IEEE Robotics & Automation Magazine - June 2020 - 41
evolution of the reference to track, Q = blockdiag (Q 1, Q 2, f,
Q T ) ! R dT # dT signifies the evolution of the required tracking precision, and R = blockdiag (R 1, R 2, f, R T - 1) !
R d (T - 1) # d (T - 1) indicates the evolution of the cost on the control inputs. In (1), S u and S x are transfer matrices; see the
supplementary material for this article in IEEE Xplore for
details of the computation. This formulation corresponds to a
basic form of MPC in the Euclidean space by considering
quadratic objective functions and linear systems with velocity
commands and position states. We showed in [39] that the
reference signal to be tracked can be represented by a GMM
to form a stepwise trajectory.
Equation (1) is typically used to compute a series of
control commands that are reevaluated at each iteration
across a time window. Thus, only the first (few) commands are used in practice. In the previous formulation,
the first time step of this moving time window corresponds to the current time step in which the problem is
solved [see Figure 6(c) for an illustration of this moving
time window and the computed control commands within this time window].
Such an MPC/LQT problem can be extended to Riemannian manifolds by exploiting the tangent space of the state x 1
(the point that will introduce the least distortions). By extension of (1), we can solve at each iteration
ut = argmin
Log x1 (x) - Log x1 (n)
u
2
Q ||
+ u
2
R
= (S u< Q || S u + R) -1 S
IEEE Robotics & Automation Magazine - June 2020
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