🤖Robotics Unit 7 – Motion Planning and Trajectory Generation

Key Concepts and Terminology

• Motion planning involves determining a sequence of valid configurations that move a robot from a start to a goal position
• Configuration space (C-space) represents all possible states of a robot, considering its degrees of freedom
• Degrees of freedom (DOF) refer to the number of independent parameters that define a robot's configuration
• Obstacles in the environment are mapped into the C-space, creating C-obstacles that the robot must avoid
• Path planning algorithms generate a collision-free path from the start to the goal configuration
• Trajectory generation techniques convert the planned path into a time-parameterized trajectory, considering robot dynamics and constraints
• Holonomic robots have no non-integrable velocity constraints, while non-holonomic robots (wheeled robots) have such constraints
• Kinematics describes the motion of a robot without considering forces, while dynamics takes into account forces and torques

Fundamentals of Motion Planning

• Motion planning is a fundamental problem in robotics that deals with finding a feasible path for a robot to navigate from a start to a goal position while avoiding obstacles
• The motion planning problem is formulated in the robot's configuration space (C-space), which represents all possible states or configurations of the robot
• C-space is divided into free space (C-free) and obstacle space (C-obstacles) based on the presence of obstacles in the environment
• The dimensionality of the C-space depends on the robot's degrees of freedom (DOF), which determine its ability to move and rotate in space
• Motion planning algorithms aim to find a continuous path in C-free that connects the start and goal configurations
• The planned path must satisfy certain constraints, such as avoiding collisions with obstacles, respecting joint limits, and considering robot kinematics and dynamics
• Sampling-based motion planning algorithms (rapidly-exploring random trees, probabilistic roadmaps) efficiently explore high-dimensional C-spaces by randomly sampling configurations
• Optimization-based motion planning techniques (optimal rapidly-exploring random trees, covariant Hamiltonian optimization for motion planning) incorporate cost functions to find optimal paths based on criteria like path length or energy consumption

Path Planning Algorithms

• Path planning algorithms are used to find a collision-free path for a robot from a start to a goal configuration in its configuration space
• Sampling-based algorithms are popular for high-dimensional motion planning problems
  • Rapidly-exploring Random Trees (RRT) incrementally build a tree of configurations by randomly sampling points in C-space and connecting them to the nearest node in the tree
  • Probabilistic Roadmaps (PRM) construct a graph of collision-free configurations by randomly sampling points in C-space and connecting them with local planners
• Graph-based algorithms discretize the C-space into a graph and search for a path using techniques like A* or Dijkstra's algorithm
• Potential field methods create an artificial potential field in the C-space, with the goal configuration as an attractive potential and obstacles as repulsive potentials
• Optimization-based algorithms formulate motion planning as an optimization problem, minimizing a cost function subject to constraints
  • Covariant Hamiltonian Optimization for Motion Planning (CHOMP) optimizes a trajectory by minimizing a cost function that includes obstacle avoidance and smoothness terms
• Anytime planning algorithms (anytime repairing A*) provide suboptimal solutions quickly and improve the solution over time if more computation is available
• Hybrid planning approaches combine discrete and continuous planning techniques to handle complex environments and robot dynamics

Trajectory Generation Techniques

• Trajectory generation involves converting a planned path into a time-parameterized trajectory that satisfies robot dynamics and constraints
• Time-optimal trajectory generation aims to minimize the time taken to execute the path while respecting velocity and acceleration limits
• Minimum-jerk trajectories minimize the integral of the squared jerk (rate of change of acceleration) to produce smooth and natural motions
• Polynomial interpolation methods (cubic, quintic) fit a polynomial curve to a set of waypoints, ensuring continuity of position, velocity, and acceleration
• Trapezoidal velocity profiles consist of three phases: constant acceleration, constant velocity, and constant deceleration
• S-curve velocity profiles add jerk-limited transitions between the acceleration and deceleration phases to reduce vibrations and improve tracking accuracy
• Dynamic movement primitives (DMPs) encode trajectories as a set of differential equations, allowing for easy adaptation and generalization to new situations
• Trajectory optimization techniques (iterative linear quadratic regulator, sequential convex programming) refine an initial trajectory by minimizing a cost function subject to constraints

Obstacle Avoidance Strategies

• Obstacle avoidance is a critical component of motion planning that ensures the robot does not collide with obstacles in its environment
• Reactive obstacle avoidance methods (potential fields, vector field histograms) generate local motion commands based on sensor data without explicit path planning
  • Potential field methods create repulsive forces around obstacles and attractive forces towards the goal, guiding the robot along a collision-free path
  • Vector field histograms discretize the robot's surroundings into angular sectors and select the most promising direction for motion based on obstacle density
• Deliberative obstacle avoidance techniques integrate obstacle information into the path planning process
  • C-space expansion algorithms (Minkowski sum) enlarge obstacles by the robot's dimensions to simplify collision checking
  • Collision detection libraries (Flexible Collision Library, Bullet) efficiently test for intersections between the robot and obstacles
• Velocity obstacles represent the set of robot velocities that would result in a collision with a moving obstacle within a given time horizon
• Inevitable collision states (ICS) are robot states for which no trajectory exists to avoid a collision in the future, regardless of the control input
• Reachability analysis techniques (Hamilton-Jacobi reachability) compute the set of states from which the robot can safely avoid obstacles under bounded control inputs
• Chance-constrained obstacle avoidance methods explicitly consider uncertainty in robot motion and obstacle positions, ensuring a high probability of collision-free operation

Kinematics and Dynamics in Motion Planning

• Kinematics and dynamics play a crucial role in motion planning, as they determine the feasibility and executability of planned paths
• Forward kinematics calculates the end-effector position and orientation given the joint angles or positions
  • Denavit-Hartenberg (DH) parameters provide a systematic way to define coordinate frames and transform between them for serial manipulators
• Inverse kinematics determines the joint angles or positions required to achieve a desired end-effector pose
  • Analytical methods (geometric, algebraic) solve the inverse kinematics problem in closed form for specific robot architectures
  • Numerical methods (Jacobian pseudoinverse, cyclic coordinate descent) iteratively compute joint angles for general robot structures
• Differential kinematics relates end-effector velocities to joint velocities through the Jacobian matrix
• Dynamics describes the relationship between forces/torques and the resulting motion of the robot
  • Lagrangian formulation derives the equations of motion using the system's kinetic and potential energy
  • Newton-Euler formulation recursively computes the forces and torques acting on each link of the robot
• Motion planning algorithms that consider dynamics (kinodynamic planning) generate trajectories that satisfy the robot's equations of motion
• Optimal control techniques (linear quadratic regulator, model predictive control) generate control inputs that minimize a cost function while satisfying dynamic constraints

Real-world Applications and Case Studies

• Autonomous vehicles use motion planning algorithms to navigate in complex environments, considering obstacles, traffic rules, and vehicle dynamics
  • The DARPA Urban Challenge showcased the capabilities of autonomous vehicles in urban settings, requiring advanced motion planning and obstacle avoidance techniques
• Industrial manipulators rely on motion planning to perform tasks such as welding, painting, and assembly while avoiding collisions with the environment
  • Robotic pick-and-place operations in warehouses and factories require efficient motion planning to minimize cycle times and maximize throughput
• Surgical robots employ motion planning to safely navigate and manipulate tools within the confined spaces of the human body
  • The da Vinci surgical system uses motion planning to control multiple robotic arms and perform minimally invasive procedures
• Unmanned aerial vehicles (UAVs) use motion planning to navigate in 3D environments, avoiding obstacles and optimizing trajectories for energy efficiency
  • Quadrotor UAVs have been used for search and rescue missions, infrastructure inspection, and aerial photography, requiring advanced motion planning capabilities
• Space robotics applications, such as satellite servicing and planetary exploration, demand motion planning algorithms that can handle the unique challenges of microgravity and unknown environments
  • The Robonaut 2 humanoid robot, developed by NASA, has been used on the International Space Station to assist astronauts with tasks requiring dexterous manipulation
• Legged robots, such as humanoids and quadrupeds, use motion planning to generate stable and agile locomotion patterns over uneven terrain
  • The Boston Dynamics Atlas humanoid robot has demonstrated impressive motion planning capabilities, including parkour and backflips

Challenges and Future Directions

• Scalability of motion planning algorithms to high-dimensional configuration spaces and complex environments remains a challenge
  • Developing efficient sampling strategies and heuristics for exploring large C-spaces is an active area of research
• Integrating motion planning with perception and control to create fully autonomous systems that can operate in dynamic and uncertain environments
  • Incorporating real-time sensor data and adapting to changes in the environment require fast and reactive motion planning techniques
• Handling uncertainty in robot motion, sensor measurements, and environment models is crucial for robust motion planning
  • Probabilistic approaches, such as belief space planning and partially observable Markov decision processes (POMDPs), explicitly model and reason about uncertainty
• Generating human-like and socially acceptable motions for robots operating in human environments
  • Learning from human demonstrations and incorporating social norms and preferences into motion planning algorithms can improve human-robot interaction
• Developing motion planning algorithms that can adapt to changes in the robot's dynamics or environment, such as damage or wear
  • Reinforcement learning and adaptive control techniques can enable robots to learn and adjust their motion plans based on experience
• Extending motion planning algorithms to multi-robot systems and decentralized planning architectures
  • Coordinating the actions of multiple robots to achieve a common goal while avoiding collisions and deadlocks is a complex problem
• Integrating motion planning with task and action planning to enable robots to perform high-level, goal-directed behaviors
  • Hierarchical planning approaches decompose complex tasks into subtasks and generate appropriate motion plans for each level of abstraction
• Improving the computational efficiency and real-time performance of motion planning algorithms to enable their deployment on resource-constrained platforms
  • Parallel computing, GPU acceleration, and approximation techniques can help speed up motion planning computations