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2021-9-26 published at Occasions/ Family category

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1.Dynamics of Articulated Robots Kris Hauser CS B659: Principles of Intelligent Robot Motion Spring 2013
2.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
3.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
4.Rigid body ordinary differential equations We will express forces and torques in terms of terms of H (a function of R), 𝜔, 𝑣 and 𝜔 𝑓 = 𝑚 𝑣 𝜏 = [𝜔] 𝐻 𝜔 + 𝐻 𝜔 Rearrange… 𝑣 =𝑓/𝑚 𝜔 = 𝐻 −1 (𝜏− 𝜔 𝐻 𝜔 ) 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
5.Articulated body ODEs We will express joint torques 𝜏 in terms of terms of 𝑞, 𝑞 , and 𝑞 and external forces f 𝐵 𝑞 𝑞 +𝐶 𝑞, 𝑞 +𝐺 𝑞 =𝜏+ 𝐽 𝑇 𝑞 𝑓 Rearrange… 𝑞 =𝐵 𝑞 −1 𝜏+ 𝐽 𝑇 𝑞 𝑓 −𝐶 𝑞, 𝑞 −𝐺 𝑞 An ODE in the state space x=(𝑞, 𝑞 ) 𝑑𝑥 𝑑𝑡 = 𝑑𝑞 𝑑𝑡 𝑑 𝑞 𝑑𝑡 =𝑓 𝑞, 𝑞 = 𝑞 𝐵 𝑞 −1 𝜏+ 𝐽 𝑇 𝑞 𝑓 −𝐶 𝑞, 𝑞 −𝐺 𝑞 Solve using numerical integration
6.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
7.Dynamics of Rigid Bodies
8.Kinetic energy for rigid body Rigid body with velocity v, angular velocity w about COM KE = ½ (m vTv + wT H w) World-space inertia tensor H = R HL RT w v T w v H 0 0 m I 1/2
9.Kinetic energy derivatives 𝜕𝐾𝐸 𝜕𝑣 = 𝑚 𝑣 Force (@CM) 𝑓 = 𝑑 𝑑𝑡 ( 𝜕𝐾𝐸 𝜕𝑣 )= 𝑚 𝑣 𝜕𝐾𝐸 𝜕𝜔 =𝐻𝜔 𝑑 𝑑𝑡 H = [w]H – H[w] Torque t = 𝑑 𝑑𝑡 𝜕𝐾𝐸 𝜕𝜔 = [w] H w + H 𝜔
10.Summary 𝑓 = 𝑚 𝑣 𝜏 = [𝜔] 𝐻 𝜔 + 𝐻 𝜔 Gyroscopic “force”
11.Force off of COM x F
12.Force off of COM x F Consider infinitesimal virtual displacement 𝛿𝑥 generated by F. (we don’t know what this is, exactly) The virtual work performed by this displacement is FT𝛿𝑥 𝛿𝑥
13.Generalized torque f Now consider the equivalent force f, torque τ at COM
14.Generalized torque f Now consider the equivalent force f, torque τ at COM And an infinitesimal virtual displacement of R.B. coordinates 𝛿𝑞 𝛿𝑥 𝛿𝑞
15.Generalized torque f 𝛿𝑥 𝛿𝑞 Now consider the equivalent force f, torque τ at COM And an infinitesimal virtual displacement of R.B. coordinates 𝛿𝑞 Virtual work in configuration space is [fT,τT] 𝛿𝑞
16.Principle of virtual work f 𝛿𝑥 𝛿𝑞 [fT,τT] 𝛿𝑞= FT 𝛿𝑥 Since 𝛿𝑥=𝐽 𝑞 𝛿𝑞 we have [fT,τT] 𝛿𝑞= FT𝐽 𝑞 𝛿𝑞 F
17.Principle of virtual work f 𝛿𝑥 𝛿𝑞 [fT,τT] 𝛿𝑞= FT 𝛿𝑥 Since 𝛿𝑥=𝐽 𝑞 𝛿𝑞 we have [fT,τT] 𝛿𝑞= FT𝐽 𝑞 𝛿𝑞 Since this holds no matter what 𝛿𝑞 is, we have [fT,τT] = FTJ(q), Or JT(q) F = F f τ
18.Articulated Robot Dynamics
19.Robot Dynamics Configuration 𝑞, velocity 𝑞  Rn Generalized forces u  Rm 𝑢 = 𝜏 + 𝑘 𝐽𝑘 𝑞 𝑇𝑓𝑘 Joint torques 𝜏 and external forces 𝑓𝑘 How does u relate to 𝒒 and 𝒒 ? Use Langrangian mechanics to find a link between u and 𝑞
20.Lagrangian Mechanics 𝐿 𝑞, 𝑞 =𝐾 𝑞, 𝑞 −𝑃 𝑞 The trajectory between two states ( 𝑞 0 , 𝑞 0 ), 𝑞 𝑇 , 𝑞 𝑇 is the one that minimizes the “action” 𝑆= 0 𝑇 𝐿 𝑞, 𝑞 𝑑𝑡 𝐿 𝑞, 𝑞 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
21.Lagrangian Mechanics 𝐿 𝑞, 𝑞 =𝐾 𝑞, 𝑞 −𝑃 𝑞 Minimum action condition => Euler-Lagrange equations of motion: 𝑑 𝑑𝑡 𝜕𝐿 𝜕 𝑞 − 𝜕𝐿 𝜕𝑞 =𝑢 Note that P is independent of 𝑞 , so 𝑑 𝑑𝑡 𝜕𝐾 𝜕 𝑞 − 𝜕𝐾 𝜕𝑞 + 𝜕𝑃 𝜕𝑞 =𝑢 A system of n partial differential equations
22.Sanity check: Newton’s laws Example: Point Mass Coordinates q = (x,y) Potential field P(x,y) Lagrangian: 𝐿 𝑞, 𝑞 = 1 2 𝑚 𝑥 2 + 𝑦 2 −𝑃 𝑥,𝑦 Equations of motion 𝑑 𝑑𝑡 𝑚 𝑥 + 𝜕𝑃 𝜕𝑥 =𝑚 𝑥 + 𝜕𝑃 𝜕𝑥 = 𝑢 𝑥 𝑑 𝑑𝑡 𝑚 𝑦 + 𝜕𝑃 𝜕𝑦 = 𝑚 𝑦 + 𝜕𝑃 𝜕𝑦 =𝑢 𝑦
23.Kinetic energy for articulated robot 𝐾(𝑞, 𝑞 ) = 𝑖 𝐾𝑖(𝑞, 𝑞 ) Velocity of i’th rigid body 𝑣𝑖 = 𝐽 𝑖 𝑝 𝑞 𝑞 Angular velocity of i’th rigid body 𝜔𝑖=𝐽𝑖𝑟(𝑞) 𝑞 𝐾 𝑖 = 1 2 𝑞 𝑇(𝑚𝑖𝐽𝑖𝑝𝑇𝐽𝑖𝑝 + 𝐽𝑖𝑟𝑇𝐻𝑖𝐽𝑖𝑟) 𝑞 𝐾(𝑞, 𝑞 )=½ 𝑞 𝑇 𝐵(𝑞) 𝑞 Mass matrix:symmetric positive definite
24.Derivative of K.E. w.r.t 𝑞 𝜕𝐾 𝜕 𝑞 𝑞, 𝑞 =𝐵 𝑞 𝑞 𝑑 𝑑𝑡 𝜕𝐾 𝜕 𝑞 =𝐵 𝑞 𝑞 + 𝑑 𝑑𝑡 𝐵 𝑞 𝑞 =𝐵(𝑞) 𝑞 + 𝑖 𝑞 𝑖 𝜕 𝜕 𝑞 𝑖 𝐵 𝑞 𝑞
25.Derivative of K.E. w.r.t q 𝜕 𝜕𝑞 𝐾 𝑞, 𝑞 =½ 𝑞 𝑇 𝜕 𝜕 𝑞 1 𝐵(𝑞) 𝑞 … 𝑞 𝑇 𝜕 𝜕 𝑞 𝑛 𝐵(𝑞) 𝑞
26.Potential energy for articulated robot in gravity field 𝜕𝑃 𝜕𝑞 = 𝑖 𝜕 𝑃 𝑖 𝜕𝑞 𝜕 𝑃 𝑖 𝜕𝑞 = 𝑚 𝑖 0 0 𝑔 𝑇 𝑣 𝑖 = 𝑚 𝑖 0 0 𝑔 𝑇 𝐽𝑖𝑝(𝑞) G(q) Generalized gravity
27.Putting it all together 𝑑 𝑑𝑡 𝜕𝐾 𝜕 𝑞 − 𝜕𝐾 𝜕𝑞 + 𝜕𝑃 𝜕𝑞 =𝑢 𝐵 𝑞 𝑞 + 𝑑 𝑑𝑡 𝐵(𝑞) 𝑞 – ½ 𝑞 𝑇 𝜕 𝜕 𝑞 1 𝐵(𝑞) 𝑞 … 𝑞 𝑇 𝜕 𝜕 𝑞 𝑛 𝐵(𝑞) 𝑞 + 𝐺(𝑞) = 𝑢 Group these terms together
28.Final canonical form 𝐵 𝑞 𝑞 +𝐶 𝑞, 𝑞 +𝐺(𝑞) = 𝑢 Generalized inertia Centrifugal/coriolis forces Generalized gravity Generalized forces (joint torques + external forces)
29.Forward/Inverse Dynamics Given 𝑢, 𝑞, and 𝑞 , find 𝑞 From torques to accelerations 𝑞 = 𝐵 𝑞 −1 (𝑢−𝐶(𝑞, 𝑞 )−𝐺(𝑞) ) Given 𝑞, 𝑞 , and 𝑞 , find 𝑢 From desired accelerations to necessary torques 𝑢=𝐵 𝑞 𝑞 +𝐶(𝑞, 𝑞 )+𝐺(𝑞)
30.Example: RP manipulator
31.Application: Effective Inertia If a force 𝑓 is applied to a point 𝑝 on a robot, how much will 𝑝 accelerate?
32.Application: Effective Inertia If a force 𝑓 is applied to a point 𝑝 on a robot, how much will 𝑝 accelerate? Assume a stationary system, no acceleration when no force is applied 𝑞 0 = 𝐵 𝑞 −1 𝜏−𝐶 𝑞, 𝑞 −𝐺 𝑞 =0 With the force: 𝑞 = 𝐵 𝑞 −1 𝐽(𝑞) 𝑇 𝑓
33.Application: Effective Inertia If a force 𝑓 is applied to a point 𝑝 on a robot, how much will 𝑝 accelerate? Assume a stationary system, no acceleration when no force is applied 𝑞 0 = 𝐵 𝑞 −1 𝜏−𝐶 𝑞, 𝑞 −𝐺 𝑞 =0 With the force: 𝑞 = 𝐵 𝑞 −1 𝐽(𝑞) 𝑇 𝑓 𝑝 =𝐽 𝑞 𝑞 =𝐽(𝑞)𝐵 𝑞 −1 𝐽(𝑞) 𝑇 𝑓 The matrix 𝐽 𝑞 𝐵 𝑞 −1 𝐽 𝑞 𝑇 −1 is called the effective inertia matrix 𝑓= 𝐽 𝑞 𝐵 𝑞 −1 𝐽 𝑞 𝑇 −1 𝑝 Can be infinite at singular configurations!
34.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
35.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
36.Feedforward Torques Given current 𝑞, 𝑞 , desired 𝑞 1. Estimate B, C, G 2. Compute u 𝑢=𝐵 𝑞 𝑞 +𝐶(𝑞, 𝑞 )+𝐺(𝑞) 3. Apply torques u How to compensate for errors in B,C,G? Combine feedforward with feedback. More in later classes…
37.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)
38.Forward Dynamics: Basic Intuition Downward recursion: Starting from root, compute “articulated body inertia matrix” for each link 6x6 matrix 𝐼 𝑖 𝐴 relating (𝑓,𝜏) vectors to translational/angular accelerations (a,𝛼) respectively Also need a “bias force” 𝑃 𝑖 𝑓,𝜏 = 𝐼 𝑖 𝐴 𝛼,𝑎 + 𝑃 𝑖 Upward recursion: Starting from leaves, compute accelerations on links Given (𝑓,𝜏) acting on i’th link, compute acceleration of joint i and the joint constraint forces on the i-1’th link (𝑓,𝜏) includes external forces + joint constraint forces from downward links
39.Software Both Lagrangian dynamics and Newton-Euler methods are implemented in KrisLibrary Lagrangian form is usually most mathematically convenient representation
40.Constrained Dynamics
41.Constrained Systems Suppose the system is constrained by 𝐴 𝑞 𝑞 =0 E.g., closed-chains, contact constraints, rolling constraints A is a k x n matrix (k constraints) How does 𝑞 evolve over time?
42.The Wrong Way Suppose the system is constrained by 𝐴 𝑞 𝑞 =0 E.g., closed-chains, contact constraints, rolling constraints A is a k x n matrix (k constraints) How does 𝑞 evolve over time? Wrong way: 𝑑 𝑑𝑡 𝐴 𝑞 𝑞 = 𝐴 𝑞 𝑞 +𝐴 𝑞 𝑞 =0 Solve for 𝑞 as usual, then project it onto the subspace that satisfies this equation, obtaining 𝑞 𝑐𝑜𝑛𝑠𝑡 The correct answer will be a projection, but a very specific one!
43.The Right Way… Constrained system of equations: 𝐵 𝑞 𝑞 +𝐶 𝑞, 𝑞 +𝐺 𝑞 =𝑢+𝐴 𝑞 𝑇 𝜆 (1) 𝑑 𝑑𝑡 𝐴 𝑞 𝑞 = 𝐴 𝑞 𝑞 +𝐴 𝑞 𝑞 =0 (2) Lagrange multipliers have been introduced 𝜆= 𝜆 1 ,…, 𝜆 𝑘 𝑇 𝜆 can be thought of as constraint forces Solve for n+k variables 𝑞 ,𝜆
44.Solving… Constrained system of equations: 𝐵 𝑞 𝑞 +𝐶 𝑞, 𝑞 +𝐺 𝑞 =𝑢+𝐴 𝑞 𝑇 𝜆 (1) 𝑑 𝑑𝑡 𝐴 𝑞 𝑞 = 𝐴 𝑞 𝑞 +𝐴 𝑞 𝑞 =0 (2) Solve for n+k variables 𝑞 ,𝜆 A solution must satisfy 𝑞 = 𝐵 −1 𝑢+ 𝐴 𝑇 𝜆−𝐶−𝐺 (3) solve 1 for 𝑞 𝐴 𝑞 +𝐴 𝐵 −1 𝑢+ 𝐴 𝑇 𝜆−𝐶−𝐺 =0 (4) subst (3) in (2) 𝜆= 𝐴 𝐵 −1 𝐴 𝑇 −1 𝐴 𝑞 +𝐴 𝐵 −1 𝐶+𝐺−𝑢 (5) solve for 𝜆 in (4), use − 𝐴 𝑞 =𝐴 𝑞 from (2) 𝑃 𝑞 =𝑃 𝐵 −1 (𝑢−𝐶−𝐺) (6) more manipulations.. With 𝑃=𝐼− 𝐵 −1 𝐴 𝑇 𝐴 𝐵 −1 𝐴 𝑇 −1 𝐴
45.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)
46.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
47.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
48.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 𝒒 such that 𝒒 𝑻 𝑩 𝒒 is minimized Constraint forces dissipate kinetic energy in a minimal fashion!
49.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 𝐵 𝑞 𝑞 +𝐶 𝑞, 𝑞 +𝐺 𝑞 =𝑢+𝐴 𝑞 𝑇 𝜆 (1) 𝑑 𝑑𝑡 𝐴 𝑞 𝑞 = 𝐴 𝑞 𝑞 +𝐴 𝑞 𝑞 =0 (2) Solve for n+k variables 𝑞 ,𝜆 (1), (2) are sparse systems and are solved using specialized solvers More on frictional contact later…
50.Next class Feedback control Principles App J

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