dynamics of articulated robots. rigid body dynamics the following can be derived from first...
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Dynamics of Articulated Robots
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Rigid Body Dynamics
The following can be derived from first principles using Newton’s laws + rigidity assumption
Parameters CM translation x(t) CM velocity v(t) Rotation R(t) Angular velocity (t) Mass m, local inertia tensor HL
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Kinetic energy for rigid body
Rigid body with velocity v, angular velocity KE = ½ (m vTv + T H )
World-space inertia tensor H = R HL RT
v
T
v
H 0 0 m I
1/2
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Kinetic energy derivatives
KE/v = m v Force (@CM) f = d/dt (KE/v) = m v’
KE/ = H d/dt H = []H – H[] Torque = d/dt (KE/) = [] H H ’
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Summary
f = m v’ = [] H H ’
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Robot Dynamics
Configuration q, velocity q’ Rn
Generalized forces u Rm
Joint torques/forces If m < n we say robot is underactuated
How does u relate to q and q’?
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Articulated Robots
Treat each link as a rigid body Use Langrangian mechanics to
determine dynamics of q, q’ as a function of generalized forces u
(Derivation: principle of virtual work)
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Lagrangian Mechanics
L(q,q’) = K(q,q’) – P(q)
Lagrangian equations of motion:d/dt L/q’ - L/q = u
Kinetic energy Potential energy
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Lagrangian Approach
L(q,q’) = K(q,q’) – P(q)
Lagrangian equations of motion:d/dt L/q’ - L/q = u
L/q’ = K/q’ L/q = K/q - P/q
Kinetic energy Potential energy
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Kinetic energy for articulated robot K(q,q’) = i Ki(q,q’) Velocity of i’th rigid body
vi = Jit(q) q’
Angular velocity of i’th rigid bodyi = Ji
r(q) q’ Ki = ½ q’T(miJi
t(q)TJit(q) + Ji
r(q)THi(q)Jir(q))q’
K(q,q’) = ½ q’T B(q) q’Mass matrix
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Derivative of K.E. w.r.t q’
/q’ K(q,q’) = B(q) q’ d/dt (/q’ K(q,q’)) = B(q) q’’ + d/dt B(q) q’
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Derivative of K.E. w.r.t q
/q K(q,q’) = ½ q’T /q1 B(q) q’…
q’T /qn B(q) q’
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Potential energy for articulated robot in gravity field
P/q = i Pi/q
Pi/q = mi (0,0,g)T vi = mi (0,0,g)T
Jit(q) q’
-G(q)Generalized gravity
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Putting it all together
d/dt K/q’ - K/q - P/q = u
B(q) q’’ + d/dt B(q) q’ – ½ + G(q) = uq’T /q1 B(q) q’
…q’T /qn B(q) q’
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Putting it all together
d/dt K/q’ - K/q - P/q = u
B(q) q’’ + d/dt B(q) q’ – ½ + G(q) = uq’T /q1 B(q) q’
…q’T /qn B(q) q’
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Final canonical form
B(q) q’’ + C(q,q’) + G(q) = u
Mass matrix Centrifugal/coriolis forces
Generalized gravity
Generalized forces
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Forward/Inverse Dynamics
Given u, find q’’q’’ = M(q)-1 (u - C(q,q’) - G(q) )
Given q,q’,q’’, find uu = M(q) q’’ + C(q,q’) + G(q)
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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
Faster forward/inverse dynamics for large chains (O(n) vs O(n3))
Lagrangian form still mathematically handy
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Software
Both Lagrangian dynamics and Newton-Euler methods are implemented in KrisLibrary
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Application: Feedforward control
Joint PID loops do not follow joint trajectories accurately
Include feedforward torques to reduce reliance on feedback
Estimate the torques that would compensate for gravity and coriolis forces
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Application: Feedforward control
Joint PID loops do not follow joint trajectories accurately
Include feedforward torques to reduce reliance on feedback
Estimate the torques that would compensate for gravity and coriolis forces
![Page 22: Dynamics of Articulated Robots. Rigid Body Dynamics The following can be derived from first principles using Newton’s laws + rigidity assumption Parameters](https://reader036.vdocuments.mx/reader036/viewer/2022070412/56649efd5503460f94c117d1/html5/thumbnails/22.jpg)
Feedforward Torques
Given q,q’,q’’ of trajectory 1. Estimate M, C, G 2. Compute u
u = M(q) q’’ + C(q,q’) + G(q) 3. Add u into joint PID loops