Object Trajectory Optimization With Complementarity Collision Constraints
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Solution Overview
Problem
Existing trajectory optimization algorithms for robotic systems in cluttered environments face challenges in formulating non-smooth, non-convex constraints for obstacle avoidance, leading to potential collisions and suboptimal solutions due to the complexity of differentiating distance constraints.
Innovation Solution
The formulation of obstacle avoidance constraints as convex optimization problems using complementarity conditions allows for the creation of smooth constraints, enabling the use of nonlinear programming algorithms and ensuring collision-free trajectories by transforming nested optimization problems into single optimization problems with first-order stationary conditions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If conventional trajectory planning algorithms use nonlinear programming algorithms with distance constraints for obstacle avoidance, then the optimization problem can be formulated, but the constraints become non-differentiable leading to computational difficulties
Solution Approach 1:
The patent transforms the non-differentiable distance constraint into a differentiable form by changing the mathematical parameter representation. Instead of using direct distance constraints that are non-differentiable at zero distance, the patent employs a transformed constraint formulation that maintains the obstacle avoidance requirement while ensuring continuous differentiability, enabling smooth gradient-based optimization.
2Reliability
If the optimization problem includes non-smooth, non-convex constraints for obstacle avoidance, then collision avoidance can be enforced, but multiple local minima exist and global minimum cannot be guaranteed
Solution Approach 1:
The patent inverts the conventional approach by instead of directly enforcing non-convex distance constraints, it formulates the problem in terms of maintaining a minimum separation distance through a transformed constraint structure. This inversion converts the non-convex optimization landscape into a more tractable form that still guarantees collision avoidance while improving the reliability of finding optimal solutions.
3Reliability
If explicit collision checking modules are used to ensure obstacle avoidance, then collision-free trajectories can be verified, but resource consumption and CPU time increase
Solution Approach 1:
The patent substitutes the mechanical approach of explicit collision checking modules with a mathematical constraint-based approach. By embedding the collision avoidance requirement directly into the optimization constraints, the system eliminates the need for separate collision detection algorithms, thereby reducing computational overhead and CPU time while maintaining the guarantee of collision-free trajectories.
Data Source
AI summary
A controller controls a motion of an object performing a task for changing a state of the object from a start state to an end state while avoiding collision of the object with an obstacle according to an optimal trajectory determined by solving an optimization problem of the dynamics of the object producing an optimal trajectory for performing the task subject to constraints on a solution of first-order stationary conditions modeling a minimum distance between the convex hull of the object and the convex hull of the obstacle using complementarity constraints.