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Lecture 23. Dimitar Stefanov. Wheelchair kinematics. Recapping. Rolling wheels . Instantaneous Centre of Curvature (ICC). Nonholonomic constraint. motion must be consistent. Position Estimation. (x n+1 , y n+1 ). (x n , y n ). Basic position estimation equations are:. where:.

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lecture 23

Lecture 23

Dimitar Stefanov


Wheelchair kinematics


Rolling wheels

Instantaneous Centre of Curvature (ICC)

Nonholonomic constraint

motion must be consistent

position estimation
Position Estimation

(xn+1, yn+1)

(xn, yn)

Basic position estimation equations are:


D = vehicle displacement along path;

Θ = vehicle orientation (heading).

ackerman steering
Ackerman Steering
  • The inside front wheel is rotated slightly sharper than the outside wheel (reduces tire slippage).
  • Ackerman steering provides a fairly accurate dead-reckoning solution while supporting traction and ground clearance.
  • Generally the method of choice for outdoor autonomous vehicles.
ackerman steering cont 1
Ackerman Steering (cont.1)



Ackerman equation:


Θi = relative steering angle of inner wheel;

Θo = relative steering angle of outer wheel;

l = longitudinal wheel separation;

d = lateral wheel separation.

ackerman steering cont 2
Ackerman Steering (cont.2)




ΘSA = vehicle steering angle.

synchro drive
Synchro Drive
  • Three or more wheels are mechanically coupled. All wheels have one and the same orientation and rotate in the same direction at the same speed.
  • Improved dead reckoning.
  • Synchro drives use belt, chain or gear drives.
  • Problems in steering accuracy with wear/tear
synchro drive8
Synchro Drive

Dead reckoning for synchro-drive:

  • If a steerable drive wheel and encoder is used, then we can use the Ackerman steering model.
  • Otherwise use we the differential odometry mode
tricycle problems
Tricycle Problems
  • When going uphill the center of gravity of the wheelchair tends to move away from driven wheel. Causing loss of traction.
  • As Ackerman-steered design causes surface damage.
omni directional drives
Omni-Directional Drives
  • Minimum is a 3 wheel configuration.
  • Each individual motor are driven independently, using velocity control.
omni directional drives continue
Omni-Directional Drives, continue

Let’s note the velocity of the wheelchair platform in x and y direction with Vx and Vy respectively.

beacon based localization
Beacon-based Localization
  • • Trilateration
    • – Determine wheelchair position from distance measurements to 3 or more known beacons.
  • • Triangulation
    • – Determine wheelchair position for angular measurements to 3 or more known beacons.
  • Solution to constraint equations relating the pose of an observer to the positions of a set of landmarks.
  • Usually, the problem is considered in the 2D case.
  • Passive
  • Active
  • Active triangulation (AT):
  • A controlled light source (such as a laser) is positioned at point P1.
  • A imaging detector is placed at P2.
  • The distance A is preliminary known.
  • The image detector measures the angle position of the reflected-light beam.
  • AT requires one camera or one position sensitive detector;
  • AT does not depend on the ambient lighting of the object.
active triangulation
Active triangulation
  • Photo detector
    • – one- or two-dimensional array detector such as a CCD camera or photosensitive line.

Calibration – signals are measured on two preliminary known distances between the sensors and the object.


Active rangefinder chip – an example

TRC Beacon navigation System