Dynamics of Articulated RobotsKris HauserCS B659: Principles of Intelligent Robot MotionSpring 2013

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

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 w(t)• Mass m, local inertia tensor HL

Rigid body ordinary differential equations• We will express forces and torques in terms of terms of H (a

function of R), , and

• Rearrange…

• So knowing f(t) and τ(t), we can derive c(t), v(t), R(t), and w(t) by solving an ordinary differential equation (ODE)• dx/dt = f(x)• x(0) = x0

• With x=(c,v,R,w) the state of the system• Numerical integration, also known as simulation

Articulated body ODEs• We will express joint torques in terms of terms of and and

external forces f

• Rearrange…

• An ODE in the state space x=()

• Solve using numerical integration

Numerical integration of ODEs• If dx/dt = f(x) and x(0) are known, then given a step size h,

• x(kh) xk = xk-1 + h f’(xk-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

DYNAMICS OF RIGID BODIES

Kinetic energy for rigid body

• Rigid body with velocity v, angular velocity w• KE = ½ (m vTv + wT H w)

• World-space inertia tensor H = R HL RT

wv

T

wv

H 0 0 m I

1/2

Kinetic energy derivatives

• Force (@CM)

• H = [w]H – H[w]• Torque t = = [w] H w + H

Summary

Gyroscopic “force”

Force off of COM

x

F

Force off of COM

x

F

Consider infinitesimal virtual displacement generated by F. (we don’t know what this is, exactly)The virtual work performed by this displacement is FT

𝛿𝑥

Generalized torque

f

Now consider the equivalent force f, torque τ at COM

Generalized torque

f

Now consider the equivalent force f, torque τ at COMAnd an infinitesimal virtual displacement of R.B. coordinates

𝛿𝑥

𝛿𝑞

Generalized torque

f𝛿𝑥

𝛿𝑞

Now consider the equivalent force f, torque τ at COMAnd an infinitesimal virtual displacement of R.B. coordinates Virtual work in configuration space is [fT,τT]

Principle of virtual work

f𝛿𝑥

𝛿𝑞

[fT,τT] = FT

Since we have [fT,τT] = FT

F

Principle of virtual work

f𝛿𝑥

𝛿𝑞

[fT,τT] = FT

Since we have [fT,τT] = FT

Since this holds no matter what is, we have [fT,τT] = FTJ(q),

Or JT(q) F =

F

fτ

ARTICULATED ROBOT DYNAMICS

Robot Dynamics

• Configuration , velocity Rn

• Generalized forces u Rm

• Joint torques and external forces • How does u relate to and ?

• Use Langrangian mechanics to find a link between u and

Lagrangian Mechanics

• The trajectory between two states , is the one that minimizes the “action”

• is defined such that the path minimizing S is equivalent to the one produced by Newton’s laws, subject to the constraints that the system only moves along coordinates q

Kinetic energy Potential energy

Lagrangian Mechanics

• Minimum action condition => Euler-Lagrange equations of motion:

• Note that P is independent of , so

A system of n partial differential equations

Sanity check: Newton’s laws

Example: Point Mass• Coordinates q = (x,y)• Potential field P(x,y)• Lagrangian: • Equations of motion

Kinetic energy for articulated robot

• Velocity of i’th rigid body

• Angular velocity of i’th rigid body

Mass matrix:symmetric positive definite

Derivative of K.E. w.r.t

•

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

𝜕𝜕𝑞

𝐾 (𝑞 ,�̇� )=½(�̇�𝑇 𝜕𝜕𝑞1

𝐵(𝑞) �̇�

…

�̇�𝑇 𝜕𝜕𝑞𝑛

𝐵(𝑞) �̇�)

Potential energy for articulated robot in gravity field

• G(q)

Generalized gravity

Putting it all together

Group these terms together

Final canonical form

𝐵 (𝑞) �̈�+𝐶 (𝑞 , �̇�)+𝐺(𝑞)=𝑢Generalized inertia

Centrifugal/coriolis forces

Generalized gravity

Generalized forces (joint torques + external forces)

Forward/Inverse Dynamics

• Given , , and , find • From torques to accelerations

• Given , , and , find • From desired accelerations to necessary torques

Example: RP manipulator

Application: Effective Inertia• If a force is applied to a point on a robot, how much will

accelerate?

Application: Effective Inertia• If a force is applied to a point on a robot, how much will

accelerate?• Assume a stationary system, no acceleration when no force is

applied• =0

• With the force:

Application: Effective Inertia• If a force is applied to a point on a robot, how much will

accelerate?• Assume a stationary system, no acceleration when no force is

applied• =0

• With the force:

• The matrix is called the effective inertia matrix

Can be infinite at singular configurations!

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

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

Feedforward Torques

• Given current , , desired • 1. Estimate B, C, G• 2. Compute u

• 3. Apply torques u

• How to compensate for errors in B,C,G? Combine feedforward with feedback. More in later classes…

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(n3)

for direct matrix computations)

Forward Dynamics: Basic Intuition• Downward recursion: Starting from root, compute

“articulated body inertia matrix” for each link• 6x6 matrix relating vectors to translational/angular accelerations

respectively• Also need a “bias force”

• Upward recursion: Starting from leaves, compute accelerations on links• Given acting on i’th link, compute acceleration of joint i and the

joint constraint forces on the i-1’th link• includes external forces + joint constraint forces from downward

links

Software• Both Lagrangian dynamics and Newton-Euler methods are

implemented in KrisLibrary

• Lagrangian form is usually most mathematically convenient representation

CONSTRAINED DYNAMICS

Constrained Systems

• Suppose the system is constrained by • E.g., closed-chains, contact constraints, rolling

constraints• A is a k x n matrix (k constraints)

• How does evolve over time?

The Wrong Way

• Suppose the system is constrained by • E.g., closed-chains, contact constraints, rolling

constraints• A is a k x n matrix (k constraints)

• How does evolve over time?• Wrong way:

• Solve for as usual, then project it onto the subspace that satisfies this equation, obtaining

• The correct answer will be a projection, but a very specific one!

The Right Way…

• Constrained system of equations:• (1)

• (2)

• Lagrange multipliers have been introduced

• can be thought of as constraint forces• Solve for n+k variables

Solving…

• Constrained system of equations:• (1)

• (2)

• Solve for n+k variables • A solution must satisfy

• (3) solve 1 for

• (4) subst (3) in (2)

(5) solve for in (4), use from (2)

• (6) more manipulations..

• With

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+ = (ATA)-1AT when ATA is invertible(overconstrained)• A+ = AT(AAT)-1 when AAT is invertible(underconstrained)

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

Weighted Pseudoinverse• If (AAT)-1 exists, given any positive definite weighting matrix W,

we can derive a new pseudoinverse• A# = W-1AT(AW-1AT)-1

• This is a weighted pseudoinverse• Has the property that x=A#b is a solution to Ax = b such that

• x minimizes xTWx – a weighted norm

Weighted Pseudoinverse• If (AAT)-1 exists, given any positive definite weighting matrix W,

we can derive a new pseudoinverse• A# = W-1AT(AW-1AT)-1

• This is a weighted pseudoinverse• Has the property that x=A#b is a solution to Ax = b such that

• x minimizes xTWx – a weighted norm

• Revisiting constrained dynamics…• The P projection matrix solves for such that is minimized

• Constraint forces dissipate kinetic energy in a minimal fashion!

Rigid Body Simulators• Articulated robots are often simulated as a set of connected

rigid bodies (Open Dynamics Engine, Bullet, etc)• Connections give rise to constraints in the dynamics

• (1)

• (2)

• Solve for n+k variables • (1), (2) are sparse systems and are solved using specialized

solvers

• More on frictional contact later…

Next class• Feedback control

• Principles App J