Holonomic Robots In robotics the concept of holonomy is often used The term holonomic is used in many branches of mathematics In mobile robotics holonomic refers to the kinematic constraints of a robot chassis A holonomic has zero kinematic constraints A non-holonomic robot has some constraints Fixed and steered standard wheels impose non-holonomic constraints

Robots In Their Workspace When we think about the degrees of freedom of a robot we are not telling the whole story Not only do we have to think about the arrangement of the robot, but also the robot’s pose within its environment So it is very important to consider the robot within its workspace

Paths & Trajectories It is easy to talk about the paths we expect robots to take through their environment A path is specified in three dimensions as the robot’s x coordinate, y coordinate and rotation (θ) A trajectory involves a fourth dimension - time

Path/Trajectory Considerations Suppose we want to perform the following: Move along XI axis at a constant speed of 1m/s for 1 second Change orientation clockwise 90° in 1 second Move along YI axis at 1 m/s for 1 second Let’s see how a holonomic robot and then a non-holonomic robot would achieve this

Path/Trajectory Considerations: Holonomic Robot

Path/Trajectory Considerations: Non-Holonomic Robot

Motion Control (Kinematic Control) The objective of a kinematic controller is to follow a trajectory described by its position and/or velocity profiles as function of time Motion control is not straight forward because mobile robots are non-holonomic systems However, it has been studied by various research groups and some adequate solutions for (kinematic) motion control of a mobile robot system are available

Motion Control: Open Loop Control Trajectory divided in motion segments of 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 robot’s velocities and accelerations Does not adapt or correct the trajectory if changes of the environment occur The resulting trajectories are usually not smooth

Motion Control: Feedback Control Motion control becomes a closed-loop problem where we try to minimise the error between the robot’s current position and the position of its goal

Summary Today we looked at: Kinematic constraints imposed by robot wheel arrangments Paths & trajectories Kinematic motion control Next time we will start to look at localisation and mapping

Questions ?