7.2 Sampling-based and optimization-based planning methods
2 min read•july 25, 2024
Sampling-based and optimization-based planning methods are two key approaches in robotics. Sampling methods randomly explore the , building graphs or trees without explicit obstacle representation. Optimization methods formulate planning as a mathematical problem, finding optimal solutions with explicit obstacle representation.
Sampling-based methods like PRMs and RRTs are probabilistically complete and efficient in . Optimization-based methods, including and , produce smooth, optimal trajectories and easily incorporate dynamic constraints. Each approach has unique strengths and limitations in robotic path planning.
Sampling-Based and Optimization-Based Planning Methods
Sampling vs optimization planning methods
Top images from around the web for Sampling vs optimization planning methods
machine learning - what is the meanning of iterations of neural network ,gradient descent steps ... View original
Is this image relevant?
Gradient Descent and its Variants View original
Is this image relevant?
Gradient Descent [The Hundred-Page Machine Learning Book] View original
Is this image relevant?
machine learning - what is the meanning of iterations of neural network ,gradient descent steps ... View original
Is this image relevant?
Gradient Descent and its Variants View original
Is this image relevant?
1 of 3
Top images from around the web for Sampling vs optimization planning methods
machine learning - what is the meanning of iterations of neural network ,gradient descent steps ... View original
Is this image relevant?
Gradient Descent and its Variants View original
Is this image relevant?
Gradient Descent [The Hundred-Page Machine Learning Book] View original
Is this image relevant?
machine learning - what is the meanning of iterations of neural network ,gradient descent steps ... View original
Is this image relevant?
Gradient Descent and its Variants View original
Is this image relevant?
1 of 3
- Sampling-based methods randomly explore configuration space building graph or tree structures without explicit obstacle representation (PRMs, RRTs)
- Optimization-based methods formulate planning as mathematical optimization problem finding optimal solutions often requiring explicit obstacle representation (gradient descent, convex optimization)
Principles of sampling-based planning
- Probabilistic Roadmaps (PRM)
- Sampling phase randomly generates free space configurations connecting nearby ones with local planners
- Query phase connects start and goal to roadmap searching for path using graph algorithms (A*, Dijkstra's)
- Rapidly-exploring Random Trees (RRT)
- Incremental tree growth starts from initial configuration iteratively expanding towards random samples
- Biased exploration tends to explore large unexplored areas efficiently
- Variants include RRT-Connect growing trees from both start and goal and RRT* providing asymptotic
Concepts in optimization-based planning
- Gradient descent iteratively optimizes by following negative gradient of objective function for local optimization
- Convex optimization solves problems with convex objective function and constraints guaranteeing global optimum if exists
- represents path as waypoint sequence minimizing cost function (smoothness, obstacle avoidance)
- uses receding horizon planning optimizing short time horizons executing first control input then replanning
Comparison of planning method applications
- Sampling-based advantages
- Probabilistically complete ensuring solution if one exists
- Efficient in high-dimensional spaces (robotic arms, multi-robot systems)
- Handle complex geometric constraints well (narrow passages, cluttered environments)
- Sampling-based limitations
- May produce non-optimal paths requiring post-processing
- Performance degrades in narrow passages needing specialized sampling strategies
- Difficulty incorporating dynamic constraints (velocity limits, acceleration bounds)
- Optimization-based advantages
- Produce smooth optimal trajectories considering multiple objectives
- Easily incorporate dynamic constraints (joint limits, energy consumption)
- Handle time-varying objectives and constraints (moving obstacles, changing goals)
- Optimization-based limitations
- May get stuck in local optima requiring multiple initializations
- Computationally expensive for complex problems with many variables
- Sensitive to initial conditions affecting solution quality
Key Terms to Review (20)
A* algorithm: The A* algorithm is a popular pathfinding and graph traversal algorithm that efficiently finds the shortest path from a starting point to a target point in a weighted graph. It combines features of Dijkstra's algorithm and greedy best-first search, using both actual costs and estimated costs to guide its search, making it well-suited for various planning and navigation tasks.
Completeness: Completeness refers to the property of a planning method that guarantees finding a solution if one exists within the given constraints. In robotics and artificial intelligence, completeness is crucial because it ensures that the algorithm will explore all possible configurations or paths to determine a viable route or solution. This concept plays a significant role in both sampling-based and optimization-based methods, as well as in 3D vision where depth perception can influence decision-making and pathfinding.
Configuration Space: Configuration space is a mathematical representation of all possible states or arrangements of a robotic system, defined by the positions and orientations of its components. It helps in understanding how robots can move and interact with their environment, especially in relation to obstacles and other entities they may encounter. By mapping out this space, planners can devise strategies for movement and manipulation within the constraints posed by obstacles or goals.
Convex optimization: Convex optimization is a subfield of mathematical optimization that focuses on minimizing convex functions over convex sets. This type of optimization is significant because it guarantees that any local minimum is also a global minimum, making the problem easier to solve and analyze. The properties of convexity lead to efficient algorithms and techniques, which are particularly useful in areas like planning and trajectory generation.
Dynamic environments: Dynamic environments refer to settings where conditions and variables are constantly changing and evolving, requiring adaptive responses from robotic systems. In such environments, uncertainty can arise from factors like moving obstacles, changing terrains, or varying conditions that affect the robot's ability to navigate and perform tasks effectively.
Forward Kinematics: Forward kinematics is the process of calculating the position and orientation of a robot's end effector based on the joint parameters, such as angles and displacements. This process is crucial for understanding how movements in a robotic system relate to its physical configuration, enabling precise control and manipulation in various applications.
Gradient descent: Gradient descent is an optimization algorithm used to minimize a function by iteratively moving toward the steepest descent as defined by the negative of the gradient. This method plays a crucial role in improving the performance of algorithms, especially in applications where finding optimal solutions is necessary, such as in planning, trajectory generation, and visual tracking systems.
High-dimensional spaces: High-dimensional spaces refer to mathematical constructs that extend beyond the conventional three dimensions we experience in the physical world, incorporating a multitude of dimensions, often hundreds or thousands. In the context of robotics and planning methods, these spaces are essential for representing complex configurations and states that a robot can occupy, enabling advanced algorithms to navigate and optimize paths in a virtually limitless environment.
Importance Sampling: Importance sampling is a statistical technique used to estimate properties of a particular distribution while only having samples from a different distribution. This method focuses on sampling more frequently from the regions of the input space that contribute most to the desired outcome, which enhances the efficiency of the sampling process. It is particularly useful in high-dimensional spaces, where traditional methods may struggle to capture relevant data effectively.
Inverse Kinematics: Inverse kinematics is the process of calculating the joint parameters needed to place the end-effector of a robotic arm or manipulator at a desired position and orientation in space. This technique is essential for controlling robotic systems, as it allows for precise movement and positioning based on the goals set by a user or program.
Model predictive control (mpc): Model predictive control (MPC) is an advanced control strategy that uses a model of the system to predict future behavior and optimize control actions over a specified time horizon. It continuously updates its predictions based on new information and adjusts control inputs to achieve desired outcomes while satisfying constraints. MPC is widely used in robotics due to its ability to handle multi-variable control problems and incorporate constraints directly into the optimization process, making it relevant in planning and sensor data processing.
Optimality: Optimality refers to the quality of being the best or most effective solution to a problem under given constraints. In the context of planning methods, it involves finding the most efficient path or strategy that minimizes costs or maximizes benefits, ensuring that the chosen solution is the best among all possible alternatives.
Path Smoothing: Path smoothing is the process of refining a trajectory or path in robotics to make it more efficient and natural for a robot to follow. It typically involves reducing sharp turns and unnecessary complexity while maintaining the overall goal of reaching a destination. By applying algorithms, this technique optimizes the robot's movement, enhances stability, and improves energy efficiency, which is especially important in both sampling-based and optimization-based planning methods.
Planning time: Planning time refers to the duration required for a robotic system to generate a feasible trajectory or path to achieve its goal while avoiding obstacles and adhering to constraints. This concept is crucial in robotic motion planning as it affects how quickly and efficiently a robot can respond to changes in its environment or tasks it needs to perform. The efficiency of planning time can significantly impact the overall performance of sampling-based and optimization-based planning methods, which are two common approaches used in robotics for generating paths.
Probabilistic roadmap (PRM): A probabilistic roadmap (PRM) is a sampling-based method used in robotic motion planning to create a network of nodes that represent feasible configurations in the robot's configuration space. This technique relies on randomly sampling the configuration space, connecting these samples to form a roadmap, and then using this roadmap to efficiently find paths between start and goal configurations. PRMs are particularly useful in high-dimensional spaces where traditional methods may struggle, making them a vital tool in the field of robotics.
Rapidly-exploring random tree (RRT): A rapidly-exploring random tree (RRT) is a probabilistic algorithm used for path planning in high-dimensional spaces, enabling a robot to navigate from a start point to a goal by incrementally constructing a tree of feasible paths. This method leverages random sampling to explore the environment and efficiently finds a path, making it especially useful in complex spaces where traditional methods may struggle. RRT is key in sampling-based planning, balancing exploration and exploitation to quickly cover vast areas while seeking a valid trajectory.
Rejection Sampling: Rejection sampling is a statistical technique used to generate samples from a target probability distribution by using a proposal distribution. It works by drawing samples from the proposal distribution and accepting or rejecting these samples based on a criterion related to the target distribution. This method is particularly useful when direct sampling from the target distribution is difficult or impossible, and it forms a fundamental part of many sampling-based planning methods in robotics.
Sampling-based optimization: Sampling-based optimization is a method used in robotics and planning that seeks to find optimal solutions by randomly sampling the space of possible configurations. This approach is especially useful in high-dimensional spaces, where traditional optimization techniques may struggle. By generating samples and evaluating their performance, this method can effectively explore complex environments and improve the efficiency of planning algorithms.
Success rate: Success rate refers to the proportion of successful outcomes achieved in a given process or method, often expressed as a percentage. In the context of robotics, particularly in planning and navigation, it is crucial to evaluate how effectively an algorithm can generate valid paths that meet the desired objectives. A high success rate indicates that an algorithm efficiently navigates through the environment, while a low success rate suggests challenges in handling obstacles or achieving goals.
Trajectory optimization: Trajectory optimization is the process of finding the best path or trajectory for a robotic system to follow in order to achieve a specific goal while minimizing costs such as time, energy, or deviation from constraints. This involves analyzing the motion dynamics and constraints of the system, which is crucial for effective control and performance.