A Unifying Variational Framework for Gaussian Process Motion Planning

To control how a robot moves, motion planning algorithms must compute paths in high-dimensional state spaces while accounting for physical constraints related to motors and joints, generating smooth and stable motions, avoiding obstacles, and preventing collisions. A motion planning algorithm must therefore balance competing demands, and should ideally incorporate uncertainty to handle noise, model errors, and facilitate deployment in complex environments. To address these issues, we introduce a framework for robot motion planning based on variational Gaussian Processes, which unifies and generalizes various probabilistic-inference-based motion planning algorithms. Our framework provides a principled and flexible way to incorporate equality-based, inequality-based, and soft motion-planning constraints during end-to-end training, is straightforward to implement, and provides both interval-based and Monte-Carlo-based uncertainty estimates. We conduct experiments using different environments and robots, comparing against baseline approaches based on the feasibility of the planned paths, and obstacle avoidance quality. Results show that our proposed approach yields a good balance between success rates and path quality

A cocrystal of 3α-hy­droxy­tirucalla-8,24-dien-21-oic acid and 3β-fluoro­tirucalla-7,24-dien-21-oic acid (0.897:0.103)

The title compound, 0.897C30H48O3.0.103C30H47O2F is a co-crystal of two triterpenes isolated from the resin of Canarium schweinfurthiiand Engl. Both triterpenes consists of four trans-fused rings having chair/half-chair/half-chair and envelope conformations. The mol­ecular conformations are stabilized by intra­molecular C—H⋯O hydrogen bonds, forming rings of S(7) graph-set motif. In the crystal, mol­ecules are linked by inter­molecular O—H⋯O and C—H⋯O inter­actions, forming sheets parallel to (001). All atoms. excepting the axially-oriented hydroxyl group in the major component and the equatorially-oriented fluorine atom in the minor component, are overlapping

3α-Hy­droxy­tirucalla-8,24-dien-21-oic acid

The title compound, C30H48O3, a triterpene isolated from the resin of canarium schweinfurthiiand, is an isomer of the previously reported triterpene 3α-hy­droxy­tirucalla-7,24-dien-21-oic acid [Mora et al. (2001 ▶). Acta Cryst. C57, 638–640], which crystallizes in the same trigonal space group. The title mol­ecule consists of four fused rings having chair, half-chair, half-chair and envelope conformations for rings A, B, C and D, respectively (steroid labelling). An intra­molecular C—H⋯O hydrogen bond generates an S(7) ring. In the crystal, mol­ecules are linked by O—H⋯O and C—H⋯O inter­actions, forming (001) sheets

Bis(2-amino­thia­zolium) succinate succinic acid

In the title compound, 2C3H5N2S+·C4H4O4 2−·C4H6O4, the thia­zolium ring is almost planar, with the maximum deviation from planarity being 0.0056 (8) Å for the C atom carrying the amine substituent. The N atom of the 2-amino­thia­zole mol­ecule is protonated. Both the anion and the acid lie across inversion centres. The crystal packing is consolidated by inter­molecular O—H⋯O, N—H⋯O and C—H⋯O hydrogen bonds. Mol­ecules are stacked down the b axis

2-Ethyl­piperidinium chloride

In the title molecular salt, C7H16N+·Cl−, the piperidinium ring adopts a chair conformation. In the crystal, the two components are connected by N—H⋯Cl and C—H⋯Cl hydrogen bonds, forming a supra­molecular double-chain structure along the c axis

Monoclinic polymorph of poly[aqua(μ4-hydrogen tartrato)sodium]

A monoclinic polymorph of the title compound, [Na(C4H5O6)(H2O)]n, is reported and complements an ortho­rhom­bic form [Kubozono, Hirano, Nagasawa, Maeda & Kashino (1993 ▶). Bull. Chem. Soc. Jpn, 66, 2166–2173]. The asymmetric unit contains a hydrogen tartrate anion, an Na+ cation and a water mol­ecule. The Na+ ion is surrounded by seven O atoms derived from one independent and three symmetry-related hydrogen tartrate anions, and a water mol­ecule, forming a distorted penta­gonal–bipyramidal geometry. Independent units are linked via a pair of inter­molecular bifurcated O—H⋯O acceptor bonds, generating an R 2 1(6) ring motif to form polymeric two-dimensional arrays parallel to the (100) plane. In the crystal packing, the arrays are linked by adjacent ring motifs, together with additional inter­molecular O—H⋯O inter­actions, into a three-dimensional network

2,6-Difluoro­benzoic acid

In the title compound, C7H4F2O2, the dihedral angle between the benzene ring and the carboxyl­ate group is 33.70 (14)°. In the crystal structure, inversion dimers linked by pairs of O—H⋯O hydro­gren bonds occur, generating R 2 2(8) loops. The dimers are linked into sheets lying parallel to (102) by C—H⋯F hydrogen bonds

N,N′-[4,4′-Methyl­enebis(4,1-phenyl­ene)]bis­(2,6-difluoro­benzamide)

The complete mol­ecule of the title compound, C27H18F4N2O2, is generated by crystallographic twofold symmetry, with one C atom lying on the rotation axis. The dihedral angle between fluoro-substituted phenyl ring and the adjacent benzene ring is 10.37 (5)°. In the crystal, mol­ecules are connected by N—H⋯O and C—H⋯F hydrogen bonds, resulting in supra­molecular chains propagating along the c direction

Triethyl­ammonium 4-nitro­benzene­sulfonate

In the anion of the title molecular salt, C6H16N+·C6H4O5S−, the nitro group is twisted slightly from the benzene ring, making a dihedral angle of 3.16 (10)°. In the crystal structure, the cations and anions are linked into a two-dimensional network parallel to the ab plane by C—H⋯O and N—H⋯O hydrogen bonds

Poly[μ2-aqua-(μ3-2,5-dichloro­benzene­sulfonato)sodium]

In the title compound, [Na(C6H3Cl2O3S)(H2O)]n, the NaI ion is penta­coordinated by three dichloro­benzene­sulfonate anions and two water mol­ecules, forming a distorted trigonal-bipyramidal geometry. The NaI ions are bridged by the sulfonate groups and the water mol­ecules, leading to a polymeric layer structure parallel to the bc plane in which O—H⋯O hydrogen bonds are observed