Page 78 - Introduction to Autonomous Mobile Robots

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Mobile Robot Kinematics
equation (3.26) is a constraint over all standard wheels that their components of motion
orthogonal to their wheel planes must be zero. This sliding constraint over all standard
wheels has the most significant impact on defining the overall maneuverability of the robot
chassis, as explained in the next section.

3.2.5 Examples: robot kinematic models and constraints
In section 3.2.2 we presented a forward kinematic solution for ξ · in the case of a simple
I
differential-drive robot by combining each wheel’s contribution to robot motion. We can
now use the tools presented above to construct the same kinematic expression by direct
application of the rolling constraints for every wheel type. We proceed with this technique
applied again to the differential drive robot, enabling verification of the method as com-
pared to the results of section 3.2.2. Then we proceed to the case of the three-wheeled omni-
directional robot.

3.2.5.1 A differential-drive robot example
First, refer to equations (3.24) and (3.26). These equations relate robot motion to the rolling
and sliding constraints J β() and C β() , and the wheel spin speed of the robot’s wheels,
s
1
s
1
ϕ · . Fusing these two equations yields the following expression:
J β( s ) · J ϕ
2
1
I
C β( ) R θ()ξ = (3.28)
1 s 0
Once again, consider the differential drive robot in figure 3.3. We will construct J β()
1 s
and C β() directly from the rolling constraints of each wheel. The castor is unpowered and
1 s
is free to move in any direction, so we ignore this third point of contact altogether. The two
remaining drive wheels are not steerable, and therefore J β() and C β() simplify to J 1f
1
1
s
s
and C respectively. To employ the fixed standard wheel’s rolling constraint formula,
1f
α
β
equation (3.12), we must first identify each wheel’s values for and . Suppose that the
robot’s local reference frame is aligned such that the robot moves forward along +X R , as
⁄
shown in figure 3.1. In this case, for the right wheel α = – π 2 , β = π , and for the left
β
wheel, α = π 2 ⁄ , β = 0 . Note the value of for the right wheel is necessary to ensure
that positive spin causes motion in the +X direction (figure 3.4). Now we can compute
R
the J 1f and C 1f matrix using the matrix terms from equations (3.12) and (3.13). Because
the two fixed standard wheels are parallel, equation (3.13) results in only one independent
equation, and equation (3.28) gives