IEEE Robotics & Automation Magazine - June 2020 - 40
(EM) [37], where the Gauss-Newton procedure presented
previously is performed in the M step.
Figure 5(a) shows that a GMM computed in a single tangent space (here, at the origin of the manifold) introduces distortions resulting in poor modeling of the data. Figure 5(b)
illustrates that the proposed representation limits the distortions by encoding the local spread of the data in covariance
matrices expressed in different tangent spaces (i.e., at the centers of the Gaussians). An example application with links to
robotics is [35], where human poses are modeled using a
GMM on S d . MATLAB examples demo_Riemannian_
Sd_GMM*.m can be found in [9].
Gaussian Conditioning
As detailed in [4], we consider input and output data
jointly encoded as a multivariate Gaussian N M (n, R)
partitioned with symbols I and O (input and output).
Given an input data point x I, the conditional distriI
t Oh can be locally evaluated
bution x O x ~N M ^ nt O, R
by iterating
I -1
u = Log nt O (n O) - R OI
Log xI (n I), nt O ! Exp nt O (u),
|| R ||
K
-1
u = e / R k o Log n (n k),
-1
n
k=1
! Exp n (u),
with covariance matrix R || k transported from n k to n (see
the "Riemannian Geometry in Robotics" section for the
description of parallel transport). After convergence, the cova-1
riance is computed in the tangent space as R = ^R Kk = 1 R ||-k1h .
2
An example of the product of Gaussians on S is depicted in
Figure 1(b). A MATLAB example demo_Riemannian_Sd_
GaussProd01.m can be found in [9].
Model Predictive Control
MPC is widely employed in robotics as an adaptive control
strategy with anticipation capability. It consists of estimating a
series of control commands u across a moving time window
of size T - 1. The problem is described here as an LQT problem with velocity commands u t ! R d and an evolution
of the state x t ! R d described by a linear system
x t + 1 = A t x t + B t u t . However, the approach can be generalized to other controllers. The resulting controller is
ut = argmin
x-n
u
2
Q
+ u
2
R
= (S
IEEE Robotics & Automation Magazine - June 2020
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