Since differentiation is a linear operator, the derivative of a GP is another GP [152]. Thus we can use GPs to make predictions about derivatives, and also to make inference based on derivative information. In general, we can make inference based on the joint Gaussian distribution of function values and partial derivatives. A covariance functionk(·,·)on func- tion values implies the following (mixed) covariance between function values and partial derivatives, and between partial derivatives,
cov(fi, ∂fj ∂xdj ) = ∂k(xi,xj) ∂xdj (6.15) and cov( ∂fi ∂xdi , ∂fj ∂xej ) = ∂k(xi,xj) ∂xdi∂xej (6.16)
With n data points in D dimensions, the complete covariance of f and its D partial
derivatives involves n(D+ 1) quantities. In a typical application we may only have access to a subset of them. In that case, we simply remove the rows and columns from the joint matrix which are not needed.
Chapter 7
Online Mapping using GPIS
An accurate and efficient representation of spatial structures is crucial for successful planning and control in navigation or manipulation tasks. We explore a new method to express the structure of the environments continuously with an underlying assumption that there exist continuous surfaces of volumetric objects.
Currently the point cloud representation and the grid-based occupancy representation are widely employed [179] in robotics. Gaussian Processes (GPs) have been suggested as an alternative [44, 135] as discussed in 6.2. Unlike grid-based methods, GP does not de- pend upon a particular choice of coordinates while providing a compact representation of continuous functions. They can continuously interpolate surface structures and can be nat-
(a)
(b)
urally extended to incorporate environmental priors and derivatives as well. Specifically, GP can be used as a regression to a distance function and build a map as the zero level set of the function. In this way, the map is represented as an implicit function, for which the GP regression maintains a probabilistic estimate of the SDF to objects in the environment (Fig. 7.1). The representation itself includes distance and its gradient, which are critical information for online obstacle avoidance and trajectory planning.
7.1
Contributions
• While prior research has focused on modeling the GP without much discussion on an
online update, out study contributes a principled way to update GPIS models incre- mentally with noisy sensor measurements. This is necessary for a robot to continually plan while navigating in an uncertain environment.
• The update process we present modifies the existing map by comparing it with new ob-
servations in a Bayesian update scheme. We introduce a GP regressor on measurements and infer corresponding surface points and normals continuously. The measurement noise model considered is heteroscedastic and accounts for noise resulting from the geometric relationship between the surface and the observer’s viewpoint.
• Our GPIS implementation consists of multiple small GPs that are updated and tested
locally to mitigate the notorious complexity of the standard GP model. A spatial partitioning data structure is exploited to manage this data efficiently.
• We experimentally demonstrate the performance of our method in terms of accuracy
for mapping and localization. It is evaluated in comparison with other grid-based representations including occupancy grids and discrete SDFs.
7.2
Models
This section presents our GPIS representation as an approximate SDF, which is based on Sec. 6.4 with subtle differences from the standard GP model. Our approach is specifically based on the noisy input GP model [119], where the input x˜ of GP is corrupted by a Gaussian noisex ∼ N(0,Σx),e.i.,x˜ =x+x. Then the observed output can be written
as, y=f( ˜x−x) +y, where y ∼ N(0, σ2y) is a noise to the output variable. Then, when
its first order Taylor expansion is considered, the probability of an observation, y, can be expressed as,
P(y|f) =N(f, σ2y+∂f>Σx∂f), (7.1)
where∂f is denoted as the derivative of the function with respect to the input, which corre-
sponds to the surface normal in our problem. Σx can be modeled as an isotropic Gaussian
covariance with parameter σx, assuming surface observation noise is not directional. Also
the derivatives satisfy ||∂f||2 = 1 by definition of surface normal. Then, Eq.(7.1) can be
rewritten as,
P(y|f) =N(f, σy2+σx2). (7.2) While the standard GP considers the output noise only, having non-zero the input noise allows us to incorporate the priors of measurement noise to the surface update. The noise may be further modeled as another process as in [144]. However, in order to render the problem tractable, we assume that the noise is independent. Thus, the noise covariance of
ninput points can be written asKx=diag{σx21, σ 2 x2, ..., σ
2
xn}. Then the inference equations
for the mean f¯∗ and the variance V[f∗] of the function value given a test point x∗ can be summarized as, ¯ f∗ =k>∗(K+Kx) −1 y, V[f∗] =k(x∗,x∗)−k>∗(K+Kx) −1 k∗. (7.3)
Eq.(6.8) and Eq.(6.9). These terms represent the vector of covariances between x∗ and
n training points, the n×n covariance matrix of the training points, and the covariance
function value ofx∗ respectively. y is the target vector of the training points.
Now let us consider available training data for the function f. While surface points are
observed, we do not explicitly observe exemplars of off-surface or internal points. Some methods employ control points of the two types to express the directional information of the surface ([57]). In this study, we consider jointly modeling the SDF and the derivatives to encode the surface direction as in [44, 117]. In our implementation, a Matérn class covariance function with ν= 3/2), Eq.(6.13), is used with its first-order derivatives.