Description
Q) Trajectory planning using polynomial functions
A service robot wakes up in a room at coordinates (3, 0) at to = 0. Within the room, the robot has to
visit the point (1, 2.5) in order to collect a cookie for his friend. The robot then has to move to the other
end of the room at (9, 5) to give his friend the cookie, at tf = 5. In addition, the robot must not crash
into his friend, so his end speed must be equal to 0.
Assume the robot is holonomic i.e. he can move
along the x and y directions independently. Sample t from to to tf in increments of 0.1.
(a) Compute the trajectory for this desired motion profile using a polynomial function of degree four.
Recall that this amounts to formulating a linear system Ax = b based on the given constraints, for both
x and y, and simply solving for the coefficients of the polynomial function. Plot the obtained trajectory
indicating the start, end, and waypoint clearly. Also plot the corresponding velocity and acceleration
profiles.
(b) Next, compute the trajectory as a fifth degree Bernstein polynomial, i.e. as a linear combination of
the Bernstein basis polynomials. Such a polynomial is defined as,
f(t) = Xn
i=0
nCiτ
i
(1 − τ )
n−i
pi
where pi
is the scalar coefficient, and τ = ( t−to
tf −to
). Plot the obtained trajectory indicating the start, end,
and waypoint clearly. Also plot the corresponding velocity and acceleration profiles.
1
Assignment-5 2
(c) To complicate things, an evil robot of radius 2 units appears at (5, 4) to steal his cookie. The robot,
whose radius is 1 unit, must avoid this evil robot in order to successfully deliver the cookie. Describe
how you would compute a trajectory accordingly. [Bonus] points for implementing it.
