Closed-Kinematic Manipulator Path Planning for Collision Avoidance
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Solution Overview
Problem
Existing movement planning algorithms for robots with closed kinematics struggle to generate deterministic, collision-free trajectories efficiently, especially when dealing with complex environments and dynamic changes.
Innovation Solution
Formulating the collision-free movement planning as a dynamic optimization problem that accounts for the dynamics of the manipulator, weighted via a cost functional, and solved numerically with inequality and equality restrictions to ensure closed kinematics and avoid collisions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If probabilistic methods (PRM or RRT) are used for motion planning with closed kinematics, then the method can handle complex environments, but the probability that a random configuration satisfies the closed kinematics constraint approaches zero, leading to high computational times
Solution Approach 1:
The patent replaces probabilistic sampling methods with a deterministic optimization-based approach. Instead of randomly sampling configurations and hoping to satisfy closed kinematics constraints, the system formulates motion planning as a boundary value problem with differential algebraic equations that directly enforce closed kinematics constraints, eliminating the randomness and associated computational inefficiency.
Solution Approach 2:
The patent changes the fundamental parameters of the motion planning approach by switching from probabilistic sampling to deterministic numerical integration and optimization. The system uses differential algebraic equations and numerical solvers to directly compute trajectories that satisfy closed kinematics, transforming the problem from one of random search to one of controlled numerical computation.
2Loss of time
If probabilistic methods are used for motion planning, then computational time can be limited, but the trajectories generated are non-deterministic and unpredictable
Solution Approach 1:
The patent substitutes probabilistic algorithms with deterministic numerical methods. The motion planning is formulated as a boundary value problem solved through numerical integration of differential algebraic equations, which inherently produces deterministic results. The use of numerical solvers with fixed initial conditions and constraint equations ensures that the same input always yields the same trajectory.
Solution Approach 2:
The patent segments the motion planning problem into distinct mathematical components: differential algebraic equations for dynamics, boundary conditions for start and end states, and algebraic constraints for closed kinematics. This segmentation allows each component to be solved deterministically using appropriate numerical methods, ensuring reproducible results.
3Reliability
If optimization-based methods are used for motion planning with closed kinematics, then deterministic trajectories can be generated in finite computing time, but the formulation of a suitable optimization problem that comprehensively encompasses motion planning is required
Solution Approach 1:
The patent creates a universal optimization framework that handles multiple aspects of motion planning simultaneously. The differential algebraic equation approach naturally incorporates dynamics, closed kinematics constraints, and boundary conditions in a unified mathematical formulation, eliminating the need for separate optimization problems for each constraint type.
Solution Approach 2:
The patent introduces numerical integration and algebraic constraint solving as intermediary steps between the optimization objective and the final trajectory. These intermediaries translate the complex multi-constraint problem into a sequence of manageable numerical computations, making the overall formulation more tractable while maintaining comprehensiveness.
4Reliability
If traditional motion planning methods are used for closed kinematics, then the closed kinematics constraint must be satisfied at all times, but this represents an additional restriction on freedom of movement that makes finding collision-free paths difficult
Solution Approach 1:
The patent applies dynamics principles by formulating motion planning as a differential algebraic equation problem that inherently respects the dynamic constraints of closed kinematic chains. The numerical integration approach naturally maintains the dynamic relationships between connected bodies, allowing freedom of movement within the constraints of the closed chain rather than treating constraints as static restrictions.
Solution Approach 2:
The patent transitions from viewing closed kinematics constraints as restrictions in configuration space to treating them as natural consequences of the system's dynamic model. By formulating the problem in terms of differential algebraic equations that capture the essence of closed kinematic chains, the constraints become integral to the motion description rather than external limitations.
Data Source
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AI summary
A method (100) for collision-free movement planning for a first manipulator (28) in a closed kinematic system, having the steps of: defining a dynamic optimisation problem (12), solving the optimisation problem (12) using a numerical approach and determining a first path of movement for the first manipulator (28) on the basis of the solution to the optimisation problem (12). The dynamic optimisation problem (12) comprises a cost functional (20), which weights states (16) and manipulated variables (18) of the first manipulator (28), a dynamic system (14), which defines states (16) and manipulated variables (18) of the first manipulator (28) on the basis of time, and at least one inequation restriction (24) for a distance from collisions. Further, the optimisation problem (12) comprises at least one equation restriction (22) for the closed kinematic system.