Geometry-aware Bayesian Optimization in Robotics using Riemannian Matérn Kernels
Anonymous Author(s)
AfﬁliationAddress email
Abstract: Bayesian optimization is a data-efﬁcient technique which can be usedfor control parameter tuning, parametric policy adaptation, and structure designin robotics.
Many of these problems require optimization of functions deﬁned onnon-Euclidean domains like spheres, rotation groups, or spaces of positive-deﬁnitematrices.
To do so, one must place a Gaussian process prior, or equivalentlydeﬁne a kernel, on the space of interest.
Effective kernels typically reﬂect thegeometry of the spaces they are deﬁned on, but designing them is generally nontrivial.
Recent work on the Riemannian Matérn kernels, based on stochasticpartial differential equations and spectral theory of the Laplace–Beltrami operator,offers promising avenues towards constructing such geometry-aware kernels.
Inthis paper, we study techniques for implementing these kernels on manifolds ofinterest in robotics, demonstrate their performance on a set of artiﬁcial benchmarkfunctions, and illustrate geometry-aware Bayesian optimization for a variety ofrobotic applications, covering orientation control, manipulability optimization, andmotion planning, while showing its improved performance.
Introduction
Fast and data-efﬁcient adaptation is a key challenge in a large range of robotics applications, owingto the need to handle sensor noise, model uncertainty, and generalize to unforeseen settings.
Inthis context, Bayesian optimization [43] is a technique of growing interest in robotics due to itsability to efﬁciently solve challenging optimization problems, including controller tuning [5, 30],policy adaptation [4, 14] and robot design [39].
This makes it an attractive tool for settings wheredata-efﬁciency is of key interest and large-scale methods such as deep reinforcement learning are notpresently viable.
Bayesian optimization algorithms are generally powered by probabilistic modelsbased on Gaussian processes that rely on inductive biases encoded by kernels.
Improving performanceand scalability of such models via a number of different strategies has therefore been a central goalpursued by the Bayesian optimization and Gaussian process literatures.
Inductive bias in robotics can be introduced by exploiting domain knowledge about the task at hand.
For example, task-speciﬁc kernels were proposed by Antonova et al.
[5] by learning a distance metricvia simulated bipedal locomotion patterns to optimize gait controllers, and by Rai et al.
[35] using gaitfeature transformations to design kernels to optimize locomotion controllers.
More generic kernelswere introduced by Marco et al.
[30], Marco et al.
[31], and Wilson et al.
[50] for policy optimization,by incorporating linear-quadratic control structures, and trajectory information into the kernel design.
While these kernels can be exploited for robot control or policy search with many different tasks andsystems, their use is speciﬁc to the problems they were designed for.
A more general approach is to carefully select the domain of the function being optimized, thusincorporating information about the geometry of the search space into the optimization algorithm.
Many quantities of interest in robotics carry such geometric information: three-dimensional rotationscan be viewed as elements of the Lie group SO(3) [19] or the unit-sphere manifold S 3 [48], controlgains, inertia and manipulability ellipsoids lie in the manifold of symmetric-positive-deﬁnite matrices S d ++ , while the joint conﬁguration of a d -degree-of-freedom robot with revolute joints may be viewed as a point on a torus T d , i.e. the product of d circles [26], (see Fig.
1 for illustrated examples).
Incorporating the geometric structure of these spaces into Bayesian optimization has recently beenshown to improve its performance on a number of important tasks [25], despite the use of a naïvegeometrical approximation in the kernel design.
Geometry-aware Bayesian optimization is therefore a promising emerging tool for general use inrobotics.
One of the main difﬁculties in developing this tool is that deﬁning geometry-aware Gaussianprocess models is technically non-trivial: classic Euclidean methods ignore the parameter spacegeometry, and naïve geometric approximations do not capture the manifold’s geometric structure. Fortunately, mathematical theory and techniques for building the necessary kernels are beginningto emerge [7, 28].
In particular, the work of Borovitskiy et al.
[7] on Riemannian Matérn kernelsprovides mathematically-sound tools for calculating geometry-aware kernels by building on stochasticpartial differential equations and spectral theory of the Laplace–Beltrami operator.
This provides thekey ingredients to deploy geometry-aware Bayesian optimization on practical robotics problems.
Building on the performance improvements in Jaquier et al.
[25] and the theoretically-grounded Matérn kernels proposed in Borovitskiy et al. [7], we present a more general kernel formulation for awide range of Riemannian manifolds of interest in robotics. Our approach uses the fact that a Matérnkernel can be formulated as the integral of a squared-exponential kernel—this idea enables us todeﬁne Riemannian Matérn kernel on non-compact manifolds, generalizing the work of Borovitskiyet al.
[7] which applies only to compact manifolds.
We introduce new Riemanian Matérn kernelsfor the special orthogonal group, manifold of symmetric positive deﬁnite matrices, and hyperbolicspace, and study their use in robotic applications. This more general formulation avoids the need todetermine valid length scales as in [25], provides theoretical guarantees of positive-deﬁniteness, andmakes kernel smoothness optimization possible.
We benchmark these kernels on a set of artiﬁcialfunctions and on a number of robotics problems of interest to practitioners.
Geometry-aware Bayesian Optimization
Bayesian optimization [42] is a class of sequential search algorithms for ﬁnding a global minimizerx ∗ =
arg min x ∈X f ( x ) of an unknown objective function f deﬁned over a domain of interest X .
The function f is not observed directly: instead, at each step the algorithm must select a query pointx ∈ X at which to evaluate f .
Moreover, it is usually assumed that the algorithm only observes anoisy value y ∈ R whose expectation E ( y ) =
f ( x ) is the true function value at the chosen point.
Based on the full set of observed evaluations, the algorithm must select the next query point x .
The vast majority of Bayesian optimization algorithms employ Gaussian process models, whichrepresent the known information about the objective function using a Gaussian process priorf ∼
GP( µ, k ) , with mean function µ : X → R and symmetric-positive-deﬁnite kernelk :
X × X → R .
The Gaussian process is updated by conditioning on observed data ( x i , y i ) ni
=1using Bayes’ Rule.
The next query point is selected by optimizing a carefully-constructed acquisitionfunction deﬁned with respect to the resulting posterior distribution.
In robotics, the domain of interest X often possesses important non-Euclidean geometric structure.
For example, when tuning robot controllers, the variables of interest include end effector poseslying on R 3 × S 3 where S d denotes the d -dimensional unit sphere, robot joint postures deﬁned ona torus T d = S 1 × . . .
× S 1 , as well as gain matrices and manipulability ellipsoids belonging tothe space of symmetric-positive-deﬁnite matrices S d ++ .
In addition, when adapting parametric robot policies, some of their parameters such as mean vectors or covariance matrices lie in Riemannianmanifolds.
This inherent geometry is often overlooked in robotic applications [30, 37, 38].
In thispaper, we demonstrate that incorporating the correct geometric structure of X into the Gaussianprocess is an important potential avenue for improving the Bayesian optimization performance.
Performing geometry-aware Bayesian optimization on Riemannian manifolds requires one to addresstwo main challenges.
Firstly, deﬁning a valid Gaussian process prior is more difﬁcult than in the Euclidean case.
Secondly, numerical techniques such as training and optimizing the acquisitionfunctions may require special considerations.
We address these issues in the remainder of this section.
2.1 Gaussian Processes on Riemannian Manifolds
The key challenge in constructing Gaussian processes on non-Euclidean domains is in deﬁning avalid kernel: seemingly obvious and innocuous-looking candidate expressions such as the squaredexponential geodesic distance exp( − d g ( x , x (cid:48) ) 2 /κ ) are not valid kernels for all length scales κ > 0simultaneously [13].
In spite of this, they have been shown to be beneﬁcial for practical roboticsapplications [25]: this motivates one to develop technically sound analogues.
To tackle these issues,recent works on non-Euclidean Gaussian processes, including Lindgren et al.
[28] and Borovitskiyet al.
[7], have developed provably well-posed techniques based on stochastic partial differentialequations and spectral theory of the Laplace–Beltrami operator.
We focus on the latter work as itdoes not require one to solve stochastic partial differential equations numerically.
The general theory of Matérn Gaussian processes on compact Riemannian manifolds from Borovitskiyet al.
[7] introduces expressions for the kernels of these processes in terms of the Laplace–Beltramieigenpairs of their respective manifolds.
For more general settings including non-compact manifoldswhich arise in robotics, such as the manifold of positive deﬁnite matrices and hyperbolic space, kernelscan often be built by exploiting the connection between Riemannian squared exponential kernels andheat kernels in the sense of fundamental solutions of heat equations.
These can be connected with Matérn kernels by viewing them as certain integrals of squared exponential kernels [47].
We use thisconnection to introduce new Riemannian kernels on non-compact manifolds, and to further exploreexisting Riemannian Matérn kernels in Section 3.
2.2 Optimization of acquisition functions on manifolds
To use these Gaussian processes in a Bayesian optimization system, one must further be able determinewhere to evaluate the unknown function and collect additional data.
To do so, Bayesian optimizationminimizes an acquisition function, which resolves the explore-exploit tradeoff [42] inherent in theproblem due to lack of complete information about f .
This is done, for instance, by prioritizingpoints which are known to perform well due to having high posterior mean, while also consideringpoints whose performance is unknown due to high posterior variance.
In this way, the probabilisticuncertainty given by the posterior Gaussian process is used to resolve the tradeoff.
In the geometric setting, Bayesian optimization therefore requires minimization of real-valuedfunctions on manifold domains as an intermediate step.
To do so, we leverage optimization algorithmson Riemannian manifolds [1] to optimize the acquisition function in a geometry-aware manner.
These algorithms take advantage of the Euclidean tangent space T x X linked to each point x on themanifold X .
The optimization process typically operates on function gradients, which are intrinsicallydeﬁned on the tangent space, and projects the result of these computations back onto the manifoldwith the exponential map Exp x : T x X → X .
Notice that these different operations are determinedbased on the Riemannian metric with which the manifold is endowed.
In this paper, we employ trust region methods on Riemannian manifolds, as introduced by Absilet al.
[2], to optimize the acquisition function at each iteration of Bayesian optimization.
Therecursive process used in trust region methods on Riemannian manifolds involves the same steps asits Euclidean counterpart, namely: ( i ) the optimization of a quadratic subproblem m k trusted locallyin a region around the iterate; ( ii ) the update of the trust region parameters, typically the trust-regionradius ∆ k ; ( iii ) the iterate update, where a candidate is accepted or rejected as function of the qualityof the model m k .
For the cases where the space of interest is restricted to a subset of a Riemannianmanifold, we employ a constrained trust region algorithm proposed by Jaquier and Rozo [24] thathandles linear constraints.
This allows us to introduce physical or safety constraints that naturallyarise when interacting with physical systems and are crucial for robotics applications.
Matérn Kernels on Riemannian Manifolds
Training Gaussian processes, a crucial component of Bayesian optimization, requires us to be ableto compute the kernel point-wise.
This kernel encapsulates similarity information in the domain ofinterest: for domains with geometric structure, such as Riemannian manifolds, the kernel shouldreﬂect this structure in order to perform effectively.
We now introduce geometric kernels that capturethe structure of a number of spaces of interest in robotics, and study techniques for computing them.
As aforementioned, we focus on Riemannian Matérn kernels.
These are based on the characterizationof the widely used Euclidean Matérn Gaussian processes as solutions of a class of stochastic partialdifferential equations given by Whittle [49].
These equations naturally generalize to the Riemanniansetting, and yield kernels which are well-deﬁned for all Riemannian manifolds and hyperparametervalues, unlike [25] where valid length scales are required to guarantee positive deﬁniteness.
The maindifﬁculty of this approach is that this deﬁnition is implicit and does not give practical expressions forcomputing the kernel without further considerations.
For compact Riemannian manifolds, Borovitskiyet al.
[7] prove general expressions for Matérn kernels of the form
[Equation]
where ( λ n , f n ) are Laplace–Beltrami eigenpairs, κ is the length scale, σ 2 is the variance, ν is thesmoothness and the constants C ν , C ν are chosen so that the resulting kernels have average varianceacross the domain equal to σ 2 .
Note that despite notation C ν depends on both ν and κ .
The limitingkernel as ν → ∞ , which we denote by k ∞ ,κ,σ 2 , can be viewed as the Riemannian analog of the Euclidean squared exponential kernel.
In practice, the kernel (1) and its squared exponential limit are computed by calculating the eigenpairseither analytically or numerically, and then truncating the inﬁnite sum.
Numerical computation ofthe eigenpairs amounts to approximating the manifold with a mesh and solving the eigenproblemwith the ﬁnite element method.
Since this approach scales exponentially with manifold dimension,we do not focus on it in this work.
Fortunately, owing to their extensive use in physics and otherareas, mathematical techniques for evaluating the eigenpairs and simplifying the related expressionsare available for many manifolds of interest in robotics: examples of this can be seen in the sequel.
Although we only present the formulas for squared exponential (heat) kernels for the sake brevity,respective formulas for Matérn kernels hold as well.
The expression (1) does not apply for non-compact manifolds, although the implicit stochastic partialdifferential equation deﬁnition it is derived from is still valid.
In this case, we leverage representationsof Matérn kernels as integrals of squared exponential kernels, also known as heat kernels in themathematical literature.
In the Euclidean case, observe that, up to a normalizing constant,
[Equation]
This can be obtained by writing the spectral measure of the Matérn kernel, the T distribution, as agamma mixture of Gaussians.
This relationship also holds in the compact Riemannian case (see
Appendix B for details).
More generally, it can be considered as the deﬁnition of a Matérn kernel inany setting where a suitable notion of a squared exponential or heat kernel is available, which provesto be particularly useful in non-compact settings.
Moreover, as shown in Appendix B, this kernelis positive deﬁnite whenever the respective heat kernel is positive deﬁnite.
Therefore, it sufﬁces toobtain expressions for the squared exponential kernel: analogous expressions for the Matérn kernelfollow by integrating over length scales via (2).
We thus focus on the heat kernel henceforth.
These general expressions for Riemannian Matérn kernels allow us to build theoretically-soundkernels for the torus and sphere. More importantly, we can introduce Riemannian formulations forkernels on non-compact manifolds such as the manifold of symmetric positive deﬁnite matrices, and H d . We now turn our attention to manifolds of key importance in robotics applications.
Torus.
The torus T d is deﬁned as a product of circles T d = S 1 × . . .
× S 1 , which naturally arisesas, for example, the space of joint conﬁguration of a d -degree-of-freedom robot.
Functions on a torus f : T d → R can be viewed as 1 -periodic functions f : R d → R .
Using this identiﬁcation, the
Laplace–Beltrami eigenfunctions on T d are exactly the eigenfunctions of the Euclidean Laplacian,which are 1 -periodic.
Speciﬁcally, they correspond to the sines and cosines whose frequency is aninteger multiple of 2 π .
Substituting this into the general formula yields
[Equation]
where the sum is over vectors of integers τ , x , x (cid:48) ∈ [0 , 1) d are the representation of elements of
T d as vectors of angles divided by 2 π , i.e., such that (cid:0) e 2 πix 1 , . .
, e 2 πix d (cid:1) and (cid:0) e 2 πix (cid:48) 1 , . . .
, e 2 πix (cid:48)
d (cid:1)are the corresponding elements of T d .
Here, C ∞ is again the normalizing constant chosen so thatk ( x, x ) = σ 2 .
The key difference with (1) is that the eigenpairs are now parameterized by vectors ofintegers, slightly simplifying the expression.
A simple proof of this expression is given in Appendix B.
Sphere.
The sphere manifold occurs in a number of settings: in particular, unit quaternionscommonly used in robotics to represent the orientation of rigid bodies are a common example of thethree-dimensional sphere S 3 .
On the d -dimensional sphere S d the squared exponential kernel is
[Equation]
where C ∞ is a normalizing constant which ensures k ∞ ,κ,σ 2 ( x, x ) = σ 2 , c n,d are explicit constants,
C ( · ) n are the Gegenbauer polynomials, and d g is the geodesic distance on the sphere [7].
This kernelis obtained by noting that the eigenfunctions of the spherical Laplace–Beltrami operator are thespherical harmonics, the sums of which may be reformulated through the Gegenbauer polynomials—the resulting algebra, along with combining repeated eigenvalues, leads to the desired expression.
This expression is a signiﬁcant simpliﬁcation compared to the general case: note in particular that thekernel factorizes and can be written as a function only of the geodesic distance d g .
Special orthogonal group.
The special orthogonal group SO( d ) is a compact Riemannian manifold 2 of rotations of d -dimensional Euclidean space, most often represented as the set of orthogonald × d matrices of determinant 1 .
SO(3) is widely used to represent orientations in the 3 -dimensionalphysical space.
This manifold may likewise be useful as a building block for representing othergeometric objects.
For instance, the product R d × SO( d ) may be used as a proxy to representsymmetric matrices via a non-unique decomposition A = UDU (cid:62) , where D = diag( λ 1 , . . .
, λ d ) ,λ i ∈ R and U ∈ SO( d ) .
A kernel on R d × SO( d ) may be designed as a product of kernels on R dand on SO( d ) .
By restricting the Euclidean part of the product kernel to R d ≥ 0 or R d> 0 , we may obtaina kernel over positive (semi)deﬁnite matrices.
By leveraging the group structure of SO( d ) it is possible to express the corresponding heat kernel interms of group characters .
Speciﬁcally, we have that
[Equation]
(5) where the sum is over a certain set of tuples of nonnegative integers π called the highest weights and
C ∞ is the normalizing constant ensuring that k ∞ ,κ,σ 2 ( X , X ) =
σ 2 (see Appendix B).
Here, λ π are
Laplacian eigenvalues, χ π are certain functions called characters that depend only on eigenvalues oftheir argument, and d π = χ π
( I ) .
Partial sums of (5) corresponding to the most signiﬁcant eigenvaluesmay be computed in closed form by means of the Weyl character formula and Freudenthal’s formulafor the eigenvalues of the Casimir operator.
We provide technical details concerning this formulaand its practical implementation in Appendix B and in the supplemented code respectively, whiledeferring a thorough discussion to future work.
Hyperbolic space.
The hyperbolic space H d is the unique simply-connected complete ddimensional Riemannian manifold with a constant negative sectional curvature − 1 .
An important property of this space is the exponential rate of growth of the volume of a ball with respect to itsradius.
Because of this, it is often used to hold embeddings of hierarchical data, such as trees orgraphs [23, 34].
Although its potential to embed discrete data structures into a continuous space iswell known in the machine learning community, its application in robotics is presently scarce.
In
Section 4, we provide a toy example featuring a simple path planning on a 2D grid to illustrate howthe hyperbolic manifold can be exploited in robotics.
Hyperbolic space is non-compact, thus the technique of Borovitskiy et al.
[7] is unable to representthis kernel.
Fortunately, explicit formulas for the corresponding heat kernel in arbitrary dimension dare present in the literature.
Take x , y ∈ H d and denote by ρ = dist H d ( x , y ) the geodesic distancebetween x and y in H ( d ) .
Then we have
[Equation]
for d = 2 and d = 3 , and the following recurrence relation, known as Millson’s formula, for d > 3 :
[Equation]
These expressions are introduced in Grigoryan and Noguchi [18].
Manifold of positive deﬁnite matrices.
Positive deﬁnite matrices occur naturally in many physicsrelated problems.
In robotics, they are used to represent the stiffness and manipulability ellipsoids [3,52] or as control gain matrices [30].
These matrices strongly affect the performance of robot motionand force controllers, and they are variables of interest for task-compatible trajectory optimization.
Symmetric positive deﬁnite (SPD) matrices may be endowed with different Riemannian structures.
For instance, A (cid:55)→ exp( A ) and A (cid:55)→ AA (cid:62) deﬁne one-to-one maps between symmetric matricesor lower triangular matrices with positive elements on the diagonal and SPD matrices, allowingone to transfer Euclidean structure and kernels to SPD matrices.
Though simple and natural, these
Riemannian structures are often too simplistic and poorly capture the geometry of SPD matrices [12].
Another approach is to consider the map
A (cid:55)→ AA (cid:62) as a function from the space GL( d ) of invertiblematrices to S d ++ .
This map is not one-to-one: speciﬁcally, AA (cid:62) =
BB (cid:62) if and only if A = BQwhere Q is an orthogonal matrix — a proof is given in Appendix B.
To make this relation unique,we need to switch the domain to the quotient space of GL( d ) by the equivalence relation of beingequal up to an orthogonal matrix.
The resulting space GL( d ) / O( d ) is a Riemannian manifold.
The
Riemannian structure deﬁned this way is not Euclidean and the resulting manifold is non-compact,hence the method of Borovitskiy et al.
[7] does not apply here either.
Still, explicit formulas forthe corresponding heat kernels can be found in literature for cases of d = 2 and d = 3 , which arecommon dimensions in robotics problems.
Speciﬁcally, for d = 2 there is a relatively simple formula.
Take X , Y ∈ S 2 ++ , denote by H 1 ≥ H 2the singular values of the matrix XY − 1 , and α = H 1 − H 2 , then, by Sawyer [40], we havek ∞ ,κ,σ 2 ( X , Y ) =
σ
[Equation]
where the one-dimensional integral on the right-hand side may be evaluated numerically and C ∞ is anormalizing constant ensuring that k ∞ ,κ,σ 2 ( X , X )
=
σ 2 .
Additional details are given in Appendix B.
Product kernels on manifolds.
If the Riemannian manifold M of interest is the product of dmanifolds M = M 1 × . . .
× M d , where each factor M j is equipped with a Matérn kernel k M j ν,κ,σ 2 ,then we can introduce the following product Matérn kernel as a kernel on the product manifold M .
This is done by writing
[Equation]
where κ = ( κ 1 , . .
, κ d ) , x = ( x 1 , . .
, x d ) and x (cid:48) =
( x (cid:48) 1 , . . .
, x (cid:48) d ) .
Note that this enables automaticrelevance determination in the product Riemannian setting.
Because of this, it is an alternativeand more expressive way to deﬁne a kernel on, for example, a torus or other product manifold.
The product of manifolds naturally arises when representing the pose of rigid bodies such as theend-effector of a robotic manipulator, which lies on R 3 × S 3 .
Experiments
We evaluate the performance of geometry-aware Bayesian optimization via Riemannian Matérnkernels on a set of benchmark functions and robotics settings, using expected improvement [32]as the acquisition function.
Each optimization process is repeated 30 times with 5 random initialsamples.
Our implementations employ GPyTorch [15], BoTorch [6] and Pymanopt [46] as well as the
Robotics Toolbox [10].
Cost parameter values and additional experiments are given in Appendix A.
4.1 Benchmarks
We ﬁrst study performance using benchmark test functions projected onto the manifolds S 5 , SO(3) ,
S 2 ++ , and H 3 .
The search domain X is given by the full manifold, except for S d ++ where we restrict tomatrices with eigenvalues λ i ∈ [0 . 001; 5] .
For each manifold, we compare geometry-aware Bayesianoptimization against classical Bayesian optimization with Euclidean kernels using a constrainedoptimization on the Euclidean space to minimize the acquisition function.
For the sphere and SPDmanifolds, we additionally compare against the scheme used by Jaquier et al.
[25] featuring a naïve Riemannian squared exponential kernel, where Euclidean distance is replaced by the geodesic distance.
For the SPD manifold, we consider two Riemannian kernels, namely (8), and a product of kernels 280 R 2 × S 2 which is applied on the eigendecomposition of SPD matrices.
We also compare against analternative implementation using the Cholesky decomposition of an SPD matrix X = LL T , so thatthe resulting parameter is the vectorization of the lower triangular matrix L .
Figure 2 shows performance results.
The left graphs show the evolution of the median for the different
Bayesian optimization approaches, and the right graphs display the distribution of the logarithm ofthe simple regret of the ﬁnal recommendation x N after 200 iterations.
We observe that geometryaware algorithms generally match or outperform their Euclidean counterparts.
These differencesare more pronounced in higher dimensions, and for manifolds which more signiﬁcantly depart fromthe Euclidean setting due to presence of curvature or other geometric features, with the hyperbolicmanifold H 3 showing the most pronounced difference.
Surprisingly, the naïve Riemannian squaredexponential kernel—which for certain length scales may become ill-deﬁned—performs competitivelyin some settings: this may occur because small length scales keep it out of the problematic regime,and its simple form introduces less numerical error than series truncations or other approaches.
Asexpected, geometry-aware Matérn kernels ( ν = 2 . 5 ) outperform the squared exponential ones for thetest functions with uneven landscapes, such as the Ackley function in Figure 2).
Overall, we see thattaking geometry into account generally results in faster, more data-efﬁcient convergence and lowervariance, at cost of more complex algorithmic implementation.
4.2 Robotics Experiments
Here, we evaluate the performance of geometry-aware Bayesian optimization on several simulatedrobotics scenarios.
In the ﬁrst experiment, we use Bayesian optimization as an orientation sampler aiming at satisfying the requirements deﬁned by a cost function, similarly to the experiment presentedin Jaquier et al.
A velocity-controlled robot samples an orientation reference x ≡ ˆ q arounda prior orientation ˜ q .
The objective of the robot is to minimize the error between the prior and thecurrent end-effector orientation with low joint torques and an isotropic manipulability ellipsoid, i.e.,f ( q ) =
w q dist S 3 (˜ q , q ) + w τ (cid:107) τ
(cid:107) 2 + w M cond
( M ) , where q is the current end-effector orientation,and cond ( M ) is the condition number of the velocity manipulability ellipsoid.
Next, we consider a task-compatible manipulability optimization scenario, which is a commonproblem in robotics where most solutions ignore the geometry of the manipulability ellipsoids [20,27].
Here, a 8 -degree-of-freedom planar robot is required to track a desired Cartesian velocitytrajectory leading to a vertical line, while tracking a desired manipulability ellipsoid in its nullspace.
Optimization aims at ﬁnding the desired manipulability x ≡ ˆ M ∈ S 2 + + so that the end-effectoracceleration ¨ p is minimized and the robot tracks the desired manipulabity with a high precision, i.e.,f ( ˆ M ) =
w ¨ p (cid:107) ¨ p (cid:107) + w M dist S 2++ ( M , ˆ M ) with M the current manipulability.
Finally, we consider a simple path planning problem on R 2 for a robot represented by a point particle.
Our goal is to ﬁnd a collision-free path between two points in the environment.
To do so, we leveragegeometry-aware Bayesian optimization to estimate paths for sampling, similarly to Marchant and
Ramos [29], while introducing a novel perspective to the path search on a hyperbolic manifold..
Speciﬁcally, we discretize the robot environment by deﬁning a grid on R 2 with 4 -connected nodes,represented by a connected undirected graph G =
( V, E ) , of n vertices V and edges E .
A circleshaped obstacle with unknown location occupies this environment: the robot ﬁnds this by sensingcollisions while testing the sampled path.
The graph G is mapped to H 2 by learning embeddingsin the Lorentz model of hyperbolic geometry, similarly to Nickel and Kiela [33].
At each iteration,the algorithm samples a set of m points { x 1 , . . .
x m } ∈ H 2 representing a "hyperbolic" path of mnodes.
These "hyperbolic" nodes are then mapped back to the grid in R 2 , in order for the robotto navigate the environment using the resulting path.
An optimal path corresponds to the shortestcollision-free trajectory from a ﬁxed starting robot location to the goal x g .
We minimize the costf ( x 1 ,...,m ) =
w g dist x g ( x 1 ,...,m ) + w d dist t ( x 1 ,...,m ) , where dist x g ( · ) is the distance to the goalx g (zero in no-collision cases), and dist t ( · ) is the total travel distance.
Figure 3 shows performance results.
Here, we also see a moderate improvement in performance whenusing geometry-aware Bayesian optimization, mirroring the results on benchmark functions.
Theseare least pronounced for the orientation experiment, where the low-dimensional nature may make the
Euclidean and Riemannian kernels very similar, limiting the effect of geometry, and most pronouncedin the manipulability and path planning problems, which depart more from the Euclidean setting.
Conclusion
In this work, we study geometry-aware Bayesian optimization for robotics problems, and proposetechniques for computing geometry-aware kernels on a number of Riemannian manifolds that occurin robotics applications.
This is achieved by studying how general expressions for Matérn kernelsbased on stochastic partial differential equations simplify for the given manifolds of interest.
Thisprovide practitioners with explicit formulas, enabling geometry -aware Bayesian optimization to be implemented in novel settings.
We demonstrate our techniques both in benchmark settings to showcase improved performance and on robotics examples to showcase potential use cases.
We hope these contributions enable greater use of geometry among robotics practitioners