Structure and Synthesis of Robot Motion

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1 Structure and Synthesis of Robot Motion Dynamics: Constraints, Continua, etc. Subramanian Ramamoorthy School of Informatics 5 February, 2009

2 Recap Last time, we discussed two major approaches to describing rigid body motion Newton Euler: Compute forces/torques and describe motion in terms of their balance Lagrangian: Define an energy functional and describe motion in variational terms I mentioned the idea of constraints to motion This lecture builds on previous one, by providing a few more advanced topics. 05/02/2009 Structure and Synthesis of Robot Motion 2

3 Constraints They are everywhere in robotics! 05/02/2009 Structure and Synthesis of Robot Motion 3

4 Understanding Constraints Bead on Wire How does this bead move along the loop due to applied forces? Along the circle, it can slide freely However, it should never come off by pulling A simple proposal: Attach the bead to the loop via a spring What happens if spring is too soft? Bead slowly wobbles in a goopy way What happens if spring is too hard? Bead vibrates in an undesirably hard way 05/02/2009 Structure and Synthesis of Robot Motion 4

5 Another Way to Deal with Constraint The position of bead may be described parametrically: q = r[cos, sin ] This 1-DOF constraint must always be met q Warning: A lot of care is required to setup such problems see Ch 12 of Choset et al. 05/02/2009 Structure and Synthesis of Robot Motion 5

6 Constraints Often Yield DAEs DAE: Differential Algebraic Equations General structure: x f ( x, z, t) 0 g( x, z, t) The idea is that the ODE for x(t) depends on additional algebraic variables z(t) and the solution must satisfy the additional algebraic constraint! This makes is hard for traditional ODE solvers Jacobians can be singular The solution is related to what we saw with the bead-on-loop example but the details are a bit more involved In practice, make use of pre-existing solvers! 05/02/2009 Structure and Synthesis of Robot Motion 6

7 Modelling Other Effects: Dissipation So far, we haven t talked about effects like friction or energy dissipation A simplistic way to bring them into equations of motion is to just add terms for them we d like to be more principled How to bring dissipation into the Lagrangian model? 05/02/2009 Structure and Synthesis of Robot Motion 7

8 Modelling Dissipative Forces 05/02/2009 Structure and Synthesis of Robot Motion 8

9 Dissipation - Example Imagine a block sliding down a wedge How to model the friction? 05/02/2009 Structure and Synthesis of Robot Motion 9

10 Next Step: Towards Continuous Objects Most of our attention has been on systems consisting of (relatively few) links of rigid bodies How do these methods scale towards `realistic objects that are flexible and deformable? How do these methods apply to large ensembles (e.g., team of distributed robots or avatars)? - These are some of the frontier problems in robotics and other areas involving computational motion - These are also vast areas of study on their own, so we ll focus on two things: deformation and flows 05/02/2009 Structure and Synthesis of Robot Motion 10

11 Recap: Linear Transformations A rotation matrix is a specific example of a more general class of linear transformations: Can achieve many types of distortions : Using diagonal elements, describe axis-wise scaling A negated diagonal element leads to mirroring Off-diagonal elements lead to shearing 05/02/2009 Structure and Synthesis of Robot Motion 11

12 An Extension In general, we could use tensors to describe transformations A benefit of using such notation is that we can also naturally talk about higher derivatives 05/02/2009 Structure and Synthesis of Robot Motion 12

13 What is a Tensor, Really? It measures distortion of space in terms of distortion of coordinate frames If you measure angles between old and new coordinate frames as the underlying quantity (object) is being transformed then you can describe the process well 05/02/2009 Structure and Synthesis of Robot Motion 13

14 Using Tensors to Describe Deformations 05/02/2009 Structure and Synthesis of Robot Motion 14

15 Describing Fluid Flows Similar ideas can be used to describe fluids Consider this volume within a fluid The motion of the fluid is defined by conditions on balance of forces The pressure balance may be described (in equilibrium) as: 05/02/2009 Structure and Synthesis of Robot Motion 15

16 A Bit More Realistic The equation in previous slide only make sense in a very static setting (may not be of most interest) In fact, there are viscous forces that drag a fluid and then there may be resulting acceleration Rewritten somewhat differently (a version of Navier-Stokes), 05/02/2009 Structure and Synthesis of Robot Motion 16

motion of crowds and ensembles [Pimenta et al.

17 What does Navier-Stokes have to do with Robotics?! Sometimes such fluid models are very useful ways to describe the motion of crowds and ensembles [Pimenta et al., ICRA 2008] [Treuille et al., ACM SIGGRAPH 2006] 05/02/2009 Structure and Synthesis of Robot Motion 17

18 Another Application: Sometimes we wish to Manipulate Flexible Objects 05/02/2009 Structure and Synthesis of Robot Motion 18

strategies to enforce movements then you may find more compact, but conceptually more

19 In Practice People in animation often treat many of these things as just very large springmass (rigid body) systems In robotics, especially if you were designing control strategies to enforce movements then you may find more compact, but conceptually more involved, models useful! [Selle et al., ACM SIGGRAPH 2008] 05/02/2009 Structure and Synthesis of Robot Motion 19

20 In Practice Finite Elements Often, problems involving deformation and flows are handled using methods such as finite elements Here is the basic idea Consider a process that can be described in terms of differential equations (Note that we ve left the form of dynamics open some combination of position, velocity, acceleration terms) Discretize the domain: 05/02/2009 Structure and Synthesis of Robot Motion 20

21 Finite Elements, contd. Define an interpolation scheme within each element Based on this, define a large matrix system to describe motion 05/02/2009 Structure and Synthesis of Robot Motion 21

22 In Practice Particle Systems Particle system is an array of particles. Decouple system from solver: Translate into a generic position and acceleration vector 1. Set current state (positions and velocities) 2. Get current state (positions and velocities) 3. Compute accelerations f(x, t) Integration uses only these methods to simulate evolution of a particle system 05/02/2009 Structure and Synthesis of Robot Motion 22

23 In practice Particle Systems AnimateParticles(n, y 0, t 0, t f ) { y = y 0 t = t 0 DrawParticles(n, y) while(t!= t f ) { f = ComputeForces(y, t) dydt = AssembleDerivative(y, f) //there may be multiple force fields {y, t } = ODESolverStep(6n, y, dy/dt) DrawParticles(n, y) } } 05/02/2009 Structure and Synthesis of Robot Motion 23

24 Particle Animation [Reeves et al. 1983] Start Trek, The Wrath of Kahn 05/02/2009 Structure and Synthesis of Robot Motion 24

25 Summary We discussed a few special topics involving dynamics We talked about modelling constraints and including them in equations of motion Then we discussed (in outline form) models of deformations and flows Such models are useful in a variety of areas ranging from distributed robotics to complex manipulation problems Everything in this lecture is literally just a sampler read further if you see yourself needing these topics! 05/02/2009 Structure and Synthesis of Robot Motion 25

2.7 Cloth Animation. Jacobs University Visualization and Computer Graphics Lab : Advanced Graphics - Chapter 2 123

2.7 Cloth Animation. Jacobs University Visualization and Computer Graphics Lab : Advanced Graphics - Chapter 2 123 2.7 Cloth Animation 320491: Advanced Graphics - Chapter 2 123 Example: Cloth draping Image Michael Kass 320491: Advanced Graphics - Chapter 2 124 Cloth using mass-spring model Network of masses and springs