Slides3

Kinematics Model: 

Kinematics Model robot speed wheel speeds steering angles steering speeds geometric parameters of the robot (configuration coordinates). forward kinematics Inverse kinematics very difficult to solve 3.2.1

Robot Position: 

Initial frame: Robot frame: Robot position: Mapping between the two frames Example: Robot aligned with YI Robot Position 3.2.1

Example: 

Example 3.2.1

Wheel Kinematic Constraints: Assumptions: 

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) 3.2.3

Fixed Standard Wheel: 

Fixed Standard Wheel 3.2.3

Example Fixed Standard Wheel: 

Example Fixed Standard Wheel 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: 3.2.3 Rolling constraint Change in orientation Sliding constraint

Steered Standard Wheel: 

Steered Standard Wheel 3.2.3

Castor Wheel: 

Castor Wheel 3.2.3

Swedish Wheel: 

Swedish Wheel 3.2.3

Spherical Wheel: 

Spherical Wheel 3.2.3

Robot Kinematic Constraints: 

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 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 3.2.4

Mobile Robot Maneuverability: 

Mobile Robot Maneuverability The maneuverability of a mobile robot is the combination of the mobility available based on the sliding constraints plus additional freedom contributed by the steering Three wheels is sufficient for static stability additional wheels need to be synchronized this is also the case for some arrangements with three wheels It can be derived using the equation seen before Degree of mobility Degree of steerability Robots maneuverability 3.3

Mobile Robot Maneuverability: Degree of Mobility: 

Mobile Robot Maneuverability: 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 projection matrix Null space of is the space N such that for any vector n in N Geometrically this can be shown by the Instantaneous Center of Rotation (ICR) 3.3.1

Mobile Robot Maneuverability: Instantaneous Center of Rotation: 

Mobile Robot Maneuverability: Instantaneous Center of Rotation Ackermann Steering Bicycle 3.3.1

Mobile Robot Maneuverability: More on Degree of Mobility: 

Mobile Robot Maneuverability: More on 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 3.3.1

Mobile Robot Maneuverability: Degree of Steerability: 

Mobile Robot Maneuverability: 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 3.3.2

Mobile Robot Maneuverability: Robot Maneuverability: 

Mobile Robot Maneuverability: 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 and can be set to any point on the plane 3.3.3

Mobile Robot Maneuverability: Wheel Configurations: 

Mobile Robot Maneuverability: Wheel Configurations Differential Drive Tricycle 3.3.3

Five Basic Types of Three-Wheel Configurations: 

Five Basic Types of Three-Wheel Configurations 3.3.3

Synchro Drive: 

Synchro Drive 3.3.3

Mobile Robot Workspace: Degrees of Freedom: 

Mobile Robot Workspace: Degrees of Freedom Maneuverability is equivalent to the vehicle’s degree of freedom (DOF) But what is the degree of vehicle’s freedom in its environment? A car can reach any point in the plane at any orientation. What is the workspace of a vehicle? How is the vehicle able to move between different configurations? Admissible velocity space Independently achievable velocities of the vehicle = differentiable degrees of freedom (DDOF) = Bicycle: DDOF=1; DOF=3 Omnidrive vehicle: DDOF=3; DOF=3 3.4.1

Mobile Robot Workspace: Degrees of Freedom, Holonomy: 

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 paths Holonomic Robots A holonomic kinematic constraint can be expressed as 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 3.4.2

Mobile Robot Kinematics: Non-Holonomic Systems: 

Mobile Robot Kinematics: Non-Holonomic Systems Non-holonomic systems differential equations are not integrable to the final position. the measure of the 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. s1=s2 ; s1R=s2R ; s1L=s2L but: x1 = x2 ; y1 = y2 3.4.2

Non-Holonomic Systems: Mathematical Interpretation: 

Non-Holonomic Systems: Mathematical Interpretation A mobile robot is running along a trajectory s(t). At every instant of the movement its velocity v(t) is: Function v(t) is said to be integrable (holonomic) if there exists a trajectory function s(t) that can be described by the values x, y, and q only: This is the case if With s = s(x,y,q) we get Condition for integrable function 3.4.2

Non-Holonomic Systems: The Mobile Robot Example: 

Non-Holonomic Systems: The Mobile Robot Example In the case of a mobile robot where and by comparing the equation above with we find Condition for an integrable (holonomic) function: the second (-sinq=0) and third (cosq=0) term in the equation do not hold! 3.4.2

Path / Trajectory Considerations: Omnidirectional Drive: 

Path / Trajectory Considerations: Omnidirectional Drive 3.4.3

Path / Trajectory Considerations: Two-Steer: 

Path / Trajectory Considerations: Two-Steer 3.4.3

Motion Control (Kinematic Control): 

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. Most controllers are not considering the dynamics of the system 3.6

Motion Control: Open Loop Control: 

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 3.6.1

Motion Control: Feedback Control, Problem Statement: 

Motion Control: Feedback Control, Problem Statement Find a control matrix K, if it exists with kij=k(t,e) such that the control of v(t) and w(t) drives the error e(t) to zero. 3.6.2

Motion Control: Kinematic Position Control: 

Motion Control: Kinematic Position Control The kinematic of a differential drive mobile robot described in the initial frame {xI, yI, q} is given by, where x and y are the linear velocities in the direction of the XI and YI of the initial frame. Let a denote the angle between the XR axis of the robots reference frame and the vector x connecting the center of the axle of the wheels with the final position. 3.6.2   ^

Kinematic Position Control: Coordinates Transformation: 

Kinematic Position Control: Coordinates Transformation Coordinates transformation into polar coordinates with its origin at goal position: System description, in the new polar coordinates 3.6.2 for for

Kinematic Position Control: Remarks: 

Kinematic Position Control: 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. 3.6.2

Kinematic Position Control: The Control Law: 

Kinematic Position Control: The Control Law It can be shown, that with the 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. 3.6.2

Kinematic Position Control: Resulting Path: 

Kinematic Position Control: Resulting Path 3.6.2

Kinematic Position Control: Stability Issue: 

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 = x and the characteristic polynomial of the matrix A of all roots have negative real parts. 3.6.2