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+title = "A Unifying Variational Framework for Gaussian Process Motion Planning"
+[extra]
+authors = [
+    {name = "Lucas Cosier", url = "https://luke-ck.github.io", star = true},
+    {name = "Rares Iordan", star = true},
+    {name = "Sicelukwanda Zwane"},
+    {name = "Giovanni Franzese"},
+    {name = "James T. Wilson"},
+    {name = "Marc Peter Deisenroth", url = "https://deisenroth.cc/"},
+    {name = "Alexander Terenin", url = "https://avt.im/"},
+    {name = "Yasemin Bekiroğlu", url = "http://yaseminb.github.io" },
+]
+venue = {name = "AISTATS", date = 2023-05-02, url = "http://aistats.org/aistats2024/"}
+buttons = [
+    {name = "Paper", url = "https://arxiv.org/abs/2309.00854"},
+    {name = "PDF", url = "https://arxiv.org/pdf/2309.00854"},
+    {name = "Code", url = "http://github.com/luke-ck/vgpmp"},
+]
+katex = true
+large_card = true
++++
+
+In robot motion planning, one must compute paths through high-dimensional state spaces, while adhering to the physical constraints of motors and joints, ensuring smooth and stable movement, and avoiding obstacles.
+This requires a balance of competing computational factors, particularly if one wants to handle this in the presence of uncertainty, including noise, model error, and other complexities arising from real-world environments.
+We present a motion planning framework based on variational Gaussian Processes, which supports a flexible family of motion-planning constraints, including equality-based and inequality-based constraints, as well as soft constraints through end-to-end learning.
+Our framework is straightforward to implement, and provides both interval-based and Monte-Carlo-based uncertainty estimates.
+We evaluate it on a number of different environments and robots, compare against baselines approaches based on feasibility of sampled motion plans and obstacle avoidance quality.
+Our results show the proposal achieves a reasonable balance between the motion plan's rate of success and path quality.
+
+# Applying Variational Gaussian Processes to Motion Planning
+
+We begin with a motion planning framework, which we call *variational Gaussian process motion planning (vGPMP)*.
+This framework is based on variational Gaussian processes, which were originally introduced for scalability:[^vfe][^gpbd] here, we instead apply them to create a straightforward way to parameterize motion plans.
+Let `$\mathcal{T}$` represent time: our motion plan is a map `$f: \mathcal{T} \to \mathbb{R}^d$`, where the output space represents the position of each of the robot's joints.
+We parameterize `$f$` as a posterior Gaussian process, conditioned on `$f(\boldsymbol{z}) = \boldsymbol{u}$`, where `$\boldsymbol{z}$` is a set of inducing locations `$\boldsymbol{z} \in \mathcal{T}^m$`, and `$\boldsymbol{u}$` are robot joint states at times `$\boldsymbol{z}$`. 
+We interpret `$(z_j,u_j)$`-pairs as *waypoints* through which the robot should move.
+Our precise formulation in the paper also includes a bijective map which accounts for joint constraints: we suppress this here for simplicity.
+
+To draw motion plans, we apply *pathwise conditioning*,[^efficient-sampling][^pathwise-conditioning] and represent posterior samples as
+
+```
+$$
+(f \mid \boldsymbol{u})(\cdot) = f(\cdot) + \mathbf{K}_{(\cdot),\boldsymbol{z}} \mathbf{K}_{\boldsymbol{z},\boldsymbol{z}}^{-1} (\boldsymbol{u} - f(\boldsymbol{z})).
+$$
+```
+
+where the inducing points `$\boldsymbol{z}$` and the variational distribution `$q(\boldsymbol{u})$` of the values `$\boldsymbol{u}$` represent the distribution of possible motion plans.
+Computing the motion plan therefore entails optimizing these parameters with respect to an appropriate variational objective.
+Once optimized, in practice we can sample from the posterior using efficient sampling, that is, by first approximately sampling the prior `$f(\cdot)$` using Fourier features, then transforming the sampled prior motion plans into posterior motion plans. 
+This procedure allows us to draw random curves representing the posterior in a way that *resolves the stochasticity once in advance* per sample, after which we can evaluate and differentiate the motion plan at arbitrary time points without any additional sampling.
+Compared to prior work such as GPMP2 and its variants,[^gpmp][^gpmp2][^igpmp2] we support general kernels and avoid relying on specialized techniques for stochastic differential equations, thereby enabling explicit control of motion plan smoothness properties.
+Additionally, in contrast with prior work,[^gvi] our formulation bypass the need to use interpolation to evaluate the posterior in-between a set of pre-specified time points.
+
+Following the framework of variational inference, the resulting variational posterior can be trained by solving the optimization problem 
+
+```
+$$
+\min \mathbb{E}_{q(\boldsymbol{u})} \mathbb{E}_{p(\boldsymbol{f}\mid\boldsymbol{u})} \log p(\boldsymbol{e}\mid\boldsymbol{f}) - D_{\operatorname{KL}}(q(\boldsymbol{u})\mid\mid p(\boldsymbol{u}))
+$$
+```
+
+where `$p(\boldsymbol{f}\mid\boldsymbol{u})$` is the *log-likelihood* term which in motion planning is used to encode soft constraints.
+To apply hard equality-based constraints, we use the fact that the variational posterior is Gaussian, and apply conditional probability to an appropriate set of $(z_i,u_i)$-pairs.
+We can also apply hard inequality-based constraints, such as joint constraints, and describe this further in the sequel.
+
+This objective reveals Gaussian-process-based motion planning algorithms to be *stochastic generalizations* of optimization-based planners such as STOMP[^stomp], CHOMP[^chomp], and other variants: compared to these, the main difference is the presence of the Kullback--Leibler divergence, which prevents the posterior from collapsing to a single trajectory, thereby including uncertainty.
+For the log-likelihood, we use
+
+```
+$$
+p(\boldsymbol{e}\mid\boldsymbol{f}) = \exp\left(-\frac{1}{2} \left(\underset{\text{collision term}}{\undergroup{\left\|\operatorname{h}_\varepsilon(\operatorname{sdf}_s(\operatorname{k}_{\operatorname{fwd}}(\sigma(\boldsymbol{f}))))\right\|_{\mathbf\Sigma_{\operatorname{obs}}}^2}} + \underset{\text{soft constraint term}}{\undergroup{\left\|\operatorname{c}(\sigma(\boldsymbol{f}))\right\|^2_{\mathbf\Sigma_{\operatorname{c}}}}} \right)\right)
+$$
+```
+
+which works as follows.
+For the collision term, we first approximate a robot by a set of spheres, and compute the signed distance field `$\operatorname{sdf}_s$` for each sphere.
+This is done by composing the forward kinematics map `$\operatorname{k}_{\operatorname{fwd}}$` with the (inverse) constraint map `$\sigma$` which enforces a set of box constraints to account for joint limits.
+Then, we compute the hinge loss `$\operatorname{h}_\varepsilon(x) = \max(-x + \varepsilon, 0)$`, where `$\varepsilon$` is the *safety distance* parameter, and calculate its squared norm with respect to a diagonal scaling matrix `$\mathbf\Sigma_{\operatorname{obs}}$` which determines the overall importance of avoiding collisions in the objective.
+The soft constraint term, which can be used to encode desired behavior such as a grasping pose, is handled analogously.
+Compared to prior work,[^gpmp][^gpmp2][^igpmp2][^gvi] one of the key differences is the introduction of `$\sigma$`, which guarantees that joint limits are respected without the need for clamping or other post-processing-based heuristics.
+
+# Experiments
+
+Below, we provide a side-by-side comparison with our framework, vGPMP, with the Gaussian-process-based baseline GPMP2, on the WAM robot.
+This comparison shows that, on this example, vGPMP produces longer paths, which are more conservative with respect to collision avoidance.
+
+**TODO: paths**
+
+Next, we demonstrate that our framework can enable one to incorporate a grasping pose without further tuning or changing the underlying model assumptions. 
+This amounts to adding appropriate terms to the soft constraints, and enables the Franka robot, in simulation, to pick up a can.
+Further adjustments to can be made to allow the robot to pick up the can and move it to desired places by conditioning end-effector joints to open and close at specific time steps.
+
+**TODO: figure 2**
+
+Finally we present a demonstration of the Franka robot avoiding boxes.
+Here, our framework allows sampled paths to be computed at any required resolution without additional interpolation, enabling fine-grained control over smoothness.
+
+**TODO: VIDEO**
+
+# Conclusion
+
+We present a unifying framework which simplifies motion planning with Gaussian processes by applying a formulation based on variational inference.
+Through the use of this inducing-point-based framework and pathwise conditioning, we support general kernels that provide explicit control over motion plan smoothness properties.
+Our computations are straightforward to implement, and avoid the need for interpolation, clipping, and other post-processing.
+The framework also connects Gaussian-process-based motion planning algorithms with optimization-based approaches.
+We evaluate the approach on different robots, showing that it accurately reaches target positions while avoiding obstacles, providing uncertainty, and increasing clearance compared to baselines.
+We demonstrate the approach on a real robot, executing a randomly sampled trajectory while avoiding obstacles.
+
+[^vfe]: M. Titsias. Variational learning of inducing variables in sparse Gaussian processes. AISTATS 2009.
+
+[^gpbd]: J. Hensman, N. Fusi, and N. Lawrence. Gaussian Processes for Big Data. UAI 2013.
+
+[^efficient-sampling]: J. T. Wilson, V. Borovitskiy, A. Terenin, P. Mostowsky and M. P. Deisenroth. Efficiently Sampling Functions from Gaussian Process Posteriors. ICML 2020.
+
+[^pathwise-conditioning]: J. T. Wilson, V. Borovitskiy, A. Terenin, P. Mostowsky and M. P. Deisenroth. Pathwise Conditioning of Gaussian Processes. JMLR 2021.
+
+[^gpmp]: M. Mukadam, X. Yan, and B. Boots. Gaussian process motion planning. ICRA 2016.
+
+[^gpmp2]: J. Dong, M. Mukadam, F. Dellaert, and B. Boots. Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs. RSS 2016.
+
+[^igpmp2]: M. Mukadam, J. Dong, X. Yan, F. Dellaert, and B. Boots. Continuous-time Gaussian Process Motion Planning via Probabilistic Inference. IJRR 2018.
+
+[^gvi]: H. Yu and Y. Chen. A Gaussian Variational Inference Approach to Motion Planning. RAL 2023.
+
+
+[^stomp]: M. Kalakrishnan, S. Chitta, E. Theodorou, P. Pastor, and S. Schaal. STOMP: Stochastic trajectory optimization for motion planning. ICRA 2011.
+
+[^chomp]: M. Zucker, N. Ratliff, A. Dragan, M. Pivtoraiko, M. Klingensmith, C. Dellin, J. A. Bagnell, and S. Srinivasa. CHOMP: Covariant Hamiltonian Optimization for Motion Planning. IJRR 2013.
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