1. Articulated Body Dynamics The Basics Comp 768 October 23, 2007 Will Moss

2. October, 23 20072 Overview Motivation Background / Notation Articulate Dynamics Algorithms – Newton-Euler Algorithm – Composite-Rigid Body Algorithm – Articulated-Body Algorithm (Featherstone)‏ – Lagrange Multiplier approach (Baraff)‏

3. Originally a problem from robotics – Given a robotic arm with a series of joints that can apply forces to themselves (called motors)‏, find the forces to get the robot arm into the desired configuration October, 23 20073 History

4. October, 23 20074 Applications Computer Graphics – Humans, animals, birds, robots, etc. – Wires, chains, ropes, etc. – Trees, grass, etc. – Many more http://vrlab.epfl.ch/~alegarcia/VHOntology/long.html

5. October, 23 20075 Basics An articulated body is a group of rigid bodies (called links) connected by joints Multiple types of joints – Revolute (1 degree of freedom)‏ – Ball joint (3 degrees of freedom) – Prismatic, screw, etc.

6. October, 23 20076 Notation In rigid body dynamics we had two equations – F s is the vector of spatial forces – I s is the spatial inertia matrix (6 x 6) – a s is the spatial acceleration This is called spatial algebra – Combines the linear and angular components of the physical quantities into one 6 dimensional vector

7. October, 23 20077 Notation Transitioning this to articulated bodies – Q i is the force on link i – H is the joint-space inertia matrix (n x n) – are the coordinates, velocities and accelerations of the joints – C term produces the vector of forces that produce zero acceleration

8. October, 23 20078 Forward vs. Inverse Dynamics Inverse Dynamics – The calculation of forces given a set of accelerations Forward Dynamics – The calculation of accelerations given a set of forces

9. October, 23 20079 Algorithms Inverse Dynamics – Newton-Euler Algorithm Forward Dynamics – Composite-Rigid-Body Algorithm – Articulated-Body Algorithm – Lagrange Multiplier Algorithm

10. October, 23 200710 Newton-Euler Algorithm Goal – Given the accelerations and velocities at the joints, find the forces required at the joints to generate those accelerations Recursive approach – Finds the accelerations and velocities of link i in terms of link i - 1

11. October, 23 200711 Newton-Euler Algorithm Method 1.Calculate the velocities and accelerations at each link 2.Calculate the required net force acting on each link to generate those accelerations 3.Calculate the joint forces required to generate the net forces on each link

12. October, 23 200712 Newton-Euler Algorithm 1. Find the velocities and accelerations of the links

13. October, 23 200713 Newton-Euler Algorithm 2. Find the forces on each link

14. October, 23 200714 Newton-Euler Algorithm 3. Find the forces on the joints This can be reformulated in link coordinates to speed up the calculation Runs in O(n)

15. October, 23 200715 Forward vs. Inverse Dynamics Inverse Dynamics – The calculation of forces given a set of accelerations Forward Dynamics – The calculation of accelerations given a set of forces

16. October, 23 200716 Composite-Rigid-Body Algorithm – Q is the vector of the forces on the links – H is the joint-space inertia matrix (n x n) – C vector of forces that produce zero acceleration – Algorithm – Calculate the elements of C – Calculate the elements of H – Solve the set of simultaneous equations

17. October, 23 200717 Composite-Rigid-Body Algorithm Solve for C – Setting the acceleration to zero, we get – We can, therefore, interpret C as the forces which produce no acceleration – We can use a forward-dynamics solver (like Newton- Euler) to solve for the forces given the position, velocity and an acceleration of zero

18. October, 23 200718 Composite-Rigid-Body Algorithm Solve for H – If we set C to 0, we observe that is the vector of joint forces that will impart an acceleration of onto a stationary robot Therefore, the i th column of H is the vector of forces required to produce a unit of acceleration about joint i and no other acceleration. – Treat the links i…n as a rigid-body with inertia defined by – Treat the links from 1…i-1 are therefore unmoving

19. October, 23 200719 Composite-Rigid-Body Algorithm Solve for H (cont.) – Given that – Since none of the links from 1 … i-1 are moving, every joint transmits onto the subsequent link, so we can solve for H by solving – Which is a complete solution for H since it is symmetric – Runs in O(n 2 )

20. October, 23 200720 Composite-Rigid-Body Algorithm Once you have H and C, solve the system of equations using any solver – O(n 3 ), but the constant is small enough that for n less than ~12 the O(n 2 ) term dominates Like Newton-Euler, this can be reformulated in link coordinates – Faster for n ≤ 16

21. October, 23 200721 Articulated-Body Algorithm (Re)consider the equation of motion of an articulated body This is true for any link in the articulated body

22. October, 23 200722 Articulated-Body Algorithm Consider an articulate robot as a single joint attached to an articulated body – The problem simplifies to the forward dynamics of a one-joint robot (much simpler than the general case) – The first joint is simply a one-joint robot – The second joint is a one-joint robot with a moving base (slightly more complicated, but still much simpler that the general case) – Solving this requires two tasks Solving the one-joint robot forward dynamics problem Finding the articulated-body inertias (I) and bias forces (p)

23. October, 23 200723 Articulated-Body Algorithm Solving the one-joint robot problem

24. October, 23 200724 Articulated-Body Algorithm Finding the articulated-body inertia (I A ) and bias force (p) Where is called the velocity-product force and is defined to be

25. October, 23 200725 Articulated-Body Algorithm These formulas can be reformulated recursively, so allow us to find and in terms of only and Our algorithm is then – Calculate the series of articulated body inertias and bias forces – Using these inertias and bias forces, calculate the joint accelerations Since these are both defined recursively, they each take O(n), making the entire algorithm O(n )

26. October, 23 200726 Lagrange-Multiplier Method The preceding methods are reduced- coordinate formulations – These methods remove some of the dof’s by enforcing a set of constraints (a joint can only rotate in a certain direction constraining the motion of the joint and the link) – Finding a parameterization for the generalized coordinates in terms of the reduced coordinates is not always easy The Lagrange Multiplier Method considers all the d.o.f.’s of the system

27. October, 23 200727 Lagrange-Multiplier Method Consider the equation of motion of i bodies – M describes the mass properties of the system is an d x d matrix where d is the number of dof’s of body i when not constrained Also consider a constraint i that removes m dof’s from the system, we can write it as – Where each j ik is a m x d matrix that represents the constraint on link k where d is again the number of dof’s of body k and m is the number of dof’s removed by the constraint

28. October, 23 200728 Lagrange-Multiplier Method To simplify the notation, we replace the q individual constraint equations With – Where J is a q x n matrix of the individual j ik matrices Where q is the total number of constraints on the system and n is the number of bodies – c is a q dimensional vector

29. October, 23 200729 Lagrange-Multiplier Method Just as we did when solving the constrained particle dynamics problems, we require that the constraint does no work. This results is a constraint force of the form: – Where λ i is an m (dof’s removed by constraint i) dimensional column vector and is referred to as the Lagrange multiplier The problem is now just to find a λ so the constraint forces and any external forces satisfy the constraints

30. October, 23 200730 Lagrange-Multiplier Method If we introduce an external force acting on the system and combine the equations, we get Solving for and plugging into our constraint equation, we get For constraints that act on two bodies, the matrix system is tightly banded and can be solved in O(n) – Using banded Cholesky decomposition, for example

31. October, 23 200731 Lagrange-Multiplier Method For more complicated constraints, is no longer sparse and we reformulate the equation as If we required acyclic constraints, then sparse- matrix theory tells us that has perfect elimination order – This means that if we factor into three matrices LDL T, L will be as sparse as H and can be computed in O(n) – We can then solve for λ, by solving each piece of LDL T separately, each also in O(n) time and combining the solutions

32. October, 23 200732 Summary Inverse Dynamics – Newton-Euler is the standard implementation in O(n) Forward dynamics – Composite-rigid-body algorithm is simpler and faster for n < 9, runs in O(n 3 ) – Articulated-body algorithm is faster for n > 9, runs in O(n) – Lagrange multiplier method is somewhat simpler than ABA and speed is comparable, runs in O(n)

33. October, 23 200733 References / Thanks R. Featherstone, Robot Dynamics Algorithms, Boston/Dordrecht/Lancaster: Kluwer Academic Publishers, 1987. D. Baraff, "Linear-Time Dynamics using Lagrange Multipliers," Proc. SIGGRAPH '96, pp. 137-146, New Orleans, August 1996. Thanks to Nico for his slides from last year