Page 71 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

46 A QUICK SKETCH OF MAJOR ISSUES IN ROBOTICS

the fact of maximum attainable velocity. (b) Accelerations (and hence torques)
cannot be independently specified at the ends of the trajectory. This significantly
limits our freedom in specifying the pattern of robot motion. The problem can
be fixed via additional constraints, namely by specifying accelerations α at the
path’s beginning and end, α(a) and α(b). Now we have a total of six constraints;
to meet them, a minimum of fifth-order polynomial is needed:

t
2
3
θ(t) = (1 − t) θ a + (3θ a + ω a )t + (α a + 6ω a + 12θ a )
2
(1 − t) 2
+ t 3 θ b + (3θ b − ω b )(1 − t) + (α b − 6ω b + 12θ b ) (2.22)
2
Straight-Line Interpolation. Achieving a straight-line motion with the arm shown
in Figure 2.12 may be tricky. The reason for that is that the arm’s rotating joints
move links’ endpoints on circular curves. This means that we cannot have an ideal
straight line, and so we need to synthesize it from curves. The more those curves
and the shorter they are, the better our straight-line approximation. To accomplish
this, we will calculate a number of points along the desired straight line, and will
force the arm endpoint through those points. Between those guaranteed points, the
arm will move as it pleases; more precisely, the arm’s own control will be linearly
interpolating points between our specified points. The interpolation is done in terms
of joint angles, or, as we call it, in the arm’s joint space (or configuration space).
To summarize, the straight-line interpolation of added points is to be done in
Cartesian space, and the arm’s own interpolation between those given points takes
place in joint space. Assume, for example, that the θ 1 and θ 2 angles of points
π
π
π
p a and p b in Figure 2.12 are p a = (0, ) and p b = ( , ). This corresponds,
6 3 2
respectively, to these Cartesian coordinates:
√

3 1
p a = l 1 + l 2 , l 2
2 2
√ √ √

1 3 3 3
p b = l 1 − l 2 , l 1 + l 2
2 2 2 2
If we provide the robot with only starting and ending points (p a ,p b ) of the
desired straight-line path (p a , p c , p b ) (Figure 2.13), then, given the robot’s joint
space linear interpolation procedure, it will produce instead the curve (p a , p j , p b ).
The midpoints of the joint and Cartesian paths indicate the extent of deviation of
√
1
the actual path from the desired path: in Cartesian terms, p j = ( 3 l 1 ,( l 1 + l 2 ))
√ 2 2
3
1
and p c = ( l 1 ,( 3 l 1 + l 2 )).
4 4 2
A reasonable idea then is to further approximate the desired straight-line path
by forcing the robot endpoint through more intermediate points along the straight
line. This is called the bounded deviation paths technique [10]; the added points
are called knots. The process starts with two knots, the initial and ending points.