Robot Arm Trajectory Optimization With Differentiable Obstacle Constraints
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Solution Overview
Problem
Conventional trajectory planning methods for robotic manipulators face challenges in optimizing paths that avoid obstacles while minimizing time, as they often result in conservative or infeasible trajectories due to overestimation of physical extents and nonsmooth, non-convex constraints, leading to multiple local minima and a lack of guarantee for global minimum solutions.
Innovation Solution
The approach involves formulating trajectory optimization with continuous-time collision avoidance using geometric constraints as first-order differentiable functions, allowing for the incorporation of state, input, and geometric constraints within nonlinear optimization, which enables the generation of collision-free and optimal trajectories without explicit collision checking.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional trajectory planning methods use circumscribing of obstacles and robot arm by convex objects, then obstacle avoidance is achieved, but the physical extent is overestimated leading to conservative or infeasible trajectories
Solution Approach 1:
The robot arm is segmented into multiple links, and the convex hull is computed for each link separately rather than for the entire arm. This segmentation allows for more precise representation of the actual physical extent of each component, reducing overestimation while maintaining collision avoidance guarantees.
Solution Approach 2:
The method applies local convex hull computation to each specific link of the robot arm rather than a global approximation. This local quality approach ensures that the geometric constraints accurately reflect the actual shape and position of each link, improving trajectory quality without sacrificing safety.
2Reliability
If conventional methods use nonsmooth, non-convex constraints for obstacle avoidance, then collision detection is enforced, but the optimization problem becomes challenging with multiple local minima and no guarantee of global minimum
Solution Approach 1:
The patent introduces an intermediary function - the signed distance function - that acts as a mediator between the complex nonsmooth collision constraints and the optimization algorithm. This intermediary provides a smooth, differentiable representation of the collision avoidance requirement, enabling the use of efficient nonlinear programming algorithms while maintaining the same safety guarantees.
Solution Approach 2:
The method changes the parameterization of the constraint from a nonsmooth, non-convex form to a smooth, convex form using the signed distance function. This parameter transformation converts the optimization landscape into one with a single global minimum, eliminating local minima issues while preserving the collision avoidance requirement.
3Device complexity
If point mass representation is used for the robot, then the optimization problem is simplified, but the geometry of the robotic device is not represented leading to conservative trajectories
Solution Approach 1:
The method transitions from a static point mass model to a dynamic multi-link model where each link has its own geometry and motion characteristics. The convex hull is computed dynamically for each link based on its current configuration, allowing the model to accurately represent the robot's actual geometry while maintaining computational tractability through efficient convex optimization.
Data Source
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AI summary
A method and system to determine trajectories including a path with a starting pose and an ending pose over a sequence of time intervals while satisfying dynamic constraints and geometric constraints on a robot arm having links. Determine the dynamic constraints based on dynamics of each link, and determine the geometric constraints as a first-order differentiable function by generating a coordinate grid of an environment, to determine Cartesian coordinates of an obstacle and end-points of each link. Use a multi-link dynamics model to formulate a non-linear optimization (NLO) program for the robot arm, the NLO program uses as inputs the starting and ending pose, the dynamic and geometric constraints, to optimize an objective function of the trajectories while satisfying the dynamic and geometric constraints and robot arm operating constraints, for each time interval. Control the motors of the robot drive to move the robot arm based upon the trajectories.