Mobile Robot Kinematics Wheeled Types 1 Conventional wheels

Mobile Robot Kinematics

Wheeled Types 1. Conventional wheels 1) Powered fixed wheel • Driven by motor mounted on a fixed position • Axis of rotation is fixed w. r. t the platforms coordinate 2) Castor wheel • Not powered • Rotate freely about an axis perpendicular to its axis of rotation 3) Powered steering wheel • Driven by motor and can be steered ü Higher load capacity ü Higher tolerance for ground irregularities ü Not omnidirectional (Nonholonomic) 2

Wheel Types 2. Special wheels 1) Universal wheel 2) Mecanum wheel 3) Ball wheel ü Greater maneuvarability 3

Drive Types • Drives of WMR (wheeled mobile robot) – – – Differential drive Tricycle Omnidirectional Syncro drive Ackerman steering Skid steering 4

Drive Types • Differential drive – 2 powered fixed wheels + 1 or 2 castor wheels ICC (instantaneous center of curvature) 5

Drive Types • Tricycle – 1 wheel which is both driven and steered + 2 unpowered fixed wheels 6

Drive Types • Omnidirectional – 3, 4 or more omnidirectional wheels (universal/mechanum) 7

Drive Types • Synchro drive – 3 or more wheels that are mechanically coupled – Steering synchronization – 1 drive motor & 1 steering motor 8

Drive Types • Ackerman steering – Standard steering used in cars – 2 combined driven rear wheels + 2 combined steered front wheels – Rotation axes of wheels intersect at the same ICR (instantaneous center of rotation) or 9

Drive Types • Ackerman steering – For parallel parking, a considerable maneuvering is required Inaccessible areas of vehicle 10

Drive Types • Skid steering – Special implementation of differential drive in track form – Increased maneuverability in uneven terrains – Higher friction and multiple contact points with terrain 11

Maneuverability • 12

Maneuverability • 13

Degrees of Freedom • 14

Wheel Drive Configurations 15

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Direct and Inverse Kinematics • Direct (forward) kinematics • Inverse kinematics 20

Direct and Inverse Kinematics • Differential Kinematics Jacobian matrix ( m x n ) if m = n 21

Direct and Inverse Kinematics • Generalized inverse (m ≠ n) 1) m > n (overspecified) if rank(J) = n 2) m < n (underspecified) if rank(J) = m 22

Nonholonomic Constraints • Holonomic constraint Kinematic constraint if it can be expressed in the form • Nonholonomic constraint Kinematic constraint if it cannot be expressed in the form Ø Constraint in the form If integrable → holonomic constraint If not integrable → nonholonomic constraint 23

Nonholonomic Constraints (Example) unicycle Kinematic constraint : Cannot move sideways → Not integrable → Nonholonomic constraint 24

Nonholonomic Constraints (Example) Unsteered cart Kinematic constraint 1: cannot move sideways Kinematic constraint 2: cannot turn → Integrable → Holonomic constraint 25

Nonholonomic Constraints • Holonomic does not mean unconstrained 26

Nonholonomic Constraints • In robotics (ex) differential drive WMR: n=3, k=2 => n-k=1 nonholonomic constraint 27

Nonholonomic Constraints • Pfaffian constraints 28

Nonholonomic WMR • • Unicycle Differential drive WMR Tricycle WMR Car-like WMR 29

Unicycle • Geometry • Kinematics nonholonomic constraint 30

Unicycle • Affine system dynamic equation x : state vector u : input (or control) vector m=2, n=3: underactuated system 31

Differential Drive WMR • Geometry (nonslippage assumption) → Since we have 32

Differential Drive WMR • Kinematic equation • Nonholonomic constraint • Affine form Rotation of right wheel Rotation of left wheel 33

Tricycle WMR • Geometry linear angular velocity steering angular velocity • Kinematic equation 34

Car-Like WMR • Geometry driving velocity steering velocity Rear-wheel driving 35

Car-Like WMR • Kinematic equation • Nonholonomic constraint (rear wheel) (front wheel) =0 36

Omnidirectional WMR • With universal wheels • With mecanum wheels 37

WMR with Universal Three-Wheel • Geometry Total velocity of wheel i where 38

WMR with Universal Three-Wheel • Kinematic equation 39

WMR with Mecanum Four-Wheel • Geometry Wheel velocity vector 40

WMR with Mecanum Four-Wheel • Kinematic equation 41

WMR with Mecanum Four-Wheel • Four-wheel case 42