Page 76 - Introduction to Autonomous Mobile Robots

61
Mobile Robot Kinematics
wheel is clearly omnidirectional and places no constraints on the robot chassis kinematics.
Therefore equation (3.21) simply describes the roll rate of the ball in the direction of motion
v of point of the robot.
A
A
·
·
l
sin ( α + β) – cos ( α + β) –()cos β R θ()ξ – rϕ = 0 (3.21)
I
By definition the wheel rotation orthogonal to this direction is zero.

·
cos ( α + β) sin ( α + β) lsin β R θ()ξ I = 0 (3.22)

As can be seen, the equations for the spherical wheel are exactly the same as for the fixed
standard wheel. However, the interpretation of equation (3.22) is different. The omnidirec-
tional spherical wheel can have any arbitrary direction of movement, where the motion
β
direction given by is a free variable deduced from equation (3.22). Consider the case that
the robot is in pure translation in the direction of Y R . Then equation (3.22) reduces to
sin ( α + β) = 0 , thus β = – α , which makes sense for this special case.

3.2.4 Robot kinematic constraints
Given a mobile robot with M wheels we can now compute the kinematic constraints of the
robot chassis. The key idea is that each wheel imposes zero or more constraints on robot
motion, and so the process is simply one of appropriately combining all of the kinematic
constraints arising from all of the wheels based on the placement of those wheels on the
robot chassis.
We have categorized all wheels into five categories: (1) fixed and (2)steerable standard
wheels, (3) castor wheels, (4) Swedish wheels, and (5) spherical wheels. But note from the
wheel kinematic constraints in equations (3.17), (3.18), and (3.19) that the castor wheel,
Swedish wheel, and spherical wheel impose no kinematic constraints on the robot chassis,
since ξ · can range freely in all of these cases owing to the internal wheel degrees of free-
I
dom.
Therefore only fixed standard wheels and steerable standard wheels have impact on
robot chassis kinematics and therefore require consideration when computing the robot’s
kinematic constraints. Suppose that the robot has a total of standard wheels, comprising
N
N f fixed standard wheels and N s steerable standard wheels. We use β t() to denote the
s
variable steering angles of the N s steerable standard wheels. In contrast, β f refers to the
orientation of the N fixed standard wheels as depicted in figure 3.4. In the case of wheel
f
spin, both the fixed and steerable wheels have rotational positions around the horizontal
axle that vary as a function of time. We denote the fixed and steerable cases separately as
ϕ t() and ϕ t() , and use ϕ t() as an aggregate matrix that combines both values:
f
s