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Dynamics of Articulated Robots Kris Hauser CS B659: Principles of Intelligent Robot Motion Spring 2013

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Agenda Basic elements of simulation Derive the standard form of the dynamics of an articulated robot in joint space Also works for humans, biological systems, non-actuated mechanical systems … Featherstone algorithm: fast method for computing forward dynamics (torques to accelerations) and inverse dynamics (accelerations to torques) Constrained dynamics

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Rigid Body Dynamics The following can be derived from first principles using Newton’s laws + rigidity assumption Parameters CM translation c(t) CM velocity v(t) Rotation R(t) Angular velocity (t) Mass m, local inertia tensor H L

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Rigid body ordinary differential equations

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Articulated body ODEs

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Numerical integration of ODEs If dx/dt = f(x) and x(0) are known, then given a step size h, x(kh) x k = x k-1 + h f’(x k-1 ) gives an approximate trajectory for k 1 Provided f is smooth Accuracy depends on h Known as Euler’s method Better integration schemes are available (e.g., Runge-Kutta methods, implicit integration, adaptive step sizes, etc) Beyond the scope of this course

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DYNAMICS OF RIGID BODIES

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Kinetic energy for rigid body Rigid body with velocity v, angular velocity KE = ½ (m v T v + T H ) World-space inertia tensor H = R H L R T vv T vv H 0 0 m I 1/2

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Kinetic energy derivatives

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Summary Gyroscopic “force”

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Force off of COM x F

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x F

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Generalized torque f Now consider the equivalent force f, torque τ at COM

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Generalized torque f

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f

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Principle of virtual work f F

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f F fτfτ

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ARTICULATED ROBOT DYNAMICS

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Robot Dynamics

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Lagrangian Mechanics Kinetic energyPotential energy

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Lagrangian Mechanics A system of n partial differential equations

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Sanity check: Newton’s laws Example: Point Mass

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Kinetic energy for articulated robot Mass matrix: symmetric positive definite

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Derivative of K.E. w.r.t q

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Potential energy for articulated robot in gravity field Generalized gravity

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Putting it all together Group these terms together

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Final canonical form Generalized inertia Centrifugal/ coriolis forces Generalized gravity Generalized forces (joint torques + external forces)

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Forward/Inverse Dynamics

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Example: RP manipulator

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Application: Effective Inertia

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Can be infinite at singular configurations!

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Application: Feedforward control Feedback control: let torques be a function of the current error between actual and desired configuration Problem: heavy arms require strong torques, requiring a stiff system Stiff systems become unstable relatively quickly

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Application: Feedforward control Solution: include feedforward torques to reduce reliance on feedback Estimate the torques that would compensate for gravity and coriolis forces, send those torques to the motors

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Feedforward Torques

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Newton-Euler Method (Featherstone 1984) Explicitly solves a linear system for joint constraint forces and accelerations, related via Newton’s equations No matrix larger than 6x6 Faster forward/inverse dynamics for large chains (O(n) vs O(n 3 ) for direct matrix computations)

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Forward Dynamics: Basic Intuition

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Software Both Lagrangian dynamics and Newton-Euler methods are implemented in KrisLibrary Lagrangian form is usually most mathematically convenient representation

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CONSTRAINED DYNAMICS

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Constrained Systems

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The Wrong Way

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The Right Way…

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Solving…

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Back to Pseudoinverses A pseudoinverse A # of the matrix A is a matrix such that A = AA # A A # = A # AA # Generalizes the concept of inverse to non-square, noninvertible matrices Such a matrix exists (in fact, there are infinitely many) The Moore-Penrose pseudoinverse, denoted A +, can be derived as A + = (A T A) -1 A T when A T A is invertible(overconstrained) A + = A T (AA T ) -1 when AA T is invertible(underconstrained)

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Properties Note connection to least-squares formula Ax=b => x = A + b If system is overconstrained, this solution minimizes ||b-Ax|| 2 If system is underconstrained, this solution minimizes ||x|| 2 Note that (I-AA+)Ay = 0 is always satisfied (I-AA+) is a projection matrix

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Weighted Pseudoinverse If (AA T ) -1 exists, given any positive definite weighting matrix W, we can derive a new pseudoinverse A # = W -1 A T (AW -1 A T ) -1 This is a weighted pseudoinverse Has the property that x=A # b is a solution to Ax = b such that x minimizes x T Wx – a weighted norm

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Weighted Pseudoinverse

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Rigid Body Simulators

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Next class Feedback control Principles App J

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