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Introduction To Robotics
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1) Outward Recursion – Kinematic Computation
Review Recursive Inverse Dynamics Inverse Dynamics – Known joint angles compute joint torques 1) Outward Recursion – Kinematic Computation Known Compute From 0 to n, recursively based on geometrical and differential relationship associated with each link. 2) Inward Recursion – Dynamics Computation Compute wrench wi based on wi+1 and kinematic quantities obtained from 1) From n+1 to 0, recursively using Newton-Euler equation
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Each link – 6-DOF; Within the system – 1-DOF
Review The Natural Orthogonal Compliment Each link – 6-DOF; Within the system – 1-DOF 5-DOF constrained Kinematic Constraint equation T : Natural Orthogonal Complement (Twist Shape Function)
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Review Natural Orthogonal Complement (cont'd)
Use T in the Newton-Euler Equation, the system equation of motion becomes: where Consistent with the result obtained from Euler-Lagrange equation Generalized inertia matrix Active force Dissipative force Gravitational force Vector of Coriolis and centrifugal force
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Natural Orthogonal Complement
Constraint Equations & Twist-Shape Matrix 1) Angular velocity Constraint Ei : Cross-product matrix of ei 2) Linear Velocity Constraints ci = ci-1+ i-1 + i Differentiate: Oi+1 Oi-1 Oi O Ci-1 Ci ci-1 c i-1 i
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Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint Equations (6.63) and (6.64) pertaining to the first link:
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Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint 6n 6n matrix
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Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint Define partial Jacobian 6 n matrix with its element defined as Mapping the first i joint rates to ti of the ith link
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Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint
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Natural Orthogonal Complement
Constraint Equation and Twist Shape Matrix – R Joint Easy to verify Recall
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Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint Oi-1 Oi O'i Oi+1 Ci-1 Ci i-1 ci-1 i ci bi ai ai+1
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Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint Regroup (6.74a) and (6.77):
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Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint If the first joint is prismatic, then where Define partial Jacobian
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Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix Compute If kth joint is prismatic, then
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Natural Orthogonal Complement
Noninertial Base Link Include it in the joint rate vector - 6(n+1) The generalized velocity: Twist of base link (6-DOF)
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Forward Dynamics Overview
Purpose of forward dynamics – Simulation, Model-based control Method – Solving Ordinary Differential equation (System E.O.M): Dissipative force Gravitational force EE Static wrench acting on the joint Generalized Inertia Matrix Inertial Force Active working wrench
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Forward Dynamics Problem Description Known: at To find: at
Solution: Integration to compute at Need to compute I, , and
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Forward Dynamics Computation Procedure (1) Compute I
Using T, the Natural Orthogonal Complement Recall M – Positive Semi-Definite Factoring:
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Forward Dynamics Computation Procedure
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Forward Dynamics Computation Procedure (2) Compute
Rewrite system equation as the problem can be solved as an inverse dynamics problem using the recursive algorithm. Know current compute Torque required to produce the current joint angles and rates when joint acceleration and dissipative force vanish
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Forward Dynamics Computation Procedure (3) Solving Equations
Cholesky decomposition of the generalized inertia matrix Solving two linear systems of equations Alternative solution
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Planar Manipulator Fundamentals Basic definitions in 2-D System level:
Newton-Euler Equation in 2-D Matrix forms: Element level: System level: Scalar Scalar 2-D Vector 2-D Vector
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Planar Manipulators Fundamentals
Constraint equations/Natural Orthogonal Compliment K – 3n3n matrix T – 3nn matrix Equation of Motion
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Planar Manipulators Example
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Planar Manipulators Example Solution: Angular velocities:
Twist-Shape matrix
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Planar Manipulators Example
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Planar Manipulators Example The inertial matrix Elements
Generalized Inertial Matrix
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Planar Manipulators Example Twist Shape Matrix Rate
Let represent (i,j) entry of
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Planar Manipulators Example Now define
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Planar Manipulators Example Gravity wrench
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Planar Manipulators Example Final form
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Dynamic Model Review Summary Dynamic Model of a system
Euler-Lagarange Equation (System Level Model) Apply Kinematic constraint conditions in terms of K and T Newton-Euler Equation (Element Level Model – Uncoupled)
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Gravity Term in E.O.M Model Gravitational Force
Incorporate gravity into recursive inverse dynamics algorithm Using the natural orthogonal complement T No change in the algorithm.
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Dissipative Term in E.O.M
Model Friction Forces Viscous Friction – Solid vs viscous fluids Coulomb Friction – Solid vs Solid (Dry friction) (1) Viscous Friction Velocity field v = v(r, t) v vanishes at the interface surface v Symmetric Skew-Symmetric
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Dissipative Term in E.O.M
Model Friction Forces Only the symmetric part of the gradient is responsible for power dissipation - Viscosity coefficient of the fluids For revolute joint pair velocity field can be modeled as pure tangential The dissipative function At each joint System level
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Dissipative Term in E.O.M
Model Friction Forces The dissipative force (2) Coulomb Friction Simplified model Constant determined experimentally (R – Force; P – Torque) Dissipative function: At joint i Overall
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Dissipative Term in E.O.M
Model Friction Forces Property: Lower relative speed -> Coulomb friction is high High relative speed -> Coulomb friction is low Enhanced model:
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Course Review Overview of Robotics
Analysis for robot structure design, operation procedure planning, singularity/work space analysis Kinematics (Static) Mechanics Math model for control and simulation, dynamical analysis Dynamics Force/torque control Control Robotics (Chapter 1) Trajectory/position control Computer Vision Artificial Intelligence
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Course Review Robotics Topics
Basic definition & Notation – DH Notation, Coord. Trans, .. Geometrical – Closed form solution Kinematics(Chapter 4) General Analysis Differential – Jacobian analysis, vel, acceleration Direct Kinematic Problem Specific problems Inverse Kinematic Problem Static (Chapter 4) Mapping between EE wrench and joint torque Basic definition & Notation – Mass/inertia matrices, twist, wrench Dynamics (Chapter 6) General Analysis – Multi-body dynamics(N-E, E-L), constraints (K, T) Forward Dynamics – Solve ODE Specific problems Inverse Dynamics – Recursive algorithm
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Course Review Analysis & Modelling Tools
Linear Transformation – Q, Coordinate Trans: [p]1 = a + Q[p]2 Math Tools (Chapter 2) Invariant Concept – vect(Q), tr(Q) Motion - twist Definition & Concept Force/Torque - wrench Mass property – m, I, c Rigid-body Mechanics (Chapter 3) Analysis – Screw motion (screw axis, pitch, properties) Newton-Euler – body level E.O.M Euler-Lagarange – System level
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Office Hour Next Week Mon/Tues (Dec 6, 7) 17:00-18:00 MD 457
Assignment #4 Due on Dec 6. Submit your assignment during the office hour and get the solution. Final Exam (Open Book): 14:00 – 17:00 Dec 8, 2004
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