"Optimal Trajectories of Curvature-Constrained Motion in the Hamilton-Jacobi Formulation"

Authors:  Ryo Takei & Richard Tsai (UCLA, 2013)

Presenter:  William Pope

Motivation

  • Tree-based online (PO)MDP solvers use rollout simulations to initialize new nodes
    • Reward is sparse in navigation scenarios, so need a good rollout policy to guide
    • Value estimate must be generated quickly to build search tree online
  • HJB provides on-demand optimal trajectories from any point in space, used for rollout in pedestrian navigation problem

Multi-Query Motion Planning

  • Can use multi-query planning as policy for rollout simulations
  • Single run of planner algorithm provides complete trajectories on-demand from any point in state space
  • Common multi-query methods:
    • Fast marching method (FMM)
    • Probabilistic roadmaps (PRM)

PRM

FMM

However: these methods don't work well for systems with differential constraints

Curvature-Constrained Dynamics

  • Simple nonlinear kinematic model used to approximate motion of a car
  • Assumes instantaneous changes in speed and steering angle
\dot{x} = f(x,u) = \begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} u_{v} \cos{\theta} \\ u_{v} \sin{\theta} \\ u_{v}\frac{1}{l}\tan{u_{\phi}} \end{bmatrix}
u = \begin{bmatrix} u_{v} \\ u_{\phi} \end{bmatrix}
x = \begin{bmatrix} x \\ y \\ \theta \end{bmatrix}

Reeds-Shepp Curves

  • For vehicle with car-like steering, shortest path between two states will always be a series of straight lines connected by minimum-radius curves, driven at max speed
U_{opt} = \begin{Bmatrix} (\max(u_{v}), 0.0) & (\min(u_{v}), 0.0) \\ (\max(u_{v}), \max(u_{\phi}) & (\min(u_{v}), \max(u_{\phi}) \\ (\max(u_{v}), \min(u_{\phi}) & (\min(u_{v}), \min(u_{\phi}) \end{Bmatrix}
  • So for first-order dynamics, continuous-time optimal action will always be 1 of 6:

Hamilton-Jacobi-Bellman Equation

  • PDE for finding global optimal control of a system (Bellman, 1950s)
  • Value function: minimum cost-to-go over given time interval
    • C – cost rate function
    • D – terminal value
V(x(t_{0}),t_{0}) = \min_{u}\{\int_{t_{0}}^{t_{f}} C(x(\tau),u(\tau)) d\tau + D(x(t_{f}))\}
V(x(t_{0}),t_{0}) = \min_{u}\{V(x(t_{0}+dt),t_{0}+dt) + \int_{t_{0}}^{t_{0}+dt} C(x(\tau),u(\tau)) d\tau\}
  • Rewriting with dynamic programming principle:

Hamilton-Jacobi-Bellman Equation

  • Applying Taylor series expansion to right side:
V(x(t),t) = \min_{u}\{V(x(t+dt),t+dt) + \int_{t}^{t+dt} C(...) d\tau\}
V(x(t),t) = \min_{u}\{V(x(t),t) + \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial x}dx + \int_{t}^{t+dt} C(...) d\tau\}
V(x(t),t) = V(x(t),t) + \frac{\partial V}{\partial t}dt + \min_{u}\{\frac{\partial V}{\partial x}\dot{x}dt + \int_{t}^{t+dt} C(...) d\tau\}
0 = \frac{\partial V}{\partial t} + \min_{u}\{\frac{\partial V}{\partial x}\dot{x} + \frac{1}{dt}\int_{t}^{t+dt} C(...) d\tau\}
0 = \frac{\partial V}{\partial t} + \min_{u}\{\frac{\partial V}{\partial x}f(x,u) + C(x,u) \}
  • Hamilton-Jacobi-Bellman partial differential equation:

Solving HJB for Optimal Trajectories

  1. Offline:
    1. Given map of environment, discretize state space
    2. Assign known value to target set
    3. Solve PDE for value function V(x,y,θ) using finite difference method iterated over grid nodes
  2. Online:
    1. Use gradient descent to generate optimal path from current leaf node
    2. Simulate rollout using optimal path, return value to search tree

Solving HJB for Optimal Trajectories

Value function from solving HJB PDE:

Optimal paths from gradient of value function:

Sliced at θ=50°

Solving HJB

  • Apply system information to HJB PDE:
0 = \frac{\partial V}{\partial t} + \min_{u}\{\frac{\partial V}{\partial x}f(x,u) + C(x,u) \}
\frac{\partial V}{\partial t} = 0
C(x,u) = 1
  • Value function doesn't change over time:
  • Cost = time elapsed, cost rate:
-1 = \min_{u}\{\frac{\partial V}{\partial x} \cdot f(x,u) \}
-1 = \min_{(u_{v},u_{\phi})}\{V_{x}(u_{v}\cos{\theta}) + V_{y}(u_{v}\sin{\theta}) + V_{\theta}(u_{v}\frac{1}{l}\tan{u_{\phi}}) \}

Solving HJB

  • Finite difference method (FDM) – numerical method for approximating derivatives
    • Forward/backward:
V_{x} = \frac{V_{i+1,j,k} - V_{ijk}}{h_{xy}}
V_{x} = \frac{-(V_{i-1,j,k} - V_{ijk})}{h_{xy}}
\text{PDE: } -1 = \min_{(u_{v},u_{\phi})}\{V_{x}(u_{v}\cos{\theta}) + V_{y}(u_{v}\sin{\theta}) + V_{\theta}(u_{v}\frac{1}{l}\tan{u_{\phi}}) \}
  • Upwind scheme
    • Value at given state only depends on value of "upwind" states (closer to target)
    • In FDM, need to pull value from upwind states
      • Upwind direction is determined by state/action
i_{uw} = i + sgn(\dot{x}) \\ j_{uw} = j + sgn(\dot{y}) \\ k_{uw} = k + sgn(\dot{\theta})
V_{x} = \frac{sgn(\dot{x})(V_{i_{uw},j,k} - V_{ijk})}{h_{xy}}

Solving HJB

  • Plugging in upwind FDM:
-1 = \frac{sgn(\dot{x})(V_{i_{uw}} - V)}{h_{xy}}(u_{v}\cos{\theta}) + \frac{sgn(\dot{y})(V_{j_{uw}} - V)}{h_{xy}}(u_{v}\sin{\theta}) + \frac{sgn(\dot{\theta})(V_{k_{uw}} - V)}{h_{\theta}}(u_{v}\frac{1}{l}\tan{u_{\phi}})
V_{i,j,k} = \frac{\frac{h_{xy}}{u_{v}} \text{ } + \text{ } V_{i_{uw},j,k} \text{ } s_{\dot{x}}\cos{\theta_{k}} \text{ } + \text{ } V_{i,j_{uw},k} \text{ } s_{\dot{y}}\sin{\theta_{k}} \text{ } + \text{ } V_{i,j,k_{uw}} \text{ } s_{\dot{\theta}}(\frac{h_{xy}}{h_{\theta}l})\tan{u_{\phi}}}{s_{\dot{x}}\cos{\theta_{k}} \text{ } + \text{ } s_{\dot{y}}\sin{\theta_{k}} \text{ } + \text{ } s_{\dot{\theta}}(\frac{h_{xy}}{h_{\theta}l})\tan{u_{\phi}}}
\text{PDE: } -1 = \min_{(u_{v},u_{\phi})}\{V_{x}(u_{v}\cos{\theta}) + V_{y}(u_{v}\sin{\theta}) + V_{\theta}(u_{v}\frac{1}{l}\tan{u_{\phi}}) \}
  • Solution:

Solving HJB

  • Implementation:
    1. ​Set initial value at every node in grid (target: 0, else: large)
    2. Iterate through all nodes in free space
      1. Calculate V_ijk through FDM with upwind neighbors for each of 6 possible optimal actions, keep lowest value
    3. Repeat sweeps until all nodes have converged

Step 22

Step 66

Results

h_{xy} = 0.25 \text{ m}
h_{xy} = 0.6 \text{ m}
h_{xy} = 0.1 \text{ m}

Results

h_{xy} = 0.25 \text{ m} \\ T_{path} = 7.66 \text{ s}
h_{xy} = 0.6 \text{ m} \\ T_{path} = \text{n/a}
h_{xy} = 0.1 \text{ m} \\ T_{path} = 7.45 \text{ s}

Results

  • Runtime tracks number of nodes
  • For given environment, need ~43 sec to calculate value at usable fidelity
  • At lower resolution, gradient descent is unable to find usable paths

Issues

  • Realistic scenarios may pose challenges in computation time:
    • Larger environments
    • Higher order dynamical models
    • Lower power computing
  • PDE solver is inflexible to changes in environment:
    • Moving goal or adding obstacles requires completely recomputing HJB

 

  • Improvements:
    • Adaptive mesh refinement
    • Splitting methods
    • Approximate value methods

Next Steps

  • Past work:
    • HJB generator loaded on NUC
    • HJB path planner implemented in ROS

 

  • Current/future work:
    • Refine existing HJB solver
    • Investigate improvements
    • Add velocity to state space

Copy of "Optimal Trajectories

By Zachary Sunberg

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