Robotics Review Saurabh Gupta
Robotic Tasks Manipulation
Typical Robotics Pipeline State Low-level Observations Planning Control Estimation Controller
Typical Robotics Pipeline State Low-level Observations Planning Control Estimation Controller Manipulation Grasp Motion Observed Images 6DOF Pose Planning
Robot Navigation Goal “Go 300 feet North, 400 feet East” “Go Find a Chair” Robot with a first Dropped into a novel Navigate person camera environment around
State Low-level Planning Control Observations Estimation Controller Mapping Observed Images Geometric Reconstruction Planning Hartley and Zisserman. 2000. Multiple View Geometry in Computer Vision Thrun, Burgard, Fox. 2005. Probabilistic Robotics Canny. 1988. The complexity of robot motion planning. Kavraki et al. RA1996. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. Lavalle and Kuffner. 2000. Rapidly-exploring random trees: Progress and prospects. Path Plan Video Credits : Mur-Artal et al., Palmieri et al.
Typical Robotics Pipeline State Low-level Observations Planning Control Estimation Controller Observed Images 6DOF Pose Geometric or Semantic Maps Observed Images
Typical Robotics Pipeline State Low-level Observations Planning Control Estimation Controller
Understand how to move a robot Video from Deepak Pathak.
Terminology Fore Arm • Link Elbow 𝜄 4 Block • Joint • End Effector 𝜄 5 𝜄 6 𝜄 3 • Base Upper Arm • Sensors Shoulder 𝜄 2 𝜄 1 Base Slide from Dhiraj Gandhi.
Spaces ( θ 1 , θ 2 , θ 3 , …) Work Space Configuration Space Task Space
Configuration Space ! obstacles$ " configuration$space$obstacles Configuration$ Space Workspace (2$DOF:$translation$only,$no$rotation) free$space obstacles Slide from Pieter Abbeel.
Configuration Space Another Example Elbow Angle Elbow Shoulder Angle Shoulder Slide from Pieter Abbeel.
How to move your robot? 1. Task space to Configuration space 2. Configuration space trajectory (dynamically feasible) Initial 3. Trajectory tracking configuration ( x , y , θ )
Configuration Space to Task Space Forward Kinematics 𝜄 4 𝜄 5 𝜄 6 𝜄 3 𝜄 2 𝜄 1 Base Slide from Dhiraj Gandhi.
Configuration Space to Task Space Forward Kinematics 𝑍 ( P x , P y , P θ ) 𝑄 𝑚 2 s 𝑗𝑜 ( 𝜄 1 + 𝜄 2 ) 𝑚 2 𝜄 2 P x = l 1 cos ( θ 1 ) + l 2 cos ( θ 1 + θ 2 ) + P y = l 2 sin ( θ 1 ) + l 2 sin ( θ 1 + θ 2 ) 𝑚 1 sin ( 𝜄 1 ) 𝑚 1 P θ = θ 1 + θ 2 𝑄 𝑧 = 𝑌 𝜄 1 + 𝑄 𝑦 = 𝑚 1 cos ( 𝜄 1 ) 𝑚 2 cos ( 𝜄 1 + 𝜄 2 ) Slide from Dhiraj Gandhi.
Configuration Space to Task Space Forward Kinematics 𝜄 4 𝜄 5 𝜄 6 𝜄 3 𝜄 2 𝜄 1 Slide from Dhiraj Gandhi.
Configuration Space to Task Space Forward Kinematics 𝑎 𝐶 P 𝑍 𝐶 𝑎 𝐵 𝑍 𝐵 𝑌 𝐶 𝑄 𝐵 = 𝑈 𝐵 𝑄 𝐶 𝑌 𝐵 𝐶 𝑠 11 𝑠 21 𝑠 31 Δ 𝑦 𝑠 12 𝑠 22 𝑠 32 Δ 𝑧 𝑈 𝐵 𝐶 = 𝑠 13 𝑠 23 𝑠 33 Δ 𝑨 0 0 0 1 Slide from Dhiraj Gandhi.
Configuration Space to Task Space Forward Kinematics 1 𝑈 𝑜 − 2 𝑈 𝑜 − 1 𝑜 − 1 ( 𝜄 𝑜 ) ( 𝜄 𝑜 ) 𝑈 0 1 ( 𝜄 1 ) 𝑈 2 ( 𝜄 2 )… 𝑜 Slide from Dhiraj Gandhi.
Configuration Space to Task Space Forward Kinematics 1 𝑈 𝑜 − 2 𝑈 𝑜 − 1 𝑜 − 1 ( 𝜄 𝑜 ) ( 𝜄 𝑜 ) 𝑈 0 1 ( 𝜄 1 ) 𝑈 𝑈 = 2 ( 𝜄 2 )… 𝑜 𝑠 31 Δ 𝑦 𝑠 11 𝑠 21 Maps configuration 𝑠 12 𝑠 22 𝑠 32 Δ 𝑧 space to work space = 𝑠 13 𝑠 23 Δ 𝑨 𝑠 33 0 0 0 1 x f ( θ ) = = 𝑔 ( 𝜄 1 , 𝜄 2 , . . , 𝜄 𝑜 − 1 , 𝜄 𝑜 ) Slide adapted from Dhiraj Gandhi.
Task Space to Configuration Space Forward Kinematics Inverse Kinematics Numerical IK Solve for in: θ d x d − f ( θ d ) = 0 Analytical IK • Robot Specific • Fast x = f ( θ ) • Characterize the solution space Maps configuration Find configuration(s) that map space to work space to a given work space point Slide adapted from Dhiraj Gandhi, Modern Robotics
Task Space to Configuration Space Analytical Inverse Kinematics 𝑍 𝑄 𝑚 2 𝐵 2 𝑄 ′ 𝜄 2 𝐵 𝑚 1 𝐵 1 𝜄 1 𝑃 𝑌 Slide from Dhiraj Gandhi.
How to move your robot? 1. Task space to Configuration space 2. Configuration space trajectory (dynamically feasible) Initial Desired 3. Trajectory tracking configuration configuration ( x , y , θ )
How to move your robot? 1. Task space to Configuration space 2. Configuration space trajectory Initial Desired 3. Trajectory tracking configuration configuration ( x , y , θ )
Path Planning Configuration Space Initial Goal Feasible State + + With Obstacles Config. Config. Trajectory Picture Credits : Palmieri et al.
Path Planning 1. Complete Methods 2. Grid Methods 3. Sampling Methods 4. Potential Fields 5. Trajectory Optimization
Probabilistic Roadmaps Space$ ℜ n forbidden$ space Free/feasible$space Slide from Pieter Abbeel.
Probabilistic Roadmaps Randomly Sample Configurations Slide from Pieter Abbeel.
Probabilistic Roadmaps Randomly Sample Configurations Slide from Pieter Abbeel.
Probabilistic Roadmaps Test Sampled Configurations for Collisions Slide from Pieter Abbeel.
Probabilistic Roadmaps The collision-free configurations are retained as milestones Slide from Pieter Abbeel.
Probabilistic Roadmaps Each milestone is linked by straight paths to its nearest neighbors Slide from Pieter Abbeel.
Probabilistic Roadmaps Paths that undergo collisions are removed Slide from Pieter Abbeel.
Probabilistic Roadmaps The collision-free links are retained as local paths to form the PRM Slide from Pieter Abbeel.
Probabilistic Roadmaps The start and goal configurations are included as milestones g s Slide from Pieter Abbeel.
Probabilistic Roadmaps The PRM is searched for a path from s to g g s Slide from Pieter Abbeel.
Probabilistic Roadmaps Challenging to link milestones. Slide from Pieter Abbeel.
Probabilistic Roadmaps Challenging to link milestones. Collision checking can be slow. start goal All straight line paths may not be feasible, or a good measure of distance between states. Slide from Pieter Abbeel, Modern Robotics.
Rapidly Exploring Random Trees (RRTs) Kinodynamic planning Build up a tree through generating "next states" in the tree by executing random controls. Slide from Pieter Abbeel.
Rapidly Exploring Random Trees (RRTs) Build up a tree through generating "next states" in the tree by executing random controls. ! SELECT_INPUT ( x rand ,$ x near ) ! Two$point$boundary$value$problem ! If$too$hard$to$solve,$often$just$select$best$out$of$a$set$of$control$sequences.$$ This$set$could$be$random,$or$some$well$chosen$set$of$primitives. Slide from Pieter Abbeel.
Rapidly Exploring Random Trees (RRTs) Build up a tree through generating "next states" in the tree by executing random controls. Slide from Pieter Abbeel.
Rapidly Exploring Random Trees (RRTs) Build up a tree through generating "next states" in the tree by executing random controls.
How to move your robot? 1. Task space to Configuration space 2. Configuration space trajectory Initial Desired 3. Trajectory tracking configuration configuration ( x , y , θ )
How to move your robot? 1. Task space to Configuration space 3. Trajectory execution 2. Configuration space trajectory Initial Desired configuration configuration ( x , y , θ )
Trajectory Execution x ref Dynamically feasible trajectory t from planner What control commands should I apply in order to get the robot to robustly track this trajectory? Robot state x t Robot location, or joint angles. Control sequence u t Velocities, torques. Dynamics function State evolution as we apply control. x t +1 = f ( x t , u t ) Cost function ∑ ∥ x t − x ref t ∥ t Low-level control can be formulated as an optimization problem.
Trajectory Execution Feedback Control forces motions and and desired torques forces behavior dynamics of controller arm and environment Figure from Modern Robotics.
Low-level Control Linear Quadratic Regulator Simplifying assumptions: ⇒ Exactly solved using Linear dynamics, quadratic cost. dynamic programming. Slide from Pieter Abbeel.
Linear Quadratic Regulator Cost if the system is in state x, J i ( x ) and we have i steps to go. Cost if the system is in state x, J i +1 ( x ) and we have i+1 steps to go. = min u x T Qx + u T Ru + J i ( Ax + Bu )
Linear Quadratic Regulator Slide from Pieter Abbeel.
Linear Quadratic Regulator In summary: J 1 (x) is quadratic, just like J 0 (x). Update is the same for all times and can be done in closed form for this particular continuous state-space system and cost! Slide from Pieter Abbeel.
Linear Quadratic Regulator Slide from Pieter Abbeel.
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